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Comput. Methods Appl. Mech. Engrg. 197 (2008) 1890–1905 www.elsevier.com/locate/cma
A computational framework for derivative-free optimization of cardiovascular geometries Alison L. Marsden a,*, Jeffrey A. Feinstein b, Charles A. Taylor c a
Mechanical and Aerospace Engineering, University of California, San Diego, USA b Pediatrics Department, Stanford University, USA c Bioengineering Department, Stanford University, USA
Received 12 December 2006; received in revised form 4 December 2007; accepted 7 December 2007 Available online 31 December 2007
Abstract Predictive simulation and optimization-based design tools have great potential to improve the design of surgeries and interventions used in cardiovascular medicine. The present work builds upon recent advances in blood flow simulation capabilities to develop tools for optimization. A framework for coupling optimal shape design to time-accurate three-dimensional blood flow simulations in idealized cardiovascular geometries is presented. The optimization method employed is a tailored version of the surrogate management framework, a method that was developed previously for expensive functions with little or no gradient information. In this method, we employ a derivative-free approach using surrogates for increased efficiency together with mesh adaptive direct search to guarantee convergence to a local minimum. The optimization procedure has been fully automated to include the generation and parameterization of three-dimensional solid geometries, mesh generation, finite element flow computation, post processing and optimization. This methodology is demonstrated on three model problems that are representative of typical cardiovascular geometries: a stenosis, a vessel bifurcation modeled on Murray’s law, and an end-to-side anastomosis. These problems are used to illustrate the importance of the choice of cost function and the performance of the optimization algorithm. This framework for treatment optimization will be applicable to a wide range of cardiovascular surgical or catheter-based interventions in future work. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Optimization; Surrogate; Cardiovascular; Murray’s law; Hemodynamics; Finite elements
1. Introduction The coupling of shape optimization to cardiovascular blood flow simulations has potential to improve the design of current surgeries and to eventually enable the optimization of surgical designs for individual patients. Recent work has demonstrated that numerical simulation of blood flow is a viable tool for surgical planning and device design as well as for understanding mechanisms and progression of cardiovascular disease. Simulations have played a key role in understanding hemodynamics of bypass grafting [29,36,57], cardiovascu*
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lar treatment planning [69], atherosclerosis in the carotid artery [56,50] and the abdominal aorta [71], cerebrovascular flow [12,62,27], the effects of exercise on aortic flow conditions [67,73], congenital cardiovascular disease [43,49,48, 28], and coronary stents [31,32]. Key advances in cardiovascular simulation capabilities in recent years also include image-based modeling [76,69,70,72,51,64,63,13], the development of boundary conditions that allow for physiologic pressure levels [75,33,19] and efficient fluid-structure interaction algorithms [17,58,20]. To date, the vast majority of blood flow simulations have considered either fixed geometries or have examined a small number of geometry variations using a ‘‘trial-anderror” approach to surgical design. The medical field has also relied on a trial-and-error approach when developing
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new surgical designs. In contrast, traditional engineering fields such as aeronautics and automotive design commonly rely on optimal shape design and simulation in their design process. Cardiovascular medicine could benefit from employing similar predictive design tools. To produce clinically relevant design results using optimization, several challenges must first be overcome. First, simulation results must be physiologically accurate and methods thoroughly validated. Second, appropriate measures of performance (cost functions) for cardiovascular designs must be defined based on physiologic information. Third, the choice of optimization method must be appropriate for expensive, time-dependent, three-dimensional fluid mechanics problems. And fourth, new tools should be developed for the efficient parameterization of patientspecific geometries. Some progress has been made on the first, second and fourth items, but the issue of optimization methodology for unsteady 3D blood flow problems on unstructured meshes has not been previously addressed. The focus of this paper will therefore be on second and third points. We will introduce an efficient and flexible optimization framework for unsteady, computationally expensive blood flow problems. We will also discuss issues of cost function choice and numerical methods for blood flow simulations. One general distinction among optimization techniques is between gradient-based methods and derivative-free methods. The choice of method for a particular problem depends on factors such as the cost of evaluating the function, the availability of gradient information, the level of noise in the function, and the complexity of implementation. Coupling optimization algorithms to complex fluid mechanics simulations (such as blood flow problems) poses several important challenges. To perform physiologically realistic cardiovascular design, each cost function evaluation requires a time-dependent, three-dimensional solution of the Navier–Stokes equations. These problems are computationally expensive to evaluate, and obtaining gradient information with traditional methods poses formidable challenges. First, the combination of steps required in coupling shape optimization with CFD (meshing, geometry generation and Navier–Stokes solution) makes analytic determination of gradients impossible without specialized numerical methods (e.g. differentiable mesh movement schemes). For example, the issue of unstructured meshing and complex geometries in optimization for steady flow aerodynamics has been examined by Burgreen, Peraire, and others [11,54,15,16]. Due to these challenges, gradient information is generally obtained numerically using adjoint solutions or directly using finite difference methods. Important strides have been made using adjoint solvers in the work of Jameson and others [23,24] for efficiently obtaining gradient information in aerodynamics problems, however challenges remain for unsteady flow problems due to the need to store large time histories. Gradients obtained directly
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using finite differences are often prohibitively expensive for large problems and can be easily drowned out by numerical noise. In spite of these challenges, there has been some important progress in applying gradient-based optimization methods to blood flow problems, mostly focusing on 2D and/or steady flow problems. This work includes examination of non-Newtonian effects in shape optimization by Abraham, Behr and Heinkenschloss [2,1], optimization of blood pump components [5,10] and work by Quarteroni and others on shape design for arterial bypasses [59,60,3,4]. However, applying traditional gradient methods in a fully time-dependent setting using unstructured solvers, complex geometries and constraints, as are common in cardiovascular flow problems, remains a significant challenge. With these issues in mind, the present work presents a framework for derivative-free optimization. Derivative-free methods include pattern search methods, approximation models, response surfaces and evolutionary algorithms. There are several considerations in choosing a tractable optimization method for blood flow problems. The primary concern is the computational expense of the function evaluations, especially when each function evaluation requires a three-dimensional solution of the time-dependent Navier–Stokes equations. Other considerations are availability of gradient information, and convergence properties of the optimization method. Previous work in derivative-free optimization and fluid mechanics has successfully applied the surrogate management framework to the constrained optimization of an airfoil trailing-edge for suppression of vortex shedding noise in laminar flow [44,45] and for the suppression of broadband noise in turbulent flow [46]. In addition, the application of derivative-approximating trust region methods to several steady flow fluid mechanics problems has been explored by Lehnhauser and Shafer [34]. Building on these recent advances, we are now in a position to couple blood flow simulations to optimization algorithms for the purpose of designing new surgeries and treatments. In this work, we present a framework for derivative-free optimization of idealized cardiovascular geometries using the surrogate management framework (SMF) [9,61] together with mesh adaptive direct search methods [8]. The SMF method relies on convergence theory of derivative-free pattern search methods and incorporates surrogate functions for improved efficiency. Three idealized but representative cardiovascular optimization model problems are presented. The first two cases (a stenosis and a bifurcation) are problems for which we have some basis to predict the outcome of the optimization. The third case is the optimization of an end-to-side anastomosis, for which we explore the relationship between angle of incidence of the anastomosis and the ratio of the outlet resistance values. These problems demonstrate the potential of the optimization framework for cardiovascular treatment planning.
