surgical design for the fontan procedure using computational fluid
TRANSCRIPT
UC San DiegoUC San Diego Electronic Theses and Dissertations
TitleSurgical design for the Fontan procedure using computational fluid dynamics and derivative-free optimization
Permalinkhttps://escholarship.org/uc/item/86b0t9qv
AuthorYang, Weiguang
Publication Date2012-01-01 Peer reviewed|Thesis/dissertation
eScholarship.org Powered by the California Digital LibraryUniversity of California
UNIVERSITY OF CALIFORNIA, SAN DIEGO
Surgical Design for the Fontan Procedure Using Computational FluidDynamics and Derivative-free Optimization
A dissertation submitted in partial satisfaction of the
requirements for the degree
Doctor of Philosophy
in
Engineering Sciences (Mechanical Engineering)
by
Weiguang Yang
Committee in charge:
Professor Alison L. Marsden, ChairProfessor Yuri BazilevsProfessor Juan C. del AlamoProfessor Jeffrey A. FeinsteinProfessor Juan C. LasherasProfessor Andrew D. McCullochProfessor Beth J. Printz
2012
Copyright
Weiguang Yang, 2012
All rights reserved.
The dissertation of Weiguang Yang is approved, and it is
acceptable in quality and form for publication on micro-
film and electronically:
Chair
University of California, San Diego
2012
iii
DEDICATION
To my parents
iv
TABLE OF CONTENTS
Signature Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
Vita and Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi
Abstract of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii
Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Single ventricle heart defects (SVHD) and surgical palli-
ations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Magnetic resonance imaging (MRI) for single ventricle
heart defects . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Computational fluid dynamics for single ventricle heart
defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5 In-vitro flow experiments for single ventricle heart defects 121.6 A novel Y-graft and optimal design . . . . . . . . . . . . 151.7 Outline of the thesis . . . . . . . . . . . . . . . . . . . . 18
Chapter 2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.1 Finite element methods (FEM) for blood flow problems . 21
2.1.1 FEM for convection-dominated flow . . . . . . . 212.1.2 Stabilized FEM for Navier-Stokes equations . . . 262.1.3 Boundary conditions . . . . . . . . . . . . . . . . 29
2.2 Surrogate management framework (SMF) . . . . . . . . . 342.2.1 Surrogate models . . . . . . . . . . . . . . . . . . 372.2.2 Mesh adaptive direct search (MADS) . . . . . . . 41
Chapter 3 Constrained Optimization of an Idealized Y-graft Model . . . 433.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.1.1 Model construction and parameterization . . . . . 443.1.2 Flow simulation and boundary conditions . . . . . 483.1.3 Unconstrained optimization . . . . . . . . . . . . 50
v
3.1.4 Polling strategies . . . . . . . . . . . . . . . . . . 513.1.5 Constrained optimization . . . . . . . . . . . . . . 533.1.6 Choice of cost function and constraints for Fontan
optimization . . . . . . . . . . . . . . . . . . . . . 553.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2.1 Unconstrained optimization . . . . . . . . . . . . 593.2.2 Polling comparison . . . . . . . . . . . . . . . . . 663.2.3 Constrained optimization . . . . . . . . . . . . . . 68
3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 713.4 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . 763.5 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . 77
Chapter 4 Hemodynamic Evaluations for traditional and Y-graft FontanGeometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.1.1 Geometrical model construction . . . . . . . . . . 804.1.2 Flow simulation and boundary conditions . . . . . 834.1.3 Determination of performance parameters . . . . 84
4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.2.1 Hepatic flow distribution . . . . . . . . . . . . . . 874.2.2 SVC pressure . . . . . . . . . . . . . . . . . . . . 894.2.3 Power loss . . . . . . . . . . . . . . . . . . . . . . 924.2.4 Wall Shear Stress . . . . . . . . . . . . . . . . . . 924.2.5 Averaged results . . . . . . . . . . . . . . . . . . . 94
4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 954.3.1 Hepatic flow distribution . . . . . . . . . . . . . . 964.3.2 Power loss . . . . . . . . . . . . . . . . . . . . . . 984.3.3 SVC pressure . . . . . . . . . . . . . . . . . . . . 984.3.4 Wall shear stress . . . . . . . . . . . . . . . . . . 994.3.5 Ranking . . . . . . . . . . . . . . . . . . . . . . . 100
4.4 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . 1024.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 1034.6 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . 104
Chapter 5 Y-graft optimal design for improved hepatic flow distribution . 1055.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.1.1 Geometrical model construction . . . . . . . . . . 1075.1.2 Flow simulation and boundary conditions . . . . . 1125.1.3 Optimization algorithm . . . . . . . . . . . . . . . 114
5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155.2.1 Idealized cases . . . . . . . . . . . . . . . . . . . . 1155.2.2 Patient-specific cases . . . . . . . . . . . . . . . . 119
5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 124
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5.3.1 Idealized cases . . . . . . . . . . . . . . . . . . . . 1245.3.2 Patient-specific cases . . . . . . . . . . . . . . . . 1285.3.3 Technical considerations for Y-graft implantation 130
5.4 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . 1315.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 1335.6 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . 134
Chapter 6 Simulations and validation for the first cohort of Y-graft Fontanpatients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1366.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.1.1 Surgical technique and clinical data acquisition . . 1376.1.2 Model construction . . . . . . . . . . . . . . . . . 1386.1.3 Flow simulation and boundary conditions . . . . . 140
6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1426.2.1 Simulation vs. lung perfusion . . . . . . . . . . . 1426.2.2 Longitudinal HFD . . . . . . . . . . . . . . . . . 1436.2.3 HFD estimation without in vivo flow conditions . 1446.2.4 Thrombus investigation . . . . . . . . . . . . . . . 144
6.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 1486.4 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . 1556.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 1566.6 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . 157
Chapter 7 Conclusions and future work . . . . . . . . . . . . . . . . . . . 1587.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 1587.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . 161
7.2.1 Pre-operative prediction and assessment . . . . . 1617.2.2 Patient specific optimal design . . . . . . . . . . . 1627.2.3 Validation against 4D MRI . . . . . . . . . . . . . 163
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
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LIST OF FIGURES
Figure 1.1: Staged surgical palliations for SVHD. (a) Norwood procedure:a modified BT-shunt is created to channel aortic flow to thepulmonary arteries (PAs) (b) Glenn procedure: the superiorvena cava (SVC) is disconnected from the heart and reimplantedinto the PAs. (c) Fontan procedure: the inferior vena cava(IVC) is connected to the PAs via a lateral tunnel (LT) or anextracardiac (EC) Gore-Tex tube. Figures are reproduced withpermission from Gaca et al., Radiology, 2008;247:617-631. . . . 3
Figure 1.2: An illustration of PAVMs in a Glenn patient. This figure isreproduced with permission from Duncan et al., Annals of Tho-racic Surgery, 2003;76:1759-1766. . . . . . . . . . . . . . . . . 6
Figure 1.3: A sketch for two novel designs. (a) A dual-bifurcation designproposed by Soerensen et al.1 bifurcates the IVC and SVC flow.(b) A Y-shaped graft proposed by Marsden et al.2 bifurcatesthe IVC flow only. . . . . . . . . . . . . . . . . . . . . . . . . . 16
Figure 2.1: Solutions for the 1D steady convection-diffusion problem (2.1)with f = 0, g(0) = 0, g(10) = 1 using Galerkin, exact arti-ficial diffusion (EAD) and streamline-upwind-Petrov-Galerkin(SUPG) schemes. . . . . . . . . . . . . . . . . . . . . . . . . . 25
Figure 2.2: Solutions for the 1D steady convection-diffusion problem (2.1)with f = −16 a
10
(−2 + 4 x
10
), g(0) = 0, g(10) = 1 using Galerkin,
exact artificial diffusion (EAD) and streamline-upwind-Petrov-Galerkin (SUPG) schemes. . . . . . . . . . . . . . . . . . . . . 27
Figure 2.3: A spatial domain is divided into a 3D domain Ω modeled byNavier-Stokes equations and a downstream Ω′ modeled by lumpparameter models. The DtN outflow boundary conditions areprescribed on the boundary ΓB that separates Ω and Ω′. . . . 30
Figure 2.4: A three element Winkessel model. . . . . . . . . . . . . . . . . 33Figure 2.5: Flowchart of SMF using MADS. Search and poll steps are exe-
cuted alternately according to whether a design point that im-proves the current best cost function is found. . . . . . . . . . 36
Figure 3.1: Model parametrization showing the six design parameters usedfor shape optimization (a), and the resting pulsatile IVC andSVC flow waveforms used for inflow boundary conditions (b). . 46
Figure 3.2: Pulsatile waveforms for the rest, 2X and 3X exercise cases. Tosimulate two exercise levels, the IVC flow rate was increased by2 and 3 times; SVC flow was unchanged. . . . . . . . . . . . . 50
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Figure 3.3: The shape optimization procedure is made up of a series ofautomated sub-steps from model construction to the input ofthe cost function value into the optimization algorithm. . . . . 51
Figure 3.4: Example of a filter for the constrained optimization problem.The filter shown in (a) is improved when a dominating point isfound, producing the filter shown in (b). . . . . . . . . . . . . 56
Figure 3.5: Time-averaged shear stress magnitude (dynes/cm2) of the opti-mal shape over one respiratory cycle during the rest conditionusing pulsatile waveform. . . . . . . . . . . . . . . . . . . . . . 58
Figure 3.6: Unconstrained optimization results using steady inflow condi-tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Figure 3.7: Convergence history for the unconstrained optimization understeady inflow conditions. . . . . . . . . . . . . . . . . . . . . . . 61
Figure 3.8: Mean pressure for the optimal shapes under the rest, 2X and3X exercise levels using pulsatile waveforms. . . . . . . . . . . . 63
Figure 3.9: Instantaneous velocity magnitude on the centerline cut plane ofthe optimal shapes using pulsatile waveforms for the rest, 2X,and 3X cases with unconstrained optimization. . . . . . . . . . 64
Figure 3.10: Velocity vectors at peak IVC inflow for the exercise and restoptimal shapes at exercise conditions. Compared with the ex-ercise optimal design (left), the large graft of the rest optimaldesign (right) results in more flow separation and causes moreenergy loss. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Figure 3.11: Convergence history for 5 LTMADS and 1 OrthoMADS in-stances with poll only under the 3X steady inflow condition. . . 67
Figure 3.12: Convergence history for 5 LTMADS and 1 OrthoMADS in-stances with search and poll together under the 3X steady inflowcondition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Figure 3.13: Final results of the constrained optimization plotted as costfunction J vs. constraint function H for the rest case. Themodel in the upper left corner is the best feasible design andthe model in right bottom corner is the design with highestenergy efficiency. Differences in shape among these models showa strong effect of the WSS constraint for the rest case. . . . . . 70
Figure 3.14: Final results of the constrained optimization plotted as costfunction J vs. constraint function H for the 2X exercise case.The number of feasible points is increased to 11. The bestfeasible, least infeasible and highest energy efficiency models arelisted from left to right. Different points in the filter plot havesimilar geometry. Results for the 2X exercise case show that theeffect of the WSS constraint is weakened as inflow rates increase. 71
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Figure 3.15: Final results of the constrained optimization plotted as costfunction J vs. constraint function H for the 3X exercise case.The number of feasible points is increased to 32. . . . . . . . . 72
Figure 4.1: Original Glenn models and variations of Fontan geometries forfive patients. The Y-graft includes a 20 mm trunk and two 15mm branches. The size of the tube-shaped graft is 20 mm. Pa-tients B and E have a stenosis in the LPA and RPA, respectively,denoted by arrows. . . . . . . . . . . . . . . . . . . . . . . . . 82
Figure 4.2: Based on conservation of mass, we have QRPA = QIV C · x +QSV C · y and QLPA = QIV C · (1 − x) + QSV C · (1 − y), wherex is the fraction of hepatic flow going to the RPA, and y is thefraction of SVC flow going to the RPA. . . . . . . . . . . . . . 86
Figure 4.3: Visualization of the particle tracking in the model Y-graft IIfor patient B. Particle tracking is terminated when particles arewashed from the model. . . . . . . . . . . . . . . . . . . . . . . 87
Figure 4.4: Left: Hepatic flow distribution at rest. Right: Differences (per-centage of the IVC flow) from the theoretical optima for eachdesign at rest and exercise. Note that the theoretical optimafor patient A at rest, 2X and 3X are 61/39, 70/30 and 72/28,respectively, and that a 50/50 split can not be achieved in the-ory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Figure 4.5: Time-averaged velocity vectors in the Y-graft and T-junctionmodels for patients A, D and E. In the T-junction design forpatient A, the SVC jet blocks the hepatic flow entering the LPA.In patient D, most SVC flow is directed to the RPA due to acurved SVC. In patient E, Y-graft II improves the hepatic flowdistribution by having a straight proximal branch for the RPA,in which the SVC jet blocks hepatic flow going to the RPA fromthe right branch, compared to Y-graft I. . . . . . . . . . . . . . 91
Figure 4.6: Hepatic flow distribution changes with variations in pulmonaryflow split. Patients’ original pulmonary flow splits are markedby the arrows at the x axis. The table shows the averageddeviations with respect to the original hepatic flow distributionfor a 25% change in pulmonary flow split. . . . . . . . . . . . . 92
Figure 4.7: Contours of time-averaged WSS (dynes/cm2) at rest for patientsA and C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Figure 4.8: Averaged differences from the theoretical optima and powerlosses over five patients. The best performing of the Y-graftand offset designs for patients B and E are used. The differencesbetween the Y-graft and T-junction designs are statistically sig-nificant (∗P < 0.05). . . . . . . . . . . . . . . . . . . . . . . . . 95
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Figure 5.1: Model parameterization and flared SVC anastomosis. Upperleft: Design parameters and centerlines of an idealized Y-graftFontan model. Upper right: A representative Y-graft model.Parameters DL and DR allow two branches to vary indepen-dently. Bottom left: An LPA-flared SVC anastomosis with astraight junction for the RPA side. Bottom right: A curved-to-LPA SVC anastomosis. . . . . . . . . . . . . . . . . . . . . . . 109
Figure 5.2: 1. A patient-specific Glenn model. 2. In the semi-idealizedGlenn model, the PA is approximated by uniform circular seg-mentations and the pulmonary artery branches are neglected.The PA diameter is equal to the averaged diameter of the patient-specific PA. 3. A Y-graft is implanted forming a semi-idealizedFontan model for the same patient. The design parameters forthe Y-graft areXL, XR, LIV C andDbranch. When large branchesare anastomosed, the segmentation at the anastomosis is en-larged to the graft size. Then the rest of the PA segmentationsare enlarged linearly according to the distance to the closestanastomosis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Figure 5.3: Optimal values for the HFD. Based on Equation (4.2), the the-oretical optimum for the HFD, defined as the value closest to50/50, is determined given an inflow ratio QIV C
QSVCand a pul-
monary flow split FRPA (% inflow to RPA). . . . . . . . . . . . 114Figure 5.4: A comparison of HFD and energy loss for optimal unequal and
equal-sized branches. HFDs for the unequal and equal-sizedbranches are 63/37 and 65/35 (IVC-RPA/IVC-LPA), respec-tively, but equal-sized branches perform better in reducing en-ergy loss. The pulmonary flow split is 79/21 (RPA/LPA). . . . 117
Figure 5.5: Optimal Y-grafts with equal-sized branches for a large range ofpulmonary flow splits. Theoretical optima given by Equation4.2 are achieved by using optimization. The difference from thetheoretical value is shown at each point. . . . . . . . . . . . . 117
Figure 5.6: Time-averaged flow fields of optimal Y-grafts for a straight SVC-PA junction and two types of flared SVC anastomoses. Thepulmonary flow split is 55/45 (RPA/LPA). Compared to themodel with a straight SVC-PA junction, the optimal Y-graftsfor two flared SVC anastomoses have a more distal anastomosisfor the LPA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
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Figure 5.7: HFD vs. QIV C
Qinflowfor an idealized model and a patient-specific
model (patient B). Patient B’s original inflow ratio QIV C
Qinflowis
marked by an arrow. Total inflow is kept constant in this com-parison. The idealized Y-graft is optimized for an IVC inflow-to-total inflow ratio of 45%. There is only 1% change in theY-graft model when the ratio is altered. However, the patientspecific model is more sensitive to the change of IVC inflow-to-total inflow ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Figure 5.8: a) Time-averaged velocity vector fields in the semi-idealizedand patient-specific models for patient A. b) Particle snapshotstaken at T=3s for the non-optimized and optimal models. Thebar chart shows the semi-idealized model (upper left) has a sim-ilar hepatic flow split to the patient-specific model (upper right)for the same optimal Y-graft, and that the optimized Y-graft im-proves the HFD by 79%, compared to the original non-optimizeddesign (lower left). The optimal and non-optimized branch sizesare 12.9 and 15 mm, respectively. . . . . . . . . . . . . . . . . . 121
Figure 5.9: Time-averaged velocity vector fields in the semi-idealized andpatient-specific models with and without the RUL for patientB. The Y-graft is optimized for a HFD of 50/50. Due to theeffect of the RUL, the optimized Y-graft skewed the hepatic flowby around 15% after it was implanted into the patient-specificmodel. When the RUL is excluded from the patient-specificmodel, the HFD is consistent with the idealized model prediction.122
