Optimal control in blood flow simulations

Optimal control in blood flow simulations

Telma Guerra a,b, Jorge Tiago c, Adélia Sequeira c,nQ1

a Department of Mathematics and Management, Escola Superior de Tecnologia do Barreiro (IPS), Rua Américo da Silva Marinho, 2839-001 Lavradio,Lisbon, Portugalb CMA, FCT-UNL, Portugalc Department of Mathematics and CEMAT, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisbon, PortugalQ2

a r t i c l e i n f o

Article history:Received 27 December 2013Received in revised form14 April 2014Accepted 15 April 2014

Keywords:Optimal controlNon-Newtonian shear-thinning viscosityData Assimilation

a b s t r a c t

Blood flow simulations can be improved by integrating known data into the numerical simulations. DataAssimilation techniques based on a variational approach play an important role in this issue. We proposea non-linear optimal control problem to reconstruct the blood flow profile from partial observations ofknown data in idealized 2D stenosed vessels. The wall shear stress is included in the cost function, whichis considered as an important indicator for medical purposes. To simplify, blood flow is assumed as anhomogeneous fluid with non-Newtonian shear-thinning viscosity. Using a Discretize then Optimize (DO)approach, we solve the non-linear optimal control problem and we propose a weighted cost functionthat accurately recovers both the velocity and the wall shear stress profiles. The robustness of such costfunction is tested with respect to different velocity profiles and degrees of stenosis. The filtering effect ofthe method is also confirmed. We conclude that this approach can be successfully used in the 2D case.

& 2014 Published by Elsevier Ltd.

1. Introduction

Nowadays medical researchers, bioengineers and numericalscientists join efforts with the purpose of providing numericalsimulations of the human cardiovascular system in healthy andunhealthy conditions. This multidisciplinary collaboration offersnot only an exchange of knowledge between those communitiesbut also an exchange of data information that can be used in thenumerical simulations.

The progress in medical imaging techniques, blood flow mod-eling, and computational approaches can be joined together toobtain patient specific simulations. When such tools are availableto the medical community, the advantages in therapy prediction,training and expenses reduction can be very significant. Datainformation must be included in the numerical simulations toobtain accurate results. These procedures are known in theliterature as Data Assimilation (DA) techniques. They have beenwidely used in some engineering fields, like geophysics andmeteorology, but recent work [20,21] has shown that DA can alsobe successfully used to simulate blood flow problems.

One of the most frequent diseases of the cardiovascular systemis atherosclerosis, which generates a stenosis of the blood vessels,by partially obstructing them. This reduction in the vessel diameter

changes in particular the mechanical behavior of the blood circula-tion. Although not yet fully understood, it is already known thathemodynamical characteristics, like the shear stress exerted by theblood flow on the vessel walls, can affect the progression of this andof other pathologies (see [12,31,9]).

Whole blood is a complex mixture of formed cellular elementsthat includes red blood cells (RBCs or erythrocytes), white bloodcells (WBCs or leukocytes) and platelets (thrombocytes) in anaqueous polymeric and ionic solution, the plasma, containingelectrolytes, organic molecules and various proteins. While plasmais nearly Newtonian in behavior, whole blood exhibits markednon-Newtonian characteristics, mainly explained by the erythrocytesaggregation and the formation of 3D microstructures (rouleaux) atlow shear rates, as well as by their deformability and their tendencyto align with the flow field at high shear rates [13–17]. Thesecomplex dynamic processes contribute to the shear-thinningbehavior of blood. There is also experimental evidence thatsupports the fact that blood exhibits stress relaxation, see e.g.Thurston [38], Quemada [33], and Chien et al. [18]. Moreover,Evans and Hochmuth [23] have found that the red blood cellmembrane, which is a component of blood, has stress relaxation.On the other hand, the experimental results of Thurston [39] haveshown that the relaxation time depends on the shear rate. In viewof the experimental work performed so far, a reasonable non-Newtonian fluid model for blood should capture shear-thinningand stress relaxation, with the relaxation time depending on theshear rate. The response of such fluids is still almost unknown andis nowadays a subject of active research.

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Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/nlm

International Journal of Non-Linear Mechanics

http://dx.doi.org/10.1016/j.ijnonlinmec.2014.04.0050020-7462/& 2014 Published by Elsevier Ltd.

n Corresponding author.E-mail addresses: [email protected] (T. Guerra),

[email protected] (J. Tiago), [email protected] (A. Sequeira).

Please cite this article as: T. Guerra, et al., Optimal control in blood flow simulations, International Journal of Non-Linear Mechanics(2014), http://dx.doi.org/10.1016/j.ijnonlinmec.2014.04.005i

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Rajagopal and Srinivasa [34] have developed a thermodynamicframework requiring the knowledge of how the material storesenergy and how it produces entropy, which is particularly wellsuited for describing the viscoelastic response of bodies withmultiple configurations. Within this framework rate type modelsdue to Maxwell, Oldroyd and others can be generated. Thesemodels describe shear-thinning behavior and satisfy the secondlaw of thermodynamics. In fact, the framework requires the rate ofentropy production to be non-negative. Moreover, from amongstthe class of admissible constitutive relations meeting the conditionthat the rate of entropy production is non-negative, this theoryalso requires that the model should maximize the rate of entropyproduction. Based on [34], Anand et al. [1] introduced a model thatis suitable for describing the response characteristics of blood. Thismodel contains the Oldroyd-B model as a special sub-class.