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2. Optimization method The general optimization problem with linear bound constraints is formulated as follows: minimize
J ðxÞ
subject to
x 2 X:
ð1Þ
where J : Rn ! R is the cost function, and x is the vector of design parameters. The parameter space is defined by X ¼ fx 2 Rn j l 6 x 6 ug, where l 2 Rn is a vector of lower bounds on x and u 2 Rn is a vector of upper bounds on x. In the present problem, the function J ðxÞ depends on the solution of the Navier–Stokes equations, and the cost function is computed in a post-processing step. In this work, the surrogate management framework is implemented and demonstrated for optimization in three cardiovascular model problems. In each of these problems, J is computed directly from the Navier–Stokes solution in a post-processing step, and x is a vector of parameters that describe the shape of the vessel of interest. 2.1. Surrogate management framework This section gives an overview of the steps required in the surrogate management framework (SMF), the optimization algorithm that was used in this work. The SMF method was introduced by Booker et al. [9] and belongs to a general class of pattern search optimization methods with established convergence theory. As in pattern search methods, all points that are evaluated by the SMF algorithm are restricted to lie on a mesh. In addition, SMF employs a surrogate function as a predictive tool, which can greatly increase the efficiency of the algorithm over traditional pattern search methods. The SMF algorithm is comprised of two steps, SEARCH and POLL. First, the SEARCH step uses the surrogate to identify one or more points that are likely to improve the cost function. The SEARCH step provides means for local and global exploration of the parameter space, but is not strictly required for convergence. Because the SEARCH step is not integral to convergence, it affords the user a great deal of flexibility and may be adapted by the user to a particular engineering application. Second, the POLL step provides the convergence framework for the algorithm. In this work, we have employed mesh adaptive direct search in the POLL step, as discussed in the next section. The most common choice of surrogate in the SEARCH step, and that used in this work, is an interpolating function. However a surrogate is generally defined as anything that takes the place of the expensive ‘‘true function” in optimization. Evaluation of the surrogate is generally less computationally expensive than evaluation of the true function and can be used to increase the efficiency of the algorithm. Other examples of surrogates could include reduced-order physics models, lumped parameter models, coarse mesh simulations, or neural network models.
This work applies Kriging as the surrogate function. Kriging is an interpolating surrogate originating from the field of geostatistics that relies on the use of spatial correlation functions. Because it is easily extended to multiple dimensions, it is attractive for optimization problems with many parameters. Detailed derivations of Kriging functions are available in [44,41]. In this work, the MATLAB DACE package [41] was incorporated for surrogate function building. Construction of the Kriging surrogate is performed by first generating a set of initial data points using latin hypercube sampling (LHS) [47]. Latin hypercube sampling guarantees a well distributed set of points in the parameter space, for which each input variable has all portions of its range represented. The cost function J ðxÞ is evaluated at each of the initial points fx1 ; . . . ; xm g and the Kriging surrogate function is constructed. The purpose of the surrogate is to predict the value of the function at particular locations in the parameter space, and to aid in identifying areas with low function values. As the optimization proceeds, the surrogate accuracy is improved through incorporation of new data. All points evaluated in either the SEARCH or POLL step of the SMF algorithm must lie on a mesh in the parameter space. The vectors defining the mesh directions must positively span Rn [37]. If we define D as a matrix whose columns form a positive spanning set in Rn , then the set of mesh points surrounding a point x are given by Mðx; DÞ ¼ fx þ DDz : z 2 NnD g;
ð2Þ
where D is the mesh size parameter, and nD is the number of columns in D. A positive spanning set is simply the set of positive linear combinations of the vectors making up the mesh directions. The set is considered positively independent if none of the vectors in a given set can be formed from a non-negative combination of the others [14]. A positive basis is then defined as a positively independent set whose positive span is Rn . Following the above definition, the mesh in SMF may be refined or coarsened by changing D > 0, and the mesh may be rotated from one iteration to the next. Further technical restrictions on the mesh definition are discussed in [74]. The SMF algorithm begins with a SEARCH step, after construction of the initial surrogate has been completed. In the SEARCH step, optimization is performed on the surrogate to predict the location of one or more minimizing points, and the function is evaluated at these points. The SEARCH is considered successful if an improved cost function value is found, at which point the surrogate is updated, and another search step is performed. The SEARCH is considered unsuccessful if it fails to find an improved point, and a POLL step is performed. The basis for convergence of the SMF algorithm is given by the POLL step. In the POLL step, the function is evaluated at a set of points that form a positive spanning set of directions in order to evaluate whether the current best point is a mesh local optimizer. A set of n þ 1 POLL points
