A cardiovascular-respiratory control system model ...staff.technikum-wien.at/~teschl/delay.pdf · A...

A cardiovascular-respiratory control system modelincluding state delay with application to con-gestive heart failure in humans

Jerry Batzel Contact AuthorSpecial Research Center ”Optimization and Control,University of Graz,Heinrichstraße 22, A-8010 Graz, Austria.

Susanne Timischl-TeschlFachhochschule Technikum WienHoechstaedtplatz 5, 1200 WienAustria

Franz KappelSpecial Research Center ”Optimization and Control andInstitute of MathematicsUniversity of GrazHeinrichstraße 36, A-8010 Graz, Austria.

Email: [email protected] - Contact emailEmail: [email protected]: [email protected] FAX: +43 316 380 9795contact phone: +43 316 380 8552

Journal of Mathematical Biology manuscript No.(will be inserted by the editor)

Jerry J. Batzel, Susanne Timischl-Teschl and Franz Kappel

A cardiovascular-respiratory control systemmodel including state delay with applicationto congestive heart failure in humans

the date of receipt and acceptance should be inserted later – c© Springer-Verlag2004

Abstract. This paper considers a model of the human cardiovascular-respiratorycontrol system with one and two transport delays in the state equations describ-ing the respiratory system. The effectiveness of the control of the ventilation rateVA is influenced by such transport delays because blood gases must be trans-ported a physical distance from the lungs to the sensory sites where these gasesare measured. The short term cardiovascular control system does not involve suchtransport delays although delays do arise in other contexts such as the barore-flex loop (see [46]) for example. This baroreflex delay is not considered here. Theinteraction between heart rate, blood pressure, cardiac output, and blood vesselresistance is quite complex and given the limited knowledge available of this inter-action, we will model the cardiovascular control mechanism via an optimal controlderived from control theory. This control will be stabilizing and is a reasonableapproach based on mathematical considerations as well as being further motivatedby the observation that many physiologists cite optimization as a potential influ-ence in the evolution of biological systems (see, e.g., Kenner [30] or Swan [62]). Inthis paper we adapt a model, previously considered (Timischl [63] and Timischlet al. [64]), to include the effects of one and two transport delays. We will firstimplement an optimal control for the combined cardiovascular-respiratory modelwith one state space delay. We will then consider the effects of a second delay inthe state space by modeling the respiratory control via an empirical formula withdelay while the the complex relationships in the cardiovascular control will stillbe modeled by optimal control. This second transport delay associated with thesensory system of the respiratory control plays an important role in respiratorystability. As an application of this model we will consider congestive heart fail-ure where this transport delay is larger than normal and the transition from thequiet awake state to stage 4 (NREM) sleep. The model can be used to study the

Jerry J. Batzel: SFB ”Optimierung und Kontrolle”, Karl-Franzens-Universitat,Heinrichstraße 22, 8010 Graz, Austria

Franz Kappel: Mathematics Institute and SFB ”Optimierung und Kontrolle”,Karl-Franzens-Universitat, Heinrichstraße 36, 8010 Graz, Austria

Susanne Timischl-Teschl: Fachhochschule Technikum Wien, Austria

Supported by FWF (Austria) under grant F310 as a subproject of the SpecialResearch Center F003 ”Optimization and Control”

Version August 23, 2004.

Key words: Respiratory system, Cardiovascular System, Optimal control, Delay

Mathematics Subject Classification (2000): 92C30, 49J15

2 Jerry J. Batzel, Susanne Timischl-Teschl and Franz Kappel

interaction between cardiovascular and respiratory function in various situationsas well as to consider the influence of optimal function in physiological controlsystem performance.

1. Introduction

The cardiovascular system functions to maintain adequate blood flow tovarious regions of the body. This function depends upon the interaction ofa large number of factors including blood pressure, cross-section of arteries,cardiac output, and partial pressures of CO2 and O2 in the blood. Thereare global control mechanisms that act on the entire system to maintainappropriate blood flow and these mechanisms are supplemented by localmechanisms in each vascular region which act to shunt blood to those regionswhere demand is high and away from areas where demand is low. Theoverall control process which stabilizes the system is quite complicated andnot fully elucidated. Principles of optimal control theory will be applied todesign a control mechanism for this system. For further details about thecardiovascular system and control see, e.g., Rowell [55].

When breathing is not under voluntary control or subject to neurolog-ically induced changes, the human respiratory control system varies theventilation rate in response to the levels of carbon dioxide CO2 and oxy-gen O2 in the body (via partial pressures PaCO2

and PaO2). This chemical

control system depends upon information fed back from two sensory siteswhich monitor the blood gas levels (producing a negative feedback controlloop). These sensory sites at which the blood gas levels are measured are aphysical distance from the lungs (where blood gas levels are adjusted) andthus there are transport delays (which vary depending on blood flow) in thenegative feedback loop. Under normal conditions (even with delays in thefeedback control loop) the control system is sufficiently stable to maintainblood levels of these gases within very narrow limits. See, e.g., [11] or [14]for more information on this system.

There are a number of links between the respiratory and cardiovascularsystems. Function of the respiratory system depends on blood flow throughthe lungs and tissues. The amount of oxygen O2 transported to the tissuesand carbon dioxide CO2 transported away from the tissues depends oncardiac output Q and blood flow F through the pulmonary and systemiccircuits. Q and F depend in turn upon heart rate H , stroke volume Vstr ,resistance in the vascular system R, and blood pressure P . Arterial bloodpressure Pas is controlled via the baroreceptor negative feedback loop whichhas important effects on H , Vstr, R, and hence Q. Systemic resistance whichimpacts blood pressure is also influenced by local metabolic control actingon the resistance of the blood vessels of various tissues. This local control isin turn influenced by local concentrations of CO2 and O2, thus illustratinganother important link between the two systems. The effect of concentrationof O2 on the resistance of the systemic blood vessel is included in this model.Furthermore, PaCO2

and PaO2can affect cardiac output and contractility as

Cardiovascular-respiratory control system 3

well (see, e.g., Richardson et al. [54]). Neither these blood gas effects norsynchronization of heart rate and ventilation are included in this model.

An optimal control approach will be used to model the complex inter-actions in the cardiovascular-respiratory control system. The cardiovascu-lar and respiratory controls are represented by a linear negative feedbackcontrol which minimizes a quadratic cost functional defining optimal per-formance. Reasons and motivation for incorporating an optimal control ap-proach is given in Section 3. This modeling approach was previously appliedby Kappel and Peer [24] and Timischl [63] to study transition from rest toexercise under a constant ergometric workload and the role of pulmonaryresistance during exercise.

The equations describing the state of the system are developed followingthe ideas of Khoo et al. [31], Grodins et al. [15], and Grodins [16,17] andKappel and coworkers [24,28,48].

The model can also be used to study difficult to measure parameters(such as pulmonary resistance) in other conditions as well such as congestiveheart failure.

2. Model equations with delay

The general model equations including delays are given in equations (1) to(14). Symbols are defined in Tables 1 and 2. The respiratory component ofthe model is defined by equations (1) to (5) and is based on equations givenin Khoo et al. [31]. Two compartments, a lung compartment and a generaltissue compartment, are used to model the respiratory component of thesystem (see Figure 1).

The lung compartment equation (1) represents a mass balance equationfor CO2 and equation (2) similarly represents a mass balance equation forO2. The mass balance equations for CO2 and O2 in the tissue compartmentare given by Equations (3) and (4). Equation (5) tracks CO2 in the brainwhich is needed as input to the central respiratory sensor (see Section 10).We note that the brain is considered as part of the general tissue compart-ment. Transport delays appear in the mass balance equations as it takestime for tissue venous blood to reach the lungs and vice versa. The com-partment blood gas levels are adjusted by the ventilation rate VA which willbe further discussed in Section 3.

Note that the role of VA in the state equations for the lung compartment(1) and (2) is that of effective ventilation reflecting net ventilation after deadspace effects are removed.

Among the assumptions incorporated in the model we mention that themodel is an average flow model and thus ventilation represents minute ven-tilation and cardiovascular flow is non-pulsatile. Given the time scales andfocus of this study, these assumptions are reasonable. Other assumptionsare given in the appendix.

In passing we note that alveolar minute ventilation does not reflect mod-ulation of ventilation by the rate or depth of breathing which can influence


stability (see Batzel and Tran 2000 [1–3]) and that pulsatility in blood flowcan influence the distribution of blood and play a role in the baroreflexcontrol.

VACO2PaCO2

(t) = 863Fp(t)(CvCO2(t− τV )− CaCO2

(t)) (1)

+ VA(t)(PICO2− PaCO2

(t)),

VAO2PaO2

(t) = 863Fp(t)(CvO2(t− τV )− CaO2

(t)) (2)

+ VA(t)(PIO2− PaO2

(t)),

VTCO2CvCO2

(t) = MRCO2 + Fs(t)(CaCO2(t− τT )− CvCO2

(t)), (3)

VTO2CvO2

(t) = −MRO2 + Fs(t)(CaO2(t)− CvO2

(t− τT )), (4)

VBCO2CBCO2

(t) = MRBCO2+ FB(t)(CaCO2

(t− τB)− CBCO2(t)), (5)

casPas(t) = Ql(t)− Fs(t), (6)

cvsPvs(t) = Fs(t)−Qr(t), (7)

cvpPvp(t) = Fp(t)−Ql(t), (8)

Sl(t) = σl(t), (9)

Sr(t) = σr(t), (10)

σl(t) = −γlσl(t)− αlSl(t) + βlH(t), (11)

σr(t) = −γrσr(t)− αrSr(t) + βrH(t), (12)

H(t) = u1(t), (13)

VA(t) = u2(t). (14)

Table 1. Respiratory symbols

Symbol Meaning unit

Ca concentration of blood gas in arterial blood lSTPD · l−1

Cv concentration of blood gas in mixed venous blood lSTPD · l−1

MR metabolic production rate lSTPD ·min−1

Pa partial pressure of blood gas in arterial blood mmHgPv partial pressure of blood gas in mixed venous blood mmHgPI partial pressure of inspired gas mmHgB brain compartment -

u2 control function, u2 = VA lBTPS ·min−2

VA alveolar ventilation lBTPS ·min−1

VA time derivative of alveolar ventilation lBTPS ·min−2

VA effective gas storage volume of the lung compartment lBTPS

VT effective tissue gas storage volume lCO2,O2 carbon dioxide and oxygen respectively -

τ transport delay secIp, Ic cutoff thresholds mmHg


The cardiovascular component of the model is based on the work ofGrodins and coworkers [15–17] and Kappel and coworkers [24,28,48] andis described by equations (6) to (12). This component includes two circuits(systemic and pulmonary) which are arranged in series, and two pumps (leftand right ventricle). See Figure 1. Each circuit subsumes the system of ar-teries and veins, arterioles, and capillary networks under three components:a single elastic artery, a single elastic vein, and a single resistance vessel.Blood flow is assumed to be unidirectional and non-pulsatile. Thus, bloodflow and blood pressure are to be interpreted as mean values over the lengthof a pulse.