The theory developed by Rajagopal and Srinivasa [34] makesuse of the fact that the natural configuration of the materialchanges as the body produces entropy, and also that one canunload from the current configuration of the body to its naturalconfiguration through an instantaneous elastic response. As bloodis modeled as a viscoelastic fluid capable of instantaneous elasticresponse, the framework developed in [34] is particularly wellsuited to develop a model for blood. This framework needs thespecification of how the body stores and dissipates energy. Theway in which the body stores energy is usually provided byassociating a specific Helmholtz potential with the body, andhow the body dissipates energy is given by a rate of dissipationfunction. However, as shown recently in [35], not all viscoelasticfluids can be described within that earlier framework. In fact, thereare (rate type) viscoelastic fluids that cannot have a specificHelmholtz potential associated with them. For certain viscoelasticfluids one cannot associate a specific Helmholtz potential but onlya Gibbs potential. Rate type fluids stemming from a Gibbspotential approach might be useful in describing the responseof blood.

Meaningful hemodynamic simulations require constitutivemodels that can accurately capture the rheological response ofblood over a range of physiological conditions. Despite its complexstructure and behavior, in sufficiently large vessels blood can bemodeled as an homogeneous fluid and, depending on the size ofthe blood vessels and the flow behavior, it is approximated as aNavier–Stokes fluid or as a non-Newtonian fluid. In normalconditions the flow of blood has a Newtonian behavior in mostparts of the arterial system. However, in the venous system or insome disease states, namely if the arterial geometry has beenaltered to include regions of recirculation, detected for instance inintracranial aneurysms or downstream of a stenosis, related to theexistence of atherosclerosis, more complex constitutive modelsshould be used. In such cases it is important the study of non-Newtonian blood flow behavior, including the shear-thinningviscosity, thixotropy, viscoelasticity, or even the yield stress. Asalready mentioned, this behavior is related to specific properties ofthe red blood cells, namely their tendency to form three-dimensional structures at low shear rates and to disaggregateand align in the blood flow direction at high shear rates.

Reliable experimental measurements of velocity, shear stressand interactions between the cellular components of blood andplasma are essential and need further improvement to developmicrostructural blood flow models, appropriate in smaller vesselsin which the cell and lumen sizes are comparable. For a detaileddiscussion of the physical properties of blood and correspondingmathematical models see for instance [36] and references therein.

In this paper we focus on a simple blood flow model that onlycaptures its shear-thinning rheological behavior. In this sense, wepropose and validate a DA method, based on a variationalapproach [20], to numerically reconstruct the blood flow in a 2D

idealized stenosis, governed by a generalized Newtonian modelwith shear-thinning viscosity. In the DA method we take intoaccount the Wall Shear Stress (WSS) and verify that this leads to abetter precision in a posteriori measurements. We validate therobustness of the method with respect to the stenosis degree andto different flow profiles. Additionally, we verify the filtering effectof the DA process.

The paper is organized as follows. Section 2 is devoted to themathematical model that will be used later in the numericalsimulations. In Section 3 we describe the DA procedure as avariational frame, which is our choice for DA implementationand we state the derived control problem formulation. Section 4 isessentially devoted to the discretization of the optimal controlproblem using DO approach. Numerical results related to thechoice and robustness of the cost function parameters are reportedin Section 5. Finally, the conclusions are summarized in Section 6.

We emphasize the fact that this is a preliminary study, thatcould also be extended to a more complex blood flow modelincluding shear-thinning and stress relaxation behavior, with therelaxation time depending on the shear rate, as the modeldeveloped by Anand et al. [1]. Moreover, the biochemistry is alsovery significant in blood flow simulations and the optimal controlof biochemical reactions is of paramount importance, for instancein blood coagulation disorders as those mentioned in [2–5]. Thispaper does not concern the application of the DA method tocontrol biochemistry but we think it would be very interesting toconsider later this extension as well.

2. Dynamic equations and viscosity models

Along this work, the blood flow is modeled as an incompres-sible and laminar fluid whose dynamic equations derive fromclassical mechanics principles: the conservation of mass and oflinear momentum. These equations describe the fluid motion in adomain of interest Ω� R2 and can be written as

ρ∂y∂t

þðy �∇Þy� �

�divðτðDyÞÞþ∇p¼ f in Ω

div y¼ 0 in Ω

8><>: ð1Þ

with suitable boundary conditions. The control and state variablesare constrained to satisfy a system with shear dependent viscositywhich decreases with increasing shear rate (shear-thinning fluid).The unknowns are the velocity field y and the pressure p. Thedensity of the fluid is represented by ρ and f is the given bodyforce. The viscous stress tensor is represented by

τ¼ 2μð_γ ÞDywhere μ is the viscosity, _γ is the shear rate given by

_γ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12 ð∇yþð∇yÞT Þ : ð∇yþð∇yÞT Þ

q¼

ffiffiffi2

pDy ;j��

and D is the strain tensor

Dy¼ 12 ð∇yþð∇yÞT Þ;

that is, the symmetric part of the velocity gradient.For a Newtonian flow, μ is a constant and τ is a linear function

depending on D y. This property is not verified when the fluid hasa non-Newtonian flow behavior and μ is a non-linear function ofthe stress tensor. This leads to difficulties in the treatment of thenon-linearities in addition to the convective term that has also anon-linear nature. Here we compare blood flow simulations for a2D generalized Newtonian model with shear-thinning viscosity inthe stationary case. Viscosity functions can be written in thegeneral form

μð_γ Þ ¼ μ1þðμ0�μ1ÞFð_γ Þ ð2Þ

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where μ0 and μ1 are the asymptotic viscosity values at zero and atinfinite shear rates, respectively, and Fð_γ Þ is a bounded functiondepending on the shear rate. The function F is such that

lim_γ-0

FðxÞ ¼ 1 and lim_γ-þ1

FðxÞ ¼ 0:

Different choices of F correspond to different blood flow models.The choice of the correct viscosity blood is important to obtainaccuracy in blood flow simulations and the viscosity parametersrelated to each model have to be chosen according to the specificmodel and taking into account physical parameters like thetemperature, hematocrit (percentage of red blood cells per unitvolume) and the healthy or unhealthy state of the individual donor[36,41].