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are required to generate a positive basis, where n is the number of optimization parameters. In this work, the poll set is chosen using mesh adaptive direct search (MADS) [8], described in the next section. It should be noted that as soon as an improved point is found, it is no longer necessary to evaluate the remainder of the set of POLL points, thereby reducing the number of required function evaluations. If the POLL produces an improved point, then a SEARCH step is performed on the current mesh. Otherwise, if no improved points are found, then the current best point is defined to be a mesh local optimizer as in [6]. This terminology is to acknowledge that it may not be a local minimizer of the objective function on the mesh since the POLL set is typically a subset of all the points in the mesh adjacent to the POLL center. For greater accuracy, the parameter space mesh may be refined, at which point the algorithm continues with a SEARCH. Convergence is reached when a local minimizer on the mesh is found, and the mesh has been refined to the desired accuracy. Each time new data points are found in a SEARCH or POLL step, the data is added to the surrogate and it is updated. The steps in the algorithm are summarized below, where the set of points in the initial mesh is M 0 , the mesh at iteration k is M k , and the current best point is xk . 1. SEARCH (a) Identify a finite set T k of trial points on the mesh Mk. (b) Evaluate J ðzÞ for all trial points z 2 T k M k . (c) If for any trial point in T k , J ðzÞ < J ðxk Þ, a lower cost function value has been found, and the SEARCH is successful. Increment k and go back to (a). (d) Else, if no trial point in T k 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 X k as the set of mesh points adjacent to xk in these directions. (b) If J ðxpoll Þ < J ðxk Þ for any point xpoll 2 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 X k improves the cost function, POLL is unsuccessful. (i) If convergence criteria are satisfied, a converged solution has been found. STOP. (ii) Else if convergence criteria are not met, refine mesh. Increment k and go to SEARCH. Because the method has distinct SEARCH and POLL steps, convergence theory for the SMF method reduces to convergence of pattern search methods. Convergence of the SMF method is discussed at length by Booker et al. [9] and by Serafini in [61]. Pattern search convergence the-
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ory is presented by Audet and Dennis [6], Torczon [74] and Lewis and Torczon [38–40]. 2.2. Mesh adaptive direct search The three optimization problems presented in this work incorporate a new method for generating poll sets called mesh adaptive direct search (MADS), as recently described by [8]. Compared to standard generalized pattern search (GPS) polling, MADS offers greater flexibility and is capable of generating a dense set of directions. This means that as the mesh becomes infinitely fine, the number of possible poll directions approaches infinity. This results in stronger convergence theory, especially with regard to non-linear constraints. We refer the reader to [8] for more details and proof of convergence. MADS has also been applied previously in a fluid mechanics design problem by Marsden et al. [46]. MADS differs from GPS by decoupling the frame size from the mesh size and allowing these parameters to evolve independently. This difference is illustrated in Figs. 1 and 2. In addition to the mesh size parameter Dmk , the poll size parameter Dpk is introduced. The poll size parameter defines a frame around the poll center containing all points that could be used to define a poll direction. Fig. 1 shows the process of mesh refinement, and corresponding GPS poll sets in two dimensions. In this case, the number of points contained in the frame (bold line outlined box) remains constant (9 in this example) as the mesh is refined. In contrast, Fig. 2 shows a sequence of three successively finer meshes used for MADS polling. As the mesh is refined, each time by 1/4, the frame size shrinks more slowly, so that with each refinement the frame contains a greater number of candidate points that can be used to define a poll direction. We therefore see that MADS allows the number of possible poll directions increases as the mesh is refined. In many applied engineering problems such as those presented here, it is not practical to perform many mesh refinement steps due to cost considerations. As long as Dmk < 1, MADS allows a larger number of possible poll directions, offering an advantage over GPS. When a series of POLL steps is performed, a more thorough local search of the parameter space results. In addition, because MADS generates a dense set of poll directions, these directions will automatically conform to the boundary of the parameter space when polling on or near the boundary. Whereas GPS requires 2n poll points near the boundary to guarantee convergence, MADS requires only n þ 1. Although the problems presented here apply only simple bound constraints, it is worth noting that MADS offers stronger convergence in problems with non-linear constraints. In a problem with non-linear constraints, the boundary of the feasible region is unknown. Because GPS polling is limited in aggregate to a finite set of directions, it is not guaranteed that the polling directions will conform to the boundary, and may miss the descent direc-
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Fig. 1. Example of GPS frames P k ¼ fxk þ Dmk d : d 2 Dk g ¼ fp1 ; p2 ; p3 g for different values of Dmk ¼ Dpk . In all three figures, the mesh M k is the intersection of all lines.
Fig. 2. Examples of MADS polling. As the mesh is refined, the set of possible poll points grows.
tion unless conforming directions are added. In contrast, MADS polling guarantees that a poll point will eventually land in the descent direction because the aggregate poll directions will fill the space as the number of mesh refinements approaches infinity. This difference makes it possible to use the barrier approach to constraints together with MADS polling, in which the cost function value is simply set to infinity if constraints are violated. While the focus of the present work is on unconstrained optimization, constraints will arise in cardiovascular applications in future work. These could include constraints on surgical feasibility, resistance to flow, performance during exercise, or bounds on wall shear stress values. 3. General implementation To have a fully automated optimization framework, evaluation of the cost function must be coupled to the optimization algorithm in a fully automated fashion. In this work, a single cost function evaluation is a multi-step process that includes automated geometry generation, meshing, running the flow solver and performing a postprocessing step to evaluate the cost function value. Fig. 3 shows the steps that must be linked together to form an automated loop. Automation was achieved through a series of scripts that called external programs (such as the flow solver) and file input/output to pass information from one
Parameters
Optimization
Surface lofting
Geometric solid model
Cost funtion
FE mesh
Flow solver
Fig. 3. Optimization procedure. The boxes shown represent steps that must be fully linked to have an automated optimization framework.