Mass balance equations for blood flowing through the systemic arteryand vein components are given by equations (6) and (7) respectively. Equa-tion (8) gives the mass balance equation for the pulmonary venous compo-nent. Under the assumption of a fixed blood volume V0, the equation forthe pulmonary arterial pressure can then be derived from the other cardio-vascular compartment pressures:

Pap(t) =1

cap(V0 − casPas(t)− cvsPvs(t)− cvpPvp(t)). (15)

Table 2. Cardiovascular symbols

Symbol Meaning Unit

α coefficient of S in the differential equation for σ min−2

Apesk Rs = ApeskCvO2mmHg ·min ·l−1

β coefficient of H in the differential equation for σ mmHg ·min−1

ca arterial compliance l ·mmHg−1

cv venous compliance l ·mmHg−1

F blood flow perfusing compartment l ·min−1

H heart rate min−1

γ coefficient of σ in the differential equation for σ min−1

Pas mean blood pressure in systemic arterial region mmHgPvs mean blood pressure in systemic venous region mmHgPap mean blood pressure in pulmonary arterial region mmHgPvp mean blood pressure in pulmonary venous region mmHgQ cardiac output l ·min−1

R resistance in the peripheral region of a circuit mmHg ·min ·l−1

S contractility of a ventricle mmHgσ derivative of S mmHg ·min−1

u1 control function, u1 = H min−2

Vstr stroke volume of a ventricle lV0 total blood volume ll,r left and right heart -p,s pulmonary and systemic circuits -


Lungs

PaO2

PaCO2

PaO2

PaCO2

PaCO2 PaO2

PvCO2PvO2

PvCO2

PvCO2

PvO2

PvO2

PvpPap

Pvs Pas

Qr Ql

Fs

Fp

Tissue

Controller

Auto-regulation

VA

H

PvO2

Rs

O2MR CO2MR

debitvalues

Fig. 1. Model block diagram

Blood flow F , which appears in equations (6) through (8) is related to bloodpressure via a form of Ohm’s law

Fs(t) =Pas(t)− Pvs(t)

Rs(t), (16)

Fp(t) =Pap(t)− Pvp(t)

Rp, (17)

where Pa is arterial blood pressure, Pv is venous pressure, and R is vascularresistance. Details can be found in [24,28,48].

As mentioned above, cardiac output Q is defined as the mean blood flowover the length of a pulse,

Q(t) = H(t)Vstr(t), (18)


where H is the heart rate and Vstr is the stroke volume. Subindices l andr are used to distinguish between left and right ventricle. Subindices s andp represent systemic and pulmonary circuits respectively. A complex rela-tionship between stroke volume and blood pressure is given in Kappel andPeer [24] which reflects the Frank-Starling law and the basic relation

Vstr(t) = S(t)cPv(t)

Pa(t). (19)

Here S denotes the contractility, Pv is the venous filling pressure, Pa is thearterial blood pressure opposing the ejection of blood, and c denotes thecompliance of the relaxed ventricle.

The Bowditch effect, which describes the observation that contractil-ity Sl (respectively Sr) increases if heart rate increases, is introduced viaEquations (9) through (12). This relation is essentially modeled via a secondorder differential equation. For details see Kappel and Peer [24].

Equations (13) and (14) define the variation of heart rate (H(t)) andvariation in ventilation (VA(t)) as mathematical control variables. The func-tions u1(t) and u2(t) will be derived using an optimality criterion which actsto minimize deviations in several quantities including these variations H(t)and VA(t) (see Section 3). Thus, limits are placed on the magnitude andvariation of the changes in the physiological controls H(t) and VA(t) whichreflects an assumption of minimal energy expense effort as an optimal con-trol criterion for the physiological control process.

Local metabolic autoregulation of systemic resistance is modeled usingthe assumption that systemic resistance Rs depends on venous oxygen con-centration CvO2

. Thus Rs is described by

Rs(t) = ApeskCvO2(t), (20)

where Apesk is a parameter. This relationship was introduced by Peskin[49] and is based on work on autoregulation by Huntsman et al. [22]. Theabove relationship was also used in Kappel and Peer [24]. Essentially, thisequation describes an important local constriction/relaxation mechanismacting on small vascular elements in response to local oxygen concentrationCvO2

(some tissues respond also to CvCO2). Global changes in Rs will be

discussed in Section 7. Delay in the control process of global resistance isnot analyzed in this paper.

Links between the respiratory and cardiovascular components can beseen in the equations. The respiratory mass balance equations include ex-pressions for the blood flows Fs and Fp. Levels of CvO2

which influencessystemic resistance via equation (20) is in turn affected by the respira-tory system. Heart rate H and ventilation rate VA influence both systemsthrough the control functions u1 and u2, while Pas, PaCO2

, and PaO2affect

the dynamical behavior through the cost functional.


3. Control of the system

The control for the cardiovascular and respiratory system will be designedto transfer the state of the organism from an initial perturbation (initialstate) to a final steady state in an optimal way that will be defined below.Given the complex and interrelated nature of the control systems discussedhere, the interaction of the various control effects will be represented as astabilizing control derived from optimal control theory. This approach todesigning a stabilizing control is motivated primarily by mathematical con-siderations and is reasonable given the lack of detailed information aboutparticular control interactions. Furthermore, this approach can provide in-formation on the nature and function of the controller as well as to helpidentify and study key controlling and controlled quantities.

As mentioned above, this approach is motivated primarily on mathe-matical grounds, but such a control derived from optimal control theory isfurther motivated by the view that optimal function likely plays a role inphysiological design. See, for example, Kenner [30] or Swan [62]. Minimizingstress on the system either by avoiding extreme actions or inefficient oper-ating states would represent such an optimal design criterion. The degree towhich physiological systems behave optimally is an open question of greatinterest.

In the model we focus on two physiological quantities which influencethe system: heart rate H and the ventilation rate VA. In the cardiovascularsystem, H is adjusted via the baroreceptor control loop and VA is adjustedby the respiratory control loop. These quantities are varied so that themean arterial blood pressure Pas and the partial pressures of carbon dioxidePaCO2

and oxygen PaO2in arterial blood are stabilized (along with the whole

system) to their steady state operating points when an initial perturbationoccurs. Parameters are chosen to define the steady state values which willbe the operating points (final steady state) as well as to derive the initialperturbed condition of the system.

The cost function we will use enforces the condition that the transi-tion from initial condition to final steady state is optimal in the sense thatPas, PaCO2

, and PaO2are stabilized such that the cumulative deviations of

these quantities from their final steady state values are as small as possible,while the presence of u1(t) and u2(t) in the cost functional implements thefurther restriction that excessive heart rate and ventilation change are re-stricted (effort is efficient). In this way, the stabilizing feedback control canbe considered also as an optimizing feedback control.

In the mathematical setting for this problem, it is the variations in heartrate (H(t)) and ventilation (VA(t)) that represent the control functions u1(t)and u2(t). By including u1(t) and u2(t) in the cost functional, limits areplaced on the degree to which H and VA can be varied to stabilize thesystem, a reasonable physiological constraint which also reflects an efficiencyof effort. The calculated control acts in the optimal way as defined by the


cost functional to transfer the system from one state (initial condition) toanother (steady) state.

The control problem is then formulated as follows: We determine controlfunctions u1 and u2 that transfer the system from one state to another suchthat the cost functional

∫ ∞

0

(qas(Pas(t)− P feas )2 + qc(PaCO2

(t)− P feaCO2)2 (21)

+qo(PaO2(t)− P feaO2

)2 + q1u1(t)2 + q2u2(t)2)dt

is minimized under the restriction of the model equations:

x(t) = f(x(t), x(t − τT );W s) +B u(t), x0 = φ.

y(t) = Dx(t).(22)

where x(t) ∈ R14 is given by

x(t) = (PaCO2, PaO2

, CvCO2, CvO2

, CBCO2, Pas, Pvs, Pvp, Sl, Sr, σl, σr , H, VA)T .

The vector f represents the system equations, W s represents the vectorof associated weights in the cost functional, and y(t) is a vector whichrepresents the observation of controlled values. The delay τT ∈ R+ is afixed point delay and the initial condition is a function φ ∈ C where Cdenotes C([−τT , 0],R14). The positive scalar coefficients qas, qc, qo, q1, andq2 determine how much weight is associated to each term in the integrand.Superscript ”fe” refers to the final equilibrium or steady state to whichthe system is transfered by the control. We note that partial pressures andconcentrations are interchangeable according to the dissociation formulas.We use concentrations in some state equations to simplify the form of theequations. For further information related to applications of optimal controltheory in biomedicine see, e.g., Swan [62] or Noordergraaf and Melbin [45],and for general reference on mathematical control theory see Russell [56].

4. Effects of the weights

In the simulations presented in this paper the weights associated with thequantities in the cost functional have values all set equal to one with theexception of qo, the weighting factor of PaO2

, which is set to 0.3. The moti-vation for a smaller weight for qo is that only large deviations in PaO2

act tosignificantly alter ventilation because, as can be seen from the adult oxyhe-moglobin saturation curve, there is a significant reserve of oxygen. Ventila-tory control response to PaO2

will be more pronounced only at lower levelsof PaO2

. In normal operating conditions a deviation of 1 mmHg in PaCO2

produces a larger percentage change in ventilation than does a proportionalmmHg deviation in PaO2

, thus suggesting that PaCO2is the primary focus of

control. These factors are also expressed in empirical relationships betweenPaO2

, PaCO2, and VA given by Wasserman et al. [67] or Khoo et al. [31] (see


Section 10). For these reasons, a smaller weight is associated with deviationsin PaO2

in the cost functional. In regards to the other weights it appearsthat the system reacts in a more sensitive way to deviations in the othervariables than to deviations in PaO2

. Because we lack more information, wetake the same weight for all variables except PaO2

. In order to get moreinformation on these weights one would have to parameter estimation onthe basis of data obtained from appropriate tests which we plan to pursue.

In simulation studies [64] it was found that respiratory and cardiovascu-lar quantity cross-interaction through the cost functional is minimal. Con-sidering the respiratory component weights in isolation, simulations indicatethat the initial drops in VA and PaO2

at the transition to sleep (see Figures(2) thru (7)) are much more extreme when the PaO2

weight is small. Whenequal weights are given to PaO2

and PaCO2the initial undershoot of the final

steady states is much smaller. Simulations indicate that the undershoot issmall for qo greater than 0.3. The weights chosen for these simulations arereasonable given that there is certainly a range of responses for VA to givenPaO2

and PaCO2levels and of heart rate H to arterial blood pressure Pas

levels. Parameter identification could set these values for individual cases.

5. Simulation Method

In the simulations we will consider the control which transfers the systemfrom the initial condition ”resting awake” denoted by xa to the final steadystate ”stage 4 quiet sleep” denoted by xs. We view the ”resting awake state”as an initial perturbation of the steady state ”stage 4 quiet sleep”. Thesestates are defined by parameter choices. We will consider two cases. In thefirst case we implement both the respiratory and cardiovascular controls asoptimal controls and derive formulations for both heart rate H and venti-lation rate VA which transfer the system from ”resting awake” xa to ”stage4 sleep” xs in an optimal way. Thus we do not consider explicitly the respi-ratory control sensory system and hence equation (5) is not required untilSection 10. For this case we have only the delays in transport between thelung and tissue compartments. Here τT is the transport delay from the lungsto the tissue compartment (see Grodins, [15] τT = 24 s). The transport delayfrom the tissue to the lung compartment τV is somewhat longer but due tothe relatively stable behavior of the venous side blood gases under normalconditions (situations where the state variables are in the physiologicallymeaningful range and without extreme variations) it is reasonable to con-sider τV = τT . Indeed, the dynamics of the system change almost not at all ifτv is varied, given that the venous side state variable variations are minimaland much damped compared to arterial side changes. This approximationof τv = τT is chosen to simplify computations.

