Blood Flow

C. Alberto Figueroa1, Charles A. Taylor2 and Alison L. Marsden31Departments of Biomedical Engineering and Surgery, University of Michigan, Ann Arbor, MI, USA2HeartFlow Inc., Redwood City, CA, USA3Departments of Pediatrics and Bioengineering, Stanford University, Stanford, CA, USA

1 Introduction 1

2 Overview of the Cardiovascular System 3

3 Established Methods in Blood Flow Modeling 4

4 Novel Computational Tools 13

5 Clinical Applications 17

6 Future Challenges 24

7 Related Chapters 24

Acknowledgements 24

References 24

1 INTRODUCTION

The cardiovascular system transports gases (oxygen andcarbon dioxide), nutrients, enzymes, hormones, and heatbetween the respiratory, digestive, endocrine, and excre-tory systems and the cells of the body. It is the inherentscale-linking mechanism between organ system and cellularscales. Far from a static system, the cardiovascular systemcontinually changes and adapts to meet the demands ofthe organism in healthy and diseased states. Hemodynamic(blood fluid mechanic) factors including flow rate, shearstress, and pressure forces provide the stimuli for many acuteand chronic biologic adaptations.

Local hemodynamic factors often act as stimuli totrigger changes in cardiac output and downstream vascular

Encyclopedia of Computational Mechanics Second Edition,Edited by Erwin Stein, René de Borst and Thomas J.R. Hughes.© 2017 John Wiley & Sons, Ltd.DOI: 10.1002/9781119176817.ecm2068

resistance that are essential in responding to acute changesin organ demand. For example, during exercise, the bloodflow rate established to meet the metabolic demands ofresting conditions is insufficient for the active muscles.These muscles release metabolic products that cause thelocal vessels to dilate, resulting in a decrease in the localvascular resistance and an increase in muscle blood flowby shunting blood from other tissues and organs. Theblood vessels upstream of those in the working muscle inturn dilate because of increased flow. This mechanism ishypothesized to be modulated by increased shear stresstriggering the release of nitric oxide from the endothelialcells lining the inner surface of blood vessels experiencingincreased flow. Ultimately, the dilation of the vascular bedsupplying the working muscles reduces the overall vascularresistance felt by the heart and would reduce systemic bloodpressure if the cardiac output remained constant. Instead,baroreceptors, specialized pressure-sensing cells in the aortaand carotid arteries, provide feedback to the nervous systemto increase heart rate and cardiac output to maintain bloodpressure. While blood flow rate can increase several-foldduring exercise conditions, mean blood pressure typicallychanges by less than 10–20%. In summary, the hemody-namic variables of flow rate, shear stress, and blood pressureall play a role in the response of the cardiovascular systemto acute changes in end-organ demand. Quantifying thesevariables under a range of physiologic conditions is oneof the important applications of computational methodsapplied to model blood flow.

The cardiovascular system also adapts to long-termanatomic and physiologic changes of the organism as occursduring growth, aging, injury, increased or decreased physicalactivity levels, and the onset of disease. Again, hemody-namic conditions play important roles in vascular adaptation

2 Blood Flow

in health and disease. Changes in blood velocity andpressure fields, sensed at a cellular level, initiate a cascadeof biochemical signals leading to hierarchical reorganiza-tion across molecular, cellular, tissue, and system scales(Humphrey, 2008). For example, blood vessels enlarge inresponse to chronic increases in blood flow through a mech-anism thought to be modulated by increased shear stressacting on the endothelial cells (Kamiya and Togawa, 1980;Zarins et al., 1987). In contrast to vascular enlargement inresponse to increased flow, blood vessels reduce in caliberin response to reductions in blood flow (Galt et al., 1993).In addition to shear-modulated changes in the diameter ofblood vessels due to changes in blood flow, the walls ofblood vessels get thinner as a result of decreased pressureand thicken in response to increased pressure through amechanism hypothesized to be modulated by changes intensile stress (Liu and Fung, 1989). These adaptive processesare part of a growing body of research on vascular growthand remodeling (G&R), which will be described in thefollowing sections (Figueroa et al., 2009; Sankaran et al.,2013; Valentín and Humphrey, 2009).

Computational methods have been widely applied to thequantification of hemodynamic and biomechanical factorsin relation to the genesis, progression, and clinical conse-quences of congenital and acquired cardiovascular disease.Congenital cardiovascular diseases arise from structuralabnormalities of the heart and blood vessels including septaldefects (holes between the atria or ventricles), obstructionsof the valves of the heart or major arteries leaving the heart,transposition of the major arteries exiting the heart, and inad-equate development of the right or left ventricles. In somecases, multiple defects are present or one defect can precip-itate another, as occurs when obstruction of the tricuspidvalve (between the right atrium and ventricle) interfereswith the normal development of the right ventricle. As in thecase of the normal adaptation of blood vessels in responseto changes in shear or tensile stress, hemodynamic factorsplay an important role in the development and progressionof congenital cardiovascular disease. Further, since recon-structive surgeries or catheter-based interventions used torepair congenital malformations alter the hemodynamicconditions, methods to model blood flow, coupled to cardio-vascular physiology, have increasing application in clinicaldecision-making in pediatric cardiology and surgery (Yanget al., 2011; Baretta et al., 2011; Hsia et al., 2011; Ensleyet al., 2000b).

Acquired cardiovascular disease became the leading causeof death in the industrialized world in the early twen-tieth century and contributes to roughly a third of globaldeaths. The most predominant form of acquired cardio-vascular disease, atherosclerosis, results from the chronicbuildup of fatty material in the inner layer of the arteries

supplying the heart, brain, kidneys, digestive system, andlower extremities. Interestingly, the upper extremity vesselsare typically spared of atherosclerosis. Risk factors foratherosclerosis including smoking, high cholesterol diet,physical inactivity, and obesity affect all the arteries of thebody, but notably, the disease is localized at branches andbends of the arterial tree. The observation that atheroscle-rosis occurs only in localized regions of the body has ledto the hypothesis that hemodynamic factors play a criticalrole in its development (Zarins et al., 1983; Caro et al., 1971;Friedman et al., 1981). This has motivated the application ofexperimental and computational methods to quantify hemo-dynamics and vessel wall biomechanics in human arteries.This will be discussed further in the remainder of this chapter.

In contrast to occlusive diseases such as atherosclerosisthat results in vessel narrowing, aneurysmal disease resultsin vessel enlargement and in some cases rupture. As in thecase of atherosclerosis, despite risk factors that are diffusethroughout the body, aneurysmal disease is highly local-ized and occurs predominantly in the aorta, and the iliac,popliteal, carotid, and cerebral arteries, and can occur in thecoronary arteries as a result of Kawasaki disease. The local-ization of aneurysmal disease is hypothesized to be influ-enced by hemodynamic conditions including flow stagnationand pressure wave amplification. For example, one of themost common sites of aneurysmal disease, the infrarenalabdominal aorta, is a location where blood flow is particu-larly complex and recirculating as a result of the multiplebranches that deliver blood to the organs in the abdomen(Ku, 1994; Xu et al., 1994; Taylor et al., 1998a, 1999, 2002).The resulting flow stagnation in the infrarenal abdominalaorta may enhance inflammatory processes hypothesized tocontribute to the degradation of the vessel wall. In addition,because of the reduction of cross-sectional area and progres-sive stiffening of the aorta from the heart to the pelvis, aswell as pressure reflections from downstream vessels, thepressure pulse increases in magnitude and contributes to agreater load on the wall of the abdominal aorta. Once again,the quantification of blood flow velocity and pressure fieldsmay contribute to investigations into the pathogenesis ofaneurysmal disease.

Finally, in clinical medicine, the quantification of bloodflow velocity and pressure fields is becoming increasinglyimportant in the diagnosis, treatment planning, and subse-quent management of patients with congenital and acquiredcardiovascular diseases. Risk stratification may be enabledusing computational modeling to augment clinical imagingstudies, providing hemodynamic data that may be otherwiseunattainable. One such technology, FFRCT , which is alreadygaining traction in the clinic for the evaluation of functionalsignificance of coronary lesions, was recently approvedfor clinical use by the US Food and Drug Administration

Blood Flow 3

(FDA). Medical, interventional, and surgical therapies usedto increase or restore blood flow to compromised organs andtissues or isolate aneurysmal regions from pressure forcesmay benefit from quantitative data provided by computa-tional methods.

This chapter is focused on computational methods forquantifying blood flow velocity and pressure fields in humanarteries and is organized as follows. Section 2 provides aconcise overview of important concepts from cardiovascularanatomy and physiology. Section 3 summarizes severalwell-established methods in blood flow modeling, includingthe paradigm of image-based modeling, zero-dimensional(0D) (i.e., lumped parameter) models, one-dimensional(1D) theory, three-dimensional methods, and fluid–structureinteraction (FSI) formulations. Section 4 explores novelcomputational tools for blood flow modeling, includingformulations for tissue G&R, methods for parameter estima-tion and uncertainty quantification (UQ), and optimization.Section 5 focuses on several important clinical applications,namely, pediatric cardiology, coronary artery disease, andsimulation of transitional hemodynamic stages. Finally,Section 6 describes some of the important areas of futuredevelopment and challenges anticipated for computationalmodeling of cardiovascular blood flow.

2 OVERVIEW OF THECARDIOVASCULAR SYSTEM

The circulatory system is the primary means of transportfor oxygen, nutrients, and hormones in organisms. The keyelements of all closed circulatory systems are a circulatingfluid (blood), a pumping mechanism (heart), a distributionsystem (arteries), an exchange system (capillaries), and acollection system (veins). Note that some animals, notablyinsects, and many mollusks, eject a blood-like fluid directlyinto their body cavities. This open circulatory system, oper-ating under low pressures and small velocities, is a relativelyinefficient transport system that poses great difficulties forthe organism in achieving a differential distribution of bloodflow to various organs and tissues (Kay, 1998). Here, wefocus our attention on mammalian circulatory systems. Theinterested reader is referred to the following books (Kay,1998; Kardong, 2002) for further discussion of circulatorysystems in other animals.