In [24], the authors used known estimates to convert data andobtained realistic viscosity values at 37 1C. Comparing data fordifferent viscosity models, they obtained the constants for eachone using non-linear least squares fitting of the viscosity data.Despite the absence of data for μ0, the authors concluded that theCarreau and the Cross models fit very well the experimental data.Based on their conclusions, for the 2D stenosis case, we considered

Fð_γ Þ ¼ 1

ð1þðλ_γ ÞbÞa;

with the parameters a; b; λ40. Hence (2) becomes

μð_γ Þ ¼ μ1þ μ0�μ1ð1þðλ _γ ÞbÞa

; ð3Þ

which is usually known as the generalized Cross model.

3. The DA method

In this section we describe our approach to the DA problem.The main goal is to obtain numerical solutions for problem (1)which coincide, within a certain error, to observed data measuredin certain parts of the domain. Different techniques may be usedfor this purpose, namely the so-called variational approach, and, atthe discrete level, matrix updating or domain splitting. In [22] thevariational approach gives better results than the two others.

The variational approach is based on the assumption that weare free to adjust or control some of the model parameters andchose among the corresponding solution, according to somecriteria. In the simplest frame such criteria correspond to thedifferenceZΩpart

jy�ydj2 dx

between the obtained solution y and the data yd measured over asubdomain Ωpart. By taking the adjustable parameters as a controlfunction u defined on Ωc, possibly different from Ωpart, we are thenfacing an optimal control problem which also corresponds to aninverse problem. It is known that, when dealing with linearmodels, the possible ill-posedness (non-uniqueness) of suchinverse problem can be avoided by adding to the criteria aregularizing term likeZΩc

juj2 dx orZΩc

j∇uj2 dx:

Using such type of cost functions allows to prove, within certainassumptions, the uniqueness of solution for the correspondingoptimal control problem.

Here we assume that the 2D domain Ω represents an arterytruncated by two artificial boundaries Γin and Γout, the inlet andoutlet respectively, as represented in Fig. 1.

The control function corresponds to the inlet velocity fieldprofile. Therefore, the gradient in the regularizing term must be

the surface tangential gradient ∇su. We also assume that the datacorresponds to the velocity observed in Ωpart �Ω. Additionally, wecan assume to know the WSS in a certain part of the boundaryΓwall. In fact, as mentioned in Section 1, the WSS is an importantindicator for monitoring certain pathologies. Therefore, it is desir-able that the DA method can accurately reconstruct the desiredWSS. In this sense we also include the differenceZΓwall

jw�wdj2 ds

between the obtained WSS magnitude w and wd, the registeredWSS magnitude. Note that the WSS is the tangential component ofthe stress exerted by the fluid in the vessels wall and is given by

WSS¼sn�ðsn � nÞn; ð4Þwhere n is the outward normal to the wall surface and sn ¼sn isoften called the normal component of the stress tensor s. Assum-ing s¼ pI�τ, (4) can be written as

WSS¼ μð∇yþð∇yÞT Þn�μ½ðð∇yþð∇yÞT ÞnÞ � n�n: ð5ÞThe DA method consists in solving the optimal control problem

minu

Jðy;uÞ ¼w1

ZΩpart

jy�ydj2 dxþw2

ZΓwall

jw�wdj2 dsþw3

ZΓin

j∇suj2 ds

ð6Þsubject to

�div τþρðy �∇Þyþ∇p¼ f in Ω

div y¼ 0 in Ω;

y¼ 0 on Γwall

y¼ u on Γin

�sn¼ 0 on Γout :

8>>>>>><>>>>>>:

ð7Þ

The weights w1, w2 and w3 should be chosen in such a way thatproblem (6) and (7) admit a solution (preferably unique).

As far as we know, the existence of a solution to problem (6)and (7) has not been studied, even for the quadratic cost functionalobtained by taking w2 ¼ 0. In fact, optimal control problems fornon-Newtonian fluids have been studied only very recently. Forthe two-dimensional steady case we mention [10], where boundarycontrol of quasilinear elliptic equations is considered, and [11,37]both for distributed control. For the two-dimensional unsteady case,we refer to [40], and to [30] for three-dimensional coupled modifiedNavier–Stokes and Maxwell equations. Furthermore, for distributedcontrol on three-dimensional domains we cite [6,28]. None of thecited authors focused on the boundary control problem for the non-linear system (7).

In spite of lacking an appropriate theoretical frame, we assumeenough regularity on the problem variables and we propose anumerical approach to solve the problem. This will be the subjectof the next section.

4. Discretization of the main problem

The numerical solution of an optimal control problem can beobtained by either the Discretize then Optimize (DO) or the

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wall

wall

in out

Fig. 1. Computational domain Ω. (For interpretation of the references to color inthis figure caption, the reader is referred to the web version of this paper.) Q3

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Optimize then Discretize (OD) approaches. There is still somecontroversy concerning the performance of each one of thoseapproaches. Here we follow the work [20] where the DO approachis used (see, for instance, [29] for some comments on this issue).

The DO approach consists in first discretizing the optimalcontrol problem and then solve the finite dimensional optimiza-tion problem resulting from the discretization. For the first step weuse typical FEM. We first address the weak formulation of thesystem in (7).