box to another. Each step in the automated loop is described below. Optimization was performed using the SMF method with MADS polling, as described in the previous sections. In all examples in this work, two or three mesh refinements were done in the parameter space and convergence was reached when a poll step failed on the final mesh refinement level. Each search step in the optimization algorithm included two function evaluations. The first was the minimum predicted by the Kriging surrogate function. The second search evaluation was a point from the poll set that was predicted by the surrogate to have the lowest function value. Geometric models were created automatically by generating a series of rings that were lofted together with splines
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using custom software [76,77]. In all cases, the geometry was cylindrical and defined according to an analytical parameterization, for example using the angle between intersecting cylinders or the radius values of the rings. The specific parameter definitions for each of the three model problems will be discussed in the next section. Once a solid model was generated, a tetrahedral mesh was generated for each geometry considered by the optimization algorithm. Next, to simulate blood flow, a custom finite element solver was used in which the time-accurate Navier–Stokes equations are discretized with piecewise linear-in-space elements, second order accuracy in time, and stabilization. Further details on the finite element method can be found in work of [70,72,69,68,25,75]. Verification of the methods employed using analytical solutions can be found in [72], in vitro validation in [72,30], and preliminary in vivo validation in [29]. A Newtonian approximation for the viscosity was assumed with a value of 0.04 g/(cm s) and the density of blood was 1.06 g/cm3. A rigid-wall approximation was employed. The time-step for all cases was chosen according to the CFL (Courant, Friedrichs, and Lewy) condition. All inflow profiles were mapped to a parabolic velocity profile. Resistance boundary conditions [75] were applied at the outlets of all models using values chosen to match typical physiologic pressure levels. Using the results of the flow simulation, the cost function was computed as a post-processing step. In the next three sections we present three model problems in which optimal shape design is applied to idealized cardiovascular geometries. These are a carotid artery stenosis, a vessel bifurcation, and a femoral artery end-to-side anastomosis. Details on simulations, geometry parameterization and cost function definitions for all three cases are outlined here and the results for each case are described in Section 5. 4. Model problem definitions and methods
bounds on the parameters defining the stenosis geometry were 0:1 < r1 < 0:5 cm and 0:1 < r2 < 0:5 cm, thereby allowing for both stenosis and aneurysm shapes in the design space. Simulations of flow through this geometry were run using a finite element mesh size of approximately 200,000 elements. In this example a steady inflow condition was used with a flow rate of 4.4 cc/s. The resistance value applied at the outflow boundaries was chosen to achieve a pressure level of approximately 68 mmHg. These values are based on a typical radius, diastolic flow rate, and diastolic pressure level for a human carotid artery. We compare results for two stenosis optimization cases that use cost functions based on energy efficiency and wall shear stress, respectively. The first case aims to maximize energy efficiency. Energy efficiency is computed by integrating the energy flux over the inlets and outlets of the model. Energy efficiency has been used previously as a measure of performance in simulations of the Fontan, a surgery performed for severe congenital heart defects [43]. The importance of computing the exact energy integral was discussed in [21]. By defining a control volume around the model, the Reynolds transport theorem is used to integrate energy flux over model inlets and outlets. The energy dissipation (neglecting gravitational effects) is given by N in Z N out X X 1 Ediss ¼ p þ qu2 u dA 2 INi i¼1 i¼1 Z 1 2 p þ qu u dA ð3Þ 2 OUTi and the energy efficiency is Eeffic ¼ Eout =Ein ;
In the first model problem we optimize the stenosis shape shown in Fig. 4. The motivation behind this example is to verify that the optimization algorithm is successful in identifying a straight vessel as the optimal solution. The geometry is parameterized with two variables, r1 and r2 , the radius of rings that make up the stenosis region. In our model, the outer radius of the stenosis model was fixed at rv ¼ 0:3 cm, the size of a typical carotid artery. The
ð4Þ
where Ein is the first term in Eq. (3) and Eout is the second term. A cost function to minimize energy losses is thus defined by J ¼ 1 Eeffic :
4.1. Model problem 1: Carotid artery stenosis
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ð5Þ
Wall shear stress and related quantities such as oscillatory shear index (OSI) are also recognized as important factors in vascular biology [55,67], the localization of disease [79,26] and efficacy of interventions [42]. To demonstrate the influence of the choice of cost function on the optimization results, we define an alternate cost function based on uniformity of wall shear stress J¼
maxðsw Þ minðsw Þ ; meanðsw Þ
Fig. 4. Parameterization of stenosis geometry.