Blood is a suspension of cells in plasma. The concentrationof cells, termed the hematocrit, is approximately 50% byvolume but may vary in diseased states. We can distinguishthree major types of cells on the basis of their function inthe circulation: erythrocytes, leukocytes, and thrombocytes.Erythrocytes or red blood cells are the most abundant cellsin the blood stream (and the body) and are the primary

transport vessels for oxygen and carbon dioxide. It is esti-mated that of the approximately 100 trillion cells in the body,fully one-fourth are erythrocytes. Mammalian erythrocytesare disk shaped (due to the absence of a nucleus) with adiameter of approximately 8μm and a thickness of 2μm.Leukocytes or white blood cells represent less than 1% ofblood cells but have a critical role in producing antibodiesand identifying and disposing of foreign substances. Mono-cytes and lymphocytes are two particular types of leukocytesthat have been implicated in the early stages of atheroscle-rosis and support the current view of atherosclerosis asa chronic inflammatory disease (Ross, 1999). The thirdmajor category of blood cells, thrombocytes or platelets,represents approximately 5% of blood cells. Thrombocytesinteract with the protein fibrinogen to form fibrin, a meshthat traps cells to aid in the healing process. Finally, theplasma consists of approximately 90% water, 8% plasmaproteins, 1% inorganic substances, and 1% emulsified fat.Blood plasma comprises approximately 20% of the entireextracellular fluid of the body.

Of central importance in the cardiovascular system, theheart propels blood through the systemic and pulmonaryarteries and receives it from the systemic and pulmonaryveins. In the normal heart, during the diastolic phase ofthe cardiac cycle, blood entering the atria flows into theventricles as the ventricles relax and pressures fall belowatrial pressures. Blood flowing from the right atria passesthrough the tricuspid valve into the right ventricle, whileblood travels from the left atria to the left ventricle throughthe mitral valve. During the systolic phase of the cardiaccycle, the rising pressure in the contracting left ventricleexceeds aortic pressure and blood is ejected from the leftventricle through the aortic valve into the ascending aorta.Simultaneously, once the rising right ventricular pressureexceeds the main pulmonary artery pressure, blood flowsthrough the pulmonary valve into the pulmonary arteries.Blood fills the right atria from the superior and inferiorvena cava and the left atria from the pulmonary veins asthe blood exiting the atria during diastole reduces atrialpressure relative to the venous filling pressure. The cardiaccycle, once complete, repeats itself again, as it will another35 million times each year for the average adult. A usefuldescription of the relationship between ventricular pressureand flow during the cardiac cycle, the pressure-volume loop,is shown in Figure 1. Clearly, changes in left ventricularvolume during ejection directly correspond to the aorticflow. The area inside the left ventricular pressure–volumeloop corresponds to the work performed by the ventricle onthe blood. As discussed in the following section, lumpedparameter models of the heart are often developed with theaim of replicating the pressure–volume loop under a range ofphysiologic states.

4 Blood Flow

0 50 100 150

Left ventricular volume (ml)

40

80

120

Left

vent

ricul

ar p

ress

ure

(mm

Hg)

A

B

C

D

Isovolumicrelaxation

Isovolumiccontraction

Diastolic filling

Ejection

Figure 1. Left ventricular pressure–volume loop. The systolicphase of the cardiac cycle starts with isovolumic contraction (A–B)followed by ejection (B–C). Diastole begins ith isovolumic relax-ation (C–D) followed by diastolic filling (D–A).

The final three key elements of all closed circulatorysystems are the distribution system (arteries), exchangesystem (capillaries), and collection system (veins). Forthe purposes of this chapter, we focus our attention onthe arterial system, but note that consideration of bloodflow in the capillary bed necessitates consideration of thecellular nature of blood, and models of venous blood flowneed to incorporate the physics of flow in collapsible tubessubject to low venous pressures. Turning our attention to thearterial system, we note that, in fact, there are two arterialsystems in the body, the systemic and pulmonary arteries.For normal children and adults, these systems are in series,and the volumetric flow through each system is the same.However, because of a higher vascular resistance, bloodpressures are approximately six times higher in the systemiccirculation compared to the pulmonary circulation. Botharterial systems are often described using fractal geometry,and while there is some debate as to whether different partsof the vascular system are truly fractal in nature, at thevery least, the arterial (and venous) systems are space filling,branching continuously to ever smaller vessels until the levelof the capillaries is reached. Descending the vascular tree,the cross-sectional area increases, resulting in a decreasein the resistance of the vascular bed as compared to anarea-preserving branching network. Alternate approacheshave been used to describe the branching patterns of arteries,ranging from symmetric and asymmetric binary fractal trees(Olufsen, 1999) to diameter-defined Strahler systems that

accommodate branching systems with small side branchescoming off a larger trunk as frequently seen in the arterialsystem (Kassab and Fung, 1995; Choy and Kassab, 2008;Spilker et al., 2007). Regardless of the method for describingthe vascular anatomy, the direct representation of all thevessels from the major arteries to precapillary arterioles ina computational model is a daunting, and often intractable,task resulting in tens to hundreds of millions of vessels.We first consider methods that lump all these vessels into asmall number of components and then describe methods toincorporate anatomically representative models of the entirevascular system.

3 ESTABLISHED METHODS IN BLOODFLOW MODELING

3.1 Image-based modeling

Analysis and simulation of blood flow necessitates a combi-nation of image analysis and model construction, boundarycondition (BC) and material property selection, accuratesolution of the governing equations, physiology models,and high-performance computing. Patient-specific cardio-vascular simulations typically begin with 3D reconstructionof a portion of the vascular anatomy from CT or MRI imagedata and progress through stages of meshing, BC and param-eter assignment, hemodynamic simulation, and flow anal-ysis (Wilson et al., 2001; Updegrove et al., 2016; Tayloret al., 1998b). This produces a wealth of data from whichwe aim to extract clinically relevant information to improvepatient care. Since only a portion of a patient’s anatomycan be included in the 3D model, both due to computa-tional expense and limits of image resolution, BCs mustbe applied at inlets and outlets of the model to accuratelyrepresent the vascular network outside of the 3D domain,enabling physiologic representation of flow and pressurewaveforms.

Patient-specific modeling requires construction ofthree-dimensional models of vascular anatomy deriveddirectly from patient image data. The imaging modality istypically CT or MRI data, though ultrasound and angiog-raphy data can provide an attractive alternative. Image data istypically segmented using 2D or 3D level set or thresholdingmethods. Available packages for model construction are theopen source packages SimVascular (www.simvascular.org),CRIMSON (www.crimson.software), and ITK-SNAP orcommercial packages such as Mimics (Materialise, Leuven,Belgium) (Updegrove et al., 2016). The model constructionprocess typically employs a four-step process: (i) creationof centerline vessel paths, (ii) 2D segmentation of thelumen of each vessel, (iii) lofting segmentations to create

Blood Flow 5

1. MRI 2. Pathlines 3. Segmentation 4. Solid 5. Mesh 6. Boundary conditions 7. Simulation

S

LR

Aneurysm

LII

LEI

RII

REI

SMAARR

RR

H

0 Mean wall shear stress (dynes/cm2) 10

Figure 2. The typical steps in the patient-specific modeling process include (from left to right): importing medical image data, constructionof centerline paths and segmentations, three-dimensional model construction, unstructured mesh generation, assignment of physiologicboundary conditions, and finite-element flow simulation to extract quantities of clinical interest. (Reproduced with permission from Leset al., 2010. © Biomedical Engineering Society, 2010.)

a 3D model with multiple branches, and (iv) unstructuredmesh generation. The steps of the typical patient-specificmodeling process are shown in Figure 2. Image segmentationmethods are a challenging component of the cardiovascularmodeling process, often requiring extensive user interven-tion and manual segmentation to supplement automatedmethods. Recent efforts to improve automated segmentationalgorithms have focused on machine learning algorithmsand shape analysis and have also been extended to 3Dsegmentation methods using region growing and activecontours (Merkow et al., 2015). Continued advancementsin these areas will be crucial to enable high-throughputprocessing of models for large patient cohorts in clinicalstudies.

3.2 Zero-dimensional methods

Perhaps the simplest model one can use to model bloodflow in the cardiovascular system is a 0D lumped param-eter method, which utilizes an analogy between fluid flowin the heart and vascular system and current flow in electriccircuits to obtain systems of ordinary differential equations(ODEs) governing pressure and flow rate over time. Inthis method, pressure and flow rate replace voltage andcurrent, resistors and inductors are used to represent theviscous and inertial properties of blood, and capacitorsto represent elastic wall behavior (Westerhof et al., 1969,2009).

Using this analogy, some basic building blocks of a 0Dlumped parameter network (LPN) can be defined as follows.Making the analogy of flow rate to current, and pressure dropto voltage, the relations for a resistor (R), capacitor (C), andinductor (L), with appropriate analogies to fluid dynamics,where ΔP is pressure drop and Q is flow rate, are as follows:

V = IR ⇒ ΔP = QR (1)

V = ddt

LI ⇒ ΔP = ddt

LQ (2)

I = ddt

CV ⇒ Q = ddt

CP (3)

Making a rigid wall, Poiseuille flow assumption, the resis-tance of a section of blood vessel is

R = 8𝜇l𝜋a4

(4)

where a is the vessel radius, 𝜇 viscosity, and l the vessellength, so that resistance is inversely propotional to thefourth power of the radius. These circuit building blocks canthen be used to assemble networks of vessels and organs.At element junctions, conservation of mass is enforcedfollowing Kirchoff ’s law and continuity of pressure isassumed.

