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A Computational Model of Human Blood Clotting: Simulation, Analysis, Control, and Validation Joseph Gerard Makin Electrical Engineering and Computer Sciences University of California at Berkeley Technical Report No. UCB/EECS-2008-165 http://www.eecs.berkeley.edu/Pubs/TechRpts/2008/EECS-2008-165.html December 17, 2008
Transcript of A Computational Model of Human Blood Clotting: · PDF fileA Computational Model of Human Blood...
A Computational Model of Human Blood Clotting:Simulation, Analysis, Control, and Validation
Joseph Gerard Makin
Electrical Engineering and Computer SciencesUniversity of California at Berkeley
Technical Report No. UCB/EECS-2008-165
http://www.eecs.berkeley.edu/Pubs/TechRpts/2008/EECS-2008-165.html
December 17, 2008
Copyright 2008, by the author(s).All rights reserved.
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Acknowledgement
Thanks to Profs. Srini Narayanan and Jerome Feldman for theirsupervision of this thesis; to Profs. Bruno Olshausen and Jose Carmena forserving as readers; and to Profs. Shankar Sastry and Tom Budinger forserving as qualification examiners.
A Computational Model of Human Blood Clotting: Simulation, Analysis,Control, and Validation
by
Joseph Gerard Makin
B.S., B.A. Swarthmore College 2003
A dissertation submitted in partial satisfactionof the requirements for the degree of
Doctor of Philosophy
in
Engineering - Electrical Engineering and Computer Sciences
in the
GRADUATE DIVISION
of the
UNIVERSITY OF CALIFORNIA, BERKELEY
Committee in charge:
Professor Jerome Feldman, ChairProfessor Bruno Olshausen
Professor Jose CarmenaProfessor Srini Narayanan
Fall 2008
The dissertation of Joseph Gerard Makin is approved.
Chair Date
Date
Date
Date
University of California, Berkeley
Fall 2008
A Computational Model of Human Blood Clotting: Simulation, Analysis, Control,
and Validation
Copyright c© 2008
by
Joseph Gerard Makin
Abstract
A Computational Model of Human Blood Clotting: Simulation, Analysis, Control,
and Validation
by
Joseph Gerard Makin
Doctor of Philosophy in Engineering - Electrical Engineering and Computer Sciences
University of California, Berkeley
Professor Jerome Feldman, Chair
Complex biological systems pose many challenges to researchers, including, inter alia, choice
of computational model, with its consequences for simulation and analysis; methods of ma-
nipulating the system exogenously (control); and model validation. I attempt to address
these issues for human blood clotting. By treating the system as comprising interacting
discrete and continuous aspects, i.e. as hybrid, the entire coagulation cascade may be
simulated: Blood proteins, elemental ions, and other state elements are modeled either
as real-valued concentrations or as binary variables (present/absent); interactions are ren-
dered either as ODEs (per their chemical equations) or as discrete events. Techniques from
nonlinear control theory are then used to devise drug therapies for diseased patients. Fi-
nally, the model is used to warp variations in the input parameters—rate constants and
initial conditions—into an output space where pathologies and healthy clotting are cleanly
separated by a semi-supervised clustering analysis. This serves to validate the model as
1
well as to summarize efficiently the predicted clinical consequences of individual variations.
Professor Jerome FeldmanDissertation Committee Chair
2
To S.C.M.: who predicted it.
i
Contents
Contents ii
List of Figures v
List of Tables viii
Acknowledgments ix
1 Introduction 1
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 The Coagulation Cascade 8
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 The Cascade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.1 Intrinsic pathway . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.2 Extrinsic pathway . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.3 Common pathway . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.4 Down-regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.5 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Diseases of Coagulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Mathematical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Hybrid Systems 23
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Types of Hybrid-System Investigation . . . . . . . . . . . . . . . . . . . . . 25
ii
3.2.1 Modeling and simulation . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2.2 Verification and decidability . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.3 Controller synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Hybrid Petri Nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4 Hybrid Systems in Biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4 Simulations and Sensitivity 41
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2 Model Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2.1 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3.1 Normal clotting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3.2 Simulating hæmophilia A . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3.3 Simulating factor-V Leiden . . . . . . . . . . . . . . . . . . . . . . . 57
4.4 Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.5 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.5.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.5.3 Sensitivity and prothrombin time . . . . . . . . . . . . . . . . . . . . 64
4.5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.6 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5 Control: A First Pass 73
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.3 Application to the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.3.1 Feedback linearization . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.3.2 Step-input control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.4 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.6.1 Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.6.2 Non-regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.6.3 Controller merits and demerits . . . . . . . . . . . . . . . . . . . . . 97
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5.6.4 Significance of model assumptions . . . . . . . . . . . . . . . . . . . 98
5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6 Analysis: Stability, Model Reduction, and Control 110
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.2 Model Reduction and Bounded Tracking . . . . . . . . . . . . . . . . . . . . 111
6.2.1 Bounded tracking and minimum-phase systems . . . . . . . . . . . . 111
6.2.2 Construction of the zero dynamics . . . . . . . . . . . . . . . . . . . 112
6.2.3 Intermezzo: model reduction . . . . . . . . . . . . . . . . . . . . . . 115
6.2.4 Stability of the zero dynamics . . . . . . . . . . . . . . . . . . . . . . 118
6.3 Control: Model-Predictive and Proportional . . . . . . . . . . . . . . . . . . 119
6.3.1 State observability and feedback linearization . . . . . . . . . . . . . 119
6.3.2 Control with constraints: model prediction . . . . . . . . . . . . . . 122
6.4 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7 Validation/Learning 135
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.4.1 Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.4.2 Decision tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7.4.3 A mathematical take . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
8 Conclusions & Future Work 146
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
8.2 The Hybrid-System Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 147
8.3 Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
8.3.1 Modeling assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . 149
8.3.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
8.4 Learning/Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
8.5 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Bibliography 158
iv
List of Figures
2.1 The coagulation cascade. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.1 A hybrid system model of a thermostat; see text for details. . . . . . . . . . 24
3.2 A very simple Discrete Petri Net (DPN) with purely discrete components.The input and test arcs’ respective resource requirements were satisfied, butthe inhibitory arc’s requirement is not—so the transition is enabled. . . . . 30
3.3 The simple Discrete Petri Net (DPN) of Figure 3.2, advanced by one step:the input arc’s place has been depleted by its resource requirement (1 token),and the output arc’s place has been augmented by its weight (1 token). . . 31
3.4 A very simple Petri net with purely continuous components (CPN). The rateat which “token fluid” leaves m1 is the rate at which it accumulates in m2,which in this example is m2(m1 + 1) . . . . . . . . . . . . . . . . . . . . . . 35
3.5 A summary of the various types of arcs in an HPN; cf. Defs. 3.3.6 and 3.3.7. 37
4.1 The activation module, which models the activation of a zymogen (IN1) intoits active configuration (OUT) by a catalyst (IN2). . . . . . . . . . . . . . . 46
4.2 The binding module, which models the (reversible) binding of two compo-nents, IN1 and IN2, into the macromolecule OUT. . . . . . . . . . . . . . . 47
4.3 A hybrid Petri net module modeling the initiation of the intrinsic pathway. 48
4.4 An HPN module depicting the final stages of blood clotting: the activationof factors I and XIII, and the formation of a clot. . . . . . . . . . . . . . . . 50
4.5 The activation module of Figure 4.1, but here two of the reactions areswitched on (off) by the presence (absence) of a discrete variable. . . . . . . 53
4.6 A simulation of the concentration of thrombin (factor IIa) in nM as it varieswith time (in seconds) in the blood clotting of a normal patient. . . . . . . 56
4.7 Simulated results of the concentration (nM) of thrombin versus time (s) dur-ing the clotting process for a person with severe hæmophilia A (low curve)and for normal clotting (high curve). . . . . . . . . . . . . . . . . . . . . . . 57
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4.8 Simulated results of the concentration (nM) of thrombin versus time (s) dur-ing the clotting process for a person with factor-V Leiden (high curve) anda normal patient (low curve). . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.1 A conceptual rendering of feedback linearization. The arrangement can beviewed either as a controller in a feedback loop with the original control-affine system (solid lines), plus an output function; or as an equivalent outputfunction along with coupled linear and nonlinear systems (dashed lines), withthe former evolving independently of the latter, and the output dependingsolely on the former. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.2 Simulated control of thrombin concentration during a clotting event in apatient with factor-V Leiden. The controller was unconstrained, and wasupdated continuously, allowing arbitrarily close tracking of the desired tra-jectory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.3 The heparin input in nM/s that generated Figure 5.2. Although this inputyields perfect tracking, it is obviously unacceptable for any realistic drug-delivery mechanism, operating as it does on a continuous time scale, andusing negative inputs (see text). . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.4 Simulated control of thrombin concentration during a clotting event in apatient with factor-V Leiden, again continuously sampling. The input wasconstrained to the range 0-20 nM/s. . . . . . . . . . . . . . . . . . . . . . . 104
5.5 The input generating the output of Figure 5.4. Note that the input is con-strained to the range 0-20 nM/s. . . . . . . . . . . . . . . . . . . . . . . . . 105
5.6 Simulated control of thrombin concentration during a clotting event in apatient with factor-V Leiden, this time using a discrete controller samplingevery 0.5 s. The input was again constrained in the range between 0 and 20nM/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.7 The discrete input, constrained to lie between 0 and 20 nM/s, that generatedFigure 5.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.8 Simulated control of thrombin in a patient with moderate hæmophilia A (seetext), where the controller inputs the proper initial concentration of factorVIII at the first time step, then shuts off. . . . . . . . . . . . . . . . . . . . 108
5.9 Control of thrombin concentration in a hæmophiliac using a single step inputat time t = 0. The input is 6.92 pM/s. . . . . . . . . . . . . . . . . . . . . . 109
5.10 Control of thrombin concentration in a patient with factor-V Leiden by asingle step input at time t = 0. The input is 4.39 nM/s. . . . . . . . . . . . 109
6.1 Simulated proportional control of thrombin concentration during a clottingevent in a patient with factor-V Leiden, once more using a discrete controllersampling every 0.5 s (as in the previous chapter), and with input again con-fined to the range between 0 and 20 nM/s. . . . . . . . . . . . . . . . . . . . 121
6.2 The discrete input, constrained to lie between 0 and 20 nM/s, that generatedFigure 6.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
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6.3 Simulated proportional control of thrombin during a clotting event in a pa-tient with factor-V Leiden, again at 2 Hz, but with input constrained to therange 0-5 nM/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.4 The proportional controller’s discrete input for Figure 6.3, constrained to liebetween 0 and 5 nM/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.5 Factor-V Leiden, treated by heparin via a feedback-linearization controlleroperating at 2 Hz, and within the range 0-5 nM/s. . . . . . . . . . . . . . . 128
6.6 The discrete input generated by the feedback-linearization scheme, clampedbetween 0 and 5 nM/s, that produced Figure 6.5. . . . . . . . . . . . . . . . 129
6.7 Control of thrombin using the LMPC scheme in conjunction with feedbacklinearization. Again, the uncontrolled trajectory is a result of factor-V Lei-den, and the input rate of heparin is confined to the range 0-5 nM/s. Notethe improvement over Figure 6.5. . . . . . . . . . . . . . . . . . . . . . . . 130
6.8 The discrete input (constrained to 0-5 nN/s) generated by the LMPCer andproducing Figure 6.7. Sampling rate is 0.5 Hz . . . . . . . . . . . . . . . . . 131
7.1 An output space in which hypo- (black x’s), hyper- (light gray +’s), andnormal (gray circles) coagulation are cleanly separated by k-means, withk = 4 The white hexagram, black square, and white pentagram (resp.) arethe known exemplars of the three classes. See text for an interpretation ofthe fourth class (gray diamonds). . . . . . . . . . . . . . . . . . . . . . . . . 139
7.2 The decision tree. The class labels are M = medium, L = low, H = high,and O = outliers for the four classes of coagulation levels. . . . . . . . . . . 145
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List of Tables
2.1 Primary Coagulation Factors and their Pre-Injury Concentrations . . . . . 11
4.1 Parameter values and types used in the HPN modules of Figures 4.4 and 4.3 52
4.2 Coagulation Disorders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3 Averaged Normalized Sensitivities . . . . . . . . . . . . . . . . . . . . . . . 63
4.4 PT-Time Parameter Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.1 Chemical Reaction Set I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.2 Chemical Reaction Set II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.3 Chemical Reaction Set III . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.4 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
viii
Acknowledgments
Most people probably read the acknowledgments of a thesis in search of their own names;
a few for completeness; and the rest hoping to get an inside joke. Almost everyone, then,
will be disappointed.
My first thanks are to Prof. Srini Narayanan, who conceived of the project, took me on
to work on it, and guided me along the way. But Srini has been more than an advisor: he’s
been a friend. That means I told lots of inappropriate jokes around him, but also that he
listened to my complaints about personal life, palavered about politics and the NFL (even
if he pulled for the front-runners), drank with us, and even kept some track of the number
of miles I ran in a week.
Prof. Jerry Feldman served as my “official” advisor, and many thanks are due to him
as well. Jerry was the voice of wisdom on technical matters—he suggested the learned
step-controller of Chapter 5—as well as all the practical concerns of the graduate program.
More importantly, his “neural theory of language” is what induced me to join the group.
Thanks also to my other committee members, Drs. Jose Carmena and Bruno Olshausen,
for their time—but also especially to Prof. Carmena for filling in as a reader at the last
minute and with such alacrity; and to Bruno for the hours spent in his reading group, hon-
estly the most enjoyable of my academic life at Berkeley—and for being an all-around nice
guy. I owe the other members of my qualifying-exam committee, as well: Dr. Tom Budinger
of the Lawrence-Berkeley Lab; and Prof. Shankar Sastry, who additionally provided many
useful hours of instruction in the classroom, and the authoritative textbook on nonlinear
systems.
My peers have contributed to this thesis, too, in one way or another, and deserve a shout-
out as well. Alessandro Abate ran the bio-hybrid-systems reading group, and gave me plenty
of useful advice on control theory. I owe Jeff Doyon for my tuition on chemical kinetics
and for our profitable conversations on research (mine and his both)—mostly conducted on
long runs. Matt Gedigian managed (inexplicably) to make most of my campus talks, and
read drafts of my papers. And thanks to my officemates: Steve Sinha, with whom I had
ix
many enjoyable conversations about politics, and a few about technical matters; and Leon
Barrett, with whom I had many useful conversations on technical matters, and a few about
politics. The remainder of my thanks for fellow students must be divided among the rest of
our research group—Nancy Chang, John Bryant, and Eva Mok—and attributed to Friday
evenings at the bar.
Without my parents, Thomas and Donita Makin, I should not be here at all. My debt
to them is beyond evaluation.
Penultimate thanks are to the authors of the TEX template for this thesis and of the
ucthesis.cls class file! and to the indispensable Ruth Gjerde, who is solely responsible for
my having filed just about every piece of department paperwork, and with whom I always
had a pleasant chat.
Finally, my largest debt of gratitude for this dissertation is to Professor Lynne Molter
of Swarthmore College, who first taught me the rudiments of control theory, who first
employed me as a researcher, who encouraged my interest in electrical engineering and in
graduate studies, and who remains a friend.
x
xi
Chapter 1
Introduction
1.1 Overview
It has long been a dream of science to apply the leverage of its more mathematical
disciplines to its biological, especially medical, investigations. Biology is messy, however,
and will not be shoehorned into the same kind of mathematical formalization as (say)
physics or information theory or even chemistry. Yet a mathematical approach may very
well help us better to understand a biological process, or to manipulate it, or to connect
changes in its structure to changes in its function and make predictions on that basis; and
hence to diagnose and treat diseases. This thesis, then, labors in the service of a dream,
but not a vain one.
The object of this study is human blood clotting, a process exhibiting complexity typ-
ical of biological systems: no single set of differential equations or discrete-process model
can completely describe the coagulation cascade (at least not given the current state of
knowledge); yet on the other hand it is obviously of crucial clinical importance. Disorders
are various and some not uncommon: hæmophilia A affects about one in ten thousand
individuals; protein-C deficiency one in five hundred; von Willebrand disease about one in
a hundred; and the recently (1994) identified factor-V Leiden may be as common as one in
twenty individuals (Crookston; King). (A detailed description of the blood-clotting process
1
and its disorders appears in Chapter 2.) Clinicians would like to understand these diseases
better, as well as to treat them effectively.
The complexity of the coagulation cascade limits the usefulness of qualitative descrip-
tions of the roles of its proteins and reactions. Such limitations apply a fortiori to any
predictions of the effects of changes to these proteins and their reactions that are based
(solely) on a set of qualitative descriptions. Here the usefulness of a computational model
comes into view: alterations of initial protein concentrations, or of reaction rates, or of
the presence or absence of certain reaction-enabling ions (like calcium) can be simulated in
a such a model, and the consequences directly observed—at least insofar as the model is
accurate.
But how should one model a system that is not completely describable in terms of
continuous processes and states, and in which qualitative information is to play a role?
The approach of this thesis is to treat coagulation as comprising interacting discrete and
continuous aspects, i.e. as hybrid. Choosing this formalism exacts a price in analyzability,
but it pays off in the form of a complete model of the coagulation cascade, in contrast
to other models (see Chapter 2). That is, many of the mathematical tools which exist
for analyzing purely discrete or purely continuous systems (stability, location of equilibria,
controllability and observability, etc.) do not exist for hybrid systems—the theory of which
is described in detail in Chapter 3; but homogeneous models cannot (properly) embrace the
breadth of the coagulation process. To capture the entire system, then, I have constructed
a hybrid-system model, which along with its simulations appears in Chapter 4. As for
the deficiencies of analysis, we shall see that for a large class of cases, they can in fact be
circumvented.
We are interested, to repeat, in treating diseases as well as in understanding them (and
non-pathological clotting), and here too the current state of the art would benefit from
a mathematical model. So, for example, the hypercoagulatory disorder factor-V Leiden,
which serves as a principal case study in this paper, is usually treated by daily visits to the
doctor’s office, where blood is drawn and protein concentrations are measured, after which
an anti-coagulant (like heparin) is administered. If certain protein concentrations exceed
2
ranges which clinical experience has found to be unsafe, the dosage is increased or decreased
as appropriate; otherwise yesterday’s dosages are re-administered.
This kind of treatment could certainly be improved upon. To begin with, the measure-
ment interval (one day) is rather large; and the method is “global” rather than local: that
is, treatment is effected by changing overall protein concentrations—throughout the entire
blood stream—rather than at the site of a thrombotic event. The difficulty here is techno-
logical, however, rather than mathematical—solution requires the appropriate sensors and
drug delivery mechanisms—which difficulty this document will ignore; in fact, this thesis
will sometimes assume that these issues are resolvable. The more interesting shortcoming of
present treatments is that they prescind away from all but the rudest qualitative knowledge
of blood clotting: too much of this protein implies a need for an anticoagulant; too little of
that protein demands a procoagulant. In Chapters 5 and 6 I shall show how a mathematical
model of coagulation can be exploited to allow much finer control of clotting, using some
tools from computer science and nonlinear control theory.
I have already alluded to the question of the model’s fidelity, which motivates the
remaining portions of this thesis. It must first be said that no complete comparison between
the data produced by the model and actual in vivo clotting is possible, since (as mentioned
above) the sensors for measuring all the protein concentrations at the site of an injury in
real time do not exist. This is not, of course, to say that the parameters (rate constants)
and structure (reactions) of this model are not the result of empirical investigation; indeed,
they are, but of countless (painstaking) experiments on the interactions of pairs or triads
or perhaps a few blood proteins. These experiments are, first of all, perforce in vitro, so
measured rate constants might vary from their in vivo cousins in virtue of (e.g.) a different
prevailing ion concentration, or of the substitution of a fluid reaction for one on a substrate.
They, secondly, depend on hypotheses about which proteins effectively interact with each
other. These considerations ensure that the model will have errors.
Since the data do not exist for a complete in vivo validation, then, how are we to
interpret the results? The simulations from the hybrid-system model (Chapter 4) should
be interpreted as demonstrating the utility of such a model, not as providing gold-standard
3
simulation results; as well as providing a mode of comparison for future clinical data and
hence a method for refining the model. The control schemes of Chapters 5 and 6 are
of course not intended to provide precise numerical values for the controllers; indeed, the
feedback controllers are intended to obviate exactly the need for such model precision (since
uncertainty will anyway be introduced, even in the case of a “perfect model,” by inter-
individual differences). A fuller discussion of the effect of model errors on the control
schemes appears in Chapters 5 and 6.
What other kinds of information can profitably be reaped from an imperfect model?
One such is the sensitivity of the model to changes in its parameters, i.e. the differential
changes in state over time from the nominal (unperturbed) trajectory, introduced by (static)
changes in rate constants. Such results do not depend on exact values of rate constants; in
fact, the sensitivity of the system reveals just how much errors in rate constants matter to
overall results. Sensitivity analysis also serves inter alia a clinical purpose, revealing where
in the clotting cascade pharmaceutical intervention will have the most effect with the least
effort. A sensitivity analysis of a large portion of the clotting cascade appears in Chapter
4.
Finally, this thesis proposes a method, exploiting some well-established machine-learning
techniques, for making use of what clinical data do exist, for simultaneously validating and
refining the model. The trick is to rely on the model where correct and on “gold standard”
data where they exist. In brief, the model is interpreted as a map from an input space
into an output space, where the former consists of, e.g., initial condition values, or again
of rate-constant values; and the latter consists of (say) thrombin concentrations at various
intervals during the course of a clot event, as predicted by the model. The output data
can then be partitioned in a semi-supervised manner, the labeled data consisting of known
inputs (e.g. initial conditions) and their corresponding outputs (e.g. “hypercoagulatory”
vs. “normal” vs. “hypocoagulatory”); and the unlabelled data comprising the remaining
input/output pairs. This technique is explored in the penultimate chapter (7).
A final chapter (8) sums up the contributions of this thesis; laments the loose ends it
4
failed to tie up; and recommends some of these as well as some additional threads to future
investigators.
1.2 Notation
The mathematical notation used in this thesis is standard; in fact, it is perhaps overly
punctilious, reflecting the author’s fondness for consistency and unambiguousnesss.
Lowercase letters from the end of the Latin alphabet (u - z) are used to represent
variables, with italic script for scalars and bold invariably reserved for vectors. In the
context of a control system, x, u, and y are always the state, input, and output, respectively.
Following the literature on feedback linearization, the Greek letters ξ and η are also used to
represent the state, after a change of variables. However, Greek-letter vectors and scalars
are not distinguished by font, since the difference between bold and non-bold Greek letters
is imperceptible in this typeface. The letter t is of course always reserved for time.
Real-valued functions, whether scalar- or vector-valued, are usually taken, as conven-
tionally, from the lowercase Latin letters f through h, plus r and s. Vector-valued functions
and vector fields are bolded as well, the difference between the two being indicated by the
argument font; hence f(x) and f(x), respectively. When these letters have been exhausted,
functions resort to the back of the Greek alphabet, capitalization there indicating vector
outputs. Other kinds of maps (i.e. not exclusively involving real numbers) occur in the
material on Petri nets, and there letters were chosen mostly on alliterative or mnemonic
considerations.
Integers are represented by the lowercase Latin letters from i to q—excluding l and o
for their likeness to 1 and 0, and k for its use as a generic rate constant—with n usually
reserved for the dimension of the state and q for the strict relative degree of the system.
(Constant) matrices and vectors are represented with capital and lowercase letters,
respectively, from the beginning of the Latin alphabet. Vectors are again bolded. In the
context of linear time-invariant systems, the usual conventions are respected: A is the state
5
matrix and B (b) is the input matrix (vector). Constant scalars are usually drawn from the
beginning of the Greek alphabet, although feedback gains are represented (per convention)
with K and the sampling interval is always symbolized T .
Calligraphic script is reserved for sets, which use capital Latin letters. Elements of
sets are then represented with the corresponding lowercase letter. Excepted are the well-
known number sets, which are rendered in blackboard bold: N,Z,R, and C for the naturals,
integers, reals, and complex numbers, respectively. The natural numbers are taken to
include 0; restrictions to the positive or negative subsets are indicated by a superscripted
+ or −. As usual, C0 is the set of continuous functions. The calligraphic N (·) is not used
for sets but to denote the nullspace of its argument.
Subscripts denote elements of a matrix or vector: di is the ith column of D; xj is the jth
element of x. Plain numerical superscripts on the other hand may indicate exponentiation,
a recursive operation, or simply a numbering, depending on context.
Differentiation is expressed as follows. Time derivatives use Newton’s notation: one,
two, or three dots over a variable for the corresponding number of derivatives, and a paren-
thetical superscripted numeral for higher derivatives. Leibniz’s notation is used for all other
standard derivatives, including the Jacobian (total derivative): ∂f/∂x—except the gradient,
which is represented with the nabla; hence ∇xφ is the row vector [∂φ/∂x1, ..., ∂φ/∂x1] of
partial derivatives.
The Lie derivative of a (scalar-valued) function h along trajectories of the vector field
f , that is (∇xh)f(x), gets its own symbol, Lfh(x). The Lie derivative of this object, say
along trajectories of g, i.e. Lg(Lfh(x)), dispenses with parentheses and is rendered sim-
ply LgLfh(x). Iterated Lie derivatives along the same vector field are written even more
concisely as Lifh(x), where i is the number of iterations.
Square brackets are put to various uses: In the usual cases they indicate vectors or
matrices; around the name of a chemical (usually a blood protein) they denote concentration
thereof; and in the context of [·, ·], they are the Lie bracket operation; that is,
[f ,g] :=∂g∂x
f − ∂f∂x
g.
6
Finally:∑
i is used for summations; a superscripted T indicates the matrix transpose;
∈ denotes set inclusions; I is reserved for the identity matrix; rk(·), Sp·, and spec· are
the matrix rank, span, and spectrum (eigenvalues), respectively, of their arguments. Except
in the case of the gradient or when otherwise explicitly stated, all vectors are assumed to
be columns.
7
Chapter 2
The Coagulation Cascade
2.1 Introduction
This chapter provides a detailed description of the biology of blood clotting, and can-
vasses previous attempts to model it mathematically. Much of our knowledge of the clotting
process comes in the form of chemical reactions that can be translated without remainder
into differential equations, but some reactions are known only qualitatively; and still other
information is essentially bivalent, like whether or not a reaction can take place in the ab-
sence of a certain ion. The heterogeneous nature of these descriptions is stressed in what
follows, since it motivates the hybrid-system model.
2.2 Preliminaries
Human blood clotting is a complicated biochemical and (arguably) rheological control
process whose normal function is to minimize the blood loss induced by vascular trauma.
Clotting is then one aspect of hemostasis—the other aspects being widening of the rele-
vant blood vessels (vasodilation) and clot dissolution (fibrinolysis)—that is, the dynamic
maintenance of balance between excessive bleeding (hæmophilia) and excessive clotting
(thrombophilia).
8
Healthy blood circulates with a set of proteins called clotting or coagulation factors,
most of which are inactive enzyme or cofactor precursors. The clotting process normally
begins when a blood vessel breakage precipitates the modification of some of these proteins,
transforming them from their unactivated (zymogen) to their activated (enzymatic) forms.
As concentrations of the latter increase, these enzymes trigger the activation of other clotting
factors—sometimes requiring first the activation of their cofactors—and so on, giving rise
to a cascade-like series of activations. (As we shall see in Chapters 3 and 4, representing
these cascades perspicuously is part of the motivation for using hybrid Petri nets, in which
this structure is immediately apparent; whereas it is very difficult indeed to read this kind
of description off a list of ODEs.) Ultimately, the glycoprotein fibrinogen is converted into a
fiber-like form (aptly named “fibrin”), which binds to the site of injury along with platelets
and another clotting factor (XIIIa), producing a clot and sealing the damaged vessel. Along
the way, other proteins serve to inhibit the activation of clotting factors and still others to
dilate the blood vessel (“vasodilators”).
The coagulation factors are listed in Table 2.1. Many of these factors have a Roman-
numerical designation, their active forms indicated by an appended letter “a.” (There is
no “factor VI” because this name was historically annexed to activated factor V.) These
are all, in some sense, pro-coagulant: a deficiency in any of these factors hinders clotting.
Prekalikrein and high molecular-weight kininogen are also pro-coagulant in this sense. On
the other side are the down-regulating proteins: antithrombin (sometimes “antithrombin
III”), proteins C and S, thrombomodulin (Tm), and tissue factor pathway inhibitor.
Factors IIa, VIIa, IXa, Xa, XIa, XIIa, as well as activated protein C (APC) and
kallikrein, are serine proteases, enzymes which function by cleaving peptide bonds in other
proteins and which have serine at their active sites. Factor XIII is also an enzyme, but
a transferase rather than a serine protease—in fact, the only enzyme of the cascade that
is not a serine protease.1 Factors Va and VIIIa, along with tissue factor (TF), protein S
(PS), thrombomodulin, and high molecular-weight kininogen (HMWK), are cofactors, re-
quired for certain enzymatic reactions (see below). Antithrombin (AT) is a serine protease1A transferase is an enzyme that promotes the transfer of a functional group from one molecule to another.
9
inhibitor (serpin), whereas tissue factor pathway inhibitor is a different kind of inhibitor
(Kunitz-type). The importance of this distinction for our purposes is that serpins inhibi-
tion is irreversible. This leaves so-called factor IV (this name is rarely used), which is not a
protein at all but calcium ions; and factor I, fibrinogen, a glycoprotein whose cleavage into
fibrin marks the terminal event of the coagulation cascade.
2.3 The Cascade
Under normal circumstances, the coagulation cascade is simultaneously initiated by two
different mechanisms whose resulting “pathways” meet at the activation of factor X into
factor Xa (thrombokinase). More recently, it has become clear that all three pathways
interact a good deal during the clotting process. In fact, the pathway appellations are in
some respects merely holdovers from early, more limited models of blood clotting; they
are presently retained primarily to indicate initiation mechanism. The traditional nomen-
clature also prevails because it provides a way of bracketing parts of the cascade for ease
of understanding, and because the distinctions still provide a valid and instructive way to
conceptualize the clotting cascade (i.e. in spite of their interactions). A more or less tradi-
tional description of the cascade in terms of pathways, then, follows. Its graphical depiction
appears in Figure 2.1.
2.3.1 Intrinsic pathway
The intrinsic pathway is triggered when vascular cell damage exposes blood plasma to
the negatively charged surface of the subendothelial tissue2—for which reason the intrinsic
pathway is also known as the “contact system.” Hageman factor (factor XII) binds to this
surface, which induces a conformation change in the bound proteins, transforming them
into their active (protease) state.
Historically, this was believed to be the only method of activation for the intrinsic path-
way. It is now, however, also believed that vascular injury precipitates a rise in plasma2The endothelium is the layer of cells that line the interior walls of blood vessels.
10
Fact
or
(ab
br.
)T
rivia
lN
am
eA
ctiv
ate
dF
orm
Init
ial
Con
c.(n
M)
Sou
rce
Ifi
bri
nogen
fib
rin
(Ia)
6000-1
3000
(Halk
ier,
1991)
IIp
roth
rom
bin
thro
mb
in(I
Ia)
1400
(Bu
ngay
etal.,
2003)
III
tiss
ue
fact
or
—0.0
05
a(B
un
gay
etal.,
2003)
IVca
lciu
mio
ns
—1.2
x10
6(G
olt
zman
,2005)
Vp
roacc
eler
inacc
eler
in(V
a)
20
(Bu
ngay
etal.,
2003)
VII
pro
conver
tin
conver
tin
(VII
a)
10/0.1
b(B
un
gay
etal.,
2003)
VII
Ianti
hæ
mop
hilia
cfa
ctor
AV
IIIa
0.7
(Bu
ngay
etal.,
2003)
IXp
lasm
ath
rom
bop
last
inco
mp
on
ent
IXa
90
(Bu
ngay
etal.,
2003)
XS
tuart
-Pro
wer
fact
or
thro
mb
okin
ase
(Xa)
170
(Bu
ngay
etal.,
2003)
XI
pla
sma
thro
mb
op
last
inante
ced
ent
XIa
30
(Bu
ngay
etal.,
2003)
XII
Hagem
an
fact
or
XII
a500
(Halk
ier,
1991)
XII
Ifi
bri
nst
ab
iliz
ing
fact
or
pla
sma
tran
sglu
tam
inase
(XII
Ia)
70
(Halk
ier,
1991)
pre
kallik
rein
(PK
)F
letc
her
fact
or
kallik
rein
500
(Halk
ier,
1991)
pro
tein
C(P
C)
—A
PC
60
(Bu
ngay
etal.