Figure 5.10: a) Time-averaged velocity vector fields and HFD for patient B.The Y-graft in the semi-idealized model (upper left) is opti-mized for a hepatic flow split of 65/35 (RPA/LPA) to accountfor the overestimation of the RPA hepatic flow in the semi-idealized model. b) Particle snapshots taken at T=3s for thenon-optimized and optimal models. The bar chart shows theoptimal Y-graft improves the performance by 94% achieving aneven HFD in the patient-specific model (upper right), comparedto the non-optimized design (lower left). The optimal and non-optimized branch sizes are 16 and 15 mm, respectively. . . . . . 123
Figure 6.1: Post-operative MRI/CT images and models. Since patientsYF5 developed thrombus in the left branch, an unblocked Y-graft was reconstructed for study. . . . . . . . . . . . . . . . . 139
Figure 6.2: a) Comparison between early post-operative simulation-derivedHFD and lung perfusion data for patients YF1, YF2 and YF3.HFD in the early post-operative stage was quantified by simu-lation and lung perfusion. b) Changes in HFD from the earlyto six-month post-operative stages derived from simulations. . 143
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Figure 6.3: Time-averaged WSS magnitude for patients YF1, YF2 YF3 andYF5 in the early post-operative stage. YF5 R14-L10 and YF5R12-L12 are two modified Y-graft designs for patient YF5. Inthe baseline model for patient YF5, a distal anastomosis for theleft branch and a highly skewed pulmonary flow split resultedin larger low WSS area in the left branch. In model R12-L12,the WSS in the left branch was enhanced due to a proximalanastomosis that allowed SVC flow to wash the left branch. . . 147
Figure 6.4: Percentage of low WSS region for two branches in patients YF1,YF2, YF3 and YF5. For each threshold value τ , the low WSSarea relative to each branch surface was computed. . . . . . . 147
Figure 6.5: Percentage of low WSS region for patient YF5’s modified Y-grafts. In model R12-L12, the low WSS area in the left branchcan be effectively minimized by using a proximal anastomosisin which the SVC jet impinged the wall and the impact of theSVC jet on the WSS was reduced with increasing LPA flow.Compared to the baseline model (Figure 6.4), model R14-L10has a similar percentage of low WSS area for the left branchin the early post-operative stage for threshold values below 2dynes/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
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LIST OF TABLES
Table 3.1: Bounds on the design parameters for the idealized model. Nega-tive values for ΔR and ΔL indicate inward convex branches andpositive values denote outward concave branches. Bounds werechosen to be consistent with MRI data from a typical patient. . 47
Table 3.2: Values of the four constant geometric parameters used in modelconstruction, taken from MRI data of a typical Fontan patient. 47
Table 3.3: Mean flow rates, Re in the IVC and SVC and resistance dropsat rest and two levels of simulated exercise. . . . . . . . . . . . . 49
Table 3.4: Optimal parameters, cost function values and number of evalu-ations for the unconstrained optimization using different inflowconditions. Parameters that lie on the boundary are in bold. . . 64
Table 3.5: Comparison results for 5 LTMADS and 1 OrthoMADS instanceswith poll only under the 3X steady inflow condition. Parametersthat lie on the boundary are in bold. . . . . . . . . . . . . . . . 67
Table 3.6: Comparison results for 5 LTMADS and 1 OrthoMADS instanceswith search and poll together under the 3X steady inflow condi-tion. OrthoMADS found the best solution among 5 instances ofLTMADS, with relative precision 0.1%. Parameters that lie onthe boundary are in bold. . . . . . . . . . . . . . . . . . . . . . . 68
Table 3.7: Comparison of the best feasible parameter and the highest energyefficiency points for the constrained optimization. Parametersthat lie on the boundary are in bold. . . . . . . . . . . . . . . . 73
Table 4.1: MRI inflow rates, MRI outflow splits and the theoretical optimalhepatic flow splits (TOHFS) at rest for the five study patients. . 86
Table 4.2: Mean SVC pressure (mmHg), power loss (mW) and mean (inspace) WSS magnitude (dynes/cm2) on the IVC graft for theFontan models. Compared to the best Y-graft design for thesame patient, increases in power loss for the T-junction and offsetdesigns are also shown. . . . . . . . . . . . . . . . . . . . . . . . 93
Table 4.3: Ranking of energy loss and hepatic flow distribution for eachpatient. The ranking of the hepatic flow distribution is based onthe differences from the theoretical optima. In patients C andD, there are two designs tied for the hepatic flow distribution. . 101
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Table 5.1: Bounds on the design parameters for the semi-idealized model.XR and XL are measured from the SVC-PA junction to the rightand left anastomosis points, respectively. Since the PA path is aparametric spline S(t), the anastomosis location can be changedby varying the spline parameter t. In our previous study,3 patientspecific models employed a uniform 20-15-15 mm Y-graft. Tooptimize the graft size, the branch diameter was allowed to varybetween 12 and 16 mm. . . . . . . . . . . . . . . . . . . . . . . 110
Table 5.2: Mean pulsatile inflow rates, pulmonary flow splits and pressure.A respiratory model4 was superimposed to the IVC flow acquiredfrom PCMRI for each patient following our previous work. Norespiratory model was added to the SVC input. The flow ratesused for the idealized model were taken from a typical Fontanpatient.4,5 We varied the RPA/LPA flow split in the idealizedmodel for different conditions and set a Fontan pressure (centralvenous pressure) of 12 mmHg. For patients A and B, pulmonaryflow splits and pressure data were taken from MRI and catheter-ization prior to the Fontan procedure. Transpulmonary gradient(TPG) is the mean pressure difference between the SVC and theleft atrium.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Table 5.3: Geometric parameters and power loss for patient specific models.In patient A, the optimal XR was reduced resulting in a moreproximal anastomosis for the right branch. In patient B, the rightanastomosis is more distal while the left one is more proximalin the optimized model. In both cases, a smaller branch sizeresulted in more power loss. . . . . . . . . . . . . . . . . . . . . 122
Table 6.1: Patients’ flow conditions used in simulations. The vena cava flowand pulmonary flow split were measured by PC-MRI except forpatients YF4 and YF6, who had CT imaging. We use the for-mat, RPA/LPA, to present pulmonary flow split. For patientsYF1, YF2 and YF3, “early” and “6 month” denote measure-ments taken in the early (< 1 month) and 6 month post-operativestages, respectively. For patient YF5, pre-operative and 3 monthpost-operative measurements were performed. . . . . . . . . . . 141
xv
Table 6.2: Mean WSS magnitude for Y-graft branches and HFD. Comparedto other patients, patient YF5 had low WSS in the left branchin the early post operative stage but the WSS in the left branchincreased in the 3 month post-operative stage in which the pul-monary flow split changed from 81/19 to 54/46. The mean WSSfor patient YF3 is low due to a lower cardiac output. The mod-ified Y-grafts for patient YF5 increased mean WSS in the leftbranch in the early post-operative stage compared to the origi-nal Y-graft. All Y-graft designs for patient YF5 skewed hepaticflow to the RPA with PAVMs in the early post-operative stagebut the HFD was improved in the 3 month post-operative stage. 145
xvi
ACKNOWLEDGEMENTS
When I started writing the acknowledgements, I realized that words fail me
in expressing express my gratitude towards my adviser, Professor Alison Marsden.
I would not have been able to finish my Ph.D. study without her constant support
and guidance. I am grateful to Professor Marsden for giving me an opportunity to
work in a new area with great clinical impact in which CFD and optimization could
be used to save lives. Professor Marsden’s trust and patience created a unique
environment in the lab allowing me to explore the problems to my satisfaction.
Professor Marsden provided me with a platform where I can work with a group of
fantastic people and learn from them. Doing research is not always straightforward.
Professor Marsden’s encouragement and guidance helped me overcome a lot of
difficulties and made things that were hard a lot easier. I am honored to be her
student.
Dr. Jeffrey Feinstein is one of our clinical collaborators. Our research is
driven by needs from clinicians and patients. Without his collaboration and guid-
ance, this work would not have been possible. I appreciate the medical knowledge
and advice he offered. I acknowledge Professor Shawn Shadden, Dr. Irene Vignon-
Clementel, Professor Charles Audet, Professor Sebastien Le Digabel, Professor
John Dennis, Dr. Nathan Wilson and Professor Charles Taylor for sharing their
expertise in numerical simulation, optimization and modeling. The tools devel-
oped by them have been of great help and have been the core tools of my study.
xvii
I would also like to thank Dr. V. Mohan Reddy and Dr. Frandics Chan. With-
out Dr. Reddy’s masterly surgical skills, the Y-graft design would not have been
implemented. Dr. Chan’s MRI data were crucial for the patient specific study.
Through my stay at UCSD I have had the chance to learn from a lot of
people. I am thankful to all of them. In particular, I greatly appreciate the
advice and help from my other committee members, Professor Juan Lasheras,
Professor Juan Carlos del Alamo, Professor Yuri Bazilevs, Professor McCulloch
and Dr. Beth Printz. I was inspired by Professor Juan Lasheras’ broad knowledge
from mechanics to biology and his attitude to research. I benefit from Professors
Bazilevs and del Alamo’s courses on finite element methods and turbulence. In
addition, I wish to thank Professor Yaosong Chen at Peking University for his
advice and help.
For the people mentioned above who taugh me, guided me, inspired me,
and encouraged me, I would like to quote an old Chinese saying to express my
gratitude and respect —“be a teacher for one day, be a father for all life”.
I appreciate all the help I have received from my labmates: Dr. Sethu-
raman Sankaran, Dibyendu Sengupta, Mahdi Esmaily Moghadam, Chris Long,
Matt Bockman, Abhay Ramachandra, Dr. Ethan Kung and Jessica Oakes. It was
always fun and helpful to learn from them.
This work was supported by the American Heart Association, a Burroughs
Wellcome Fund Career Award at the Scientific Interface, a Leducq Foundation Net-
work of Excellence grant and a UCSD Kaplan dissertation fellowship. I appreciate
xviii
the funding from all these agencies, which made this work possible. This disser-
tation has resulted in the following papers that have been published, accepted
or is being prepared for publication. The dissertation author was the primary
investigator and author of these publications.
Chapter 3
Yang, W., Feinstein, J. A. and Marsden, A. L. Constrained Optimization of an
Idealized Y-shaped Baffle for the Fontan Surgery at Rest and Exercise. Comput.
Meth. Appl. Mech. Engrg. 2010;199:2135-2149.
Chapter 4
Yang, W., Vignon-Clementel, I. E., Troianowski, G., Reddy, V. M., Feinstein,
J. A. and Marsden, A. L. Hepatic blood flow distribution and performance in
traditional and Y-graft Fontan Geometries: A Case Series Computational Fluid
Dynamics Study. J. Thorac. Cardiovasc. Surg. 2012;143: 1086-1097.
Chapter 5
Yang, W., Feinstein, J. A., Shadden, S. C., Vignon-Clementel, I. E. and Marsden,
A. L. Optimization of a Y-graft Design for Improved Hepatic Flow Distribution in
the Fontan Circulation. J. Biomech. Engrg., accepted.
Chapter 6
xix
Yang, W., Chan, F. P., Feinstein, Reddy, V. M., Marsden, A. L., and Feinstein,
J. A. Flow Simulations and Validation for the First Cohort of Y-graft Fontan
Patients., in preparation.
In addition, I would like to appreciate the support and love from my family
members. My father Zhi Yang taught me to appreciate the beauty of engineering
since I was a child. Those interesting stories he told significantly influenced my
choice of major. I am grateful to my mother Qian Wei for raising and supporting
me. I apologize for not spending more time with her during nine years of college
and graduate school study. My grandparents Yongzhong Yang, Mingqiong Dai,
Peimin Wei and Sixia Cai, uncles Hong Yang, Xin Yang, Yi Wei and Xin Wei,
aunts Zhaoxia Sun, Su Zhao, Bo Qu and Jing Xie deserve my special thanks. I
would like to particularly thank my uncle Xin Yang and aunt Su Zhao for their
tremendous support during my graduate school. The soccer games with my uncle
Xin Yang on Saturday afternoon let me take a break from work and added a good
balance to my life.
Finally, I wish to thank my good friends Rong Jiang, Peng Wang, Matt
de Stadler, Zhangli Peng and On Shun Pak for those lively conversations and for
providing both technical and personal help time and again.
xx
VITA
2007 Bachelor of Engineering, Peking University, China.
2009 Master of Science, University of California, San Diego, USA.
2007-2012 Research Assistant, University of California, San Diego, USA.
2012 Doctor of Philosophy, University of California, San Diego,USA.
JOURNAL PUBLICATIONS
Yang, W., Feinstein, J. A. and Marsden, A. L. Constrained Optimization of anIdealized Y-shaped Baffle for the Fontan Surgery at Rest and Exercise. Comput.Meth. Appl. Mech. Engrg. 2010;199:2135-2149.
Yang, W., Vignon-Clementel, I. E., Troianowski, G., Reddy, V. M., Feinstein,J. A. and Marsden, A. L. Hepatic blood flow distribution and performance intraditional and Y-graft Fontan Geometries: A Case Series Computational FluidDynamics Study. J. Thorac. Cardiovasc. Surg. 2012;143: 1086-1097.
Yang, W., Feinstein, J. A., Shadden, S. C., Vignon-Clementel, I. E. and Marsden,A. L. Optimization of a Y-graft Design for Improved Hepatic Flow Distribution inthe Fontan Circulation. J. Biomech. Engrg., accepted.
Yang, W., Chan, F. P., Feinstein, Reddy, V. M., Marsden, A. L., and Feinstein,J. A. Flow Simulations and Validation for the First Cohort of Y-graft FontanPatients., in preparation.
AWARDS
Kaplan Dissertation Fellowship, University of California, San Diego (2011).
xxi
ABSTRACT OF THE DISSERTATION
Surgical Design for the Fontan Procedure Using Computational FluidDynamics and Derivative-free Optimization
by
Weiguang Yang
Doctor of Philosophy in Engineering Sciences (Mechanical Engineering)
University of California, San Diego, 2012
Professor Alison L. Marsden, Chair
Single ventricle heart defects are among the most serious forms of congenital
heart disease. For hypoplastic left heart syndrome (HLHS), a three-staged surgical
course, consisting of the Norwood, Glenn, and Fontan surgeries is performed. In
the extracardiac Fontan procedure, the inferior vena cava (IVC) is connected to the
PAs either via a Gore-Tex tube. Serious clinical challenges remain despite post-
operative survival rates upwards of 90%. A novel Y-shaped graft has been proposed
to replace current tube-shaped grafts showing promising preliminary results.
To refine the Y-graft design and further study the hemodynamic perfor-
mance of the Y-graft, a 3D time-dependent finite element flow solver was coupled
to a derivative free optimization algorithm using surrogate management framework
xxii
(SMF) and mesh adaptive direct search (MADS). In the first part of this disser-
tation, an idealized Y-graft model was parameterized and optimized for energy
efficiency. Constrained optimization with a wall shear stress (WSS) constraint was
performed in order to study the risk of thrombosis.
In the second part of this dissertation, patient specific Glenn models were
virtually converted into Fontan models by implanting Y- and tube-shaped grafts for
comparison. Particular attention was paid to the hepatic flow distribution (HFD),
a clinical parameter that plays an important role in the formation of pulmonary
arteriovenous malformations (PAVMs).
In a third study, we coupled Lagrangian particle tracking to an optimal
design framework to study the effects of boundary conditions and geometry on
HFD. Two patient-specific examples showed that optimization-derived Y-grafts
effectively improved HFD, compared to initial non-optimized designs.
Based on our preliminary simulation results, the Y-graft has been translated
into use in a clinical pilot study. Post-operative flow simulations showed good
agreement with the lung perfusion data measured in the clinic. The development
of thrombosis in one patient’s Y-graft was investigated from a hydrodynamic point
of view. Results suggested that low WSS area and flow stasis should be taken into
account in the surgical design for improved HFD.
To our knowledge, this is the first study to apply formal optimization to
the Fontan surgical design. Findings in this dissertation may provide guidelines
for the future Y-graft surgeries.
xxiii
Chapter 1
Introduction
1.1 Single ventricle heart defects (SVHD) and
surgical palliations
The American Heart Association’s statistics show that congenital heart
defects (CHD) are the number one cause (>24%) of death from birth defects.6
In 2004, the hospital cost for CHD was $2.6 billion.6 Single ventricle heart de-
fects including hypoplastic left heart syndrome (HLHS), pulmonary atresia/intact
ventricular septum and tricuspid atresia are among the most severe CHD malfor-
mations. The prevalence of HLHS is about 2.39 per 10,000 live birth.7 In patients
with such defects, inadequate blood flow to the lungs results in caynosis after birth.
The patients uniformly die without treatment. Usually a three-staged surgery is
performed. The first stage consists of establishing stable sources of aortic and pul-
1
2
monary blood flow, in a Norwood procedure or variant thereof. This procedure is
typically performed in the first week of life. In the second stage, the bidirectional
Glenn procedure, the superior vena cava (SVC) is disconnected from the heart and
reimplanted into the pulmonary arteries (PAs) at about 4-6 months of age. In
the third and final stage, the Fontan procedure, the inferior vena cava (IVC) is
connected to the PAs 2-4 years after the first stage. Therefore, the Fontan proce-
dure is also called the total cavopulmonary connection (TCPC). The first Fontan
procedure was performed by Fontan and Baudet for repairing tricuspid atresia.8
In the classic Fontan procedure, the SVC was connected to the right PA and the
IVC blood flow was directed to the left PA.8 Then, an intracardiac baffle (lateral
tunnel) formed by the sinus venarum and a prosthetic patch was used to channeled
the IVC flow to the PAs. Since the 1990s, a Gore-Tex conduit has been used widely
to connect the IVC to the PAs forming an extracardiac connection.9 Figure 1.1
illustrates the surgical palliations for SVHD.
