Optimization in Cardiovascular Modeling

FL46CH21-Marsden ARI 23 September 2013 10:30

RE V I E W

S

IN

AD V A

NC

E

Optimization in CardiovascularModelingAlison L. MarsdenDepartment of Mechanical and Aerospace Engineering, University of California,San Diego 92093; email: [email protected]

Annu. Rev. Fluid Mech. 2014. 46:519–46

The Annual Review of Fluid Mechanics is online atfluid.annualreviews.org

This article’s doi:10.1146/annurev-fluid-010313-141341

Copyright c© 2014 by Annual Reviews.All rights reserved

Keywords

cardiovascular fluid mechanics, hemodynamics, surrogates, pattern search,blood flow

Abstract

Fluid mechanics plays a key role in the development, progression, and treat-ment of cardiovascular disease. Advances in imaging methods and patient-specific modeling now reveal increasingly detailed information about bloodflow patterns in health and disease. Building on these tools, there is now anopportunity to couple blood flow simulation with optimization algorithmsto improve the design of surgeries and devices, incorporating more infor-mation about the flow physics in the design process to augment currentmedical knowledge. In doing so, a major challenge is the need for efficientoptimization tools that are appropriate for unsteady fluid mechanics prob-lems, particularly for the optimization of complex patient-specific modelsin the presence of uncertainty. This article reviews the state of the art inoptimization tools for virtual surgery, device design, and model parameteridentification in cardiovascular flow and mechanobiology applications. Inparticular, it reviews trade-offs between traditional gradient-based methodsand derivative-free approaches, as well as the need to incorporate uncertain-ties. Key future challenges are outlined, which extend to the incorporationof biological response and the customization of surgeries and devices forindividual patients.

519

Review in Advance first posted online on October 2, 2013. (Changes may still occur before final publication online and in print.)

Changes may still occur before final publication online and in print

Ann

u. R

ev. F

luid

Mec

h. 2

014.

46. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

by W

IB62

42 -

Uni

vers

itaet

s- u

nd L

ande

sbib

lioth

ek D

uess

eldo

rf o

n 10

/08/

13. F

or p

erso

nal u

se o

nly.


Thrombosis: theformation of a bloodclot inside a bloodvessel, obstructingblood flow through thecirculatory system

1. INTRODUCTION

The introduction of new cardiovascular surgical techniques and medical devices has traditionallyrelied on a build-and-test approach, using clinical trials, surgeon experience, and retrospectiveevaluation of patient outcomes to judge success. Although impressive advances have been madein medical imaging, providing increasingly detailed anatomy and, in some cases, time-resolvedflow information, imaging on its own is not a predictive tool. Cardiovascular simulations canaugment imaging methods, systematically predicting surgical outcomes and device performanceat little or no risk to the patient, and on a patient-specific basis. Although the medical field hasreadily embraced the customization of orthopedic devices, including knee and hip replacements,customization for cardiovascular applications is not yet routine. Optimization in conjunction withpatient-specific modeling will enable customized treatment planning for individual patients andaccelerate the acceptance of these tools for clinical use.

Optimization is the identification of the best-available value of an objective function in a defineddomain, given a set of parameters and constraints. In the setting of cardiovascular disease, opti-mization may be applied to improve surgical techniques and design medical devices or for modelparameter identification. Shape optimization is performed to identify surgical geometries such asbypass grafts leading to favorable hemodynamic conditions, for example, by decreasing cardiacwork load, reducing areas of flow stagnation, improving flow distribution, or increasing oxygendelivery. Considerations for device optimization, such as stents and blood pumps, may also includeoutput, efficiency, reliability, and minimization of blood damage. In all cases, perhaps the mostchallenging aspect of applying optimization to a mechanobiological system is the identificationof appropriate cost functions to measure performance. In addition to shape optimization appli-cations, parameter identification in cardiovascular models is paramount for improving modelingmethods. Optimization coupled to cardiovascular simulation offers a powerful tool for systematictesting of new surgical and device designs, as well as improvements to modeling methods.

Numerical simulation methodology for cardiovascular disease is increasingly sophisticated(Taylor et al. 1998, Ku et al. 2002, Lagana et al. 2002, Steinman 2002, Taylor et al. 2002, Chenget al. 2004, Figueroa et al. 2006, Tang et al. 2006, Vignon-Clementel et al. 2006, Zhang et al.2007, Bove et al. 2008, Shadden & Taylor 2008, Bazilevs et al. 2009). Models are typically con-structed from magnetic resonance imaging (MRI) or computed tomography (CT) data, includingdetailed anatomy with multiple vessel bifurcations and outlets. Boundary conditions for simula-tions are often derived from two-dimensional (2D) or 4D phase-contrast MRI measurements offlow. Outflow boundary conditions are particularly challenging, and recent advances in multi-scale lumped-parameter modeling, 1D wave propagation equations, and numerical stability haveincreased physiological realism. FSI simulations provide wall deformations, stresses, and strainsthrough solving a coupled fluid-solid problem.

The above advances have provided new insights into disease processes in applications such asabdominal aortic aneurysms, cerebral aneurysms, multiscale modeling of single-ventricle heartpatients, coronary aneurysms in Kawasaki disease, bypass grafts, coarctation of the aorta, andprediction of thrombosis (de Leval et al. 1988, Lei et al. 2001, Lagana et al. 2002, Castro et al.2006, Bove et al. 2008, Dasi et al. 2008, LaDisa et al. 2011). Device simulations have included theevaluation of wall shear stress (WSS) patterns in stents, ventricular assist devices, and pacemakerlead placement (LaDisa et al. 2004, Lonyai et al. 2010).

Coupling optimization to cardiovascular simulations has the potential to increase clinical utilityby improving current surgical designs and enabling customization for individual patients. How-ever, early work in surgery and device optimization was primarily limited to small-scale 2D and/orsteady-flow problems and a handful of trial-and-error designs (Abraham et al. 2005b; Agoshkov

520 Marsden


Ann

u. R

ev. F

luid

Mec

h. 2

014.

46. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

by W

IB62

42 -

Uni

vers

itaet

s- u

nd L

ande

sbib

lioth

ek D

uess

eldo

rf o

n 10

/08/

13. F

or p

erso

nal u

se o

nly.


Pattern search:a class of optimizationmethods that do notrequire gradients, withconvergence theorybased on theconstruction ofpositive bases

et al. 2006a,b; Quarteroni & Rozza 2003). The coupling of optimization algorithms to bloodflow simulations is particularly challenging because each cost function evaluation requires an un-steady, 3D solution of the Navier-Stokes equations on multiple processors. These evaluations arecomputationally expensive, and gradient information is often difficult to obtain.

In this review, I briefly outline current methods and challenges in cardiovascular simulation. Ithen present available methods for cardiovascular optimization, with a particular focus on a classof derivative-free pattern search optimization methods with well-developed convergence the-ory. Challenges are outlined in cost function identification, model parameterization, uncertaintyquantification (UQ), and the coupling of computational fluid dynamics (CFD) with optimizationalgorithms. Finally, I discuss the application of these methods in several cardiovascular applica-tions, including virtual surgery, parameter identification, and device design.

2. CURRENT METHODS FOR CARDIOVASCULAR SIMULATION

Blood flow is governed by the incompressible Navier-Stokes equations for conservation of massand momentum. Although blood is known to be a non-Newtonian shear-thinning fluid at smallscales, a Newtonian assumption generally holds in large vessels, and the reader is referred to otherrelevant discussions of non-Newtonian effects in blood flow (Haynes & Burton 1959, Gijsenet al. 1999). The numerical simulation of hemodynamics typically requires complex geometries,often with fluid-structure interaction (FSI), and time-dependent pulsatile solutions. For thesereasons, finite-element methods have been widely used in the cardiovascular literature because oftheir ability to easily handle complex geometries. Most recent work has used stabilized [SUPG(streamline upwind/Petrov-Galerkin)] methods with linear elements, but higher-order spectralelements have also been employed (Karniadakis & Sherwin 2005). Unstructured finite-volumemethods with conservative numerical schemes also offer an attractive alternative and have beenshown to more accurately capture cycle-to-cycle variations in unsteady flow in aneurysms (Valen-Sendstad et al. 2011).

2.1. Model Construction and Segmentation

The term patient-specific modeling refers to the construction of 3D models of vascular anatomyderived directly from patient image data. Depending on the problem, CT or MRI data are typ-ically used, although ultrasound and angiography data can provide an attractive alternative. Im-age data are typically segmented with 2D or 3D level set or thresholding methods using opensource packages such as SimVascular (http://simtk.org) (Schmidt et al. 2008) or ITK-SNAP(Yushkevich et al. 2006) or commercial packages such as Mimics (Materialise, Leuven, Bel-gium). Image-segmentation methods are a challenging component of the cardiovascular modelingprocess, often requiring extensive user intervention and manual segmentation when automatedmethods fail. Recent efforts to improve automated segmentation algorithms have focused onmachine learning algorithms and shape analysis. These improvements will be crucial to enablehigh-throughput processing of models for large patient cohorts, which is needed in the clinicalsetting to produce statistically significant results. Figure 1 shows the typical steps for model con-struction and simulation for a patient-specific model of the aorta and coronary arteries in a patientwith Kawasaki disease (Wilson et al. 2001, Sengupta et al. 2012).

2.2. Fluid-Structure Interaction

Recent advances in FSI include the coupled momentum method (Figueroa et al. 2006), whichenables efficient simulations within the limit of small deformations, and the arbitrary Lagrangian

www.annualreviews.org • Optimization in Cardiovascular Modeling 521


Ann

u. R

ev. F

luid

Mec

h. 2

014.

46. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

by W

IB62

42 -

Uni

vers

itaet

s- u

nd L

ande

sbib

lioth

ek D

uess

eldo

rf o

n 10

/08/

13. F

or p

erso

nal u

se o

nly.

http://simtk.org


Patient image data Custom 3D model Flow simulationAdapted mesh

a b c d

Figure 1Steps in the process of patient-specific modeling for a patient with coronary aneurysms caused by Kawasakidisease: (a) volume-rendered CT image data, (b) patient-specific model with aorta and coronary arteries,(c) adapted mesh, and (d ) contours of wall shear stress from simulation results.

Lumped-parameternetwork (LPN):a reduced-order modelmade up of circuitelements in whichcurrent and voltage areanalogous to flow andpressure

Eulerian (ALE) method, which allows for larger deformations and mesh motion (Bazilevs et al.2008, 2009). ALE methods are particularly attractive for membrane buckling, valves, and ventricu-lar mechanics but often incur higher computational cost. Immersed boundary methods, originallymotivated by the need for methods to handle cardiac fluid mechanics, are also widely used forventricular fluid mechanics and medical device simulations (Peskin 1977).