One of the most interesting applications of the lumpedparameter method is in constructing a model of the heart thatcan reproduce pressure–volume data and cardiac function. Inthis approach, the contraction of the heart is modeled usingtime-varying capacitors and unidirectional flow throughheart valves is modeled using diodes. The maximum pres-sure developed in the ventricles is directly related to the“afterload” posed by the vascular system. Here, simplelumped parameter models, such as the RCR circuit (wherea relatively small upstream resistor is connected in seriesto a parallel arrangement of a capacitor and downstreamresistor), have been used to model the effect of arterialcompliance and a high resistance distal vascular bed onpressure and flow waveforms. This approach is particularlyuseful in developing in vitro mock circulatory systems forexperimental flow studies (Kim et al., 2009; Vukicevic et al.,2013).

6 Blood Flow

−100

0 1 2 3Time (s)

0 1 2 3Time (s)

10

20

30F

low

(m

l s−1

)

−100

10

20

30

Flo

w (

ml s

−1)

Upper body

Lungs

Atrium Ventricle

Liver

Kidneys

Intestine

Aorta

Legs

InferiorVena cava

(a) (b)

Figure 3. A patient-specific model of the Fontan surgery with a Y-graft configuration illustrating the use of open-loop (a) and closed-loop(b) boundary condition configurations for multiscale modeling.

In addition to cardiac models, lumped parameter modelsprovide a surprisingly accurate representation of theclosed-loop circulatory physiology. Lumped parametermodels have been used to provide BCs for other analysismethods and are an essential component in efforts to develop“closed-loop” circulatory models (Esmaily-Moghadamet al., 2013b; Lagana et al., 2002; Sankaran et al., 2012;Corsini et al., 2011). The RCR (Windkessel) circuit iscommonly adopted as an outlet BC to model the distalvasculature with one capacitor modeling vessel complianceand two resistors modeling proximal and distal pressuredrops. Diodes and other specialized components may alsobe included to model heart valves and elastance of heartchambers. As some circuit components are time dependent,for example, due to the presence of inductors and capacitors,the circuit is represented by a set of time-dependent ODEs.Combining these equations produces a single ODE withorder equal to the number of time-dependent componentsin the circuit. For simple circuits with low-order ODE,an analytical solution can be obtained, as is the case with

simple resistors, RCR (Windkessel) circuits, and coronaryartery models, which are commonly used as outlet BCsin an open-loop configuration. Models can be expandedto a full closed-loop networks that can then be coupled tothe 3D domain. A closed-loop lumped-parameter modelhas the advantage that the effects from the global circu-lation are fully coupled to influence the overall simulatedphysiology. Figure 3 shows an example of closed-loopand open-loop multiscale simulation setups. Two limi-tations of lumped parameter models are worth noting.First, since geometry is not explicitly defined, wave prop-agation effects are not incorporated in lumped parametermodels. Second, the determination of model parametersis generally based on the behavior of the entire system orsubsystem, not on first principles as in the case of methodsthat emanate from initial-boundary-value problems. As aresult, changes to components of the system, for example,a single artery, or an entire vascular bed, are not easilymodeled.

Blood Flow 7

3.3 One-dimensional methods

Nonlinear and linear 1D methods can accurately describepulse wave propagation phenomena in extensive vascularnetworks while keeping the computational cost down. 1Dmethods are based on the assumptions that the dominantcomponent of blood flow velocity is oriented along the vesselaxis and that pressure can be assumed constant over thecross section of the vessel. For the flow of a Newtonianfluid in a deforming, impermeable, elastic domain, theseequations consist of the following: (i) a continuity equation,(ii) a single axial momentum balance equation, and (iii) aconstitutive equation (also known as tube law), together withsuitable initial and BCs. Originally developed by Hughesand Lubliner (1973), the most commonly used form of thenonlinear continuity and momentum balance 1D equationsof blood flow is as follows (Sherwin et al., 2003; Peiró et al.,2009): ⎧⎪⎨⎪⎩

𝜕A𝜕t

+ 𝜕(AU)𝜕x

= 0

𝜕U𝜕t

+ U 𝜕U𝜕x

+ 1𝜌f

𝜕P𝜕x

=f

𝜌fA

(5)

where x is the axial coordinate along the vessel, t the time,A(x, t) the cross-sectional area of the lumen, U(x, t) the axialblood flow velocity averaged over the cross section, P(x, t)the blood pressure averaged over the cross section, 𝜌f thedensity of blood assumed to be constant, and f (x, t) thefrictional force per unit length. These equations assume avalue of one for the momentum correction factor in theconvection acceleration term of equation (5), following thework of Stergiopulos et al. (1992).

Often, the velocity profile is assumed to be constant inshape and axisymmetric. The axial velocity (u) can bedescribed by a polynomial:

u(x, 𝜉, t) = U(x, t)𝜁 + 2𝜁

[1 −

(𝜉

r

)𝜁](6)

where r(x, t) is the lumen radius, 𝜉 the radial coordinate,and 𝜁 the polynomial order. 𝜁 = 9 has been shown toprovide a good compromise fit to time-resolved experimentaldata (Smith et al., 2002). If we consider an axisymmetricvessel, the frictional force per unit length yields f (x, t) =2𝜇𝜋r 𝜕u

𝜕𝜉|𝜉=r. For the velocity profile given by equation (6), we

have f = −2(𝜁 + 2)𝜇𝜋U in which the local f is proportionalto the local flow. Note that 𝜁 = 2 leads to the Poiseuille flowresistance f = −8𝜇𝜋U. An explicit algebraic relationshipbetween P and A (or tube law) is also required to close eqau-tion (5) and account for the FSI of the problem. In general, wehave P = (A(x, t); x, t), where the function depends onthe model used to describe the dynamics of the arterial wall.

In the 1D model, each cross section is assumed to deformaxisymmetrically and independently from the others. A rela-tionship between circumferential hoop stress, T𝜃 , and radialdisplacement can then be invoked: displacement can then beinvoked:

T𝜃 =E

1 − 𝜈2r − rd

rd(7)

where rd is the radius at diastolic pressure (Pd). ApplyingLaplace’s law, T𝜃 = (P − Pd)r∕h, and assuming that 1∕r canbe approximated by 1∕rd, we obtain the following constitu-tive equation (Alastruey et al., 2011):

P = Pd +43

Ehr − rd(rd)2

= Pd +𝛽

Ad(√

A −√

Ad),

𝛽 = 43

√𝜋Eh (8)

where Ad(x) is the luminal area at diastolic pressure. Inequation (8), both E and h are functions of the position x.The pulse wave speed c(x, t) is related to A through

c =

√𝛽

2𝜌fAdA1∕4 (9)

The initial area A0 is calculated by replacing P = 0 and A =A0 in equation (8), which leads to

A0 = Ad(

1 −√

AdPd𝛽

)2(10)

When solving 1D simulations in a network of vessels, onemust involve matching conditions at bifurcations by takinginto account the correct propagation of the characteristicinformation and neglecting energy losses (Alastruey et al.,2008). Nonlinear 1D formulations naturally account fornonlinear advective losses in the flow and are able to appro-priately describe pulse wave propagation in the cardiovas-cular system. These formulations are particularly well suitedfor modeling blood flow in the larger arteries. In addition toaccounting for pulse wave propagation, they also enable asimple framework to utilize viscoelastic constitutive models(Valdez-Jasso et al., 2011; Raghu et al., 2011), a trait ofrecognized importance in the behavior of the arterial system(Holenstein et al., 1980).

The 1D theory, due to its low computational cost, has beenapplied to model hemodynamics in large arterial networks(Reymond et al., 2009; Blanco et al., 2015) and, in particular,to understand pulse wave propagation in human and animalcirculations (Matthys et al., 2007; Alastruey et al., 2011).Recently, formulations have been derived that enable a moreaccurate characterization of quantities such as wall shearstress (WSS), by releasing the common assumption in 1D

8 Blood Flow

methods in which velocity and flow waveforms are in phase(Bessems et al., 2007). Furthermore, studies have confirmedthat 1D simulations produce flows and pressure waveformsthat compare well with those obtained with 3D formulations,in both idealized (Xiao et al., 2014) and even subject-specificgeometries (Alastruey et al., 2016).

However, 1D methods are not well suited to study situ-ations in which the flow is complex due to curvature,branching, and stenoses. Here, the axisymmetric assump-tion, fundamental to any 1D models, simply breaks down.There have been attempts to address this limitation bydeveloping empirically derived losses to model the effectsof stenoses (Seeley and Young, 1976; Young and Tsai,1973; Stergiopulos et al., 1992; Olufsen, 1999). However,these approaches introduce ad hoc parameters that must becalibrated for each specific problem. Therefore, in situationsin which the flow is complex, as is commonly the casein disease conditions (e.g., stenosis, aneurysms, fistulas,and shunts), or in situations in which a medical device isdeployed in the system, 3D methods must be utilized inorder to describe such flows.