In the second case we will incorporate VA into the state equations viaan empirical formula with delay relating VA to levels of PaCO2

, PaO2, and

PBCO2. These delays are in reality state dependent and nonconstant since

they depend upon blood flow Fs(t) which in turn is affected by cardiac


output Q(t), systemic resistance, and indeed the blood gases PaCO2and

PaO2. However the decrease in cardiac output during the transition to stage

4 sleep is about 10% and we will assume the delays are constant.The equilibrium equations for the system (1) to (14) determine a two-

degree of freedom set of steady states. Thus it is necessary to choose steadystate values of two state variables as parameters when calculating the awakeand sleep steady states for the system. In general we choose values for PaCO2

and H . These quantities are chosen as the parameters for the equilibriabecause PaCO2

is tightly controlled independently of the special situationand H is easily and reliably measured.

In summary, the transition from the ”resting awake” steady state to”stage 4 (NREM) sleep” is simulated by carrying out the following steps:

1. Compute the steady states ”resting awake” xa and ”stage 4 sleep” xs.The steady states ”awake ” and ”sleep” are defined by a set of parameterchanges to be discussed in Section 7.

2. The control functions u1 and u2 which transfer system (22) from theinitial steady state ”awake”, xa, to the final steady state ”sleep”, xs, arefound as follows. We consider the linearized system around xs with initialcondition x(0) = xa, and the cost functional equation (21). The controlfunctions u1 and u2 are then computed such that the cost functionalis minimized subject to the linearized system. This is accomplished bysolving an associated algebraic matrix-Riccati equation which is used todefine the feedback gain matrix. In particular, u1 and u2 are given asfeedback control functions.

3. This control is used to stabilize the nonlinear system (22) defined byequations (1) to (14). The control will be suboptimal for the nonlinearsystem in the sense of Russell [56] and stabilizing.

6. Analytical considerations

We consider first the case where both heart rate H and the ventilationrate VA are modeled as optimal controls. We give the mathematical settingfor the system with one delay. In this case, since VA is defined by optimalcontrol we don’t need equation (5). We carry it along here for reference inthe two delay case. The nonlinear system described by Eq. (1) to Eq. (14)with one constant point delay is represented by the vector system (23) as

x(t) = f(x(t), x(t − τT );W s) +B u(t), x0 = φ

y(t) = Dx(t)(23)

where x(t) ∈ R14 is given by

x(t) = (PaCO2, PaO2

, CvCO2, CvO2

, CBCO2, Pas, Pvs, Pvp, Sl, Sr, σl, σr , H, VA)T .

The vector W s represents the associated weights for the sleep steady state(in general, we use the same weights for ”awake” and ”sleep” states. Theinitial condition function φ ∈ C will be chosen as a constant function,


φ = xa, where xa is equal to the initial steady state vector ”awake” fromwhich the system will be transferred to the final steady state sleep xs bythe control. As controlled variables we have the observations

y(t) = Dx(t) = (PaCO2(t), 0.3PaO2

(t), Pas(t))T , (24)

where D ∈M3,14(R) is the weighting on the observations given by

D =

1 0 0 0 0 0 . . . 00 .3 0 0 0 0 . . . 00 0 0 0 0 1 . . . 0

(25)

The control u(t) ∈ R2 denotes the vector

u(t) = (u1(t), u2(t))T = (H, VA)T

where B ∈M2,14(R). Explicitly, we compute a feedback control

u(t) = −Fm x(t) (26)

where Fm is the feedback gain matrix.We want to stabilize system (23) around the stage 4 sleep equilibrium

xs. Therefore our first step is a shift in the origin of the state space byintroducing new vector variables ξ and η

ξ(t) = x(t) − xs,η(t) = y(t)− ys. (27)

Next, we approximate ξ linearly. To this aim we replace x in (23) by ξand make a Taylor expansion around xs for fixed time t. We are treatingx(t− τT ) as an independent variable for the expansion. This yields

x(t) = ξ(t) = f(xs + ξ(t), xs + ξ(t− τT );W s) +B u

= A1 ξ(t) +A2 ξ(t− τT )) +B u+ o(ξ).(28)

Here o(·) denotes the Landau symbol (h(x) = o(k(x)) :⇔ ‖h(x)‖/‖k(x)‖ −→0 as x −→ ∞). Note that the original state equations were already linearwith respect to the control u. The matrices Ai ∈M14,14(R), i = 1, 2 are theJacobians of f with respect to x(t), and x(t − τT ), respectively, evaluatedat x = xs,

A1 =∂f

∂x(t)(xs;W s),

A2 =∂f

∂x(t− τT )(xs;W s).

(29)

Analogously,

η(t) = (PaCO2(t)− P saCO2

, 0.3(PaO2(t)− P saO2

), Pas(t)− P sas)T

= D ξ(t).(30)


By neglecting terms of order o(ξ) we derive linear approximations ξ`(t)and η`(t) for ξ(t) and η(t), respectively,

ξ`(t) = A1ξ`(t) +A2 ξ`(t− τT ) +B u(t),

η`(t) = Dξ`(t),

ξ`(0) = xa − xs.(31)

This is a special case of the general linear hereditary control system

x(t) = Lxt +Bu(t), t ≥ 0,

y(t) = Dx(t).(32)

Here xt(s) = x(t + s), −h ≤ s ≤ 0, h > 0, where x(t) ∈ R14, u(t) ∈R2, and y(t) ∈ R3. Also B ∈ M2,14(R) and D ∈ M3,14(R). In the abovecase Lxt = A1xt(0) + A2xt(−τT ) and h = τT . With this setting we applythe results on approximation of feedback control for delay systems usingLegendre polynomials found in Kappel and Propst [26] (see also [27]). In thisapproach the control is found for approximating systems defined on finitedimensional subspaces of Rn x L2 [−h, 0] utilizing Legendre polynomials. Forthis approach we use the first 5 Legendre polynomials. Thus, the calculatedcontrol will be an approximate control for the actual system. In the abovepaper it was shown that the control for the approximating system convergesto the control for the actual system as the approximating system convergesto the actual system in an appropriate sense.

7. Modeling Sleep

During sleep, as a result of physiological changes in the body (sometimesreferred to as the withdraw of the ”wakefulness drive”), the ventilatorycontrol system is less effective for a given level of blood gases. For example,lower muscle tone during quiet sleep affects the reaction of the respiratorymuscles to control signals. This reduction in responsiveness results in VAfalling as one transits from the ”awake” state through stage 1 to stage 4quiet or NREM sleep. The net effect is a decrease in PaO2

and an increasein PaCO2

(see Shepard [59]) even though metabolic rates also fall. See, e.g.,Krieger et al. [38], Batzel and Tran [1], or Khoo et al. [35] for further details.

In sleep, general sympathetic activity is reduced and heart rate andblood pressure fall. Cardiac output is generally reduced though the degreeof reduction varies with situation and individual. See, e.g., Somers et al.[60], Mancia [41], Podszus [52] and Shepard [59].

Research suggests (cf., eg., Mancia [41], Podszus [52], Bevier et al. [5],and Somers et al. [60]), that peripheral resistance, as well as, perhaps, strokevolume are reduced during NREM-sleep. The reduction of sympathetic ner-vous system activity (see [60]) in the transition from quiet awake to NREMsleep would trigger these changes.

Given the reduction in sympathetic activity, a reduction in peripheralresistance is a reasonable consequence and, in general, such a reduction in


sympathetic activity should also impact contractility, resulting in a reducedstroke volume. It appears that counter influences such as an increase incardiac filling pressure (and end diastolic volume) resulting from being inthe supine position would tend to raise stroke volume. In the simulationshere, while not explicitly modeling the influence of position, we do includethe effect of the reduction in contractility caused by reduced sympatheticactivity. As a result a small drop in stroke volume is observed. However, themajor influence on Q is the drop in H .

The effect of reduced sympathetic activity on systemic resistance inNREM sleep is implemented as follows. In the model in the awake reststate, variation in Rs is only produced by the local control described above(with the base line global resistance fixed by the parameter Apesk). Thislocal control would not disappear during sleep but global influence on sys-temic resistance by reduced sympathetic activity should result in a modestreduction in systemic resistance. To model this change, note that Apeskwhich appears in relation (20), acts as a gain constant, relating oxygen con-centration and resistance. This constant will be reduced to model the globalreduction due to sympathetic effects. Further research will include a morecomplete picture of the two contributing and interacting controllers of Rs,one local, and one global. Based on steady state relations in the model, thesympathetic influence on contractility is modeled via a reduction in param-eters βl in equation (11) and βr in equation (12). In the simulations givenhere, Apesk is reduced by 5%, and in the contractility equations (Bowditcheffect), the parameters βl and βr are reduced by 10% in the NREM-steadystate (thus reducing contractility). Tables such as 3 and 4 list these valuechanges. In summary, the steady state ”sleep” is implemented by the fol-lowing parameter changes (recall PaCO2

and H are chosen as parameters forthe system):

– lower heart rate H ,– higher PaCO2

concentration in arterial blood,– lower O2 demand (MRO2 ) and lower CO2 production (MRCO2),– decrease in Rs by reducing Apesk ,– decrease in contractility by reducing βl and βr.

Once the parameters are chosen, we implement the steps outlined in Sec-tion 5.

The transition to stage 4 sleep is in reality not instantaneous but takessome minutes. We consider a transition time of 3-4 minutes. We include forthe dynamic simulation a time dependent decrease in the metabolic ratesover the transition time to stage 4 sleep and the same for the changes incontractility and systemic resistance. We still implement the control func-tions u1 and u2 calculated for a time-independent linear system thoughthese changes are time dependent. This further reduces the optimality butthe thereby obtained (suboptimal) control still stabilizes the system and isuseful for dynamic studies.


The sleep dependent changes in contractility (βl and βr), the metabolicrates, and resistance (Apesk) are assumed to be mostly accomplished bystages one and two. This assumption is made for purposes of exploringthe dynamics of transition and because not much is available in the liter-ature about the actual time course of these parameter changes changes insleep transition. With this model it is possible to explore various parameterchange time courses.

8. One delay simulations

In the first simulation we consider the transition between xa and xs fora normal adult with slightly elevated Rs. We assume that heart rate Hfalls from 75 to 68 bpm and that PaCO2

rises from 40 mmHg to 44 mmHg.We will apply the calculated control for the linear system to the nonlinearsystem and in this case, the control will be suboptimal but stabilizing. Allcalculations are performed using Mathematica 3.0 Tool boxes. The Mathe-matica package NDelayDSolve by Allen Hayes gives the numerical solutionof delay-differential equations.

Tables 3 and later parameter tables give the chosen parameters used formodeling given conditions such as the ”resting awake state”, ”NREM sleepstate” or ”congestive heart failure state”. Table 4 and later steady statevariable tables give the steady states computed from the model with thechosen parameters. In this case Tables 3 and 4 give values for resting awakeand stage 4 sleep with optimal control for a normal adult. Tables 18 to 20in the appendix give some comparison values from the literature.