2.3. The Impact of Boundary Conditions

The choice of boundary conditions is of paramount importance in cardiovascular simulations.Prior studies have demonstrated drastic differences in solutions with different boundary conditionchoices (Vignon-Clementel et al. 2006, Balossino et al. 2009). The use of zero-pressure boundaryconditions, for example, can lead to unrealistic flow distributions in models with multiple outletsor inaccurate wall-deformation predictions in FSI simulations. Inflow boundary conditions typ-ically impose a velocity profile on the inflow face of the model. Although plug flow or parabolicprofiles are common, the most accurate choice in large vessels is generally the analytic solution ofWomerseley (1957) for pulsatile flow in a rigid or elastic tube (Zamir 2000).

In an open-loop configuration, one imposes an inlet flow or pressure waveform together withoutlet models of the distal vasculature. Outlet models can consist of resistance or impedance condi-tions, lumped-parameter networks (LPNs), or the 1D equations of blood flow. In lumped models,one makes an analogy to an electrical circuit and lumps the resistive, elastic, and inertial propertiesof blood flow through the vessels into electrical elements. One then solves the associated set ofordinary differential equations (ODEs) governing the electrical circuit. Simple circuit models suchas resistors, Windkessel (RCR), or coronary models can be solved analytically and implementedusing an implicit monolithic coupling approach (Vignon-Clementel et al. 2006).

In a closed-loop configuration (Figure 2), one typically uses an LPN to capture the interplaybetween the 3D domain and the dynamics of the circulation. The values of circuit elements aretuned to match physiologic and clinical values. The closed-loop configuration poses numericalchallenges because the ODE network must be solved numerically and coupled to the 3D domain.Iterative coupling can be performed using explicit methods, as implemented in prior work usingcommercial solvers, with the drawback that it can be time-step restricted. Recent advances in

522 Marsden


Ann

u. R

ev. F

luid

Mec

h. 2

014.

46. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

by W

IB62

42 -

Uni

vers

itaet

s- u

nd L

ande

sbib

lioth

ek D

uess

eldo

rf o

n 10

/08/

13. F

or p

erso

nal u

se o

nly.


UPPER BODY

ATRIUM VENTRICLE AORTA

INTESTINE

KIDNEYS

LIVER

LEGS

LUNGS

a

b

–10

0

0

10

20

30

1 2 3

Flo

w (m

L/s)

Time (s)

–10

0

0

10

20

30

1 2 3

Flo

w (m

L/s)

Time (s)

INF

ER

IOR

VE

NA

CA

VA

Figure 2Typical boundary condition specification for a patient-specific model of the Fontan surgery. (a) Open-loop boundary conditions withprescribed inflow and Windkessel (RCR) outflow conditions. (b) Multiscale closed-loop boundary conditions using a lumped-parameternetwork. Blocks representing different portions of the anatomy (e.g., heart, lungs) are shown.

coupling algorithms have produced implicit methods that are both modular and efficient, allowingfor the solution of more complex models and networks in reasonable time. Using this approach,one can treat inlets and outlets as either Neumann or Dirichlet boundaries (Esmaily Moghadamet al. 2013).

Another important issue is that of numerical instabilities due to backflow. Flow reversal occursnormally, for example, during diastole in the descending aorta. This issue also arises when enforc-ing a Neumann condition on an inflow face of a model, such as when the inflow is connected to anLPN. Although this is an often-undiscussed problem in cardiovascular simulation, backflow canlead to simulation divergence if boundaries are not properly treated. Methods to avoid divergenceinclude adding outlet extensions to dissipate flow structures before they exit the domain; enforcinga known velocity profile using a Lagrange multiplier constraint (Kim et al. 2009); and enforcingthat the velocity vectors be normal to the outflow face, which is a common option in commercialsolvers. Alternately, stable solutions can be achieved using outlet stabilization, in which a smalltraction force is added only on nodes where there is reversed flow, which has stabilized the flow



Ann

u. R

ev. F

luid

Mec

h. 2

014.

46. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

by W

IB62

42 -

Uni

vers

itaet

s- u

nd L

ande

sbib

lioth

ek D

uess

eldo

rf o

n 10

/08/

13. F

or p

erso

nal u

se o

nly.


with minimal intrusion into the flow physics and with no additional computational cost (EsmailyMoghadam et al. 2011).

3. COUPLING SIMULATIONS WITH OPTIMIZATION

Although optimization has been widely used in traditional disciplines such as aeronautical andautomotive design, its application to cardiovascular medicine is still relatively recent. Shapeoptimization can be used for virtual surgery design, in which a patient-specific model is optimizedto minimize a disease-related cost function derived from the flow field. Examples of surgeryoptimization include identifying optimal angles and radii for bypass grafts and finding the optimalshape of complex surgical connections for single-ventricle heart patients. Optimization can also beused in the design of medical devices, such as stents, ventricular assist devices, and coil placementfor cerebral aneurysms. Outside of shape optimization, one can also apply optimization in thesetting of parameter identification for cardiovascular modeling. For example, optimization canbe used to identify parameters in cardiovascular growth and remodeling (G&R) or to determinematerial properties from medical image data for FSI. Below key methodologies are outlined forcoupling optimization algorithms to cardiovascular simulations, and then examples are providedin each of these three application areas.

3.1. Optimization Methodologies

The primary distinction between optimization methods is whether they are gradient based orderivative-free. Factors contributing to this choice include the availability of cost function gradi-ents, the computational cost of the function evaluations, the level of noise and discontinuities inthe function, the complexity of implementation, the number of design parameters, convergenceproperties, efficiency, and scalability. In surgical shape optimization, each cost function evalua-tion requires a time-dependent, 3D solution of the Navier-Stokes equations. These problems arecomputationally expensive to evaluate, often requiring postprocessing steps.

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

minimize J(x)subject to x ∈ �,

(1)

where J : Rn → R is the cost function, and x is the vector of design parameters. The parameter

space is defined by � = {x ∈ Rn|l ≤ x ≤ u}, where l ∈ R

n is a vector of lower bounds on x,and u ∈ R

n is a vector of upper bounds on x. In a cardiovascular shape optimization problem,the function J(x) depends on the solution of the Navier-Stokes equations, and the cost functionvalue is computed in a postprocessing step.

Optimization routines are linked with the model construction and flow solver components viaautomated scripts. To perform a single cost function evaluation, one generally requires meshing,model generation, Navier-Stokes solution, and postprocessing. This makes the analytic determi-nation of gradients a formidable challenge without specialized numerical methods (e.g., differen-tiable mesh movement schemes). The issue of unstructured meshing and complex geometries inoptimization for steady-flow aerodynamics has been examined by Burgreen, Peraire, and others(Burgreen & Baysal 1996; Newman & Taylor 1996; Elliot & Peraire 1997, 1998).

3.1.1. Gradient-based optimization. When a gradient-based method is used, gradients aregenerally obtained numerically using adjoint solutions or finite-difference methods. Importantstrides have been made using adjoint solvers in the work of Jameson (1988; Jameson et al. 1998)

524 Marsden


Ann

u. R

ev. F

luid

Mec

h. 2

014.

46. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

by W

IB62

42 -

Uni

vers

itaet

s- u

nd L

ande

sbib

lioth

ek D

uess

eldo

rf o

n 10

/08/

13. F

or p

erso

nal u

se o

nly.


Hemolysis: damageto the red blood cellsresulting in ruptureand release of contentsinto surrounding fluid

for efficiently obtaining gradient information in aerodynamics problems. The more recent work ofWang and colleagues (2009) has shown promise in the application of adjoints to unsteady problems.However, challenges remain because of the need to store large time histories. In addition, theinclusion of postprocessing steps in the adjoint solution remains a challenge, particularly when theyrequire a secondary simulation, such as in residence time or hemolysis computations. Gradientsobtained directly using finite differences are often prohibitively expensive for large problems andcan be easily drowned out by numerical noise.

In spite of these challenges, there has been important progress in applying gradient-basedoptimization methods to cardiovascular problems, mostly with 2D geometries and/or steady flow.Non-Newtonian effects in shape optimization have been examined by Abraham et al. (2005a,b).The optimization of blood pump components has been carried out by Antaki and colleagues (1995;Burgreen et al. 2001), and Rozza and coworkers (Quarteroni & Rozza 2003, Rozza 2005, Agoshkovet al. 2006a,b) have applied optimization to shape design for arterial bypasses. However, it remainsa challenge to apply gradient methods in a fully time-dependent setting using unstructured solvers,complex geometries, and constraints, which are common in cardiovascular flow problems.

3.1.2. The surrogate management framework. Derivative-free optimization methods can of-fer a promising alternative to gradient-based methods. The surrogate management framework(SMF) (Serafini 1998, Booker et al. 1999) is a derivative-free pattern search optimization methodthat relies on surrogates for increased efficiency. Previous work successfully applied SMF to theconstrained optimization of an airfoil trailing edge for the suppression of vortex shedding noisein laminar flow (Marsden et al. 2004a,b) and for the suppression of broadband noise in turbulentflow (Marsden et al. 2007). In addition, Lehnhauser & Shafer (2005) have explored the applicationof derivative-approximating trust region methods to steady-flow fluid mechanics problems.

The main idea behind the SMF method is to increase efficiency by using a surrogate function tostand in for an expensive function evaluation, while also benefiting from the convergence propertiesof pattern search methods. In contrast to genetic algorithms, this class of pattern search methods isone of the only derivative-free methods with established convergence theory (Audet 2004; Audet& Dennis 2004, 2006). Advantages of SMF are its nonintrusive nature, ease of implementation,efficiency, and parallel structure.

The SMF algorithm typically consists of a search step, employing a Kriging surrogate functionfor improved efficiency, together with a poll step to guarantee convergence to a local minimum(Simpson et al. 1998, Lophaven et al. 2002). The exploratory search step uses the surrogate toselect points that are likely to improve the cost function but is not strictly required for convergence.Convergence is guaranteed by the poll step, in which points neighboring the current best pointon the mesh are evaluated in a positive spanning set of directions to check if the current best pointis a mesh local optimizer.

All points evaluated in either the search or poll step of the SMF algorithm must lie on a meshin the parameter space. The vectors defining the mesh directions must positively span R

n (Lewis& Torczon 1996). If we define D as a matrix whose columns form a positive spanning set in R

n,then the set of mesh points surrounding a point x is given by

M (x,�) = {x + � Dz : z ∈ NnD}, (2)

where � is the mesh size parameter, and nD is the number of columns in D. A positive spanningset is simply the set of positive linear combinations of the vectors making up the mesh directions(Davis 1954). A set of n + 1 poll points is required to generate a positive basis, where n is thenumber of optimization parameters. Following the above definition, the mesh in SMF may be



Ann

u. R

ev. F

luid

Mec

h. 2

014.

46. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

by W

IB62

42 -

Uni

vers

itaet

s- u

nd L

ande

sbib

lioth

ek D

uess

eldo

rf o

n 10

/08/

13. F

or p

erso

nal u

se o

nly.


GPS polling

MADS polling

No

Yes

No

Improved?

Convergent?

InitializationLHS

Search

Stop

PollA set of positive

spanning directions

Refinemesh

No

Yes

Yes

Improved?

p1

p2 xk

p3

p1

p2

xkp3

p1

p2xk

p3

p3

p1 xk

p2

p1

p2

xk

p3p1

p2xkp3

Figure 3Surrogate management framework optimization algorithm, showing a Kriging surrogate model for thesearch step, and two options for the poll step, generalized pattern search (GPS) and mesh adaptive directsearch (MADS). Abbreviation: LHS, Latin hypercube sampling.

Surrogate model:an inexpensiveapproximation thatstands in for anexpensive functionevaluation

refined or coarsened by changing � > 0, and the mesh may be rotated from one iteration to thenext (Torczon 1997).

Convergence is reached when a local minimizer on the mesh is found, and the mesh has beenrefined to the desired accuracy. Each time new data points are found in a search or poll step, thedata are added to the surrogate, and it is updated. The steps in the algorithm are summarizedbelow and in Figure 3, where the set of points in the initial mesh is M0, the mesh at iteration k isMk, 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 ⊂ M k.c. If for any trial point in Tk, J(z) < J(xk), a lower cost function value has been found, and

the search is successful. Increment k and go back to step a.d. Else, if no trial point in Tk improves the cost function, search is unsuccessful. Increment

k and go to poll.

2. Poll

a. Choose a set of positive spanning directions, and form the poll set Xk as the set of meshpoints adjacent to xk in these directions.

526 Marsden


Ann

u. R

ev. F

luid

Mec

h. 2

014.

46. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

by W

IB62

42 -

Uni

vers

itaet

s- u

nd L

ande

sbib

lioth

ek D

uess

eldo

rf o

n 10

/08/

13. F

or p

erso

nal u

se o

nly.


Anastomosis:surgical connectionbetween blood vessels

Fontan surgery: thethird surgery in thestaged palliation ofsingle-ventricle heartdefects; the vena cavaeare anastomosed to thepulmonary arteries

b. If J(xpoll) < J(xk) for any point xpoll ∈ X k, then a lower cost function has been found,and the poll is 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 methodreduces to the convergence of pattern search methods. The reader is referred to the extensiveliterature summarizing the relevant mathematical convergence theory (Torczon 1997; Serafini1998; Booker et al. 1999; Lewis & Torczon 1999, 2000, 2002; Audet & Dennis 2003).

There has also been recent success with a new polling method called mesh adaptive directsearch (MADS) (Audet & Dennis 2006). Compared to standard generalized pattern search (GPS)polling, MADS offers greater flexibility and is capable of generating a dense set of directions. Thisresults in stronger convergence theory, especially with nonlinear constraints (Audet & Dennis2006). MADS has been applied previously in fluid mechanics design by Marsden et al. (2007).Figure 3 illustrates poll sets and parameter space mesh refinement using both GPS and MADSfor the case of two design parameters. The SMF method can be easily extended to constrainedoptimization using a filter (Fletcher & Leyffer 2002) or a barrier approach. Filters are incorporatedinto pattern search methods (Audet & Dennis 2004) and have been employed in an aeronauticsapplication using MADS (Marsden et al. 2007).

3.2. Coupling with Numerical Simulations

An automated framework is required to couple the cardiovascular solver to the optimization al-gorithm (Marsden et al. 2008, Yang et al. 2010). In this process, geometry parameterization,model generation, meshing, flow simulation, and postprocessing for cost function computationare performed in an automated loop. The SMF method has been applied to several idealized car-diovascular models in recent work, including a stenosis, a vessel bifurcation modeled on Murray’slaw, an end-to-side anastomosis, and a Y-graft Fontan surgery design.

Figure 4 shows the typical procedure for applying SMF to a cardiovascular shape optimizationproblem, with all steps performed via automated scripts. First, the optimization algorithm producesa set of design parameters to define the model geometry. In the example shown, these are theangles of attachment, radius, and attachment locations of a graft to be implanted. Then themodel is automatically generated using these design parameters, and a mesh is generated. A flowsimulation is then performed and postprocessed to extract a predefined cost function, which mayor may not require the use of additional postprocessing simulations, for example, to performLagrangian particle tracking in a residence time computation (Esmaily Moghadam et al. 2012).

3.3. Geometry Parameterization

Unique challenges arise when parameterizing cardiovascular geometries. Patient-specific geome-tries constructed from medical image data often have multiple branches and connections andcannot be easily represented analytically. Manipulation of these models can be difficult to auto-mate, depending on the original mode of construction (e.g., 2D or 3D segmentation methods).These challenges are distinct from traditional optimization applications, which typically deal withprecisely defined parts [e.g., in computer-aided design (CAD) models] or combinations of math-ematically simple shape primitives.

For the above reasons, initial work on cardiovascular optimization employed idealized geome-tries that could be parameterized analytically (Quarteroni & Rozza 2003; Agoshkov et al. 2006a,b;



Ann

u. R

ev. F

luid

Mec

h. 2

014.

46. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

by W

IB62

42 -

Uni

vers

itaet

s- u

nd L

ande

sbib

lioth

ek D

uess

eldo

rf o

n 10

/08/

13. F

or p

erso

nal u

se o

nly.


min J(a) with SMF

Poll

Search

Start

Poll

0D3D

Set of parameters

Create the 3D model

Mesh the 3D model

Do a coupled simulation

Find the objective function

Optimization code

Search

End

Refinementp1

p2 xkp3

Figure 4Steps required for automated shape optimization in an idealized model of the Blalock-Taussig shunt surgery to treat single-ventricleheart patients: geometry parameterization, meshing, flow simulation, and optimization algorithm to produce new design parameter set.Abbreviation: SMF, surrogate management framework.

Marsden et al. 2008). For example, using 2D segmentation methods, one can translate, rotate, orexpand the segmentations to create a desired geometry according to a set of analytical parameters.As parameter values change, such as graft radii, positions, and angles, new designs are automaticallygenerated using a script.

Whereas analytic manipulation is effective for simple geometries, manipulating patient-specificmodels is more challenging because the union of intersecting vessels is usually not well definedover a wide range of parameter values. For example, moving a vessel to a different location orincreasing its size may result in a discontinuous surface. This limits the use of analytical tools toidealized geometries with few design parameters. Conversely, the manipulation of patient-specificmodels by acting directly on the segmentations can require laborious user intervention, limitingthe number of geometries that can be tested.

There are many promising alternatives for the parameterization of complex models. One ap-proach is to represent the model surface using nonuniform rational B-splines (NURBS), in whichcase the model can be manipulated by acting on the NURBS function. This method has the advan-tage of guaranteeing a smooth surface but the potential disadvantage of producing an unreasonablylarge number of parameters resulting from numerous NURBS surface patches. In addition, global

528 Marsden


Ann

u. R

ev. F

luid

Mec

h. 2

014.

46. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

by W

IB62

42 -

Uni

vers

itaet

s- u

nd L

ande

sbib

lioth

ek D

uess

eldo

rf o

n 10

/08/

13. F

or p

erso

nal u

se o

nly.


design parameters such as the graft radius, angle, and position of attachment or flare are oftenmore practical than local surface parameters in the surgical context.

Another recent approach has been the development of a human-shape interaction system,called SURGEM (Sundareswaran et al. 2007, Pekkan et al. 2008). This real-time interactive toolis based on two-hand, free-form manipulation and incorporates haptic tools for direct by-handmanipulation by the surgeon. The SURGEM tools were motivated by virtual sculpturing in whicha user can grab a portion of the shape and then pull, push, twist, and bend it. Only the portion ofthe shape in the vicinity of the grabbed point is affected, and the effect is lessened with distancethrough a specified decay profile. Although this tool provides a high level of user interaction, itdoes not involve the direct definition of design parameters and therefore has not yet been appliedin conjunction with formal optimization methods.

The computer graphics field has developed tools for model morphing and parameterization thatare applicable to cardiovascular models but have not yet been applied in this context. Hierarchicalmethods for subdivision are attractive as they require only a limited number of design parametersto manipulate a shape, while retaining detailed surface features. These methods offer topologicalgenerality, multiresolution capability, and uniformity of representation (Zorin et al. 1997; Ciraket al. 2000, 2002; Bekkers & Taylor 2008). In addition, isogeometric methods, developed forfinite-element simulations with FSI, allow for uniform representation of design variables andshape function definition for finite-element simulations (Zhang et al. 2007, Bazilevs et al. 2008).

Ultimately, there is a need for parameterization methods that can be interfaced with modelsconstructed in a variety of software packages, both commercial and open source. For this rea-son, methods that can act directly on the surface mesh, rather than the underlying constructsor segmentations, are attractive. Physics-based methods, in which one uses physical principlessuch as mechanical deformation, are a particularly attractive approach to deform models whilemaintaining continuous surface representations.

3.4. Choice of Cost Function and Constraints

Perhaps the most challenging aspect of cardiovascular optimization is the choice of cost function.This choice is highly disease or device specific and ideally should relate directly to clinical outcomesand biological response. For example, in modeling single-ventricle heart patients, cost functionchoices have aimed to reduce energy loss (to reduce cardiac workload) or improve hepatic flowdistribution to the lungs (to reduce pulmonary arteriovenous malformations) (Yang et al. 2010,2012, 2013). Additional constraints have been added to limit areas of low WSS, assumed to belinked to thrombotic risk. In bypass graft optimization, cost functions have aimed to reduce areasof low WSS, WSS gradients, and oscillatory shear index because of demonstrated biological linksto atherosclerosis localization and thrombosis (Sankaran & Marsden 2010).

A particular challenge is the reduction of thrombotic risk, which arises in both surgical anddevice applications. The formation of thrombus is a complex and multifaceted process in whichbiochemical response and hemodynamics are closely intertwined. Recent progress has shownsuccess with mathematical modeling of the thrombosis process, in which a set of ODEs governingblood chemistry is coupled with the Navier-Stokes equations. However, because of the highlymultiscale nature of these systems, it is not yet practical to model the full chemistry processin a large-scale patient-specific simulation. There is therefore a need for simpler surrogates forthrombotic risk that can be derived directly from the flow field or with reduced-order post-processing steps. There are several methods available for computing areas of flow stagnation andparticle residence time that may serve as surrogates for thrombotic risk, including Lagrangianparticle tracking and nondiscrete methods (Shadden & Taylor 2008).



Ann

u. R

ev. F

luid

Mec

h. 2

014.

46. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

by W

IB62

42 -

Uni

vers

itaet

s- u

nd L

ande

sbib

lioth

ek D

uess

eldo

rf o

n 10

/08/

13. F

or p

erso

nal u

se o

nly.