3.4 Three-dimensional methods

Although the suspension fluid, plasma, is largely composedof water, the large concentration of cells results in anon-Newtonian rheological behavior for whole blood.Specifically, blood exhibits a shear-thinning behavior wherethe viscosity at any shear rate increases with an increasedpercentage of cells (Fung, 1984). Despite this, it is gener-ally accepted that blood flow in the large vessels can berepresented as an incompressible fluid whose constitu-tive behavior is usually approximated by a Newtonianmodel. Arterial blood flow has been traditionally repre-sented using the Incompressible Navier–Stokes equationsin a fixed Eulerian frame of reference. However, bloodvelocity and pressure fields can be greatly influenced bythe motion of external or internal vascular structures, suchas the contracting cardiac muscle, moving heart valves,or deforming large arteries. In these situations, one mustcharacterize the mechanical behavior of the moving vascularstructure and its interactions with the blood flow. Dependingon whether the deformability of the vascular structures isconsidered or not, one must choose between adopting a rigidwall and an FSI formulation. Rigid wall analyses are signif-icantly less computationally expensive and provide usefulinformation when the goal is to characterize transport, WSSand derivative quantities, and overall flow and pressure. Incontrast, when studying processes in which the interactionbetween blood and vascular structure is important, suchas hypertension, aneurysms, or a variety of device–vesselinteractions, FSI formulations must be utilized.

3.4.1 Rigid wall formulations

The governing equations for the three-dimensional theoryof blood flow under the assumptions of an incompress-ible, homogeneous, Newtonian fluid flow in a fixed Eulerianframe consist of the Navier–Stokes equations and suitableinitial and BCs:

𝜌(v̇ + v ⋅ ∇ v) = −∇ p + ∇ ⋅ 𝝉 f + b in Ωf∇ ⋅ v = 0 in Ωf

v(x, t = 0) = v0 in Ωfv = g on Γg

(−pI + 𝝉 f) ⋅ n = h on Γh

(11)

where Ωf denotes the domain where the flow takes place, vthe fluid velocity, p the pressure, b a body force, n a boundaryunit normal, 𝜌 the density, and 𝝉 f = 𝜇(∇ v + (∇ v)T) theviscous stress tensor for a Newtonian fluid. Γg and Γhare the Dirichlet and Neumann portions of the boundary,respectively.

These equations are solved numerically using either finiteelements or finite volume methods. In the past two decades,the number of contributions to the field in which the 3DNavier–Stokes were solved to simulate hemodynamic prob-lems can be counted by the thousands, in areas ranging fromdisease research, to medical device design and performanceevaluation, to noninvasive diagnostics, and even virtualsurgical planning. Commercial packages have made theseformulations readily accessible to a wide range of users.However, these packages often limit the user in terms of bothproper mesh refinement strategies and, most importantly,the ability to include sophisticated inflow and outflow BCsthat can meaningfully represent the hemodynamics of thesystem at hand (Section 3.5).

Perktold in the late 1980s and early 1990s was a pioneerin the use of computational methods to solve the 3DNavier–Stokes equations to represent blood flow in humanarteries (Perktold and Hilbert, 1986; Perktold et al., 1987,1991a,b; Perktold and Peter, 1990). Taylor, Hughes andZarins then pioneered the paradigm of “image-based”modeling, in which geometrical models of the vesselsof interest are extracted from medical image data andcomputational methods used to solve the incompressibleNavier–Stokes equations (Taylor et al., 1996, 1998b).

The methods have been used successfully to study hemo-dynamics in abdominal aortic aneurysms (Les et al., 2010;Tang et al., 2006), the pulmonary arteries (Tang et al., 2011),cerebral aneurysms (Cebral et al., 2005a,b; Sforza et al.,2009), the Fontan circulation (Marsden et al., 2007, 2009),aortic coarctation (LaDisa et al., 2011a,b; Coogan et al.,2013), and aortic dissection (Dillon-Murphy et al., 2016).Lagrangian analysis has been used to characterize coherentstructures within cardiovascular flows (Shadden and Taylor,

Blood Flow 9

CT data

CFD 4D PC-MRI CFD 4D PC-MRI

Potentialtear

Potentialtear

AA

B

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Flo

w (

cm3

s−1 )

00 0.65 1.3

Time (s)

230

460

37.5

0 cm s−1 75

(a) (b) (c)

Figure 4. Comparison of CFD data and acquired 4D PC-MRI flow data at peak systole and mid-diastole in a patient with Type B aorticdissection. In aortic dissection, the aorta is divided into two different passages, a true lumen and a false lumen. Multiple communicationtears may connect the two passages, creating a complex hemodynamic environment. (a) The computed tomography image data at twolocations shows suspected connecting tears between the true and false lumina (Dillon-Murphy et al., 2016). panels b and c compare CFDand 4D PC-MRI velocity patterns at peak systole and mid diastole, respectively. (Dillon-Murphy, https://link.springer.com/article/10.1007/s10237-015-0729-2. Used under CC BY 4.0 https://creativecommons.org/licenses/by/4.0/.)

2008). In summary, the development of advanced imagingmodalities, with higher spatial and temporal resolution, andeven time-resolved volumetric velocity information (e.g., 4DPC-MRI Markl et al., 2007), make for a closer integrationbetween medical data and computation of hemodynamics(Figure 4).

Recently, the field has witnessed the first commercial appli-cation of 3D computational methods for blood flow. Heart-flow, Inc. received FDA approval in 2014 for a workflow inwhich the severity of coronary artery disease can be assessedusing computed tomography data to reconstruct the mainvessels of the coronary tree of a patient, and computer simu-lations of blood and pressure (Taylor et al., 2013) to estimatethe so-called FFRCT (fractional flow reserve), a normalizedmetric of pressure gradients through a stenotic vessel underconditions of maximum flow.

It is apparent that 3D methods for blood flow in rigiddomains constructed from image data are in a very advancedstage of maturity. Provided that appropriate inflow andoutflow BCs are specified via multiscale or multiresolu-tion methods (Section 3.5) and that proper care is given toachieving mesh-independent solutions, these formulationshave been widely effective in situations in which the goal is tocharacterize quantities such as pressure, pressure gradients,flow distributions, WSS and its derivatives, and transport.

3.4.2 Fluid–structure interaction formulations

Blood velocity and pressure fields can be greatly influencedby the motion of external or internal vascular structures, suchas the contracting cardiac muscle, moving heart valves, ordeforming large arteries of the body. In these situations, onemust characterize the mechanical behavior of the movingvascular structure (usually in a Lagrangian frame of refer-ence) and its interactions with the blood flow, defining anFSI problem. We next provide an overview of the differentfamilies of FSI methods that have been used to model cardio-vascular FSI problems.

Arbitrary Lagrangian–Eulerian formulations. In the arbi-trary Lagrangian–Eulerian (ALE) formulation (Hugheset al., 1981; Le Tallec and Mouro, 2001), the Navier–Stokesequations are written in a moving reference frame thatfollows the motion set by the vascular structure (Figure 5).At this interface, kinematic (continuity of velocities) anddynamic (equilibrium of tractions) compatibility condi-tions must be satisfied between fluid and structure. In ALEformulations, the motion of computational grid in whichthe flow problem is described is arbitrary and defined bya grid velocity vG = (

𝜕x𝜕t)𝝌

. This velocity is defined by theLagrangian motion of the vascular structure at the fluid–solidinterface Γsx, but it needs to be propagated to all grid points

10 Blood Flow

Reference domain: Ωχ

Materialdomain: ΩX

x = ϕ(χ, t)X = ψ(χ, t)

Vessel wall particleBlood flow particleLagrangian nodeALE node

x = φ(X, t)Spatial

domain: Ωx

Ωsχ

ΩsX

Γsχ

Ωsx Γsx

Figure 5. Different configurations used in the ALE formulation.

in the interior of the flow domain. In an ALE formulation,one must therefore solve the following three-field problem(Figure 5):

1. The Navier–Stokes equations of motion of a fluid ina moving spatial domain Ωx, to represent the bloodmotion,

𝜌

(𝜕v𝜕t 𝝌

+ (v − vG) ⋅ ∇ xv)

= −∇ xp+ ∇ x ⋅ 𝝉 f + b in Ωx

∇ x ⋅ v = 0 in Ωx(12)

2. The elastodynamics equations of motion of the vascularstructure written in a Lagrangian frame of reference withrespect to some initial configuration,

𝜌s0

(𝜕v𝜕t X

+ ∇ X(FS))= b0 in ΩsX (13)

3. The motion of the computational grid for the blood flowdomain, defined by an arbitrary mapping x = 𝜙(𝝌 , t) thatmatches the motion of the structure at the interface Γsx.

The ALE formulation is a boundary-fitting technique,in which the fluid–solid interface is accurately captureddue to continuous updating of the fluid grid. However, inproblems with large deformations in the vascular structure,ALE formulations may be time consuming. Modular (e.g.,staggered) and nonmodular (e.g., monolithic) approachesfor the solution of the coupled algebraic system resultingfrom the space–time discretization of the FSI problemhave been proposed. Modular preconditioners allow for theuse of independent, specialized fluid and structure solverscoupled via a relatively simple iterative scheme. These

algorithms may exhibit poor convergence in cardiovas-cular problems, in which the density of the fluid 𝜌 andthe structure 𝜌s are comparable. Nonmodular precondi-tioners require a more elaborated coupling between thefluid and solid solvers but result in faster, more stablealgorithms.

ALE formulations were first used in cardiovascular appli-cations by Perktold and Rappitsch (1995) and Prosi et al.(2004). More recently, Gerbeau et al. (2005), Gerbeau andVidrascu (2003) studied hemodynamics in FSI models ofcerebral aneurysms and carotid bifurcations, Bazilevs et al.(2006), Zhang et al. (2007) performed patient-specific FSIsimulations using an isogeometric framework, and Wolterset al. (2005) and Scotti et al. (2005) investigated FSI inpatient-specific abdominal aortic aneurysm models.