Table 3. Optimal control parameters: normal adult sleep transition

Parameter Awake Sleep

Apesk 147.16 139.80βl 85.89 77.30βr 2.083 1.87H 75.0 68.0MRCO2 0.266 0.224MRO2 0.310 0.260PaCO2

40.0 44.0τT 24.0 24.0

Figures (2) thru (7) give the dynamics of the system produced by thecontrol for the optimal control case.

Using the above parameter assumptions, the steady state values for”resting awake” and ”stage 4 sleep” are calculated from the model and givenin Table 4. The qualitative changes in steady state values derived from themodel agree with observed behavior of the cardiovascular-respiratory con-trol system. Quantitatively, the simulated values fall within cited ranges


Table 4. Optimal control steady states: normal adult sleep transition

Steady State Awake Sleep

H 75.00 68.00Pas 101.7 87.04Pap 17.18 16.00Pvs 3.619 3.912Pvp 7.477 7.486PaCO2

40.0 44.0PaO2

103.38 98.92PvCO2

48.29 51.95PvO2

34.46 35.23Ql 4.938 4.335Qr 4.938 4.335Rs 19.86 19.18Sl 71.999 58.751Sr 5.488 4.478

VA 5.739 4.393Vstr,l 0.0659 0.0637Vstr,r 0.0659 0.0637

1 2 3 4 5minutes

38

40

42

44

46

mm Hg

PACO2

1 2 3 4 5minutes

85

90

95

100

105

110mm Hg

PAO2

Fig. 2. Optimal control: normal case

1 2 3 4 5minutes

46474849505152mm Hg

PVCO2

1 2 3 4 5minutes

34.234.434.634.8

3535.235.4

mm Hg

PVO2


(see below) of commonly reported values for the physiological conditions weare considering. We note that there exists a variety of response combina-tions for various individuals requiring a parameter identification if specificdata is compared. The model predicts decreases in Pas and VA as experi-mentally observed in the sleep state (see, e.g., Krieger et al. [38], Phillipson


1 2 3 4 5minutes

85

90

95

100

105mm Hg

PAS

1 2 3 4 5minutes

3.2

3.4

3.6

3.8

4mm Hg

PVS


1 2 3 4 5minutes

15.5

16

16.5

17

17.5

18mm Hg

PAP

1 2 3 4 5minutes

7.4257.45

7.4757.5

7.5257.55

7.575

mm Hg

PVP


1 2 3 4 5minutes

0.063

0.064

0.065

0.066

0.067

0.068liters

VSTR

1 2 3 4 5minutes

3.5

4

4.5

5

5.5litersper min

QL


[50], Podszus [52], Somers et al. [60], and Mateika et al. [43]). Decreases inQ and stroke volume as reported in Shepard [59] or Schneider et al. [57]are indicated by the model. The drop in PaO2

and increase in PaCO2is con-

sistent with data provided in Koo et al. [37], Phillipson [50], and Shepard[59]. Further, the model reflects the drop in systemic resistance as well aspredicting an increase for Pvs. See Tables 18 and 20 in the appendix for asummary of state values derived from research literature for the awake andNREM sleep states.

Using the parameter values and steady states from Tables 3 and 4 wecalculate the controls u1 and u2 which transfer the system from xa to xs.Reference data can be found in Burgess et al. [8] and Bevier et al. [5] forthe dynamic time course of various state transitions. The data provided inBurgess et al. [8] suggest a disproportionate drop in H during the initialphase of sleep onset. Model simulations also show that the controlH declines


1 2 3 4 5minutes

3.54

4.55

5.56

liters per minute

VA.

º

1 2 3 4 5minutes

62.565

67.570

72.575

77.580bpm

H


with the largest part of the decline occurring during the initial stage of sleeponset consistent with the results in Burgess et al. [8]. In contrast, there isa continual smooth decline in Vstr,l and Q, more or less unaffected by sleepstage, The model can be used to explore various parameter effects on thesetime courses.

As can be seen by comparison with the simulations presented in [64]little dynamical difference is produced by introduction of delay into themass balance equations of the respiratory system. Indeed, even if the delayis increased by a factor of 15, no significant differences in dynamics appears.

The important delay introducing dynamic instability into the system isthe delay in the feedback control loop of the respiratory control system (seee.g. [2,3]). We will consider this in Section 10. An important physiologicalcondition which increases transport delay in the feedback control loop isfound in congestive heart failure.

9. Congestive heart condition and transport delay

Heart failure is a generic term covering a number of physiopathologies inheart performance which result in a decrease in general blood flow. Thiscondition is often referred to as congestive heart failure to focus on a mainconsequence, namely pulmonary or systemic edema.

Chronic heart failure is to be distinguished from ”heart attack” whichresults in blockage in coronary blood flow or blockage in ventricular or atrialflow. Heart failure can be categorized in a number of ways: forward versusbackward, left versus right, systolic versus diastolic, and low output versushigh output. These classifications are not uniformly consistently appliedbut they are useful in focusing on specific features of heart failure. Giventhe inherent connectedness of the circulation such divisions are to someextent artificial and indeed these distinctions can overlap. For example, inthe forward/backward division:

– forward failure focuses on reduced blood delivery, reduced ejection ofblood from the ventricles and insufficient Q for metabolic needs.

– backward failure focuses on reduced filling of the ventricles, reducedemptying of the venous system, or reduced Q unless high ventricularpressures exist.


On the other hand, in the division systolic/diastolic:

– systolic failure focuses on insufficient systolic action, often impaired con-tractility, and consequently reduced ejection fraction and Q.

– diastolic failure refers to the impairment of ventricular filling withoutnecessarily an impairment of ejection fraction.

These classifications are subdivided into left and right heart categoriesand ”typical” clinical heart failure is due to impairment of left ventricularfunction and reflects systolic disfunction and forward failure. Causes of heartfailure include any condition which reduces heart performance such as:

– myocardial damage which weakens the myocardial muscle,– insufficient coronary blood flow,– reduced myocardial contractility.

In general, heart failure implies the consequence that the heart fails toprovide sufficient blood flow to meet the metabolic needs of the body. Inmost cases, this means that the heart exhibits a deterioration of the heart’spumping ability. Pumping impairment that is due to a reduction in con-tractility is a consequence of the heart muscle being damaged or weakenedin some way.

In chronic heart failure, there is a progressive deterioration in heartfunction over time which is the reason the condition is so serious. The con-dition becomes progressively more sever due to the compensatory mecha-nisms which try to maintain normal cardiovascular function. In a left heartfailure scenario, for example, if heart muscle is damaged so that contractil-ity is reduced, stroke volume and cardiac output will be decreased. Arterialblood pressure falls due to the impaired pumping efficiency of the heart. Thebaroreceptors, sensing reduced pressure, trigger compensatory sympatheticsystem activity and vasoconstriction. These responses can produce signifi-cant elevation of afterload, which can further reduce stroke volume. Overtime, the added stress to the heart results in damaging cardiac muscle com-pensatory changes (remodeling) which further weakens heart function. Thusthe deterioration in heart function is self-reinforcing. This form of heart fail-ure is referred to as chronic in contrast to acute heart failure which is theresult of heart damage occurring over a short time frame.

The kidneys may also respond to reduced cardiac function by induc-ing fluid retention to increase blood volume. This compensatory responseis triggered by the perceived reduction in circulating blood volume andacts to raise blood pressure. This fluid retention will increase preload orfilling pressure but the increased pressure and excess fluids can cause pul-monary or systemic fluid congestion and edema. In left ventricular failure,the reduced left ventricular function results in blood accumulating in thepulmonary venous system (raising pulmonary venous pressure) and can re-sult in significant pulmonary congestion and difficulty in breathing. Hence,the term ”congestive heart failure” is often used, though not every form ofheart failure exhibits this quality.


Given the interconnectedness of the circulatory system, progressive dete-rioration in one ventricle can lead to the other ventricle becoming impairedand the occurrence of simultaneous left and right heart failure. Diastolicand systolic failure can also occur together. In this paper we will refer toleft ventricular failure as that clinical condition of failure of forward blooddelivery due to reduced systolic function and reduced contractility of theheart tissue.

An example of stable pressure states for four types of clinical heartfailure are given in Table 5. For example, in clinical left heart failure (leftventricular failure):

– cardiac output is reduced 20 to 50%;– significant elevation in pulmonary venous pressure occurs;– modest elevation in pulmonary arterial pressure is observed;– modest elevation in systemic venous pressure occurs;– no significant change in systemic arterial pressure is found.

Table 5. Congestive heart failure pressure changes at left/right ventricular(LV/RV) failure and left/right backward (LB/RB) failure.

Condition Pas Pvs Pap Pvp

LV failure no change ↑ moderately ↑ moderately ↑ significantlyRV failure no change ↑ significantly small change small changeLB failure no change no change no change ↑RB failure no change ↑ significantly no change no change

The impairment of the heart’s pumping action will be modeled by areduction in contractility and consequently the ejection fraction. Given theeffects of remodeling of the heart tissue due to the stress on the system, wecould also include a reduction in the ventricular compliance parameter inEq. (19) as a contributing factor to the reduction of stroke volume.

Tables 6 and 7 give the parameters and computed steady states for rest-ing awake and stage 4 sleep for serious chronic left ventricular heart failure.In this model, the left contractility Sl is reduced by 65% from normal. Giventhat there is little change in arterial blood pressure, this implies a similarreduction in ejection fraction consistent with clinical observations found inNiebauer et al. (1999) [44]. We also assume a small drop in right contrac-tility of 8% from normal. Due to the compensatory mechanism describedabove, the systemic resistance Rs parameter, Apesk , is increased by 35%.Heart rate H is increased by 15%. We set PaCO2

to 40.5 mmHg, at the upperend of values reported in Javaheri (1999) [23]. Pulmonary resistance Rp isalso increased by 10% (see, e.g., Moraes et al. (2000) [12]). In this chroniccondition water retention and other mechanisms act to increase total bloodvolume and we assume V0 is increased by 25%. The increase in V0 acts to


raise Pvs. These assumptions are consistent with the observations in Parm-ley [47] as well as Chiariello and Perone-Filardi [10]. As a consequence ofthese changes Q decreases by 20% and hence transport delay is increasedby about 25% (we ignore the effects of the cerebral blood flow). The cardio-vascular steady state values for this case and further simulations presentedlater (see Section 11) can be compared with values presented in Tsuruta etal. (1994) [66]. In that paper, a model is developed and used to identify car-diovascular parameters relating to the four classes of severity of heart failureas defined by the New York Heart Association. The parameter estimationdepended on steady state values of H , Pas, Pap, Pvs, Pvp, V0, and cardiacoutput. The values for these state variables in the four classes depended oninterpolation from certain known values. Among the parameters which wereidentified were the vascular resistances.

Table 6. Optimal control parameters: chronic left ventricular heart failure sleeptransition


Apesk 198.7 188.7βl 25.8 23.19βr 1.67 1.50Rp 2.16 2.16V0 6.25 6.25H 86.02 78.02PaCO2

40.5 44.5τT 30.0 30.0

In contrast, Tables 8 and 9 give the parameters and computed steadystates for resting awake and stage 4 sleep for left heart failure where thereis no increase in blood volume V0 as might be the case when there is acuteheart failure. A small decrease in Sr is assumed. In this case, no change isassumed in Rp or PaCO2

. In this case, no increase in Pvs is seen but rathera drop occurs.