,2003)
pro
tein
S(P
S)
——
300
(Bu
ngay
etal.,
2003)
anti
thro
mb
in(A
T)
——
3400
(Bu
ngay
etal.,
2003)
thro
mb
om
od
ulin
(Tm
)—
—1
(Bu
ngay
etal.,
2003)
tiss
ue
fact
or
path
way
aka
LA
CI,
EP
I—
2.5
(Bu
ngay
etal.,
2003)
inh
ibit
or
(TF
PI)
hig
hm
ole
cula
r-w
eight
Fit
zger
ald
fact
or
—1000
(Halk
ier,
1991)
kin
inogen
(HM
WK
)
aU
pon
blo
od
ves
sel
rup
ture
;ci
rcu
lati
ng
con
centr
ati
on
is0
nM
.
bU
nact
ivate
dan
dact
ivate
dco
nce
ntr
ati
on
sre
spec
tivel
y,of
fact
or
VII
.
Tab
le2.
1:P
rim
ary
Coa
gula
tion
Fact
ors
and
thei
rP
re-I
njur
yC
once
ntra
tion
s
11
Figure 2.1: The coagulation cascade.
12
zinc concentration (Røjkjær and Schmaier, 1999), (Shariat-Madar et al., 2002), which in
turn enables the binding of HMWK and prekallikrein, and the activation of complexed
prekallikrein into its enzymatic form. Both these events are understood as threshold medi-
ated: zinc levels must exceed about 3µM for the binding reaction and 5µM for the activation
of kallikrein. The latter reaction is enhanced by the presence of factor XIIa. Even in the
absence of zinc, however, factor XIIa can activate the uncomplexed species of prekallikrein,
albeit much more slowly than the activation of prekallikrein in the PK:HMWK complex.
The serine protease kallikrein can also activate factor XII, and thus induces a positive
feedback loop. Since this reaction takes place in plasma rather than on the endothelial wall,
it is termed the “fluid-phase” activation—the negatively charged surface inducing “solid-
phase” activation. The reaction kinetics do indeed differ, since solid-phase reactions are
highly influenced, as fluid-phase reactions are not, by the geometry of surfaces.
A sufficient quantity of factor XIIa suffices to activate factor XI. Here the intrinsic
pathway starts to interact with the other pathways, since factor XI can also be activated
by feedback from the so-called common pathway (see below). Whatever its method of
activation, factor XIa (in the presence of calcium ions) cleaves factor IX, which in turn
activates factor X. This latter reaction, however, is extremely slow in the absence of factor
IXa’s cofactor, factor VIIIa. (In fact, following (Bungay et al., 2003), the present study
neglects this reaction altogether.) But factor VIII is activated by the common pathway—so
the intrinsic pathway alone is insufficient to generate blood clots.
2.3.2 Extrinsic pathway
The extrinsic pathway is initiated when the ruptured blood vessel releases tissue factor
(TF) into the plasma (for which reason it is now sometimes called the tissue-factor pathway),
which subsequently binds with unactivated factor VII (proconvertin). Now, the TF:VII
complex is inert until activation by factor Xa, and we have just finished saying that the latter
cannot be activated by the intrinsic pathway alone. The mystery is resolved by allowing
that some (small) fraction of circulating factor VII is in enzymatic (VIIa) form (Bungay
13
et al., 2003). This allowance presents no embarrassment for the non-clotting situations,
since factor VIIa is inert in the absence of its cofactor, tissue factor. Again in the presence
of calcium, the TF:VIIa complex can indeed activate factor X, as well as factor IX (reaching
over into the intrinsic pathway); and Xa in turn reciprocally activates the TF:VII complex.
This concludes the work of the extrinsic pathway.
In light of the inability of the intrinsic pathway alone to induce clotting, what might
we say about the roles of the two initiation pathways? It is now widely believed that
the extrinsic pathway serves to “kick-start” the intrinsic pathway into action via feedback
from thrombin (IIa) and thrombokinase (Xa) activating factors VIII and XI. Thus the
extrinsic pathway directly achieves minimal thrombin production but does so on the order
of seconds, whereas the intrinsic pathway generates large quantities of thrombin but on
the order of minutes (Viera Stvrtinova, 1995). (Indeed—to anticipate—this qualitative
description is congruent with the simulations from the model of this thesis: Activation by
the intrinsic pathway alone, including an equation for the [slow] activation of factor X by
factor IXa, results in a significant [i.e. normal] amount of thrombin; whereas a deficiency
of the intrinsic pathway’s factor VIII, as in hæmophilia A, results in about 1% of normal
thrombin generation.)
2.3.3 Common pathway
The common pathway begins at the activation of factor X and terminates ultimately in
the production of a fibrin clot. As we have already seen, factor Xa feeds back to activate
the TF:VII complex in the extrinsic pathway and to activate factor VIII in the intrinsic
pathway, which subsequently binds to factor IXa. These two complexes from their respective
pathways feed forward to generate more factor Xa, though again both reactions require
calcium ions. Factor X also activates factor V in the common pathway, and proceeds to
bind to it.
This complex, Xa:Va, activates the most important protein of the cascade, thrombin
(factor IIa) as follows. First, the complex binds with prothrombin (factor II). Then, in
14
the presence of calcium ions, the prothrombin part of the complex is transformed into the
intermediate (between zymogen and full enzyme) protein meizothrombin (mIIa). Finally,
the complex dissociates into Xa:Va and the protease thrombin (factor IIa). (For simplicity,
meizothrombin is omitted from Figure 2.1)
Thrombin initiates the final portion of the clotting cascade, but it also participates
in three feedback loops. It activates factors VIII and XI (in the intrinsic pathway), and
factor V in the common pathway. The Xa:Va:mIIa complex can also dissociate without
transforming meizothrombin to thrombin, in which case the free mIIa proteins can also
feed back to activate factors VIII and V, albeit at a slower rate than their activation via
thrombin.
Thrombin induces clotting by cleaving the soluble protein fibrinogen (factor I) into the
insoluble protein fibrin, which then spontaneously polymerizes into a mesh. Concurrently,
thrombin also converts factor XIII into its activated form, which stabilizes the fibrin mesh
by cross-linking with it. As platelets aggregate into the cross-linked mesh, the clot, bleeding
is stopped. Both the activation of factor XIII and fibrin are known to take place at certain
thresholds of thrombin.
2.3.4 Down-regulation
In addition to all the interactions lately discussed, thrombin also participates in its own
down-regulation. In complex with the cofactor thrombomodulin, thrombin activates the
inhibitor protein C; which, along with its cofactor, protein S, irreversibly inactivates factors
Va and VIIIa by cleaving the appropriate amino acids. The serpin antithrombin also serves
as an inhibitory “sink” by irreversibly binding up free factor XIa, meizothrombin, factor
Xa, and thrombin.
The final inhibitor we consider in our model is tissue factor pathway inhibitor, which
binds up free factor Xa, neutralizing it. This complex (TFPI:Xa) has further inhibitory
powers, however, soaking up the TF:VIIa complex. Both of these bindings are, however,
15
reversible, so they should be thought of not so much as sinks but as negative feedback loops,
preventing the concentrations of Xa and TF:VIIa from increasing overmuch.
2.3.5 Completeness
This concludes the elements of blood clotting that are covered by the model of this
thesis. Now, I have called my model “complete,” and it is insofar as the (time) evolution of
the coagulation cascade is concerned; but we have so far ignored (1) spatial considerations,
and (2) the ancillary systems of vasodilation and fibrinolysis. We consider the latter first.
Vasodilation is less well-understood than blood clotting, but the outlines are clear
enough. High molecular-weight kininogen, under the influence of factor XIIa and kallikrein,
releases the vasodilator bradykinin. Likewise, a low molecular-weight kininogen is cleaved
by so-called tissue kallikrein to release another kinin, kallidin. The kinins induce the relax-
ation of the local smooth muscle, dilating the blood vessel in the area. Collectively, these
proteins and their interactions are known as the kinin-kallikrein system.
The fibrinolytic system is initiated by the injury-induced release of the serine proteases
tissue-plasminogen activator (tPA) and urokinase from the vascular wall. Both (the former
more efficaciously) cleave the fibrin-bound zymogen plasminogen into plasmin, yet another
serine protease, which degrades fibrin. Urokinase and tPA are in turn down-regulated
by the serpins plasminogen activator inhibitor 1 and 2; while plasmin is inactivated by
the serpin α2-antiplasmin. Thrombin also plays a role in down-regulation by activating
the appropriately named thrombin-activatable fibrinolysis inhibitor (TAFI). It functions by
rendering fibrin a less potent cofactor for the activation of plasminogen.
These two pathways were ignored because the author did not want to needlessly com-
plicate the model; and this choice is justified by the fact that neither one feeds back to
affect the coagulation cascade, either in the form of explicit feedback loops or (as far as
is known) in the form of substrate competition. Their exclusion, then, should in no way
affect our simulations of clot formation or thrombin generation under various physiological
conditions; nor the control thereof. Nevertheless, as we shall see in the Chapter 4, the model
16
was constructed in such a way as to allow the easy incorporation of addition elements; and
furthermore, the addition of these pathways could be constructed from the same primitives
as the rest of the model: a series of possibly threshold-controlled bindings and activations.
The other major feature we have ignored until now is the effect of spatial arrangements
on the reactions. Since certain reactions take place on the endothelial wall, others on the
surface of platelets, and still others extracellularly, it has been claimed that a faithful model
of blood clotting must separate these reactions from each other, modeling their interaction
through spatial-diffusion equations (Panteleev et al., 2006). Likewise, von Willebrand fac-
tor, a blood glycoprotein with which (unactivated) factor VIII circulates in complex, and
which protects the latter from degradation, binds to platelets most efficiently under high
shear stress, which is obviously a function of spatial considerations (e.g. diameter of the
vasculature at the site of injury). Or again, clot dissolution is a matter of rheology (i.e., in-
volving flows, particularly of substances with varying viscosities) as well as of biochemistry.
These “spatial effects,” as I have called them, have two unfortunate consequences for
models. The first is that accounting for diffusion requires transforming the ordinary differen-
tial equations governing the time-evolution of protein concentration into partial differential
equations governing spatial evolution, as well (by introducing partial derivatives with re-
spect to each of the three spatial dimensions). The second unfortunate consequence is that
interactions on surfaces rather than in fluids vitiates the law of mass action (see Section 2.5
below)—although the models of coagulation discussed below which do model spatial effects
seem not to have appreciated this fact—meaning that (in short) reactions may not be sim-
ply proportional to the (volumetric) concentrations of reactants. We justify the exclusion
of these effects from the present model, then, on the grounds that (1) they have generally
been neglected in the other models in the literature (again see Section 2.5 below); (2) many
of the data we use for validation come in fact from an in vitro environment, where spatial
effects do not play any significant role; (3) while not completely accurate, the fluid-based
reactions are nevertheless an approximation of the actual in vivo process; and finally (4)
although PDEs are simulatable, restricting the model to ODEs provides great traction in
analysis and control, as we shall see in Chapters 5 and 6.
17
Finally, and obviously, in light of the preceding, we do not mean by “complete” “free
from any error.” In particular, the rate constants (and other parameters used in the model)
are empirical, with both the negative and positive connotations of that word: They are
indeed drawn from the literature, each one the result of the detailed measurements of
some biochemical experiment or other, but there is great variance among them. Some rate
constants vary by as much as orders of magnitude from one study to the next (Panteleev
et al., 2006) (or cf. the rate constants of (Bungay et al., 2003) and (Luan et al., 2007),
and their references). Thus we stress that the model, while it matches such experimental
data as exist for the whole cascade, is underdetermined by those data. The point of this
dissertation is to show how what information we do know, about any part of the cascade,
can be incorporated into a single model; how the simulations of this model can then be used
with clinical experiments by iterating between the two to refine, seriatim, our knowledge
of coagulation and the model; and furthermore to use this information to design clinical
interventions by treating the cascade and a drug as together composing a control system.
2.4 Diseases of Coagulation
Many things can go wrong with the coagulation system and there are correspondingly
many diseases. Two in particular play a large role in this dissertation, as the subject of our
simulations and control schemes; we discuss these first. In what follows we draw primarily
on (Crookston) and (King).
The best-known blood-clotting disorder is almost certainly hæmophilia, acquaintance
with which dates from the Biblical era (Rosner, 1995), although its modern diagnosis waited
until the nineteenth century. Its most common incarnation is factor-VIII deficiency, i.e.
hæmophilia A, which affects one in about every five to ten thousand individuals, almost
all of them male since it is an X-linked trait. Treatment can be either prophylactic or in
response to an injury, but in either case involves the injection of some replacement form of
factor VIII (either isolated from serum or “recombinant,” i.e. bioengineered). Symptoms
obviously include prolonged bleeding, but also joint and muscle hemorrhage. The severity
18
of the diseases varies with the extent of the deficiency of factor VIII: 5% of normal levels
has mild consequences; 1-5%, moderate; less than 1%, severe.
A less well known but in fact much more common bleeding disorder is the recently
discovered factor-V Leiden, the heterozygous form of which may affect as much as five
percent of the white population. Factor-V Leiden is a genetic disorder in which the amino
acid arginine505 of the heavy chain of factor V is replaced by glutamine505, which effectively
disables APC-mediated cleavage of factor V at this point. (There are in fact other forms of
APC resistance, but factor-V Leiden accounts for the overwhelming majority [95%] of cases.)
The consequence is hypercoagulatory behavior, increasing the odds of venous thrombosis
(clotting within a vein) by as much as a factor of ten for heterozygotes and of 80 for
homozygotes. Factor-V Leiden was first identified in 1994, and perhaps as a result is not
treated prophylactically—though of course its thromboemboli are treated, according to the
usual protocol, viz. administration of an anticoagulant like warfarin or heparin. Estrogen-
based oral contraceptives are also known to increase hypercoagulatory risk.
Hæmophilia B is less common than its aforementioned cousin, affecting one male in
about thirty thousand, and resulting from a deficiency of factor IX; its severity varies just
as hæmophilia A, mutatis mutandis for factors. Hæmophilia C, i.e. factor XI deficiency,
is even less common in the general population (one in one-hundred thousand), although
it is fairly prevalent among the Ashkenazim. The most common hemorrhagic disorder is
not any of the hæmophilias, however, but von Willebrand’s disease, a deficiency (either in
quantity or quality) of von Willebrand factor, affecting upwards of 2% of the population.
Von Willenbrand’s disease discourages clotting by hindering platelet adhesion; but also by
failing to protect the zymogen factor VIII from degradation in pre-clotting conditions, which
symptom sometimes disguises the disease as hæmophilia A.
There are of course many other coagulation diseases, for which the interested reader is
encouraged to consult (King) and especially (Crookston); but they are not relevant to this
thesis.
19
2.5 Mathematical Models
Many mathematical models of blood clotting have been constructed, most in lesser detail
than the present study but a few covering aspects of the process that I have neglected. We
discuss the relevant exemplars.
Most models, like ours, are concerned primarily with the coagulation cascade (i.e. ig-
nore spatial and rheological effects). These approaches generally model the time evolution
of protein concentrations with a series of nonlinear, ordinary differential equations, deriving
from the law of mass action, the Michaelis-Menten equations, or other chemical considera-
tions. Due to the extremely complicated nature of blood clotting, these attempts usually
focus on small subsets of the entire process. So, for example, Butenas et al. (2004) model
only the interactions of coagulation factors II, IX, and X; Panteleev et al. (2002) and Adams
et al. (2002) model the extrinsic pathway; Qiao et al. (2004) model a portion of the common
pathway and a small part of the intrinsic pathway; and Xu et al. (2002) and Leipold et al.
(1995) model some of the interactions of factors II, V, VII, VIII, IX, and X.
The most complete and ambitious models of the cascade are (Bungay et al., 2003),
and (Luan et al., 2007), which use systems of 73 and 98 (respectively) coupled, nonlinear,
differential equations to describe all of the extrinsic pathway, a large portion of the (ulterior)
intrinsic pathway, and the common pathway up through thrombin production. Luan et al.
(2007) include more variables as a consequence of including the effects of platelets, but
otherwise the models cover essentially the same portions of the coagulation cascade.
Note the absence of the early intrinsic pathway—HMWK, PK, factor XII—from any of
these models, which is at least in part a consequence of its being poorly understood. Now,
Kogan et al. (2001) do build a model of the intrinsic and common pathways in the service of
matching the results of a clinical test, the so-called activated partial-thromboplastin time.
However, they cannot in fact match clinical aPTT times, nor are their choices of intrinsic-
pathway chemical reactions uncontentious: e.g., they do not include HMWK, nor is there
mention of zinc concentrations (see above, Section 2.3.1). Thus in the model of this thesis,
20
as we shall see in Chapter 4, the early intrinsic pathway was modeled qualitatively as a
hybrid system.
An important distinction among these ODE models is whether they rely strictly on
the law of mass action or whether, on the contrary, they make use of the Michaelis-Menten
approximation. The former states that the rate at which (elementary) reactions take place is
proportional to the product of the chemical activities of the reactants. The proportionality
constant is empirically determined and called (appropriately) the “rate constant” for that
reaction. Now, the law of mass action is true of every (free-diffusion) reaction3 and at the
protein-concentration levels of blood clotting, we can substitute the word “concentrations”
for “chemical activities” in the formulation just given. That is not to say, however, that the
rate of product formation in, e.g., an enzymatic reaction is proportional to the product of the
concentrations of the enzyme and substrate, because such reactions involve an intermediate
reaction in which an enzyme-substrate complex forms. This is written:
E + Skon−−−−koff
E : S kcat−−→ E + P (2.1)
where E an enzyme (say, factor Xa), S is a substrate (say, factor V), E:S is the aforesaid
complex, P is the product (in this case, factor Va), and koff, kon, and kcat are the rate
constants. Nevertheless, the law of mass action holds for each elementary reaction, so here
generation of the product proceeds at a rate proportional to the concentration of the E:S
complex, rather than those of the initial reactants.
Now, on the other hand, in many enzymatic reactions, the substrate concentration
greatly exceeds the enzyme concentration (high enzyme efficacy making only a compara-
tively small amount necessary for the reaction), in which case the change in concentration of
the intermediate enzyme-substrate complex is approximately zero, and the rate of product
formation (v) can be written more simply as:
v = kcat[E]0[S]
Km + [S], (2.2)
with Km = koff+kcat
kon. These so-called Michaelis-Menten kinetics can simplify our equations,
and offer another advantage as well: chemists often do not bother to determine the on- and3In fact, the law of mass action can be derived from statistical mechanics
21
off-rates separately, measuring instead the more easily acquired Km. (An instance of this
will show up in Chapter 5.) However, as we have seen, this approximation is only valid
when the substrate is in great excess over the enzyme, which we cannot guarantee over
the course of the coagulation process. Furthermore, adoption of MM kinetics violates the
conservation of mass implicit in mass-action kinetics, which conservation makes for some
nice mathematical properties about which we shall have something to say in Chapter 6.
It was mentioned above (Section 2.3.5) that the model of this dissertation ignores the
spatial effects on reactions. Recent models by two groups have explicitly included these
effects: Panteleev et al. (2006) include spatial diffusion in their model of the extrinsic
pathway and parts of the common pathway. The authors do not restrict themselves to
mass-action kinetics, using instead MM kinetics and another, similar approximation where
appropriate. The diffusion terms of course result in partial, rather than ordinary, differential
equations, as we have lately appreciated. A similar approach, modeling a more or less
coextensive portion of the cascade, appears in the thesis (Mohan, 2005), which also includes
a viscoelastic-fluid model of flow across the clot. The goal of neither of these models is to
capture the entire coagulation process, but rather to render existing, incomplete models
more realistic.
There are also computational (one hesitates to say “mathematical”) models of blood
clotting which abjure equations altogether. Mounts and Leibman (1997) use a variant of the
Petri-net formalism (as I do: see Chapter 3) to describe the coagulation cascade. Since (or-
dinary) Petri nets have a discrete state space, the tokens (again see the description of Petri
nets in Chapter 3) of the model are used to represent some number of molecules of a given
substance. The authors describe their model as “qualitative,” but in fact their discretiza-
tion is fine enough to give rather precise results. Finally, in a similar vein, Signorini and
Gruessay (2004) take an object-oriented approach, specifying the structure and dynamics of
the cascade by treating each factor as a message-passing object. The order of the message
passing is specified by an “activity diagram,” which in fact is more or less conceptually
identical to a Petri net, with mechanisms for modeling concurrency and synchronization.
22
Chapter 3
Hybrid Systems
3.1 Introduction
Classical control theory and system modeling have focused on systems with purely
continuous dynamics and those with purely discrete dynamics. However, many real-world
systems necessarily involve both continuous and discrete components, or are best modeled
as interacting continuous and discrete subsystems. These systems are called hybrid sys-
tems, and have motivated a great deal of research in the last two decades. What follows is
a somewhat pedantic introduction for readers completely unfamiliar with hybrid systems;
initiates should skip to Section 3.3 on hybrid Petri nets. Readers looking for more infor-
mation are encouraged to download the excellent but as-yet unpublished book on hybrid
systems, (Lygeros et al., 2001).
A system may be considered hybrid either because the state space consists of both
continuous and discrete components, or because the dynamics manifest both continuous-
time and discrete-time behaviors. Consider, for example, the by now well known example of
a thermostat. (The following treatment of the classic thermostat model was adapted with
little change from an example in (Lygeros, 2004).) Suppose the change in temperature is
governed by one differential equation when the heat is on,
x = α(85− x); (3.1)
23
Figure 3.1: A hybrid system model of a thermostat; see text for details.
but when the state (temperature) crosses a certain threshold, say 75F, the heat is turned
off and a new differential equation applies:
x = −αx. (3.2)
When the temperature falls below 70F, the heat goes back on and Eq. 3.1 again
applies. Thus the temperature increases exponentially toward 85F until it hits a boundary
in the state space, at which point the heat is turned off and the temperature declines, again
exponentially, toward zero degrees. The various regions where different sets of continuous-
time dynamics obtain may be modeled as different discrete states of the system; hence, the
entire network may be thought of as the first type of hybrid system, where the state is
jointly defined by a continuous variable (the temperature) and a discrete state (indicating
whether or not the heat is on).
The model is illustrated in Figure 3.1. The system is very much like a finite state
machine, but additionally a set of continuous dynamics (a differential equation) is associated
with each of the discrete states. The arrow labeled x = 72 indicates that the initial state is
(x = 72, q = off), where the pair (x, q) defines the state, with x ∈ X , the continuous state
space, and q ∈ Q, the discrete state space. The arrows connecting discrete states are, as
24
in finite state machines, called “edges,” and are written (q, q′), where the edge starts at q
and ends at q′. The edges are labeled with the so-called guard conditions (x = 70, x = 75):
given the state (x, q), if x belongs to the set specified by the guard associated with the edge
(q, q′), then the system may transition to the discrete state q′. The continuous variable may
in general also be reset at a discrete transition, but all of the reset maps of the thermostat
model are the identity map.
Finally, a domain D(q) (x ≥ 70, x ≤ 75) is associated with each discrete state. When
the continuous state reaches the boundary of the domain, a discrete transition is forced—
unless no guard condition is satisfied, in which case the system is “blocked” and stops.
Alternatively, the guard may be enabled before the trajectory reaches the boundary of
the domain, in which case the discrete transition may but need not occur, and the model
becomes non-deterministic. This point is made for the sake of generality; in the thermostat
example, the guard conditions and domain have been written so as to preclude both blocking
and non-determinism. The same is true of the blood clotting model presented in this paper.
The thermostat may be modeled as a hybrid system because, again, it has discrete
states as well as continuous states and evolves in continuous time. However it should be
noted that hybrid systems may also arise in the interaction of discrete-time systems with
continuous dynamics. So, for example, we may wish to model the interaction of a purely
continuous system like a chemical batch process with a digital controller which has an
essentially continuous state space but evolves in discrete time. The blood clotting model of
this paper is of the first type.
3.2 Types of Hybrid-System Investigation
Hybrid systems arise in numerous other contexts, among them chemical batch processes
(Engell et al., 2000); road-traffic controllers (Czogalla et al., 2002), (Horowitz and Varaiya,
2000); air traffic control (Oishi et al., 2002); robotic control (Fierro et al., 2001),(Schlegl
et al., 2002); automotive applications (Pettersson and Lennarton, 2003), (Antsaklis and
Koutsoukos, 2003); embedded systems (Neuendorffer, 2004); and biological systems (Ghosh
25
and Tomlin, 2001),(Chen and Hofestadt, 2003),(Matsuno et al., 2000),(Matsuno et al.,
2003), (Jong et al., 2003). In all of these examples, there is a variety of questions we may
be interested in asking, and which recent research has attempted to find ways of answering.
We provide a brief overview here in order to give the reader a flavor of the field.
3.2.1 Modeling and simulation
Perhaps the most obvious of these is modeling and simulation. For hybrid systems in
particular, formal analysis is restricted to the simplest of systems or to highly circumscribed
cases, so a powerful alternative is to construct a model of the system and simulate its
behavior rather than perform an exhaustive analysis. However, whereas a great variety of
modeling and simulation tools exists for purely continuous or purely discrete dynamics, only
recently have such tools been developed specifically for hybrid systems. The extent of the
novel modeling and simulation issues associated with hybrid systems are beyond the scope
of this paper, but three should be mentioned because of their relevance and their ubiquity.
The first is choice of representation. Whereas purely discrete and purely continuous
dynamics have fairly well-established representations from the computer science and control
theory disciplines (respectively), a standard framework for modeling hybrid systems has
not yet arisen. Hybrid systems also come in a host of flavors, and ideally a simulation
or modeling program should be able to accommodate all of these. Lygeros (2004) has
emphasized that a hybrid-system modeling language should be descriptive, in the sense of
being able to model a wide range of continuous and discrete dynamics and their interactions,
and to accommodate stochasticity in a variety of contexts; composable from smaller units
into larger networks; and abstractable, in the sense of being able to cash out composite
model specifications in terms of component specifications as well as determine composite-
level behavior via knowledge of component behavior.
The second simulation concern is the development of accurate and efficient numerical
integration techniques. In a familiar problem from the hybrid-systems literature, imprecise
numerical integration triggers a discrete event—and hence, perhaps, a new set of differential
26
(state) equations—in the simulation, where no such event occurs in the actual system. Pu-
tative solutions which simply shrink the step size, however, can greatly increase simulation
time, and moreover are not per se a guarantee of eliminating this type of simulation error.
This issue will arise again in the context of the present study (see Section 8.2).
The third and final simulation issue to be discussed here is the problem of so-called
“Zeno” systems, in which trajectories of the system take an infinite number of discrete
transitions in finite time. In such cases, simulation time “stops,” and the simulation termi-
nates only when the system hangs. The well-known bouncing-ball hybrid system exhibits
this type of behavior: the “moving up” and “moving down” state alternate increasingly
faster, according to a geometric series that converges in the limit but will never converge
in simulation. Even in non-Zeno systems, existence and uniqueness of solutions is not in
general guaranteed for hybrid systems, and special care must be taken in their simulation.
3.2.2 Verification and decidability
A second type of question we may be interested in asking about a particular hybrid
system is known as verification. Verification is the formal proving of certain (interesting)
system properties, given the system and a range of inputs. There are two variations: algo-
rithmic verification (“model checking”) and deductive verification, where the former uses
search techniques and the latter involves the construction of a formal proof (Kowalewski,
2002). In both cases, the property of interest is usually reachability. In general, however,
reachability analyses on hybrid systems are prohibitively difficult. Results have been con-
fined to small classes of highly circumscribed systems (e.g., timed automata and rectangular
automata (Henzinger et al., 1998)). This problem, too, will return to haunt our model. Ver-
ification is also sometimes performed vis-a-vis the stability of the system; we may want to
ask, say, if the closed-loop system is asymptotically stable (Lygeros, 2004).
It has also proven useful for analysis techniques, particularly reachability, to perform
abstractions on systems; i.e., to replace a complicated hybrid system model with a less
complicated one in which properties previously difficult or impossible to prove are rendered
27
tractable. The general procedure is to construct a simplified model which contains the
behavior of the original system as well as some new behavior, an artifact of the abstraction.
This system is then tested for some appropriately abstracted version of the original property
to be tested (e.g. reachability); for example, if a state can be shown to be unreachable in
the abstracted system, then it has been shown to be unreachable as well in the original
system (Kowalewski, 2002),(Antsaklis and Koutsoukos, 2003).
A related question is that of decidability: whether a problem can be answered, affirma-
tively or negatively, by some algorithm in finite time. Specifically, in the present case, the
question is whether or not a system can be verified in finite time; and more specifically,
whether or not the reachability of some state(s) or region in the state space can be com-
puted in finite time. It should be noted that reachability analyses can be fruitfully pursued
even for undecidable systems, since the particular (say) state about which reachability is to
be determined may be decidable, even though reachability in general is not (Kowalewski,
2002),(Henzinger et al., 1998).
3.2.3 Controller synthesis
A third type of question that can be asked about hybrid systems concerns controller
synthesis. Controller specifications in hybrid systems are often given in terms of temporal
logic. For instance, we may require that all trajectories of a system remain within a set of
states F ⊆ X×Q, where X and Q are the continuous and discrete state spaces, respectively.
The condition is written as
((q, x) ∈ F), (3.3)
where q ∈ Q and x ∈ X are the discrete and continuous state, respectively. Or we may
insist that the trajectory eventually reach some set of states F , written
♦((q, x) ∈ F). (3.4)
Designing a controller then amounts to picking a set of inputs from the input space for each
state of the system such that the specification of interest is satisfied. There are various
techniques for performing this task. For example, both the theory of optimal control and
28
game-theoretic approaches can be used to derive the Hamilton-Jacobi partial differential
equations whose solutions are the boundaries of the reachable sets. These can then be
solved approximately, providing (real-time) feedback-control laws which provably satisfy
the specifications (Lygeros et al., 2001).
3.3 Hybrid Petri Nets
There are numerous hybrid system modeling languages (see, for example, (Alur
et al.),(Chutinan et al.),(Dang and Maler),(Henzinger et al.),(Miyano et al.), (Lee
et al.),(Mat),(Daws et al.),(de Moura et al.),(Deshpande et al.), (Drath, 2002), and (Larsen
et al.)) and in choosing among them the following constraints were considered. First and
foremost, of the different types of hybrid-system investigations canvassed above, our present
focus is primarily simulation. (Why are we unconcerned with the control problem, the im-
portance of which, in connection with disease treatment, we saw in Chapter 1? We discuss
this in Chapters 5 and 6, but to anticipate: the high dimension of the state space precludes
the standard hybrid-systems control techniques, so an alternative based on decomposing
the system is pursued instead.)
A second major design focus was ensuring that the model be easily and intuitively
modifiable, not just in parameter settings like concentrations of proteins, but in structure
as well, so that new (say) proteins and reactions can be incorporated painlessly into it. An
ideal model will also provide a perspicuous representation of the system of interest. This
is especially significant in the present case since is intended in part for use in a clinical
setting by biologists who may have little or no familiarity with programming languages.
This constraint obviously militates in favor of a graphical modeling framework.