Immersed Boundary Method formulations. The immersedboundary method (IBM) was first introduced by Peskin(1972, 2003) in the context of finite differences. The IBMis a non-boundary-fitting formulation in which the structurewas represented by a set of nonconforming, interconnectedelastic boundary points embedded in the fluid domain, whichis kept fixed throughout the computations. IBM formu-lations have been used in cardiovascular applications byLemmon and Yoganathan (2000) to examine left ventric-ular dysfunction; Watton et al. (2007) to study prostheticmitral valves; Griffith et al. (2009) to investigate natural andprosthetic heart valves; and Vigmond et al. (2008) to developa whole heart electro-mechano-fluidic computational frame-work. On the microscales, the IBM has also been applied tomodeling the interactions of red blood cells and plasma inthe meso-circulation (Bagchi, 2007).

Closely related to the IBM, the fictitious domain method,first developed by Glowinski et al. (1997) in the context offinite elements, introduced Lagrange multipliers to constrainthe motion of the fluid and the solid at the interface. Baai-jens (2001) subsequently developed an extension suitable forslender structures and performed cardiovascular FSI simula-tions of the aortic valve (De Hart et al., 2003, 2004). van deVosse et al. (2003) and van Loon et al. (2003). also utilized acombination of ALE and fictitious domain methods to simu-late dynamics of heart valves.

Coupled Momentum Method. Figueroa et al. (2006) devel-oped a “coupled-momentum method” in which the elasto-dynamics equations are coupled with the fluid through afictitious body force that drives the motion of the vessel wall,which is modeled as a thin-walled membrane. This mono-lithic method, coupled with multidomain formulations forinflow and outflow BCs (Vignon-Clementel et al., 2006),enables FSI simulations with small increase in computationalcost compared to rigid wall models (LaDisa et al., 2011a,b;Coogan et al., 2013). It is particularly well suited to studypulse wave propagation in complex 3D vascular models built

Blood Flow 11

Flow(ml s−1)

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Renal R

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Brachial L

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Abd. aorta

Com. Iliac L

Ext. Iliac L

Femoral L

Tibial L

Flow and pressure atdistal locations

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1 to 2 3.93

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6 to 7 7.87

7 to 8 8.29CF-PWV

Pulse wave velocity

8.81

0 0.5 1

80

120

160

0

15

30

Figure 6. Pressure and flow waves at multiple sites in a full-body scale FSI model built from computed tomography. Table shows pulsewave velocity values obtained by measuring the time between the “foot” of the pressure waves at two sites. The calculated carotid-to-femoralpressure wave velocity (CF-PWV) is representative of typical values found in experimental measurements. (Reproduced with permissionfrom Xiao et al., 2013. © Elsevier, 2013.)

12 Blood Flow

from both human (Xiao et al., 2013) and animal image data(Cuomo et al., 2015). Figure 6 shows 3D hemodynamicsin a full-body vascular model consisting of the 84 largestarteries in the human. The model has spatially varying tissueproperties that include higher compliance in the ascendingaorta and smaller compliance in distal aorta and peripheralvessels. The model reproduces accurately regionally flowdistributions, pressure and flow waveforms, and pulse wavevelocity.

3.5 Multi-scale modeling

Because an image-based model is restricted to a certainregion of the vascular anatomy and limited by image reso-lution to large and mid-sized vessels, it is crucial to employmultiscale modeling methods that capture organ-levelresponses of the circulation as well as resistance, impedance,and wave propagation effects of the distal vascular and capil-lary beds. In the following paragraphs, we describe recentdevelopments toward modeling methods that capture theseeffects, primarily through the use of closed-loop lumpedparameter networks and heart models.

The choice of BCs is of paramount importance in cardio-vascular simulations, as the local flow dynamics are greatlyinfluenced by conditions upstream and downstream of the3D model. Numerous studies have demonstrated drasticdifferences in flow solutions with different BC choices,even with simple geometries, and particularly in modelswith multiple outlets (Vignon-Clementel et al., 2006,2010). Commonly used outlet BCs such as zero pressure orzero traction, while easiest to implement, are well knownto lead to unrealistic solutions, in part because of theirinability to capture physiologic levels of pressure. Thesemethods should not be used for FSI problems where thewall deformation depends directly on the pressure levelin the vessel. Vascular resistance in arterioles and capil-laries is largely responsible for determining blood pressurelevels in large arteries. The same vascular resistances arealso responsible for regulating the distribution of bloodflow to different regions of the body. A fluid dynamicsimulation of large arteries, without consideration of thesmaller downstream vessels, neglects these importanteffects.

A typical choice of inlet BC is to impose a prescribed pres-sure or flow waveform. Typical choices for outlets are zeropressure or zero traction conditions, resistance or impedanceconditions, reduced order models which can be open orclosed loop, or reduced order 1D wave propagation equations(Vignon-Clementel et al., 2006, 2010; Formaggia et al.,2003). A closed-loop approach can also be taken, in whichall boundaries of the 3D model are coupled to a lumped

parameter network. These BCs are imposed via a Dirichletcondition

u = g, x ∈ Γg (14)

or a Neumann condition

T ⋅ n = h, x ∈ Γh (15)

in which Γg and Γh denote Dirichlet and Neumann bound-aries, respectively.

Note that g and h are a prescribed function of x and t foruncoupled BCs (e.g., zero traction), whereas they are also afunction of u and p for complex reduced order models (e.g.,closed-loop heart models).

Here, we discuss BCs coming from closed-loop lumpedparameter networks. To capture the interaction betweenthe local 3D domain and the global circulation, the 3DNavier–Stokes solver must be coupled to a reduced orderLPN model. In a closed-loop configuration, the 3D domaininflow is extracted from the 0D domain solution, incor-porating the responses of the heart and global physiologyto altered flow conditions at the 3D–0D interface. Ina closed-loop scenario, the 3D domain presents itselfto the 0D model as a set of time-varying resistance,inductance, and capacitance (in the case of deformablewall simulation), which are functions of the 3D domainhemodynamics. The 0D model behavior is affected bythis 3D model behavior and thus provides BCs that arefully coupled to the 3D domain, forming a completefeedback loop.

Closed-loop multiscale models have been particularlyuseful in modeling coronary flows, as well as complexsurgeries in pediatric cardiology applications (Migliavaccaet al., 2003; Kung et al., 2013; Sankaran et al., 2012).For example, the effects of a surgical shunt in singleventricle patients can be investigated using such a model,which enables considerations of how the shunt resistanceinfluences the balance of blood flow through various path-ways in light of systemic resistances and heart behavior(Esmaily-Moghadam et al., 2012). After defining thephysics of the 3D and reduced-order domains, the coupledsystem must be solved by a numerical method that is bothstable and modular. A summary of recent methods forstable and efficient coupling can be found in Marsden andEsmaily-Moghadam (2015).

Depending on the complexity and order of the ODEnetwork governing the LPN, one may use either a mono-lithic or a coupled approach for solution of the 0D–3Dcoupled system. A monolithic approach can be used whenthe analytical solution of the ODE of 0D model is knownand can be implemented directly in the numerical solver

Blood Flow 13

and the entire problem solved simultaneously. The mono-lithic approach is now widely accepted with details describedin foundational work by Vignon-Clementel et al. (2006)and is best suited to simple models, at most containing fewtime-dependent components, such as resistance and RCRcircuits.

For more complex circuit networks, obtaining the analyt-ical solution of a high-order nonlinear ODE is not practical,and the system must be solved numerically to obtain theDirichlet-to-Neumann relationship. Use of the monolithicapproach is no longer practical in this case, and parti-tioned approaches have been introduced by several groupsEsmaily-Moghadam et al. (2013b), Ismail et al. (2013), andKuprat et al. (2013). The key concept of the partitionedapproach is to have a separate solver for the 0D domainthat is coupled to the 3D solver through a well-definedinterface. Numerical solution of the ODEs in the 0Dsolver, for example, with a Runge–Kutta time-steppingscheme, enables simulation of circuits with multipletime-dependent and nonlinear components and multipleorgan blocks.

Recent studies have also addressed issues of numericaldivergence due to back flow, as well as specialized precon-ditioners (Esmaily-Moghadam et al., 2011, 2013a). Whilerecent work has largely focused on 0D LPN models as thechoice of reduced-order model, we note that multiscalemodeling methods can be extended to 1D models byreplacing the ODE governing the 0D model with partialdifferential equations governing the 1D model, whilekeeping the coupling scheme unchanged.

4 NOVEL COMPUTATIONAL TOOLS

4.1 Tissue growth and remodeling

Blood vessels exhibit remarkable ability to adapt throughoutlife adaptations in development and aging; to injury suchas in wound healing and vasospasm; in disease, such asatherosclerosis and aneurysms; and lastly, adaptations tochronic increase in flow in endurance training, or to hyper-tension. These adaptations result in changes in vessel shapeand material properties and depend on an array of factorssuch as gene pathways, biochemical processes in the vesselwall, and, as numerous findings over the past few decadesindicate, the mechanical environment of the vessel. Manyobservations at the subcellular, cellular, and even cell matrixlevels point to the existence of a “preferred” or “home-ostatic” mechanical state across multiple space and timescales. Thus, it is thought that vascular G&R happens inresponse to a significant alteration of the homeostatic stateof a vessel (Humphrey, 2008). Changes in the shape and

mechanical properties of blood vessels are ultimately relatedto changes in its constituents: resident cells (smooth muscleand endothelial cells) and the extracellular matrix. The mainremodeling agents in the vessel wall are the endothelial cells,vascular smooth muscle cells, and fibroblasts.