Let us assume that H, Q and U¼ γΓinðHÞ are Hilbert spaces

chosen in such a way that, for yAH, pAQ , uAU and f sufficientlysmooth, problem (6) and (7) are well posed. Here γΓin

representsthe trace function to Γin. Also, we consider the spaces of testfunctions corresponding to y and p to be H0 ¼ fvAH : vj∂ΩD

¼ 0gand Q, respectively, where ∂ΩD ¼ Γin \ Γwall. Multiplying bothequations by suitable test functions and integrating by parts weobtainR

ΩτðDyÞ : ∇v dxþRΩðρðy � ∇ÞyÞ � v dx�RΩp div v dx¼ RΩf � v dxRΩq div y dx¼ 0

(

ð8Þ

for all vAH0 and qAQ .Let us consider the finite dimensional approximating subspaces

Vh �H, Vh;0 ¼ fvhAVh : vhj∂ΩD¼ 0g �H0, Qh � Q and Uh ¼ γΓin

ðVhÞwith h40, dimðVhÞ ¼Ny, dimðQhÞ ¼Np and dimðUhÞ ¼Nu. We thentake the finite dimensional approximations

yh ¼ ∑Ny

j ¼ 1yjϕjAVh; ph ¼ ∑

Np

k ¼ 1pkψkAQh ð9Þ

where ϕj and ψk are the shape functions belonging to Vh and Qh,respectively. The coefficients yj and pk are unknown values to bedetermined.

We reach the following approximated problem: find yhAVh

and phAQh such thatRΩτðDyhÞ : ∇vh dxþ

RΩðρðyh �∇ÞyhÞ � vh dx�

RΩph div vh dx¼

RΩf � vh dx;R

Ωqh div yh dx¼ 0;

(

ð10Þ

for all vhAVh;0 and qhAQ . If we choose Vh and Qh compatible (seee.g. [19]) then it is possible to prove the existence of a solution ofthe approximated problem and to check that ðyh; phÞ converges tothe solution ðy; pÞ of (8).

Here we use the well-known spaces corresponding to theTaylor–Hood elements (P2–P1), both for shape and test functionsto built the finite element spaces Vh and Qh, respectively.

First, let us consider the convective termZΩðρðyh � ∇ÞyhÞ � vh dx:

Replacing yh by the corresponding finite approximation (9) and vhby the test functions ϕi; i¼ 1…Ny, we obtain

ZΩ

ρ ∑Ny

j ¼ 1yjϕj � ∇

!∑Ny

k ¼ 1ykϕk � ϕi

" #dx; i¼ 1…Ny:

This can be written as

∑Ny

j ¼ 1yj ∑

Ny

k ¼ 1yk

ZΩðρϕj �∇Þϕk � ϕi dx

!i ¼ 1…Ny

¼ GðYÞY ;

where Y ¼ ðy1;…; yNyÞT and

GðYÞ ¼ ∑Ny

k ¼ 1yk

ZΩðρϕj �∇Þϕk � ϕi dx:

We now turn our attention to the viscous stress tensor τ. Theterm we want to discretize isZΩτðDyhÞ : ∇vh dx

where the tensor is written as

τ¼ 2μðjDyjÞDy;and μð�Þ is given by the Cross model for viscosity described by (3).

Replacing yh by its corresponding finite approximation we canwriteZ

Ω2μ ∑

Ny

j ¼ 1yjDϕj

��

!∑Ny

k ¼ 1ykDϕk : ∇ϕi dx

!i ¼ 1…Ny

¼Q ðYÞ:

Proceeding in a similar way for both linear termsRΩph div vh dx

andRΩqh div yh dx, system (10) becomes

Q ðYÞþGðYÞYþBTP ¼ F

BY ¼ 0

(ð11Þ

where the elements of matrix B are given by

bij ¼ZΩψ i div ϕj dx; i¼ 1…Np; j¼ 1…Ny:

As for the cost function (6), assuming that we fix a mesh for thedomain Ω, we refer to the basis functions associated to the nodesin Ωpart as ðϕiÞi ¼ 1…No

and to those on Γin, as ðϕiÞi ¼ 1…Nu. The

approximated control function in Uh is then defined by

uh ¼ ∑Nu

j ¼ 1ujϕj:

To discretize the first term we replace both y and yd by theirrespective finite dimensional approximations yh and ydh given by

yhd ¼ ∑N0

i ¼ 1ydiϕi:

We then obtainZΩpart

∑No

i ¼ 1ðyi�ydiÞϕi � ∑

No

j ¼ 1ðyj�ydjÞϕj

" #dx

¼ZΩpart

∑No

i ¼ 1ðyi�ydiÞ ∑

No

j ¼ 1ðyj�ydiÞðϕi � ϕjÞ dx

¼ ∑No

i ¼ 1ðyi�ydiÞ ∑

No

j ¼ 1ðyj�ydiÞ

ZΩpart

ϕi � ϕj dx

¼ ðY�YdÞTMðY�YdÞ¼ ðY�Yd;Y�YdÞM¼ ‖Y�Yd‖2Ny

ð12Þ

where ‖ � ‖Ny is the norm induced by the inner product ð�; �ÞM ,YdARNy is composed of ydi in the components corresponding tothe observations nodes, and null elsewhere. The Ny � Ny matrix Mis null everywhere, except for the elements

mij ¼ZΩpart

ϕi � ϕj dx; i; j¼ 1…No

which also correspond to the observation nodes.We now turn our attention to the term related to the WSS.

From (5) and taking

w¼ 2μðjDyjÞDyn�½2μðjDyjÞðDynÞ � n�nwe get

w¼ ffiffiffiffiffiffiffiffiffiffiffiffiw �wp

:

Since w will be evaluated on the no-slip boundary, Dy must beapproximated by an expression D(Y) depending on the neighbor-hood elements where the velocity does not vanish. Such expression

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can be obtained using finite differences or an inheritance basedaveraging method [42]. Then the approximation wh can be writtenas

whðYÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwhðYÞ �whðYÞ

pwhere

whðYÞ ¼ 2μðDðYÞÞðDðYÞn�ðDðYÞn � nÞnÞ:A similar procedure for wd (WSS) is used to approximateZΓwall

jw�wdj2 ds

byZΓwall

jwh�whd j2 ds¼WðYÞ ð13Þ

where W, obtained by a Gaussian Quadrature rule, depends non-linearly on Y.