A third and final consideration is that the clotting cascade consists in large part of
numerous similar reactions among different clotting factors. The reactions often involve
reactants which play no roles in any other reactions or events. These two facts suggest an
object-oriented approach to modeling, since this affords methods of abstraction, reuse, in-
29
Figure 3.2: A very simple Discrete Petri Net (DPN) with purely discrete components. The
input and test arcs’ respective resource requirements were satisfied, but the inhibitory arc’s
requirement is not—so the transition is enabled.
formation hiding, and inheritance. (For another object-oriented approach to blood clotting,
see (Signorini and Gruessay, 2004).)
A modeling language which meets all of these constraints is the hybrid Petri net (HPN).
Petri nets tout court are a graphical modeling formalism for distributed discrete-event sys-
tems which generalize automata theory to allow notions of concurrency and resource con-
sumption; their hybrid cousins are a further generalization to include notions of continuous
state and continuous time. We begin with an informal description of Petri nets, followed
by a formal exposition of the network semantics, and then move on to HPNs.
An ordinary Petri net comprises three kinds of components: transitions, places, and
directed arcs (see Figure 3.2). Places (drawn as circles) represent discrete quantities in
virtue of the number of “tokens” (n) they carry (n ∈ N), and events are modeled as firings
of transitions (drawn as black rectangles). In the figure, tokens are drawn as little black
circles which live in places. The distribution of tokens over all the places in the net is called
the marking of the net. The marking changes when transitions fire and tokens are consumed
from the input places and produced at the output places.
A transition is enabled if and only if each place connected via an input arc contains as
30
Figure 3.3: The simple Discrete Petri Net (DPN) of Figure 3.2, advanced by one step: the
input arc’s place has been depleted by its resource requirement (1 token), and the output
arc’s place has been augmented by its weight (1 token).
many tokens as the “resource requirement” of its corresponding arc. If an arc is unlabeled,
its resource requirement is assumed to be unity. When a transition is enabled, it “fires,”
meaning that tokens are consumed from input places according to the resource requirement,
and produced in the output place(s) according to the weight(s) associated with the outgoing
arc(s); since the outgoing arc is unlabeled in the figure, the output is assumed to be a single
token.
Petri Nets are often outfitted additionally with “test” input arcs, which function ex-
actly like normal arcs except that tokens are not consumed when a transition fires; and
“inhibitory” input arcs, which enable transitions just in the case that the input place has
fewer tokens than the resource requirements (again with no corresponding token consump-
tion). These also appear in Figure 3.2. The test arc is drawn as a dashed arrow; since the
preceding place contains two tokens, it satisfies the resource requirement given by the test
arc’s weight. Similarly, the input place to the inhibitory arc, drawn as an arrow ending in
an open circle, contains fewer than three tokens, so it does not inhibit the transition. The
firing is determined by the conjunction of these conditions; the Petri net in Figure 3.2 will
thus transition to the one shown in Figure 3.3.
31
Definition 3.3.1. Discrete Petri Net (DPN). A DPN is a tuple (P,S,A,W,M0),
where:
• P is a set of discrete places.
• S is a set of discrete transitions.
• A is a set of weighted directed arcs which connect places to transitions and transitions
to places; i.e. A ⊆ (P × S) ∪ (S × P).
• W : A 7→ N+ maps each arc a ∈ A to a weight w from the positive integers.
• M0: P 7→ N is the initial marking of the network, which gives the original token
distribution in the network via a map from each place p ∈ P in the net to the natural
numbers.
The meaning of an arc differs according to whether it connects places to transitions or
vice versa, and on which of three flavors a member of the former set comes in: test arcs E ,
inhibitory arcs I, or resource arcs R.
Definition 3.3.2. DPN Arcs. A = ∗A ∪ A∗, where
• ∗A ⊆ (P × S) are input arcs, and
• A∗ ⊆ (S × P ) are output arcs.
Furthermore, ∗A = E ∪ I ∪ R
DPNs have a well specified real-time execution semantics where the next state function
is specified by the firing rule. In order to simulate the dynamic behavior of a system, a
marking of the DPN is changed according to the following firing rule:
Definition 3.3.3. DPN Execution Semantics.
• A transition s ∈ S is said to be enabled if:
32
1. the source place p of each inhibitory arc i ∈ I of s has (strictly) fewer tokens
than wps, and
2. the source place of each test and resource arc (e ∈ E and r ∈ R, respectively)
contains at least wps tokens;
where in each case wps is the weight of the input arc from each source place p to s.
• The firing of an enabled transition, s, removes wps tokens from the source, p, of each
resource arc, and places wsp tokens in each output place p.
Notice that the semantics associated with the enable condition make reference only to
the input arcs ∗A, whereas the firing semantics invoke both input and output arcs. It should
also be pointed out that the definition implies that transitions which have no input arcs are
always enabled.
Transition firings take place as soon as their resource requirements are satisfied; i.e.,
time does not pass, even through successive transitions; or we might say, more accurately,
that so far the DPN has no time concept. Of course, we may want to model not simply
the sequencing of events but the time it takes for the events to transpire. In this case we
can assign delay times to the transitions: a transition with a delay of τ seconds will fire
exactly τ seconds after its resource requirements have been met. If in the meantime the
resources are depleted and the requirement is no longer fulfilled, then the transition will not
fire. Time flows whenever none of the transitions is firing, which means that each transition
is either not enabled or is enabled but experiencing a delay. More formally, we need to
augment the execution semantics as follows:
Definition 3.3.4. Time. A DPN may be augmented with a time concept, where time
t ∈ R. Time stops running (i.e. increasing from t0 = 0) whenever a (discrete) transition
fires.
Definition 3.3.5. Transition delay. Associate to each transition sj ∈ S a delay τj ∈ R.
The firing of an enabled transition takes place at time t∗ + τ , where the transition was
enabled at time t∗ and remained enabled throughout the interval [t∗, t∗ + τ ].
33
Assigning a delay of zero seconds to a transition restores the original execution seman-
tics, i.e. makes firing take place as soon as the transition is enabled.
In the present study, the Petri net language was additionally required to represent
continuous-time events and a continuous state space. Hybrid Petri nets meet these re-
quirements by providing, respectively, continuous transitions and continuous places with
real-valued “token fluid.”
The definition deserves some preparatory remarks: (1) In contrast to a DPN, an HPN
marks its continuous places with real numbers, in addition to assigning integers to the
discrete places. (2) No weights are assigned to arcs which link continuous places to con-
tinuous transitions, or which link continuous transitions to continuous places (since input
and output are governed by the differential equation associated with the transition). Fur-
thermore, the weight maps assign real or natural numbers where appropriate. (3) Neither
resource arcs nor output arcs can join discrete places with continuous transitions (since the
places’ requirement of discrete state is incompatible with the transitions’ requirement of
continuous state change). In contrast, continuous places and transitions can only be joined
by resource or output arcs. Figure 3.5 summarizes these constraints. (4) The meanings
of the arc-starring convention and of E , I, and R are the same as above. Notice that the
continuous input arcs of ∗Ac are not members of any of these sets.
Formally:
Definition 3.3.6. Hybrid Petri Net (HPN). A hybrid Petri net is a tuple
(P,S,A,Wc,Wd,Mc0,Md0, T, F ), where:
• P = Pc ∪ Pd is a set of continuous places and discrete places.
• S = Sc ∪ Sd is a set of continuous transitions and discrete transitions.
• A ⊆ (P × S) ∪ (S × P ) is a set of weighted directed arcs. Furthermore,
– Ad = a ∈ A|a ∈ (Pd × Sd) ∪ (Sd × Pd) ⊆ (E ∪ I ∪ R ∪Ad∗);
– Adc = a ∈ A|a ∈ (Pd × Sc) ∪ (Sc × Pd) ⊆ (E ∪ I);
34
Figure 3.4: A very simple Petri net with purely continuous components (CPN). The rate
at which “token fluid” leaves m1 is the rate at which it accumulates in m2, which in this
example is m2(m1 + 1)
– Acd = a ∈ A|a ∈ (Pc × Sd) ∪ (Sd × Pc) ⊆ (E ∪ I ∪ R ∪Acd∗);
– Ac = a ∈ A|a ∈ (Pc × Sc) ∪ (Sc × Pc); and
– A = Ad ∪ Adc ∪ Acd ∪ Ac.
• Wc : Acd 7→ R+ is the continuous weight map.
• Wd : (Ad ∪ Adc) 7→ N+ is the discrete weight map.
• Mc0: Pc 7→ R is the initial token-fluid marking of the network.
• Md0: Pd 7→ N is the initial token marking of the network.
• T : Sd 7→ (R+∪0) maps each discrete transition s ∈ Sd to a non-negative real-valued
delay τ .
• F : Sc 7→ C0 maps each continuous transition s ∈ Sc to a function f(m1, ...,mn) of
the network markings, from the space of continuous functions.
The fundamental addition to the execution semantics is continuous token-fluid flow. As
shown in Figure 3.4, a continuous transition fires continuously according to a “firing speed”
(i.e. a differential equation) which defines the speed of consumption and production of fluid
from the various input and output places, respectively, associated with it. Token fluid leaves
the input places at this rate and enters the output places at the same rate.
Definition 3.3.7. HPN Execution Semantics.
• The enabling of an HPN is identical to that of a DPN, given in Def. 3.3.3.
35
• The firing for all arcs asp ∈ (Ad ∪ Adc ∪ Acd) is identical to that of a DPN, given in
Def. 3.3.3.
• The firing for all continuous input arcs ∗aps ∈ ∗Ac from place p to transition s is
given by the equation
dmp(t)dt
= −fs(m1(t), ...,mn(t)), (3.5)
where mp(t) ∈ Mc is the marking associated with the input place p, and fs is the
function associated with the transition s.
• The firing for all continuous output arcs a∗sp ∈ Ac∗ is given by the equation
dmp(t)dt
= fs(m1(t), ...,mn(t)), (3.6)
where mp(t) ∈ Mc is the marking associated with the output place p, and fs is the
function associated with the transition s.
The reader should take care to note that the enable semantics from Def. 3.3.3 apply
only to arcs ∗a ∈ E ∪ I ∪ R = (∗Ad ∪ ∗Adc ∪ ∗Acd), and that as a consequence, HPN
transitions which are fed only by continuous input arcs ∗a ∈ ∗Ac or by no arcs at all are
always enabled. Additionally, Def. 3.3.3 refers only to “tokens,” but in Def. 3.3.7 it is
assumed that this be taken as either tokens or token fluid, as the case may be.
3.4 Hybrid Systems in Biology
We conclude with a brief survey of some (related) recent work in applying hybrid models
to biological systems.
In (Ghosh and Tomlin, 2001), a model of cell differentiation is constructed. The authors
represent the spatiotemporal evolution of protein concentrations within a cell by ordinary
differential equations, derived from a simplified form of the chemical kinetics. These dynam-
ics are turned on and off by discrete switches, which are triggered by protein concentrations
reaching set thresholds. In both these respects, this model is like the blood-clotting model
36
Figure 3.5: A summary of the various types of arcs in an HPN; cf. Defs. 3.3.6 and 3.3.7.
37
of this thesis. On the other hand, the model assumes that cells interact only with their
six neighbors in their planar hexagonal array. The authors produce various simulations for
reasonably sized grids, for comparison with biological data; and prove the existence and
attraction of certain equilibria; but the latter results are restricted to a two-cell network.
This is a consequence of the restriction, alluded to above, that high state dimensionality
imposes on reachability results. In fact (and interestingly), the authors claim that this
restriction can be overcome with a model checker (see Section 3.2.2 above), since the con-
tinuous dynamics of their model admit analytic solution. This is certainly not the case in
the coagulation model; i.e. no analytic solutions to its ODEs are possible.
Chen and Hofestadt (2003) present a methodology for the modeling of intracellular
metabolic networks and their regulation by genes, including a case study on the urea cycle.
The paper advocates a hybrid model for these metabolic networks in virtue of its ability
to incorporate qualitative and quantitative data into a single framework—a consideration
which (among others), as discussed in the introduction to this thesis, motived this author’s
adoption of a hybrid model. Here, once again, the continuous dynamics are generally a
consequence of chemical kinetics, with continuous state variables representing species (not,
however, necessarily protein) concentrations. The authors also consider spatial dynamics,
however, including diffusion transportation, which is neglected in the coagulation model
(for reasons that will be explained in the next chapter, 4). The discrete aspects coincide
with the gene regulation, since (1) these processes are less well understood, and (2) many
of them anyway very closely resemble switches. Finally, and again for very similar reasons,
Chen and Hofestadt (2003) choose hybrid Petri nets as their modeling language.
Similar approaches are taken in the work of Matsuno and friends, (Matsuno et al.,
2000) and (Matsuno et al., 2003). In (Matsuno et al., 2000), genetic regulatory networks are
considered, in particular the genetic switching of the lambda phage, a virus particle, between
lysis and lysogeny (two different methods of virus reproduction). Again HPNs are employed.
Here, the authors additionally stress the ability of HPNs easily to model stochastic behavior
and to be constructed hierarchically. In (Matsuno et al., 2003), the scope is broadened to
biopathways in general; case studies of circadian-rhythm regulation and Fas-ligand-induced
38
apoptosis (programmed cell death) are modeled, again with a variant on the hybrid Petri
net. Here the authors also stress the advantage in terms of perspicuousness of an HPN over
ODEs for representing large networks of cascades—another consideration that influenced
the present blood-clotting model, coagulation being traditionally described precisely as a
series of cascades. In these papers, as in (Chen and Hofestadt, 2003), large systems are
modeled, and so the inevitably large state space restricts the researchers to simulations, i.e.
away from any analysis.
A very different approach to modeling genetic regulatory networks as hybrid systems
appears in (Jong et al., 2003). There, the authors again consider the threshold-controlled
time evolution of protein concentrations according to differential equations, but assume
that only the orderings of the thresholds are known. Similarly, the rate constants governing
degradation and accumulation of proteins are specified only by inequalities. Specifically,
since the ODEs within each threshold-bounded domain are linear and uncoupled, the equi-
libria within each domain can be expressed explicitly in terms of the rate constants; which
equilibria are then specified only to lie between some pair of the thresholds. This turns
out to be enough information for an algorithm to generate the “qualitative states” of the
system (equilibria and cycles), and the transitions between them.
The theory of hybrid systems can be exploited for the control of biological systems, as
well. The chemical kinetics of many biological systems can be translated into differential
equations which are, in general, multi-affine: i.e., polynomial in their state variables, with
the proviso that the degree of any of those variables is no greater than one.1 A recent set
of papers (Belta et al., 2002, 2004) supplies necessary and sufficient conditions for steering
of multi-affine systems on N -dimensional rectangles; and a recipe for deriving the control
law to effect this steering. The several dynamics in their several rectangles, then, together
compose a hybrid system.
This clever scheme was considered for control of the blood-clotting model, so more
details are in order. The control task of steering the system through a particular facet1The idea is that, picking for a moment a single variable xi and holding all the others constant, the
system is affine in that variable: f(c1, c2, ..., xi, ci+1, ...cn) = mxi + b.
39
(an [n− 1]-dimensional face), F ∗, of an n-dimensional polytope is equivalent to steering it
through the corresponding facet of the n-dimensional hypercube (since the polytope can
be transformed into a hypercube by an affine transformation of the original state variables,
which transformation will evidently result in another multi-affine system); and this control
task in turn can be be specified in terms of necessary and sufficient conditions on the
differential equations f(x) + g(x)u merely at the vertices of the polytope (rather than
anywhere within the rectangle). In particular, if the vector fields can be shown to point in
the “correct” direction at each vertex—i.e., in the direction of the outward-pointing vector
normal to F ∗ and away from (or parallel to) all of the other facets—then the trajectory
of the state vector, starting from any point within the hypercube, will exit the rectangle
through F ∗. The goal is to choose the input u so as to guarantee that the state remains in
“safe” or acceptable regions. (This scheme is inappropriate for our model, unfortunately;
see Chapter 8.)
Finally, although the present study does not include stochastic processes, we mention
here for completeness two hybrid-system models that do. In (Hu et al., 2004), the produc-
tion of the antibiotic subtilin by the bacterium Bacillus subtilis is modeled via differential
equations, each particular to the current state of a Markov chain. Alternatively, stochas-
ticity enters the model of (Joshi et al., 2004) in the following way: Chemical kinetics, as we
have been stressing, are translatable into differential equations describing the time evolu-
tion of concentrations. This translation, however, assumes that there are enough particles
(proteins, etc.) that the stochastic effects are “washed out”; whereas we may be concerned
with very small quantities, in which case accuracy requires treating the reaction as a se-
ries of discrete, random events. Joshi et al. (2004) proposes a method for dynamically
partitioning the reactions and their corresponding reactants into discrete and continuous
sets, the evolution of the latter being governed by deterministic ODEs and the former by
a continuous-time Markov chain. This method allows accurate and efficient simulation of
chemical reactions with simultaneous fast and slow dynamics. The blood clotting model
of this dissertation, however, will be concerned only with (relatively) large quantities of
proteins, obviating the need for such a technique.
40
Chapter 4
Simulations and Sensitivity
4.1 Introduction
As we saw in Chapter 2, the process of blood coagulation in mammals is complicated,
and involves the interaction of more than a dozen coagulation factors as well as a number of
proteins from the kinin-kallikrein system and protein inhibitors. Attempts to model coagula-
tion mathematically therefore usually focus on a smaller subset of interactions, perhaps one
of the so-called pathways or just a portion of one of them (cf. (Panteleev et al., 2002),(Ko-
gan et al., 2001),(Bungay et al., 2003), (Leipold et al., 1995),(Qiao et al., 2004),(Butenas
et al., 2004) and (Pohl et al., 1994)). Such models generally consist of a set of coupled, usu-
ally nonlinear, differential equations governing the time evolution of protein concentrations.
Although on one level of analysis all of the processes in blood clotting comprise discrete
events, like cleavage of chemical bonds, formation of bonds, and the like; nevertheless, at
the scale of interest it is concentrations that matter, and these exhibit continuous dynamics.
There are, however, at least two reasons why this methodology is inadequate for mod-
eling the entire coagulation cascade. First, there is reason to believe (Halkier, 1991) that
certain events in the cascade are better modeled as “switched.” So, e.g., many of the chem-
ical reactions of the cascade require the presence of calcium ions; that is, not the speed of
41
these reactions but whether or not they take place at all is controlled by calcium.1 Simi-
larly, concentrations of free zinc ions are thought to toggle the activation of several of the
proteins of the contact activation portion of the clotting cascade (Røjkjær and Schmaier,
1999),(Shariat-Madar et al., 2002); that is, the reactions can take place only after a thresh-
old concentration of zinc has been exceeded. Relatedly, the reactions of the coagulation
cascade take place on very different time scales (which fact, inter alia, causes difficulties
for simulation: it damns the HPN simulator to operate at the granularity of the fastest
reaction, i.e. the smallest time step). Although we have not systematically purged fast
reactions from the model by replacing them with switches, some fast reactions have been
modeled thus for that reason2; and furthermore the hybrid framework pioneered here shows
how such a purge might be accomplished.
The second reason for including discrete events in the model is that the current state
of clinical knowledge is not sufficient to provide a precise description of the various param-
eters and their interactions in terms of differential equations. So, for example: the rate
constants for the various steps in factor XIII activation are not known—even the form of
the equation is unclear; the role of high molecular-weight kininogen in enabling activation
of various members of the intrinsic pathway has not been formalized; and the individual
on- and off- rates for substrate-enzyme interactions has not been determined for various
proteins.3 Certainly there is no closed-form set of differential equations for the system as a
whole. On the other hand, more coarse-grained information is available—e.g., the thrombin
concentration that is required for appreciable conversion of factor XIII into its activated
form. This information can often be incorporated in the form of discrete events toggled
(possibly) by thresholds.
Thus the alternative pursued here is to model the coagulation cascade as a hybrid system1It is perhaps possible that the speed of some of these reactions does indeed vary continuously rather
than discretely with calcium concentration—though to my knowledge this has not been reported—but thatin any case it makes no matter since either (1) intermediate levels of calcium never occur physiologically;or (2) (more likely) the change from “off” to “on” occurs too steeply, i.e. over too small a range of calciumconcentrations, for a continuous variation to be either measurable or meaningful. This latter consideration isakin to the (somewhat standard) modeling practice of replacing steep sigmoid functions with step functionswhere this simplifies things.
2See the previous footnote3These last, however, can at least be approximated by differential equations with known parameters, i.e.
using Michaelis-Mentin rather than mass-action kinetics; see the discussion in Chapter 2.
42
(HS), i.e. one consisting of interacting continuous and discrete dynamics. Hybrid systems
were discussed at length in Chapter 3; in the present case, our aim was to construct a robust
and faithful model of the coagulation process: faithful in the sense of accurately modeling
human (or generally, mammalian) blood clotting, and robust in the sense of doing so over a
wide range of parameter settings. We additionally required that our model be perspicuous
(with the biologist in mind), and easily modifiable. In light of these constraints, the model
was implemented using hybrid Petri nets (HPNs), described in Chapter 3 in excruciating
detail.4
The model serves both a specific and a more general purpose. Specifically, by accurately
simulating the blood-clotting process, the model serves as a basis for predictions: the effect
of alterations in coagulation-factor concentrations, on both the overall clotting time and on
the concentration of other factors, can be simulated effectively. These simulations can serve
as the basis for predictions about the effect of pharmacological intervention; for understand-
ing the nature of certain blood-clotting disease pathologies (e.g. the various hæmophilias,
factor-V Leiden, etc.); and for refinement of our understanding of blood clotting in general.
More generally, the model demonstrates the utility of the design methodology, viz. using
hybrid systems, and in particular hybrid Petri nets, to model cascade-like biological pro-
cesses where both discrete and continuous dynamics play a role. In virtue of its ability to
incorporate both types of dynamics, the model is able to support robust analysis and pre-
diction in cases where parts of a complex process may be known precisely (with differential
equations) while other aspects may have qualitative descriptions only (through punctuated
phase changes, discrete transitions, and threshold behaviors). This ability to reason ef-
fectively with representations of multiple granularities addresses a central requirement in
modeling complex biological processes.
The model was tested by simulating normal blood clotting, as well as various blood
clotting disorders. The resulting time to clotting and the time course of blood protein
concentrations were compared against the clinical literature, and gave consistent results.4Our current implementation is based on the Visual Object Net++ platform, a dedicated HPN mod-
eling and simulation environment (Drath, 2002), which includes a graphical language that offers a suite ofobject-oriented programming (OOP) features: hierarchical organization, inheritance and object reuse.
43
4.2 Model Implementation
The cascade consists largely of a series of very similar reaction types, as we shall see, for
which reason I used an object-oriented HPN modeling language; that is, one with constructs
for information hiding and object reuse.
The HPN version of the coagulation cascade is depicted in Figure 4.9, at the end of
the chapter. (A more readable color version of this figure—and the rest of the thesis—
can be found online(Makin, 2008).) Places at the left hand side represent unactivated
blood factors—zymogens of serine proteases, their cofactors, and various ancillary proteins,
regulatory and otherwise—at pre-injury in vivo concentrations. The figure refers to the
factors by their Roman numeral designations (if such they have); alternative names, along
with initial concentrations and their sources in the literature, appear in Table 2.1 in Chapter
2.
The boxes in the figure are obviously neither places nor transitions; they are rather
objects which themselves contain Petri nets. The majority of these objects are in fact
three continuous-time/state modules; the remaining two, the modules Init-Intrinsic and
Fibrin, are hybrid Petri nets.
4.2.1 Modules
Figure 4.9 shows the entire clotting cascade. The overall model comprises multiple
instances of four basic modules (along with a few other places and transitions), with different
parameters for different pathways, factors, and enzymes. There are two modules with purely
continuous dynamics and another two with hybrid dynamics. The two continuous modules
involve (a) blood factor activation and (b) factor-factor binding; the two hybrid modules
involve (a) the initiation of blood clotting in the intrinsic pathway and (b) the formation
of a fibrin clot. These modules are described below.
44
Blood Factor Activation
Figure 4.1 depicts one of the basic aspects of the coagulation cascade, the enzyme-
induced transformation of a blood factor (which may be either a serine protease or a glyco-
protein) from its inactive (zymogen) form to its active configuration. In fact, the coagulation
cascade consists largely of a series of such activations, where the newly activated coagu-
lation factor proceeds to activate another factor (again see Figure 2.1). The places IN1,
IN2 and OUT represent the concentration (in nanomolarity [nM], i.e. nanomoles per liter,
though the units are not conceptually relevant) of various blood factors: IN1 is the zymogen
(substrate) and OUT is its activated form; IN2 is the catalyzing enzyme; IN1:IN2 is an in-
termediate macromolecule. The places labeled with k’s are the constants of classic enzyme
kinetics: on-rates, off-rates, and catalytic rates. This reaction can also be written as a set
of differential equations; square brackets are used to remind the reader that concentrations
are being indicated:d[IN1]dt
= koff[IN1:IN2]− kon[IN1][IN2] (4.1)
d[IN2]dt
= koff[IN1:IN2]− kon[IN1][IN2]
+kcat[IN1:IN2] (4.2)
d[IN1:IN2]dt
= kon[IN1][IN2]− koff[IN1:IN2]
−kcat[IN1:IN2] (4.3)
d[OUT]dt
= kcat[IN1:IN2] (4.4)
Of course, since the variables IN1, IN2, and OUT participate in other reactions, Eqs. 4.1,
4.2, and 4.4 do not completely define the dynamics of any of these variables; the complete
governing equations may contain additional additive terms from other reactions. That is
why these three places have shaped outer rings in Figure 4.1: it indicates that they are
“published,” i.e. available to interact with other objects and hence other reactions. (The
rate constants are also published, but this is rather so that the user can easily change them
without having to open up the object of interest.)
45
Figure 4.1: The activation module, which models the activation of a zymogen (IN1) into its
active configuration (OUT) by a catalyst (IN2).
Factor-Factor Binding
The second recurring reaction is the binding of two blood factors, depicted in Figure 4.2.
Note that, as in the activation reaction (Figure 4.1), the rates constants ki are connected
to transitions via test arcs; this reflects the fact that these quantities are unchanged by the
reaction. In the present case, the governing equations are simply:
d[IN1]dt
= koff[OUT]− kon[IN1][IN2] (4.5)
d[IN2]dt
= koff[OUT]− kon[IN1][IN2] (4.6)
d[OUT]dt
= kon[IN1][IN2]− koff[OUT] (4.7)
Both of these modules appears in many instantiations throughout the model, differing
from each other only in their rate constants and their interconnections with the rest of the
network. The differential equations that they model were drawn from (Bungay et al., 2003).
46
Figure 4.2: The binding module, which models the (reversible) binding of two components,
IN1 and IN2, into the macromolecule OUT.
The Intrinsic Pathway
The Init-Intrinsic module, shown in Figure 4.3, models the initiation of blood clotting
via the intrinsic pathway. (Details of this process were drawn from (Shariat-Madar et al.,
2002) and (Røjkjær and Schmaier, 1999).) The continuous transition labeled “zinc flow
rate” models the increase in zinc concentration according to a simple first-order differential
equation (exponential growth up to an asymptote). At the same time, exposure of a nega-
tively charged surface (modeled as the binary variable m1) allows solid-phase activation of
factor XII (Hageman factor).
When zinc-ion concentration exceeds 0.3 µM, high-molecular-weight kininogen
(HMWK) binds with the plasma protein prekallikrein. The ambient zinc concentra-
tion continues to rise, meanwhile, and when it exceeds 5 µM the activation of prekallikrein
to kallikrein is enabled. This process is greatly enhanced by the presence of activated factor
XII, sufficient quantities of which enable activation via the “fast” transition.
Once activated, kallikrein participates in a feedback loop by enabling the fluid-phase
activation of factor XII, which in turn activates more kallikrein. Notice that factor XIIa can
47
Fig
ure
4.3:
Ahy
brid
Pet
rine
tm
odul
em
odel
ing
the
init
iati
onof
the
intr
insi
cpa
thw
ay.
48
activate kallikrein through either a “slow” or “fast” transition, where the former corresponds
to activation of free prekallikrein and the latter to activation of prekallikrein bound to the
surface of HMWK. The speeds of these reactions, fast and slow, are modeled by assigning
appropriate time delays to the discrete transitions.
Eventually the concentration of factor XIIa crosses the threshold for the activation of
factor XI (plasma thromboplastin antecedent, or PTA), generating a discrete quantity of
factor XIa (given by the output arc weight). A sufficient quantity of factor XIa prevents
further activation by factor XIIa, hence the inhibitor arc returning from the place XIa to the
activation and binding transition. Factor XIIa production is itself inhibited by the serpin
C1-inhibitor, which is activated by sufficient quantities of fXIIa.
The Fibrin Module
Figure 4.4 shows the Fibrin module, which models the final portion of the blood clotting
pathway: the activation of factors XIII and I, and the formation of a fibrin clot. (Data for
this module were drawn from (Muszbek et al., 1999),(Greenberg et al., 1985),(Standeven
et al., 2005),(Lewis et al., 1997), (Brummel et al., 2002), and (Mosesson, 2000); see Table
4.1 for details.) Factor XIII (fibrin stabilizing factor) normally circulates in plasma bound
to fibrinogen (factor I). When thrombin concentration exceeds a threshold, the Arg37-Gly38
bond of factor XIII is cleaved, producing factor XIIIa′. There is again a “fast” and “slow”
cleavage, modeled by transitions with identical delays but which are enabled by differently
weighted test arcs from the place IIa. The fast cleavage requires lower levels of thrombin
but additionally the presence of fibrin or fibrinogen. (The reader may convince himself that
the boxed elements labeled “OR” in Figure 4.4 do in fact enforce an or-gate of sorts.) Factor
XIII can also be cleaved at the Lys513-Ser514 bond, which renders it useless with respect to
clotting. This cleavage is however entirely inhibited by the presence of calcium ions (Ca2+),
hence the inhibitor arc from the calcium place to this transition. Meanwhile, thrombin
levels rise, and when they exceed 2.48 nM, fibrinopeptide A is released from fibrinogen
thereby activating it to fibrin.
49
Fig
ure
4.4:
An
HP
Nm
odul
ede
pict
ing
the
final
stag
esof
bloo
dcl
otti
ng:
the
acti
vati
onof
fact
ors
Ian
dX
III,
and
the
form
atio
nof
acl
ot.
50
In the presence either of calcium or fibrin (or both), the A′ and B subunits of factor
XIIIa′ dissociate. Next, again only if calcium ions are present, the active site on the A′
subunits is unmasked, resulting in the transglutaminase FXIIIa*. Finally, FXIIIa* and the
fibrin polymers cross-link to form a clot.
The place labeled m1 in the figure represents the percentage of cross-linking accom-
plished, where each token corresponds to 10%. Thus when m1 acquires ten tokens, cross-
linking is complete. Now, cross-linking is self-regulating in that it inhibits the upstream
promoter effects of fibrin and fibrinogen on factor XIII activation once about 40% of the
cross-linking has been accomplished (Muszbek et al., 1999); hence the inhibitor arc from
m1 to the “fast” transition.
Both of these HPNs made use of various thresholds, arcweights, timing delays, and
binary dependencies (i.e. switches). Table 4.1 lists all of these parameters, and their sources
in the literature, if such there be. Parameters which do not have literature sources were
either interpolated from other relevant data or, in the limit, are informed guesses. As for
the continuous portions of the network, its parameters are all rate constants; they appear
in the Chapter 5.
Finally, several of the activation reactions of the clotting cascade require free calcium
ions. Figure 4.5 reprises Figure 4.1, except that the discrete variable sw switches the binding
and catalytic reactions on and off; if the sw place holds a token (or more than one), then
the reactions may take place, otherwise they may not.