1.2 Outcomes
Despite relatively high post-Fontan survival rates upwards of 90%,10 the
5-year survival rate drops to about 80% and the long-term outlook is still unsat-
isfactory. Complications post Fontan include diminished exercise capacity, pro-
tein losing enteropathy, arteriovenous malformations, thrombosis, arrhythmias,
and heart failure.10,11 It is shown that hemodynamic performace is closely re-
3
stage 1 stage 2
stage 3
BT-shunt SVC
PAsaorta
LT Fontan EC Fontan
Gore-tex tube
Figure 1.1: Staged surgical palliations for SVHD. (a) Norwood procedure: a mod-ified BT-shunt is created to channel aortic flow to the pulmonary arteries (PAs)(b) Glenn procedure: the superior vena cava (SVC) is disconnected from the heartand reimplanted into the PAs. (c) Fontan procedure: the inferior vena cava (IVC)is connected to the PAs via a lateral tunnel (LT) or an extracardiac (EC) Gore-Tex tube. Figures are reproduced with permission from Gaca et al., Radiology,2008;247:617-631.
lated to Fontan patients’ outcomes. Reduced exercise performance after Fontan
completion is well documented. Exercise capacity is associated with oxygen con-
sumption and stroke volume.12 In contrast to healthy people, Fontan patients can
only respond to exercise by increasing their heart rate because the stroke volume is
limited.13 Although the mechanism of limited exercise capacity has not been fully
elucidated, it has been shown that reductions in peak oxygen uptake and stroke
volume were observed in Fontan patients.14–16 Giardini et al.17 showed that con-
4
version of atriopulmonary Fontan to extracardiac Fontan improved patients’ peak
oxygen uptake. Since the blood in the Fontan circulation is passively pumped to
the lung by the pressure difference between the vena cava and the left atrium, the
vascular resistance plays an important role in the regulation of cardiac output.
Animal experiments by Guyton et al.18 demonstrated that a small increase in ve-
nous resistance resulted in a significant drop in the cardiac output for a modified
circulation bypassing the right ventricle. Sundareswaran et al.19 used a computa-
tional model to correlate the Fontan geometric resistance with cardic function and
suggested that optimizing the Fontan geometry may improve exercise capacity.
Therefore, it is hypothesized that improving TCPC resistance might lower central
venous pressure and improve patients’ quality of life.13
Thrombus formation is a significant issue causing morbidity and mortality.
Thrombus can lead to chronic pulmonary embolic disease, stroke, unbalanced per-
fusion, elevation of pulmonary vascular resistance and even death.20 The incidence
of thrombosis following Fontan completion can be as high as 20% to 30%.10 For
atriopulmonary and lateral tunnel connections, there is no difference in freedom
from thrombus.20 A major adverse outcome for the extracardiac Fontan proce-
dure is the inherent risk of thrombosis in the graft. Shirai et al.21 reported a
20% incidence of thrombus formation in the conduit. The formation of throm-
bus is multifactorial including flow stagnation, hypercoagulable state and atrial
arrhythmias.10 In a study of Alexi-Meskishvili et al.,22 two out of six patients with
oversized conduits developed thrombosis, indicating a possible correlation between
5
conduit size and thrombosis, though the optimal conduit size is still unclear.
Pulmonary arteriovenous malformations (PAVMs), characterized by abnor-
mal communication between the pulmonary arteries and pulmonary veins, are an
uncommon but serious complication, occurring in as many as 25% of patients with
superior cavopulmonary anastomosis due to the exclusion of hepatic blood flow
from the pulmonary circulation.23 In the lungs with PAVMs, the PAs are dilated
and proliferated (Figure 1.2). Pulmonary flow enters the pulmonary veins without
being oxygenated in the pulmonary capillary bed, forming a “short circuit” in the
pulmonary circulation. Consequently, patients with PAVMs may develop cyanosis,
congestive heart failure, and respiratory failure.24 Although the cause of PAVMs
is not fully understood, clinical evidence shows that the absence of a hepatic fac-
tor carried in the IVC blood is a likely cause. While the incidence of PAVMs
decreases after Fontan completion, they may still persist or develop due to skewed
hepatic flow distribution, particularly in patients with heterotaxy and interrupted
IVC.25–27 PAVMs can be resolved by surgical correction of uneven hepatic flow
distribution (HFD).26–28
1.3 Magnetic resonance imaging (MRI) for sin-
gle ventricle heart defects
Computed tomography (CT) and magnetic resonance imaging (MRI) are
two major non-invasive imaging tools used in day-to-day clinical practice. Com-
6
Figure 1.2: An illustration of PAVMs in a Glenn patient. This figure is reproducedwith permission from Duncan et al., Annals of Thoracic Surgery, 2003;76:1759-1766.
pared to MRI, CT provides finer resolution with much less scanning time but
patients are exposed to ironing radiation. It has been shown that the radiation
exposure from CT scans in childhood elevates cancer risk.29 In addition, hemody-
namic information which is not available for CT can be obtained by phase con-
trast MRI (PC MRI) techniques. Therefore, MRI is the preferred modality for
diagnosing and studying congenital heart diseases, and for use in simulation stud-
ies. According to Samyn,30 cardiac MRI provides: “(1) segmental description of
cardiac anomalies, (2) evaluation of thoracic aortic anomalies, (3) non-invasive de-
tection and quantification of shunts, stenoses, and regurgitation, (4) evaluation of
conotruncal malformations and complex anatomy, (5) identification of pulmonary
and systemic venous anomalies, and importantly, (6) post-operative study and
7
evaluation of adult congenital heart disease.”
Pulmonary and caval flow distribution is determined by the pulmonary vas-
cular resistance and ventricular function.31 Abnormal pulmonary flow distribution
has been linked to risk of development of PAVMs.32 Fogel et al. quantified the
caval flow contribution to lungs in ten LT Fontan patients by labeling IVC or
SVC flow with a presaturation pulse.33 They found that a nearly even pulmonary
flow split was common in ten patients, and that 67%±12% of IVC flow went to
the LPA. In contrast, a lung perfusion study by Seliem et al.31 showed that only
27% of Glenn patients immediately prior to the Fontan procedure had symmetric
pulmonary flow distribution and Houlind et al.34 found that the mean RPA-LPA
flow ratio is 1.5 in seven LT Fontan patients. Uneven pulmonary flow distribution
in Fontan patients indicates a disparity in pulmonary vascular resistances on the
left and right. Hager et al.35 assessed pulmonary flow patterns in the Fontan
circulation. Compared to healthy volunteers, Fontan patients exhibited highly
variable waveforms without a typical patterns except for a slight late diastolic
peak.35 However, respiratory effects were excluded in the flow waveforms in this
study. The Fontan circulation is respiratory driven because the SVC and IVC are
disconnected from the heart. Most MRI drived flow data were obtained during
breath hold. To investigate actual vena caval flow in Fontan patients, Hjortdal et
al.36 performed real-time MRI measurements on 11 TCPC patients. Compared to
SVC flow, IVC flow is significantly influenced by respiratory effects during rest and
exercise, reaching the highest flow rate during inspiration.36 This study provides
8
an important basis for simulating actual pulsatile flow in Fontan patients.
Compared to CFD, a unique advantage for MRI is the capability of measur-
ing velocity fields in vivo. Be’eri et al.37 managed to obtain a velocity vector field
on a plane that includes the Fontan pathway. Sundareswaran et al.38 extended a
single plane acquisition to multiple plane acquisitions and obtained a 3D velocity
field by using divergence-free interpolation techniques. Recently, Markl et al.39 ap-
plied state-of-the-art time-resolved 3D magnetic resonance velocity mapping (4D
MRI) to four Fontan patients. Complex in vivo flow fields were revealed. In all
patients, IVC flow was evenly perfused to two lungs with RPA-skewed pulmonary
flow splits.39 Since 4D MRI is still a new technique in the development stage,
quantifying small structures and highly dynamic flow patterns is limited.39 In
addition, quantification of wall shear stress by MRI is still challenging due to near-
wall resolution issues. However, these promising imaging techniques will facilitate
the study of Fontan hemodynamics together with computational fluid dynamics
(CFD).
1.4 Computational fluid dynamics for single ven-
tricle heart defects
With the advent of medical imaging processing, CFD for blood flow entered
an era of image-based simulation in the 1990s.40, 41 X-ray, magnetic resonance
and ultrasound imaging derived geometries allow one to better characterize local
9
hemodynamics. With PC-MRI, patients’ flow information can be incorporated into
simulations achieving a patient-specific modeling.42 Although most CFD studies
for blood flow focus on adult arterial vessels, there has been a growing interest in
modeling blood flow in children with congenital heart diseases. In the pioneering
work of Dubini et al.,43,44 blood flow in Fontan models with T-junction and offset
connections was simulated. Their studies revealed that the offset design reduced
energy loss compared to the T-junction design. Subsequently, surgeons adopted
their designs connecting the graft with an offset relative to the SVC in order to
reduce energy loss due to the caval flow collision. Following the work of Dubini et
al.,43 a series of numerical studies were carried out showing that the Fontan geom-
etry plays an important role in Fontan patients’ hemodynamic performance.2, 45–47
The impacts of geometric factors including the offset value, baffle size, PA diame-
ter and connection flaring on energy loss were emphasized in most studies.45, 47, 48
The PA size was found to be the strongest correlate for energy dissipation.48
The limitations of early work include the use of idealized models, steady
inflow conditions and focus on energy loss as the sole parameter of interest.43,47
Migliavacca and colleagues were the first to achieve a flow simulation with image
derived realistic models.45 Whitehead et al.46 studied power loss in ten patient spe-
cific models, but steady inflow conditions is not physiologically realistic and may
have resulted in inaccurate estimates of quantities of interest compared to un-
steady conditions.4,49 The pulsatility of caval flow in Fontan patients is reduced as
the IVC and SVC are disconnected from the right atrium. But respiratory effects
10
on the IVC flow are pronounced.36 Since the MRI acquisition is often cardiac-
gated and resulting measurements do not account for the effect of respiration,4,50
Marsden et al. introduced a respiratory model for pulsatile Fontan hemodynamic
simulations and showed nonnegligible differences in energy efficiency and pressure
drop. Later on, multiple physiologically relevant parameters were used to com-
prehensively evaluate patients’ hemodynamic performance.2, 4, 51 Results showed
that rankings of competing designs were sensitive to the choice of parameter under
consideration. For example, models with high energy efficiency can result in highly
uneven hepatic flow distribution and patients with low Fontan pressure.
Recently increasing attention has been paid to the quantification of HFD
due to a close connection with PAVMs. Bove and associates showed that the in-
tracardiac Fontan models resulted in more even HFD than extracardiac Fontan
models due to a better mixing of caval flow.52 Dasi et al.53 used patient specific
models with steady inflow conditions and showed that the HFD correlated with the
IVC offset for extracardiac models. Yang et al.3 compared the hemodynamic per-
formance of the T-junction, offset and Y-graft designs in multiple patient specific
models with an emphasis on HFD. It has been shown that overall the Y-graft de-
signs resulted in better HFD than traditional tube-shaped grafts but the geometry
and pulmonary flow splits significantly influence HFD.
Outflow boundary conditions are as important as inflow boundary condi-
tions in determining blood flow patterns. Applying proper boundary conditions
is essential to obtaining physiologic results. Vignon-Clementel et al. illustrated
11
the difference in pulmonary flow distribution caused by using zero pressure and
resistance boundary conditions.50 For pediatric hemodynamic simulations, it is
difficult to obtain outflow rates at multiple branches due to resolution issues and
difficulties in synchronizing the measured outflow data.50 Since constant pressure
at the outlets may alter the flow field and pressure distribution, time-dependent
and physiological boundary conditions are preferred.50 Recently, residence, RCR
(resistor-capacitor-resistor) and other lumped parameter boundary conditions that
use a circuit analogy to model the circulation have been incorporated into flow
simulations for single ventricle heart diseases.2, 4, 5, 54 Migliavacca and colleagues
coupled CFD into a closed-loop network such that a 3D flow simulation is per-
formed in the TCPC and 0D circuit simulations are performed for the rest of the
vascular system.54 This multiscale lumped parameter network provides a tool to
model the influences of local geometry and physiologic changes on global systemic
and cardiac parameters.
In most CFD simulations for SVHD, a rigid wall assumption is used.13
Orlando et al.55,56 first considered the effect of compliant walls on energy loss,
reporting a 10% increase. Bazilevs et al.57 applied the state of art fluid structure
interaction (FSI) to a complex patient specific Fontan model and showed that pres-
sure and wall shear stress were overpredicted in the simulation with rigid walls.
Recently, Long et al.58 performed Fontan FSI simulations with variable wall prop-
erties and showed that the differences in energy loss and hepatic flow distribution
were small. Thus, anatomically realistic models with a rigid wall assumption are
12
still adequate to evaluate Fontan patients’ hemodynamic performance.
1.5 In-vitro flow experiments for single ventricle
heart defects
In-vitro flow experiments for modeling the Fontan circulation began ear-
lier than numerical flow simulations. de Laval and colleagues’ pioneering in-vitro
hydrodynamic study dates back to 1988.59 They found that a valveless chamber
caused turbulence and increased resistance to steady flow. Their in-vitro exper-
iments supported the replacement of the atriopulmonary connection by the new
TCPC surgical methods. To our knowledge, this was the first study that used
fluid mechanics to improve the Fontan surgical procedure. Later on, in-vitro ex-
periments of Low et al.60 showed that the lateral tunnel TCPC reduced energy loss
compared to the atriopulmonary connection. Since the lateral tunnel and extrac-
ardiac connections gained popularity in the 1990s, the atriopulmonary connection
was less well studied in in-vitro experimental studies for the Fontan procedure.
Sharma et al.61 studied the effect of caval offset and varying pulmonary flow split
on energy loss using idealized glass models. They concluded that 1 to 1.5 diameter
offset with an even pulmonary flow split resulted in lower energy loss.61 Based
on the study of Sharma et al.,61 idealized models with different vessel sizes and
anastomoses were tested in a similar manner.62, 63 Lardo et al.64 performed the
Fontan procedure with three variants (intracardiac lateral tunnel, extracardiac lat-
13
eral tunnel and extracardiac conduit) on fresh explanted sheep heart preparations
and compared energy efficiency in an in-vitro flow loop. The extracardiac conduit
was shown to have higher energy efficiency than two tunnel configurations and a
further reduction of 36% in energy loss was observed in the offset configuration.64
Compared to idealized models, a patient specific model can result in more complex
and unsteady flow structures even with steady inflow conditions.65
In addition, in-vitro flow experiments are valuable to validate numerical
simulations. Ryu et al.47 compared CFD solutions to in-vitro experiments show-
ing CFD gave similar flow field and power loss values in an idealized model. Khu-
natorn et al.66 further compared velocity fields in idealized TCPC models with
steady inflow conditions. The differences between PIV and CFD for the mean
axial velocity were within 20% but significant differences were shown in the sec-
ondary flow patterns. Pekkan and coworkers compared CFD derived power loss
to PIV data for a patient specific model showing close agreement under steady
inflow conditions.65,67 The discrepancy increased with increasing cardiac output
indicating a challenge for the flow solver to model turbulence transition. However,
the Reynolds number in the Fontan circulation is usually lower than 2000 in most
cases.67
While most in-vitro flow studies focused on energy loss, Walker et al.68,69
investigated the effects of connection geometry on hepatic flow distribution by
injecting dye into the IVC flow. It has been shown that the straight connection
resulted in relatively less sensitive hepatic flow distribution to the pulmonary flow
14
split because of a better mixing though it dissipated more energy compared to
the offset, flared and curved designs. However the actual hepatic flow distribution
in the T-junction design can vary significantly from patient to patient due to the
effects of non-idealized geometries and uneven pulmonary flow splits.3
Steady inflow conditions were used in the in-vitro studies mentioned above.
Although useful conclusions may still be drawn from steady flow experiments, the
flow field is altered. Therefore, besides realistic anatomic geometry, physiologic
flow conditions are needed for in-vitro flow experiments to study Fontan hemo-
dynamics. Figliola and colleagues have developed mock flow circuits that allow
one to simulate the Fontan circulation with physiologic realism.70 In mock flow
circuits, resistance, compliance and inertance are implemented by capillary tubing
or honeycomb matrix, variable air volume chambers, and lengths of flow tubing,
respectively.70 Similar to multiscale numerical modeling for the Fontan circula-
tion,54 upper body, hepatic and lower body compartments and the two pulmonary
branches were modeled by mock flow circuits, and the TCPC test section consists
of a MRI-derived Fontan model.70 This system is particularly useful to duplicate
clinical reports, perform patient specific in-vitro experiments and validate numer-
ical simulations.
15
1.6 A novel Y-graft and optimal design
In the previous sections, we have shown that the Fontan geometry plays
an important role in the hemodynamic performance, and that fluid mechanics
promoted the surgical community’s shift from the atriopulmonary connection to
TCPC and from the extracardiac T-junction to the offset design. Although sur-
gical modifications with other advances in pediatric cardiology and intensive care
medicine greatly improved early postoperative survival rates, clinicians and engi-
neers have not stopped pursuing better surgical designs for Fontan patients.10,13
In recent years, a Y-shaped graft has been proposed by two research groups. So-
erensen et al.1 introduced a dual-bifurcation design that bifurcates the SVC and
IVC flow (Figure 1.3a). Steady simulations with an idealized model showed that
energy dissipation was improved by reducing direct flow collision.1 However, this
design not only introduces extra artificial materials but also increases the surgical
technical difficulty. Therefore, no further study on the dual bifurcation design has
been made. Marsden et al.2 used a Y-shaped graft to replace the tube-shaped
graft for the IVC flow only (Figure 1.32). A patient specific model shows promis-
ing results , demonstrating that the Y-graft design improved the energy efficiency,
SVC pressure and HFD, compared to the tube-shaped graft.