Finally, for the regularization term we haveZΓin

∑Nu

i ¼ 1ui∇sϕi

��2

ds¼ZΓin

∑Nu

i ¼ 1ui∇sϕi : ∑

Nu

j ¼ 1uj∇sϕj

" #ds

¼ ∑Nu

i ¼ 1ui ∑

Nu

j ¼ 1uj

ZΓin

∇sϕi : ∇sϕj ds¼UTAU

¼ ðU;UÞA ¼ ‖U‖2Nuð14Þ

where ‖ � ‖Nu is the norm induced by the inner product ð�; �ÞA and Ais a symmetric Nu � Nu matrix where each element is given by

aij ¼ZΓin

∇sϕi : ∇sϕj ds:

Taking into account (11)–(14), the discrete version of the controlproblem (6) and (7) becomes

minU

JðYðUÞ;UÞ ¼w1‖Y�Yd‖2Nyþw2WðYÞþw3‖U‖2Nu

ð15Þ

subject to

Q ðYÞþGðYÞYþBTP ¼ F

BY ¼ 0:

(ð16Þ

Vector Y can, in fact, be rewritten as Y ¼ ðYU ;UÞ where YUrepresents the uncontrolled velocity coefficients which dependon the inlet velocity coefficients U.

4.1. Sequential quadratic programming (SQP)

To solve problem (15) and (16) which is non-linear both withrespect to the cost function and constraints we use a particularSequential Quadratic Programming (SQP) approach implementedin the package Sparse Non-linear Optimization (SNOPT) which isbriefly described here (see [25,43] for more details).

First, a major step is required to approximate the non-linearproblem by a quadratic programming problem (QP). The con-straints of such approximation correspond to the linearization of(16). The quadratic cost is a second order approximation to themodified Lagrangian function associated to (15) and (16). TheHessian matrix is replaced by a quasi-Newton approximationobtained by a BFGS update. The second step consists in solvingthe (QP) problem which is done by a reduced-Hessian iterativemethod implemented in SQOPT [27]. The solution of the approxima-tion (QP) is used to compute a descent direction for (15) and (16).A line search method is then executed to find the new solution of thenon-linear problem.

The strength of this approach is the fact that it was tested inlarge scale non-linear optimization problems with up to 40,000variables (see [25]).

5. Numerical results

As known data yd and wd we used synthetic data numericallygenerated. For this purpose, we consider a 2D stenosed vessel (seeFig. 1) with length equal to L¼2R, maximum diameter D¼ 2R andminimum diameter d¼R with R¼3.1 mm, which is close tophysiological measures [7]. For the fluid model we consider thegeneralized Cross model, as described in Section 2, with boundaryconditions defined in (7). The parameters for the generalized Crossmodel were chosen as in [7]:

μ0 ¼ 1:6e�1 Pa s; μ1 ¼ 3:6e�3 Pa s

a¼ 1:23; b¼ 0:64; λ¼ 8:2 s

ρ¼ 1059 kg=m3:

To construct and to treat our working domains we resort toCOMSOL Multiphysics [44]. Using the available options for LaminarFlow, we built a mesh composed by 3398 triangular and quad-rilateral elements with 15,986 degrees of freedom (dofs) for thevelocity. Quadrilateral elements were used to refine the meshalong the no-slip boundaries, the regions of boundary layers. Theminimum element quality registered was 0.2628 and the averageelement quality was 0.8262 (note that 0 corresponds to a degen-erate element and 1 corresponds to a symmetric element).

The fluid model was solved by imposing a Poiseuille velocityprofile at the inlet with a maximum velocity given by U0 ¼0:0993 m=s, which corresponds to the Reynolds number equal to120. The non-linear problem was treated with Newton's methodwhile each resulting linear systemwas solved using the direct solverPARDISO. We used the Nested Dissection Method, as a preorderingalgorithm, to minimize the zero elements after the factorization.Such methods are also available in COMSOL Multiphysics.

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101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566

676869707172737475767778798081828384858687888990919293949596979899

100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132

Fig. 2. Velocity field magnitude highlighting the recirculation region downstream the stenosis obtained with the direct problem. Left: non-Newtonian case. Right:Newtonian case.






Fig. 2 represents the velocity field magnitude obtained asdescribed before, where we highlight the recirculation region thatappears downstream the stenosis. We compare the resultsobtained with the Cross model with those for the Newtonian casewhere the viscosity is equal to μ1. These results are consistentwith those obtained in [7], even for a different stenosed vessel.

Fig. 3 represents the magnitude of the WSS for the non-Newtonian case. As expected, we observe a high increase of themagnitude value in the region of partial obstruction of the flow,when comparing with the remaining part of the vessel.

5.1. Choice of parameters and validation

The cost function (6) plays an important role in the perfor-mance of the DA method. It depends both on the observed domainΩpart and on the weighting parameters w1;w2 and w3. The correctchoice of the observed domain is an important issue (see [8]). Ithas been considered for the discrete linearized Navier–Stokesequations in [20], where Ωpart was taken as a finite number ofpoints. Here we do not deal with this issue, and we assume thatyd ¼ ydðx; yÞ is fully determined over Ωpart, corresponding to thevertical blue lines represented in Fig. 1. We will be concerned,however, with the choice of the weighting parameters for the costfunction.

It is well known that a proper balance of the terms in aquadratic type cost function (w2 ¼ 0) can be treated by Tikhonov's

regularization techniques (see [20] for some applications). But, asfar as we know, such techniques cannot be easily applied to a non-linear cost function like (15). Therefore, we present an heuristicnumerical study to investigate which set of parametersðw1;w2;w3Þ provides a better approximation of the data, includingthe velocity, the WSS and the inlet profile, through the optimalcontrol variable.