Figures (8) thru (13) give the dynamics of the control for the optimalcontrol case of acute left heart failure.

10. Modeling two delays in the state space

Previously we have modeled the transition to sleep considering ventilationas an optimal control. We are now going to use formula (33) which describesan empiric relation between VA and the blood gas partial pressures PaCO2

,

PBCO2, and PaO2

. Thus VA becomes incorporated into the state equationsand we can consider the transport delay in this control. We consider a singledelay for both the peripheral and central controls. This is reasonable as thetransport delay in the central controller is only about 15% more than the


Table 7. Optimal control steady states: chronic left ventricular heart failure sleeptransition



40.5 44.5PaO2

102.8 98.35PvCO2

50.78 54.46PvO2

29.73 30.36Ql 3.98 3.46Qr 3.98 3.46Rs 23.79 23.02Sl 24.77 20.22Sr 5.04 4.11

VA 5.67 4.34Vstr,l 0.0463 0.0444Vstr,r 0.0463 0.0444

Table 8. Optimal control parameters: acute left heart failure sleep transition


Apesk 198.7 188.7βl 25.8 23.19βr 1.77 1.59Rp 1.965 1.965V0 5.0 5.0H 86.02 78.02PaCO2

40.0 44.5τT 32.0 32.0

peripheral controller delay (Khoo, [31], τp = 6s and τc = 7s). Furthermore,it is the peripheral control which is responsible for instability in the control(Khoo et al. [31], Batzel and Tran, [2,3]). A relationship describing thedependence of VA on PaCO2

, PaO2and PBCO2

is given by

VA(t) = Gpe−0.05PaO2

(t−τp)max(0, PaCO2

(t− τp)− Ip) (33)

+Gc max(0, PBCO2(t)− MRBCO2

KCO2FB− Ic).

The first term above describes the effect on ventilation of the blood gasesPaCO2

and PaO2as measured by peripheral sensors located in the carotid

artery. This will be referred to as the peripheral control. The second termdescribes the effect of the brain CO2 level (PBCO2

) and will be referred toas the central control. This formula taken from Khoo et al. [31] is based on


Table 9. Optimal control steady states: acute left heart failure sleep transition



40.0 44.5PaO2

103.38 98.35PvCO2

51.06 55.17PvO2

28.12 28.83Ql 3.70 3.23Qr 3.70 3.23Rs 22.64 22.00Sl 24.77 20.22Sr 5.35 4.37

VA 5.739 4.34Vstr,l 0.0430 0.0414Vstr,r 0.0430 0.0414

1 2 3 4 5minutes

38

40

42

44

46

mm Hg

PACO2

1 2 3 4 5minutes

85

90

95

100

105

110mm Hg

PAO2

Fig. 8. Optimal control: acute left heart failure case

experimental observations such as presented in the Handbook of Physiology[14]. A transport delay τp between the lungs and peripheral control appearsin this equation. Note that Ip and Ic denote cutoff thresholds, so that therespective ventilation terms become zero when the quantities fall below thethresholds.

Ventilatory dead space effects are accounted for by defining the quantityVA = K · VE where VE is minute ventilation and K is a constant smallerthan one. In this way effective ventilation is reduced by a fixed dead spacepercent which corresponds to modeling change in ventilation as a changein rate of breathing. This is implemented here by a scale reduction in thecontrol gains Gc and Gp. See, e.g., Batzel and Tran [3].

The optimal control now only models the cardiovascular control. Therespiratory control is given by an empirical formula with delay.


1 2 3 4 5minutes

44464850525456mm Hg

PVCO2

1 2 3 4 5minutes

27.5

28

28.5

29

29.5mm Hg

PVO2


1 2 3 4 5minutes

65

70

75

80

85

90mm Hg

PAS

1 2 3 4 5minutes

2.22.42.62.8

33.23.4

mm Hg

PVS


1 2 3 4 5minutes

17

18

19

20

mm Hg

PAP

1 2 3 4 5minutes

11.4

11.6

11.8

12

12.2

12.4

mm Hg

PVP


A model describing the transition to sleep was given by Khoo et al. [35].During sleep, ventilatory drive is diminished by reducing the sleep gain fac-tor Gs and there is an increase in a shift term Kshift altering the operating

point of ventilation. The effective drive during sleep Vsleep is described by

Vsleep(t) = Gs(t)[max(0, Vawake(t)−Kshift(t))]. (34)

The time dependencies for Gs and Kshift reflect the smooth changein these parameters that occurs in the transition from ”awake” state to


1 2 3 4 5minutes

0.038

0.04

0.042

0.044

0.046liters

VSTR

1 2 3 4 5minutes2.4

2.62.8

33.23.43.63.8

4liters per min

QL


1 2 3 4 5minutes

3.54

4.55

5.56

liters per minute

VA.

º

1 2 3 4 5minutes

72.575

77.580

82.585

87.590bpm

H


”stage 4 quiet sleep ”. Gs is set during the awake state at 1 and reducessmoothly to a minimum (normally 0.6) at stage 4 sleep. Kshift begins at 0and increases to a maximum (normally about 4 mmHg) by the beginning ofstage 1 sleep. These changes reflect the reduction in the normal ventilatoryresponse Vawake as a result of physiological changes during sleep. Once stage4 sleep is reached these values are constant. For these simulations we use abase line transit time to ”stage 4 sleep” to be three minutes. The changesin Gs and Kshift will be modeled by incorporating exponential functionswhich change smoothly through the various sleep stages between awake andstage 4 NREM sleep. The parameters in these expressions can be adjustedto simulate an essentially linear decrease over the entire transition fromawake to stage 4 sleep or bias the decrease to the early stages of sleep.

Similar decreases for the metabolic rates, sleep contractility, and sys-temic resistance reflecting the physiological changes during sleep transition(discussed above) are incorporated. The system is nonautonomous, howeverwe still implement the control functions u1 and u2 as calculated for a time-independent linear system around the final steady state ”stage 4 sleep”.This reduces the optimality of the control for the original nonlinear systembut the thereby obtained (suboptimal) control still stabilizes the system andis useful for dynamic studies.


Thus we now determine a control function u1 such that the cost func-tional

∫ ∞

0

(qas(Pas(t)− P sas)2 + q1u1(t)2

)dt (35)

is minimized under the restriction

x(t) = f(x(t), x(t − τp), x(t− τT );W s) +B u(t), x0 = φ.

y(t) = Dx(t)(36)

Clearly τp < τT and, again, φ ∈ C where C denotes C([−τT , 0],R14).

In an analogous fashion with the one delay case we form the linearizedsystem now expanding the system using two delays.

x(t) = ξ(t) = f(xs + ξ(t), xs + ξ(t− τp), xs + ξ(t− τT );W s) +B u

= A1 ξ(t) +A2 ξ(t− τp) +A3 ξ(t− τT ) +B u+ o(ξ).(37)

The matrices Ai ∈M14,14(R), i = 1, 2, 3 are the Jacobians of f with respectto x(t), x(t − τp), and x(t− τT ) , respectively, evaluated at x = xs,

A1 =∂f

∂x(t)(xs;W s),

A2 =∂f

∂x(t− τp)(xs;W s),

A3 =∂f

∂x(t− τT )(xs;W s).

(38)

Analogously,

η(t) = (Pas(t)− P sas)T = Dξ(t). (39)

By neglecting terms of order o(ξ) we get linear approximations ξ`(t) andη`(t) for ξ(t) and η(t), respectively,

ξ`(t) = A1ξ`(t) +A2 ξ`(t− τp) +A3 ξ`(t− τT ) + B u(t),

η`(t) = Dξ`(t),

ξ`(0) = xr − xs.(40)

Again, we apply the results on approximation of feedback control for delaysystems using Legendre polynomials found in Kappel and Propst [26].


Table 10. VA empirical control parameters: normal adult sleep transition


Gc 1.44 1.44Gp 30.24 30.24Gs 1.0 0.6Kshift 0 4.2IC 35.5 35.5IP 35.5 35.5H 75.02 68.02MRCO2 0.266 0.224MRBCO2

0.042 0.040MRO2 0.310 0.260Apesk 147.16 139.80βl 85.89 77.30βr 2.083 1.874τp 7.8 7.8τT 24.0 24.0S 4 transit - 3 min

Table 11. VA empirical control steady states: normal adult sleep transition



39.16 42.67PaO2

104.37 100.47PvCO2

47.44 50.62PvO2

34.50 35.30PBCO2

47.23 50.34Ql 4.938 4.334Qr 4.938 4.334Rs 19.88 19.21Sl 72.02 58.77Sr 5.49 4.48VA 5.86 4.53Vstr,l 0.0658 0.0637Vstr,r 0.0658 0.0637

11. Simulations with two delays

Tables 10 and 11 give the parameters and computed steady states for restingawake and stage 4 sleep for a normal adult with borderline elevated Rsvalues and with moderate sleep transition profile. From this point on, weare using the empirical control for VA while maintaining the optimal controlfor the cardiovascular system. We will refer to this case as the VA empiricalcase. In all figures, we exhibit simulations for the first few minutes to focus


on the early transition dynamics in detail. Simulations of longer durationclearly show the stabilizing influence of the control.

Figures (14) thru (20) give the dynamics of the control for the normal VAempirical control case. For this case the transition to stage 4 sleep is assumedto be 3 minutes. Furthermore, the shift Kshift is assumed to occur by stage1 (one fourth of the transition time) as in [35]. Gs reduces smoothly fromstage 1 to stage 4 with most of the change occurring in the first two stages.As in the optimal case the reductions in βl, βr, metabolic rates, and Apeskare also assumed to be significantly reduced by stage 1. This assumption ismade for purposes of exploring the dynamics of transition and because notmuch is known about the actual time course of these parameter changes insleep transition.

1 2 3 4 5minutes

39

40

41

42

43

44mm Hg

PACO2

1 2 3 4 5minutes

7580859095

100105110mm Hg

PAO2

Fig. 14. VA empirical control: normal case

1 2 3 4 5minutes

46474849505152mm Hg

PVCO2

1 2 3 4 5minutes

34.234.434.634.8

3535.235.4

mm Hg

PVO2


Figures (21) thru (24) give the dynamics of the control for the normalVA empirical control case with a reduced sleep transition time. For thiscase the transition to stage 4 sleep is assumed to be 2 minutes. The shiftKshift is again assumed to occur by stage one (one fourth of the transitiontime) and the reductions in Gs, βl, βr and the metabolic rates are assumedto be reduced as in the previous case. The shift Kshift is increased to 5.2and the gain Gs at stage 4 is 0.4. The quicker transition time and largershift create a deeper drop in the ventilation rate with sleep onset than in


1 2 3 4 5minutes

85

90

95

100

105mm Hg

PAS

1 2 3 4 5minutes

3.2

3.4

3.6

3.8

4mm Hg

PVS


1 2 3 4 5minutes

15.5

16

16.5

17

17.5

18mm Hg

PAP

1 2 3 4 5minutes

7.4257.45

7.4757.5

7.5257.55

7.575

mm Hg

PVP


1 2 3 4 5minutes

4.2

4.4

4.6

4.8

5liters per min

FS

1 2 3 4 5minutes

18

18.5

19

19.5

20

mm Hg min per lit

RS


the previous case. This behavior will be compared now with the congestiveheart condition case which includes an increased delay time.