Although the preliminary results for the Y-graft design support the re-
placement of the T-junction with the Y-graft design, several problems were still
unclear and further studies were needed to refine the design before translating the
16
SVC
IVC
RPA LPA
SVC
IVC
RPA LPA
a)
b)
Figure 1.3: A sketch for two novel designs. (a) A dual-bifurcation design proposedby Soerensen et al.1 bifurcates the IVC and SVC flow. (b) A Y-shaped graftproposed by Marsden et al.2 bifurcates the IVC flow only.
Y-graft into clinical use. Does the Y-graft outperform the traditional designs in
all patients with a variety of anatomic structures and flow conditions? What is
the optimal design for the Y-graft? Does a one-size-fits-all designs exist? Which
metrics should we use to evaluate the hemodynamic performance of the Y-graft
design? The studies in this thesis are motivated by these questions.
Optimal design has been used widely in a series of engineering problems.
In structural mechanics, one may want to know an optimal material distribution
to maximize the stiffness of a structure.71 In a thermal insulation system, one may
17
want to design a heat intercepts with minimum heat flow flux.72 In operations
research, a famous optimization example is the traveling salesman problem that
finds the shortest route to visit all cities once and return to the original. In
fluid mechanics, shape optimization is employed in harbor design, microfluidic
technology, combustion and other applications.73, 74 Aircraft design is one of the
most important research activities in shape optimization for fluid mechanics, which
is driven by considerable benefits brought by optimization.75
Optimization techniques can be categorized into two groups: gradient-based
and derivative-free algorithms. In gradient-based methods, derivatives are usually
obtained by adjoint or finite difference methods.76 Since the cost of obtaining gradi-
ent information by using adjoint methods is independent of the number of design
variables, it is widely used in aerodynamic design, as pioneered by Jameson.77
However, several difficulties limit the application of adjoint methods. First, ad-
joint methods are difficult to implement and not portable to different solvers. Sec-
ond it is challenging to apply adjoint methods to time-accurate problems. Third,
noisy or unavailable derivatives make adjoint methods inapplicable.76 In contrast,
derivative-free methods including pattern search, response surfaces and evolution-
ary algorithms are non-intrusive and suitable for non-differentiable cost functions
at a price of increased cost. The surrogate management framework (SMF) has
been successfully applied to an expensive trailing-edge noise reduction problem by
Marsden et al..78–80 Despite large applications of optimization in engineering prob-
lems, shape optimal design is still relatively new to bioengineering, particularly to
18
flow simulation based surgical design in which trial and error methods are used.
To our knowledge, SMF had not been used in bioengineering problems previously
before Marsden and colleagues applied it to a few idealized biofluid problems.76
Since there has been considerable work in the development of shape optimization
tools for engineering fluid mechanics, we believe that formal optimization tools
which systematically sort out optimal solutions from numerous candidates can fa-
cilitate discovery of new surgical designs and patients will eventually benefit from
the applications of optimization.
1.7 Outline of the thesis
The work presented in this thesis evaluated the hemodynamic performance
of the Y-gaft Fontan design and identified a series of optimal designs under different
conditions. Chapter 2 briefly introduces the methods used in this work. Chapter
3 presents a constrained optimization study of an idealized Fontan model. Energy
efficiency is chosen as the objective function. The optimal shapes for steady and
unsteady flow conditions were identified. A trade-off relationship between the en-
ergy efficiency and low WSS area was revealed. Chapter 4 extends idealized models
to patient specific models. Multiple patient specific models with T-junction, offset
and Y-graft designs were studied with particular attention paid to the HFD. In
Chapter 5, we applied optimization techniques to study the necessity of unequal-
sized branches for the Y-graft design and the influences of geometric characteristics
19
and boundary conditions on HFD using an idealized model. Two underperform-
ing patient specific Y-graft Fontan models were optimized to achieve even HFD.
Chapter 6 presents post-operative hemodynamics evaluations for the first cohort
of the Y-graft Fontan patients and in-vivo validation results. One adverse event,
in which a patient developed thrombosis post Fontan, was studied providing data
and principles for future Fontan surgical design. Finally, the major results of this
work are summarized and related future work is discussed in Chapter 7.
Chapter 2
Methods
In this chapter, we briefly introduce the numerical methods used in this
work. Finite element methods (FEM) were used to simulate blood flow and the
derivative-free surrogate management framework (SMF) together with mesh adap-
tive direct search (MADS) was used to identify optimal solutions. Although the
literature for FEM and optimization is vast, fundamental concepts are introduced
here for the sake of completeness. The contents of this chapter are primarily drawn
from the textbooks and studies of Hughes et al.,81, 82 Zienkiewicz et al.,83 Donea et
al.,84 Vignon-Clementel et al.85 and Audet et al.86
20
21
2.1 Finite element methods (FEM) for blood flow
problems
2.1.1 FEM for convection-dominated flow
The concept of finite elements originated from the work by Hrennikoff (1941)
and Courant (1942).87 Argyris, Clough and Zienkiewicz were the pioneers who de-
veloped FEM for structural analysis in the 1950s-1960s.88 The rigorous mathemat-
ical foundation was laid by Strang and Fix in the 1970s.89 It is worth mentioning
that Kang Feng, a Chinese mathematician, independently developed a FEM the-
ory so-called “Finite difference method based on variation principle” in 1965 in
parallel with the developments in the West90 and made original contributions to
the natural integral operator for the natural boundary reduction, which is also
known as the Dirichlet-to-Neumann (DtN) map.91–93
Unlike the finite difference method, FEM starts with a weak or variational
form that is converted from the strong form or classical form of the governing
equations using the principles of virtual work. After obtaining an equivalent weak
form of the problem, the weak form can be converted to a system of equations
and then discretized into a matrix form which can be solved numerically. Al-
though FEM achieved great success in solid mechanics, it historically encountered
some difficulties in convection-dominated flow problems. Let us use a 1D steady
convection-diffusion equation to illustrate the difficulties caused by using standard
22
Galerkin methods.
The strong form of the steady convection-diffusion problem, (S), is stated
as follows:
(S)
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
Given f : Ω → R, find u : Ω → R, such that
a · ∇u− κ∇2u = f on Ω
u(0) = g0 on Γ
u(L) = gL on Γ
(2.1)
where a is the convection velocity, κ is the diffusion coefficient and f is the force
term. Then, we define a collection of trial solutions, S, and a collection of weighting
functions, V , as follows:
S ={u|u ∈ H1, u(0) = g0 and u(L) = gL
}, (2.2)
V ={w|w ∈ H1, w(0) = 0 and w(L) = 0
}, (2.3)
where H1 denotes a class of Sobolev spaces, in which functions and their first
derivative are square integrable.
We now multiply equation (2.1) by w and integrate the diffusion term by
parts. Let u,x = du/dx and (w, u) =∫ΩwudΩ. We obtain the weak form (W ) as
follows:
23
(W )
⎧⎪⎪⎪⎨⎪⎪⎪⎩Given f : Ω → R, find u ∈ S such that for all w ∈ V
(w, au,x) + (κu,x, w,x) = (w, f) ,
(2.4)
Equation (2.4) can now be discretized by the Galerkin method. We approximate
u and w by uh and wh such that
u ≈ uh =
n∑A=1
dANA + g0N0 + gLNn+1, (2.5)
w ≈ wh =
n∑A=1
cANA, (2.6)
where NA’s are shape functions that satisfy
NA(0) = 0, A = 1, 2, ..., n+ 1, (2.7a)
NA(L) = 0, A = 0, 2, ..., n (2.7b)
N00 = 1 (2.7c)
Nn+1(L) = 1. (2.7d)
(2.7e)
We can derive a system of linear algebraic equations by substituting equations
24
(2.5) into equation (2.4).
n∑B=1
[(NA, aNB,x
)+
(NA, κNB,x
)]dB
= (NA, f)−(NA, aNn+1,x
)−
(κNn+1, NA,x
), for A = 1, 2, 3...n (2.8)
Given a mesh size h, the mesh Peclet number Pe is defined as ah2κ. For
a large Pe number, which indicates the flow is convection dominated, the stan-
dard Galerkin method is not stable. Considering a 1D steady convection-diffusion
problem with a zero force term, f = 0 and Dirichlet boundary conditions, g(0) =
0, g(L) = 1 in a dimensionless domain, L = 10, it can be proven that the Galerkin
scheme is equivalent to the central difference method which introduces negative
numerical diffusion for the diffusive term and results in unstable solutions.84 An
upwind scheme was proposed to eliminate the oscillation and achieve exact nodal
solutions. The optimal upwind scheme is equivalent to solving a modified equation
with artificial diffusion using a Galerkin method:
a · ∇u− (κ+ κ)∇2u = 0, (2.9)
κ = βah
2,
β = cothPe− 1/Pe,
25
where β is a parameter to control the magnitude of artificial diffusion.
0 1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
GalerkinEADSUPGEaxct Sol.
a=1 κ=1
0 1 2 3 4 5 6 7 8 9 10−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
x
u
a=10 κ=1
x
u
0 1 2 3 4 5 6 7 8 9 10−5
−4
−3
−2
−1
0
1
x
u
a=100 κ=1
Figure 2.1: Solutions for the 1D steady convection-diffusion problem (2.1) withf = 0, g(0) = 0, g(10) = 1 using Galerkin, exact artificial diffusion (EAD) andstreamline-upwind-Petrov-Galerkin (SUPG) schemes.
Figure 2.1 shows that the standard Galerkin method is unstable for con-
vection dominated flow. However, the upwind scheme is not consistent, resulting
26
in incorrect solutions when the forcing term is present (Figure 2.2). Brooks and
Hughes proposed a stabilized consistent scheme called streamline-upwind-Petrov-
Galerkin (SUPG) to overcome these difficulties.81 The idea of SUPG is to apply a
modified weighting function w = w+p to all terms in order to achieve a consistent
formulation. For equation (2.4), the corresponding SUPG scheme is
(wh, auh
,x
)+
(wh
,x, κuh,x
)+
nel∑e=1
∫Ωe
p(auh
,x − kuh,xx − f
)dx =
(wh, f
), (2.10)
p = aw,xτ,
τ = κ/ ‖a‖2 ,
where∑nel
e=1 and Ωe denote a summation from the 1st element to the nelth element
and the domain of the eth element. Figure 2.2 shows that the SUPG overcomes
the shortcoming of the upwind scheme.
2.1.2 Stabilized FEM for Navier-Stokes equations
In the previous section, we have shown that the standard Galerkin method
results in unstable solutions for convection-dominated flow problems. The SUPG
scheme suppresses the oscillatory phenomena in a consistent manner. In this sec-
tion, the SUPG formulation is applied to incompressible viscous fluid flows. The
incompressible Navier-Stokes equations for a Newtonian fluid can be written as
27
0 1 2 3 4 5 6 7 8 9 10−1
0
1
2
3
4
5
6
7
8
x
u
GalerkinEADSUPGExact Sol.
a=10 κ=1
Figure 2.2: Solutions for the 1D steady convection-diffusion problem (2.1) withf = −16 a
10
(−2 + 4 x
10
), g(0) = 0, g(10) = 1 using Galerkin, exact artificial diffusion
(EAD) and streamline-upwind-Petrov-Galerkin (SUPG) schemes.
(S)
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
Given f : Ω× (0, T ) → R3, find u(x, t) and p(x, t) : Ω → R3, such that
ρu,t + ρu · ∇u = −∇p +∇ ·T+ f in Ω× (0, T )
T = μ(∇u+∇uT
)∇ · u = 0 in Ω× (0, T )
u(x, 0) = u0(x) in Ω
u = g in Γg × (0, T )
n · (−pI+T) = h in Γh × (0, T )
(2.11)
where ρ, μ, u, p, T, and f are density, dynamic viscosity, velocity, pressure, devi-
28
atoric stress tensor and body force, respectively. The boundary is divided into Γg
and Γh (Γg ∪Γh = Γ and Γg ∩Γh = ∅ ), in which the velocity (Dirichlet condition
) and traction (Neumann condition) are prescribed, respectively.
The trial solution and weighting function spaces for the semi-discrete for-
mulation are defined as
S ={u|u (x, t) ∈ H1 (Ω)3 , t ∈ [0, T ] ,u (x, t) = g on Γg
}, (2.12)
W ={w|w (x, t) ∈ H1 (Ω)3 , t ∈ [0, T ] ,w (x, t) = 0 on Γg
}, (2.13)
P ={p|p (x, t) ∈ H1 (Ω) , t ∈ [0, T ]
}. (2.14)
The weak form for equation (2.11) using the stabilized finite element method is :
29
(W )
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
Given f : Ω× (0, T ) → R3 and g : Γg × (0, T ) → R�
find u(x, t) ∈ S and p(x, t) ∈ W such that ∀ w ∈ W and q ∈ P
BS (w, q;u, p) = 0
BS (w, q;u, p) = BG (w, q;u, p)
+∑nel
e=1
∫Ωe
(u · ∇w · τmR+∇ ·wτc∇ · u) dx
+∑nel
e=1
∫Ωe
[w · (−τmR · ∇u) + (R · ∇w) · (τR · ∇u)] dx
+∑nel
e=1
∫Ωe
∇q · τmρRdx
BG (w, q;u, p) =∫Ω[w · (ρu,t + ρu · ∇u− f) +∇w : (−pI+T)] dx
−∫Ω∇q · udx−
∫Γh
w · (−pI+T) · nds+∫Γqu · nds
where R is the residual vector of the momentum equation in Equation (2.11)
and τm, τc and τ are stabilization parameters that are constructed to achieve exact
solutions in the case of 1D model problems .81, 85, 94 The details of these parameters
can be found in .81,85, 94 A second-order accurate generalized α-method is used to
integrate the above semi-discrete equations with respect to time.94, 95
2.1.3 Boundary conditions
Dirichlet and Neumann boundary conditions were used in the problem
stated above. A no-slip boundary condition is imposed on the walls. Since the
30
inlet flowrates usually can be obtained by PC-MRI, Dirichlet boundary condi-
tions were used at the inlets by mapping parabolic or Womersley profiles. Outlet
boundary conditions are critical to obtain physiologically relevant solutions. In
most cases, the pressure or velocity measurements are not available at the outflow
boundaries. In addition, it is infeasible to model the entire vascular system in 3D
flow simulations. The 3D model is truncated at the boundaries of a domain of
interest. Thus, it is challenging to prescribe correct Dirichlet or Neumann outlet
boundary conditions directly. Alternatively, a model that prescribes a pressure-
flow relationship accounting for the downstream vasculature is preferred for the
outflow boundary conditions.96
Figure 2.3: A spatial domain is divided into a 3D domain Ω modeled by Navier-Stokes equations and a downstream Ω′ modeled by lump parameter models. TheDtN outflow boundary conditions are prescribed on the boundary ΓB that separatesΩ and Ω′.
Lumped parameter models including resistance, impedance and three el-
ement Windkessel models (RCR) are standard choices for cardiovascular simu-
lation boundary conditions. Lumped parameter models quantitatively describe
the pressure-flow relationship but lack detailed local flow information. Therefore,
lumped parameter models are suitable for modeling downstream vascular beds.
31
Vignon-Clementel et al.85 incorporated resistance, impedance and RCR models
into 3D finite element flow simulations, using the DtN map to achieve physiologic
flow conditions.85,97 In the work of Vignon-Clemental et al.,85 a spatial domain Ω
is discretized into a 3D numerical domain Ω and a downstream lumped parameter
domain Ω′ such that Ω ∩ Ω′ = ∅ and Ω ∪ Ω′ = Ω as shown in Figure 2.3. Then, u
is decomposed as
u = u+ u′ with u|Ω′ = 0,u′|Ω = 0 and u|ΓB= u′|ΓB
(2.15)
Other variables and weighting functions can be decomposed in a similar way. Since
variables and weighting functions for the domain Ω vanish on the domain Ω′ and
vice versa, the variational form for domain Ω becomes:
BG(w, q;u, p) =
∫Ω
w · (ρu,t + ρu · ∇u− f) +∇w :(−pI+ T
)dx
−∫Γh
w ·(−pI+ T
)· nds
+
∫Ω′w′ · (ρu′
,t + ρu′ · ∇u′ − f) +∇w′ : (−p′I+T′) dx
−∫Γ′
h
w′ · (−p′I+T) · n′ds−∫Ω
∇q · udx
+
∫Γ
qu · nds−∫Ω′∇q′ · udx+
∫Γ′q′u′ · n′ds (2.16)
Since Ω∩Ω′ = ∅, the third and seventh terms in the right hand side (RHS) have no
contribution to domain Ω. Therefore, the variational form for the 3D subdomain
32
Ω is shown as follows:
BG(w, q;u, p) =
∫Ω
w · (ρu,t + ρu · ∇u− f) +∇w :(−pI+T
)dx
−∫Γh
w ·(−pI+ T
)· nds−
∫ΓB
w′ · (−p′I+T) · n′ds
−∫Ω
∇q · udx+
∫Γ
qu · nds+∫ΓB
q′u′ · n′ds (2.17)
In Equation (2.17), the third and sixth integrals are the boundary terms that
connect to the downstream domain Γ′. Since variables and weighting functions
for domains Ω and Ω′ are equal, and n′ = −n on the boundary ΓB, the only
unknown term is the pressure p′ in the third integral. Lumped parameter models
approximate p′ on the boundary by specifying a pressure-flow relationship.