In order to solve each problem of type (15) and (16) we use theSQP method implemented in SNOPT, as briefly described inSection 4.1. To compute the gradient of the cost function withrespect to the control variables we used the adjoint method [42].The optimality tolerance, which determines the accuracy of theapproximation of first-order optimality conditions [26], was fixedto 10�6. For the QP solver, we imposed that the Cholesky factor ofthe reduced Hessian should be kept. The control problems weresolved using the mesh described above and taking 37 degrees offreedom for the control variable imposed on the inlet boundary.

We can observe the quantitative results obtained in thesimulations in Tables 1–3, where we take Y ¼ y�yd, W ¼w�wd

and CF to represent the final cost function value. We can check thatboth the velocity and the WSS are better approximated when allthe parameters are non-null.

As expected, in Table 1 we notice that better results for thevelocity field and WSS are attained when w1 increases. Suchbehavior of w1 requires a growing number of SNOPT iterations tosolve (15) and (16). Therefore, we only present results obtained for

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101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566

676869707172737475767778798081828384858687888990919293949596979899

100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132

Fig. 3. WSS magnitude obtained for the direct problem.

Table 1Values of the cost function parameters for fixed w2 and w3 and varying w1.Q4

RΩpart

jY j2 RΓwall

jWj2 RΓin

j∇uj2 CF

ð0;106 ;10�3Þ 4:152000� 10�7 1:838053� 10�13 8.190109 0.0081902934

ð105;106 ;10�3Þ 2:049161� 10�11 2:284326� 10�16 8.477968 0.0084800177

ð106;106 ;10�3Þ 2:196835� 10�13 6:982916� 10�18 8.481701 0.0084819206

ð107;106 ;10�3Þ 2:225967� 10�15 1:029709� 10�19 8.482097 0.0084821196

Table 2Values of the cost function parameters for fixed w1 and w3 and varying w2.

RΩpart

jYj2 RΓwall

jWj2 RΓin

j∇uj2 CF

ð106;0;10�3Þ 2:229353� 10�13 4:493504� 10�13 8.481696 0.0084819189

ð106;105 ;10�3Þ 2:198003� 10�13 6:444382� 10�16 8.481701 0.0084819206

ð106;107 ;10�3Þ 2:196719� 10�13 7:027137� 10�20 8.481701 0.0084819206






less than 400 iterations. If we consider w1 ¼ 0, the resulting costfunction has the worst performance for the velocity field approx-imation and one of the worst for WSS. Maintaining w1 and w3 andincreasing w2, in Table 2 we observe better results for WSS andalso for the velocity field, but less evident. We conclude that theexistence of the term

RΩpart

jy�ydj2 is crucial for good velocity andWSS approximations. We also observed that better results areobtained when the term

RΓwall

jw�wdj2 is present in the costfunction.

Let us now consider the so-called regularization termRΓinj∇uj2.

By observing Table 3, it is clear that the cost function value ishighly affected by this term, as we can see when we change thevalue of w3, maintaining w1 and w2. We can also observe that ifw3 ¼ 0 we have good approximations for the velocity field andWSS, once the minimization only concerns the remaining terms.However, the absence of a regularization term may possiblyinduce a spurious minimum (see for instance [8]). This is the casehere as we can confirm in Fig. 5, observing the results on the right-hand side. The optimal control for w3 ¼ 0 is highly irregular and far

from the parabolic profile. Thus, although it corresponds to aminimum for the discrete control problem, it does not representan admissible control function for our DA problem.

Looking at the three tables we conclude that, within the testswe have considered (less than 400 SNOPT total iterations), the setof parameters corresponding to better approximations isð107;107;10�3Þ. However we need 381 iterations to find theoptimal solution, which corresponds to a large computation time.To set a reasonable balance between accuracy and computationaleffort, and since the weights

ðw1;w2;w3Þ ¼ ð106;106;10�3Þ ð17Þalso perform very well, while taking only 281 iterations tominimize, we set these values as our reference choice for the DAproblem. Therefore, in the remainder, we will focus in showinghow the set of parameters (17) can perform better than the limitcases, when one of the weights vanishes. This is the purpose ofFigs. 4 and 5 where we represent the optimal controls versus theinlet profile used to generate yd.

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101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566

676869707172737475767778798081828384858687888990919293949596979899

100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132

Table 3Values of the cost function parameters for fixed w1 and w2 and varying w3.

RΩpart

jYj2 RΓwall

jWj2 RΓin

j∇uj2 CF

ð106 ;106 ;10�5Þ 7:794634� 10�14 1:479646� 10�16 8.529495 8:53730414� 10�5

ð106 ;106 ;10�4Þ 2:224927� 10�15 1:898105� 10�21 8.482097 8:48211969� 10�4

ð106 ;106 ;10�2Þ 2:052022� 10�11 2:194004� 10�14 8.477961 0.0848001567

ð106 ;106 ;10�1Þ 1:727427� 10�9 2:725338� 10�11 8.445729 0.8463276594

ð106 ;106 ;0Þ 6:029549� 10�14 9:237606� 10�17 171.890855 6:03878696� 10�8

ð106 ;106 ;101Þ 4:600914� 10�6 3:811922� 10�6 6.645082 74.863651813

Fig. 4. Left: Control for ð106 ;106;10�3Þ. Right: Control for ð106;0;10�3Þ.

Fig. 5. Left: Control for ð0;106 ;10�3Þ. Right: Control for ð106 ;106 ;0Þ.






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101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566

676869707172737475767778798081828384858687888990919293949596979899

100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132

Table 4Relative and absolute errors for different parameters computed in Ωpart.