4. ACCOUNTING FOR UNCERTAINTIES

Uncertainties arise from numerous sources in cardiovascular simulation and design and can gen-erally be classified into two groups: (a) uncertain input parameters and (b) uncertain design pa-rameters. Uncertainties of the first type arise, for example, from noise in the image data, materialproperty values, boundary condition parameters, non-Newtonian models, and user variability inmodel construction. Uncertainties of the second type arise in shape optimization problems inwhich the design parameters themselves are uncertain. For example, in the optimization of asurgical geometry, one may wish to account for a fudge factor in the surgical implementation.Cardiovascular simulations should account for uncertainties of the first type, and statistics andconfidence intervals should be produced to provide error bars on simulation results. In optimiza-tion, one may need to account for one or both types of uncertainties, which results in a need toperform robust optimization. In this section, we briefly review recent methods that address boththese needs.

4.1. Uncertainty Quantification in Simulations

Before simulations can be reliably used in the clinic, it is essential to quantify the level of confi-dence that can be placed on simulation outputs. Traditionally, a single simulation is performed,and results such as shear stresses and velocities are deterministically quantified. However, inputuncertainties to the model, such as those listed above, result in a range of possible simulationoutputs that can be quantified statistically. Manually perturbing these parameters using a design-of-experiments strategy quickly becomes computationally intractable for problems with a largenumber of parameters.

Recent work has applied stochastic collocation to handle uncertainties and quantify outputsusing probability density functions (PDFs) and confidence intervals (Ghanem & Spanos 1991,Babuska et al. 2010, Xiu & Hesthaven 2005, Najm 2009). This method offers substantiallybetter convergence than traditional Monte Carlo methods (Klimke 2006, Ganapathysibrama-nian & Zabaras 2007, Sankaran 2009), is nonintrusive to implement, and is highly paralleliz-able. Instead of dealing with PDFs directly, we use the concept of stochastic spaces, defined byξ = [ξ 1, ξ 2, . . . , ξ N ], where ξ i represents either uniform or normally distributed random vari-ables. In the collocation scheme, the stochastic space is approximated using mutually orthogonalinterpolating functions. The stochastic space can then be queried at any point, and PDFs can beconstructed. This method is specifically designed for UQ in large-scale simulations, such as CFDsimulations in complex geometries.

A hierarchical approach is used to refine the depth of interpolation until convergence of theoutput PDF is achieved. In practice, the depth of interpolation allows for flexibility because it canbe adjusted according to computational expense. Statistics are generated on the set of simulationoutputs, and confidence intervals are computed for specified levels (e.g., 95%, 99%). Because thisapproach is nonintrusive, existing computational solvers can be used and UQ can be performedtrivially in parallel machines. There has also been recent interest in applying UQ methods to fluidmechanics problems (Najm 2009, Witteveen & Iaccarino 2013).

4.2. Incorporating Uncertainties for Robust Optimization

When performing robust design, one aims to identify optimal solutions that are robust to inputuncertainties. Uncertain parameters in this setting may arise in both model parameters and in thedesign variables themselves. The cost function must be modified from the deterministic case and

530 Marsden


Ann

u. R

ev. F

luid

Mec

h. 2

014.

46. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

by W

IB62

42 -

Uni

vers

itaet

s- u

nd L

ande

sbib

lioth

ek D

uess

eldo

rf o

n 10

/08/

13. F

or p

erso

nal u

se o

nly.


is generally defined as

J =N c∑i=1

⎛⎝αi E(Ci ) +

N m∑j=1

βi j M j (Ci )

⎞⎠, (3)

where J represents the robust objective function, Ci are quantities derived from hemodynamicsthat are to be minimized, αi and βi j represent weights that are attached to the different costfunctions, E(·) denotes the expectation operator, and M j (·) represents the j-th statistical moment.Nc and Nm represent the number of cost functions and number of statistical moments, respectively.The second moment can be used to capture standard deviation, the third moment (or cumulant)can be used to enforce symmetry, and so on. The standard deviation is key to enforcing that theobjective function is robust to fluctuations in the design variables. This term is essentially a penaltyfor large steepness in the uncertain design variables or uncertain input parameters. E(·) and M(·)are operators over the stochastic space that can contain both input parametric uncertainties anddesign variables.

Combining the previously described SMF and UQ tools, Sankaran and colleagues (2010,2011; Sankaran 2009) developed, and proved convergence of, a new stochastic SMF optimiza-tion algorithm. This method has been applied to choose surgical designs that are robust tofluctuations in physiologic parameters and surgical implementation. Although this method hasbeen successfully demonstrated in idealized models (e.g., in bypass graft optimization), it hasnot yet been applied in the context of patient-specific models. Advances in model parameter-ization techniques will enable more detailed analysis of uncertainties arising from the modelconstruction process in the future. The next section discusses the application of deterministic andstochastic optimization in the context of key applications in cardiovascular surgery and devicedesign.

5. APPLICATIONS IN CARDIOVASCULAR DISEASE

5.1. Surgical Planning

Optimization has been applied to cardiovascular surgical planning in both adult and pediatricdisease applications. Because of challenges of model parameterization, computational cost, andobtaining sensitivities, much of this work has been limited to steady-flow and idealized geometries.However, advances in adjoint methods, automatic differentiation, and derivative-free optimizationhold promise for extension to more realistic cases. A hybrid approach has also been used, in whichan idealized graft design is implanted into a patient-specific model (Yang et al. 2013). In addition,parametric studies have provided insights into the relationship between flow and geometry, evenwhen a full optimization has not been performed (Sankaran et al. 2012).

There is particular interest in the optimization of bypass grafts, both in the coronary arteries andin the peripheral vasculature (Quarteroni & Rozza 2003, Dur et al. 2011). It is well accepted thataltered flow conditions, such as separation and stagnation, flow reversal, and low and oscillatoryshear stress, are important factors in the development of arterial disease (Bassiouny et al. 1992,Malek et al. 1999). A range of cost functions has been proposed to assess bypass graft performance,generally aiming to reduce the risk of atherosclerosis, plaque deposition, intimal hyperplasia, andthrombosis. The choice of cost function is perhaps the biggest challenge as it is inherently linkedto the biomechanical response of the vessel wall to changing flow conditions, which remainsan active area of research. Proposed cost functions have included the area of low WSS, WSSgradient, alternating flow index, gradient oscillatory number, and combinations thereof (Ojha1994, Lei et al. 1997, Manthaa et al. 2006).



Ann

u. R

ev. F

luid

Mec

h. 2

014.

46. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

by W

IB62

42 -

Uni

vers

itaet

s- u

nd L

ande

sbib

lioth

ek D

uess

eldo

rf o

n 10

/08/

13. F

or p

erso

nal u

se o

nly.


Stochastic Deterministic

756

5

4

3

70

65

60

55

50

45

40

35

30

257570656055504540353025

θ2

θ2

θ1

θ1

7570656055504540353025θ1

2.4

2.2

2.0

1.8

1.6

1.4

1.2

1.0

0.8

v rsten

SD SDDeterministicoptima

Stochasticoptima

Figure 5Optimization of bypass graft using the stochastic surrogate management framework method, demonstrating the differences in optimalsolution between the deterministic and stochastic cases. We observe that the optimal solution has moved from an area of high standarddeviation to low standard deviation for both design variables (left) and uncertain input variables (right), when uncertainties are includedin optimization. Colors correspond to contours of standard deviation (SD).

Early work in cardiovascular shape optimization employed shape perturbation methods withan adjoint formulation in idealized models of bypass grafts. These methods have been successfullyapplied to bypass graft design using the L2 norm of the vorticity as the cost function in generalizedsteady Stokes flow (Agoshkov et al. 2006b) and unsteady Stokes flow (Agoshkov et al. 2006a). Morerecently, the same authors combined reduced basis methods with free-form deformation to allowfor more complex shapes with a reasonable number of control points and reduced computationalcost (Manzoni et al. 2011). These methods were applied in Stokes flow, with favorable comparisonsbetween the reduced model and the full finite-element simulation. This combination of severalreduction techniques allows for more versatility in geometrical parameterizations while limitingthe number of parameters and achieving significant computational savings.

Deterministic and stochastic SMF methods have been applied in unsteady flow to bypassgraft optimization, with uncertainties in proximal and distal anastomosis angles, stenosis radius,and inflow velocity. Results demonstrated significant differences between deterministic andstochastic optimal solutions. Stochastic optima moved away from areas of high standard deviationcompared to deterministic optima. This example, illustrated in Figure 5, demonstrates the

532 Marsden


Ann

u. R

ev. F

luid

Mec

h. 2

014.

46. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

by W

IB62

42 -

Uni

vers

itaet

s- u

nd L

ande

sbib

lioth

ek D

uess

eldo

rf o

n 10

/08/

13. F

or p

erso

nal u

se o

nly.


efficient application of SMF with adaptive stochastic collocation for optimization in 3D unsteadyNavier-Stokes flow (Sankaran & Marsden 2010). Future work should apply these methods tomore realistic geometries with more design variables.

Optimization has also been used in the design of surgeries for single-ventricle heart disease,a form of congenital heart disease that is uniformly fatal without treatment and that typicallyrequires a series of three surgeries starting in the neonatal period. Optimization using SMF hasuncovered links among the graft position, cardiac output, and coronary oxygen delivery in the firststage of single-ventricle repair, the Blalock-Taussig shunt (Esmaily Moghadam et al. 2012). Inthis study, optimization was performed on a 3D model of the aorta and pulmonary arteries, usinga closed-loop LPN. The graft geometry was parameterized and implanted into the model usingautomated scripts, and the 3D unsteady Navier-Stokes equations were solved in each functionevaluation on a parallel architecture (Figure 4).

The SMF method has also been used to optimize the design of a novel Y-shaped graft for theFontan surgery, the third stage of single-ventricle repair. Figure 6 shows the evolution of theY-graft design from the initial concept to clinical translation. Optimization was first performedon idealized models, using six to seven shape design parameters with energy loss as the costfunction and a constraint on the WSS. Subsequent studies aimed to improve the distribution ofhepatic flow to the lungs. Optimization was also performed on semi-idealized models derived frompatient-specific models of the Y-graft surgery. Results of optimization were verified using the fullpatient-specific models, and improvement was demonstrated in previously underperforming cases(Yang et al. 2010, 2012, 2013).

Optimization with SMF in single-ventricle heart disease demonstrated the efficiency of themethod for unsteady blood flow problems, generally requiring 50–150 function evaluations for5–10 parameter runs. Two competing methods for polling were compared: orthogonal MADSwith 2n poll points and lower triangular MADS with n + 1 poll points. Despite the increased costinvolved in each individual polling step, the two methods were competitive in terms of overallcomputational cost.