Resistance model
Resistance is the simplest pressure-flow relationship which defines a con-
stant ratio of pressure to flowrate R = P/Q. Assuming the pressure is constant
over the cross sectional area of the boundary ΓB, we have:
p′ − n′ ·T′ · n = −R
∫ΓB
u′ · n′ds = R
∫ΓB
u · nds (2.18)
Therefore, the unknown Neumann boundary condition can be approximated
as
33
(−p′I+T′) ≈ −R
∫ΓB
u · ndsI− n · T · nI+T (2.19)
Windkessel RCR model
Instead of using a single resistance, a Winkessel RCR model uses a capacitor
and two resistors to model the downstream domain (Figure 2.4). The proximal
resistance R and capacitance C represent the major downstream arteries. The
distal resistance R represents the distal vascular bed. The RCR circuit ordinary
differential equation is:
P +RdCdP
dt= (R +Rd)Q+RRdC
dQ
dt+ Pd +RdC
dPd
dt(2.20)
RdR
C
P, Q
Pd(t)
Pd(t)
Figure 2.4: A three element Winkessel model.
Similarly, we use the pressure-flow relationship in Equation (2.20) to ap-
proximate the unknown Neumann boundary conditions in Equation (2.17).
34
p′ − n′ ·T′ · n′ =[P ′(0) +R
∫ΓB
u′(0) · n′ds− P′d(0)
]e−t/(RdC) + P
′d(t)
− R
∫ΓB
u′(t1) · n′ds−∫ t
0
e−(t−t1)/(RdC)
C
∫ΓB
u′(t1) · n′dsdt1 (2.21)
Therefore, the Neumann boundary conditions can be approximated as
(−p′I+T′) ≈(−R
∫ΓB
u(t1) · nds)I
−(∫ t
0
e−(t−t1)/(RdC)
C
∫ΓB
u(t1) · ndsdt1)I− n · I · nI
−[(
P ′(0)−R
∫ΓB
u · nds− P′d(0)
)e−t/RdC + P
′d(t)
]I+ T (2.22)
2.2 Surrogate management framework (SMF)
For many simulation-based optimization problems with multiple design pa-
rameters, it is prohibitively expensive to perform a large number of simulations to
identify an optimal solution. Despite inevitable errors, surrogate models, which
approximate or interpolate the true cost functions based on a limited set of known
data, may predict trends in the true cost function, reducing unsuccessful trials
when gradient information is not available. Generally, surrogate models can be
polynomials such as Lagrangian interpolation or statistics-based models such as
Kriging, which is also known as Gaussian process regression. Compared to the true
35
cost function, surrogate models are usually cheap to evaluate. Thus, the methods
used to search the minimum of surrogate models are less restricted.
SMF, proposed by Booker et al.,98 incorporates a surrogate model into
pattern search algorithms to improve search efficiency with a convergence proof
provided by pattern search algorithms. The SMF consists of the following steps.
First, we employ latin hypercube sampling (LHS) to construct a well distributed
initial set in the parameter space.99 An initial surrogate is built by evaluating
the cost function at each design point in the initial set. We define a mesh in
the parameter space, and all points subsequently evaluated by the algorithm are
restricted to lie on this mesh. The search and poll step are executed alternately,
depending on whether a design point that improves the current best cost function
is found. Figure 2.5 shows a flowchart of the SMF algorithm using mesh adaptive
direct search (MADS) which is a type of pattern search algorithms.86 In the
search step, the minimum of the current surrogate function, the incumbent point,
is evaluated to calibrate the model prediction. Search steps are performed until
they fail to find an improved point, at which time a poll step is performed. In
the poll step, points neighboring the current best point are evaluated in a set of
positive spanning directions.86 Letting Dk be the positive spanning set in Rn at
iteration k, the polling points Pk are given by
Pk = {xk +Δmk d : d ∈ Dk} , (2.23)
36
where xk is the current best point and Δmk is the mesh size in parameter space at
iteration k, defined to be equal to 4−l, l = 0, 1, 2, 3....
If the poll step is successful, the algorithm returns to the search step. If the
poll step fails to find an improved point, the mesh is refined and we return to the
search step. When the parameter space mesh has been refined to a size smaller than
the minimum size set by the user, the optimization algorithm will stop and output
the optimal solution. The poll step guarantees convergence to a local minimum
of the function, following previously published convergence proofs.98, 100–102 This
indicates that SMF will converge regardless of the search strategy under certain
conditions .98 In addition, the rule to update the mesh size is flexible. The mesh
size can be returned to the previous size or kept at the current size if the search
or poll step is successful.103
Ini�aliza�onLHS Search
PollA set of posi�ve
spanning direc�on
Stop
Improved?
Improved?
Yes
No
Yes
Yes
NoRefine mesh
Figure 2.5: Flowchart of SMF using MADS. Search and poll steps are executedalternately according to whether a design point that improves the current best costfunction is found.
37
2.2.1 Surrogate models
Kriging is part of a class of statistical interpolation techniques for a random
field, pioneered by geostatisticians Krige and Matheron .104–106 Later on, Kriging
was applied to deterministic and random simulation models with multiple inputs
.107–111 Generally speaking, a Kriging model is a linear combination of n observa-
tions with n weights such that the variance of prediction is minimized. Since the
actual function is unknown, the error at each estimated point is modeled based on
probability theory. Therefore, the key to creating a Kriging model is to determine
the combination of weights for a given data set. Following Lophaven et al. ,112 we
briefly introduce the construction of Kriging.
Assuming we have m pairs of design sites S = [s1 · · · sm]T with si ∈ Rn
and responses Y = [y1 · · · ym]T, the Kriging predictor y(x) for x ∈ Rn is a linear
combination of known responses
y(x) = cTY, (2.24)
where c = c(x) ∈ Rm.
Then, we assume the stochastic process Y can be modeled by a linear
combination of p regression models and a random function:
Y = Fβ + Z, (2.25)
38
where F = [Fij ] = [fj(si)] ∈ Rmp is the basis function matrix, β = [β1 · · ·βp]T
is the coefficient vector, and Z = [z1 · · · zm]T is a random function vector. For
x = [x1 · · ·xn]T, the regression models with polynomials of orders 0 to 2 are defined
as
Constant: f1(x) = 1, p = 1, (2.26a)
Linear: f1(x) = 1, f2(x) = x1, . . . , fp(x) = xn, p = n + 1 (2.26b)
Quadratic: f1(x) = 1, . . . , fn+1(x) = xn, . . . , fn+2(x) = x21, . . . ,
f2n+1(x) = x1xn, . . . fp(x) = x2n, p =
1
2(n + 1)(n+ 2). (2.26c)
Therefore, y(x) is modeled by
y(x) = f(x)Tβ + z. (2.27)
Using Equations (2.24),(2.25) and (2.27), we obtain an expression for the error:
y(x)− y(x) = cTY − y(x)
= cT(Fβ + Z)− (f(x)Tβ + z)
= cTZ − z + (FTc− f(x))Tβ (2.28)
39
Imposing the unbiased condition, we require FTc(x) = f(x). Thus, the mean
squared error (MSE) is
E[(y(x)− y(x))2]
=E[(cTZ − z)2]
=E[z2 + cTZZTc− 2cTZz] (2.29)
The correlation matrix for the random function z between two design sites is define
as follows,
Rij = R(θ, si, sj), i, j = 1, . . . , m, (2.30)
where θ is a correlation parameter. A vector of correlations between an unknown
point x ∈ Rn and known design sites can be defined as follows,
rx = [R (θ, s1, x) . . .R (θ, sm, x)]T . (2.31)
Thus we have E[z2] = σ2, E[Zz] = σ2r, E[ZZT] = σ2R, and Equation (2.29)
becomes
E[(y(x)− y(x))2] = σ2(1 + cTRc− 2cTr). (2.32)
Correlation models are user defined. A Guassian correlation function is chosen in
40
this work. Applying Lagrange multipliers to Equation (2.32) to minimize the MSE
with the unbiased condition, the Lagrangian and its gradient with respect to c are
L(c, λ) = σ2(1 + cTRc− 2cTr)− λT(FTc− f), (2.33)
L′(c, λ) = 2σ2(Rc− r)− Fλ. (2.34)
When the MSE is minimized, (2.34) is zero. Thus we obtain a linear system
Rc− Fλ
2σ2= r (2.35a)
FTc = f (2.35b)
Solving Equation (2.35), we get an expression for c in terms of F , R and r
c = R−1[r − F (FTR−1F )−1(FTR−1r − f)]. (2.36)
Substitute (2.36) into (2.24), we have
y(x) = rTR−1Y − (FTR−1r − f)T(FTR−1F )−1FTR−1Y. (2.37)
41
2.2.2 Mesh adaptive direct search (MADS)
MADS was developed by Audet and Dennis to improve on the convergence
theory of previous GPS polling methods, particularly in the presence of nonlinear
constraints. First, we define a mesh
Mk = x+Δmk Dz : x ∈ Vk, z ∈ NnD , (2.38)
where k denotes the number of iterations, Vk ⊂ Rn is the set of all points with
known cost function values, Δmk ∈ R+ is the mesh size parameter, and D ∈ Rn×nD
is a set of positive spanning directions. At least n + 1 and at most 2n vectors
are required to form a positive basis. The polling points for a polling center xk at
iteration k can be defined as
Pk = {xk +Δmk d : d ∈ Dk ⊂ D} . (2.39)
Pattern search algorithms are differentiated by the choice of positive spanning
direction Dk. The original variant of SMF used generalized pattern search (GPS)
for polling. The drawback of the GPS algorithm is that the polling directions are
chosen from a fixed finite set D whose columns contains all possible combinations
of -1, 0, 1 except [0 0]T. For example, D =
⎡⎢⎢⎣ 1 0 −1 0 1 1 −1 −1
0 1 0 −1 1 −1 −1 1
⎤⎥⎥⎦,
when n = 2. Since the polling distance is given by Δmk ‖d‖∞, the polling parameter
Δpk, which determines the magnitude of the polling distance, is equal to the mesh
42
size parameter Δmk . If the polling frame shrinks more slowly than the mesh size, the
number of polling candidates will be increased. For MADS, Dk is constructed such
that Δmk ‖d‖ ≤ Δp
k max {‖d′‖ : d′ ∈ D} and limk→+∞ inf Δmk = limk→+∞ inf Δp
k =
0.
Audet et al.86 proposed an instance of MADS called LTMADS that uses
a lower triangluar matrix (LT) to construct positive spanning directions Dk. The
procedure is summarized below.
1. Construct a LT matrix B in which the diagonal terms are −1√Δm
k
or 1√Δm
k
and
lower entries are integers randomly chosen with equal probability in the open
interval ] −1√Δm
k
, 1√Δm
k
[.
2. Permute the lines and rows of B randomly. Letdq =[Bi1,jq , Bi2,jq , . . . , Bin,jq
]Tfor q = 1, . . . , n, where i1, i2, . . . , in and j1, j2, . . . , jn are random permutations
of the set 1,2,. . . ,n.
3. Construct a positive basis:
(a) n+ 1 directions: Dk = [d1, d2, . . . , dn+1], where dn+1 = −∑n
i=1 di.
(b) 2n direction: Dk = [d1, . . . , dn, dn+1, . . . , d2n] = [d1, . . . , dn,−d1, . . . ,−dn]
Since the entries in Dk are chosen from [ −1√Δm
k
1√Δm
k
], we have ‖d‖∞ = 1√Δm
k
and
‖d‖∞ ≤ n√Δm
k
and for 2n and n+1 positive spanning directions respectively. Thus
the polling frame defined by ‖Δmk d‖∞ = Δm
k ‖d‖∞ is bounded by n√
Δmk = Δp
k.
It has been shown the set of polling directions generated by LTMADS over all
iterations is dense.86
Chapter 3
Constrained Optimization of an
Idealized Y-graft Model
To extend the previous framework,76 we now propose a model problem in
which we optimize a new Y-graft design for the Fontan procedure, a surgery used
to treat single ventricle heart defects. We perform constrained optimization of an
idealized Y-graft model with multiple design parameters under pulsatile rest and
exercise flow conditions.
In this work, formal shape optimization with established convergence the-
ory was performed on an idealized Y-graft geometry that was parameterized using
six design parameters. To our knowledge, this work represents the first use of
formal optimization algorithms for Fontan surgery design to date. The goal of this
study is to demonstrate the feasibility of applying optimization to Y-graft design,
to compare the optimal shapes with those tested previously, to identify optimal
43
44
parameters for an idealized Y-graft Fontan model, and to assess the impact of
different flow conditions and constraints on the resulting optimal design. Results
from this work demonstrate that the SMF framework is efficient and robust for
pulsatile 3D problems with unsteady flow and constraints. While it is unlikely
the idealized model used in this work will be sufficient to make a conclusive sur-
gical recommendation, this work will lay the foundation for future patient specific
optimization that may assist in improving patient outcomes.
3.1 Methods
3.1.1 Model construction and parameterization
Idealized 3-D solid models of the Fontan geometry were generated using a
customized version of the open-sourced Simvascular software environment.113,114
The geometries of the IVC, Y-graft, PAs and SVC were defined analytically, and
constructed by lofting together a series of circles and ellipses with prescribed radii.
The model was parameterized with six parameters (Figure 3.1), each of which
could be systematically varied to generate a range of Y-graft designs. Models
were generated automatically using a computer script which takes the six design
parameters as inputs and produces as output a three dimensional solid model. The
design parameters Dbranch, LIV C , XLPA and XRPA are defined as the diameter of
branches, length of the IVC trunk, and positions where left and right branches
connect to the LPA and RPA, respectively. Parameters ΔR and ΔL, which control
45
the curvature of the two branches, are the perpendicular distances to the dashed
lines C0C1 and C0C2. The centerline paths of the branches C0C3C1 and C0C4C2 are
defined by interpolating two cubic Hermite spline functions passing through control
points C0, C1, C2, C3 and C4, marked with circles shown in Figure 3.1a. Given
LIV C , XLPA, XRPA, the end points of the two graft branches, the positions of C0,
C1 and C2, are fixed. The branch curvatures are controlled by manipulating the
control points C3 and C4, which are the midpoints of curves C0C3C1 and C0C4C2,
respectively. After the centerlines of the idealized Y-graft model are determined, a
series of circles perpendicular to the centerlines are lofted together to form a solid
model. Since the sizes of the IVC, graft, PAs and SVC were different, interpolation
was used to smoothly connect the geometry at each junction.
In addition to the six design parameters described above, four other con-
stant parameters define the model geometry. The diameters of the SVC, IVC, PAs
and the distance between the PAs and bottom of the graft, which are denoted
by DSV C ,DIV C , DPA and LPA−IV C , respectively, were measured from MRI image
data of a typical 4-year-old male Fontan patient with a traditional T-junction graft.
Tables 3.1 and 3.2 list the bounds of the design parameters and constants. The
maxima of XLPA and XRPA were based on the distances between the SVC and
the first branches of LPA and RPA as measured by the image data. Other bounds
for parameters were chosen to give reasonable flexibility in the design space while
maintaining surgical feasibility.
46
−10 −8 −6 −4 −2 0 2 4 6 8 100
2
4
6
8
10
12
XLPAXRPA
C2
ΔL
C4
C0
LIVC
ΔR C3
C1
(a) Centerline paths of the model. Circles C0, C1, C2, C3, C4
are the control points of the two branches. Parameters ΔR andΔL, which control the curvatures of the two branches, denotethe perpendicular distance to the dashed line C0C1 and lineC0C2. The other four design parameters are XLPA, XRPA,LIV C and Dbranch. Dbranch is marked in Figure 3.1b.
Outflow RPA
0 0.5 1 1.5 2 2.5 3−10
0
10
20
30
40
50
60
time(s)
IVC
flo
wra
te(c
c/s
)
0 0.5 1 1.5 2 2.5 3−10
0
10
20
30
40
50
60
time(s)
SV
C fl
ow
rate
(cc
/s)
Inflow IVC
Inflow SVC
Outflow LPA
Dbranch
(b) Solid model and pulsatile flow waveform applied at theinlets of IVC and SVC. Flow directions are denoted witharrows. Resistance boundary conditions are used at theoutlets of the PAs.
Figure 3.1: Model parametrization showing the six design parameters used forshape optimization (a), and the resting pulsatile IVC and SVC flow waveformsused for inflow boundary conditions (b).
47
Table 3.1: Bounds on the design parameters for the idealized model. Negativevalues for ΔR and ΔL indicate inward convex branches and positive values denoteoutward concave branches. Bounds were chosen to be consistent with MRI datafrom a typical patient.
Design parameters Max (cm) Min (cm)
LIV C 4.0 2.0Dbranch 2.4 1.2XRPA 3.0 1.0ΔR 0.1 -0.3
XLPA 6.0 1.0ΔL 0.1 -1.0
Table 3.2: Values of the four constant geometric parameters used in model con-struction, taken from MRI data of a typical Fontan patient.
Constant parameters cm
DSV C 1.2DIV C 2.0DPA 1.2
LPA−IV C 7.0
48
3.1.2 Flow simulation and boundary conditions
To simulate blood flow, a stabilized finite element solver94, 115 from the Sim-
vascular software package was used to solve the time-dependent 3D Navier-Stokes
equations. The simvascular flow solver has been developed from PHASTA(Parallel,
Hierarchical, Adaptive, Stabilized, Transient Analysis),115 which uses the SUPG
(Streamline-Upwind Petrov-Galerkin) formulation81 and a generalized α-method95
with second order accuracy in time. To tailor PHASTA for blood flow simulations,
developments in the areas of boundary conditions85 and fluid-solid interaction116
have been made. The solver has been previously validated using comparisons with
analytical solutions and pulsatile flow experiments.94, 117, 118 We employ a rigid-
wall and Newtonian approximation in this work. The viscosity was set to 0.04
g/(cm · s) and the density of blood was assumed to be 1.06 g/cm3. Resistance
boundary conditions85 were applied at the outlets of the PAs to model the resis-
tance of the downstream pulmonary beds. Resistance values for the resting flow
conditions were chosen such that the mean IVC pressure would be 12 mmHg, a
standard clinical value, and flow split of the RPA/LPA would be approximately
55%/45%.