The Complete Cascade
With the component modules explained, we can now turn to the entire coagulation
cascade, referring to Figures 4.9 and 2.1 throughout. To reiterate, the cascade is initiated
through both the intrinsic and extrinsic pathways. The initiation of the former has been
explained; the latter pathway (see the lower left corner of Figure 4.9) begins with the binding
of tissue factor (TF) to lipid-bound factor VII, in both its activated and unactivated forms.
(For the sake of brevity, the many lipid binding reactions will not be mentioned explicitly).
51
Para
met
erT
yp
eD
escr
ipti
on
Valu
eSourc
e
[Zn
2+
]arc
wei
ght
thre
shold
for
HM
WK
:PK
bin
din
g0.3µ
M(S
hari
at-
Madar
etal.,
2002)
[Zn
2+
]arc
wei
ght
thre
shold
for
PK
act
ivati
on
on
cells
5µ
M(R
øjk
jær
and
Sch
maie
r,1999)
[Zn
2+
]arc
wei
ght
thre
shold
inhib
itio
nof
HM
WK
:PK
bin
din
g10µ
M(S
hari
at-
Madar
etal.,
2002),
(Røjk
jær
and
Sch
maie
r,1999)
[Zn
2+
]arc
wei
ght
thre
shold
for
FX
IIfluid
-phase
act
ivati
on
10µ
M(S
hari
at-
Madar
etal.,
2002)
[Zn
2+
]arc
wei
ght
thre
shold
for
FX
I:H
MW
Kbin
din
g10µ
M(S
hari
at-
Madar
etal.,
2002)
Zin
cflow
rate
conti
nuous
transi
tion
rate
of
zinc
ion
acc
um
ula
tion
˙[Zn
]=
20−
[Zn
]ex
trap
ola
ted
from
intr
in.
(diff
eren
tial
equati
on)
path
way
tim
edata
[IIa
]arc
wei
ght
thre
shold
for
fast
clea
vage
of
Arg
37-G
ly38
bond
1.5
6nM
inte
rpola
ted
from
(Bru
mm
elet
al.,
2002)
[IIa
]arc
wei
ght
thre
shold
for
slow
clea
vage
of
Arg
37-G
ly38
bond
90nM
inte
rpola
ted
from
(Bru
mm
elet
al.,
2002)
[IIa
]arc
wei
ght
thre
shold
for
rele
ase
of
fibri
nop
epti
de
A2.4
8nM
inte
rpola
ted
from
(Bru
mm
elet
al.,
2002)
[IIa
]arc
wei
ght
thre
shold
for
rele
ase
of
fibri
nop
epti
de
B3.2
8nM
inte
rpola
ted
from
(Bru
mm
elet
al.,
2002)
[XII
I]arc
wei
ght/
del
ayam
ount
of
fXII
Icl
eaved
per
unit
tim
e:
fast
clea
vage
1nM
/3s
inte
rpola
ted
from
(Bru
mm
elet
al.,
2002)
slow
clea
vage
0.3
3nM
/3s
inte
rpola
ted
from
(Bru
mm
elet
al.,
2002)
cross
-lin
kin
garc
wei
ght
per
centa
ge
of
cross
-lin
kin
gat
whic
hpro
mote
reff
ect
of
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52
Figure 4.5: The activation module of Figure 4.1, but here two of the reactions are switched
on (off) by the presence (absence) of a discrete variable.
Now, factor VII may be activated by factor Xa, but this requires completion of either the
intrinsic or extrinsic pathways, both of which, as we shall see, require the activation of factor
VII. Thus it is generally accepted (Bungay et al., 2003) that some very small quantity of
factor VIIa (0.1 nM) exists in the blood stream prior to vascular injury. This does not
imply spontaneous activation of the coagulation system since without tissue factor the rest
of the pathway cannot proceed.
The TF:VIIa complex proceeds to bind with lipid-bound factor X (lower right corner of
Figure 4.9), which complex in turn transforms (see the subsequent continuous transition)
into TF:VIIa:Xa via cleavage of a bond in the factor X component. The complex then dis-
sociates (the subsequent transition) into factor Xa (i.e., its activated form, thrombokinase)
and the TF:VIIa complex. Factor Xa then feeds back to activate lipid-bound factor VII
into VIIa as well as the TF:VII complex into TF:VIIa. Factor Xa also binds with a protein
called tissue factor pathway inhibitor (TFPI) (bottom right of Figure 4.9), which inhibits
the extrinsic pathway by binding to free TF:VIIa complex and removing it from further
reactions. Activation of factor X demarcates the traditional ending point of the extrinsic
pathway.
As we have already seen in the Init-Intrinsic module, the exposure of HMWK,
kallikrein, and factor XII to an electro-negative surface results ultimately in the activa-
53
tion of factor XI to XIa. Factor XIa can then activate factor IX to IXa (middle right of the
figure), but the cascade can proceed no further until factor Xa resulting from the action of
the extrinsic pathway activates lipid-bound factor VIII. This comports with the accepted
theory that the extrinsic pathway serves to kick-start the intrinsic pathway into action (see
above). The activated form of factor VIII (VIIIa) forms a complex with factor IXa, which
in turn activates more (lipid-bound) factor X. This marks the end of the intrinsic pathway.
The common pathway of the coagulation cascade involves factors V, II, and X; plus
the proteins antithrombin, thrombomodulin, protein S, and protein C (see the upper half
of the figure). Lipid-bound factor V is activated by factor Xa. The activated form, Va,
then forms a complex with Xa, which in turn binds to lipid-bound factor II. This Xa:Va:II
complex spontaneously converts (via the continuous transition) to Xa:Va:mIIa, where mIIa
is meizothrombin, an intermediate form of factor II. This complex then dissociates either
into Xa:Va and the activated form of factor II, thrombin (via the continuous transition), or
into Xa:Va and mIIa (via the Va:Xa:mIIa reversible binding module). Both meizothrombin
and thrombin feed back to activate more factor V. Thrombin also feeds into the intrinsic
pathway to activate factor VIII, as does meizothrombin; and to activate factor XI.
The common pathway is inhibited by antithrombin, thrombomodulin, and proteins C
and S. Thrombin binds to thrombomodulin (upper right corner), effectively removing it
from further catalytic reactions. This complex has a further inhibitory role, however, in
activating lipid-bound protein C. Activated protein C (APC) binds with lipid-bound protein
S, and the APC:PS complex feeds back to inactivate both VIIIa in the intrinsic pathway
and Va in the common pathway. Factors Xa, mIIa, and IIa (thrombin), meanwhile, are
irreversibly bound by antithrombin, removing them completely from further reactions.
Thrombin’s final role, of course, is to activate factors I and XIII, but this has already
been detailed above in the description of the Fibrin object.
54
4.3 Results
We now present the results of various simulations. We start with the normal clotting
pathway, and then simulate some disorders of blood clotting as changes to either the relevant
coagulation factors and associated protein complexes—i.e. the initial values of places in the
network—or to the structure of the network. The results shown for the abnormal cases are
for thrombin levels, but of course simulations produce time-course results for all the places
in the network. Thus the impact of specific diseases or combinations of diseases on the
entire clotting cascade may be examined. This includes both qualitative and quantitative
aspects, like coagulation with versus without therapeutic intervention, and like the effect of
specific dosage levels of an intervention, respectively.
4.3.1 Normal clotting
We choose to examine thrombin because it is the most important enzyme product of
the coagulation cascade: it participates in far more reactions than any of the other factors,
including both feedforward and feedback regulation, and is essential for normal blood clot-
ting. Time to thrombin activation is consequently one of the major parameters measured
in clinical tests. To evaluate the baseline performance of our model, we set the initial con-
centrations as shown in Table 2.1 and compared the time course of thrombin production in
our computational simulation with results reported in the clinical literature (Halkier, 1991;
Bungay et al., 2003). Figure 4.6 shows the concentration of thrombin produced under nor-
mal conditions upon triggering of the clotting cascade. The value of thrombin is shown in
nM, as a function of time, in seconds. Consistent with previously reported clinical studies,
thrombin concentration using the model peaks at about 160 nM around 100 seconds (cf.
(Undas et al., 2001)). A complete clot (i.e. 100% cross-linking) occurs after 105 seconds,
which is again a reasonable figure.
55
Figure 4.6: A simulation of the concentration of thrombin (factor IIa) in nM as it varies
with time (in seconds) in the blood clotting of a normal patient.
4.3.2 Simulating hæmophilia A
Now consider a simulation of hæmophilia A, a disease of the clotting system which
results in excessive bleeding. The cause of the pathology is a deficiency of coagulation
factor VIII, which can range from mild to severe (see Table 4.2).
Table 4.2: Coagulation Disorders
Condition Cause Result Notes
Hæmophilia A deficiency excessive severity:
of fVIII bleeding > 5% (mild)
1− 5% (moderate)
< 1% (severe)
Factor V APCR in fV hypercoag. APC resistance
Leiden ≈ 5% of population
The coagulation disorders simulated in this chapter.
56
Figure 4.7: Simulated results of the concentration (nM) of thrombin versus time (s) during
the clotting process for a person with severe hæmophilia A (low curve) and for normal
clotting (high curve).
To simulate the clotting disorder, therefore, the continuous HPN place VIII, representing
the initial (pre-injury) plasma concentration of clotting factor VIII, is set to 0.035 nM, which
corresponds to the borderline between mild and moderate hæmophilia (cf. Tables 2.1 and
4.2), and the simulation is re-run. The result is shown in Figure 4.7: thrombin concentration
peaks later (at 120 seconds) and much lower (at just over 6 nM), which is congruent with
clinical observations. The maximum cross-linking achieved is 40%, which again is consistent
with the disease pathology.
4.3.3 Simulating factor-V Leiden
Factor-V Leiden is a coagulation disorder characterized by a condition called activated
protein C resistance (APCR), in which a genetic mutation in the factor-V gene renders the
resulting factor-V protein resistant to inactivation by activated protein C (APC). (Recall,
following Figure 2.1, that the function of APC is to inactivate factor Va and factor VIIIa.)
Factor V is a procoagulant, so the consequence of its slower rate of inactivation is generally
a thrombophilic (propensity to clot) state. The disorder has in fact only recently been doc-
57
Figure 4.8: Simulated results of the concentration (nM) of thrombin versus time (s) during
the clotting process for a person with factor-V Leiden (high curve) and a normal patient
(low curve).
umented: APCR was first described in 1993; factor-V Leiden was subsequently discovered
in 1994.
More sophisticated ways of simulating factor-V Leiden are available within the present
model—and in fact I shall demonstrate one such in the context of the control schemes in
the next chapter—but for now, we choose simply to remove the module by which APC
inactivates factor V (see Section 4.2.1). As expected, the result (Figure 4.8) is an increase
in the amount of thrombin produced, congruent with clinical observations (cf., e.g., (van’t
Veer et al., 1997)). This simulation of factor-V Leiden does not predict a shortened clotting,
and this, too, is in agreement with clinical results.
4.4 Decomposition
We noted at the outset that the primary goal of the HS model was simulation: hybrid-
system theory affords far fewer and less powerful analytical tools than the complementary
methods for purely discrete and purely continuous systems. We should like to exploit some
58
of those methods in the remainder of this thesis—for sensitivity analysis, control, and model
reduction. Therefore, we now propose a decomposition of the system into interacting sub-
systems, with the aim of applying our control techniques to the purely continuous portions,
and then proving the efficacy of these techniques on the overall process by applying a
reachability analysis to the other subsystems.
From Figure 4.9, it is apparent that the Init-Intrinsic module interacts with the other
(continuous) modules only via factor XIa; and that the Fibrin module interacts with these
modules only via thrombin. In the latter, moreover, interaction is only in one direction:
because thrombin affects the Fibrin module only by activating certain (discrete) transitions,
no thrombin is actually consumed by the fibrin module. Now, it must be noted that this is
not strictly correct, since some amount of thrombin complexes with both factors I and XIII;
however, we assume that very little of either complex forms, as in (Khanin et al., 1998). The
Init-Intrinsic module, on the other hand, affects and is affected by the purely continuous
modules (through the evolution of factor XIa). However, deficiencies of the clotting factors
of this early portion of the intrinsic pathway do not have hemorrhagic consequences(Adcock
et al., 2002), (which is in agreement with the present model).
Armed with these considerations, we restrict our attention from here on to the continu-
ous subsystem of Figure 4.9 which comprises all those parts of the cascade not included in
the Init-Intrinsic or Fibrin modules. Now, to prove that control of these ODEs suffices
to induce healthy clotting, we should at least have to show that the amount of factor XIa
produced by the Init-Intrinsic module is negligible compared to the amount produced by
reciprocal activation by factors Xa and IIa. In fact we might be interested in showing for
what range of initial conditions in the Init-Intrinsic module this result obtains. Similarly,
a proof would require us to show that the thrombin trajectories we control are sufficient
to produce a clot, and within a reasonable amount of time. This question, too, might be
posed in terms of ranges: what range of thrombin is sufficient to effect this end?
Both of these questions are questions of reachability. As we have alluded to previously,
and shall spell out in more detail in the next chapter, reachability on large hybrid systems
is very expensive in computational terms, often prohibitively so. A more feasible approach
59
in the present case is to discretize the state space of these two HPN modules and perform a
discrete reachability analysis. In fact, reachability in (discrete) Petri nets often (depending
on the structure of the network) has a very simple solution (Bause and Kritzinger, 1996).
Whether or not such a discretization is feasible, and whether the resulting network is indeed
amenable to the normal reachability algorithms, are questions we leave to the final chapter,
8.
The idea, then, is to steer model thrombin concentrations, with the hope that it can be
proven that these trajectories suffice to induce proper clotting. Lacking such a proof (and
not wanting to rely on hope), we will throughout aim at the tightest control possible: that
is, we will attempt to force thrombin concentrations in diseased-patient models to match
the trajectory of thrombin during a healthy clotting event. This may be overkill in the
sense that looser criteria might be adduced, via the reachability analysis lately outlined,
to guarantee proper clotting, but it is certainly sufficient (at least insofar as the model is
correct).
Finally, if our controller is going to neglect these two HPN modules, then certainly the
input must not interact with them. Fortunately, the major drugs for the major clotting
problems do not. In the following chapters, we focus on the two most common bleeding dis-
orders, factor-V Leiden and hæmophilia A. The latter is treated with (recombinant) factor
VIII, which lies in the purely continuous subsystem; and the former is treated with heparin,
which—as we shall see—increases the efficacy of antithrombin in inhibiting thrombin and
factor Xa, all of which proteins live in the purely continuous portion of the model.
4.5 Sensitivity Analysis
What if our model is wrong? In particular, what if the rate constants in the ODEs, which
after all were determined empirically, are in error? How much will such an error affect the
model? Alternatively, suppose we wanted to change the course of coagulation by changing
a reaction rate; where shall we get the most bang for our buck? Or again, if a blood-clotting
disease alters certain rate constants (as e.g. factor-V Leiden does), how dramatic will the
60
effect be on the concentrations of blood proteins? These questions are all variations on the
same theme, namely finding how sensitive a model is to its parameters; and we should like
to compute these sensitivities not simply for this or that parameter, but for the model as a
whole, i.e. for all the rate constants. In what follows we restrict ourselves to a large portion
of the cascade which can be described entirely by a set of ordinary differential equations;
however, the sensitivity analysis can (in principle) and should be extended to the HPNs,
as well (cf. for example (Hiskens and Pai, 2000)). The ODEs analyzed here are, then, the
completely continuous subsystem discussed in the previous section.
4.5.1 Methods
We follow the technique employed on another model of blood clotting, Luan et al. (2007),
which we recapitulate here for completeness. The raw sensitivity σij(t) is the change in state
i (blood-protein concentration) with a change in the jth parameter value (rate constant):
σij(t) :=∂xi∂kj
∣∣∣∣∣t
(4.8)
Note that sensitivities are time-varying. We can find the ODE governing the raw sensitivities
by notingd
dt
∂x∂kj
=∂
∂kjx =
∂f∂kj
+∂f∂x
∂x∂kj
.
Thus the vector of sensitivities σj(t) = [∂x/∂kj ]|t obeys the differential equation
σj = A(x(t),k)σj + bj(x(t),k), (4.9)
where
A(x(t),k) =∂f∂x
∣∣∣∣∣x∗(t),k∗
bj(x(t),k) =∂f∂kj
∣∣∣∣∣x∗(t),k∗
,
i.e. the derivatives evaluated along the nominal trajectory of x and for the numerical
value of k. The matrix A and vector b can be determined analytically (although they
are only estimated in (Luan et al., 2007)), but their value at any given point in time
must be evaluated based on the numerical simulation of the governing ODE x = f(x).
Additionally, the differential equation (4.9) admits no analytic solution, so the two systems
are simulated in tandem. Now, the differentially perturbed trajectory must have the same
61
initial conditions x0, which is tantamount to insisting that [∂x/∂kj ]|t=0 = 0; so Eq. 4.9 is
simulated with zeroed initial conditions. This procedure is then repeated for all |k| rate
constants.
Having computed σij(t) for each parameter, state, and time sample, we proceed to
combine these values into |k| overall sensitivities, one for each parameter, taking a Euclidean
norm across both time and state, and scaling the contribution of each state variable by (the
inverse of) its magnitude. The result is then scaled by the nominal parameter value, under
the assumption that variations in them will be proportional to their magnitude. Thus the
overall sensitivity for parameter kj is the unitless quantity
σj = kj
[ tf/T∑m=1
n∑i=1
(1x∗i
∂xi∂kj
)∣∣∣∣2t=mT
]1/2
, (4.10)
where T is the sampling interval, tf is the final time of the simulation, and n is (as ever)
the dimension of the state space.
Now, we are interested in the sensitivity of the system not only along the orthodox
trajectory predicted by our model, but for nearby trajectories as well; after all, our rate
constants may be wrong. So, again following (Luan et al., 2007), we simulate Eq. 4.9 for
a range of different parameter values. On each of 200 trials we independently draw each
parameter kj from a uniform distribution stretching from 50% to 150% of its orthodox value
and re-solve (numerically) the two systems of differential equations. The resulting overall
sensitivity vectors σ are normalized at each trial into a [0, 1] range, and averaged over trials.
4.5.2 Results
Table 4.3 shows the 25 most sensitive rate constants, ranked according to average nor-
malized sensitivity σj . We note that 200 trials appears to be sufficient for the σj to converge.
62
Table 4.3: Averaged Normalized Sensitivities
Ranking Reaction Avg. sensitivity ± variance
1 Xa:Va:II Xa:Va:mIIa 0.87 ± 0.028
2 Xa:Va + IIL Xa:Va:II 0.72 ± 0.040
3 TF:VIIa:X TF:VIIa:Xa 0.71 ± 0.046
4 TF + VIIaL TF:VIIa 0.65 ± 0.040
5 XaL + VaL Xa:Va 0.65 ± 0.039
6 Xa:Va + IIL Xa:Va:II 0.62 ± 0.037
7 V:mIIa VaL + mIIaL 0.56 ± 0.030
8 XaL + VaL Xa:Va 0.53 ± 0.026
9 XL + TF:VIIa TF:VIIa:X 0.52 ± 0.031
10 XL + TF:VIIa TF:VIIa:X 0.51 ± 0.029
11 XIa:IX IXaL + XIa 0.48 ± 0.023
12 PCL + IIa:Tm IIa:Tm:PC 0.47 ± 0.019
13 XI:IIa XIa + IIa 0.46 ± 0.022
14 IXaL + VIIIaL IXa:VIIIa 0.46 ± 0.021
15 XI + IIa XI:IIa 0.46 ± 0.022
16 VL + mIIaL V:mIIa 0.43 ± 0.018
17 XL + IXa:VIIIa IXa:VIIIa:X 0.43 ± 0.017
18 VIIIaL + APC:PS APC:PS:VIIIa 0.41 ± 0.017
19 IIa:Tm:PC APCL + IIa:Tm 0.40 ± 0.017
20 VIII:mIIa VIIIaL + mIIaL 0.39 ± 0.014
21 XI + IIa XI:IIa 0.38 ± 0.017
22 VL + mIIaL V:mIIa 0.38 ± 0.014
23 IXL + XIa XIa:IX 0.37 ± 0.012
24 IXa:VIIIa:X XaL + IXa:VIIIa 0.36 ± 0.014
25 APC:PS:VIIIa VIIIaiL + APC:PS 0.35 ± 0.011
Averaged, normalized sensitivities (see Methods) for the 25 most influential rate constants. The rate
constant is given by its associated chemical reaction.
63
How does this compare to the sensitivity analysis of (Luan et al., 2007), which models
more or less the same portions of the cascade, but including a few more equations for
platelet binding? First of all, these platelet reactions account for about half their list of
most “fragile” (sensitive) reactions—which is somewhat curious, since platelet disorders of
coagulation are not common. Second, the reactions involved in Xa:Va activation of thrombin
are on average more sensitive in our model than the reactions for activation of factor X by
the TF:VIIa complex; whereas the converse is true in the Luan et al. (2007) model. In
both, however, these reactions (and their related platelet reactions in (Luan et al., 2007))
dominate the list.
4.5.3 Sensitivity and prothrombin time
One of the (two) most common clinical tests for a clotting problem is the prothrombin-
time (PT) test, a measure of the time required for TF-initiated coagulation of blood plasma.
The test is performed by first drawing blood into a test tube containing citrate, which soaks
up the free calcium ions and thus prevents the main clotting reactions. The blood is then
centrifuged and the plasma separated from the blood cells, after which an excess of calcium
and tissue factor are added. Clotting time is measured from the addition of TF to the
first visible signs of clot formation in the tube, i.e. when about 10 nM thrombin has been
produced (Mann et al., 2003). The healthy range for this time is considered to be 12-15
seconds.
Now, this set-up is very close to the simulations of this chapter, but in the PT test,
clotting is initialized by a saturating amount of tissue factor (20 nM, vs. 0.005 nM in
the simulations above). However, in our model, this results in a clot time around 30
seconds. Moreover, the differences of the present study with the PT test all suggest a shorter
rather than a longer clot time: our model includes phospholipids (recall that the PT test is
performed on purified plasma), which enhance reactions rates; as well as another initiation
mechanism for clotting, the intrinsic pathway. Why do we find this large disparity between
clot times, then? If we assume the structure of the model is correct, then the difference must
be caused by parameter errors. And this seems rather plausible, since literature values for
64
rate constants vary by upwards of orders of magnitude (Panteleev et al., 2006) (or cf. the
rate constants of (Bungay et al., 2003) and (Luan et al., 2007), and their references).
How, then, might we change the parameters as little as possible to achieve a PT of
(say) 13 seconds? A principled approach will exploit the sensitivities of the system; in
particular, we might try to compute the sensitivity of thrombin over the first p seconds to
changes in the rate constants, i.e. ∂z/∂kj , where z is the concentration of thrombin; and
then change only the most sensitive parameters, moving them in the direction of greater
thrombin. Without an explicit solution to this maximization problem, this would amount
to a kind of gradient ascent, changing parameters incrementally in the direction of greatest
thrombin increase.
Unfortunately, true gradient ascent would require the recalculation of the thrombin
sensitivity ∂z/∂kj at every step through the ascent, and as we have seen, Eq. 4.9 has no
analytic solution; and furthermore the numerical solution is quite expensive, computation-
ally, since it requires simulation of a (|k|+ |x|)-dimensional system, where (|k|+ |x|) ≈ 200.5
Rather than repeatedly recalculate the gradient, then, we employ a similar method to the
one lately described, viz. averaging the sensitivity over a range of different parameter values,
each randomly perturbed from the “correct” parameters (see above). This time, however,
we care about the sign of the sensitivity (it tells us which whether to increase or decrease
the parameter), so we dispense with the squaring in Eq. 4.10 and simply average the results
of the several unnormalized trials:
s[IIa]j = kj
(1x∗i
∂xi∂kj
)∣∣∣∣t=t∗
, (4.11)
where t∗ is the desired clot time, 13 seconds. Now we can use this thrombin-sensitivity vector
s[IIa] as an approximation of the gradient throughout. Also note that although we have
again neglected the discrete parts of the network, we can do so with some confidence, since
(1) the PT test ignores the intrinsic pathway, and (2) we can estimate clotting time from
thrombin alone as long as we bear in mind that this estimate assumes healthy functioning5Now, Eq. 4.9 is a linear, time-varying differential equation, for which an analytic solution exists in
the form of a matrix power series (Blair, 1971). However, the early terms of the series are so large as tointroduce rounding errors into the low (and ultimately most meaningful) bits of the sum, rendering thetechnique useless for the present ODE.
65
of fibrin and factor XIII (but after all, in these simulations we are assuming that the entire
clotting process is healthy). We therefore follow (Mann et al., 2003) and take 10 nM of
thrombin to correspond to the formation of a visible clot in the test tube.
4.5.4 Results
In all experiments, the initiating dose of TF is 20 nM, as in the actual PT test (Mann
et al., 2003). Now, with our estimate of the thrombin gradient in hand, one question we
might ask is: How few parameters can we alter and still effect the desired PT, given an
upper limit on those parameter shifts? Table Table 4.4 shows the results for three different
upper limits.
Table 4.4: PT-Time Parameter Shifts
Max Shift (%) Min Params
60 12
50 19
40 70
Minimum number of parameters required to shift in order to achieve a PT time of 13 seconds, for
three different limits on the maximum parameter shift.
Two interesting features of these results should be noted. First, entertaining parameter
variances of the reasonable (in light of the dispersion of literature values) figure of 60%
enables the model to match clinical PT times with the shifting of just 12 (out of over 100)
parameters. This shows that very few parameters can do a lot of work. Contrariwise, forcing
all rate constants to live within 40% of their nominal values precipitously increases this
minimum number of rate constants, up to about two-thirds of them. Thus the relationship
between the maximum shift and minimum number of parameters is evidently nonlinear;
and in particular the latter moves from a small minority of the total number of parameters
to a sizable majority as the maximum shift is drawn down under the 50% mark.
66
Before we leave this small investigation, we reverse the question and ask how small we
can make the maximum parameter shift if we let, say, all of the parameters vary. The
answer is 38%—which, considering that 40% shifts require the movement of about 40 fewer
rate constants, and 50% shifts require alteration of 90 fewer, indicates how very little effect
these remaining constants have on thrombin levels—at least at t = 13 seconds.
4.6 Discussion and Conclusions
The modeling framework and software program described provides a robust, faithful,
interactive, and graphical computer simulation of the entire coagulation process. The frame-
work is faithful in that it accurately models mammalian blood clotting; robust in the sense
of doing so over a wide range of parameter settings; interactive in that the system operation
and parameter settings can be interactively changed while the software program is execut-
ing, and hypothetical “what-if” simulations performed; and graphical in that the model is
a formal graphical structure that supports visualization of the clotting process as well as
exact quantitative analysis.
The simulations of factor-V Leiden and hæmophilia A were consistent with clinical
results, and to that extent vindicate both the present model and its parameters as well
as the methodology, i.e. a HS approach. Of course, consistency is not tantamount to
quantitative identity, but here the obstacle lies not with the model so much as the state of
the clinical data. As Wagenvoord et al. (2006) has pointed out, most thrombin curves can
be fit by an equation with a mere four parameters, so these data vastly underdetermine the
parameters of the model. Including clot time does not substantially decrease the number
of free parameters.
We interpret our simulations, then, as having demonstrated a method for incorporating
qualitative or otherwise discrete information into a single model, which model is consistent
with clinical findings; but I do not claim the model to be error-free. On the other hand, the
model also provides a platform for testing clinical hypotheses, quantitative or qualitative,
about the coagulation cascade, since consistency with known data is a necessary (if not
67
sufficient) criterion for the correctness of those hypotheses. This provides a method for
model refinement, as well. Furthermore, inasmuch as this kind of model can indeed capture
all the relevant details of coagulation, the analysis techniques demonstrated in this chapter
and those to come will apply equally well to a “final,” error-free model.
In anticipation of the control techniques of the next two chapters, we discussed how
two hybrid subsystems could be decomposed from the rest of the model, leaving a large
system of ODEs—which will be augmented in the next chapter by additional differential
equations for the interaction of heparin with the system and for a more accurate simulation
of factor-V leiden. This decomposition allows us to treat our design of clinical interventions
as a problem in nonlinear control theory.
Finally, we saw that (restricting our view to the continuous subsystem) relative changes
in protein concentrations are most sensitive to changes in the rate constants involving
activation of thrombin by the Xa:Va complex and activation of Xa by the TF:VIIa complex.
We also saw that these sensitivities could be exploited to change the model in the most
parsimonious way in order to match the results of the PT test: shifting a mere 12 rate
constants by as little as 60% sufficed to achieve a reasonable 15 seconds for this test.
On the other hand, it turns out that using these parameters in the original clotting
simulations of this chapter vitiates their agreement with clinical results. That suggests that
something more subtle than rate constant alterations is responsible for the discrepancy in
PT-test results.
68
Figure 4.9: The HPN model
69
Figure 4.9: The HPN model
70
Figure 4.9: The HPN model
71
Figure 4.9: The HPN model
72
Chapter 5
Control: A First Pass
5.1 Introduction
The complexity of the coagulation process that has been stressed so far renders treat-
ment of its disorders difficult. Indeed, current treatments make no attempt to manipulate
coagulation in real time; rather, periodic (on the order of one day to one week) measure-
ments and interventions are made with the aim of keeping certain known risk factors within
safe ranges. This limitation is a consequence of two facts: first, the state of current sensor
technology, which precludes real-time measurements of blood proteins (and hence real-time
feedback control); and second, the lack of theoretical techniques for such control. This chap-
ter addresses the second of these limitations by applying techniques from computer science
and from mathematical control theory and demonstrating their validity—and limitations—
both mathematically and in a series of simulations. By providing part of the solution to
the theoretical problem of controlling blood clotting, I hope to provide an impetus for the
development of the relevant sensor technologies.
The simulations of Chapter 4 demonstrate the utility of the model for certain purposes,
namely prediction, theoretical investigation, and sensitivity analysis. However, ultimately
we should like to use it as a basis for the real-time control of blood-clotting. Now, as
we saw in Chapter 3, the standard technique for control of hybrid systems is numerical
73
solution of a set of partial differential equations—the Hamilton-Jacobi equations. However
it turns out that for systems of dimension greater than about five, solution of the equations
is computationally infeasible (a consequence of the curse of dimensionality) (Mitchell et al.,
2001). Unfortunately, our system has upwards of 100 state variables, so this technique is
patently unworkable.
The approach taken here is to consider only a subsystem of the model, a set of nonlinear
ODEs which were originally drawn largely from (Bungay et al., 2003) and (Hockin et al.,
1999). In fact, the model can be decomposed into this (purely continuous) subsystem and
two hybrid systems, the three interacting only through two state variables, as we have seen
in Chapter 4, so that control of clot formation in the complete model can be approached
as a problem of continuous (nonlinear) control. Thus in the present study we content
ourselves with deriving and simulating the control techniques for the system of ODEs alone.
Specifically, the control task is to steer the concentration of thrombin (activated blood
factor II) along a desired trajectory during a clotting event, by controlled rate of injection
of an exogenous pharmaceutical (e.g. heparin). I demonstrate two different techniques
to effect this control: the first, following (Sastry, 1999), using nonlinear feedback and a
change of variables to partially linearize the ODEs, and then controlling the linearized,
single-input/single-output (SISO) system using standard methods; and the second based
on step-input control. In the simulations, the input is either the anti-coagulant heparin or
(recombinant) factor VIII, and the output is thrombin, the most important enzyme product
of the coagulation cascade (see below).
We present the theory of the control techniques first, along with some preparatory results
on alternative approaches, before presenting a series of simulations. We then simulate the
clotting process in a patient with the pro-coagulatory disorder factor-V Leiden and in a
patient with moderate hæmophilia A, and then repeat the simulation under application of
the control techniques. Finally, we discuss both the relevance of this control task to the
overall task of controlling blood clotting, as well as implementation issues, and then propose
an alternative technique to remedy some of the defects of the present approach.