ð6Þ
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where maxðsw Þ and minðsw Þ are the maximum and minimum values of wall shear stress found on the model walls, respectively, and meanðsw Þ is the integral of wall shear stress on the model walls divided by the total wall surface area. 4.2. Model problem 2: vessel bifurcation Our second model problem is a computational reproduction of Murray’s law. In 1926, Murray solved a simple analytical optimization problem to come up with a law governing the branching patterns of vessels. Known as ‘‘Murray’s law,” the result is that the radii of branching vessels are governed by a power law rap ¼ rad 1 þ rad 2
ð7Þ
where rp is the radius of the parent vessel and rd 1 and rd 2 are the radii of the daughter vessels. Murray found a value of a ¼ 3 in the above expression and the steps of his derivation are outlined below. A symmetric bifurcation model with two branches ðrd 1 ¼ rd 2 Þ is shown in Fig. 5. To arrive at the above expression, Murray proposed a cost function made up of a power loss term and a second term measuring the metabolic cost of maintaining the blood volume so that J ¼ DpQ þ b V
ð8Þ
where Dp is the pressure drop in a vessel of circular cross section, V is the vessel volume, Q is the flow rate, and b is a metabolic constant. He made the assumption of Poiseuille’s solution (fully developed parabolic velocity profile everywhere) in a single vessel to obtain J¼
8Q2 ll þ blpr2 pr4
ð9Þ
where l is the viscosity of blood and l is the length of the vessel. Minimizing the above expression he then obtained
dJ 32Q2 ll ¼ þ 2blpr ¼ 0 dr pr5
ð10Þ
resulting in Q ¼ kr3
12
ð11Þ
pb where k ¼ 16l . Together with conservation of mass for a vessel with two branches, Murray arrived at the power law expression in Eq. (7). Fixing the ratio of parent and daughter vessels, Murray then went on to compute the optimal branch angle in a second paper [52], arriving at an optimal angle of 37.5°. Our objective in this example is to reproduce Murray’s optimization problem computationally using a threedimensional vessel geometry and a time-dependent solution of the Navier–Stokes equations. Based on Murray’s problem, we have the following cost function:
J ¼ Ediss þ bV ;
ð12Þ
where the energy dissipation Ediss , is computed using the energy fluxes on the inlet and two outlets of the model according to Eq. (3). Unlike in Murray’s analytic solution, a value for the constant b must be determined in order to perform the optimization. Published values for b are shown in Table 1. The values published by [53,78] have been found using Eq. (10) with typical values of flow rates, blood pressure and blood viscosity for a human. The value published by [66] was based on measured oxygen consumption rates for red and white blood cells in rats. The range of b values shown in the table covers nearly two orders of magnitude. It is fortuitous that this constant drops out in the analytical solution to Murray’s problem (Eq. (7)). However the value of b is clearly paramount to obtaining the correct solution when using the full cost function because it determines the relative importance of the energy dissipation term and the metabolic cost. For
14 12 10 8 6 4 2
0
0.2
0.4
0.6
0.8
1
Fig. 5. Geometry parameterization for bifurcating vessel. Optimization is performed using rp , rd , and h as parameters. Right: examples of resulting geometry for h = 20, 30 and 45.
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Table 1 Published values for constant b in Murray’s birfurcation cost function Source
b (erg/cc s)
Murray [52] Zamir [78] Taber [66]
19,300 7110 778
example, if the second term is large enough the optimization will minimize the volume of the vessel, and energy losses in the fluid will not play a role. On the other hand, if the value of b is very small, the optimization will minimize energy losses. A three-dimensional computational version of Murray’s problem is shown in Fig. 5. Modeled on Murray’s geometry, we consider a blood source S and two fixed locations A and B that require blood delivery. The bifurcation model has three geometric parameters: rp , the radius of the parent vessel, rd , the radius of both daughter vessels, and h the angle between the vertical axis and the branch vessels. Keeping points A and B fixed, the length of the all vessels changes with the angle. Unlike Murray’s original problem, the computational framework allows us to optimize for the radii and angle simultaneously. Three examples of shapes in the parameter space are shown on the right of Fig. 5. A typical carotid artery flow waveform is shown in Fig. 5. A series of steady flow optimization cases are presented in Section 5 that use the mean flow rate, 6.46 cc/s, of this waveform as an inflow boundary condition. Optimization using the pulsatile time-dependent Navier–Stokes solution was also carried out, in which case the waveform is applied at the model inlet face. The cost function value for this case is averaged over one cardiac cycle after the simulation has been run sufficiently long to minimize the effect of initial transients. Mesh sizes for these cases were between 150,000 and 200,000 elements and a time step size of 0.01 s was used. Bounds on the parameter values for the optimization were set as follows: 0:3 6 rp 6 0:4, 0:2 6 rd 6 0:3 and 20 6 h 6 60 .
Fig. 6. Geometry parameterization for end-to-side anastomosis model.
[42,22]. Computational studies have examined flow fields and wall shear stress maps and have quantified the effect of geometry changes on shear stress, shear stress gradients, oscillatory shear index, and near wall residence times [22,36,35]. An overview of typical treatment and waveforms for femoral artery bypass grafts can be found in [65]. In this example, we optimize the attachment angle and graft radius of an idealized end-to-side anastomosis model. The geometry parameterization for this model is shown in Fig. 6. Flow rates for these simulations were extracted from doppler-ultrasound data of the femoral artery of a healthy adult male subject. Steady simulations for this case were run using a mesh size of approximately 450,000 elements, a time step size of 0.01 s, and an inflow rate of 2.5 cc/s. The range of parameter values allowed by the optimization was 0:2 < ra < 0:35 and 20 < h < 90 . Resistance boundary conditions were used with values chosen to achieve a mean pressure level of 100 mmHg. In defining a cost function for this problem, our goal is to reduce localized areas of low wall shear stress. We therefore define the cost function to be the area integral over the model surface for which the wall shear stress value falls below a threshold value Z 1 if s 6 scrit J¼ f dS; f ¼ ð13Þ 0 otherwise: S To ensure that the integral only includes localized areas of low shear, the choice of threshold value will depend on flow conditions and is discussed in Section 5.
4.3. Model problem 3: end-to-side anastomosis 5. Results The need for end-to-side anastomoses arises in many types of bypass surgeries including coronary artery bypass grafting and femoral artery bypass grafting. In this example we construct an idealized model of a Femoro-popliteal bypass graft to demonstrate the potential role of optimization in designing bypass grafts and numerous other treatments to improve hemodynamics. Femoro-popliteal bypass surgery is used to bypass blockages in the femoral artery. The bypass graft is typically either native venous vessel or synthetic graft. A common problem, especially with synthetic grafts, is graft failure due to development of thromboses or intimal hyperplasia at the distal anastomosis. Several groups have established correlations between areas of low wall shear stress and intimal thickening using experimental techniques
5.1. Model problem 1: carotid artery stenosis Results from the stenosis optimization problem using the efficiency cost function (Eq. (5)) are shown in Fig. 7. The right plot shows the flow solution in the optimal vessel shape (velocity, pressure and wall shear stress) and the left plot shows the history of cost function reduction during optimization. Surprisingly, the optimal shape using this cost function is slightly aneurysmal. While this result was non-intuitive at first, examination of the wall shear stress on the vessel wall demonstrates that lower wall shear stress occurs at the wall in the aneurysmal region. Because the flow is steady and the Reynolds number is low, the lower shear stress in this region results in a smaller pressure drop
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Fig. 7. Results of stenosis optimization problem using efficiency cost function. Left: Cost function vs. number of function evaluations shows convergence history of optimization. Right: optimal shape is slightly aneurysmal.