While many previous Fontan simulations have employed steady inflow con-
ditions at rest, recent work has demonstrated the importance of including pul-
satile flow, respiration, and exercise effects.4, 46, 51 In this study, steady and pul-
satile inflow profiles were applied at the IVC and SVC inflow faces, and mapped to
49
Table 3.3: Mean flow rates, Re in the IVC and SVC and resistance drops at restand two levels of simulated exercise.
Exercise level IVC (cc/s) SVC (cc/s) Resistance drop ReIV C(mean) ReSV C(mean)
rest 17.87 14.07 0% 302 3872X exercise 35.74 14.07 5% 603 3873X exercise 53.61 14.07 10% 905 387
parabolic velocity profiles. Pulsatile flow values were obtained from phase-contrast
MRI data of a typical Fontan patient.36 Following our previous work,4 a respira-
tory model was superimposed on the IVC inflow waveform. To simulate exercise,
the IVC flow rate was increased by 2 and 3 times (referred to as 2X and 3X in
the text) the resting flow value and the SVC rate was kept unchanged.36 Resis-
tance values were decreased by 5% and 10% to simulate vasodilation for the 2X
and 3X exercise levels, respectively.4, 119 The SVC and IVC pulsatile waveforms
with respiration for the three cases are shown in Figure 3.2, while the mean flow
rates, resistance drops and Reynolds number (Re) are listed in Table 3.3. Mesh-
Sim (Simmetrix Inc., Clifton Park, NY) is integrated into Simvascular to generate
tetrahedral meshes using an iterative mesh optimization algorithm to improve the
mesh quality.113,120In this work, meshes were generated automatically with approx-
imately 300,000 elements. The number of elements was increased by about 30-70%
for the pulsatile exercise cases to ensure that quantities of interest did not vary
significantly. The time step was chosen according to the CFL (Courant, Friedrichs,
and Lewy) condition, and it varied from 0.01s to 3.3× 10−4s. Typical 3X exercise
simulations under steady and pulsatile inflow conditions required approximately
2.9× 102 and 4.1× 103 CPU seconds with 52 processors, respectively.
50
0 0.5 1 1.5 2 2.5 3
0
20
40
60
80
100
t (s)
flow
rate
(cc/
s)
IVC restSVC rest
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
0
20
40
60
80
100
t (s)
flow
rate
(cc/
s)
IVC 2XSVC 2X
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0
20
40
60
80
100
t (s)
flow
rate
(cc/
s)
IVC 3XSVC 2X
Figure 3.2: Pulsatile waveforms for the rest, 2X and 3X exercise cases. To simulatetwo exercise levels, the IVC flow rate was increased by 2 and 3 times; SVC flowwas unchanged.
3.1.3 Unconstrained optimization
Following the method outlined in Marsden et al.,76 we consider the opti-
mization problem,
minimize J(x),
subject to x ∈ Ω, (3.1)
where J : Rn → R is the cost function, and x is the vector of parameters.
SMF introduced in Chapter 2 was used to minimize the cost function. To
fully automate the shape optimization procedure, the following sub steps are linked
in our framework: model generation, meshing, cost function evaluation (flow sim-
ulation and post processing) and data transfers. A series of custom scripts were
created and linked with Simvascular to implement and link the sub steps above
so that the optimization procedure did not require any user intervention. These
scripts call specific sub-functions inside Simvascular to automatically generate the
51
Parameters
Op�miza�on Post processing Flow simula�on
Model construc�on1. Create the centerline of vessels2. Taking circles perpendicularly3. Lo� and generate solid model 4. FE mesh genera�on
Figure 3.3: The shape optimization procedure is made up of a series of automatedsub-steps from model construction to the input of the cost function value into theoptimization algorithm.
prescribed model, and run the required flow simulation (including meshing, bound-
ary condition prescription, and post-processing), and return a cost function value.
The optimization code was written in Matlab, compiled with the Matlab compiler,
and was called automatically from within the Simvascular scripts. Figure 3.3 shows
the sub steps of the optimization procedure, where each box represents an auto-
mated sub step. The boxes and arrows in Figure 3.3 form a loop that repeats until
the convergence criteria are satisfied.
3.1.4 Polling strategies
The polling directions used in SMF must form a positive spanning set.
In this work we evaluate two competing polling strategies in order to determine
the sensitivity of the optimal solution to the polling strategy used and to com-
pare computational efficiency. In our previous work and most optimization results
presented in this paper, n + 1 positive spanning directions were employed in the
52
polling step using the LTMADS method, which has been introduced in Chapter
2.86 LTMADS generally performs well and requires fewer function evaluations than
traditional generalized pattern search (needing 2n poll directions). However, due
to the random permutation required, the results of optimization using LTMADS
are not deterministic and large polling angles may arise.
Recently Abramson et al.121 proposed a new polling strategy called Or-
thoMADS to generate deterministic and orthogonal polling directions that make
results repeatable using 2n poll points. In practice, the only difference between
LTMADS and OrthoMADS is the method used to generate the positive spanning
basis, D. The procedure used to construct the 2n orthogonal basis in Abramson et
al.121 is summarized below, and we refer the reader to this work for further details.
1. Choose a Halton direction ut, where t is the tth direction in the Halton se-
quence.
2. Find a scalar αt,l such that ‖qt(αt,l)‖ is as close as possible to 2|l|/2, where
qt(αt,l) = round(αt,l
2ut−e‖2ut−e‖
), l is the mesh parameter defined in equation
(2.23) and e is the vector whose components are all equal to 1.
3. Apply Householder transformation.
(a) T = ‖qt(αt,l)‖2(In − 2vvT ), where v =qt(αt,l)
‖qt(αt,l)‖ and In is the identity
matrix of dimension n.
(b) D = [T ,−T ].
While the bulk of results presented here use the LTMADS method, we also present
53
a thorough comparison of optimization results and efficiency using LTMADS and
OrthoMADS. Comparisons were made by performing multiple runs of LTMADS
and comparing to a single deterministic run of OrthoMADS.
3.1.5 Constrained optimization
The SMF method can be extended to the case of constrained optimiza-
tion in a straightforward manner using a filter method. The filter method was
introduced by Fletcher and Leyffer,122 and has been successfully applied to pat-
tern search algorithms86 and aeronautic constrained optimization problems in our
previous work.79,80 One of the advantages of the filter method is that it not only
identifies the optimal solutions which satisfy constraints, but it also reveals trade-
offs between the cost function and the constraint violation. The filter method can
thus provide a range of choices to help users such as surgeons and clinicians make
comprehensive decisions.
We consider the general constrained optimization problem
minimize J(x),
subject to x ∈ Ω, C(x) ≤ 0, (3.2)
where J : Rn → R is the cost function, and x is the vector of parameters. The
constraints are given by m functions ci : Rn → R, i = 1,2,...,m such that C(x)
54
= (c1(x),...,cm)T . To evaluate how closely the design satisfies the constraints, a
constraint violation function H is defined that can be formed by a single constraint
function or the weighted sum of multiple constraint functions. The bounds on the
parameter space are defined by a polyhedron in Rn denoted by Ω (Table 3.1).
In reviewing the algorithm for the constrained optimization problem, we
first define some basic concepts for completeness.103, 122
Definition 1. A point x is feasible if H(x) = 0, where H(x) is the constraint
violation function.
Definition 2. A point x is infeasible if H(x) > 0.
Definition 3. An infeasible point x′ is dominated or filtered by a point x if the
following conditions hold: H(x) ≤ H(x′) and J(x) ≤ J(x′).
Definition 4. An infeasible point x′ is undominated or unfiltered if there is no
dominating infeasible point.
Definition 5. A point x is called the least infeasible point if
H(x) = min {H(x′),x′ ∈ U}, where U is the set of undominated points.
Figure 3.4 is an example of a filter. Each point in Figure 3.4, plotted in the form of
cost function value J vs. constraint violation value H , represents a combination of
design parameters. In Figure 3.4a, points marked with either a triangle or a circle
are undominated points that form the filter. The filter excludes the dominated
points marked with stars because for each star point, there is at least one point with
55
a lower cost function and a smaller constraint violation in the filter. During the
optimization, the filter will be improved if a new undominated point is identified.
For example, a point with J = 0.4 and H = 0.15 will improve the current filter in
Figure 3.4a such that the improved filter is the one shown in Figure 3.4b. The filter
is continuously updated in this manner during the optimization until convergence
criteria are satisfied.
The basic algorithm for constrained optimization is the same as that of
the unconstrained optimization. The difference lies in the criteria that make the
search or poll step successful. In the constrained optimization, the search or poll
step is considered successful if a new undominated point is identified. The rest of
the SMF implementation remains the same.
3.1.6 Choice of cost function and constraints for Fontan
optimization
Low exercise capacity is a typical outcome after the Fontan procedure, and
Fontan patients’ exercise capacity typically decreases progressively over time.16
It has been previously hypothesized that the hemodynamic energy loss may be
related to diminished exercise capacity because a higher energy loss will impose a
larger work load on the heart.2, 46, 123 Based on this and our previous work, we have
chosen energy loss as the primary cost function for the Y-graft optimization, with
the acknowledgement that there are several other candidates for clinically relevant
56
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
H
J
dominated pointsfeasible pointsbest feasible pointleast infeasible pointundominated points
(a) Original filter example. Points are shown in a plot of cost function value J vs.constraint violation value H .
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
H
J
dominated pointsfeasible pointsbest feasible pointleast infeasible pointundominated points
(b) Improved filter example. A non-dominated point withH = 0.15 and J = 0.4 improvedthe filter forming a new filter.
Figure 3.4: Example of a filter for the constrained optimization problem. The filtershown in (a) is improved when a dominating point is found, producing the filtershown in (b).
57
cost functions that should also be considered in future work. Using the Reynolds
transport theorem, energy efficiency can be calculated by integrating the energy
flux over all inlets and outlets using the Equations (3.3) shown below. The energy
dissipation (neglecting gravitational effects) is given by
Ediss =
Ein︷ ︸︸ ︷−
Nin∑i=1
∫Ai
(p+1
2ρu2)u · dA−
Eout︷ ︸︸ ︷Nout∑i=1
∫Ai
(p+1
2ρu2)u · dA, (3.3)
where u is the velocity, p is the pressure, ρ is the density, Nin and Nout are the
number of model inlets and outlets, respectively, and Ai is the area of the ith inlet
or outlet. The energy efficiency is then
Eeffic = Eout/Ein. (3.4)
The efficiencies are the mean values over one respiratory cycle. Thus, cost function
J can be defined by
J = 1− Eeffic. (3.5)
Thromboembolic complications, which occur in 20-30% of cases, are another
common morbidity of the Fontan surgery.10, 124 For the extracadiac Fontan, the
incidence of conduit thrombosis is about 3-10%.20, 22, 125 Using a patient specific
model, Marsden et al.2 showed that although the 12mm Y-graft design has higher
energy efficiency, it has slightly larger regions of low WSS. Although the etiology
58
Figure 3.5: Time-averaged shear stress magnitude (dynes/cm2) of the optimalshape over one respiratory cycle during the rest condition using pulsatile waveform.
of thromboembolic complications is not well understood, some studies suggest
that shear stress plays a role in the formation of thrombosis.126 In the regions of
low WSS, it is hypothesized that higher particle residence time leads to increased
platelet aggregation and more adherence to the wall, increasing the probability of
clot formation.127 Previous studies have also shown that prothrombotic substances
may accumulate due to the flow or flow stasis.128–130 Here we hypothesize that the
formation of thrombosis is related to areas of low WSS. In order to limit the risk
of thrombosis in the proposed Y-graft design, we optimize to reduce energy losses
with a constraint on the time-averaged WSS. The constraint violation function is
defined by integrating the total surface area on the model for which the WSS falls
below a critical value, τcrit. The constraint violation is defined by
H =
∫S
fdS, f =
⎧⎪⎪⎪⎨⎪⎪⎪⎩1 if τ ≤ τcrit,
0 otherwise.
(3.6)
Figure 3.5 shows the time-averaged WSS contours of the optimal design
59
using the pulsatile waveform for the rest case. We observe that most of the low
WSS areas are located on the outer wall of the bifurcation near the IVC trunk and
higher WSS appears in the PAs, that have a relatively small diameter. Based on
the WSS contours of large graft designs, we chose 0.5 dynes/cm2 as the critical
value in order to minimize the deep blue regions in the WSS contours.
3.2 Results
3.2.1 Unconstrained optimization
Idealized Y-graft models were optimized for six different cases. Initial op-
timization was performed with no constraints, using energy efficiency as the cost
function. In the first group, steady inflow boundary conditions were used to sim-
ulate rest and two levels of exercise using 2 times and 3 times the IVC flow rate.
In the second group, pulsatile waveforms were applied using the respiration model
in the IVC. Figure 3.6 shows the surface pressure and velocity magnitude of the
optimal shapes for the three steady inflow cases. The resulting optimal param-
eters are summarized in Table 3.4. To reduce the total number of cost function
evaluations, optimal parameters for the rest case were added to the initial set at
the start of the 2X exercise optimization case. Similarly, optimal parameters from
the two previous cases were added to the initial set in the 3X exercise optimization
case.
Not surprisingly, the optimal parameters from the rest case are not optimal
60
under exercise flow conditions. Examining the parameters, the optimal shapes
for both the rest and 2X exercise cases have larger diameter branches than the
IVC trunk. The optimal size of the Y-graft branches is shown to decrease with
increasing exercise level, becoming smaller than the IVC trunk for the 3X exercise
case. One possible reason is that the large graft results in a smaller pressure drop
at low flow rates, but that flow separation leads to more energy loss at higher flow
rates.
(a) Pressure contours for the optimal shapes under the rest, 2X and 3X exercise levels.
(b) velocity magnitude on the centerline cut plane for the optimal shapes under the rest,2X and 3X exercise levels.
Figure 3.6: Unconstrained optimization results using steady inflow conditions.
Little difference was found between the optimal branch diameters with
steady and pulsatile (23.4 mm vs. 22.8 mm) inflow conditions at rest. How-
ever for the 2X exercise case the pusatile waveform produced a different optimal
solution than the corresponding steady case. This is likely due to higher energy
61
10 20 30 40 50 60 70 80 90 1000.011
0.0115
0.012
0.0125
0.013
0.0135
0.014
0.0145
number of evaluations
J
rest
2X
3X
Figure 3.7: Convergence history for the unconstrained optimization under steadyinflow conditions.
dissipation and a higher peak flow rate in the pulsatile case. The optimal branch
diameter is 18 mm in the 2X pulsatile case, which is smaller than the IVC trunk
(20 mm). The optimal parameters for the 3X exercise case are identical to the 2X
exercise case. This suggests that the optimal parameter set for the 2X exercise
case is also a local minimum in the 3X exercise case.
Figure 3.7 shows the convergence history for the three steady inflow cases.
The cost function was improved by 12%, 3% and 6% for the rest, 2X and 3X
exercise cases, respectively, compared with the best cost function in the initial set.
Since the optimal parameters for the previous cases were added to the initial sets
of the exercise cases, the cost function improvements for 2X and 3X exercise cases
are not as large as those for rest case. The total number of function evaluations,
48 and 65 for the 2X and 3X exercise cases, respectively, was less than that of the
62
rest case, which required 97 evaluations. In the second group using pulsatile inflow
conditions, the rest, 2X and 3X exercise cases required 74, 35 and 35 function
evaluations, respectively. The optimal designs for the 2X and 3X pulsatile cases
are identical owing to the fact that the 2X optimal solution was included in the
initial set for the 3X case, and no further improvement was found during the
optimization.
Figures 3.8 and 3.9 show the mean pressure and instantaneous velocity
magnitude of the optimal shapes using pulsatile inflow waveforms. In Figure 3.9
we observe that jets are formed in the IVC and bifurcated by the inner wall of the
two Y-shaped branches. The velocity is lower near the outer walls of the branches,
and flow separation and recirculation are observed, especially in the large graft.
Figure 3.10 shows more details of the velocity fields in the large and small
grafts. To illustrate why the optimal shape from the rest case is not optimal
at exercise, simulations were run using the optimal shape from the rest case with
both the 2X and 3X pulsatile inflow conditions. Compared with the optimal design
(small graft) for the 2X and 3X exercise cases, the large graft design has larger
separation regions near the outer walls of the branches. Making the assumption
that larger flow separation will result in higher energy dissipation compared to the
rest case explains why the optimal shapes for the 2X and 3X exercise cases have
smaller branches.
Although the optimal graft size differs with inflow conditions, there are
some common characteristics shared by all optimal shapes. First, XRPA and XLPA
63
Figure 3.8: Mean pressure for the optimal shapes under the rest, 2X and 3Xexercise levels using pulsatile waveforms.
almost reach their maximum bound (XRPA ≈ 3 cm and XLPA ≈ 6 cm) producing
a large angle between the branches. Second, ΔR and ΔL are close to their negative
maximum bound of -0.3 and -1.0, respectively, corresponding to convex curvatures.