From a computational standpoint, two main methodologieshave been proposed to study cardiovascular tissue G&R: akinematic growth approach (Taber, 1998; Rachev, 2000) anda constrained mixture approach (Humphrey and Rajagopal,2002; Watton et al., 2004; Kuhl and Holzapfel, 2007). Kine-matic growth formulations model the G&R without repre-senting the underlying biochemomechanical mechanismsmodulating growth. On the other hand, constrained mixturemodels incorporate the response of smooth muscle andcollagen fibers to changes in mechanical loading (or poten-tially other biochemical stimuli). Each constituent can beproduced and removed over time. The total mass of themixture M(𝜏) at any given time 𝜏 ∈ [0, s] is given by abalance of production and survival functions (Baek et al.,2006):

M(s) =∑

i

Mi(s) = 1J(s)

∑i

{Mi(0)Qi(s)

+∫s

0mi(𝜏)qi(s − 𝜏)J(𝜏)d𝜏

}(16)

where J(𝜏) is the determinant of the deformation gradientF(𝜏), Qi(s) the fraction of the ith constituent that waspresent at time 0 and still remains at time s, mi(𝜏) thetrue rate of production of the ith constituent at time 𝜏 perunit area, and qi(s − 𝜏) its survival function, that is, thefraction of constituent i produced at time 𝜏 that remains attime s. The mass fractions of each constituent i 𝜙i(s) (i =elastin, collagen1, collagen2, etc.) are given by

𝜙i(s) = Mi(s)

M(s)such that

∑i

𝜙i(s) = 1 ∀ s (17)

For an elastic body, the Cauchy stress T for the mixture canbe given by sum of passive and active strain energy functioncontributions:

T(s) = 2J(s)

F(s)𝜕wR(s)𝜕C(s)

F(s)T + T(act)(s) (18)

where C = FTF. The passive contribution to strain energy ofthe mixture per unit reference area wR is defined as the sum ofthe energy stored in the elastin-dominated amorphous matrixand the multiple families of collagen, namely

wR(s) = weR(s) +∑

k

wkR(s) (19)

14 Blood Flow

The strain energy for the elastin-dominated matrix is typi-cally assumed to be neo-Hookean. For collagen, the form ofthe strain energy function is assumed identical for all fiberfamilies and is typically given by an exponential Holzapfelform (Holzapfel et al., 2000). This is an illustrative exampleof a linearized form for the rate of mass production (Figueroaet al., 2009):

mk(s) = M(s)M(0)

(k𝜎(𝜎k(s) − 𝜎h) + k𝜏(𝜏w(s) − 𝜏hw) + mkbasal)(20)

where k𝜎 and k𝜏 are scalar parameters that control thestress-mediated G&R and mkbasal is a basal rate of massproduction for the kth fiber family. At the homeostatic state,therefore, the production rate follows the basal rate.

In Figueroa et al. (2009), Humphrey, Figueroa andcolleagues developed a “fluid-solid-growth” (FSG) formu-lation in which the hemodynamic forces acting during thecardiac cycle pin an FSI simulation provide the loads for thevascular G&R formulation, defined on a timescale of weeksto months, which in turn provides the updated geometry andmaterial properties for the hemodynamic simulation. ThisFSG framework was used to study the long-term evolutionof a basilar artery aneurysm under different mechanicalstimuli.

4.2 Parameter estimation and uncertaintyquantification

4.2.1 Parameter estimation via filtering techniques

A primary challenge in constructing truly patient-specifichemodynamic models is the calibration of numerousparameters required to define both the mechanical propertiesof the vascular model (e.g., distribution of stiffness andperivascular tissue support) and in the inflow and outflowBCs so that the numerical predictions are consistent withclinical data. For 3D computational models, the high costof a single forward simulation necessitates an efficientparameter estimation strategy that minimizes the number ofmodel evaluations.

While information on flow and wall motion can be readilyobtained from magnetic resonance imaging and computedtomography, reliable information on pressure can only beobtained via invasive catheterization. Lack of pressure datathus hinders knowledge of vascular stiffness and resistance.Typically, estimation of stiffness and resistance is done iter-atively using traditional optimization approaches. This stepis the most computationally expensive.

Traditional approaches for parameter estimation relied ondefining a cost function constructed from the differencebetween data and simulation results, and performing classic

optimization. This approach is quite expensive and is notparticularly well suited for time-dependent problems.

Recently, data assimilation strategies based on Kalmanfilter theory (Julier et al., 2000), in particular reduced-orderunscented Kalman filter (ROUKF) formulations, have beenused to estimate parameters in a number of cardiovas-cular and cardiac electrophysiology applications (Chapelleet al., 2013; Marchesseau et al., 2013; Xi et al., 2011;Moireau et al., 2012) in patient-specific vascular models.Unlike classic optimization, the Kalman filter is a sequentialapproach well suited to handle dynamical systems in whichtime-resolved measurements (data) are available. Given anevolution equation

Ẋ = A(X) (21)

where X is the state and A a model operator (e.g.,Navier–Stokes), the Kalman filter adds a correctionK(Z − H(X)), where Z − H(X) is the difference between ameasurement Z and a model prediction H(X) (i.e., an inno-vation), and K is a gain operator. In addition to parameterestimation, the Kalman filter has perhaps a more interestingapplication in model refinement: indeed, the evolutionequation

Ẋ = A(X) + K(Z − H(X)) (22)

can be seen as a balance between model (which mightbe fundamentally inadequate) and gain-regulated data. Thisframework enables the assessment of the suitability of agiven model by identifying situations in which the estimatedmodel parameters do not converge to steady values.

These techniques, albeit expensive (e.g., in the ROUKF,one must run p + 1 forward problem simulations to estimatep parameters), could make the process of parameter esti-mation completely automatic and operator independent, ahighly critical feature in order to facilitate clinical translationof computational tools.

4.2.2 Uncertainty quantification

The adoption of simulation tools to predict surgical outcomesis increasingly leading to questions about the variability ofthese predictions in the presence of uncertainty associatedwith the input clinical data. A pitfall of current simulationmethods is that they often fail to acknowledge or quantifythe numerous sources of uncertainty involved in the clinicaldata assimilation and modeling process. As simulation dataare increasingly incorporated into disease research, clinicaltrials, and the FDA approval process, there is a pressingneed to establish strict guidelines for assessing the impactof uncertainty on simulation predictions.

A typical CV simulation is a multistep process consistingof 3D model reconstruction from medical image data, mesh

Blood Flow 15

generation, tuning BCs to match physiologic and clinicaldata, flow simulation, and feedback to clinicians. Thisprocess involves numerous sources of uncertainty. Uncer-tainties stem from assimilation of clinical measurements(heart rate and blood pressure), echocardiography data(stroke volume, ejection fraction, and cardiac output), phasecontrast and other MRI data, and cardiac catheterizationdata. Additional uncertainties stem from vessel-wall materialproperties and physiologic data. Noise, artifacts, and limitedtemporal and spatial resolution in CT and MRI image dataalso lead to uncertainty in the image segmentation process.These input uncertainties propagate nonlinearly through themodeling process, resulting in predictions that should bereported with associated statistics and confidence intervalson a patient-specific basis.

Solver performance is of utmost importance whenperforming optimization and UQ, both of which requirerunning numerous simulations either in parallel or in succes-sion. As CV simulations typically require multiple hoursof run time on a large parallel cluster, Monte Carlo-likestrategies quickly become computationally intractablefor large problems. Maintaining tractability thus requiresboth optimized solver performance, for example, usingspecialized coupling and preconditioning methods, andefficient algorithms to limit the required number of functionevaluations for convergence.

Recent work has applied stochastic collocation for UQ incomputational fluid dynamics (CFD), providing probabilitydensity functions (PDFs) and confidence intervals on simula-tion outputs (Ghanem and Spanos, 1991; Xiu and Hesthaven,2005; Babuška et al., 2007). Collocation methods generally

offer substantially better convergence than traditional MonteCarlo methods, are nonintrusive to implement, and are highlyparallelizable. In the collocation scheme, stochastic space isapproximated using mutually orthogonal interpolating func-tions and can then be queried at any point and PDF’s can beconstructed. This method is specifically designed for UQ inlarge-scale simulations, such as CFD simulations in complexgeometries. Efficient application of UQ with stochasticcollocation using sparse grids has been demonstrated foridealized and patient-specific cardiovascular models usinga limited number of stochastic parameters (Sankaran andMarsden, 2010, 2011; Sankaran et al., 2010). These studiesemployed Smolyak sparse grids to handle parametric uncer-tainty and quantify predictive uncertainty in terms of PDFsand confidence intervals (Babuška et al., 2007; Xiu andHesthaven, 2005; Ghanem and Kruger, 1996; Najm, 2009).

Recent work has also introduced methodologies forfull propagation of uncertainty from clinical data tovirtual surgery predictions in single-ventricle palliation,as illustrated in Figure 7. A recent study characterized thepreoperative clinical uncertainty related to indirect wedgecatheter measurements of mean pulmonary artery pressureand MRI-derived right pulmonary flow split. InverseBayesian estimation was then used to infer the distributionsof boundary resistance modes, leading to the expected clin-ical PDFs and used stochastic collocation on sparse grids topropagate these BC distributions to the postoperative results.The effect of relieving the left pulmonary artery stenosis wasalso quantified both in terms of change in average resultsand associated confidence intervals (Schiavazzi et al., 2015).

0.02.04.06.08.0

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Pre op modelPost op model

Puls no stenosis modelPuls with stenosis model

0.00.10.20.30.40.5

6 7 8 9 10 11 12 13 14 15 16

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Average pulmonary pressure (mm Hg)

Pre op modelPost op model

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F

Mean RPA to LPA pressure drop (mm Hg)(a) (b)

Pre op modelPost op model

Puls no stenosis modelPuls with stenosis model

Pre-op Post-op Puls withstenosis

Puls nostenosis

Figure 7. (a) Distribution of identified pre- and predicted postoperative clinical indicators for Stage II single-ventricle palliation surgery.(b) Standard deviation of local velocity results for pre- and postoperative models.