Table 2 Comparison of optimization results for stenosis model problem using energy efficiency and wall shear stress cost functions Cost function
rv
r1
r2
Energy efficiency Uniform wall shear
0.3 0.3
0.340 0.3
0.372 0.3
and less energy dissipation. The parameter values for the optimal solution are listed in Table 2. Fig. 8 shows the results of optimization using the wall shear stress cost function defined in Eq. (6). This cost function was defined to reduce the amplitude of shear stress variation over the surface of the model. Our results verify that the optimal shape is a perfectly straight vessel (Table 2). The left of Figs. 7 and 8 show the cost function history for each of the two cost functions. The plots show all cost function improvements resulting from either search or poll steps. Convergence of the optimization in each case was
reached after three refinements of the parameter space mesh. Using the wall shear stress cost function, the final solution was reached in the first iteration of the optimization and fewer function evaluations were required to reach convergence. These results illustrate the importance of choosing a cost function that relates to the physiology of the problem. It is not surprising that the two cost functions produced different results. Our results demonstrate that, with the appropriate cost function choice, optimization is successful in identifying a straight vessel. This agrees with engineering intuition that a straight vessel should be the optimal solution, and that deviations from this generally represent disease states, either as aneurysms or stenoses. 5.2. Model problem 2: vessel bifurcation Optimization results for the vessel bifurcation problem are summarized in Table 3. Murray’s analytical solution
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Fig. 8. Results of stenosis optimization problem using wall shear stress cost function. Left: Cost function vs. number of function evaluations shows convergence history of optimization. Right: optimal shape is a straight vessel.
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volume of the optimal solutions for a range of b values. This result confirms that larger values of b result in optimal solutions with smaller volume and higher energy dissipation and vice versa. At the lower extreme, we observe that a value of b ¼ 0 produces an optimal shape with the largest possible volume within the parameter bounds allowed by the optimization. In this case the optimization is minimizing the energy dissipation and the weight on the volume term is zero. This is consistent with Poiseuille’s solution for flow through a tube. In that case minimizing the pressure losses will increase the radius of the tube since pressure losses are inversely proportional to r4 . Fig. 10 compares the optimal solutions obtained using values of b reported by Murray (left column), Zamir (center) and Taber (right). In the top row we point out that the velocity in the parent vessel is much lower when optimizing with Taber’s value due to the larger radius. As expected, the flow profile deviates significantly from a Poisseuille solution near the bifurcation. The bottom row of the figure shows pressure results, and we confirm that a mean pressure of about 100 mmHg is achieved for all three models owing to the use of resistance boundary conditions. We also confirm that the pressure drop is larger using Murray’s value of b than in the other two cases because the dissipation term was not weighted as heavily in this case. The last line of Table 3 gives results of the optimization using a pulsatile inflow condition with the waveform shown in Fig. 5. We note that for the pulsatile case, Taber’s value of b ¼ 778 results in reasonable agreement with Murray’s law, producing a value of a ¼ 3:43 and an angle of h ¼ 29:56 . Using pulsatile inflow conditions, more energy is dissipated compared to the steady flow case with the same mean flow rate. Thus, the first (energy dissipation) term in the cost function is weighted more heavily in the optimization compared to the steady inflow case. This is consistent with our observation that a smaller value of b was needed in the pulsatile flow case to achieve an optimal solution with similar values of dissipation and volume as in the best steady flow case.
Table 3 Comparison of optimization results for single vessel bifurcation using several cost functions as the value of b is varied Cost function
rp (cm)
rd (cm)
a
h
Diss (erg)
Vol (cm3)
Murray’s law Ediss + 19,300 V (Murray) Ediss + 7110 V (Zamir) Ediss + 3166 V (mean of Zamir and Taber) Ediss + 778 V (Taber) Ediss + 0 V cardiac cycle mean: Ediss + 778 V
– 0.3
– 0.2
3 1.71
37.5° 20°
– 19,969
– 3.168
0.3 0.33
0.235 0.26
2.83 2.91
48° 40°
10,402 7011
3.959 4.77
0.399 0.4 0.361
0.3 0.3 0.295
2.41 2.41 3.43
52.9° 60° 29.56°
3338 3235 4247
1899
6.87 7.07 5.9
A larger value of b increases the relative importance of the volume term in optimization. Steady flow cases use mean flow value, unsteady cases use cardiac waveform.
is listed in the first line of the table. The table lists results from the optimization using a range of values of the metabolic constant b. The values of b include those reported by [53,78,66] as well as the case for which b ¼ 0. The second line in the table reports the optimal solution of the vessel bifurcation problem using the value of b from Murray’s 1926 paper. Recall that in Murray’s analytical solution the metabolic constant b dropped out. However, Murray nevertheless estimated its value. Using this value, it is interesting to note that we do not obtain agreement with Murray’s analytical solution. In fact, this value of b is large enough that the optimal solution corresponds to the minimum volume configuration of the model that is allowed by our definition of parameter space bounds. The value of a ¼ 1:71 obtained with optimization does not agree with Murray’s value of a ¼ 3 for this value of b. As the value of b is decreased we observe that the best agreement with Murray’s result of a ¼ 3 is obtained using b = 3166 erg/cc s, the average of the values reported by Zamir and Taber. Using this value, the optimal solution results in a ¼ 2:91 and an angle of h ¼ 40 . Fig. 9 consists of two plots showing the trade-off in energy dissipation and
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Fig. 9. Trade-off in dissipation and volume as metabolic parameter b is increased.