Tables 12 and 13 give the parameters and computed steady states forresting awake and stage 4 sleep for the left ventricular heart failure case.In this model, the left contractility Sl is reduced by 65% from normal. Forejection fraction values in heart failure see Niebauer et al. (1999) [44].

We also assume a small drop in right contractility of 8% from normal.Systemic resistance Rs is increased by 35% and pulmonary resistance Rp by10% ([12]). Heart rate H is increased by 15%. The PvCO2

increase and PvO2

decrease are a consequence of the reduced cardiac output. In this chroniccondition water retention and other mechanisms act to increase total bloodvolume and we assume V0 is increased by 23%. The increase in V0 acts toraise Pvs. As a consequence of these changes Q decreases by 20% and hencetransport delay is increased by about 25%.


1 2 3 4 5minutes

0.063

0.064

0.065

0.066

0.067

0.068liters

VSTR

1 2 3 4 5minutes

3.5

4

4.5

5

5.5liters per min

QL


1 2 3 4 5minutes

3.54

4.55

5.56

liters per minute

VA.

º

1 2 3 4 5minutes

68

70

72

74

76bpm

H


1 2 3 4 5minutes

39

40

41

42

43

44mm Hg

PACO2

1 2 3 4 5minutes

7580859095

100105110mm Hg

PAO2

Fig. 21. VA empirical control: fast sleep case

Tables 14 and 15 give the parameters and computed steady states forresting awake and stage 4 sleep where the congestive heart condition involvessignificant reduction in contractility of both the left and right ventricle.Often it is the case that deterioration on one side of the heart (here the leftside) will eventually extend to deterioration of function on the other side[58]. We maintain the CO2 ventilation thresholds simulating an operatingpoint of PaCO2

in the middle range of values given in Javaheri (1999) [23].In general, PaCO2

levels in congestive heart patients are little changed fromlevels found in normal individuals even when there is reduced exchangeefficiency in the lungs due to congestion. See, e.g., Sullivan et al. (1988)[61].

As a consequence of the reduced cardiac output the transport delay isnow increased by 50%. For comparative state values in the case of severecongestive heart failure, see Bruschi et al. (1999) [7] and Bocchi et al. (2000)


1 2 3 4 5minutes

46474849505152mm Hg

PVCO2

1 2 3 4 5minutes

34.234.434.634.8

3535.235.4

mm Hg

PVO2


1 2 3 4 5minutes

15.5

16

16.5

17

17.5

18mm Hg

PAP

1 2 3 4 5minutes

7.4257.45

7.4757.5

7.5257.55

7.575

mm Hg

PVP


1 2 3 4 5minutes

2

3

4

5

6liters per minute

VA.

º

1 2 3 4 5minutes

68

70

72

74

76bpm

H


[6] for values of Q, Hanly et al. (1993) [21] for values of transport delay, andBocchi et al. (2000) [6] for comparative values of H , Rs, and Vstr. See alsoTable 22 in the appendix and Tsuruta et al. (1994) [66] and Hambrecht etal. (2000) [20] for comparative Rs values. Arterio-venous oxygen contentdifference for the severe CHF case is consistent with Kugler et al. (1982)[39]. The very low contractility implies (given the small change in pressure)an ejection fraction consistent with clinical observations found in Niebaueret al. (1999) [44] for very severe heart failure cases.

It is well known that delays in feedback control can create instabilityin a control system. In congestive heart failure, the reduced cardiac outputinduces an increased transport delay which will reduce the efficiency of thecentral and peripheral controllers of ventilation. This reduced efficiency isdue to the increased time it takes for blood gases to be transported from thesite where these blood gas levels are adjusted (the lungs) to the sensory sites


Table 12. VA empirical control parameters: left ventricular heart failure sleeptransition


Apesk 198.66 188.73βl 25.77 23.19βr 1.67 1.50Rp 2.16 2.16V0 6.25 6.25H 86.02 78.02IC 35.5 35.5IP 35.5 35.5Gs 1.0 0.6Kshift 0 4.2τp 9.75 9.75τT 30.0 30.0

Table 13. VA empirical control steady states: left ventricular heart failure sleeptransition



39.16 42.67PaO2

104.37 100.47PvCO2

49.44 52.63PvO2

29.77 30.43PBCO2

47.23 50.34Ql 3.98 3.46Qr 3.98 3.46Rs 23.82 23.06Sl 24.77 20.22Sr 5.035 4.11

VA 5.86 4.53Vstr,l 0.046 0.044Vstr,r 0.046 0.044

where these levels are measured. One form of respiratory instability associ-ated with CHF is a form of periodic breathing (PB) known as Cheyne-Stokesrespiration (CSR). This form of involuntary respiration involves periods ofregular waxing and waning of tidal volume interspersed with central apnea(CA). Cheyne-Stokes respiration seems to be a complicating factor for CHFbut the actual mechanisms inducing CSR in congestive heart patients arestill under active investigation. The increased feedback delay due to reducedcardiac output, in conjunction with other factors may be sufficient to con-tribute to the onset, characteristics, or persistence of central sleep apnea,


Table 14. VA empirical control parameters: left and right ventricular heart failuresleep transition


Apesk 250.17 237.7βl 12.88 11.60βr 1.46 1.31Rp 2.16 2.16V0 6.9 6.9H 92.02 80.02IC 35.5 35.5IP 35.5 35.5Gs 1.55 0.465Kshift 0 5.5τp 11.6 11.6τT 36.0 36.0S 4 transit - 2 min

Table 15. VA empirical control steady states: left and right ventricular heartfailure sleep transition



37.95 44.44PaO2

105.77 98.41PvCO2

50.25 56.81PvO2

25.74 25.49PBCO2

46.02 52.12Ql 3.33 2.79Qr 3.33 2.79Rs 26.22 24.67Sl 13.25 10.37Sr 4.71 3.69VA 6.05 4.35Vstr,l 0.0362 0.0348Vstr,r 0.0362 0.0348

PB, or CSR associated with CHF. See, e.g., Hall et al. (1996) [19], Pinnaet al. (2000) [51] and Cherniack (1999) [9]. For analytical results see Batzeland Tran (2000) [3].

Figures (25) thru (30) give the congestive heart failure dynamics for theVA empirical control case simulating transition to sleep with fast transitionparameters and in this case we also assume an awake feedback gain which is50% higher than normal which in effect increases CO2 sensitivity. Increasesin CO2 sensitivity have been reported in cases of central sleep apnea in


heart failure. See, e.g., Topor et al. (2001) [65] and Javaheri (1999) [23].We simulate a quick transition to sleep with sleep gain Gs reduced by 70%in stage 4 sleep as compared to the normal reduction of 40%. We furtherincrease the shift term Kshift by 33% (to 5.5 mmHg). Fast sleep onset timescan occur in patients with fragmented sleep cycles as occurs when multiplesleep apneas induce arousal and sleep disturbance. See, e.g., Bennet et al.(1998) [4]. The change in sleep control parameters in this case simulates anincreased influence of the sleep state on control effectiveness.

These deviations in standard operating points drive ventilation to nearapnea and exhibit one mechanism for inducing Cheyne-Stokes respiration(CSR) and central sleep apnea. The oscillatory behavior is induced by theincreased delay as can be observed by comparing this case with the fasttransition case for a normal adult represented in Figures (21) thru (24).Javaheri (1999) [23] and Quaranta et al. (1997) [53] indicate that a num-ber of factors influencing control stability (such as higher CO2 sensitivityand circulation delay) may contribute to central sleep apnea and CSR incongestive heart patients. Lorenzi-Filho et al. (1999) [40] report that re-ductions in PaCO2

sensed at the peripheral chemoreceptors can also triggercentral apneas during Cheyne-Stokes respiration. It is clear that the inter-action of various respiratory factors can act in complex ways to influencethe production of CSR and apnea in congestive heart failure.

The larger reduction in sleep control gain Gs in this simulation actuallyacts to reduce the magnitude of the oscillatory cycles. On the other hand,simulations indicate and, in general, theory confirms that the higher con-trol gain (CO2 sensitivity) prolongs and exaggerates oscillatory behavior.Likewise, a longer time course in the reduction in control gain from stage1 to stage 4 sleep (thus maintaining higher gain for a longer time) wouldcontribute to unstable behavior. It is the degree and speed of the shiftKshift that is responsible for the initial steep drop in ventilation which cantrigger apnea and repetitive cycles similar to CSR and the increased delayreinforces and perpetuates the oscillatory behavior.

1 2 3 4 5minutes

37.540

42.545

47.550mm Hg

PACO2

1 2 3 4 5minutes

70

80

90

100

110mm Hg

PAO2

Fig. 25. VA empirical control: severe left and right ventricular failure sleep case


1 2 3 4 5minutes

45

50

55

60

65

mm Hg

PVCO2

1 2 3 4 5minutes

24.5

25

25.5

26

26.5

27mm Hg

PVO2


1 2 3 4 5minutes

65707580859095mm Hg

PAS

1 2 3 4 5minutes

3.23.43.63.8

44.24.4

mm Hg

PVS


1 2 3 4 5minutes

24

25

26

27

28mm Hg

PAP

1 2 3 4 5minutes

18.518.75

1919.2519.5

19.7520mm Hg

PVP


12. Conclusion

In this paper we have considered a model of the cardiovascular-respiratorycontrol system with constant state equation delays. The model utilizes anoptimal control approach to represent the complex control features of thecardiovascular component in this system. The respiratory control is consid-ered both from an optimal control approach and from an empirical approachwhich introduces a respiratory feedback delay into the state equations. Themodel was applied to study the transition from the awake state to NREMsleep for normal individuals and for individuals suffering from congestiveheart problems. The model steady states are consistent with observationboth for the normal and congestive heart states. The dynamical simulationsshow that the transport delay between respiratory compartments does notcontribute to instability even at large delays. However, the transport delay


1 2 3 4 5minutes

0.034

0.035

0.036

0.037

0.038liters

VSTR

1 2 3 4 5minutes

2.6

2.8

3

3.2

3.4

3.6liters per min

QL


-1 1 2 3 4 5minutes

2

4

6

8liters per minute

VA.

º

1 2 3 4 5minutes

75

80

85

90

95

bpm

H


to the peripheral sensor is significant and can result in Cheyne-Stokes typerespiration for a severe congestive heart condition with certain respiratoryparameters during the transition to NREM sleep.

References

1. J. J. Batzel and H. T. Tran, Modeling Instability in the Control System forHuman Respiration: applications to infant non-REM sleep, Appl. Math. Com-put. 110, pp. 1-51, 2000.

2. J. J. Batzel and H. T. Tran, Stability of the human respiratory control system.Part1: Analysis of a two dimensional delay state-space model, J. Math. Biol.41(1), pp. 45-79, 2000.

3. J. J. Batzel and H. T. Tran, Stability of the human respiratory control system.Part2: Analysis of a three dimensional delay state-space model, J. Math. Biol.41(1), pp. 80-102, 2000.

4. L. S. Bennett, B. A. Langford, J. R. Stradling, and R. J. Davies, Sleep frag-mentation indices as predictors of daytime sleepiness and nCPAP response inobstructive sleep apnea, Am. J. Resp. Crit. Care Med. 158(3), pp. 778-786,1998.