Errors ð0;106 ;10�3Þ ð106;106 ;10�3Þ ð106 ;0;10�3Þ ð106 ;106;0Þ

Re 0.037254 2:709809� 10�5 2:729791� 10�5 1:419653� 10�5

Ae 6:443602� 10�4 4:687041� 10�7 4:721603� 10�7 2:455514� 10�7

Fig. 6. Left: Velocity field of the direct problem highlighting the recirculation region. Right: Velocity field of the controlled problem highlighting the recirculation region.

Fig. 7. Left: Velocity streamlines of the direct problem. Right: Velocity streamlines of the controlled problem.

Fig. 8. Left: Velocity field profile of the direct problem. Right: Velocity field profile of the controlled problem.

Fig. 9. Left: Pressure contour of the direct problem. Right: Pressure contour of the controlled problem.






To compare the effect of the different choices of parameters,not only in the minimization of each term, but in the quality of thevelocity approximation, we also report in Table 4 both relative andabsolute errors given by

Re ¼‖y�yd‖2‖yd‖2

and Ae ¼ ‖y�yd‖2;

where ‖ � ‖2 is the L2-norm.We can easily confirm that the worst cost function is the one

for which the velocity term is missing, and that it is better to havethe WSS term than not having it.

We conclude that solving the control problem with (17) is thebest option, within a certain computational effort limit. To

illustrate, we represent in Figs. 6–10 the velocity, the pressure,and the WSS magnitude corresponding to the optimal solution.We compare these results with the solution from which we tookthe observations yd and wd.

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101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566

676869707172737475767778798081828384858687888990919293949596979899

100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132

Fig. 10. WSS magnitude obtained with the controlled problem.

Fig. 11. Power law velocity field inlet profile (18): left: with exponent 4; right: with exponent 9.

Table 5Results for the Power-Law velocity field inlet profile (18), with exponent 4.

ξ¼ 4RΩpart

jY j2 RΓwall

jW j2 RΓin

j∇uj2 CF

ð0;106;10�3Þ 1:017793� 10�6 2:595413� 10�13 13.747798 0.0137480578

ð106 ;106 ;10�3Þ 2:083886� 10�12 1:033918� 10�12 14.513515 0.0145166331

ð106 ;0;10�3Þ 7:946703� 10�11 7:098585� 10�8 14.168067 0.0142475338

ð106 ;106 ;0Þ 6:535708� 10�12 1:204306� 10�11 392.3837 1:8578771535� 10�5

Table 6Results for the Power-Law velocity field inlet profile (18), with exponent 9.

ξ¼ 9RΩpart

jY j2 RΓwall

jW j2 RΓin

j∇uj2 CF

ð0;106;10�3Þ 1:001349� 10�5 3:259272� 10�13 20.106642 0.020106968

ð106 ;106 ;10�3Þ 7:801032� 10�11 2:094571� 10�11 29.855718 0.0299546736

ð106 ;0;10�3Þ 7:960777� 10�10 1:294974� 10�6 24.151194 0.0249472714

ð106 ;106 ;0Þ 1:582331� 10�15 7:475844� 10�22 565.871393 1:582331� 10�9

Table 7Relative and absolute errors for exponent 4, computed in Ωpart.

ξ¼ 4 ð0;106 ;10�3Þ ð106 ;106;10�3Þ ð106;0;10�3Þ ð106 ;106 ;0Þ

Re 0.048416 6:927796� 10�5 4:278107� 10�4 1:226887� 10�4

Ae 0.001009 1:443567� 10�6 8:914428� 10�6 2:556503� 10�6






We can see, from these results, that the original solution wasperfectly reconstructed from the observations.

5.2. Robustness of the parameters

In this section we show the robustness of the proposedparameters given by (17) in the DA method, with respect to

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101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566

676869707172737475767778798081828384858687888990919293949596979899

100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132

Table 8Relative and absolute errors for exponent 9, computed in Ωpart.

ξ¼ 9 ð0;106 ;10�3Þ ð106 ;106 ;10�3Þ ð106;0;10�3Þ ð106 ;106;0Þ

Re 0.134635 3:757856� 10�4 0.0012 1:692438� 10�6

Ae 0.003164 8:832345� 10�6 2:821485� 10�5 3:977852� 10�8

Fig. 12. Power law velocity field inlet profile with exponent 4; upper left corner: ð106 ;106 ;10�3Þ, upper right corner: ð106 ;0;10�3Þ, bottom left corner: ð0;106 ;10�3Þ, bottomright corner ð106 ;106;0Þ.

Fig. 13. Power law velocity field inlet profile with exponent 9; upper left corner: ð106 ;106;10�3Þ, upper right corner: ð106;0;10�3Þ, bottom left corner: ð0;106;10�3Þ, bottomright corner: ð106 ;106;0Þ.






different velocity field inlet profiles and to different stenosisdegrees.

5.2.1. Robustness to data inlet profileWe obtained different data using a Power-Law inlet profile (see

[32]) given by

ξþ2ξ

U1 1� y�RR

� �ξ !

: ð18Þ

Note that, for ξ¼2, expression (18) gives the particular case ofPoiseuille flow. In our simulations we have considered ξ¼4 andξ¼9 with profiles represented in Fig. 11. Constant U1 was chosenin such a way that the Reynolds number is equal to 120, as above.We tested ξ¼9 which is commonly considered in the simulationsof blood flow in arteries [32], and ξ¼4 which is an intermediatevalue between 2 and 9.

Like in the previous section we compare the set of parameters(17) with the limit cases. The quantitative results are thosereported in Tables 5 and 6. Similar conclusions can be inferredfor both profiles. The importance of the WSS term in the costfunction is more expressive when compared with the parabolicprofile. To confirm these conclusions, see Tables 7 and 8 whererelative and absolute errors are compared.

Furthermore, the only cost function that provides an adjust-ment of the control function with the original inlet profile isprecisely the one defined by (17). We can also see that suchadjustment is less precise for ξ¼9. Figs. 12 and 13 illustrate theseconclusions. They show the adjustment of the control function, for

each inlet profile, computed with the different sets of parametersdescribed above.