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Fig. 10. Contours of velocity magnitude, cm/s (top row) and pressure, mmHg (bottom row) for the optimal shapes using values for the metabolic constant reported by Murray (left, b ¼ 19; 300), Zamir (middle, b ¼ 7110), and Taber (right, b ¼ 778).
5.3. Model problem 3: end-to-side anastomosis Results for the end-to-side optimization problem are summarized in Table 5 Optimization was carried out for three different flow conditions in the native vessel. Outflow resistance values were varied to simulate three distal/proximal flow split scenarios 50/50, 70/30 and 90/10. For all three cases we have assumed complete occlusion in the femoral artery and therefore no competitive flow coming from upstream at the proximal outlet. The first case provides a test case for optimization, and the second two cases are representative of clinical scenarios in which a small amount of flow is needed to feed arteries proximal to the anastomosis. Using a mean blood pressure value of 100 mmHg, the total resistance is Rtot ¼ P =Q ¼ 53; 333 dynes s=cm5 . The outlet resistance values for these flow splits are shown in Table 4 where R1 corresponds to the upstream (proximal) outlet and R2 corresponds to the downstream (distal) outlet. Optimization was performed for each of the three flow split cases using the cost function defined by Eq. (13). The threshold value used in the cost function is chosen for each of the three flow split cases using the Poiseuille solution for that case. To ensure that the regions of low wall shear stress measured by the cost function remain localized, the threshold value scrit was defined to exclude low flow regions on the proximal end of the model. There-
Table 4 Resistance values for three different end-to-side anastomosis distal/ proximal flow splits: 50/50, 70/30 and 90/10 Flow split
R1 (dynes s/cm5)
R2 (dynes s/cm5)
50/50 70/30 90/10
106,666 177,776 533,332
106,666 76,190 59,259
fore, the threshold value is based on the flow to the proximal outlet as follows: du 4lQ ¼ 3 ; ð14Þ scrit ¼ l dr w pr0 where Q is the flow rate in the proximal direction through the native vessel and r0 is the vessel radius. Values for scrit are given in the last column of Table 5. These values compare reasonably well with values of low shear regions given in [42]. Optimal anastomosis shapes and cost function histories for the three flow split cases are shown in Fig. 11 through Fig. 13. In the case of a 50/50 flow split, the optimal graft angle is 90°. This agrees with expectations that equal flow to both outlets of the model should result in a symmetric optimal geometry. Contours of wall shear stress for this case are shown on the right side of Fig. 11. Convergence for all three flow split cases was reached in less than 35
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ra (cm)
h
J (cm2)
scrit (dynes/cm2)
50/50 70/30 90/10
0.329 0.345 0.222
90° 50.5° 20°
1.427 0.849 0.232
1.48 0.89 0.29
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However the results of the optimization clearly demonstrate that when keeping the angle fixed at 20°, a smaller graft size results in a lower cost function value. 6. Discussion
function evaluations. For all cases the optimization was stopped after two mesh refinements in the parameter space. Optimization results for the 70/30 and 90/10 cases are shown in Figs. 12 and 13. As the percentage of flow to the distal outlet is increased the optimal graft angle decreases to 50.5° and then to 20°. The trends in graft size and angle with distal flow fraction are shown in Fig. 14. While there is a clear trend that optimal graft angle decreases with increasing distal flow, the trend in graft size is less clear. In clinical practice it is currently considered best to match the graft size to the size of the native vessel.
We have presented a framework for optimization of cardiovascular geometries that has potential to impact surgical planning and treatment design in future work. The surrogate management framework was successful in optimizing three representative cardiovascular geometries with a reasonable computational cost. A modification of the surrogate management framework to incorporate MADS polling improves the convergence properties and efficiency of the optimization algorithm. The first model problem demonstrated that optimization was successful in obtaining a straight vessel by optimizing a stenosis. This problem underscores the importance of choosing a function that is physiologically relevant to the problem. Our results also showed that optimization, as
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Fig. 11. Results of end-to-side optimization problem using a 50/50 flow split with area-based cost function Left: Cost function vs. number of function evaluations shows convergence history of optimization. Right: optimal shape has a 90° angle.
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Fig. 12. Results of end-to-side optimization problem using a 70/30 flow split with area-based cost function Left: Cost function vs. number of function evaluations shows convergence history of optimization. Right: optimal shape has a 50.5° angle.
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Fig. 13. Results of end-to-side optimization problem using a 90/10 flow split with area-based cost function Left: Cost function vs. number of function evaluations shows convergence history of optimization. Right: optimal shape has a 20° angle.
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Fig. 14. Left: optimal graft angle as a function of distal flow percentage. Optimization results demonstrate that attachment angle should decrease as distal flow increases. Right: optimal graft radius as a function of distal flow percentage. Results do not indicate a clear trend for optimal graft size.
opposed to a trial-and-error approach, can often lead to results that are non-inuitive at first. Using the efficiency cost function, we demonstrated that the optimal shape had a slight aneurysm and that this shape resulted in a smaller pressure drop than the straight vessel. Using a cost function based on uniformity of wall shear stress, the optimization algorithm quickly converged to a straight vessel solution. The second model problem was a computational study of Murray’s law. Murray’s law has been an active area of research since its introduction in 1926. However, the present work represented the first time that a computational approach was used together with formal optimization methods to reproduce Murray’s original problem. Using this approach, the trade-off between the two terms in Murray’s cost function (the energy dissipation and the metabolic cost) became important for the first time since previous analytical solutions had no dependance on the relative weighting of these two terms. In this work we have demonstrated that the optimal solution is extremely sensitive to the value of the constant b that determines this relative weighting.