16 Blood Flow

Figure 7 shows how the first two statistical moments ofclinical quantities of interest have been affected by surgery,showing that postoperative flow split following surgicalremoval of left pulmonary stenosis becomes more difficultto predict due to increased sensitivity to variation in theBCs. Conversely, estimation of the pressure drop becomesmore reliable. Figure 7 also shows the spatial distributionof variance in the velocity field for pre- and postoperativemodels. These methods can also be used to target-specificareas of data collection for model improvement. Recently,methods combining multiresolution representations andsparsity-promoting recovery algorithms have shown promisefor reducing the number of model evaluations needed tocompute accurate response statistics for multiscale CVmodels (Schiavazzi et al., 2014).

4.3 Optimization

Coupling optimization to cardiovascular simulations haspotential to improve surgical designs and enable individ-ualized treatment planning. Optimization methods werefirst introduced in the field of cardiovascular modelingprimarily in two-dimensional and steady flow problems(Agoshkov et al., 2006a,b; Abraham et al., 2005; Quar-teroni and Rozza, 2003) and have recently been expandedto complex, three-dimensional, unsteady simulations(Esmaily-Moghadam et al., 2012; Yang et al., 2010).The coupling of optimization algorithms to blood flowsimulations is particularly challenging because each costfunction evaluation requires an unsteady, 3D solution ofthe Navier–Stokes equations on multiple processors. Theseevaluations are computationally expensive, and gradientinformation is often difficult to obtain.

Shape optimization can be used for virtual surgery designin which a patient-specific model is optimized to mini-mize a disease-related cost function derived from the flowfield. Examples of surgery optimization include identifyingoptimal angles and radii for bypass grafts and finding theoptimal shape of complex surgical connections for singleventricle heart patients (Yang et al., 2013; Sankaran andMarsden, 2010). Optimization can also be used in the designof medical devices, including stents, ventricular assistdevices, and coil placement for cerebral aneurysms (Longet al., 2013; Gundert et al., 2012a,b). Outside of shapeoptimization, one can also apply optimization in the settingof parameter identification for cardiovascular modeling. Forexample, optimization can be used to identify parameters incardiovascular G&R or in determining material propertiesfrom medical image data for FSI (Sankaran et al., 2013).

When choosing an optimization method, the primarydistinction is between gradient-based and derivative-free

methods. Factors contributing to this choice include theavailability of cost function gradients, computational costof the function evaluations, the level of noise and discon-tinuities in the function, complexity of implementation,the number of design parameters, convergence properties,efficiency, and scalability.

A general optimization problem may be formulated withlinear bound constraints as follows:

minimize J(x)

subject to x ∈ Ω (23)

where J ∶ ℝn → ℝ is the cost function and x the vectorof design parameters. The parameter space is defined byΩ = {x ∈ ℝn|l ≤ x ≤ u}, where l ∈ ℝn is a vector of lowerbounds on x and u ∈ ℝn a vector of upper bounds on x.In a cardiovascular shape optimization problem, the func-tion J(x) depends on the solution of the Navier–Stokesequations, and the cost function value is computed in a post-processing step.

Derivative-free algorithms such as the surrogate manage-ment framework (SMF) have shown particular promise incardiovasclar blood flow simulation due to their flexibleimplementation and ability to search globally in the param-eter space. The main idea behind the SMF method is toincrease efficiency using a surrogate function to “stand in”for an expensive function evaluation, while also benefitingfrom the convergence properties of pattern search methods(Audet and Dennis, 2004, 2006; Audet, 2004).

The SMF algorithm typically consists of a SEARCH step,employing a Kriging surrogate function for improved effi-ciency, together with a POLL step to guarantee convergenceto a local minimum (Lophaven et al., 2002; Simpson et al.,1998). The exploratory SEARCH step uses the surrogate toselect points that are likely to improve the cost function butis not strictly required for convergence. Convergence is guar-anteed by the POLL step, in which points neighboring thecurrent best point on the mesh are evaluated in a positivespanning set of directions to check if the current best point isa mesh local optimizer.

All points evaluated in either the SEARCH or POLL step ofthe SMF algorithm must lie on a mesh in the parameter space.The vectors defining the mesh directions must positively spanℝn (Lewis and Torczon, 1996). If we define D as a matrixwhose columns form a positive spanning set in ℝn, then theset of mesh points surrounding a point x are given by

M(x,Δ) = {x + ΔDz ∶ z ∈ ℕnD} (24)

where Δ is the mesh size parameter and nD the numberof columns in D. A positive spanning set is simply the set

Blood Flow 17

of positive linear combinations of the vectors making upthe mesh directions (Davis, 1954). A set of n + 1 POLLpoints are required to generate a positive basis, where nis the number of optimization parameters. Following theabovementioned definition, the mesh in SMF may be refinedor coarsened by changing Δ > 0, and the mesh may berotated from one iteration to the next (Torczon, 1997).

Convergence is reached when a local minimizer on themesh is found, and the mesh has been refined to the desiredaccuracy. Each time new data points are found in a SEARCHor POLL step, the data is added to the surrogate and it isupdated. The steps in the algorithm are summarized below,where the set of points in the initial mesh is M0, the mesh atiteration k is Mk, and the current best point is xk.

1. SEARCH(a) Identify a finite set Tk of trial points on the mesh Mk.(b) Evaluate J(z) for all trial points z ∈ Tk ⊂ Mk.(c) If for any trial point in Tk, J(z) < J(xk), a lower cost

function value has been found, and the SEARCH issuccessful. Increment k and go back to (a).

(d) Else, if no trial point in Tk improves the cost func-tion, SEARCH is unsuccessful. Increment k and goto POLL.

2. POLL(a) Choose a set of positive spanning directions, and

form the poll set Xk as the set of mesh points adjacentto xk in these directions.

(b) If J(xpoll) < J(xk) for any point xpoll ∈ Xk, then alower cost function has been found and the POLLis successful. Increment k and go to SEARCH.

(c) Else, if no point in Xk improves the cost function,POLL is unsuccessful.(i) If convergence criteria are satisfied, a converged

solution has been found. STOP.(ii) Else if convergence criteria are not met, refine

mesh. Increment k and go to SEARCH.

Because the method has distinct SEARCH and POLL steps,convergence theory for the SMF method reduces to conver-gence of pattern search methods. We refer the reader tothe extensive literature summarizing the relevant mathemat-ical convergence theory (Booker et al., 1999; Serafini, 1998;Audet and Dennis, 2003; Torczon, 1997; Lewis and Torczon,1999, 2000, 2002).

In the context of blood flow simulation, optimization hasbeen used in the design of bypass grafts (Sankaran andMarsden, 2010) as well as surgeries for single ventricleheart patients. Optimization has uncovered links among graftposition, cardiac output, and coronary oxygen delivery inthe first stage of single ventricle repair, the Blalock–Taussig(BT) shunt (Esmaily-Moghadam et al., 2012). In this chapter,

optimization was performed on a 3D model of the aorta andpulmonary arteries, using the closed-loop-lumped parameternetwork discussed earlier. The graft geometry was parame-terized and implanted into the model using automated scripts,and the 3D unsteady Navier–Stokes equations were solvedin each function evaluation on a parallel architecture. Atypical example of the implementation of shape optimizationis shown in Figure 8.

5 CLINICAL APPLICATIONS

5.1 Pediatric cardiology

Multiscale modeling is of particular importance in pediatriccardiology due to the strong interactions between surgicalmethods and global physiology of the patient, includingcardiac performance. Single ventricle heart defects are aparticularly challenging class of congenital heart defects andpresent a complex physiology requiring three open-chestsurgeries starting in the neonatal period and culminatingwith the Fontan surgery (Fontan and Baudet, 1971). Firstapplied to single ventricle physiology by Dubini, Migli-avacca, and Pennati, multiscale modeling extends standardsimulation tools to predict not only local hemodynamics suchas energy loss but also global quantities of clinical interestincluding ventricular work load, pressure volume loops,systemic and pulmonary pressure levels, and oxygen delivery(Leval and Dubini, 1996; Dubini et al., 1996; Lagana et al.,2002).

Early work on modeling single ventricle physiologyfocused on power loss as the primary metric of performancefor comparing surgical geometries. Initial simulation studiesdemonstrated improved energy efficiency in Fontan surgicalgeometries with an offset between the superior and inferiorvena cava (SVC and IVC), leading to widespread adoptionof the offset concept in the surgical community (Migliavaccaet al., 2003; Dubini et al., 1996). Subsequent studies wenton to establish links between the geometry of the Fontanjunction and hemodynamic performance (Migliavacca et al.,2003; Petrossian et al., 2006; Marsden et al., 2007; White-head et al., 2007; DeGroff, 2008; Soerensen et al., 2007).Some have also suggested that high power loss is likelylinked to poor clinical outcomes and high ventricular work-loads and that energy reduction is of primary importancein surgical design (Ensley et al., 2000a; Healy et al., 2001;Ryu et al., 2001), though other studies have showed littleinfluence of power loss on clinically relevant parameters(Kung et al., 2013; Baretta et al., 2011).

There have been a number of important studies illustratingthe use of multiscale modeling and surgical planning inpediatric cardiology. Closed-loop models have been used

18 Blood Flow

Start

Poll

0D3D

Do a coupled simulation

Mesh the 3D model

Create the 3D model

set of parameters

Find the objective function

End

Refinement

Search

Optimization code

Min J(a) with SMF

Poll

Search

0.24

0 0

Anastomosis location BA Anastomosis

location PA

0.1 0.10.2 0.2

0.3 0.30.4

0.40.5

0.50.6

0.60.7

0.7

0.80.8

0.90.9

11

J 1

0.250.260.270.280.29

0.30.310.320.330.34

p1

p2

p3

xk

Outletanastomosis

angle

A

PA

cgs100

0

755025

Figure 8. Steps required for automated shape optimization using the surrogate management framework in an idealized model of theBlalock–Taussig shunt surgery to treat single ventricle patients: (1) geometry parameterization, (2) meshing, (3) flow simulation, and (4)optimization algorithm to produce new design parameter set.