Finally, in Table 9 we compare the errors for the differentvelocity field inlet profiles, obtained with the choice of parameters(17). We observe that the DA problem solved with any of the threeinlet conditions gives good approximations of the velocity field.

Nevertheless, the closer we are to the Poiseuille inlet profile,the better relative and absolute errors are obtained.

5.2.2. Robustness as a function of the stenosis degreeIn this section we compare the control problem solutions

obtained for the cost function with parameters (17) in fourgeometries, three stenosed channels with different degrees and astraight channel. The four geometries are represented in Fig. 14.

The aim of this study is to show that our choice of parametersgiven by (17) obtained in Section 5.1 is robust with respect to thechanges imposed in the different geometries. In fact, even chan-ging the geometry, we can see that a numerical solution foroptimal control problem can be found, and that the originalvelocity is very well approximated, as shown by the relative andabsolute errors in Table 10.

5.3. Effect of noise filtering

In this section we study the effect of our DA method in filteringthe noise that can be added to the observed data yd. In realapplications this can be due to many different reasons, such as thelack of accuracy of the observation devices or to intrinsic errors ofthe model. Here noise samples were generated in a random way,normally distributed, with zero mean and standard deviationequal to s1 ¼ 0:05U0=3, s2 ¼ 0:1U0=3 and s3 ¼ 0:2U0=3, respec-tively. For the data we use the reference stenosis geometry withradius and the velocity profile as described in Fig. 2 (non-New-tonian case). However we include also in Ωpart the inlet boundary.

To understand the filtering effect of the formulation (15) and(16) we compare the solution obtained with our reference set ofparameters (17) with a direct approach. This approach consists in

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101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566

676869707172737475767778798081828384858687888990919293949596979899

100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132

Table 9Relative and absolute errors for different inlet profiles, computed with the set ofparameters (17) in Ωpart.

Errors Poiseuille Power-Law ξ¼ 4 Power-Law ξ¼ 9

Re 2:709809� 10�5 6:927796� 10�5 3:757856 � 10�4

Ae 4:687041� 10�7 1:443567� 10�6 8:832345 � 10�6

Fig. 14. Upper left corner: Stenosis with d¼3R/4. Upper right corner: Stenosis with d¼R. Bottom left corner: Stenosis with d¼5R/4. Bottom right corner: Straight channel.In all cases L¼10R and D¼2R. Dofs for velocity: About 10,000.






obtaining the numerical solution of the non-Newtonian model (1),by using a noisy parabolic profile, as inlet boundary condition.Such technique can be seen as the computational least expensivefiltering technique, when we wish to recover the solution down-stream the observations. We remark, however, that it could not beused if we pretended to reconstruct the solution upstream theobserved data. A similar comparison was done in [21] for theNavier–Stokes equations.

The differences between the solutions obtained by noisefiltering, using both methods, are presented in Table 11. As anindex of precision, we computed a posteriori, the relative error Reand the absolute error Ae in the entire domain.

The conclusions we have for the non-Newtonian problems aresimilar to those obtained in [21] for the Newtonian problem.

We can observe, as expected, that higher values for noisecorrespond to worst values for Re and Ae

6. Conclusion

In this work we proposed a DA method based on a variationalapproach to numerically reconstruct non-Newtonian shear-thinningblood flow behavior on different two-dimensional geometries.

We introduced a new cost function which takes into accountthe role of the WSS in the optimization process and we concluded,by testing the parameters w1, w2 and w3, that this modificationprovides better results for both the velocity field and the WSS. Theimportance of the presence of a regularization term in the costfunction was also tested. We observed that the absence of thisterm gives a completely irregular control corresponding to aspurious minimum. We also proposed a set of parameters as areasonable choice when balancing the approximation accuracyand the computational effort.

The method was validated by testing the robustness withrespect to changes in the geometry as well as to different inletprofiles. In each one of the cases the cost function was minimizedand we obtained very good approximations for the velocity fieldand the WSS.

Finally, we tested our method as a noise filter adding threedifferent amounts of noise to the data solution, including the inletboundary and we compared with the Direct filtering method,where the noise is added to the inlet boundary condition. The

conclusions are consistent with the results obtained in [21] for theNewtonian case. While correcting the noisy measurements, a DAapproach gives better results than a Direct approach.

Acknowledgments

This work has been partially supported by FCT (Portugal)through the Research Centers CMA/FCT/UNL, CEMAT-IST, the grantSFRH/BPD/66638/2009 and the projects PTDC/MAT109973/2009and EXCL/MAT-NAN/0114/2012.

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101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566

676869707172737475767778798081828384858687888990919293949596979899

100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132

Table 10Comparison between three different stenosis and a straight channel, regarding the cost function value, as well as the relative Re and absolute Ae errors computed in Ωpart.

Values Stenosis d¼3R/4 Stenosis d¼R Stenosis d¼5R/4 Channel

CF 0.0084819835 0.0084819208 0.0084818721 0.0084818125Re 2:068563 � 10�5 2:709878� 10�5 3:19736� 10�5 3:803092� 10�5

Ae 3:960592 � 10�7 4:687051� 10�7 5:176455� 10�7 5:703546� 10�7

Table 11Comparison between relative and absolute errors computed in Ω for the controlledproblem (CP) and the direct problem (DP) with different noise.

Noise Problem type Re Ae

s ¼ 0:05U0

3CP 0.001968 2:193509e�6

DP 0.004606 5:134504e�6

s ¼ 0:1U0

3CP 0.004974 5:545074e�6

DP 0.009214 1:027040e�5

s ¼ 0:2U0

3CP 0.009164 1:021493e�5

DP 0.018324 2:049306e�5






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