A negative maximum value of ΔR and ΔL makes the divider that bifurcates the
IVC flow as sharp as possible for given values of XRPA and XLPA. It is likely that
more kinetic energy will be lost if the jet from the IVC impinges on an obtuse
divider and reflects. This agrees with our findings that designs with a wide span
and inward convex branches have higher energy efficiency than those with straight
branches and an obtuse divider.
If XRPA and XLPA are close to the lower bound of 1 cm, the two branches
will be merged into one common branch forming a T-junction or offset model.
Computational results demonstrate that the optimization algorithm leads to a Y-
graft design rather than a T-junction or offset in order to minimize energy loss.
This agrees well with our previous findings that demonstrated lower energy loss in
a patient specific Y-graft model.
64
(a) Instantaneous velocity magnitude of the optimal shape for the rest case. A respiratorycycle for the rest case is 2.86s.
(b) Instantaneous velocity magnitude of the optimal shape for the 2X case. A respiratorycycle for the 2X exercise case is 1.71s.
(c) Instantaneous velocity magnitude of the optimal shape for the 3X case. A respiratorycycle for the 3X exercise case is 1.33s.
Figure 3.9: Instantaneous velocity magnitude on the centerline cut plane of theoptimal shapes using pulsatile waveforms for the rest, 2X, and 3X cases withunconstrained optimization.
Table 3.4: Optimal parameters, cost function values and number of evaluations forthe unconstrained optimization using different inflow conditions. Parameters thatlie on the boundary are in bold.
inflow conditions J number of evaluations LIV C (cm) Dbranch(cm) XRPA (cm) ΔR (cm) XLPA (cm) ΔL (cm)
steady rest 0.0114 97 3.04 2.34 3.0 -0.3 6.0 -1.0steady 2X exercise 0.0119 48 2.95 2.02 3.0 -0.237 6.0 -1.0steady 3X exercise 0.0134 65 2.58 1.90 2.89 -0.29 6.0 -0.99
pulsatile rest 0.0122 74 3.58 2.28 2.2 -0.060 4.0 -0.34pulsatile 2X exercise 0.0137 35 3.5 1.8 3.0 -0.3 6.0 -1.0pulsatile 3X exercise 0.0156 35 3.5 1.8 3.0 -0.3 6.0 -1.0
65
(a) Comparison of velocity vector fields at peak IVC inflow for the rest (left)and 2X exercise (right) optimal shapes under the 2X pulsatile inflow condition,colored by velocity magnitude (cm/s).
(b) Comparison of velocity vector fields at peak IVC inflow for the rest (left) andthe 3X exercise (right) optimal shapes under the 3X pulsatile inflow condition,colored by velocity magnitude (cm/s).
Figure 3.10: Velocity vectors at peak IVC inflow for the exercise and rest optimalshapes at exercise conditions. Compared with the exercise optimal design (left),the large graft of the rest optimal design (right) results in more flow separationand causes more energy loss.
66
3.2.2 Polling comparison
To assess the robustness of the solution to the choice of polling method,
and assess the efficiency of the algorithm, two polling strategies were compared.
Unconstrained optimization using both LTMADS and OrthoMADS was performed
on the idealized Y-graft model. We used the same cost function J as above and
ran simulations under the 3X steady inflow condition. Taking into account the
computational expense, we performed five runs of LTMADS to compare with an
OrthoMADS instance.
First, only the poll step was employed and the search step was turned off.
Results are shown in Figure 3.11 and Table 3.5. We followed the same standard
as in Abramson et al.121 to evaluate the result given by OrthoMADS. The score
m indicates that the result given by OrthoMADS is as good or better than m
out of five LTMADS instances, with a relative precision of 0.1%. Thus the score
for OrthoMADS will be 5 if OrthoMADS gives the best result compared with all
five runs of LTMADS. According to table 3.5, OrthoMADS scored 3 and required
12% more function evaluations than the average number of evaluations required by
LTMADS. Considering that OrthoMADS employed 2n (n = 6) polling directions
and LTMADS here used n + 1 directions, OrthoMADS behaved remarkably well
in this case. In addition, Figure 3.11 shows that OrthoMADS found its best point
more quickly than all five LTMADS runs but took more function evaluations to
reach the convergence criteria because of the 2n polling directions.
67
10 20 30 40 50 60 70 80 90 100
0.0134
0.0136
0.0138
0.014
0.0142
0.0144
0.0146
0.0148
0.015
number of evaluations
J
LT1
LT2
LT3
LT4
LT5
ortho
Figure 3.11: Convergence history for 5 LTMADS and 1 OrthoMADS instanceswith poll only under the 3X steady inflow condition.
Table 3.5: Comparison results for 5 LTMADS and 1 OrthoMADS instances withpoll only under the 3X steady inflow condition. Parameters that lie on the bound-ary are in bold.
case J number of evaluations score LIV C (cm) Dbranch(cm) XRPA (cm) ΔR (cm) XLPA (cm) ΔL (cm)
LTMADS1 0.01341680 80 2.0 1.875 3.0 -0.3 6.0 -1.0LTMADS2 0.01340101 76 2.0 1.894 3.0 -0.3 5.9219 -1.0LTMADS3 0.01349008 77 2.0 1.95 3.0 -0.3 6.0 -0.9313LTMADS4 0.01344364 93 2.0 1.931 2.875 -0.3 6.0 -1.0LTMADS5 0.01344225 87 2.0 1.8 3.0 -0.3 6.0 -1.0average 0.01343936 83
OrthoMADS 0.01344225 93 3 2.0 1.95 3.0 -0.3 6.0 -1.0
In the second comparison, we added the search step back into the opti-
mization. Due to the search step, the number of function evaluations was reduced
by approximately 27% and 24% for LTMADS and OrthoMADS, respectively, as
shown in Figure 3.12 and Table 3.6. In this comparison, OrthoMADS and 3 in-
stances of LTMADS found identical solutions and the other 2 LTMADS instances
were worse than OrthoMADS. Thus, OrthoMADS scored 5 when the search step
was added to the optimization.
68
10 20 30 40 50 60 70 80
0.0134
0.0136
0.0138
0.014
0.0142
0.0144
0.0146
0.0148
0.015
number of evaluations
J
LT1
LT2
LT3
LT4
LT5
ortho
Figure 3.12: Convergence history for 5 LTMADS and 1 OrthoMADS instanceswith search and poll together under the 3X steady inflow condition.
Table 3.6: Comparison results for 5 LTMADS and 1 OrthoMADS instances withsearch and poll together under the 3X steady inflow condition. OrthoMADS foundthe best solution among 5 instances of LTMADS, with relative precision 0.1%.Parameters that lie on the boundary are in bold.
J number of evaluations score LIV C (cm) Dbranch(cm) XRPA (cm) ΔR (cm) XLPA (cm) ΔL (cm)
LTMADS1 0.01336113 55 3.0 1.8 3.0 -0.3 6.0 -1.0LTMADS2 0.01380732 57 3.375 1.8 2.875 -0.3 6.0 -0.5875LTMADS3 0.01336574 66 3.0 1.8 3.0 -0.3 6.0 -1.0LTMADS4 0.01337445 50 3.0 1.8 3.0 -0.3 6.0 -1.0LTMADS5 0.01338340 76 2.875 1.8 3.0 -0.3 6.0 -1.0average 0.01345840 61
OrthoMADS 0.01336049 71 5 3.0 1.8 3.0 -0.3 6.0 -1.0
3.2.3 Constrained optimization
Constrained optimization using the filter method was performed for the
rest, 2X and 3X cases using pulsatile inflow conditions. The mesh size Δmk was
not allowed to increase and n + 1 polling directions were employed. To reach
convergence, 65, 42 and 62 function evaluations were required for the rest, 2X
and 3X cases, respectively. Figure 3.13 shows the final results plotted as cost
function J vs. constraint violation function H for the constrained optimization at
69
rest. The models in Figure 3.13 correspond to the best feasible point and three
undominated points that form the filter. Three feasible points were identified for
the rest case. The best feasible point (marked by a large square) satisfies the
constraint and has the lowest cost function value. The lowest undominated point
is the design with the highest energy efficiency during the optimization. Figure
3.13 shows a clear trend that the best feasible point has smaller sized branches
merging together compared with the unconstrained results. As the conduit size
and bifurcation angle increase, the cost function is decreased but the constraint
violation function is increased. The model shown at the bottom right corner of
Figure 3.13 has the highest energy efficiency, and, not surprisingly, is similar to the
optimal shape from the unconstrained optimization for the rest case. The model
has large radius, wide span branches with a slightly smaller branch size than the
unconstrained optimization result. These results showed that the WSS constraint
has a strong effect in the constrained optimization for the rest case. In order to
minimize the areas of low WSS, the Y-graft became narrow with close branches.
We observe that a large conduit size and wide span branches cause considerable
areas of low WSS located on the outer wall of the bifurcation, which corresponds
to decreased energy dissipation.
With increasing flow rate, it was expected that the effect of the WSS con-
straint would diminish and more feasible points would be identified. Results for
the 2X exercise case, shown in Figure 3.14, are consistent with our expectations.
Eleven feasible points were identified. In the 2X exercise case, there is no substan-
70
0 2 4 6 8 10 120.014
0.016
0.018
0.02
0.022
0.024
0.026
H
J
dominated pointsthe best feasible pointfeasible pointsthe least infeasible pointundominated points
Figure 3.13: Final results of the constrained optimization plotted as cost functionJ vs. constraint function H for the rest case. The model in the upper left corneris the best feasible design and the model in right bottom corner is the design withhighest energy efficiency. Differences in shape among these models show a strongeffect of the WSS constraint for the rest case.
tial difference in geometry among these three points. Table 3.7 lists the parameters
of the best feasible and highest energy efficiency models for all constrained opti-
mization cases.
When the IVC flow rate increased to 3X, the WSS constraint became even
easier to satisfy and 32 feasible points were identified. As shown in Figure 3.15a,
points are tightly clustered near the axis corresponding to zero constraint violation.
We observe that the difference between the best feasible point and highest energy
efficiency point for the 3X exercise case becomes even less pronounced than the 2X
exercise case, but the size of branches is decreased, as shown in Table 3.7. All the
models in Figure 3.15b have wide-span and inward branches that are characteristic
of unconstrained optimization.
71
0 0.1 0.2 0.3 0.4 0.5 0.6
0.016
0.018
0.02
0.022
0.024
0.026
0.028
H
J
dominated pointsthe best feasible pointfeasible pointsthe least infeasible pointundominated points
Figure 3.14: Final results of the constrained optimization plotted as cost functionJ vs. constraint function H for the 2X exercise case. The number of feasiblepoints is increased to 11. The best feasible, least infeasible and highest energyefficiency models are listed from left to right. Different points in the filter plothave similar geometry. Results for the 2X exercise case show that the effect of theWSS constraint is weakened as inflow rates increase.
3.3 Discussion
In this work, we have coupled an efficient derivative-free optimization al-
gorithm with a 3D Navier-Stokes cardiovascular flow solver to optimize the shape
of a newly developed surgical design for the Fontan procedure. The application
of the constrained optimization to the Y-graft has demonstrated that the SMF
method can be efficiently coupled to pulsatile, 3D, cardiovascular simulations with
a reasonable number of design parameters. Compared with our previous work, this
work increased geometric complexity and number of design parameters, and added
constraints to the cardiovascular optimization framework. While computational
cost is still an issue, Y-graft optimization was performed efficiently, requiring at
72
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80.016
0.018
0.02
0.022
0.024
0.026
0.028
0.03
0.032
H
J
dominated pointsthe best feasible pointfeasible pointsthe least infeasible pointundominated points
(a) Final filter for optimization with the 3X exercise condition.Points are tightly clustered near the axis corresponding to zeroconstraint violation. There are 32 feasible points.
0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055
0.0164
0.0165
0.0166
0.0167
0.0168
H
J
dominated pointsthe best feasible pointfeasible pointsthe least infeasible point undominated points
(b) A close-up of the filter for the 3X exercise case. Compared to the 2X exercisecase, there is even less difference between the best feasible point and undominatedinfeasible points.
Figure 3.15: Final results of the constrained optimization plotted as cost functionJ vs. constraint function H for the 3X exercise case. The number of feasible pointsis increased to 32.
73
Table 3.7: Comparison of the best feasible parameter and the highest energy effi-ciency points for the constrained optimization. Parameters that lie on the bound-ary are in bold.
inflow conditions LIV C (cm) Dbranch(cm) XRPA (cm) ΔR (cm) XLPA (cm) ΔL (cm) J H
rest best feasible 4.0 1.5 1.0 0.0 1.0 -0.725 0.0229 0rest highest energy efficiency 3.0 2.1 3.0 -0.3 4.75 -1.0 0.0140 8.5030
2X best feasible 3.5 1.8 3.0 -0.2 3.5 -0.175 0.0161 02X highest energy efficiency 3.5 1.8 3.0 -0.2 4.75 -0.175 0.0154 0.0140
3X best feasible 4.0 1.5 2.6 -0.3 5.7 -0.86 0.0164 03X highest energy efficiency 3.9 1.5 2.6 -0.3 6.0 -0.90 0.0163 0.0193
most 97 function evaluations for the unconstrained case (using LTMADS) and 65
evaluations for the constrained optimization case.
The unconstrained SMF method was tested using two polling strategies,
LTMADS and OrthoMADS, under the 3X steady inflow condition. Despite an
increased cost for each individual poll step, OrthoMADS behaved well overall.
OrthoMADS gave the best or one of the best repeatable results among the 5 runs
of LTMADS with n + 1 polling directions, with only a 12-16% increase in the
number of function evaluations using polling alone. Addition of the search step
saved 24% of the function evaluations for OrthoMADS compared to polling alone,
illustrating that the surrogate function is essential in improving efficiency of SMF.
OrthoMADS as an alternative polling strategy will be most suitable when the
computational cost can be reduced efficiently using a good quality surrogate, and
the number of parameters is not to large. The polling comparison confirmed that
the SMF method is robust and efficient, producing similar results with two different
polling strategies in both cost function and number of function evaluations.
Shape optimization with and without constraints has identified several in-
74
teresting trends in the resulting optimal Y-graft under various flow conditions.
Results from the unconstrained optimization cases revealed that optimal Y-graft
designs differed between rest and exercise, as well as steady and pulsatile flow.
This further illustrates that steady flow simulations alone are not sufficient to test
new surgical designs, and that pulsatility will likely produce different conclusions.
Under low flow rates, optimal shapes tend to have graft branches that are larger
than the IVC. However, under exercise conditions the maximum graft size does not
necessarily have the highest energy efficiency and larger areas of flow separation are
observed. The observed trend is that optimal designs for the exercise conditions
tend to have smaller graft branches than those for rest conditions. Additionally,
the optimal shape for the 2X exercise case is also a local optimum for the 3X cases.
We employed a filter method for constrained optimization that illustrated
the trade-off between the energy cost function and the WSS constraint. A large
graft branch size and wide span branches resulted in high energy efficiency but
larger areas of low WSS. Results from the three constrained optimization cases
showed that the impact of the WSS constraint decreases with increasing exercise
level. While the WSS constraint forced the branches close together in the rest
case, the Y-graft was clearly the design of choice at exercise. Because patients
(particularly children) spend a large percentage of time in a mild exercise state, the
choice of design should ultimately combine both rest and exercise considerations.
The critical WSS value we used in this study has not yet been validated due
to lack of necessary biological and clinical data related to thrombosis in the Fontan
75
graft. However, results of the constrained optimization have clearly demonstrated
that the WSS constraint and exercise conditions can influence the Fontan geometry.
The examples with the WSS constraint demonstrate that constrained optimization
allows us to consider more than one factor in the surgical design. Further studies
are needed to validate the link between the formation of thrombosis and areas of
low WSS. The critical WSS value which may cause thrombosis is important for
Fontan patients, but is still unknown and may differ from one patient to another. In
addition, high shear can also increase platelet aggregation.131 As more clinical data
is obtained and correlated with simulation results, more accurate knowledge of the
mechanical factors that cause thrombosis will be determined, and this information
should be incorporated into future optimization work.
The optimization results presented here agree with previous findings that
the Y-graft is likely to offer superior performance compared to traditional designs
that are currently used. Compared with a trial-and-error design method, opti-
mization offers improved confidence that the Y-graft is a promising design for the
Fontan surgery, due to the known convergence theory and global search nature
of the SMF optimization algorithm. Despite this, clinical implementation of the
Y-graft design will need to take into account several additional clinical and surgi-
cal considerations. Our previous work has confirmed that there is sufficient space
anatomically for the Y-graft by embedding the Y-graft model in the image data
to ensure that it did not overlap with anatomic features.2 However, a large Y-
graft branch size may result in technical difficulties in the actual operation. For
76
example, oversized grafts may be more challenging to connect smoothly with the
PAs in surgical practice. The results presented in this work demonstrate that the
choice of Y-graft design should balance energy loss, areas of low WSS and technical
feasibility. Thus, a design with slightly smaller branches than the IVC can achieve
a balance between energy efficiency at different exercise levels and the potential
thrombosis risks. However, weighting of these factors should be investigated and
related to clinical data in future work. Future work should also consider additional
clinical factors as cost functions or constraints in multiple objective optimization,
including flow distribution to the right and left PAs, oxygen saturation levels, and
technical difficulty.
3.4 Limitations
A major limitation of this study is the use of idealized cylindrical models
that we believe results in unrealistically high energy efficiency, and small differences
in efficiency between models. Since this study is a first step towards applying
a formal optimization method to the Fontan procedure, and parameterization of
patient specific models presents significant challenges, optimization on an idealized
model was a necessity in the present work. Moreover, our previous patient specific
simulations indicate that variations of the graft geometry (9 mm vs. 12 mm)
and inflow conditions (rest vs. exercise) can cause far more significant efficiency
differences that those observed in this work. We therefore expect that differences
77
between models will be significantly accentuated in the patient specific cases, and
this will be a focus of future study.