74
5.2 Theory
The model to be controlled is a system of about 100 coupled, nonlinear ODEs, of the
most general form,
x = f(x, u) (5.1)
where u is the (single) control variable (say, an anti- or pro-coagulant). The control task
is to force one of the state variables (thrombin) to track a desired trajectory. In fact, the
ODEs can be expressed, as we shall see, in a less general form; and the approach described
in this chapter will be to exploit some of the peculiarities of the system which distinguish
it from the most general case.
Now, control of linear systems is comparatively easy, so a standard approach is to design
the controller around a linear approximation to the true system, found by considering only
the first term of the Taylor expansion, the Jacobian [∂f/∂x](x0), near an equilibrium point
x0. Unfortunately, the Jacobian in our model is singular, so (by the Hartman-Grobman
theorem) the linearization is not guaranteed to approximate the true system.
Alternatively, the system may be exactly linearized (as opposed to merely approximated
by a linear system) by choosing the appropriate nonlinear feedback u = ψ(h(x)) and looking
at the system through a change of variables ξ = Φ(x):
ξ = Aξ + bv,
y = ξ1, (5.2)
where in fact (A,b) are in controllable canonical form, so the system is completely control-
lable. Here h(x) is an output function which reflects our observations of the state variables,
and v is a synthetic input related to the true input u by a known function. From this per-
spective we can ask whether or not, given the dynamics in Eq. 5.1, there exists an output
function rich enough to support the linearization. The answer is particularly straightforward
if, as in our case, the ODEs are affine in the control, i.e. can be written as:
x = f(x) + g(x)u, (5.3)
75
in which case necessary and sufficient conditions for the linearization can be given in the
form of conditions on the vector fields f(x) and g(x) and iterated Lie brackets thereof.
Specifically, the matrix of vector fields
[g(x), adfg(x), ..., adn−2f g(x), adn−1
f g(x)], (5.4)
often referred to as the strong-accessibility distribution, must have full rank (n) in the region
of interest; and the set of vector fields
g(x), adfg(x), ..., adn−3f g(x), adn−2
f g(x) (5.5)
must be involutive 1 in the region. Here, by recursive definition,
adkf g := [f , adk−1f g], k > 0
adkf g := g, k = 0.
If these conditions are not met and full-state linearization is not possible, one may
attempt to partially linearize the system. Here, the linearization is carried out with respect
to some given output function, which will admit the formulation of some number q of new
state variables with linear dynamics. Of course, if full-state linearization is not possible,
then q, called the relative degree of the affine-control system, is strictly less than n, the
dimension of the state space.
Consider again the differential equations of our system as the affine-control system of
Eq. 5.3, where this time h(x) = y, the variable we wish to control. Differentiating the
output with respect to time yields:
y = (∇xh)x (5.6)
= (∇xh)f(x) + (∇xh)g(x)u
= Lfh(x) + Lgh(x)u,
where Lfh(x) := (∇xh)f(x), the Lie derivative of the function h along the vector field given
by f . Now if Lgh(x) is nonzero for all x, then the system is said to have a strict relative
1A set of vector fields is involutive if the Lie bracket of any two of those vector fields is within the spanof the original set, where the Lie bracket [f ,g] of two vector fields is defined as [∂g/∂x]f − [∂f/∂x]g.
76
degree of one, and we can force it to track a desired trajectory yd by choosing:
ud =1
Lgh(x)(yd − Lfh(x)), (5.7)
and making sure that the initial conditions match (y(0) = yd(0), y(0) = yd(0)). Changing
variables according to ξ = Φ(x) := h(x) and defining for the nonce yd =: v, a synthetic
input, we see a one-dimensional linear system ξ = 0 · ξ + v and an (n − 1)-dimensional
nonlinear system η = λ(ξ, η). (The variable v is called a synthetic input in virtue of its
role as the input to this linear system.) Notice that the linear system is decoupled from the
nonlinear one, in the sense that ξ is not a function of η.
If, on the other hand, Lgh(x) is zero for all x, then we differentiate y a second time to
get:
y = L2fh(x) + LgLfh(x)u, (5.8)
and, if for all x, LgLfh(x) 6= 0 (i.e. the system has strict relative degree two), choose:
ud =1
LgLfh(x)(yd − L2
fh(x)), (5.9)
this time making sure that y(0) = yd(0) as well. Choosing ξ = Φ(x) := [h(x), Lfh(x)]T
again yields a linear system, this time two-dimensional, which is decoupled from the re-
maining (n− 2)-dimensional nonlinear system.
In general, for a system of strict relative degree q, the control law is
ud =1
LgLq−1f h(x)
(v − Lqfh(x)), (5.10)
where we choose the synthetic input v = y(q)d and set all the initial conditions
yd(0), yd(0), ..., y(q)d (0) appropriately. Note that, if q = n, the dimension of the state space,
then
Φ(x) := [h(x), Lfh(x), ..., Ln−1f h(x)]T (5.11)
is a valid change of coordinates (locally diffeomorphic), which in fact puts the system into
controllable canonical form—i.e., (Eq. 5.3) is fully linearizable by state feedback. The
procedure is illustrated in Figure 5.1.
77
Finally: we have so far ignored the case where LgLq−1f h(x) is zero for some values of x,
and nonzero for others. In this case the relative degree of the system is not well-defined,
and complications ensue. (We shall see in the sequel that this is indeed the case in the
present system.)
5.3 Application to the model
5.3.1 Feedback linearization
Full-state linearization as lately described faces a significant issue when applied to the
present model. Although verifying the two conditions associated with (5.4) and (5.5) is
mathematically straightforward, it is computationally intensive. Computing the vector
fields requires the computation of two Jacobians for every Lie derivative, although one of
them, ∂f/∂x, need only be computed once. However, each of the approximately 100 vector
fields of (5.4) needs to be computed, and computation of each of the associated Jacobian
matrices entails 1002 derivatives of polynomials. Even neglecting the multiplication and
addition operations, this brings the total to a million computations—each of which is a
(symbolic) derivative of a polynomial, increasing the total number of computations yet
more. And most unfortunately of all, each derivative (at least potentially) generates more
variables, via the chain rule of differentiation, so the polynomials have increasingly more
terms in later vector fields.
Apart from computational difficulties, it remains to construct the output function, h(x).
No mechanical procedure for this construction exists.
In fact, we can circumvent the computational obstacles: Bastin and Levine (1993) have
shown that, in the case of reaction systems like the present one, the strong-accessibility
rank is bounded from above by the rank of a matrix D (the “accessibility matrix”) which
is independent of x, and is computed very simply. The existence of such a matrix depends
once again on being able to express the governing ODEs in an even more specific form. In
78
particular, Eq. 5.3 can be written
x = Cr(x) + bu, (5.12)
where each element in the vector r(x) is a monomial corresponding to one of the chemical
reactions of the system (i.e one of the arrows in the chemical formulae); and b is a constant
vector, i.e. does not depend on the state. The present case is a special case even of the one
discussed in (Bastin and Levine, 1993), since our model includes neither inflow nor outflow
of reactants. For this system, the accessibility matrix is the augmented matrix D := [b, C],
which we now show.
Consider a vector field s(x) in the span of D, that is
s(x) =∑i
diφi(x), (5.13)
for arbitrary functions of x, φi, and where di are the columns of D as defined above. Then
the Lie bracket of the drift vector field f with s is
[f , s] =∂s∂x
f(x)− ∂f∂x
s(x)
=∑i
di(∇xφi)f(x)− C ∂r∂x
s(x). (5.14)
Now, since (∇xφi)f(x) is just another arbitrary function of x, the first term is within the
span of D (compare this term with Eq. 5.13). Similarly, the second term lies within the
span of C and thus a fortiori lies within the span of D. Thus we see that the Lie bracket
of f , the drift term of our ODE, with any vector field in the span of D, yields a vector field
which is itself in the span of D. Obviously g ∈ SpD, and hence ad1f g = [f ,g] is as well,
by the fact just noted. And now since ad1f g ∈ SpD, then so is ad2
f g = [f , ad1f g]—and so
on, extending the claim to all the vector fields in the strong-accessibility distribution. Thus
we can say:
∀x, rk(D) ≥ rk([g(x), adfg(x), ..., adn−2f g(x), adn−1
f g(x)]). (5.15)
Crucially, the rank of D is easy to compute, being but a constant matrix, and we avoid
crunching through the iterated Lie derivatives of the right-hand side.
79
For a hæmophiliac patient, we use factor VIII as the single input, giving rk(D) = 83 <
n = 98—so the system is not fully linearizable by state feedback; i.e. there is no single-
variable output function h(x) that can be constructed to support such feedback. In the case
of factor-V Leiden, several reactions corresponding to the inactivation of factor Va, as well
as the associated state variables (namely, the inactivated forms of FVa), are removed from
the model. (The modeling of factor-V Leiden is explained at length below.) Treatment
with heparin in turn requires the addition of several reactions and state variables. However,
none of these changes, nor the different control vector field b—with its single nonzero entry
in a different row—suffice to eliminate the deficit between the rank of D and the dimension
of the state: rk(D) = 73 < n = 89. The meaning of this deficit we shall discuss later.
So the system is not completely controllable. However, there is reason to believe that
clotting can be controlled just by controlling the concentration of thrombin (activated factor
II): Clotting occurs downstream in the coagulation cascade of the ODEs of this study, and
interacts with them only via thrombin. (See the previous chapter for the full justification.)
Thus we attempt to partially linearize the system of Eq. 5.3 with thrombin the output
y = h(x). Indeed, using heparin as the control variable and augmenting the system with
the appropriate terms from the heparin chemical reactions, the strict relative degree of the
system is two.2 Alternatively, using factor VIII as the input (for hæmophilia) yields a strict
relative degree of three.
However, the relative degree for neither of these systems is well-defined; in particular, the
relevant Lie derivative is zero at the initial condition, x0. We circumvent this difficulty by
allowing the plant to run uncontrolled until the concentration of thrombin is some distance
δ from the origin, i.e. until the state has drifted away from the singularity at which the
relative degree changes, before turning the controller on. The parameter δ is tuned manually2Lacking complete controllability, is there something the system has gained? I.e., is there a trade-off in
the area? An affirmative answer can be given if one considers the computational complexity: In order tocompute the feedback law, we need only calculate (two) twice-iterated Lie brackets; whereas if the systemwere completely controllable, the two vector fields would require the calculation of n-iterated Lie brackets.In a system as complicated as ours, this would be a serious problem, even given a simple output function.Additionally, we need only specify two, rather than n, initial conditions. This question is even more relevantwhen an output function which will provide strict relative degree n is available.
80
for ideal tracking. (The non-regularity of the system and its consequences are discussed in
more detail in Section 5.6.2 below.)
Error Correction
Now, in practice, small numerical discrepancies arise—e.g. from a mismatch between the
plant and the model, or from numerical approximations—so that the trajectory produced
by the control law of Eq. 5.10 deviates from the desired trajectory. Compensation is made
in the form of additional elements in the feedback loop, usually some variant on the PID
controller. A proportional and an integral term are generally used (Henson and Seborg,
1996), but here we content ourselves with proportional and derivative terms (as many as
the strict relative degree of the system) since (1) the integral term did not seem to improve
tracking in our simulations; and (2) it can be shown (Sastry, 1999) that, if the system of
Eq. 5.3 is globally exponentially minimum-phase, then this control law guarantees bounded
trajectory tracking, i.e. the tracking error and its derivatives tend asymptotically to zero.
That is, we replace the synthetic input v in Eq. 5.10 with:
v(t) = y(q)d (t) + [Kp K
1d · · · K
q−1d ][yd − ξ], (5.16)
where yd is the vector of the desired output and its derivatives:
yd = [yd yd · · · y(q−1)]T,
and ξ is the linearized state, which in our system corresponds to the actual output y and
its derivatives. Determining whether the system is in fact globally exponentially minimum
phase, however, requires some discussion, which we defer to Chapter 6.
The gains were calculated by treating the system as a linear-quadratic regulator, with—
unless otherwise noted—state-penalty matrix set to the identity matrix and the input
penalty set to unity. However, control of the nonlinear system was observed to be fairly
insensitive to the choice of cost parameters.
81
Discretization
Finally, practical application will also require discretization of the controller. Now, the
sampling rate of the controller affects the controllability of the system: under a sufficiently
low rate the theory lately outlined will fail to achieve exact output tracking. We demonstrate
a working discrete controller in the simulations below.
5.3.2 Step-input control
A much simpler control technique was also applied; the rationale for it is discussed below.
This strategy is predicated on the assumption that a single step input might suffice to force
the system to track the desired thrombin trajectory “reasonably well.” More precisely, the
assumption is that the step input that matches both the desired peak concentration of
thrombin and the occurrence of this peak will result in a thrombin trajectory that deviates
very little over its entirety from its desired counterpart. Therefore this technique was
implemented by performing a parameter search over repeated trials for that step input
which would minimize the peak-concentration and peak-time discrepancies. Thus the input
u was modified on successive trials according to:
u = u+ α(maxty(t) −max
tyd(t)) + β(argmax
ty(t) − argmax
tyd(t)), (5.17)
where α and β—positive for anti-coagulant inputs and negative for pro-coagulants—scale
the relative contribution of each term, and were adjusted by hand.
5.4 Methods
We draw our chemical reactions from three main sources, the first two of which collect
their own data from a number of primary sources (the last is itself a primary source):
(Bungay et al., 2003), (Hockin et al., 1999), and (Olson, 1988). A description of each
follows.
Forty-eight chemical reactions for normal blood clotting, with no exogenous interven-
tion, were taken directly from (Bungay et al., 2003), from which also the numerical values
82
of the rate constants were supplied. They are reproduced here in Table 5.1. The blood
factors are referred to by their Roman-numeral designations, a lowercase “a” denoting the
activated form. Other abbreviations include: mIIa for meizothrombin, LBS for the concen-
tration of lipid binding sites, PS for protein S, PC for protein C, APC for its activated form,
TFPI for tissue-factor-pathway inhibitor, Tm for thrombomodulin, AT for antithrombin,
and TF for tissue factor. A subscripted “i” indicates the inactivated form of an enzyme,
and a subscripted “L” indicates the lipid-bound form.
Table 5.1: Chemical Reaction Set I
No. Reaction kon (nM−1 s−1) koff (s−1) kcat (s−1)
1 II + LBS IIL 0.0043 1 -
2 mIIa + LBS mIIaL 0.05 0.475 -
3 V + LBS VL 0.05 0.145 -
4 Va + LBS VaL 0.057 0.17 -
5 VII + LBS VIIL 0.05 0.66 -
6 VIIa + LBS VIIaL 0.05 0.227 -
7 VIII + LBS VIIIL 0.05 0.1 -
8 VIIIa + LBS VIIIaL 0.05 0.335 -
9 IX + LBS IXL 0.05 0.115 -
10 IXa + LBS IXaL 0.05 0.115 -
11 X + LBS XL 0.01 1.9 -
12 Xa + LBS XaL 0.029 3.3 -
13 APC + LBS APCL 0.05 3.5 -
14 PS + LBS PSL 0.05 0.2 -
15 VIIIai + LBS VIIIai,L 0.05 0.335 -
16 PC + LBS PCL 0.05 11.5 -
17 TFL + VIIaL TF:VIIaL 0.5 0.005 -
18 TFL + VIIL TF:VIIL 0.005 0.005 -
19 TF:VIIaL + IXL TF:VIIa:IXL
→ TF:VIIaL + IXaL 0.01 2.09 0.34
20 TF:VIIaL + XL TF:VIIa:XL → TF:VIIa:XaL 0.1 32.5 1.5
21 TF:VIIa:XaL → TF:VIIaL + XaL - 1 -
22 TF:VIIL + XaL TF:VII:XaL → TF:VIIaL + XaL 0.05 44.8 15.2
(continued on next page)
83
Table 5.1 – continued
No. Reaction kon (nM−1 s−1) koff (s−1) kcat (s−1)
23 IXaL + VIIIaL IXa:VIIIaL 0.1 0.2 -
24 XaL + VaL Xa:VaL 1 1 -
25 IXa:VIIIaL + XL IXa:VIIIa:XL
→ IXa:VIIIaL + XaL 0.1 10.7 8.3
26 VL + XaL V:XaL → VaL + XaL 0.1 1 0.043
27 VIIIL + XaL VIII:XaL → VIIIaL + XaL 0.1 2.1 0.023
28 VL + IIa V:IIaL → VaL + IIa 0.1 6.94 0.23
29 VIIIL + IIa VIII:IIaL → VIIIaL + IIa 0.1 13.8 0.9
30 Xa:VaL + IIL Xa:Va:IIL 0.1 100 -
31 Xa:VaL + mIIaL Xa:Va:mIIaL 0.1 66 -
32 Xa:Va:IIL → Xa:Va:mIIaL - - 13
33 Xa:Va:mIIaL → Xa:VaL + IIa + LBS - - 15
34 VIIL + XaL VII:XaL → VIIaL + XaL 0.05 44.8 15.2
35 XI + IIa XI:IIa → XIa + IIa 0.1 10 1.43
36 APC:PSL + VIIIaL APC:PS:VIIIaL
→ APC:PSL + VIIIai,L 0.1 1.6 0.4
37 TFPI + Xa TFPI:Xa 0.016 0.000333 -
38 TFPI:Xa + TF:VIIaL TFPI:Xa:TF:VIIaL 0.01 0.0011 -
39 IXa + AT → IXa:AT 4.9 ×10−7 - -
40 Xa + AT → Xa:AT 2.3 ×10−6 - -
41 IIa + AT → IIa:AT 6.83 ×10−5 - -
42 VL + mIIaL V:mIIaL → VaL + mIIaL 0.1 6.94 1.035
43 VIIIL + mIIaL VIII:mIIaL → VIIIaL + mIIaL 0.1 13.8 0.9
44 IIa + TmL IIa:TmL 1 0.5 -
45 IIa:TmL + PCL IIa:Tm:PCL → IIa:TmL + APCL 0.1 6.4 3.6
46 mIIa + AT → mIIa:AT 6.83 ×10−6 - -
47 APCL + PSL APC:PSL 0.1 0.5 -
48 XIa + IXL XIa:IXL → XIa + IXaL 0.01 1.417 0.183
The chemical reactions and rate constants drawn from (Bungay et al., 2003). They govern all the
reactions except factor Va inactivation and the interaction of heparin with the system.
The simple mechanism of factor Va inactivation of the Bungay et al. (2003) model
84
consisted of a single chemical equation. We substituted for this equation 3 the model of
factor-Va inactivation of (Hockin et al., 1999), which was designed specifically to match data
for factor-V Leiden. These 34 chemical equations are listed in Table 5.2. As discussed in
Chapter 2, factor-V Leiden is a genetic disorder in which the amino acid Arg505 of the heavy
chain of factor V is replaced by Gln505, which effectively disables APC-mediated cleavage
of factor V at this point. Hockin et al. (1999) were able to match data for inactivation
of FVaLEIDEN by assuming that the form of factor V cleaved at Arg505, Va5, has no effect
on clotting, i.e. is “inactivated.” (This is so even despite its reported 60% efficacy in
complex with factor Xa as prothrombinase. Since it has no overall effect, and for the sake
of simplicity, we treated Va5L as inert.)
We made a further simplification in the incorporation of this model: Hockin et al. (1999)
model the inactivation of factor Va by APC, rather than the more efficacious APC:PS com-
plex, as in (Bungay et al., 2003). Now in fact, uncomplexed APC competes with APC:PS
for binding with factor Va, but Bungay et al. (2003) leaves out this competition—i.e., only
APC:PS is modeled as binding with Va—presumably because, under normal conditions,
very little free APC circulates: the vast majority of it is immediately bound up by protein
S. We follow this expedient in appropriating the Hockin et al. (1999) model, replacing APC
with APC:PS in all the relevant chemical equations. Doesn’t this substitution require an
alteration of the rate constants? After all, APC:PS is supposed to be more efficacious.
Yes: and here we follow the results of (Egan et al., 1997), who found that the increased
efficacy of the complex results from a factor of three increase in the cleavage rate of Va at
Arg306.4 In the interest of perspicuity, the factor of three and the original rate constant
from (Hockin et al., 1999) are listed explicitly in Table 5.2. Of course, if we wanted to
model (e.g.) protein-S deficiency, we should need to put the equations for (uncomplexed)
APC inactivation of FVa into the model.3For completeness: in fact, we used all of the chemical equations from Bungay et al. (2003) except two:
the deactivation of FVaL by APC; and the unbinding of FVaiL, the lipid-bound, inactivated form of factorVa, from its phospholipid surface—since, after all, FVaiL never shows up in our model, being replaced byvarious cleaved forms of factor Va.
4Previously it had been reported that the factor was twenty rather than three, but as Egan et al. (1997)notes, this would abolish the thrombophilic effect of factor-V Leiden: In the presence of protein C, thecleavage at Arg306 would dominate the cleavage at Arg505, so that the disabling of the latter in FVL wouldhave no overall effect on clotting.
85
Does this complicated replacement for the single equation of (Bungay et al., 2003) yield
similar results under normal (healthy) conditions? In fact, there is a significant change:
under the more complicated model, thrombin peaks during normal clotting around 90 nM,
whereas under Bungay et al. (2003)’s model, it peaked at almost 40 nM (though, curiously,
the time of the peak is the same in both models). Now Bungay et al. (2003) verifies their
model by comparing it to clinical observations of thrombin curves under various physio-
logical conditions, viz. Figure 5 of (Butenas et al., 1999). However, oddly enough, these
conditions assume no protein C or protein S, so Bungay et al. (2003)’s simulations do not
actually verify this part of their model. In point of fact, Bungay et al. (2003)’s “verifi-
cations” are anyway qualitative: the thrombin curves from (Butenas et al., 1999), while
peaking at similar concentrations, are somewhat delayed compared with those of (Bungay
et al., 2003). Resolution of this issue, too, we defer to the discussion section below. For
now it should be said that the more detailed model of (Hockin et al., 1999) is probably
more accurate than the single equation of (Bungay et al., 2003), so we proceed with some
confidence.
The heretofore unexplained abbreviations appearing in the table are as follows. The
numbers 3,5, and 6 denote the site or sites at which the species of factor Va has been cleaved,
respectively Arg306, Arg505, and Arg662. Thus Va53 is factor Va cleaved at both Arg306 and
Arg505; and VaA3 is factor V’s A2 peptide chain, cleaved at Arg306. LC denotes the light
peptide chain of factor Va, and HC its heavy chain; and VaLC is the light chain and A1
peptide chain. The subscripted L again denotes the lipid-bound species.
Table 5.2: Chemical Reaction Set II
No. Reaction kon (nM−1 s−1) koff (s−1) kcat (s−1)
49 LCL + HC VaL 2.63× 10−6 1.72× 10−5 -
50 LCL + HC5 Va5L 2.63× 10−6 1.72× 10−5 -
51 LCL + HC3 Va3L 2.63× 10−6 1.72× 10−5 -
52 LCL + HC53 Va53L 2.63× 10−6 1.72× 10−5 -
53 LCL + HC36 Va36L 2.63× 10−6 1.72× 10−5 -
54 LCL + HC56 Va56L 2.63× 10−6 1.72× 10−5 -
(continued on next page)
86
Table 5.2 – continued
No. Reaction kon (nM−1 s−1) koff (s−1) kcat (s−1)
55 LCL + HC536 Va536L 2.63× 10−6 1.72× 10−5 -
56 LCL + APC:PS L LC:APC:PSL 2.63× 10−6 1.72× 10−5 -
57 VaL + APC:PSL Va:APC:PSL 0.1 0.7 -
58 Va5L + APC:PSL Va5:APC:PSL 0.1 0.7 -
59 Va3L + APC:PSL Va3:APC:PSL 0.1 0.7 -
60 Va53L + APC:PSL Va53:APC:PSL 0.1 0.7 -
61 Va36L + APC:PSL Va36:APC:PSL 0.1 0.7 -
62 Va56L + APC:PSL Va56:APC:PSL 0.1 0.7 -
63 Va536L + APC:PSL Va536:APC:PSL 0.1 0.7 -
64 VaLCL + APC:PSL VaLC:APC:PSL 0.1 0.7 -
65 VaLCL + VaA3 Va3L 2.57× 10−6 0.028 -
66 VaLCL +VaA53 Va53L 2.57× 10−6 0.028 -
67 VaLCL +VaA36 Va36L 2.57× 10−6 0.028 -
68 VaLCL +VaA356 Va536L 2.57× 10−6 0.028 -
69 VaLC:APC:PSL + VaA3 Va3:APC:PSL 2.57× 10−6 0.028 -
70 VaLC:APC:PSL + VaA53 Va53:APC:PSL 2.57× 10−6 0.028 -
71 VaLC:APC:PSL + VaA36 Va36:APC:PSL 2.57× 10−6 0.028 -
72 VaLC:APC:PSL + VaA356 Va536:APC:PSL 2.57× 10−6 0.028 -
73 Va:APC:PSL Va3:APC:PSL - - 3*0.064
74 Va5:APC:PSL Va53:APC:PSL - - 3*0.064
75 Va5:APC:PSL Va56:APC:PSL - - 5.2× 10−4
76 Va3:APC:PSL Va36:APC:PSL - - 5.2× 10−4
77 Va53:APC:PSL Va536:APC:PSL - - 5.2× 10−4
78 Va56:APC:PSL Va536:APC:PSL - - 3*0.064
79 Va:APC:PSL Va5:APC:PSL - - 1
80 Va3:APC:PSL Va53:APC:PSL - - 1
81 Va36:APC:PSL Va536:APC:PSL - - 1
The chemical reactions and rate constants of the Hockin et al. (1999) model of factor-Va deactivation.
Factor-V Leiden is modeled as knocking out the reactions which cleave factor V at the Arg505 site.
Proteins cleaved at this site are marked by a 5. See text for abbreviations.
A word on the lipid-binding: The reader may notice that not all of the proteins sub-
87
scripted with an L have equations for dissociating from their lipid substrates. In this we
are for the most part following (Bungay et al., 2003), in which only the simple proteins,
but not the compounds, are allowed to dissociate. For the remainder, we have followed
Hockin et al. (1999) in not allowing the uncomplexed proteins LCL, Va5L, Va3L, Va53L,
Va36L, Va56L, Va536L, and VaLCL to dissociate from their substrates. Now, in fact, the
experiments in Hockin et al. (1999) assume a saturating amount of lipids; whereas in our
model, the phospholipid concentration is something less than saturating. Technically, then,
these proteins should be allowed to dissociate. However, assuming on- and off-rates equal
to that for (uncleaved) factor Va, these dissociations have an entirely negligible effect even
on the concentrations of the FVa-variants themselves, let alone on thrombin levels—which
is to be expected, since phospholipids never fall below about 1500 nM throughout the sim-
ulations, at which level binding forces vastly predominate over unbinding forces (cf. the
rate constants of chemical reaction 4). Notice, also, that the heavy-chain variants of factor
Va are not lipid bound: the binding site for lipids lies in the light chain of factor V Hockin
et al. (1999).
Finally, on this head, a correction was required: Bungay et al. (2003) claim that throm-
bin is released from the lipid surface when it is activated. However, conservation of mass
requires these lipid binding sites to be returned to the pool of uncomplexed LBS; thus in
chemical reaction 33 I have added an LBS term that was missing from (Bungay et al., 2003).
Of course, as I have just finished saying, the concentration of binding sites is in excess over
the other proteins, so this addition makes no discernible difference to the simulations.
The form of the heparin equations comes from (Olson, 1988) and appear in Table 5.3.
The abbreviated proteins are heparin, thrombin (activated factor II), and antithrombin;
(IIa-AT) is a stable complex which can no longer dissociate into inhibitor and protease.
Such are the forms of the heparin reactions; however, to my knowledge, the exact values
of the on- and off-rates under physiological conditions have never been determined. Olson
(1988) does, however, give estimates of the ratios of off- to on-rates; the following expedient
was therefore adopted: On-rates were set to an intermediate value within the biologically
normal range for enzyme-substrate reactions (see for example Table 4.4 in (Fersht, 1999),
88
but also (Bungay et al., 2003)), viz. 0.1 s−1 nM−1. The corresponding off-rates were then
computed by multiplying the aforementioned ratios by the on-rates, i.e. 0.1. Again, this
is represented explicitly in the table. This approximation does not vitiate the theoretical
apparatus, but it does have numerical consequences; we take up this question again in the
discussion below.
The final rate constant is a catalytic rate, and as such is taken directly from (Olson,
1988). However, we also deviate in one other particular from that paper: Whereas Bungay
et al. (2003) model AT binding to IIa as irreversible, with a very slow on-rate, Olson (1988)
considers the reaction to be reversible, with a very high off-to-on ratio. These mechanisms
are certainly not equivalent, and we use the more recent mechanism from (Bungay et al.,
2003). However, in any case, this reaction will be dominated by the much higher affinity
reaction between AT and heparin, the primary path to the Hep:AT:IIa complex, so the
choice has little practical consequence.
Finally: heparin is also thought to facilitate the inactivation of factor Xa, but once
again for simplicity these interactions have been neglected.
Table 5.3: Chemical Reaction Set III
No. Reaction kon (nM−1 s−1) koff (s−1) kcleavage (s−1)
82 Hep + IIa Hep:IIa 0.1 0.1 ∗ 3× 104 -
83 Hep + AT Hep:AT 0.1 0.1 ∗ 2× 102 -
84 Hep:IIa + AT Hep:AT:IIa 0.1 0.1 ∗ 20 -
85 Hep:AT + IIa Hep:AT:IIa 0.1 0.1 ∗ 3× 103 -
86 IIa:AT + AT Hep:AT:IIa 0.1 0.1 ∗ 0.6 -
87 Hep:AT:IIa → IIa-AT - - 5
The chemical reactions of the heparin reactions. The equations were drawn from (Olson, 1988) but
the rate constants were estimated as in the text.
Table 5.4 lists all the non-zero initial concentrations. The concentration of lipid binding
89
sites and the initial activating amount of tissue factor were chosen to match (Butenas et al.,
1999), as in (Bungay et al., 2003). All other concentrations are normal physiological values.
Table 5.4: Initial Conditions
Species Conc. (nM) Species Conc. (nM) Species Conc. (nM)
Tissue Factor 0.005 Factor VIII 0.7 TFPI 2.5
Factor II 1400 Factor IX 90 Antithrombin 3400
Factor V 20 Factor XI 30 Protein C 60
Factor VII 10 Factor X 170 Protein S 300
Factor VIIa 0.1 Thrombomodulin 1 LBS 3396
The non-zero initial conditions. All other proteins are initialized at zero.
The chemical reactions of Tables 5.1, 5.2, and 5.3 are equivalent to a set of coupled,
nonlinear, ordinary differential equations. For completeness we illustrate the transformation
for the first chemical equation, II + LBS IIL. Then
d[II]dt
= −kon[II][LBS] + koff[IIL]
d[LBS]dt
= −kon[II][LBS] + koff[IIL]
d[IIL]dt
= kon[II][LBS]− koff[IIL],
where square brackets denote (time-varying) concentration. More succinctly, as in Eq. 5.12,
we may write:
d
dt
[II]
[LBS]
[IIL]
=
−1 1
−1 1
1 −1
kon[II][LBS]
koff[IIL]
. (5.18)
Incorporating the other chemical reactions amounts to adding monomials to each ODE as
necessary, and of course adding an ODE for each new chemical introduced; or, alternatively,
to adding new columns to the numerical matrix, C, for each new rate constant, and adding
a new row for each new chemical.
All simulations were performed in Matlab (Mat) using the stiff ODE solver ode15s.
90
Computation of the strict relative degree of the system were also performed in Matlab,
using the symbolic toolbox.