5.2. Parameter Identification

Methods of parameter identification and state estimation offer a means to obtain patient-specificinformation using control theory methods. These methods maximize information extraction frommedical image data, as in the context of material property identification for FSI simulations. Inaddition, formal optimization and UQ can efficiently identify sets of unknown model parameters,as in the context of G&R simulations.

5.2.1. Material parameters for fluid-structure interaction. Viscoelastic support has beenshown to be a key modeling ingredient to accurately represent the effect of surrounding tissuesand organs in a fluid-structure vascular model. In recent work, Moireau et al. (2011, 2012)proposed a complete methodological chain for the identification of the corresponding boundarysupport parameters using patient image data. Here, the authors considered distance maps ofmodel-to-image contours as the discrepancy driving the data-assimilation approach and aimedto minimize these distances through control theoretic techniques. This approach relied on acombination of (a) state estimation based on the synthetic discrimination function filteringmethod, which is well suited to handle measurements extracted from image data sequences,and (b) parameter estimation based on a reduced-order unscented Kalman filtering method,which can be readily parallelized. Results demonstrated that the computational effectiveness ofthe complete estimation chain was comparable to that of a direct simulation. This framework



Ann

u. R

ev. F

luid

Mec

h. 2

014.

46. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

by W

IB62

42 -

Uni

vers

itaet

s- u

nd L

ande

sbib

lioth

ek D

uess

eldo

rf o

n 10

/08/

13. F

or p

erso

nal u

se o

nly.


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40.055

0.060

0.065

0.070

0.075

0.080

0.085

Filtered pointLeast infeasible pointUnfiltered point

H

J

Initial study Optimization study

c d

Nonoptimized Optimized0

10203040

506070

80

90100

ba

Right Left

9 mm 12 mm

18 mm

He

pa

tic

flo

w (

%)

IVC-RPAIVC-LPA

Clinical pilot study

D = 12.75 1614.13 15

Figure 6The application of optimization in a simulation-derived design for a new Y-graft Fontan surgical method. (a) Concept development andinitial patient-specific modeling study. (b) Optimization of a semi-idealized model demonstrating improved hepatic flow in anunderperforming patient. (c) Example of a filter from a constrained optimization with energy loss as an objective and hepatic flowdistribution as a constraint. (d ) Clinical translation in an initial pilot study, with resulting imaging data.

was successful in a realistic case study of hemodynamics in the thoracic aorta (Figure 7). Theestimation of the boundary support parameters proved successful, showing that direct modelingbased on the estimated parameters was more accurate than with a previous manual calibration.

5.2.2. Growth and remodeling. Optimization also provides a means to select model parametersgoverning the G&R of vessel walls. Computational models for vascular G&R are used to predictthe long-term response of vessels to changes in pressure, flow, and mechanical loading conditions

534 Marsden


Ann

u. R

ev. F

luid

Mec

h. 2

014.

46. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

by W

IB62

42 -

Uni

vers

itaet

s- u

nd L

ande

sbib

lioth

ek D

uess

eldo

rf o

n 10

/08/

13. F

or p

erso

nal u

se o

nly.


SolidContours with estimated supportContours with calibrated support

Fluid

Pressure with estimated support (mmHg)Flow with estimated support (cc/s)Segmented contours

Systole

SpineSpine

Spine

π1

π2

π3

140

120

100

80

60

160

120

80

40

0

0 0.3 0.6 0.9

140

120

100

80

60

360

270

180

90

00 0.3 0.6 0.9

140

120

100

80

60

40

30

20

10

00 0.3 0.6 0.9

Pre

ssu

re (

mm

Hg

)

Flo

w (cc/s)

Pre

ssu

re (

mm

Hg

)

Flo

w (cc/s)

Pre

ssu

re (

mm

Hg

)

Flo

w (cc/s)

Time (s)

Time (s)

Time (s)

Figure 7(Left) Comparison of segmented (dashed green) versus simulated (blue for estimated values and pink for manually calibrated values) lumencontours at peak systole. (Right) Computed fluid pressures and velocities at inlet and outlets. Figure adapted with permission fromMoireau et al. (2012).



Ann

u. R

ev. F

luid

Mec

h. 2

014.

46. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

by W

IB62

42 -

Uni

vers

itaet

s- u

nd L

ande

sbib

lioth

ek D

uess

eldo

rf o

n 10

/08/

13. F

or p

erso

nal u

se o

nly.


(Humphrey 2008, Figueroa et al. 2009, Valentin & Humphrey 2009). An accurate prediction ofthese responses is essential for understanding numerous disease processes. Such models requirenumerous input parameters, including material properties and growth rates, which are oftenexperimentally derived, and inherently uncertain. Whereas early parameter identification for thesemethods used a brute-force approach, systematic UQ in G&R models offers a more efficientmethod for parameter identification. Recent work has introduced an efficient framework for UQand optimal parameter selection in the G&R model of Humphrey and colleagues using SMFwith adaptive stochastic collocation (Sankaran et al. 2013). Near-linear scaling with the numberof parameters was demonstrated. Robust optimization was used to determine optimal arterial wallmaterial properties. Results showed that an artery can achieve optimal homeostatic conditionsover a range of alterations in pressure and flow. The robustness of the solution was enforcedby including uncertainty in loading conditions. It was then shown that homeostatic intramuraland WSS were maintained for a wide range of material properties. These tools provided the firstsystematic and efficient framework to quantify uncertainties and optimally identify G&R modelparameters (Sankaran et al. 2013).

5.3. Medical Device Design

Whereas virtual surgery applications present the challenge of implementation uncertainty, medicaldevices can be optimized and machined with precision, making them particularly amenable to theoptimization techniques described above. However, the field of medical device design still lagsbehind traditional engineering industries in the use of simulation for design optimization. Thissection outlines two examples in which optimization is coupled with flow simulations of medicaldevices to optimize (a) the design of cardiovascular stents and (b) the design of blood pumpcomponents.

It is known that coronary stent design influences local patterns of WSS that are associated withneointimal growth, restenosis, and the endothelialization of stent struts. Initial CFD studies onstents were limited to a small number of geometries to identify stent designs that reduce alterationsin near-wall hemodynamics. However, recent studies have begun to apply formal optimizationstrategies to stent strut design (Gundert et al. 2012a,b; Pant et al. 2011; Masoumi Khalil Abadet al. 2012).

In one such approach, Gundert et al. (2012a,b) proposed a methodology that couples CFDwith 3D shape optimization for stent design using the SMF method (Figure 8). The optimizationprocedure was fully automated, such that solid model construction, anisotropic mesh generation,CFD simulation, and WSS quantification did not require user intervention. This method wasapplied to determine the optimal number of circumferentially repeating stent cells for slotted-tube stents with various diameters and intrastrut areas. Optimal stent designs were defined as thoseminimizing the area of low intrastrut time-averaged WSS. It was determined that the optimalnumber of cells was dependent on the intrastrut angle with respect to the primary flow direction.Furthermore, stent designs with an intrastrut angle of approximately 40◦ minimized the area oflow time-averaged WSS regardless of vessel size or intrastrut area.

The second medical device application highlighted here is the shape optimization of artificialblood pumps, which presents a number of unique challenges. The microstructural properties ofblood affect both the choice of the design objectives, such as minimizing blood damage and clotting,and the possible need to account for non-Newtonian effects. The recent work of Behr and col-leagues (Behbahani et al. 2009) aims to address the issue of objective functions that can be correlatedwith the accumulation of blood damage along flow pathlines and the influence of the constitutivemodel (Newtonian, generalized Newtonian, and viscoelastic) on the resulting optimal shapes.

536 Marsden


Ann

u. R

ev. F

luid

Mec

h. 2

014.

46. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

by W

IB62

42 -

Uni

vers

itaet

s- u

nd L

ande

sbib

lioth

ek D

uess

eldo

rf o

n 10

/08/

13. F

or p

erso

nal u

se o

nly.


0 0.25 0.5

50

Flo

w (

ml/

min

)

Time (s)

Generate stent model for a givenparameter (e.g., θ ) a

Implant stent model in vessel usinga Boolean subtractionb

Discretize model into anisotropicfinite element meshc

Prescribe boundary conditions tothe modeld

Solve for index of interest (e.g.,time-averaged WSS) using CFDand compute the cost function

e

Update the surrogate function andcontinue the optimization until theminimum is found

f

θ

0.75

0.5

0.25

0.0

Nor

mal

ized

time-

aver

aged

WSS

Co

st

Number of circumferential repeating units

0.8

0.7

0.6

0.5

0.4 88° 62° 43° 30° 20° 7°

2 3 4 5 6 7 8 9 10 11 12 13

Rc Rd

C

Figure 8The optimization of coronary stents showing the steps for stent strut parameterization, model construction, and function evaluationusing computational fluid dynamics (CFD) (upper panel ) and the resulting cost function for different stent designs showing minimumwall shear stress (WSS) (lower panel ).

In this example, an axial left-ventricular assist device similar to the current DeBakey left-ventricular assist device from MicroMed Cardiovascular (Houston, Texas) is examined for possibleshape modifications to increase its biocompatibility (Probst 2013). Figure 9 shows part of thealready discretized surface mesh of the pumping chamber of the device, with the surface meshelements shaded according to the displacements allowed during the optimization process. Thewhite elements are not deformable and surround the impeller and upstream regions of the device.Green surface elements belong to the diffuser portion of the pump and move as a rigid body alongthe device axis, controlled by a single shape parameter. Blue surface elements are deformed toaccommodate the axial displacements of the diffuser. The Open Flipper toolkit (Botsch & Kobbelt2004) is used to deform the elements in a manner consistent with the original CAD geometry.Figure 10 shows the dependence of two objective functions: one evaluating the integral of the shearrate over the flow volume, which is related to blood damage, and one measuring the pressure headacross the pump, which is related to pump hydraulic performance. The optimal shape parameterwill depend on the relative weighting of these two quantities. Figure 11 shows the distribution ofthe scalar shear rate measure for two values of the shape parameter. A trust-region optimizationscheme is used effectively in this case to find the optimal parameter values.



Ann

u. R

ev. F

luid

Mec

h. 2

014.

46. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

by W

IB62

42 -

Uni

vers

itaet

s- u

nd L

ande

sbib

lioth

ek D

uess

eldo

rf o

n 10

/08/

13. F

or p

erso

nal u

se o

nly.


Figure 9Model of a DeBakey left-ventricular assist device. The handle region ( green) and the modeling region (blue)are selected in Open Flipper in order to modify the gap width between the impeller and diffuser.

0.96

0.97

0.98

0.99

1.00

1.01

1.02

–0.2 –0.1 0 0.1 0.2

Y

X

Pressure headShear

Figure 10Modification of the gap width between the impeller and diffuser and resulting objective function values forthe shear (red diamonds) and pressure head (blue diamonds). Values are normalized by the respective value forthe initial design.