Additionally, the computational solution was not restricted to a Poiseuille flow assumption as in Murray’s work, but used fully three-dimensional, pulsatile solutions to the Navier–Stokes equations. Using pulsatile inflow conditions, we obtained reasonable agreement with Murray’s law using the metabolic constant reported by [66] that was based on oxygen consumption rates in rats. Our results suggest that there may not be a single value of b that governs branching laws throughout the vascular system, but that the relative importance of the two terms in the cost function is likely to change with Reynolds number, or with constraints imposed on branching patterns in different locations of the vascular system. It is our opinion that future work should also consider alternative cost functions based on local vessel wall response to fluid mechanical forces. The third model problem we examined was the optimization of an end-to-side anastomosis. This problem demonstrated the potential of the SMF optimization framework to be used in cardiovascular treatment planning. While the geometry was simplified compared to patient-specific geometries, it enabled us to examine trends
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in optimal angle and graft size as the flow ratio between the two outlets was varied. The results for the optimal angle agreed with the predicted outcome that the angle should decrease as the percent of flow to the distal outlet increases. However, results for the optimal graft size were less intuitive. Current clinical practice is to try to match the size of the graft with the size of the native vessel, which is 7 mm diameter in our example. In contrast, the optimal solution for the 90/10 flow split case was to use a much smaller 4 mm diameter graft. Closer examination of the solution verifies that a larger graft size at the same angle produces a larger area of low shear stress below the threshold value and therefore a higher cost function value. However, it is possible that a cost function with a higher threshold value would produce a different optimal solution. In addition, a cost function that includes resistance to flow of the bypass graft would be expected to yield a larger diameter graft. Future work should explore the effect of cost function choice on the optimal graft size in attempt to reconcile this result with standard clinical practice. These three problems have raised several important issues. First, we have demonstrated in all three problems that the results of optimization can vary considerably depending on the choice of cost function. These results underscore the importance of choosing a cost function that has physiologic relevance and of tailoring the cost function for a particular problem. This is especially important for cardiovascular applications for which cost function choices may not be as straightforward as in more traditional engineering problems. The optimization framework we have presented offers flexibility to extend these methods to many other cardiovascular problems. This flexibility allows for a wide range of cost functions defined by the user, the addition of constraints, parallelization, and the capability to optimize pulsatile, time-dependent flow problems. For example, the end-to-side anastomosis problem could be easily extended to the pulsatile flow case using a cost function based on the oscillatory shear index (OSI) [67]. Optimization using pulsatile or time-dependent inflow conditions does not require any modifications to the optimization algorithm. 6.1. Limitations and future work The methods we have presented should be generalized to the case of constrained optimization in future work. The surrogate management framework can be easily extended to constrained optimization using a filter, as introduced by [18]. Filters were incorporated into pattern search methods by [7] and have been employed in an aeronautics application using MADS by [46]. The use of MADS in the polling step of the SMF method allows for more flexibility in the addition of constraints as well as stronger proofs of convergence [8]. Future work should also take advantage of the potential to parallelize the optimization algorithm. For example,
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when multiple points are evaluated in a search or a poll step, these function evaluations should be performed in parallel. Using the SMF algorithm, or any pattern search algorithm, the cost of the poll steps increases linearly (requiring n þ 1 points) with the number of parameters. Parallelization of the poll step could help mitigate cost for problems with more parameters. An important limitation of this work was the use of cylindrical and idealized geometries. Future work should optimize patient-specific geometries, providing that these geometries can be parameterized with a reasonable number of variables. This will require development of tools for deforming complex geometries by judiciously selecting parameters to describe the geometry. Acknowledgements This work was supported by the National Science Foundation under grant number 0205741 and the Vera Moulton Wall Center for Pulmonary Vascular Disease at Stanford University. Alison Marsden was supported by a postdoctoral fellowship from the American Heart Association, a Stanford University Medical School Dean’s Fellowship and a Burroughs Wellcome Fund Career Award at the Scientific Interface. The authors wish to thank Erik Bekkers, Nathan Wilson, Alberto Figueroa and Irene Vignon-Clementel for input and support in blood flow modeling as well as John Dennis and Charles Audet for sharing their optimization expertise. References [1] F. Abraham, M. Behr, M. Heinkenschloss, Shape optimization in steady blood flow: a numerical study of non-Newtonian effects, Comput. Meth. Biomech. Biomed. Engrg. 8 (2005) 127– 137. [2] F. Abraham, M. Behr, M. Heinkenschloss, Shape optimization in unsteady blood flow: a numerical study of non-Newtonian effects, Comput. Meth. Biomech. Biomed. Engrg. 8 (2005) 201–212. [3] V. Agoshkov, A. Quarteroni, G. Rozza, Shape design in aortocoronaric bypass using perturbation theory, SIAM J. Numer. Anal. 44 (2006) 367–384. [4] V. Agoshkov, A. Quarteroni, G. Rozza, A mathematical approach in the design of arterial bypass anastomoses using unsteady Stokes equations, J. Scient. Comput. 28 (2006) 139–161. [5] J.F. Antaki, O. Ghattas, G.W. Burgreen, B. He, Computational flow optimization of rotary blood pump components, Art. Org. 19 (1995) 608–615. [6] C. Audet, J.E. Dennis Jr., Analysis of generalized pattern searches, SIAM J. Opt. 13 (3) (2003) 889–903. [7] C. Audet, J.E. Dennis Jr., A pattern search filter method for nonlinear programming without derivatives, SIAM J. Opt. 14 (4) (2004) 980– 1010. [8] C. Audet, J.E. Dennis Jr., Mesh adaptive direct search algorithms for constrained optimization, SIAM J. Opt. 17 (1) (2006) 2–11. [9] A.J. Booker, J.E. Dennis Jr., P.D. Frank, D.B. Serafini, V. Torczon, M.W. Trosset, A rigorous framework for optimization of expensive functions by surrogates, Struct. Opt. 17 (1) (1999) 1–13. [10] G.W. Burgreen, J.F. Antaki, Z.J. Wu, A.J. Holmes, Computational fluid dynamics as a development tool for rotary blood pumps, Art. Org. 25 (2001) 336–340.
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