5.5 Results
We first simulate coagulation in a patient with the hypercoagulatory disorder factor-V
Leiden, with and without intervention by the anti-coagulant heparin, as well as normal
(nonpathological) clotting. Initiation of the clotting event is assumed to take place via the
intrinsic pathway and is therefore modeled by initializing tissue factor at a concentration
of 5 pM (following (Bungay et al., 2003)).
Figure 5.2 shows the thrombin profile during normal clotting (dark blue), clotting in a
“patient” with factor-V Leiden (light blue), and that clotting in the same patient but with
the feedback-linearizing controller (green). Exact output tracking has been achieved; how-
ever, several defects are immediately obvious. First, the controller operates continuously,
i.e. updating as often as the numerical simulation of the ODEs does, whereas any practical
controller must be digital. The input, appearing in Figure 5.3, is obviously not feasible for
any imaginable drug-delivery mechanism. In fact, to avoid numerical errors, the input was
clamped at 100 nM/s. (I explain this issue in the discussion section below.) Finally, and
relatedly, a realistic controller will have a maximum and minimum input rate; certainly, the
rate cannot be negative, since this implies withdrawal of heparin from the site of the injury.
The present controller was allowed to do just that.
We address the final defect first. Note, however, that the theory outlined above does
not guarantee the success of any these remedies. As lately noted, the input cannot be a
negative number, nor can it be much higher than a single-digit nanomolarity per second.5
We impose an upper bound (somewhat arbitrarily) at 20 nM/s and a lower bound of zero
by “squashing” inputs outside this range, i.e. setting any u calculated from Eq. 5.10 which5Continuous intravenous infusions over a 24-hour period total between 20,000 and 40,000 IU (International
units), which comes to about 10 pM/s Since our treatment lasts only over the course of a few minutes, wecan presumably use higher doses over this short interval without overmuch hæmophilic risk downstream.
91
exceeds one of these bounds to the value of the bound. The result appears in Figure 5.4,
and the counterpart input in Figure 5.5.
There is indeed some degradation from the unconstrained controller; in particular, the
controller overshoots the peak trajectory somewhat, and is unable to compensate for the
final undershoot. These defects are respectively the consequences of the maximum and
minimum constraints; but the latter at least might be remedied by some form of anticipation;
that is, if the controller could predict the final undershoot, it might ease off the input earlier.
This possibility is explored in the discussion below. As for the overshoot, it can be eliminated
by raising the maximum input to 50 nM/s (graph not shown). However, in the absence of
an established maximum, it might be safer to try to address this issue with an anticipatory
control as well.
The corresponding input in Figure 5.5 still exhibits some chatter, naturally, since it is
allowed to vary continuously. That shortcoming is rectified by discretizing the controller,
which results in the output and input of Figures 5.6 and 5.7, respectively. Here the out-
put again overshoots the desired thrombin trajectory, but also undershoots the pre-peak
trajectory, evidently because it was unable to recover from the initial input; and, for sim-
ilar reasons, undershoots the post-peak thrombin curve slightly more than its counterpart
continuous controller (Figure 5.4).
A smaller controller step yields, as expected, superior results (not shown); and raising
the input maximum again eliminates the overshoot. In the limit, of course, we should be
able to reproduce the results of the continuous controller. However, and for that very reason,
the smaller the step, the more unrealistic the controller. And then, in any case, we shall
presumably not be able to eliminate the undershoot that the continuous controller exhibits,
since that results from the input minimum; nor, if we take seriously the input maximum of
20 nM/s, shall we be able to eliminate the overshoot.
Whether the controlled trajectory of Figure 5.6 presents a hypercoagulatory risk is
something of an open question: Rapid product formation “downstream” in the cascade—i.e.
the formation of fibrin from fibrinogen and activation of factor XIII—requires concentrations
92
of thrombin less than 2 nM (Brummel et al., 2002), so it is not clear how important the
exact trajectory of the thrombin curve is. (Nevertheless, it may be important, and so we
propose a more sophisticated approach below.)
We turn to “moderate” hæmophilia A, at which native levels of the zymogen factor VIII
are about 2.5% of normal levels. Now, if indeed we have a controller that can be switched
on precisely at the onset of a clotting event (perhaps by the release of tissue factor), and can
apply the control locally—assumptions under which we have been operating so far—then
feedback linearization is overkill: We can simply dump in the healthy initial concentration
of factor VIII in the first controller time step, and then turn the controller off. Using again
a control sampling rate of 2 Hz, this technique yields the near-perfect results of Figure
5.8. Here the nonzero input at the first time step (not shown) is x(0)/T , where x(0) is the
desired initial concentration of factor VIII, 0.7 nM, and T is the controller time step, 0.5s.
An added boon of this technique is that it requires no sampling at all: the controller would
not require sensors.
Now, the simple, step controller discussed in Section 5.3.2 above operates on similar
principles, although it makes no assumptions as to the controller operating frequency. Ap-
plying the simple learning algorithm of Eq. 5.17 with α = 1 × 10−4 and β = 1 × 10−5
yields the trajectory in Figure 5.9. The constant input (not shown) of factor VIII is the
very low rate of 6.92 pM/s. The price we pay for the simplicity of this control scheme is
the overshoot on the back half of the trajectory.
Finally, we ask how well we can manage factor-V Leiden with the step controller. In
fact, we can do quite well. Choosing α = 0.05 and β = 0.01 in Eq. 5.17, the input will
settle on u = 4.39 nM/s, at which rate the peak concentration and its occurrence can be
very nearly matched (Figure 5.10). The only question that remains is how significant the
undershoot is from a clinical perspective.
93
5.6 Discussion
5.6.1 Controllability
We have shown via an algebraic criterion that our system of ODEs is not full-state
linearizable. As a matter of fact, we can say more. It follows from the method of trans-
forming chemical equations into ODEs that every reversible reaction will introduce into C
two columns that are additive inverses of each other. Thus, while the rank of C is obviously
upper-bounded by the number of unidirectional reactions—i.e. the number of arrows in all
our chemical equations, or again the number of rate constants—since this is the number
of columns in C, it is also upper-bounded by the (generally smaller) number of uni- or
bidirectional reactions—i.e. counting each pair of arrows as just one reaction; henceforth
simply called “reactions.” If this number falls short of the dimension of the state space, then
full-state linearization requires the deficit to be covered by the control vector fields—which,
in the case of a single-input, can provide additional rank of at most one. So in chemical
systems of mass-action kinetics with no outflow, a single output, and a single input, a nec-
essary (though insufficient) condition for full-state linearization is that there be at least as
many reactions as state variables.
We can translate this result into a statement about the controllability of the system.
The rank of the strong-accessibility distribution (at some point x0) is the dimension of the
locally accessible manifold (from x0); or alternatively, the difference between this rank and
the dimension of the state, n, is the number of uncontrollable modes of the system. So
the dimension of the locally accessible manifold is upper-bounded by the total number of
reactions (again, counting bidirectional reactions just once).
Considering the present system, we find that without the heparin reactions there are
96 reactions and 98 state variables. Assuming factor-V Leiden knocks out some reactions
and some state variables, leaving 77 and 84, respectively. Now, we are obviously interested
in pharmaceutical interventions in addition to heparin (a whole host of pro-coagulants,
for instance). But it can now be claimed that in order to even stand a chance of fully
linearizing the system with one of these interventions as the (sole) input, or equivalently of
94
fully controlling the system, the drug must interact with the system via at least six more
reactions than the number of variables that these reactions introduce. (We say six because
the control vector field may not be in the span of C and hence cover the remaining dimension;
recall Eq. 5.15.) And indeed, heparin introduces only six reactions for its five new state
variables, yielding 83 and 89, respectively. So even though there are more unidirectional
reactions (rate constants, columns of C) than state variables (proteins, rows of C), we know
that there are uncontrollable modes even without constructing C simply by noting that the
number of reactions falls short of the number of state variables.
Finally on the topic of full-state linearization, it can be shown that outflow of the blood
factors—in particular, heterogeneous outflow—can increase the rank of the accessibility
matrix, and hence of the strong-accessibility distribution (Bastin and Levine, 1993). For
simplicity, the model as it stands neglects outflow, but in fact in vivo coagulation will
perforce have some flow (both in and out) of clotting factors, and if it be not negligible over
the time scale of interest, this may provide for greater controllability of the system.
5.6.2 Non-regularity
The coagulation system under application of heparin is, we have been told, non-regular,
i.e. the relative degree is not constant across the state space. What consequences ensue for
control?
It appears that the technique of allowing the system to drift away from the singularity
has no adverse effects on control. We describe now in detail why this should be so. The
feedback nonlinearity that shows up in the denominator of the control law, Eq. 5.10, is in
the case (q = 2):
LgLfh(x) = −kon[IIa](t), (5.19)
inducing a singularity precisely when concentrations of thrombin (factor IIa) are zero. How-
ever, during a clotting event, thrombin concentrations will always be greater than zero, so
the state is safely bounded away from the singularity. We could make this bound precise
by choosing some δ in thrombin-concentration space below which we don’t care to regulate
95
the thrombin trajectory. And the simulations have demonstrated that in the concentration
range of interest—viz., on the order of nanomolarity—the state is not near enough to the
singularity to introduce numerical errors into the control.
Now, on the other hand, some (too large) choice of δ would violate the initial-condition
criteria too grossly for the controller to be able to recover. That it does recover at all is
a consequence (presumably) of the error correction terms added to the control law in Eq.
5.16. Again, the simulations have demonstrated that the controller can indeed recover as
late as 60 seconds into the clotting event.
We saw in the Results section above that the unconstrained (continuous) controller led
to numerical errors in the ODE simulation; which is why the data shown in Figures 5.2 and
5.3 were produced with input clamped at 100 nM/s. These errors are indeed the consequence
of the non-regularity of the system, but a peculiar consequence that could not arise in a
realistic controller: Since the heparin input rate was allowed to assume negative values, the
concentration of heparin itself as well as other state variables could be driven below zero.
And of course pushing thrombin levels through zero means operating the controller in the
region of the singularity. Lower-bounding the input at zero, as we must in any realistic
controller and as we do in the other simulations, eliminates this threat to the numerical
stability of the controller.
Are there not more theoretically sound, less heuristical, bases for avoiding the singular-
ity? There are, but these (Tomlin and Sastry, 1997),(Hauser et al., 1989) involve variations
on the following theme: compute the relative degree of the system at the singularity, and use
the corresponding control law (again Eq. 5.10, but with a higher strict relative degree, q) in
the region of the singularity. The problem with this type of technique for our model is that
the relative degree at the singularity is higher than the ability of our machine to compute
iterated Lie derivatives; that is, we are back to the same, apparently insuperable, compu-
tational obstacle that prevented direct computation of the rank of the strong-accessibility
distribution. Fortunately, the simulations presented here evidently demonstrate the super-
fluity of these more sophisticated methods.
96
5.6.3 Controller merits and demerits
How precise does our control need to be, after all? The answer is not known, but there
is some suggestion that the required amount of thrombin for clotting is much less than
the actual peak concentration: Brummel et al. have demonstrated in an in vitro study
that less than 2 nM of thrombin is required for rapid product formation downstream in
the cascade(Brummel et al., 2002). Certainly this suggests that step inputs suffice to steer
hæmophilia-A and factor-V-Leiden patients through safe clotting, given the qualitative
matches of Figures 5.10 and 5.9. In case these matches are not sufficient, however, or
more critically in case the upper bound on the input needs to be lowered, we propose in
the following section a model-predictive approach to remedy the defects of the feedback-
linearizing controller.
However, the step controllers have another enormous advantage over the feedback-
linearizing control: they require no sampling, only initiation at the inception of a clotting
event. There is not currently a method for measuring blood-protein concentrations in vivo in
real time—certainly not at the required 2 Hz sampling rate. (On the other hand, thrombin
can be measured in real-time ex vivo; see (Hemker et al., 2002).) We interpret our results
on feedback linearization, then, as showing that if indeed a greater degree of accuracy is
required for thrombin tracking, then it would be fruitful to devise the appropriate sensors
for real-time sampling, since they would make possible this suitable technique. Now, naıvely
constraining the input vitiates the accuracy of this technique (Figures 5.4 and 5.6), hence
the need for the model-predictive controller.
On the other hand, two significant advantages of the feedback controllers must be
stressed. First of all, they require no training. Patient-to-patient variation and model errors
(e.g. in rate constants, which vary fairly widely from study to study) make pre-training on
a model insufficient for the development of actual treatments, though it certainly provides
a good starting point. So in the limit, the training of the step-controller approaches the
actual (barbarous) state of the art, namely testing the patient once a day and modifying
doses accordingly. Second, and perhaps more significantly, the feedback controllers operate
97
fairly robustly over a wide range of delays in turning on. So, e.g., in the case of constrained,
discrete control (the most plausible scenario), turning on the controller a full minute after
the release of tissue factor increases the overshoot by less than 5%, with the rest of the
trajectory tracking essentially identical (data not shown). The issue here is how similar
the state at (e.g.) t = 60s is to the desired initial conditions—the closer, the more easily
the controller can recover—and as is clear from the plots above, thrombin concentrations
(presumably, inter alia) do not increase significantly until about 85 seconds.
One envisages, then, the following real-world scenario in which this type of controller
would be particularly useful: An individual suffering from factor-V Leiden cuts herself,
and—within the first minute—applies to the site of injury a device which injects heparin,
in response to the local concentration of all (or some subset of) the other blood proteins.
5.6.4 Significance of model assumptions
Three model assumptions require elaboration. First, both the feedback and step con-
trollers show some sensitivity to the choice of rate constants. An earlier version of the model,
for instance, estimated both on- and off-rates for the heparin reactions, again setting the
former to 0.1 nM/s but also fixing the latter at 10s−1, and achieved essentially perfect
tracking with the same input limits and sampling rate. As I have said repeatedly, reported
rate constants in the blood clotting literature vary somewhat widely, so this consideration
is quite relevant. However, even if we assume the affinities in (Olson, 1988) are correct, as
in the present model, there is still the matter of our assumption of the on-rates (recall that
the unknown heparin on- and off-rates were estimated from their ratio, fixing the on-rate
in the middle of the reasonable physiological range at 0.1 nM/s and then computing the
off-rate from their known ratio; see Section 5.4).
The effect on the controller turns out to be quite small. Similar tracking results (not
shown) can be achieved by both controllers for rate-constant values which span three orders
of magnitude, with kon from 0.01−1 nM/s and koff set to maintain the desired ratio. Values
98
outside this range are unlikely, but there the step controller can match the thrombin peak’s
timing and height only on pain of greatly distorting the subsequent trajectory.
What about model errors in the structure, i.e. the chemical reactions, as opposed to the
parameters of the model? Discussion of this aspect is more involved and will be deferred to
the following chapter (and there discharged the remaining promissory note on model errors
issued in the Introduction to this thesis, Chapter 1).
The final model assumption adverted to at the beginning of this section is that of locality:
the supply of unactivated (zymogen) blood factors was treated as limited. This choice was
made to conform to the model of (Bungay et al., 2003) as well as to the in vitro results of
(Butenas et al., 1999). In reality, however, zymogens are replenished by circulating blood,
and activated proteins are similarly removed. On the other hand, the limited amount of
lipids also restricts the amount of each zymogen which can ever take part in the reaction,
since nearly all of the clotting reactions take place on a phospholipid surface. Congruently,
simulations (not shown) indicate that even if the zymogens are modeled as inexhaustible,
the thrombin curve remains qualitatively the same. We proceeded then with the locality
assumption, in order to avoid the complexities of an added circulation model.
5.7 Conclusions
Regulation of many clotting disorders requires daily visits to a doctor both for the ad-
ministration of blood thickeners or thinners, and for assaying blood-protein concentrations
to determine dosages. In exceptional circumstances, continuous intravenous administration
is necessary. Neither of these alternatives is desirable, and both are in some sense a conse-
quence of taking into account only the most primitive knowledge about human coagulation:
certain drugs encourage clotting, while others discourage it.
This chapter has proposed and demonstrated two control techniques that are much more
powerful in virtue of exploiting a mathematical model of a large portion of the coagulation
cascade and the effects of the relevant pharmaceuticals. Both controllers assume local
application and the ability to detect a clotting event (release of tissue factor); under which
99
assumptions both have been shown capable of regulating the time-varying concentrations
of thrombin, the key blood protein which alone of the proteins in this model determines
clot formation.
Each controller has its merits and demerits: The step controller is more critically depen-
dent on the accuracy of the model but requires no sensors; whereas the feedback controller is
robust to model changes but requires real-time (on the order of 2 Hz) sampling of a number
of protein concentrations (viz., all those involved in the feedback terms). Feedback con-
trol is also (slightly) degraded by constraining the input rate to lie within a feasible range.
We shall address both of these deficiencies of feedback linearization in the next chapter
(although not, unfortunately, at the same time), demonstrating a feedback controller that
requires observation of only one state variable, and a different (model-predictive) controller
which overcomes some of the constraint-induced tracking degradation.
The significance of both the controllers presented in this paper for clinical application
would be greatly advanced by various improvements to the model. First, as lately noted, the
rate constants for heparin interactions with the clotting process are not known; if a precise
controller is to be constructed along the lines proposed here, these must be determined.
Second, a circulation model would dispense with the somewhat dubious locality assumption,
as well as possibly afford more powerful control, since inflow and outflow can increase the
strict relative degree of the system (and hence the number of linearized variables which can
be controlled).
Finally, this chapter has made some useful general observations about chemical kinetics
and controllability. In particular, I have shown that the dimension of the locally accessible
manifold is upper-bounded by the number of reactions in a chemical system, where bi-
directional reactions are counted just once. The author has not seen this result elsewhere.
We can say then, e.g., that since full-state linearization requires that the dimension of the
locally accessibly manifold be equal to the dimension of the state, in chemical systems it
requires that the number of chemicals not exceed the number of reactions.
100
Figure 5.1: A conceptual rendering of feedback linearization. The arrangement can be
viewed either as a controller in a feedback loop with the original control-affine system (solid
lines), plus an output function; or as an equivalent output function along with coupled
linear and nonlinear systems (dashed lines), with the former evolving independently of the
latter, and the output depending solely on the former.
101
0 50 100 150 200 250 3000
10
20
30
40
50
60
70
80
90continuous control
time (s)
conc
entra
tion
(nM
)
[IIa] Desired[IIa] Uncontrolled[IIa] Controlled
Figure 5.2: Simulated control of thrombin concentration during a clotting event in a patient
with factor-V Leiden. The controller was unconstrained, and was updated continuously,
allowing arbitrarily close tracking of the desired trajectory.
102
0 50 100 150 200 250 300−80
−60
−40
−20
0
20
40
60
80
100Input
time (s)
conc
entra
tion
rate
(nM
/s)
Figure 5.3: The heparin input in nM/s that generated Figure 5.2. Although this input yields
perfect tracking, it is obviously unacceptable for any realistic drug-delivery mechanism,
operating as it does on a continuous time scale, and using negative inputs (see text).
103
0 50 100 150 200 250 3000
10
20
30
40
50
60
70
80
90continuous control
time (s)
conc
entra
tion
(nM
)
[IIa] Desired[IIa] Uncontrolled[IIa] Controlled
Figure 5.4: Simulated control of thrombin concentration during a clotting event in a patient
with factor-V Leiden, again continuously sampling. The input was constrained to the range
0-20 nM/s.
104
0 50 100 150 200 250 3000
2
4
6
8
10
12
14
16
18
20Input
time (s)
conc
entra
tion
rate
(nM
/s)
Figure 5.5: The input generating the output of Figure 5.4. Note that the input is constrained
to the range 0-20 nM/s.
105
0 50 100 150 200 250 3000
10
20
30
40
50
60
70
80
90naive constraints
time (s)
conc
entra
tion
(nM
)
[IIa] Desired[IIa] Uncontrolled[IIa] Controlled
Figure 5.6: Simulated control of thrombin concentration during a clotting event in a patient
with factor-V Leiden, this time using a discrete controller sampling every 0.5 s. The input
was again constrained in the range between 0 and 20 nM/s.
106
0 50 100 150 200 250 3000
2
4
6
8
10
12
14
16
18
20Input
time (s)
conc
entra
tion
rate
(nM
/s)
Figure 5.7: The discrete input, constrained to lie between 0 and 20 nM/s, that generated
Figure 5.6.
107
0 50 100 150 200 250 3000
5
10
15
20
25one−shot control
time (s)
conc
entra
tion
(nM
)
[IIa] Desired[IIa] Uncontrolled[IIa] Controlled
Figure 5.8: Simulated control of thrombin in a patient with moderate hæmophilia A (see
text), where the controller inputs the proper initial concentration of factor VIII at the first
time step, then shuts off.
108
0 50 100 150 200 250 3000
5
10
15
20
25step control/parameter learning
time (s)
conc
entra
tion
(nM
)
[IIa] Desired[IIa] Uncontrolled[IIa] Controlled
Figure 5.9: Control of thrombin concentration in a hæmophiliac using a single step input
at time t = 0. The input is 6.92 pM/s.
0 50 100 150 200 250 3000
10
20
30
40
50
60
70
80
90step control/parameter learning
time (s)
conc
entra
tion
(nM
)
[IIa] Desired[IIa] Uncontrolled[IIa] Controlled
Figure 5.10: Control of thrombin concentration in a patient with factor-V Leiden by a single
step input at time t = 0. The input is 4.39 nM/s.
109
Chapter 6
Analysis: Stability, Model
Reduction, and Control
6.1 Introduction
Two threads were left dangling at the end of the previous chapter. In the context of the
hypercoagulatory disorder factor-V Leiden, we explored two different types of controllers,
one based on feedback linearization; and the other applying a simple step input, whose
value was learned over repeated trials so as to match the timing and concentration of the
thrombin peak. Each has its advantages and disadvantages (see the Discussion section of
the previous chapter), but neither was able to track perfectly the thrombin trajectory, at
least with the most likely choice of input on-rates (i.e. rates at which heparin binds to
blood proteins), and especially as the constraints on the input were made more stringent.
In this chapter we propose and demonstrate a modification of the feedback-linearization
scheme to overcome this problem. While exploring that terrain we also find a very simple
controller that operates with minimal error, at least when the input upper bound is relaxed,
and moreover obviates the need for observation of almost all of the state variables (all but
one).
The second residual issue is whether the system is globally, exponentially “minimum
110
phase,” which in turn depends on the stability of the so-called zero dynamics. If this and
another, technical condition are satisfied, then the control law (5.10) guarantees bounded
tracking.1 We shall see in the sequel that this is in fact not the case, and that the success
(such as it was) of our control law, curiously, was not guaranteed—at least not by any
theorem the author knows. Along the way we adduce some nice additional facts about the
controllable subspace of the system which we exploit to effect a model reduction.
6.2 Model Reduction and Bounded Tracking
6.2.1 Bounded tracking and minimum-phase systems
In what follows we rely on (Sastry, 1999). Recall that feedback linearization separates
the system into a set of completely controllable, linear dynamics:
ξ = Aξ + bv,
y = ξ1, (6.1)
on the one hand; and a nonlinear system:
η = λ(ξ, η) (6.2)
on the other. (See again Figure 5.1 for a conceptual rendering.) Now, the nonlinear system
(6.2), when confined to the manifold where the linear dynamics (6.1) are identically zero,
is known as the zero dynamics of the system. That is, if we insist that the output y(t) is
identically zero, which implies that all the linearized states ξi are as well (since ξi is simply
the integral of ξi+1), then the remaining variables evolve according to the so-called zero
dynamics,
η = λ(0, η), (6.3)
on an (n− q)-dimensional manifold, where again q is the relative degree of the system.
It turns out that the stability of an equilibrium point of these dynamics tells us some-
thing about the stability of certain equilibrium points of the original system. In particular,1Recall that “bounded tracking” means that the error between the desired and actual output and its
derivatives tends asymptotically to zero, and that the state is bounded.
111
suppose x0 is an equilibrium of Eq. 5.3 that also produces zero output, i.e. h(x0) = 0.
Then if we define (ξ0, η0) := Φ(x0), then ξ0 = 0 for any change of coordinates (ξ, η) = Φ(x).
(This can be seen by setting the output to zero in Eq. 6.1.) Thus η0 is an equilibrium of
the zero dynamics, and furthermore by appropriate definition of Φ(x) we make sure η0 = 0.
We are now in a position to state the minimum-phase property of an affine-control
system. The latter is called exponentially minimum phase at x0 if the equilibrium point
η = 0 of the zero dynamics is exponentially stable. This is in turn equivalent to
spec∂λ∂η
(0, 0)< 0, (6.4)
that is the spectrum (eigenvalues) of the zero dynamics at the equilibrium point η = 0 are
all in the left half of the complex plane.
Now, we care about these dynamics because a theorem exists (Sastry, 1999) that the
control law (5.10) of the previous chapter produces bounded tracking, in the sense lately
defined, if the zero dynamics are defined everywhere that our trajectory tracking requires
the state to travel, and if furthermore they are indeed exponentially stable over that same
region—that is, if the affine-control system is exponentially minimum phase.2
6.2.2 Construction of the zero dynamics
So we have at hand a simple mechanical procedure for adducing a sufficient condition for
bounded tracking and furthermore giving us some insight into the dynamics of the system.
In order to apply this procedure, however, we must first construct the nonlinear subsystem
(6.2). In particular, we find n− q solutions to the PDE
(∇xηi)g(x) = 0, (6.5)
and make sure that they are independent of the ξi. Why is this condition necessary? The
nonlinear dynamics of η are defined to be independent of the input, so these functions must
be orthogonal to the input vector field, g(x). Why does this suffice? The Frobenius theorem2The zero dynamics are also required to be Lipschitz continuous, which they are, though we do not
belabor the point here. Briefly: since the state variables are confined to finite quantities, the steepness ofthe function is concomitantly limited.
112
tells us that a distribution of p vector fields in an n-dimensional space is integrable—i.e.,
admits of n − p solutions η to the above equation—if and only if it is involutive (for the
definition of which, see the footnote on page 76in previous chapter). Now g(x) alone
constitutes a one-dimensional distribution, which is trivially involutive, so we know that
there are indeed n− 1 solutions. We shall pick the n− q of them that are also independent
of the ξi.
It is not normally easy to find solutions to a large PDE of the form (6.5), but the present
situation is an exception. In the case we have been considering at length, i.e. a heparin
input to a patient with factor-V Leiden, the vector field g(x) is a constant vector of all
zeros and a single one at the index of heparin—call it r. That means that the solutions all
have the very simple form∂ηi∂xr
= 0, (6.6)
where xr is heparin. Notice that ηi = xj is a solution to this equation for all j 6= r. Now, to
ensure that (ξ, η) = Φ(x) is a valid change of coordinates (i.e., that ∂Φ/∂x has full rank),
we obviously cannot let ηi = ξi; and in particular since ξ1 is the output we wish to control,
viz. thrombin, we don’t assign thrombin to any of the nonlinear variables either. This
leaves exactly n− 2 variables xi remaining for the variables ηin−2i ; all that remains is to
make sure that they are independent of ξ2 as well.
Thus by construction, the Jacobian ∂Φ/∂x of the change of coordinates is a permuted
identity matrix, except for the second row, corresponding to ∂ξ2/∂x. For the Jacobian to
have full rank, this row must be nonzero at the rth column, where again r is the index
corresponding to heparin. And indeed,
∂ξ2
∂xr= −k[IIa], (6.7)
that is, some rate constant k times the concentration of thrombin. Thus the matrix loses
rank only at the point [IIa] = 0—but that is, unfortunately, precisely the point we care
about, since we are trying to construct the zero dynamics (which means ξi(t) = 0 ∀i, t).
However, we can avoid this singularity in the zero dynamics by shifting the original state
variables. If, that is, xm = [IIa] in the original system, then we define xm = [IIa]− δ, where
113
δ > 0. Then∂ξ2
∂xr= −k(xm + δ), (6.8)
which is zero only when xm = −δ. The zero dynamics are globally defined, then, which
is a requirement for the bounded-tracking theorem lately reviewed. And the coordinate
change (ξ, η) = Φ(x) is defined almost everywhere: everywhere except along the manifold
where the concentration of thrombin is precisely zero. This irregularity corresponds to
the non-regularity we noted in the last chapter, i.e. the failure of the relative degree to
be constant across the entire state space; it, too, failed precisely at zero concentration of
thrombin. Recall from that chapter that our solution was to apply our control scheme
only after thrombin had drifted some (small) distance δ away from the singularity, so that
our control scheme was unaffected by it. Correspondingly, we only consider the change of
variables Φ(x) away from this manifold. (We might think of Eq. 6.3 as defining the “δ-
dynamics” of the system, then: the evolution of the nonlinear subsystem with the output
we really care about, thrombin, held at a concentration of δ.)
We have one more requirement, stipulated at the outset, and that is to make sure that
the equilibrium point η0 of the zero dynamics is 0. This can be accomplished by shifting the
η variables by the equilibrium point of the original system. (And of course adding constants
to η does not change its status as a solution to Eq. 6.5.)
Following the construction for a change of variables just outlined, then, our nonlinear
dynamics are:
η = λ(ξ, η) = f(Φ−1(ξ, η))− x0, (6.9)
where f(x) is just f(x) without the elements corresponding to heparin and thrombin; likewise
for x0. So we can now ask: are indeed the eigenvalues of the linearization of the zero
dynamics all strictly negative? that is, does Eq. 6.4 hold? Well, we can see without any
calculations that the answer must be “no.” We saw in the previous chapter that we can
express our ODE as:
x = Cr(x) + bu, (6.10)
where rk[b, C] < n. That means that the linearization ∂f/∂x = C∂r/∂x cannot have full
rank—and now some of these singularities carry over directly into ∂λ/∂x = ∂ f/∂x.
114
6.2.3 Intermezzo: model reduction
Is this the end of the line? No: these singularities are, as we shall see, a consequence
of constraints that in fact allow us to rewrite the dynamics in terms of a reduced system
evolving on a lower-dimensional manifold. We can then re-derive the nonlinear dynamics
(6.2) for this reduced system, which will no longer contain the singularities induced by the
rank deficiency of D = [b, C].
We begin by noting that
rk(D) =: p < n
and therefore
rk(N (DT )) = n− p,
i.e. the nullity (rank of the nullspace) of DT is n − p. Thus, there exist n − p linearly
independent (constant) vectors zin−pi such that
zTi (Cr(x) + bu) = 0
for any u and for all time. Alternatively, we write:
0 = zTi x =d
dt(zTi x),
so that integrating with respect to time we find that zTi x = ci for some constants cin−pi ;
or more specifically:
zTi x = zTi x0. (6.11)
Thus the n−p rank deficiency of the matrix D = [b, C] gives rise to exactly n−p linearly
independent algebraic constraints of the form (6.11)3, where the vectors zi form a basis for
the nullspace of DT . Each constraint can therefore be solved for one of the state variables
and substituted into Eq. 6.10, reducing the dimension of the system to p. Rather than
solve haphazardly for this nullspace4, however, we consider their physical meaning. These
constraints represent the conservation of mass inherent in the original differential equations,
enforced by using mass-action kinetics throughout (recall the discussion in Chapter 2).3In robotics, these constraints are called holonomic.4Matlab provides a command to do this in one step: Z = null(D’).
115
The idea behind conservation of mass in a chemical system is that the total amount
of (e.g.) factor XI in the system must be constant—in fact, equal to the initial amount
of factor XI—although how much of that protein is (e.g.) in complex with thrombin, or
in its activated form, or etc. may vary. Thus we could write the conservation equations a
priori—in the example just given:
[XI] + [XIa] + [XI:IIa] + [XIa:IX] = [XI]0; (6.12)
however that process is extremely tedious (and error-prone), especially for a very large
system like the present one. We present instead a mechanical procedure for recovering the
equations from D, requiring only an enumeration of the pure (i.e. uncomplexed) proteins.
Without loss of generality, we assume that the first n − p elements of x, i.e.
x1, x2, ..., xn−p, are these uncomplexed proteins. Each of these will show up in exactly
one conservation equation apiece since, after all, each contains only one type of protein.