538 Marsden


Ann

u. R

ev. F

luid

Mec

h. 2

014.

46. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

by W

IB62

42 -

Uni

vers

itaet

s- u

nd L

ande

sbib

lioth

ek D

uess

eldo

rf o

n 10

/08/

13. F

or p

erso

nal u

se o

nly.


0 2.5 5.0 7.5

Reduced gap

Increased gap

10.0 12.0

Shear rate (103 s–1)

Figure 11Shear rate (103 s−1) for slices through the DeBakey geometry at t = 88 ms. Two designs are compared: onereducing the initial gap width by 0.025 inches (top) and increasing the gap width by 0.025 inches (bottom).

6. OUTLOOK AND CHALLENGES

To produce clinically relevant design results using optimization, researchers must address severallong-term challenges. First, simulations must be physiologically accurate and thoroughly validatedagainst in vitro and in vivo data. Second, appropriate measures of performance (cost functions) forcardiovascular designs must be defined based on physiologic information and biological response.Third, the choice of optimization method must be appropriate for expensive, time-dependent, 3Dfluid mechanics problems. Finally, improved tools are needed for the efficient parameterizationof patient-specific geometries.

Although varying degrees of progress have been made in all four areas, as outlined above,challenges still remain. Simulations have been well validated against experimental in vitro data;however, validation against in vivo clinical data is still sorely lacking in the literature. This hasresulted in large part from the difficulty and ethical restrictions surrounding the acquisition ofclinical data; imaging data are collected generally only for reasons that are clinically indicated.The definition of more meaningful cost functions will continue to require close interaction amongengineers, clinicians, and biologists to elicit meaningful functions that capture physiological andbiological responses.

Optimization methodology is perhaps the most well developed of the above areas of need.However, future work should improve the efficiency of the algorithms and user interfaces. Futuredevelopments should take better advantage of parallelization. For example, when multiple pointsare evaluated in a search or a poll step in the SMF method, these function evaluations should beperformed in parallel. This will require a multilayered architecture in which function evaluationsare evaluated in parallel, each of which requires multiple processors. Tools for automatic differ-entiation should be improved to handle a larger variety of legacy solvers, as the implementationbarrier can be a detriment to new users. Adjoint-based methods offer the advantage of improvedscalability with the number of design parameters compared to derivative-free methods.



Ann

u. R

ev. F

luid

Mec

h. 2

014.

46. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

by W

IB62

42 -

Uni

vers

itaet

s- u

nd L

ande

sbib

lioth

ek D

uess

eldo

rf o

n 10

/08/

13. F

or p

erso

nal u

se o

nly.


Finally, there is a need for patient-specific model parameterization tools to extend the utility ofoptimization to virtual surgery in complex geometries. Future work should leverage developmentsfrom computer graphics, such as hierarchical image processing and morphing. In addition, CADtools and solvers should be integrated to perform seamless manipulation and simulation of ge-ometries. Recently developed tools such as Open Flipper or isogeometric analysis are a promisingapproach.

Beyond coupling optimization to CFD, the use of optimization and optimal control for pa-rameter identification promises to maximize the information we can extract from image data,including material parameters and external tissue support. Biomechanics models, such as modelsof vascular G&R, often require multiple parameters derived from experimental data with inherentuncertainty. Improved parameter selection will extend the utility of these models to a wider rangeof applications. Similarly, automated tuning of parameters in multiscale models, such as the LPNmodels shown in Figure 2, will be accelerated by optimization, while accounting for uncertaintiesin clinical data.

The synthesis and advancement of the above tools have the potential to improve treatmentsfor adult and pediatric patients suffering from both acquired and congenital heart disease.In particular, optimization will enable customized treatment planning, including surgicalplanning, device selection, medical therapy, and intervention. These tools will integrate with andaugment current imaging methods to provide predictive information (Koo et al. 2011). Finally,well-validated simulation tools will reduce patient risk and improve patient selection, therebyidentifying patients who are currently left untreated and avoiding unnecessary treatment inothers.

SUMMARY POINTS

1. Optimization in cardiovascular modeling can be used for (a) the design of surgical graftsand anastomoses, (b) the design of medical devices such as stents and blood pumps, and(c) parameter identification for model improvement.

2. The choice of cost function for cardiovascular optimization problems poses perhaps thebiggest challenge, as it should incorporate information about the biochemical, biome-chanical, and/or physiological response of a patient.

3. Coupling cardiovascular simulations to reduced-order models of the circulatory systemvia lumped parameter boundary conditions is key to properly model the dynamic interplaybetween local hemodynamics and circulatory physiology.

4. Derivative-free optimization offers an efficient and easy-to-implement alternative togradient-based optimization. The pattern search class of methods is among the fewderivative-free methods with well-established convergence theory.

5. The influence of uncertainties, stemming from clinical input data and modeling assump-tions, should be considered in the design process and is centrally important for the clinicalapplication of cardiovascular modeling. Uncertainties can arise in (a) input variables tothe simulation or (b) design variables in optimization.

6. Parameter identification methods, including Kalman filtering, are effective for (a) max-imizing information extraction (e.g., material properties) from medical image data and(b) accelerating optimal parameter selection in models of vascular growth andremodeling.

540 Marsden


Ann

u. R

ev. F

luid

Mec

h. 2

014.

46. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

by W

IB62

42 -

Uni

vers

itaet

s- u

nd L

ande

sbib

lioth

ek D

uess

eldo

rf o

n 10

/08/

13. F

or p

erso

nal u

se o

nly.


7. Optimization coupled to cardiovascular simulation has the potential to influence patientcare in, for example, (a) complex surgeries performed to treat congenital heart diseasepatients, (b) coronary artery bypass graft surgery, and (c) medical devices such as stentsand blood pumps.

FUTURE ISSUES

1. The development of improved tools for patient-specific geometric model parameteriza-tion will enable treatment planning for individual patients.

2. Future work should systematically quantify the propagation of uncertainties from clinicaldata to simulation results to enable the clinical application of cardiovascular simulation.

3. Arteries and veins change their thickness, size, and composition in response to changingflow and pressure conditions. The incorporation of well-established models of vasculargrowth and remodeling will enable predictive fluid-solid growth simulations.

4. Thrombosis is the result of a complex biochemical cascade that is not yet fully understood.The development of reduced-order mathematical models of the thrombosis process willenable improved surgical and device designs that minimize thrombotic risk.

DISCLOSURE STATEMENT

The author is not aware of any biases that might be perceived as affecting the objectivity of thisreview.

ACKNOWLEDGMENTS

I gratefully acknowledge the support of the Burroughs Wellcome Fund Career Award at theScientific Interface, the National Science Foundation, National Institutes of Health, AmericanHeart Association, and the Leducq Foundation. I thank Jeffrey Feinstein, Tain-Yen Hsia, JaneBurns, and Andrew Kahn for providing clinical expertise to guide this work. I also thank AlbertoFigueroa, John LaDisa, and Marek Behr for generously providing figures and information forapplications highlighted in this work and Charley Taylor, Nathan Wilson, and Irene Vignon-Clementel for providing software and methodology expertise.

LITERATURE CITED

Abraham F, Behr M, Heinkenschloss M. 2005a. Shape optimization in steady blood flow: a numerical studyof non-Newtonian effects. Comput. Methods Biomech. Biomed. Eng. 8:127–37

Abraham F, Behr M, Heinkenschloss M. 2005b. Shape optimization in unsteady blood flow: a numerical studyof non-Newtonian effects. Comput. Methods Biomech. Biomed. Eng. 8:201–12

Agoshkov V, Quarteroni A, Rozza G. 2006a. A mathematical approach in the design of arterial bypass anas-tomoses using unsteady Stokes equations. J. Sci. Comput. 28:139–65

Agoshkov V, Quarteroni A, Rozza G. 2006b. Shape design in aorto-coronaric bypass anastomoses usingperturbation theory. SIAM J. Numer. Anal. 44:367–84

Antaki JF, Ghattas O, Burgreen GW, He B. 1995. Computational flow optimization of rotary blood pumpcomponents. Artif. Organs 19:608–15



Ann

u. R

ev. F

luid

Mec

h. 2

014.

46. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

by W

IB62

42 -

Uni

vers

itaet

s- u

nd L

ande

sbib

lioth

ek D

uess

eldo

rf o

n 10

/08/

13. F

or p

erso

nal u

se o

nly.


Audet C. 2004. Convergence results for pattern search algorithms are tight. Optim. Eng. 5:101–22Audet C, Dennis JE Jr. 2003. Analysis of generalized pattern searches. SIAM J. Optim. 13:889–903Audet C, Dennis JE Jr. 2004. A pattern search filter method for nonlinear programming without derivatives.

SIAM J. Optim. 14:980–1010Audet C, Dennis JE Jr. 2006. Mesh adaptive direct search algorithms for constrained optimization. SIAM J.

Optim. 17:2–11Babuska I, Nobile F, Tempone R. 2010. A stochastic collocation method for elliptic partial differential equa-

tions with random input data. SIAM Rev. 52:317–55Balossino R, Pennati G, Migliavacca F, Formaggia L, Veneziani A, et al. 2009. Computational models to

predict stenosis growth in carotid arteries: Which is the role of boundary conditions? Comput. MethodsBiomech. Biomed. Eng. 12:113–23

Bassiouny H, White S, Glagov S, Choi E, Giddens DP, Zarins CK. 1992. Anastomotic intimal hyperplasia:mechanical injury or flow induced. J. Vasc. Surg. 15:708–16

Bazilevs Y, Calo VM, Hughes TJR, Zhang Y. 2008. Isogeometric fluid-structure interaction: theory, algorithmsand computations. Comput. Mech. 43:3–37

Bazilevs Y, Hsu MC, Benson DJ, Sankaran S, Marsden AL. 2009. Computational fluid-structure interaction:methods and application to a total cavopulmonary connection. Comput. Mech. 45:77–89

Behbahani M, Behr M, Hormes M, Steinseifer U, Arora D, et al. 2009. A review of computational fluiddynamics analysis of blood pumps. Eur. J. Appl. Math. 20:363–97

Bekkers EJ, Taylor CA. 2008. Multi-scale vascular surface model generation from medical imaging data usinghierarchical features. IEEE Trans. Med. Imaging 27:331–41

Booker AJ, Dennis JE Jr, Frank PD, Serafini DB, Torczon V, Trosset MW. 1999. A rigorous framework foroptimization of expensive functions by surrogates. Struct. Optim. 17:1–13

Botsch M, Kobbelt L. 2004. An intuitive framework for real-time freeform modeling. ACM Trans. Graphics23:630–34