5. W. C. Bevier, D. E. Bunnell, and S. M. Horvath, Cardiovascular functionduring sleep of active older adults and the effects of exercise, Exp. Gerontol.22(5), pp. 329-37, 1987.

6. E. A. Bocchi, A.V. Vilella de Moraes, A. Esteves-Filho, F. Bacal, J. O. Auler,M. J. Carmona, G. Bellotti, and A. F. Ramires, L-arginine reduces heart rateand improves hemodynamics in severe congestive heart failure, Clin. Cardiol.23(3), pp. 205-210, 2000.

7. C. Bruschi, F. Fanfulla, E. Traversi, V. Patruno, G. Callegari, L. Tavazzi, andC. Rampulla, Identification of chronic heart failure patients at risk of Cheyne-Stokes respiration, Monaldi Arch. Chest Dis. 54(4), pp. 319-324, 1999.


8. H. J. Burgess, J. Kleiman, and J. Trinder, Cardiac activity during sleep onset, Psychophysiology Vol. 36(3), pp. 298-306, 1999.

9. N.S. Cherniack, Apnea and periodic breathing during sleep, New Engl. J. Med.341(13), pp. 985-987, 1999.

10. M. Chiariello and P. Perrone-Filardi, Pathophysiology of heart failure,Miner Electrolyte Metab. 25(1-2), pp. 6-10, 1999.

11. R. G. Crystal, J.B. West, et al., eds., The Lung: Scientific foundations,Lippincott-Raven Press, Philadelphia, 1997.

12. D. L. Moraes, W. S. Colucci, and M. M. Givertz, Secondary pulmonary hyper-tension in chronic heart failure: the role of the endothelium in pathophysiologyand management., Circulation 102(14), pp. 1718-1723, 2000.

13. W. F. Fincham and F. T. Tehrani, A mathematical model of the human res-piratory system, J. Biomed. Eng. 5(2), pp. 125-133, 1983.

14. A. P. Fishman, N. S. Cherniack, J. G. Widdicombe, and S. R. Geiger,eds. Handbook of Physiology: The Respiratory System, Volume II, Controlof Breathing, Part 2 Am. Phys. Soc., Bethesda, Maryland, 1986.

15. F. S. Grodins, J. Buell, and A. J. Bart, Mathematical analysis and digitalsimulation of the respiratory control system, J. Appl. Physiol. 22(2), pp. 260-276, 1967.

16. F. S. Grodins, Integrative cardiovascular physiology: a mathematical modelsynthesis of cardiac and blood vessel hemodynamics, Quart. Rev. Biol. 34(2),pp. 93-116, 1959.

17. F. S. Grodins, Control Theory and Biological Systems, Columbia UniversityPress, New York, 1963.

18. A. C. Guyton, Textbook of Medical Physiology, 7th ed., W. B. Saunders Com-pany, Philadelphia, 1986.

19. M. J. Hall, A. Xie, R. Rutherford, S. Ando, J..S. Floras, and T.D. Bradley,Cycle length of periodic breathing in patients with and without heart failure,Am. J. Respir. Crit. Care. Med. 154 (2 pt. 1), pp. 376-381, 1996.

20. R. Hambrecht, S. Gielen, A. Linke, E. Fiehn, J. Yu, C. Walther, N. Schoene,and G. Schuler, Effects of exercise training on left ventricular function andperipheral resistance in patients with chronic heart failure: A randomized trial,Jama 283(23), pp. 3095-3101, 2000.

21. P. Hanly, N. Zuberi, and R. Gray, Pathogenesis of Cheyne-Stokes respira-tion in patients with congestive heart failure: relationship to arterial PCO2.,Chest 104(4), pp. 1079-1084, 1993.

22. L. L. Huntsman, A. Noordergraaf, and E. O. Attinger, Metabolic autoregula-tion of blood flow in skeletal muscle: a model, in J. Baan, A. Noordergraaf, andJ. Raines (eds.), Cardiovascular System Dynamics, pp. 400-414, MIT Press,Cambridge, 1978.

23. S. Javaheri, A mechanism of central sleep apnea in patients with heart failure,New Engl. J. Med. 341(13), pp. 949-54, 1999.

24. F. Kappel and R. O. Peer, A mathematical model for fundamental regulationprocesses in the cardiovascular system, J. Math. Biol. 31(6), pp. 611-631, 1993.

25. F. Kappel and R. O. Peer, Implementation of a cardiovascular model and al-gorithms for parameter identification, SFB Optimierung und Kontrolle, Karl-Franzens-Universitat Graz, technical report Nr. 26, 1995.

26. F. Kappel and G. Propst, Approximation of feedback control for delay systemsusing Legendre polynomials, Conf. Sem. Mat. Univ. Bari. Nr. 201, 1984.

27. F. Kappel and D. Salamon, Spline approximation for retarded systems andthe Riccati equation, Siam J.Control Optim. 25(4), pp. 1082-1117, 1987.

28. F. Kappel, S. Lafer, and R. O. Peer, A model for the cardiovascular systemunder an ergometric workload, Surv. Math. Ind. 7, pp. 239-250, 1997.

29. A. M. Katz, Physiology of the heart: 2nd edition, Raven Press, N.Y., 1992.


30. T. Kenner, Physical and mathematical modeling in cardiovascular systems,in: Clinical and Research Applications of Engineering Principles edited byN. H. C. Hwang, E. R. Gross, and D. J. Patel, University Park Press, Balti-more, pp. 41-109, 1979.

31. M. C. K. Khoo, R. E. Kronauer, K. P. Strohl, and A. S. Slutsky, Factors in-ducing periodic breathing in humans: a general model, J. Appl. Physiol. 53(3),pp. 644-659, 1982.

32. M. C. K. Khoo, A model of respiratory variability during non-REM sleep,in: Respiratory Control: A Modeling Perspective, edited by G. D. Swanson,F. S. Grodins, and R. L. Hughson, Plenum Press, New York, 1989.

33. M. C. K. Khoo, Modeling the effect of sleep-state on respiratory stability,in: Modeling and Parameter Estimation in Respiratory Control, pp. 193-204,edited by M. C. K. Khoo, Plenum Press, New York, 1989.

34. M. C. K. Khoo and S. M. Yamashiro, Models of control of breathing, in:Respiratory Physiology: an analytical approach edited by H. A. Chang andM. Paiva, pp. 799-829, Marcel Dekker, New York, 1989.

35. M. C. K. Khoo, A. Gottschalk, and A. I. Pack, Sleep-induced periodic breathingand apnea: a theoretical study, J. Appl. Physiol. 70(5), pp. 2014-2024, 1991.

36. R. Klinke and S. Silbernagl (eds.), Lehrbuch der Physiologie, Georg ThiemeVerlag, Stuttgart, 1994.

37. K. W. Koo, D.S. Sax, and G. L. Snider, Arterial blood gases and pH duringsleep in chronic obstructive pulmonary disease, Am. J. Med. 58(5), pp. 663-670, 1975.

38. J. Krieger, N. Maglasiu, E. Sforza, and D. Kurtz Breathing during sleep innormal middle age subjects, Sleep 13(2), pp. 143-154, 1990.

39. J. Kugler, C. Maskin, W. H. Frishman, E. H. Sonnenblick, and T. H. LeJemtel,Regional and systemic metabolic effects of angiotensin-converting enzyme in-hibition during exercise in patients with severe heart failure, Circulation 66(6),pp. 1256-61, 1982.

40. G. Lorenzi-Filho, F. Rankin, I. Bies, and T. Douglas Bradley, Effects of inhaledcarbon dioxide and oxygen on Cheyne-Stokes respiration in patients with heartfailure, Am. J. Respir. Crit. Care Med. 159 (5 pt.1), pp. 1490-98, 1999.

41. G. Mancia, Autonomic modulation of the cardiovascular system during sleep,New Engl. J. Med. 328(5), pp. 347-49, 1993.

42. L. Martin, Pulmonary physiology in clinical practice : the essentials for patientcare and evaluation, Mosby Press, St. Louis, 1987.

43. J. H. Mateika, S. Mateika, A. S. Slutsky, and V. Hoffstein, The effectof snoring on mean arterial blood pressure during non-REM sleep, Am.Rev. Respir. Dis. 145(1), pp. 141-146, 1992.

44. J. Niebauer, A. L. Clark, S. D. Anker, and A. J. S. Coats, Three year mor-tality in heart failure patients with very low left ventricular ejection fractions,Int. J. Cardiol. 70(3), pp. 245-247, 1999.

45. A. Noordergraaf and J. Melbin, Introducing the pump equation, in: Cardio-vascular System Dynamics: Models and Measurements, edited by T. Kenner,R. Busse, and H. Hinghofer-Szalkay, pp. 19-35, Plenum Press, New York, 1982.

46. J.T. Ottesen, Modeling of the baroreflex-feedback mechanism with time-delay,J. Math. Biol., 36(1), pp. 41-63, 1997.

47. W.W. Parmley, Pathophysiology of congestive heart failure, Am-J-Cardiol.56(2), pp. 7A-11A, 1985.

48. R. O. Peer, Mathematische Modellierung von grundlegendenRegelungsvorgangen im Herzkreislauf-System, Technical Report, Karl-Franzens-Universitat und Technische Universitat Graz, 1989.

49. C. S. Peskin, Lectures on mathematical aspects of physiology, in: Mathemati-cal Aspects of Physiology edited by F. C. Hoppensteadt, Lect. Appl. Math. 19,pp. 1-107, Am. Math. Soc, Providence, Rhode Island, 1981.


50. E. A. Phillipson and G. Bowes, Control of breathing during sleep, in: Hand-book of Physiology, Section 3: The Respiratory System, Volume II, Control ofBreathing, Part 2, edited by A. P. Fishman, N. S. Cherniack, J. G. Widdi-combe, and S. R. Geiger, pp. 649-690, Am. Phys. Soc., Bethesda, Maryland,1986.

51. G. D. Pinna, R. Maestri, A. Mortara, M. T. La Rovere, F. Fanfulla, andP. Sleight, Periodic breathing in heart failure patients: testing the hypothesisof instability of the chemoreflex loop, J. Appl. Physiol. 89(6) , pp. 2147-2157,2000.

52. T. Podszus, Kreislauf und Schlaf, in: Kompendium Schlafmedizin fur Aus-bildung, Klinik und Praxis, Deutsche Gesllschaft fur Schlafforschung undSchlafmedizin, edited by H. Schulz, Landsberg, Lech : Ecomed-Verl.-Ges.,1997.

53. A. J. Quaranta, G. E. D’Alonso, and S. L. Krachman, Cheyne-Stokes res-piration during sleep in congestive heart failure , Chest 111(2), pp. 467-473,1997.

54. D. W. Richardson, A. J. Wasserman, and J. L. Patterson Jr., General and re-gional circulatory responses to change in blood pH and carbon dioxide tension,J. Clin. Invest. 40, pp. 31-43, 1961.

55. L. B. Rowell, Human Cardiovascular Control, Oxford University Press, 1993.56. D. L. Russell, Mathematics of finite-dimensional control systems, Marcel

Dekker, New York, 1979.57. H. Schneider, C.D. Schaub, K.A. Andreoni, A.R. Schwartz, P.L. Smith,

J.L. Robotham, and C.P. O’Donnell, Systemic and pulmonary hemodynamicresponses to normal and obstructed breathing during sleep., J. Appl. Phys-iol. 83(5), pp. 1671-1680, 1997.