Model construction from imaging data

Virtual surgery Pulmonary stenosis investigation

Figure 9. Modeling process for virtual surgery, beginning with model construction from imaging data followed by virtual surgery (a) andthe patient-specific stage one models for two patients (b). The example shown compares the hemi-Fontan and Glenn surgeries in singleventricle palliation, as well as surgical correction of pulmonary stenosis.

to devise a protocol for simulating exercise physiology(Kung et al., 2014) and to compare surgical versus hybridapproaches for stage-one single ventricle palliation (Corsiniet al., 2011; Hsia et al., 2011). Mock circuits, with hydraulicelements and following the same principles in the in vitrosetting, have also been devised to validate simulations andtest surgeries and devices (Vukicevic et al., 2013).

Multiscale modeling is applied to compare competingsurgical approaches (Glenn vs hemi-Fontan) for thesecond-stage surgery in single ventricle repair in Figure 9(Kung et al., 2013). Multiple surgical options can then becompared in a well-controlled “virtual experiment” sinceparameter values remain fixed when comparing differentsurgical geometries and stenosis levels. A key point here

Blood Flow 19

is that since one prescribes circuit element values ratherthan directly prescribing flow and pressure, the closed-loopmodel allows for dynamic adjustment of flow and pressurewaveforms resulting from changes in geometry without theneed to assume static BCs.

5.2 Coronary artery disease

Coronary physiology is challenging to model because coro-nary flow is out of phase with aortic flow due to ventric-ular contraction (Hoffman and Spann, 1990). ConventionalBCs such as resistance or Windkessel (RCR) models do notcapture physiologic coronary waveforms, and many studieshave been forced to rely on assumed pressure waveforms orinvasively obtained data. The challenge of capturing coro-nary physiology has been addressed through several differentapproaches using lumped parameter networks, 1D models,and porosity models. Kim et al. introduced specialized coro-nary lumped parameter BCs in the context of patient-specificmodeling (Kim et al., 2010b,c) in which coronary outletBCs were constructed to account for ventricular contractionthrough incorporation of the myocardial pressure. An alter-nate approach is to incorporate the effects of cardiac contrac-tion through a vascular varying elastance model (KRAMSet al., 1989). In this approach, the increased stiffness of themyocardium during systole is postulated to drive the coro-nary flow by modulating the compliance and resistance ofthe embedded vessels.

The approach of Kim et al. (2010b,c) was extended bySankaran et al. (2012) to model the coronary circulation ina closed loop using the coupling methods reviewed above.This study reproduced the clinically observed diastoliccoronary flow peak and the postoperative increases in coro-nary perfusion, based on noninvasive preoperative clinicaldata. Bypass graft geometry affected local WSS, wall shearstress gradients (WSSGs), and oscillatory shear index (OSI),and results showed that higher anastomosis angles weremore optimal than lower angles, as hypothesised in earlierhuman in vivo studies. Coronary simulations have alsobeen applied recently in the study of aneurysms caused byKawasaki disease, illustrating the potential of simulations toimprove risk stratification for determining appropriate anti-coagulation therapy and treatment (Sengupta et al., 2012).Recent work has also uncovered significant differencesin hemodynamic and biomechanical parameters betweenarterial and venous grafts in patient-specific simulationsfollowing coronary artery bypass graft (CABG) surgery(Ramachandra et al., 2016). These findings may have impor-tant implications for mitigating vein graft failure post CABGsurgery. Figure 10 illustrates the use of a closed-loop LPNmodel of the coronary circulation, with contours of WSS

obtained from FSI simulations of a patient following CABGsurgery. Noninvasive patient-specific CFD studies of thecoronary circulation using MRI were recently performed(Torii et al., 2010), including dynamic motion using an ALEformulation and proximal velocity waveforms acquiredby MRI.

Recent studies have introduced a new noninvasive methodfor FFR predictions using patient-specific simulations(Taylor et al., 2013), with promising comparisons betweensimulated and clinical data reported in recent clinical trials(Nakazato et al., 2012; Koo et al., 2011; Min et al., 2012;Nørgaard et al., 2014; Miyoshi et al., 2015). These methodshave been demonstrated to significantly improve the diag-nosis of coronary artery disease and have the potential forsignificant reductions in the cost of care (Douglas et al.,2015; Hlatky et al., 2013).

Comprehensive multiscale models of the coronary circula-tion have also been provided by Smith and others, primarilyusing a coupled 1D modeling approach (Smith, 2004).Pulsatile flow in a distributed 1D coronary network isdescribed, in which regional myocardial stresses wereapplied on the vessels, calculated from an anisotropic finitedeformation model of the beating heart (Smith, 2004).Coronary models have recently been extended to modelthe interaction between coronary perfusion and myocar-dial mechanics using poroelasticity (Vankan et al., 1997;Cookson et al., 2012). Additional details on coronarymultiscale modeling can be found in the excellent review ofLee and Smith (2012).

5.3 Transitional hemodynamic stages

Virtually, all blood flow simulation work to-date has beendone under the assumption of fixed, cycle-to-cycle periodicpulsatile conditions. However, in order to provide adequateblood supply to different vascular territories and to keepblood pressure relatively constant under a wide rangeof different physiological conditions, such as changes inposture, digestion, stress, trauma, or exercise, the circula-tory system is equipped with several regulatory feedbackand feedforward mechanisms. Affecting local and globalproperties such as individual vessel tone and heart rate,these feedback mechanisms enable the regulation of pres-sure and flow throughout the body. Therefore, in orderto model changes in physiologic states, we must developcontrol system workflows in the multiscale simulationframework, in which error signals, response functions,and communication between different components of thesystem (typically represented via reduced-order models) areenabled.

20 Blood Flow

Coronary sub-circuits

Plv

Plv

Lungs

Rightventricle

Rightatrium

Leftventricle

Leftatrium

Figure 10. Example of a closed-loop lumped parameter network coupled to a patient-specific model of coronary artery bypass graft surgery(upper) and a simulated wall shear stress field (lower) (contours WSS magnitude min 0, max 15 dynes cm−2).

5.3.1 Models of global control – the baroreflex

One key regulatory mechanism is the arterial baroreflex – anegative feedback system that responds to short-term varia-tions in pressure by altering the state of the systemic circula-tion in order to maintain pressure homeostasis. The barore-flex can be broadly divided into three components: (i) thebaroreceptors cells, (ii) the vasomotor control center, and(iii) the sympathetic and parasympathetic nervous systems(Figure 11).

Connected to the vasomotor center of the brain via theafferent pathways, the baroreceptor cells modulate theirnervous firing rate when the magnitude of stretch deviatesfrom prior baseline values. Equipped with a memory of 1–2days, these cells are specialized for the role of short-termpressure regulation (Guyton, 1992). The vasomotor centerof the brain interprets the afferent nervous activity of thebaroreceptors. Altered afferent activity arising from vari-ations in pressure generates efferent activity within thevasomotor center that is transmitted to different anatomical

Blood Flow 21

Vasomotorcentre

Sympatheticpathway

Parasympatheticpathway

Arterial resistance

Heart rate and contraction

Venous compliance

Venous unstressed volume

Baroreceptors

Figure 11. Schematic of the main components of the baroreflex. Here, the filled square and filled circle symbols refer to the location of thecarotid and aortic baroreceptors, respectively (Lau and Figueroa, 2015). (Lau, https://link.springer.com/article/10.1007%2Fs10237-014-0645-x. Used under CC BY 4.0 https://creativecommons.org/licenses/by/4.0/.)

regions of the systemic circulation. This efferent activitytravels through two different pathways, known as the sympa-thetic response and the parasympathetic response, whichaim to restore blood pressure to baseline values. The sympa-thetic and parasympathetic nervous systems innervate theheart and the peripheral vessels, thereby allowing control ofheart rate, cardiac contractility, and vessel vasoconstriction.Increased sympathetic activity results in vasoconstriction ofthe peripheral vessels, increased heart rate, and increasedcardiac contractility factors that all have a restorative effecton blood pressure. Conversely, increased parasympatheticactivity decreases heart rate and cardiac contractility,thereby reducing systemic blood pressure (Guyton, 1992).Changes in sympathetic and parasympathetic activities occursimultaneously, for example, an increase in sympatheticactivity and a decrease in parasympathetic activity areeffected when an increase in blood pressure is desired. Viathe coordinated response of these activities, the baroreflexrapidly alters the hemodynamics throughout the systemiccirculation, exerting global control of the pressure on abeat-by-beat basis. The ability to transiently alter pressure

is clinically measured using an index referred to as thebaroreflex sensitivity (La Rovere et al., 2008). Definedas the change in peak systolic pressure over successivebeats, this sensitivity is a direct measure of the strengthof the baroreflex response and has been shown to be anindicator of mortality in diseased states (La Rovere et al.,2001). Several mathematical models of the baroreflex havebeen proposed thus far, with each examining differentaspects of this coupled system. So far, these approacheshave been mostly implemented using 0D (e.g., lumpedparameter network) models. Early 0D baroreflex modelsinclude those developed by Ottesen (1997) and (Ursinoet al. (1998); these two models explored the long-termstability of the negative feedback mechanism and the effec-tiveness of the baroreflex i