We define Z = [z1, z2, ..., zn−p], a basis for the nullspace of DT (i.e. the columns of Z are
the vectors zi above), with each column corresponding to a conservation equation. So, for
example, the vector zi corresponding to the conservation equation (6.12) above has ones
in those rows corresponding to XI, XIa, XIa:IIa, and XIa:IX, and zeros elsewhere. In fact,
the only column with a one in the row corresponding to XI is this one: again, XI shows up
in no other conservation equation. Thus, given the ordering lately defined, with the pure
proteins listed first in the state vector, we can partition the matrix Z as:
Z =
In−pZ2
(6.13)
where In−p is an (n−p)-dimensional identity matrix. Partitioning D comformably, we have
DT = [DT1 D
T2 ], (6.14)
so that instead of writing DTZ = 0 we may say DT1 In−p = −DT
2 Z2, or
Z2 = −(DT2 )+DT
1 , (6.15)
with the “+” denoting the pseudoinverse. Now, Eq. 6.15 can then solved in conjunction
116
with Eq. 6.11 for the uncomplexed proteins, and these equations substituted back into the
final p equations of Eq. 6.10, giving an equivalent, reduced (p-dimensional) system.
For our system, this procedure yields the following conservation equations:
TF:VIIa + TF + TF:VII + TF:VIIa:IX + TF:VIIa:X + TF:VIIa:Xa + TF:VII:Xa
+TFPI:Xa:TF:VIIa = 0.005
IIa:AT + IIL + mIIa + mIIaL + IIa + II + VIII:IIa + Xa:Va:II + Xa:Va:mIIa + VIII:mIIa
+IIa:Tm:PC + mIIa:AT + Hep:IIa + Hep:AT:IIa + V:IIa + XI:IIa + V:mIIa
+IIa:Tm + TAT = 1400
Va36:APC:PS + VaLC + VaLC:APC:PS + LCL + Xa:Va + Va3 + VL + VaL + V
+Va + Xa:Va:II + Xa:Va:mIIa + Va:APC:PS + Va3:APC:PS + LC:APC:PS + V:Xa + V:IIa
+V:mIIa + Va36 = 20
TF:VIIa + VII + VIIL + VIIa + VIIaL + TF:VII + TF:VIIa:IX + TF:VIIa:X
+TF:VIIa:Xa + TF:VII:Xa + VII:Xa + TFPI:Xa:TF:VIIa = 10.1
APC:PS:VIIIa + IXa:VIIIa:X + VIIIaL + VIIIaiL + VIII + VIIIL + VIIIa + VIIIai
+IXa:VIIIa + VIII:Xa + VIII:IIa + VIII:mIIa = 0.7
IXa:VIIIa:X + IXL + IXa + IXaL + IX + TF:VIIa:IX + IXa:VIIIa + IXa:AT + XIa:IX = 90
IXa:VIIIa:X + Xa:Va + XL + XaL + X + Xa + TF:VIIa:X + TF:VIIa:Xa + TF:VII:Xa
+VIII:Xa + Xa:Va:II + Xa:Va:mIIa + TFPI:Xa + V:Xa + VII:Xa
+TFPI:Xa:TF:VIIa + Xa:AT = 170
XIa + XI + XI:IIa + XIa:IX = 30
APC:PS:VIIIa + Va36:APC:PS + VaLC:APC:PS + PCL + APC + APCL + PC + IIa:Tm:PC
+Va:APC:PS + Va3:APC:PS + LC:APC:PS + APC:PS = 60
APC:PS:VIIIa + Va36:APC:PS + VaLC:APC:PS + PSL + PS + Va:APC:PS + Va3:APC:PS
+LC:APC:PS + APC:PS = 300
117
Tm + IIa:Tm:PC + IIa:Tm = 1
IIa:AT + AT + mIIa:AT + Hep:AT:IIa + IXa:AT + Xa:AT + Hep:AT + TAT = 3400
TFPI:Xa + TFPI + TFPI:Xa:TF:VIIa = 2.5
3 APC:PS:VIIIa + 3 VaLC:APC:PS + 3 Va36:APC:PS + PCL + 3 IXa:VIIIa:X + XIa:IX + VIIIaL
+VIIIaiL + TF:VIIa + IIL + mIIaL + VL + VaL + VIIL + VIIaL + VIIIL + IXL + IXaL + XL + XaL
+APCL + PSL + TF:VII + 2 Xa:Va + 2 V:Xa + V:IIa + 2 VII:Xa + 2 TF:VIIa:IX + 2 TF:VIIa:X
+2 TF:VIIa:Xa + 2 TF:VII:Xa + 2 IXa:VIIIa + 2 VIII:Xa + VIII:IIa + 3 Xa:Va:II + 3 Xa:Va:mIIa
+2 VIII:mIIa + IIa:Tm:PC + 3 Va:APC:PS + 3 Va3:APC:PS + 3 LC:APC:PS + 2 APC:PS + 2 V:mIIa
+TFPI:Xa:TF:VIIa + LBS + Va3 + Va36 + VaLC + LCL = 3396
HC3 + HC36 + HC = LCL + LC:APC:PS
VaA3 + VaA36 = VaLC + VaLC:APC:PS
6.2.4 Stability of the zero dynamics
Recall that by construction, each monomial in the original differential equation x =
f(x) + g(x)u is either a single state variable scaled by a rate constant (kxi), or a product of
(different) state variables scaled by a rate constant (kxixj , i 6= j). That is, the dynamics are
multi-affine, meaning that if we hold all the variables constant except one, xi, each equation
is affine in xi. In fact, since the monomials never have degree higher than two, we might call
these equations “bi-affine.” This fact, along with the nature of the conservation equations—
viz., weighted sums of state variables and a constant—implies that our substitutions will
not change the form of the equations: they will still be bi-affine, post-reduction. We can
therefore treat the reduced system exactly like the original system—and in fact, from here
on we shall take Eqs. 6.1 and 6.2 to refer to the linear and nonlinear decompositions
(respectively) of the reduced system. The new system likewise (necessarily) has the same
relative degree, and the same construction described in Section 6.2.2 yields an everywhere-
defined zero dynamics.
118
So where are the eigenvalues of its linearization? That depends on where we evaluate it.
The trouble is that the complexity of f makes it hard to solve analytically for the equilibrium
point of interest. (We are not interested in the easily calculated equilibrium at x = 0, since
in fact the system will obviously never reach that point from any activated state: at the
very least we know that the irreversible binding of AT with the proteases it inhibits will
produce nonzero quantities of those bound complexes. Nor are we interested in the stable
quiescent state, i.e. the physiological steady state prior to the introduction of tissue factor,
for similar reasons.) We approach the problem numerically, then, evaluating Eq. 6.4 at
points along the controlled trajectory. And indeed, along this trajectory the eigenvalues of
the linearization lie in the strictly negative half of the complex plane. However, as the state
approaches an equilibrium, the rank of the Jacobian collapses, i.e. some of the eigenvalues
migrate rightward to (within the numerical precision of the machine) the jω axis.
We conclude, then, that system is not exponentially minimum phase; the convergence
to the equilibrium of interest is at best asymptotic. That means that we cannot guarantee
the boundedness of our tracking scheme—at least not with the theorem at hand.
6.3 Control: Model-Predictive and Proportional
The feedback controller of the previous chapter suffered from two major infirmities:
(1) the need to observe the state more or less continuously, which in turn requires a great
deal of technology in the way of protein-concentration sensors; and (2) an ad hoc approach
to constraint handling. We address both these problems in this section—although not,
unfortunately, at the same time.
6.3.1 State observability and feedback linearization
In the discussion at the end of the previous chapter, we lamented an unfortunate re-
quirement of the feedback-linearization scheme: it may in general demand observation of
the entire state (see Section 5.6.3). We explore that requirement in more detail presently.
119
As throughout this chapter, we restrict our focus to the case of treating factor-V Leiden
with heparin.
Now we saw in Chapter 5 that a feedback-control law of the form:
ud =1
LgLq−1f h(x)
(v − Lqfh(x)) (6.16)
could be used to exactly linearize the system—if, that is, we can calculate the Lie derivatives
LgLq−1f =: a(x) and Lqfh(x) =: b(x), where in the present case q = 2. Since these are
functions of x, they require observation of the state to calculate. For practical reasons,
the fewer states the controller must measure, the better. How few can it get away with
observing?
To begin with, a(x) and b(x) are functions, in the present case, of only 32 of the 89
proteins involved in the system (or anyway our model of it)—a reduction of 57 dimensions
for free. As a matter of fact, a(x) is a function of thrombin only—the most useful and most
easily measured protein—so from here on we concern ourselves only with b(x). Now certainly
not all of these 32 proteins contribute significantly to the two terms we are belaboring—so we
now apply a very simplistic heuristic to eliminate the (practically) irrelevant ones: If setting
a state xi = 0 in the equations for a and b does not appreciably change their trajectory,
then we add xi to a list of possibly irrelevant proteins. “Appreciably” is intended here in a
least-squares sense:
tf/T∑m=1
[b(x(mT ))− b(x(mT ))∣∣xi=0
]2 < θ
tf/T∑m=1
b(x(t))2, (6.17)
for some (small) threshold θ. After running through all 32 variables, we check if zeroing all
the elements on the list at once has any appreciable effect on the model trajectories, and
lo! it does not (otherwise we should have been forced to think harder about this heuristic);
this leaves only 21 relevant proteins. Notice, moreover, that we could have chosen smarter
heuristics with a little more effort (and time): perhaps considering (e.g.) the effect on b(x)
of holding xi = ci, where ci is the initial value of xi, or some constant intermediate between
the initial and final values. We content ourselves with showing that at most 21 protein
concentrations need be measured for feedback linearization with heparin, in the case of
factor-V Leiden.
120
Simulation of the control scheme with this “reduced” control law—i.e. Eq. 6.16 but
with the 11 ineffectual variables zeroed—yields, unsurprisingly, identical figures to those in
the previous chapter. Still, we ask: can we do better? As an extreme case, we consider
measuring only the concentration of thrombin, which evidently involves forsaking feedback
linearization (at least for the time being). In particular, we shall restrict ourselves to a
strictly proportional feedback, with the gain term manually adjusted (as usual). That is:
u = Kpe(t) = Kp(yd − y). (6.18)
Quite remarkably, this control law yields excellent results. Compare Figure 6.1 with the
corresponding results for feedback linearization in Figure 5.6: if there is a difference to be
appreciated, it favors the proportional controller!
0 50 100 150 200 250 3000
10
20
30
40
50
60
70
80
90naive constraints
time (s)
conc
entra
tion
(nM
)
[IIa] Desired[IIa] Uncontrolled[IIa] Controlled
Figure 6.1: Simulated proportional control of thrombin concentration during a clotting event
in a patient with factor-V Leiden, once more using a discrete controller sampling every 0.5
s (as in the previous chapter), and with input again confined to the range between 0 and
20 nM/s.
So: at least in our favorite test case, factor-V Leiden under treatment with heparin,
121
0 50 100 150 200 250 3000
2
4
6
8
10
12
14
16
18
20Input
time (s)
conc
entra
tion
rate
(nM
/s)
Figure 6.2: The discrete input, constrained to lie between 0 and 20 nM/s, that generated
Figure 6.1.
and with “reasonable” constraints on the input rate, a very simple proportional controller
operating at 2 Hz can achieve excellent tracking results.
6.3.2 Control with constraints: model prediction
Here we pursue a different objective from that of the previous section. The controller
exhibited in the last chapter was degraded by the imposition of constraints, so we implement
a model-predictive control scheme which explicitly accounts for these constraints. Such a
scheme is considerably simplified, however, when applied to linear dynamics. Fortunately
we have at hand a technique for rendering the control problem linear, namely: feedback
linearization. So we shall employ a linear model-predictive controller (LMPCer), operating
on the linearized subsystem 6.1. I say this approach is at odds with the attempt to minimize
the number of observed variables, then, because it will require the full power of feedback
linearization.
122
Theory
Our method is adapted closely from (Kurtz and Henson, 1997) and (Henson and Seborg,
1996), so the exposition shall be brief. The basic idea of model-predictive control is to find,
not the “right” control input for the current time step, but the best series of control inputs
u = [u(t), u(t + T ), ..., u(t + (m − 1)T )] for a time horizon of mT seconds, where T is
the sampling interval and m is some integer. As usual, “best” means minimal error in a
least squares sense between the actual and desired outputs. This minimization is especially
felicitous in the (time-discretized) linear, time-invariant setting, where a single, quadratic
cost function can be written for the entire control horizon. Minimizing this cost subject
to the constraints on the input amounts to a quadratic-programming problem (QP). The
control is applied by using only the first element of the control vector u; at the next time
step, the QP is re-solved for a new control vector, this one evidently stretching one step
further than its predecessor.
More explicitly, at each time step i we wish to find:
v∗ = argminv
m−1∑j=1
[ξj − ξd((i+ j)T )]TQ[ξj − ξd((i+ j)T )] + α[vj − vd((i+ j)T )]2
(6.19)
where v = [v1, v2, ..., vm−1], a vector of synthetic inputs for the next m time steps; the
vectors [ξ1, ξ2, ..., ξm−1] are the linearized states for the next m steps that would be produced
by using these inputs; and vd(iT ) and ξd(iT ) are respectively the desired input and state at
time step i. (Those circumflexes are here meant to indicate that these variables aren’t “real,”
i.e. they may never actually be realized in the model.) That is, we penalize deviations from
the desired state with the matrix Q and deviations from the desired input with the scalar
α. (In general, one may penalize fast controller moves, (vj − vj−1)2, as well, but this turns
out not to improve performance, so we leave Eq. 6.19 uncluttered.)
Now, vd is again just the qth derivative of the desired output (where again q = 2 is the
relative degree of the system and hence the dimension of the linearized state), as with vanilla
feedback linearization; and likewise the vector ξd is the desired output itself and its first q−1
derivatives (recall that the linearized system of Eq. 6.1 consists simply of q integrators of
the input). That leaves the anticipated future states ξi: these are transformed, via the well
123
known solution to the discrete-time state-space equations, into functions of their initial state
(i.e., the state of the linearized system at time iT ), of the input v, and of the (discretized)
state and input matrices, A and b. Using this formula and some algebra, we can transform
Eq. 6.19 into the more useful form5
v∗ = argminv
12vTHv + dTv
, (6.20)
where the matrix H is computed from the parameters Q, A, b, and α; and the vector
d is calculated from these as well as ξd, vd, and the current linearized state ξi. Notice,
incidentally, that d must be recalculated at every step, whereas H need not be.
This accounts for the cost function: now to the constraints. We impose a hard constraint
on the (linearized) state, namely that ξ1 not differ from the desired output by more that
δ at any point. Notice that this is not redundant with the cost function, since its overall
minimization might very well call for a gross output error at some time steps. Finally,
the constraints on the input u must be transformed into constraints on the synthetic input
v, and here a complication arises. At the current (ith) time step the transformation is
straightforward: we simply invert the control law, Eq. 6.16—although since it may swap
the maximum and minimum, we write:
vmin = minu∈umin,umax
ua(x(iT )) + b(x(iT ))
(6.21)
vmax = maxu∈umin,umax
ua(x(iT )) + b(x(iT ))
,
with a and b as defined above. However, the constraint v will (in general) have moved by
the next time step, since a and b will have moved. Kurtz and Henson (1997) suggest two
possible solutions. The first is to pretend that the constraints don’t move, i.e. to use the
present vmin and vmax for the entire control horizon. This turns out, as a matter of fact, not
to work so well. Alternatively, but much more expensively, we can simulate the nonlinear
subsystem over the control horizon, using the previous step’s synthetic input vector v∗—and
this achieves reasonable results, as we shall see.
Recall that the linear subsystem we are trying to control is in controllable canonical form,5Our constant companion Matlab has a built-in function to solve quadratic programming problems with
cost functions of this form.
124
with the bottom row all zeros. Thus it is marginally stable (in fact, all of its eigenvalues are
located at zero), so to ensure the stability of the controlled system we again add proportional
and derivative terms to yqd in reckoning vd, as in Eq. 5.16.
Finally: if the QP proves infeasible, we scale back the control horizon by one time
step. If no control horizon suffices to render the problem feasible—and this can and does
happen—the input from the previous time step is used.
Results
In all cases we consider a patient with factor-V Leiden being treated by heparin. The
output to be controlled is, as ever, thrombin concentration. The controller operates at
2 Hz for the proportional and feedback-linearized (FL) controllers, and at 0.5 Hz for the
LMPCer. In the previous chapter, we suggested that heparin should not be injected at
a rate “much higher than a single-digit nanomolarity per second,” and then proceeded to
cap it “somewhat arbitrarily” at 20 nM/s. Here we explore what happens if the maximum
input is lowered to 5 nM/s. As before, the controller obviously cannot attain negative values
(recall that this would correspond to removing heparin from the system).6
The control horizon of the linear model-predictive controller was set at m = 40, which
at a 0.5 Hz sampling rate translates into 80 seconds. The state-penalty matrix Q was set
to 10Iq (a (q × q)-dimensional identity matrix), zeroing the cross-time-step penalties; and
the input-penalty scalar α fixed at unity. The maximum allowable output violation (δ) was
capped at 2 nM.
Now compare Figures 6.3, 6.5, and 6.7, of the proportional, FL, and model-predictive
controllers, respectively. Proportional control is evidently completely inadequate. The FL
controller, too, exhibits errors of up to 100%: here the degradation we saw in the previous
chapter (Figure 5.6) is exacerbated by the enforcement of stricter constraints. The linear
model-predictive controller, on the other hand, reduces this error by some 50%. Some6On the other hand, it certainly is possible to manipulate the system simultaneously with both a pro-
and anti-coagulant—and the theory exists for multi-input feedback linearization. This complicates themathematics, but more significantly would seriously complicate the drug delivery.
125
attempt was made to optimize the performance of this controller, but there is additionally
reason to believe that the tracking could be improved even more by, e.g., increasing the
sampling rate and extending the control horizon. Unfortunately, both of these remedies do
exact a price in terms of computation time.
Finally, we get some idea of how the LMPCer achieved its superior results by comparing
the inputs of Figures 6.6 and 6.8. In particular, the model-predictive controller starts
increasing heparin input around 45 seconds in response to the predicted thrombin excess
some 50 seconds later that it “knows” it will not be able to suppress then, given the
constraints. The FL controller obviously issues no such anticipatory correction.
0 50 100 150 200 250 3000
10
20
30
40
50
60
70
80
90naive constraints
time (s)
conc
entra
tion
(nM
)
[IIa] Desired[IIa] Uncontrolled[IIa] Controlled
Figure 6.3: Simulated proportional control of thrombin during a clotting event in a patient
with factor-V Leiden, again at 2 Hz, but with input constrained to the range 0-5 nM/s.
6.4 Discussion and Conclusions
The goal of this chapter was to set the control techniques of the previous chapter on
a firmer foundation. Toward that end, we set out with the hope of providing a stronger
126
0 50 100 150 200 250 3000
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5Input
time (s)
conc
entra
tion
rate
(nM
/s)
Figure 6.4: The proportional controller’s discrete input for Figure 6.3, constrained to lie
between 0 and 5 nM/s
theoretical justification for our application of feedback linearization, viz. to demonstrate
that the control law Eq. 6.16, in conjunction with the synthetic input of Eq. 5.16, would
force our system to track the desired trajectory in a bounded way, i.e. with errors asymptot-
ically approaching zero. Attention was restricted throughout the chapter to our workhorse
example: the model of the pathology factor-V Leiden, treated by heparin, and attempting
to force the thrombin concentrations of a clotting event to track the thrombin trajectory
during healthy clotting.
Our attempt to prove the boundedness of tracking was guided by a sufficient condition
in the form of the stability of the zero dynamics of the system. To give these dynamics even
the chance of being stable, we had to remove the “redundant” equations from the system,
i.e. the ones corresponding to algebraic constraints on the state. These turned out to
be precisely the conservation-of-mass equations (or some linear combination thereof) that
inhere in mass-action formulations of the kinetics. We provided a mechanical procedure for
recovering these equations (in their most perspicuous and useful form) from the formulation
127
0 50 100 150 200 250 3000
10
20
30
40
50
60
70
80
90naive constraints
time (s)
conc
entra
tion
(nM
)
[IIa] Desired[IIa] Uncontrolled[IIa] Controlled
Figure 6.5: Factor-V Leiden, treated by heparin via a feedback-linearization controller
operating at 2 Hz, and within the range 0-5 nM/s.
Eq. 6.10 of the system dynamics and a list of the uncomplexed proteins; and then used
these equations to reduce the system to a structurally identical (i.e. both are “bi-affine”)
one that evolves on a lower-dimensional manifold.
Now, the equilibrium point of interest in this new system is very difficult to solve for,
so a numerical procedure was employed instead, evaluating the zero dynamics along the
controlled trajectory. As the state approaches its equilibrium along this trajectory, the
rank of the Jacobian of the zero dynamics approaches zero (meaning some of its eigenvalues
are approaching the imaginary axis), so we conclude (with some surprise) that the zero
dynamics of this system are not exponentially stable. That means that, at least with the
theorem at hand, we cannot prove the boundedness of tracking (the apparently bounded
results of the previous chapter notwithstanding).
A second deficiency of the control techniques demonstrated in Chapter 5 was their state-
observation requirement; in particular, because current sensor technologies limit our ability
128
0 50 100 150 200 250 3000
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5Input
time (s)
conc
entra
tion
rate
(nM
/s)
Figure 6.6: The discrete input generated by the feedback-linearization scheme, clamped
between 0 and 5 nM/s, that produced Figure 6.5.
to measure protein concentrations in vivo in real time, we should prefer to rely on as few
state measurements as possible. A heuristic showed that the controller could get away with
observing as few as 29 (out of about 90) of these concentrations—but in fact observation of
the single protein thrombin suffices for proportional control (obviously), and proportional
control was in turn demonstrated to produce excellent tracking—at least as good as its FL
counterpart—under reasonable assumptions on the input bounds (the same assumptions
that were employed in Chapter 5).
This result was also quite surprising (to the author), so it merits a little discussion.
Examining again the equations (5.10) and (5.16) for the control, and considering the present
case of q = 2 for perspicuity, we see that they might be combined as:
u =[Kp
a(x)Kd
a(x)1
a(x)
]yd − ξ1
yd − ξ2
yd − b(x)
(6.22)
with a(x) and b(x) defined as above. Thus we can view control via feedback linearization
129
0 50 100 150 200 250 3000
10
20
30
40
50
60
70
80
90LMPC
time (s)
conc
entra
tion
(nM
)
[IIa] Desired[IIa] Uncontrolled[IIa] Controlled
Figure 6.7: Control of thrombin using the LMPC scheme in conjunction with feedback
linearization. Again, the uncontrolled trajectory is a result of factor-V Leiden, and the
input rate of heparin is confined to the range 0-5 nM/s. Note the improvement over Figure
6.5.
with the correction terms of Eq. 5.16 as simply a (sophisticated) variant on the generic
proportional-derivative (PD) control. In particular, the controller penalizes deviations in
the output error and its derivatives, but the gains are state-dependent (via a(x)). To see
this more clearly, we present an alternative derivation of this control law.
Suppose we want to design a PDD (proportional, derivative, and second-derivative)
controller for the system. The proportional input is obviously just up := K ′pe, where e =
yd− y is the output error and for some gain K ′p. Now, since derivatives require information
“from the future,” these controllers must in general estimate the error derivatives. However,
suppose instead we derive explicit functions for the derivatives in terms of the state. Then
we can write the derivative feedback as ud := K ′de = K ′d(yd − ξ2), where ξ2 is the name of
the explicit function of the state for the derivative y. Now, if we impose a penalty on the
second derivative of the error along the same lines, we find that yd is a function not just of
130
0 50 100 150 200 250 3000
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5Input
time (s)
conc
entra
tion
rate
(nM
/s)
Figure 6.8: The discrete input (constrained to 0-5 nN/s) generated by the LMPCer and
producing Figure 6.7. Sampling rate is 0.5 Hz
the state but of the input, as well, so that our equation for the second-derivative feedback
udd looks like:
udd = Kdd[yd − b(x)− a(x)(udd + ud + up)], (6.23)
with the familiar a(x) and b(x) functions from feedback linearization, and where we have
broken the input into its three components. Rearranging terms, we write:
udd =Kdd
1 + aKdd(yd − b)−
aKdd
1 + aKddud −
aKdd
1 + aKddup,
where the dependence of a and b on x has been suppressed for brevity. Solving for u =
udd+ud+up and recalling the definitions for the proportional and derivative feedback yields:
u =Kdd
1 + aKdd(yd − b) +
K ′d1 + aKdd
e+K ′p
1 + aKdde. (6.24)
We might call this the “true control law” for our PDD controller.
Now consider the case where a(x)Kdd 1. Then the true control law reduces to:
u ≈ 1a
(yd − b) +1a
K ′dKdd
e+1a
K ′pKdd
e,
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and if we pick K ′d = KddKd and K ′p = KddKp, then this is just the (adjusted) feedback-
linearization control law, Eq. 6.22!
But what if, on the contrary, a(x)Kdd is on the order of 1? This corresponds, once
again, to the regions where the strict relative degree increases, i.e. where a(x) is close to
zero. Take the extreme case, in which a(x)Kdd ≈ 0. Then the true control law 6.24 reduces
instead to:
u ≈ Kdd(yd − b) +K ′de+K ′pe,
that is, a constant gain PD controller, with an additional penalty term. In fine, application
of the true control law in the regime near the singularity eliminates the state-dependence of
the gains. Starting with Eq. 6.24, then, and proceeding with just a modicum of boldness,
we might dare to drop the derivative penalties and retain a proportional controller:
u = K ′pe.
Oddly enough, then, we have inverted our picture: Instead of treating feedback lineariza-
tion as the theoretical impetus for our controller, and subsequently adding proportional and
derivative terms to account for errors, or plant-model mismatch, or whatever; we take the
PDD as primitive and derive its control law, of which the feedback-linearization control law
is a special case, viz. the case when a(x)Kdd 1.
This little excursion suggests that we employ the so-called true control law, Eq. 6.24,
throughout. It also indicates that proportional control will work best when a(x) is quite
small. Returning to the present study, we can now say something more specific. If the
input is heparin and the output thrombin, then a(x) = −kon[IIa], where kon is the rate at
which heparin and thrombin bind. Since this term depends only upon thrombin, then the
concentrations of other proteins cannot affect the proximity to the singularity (where the
relative degree increases), and hence cannot affect the applicability of the approximations
of the true control law that we have lately discussed. That in turn suggests that, no matter
what the disease, if thrombin is to be guided by heparin, proportional control should work
well. Whether this is also true of other drugs turns on their chemical equations, and in
particular how these ramify in the a(x) term.
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We proceed to qualify that last assertion: proportional control works well in this setting
provided the input constraints are sufficiently loose. As we have seen, increasing these
constraints degrades both the proportional and simple FL controller; but we have also now
seen that a (linear) model-predictive controller prevents a great deal of this degradation.
It is not, however, perfect (see again Figure 6.7), and here we explore (briefly) one of the
reasons why.
The constraints umax, umin, on the input u map into time-varying constraints vmax(x(t)),
vmin(x(t)) on the synthetic input v, so to compute the latter requires knowledge of the future.
Pretending that vmax(x(t)) (ditto for the minimum) is constant over the control horizon
hopelessly degrades the controller, so alternatively we (following (Kurtz and Henson, 1997))
use the final (m−1) elements of the vector of inputs (call it vi−1) generated at the previous
time step to simulate x ahead over the whole control horizon and then use this hypothetical
state to calculate vmax(x(t)). Only the final (m − 1) (where m is the number of steps in
the control horizon) are used in this simulation because the first element was actually used
at that previous step—whereas, in point of fact, none of these other elements of vi−1 will
actually be used. Moreover, vi−1 was the solution to the QP when a different (previous) set
of constraints was in place, which has the following curious consequence: The vi−1 used in
simulating over the control horizon may translate (via Eq. 6.16) into a (true) input u which
violates the constraints umax, umin—even though at the previous step these same synthetic
inputs translated into an acceptable u. This is possible because the translation has changed,
i.e. our estimates of future a(x) and b(x) have changed, as a consequence of our input at
the previous time step. (Kurtz and Henson (1997) do not seem to have noticed this fact.)
The infeasibility of the last m − 1 elements of vi−1 obviously vitiates the optimality of its
first element, i.e. the synthetic control used at the previous time step. Thus, the more
quickly a(x) and b(x) change, the more the accuracy of the LMPC scheme will be reduced.
Once again, in the present case we can say something more: Since a(x) = −kon[IIa], we
should expect the LMPCer to have difficulty wherever the slope of thrombin is steep—and,
yes, that is exactly where it does go wrong (once more consult Figure 6.7). Notice that the
proportional and FL controllers, on the other hand, do not exhibit tracking error where the
133
slope increases, but rather at the moment the input reaches its upper bound—and this is
just what we should expect.
134
Chapter 7
Validation/Learning
7.1 Introduction
In the preceding chapters, we have so far demonstrated a methodology for modeling
blood clotting (hybrid Petri nets); a series of control techniques (and related analyses)
for manipulating it; and an analysis of the sensitivity of the model to its parameters.
The control techniques are robust to parameter variations—that is, their applicability is
not predicated on specific values of those parameters—except insofar as those parameters
(including the constraints on the input) degrade the feedback linearization, and even then
model-predictive control was shown to reduce (if not eliminate) this degradation. And
although some of our conclusions about control in Chapters 5 and 6 were contingent on
certain structural facts about the model (e.g., the relative degree of the system, or the
form of certain Lie derivatives), nevertheless the general conclusions about these techniques
depend on facts about the form of the ODEs which will be true in any correct model.
The choice of hybrid systems, too, and specifically hybrid Petri nets, was also justified on
grounds independent of specific parameter values. Only the conclusions on the sensitivity of
the system were tied to getting the rate constants largely right; and even here this analysis
was meant in part precisely to identify how badly errors in these constants, relative to each
other, would impugn the accuracy of the model.
135
This independence of many of the foregoing analyses from the model parameters and
even from some structural aspects is a two-edged sword, however: The extent to which we
do exploit actual parameter values determines the extent to which we can predict specific
outcomes in, e.g., a clinical setting. The problem is that we don’t fully trust those parame-
ters! On the other hand, the model has been demonstrated to agree, at least qualitatively,
with some clinical results; and furthermore the parameters have some good claim to truth,
being the results of painstaking chemical experiments. The structure we have even more
confidence in. So the question is: how can we take advantage of the model to the extent
that it is indeed accurate, and lean on “external” gold-standard data where they exist and
contradict the model’s predictions?
The method proposed in this chapter, and explored briefly, is to take a machine-learning
approach. The model is to be thought of as a map from the space of initial conditions and
parameters into an output space, in which space different system states (e.g. hypercoag-
ulatory vs. hypocoagulatory vs. normal) are (hopefully) cleanly separated, or clustered.
The goal is to modify the clustering function so as to maximize the probability that two
sets of initial conditions giving rise to the same disease state get mapped into the same
cluster, and that two sets of ICs representing different disease states get mapped into dif-
ferent clusters—all subject to some constraints on the clustering procedure (e.g. the shapes
and number of clusters). Of course, which ICs correspond to diseases is only known for a
few cases; for the others we rely on the model. What we end up with is a clusterer whose
assignments of points to clusters is influenced both by the blood-clotting model and by our
prior knowledge of IC-disease pairs. How little the clusterer leans on the prior information
gives some indication of how useful the model (and the choice of output space) is, at least
for the task at hand: predicting hyper- and hypo- coagulatory states.
It would be nice to dispense with model simulations altogether, if possible. How might
we go straight from initial conditions to disease states, using the clusterer just described?