58. F. J. Schoen, The heart-a pathophysiological perspective, Ch. 13, in RobbinsPathologic Basis of Disease, Sixth Edition, edited by R. S. Cotran et al.,W. B. Saunders, Philadelphia, 1999.

59. J.W. Shepard Jr., Gas exchange and hemodynamics during sleep,Med. Clin. North Am. 69(6), pp. 1243-64, 1985.

60. V. K. Somers, M.E. Dyken, A. L. Mark, and F. M. Abboud, Sympathetic-nerve activity during Sleep in normal subjects, New Engl. J. Med. 328(5),pp. 303-307, 1993.

61. M. J. Sullivan, M.B. Higginbotham, and F. R. Cobb, Increased exercise venti-lation in patients with chronic heart failure: intact ventilatory control despitehemodynamic and pulmonary abnormalities, Circulation 77(3), pp. 552-559,1988.

62. G. W. Swan, Applications of optimal control theory in biomedicine, MarcelDekker, Inc., New York 1984.

63. S. Timischl, A Global Model of the Cardiovascular and Respiratory System,PhD thesis, Karl-Franzens-Universitat Graz, 1998.

64. S. Timischl, J. J. Batzel, and F. Kappel, Modeling the human cardiovascular-respiratory control system: an optimal control application to the transition tonon-REM sleep Spezialforschungsbereich F-003 Technical Report 190, Karl-Franzens-Universitat, 2000.

65. Z. L. Topor, L. Johannson, J. Kasprzyk, and J. E. Remmers, Dynamic ven-tilatory response to CO2 in congestive heart failure patients with and withoutcentral sleep apnea, J. Appl. Physiol. 91(1), pp. 408-416, 2001.

66. H. Tsuruta, T. Sato, M. Shirataka and N. Ikeda, Mathematical model of thecardiovascular mechanics for diagnostic analysis and treatment of heart fail-ure: part 1 model description and theoretical analysis, Med. Biol. Eng. Comp.32(1), pp. 3-11, 1994.

67. K. Wasserman, B. J. Whipp, and R. Casaburi, Respiratory control during ex-ercise, in: Handbook of Physiology, Section 3: The Respiratory System, Volume


II, Control of Breathing, Part 2, edited by A. P. Fishman, N. S. Cherniack,J. G. Widdicombe, and S. R. Geiger, Am. Phys. Soc., Bethesda, Maryland,1986.


APPENDIX

We assume the following:

PAO2= PaO2

,PACO2

= PaCO2,

PBO2= PBvO2

,PBCO2

= PBvCO2,

PTO2= PTvO2

,PTCO2

= PTvCO2,

where v = mixed venous blood, T is tissue compartment.

Further we assume:

– The alveoli and pulmonary capillaries are single well-mixed spaces;– constant temperature, pressure and humidity are maintained in the gas

compartment;– gas exchange is by diffusion; ventilatory dead space is incorporated via

the optimal control VA for the optimal case and control gains Gc andGp for the empirical case (see text);

– the delay in the respiratory controller signal to effector muscles is zero;– delay in the baroreceptor signal to the controller and from controller to

effector muscles is zero;– metabolic rates and other parameters are constant in a given state;– pH effects on dissociation laws and other factors are ignored or incorpo-

rated into parameters;– acid/base buffering, material transfer across the blood brain barrier, and

tissue buffering effects are ignored;– no inter-cardiac shunting occurs;– intrathoracic pressure is ignored for this average flow model;– unidirectional non-pulsatile blood flow through the heart is assumed;

hence, blood flow and blood pressure have to be interpreted as meanvalues over the length of a pulse;

– fixed blood volume V0 is assumed.

The parameters for α, β, γ, as well as the compliances cas, cap, cvs,cvp, cl, and cr are chosen as in the paper by Kappel and Peer [24]. For theS-shaped O2 dissociation curve which relates blood gas concentrations topartial pressures we will use the relation

CO2(t) = K1(1− e−K2PO2 (t))2. (41)

This relation was also used by Fincham and Tehrani [13]. Khoo et al. [31]assumes a piecewise linear relationship.


For CO2, considering the narrow working range of PCO2 we assume alinear dependence of CCO2 on PCO2 ,

CCO2(t) = KCO2PCO2(t) + kCO2 . (42)

A linear relationship was also used by Khoo et al. [31]. Other parameterand steady state values from the literature are given in the following tables.Parameters normally used for simulations are marked with an asterisk.

Table 16. Parameter values (awake rest)

Parameter Value Unit Source

Gc 1.440 * l/(min ·mmHg) [31]3.2 l/(min ·mmHg) [32]

Gp 30.240 * l/(min ·mmHg) [31]26.5 l/(min ·mmHg) [32]

Ic 35.5 * mmHg [31], [32]45.0 mmHg [35]

Ip 35.5 * mmHg [31], [32]38.0 mmHg [35]

K1 0.2 lSTPD/l [13]K2 0.046 mmHg−1 [13]kCO2 0.244 lSTPD/l [31]KCO2 0.0065 lSTPD/(l ·mmHg) [31]

0.0057 lSTPD/(l ·mmHg) [35]mB 1400 g [36], p. 745

MRBCO20.042 * lSTPD/min [32]0.031 lSTPD/(min ·kg brain tissue) [35]0.050 lSTPD/min [15]0.054 lSTPD/min [13]

MRCO2 0.21 lSTPD/min [35]0.235 lSTPD/min [32]0.200 lSTPD/min [33]0.26 * lSTPD/min [36] p. 239

MRO2 0.26 lSTPD/min [35]0.290 lSTPD/min [32]0.240 lSTPD/min [33]0.31 * lSTPD/min [36] p. 239

CO2 sensitivity 2.1 +/- 1.0 * lBTPS/(min ·mmHg) [23]


Table 17. Parameter values (awake rest)

Parameter Value Unit Source

PICO20 mmHg [31], [15]

PIO2150 mmHg [31]

Patm 760 mmHg [31],[15]Rp 0.965 mmHg ·min /l [36] p. 233

1.4 mmHg ·min /l [36] p. 1441.95 * mmHg ·min /l [63]1.5-3 mmHg ·min /l [42]

RQ 0.88 - [15]0.81 - [31], [32]

0.84 * - [36] p. 239VAO2

2.5 * lBTPS [31]3.0 lBTPS [15]0.5 lBTPS [18], p. 1011

VACO23.2 * lBTPS [31]3.0 lBTPS [35], [15]

VTCO215 l [31], [35], [32]

VTO26 * l [31], [35], [32]1.55 l [18], p. 1011

VBCO20.9 * l [32]1.0 l [15]1.1 l [13]

VBO21.0 l [15]1.1 l [13]

VD 0.15 lBTPS [35],[36] p. 239

VD 2.4 lBTPS/min [36] p. 2392.28 lBTPS/min [31]

FB 0.5 l/(min ·kg brain tissue) [35], [36], p. 7450.75-0.8 * l/min [15], [13]

12-15% of Q l/min [55],p. 242

Table 18. Nominal steady state values (awake rest)

Quantity Value Unit Source

CaCO20.493 lSTPD/l [36], p. 253

CaO20.197 lSTPD/l [36], p. 253

CvCO20.535 lSTPD/l [36], p. 253

CvO20.147 lSTPD/l [36], p. 253

H 70 min−1 [36] p. 144Pap 12 mmHg [36] p. 144

15 mmHg [66] p. 410-22 mmHg [42] Chptr. 8

Pas 100 mmHg [36] p. 14493 mmHg [66] p. 4

Pvp 5 mmHg [36] p. 1448 mmHg [66] p. 4

Pvs 2-4 mmHg [36] p. 1445 mmHg [66] p. 4


Table 19. Nominal steady state values (awake rest)


PACO240 mmHg [18],p. 495, [36] p. 239

PAO2104 mmHg [18],p. 494100 mmHg [36] p. 239

PaCO240 mmHg [18],p. 495, [36], p. 253

PaO295 mmHg [18],p. 49490 mmHg [36], p. 253

PvCO245 mmHg [18],p. 49546 mmHg [36], p. 253

40-50 mmHg [42] Chptr. 8PvO2

40 mmHg [18],p. 494, [36], p. 25335-40 mmHg [42] Chptr. 8

Ql = Qr = Fp = Fs 6 l/min [31], [15]6.2 l/min [36] p. 2395 l/min [36] p. 144

4-7 l/min [42] Chptr. 8Rs 20. mmHg ·min /l [36] p. 144

11-18 mmHg ·min /l [42]

VA 4.038 lBTPS/min [13]5.6 lBTPS/min [36] p. 239

VE 8 lBTPS/min [36] p. 239Vstr,l 0.070 l [36] p. 144

Table 20. Nominal steady state values (NREM sleep)

Quantity % change Model Value Source

PACO2↑ 2-8 mmHg 44 mmHg [59]

PAO2↓ 3-11 mmHg 98.9 mmHg [59,37]

PvCO2↑ 6 % 51.9 mmHg estimate

PvO2↑ 1 % 35.2 mmHg estimate

VE ↓ 14-19 % 6 lBTPS/min [59,38]Pas ↓ 5-17 % 87.0 mmHg [43,59,60]H ↓ 10 % 68 mmHg [59,60]Q ↓ 0-10 % 4.3 l/min [59,57]VA ↓ 14-19 % 4.4 lBTPS/min [38]

MRO2 ↓ 15 % 0.26 lSTPD/min [35]MRCO2 ↓ 15 % 0.23 lSTPD/min [35]

sympathetic activity ↓ significantly - [60]Rs ↓ 5-10 % 19.2 mmHg ·min /l estimateSl ↓ 5-15 % 58.7 mmHg estimateSr ↓ 5-15 % 4.5 mmHg estimate


Table 21. Miscellaneous parameters (awake and sleep unless otherwise noted)


V0 5.0 l [24]Apesk 177.47 mmHg ·min ·l−1 [24]αl 89.47 min−2 [24]αr 28.46 min−2 [24]βl 73.41 mmHg ·min−1 [24]βr 1.78 mmHg ·min−1 [24]γl 37.33 min−1 [24]γr 11.88 min−1 [24]cap 0.03557 l ·mmHg−1 [24]cas 0.01002 l ·mmHg−1 [24]cvp 0.1394 l ·mmHg−1 [24]cvs 0.643 l ·mmHg−1 [24]cl 0.01289 l ·mmHg−1 [24]cr 0.06077 l ·mmHg−1 [24]

Table 22. Estimated state variable values for congestive heart failure categoriesas presented in Tsuruta et al. [66]

Quantity Normal Stage A Stage B Stage C Stage D

H 70 85 85 85 85Q 5.6 5.0 4.4 3.8 3.2Vstr .08 .059 .052 .045 .038Pap 15.0 19.0 23.0 27.0 31.0Pas 93 .3 93.3 93.3 93.3 93.3Pvp 8.0 12.0 16.0 20.0 24.0Pvs 5.0 5.0 5.0 5.0 5.0Rs 15.45 17.30 19.66 22.76 27.03

A cardiovascular-respiratory control system model ...staff.technikum-wien.at/~teschl/delay.pdf · A...

Documents

Transcript of A cardiovascular-respiratory control system model ...staff.technikum-wien.at/~teschl/delay.pdf · A...