Bove EL, Migliavacca F, de Leval MR, Balossino R, Pennati G, et al. 2008. Use of mathematic modeling tocompare and predict hemodynamic effects of the modified Blalock-Taussig and right ventricle-pulmonaryartery shunts for hypoplastic left heart syndrome. J. Thorac. Cardiovasc. Surg. 136:312–20

Burgreen GW, Antaki JF, Wu ZJ, Holmes AJ. 2001. Computational fluid dynamics as a development tool forrotary blood pumps. Artif. Organs 25:336–40

Burgreen GW, Baysal O. 1996. Three-dimensional aerodynamics shape optimization using discrete sensitivityanalysis. AIAA J. 34:1761–70

Castro MA, Putman CM, Cebral JR. 2006. Computational fluid dynamics modeling of intracranial aneurysms:effects of parent artery segmentation on intra-aneurysmal hemodynamics. AJNR Am. J. Neuroradiol.27:1703–9

Cheng CP, Herfkins RJ, Lightner AL, Taylor CA, Feinstein JA. 2004. Blood flow conditions in the proximalpulmonary arteries and vena cavae: healthy children during upright cycling exercise. Am. J. Physiol. HeartCirc. Physiol. 287:H921–26

Cirak F, Ortiz M, Schroder P. 2000. Subdivision surfaces: a new paradigm for thin-shell finite-element analysis.Int. J. Numer. Methods 47:2039–72

Cirak F, Scott MJ, Antonsson EK, Ortiz M, Schroder P. 2002. Integrated modeling, finite-element analysis,and engineering design for thin-shell structures using subdivision. Comput.-Aided Des. 34:137–48

Dasi L, Pekkan K, Katajima H, Yoganathan A. 2008. Functional analysis of Fontan energy dissipation.J. Biomech. 41:2246–52

Davis C. 1954. Theory of positive linear dependence. Am. J. Math. 76:733–46de Leval MR, Kilner P, Gewillig M, Bull C. 1988. Total cavopulmonary connection: a logical alternative

to atriopulmonary connection for complex Fontan operations. Experimental studies and early clinicalexperience. J. Thorac. Cardiovasc. Surg. 96:682–95

Dur O, Coskun S, Coskun K, Frakes D, Kara L, Pekkan K. 2011. Computer-aided patient-specific coronaryartery graft design improvements using CFD coupled shape optimizer. Cardiovasc. Eng. Technol. 2:35–47

Elliot J, Peraire J. 1997. Practical 3D aerodynamic design and optimization using unstructured meshes. AIAAJ. 35:1479–85

542 Marsden


Ann

u. R

ev. F

luid

Mec

h. 2

014.

46. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

by W

IB62

42 -

Uni

vers

itaet

s- u

nd L

ande

sbib

lioth

ek D

uess

eldo

rf o

n 10

/08/

13. F

or p

erso

nal u

se o

nly.


Elliot J, Peraire J. 1998. Progress towards a 3D aerodynamic shape optimization tool for the compressible high-Re Navier-Stokes equations discretized on unstructured meshes. Presented at AIAA Fluid Dyn. Conf., 29th,Albuquerque, NM, AIAA Pap. 1998-2897

Esmaily Moghadam M, Bazilevs Y, Hsia TY, Vignon-Clementel I, Marsden A. 2011. A comparison of outletboundary treatments for prevention of backflow divergence with relevance to blood flow simulations.Comput. Mech. 48:277–91

Esmaily Moghadam M, Migliavacca F, Vignon-Clementel IE, Hsia TY, Marsden AL, Modeling of Congen-ital Hearts Alliance (MOCHA) Investigators. 2012. Optimization of shunt placement for the Norwoodsurgery using multi-domain modeling. J. Biomech. Eng. 134:051002

Esmaily Moghadam M, Vignon-Clementel I, Figliola R, Marsden A. 2013. A modular numerical method forimplicit 0D/3D coupling in cardiovascular finite element simulations. J. Comput. Phys. 244:63–79

Figueroa CA, Baek S, Taylor C, Humphrey J. 2009. A computational framework for fluid-solid-growth mod-eling in cardiovascular simulations. Comput. Methods Appl. Mech. Eng. 198:3583–602

Figueroa CA, Vignon-Clementel IE, Jansen KE, Hughes TJ, Taylor CA. 2006. A coupled momentum methodfor modeling blood flow in three-dimensional deformable arteries. Comput. Meth. Appl. Mech. Eng.195:5685–706

Fletcher R, Leyffer S. 2002. Nonlinear programming without a penalty function. Math. Program. 91:239–69Ganapathysibramanian B, Zabaras N. 2007. Sparse grid collocation schemes for stochastic natural convection

problems. J. Comput. Phys. 225:652–85Ghanem RG, Spanos PD. 1991. Stochastic Finite Elements: A Spectral Approach. New York: SpringerGijsen FJH, Allanic E, van de Vosse FN, Janssen JD. 1999. The influence of the non-Newtonian properties

of blood on the flow in large arteries: unsteady flow in a 90◦ curved tube. J. Biomech. 32:705–13Gundert TJ, Marsden A, Yang W, LaDisa JF Jr. 2012a. Optimization of cardiovascular stent design using

computational fluid dynamics. J. Biomech. Eng. 134:011002Gundert TJ, Marsden AL, Yang W, Marks DS, LaDisa JF Jr. 2012b. Identification of hemodynamically

optimal coronary stent designs based on vessel caliber. IEEE Trans. Biomed. Eng. 59:1992–2002Haynes RH, Burton AC. 1959. Role of the non-Newtonian behavior of blood in hemodynamics. Am. J. Physiol.

197:943–50Humphrey J. 2008. Vascular adaptation and mechanical homeostasis at tissue, cellular, and sub-cellular levels.

Cell Biochem. Biophys. 50:53–78Jameson A. 1988. Aerodynamic design via control theory. J. Sci. Comput. 3:233–60Jameson A, Martinelli L, Pierce NA. 1998. Optimum aerodynamic design using the Navier-Stokes equations.

Theor. Comput. Fluid Dyn. 10:213–37Karniadakis GE, Sherwin SJ. 2005. Spectral /hp Element Methods for Computational Fluid Dynamics. New York:

Oxford Univ. PressKim HJ, Figueroa CA, Hughes TJR, Jansen KE, Taylor CA. 2009. Augmented Lagrangian method for

constraining the shape of velocity profiles at outlet boundaries for three-dimensional finite elementsimulations of blood flow. Comput. Methods Appl. Mech. Eng. 198:3551–66

Klimke A. 2006. Uncertainty modeling using fuzzy arithmetic and sparse grids. PhD thesis. Univ. Stuttgart,Stuttgart, Ger.

Koo BK, Erglis A, Doh JH, Daniels DV, Jegere S, et al. 2011. Diagnosis of ischemia-causing coronary stenosesby noninvasive fractional flow reserve computed from coronary computed tomographic angiograms.J. Am. Coll. Cardiol. 58:1989–97

Ku JP, Draney MT, Arko FR, Lee WA, Chan F, et al. 2002. In vivo validation of numerical predictions ofblood flow in arterial bypass grafts. Ann. Biomed. Eng. 30:743–52

LaDisa JF Jr, Figueroa CA, Vignon-Clementel IE, Kim HJ, Xiao N, et al. 2011. Computational simulationsfor aortic coarctation: representative results from a sampling of patients. J. Biomech. Eng. 133:091008

LaDisa JF Jr, Olson L, Guler I, Hettrick D, Audi S, et al. 2004. Stent design properties and deployment ratioinfluence indices of wall shear stress: a 3D computational fluid dynamics investigation within a normalartery. J. Appl. Physiol. 97:424–30

Lagana K, Dubini G, Migliavacca F, Pietrabissa R, Pennati G, et al. 2002. Multiscale modelling as a tool toprescribe realistic boundary conditions for the study of surgical procedures. Biorheology 39:359–64



Ann

u. R

ev. F

luid

Mec

h. 2

014.

46. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

by W

IB62

42 -

Uni

vers

itaet

s- u

nd L

ande

sbib

lioth

ek D

uess

eldo

rf o

n 10

/08/

13. F

or p

erso

nal u

se o

nly.


Lehnhauser T, Shafer M. 2005. A numerical approach for shape optimization of fluid flow domains. Comput.Methods Appl. Mech. Eng. 194:5221–41

Lei M, Archie J, Kleinstreur. 1997. Computational design of a bypass graft that minimizes wall shear stressgradients in the region of the distal anastomosis. J. Vasc. Surg. 25:637–46

Lei M, Giddens DP, Jones SA, Loth F, Bassiouny H. 2001. Pulsatile flow in an end-to-side vascular graftmodel: comparison of computations with experimental data. J. Biomech. Eng. 123:80–87

Lewis RM, Torczon V. 1996. Rank ordering and positive bases in pattern search algorithms. Tech. Rep. 96-71,Inst. Comput. Appl. Sci. Eng., NASA Langley Res. Cent., Hampton, VA

Lewis RM, Torczon V. 1999. Pattern search algorithms for bound constrained minimization. SIAM J. Optim.9:1082–99

Lewis RM, Torczon V. 2000. Pattern search methods for linearly constrained minimization. SIAM J. Optim.10:917–41

Lewis RM, Torczon V. 2002. A globally convergent augmented Lagrangian pattern search algorithm foroptimization with general constraints and simple bounds. SIAM J. Optim. 12:1075–89

Lonyai A, Dubin AM, Feinstein JA, Taylor CA, Shadden SC. 2010. New insights into pacemaker lead-inducedvenous occlusion: simulation-based investigation of alterations in venous biomechanics. Cardiovasc. Eng.10:84–90

Lophaven SN, Nielsen HB, Søndergaard J. 2002. DACE. A MATLAB Kriging toolbox. Version 2.0. Tech. Rep.IMM-TR-2002-12, Tech. Univ. Denmark, Copenhagen

Malek AM, Alper SL, Izumo S. 1999. Hemodynamic shear stress and its role in atherosclerosis. J. Am. Med.Assoc. 282:2035–42

Mantha A, Karmonik C, Benndorf G, Strother C, Metcalfe R. 2006. Hemodynamics in a cerebral artery beforeand after the formation of an aneurysm. AJNR Am. J. Neuroradiol. 27:1113–18

Manzoni A, Quarteroni A, Rozza G. 2011. Shape optimization for viscous flows by reduced basis methods andfree-form deformation. Int. J. Numer. Methods Fluids 70:646–70

Marsden AL, Feinstein JA, Taylor CA. 2008. A computational framework for derivative-free optimization ofcardiovascular geometries. Comput. Meth. Appl. Mech. Eng. 197:1890–905

Marsden AL, Wang M, Dennis JE Jr, Moin P. 2004a. Optimal aeroacoustic shape design using the surrogatemanagement framework. Optim. Eng. 5:235–62