This would amount to a kind of generalization. It would be even nicer if the generalization
gave us the “distance” between one initial condition and the closest IC in a different output
class. Then we could give (e.g.) the minimal change in ICs to get from an unhealthy class
136
into a healthy class. All of these considerations together suggest a decision tree (the classic
description is in (Breiman et al., 1984)). At each stage in a decision tree, the clusters in the
current branch are partitioned according to the most informative (in the sense of entropy
reduction) predictor variable. This process is continued either until the partitioning is
finished (each data vector has been assigned to a leaf in the tree) or until the entropy
reduction falls below some threshold.
7.2 Methods
Once again we focus only on the ODE portion of the HS model, but only for the
convenience of doing all the work within Matlab. There is no other obstacle to using the
full model. (As a matter of fact, availing ourselves of the entire model would allow us to
incorporate clotting time and cross-linking percentages into the output space, which should
be quite useful in separating healthy and diseased vectors.)
Only 15 of the 80 or so proteins have nonzero initial conditions, and we do not explore
the possibility of variations in the others. (As far as I know, this never occurs in vivo.)
Even so, the “curse of dimensionality” prevents dense sampling of the input space: m
samples of each of the 15 dimensions requires over a billion samples (i.e. a billion model
runs, each taking around 5 seconds) even for a relatively sparse m = 4. We rely on some
prior knowledge then, to decrease this number: Assuming that each non-standard initial
concentration of some zymogen or other is the result of some independent and rare event,
the chances of more than (say) two of these events happening at once (i.e. in the same
individual) are very small. Therefore, we sample only the pairwise variations in ICs, 5
samples per protein. These are distributed evenly between 2% of normal levels and 100%.
Thus, for every pair of proteins, there are 25 different initial conditions corresponding to
all the combinations of deficiencies of 2%, 26.5%, 51%, 75.5%, and 100%.
We desire an output space that will cleanly separate hyper-, hypo-, and normal
coagulation—where by “cleanly” we mean that near neighbors to each exemplar should
themselves be classified with that exemplar. Once again we turn to thrombin, in particular
137
to its concentration profiles over the first 300 seconds of a clotting event. To reduce the com-
putational load on the clustering algorithm, we use only average thrombin concentrations
over 30 second intervals; thus the data live in a ten-dimensional space. We then reduce the
dimension further by rotating the data with a singular-value decomposition and throwing
away the dimensions corresponding to small singular values. This amounts to retaining just
two dimensions (with the felicitous consequence of letting us visualize the clusters).
Clustering was performed with the standard k-means algorithm (using Matlab’s im-
plementation), allowing the starting points (“seeds”) to be selected randomly. Now, I said
above that we want to rely on prior “gold standard” data where they exist. This was im-
plemented as follows. Three “observed” vectors were selected: one for normal coagulation
(all proteins at 100% of normal levels), one for severe hæmophilia A (factor VIII at 2% of
normal pre-injury levels, all others at 100%), and one vector for antithrombin deficiency
(AT at 26.5% of normal; all others unchanged). The k-means algorithm was then initialized
with k = 3, i.e. one cluster for each of the three states (too little, too much, and just the
right amount of clotting), and then k increased and the algorithm run again until the three
“known” vectors ended up in different clusters. Obviously, if this had required a k higher
than about 6, this method would have produced uninterpretable results (what do these clus-
ters correspond to?), and we should have been forced to take a different tack. In particular,
we are ready to adjust the output space, if need be: perhaps to consider concentrations of
some other protein, or to use a nonlinear transform on the thrombin concentrations. This
sounds rather ad hoc; a more principled version of this approach is described in Section
7.4.3 below. Nevertheless, in the present case, as we shall see presently, k = 4 sufficed to
separate the three known vectors.
The decision tree was also implemented using native Matlab functions. The probability
distribution over concentrations of the fifteen nonzero proteins was assumed to be uniform
for the purposes of the entropy calculation (we explore alternatives to this below). The full
(unpruned) tree was constructed, with splitting based on Gini’s diversity index (see e.g.
(Breiman et al., 1984)).
138
7.3 Results
These results, it must be stressed, are preliminary, and are meant primarily to demon-
strate the uses of the techniques just described.
Using k = 3 clusters in the k-means algorithm pushed the hæmophiliac and normal
vectors into the same cluster, which is obviously not acceptable. However, four clusters
suffice to class all three priorly known vectors separately. These data appear in Figure 7.1.
−3500 −3000 −2500 −2000 −1500 −1000 −500 0−500
−400
−300
−200
−100
0
100
200
300
400
500
Figure 7.1: An output space in which hypo- (black x’s), hyper- (light gray +’s), and normal
(gray circles) coagulation are cleanly separated by k-means, with k = 4 The white hexagram,
black square, and white pentagram (resp.) are the known exemplars of the three classes.
See text for an interpretation of the fourth class (gray diamonds).
What are we to make of the fourth class? In all of them thrombomodulin is at or below
139
25% —and in 90% of these cases, it is at 2%. We conclude that lack of thrombomodulin
induces extreme hypercoagulation, in (literally) a class of its own.
The decision tree shown in Figure 7.2 is unpruned: branches were created until each
leaf corresponded to single class. Left branches correspond to the labels at nodes, i.e. to
concentrations being less than some amount. Notice that, since the input space was sampled
only at pairwise deficiencies of initial conditions, only two left branches are ever required
to reach a leaf. Now, we can see (among other things) that, without knowing anything
about the values of other protein concentrations, the initial concentration of protein C is
most informative; so ceteris paribus if e.g. we can only measure one variable, we ought to
measure protein C. In fact (not shown in figure), having less than 63% of protein C (breaking
at the first node in the tree) makes hypercoagulation 75% likely. (Recall, however, that this
assumes uniform priors over the sampled concentrations.) Even more interestingly, we
can see (at a glance) the implications of certain deficiencies. For example, a patient with
antithrombin deficiency no worse than 38% of normal levels (1317 nM) cannot have healthy
clotting unless native concentrations of factor XI are also below about 87% (26 nM). Finally,
the tree makes evident certain minimal adjustments to move from one class to another, i.e
the fewest number of protein concentrations to change. The hypercoagulatory consequences
of protein-C deficiency, for instance, represented at the bottom of the leftmost main trunk,
can be rectified by decreasing native tissue-factor levels below 40% (0.002 nM) (as long as
protein C concentrations are at least 38%, that is).
7.4 Discussion
7.4.1 Clustering
The results of the clustering algorithm in some sense validate the model: a principled
choice of output space puts our exemplar initial conditions into different clusters, as they
ought to be. Of course, this conclusion depends (weakly) on whether k-means is the appro-
priate clustering algorithm; and this in turn depends on whether variance is the appropriate
140
statistic to determine cluster scatter. But this is also in some sense the minimal assumption,
i.e. what we should assume in the absence of any other information on cluster scatter.
The validation is also fairly weak, in that it depends on classifying only three exemplars
properly. A more robust validation would include additional IC vectors for each of the
three classes. Knowing other initial-condition vectors that correspond to normal clotting
would be particularly helpful. A stronger validation would also check the boundaries of the
clusters.
Now, the idea was not just to classify the exemplars properly, but to lean on them
where the model failed—which we did, by using them to determine (indirectly) the number
of clusters. A principled approach (outlined below) will measure the trade-off between
these two—obviously, if the cluster choices were dictated entirely by the exemplars, the
model would not have been in any way validated—but the best we can say presently is that
both the exemplars and the model were separately necessary but insufficient to produce the
clusters. So the exemplars contribute to the clusterer—but how much extra information
does the model provide? One kind of answer is that without the model, clustering is not
even possible: the points in input space were selected uniformly, so obviously any set of
clusters would be as good as any other. The three exemplars do give some information,
but there is no reason to believe that vectors corresponding to (e.g.) factor IX deficiency
(hæmophilia B) would be placed in the same cluster as the hæmophilia A exemplar. In
the very least, much more prior information would have to be known (i.e. we should need
many more exemplars). The true test of the information provided by the model would be to
take (non-exemplar) vectors classified as belonging to class A, and then test these vectors
against, say, “hold-out” data.
7.4.2 Decision tree
The decision tree generalizes over the mappings from initial conditions to clusters in
providing summaries of these mappings in the most efficient manner. This is useful for
determining, for example, whether a deficiency of a certain clotting factor is alone enough
141
to condemn the clotting to an unhealthy cluster. The tree also tells us what factors are
most informative about disease states (assuming that the clustering is more or less correct).
So, for instance, if we had to predict clotting outcome based on only three proteins, we
ought to measure protein C, antithrombin, and thrombomodulin. It is interesting to note
that all three are inhibitors.
I have called the generalizations provided by the tree “summaries” because they do not
contain more information than that in the initial conditions with their corresponding labels,
though they do provide that information in a much more useful form. We could additionally,
however, have used non-uniform priors for the initial concentrations, corresponding to the
observed clinical occurrences of the various deficiencies, in which case the tree would reflect
a compromise between the information given by the model plus the clusterer, on the one
hand, and those priors, on the other. Predictions from this tree might be quite useful in a
clinical setting.
7.4.3 A mathematical take
Some of the methods of this chapter were rather ad hoc. What follows is a more
principled (and technical) exposition follows. The map from disease states d into parameters
θ is taken to be a probabilistic function, P (θ|d), which in general we do not know (but about
which I shall say more shortly). The function P (s|θ), although it is written as a probability,
is a deterministic map given by the model, where s is a vector in some suitably chosen output
space—perhaps, as above, the average thrombin concentrations over 10-second intervals
during the first 300 seconds of a clotting event. The suitability of our choice will turn on
how well points in this space lend themselves to clustering by coagulatory (disease) state.
Finally, a clustering function Pφ(c|s) assigns points in the output space to clusters, based
on some parameter φ—setting, e.g., cluster size, or the cluster means, or etc.
The joint probability is then:
Pφ(c, s, θ, d) = P (d)P (θ|d)P (s|θ)Pφ(c|s), (7.1)
where P (D) is the prior probabilities of diseases. We are concerned, however, with Pφ(c|d),
142
the probability of assigning a disease to a certain cluster. In fact, we shall choose the
number of clusters equal to the number of disease states, so that the goal will be to pick φ
so as to maximize the probability that, for each disease, all occurrences of that disease (i.e.,
for all corresponding parameters θ) will be mapped to the same cluster. Notice, then, that
from Eq. 7.1 we can write:
Pφ(c|d) =∫θP (θ|d)
∫sP (s|θ)Pφ(c|s)dsdθ, (7.2)
so that the objective is to find:
φ∗ := argmaxφ
∑i
Pφ(C = ci|D = di)
= argmaxφ
∑i
∫θP (θ|di)
∫sP (s|θ)Pφ(ci|s)dsdθ
.
Now, using Bayes’ rule, P (θ|d) = P (d|θ)P (θ)/P (d), and we shall assume P (θ) and P (d)
are uniformly distributed (uninformative priors). P (di|θ), on the other hand, is for certain
“observed” parameters θj (the ones which we know a priori to correspond to certain diseases)
just an indicator function of d. That is, if θj is a known parameter set for disease m, then:
P (di|θj) = 1 i = m
0 i 6= m.(7.3)
For all other θj , P (di|θj) = 1/p, where p is the number of clusters; that is, we assume nothing
about the relationship between those parameters and the disease state. Furthermore, P (s|θ)
is 1 only for the s (call it sθ) corresponding to θ, and zero elsewhere. Thus we can write
the objective function more simply as:
φ∗ := argmaxφ
∑i
∫θP (di|θ)Pφ(ci|sθ)dθ
,
with P (di|θj) as above. So φ∗ is the solution to:
∑i
∫θP (di|θ)
d
dφPφ(ci|sθ)dθ = 0. (7.4)
Solving this maximization problem for φ∗, then, gives “best” classification function
Pφ∗(ci|sθ), i.e. the one that best fits the exemplar data. Of course we shall also have
built into this function whatever assumptions we want to make about the relationship be-
tween the classes and the output space: unimodality, almost certainly, but perhaps even
143
stronger assumptions. This allows us to be quite flexible, however; for example, we may
pick a distribution with nonzero skewness so that hypocoagulation can capture even the
extreme cases that had to be classed into an outlier class in the analysis above. The most
important constraint on this function is that the derivative in Eq. 7.4 be soluble.
7.5 Conclusions
The main point of this chapter was to present some techniques for using the model of
this dissertation, even in spite of any lack of fidelity, for clinical purposes. The idea was
to use the model in a “shallow” way, that is only to cluster initial conditions into three
(coarse) bins of coagulatory risk. The clustering required some, though very small, post
hoc adjustment in order to correctly classify three known exemplars of these bins: the
number of clusters had to be extended from three to four. The final class was explained as
“ultracoagulatory,” corresponding only to cases of severe (< 30%) deficiency in the inhibitor
thrombomodulin. A decision tree for summarizing the findings was also generated, from
which could be read off the minimal proteins to change in order to move a patient from an
unhealthy class to a healthy one. Such a tree could in practice also incorporate knowledge
of the a priori probabilities of certain protein deficiencies, in which case the nodes of the
tree could be used to determine the most predictive proteins in a clinical setting.
144
Figure 7.2: The decision tree. The class labels are M = medium, L = low, H = high, and
O = outliers for the four classes of coagulation levels.
145
Chapter 8
Conclusions & Future Work
8.1 Introduction
Having come this far, we survey the landscape we have traversed; discuss some terrain I
have skirted but would revisit for a fuller exploration; and (attempt to) descry the direction
of the main paths off toward the horizon.
The goals of this dissertation were:
• to build a complete model of the coagulation cascade, i.e. one embracing all the
relevant details but also open to (facile) incorporation of more accurate parameters
or even structural elements, if necessary;
• to use this model and some tools from control theory to devise methods for control-
ling coagulation, especially in pathological cases, by clever administration of anti- or
procoagulants;
• to say something about the mathematical properties of the system (e.g. sensitivity,
stability, controllability) and their clinical ramifications;
• and to exploit in a principled way our knowledge of coagulation as embodied in the
model, even in spite of its imperfections, to make useful predictions for the clinical
setting.
146
Now, since the “relevant details” include qualitative data and discrete events, the first
goal entailed modeling the cascade as “hybrid,” i.e. consisting of interacting discrete and
continuous state spaces. Simulations of thrombin profiles and clotting times under healthy
and pathological conditions were used to verify qualitatively the fidelity of the model. We
also explored the sensitivity of the model to its parameters.
Although approaching coagulation as a hybrid system allows (more) complete simula-
tions, it limits our analysis and control techniques. However, bracketing out portions of the
model leaves us with a purely continuous system, the control of which suffices (I claim: see
Section 8.3 below for the justification of this maneuver) to control clotting. We saw four
different control techniques applied to the simulation of various diseases in this model, each
appropriate to different conditions; and along the way produced a more compact represen-
tation of the dynamics—which turned out to be at best asymptotically stable. Lastly, with
respect to control, I made some general observations about the controllability of chemical
systems, and formulated a different (more general) derivation of the main control techniques.
Finally, I proposed using the model in conjunction with some machine-learning
techniques—clustering and a decision tree—in order to generate reliable predictions about
more shallow outcomes, viz. whether a set of initial conditions produces hypercoagulatory,
hypocoagulatory, or normal clotting. The main idea was to bias the predictions of the model
using what limited clinical data exist.
Below we discuss the merits and demerits of the results of this thesis in detail.
8.2 The Hybrid-System Approach
I have said repeatedly that the simulations of Chapter 4 demonstrate the viability of
a hybrid-systems model of blood clotting, inasmuch as it produces thrombin profiles and
clotting times congruent with the clinical literature. However, it must be stressed that
congruence is a weak criterion, in particular because of the underdetermination of the
model by clinical data. That means that although I have demonstrated the sufficiency
of such a model insofar as those data are concerned, I have not shown its necessity. Of
147
course, a hybrid-systems approach can indeed capture information that cannot be forced
into ODEs, but I have not made any (falsifiable) predictions exploiting this aspect of the
model which I can also show cannot be made with an ODE model. The trouble is in large
part the dearth of clinical data. On the other hand, the model is built precisely so as to
easily incorporate clinical hypotheses, so it can be part of the solution to this problem (used
in conjunction with clinical tests). Hybrid-Petri nets are particularly useful in this regard,
being perspicuous representations.
Mention of the software platform the HPNs were built in has thus far been relegated
to a footnote, but a word is due now. The chemical equations of this model give rise
to differential equations that are “stiff,” which roughly corresponds to operating at very
different (many orders of magnitude) time scales; or, alternatively, to large eigenvalues in
the Jacobian of the ODE. Special numerical algorithms are required to avoid solving these
ODEs at the smallest time scale, and again our old friend Matlab has built-in functions
for this purpose (I used ode15s). Visual Object Net++, the HPN platform, does not.
Indeed, the algorithms used to solve stiff ODEs often react very poorly to discontinuities
(ode15s, e.g., is not recommended for use in hybrid models), so the solution to this difficulty
is not a simple matter of building a better numerical solver—nor is it within the scope of
this thesis, but it must be said that it obliges VON++ to run about a hundred times more
slowly than the equivalent simulation in Matlab.
Finally, we briefly explored the sensitivity of the system to its rate constants. These
sensitivities were then used to change the rate constants in a principled manner while
attempting to match the timing results of the PT test. It turned out that matching these
results could be achieved by changing a few (a tenth of the total number of parameters) by
60% or less; but (on the other hand) at the price of intolerably degrading the fidelity of the
original simulations.
148
8.3 Control
8.3.1 Modeling assumptions
On what grounds can we restrict our control schemes to the continuous subsystem de-
scribed in Chapters 5 and 6 and still claim to be able to control clotting satisfactorily?
This amounts to asking how we can ignore the HPN which models contact-pathway initial-
ization, and the HPN which models the final stages of clotting. As for the former: under
normal circumstances, this portion of the cascade is thought to have limited clinical signif-
icance(Adcock et al., 2002). The latter is certainly significant, so in the control schemes of
this document I have assumed a worst-case requirement of reproducing the exact thrombin
profile of the healthy clotting. However, we should prefer to relax this requirement, as well
as to provide bounds for acceptable behavior of the initializing HPN (how diseased can this
portion be without significantly affecting the downstream portion of the cascade?); we defer
this to the section below on future work.
The control schemes of this thesis assumed the following simplifications. First, the model
captures the dynamics of the cascade in plasma, i.e. ignoring the effects spatial arrangements
on reactions. The ODEs are only an approximation, then, of the true dynamics. Likewise,
I have assumed no inflow or outflow of proteins—which, interestingly, can increase the
controllability of the system; see Chapter 5—which is to some extent justified by the limited
number of lipid binding sites available at the site of injury: even as new proteins flow in, the
number of reactions dwindle as the available binding sites are exhausted. This assumption
is nevertheless not strictly correct.
Finally, I made some assumptions about the anti-coagulant heparin: It was assumed
that the major action of heparin is on thrombin, whereas in reality it acts to some extent
on factor Xa, too. Some of the rate constants for heparin were also estimated, since in the
literature only the ratio of on- to off-rates has been reported.
149
8.3.2 Conclusions
The motivation behind the control techniques was to improve upon the current state
of the art in clinical treatments, which make use of only the most basic data—roughly,
empirical correlations between administration of certain quantities of anti- and procoagu-
lants, on the one hand, and a reduction in thrombophilia and hæmophilia (resp.), on the
other. These methods demands daily visits to a clinic for blood drawing and testing, as
well as maintenance of a permanent level of some drug in the bloodstream. Both of these
requirements are obviously undesirable.
Four different control techniques were proposed and demonstrated, although two of them
turned out to be instances of a more general control law. In the absence of constraints on the
input, a learned, step-input controller could enforce reproduction of the healthy thrombin
profile in the cases of hæmophilia A and of factor-V Leiden with total errors in the region
of 5%. These controllers require no observation of the state, but their clinical application
is predicated on the (perfect) fidelity of the model.
On the other hand, and again in the absence of constraints, a feedback controller based
on feedback linearization as well as a simple proportional controller achieved the same
performance without any “training” requirement, and without being tied to the specific
rate constants of the model. In fact, the FL controller requires only that the model be
control-affine (and in fact even this requirement might be dropped (Henson and Seborg,
1996)), and the proportional controller’s requirements are even weaker.
Interestingly, we can view these two controllers as implementations of the same control
law, Eq. 6.24, operating at different regimes, dictated by proximity to the singularity where
the relative degree of the system changes. I have given a more general formulation of this
control law, deriving it via design of a PD controller. This derivation also makes it possible
to see, inter alia, feedback linearization as implementing a kind of PD controller with
state-dependent gains. Moreover, the success of the proportional controller can be seen, in
light of this control law, as a consequence of the proximity of the state to the singularity
throughout the control interval. This proximity would be the same in our model for any
150
control of thrombin with heparin, suggesting that proportional control should be effective
for any diseases employing such treatment.
Is the bounded reproduction of the thrombin trajectory guaranteed? In real life,
plant/model mismatch and sensor noise make exact tracking impossible, so a guarantee
of bounded tracking is highly desirable. However, the answer is: not by the usual theorem,
which requires exponential stability of the so-called zero dynamics (the evolution of the
nonlinear subsystem with the feedback-linearized system held at zero). These dynamics are
at best asymptotically stable, at least near the equilibrium point of interest. In proving this,
I also provided a mechanical procedure for recovering the equations for mass conservation,
inherent in mass-action kinetics, from the differential equations for the system (provided
they are written in the form Eq. 5.12), and along with it a method for reducing the system
to a more compact form.
Finally, on this head, we considered how realistic constraints on the rate of heparin
injection might degrade the previous controllers. For factor-V Leiden, tight enough con-
straints do indeed degrade the FL and proportional controllers—but the addition of (linear)
model-predictive control overcame much of this degradation. It is not perfect, however, and
I have pointed out that the errors arise (at least in part) when the controller’s prediction
of future constraints on the “synthetic” input (the input to the linearized subsystem) goes
bad—and this happens whenever the Lie derivatives used to translate from real inputs to
synthetic inputs change fast.
A final word on control: How controllable is the system in general, i.e. using any input
(rather than just heparin)? I have shown that the dimension of the locally accessible mani-
fold is upper-bounded by the number of reactions in a chemical system, where bi-directional
reactions are counted just once. This allows one to state limits on the controllability of the
system without even constructing the system of ODEs; or, alternatively, to make demands
on the number of reactions and chemicals a therapeutic intervention must introduce if
complete control of the system is to be possible.
151
8.4 Learning/Classification
A certain tension has characterized my claims about the clinical relevance of the model
of this dissertation. On the one hand, I have stressed that the current state of the art
in disease prediction and treatment rely on only the crudest knowledge—usually corre-
lations between pre-injury blood-factor concentrations (what I have been calling “initial
conditions”) and disease states or health risks. On the other hand, I have said that the
predictions made by my model (as well as others) are underdetermined by clinical data. So
the solution to the primitiveness of the medical professions’s predictions would appear to
be more refined predictions based on the use of models; but the more refined predictions
offered by a model cannot be verified, on account of the exiguousness of the clinical data!
This problem can only be solved by performing in silico experiments in conjunction with
their in vivo and in vitro counterparts—and this will take time, as well perhaps as the
development of better sensor technologies. In order to use the model as it stands, then, 1
certain compromises must be made. In particular, I suggested using the model to make the
“shallow” prediction of whether a certain initial state would correspond to a disease, but
also biasing the classification with the clinical data (“exemplar vectors” and their labels) I
have just animadverted upon.
There is evidently no hope of clustering (e.g.) hæmophilia in the input space, at least
not without many more exemplar vectors, since deficiencies of (to stick with the example)
factor VIII (hæmophilia A) cannot be generalized to deficiencies of factor IX (hæmophilia
B) by any clustering algorithm. It is the model that gives us that information—as well,
presumably, as mappings that are not known to clinicians, in contrast to the one just given.
The fact that clustering in this output space, then, separates the exemplar, is some kind of
vindication of the model with respect to this shallow criterion; even more so is its ability to
do so with very little ad hoc gerrymandering. Correct classification of three exemplar initial
condition vectors (one for each class) required only that the number of clusters be increased
from three to four. A more complete validation is proposed in the following section.1I have to stress again that the applicability of the control techniques, and the related analyses, do not
depend on the model being perfect; they require only that it not be radically wrong.
152
The importance of the decision tree as it stands is its ability to summarize efficiently our
observations about the predictiveness of certain alterations in initial conditions. Its ultimate
usefulness, however, lies in its ability to incorporate knowledge about the prior probability
of protein deficiencies into these observations. So the original HPN incorporated as much
qualitative and quantitative knowledge as we have about the mechanisms of blood clotting
into a single model; the clustering analysis allowed for incorporation of prior knowledge
about correlations between initial blood-factor deficiencies and disease states; and the deci-
sion tree enabled us to integrate knowledge about the likelihood of certain deficiencies. This
last model is only intended for what I have called “shallow” predictions, but these are nev-
ertheless useful. So, for example, if a patient has a protein-C deficiency, having only about
30 to 65% of normal pre-injury concentrations, the decision tree tells us that a reduction in
the concentrations of any of factors II, V, VIII, IX, X, XI, VIIa, or of tissue factor (consult
Figure 7.2 for the precise values of these reductions) will suffice to bring clotting into the
normal regime. Perhaps this is cheaper or easier to implement than increasing the amount
of protein C. The tree also shows the interesting result that the initial concentrations of the
three major inhibitors—protein C, thrombomodulin, and antithrombin—are the best risk
predictors—at least assuming uniform priors on all the deficiencies.
8.5 Future Work
The model is not perfect but is easily modifiable and tested. Refinement of the model
in conjunction with clinicians is therefore an obvious next step. The discrepancy between
these simulations (and those of other models!) with the PT test must also be rectified.
Thirdly, given the difficulties with stiff equations just described, the model should also be
migrated over to another platform—like PIPE (Bonet et al., 2007)—where a different ODE
solver can be implemented.
In order to relax the constraints on the output, we might prove what range of thrombin
trajectories result in healthy clotting. This amounts to a reachability analysis on the HPN
for the final portion of the system. Likewise, determination of the range of initial conditions
153
which generate a sufficiently small amount of factor XI to prevent perturbation of the con-
tinuous subsystem amounts to a reachability analysis on the initializing HPN. Reachability
on hybrid systems is computationally intensive, and in fact prohibitively so for dimensions
greater than five. Thus we might alternatively discretize these HPNs and perform a discrete
reachability analysis on them. Reachability is a solved problem for a large class of discrete
Petri nets (Bause and Kritzinger, 1996); and in fact one of the major requirements is that
the capacity of the places in the network be bounded, which conservation of mass ensures
in the present case. Relatedly, we performed our sensitivity analysis only on the ODEs,
but a similar analysis could be performed on these HPNs and hence the whole system.
(A technique for sensitivity analysis of hybrid systems with a differential-algebraic-discrete
structure is described in (Hiskens and Pai, 2000).)
In Chapter 3 we discussed a control scheme for the ODEs (Belta et al., 2002) that would
benefit from such a reachability analysis on the final (Fibrin) HPN; we return to it now in
some detail. The idea, recall, was to steer the trajectory of interest by specifying conditions
on the system of ODEs, x = f(x) + g(x)u, only at the vertices of some polytope. If these
ODEs are multi-affine in the state—and the ODEs of our model are—then specifications at
the vertices are sufficient to ensure that the trajectory exit through the desired facet of the
polytope.
The way to visualize this control scheme is to imagine a box in three-dimensional space
(imagining 80-dimensional space being out of the question), at every corner of which is
attached the base of an arrow that points in the direction of the flow of the vector fields
f(x) + g(x)u and whose length is proportional to the magnitude of the flow. We want to
set these arrows—using the only means we have, i.e. adjusting u—so that they point away
from all the faces of the box except one (for concreteness imagine it to be the “roof” of the
box). In order to achieve this we need only find controls that satisfy these requirements
on the vector fields at the corners of the box (that is the interesting result of (Belta et al.,
2002)). In the present case, the idea would be to steer the state trajectory through that
facet (hyperplane) perpendicular to the thrombin axis at which thrombin achieves some
minimum required concentration—the location of which would come from the reachability
154
analysis. This would guarantee (assuming no deficiencies of blood factors in the Fibrin
HPN) that a blood clot form—eventually (see below). Similarly, after crossing the relevant
thresholds, thrombin levels could be driven back down through the same facet using the
same control scheme (though on a different, adjacent, polytope).
At first blush it may seem quixotic to effect this change in the vector fields via manip-
ulation of u—since u, being a single input, can change only a single vector field (namely
that of heparin, or whatever pharmaceutical we’re using). However, our control problem
is unconstrained in other aspects, and hence (perhaps) more easily satisfied. In particular,
most of the supporting hyperplanes of our polytope will be more or less arbitrary: we don’t
care about the concentrations of any factors but thrombin. So we can make our box as big
(or small) as need be till the vector fields at its corners satisfy the requirements. Of course,
this isn’t quite true since the box must always live in the positive orthant—blood factors
cannot take on negative values. But this is a help rather than a hindrance, since (by the
same token) vector fields lying along the face dividing two orthants cannot point toward
the negative orthant; and so at the corners of any polytope contiguous with these faces, the
vector fields must point in the correct direction with respect to that face, viz. inside the
box. (This is of course no guarantee that the other components of the vector field point in
the correct direction; the idea is just that, though we can’t extend the box any further in
that direction, we shall never need to.)
One significant advantage of this technique over the feedback controllers of this disser-
tation is that it does not require (the currently infeasible) real-time access to blood-factor
concentrations, except for that of thrombin—and this only in the case that we need to switch
control schemes in the middle of the clotting process, e.g. if we’re trying to prevent both
hyper- and hypocoagulation (a somewhat unlikely prospect). In general, the constraints
can be computed beforehand, and we need observe nothing.
One very serious disadvantage—though perhaps one that can be remedied—is that
this scheme affords no control over temporal aspects. In fact, that is why we have such
flexibility constructing our polytope: we can afford to be prodigal in our box size because
155
we are ignoring how long the state trajectory meanders around in the state space. It is for
this reason that this scheme has not been employed in the present document.
A second very serious problem is computational load. Since the vector fields are in
general functions of the input u, the inequalities derived for each corner of the polytope will
impose constraints on that input. In particular, the constraints take the form of n systems
of 2n linear inequalities, each in m unknowns, where m is the number of inputs. At n = 80,
we have the remarkable number of (approximately) 1024 constraints, for each of 80 systems.
Luckily, for each system we can throw away a constraint as soon as we have considered it,
i.e. we keep only the strongest constraint. That means we shall never have to store more
than 76 constraint equations. Nevertheless, although this eliminates the issue of memory, it
is clear that 1022 is a show-stopping number for computation time. Some cleverness will be
required to avoid carrying out all of these computations. It may, e.g., be possible to assume
some of the constraints a priori, by (again e.g.) exploiting the mass-conserving properties
of the system.
I have only provided a preliminary investigation of the use of clustering and classification
with the model, so there is much to be done in this regard. The most useful contribution
would be cross-validation of the clustering. That is, as a true test of the clustering algorithm
and the model’s ability to classify according to the three clotting states listed above, a larger
set of exemplars should be assembled (this could be done)—perhaps one exemplar for each
of the known hyper- and hypocoagulatory diseases (assuming the disease is a matter of
initial conditions; cf. factor-V Leiden). This set should then be split into validation and
training data, the former used to form the clusters—using perhaps a constrained-clustering
algorithm: see e.g. (Davidson and Ravi, 2005)—and the latter used to determine how well
the clusters perform on unseen data.
The easiest and most obvious next step with the decision tree is to assign priors to the
distributions over factor deficiencies. In fact, these priors would also improve sampling of
the model, which could then be done in true Monte Carlo fashion, rather than the uniform,
deterministic sampling of Chapter 7.
156
Finally, an attempt should be made to control the system using the more general control
law derived in Chapter 6. This law might also be incorporated into a model-predictive
controller.
157
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