Marsden AL, Wang M, Dennis JE Jr, Moin P. 2004b. Suppression of airfoil vortex-shedding noise viaderivative-free optimization. Phys. Fluids 16:L83–86

Marsden AL, Wang M, Dennis JE Jr, Moin P. 2007. Trailing-edge noise reduction using derivative-freeoptimization and large-eddy simulation. J. Fluid Mech. 572:13–36

Masoumi Khalil Abad E, Pasini D, Cecere R. 2012. Shape optimization of stress concentration-free lattice forself-expandable Nitinol stent-grafts. J. Biomech. 45:1028–35

Moireau P, Bertoglio C, Xiao N, Figueroa CA, Taylor CA, et al. 2012. Sequential identification of bound-ary support parameters in a fluid-structure vascular model using patient image data. Biomech. Model.Mechanobiol. 12:475–96

Moireau P, Xiao N, Astorino M, Figueroa CA, Chapelle D, et al. 2011. External tissue support and fluid-structure simulation in blood flows. Biomech. Model. Mechanobiol. 11:1–18

Najm HN. 2009. Uncertainty quantification and polynomial chaos techniques in computational fluid dynamics.Annu. Rev. Fluid Mech. 41:35–52

Newman J III, Taylor A III. 1996. Three-dimensional aerodynamic shape sensitivity analysis and design optimizationusing the Euler equations on unstructured grids. Presented at AIAA Appl. Aerodyn. Conf., 14th, New Orleans,LA, AIAA Pap. 1996-2464

Ojha M. 1994. Wall shear stress temporal gradient and anastomotic intimal hyperplasia. Circ. Res. 74:1227–31Pant S, Bressloff NW, Limbert G. 2011. Geometry parameterization and multidisciplinary constrained opti-

mization of coronary stents. Biomech. Model. Mechanobiol. 11:61–82Pekkan K, Whited B, Kanter K, Sharma S, Zelicourt D, et al. 2008. Patient-specific surgical planning and

hemodynamic computational fluid dynamics optimization through free-form haptic anatomy editing tool(SURGEM). Med. Biol. Eng. Comput. 46:1139–52

Peskin CS. 1977. Numerical analysis of blood flow in the heart. J. Comput. Phys. 25:220–52

544 Marsden


Ann

u. R

ev. F

luid

Mec

h. 2

014.

46. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

by W

IB62

42 -

Uni

vers

itaet

s- u

nd L

ande

sbib

lioth

ek D

uess

eldo

rf o

n 10

/08/

13. F

or p

erso

nal u

se o

nly.


Probst M. 2013. Robust shape optimization for incompressible flow of shear-thinning fluids. PhD thesis. RWTHAachen Univ., Aachen, Ger.

Quarteroni A, Rozza G. 2003. Optimal control and shape optimization in aorto-coronaric bypass anastomoses.Math. Models Methods Appl. Sci. 13:1801–23

Rozza G. 2005. On optimization, control and shape design for an arterial bypass. Int. J. Numer. Methods Fluids47:1411–19

Sankaran S. 2009. Stochastic optimization using a sparse grid collocation scheme. Probab. Eng. Mech. 24:382–96Sankaran S, Audet C, Marsden A. 2010. A method for stochastic constrained optimization using derivative-free

surrogate pattern search and collocation. J. Comput. Phys. 229:4664–82Sankaran S, Audet C, Marsden A. 2011. A stochastic collocation method for uncertainty quantification in

cardiovascular simulations. J. Biomech. Eng. 133:031001Sankaran S, Esmaily Moghadam M, Kahn A, Tseng E, Guccione J, Marsden A. 2012. Patient-specific multiscale

modeling of blood flow for coronary artery bypass graft surgery. Ann. Biomed. Eng. 40:2228–42Sankaran S, Humphrey JD, Marsden AL. 2013. An efficient framework for optimization and parameter sensitiv-

ity analysis in arterial growth and remodeling computations. Comput. Methods Appl. Mech. Eng. 256:200–10Sankaran S, Marsden A. 2010. The impact of uncertainty on shape optimization of idealized bypass graft

models in unsteady flow. Phys. Fluids 22:121902Schmidt JP, Delp SL, Sherman MA, Taylor CA, Pande VS, Altman RB. 2008. The Simbios National Center:

systems biology in motion. Proc. IEEE Inst. Electr. Electron. Engl. 96:1266–80Sengupta D, Kahn A, Burns J, Sankaran S, Shadden S, Marsden A. 2012. Image-based modeling of hemody-

namics and coronary artery aneurysms caused by Kawasaki disease. Biomech. Model. Mechanobiol. 11:915–32Serafini DB. 1998. A framework for managing models in nonlinear optimization of computationally expensive functions.

PhD thesis. Rice Univ., Houston, TXShadden SC, Taylor CA. 2008. Characterization of coherent structures in the cardiovascular system. Ann.

Biomed. Eng. 36:1152–62Simpson TW, Korte JJ, Mauery TM, Mistree F. 1998. Comparison of response surface and Kriging models for

multidisciplinary design optimization. Presented at AIAA/USAF/NASA/ISSMO Symp. Multidiscip. Anal.Optim., 7th, St. Louis, MO, AIAA Pap. 1998-4755

Steinman DA. 2002. Image-based computational fluid dynamics modeling in realistic arterial geometries. Ann.Biomed. Eng. 30:483–97

Sundareswaran KS, de Zelicourt D, Pekkan K, Jayaprakash G, Kim D, et al. 2007. Anatomically realisticpatient-specific surgical planning of complex congenital heart defects using MRI and CFD. Conf. Proc.IEEE Eng. Med. Biol. Soc., pp. 202–5. New York: IEEE

Tang BT, Cheng CP, Draney MT, Wilson NM, Tsao PS, et al. 2006. Abdominal aortic hemodynamics inyoung healthy adults at rest and during lower limb exercise: quantification using image-based computermodeling. Am. J. Physiol. Heart Circ. Physiol. 291:H668–76

Taylor CA, Cheng CP, Espinosa LA, Tang BT, Parker D, Herfkens RJ. 2002. In vivo quantification of bloodflow and wall shear stress in the human abdominal aorta during lower limb exercise. Ann. Biomed. Eng.30:402–8

Taylor CA, Hughes TJR, Zarins CK. 1998. Finite element modeling of blood flow in arteries. Comput. Meth.Appl. Mech. Eng. 158:155–96

Torczon V. 1997. On the convergence of pattern search algorithms. SIAM J. Optim. 7:1–25Valen-Sendstad K, Mardal KA, Mortensen M, Reif BA, Langtangen HP. 2011. Direct numerical simulation

of transitional flow in a patient-specific intracranial aneurysm. J. Biomech. 44:2826–32Valentin A, Humphrey JD. 2009. Evaluation of fundamental hypotheses underlying constrained mixture

models of arterial growth and remodelling. Philos. Trans. R. Soc. A 367:3585–606Vignon-Clementel IE, Figueroa CA, Jansen KE, Taylor CA. 2006. Outflow boundary conditions for three-

dimensional finite element modeling of blood flow and pressure in arteries. Comput. Meth. Appl. Mech.Eng. 195:3776–96

Wang Q, Moin P, Iaccarino G. 2009. Minimal repetition dynamic checkpointing algorithm for unsteadyadjoint calculation. SIAM J. Sci. Comput. 31:2549–67



Ann

u. R

ev. F

luid

Mec

h. 2

014.

46. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

by W

IB62

42 -

Uni

vers

itaet

s- u

nd L

ande

sbib

lioth

ek D

uess

eldo

rf o

n 10

/08/

13. F

or p

erso

nal u

se o

nly.


Wilson N, Wang K, Dutton R, Taylor C. 2001. A software framework for creating patient specific geometricmodels from medical imaging data for simulation based medical planning of vascular surgery. In MedicalImage Computing and Computer-Assisted Intervention—MICCAI 2001, ed. WJ Niessen, MA Viergever,pp. 449–56. Lect. Notes Comput. Sci. 2208. New York: Springer

Witteveen JAS, Iaccarino G. 2013. Simplex stochastic collocation with ENO-type stencil selection for robustuncertainty quantification. J. Comput. Phys. 239:1–21

Womersley J. 1957. An elastic tube theory of pulse transmission and oscillatory flow in mammalian arteries. Rep., AirRes. Dev. Command, US Air Force, Wright Air Dev. Cent., Wright-Patterson Air Force Base, OH

Xiu D, Hesthaven JS. 2005. High-order collocation methods for the differential equation with random inputs.SIAM J. Sci. Comput. 27:1118–39

Yang W, Feinstein JA, Marsden AL. 2010. Constrained optimization of an idealized Y-shaped baffle for theFontan surgery at rest and exercise. Comput. Methods Appl. Mech. Eng. 199:2135–49

Yang W, Feinstein JA, Shadden S, Vignon-Clementel I, Marsden AL. 2013. Optimization of a Y-graft designfor improved hepatic flow distribution in the Fontan circulation. J. Biomech. Eng. 135:011002

Yang W, Vignon-Clementel I, Troianowski G, Reddy V, Feinstein JA, Marsden AL. 2012. Hepatic blood flowdistribution and performance in traditional and Y-graft Fontan geometries: a case series computationalfluid dynamics study. J. Thorac. Cardiovasc. Surg. 143:1086–97

Yushkevich PA, Piven J, Hazlett HC, Smith RG, Ho S, et al. 2006. User-guided 3D active contour segmen-tation of anatomical structures: significantly improved efficiency and reliability. Neuroimage 31:1116–28

Zamir M. 2000. The Physics of Pulsatile Flow. New York: SpringerZhang Y, Bazilevs B, Goswami S, Bajaj CL, Hughes TJR. 2007. Patient-specific vascular NURBS modeling

for isogeometric analysis of blood flow. Comput. Meth. Appl. Mech. Eng. 196:2943–59Zorin D, Schroder P, Sweldens W. 1997. Interactive multiresolution mesh editing. Proc. SIGGRAPH’97,

pp. 259–68. New York: ACM

RELATED RESOURCES

NOMAD project, providing derivative-free optimization software. http://www.gerad.ca/nomad/

SimVascular open-source project at Stanford’s Simbios center. http://simtk.orgTaylor CA, Draney MT. 2004. Experimental and computational methods in cardiovascular fluid

mechanics. Annu. Rev. Fluid Mech. 36:197–231

546 Marsden


Ann

u. R

ev. F

luid

Mec

h. 2

014.

46. D

ownl

oade

d fr

om w

ww

.ann

ualr

evie

ws.

org

by W

IB62

42 -

Uni

vers

itaet

s- u

nd L

ande

sbib

lioth

ek D

uess

eldo

rf o

n 10

/08/

13. F

or p

erso

nal u

se o

nly.

http://www.gerad.ca/nomad/

http://www.gerad.ca/nomad/

http://simtk.org

Documents

Optimization in Cardiovascular Modeling