Another limitation of this work was that the constant parameters used to
define the model geometry represent a single patient. Future work should optimize
multiple patient specific models. The influence of compliant walls and growth of
the patient also merit consideration in future optimization work. In addition, the
use of a non-Newtonian assumption for the fluid, has been shown in some previous
work132,133 to affect WSS and flow profiles.
Future work should also consider the effect of uncertainties in simulation
parameters, inflow rates, and implementation of the surgical design. Robust opti-
mization that accounts for uncertainties could identify designs that are less sensi-
tive to small changes in design parameters, thus allowing for a “fudge-factor” in
surgical implementation.134,135 Efficiency of the optimization algorithm could also
be improved by performing function evaluations in parallel for each search and poll
step, which would lead to significant savings in computational time.
3.5 Acknowledgments
This work was supported by the American Heart Association, a Burroughs
Wellcome Fund Career Award at the Scientific Interface and a Leducq Founda-
tion grant. The authors wish to thank Prof. John Dennis, Prof. Charles Audet,
Dr. Sebastien Le Digabel, cardiac surgeons Dr. Mohan Reddy and Dr. John
78
Lamberti for sharing their expertise on optimization and Fontan surgical proce-
dures. We gratefully acknowledge the use of software from the Simvascular open
source project(http://simtk.org),114 Cardiovascular Simulation, Inc., as well as the
expertise of Dr. Nathan Wilson and Prof. Charles Taylor.
Chapter 3, in full, is a reprint of the material as it appears in Numerical
Grid Generational in Yang, W., Feinstein, J. A. and Marsden, A. L. Constrained
Optimization of an Idealized Y-shaped Baffle for the Fontan Surgery at Rest and
Exercise. Comput. Meth. Appl. Mech. Engrg. 2010;199:2135-2149. The disserta-
tion author was the primary investigator and author of this paper.
Chapter 4
Hemodynamic Evaluations for
traditional and Y-graft Fontan
Geometries
The impact of geometry on Fontan hemodynamics is now widely accepted
in the engineering and clinical communities.13, 45, 50, 51 Recent advances in compu-
tational fluid dynamics (CFD), computer aided design (CAD), magnetic resonance
imaging (MRI), and fluid structure interaction methods (FSI) for blood flow mod-
eling have led to studies of Fontan hemodynamics and surgical design progressing
from idealized to patient specific models, steady to unsteady flow, and trial and
error to optimal design.2,4, 44, 57, 136
Although the preliminary studies showed the Y-graft design to improve
hemodynamics overall, it has not yet been confirmed that the superiority of the
79
80
Y-graft is universal.2 The purpose of the study in Chapter 4 is to evaluate the
potential performance of the Y-graft Fontan procedure in simulation as a step
towards clinical implementation. Multiple virtual patient models are used, and
multiple parameters51 are evaluated for each, with particular attention paid to the
hepatic flow distribution.
4.1 Methods
4.1.1 Geometrical model construction
Virtual surgery was performed on five models by implanting a Y or tube-
shaped graft into patient specific Glenn models5 constructed from MRI images. All
patients were consented as part of an institutional review board approved protocol
at Lucile Packard Children’s Hospital (Stanford University). Model construction
was performed using a custom version of the open source Simvascular software
package,114 as in our previous work.2
Geometric parameters were chosen based on our previous optimization
study.136 A Y-graft with a 20 mm diameter trunk and 15 mm diameter branches
was chosen for all patients. The T-junction and offset models were constructed
with a 20 mm diameter tube-shaped graft following common clinical practice, and
resulting models are shown in Figure 4.1.
In the virtual implantation, the centerline paths of the SVC and PA anatomy
of the original Glenn models were left unchanged. The space created by the defla-
81
tion of the right atrium (RA) during the Fontan procedure was accounted for in IVC
placement during virtual surgery. In addition, the PA segmentations are enlarged
to match the baffle size in order to avoid creating unrealistic stenoses. Although
this is purely a simulation study, all models were constructed under guidance of
a surgeon to replicate as closely as possible a realistic surgical implementation of
the intended design.
Patient B has a left PA (LPA) stenosis which is commonly observed in
Fontan patients due to aortic arch override. Three potential Y-graft designs were
proposed for this patient model. In Y-graft I, the stenosis is relieved by placing
the anastomosis of the left branch at the stenosis. In Y-graft II, the left branch
is anastomosed distal to the stenosis, without relieving it. In Y-graft III, the
left branch is also anastomosed distal to the stenosis, but the stenosis is relieved.
Patient E has heterotaxy and a right PA (RPA) stenosis. Instead of an LPA offset
connection, the baffle is anastomosed to the RPA in a mirror image of the other
patients, denoted as Offset I. The stenosis is relieved in the Y-graft I, T-junction
and Offset I designs. An LPA-offset connection without relieving the stenosis (II)
is also constructed. We redesigned the Y-graft for patient E, denoted as Y-graft
II, by bringing the left branch closer to the SVC-PA junction and less anteriorly
convex, based on the simulation results of Y-graft I.
82
A
B
C
D
E
Y-graft
Y-graft I Y-graft II Y-graft III
Y-graft
Y-graft I
Y-graft
Y-graft II
T-junction
T-junction
T-junction
T-junction
offset
offset
offset
offset
offset I offset II
T-junction
Glenn Fontan
Figure 4.1: Original Glenn models and variations of Fontan geometries for fivepatients. The Y-graft includes a 20 mm trunk and two 15 mm branches. The sizeof the tube-shaped graft is 20 mm. Patients B and E have a stenosis in the LPAand RPA, respectively, denoted by arrows.
83
4.1.2 Flow simulation and boundary conditions
Flow simulations were performed with a stabilized finite element Navier-
Stokes solver,94 assuming rigid walls and Newtonian flow with a density of 1.06g/cm3
and viscosity of 0.04 g/(cm s). MeshSim (Simmetrix, Inc., Clifton Park, NY)
was employed to generate tetrahedral meshes automatically and anisotropic mesh
adaptation was performed to ensure mesh convergence. Final meshes consisted of
approximately 1 to 1.5 million elements.
During MR imaging, phase contrast magnetic resonance imaging (PCMRI)
slices were acquired in the SVC, IVC, LPA and RPA for each Glenn patient.
The SVC waveform from PCMRI was applied directly to the SVC inflow face by
mapping it to a parabolic profile. Because the IVC flow waveforms were acquired
in the Glenn patients prior to surgery, when the IVC was still connected to the RA,
these waveforms exhibited higher cardiac pulsatility and less respiratory pulsatility
compared to typical Fontan patients.137 To account for this, the amplitude of the
IVC waveform of each Glenn patient was scaled to match typical Fontan data using
four patients from a previous study, while keeping the mean the same. Additionally,
a respiratory model4 was superimposed on the scaled IVC waveform following our
previous work.
A three element Windkessel circuit model (RCR, resistor-capacitor-resistor)85,96, 138
was applied at each outlet. Predicting changes in pulmonary resistance following
Fontan surgery remains an open question. However, since this study aims to model
84
immediate post-operative flow conditions, we have assumed that the downstream
resistances do not change significantly in the short period after the surgery. There-
fore, the same parameters for the Windkessel model were employed for the Glenn
and Fontan simulations in all patients, so that the comparison of post-operative
states among patients remains consistent.
Exercise flow conditions were simulated by increasing IVC flow and decreas-
ing outflow resistances, following our previous work.4 The mean IVC flow rate was
increased by 2 and 3 times (referred to as 2X and 3X in the text) to simulate
exercise conditions and the total resistances of each branch was decreased by 5%
and 10%, respectively.
4.1.3 Determination of performance parameters
Mean SVC pressure, wall shear stress and power loss were computed with
standard methods.2 To quantify the hepatic flow distribution in different surgical
designs, Lagrangian particle tracking139 is performed (Figure 4.3).
It is well known that the ratio of IVC/SVC flow and LPA/RPA resistances
vary widely among patients. While ideally one would like to achieve a 50/50
hepatic flow split for all patients, this will not be possible in every case. A simple
conservation of mass analysis allows us to determine the theoretical optimum for
hepatic flow distribution in a given situation.
We assume that hepatic flow is well mixed in the IVC such that the hepatic
and IVC flow distributions are the same, and that the theoretical optimum for the
85
hepatic flow distribution is the split closest to 50/50 that satisfies conservation of
mass.
Let us derive the dependence of the hepatic flow distribution on the vena
caval and pulmonary distributions (see Figure 4.2 for a schematic representation
of the Fontan configuration). Basic conservation of mass dictates that the ratio
between the RPA flow and the LPA flow can be defined as
QRPA
QLPA=
QIV C · x+QSV C · yQIV C · (1− x) +QSV C · (1− y)
=fs
1− fs, (4.1)
where Q is the flow rate, fs is the fraction of total inflow going to the RPA, x is
the fraction of hepatic flow going to the RPA, and y is the fraction of SVC flow
going to the RPA. Then, the hepatic flow distribution can be expressed by
x = fs +QSV C
QIV C
· (fs − y). (4.2)
Equation 4.2 shows that the hepatic flow distribution is a function of the
inflow rates, outflow split and percentage of SVC flow going to the RPA.
For a given hepatic flow split x, the bounds of the RPA flow are
QIV C · x ≤ QRPA ≤ QIV C · x+QSV C . (4.3)
Thus, a 50/50 hepatic flow split can not exist when QRPA < QIV C · 0.5 or
QRPA > QIV C · 0.5 + QSV C . Since QRPA is given as boundary condition, when a
86
Figure 4.2: Based on conservation of mass, we have QRPA = QIV C · x + QSV C · yand QLPA = QIV C · (1−x)+QSV C · (1− y), where x is the fraction of hepatic flowgoing to the RPA, and y is the fraction of SVC flow going to the RPA.
Table 4.1: MRI inflow rates, MRI outflow splits and the theoretical optimal hepaticflow splits (TOHFS) at rest for the five study patients.
Patient Age (yr.) BSA (m2) IVC (ml/s) SVC (ml/s) RPA/LPA flow split TOHFS (RPA/LPA)
A 4.8 0.63 13.4 14.4 81/19 61/39B 3.9 0.66 12.6 15.3 63/37 50/50C 2.8 0.56 6.3 15.2 54/46 50/50D 3.0 0.61 14.8 27.8 55/45 50/50E 3.0 0.74 12.7 19.4 70/30 50/50
50/50 split is infeasible, the theoretical optimum for the hepatic flow distribution
is defined as the value closest to 50/50 achieved by taking y = 0 or y = 1, i.e.
QIV C · x = QRPA or QIV C · x+QSV C = QRPA
Table 4.1 lists the theoretical best hepatic flow distribution for the five
patients in our study. We observe that a perfect 50/50 split can theoretically exist
for four out of the five patients, with patient A as the exception, for whom the
RPA receives more than 80% of the total inflow.
The main focus of this study is the evaluation of Fontan designs in the im-
mediate post-operative period. However, it is unlikely that pulmonary resistances
87
t=0s t=0.24s t=0.6s
t=2.4s t=4.8s t=12s
RPA LPA
Figure 4.3: Visualization of the particle tracking in the model Y-graft II for patientB. Particle tracking is terminated when particles are washed from the model.
remain constant over time for most Fontan patients due to age, growth, and remod-
eling. We therefore asses the robustness of the hepatic flow distribution to changes
in the pulmonary flow split for different surgical designs. For each patient, the
Y-graft, offset and T-junction designs were analyzed under rest conditions for a
range of pulmonary resistances to compare the robustness of different designs.
4.2 Results
4.2.1 Hepatic flow distribution
The percentages of hepatic flow to the RPA and LPA, and differences from
the theoretical optima at rest, 2X and 3X exercise conditions are shown in Figure
4.4. In patient A, the Y-graft design is closest to the theoretical optimum, even
though patient A’s right lung receives more than 80% of the total venous return.
88
To illustrate how the geometry influences the hemodynamics of hepatic flow dis-
tribution, the velocity fields for three patients are shown in Figure 4.5. In the
T-junction design of patient A, the SVC jet clearly blocks the hepatic flow enter-
ing the LPA and skews it to the RPA. In contrast, the Y-graft design effectively
mitigates this effect.
In patient B, Y-graft I hemodynamics shows that relieving the stenosis with
a proximal anastomosis allows the SVC jet to enter the baffle and hinder hepatic
flow. However, Y-graft II, which leaves the stenosis intact, achieves a nearly perfect
50/50 distribution at rest. Overall for patient B, Y-graft II has the best hepatic
distribution at rest, while the offset design performs best at exercise.
In patient C, the T-junction and offset designs skew hepatic flow to the
RPA and LPA, respectively. In both patients C and D, when the pulmonary flow
split is close to 50/50, the offset design has poor performance with a highly skewed
hepatic flow split. Overall for patient C, the Y-graft design has the best hepatic
performance though none of the designs achieve the theoretical optimum.
In patient D, although the SVC anastomosis is inclined to channel flow to
the LPA, Figure 4.5 shows that the lower section bc of the SVC is ineffective in
changing the direction of flow. Improvements are observed in both the Y-graft and
T-junction during exercise, and the Y-graft achieves a nearly 50/50 split in the 3X
exercise case. Overall for patient D, the T-junction does best at rest, while the
Y-graft does best at exercise.
In patient E, the T-junction, Offset I and Y-graft I designs skew hepatic
89
flow strongly to the RPA with a 70/30 (RPA/LPA) pulmonary flow split because
of the smooth graft and the low distal RPA resistance. For the T-junction design,
hepatic flow distribution remains constant with exercise, whereas Y-graft I pro-
gressively improves the distribution from rest to exercise. The skewed hepatic flow
is corrected in the Y-graft because the SVC jet suppresses hepatic flow entering
the RPA from the right branch (Figure 4.5). Overall for patient E, the offset II
design results in the best hepatic flow distribution at rest and exercise.
Robustness test
The hepatic flow distribution in the Y-graft, offset and T-junction designs
for three different pulmonary flow splits, and their averaged deviations for a 25%
change in flow split, are shown in Figure 4.6. The optimal design depends heavily
on the pulmonary flow split for individual patients. The Y-graft and T-junction
designs are relatively more robust in two and three patients, respectively. Although
the T-junction design underperforms for most patients with the original flow split,
deviations in hepatic flow distribution are smaller than those of the offset design.
4.2.2 SVC pressure
Pressure levels increase in all models under exercise conditions, compared
to the values at rest. The SVC pressure levels in the Y-graft are the same or
slightly lower than those in the T-junction and offset designs at rest and exercise
conditions in five patients (Table 4.2). This finding agrees with our previous work
90
0%5%
10%15%20%25%30%35%40%45%50%
Y-gra� T-junc�on Offset
rest2X3X
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50%
Y-gra� I Y-gra� II Y-gra� III T-junc�on Offset
rest2X3X
0%5%
10%15%20%25%30%35%40%45%50%
Y-gra� T-junc�on Offset
rest2X3X
0%5%
10%15%20%25%30%35%40%45%50%
Y-gra� T-junc�on Offset
rest2X3X
% IV
C flo
w
Diff
. fro
m th
eor.
opt.
Patient A
Patient E
Patient D
Patient C
Patient B
0%10%20%30%40%50%60%70%80%90%
100%
Y-gra� T-junc�on Offset Theore�cal op�mum
IVC-RPAIVC-LPA
0%10%20%30%40%50%60%70%80%90%
100%
Y-gra� I Y-gra� II Y-gra� III T-junc�on Offset Theore�cal op�mum
IVC-RPAIVC-LPA
0%10%20%30%40%50%60%70%80%90%
100%
Y-gra� T-junc�on Offset Theore�cal op�mum
IVC-RPAIVC-LPA
0%10%20%30%40%50%60%70%80%90%
100%
Y-gra� T-junc�on Offset Theore�cal op�mum
IVC-RPAIVC-LPA
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Y-gra� I Y-gra� II T-junc�on Offset I Offset II Theore�cal op�mum
IVC-RPAIVC-LPA
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50%
Y-gra� I Y-gra� II T-junc�on Offset I Offset II
rest2X3X
Figure 4.4: Left: Hepatic flow distribution at rest. Right: Differences (percentageof the IVC flow) from the theoretical optima for each design at rest and exercise.Note that the theoretical optima for patient A at rest, 2X and 3X are 61/39, 70/30and 72/28, respectively, and that a 50/50 split can not be achieved in theory.
91
a
bc
T-junction Y-graft
T-junction Y-graft
Y-graft IIY-graft I
A
E
D
Right Left Right Left
Figure 4.5: Time-averaged velocity vectors in the Y-graft and T-junction modelsfor patients A, D and E. In the T-junction design for patient A, the SVC jetblocks the hepatic flow entering the LPA. In patient D, most SVC flow is directedto the RPA due to a curved SVC. In patient E, Y-graft II improves the hepaticflow distribution by having a straight proximal branch for the RPA, in which theSVC jet blocks hepatic flow going to the RPA from the right branch, compared toY-graft I.
92
Patient EPatient D
Patient CPatient BPatient A
0%10%20%30%40%50%60%70%80%90%
100%
0% 20% 40% 60% 80% 100%
Y-gra�Offset
% inflow to RPA
% IV
C flo
w to
RPA
0%10%20%30%40%50%60%70%80%90%
100%
0% 20% 40% 60% 80% 100%
Y -gra� IY-gra� IIY-gra� IIIOffset
% inflow to RPA
% IV
C flo
w to
RPA
0%10%20%30%40%50%60%70%80%90%
100%
0% 20% 40% 60% 80% 100%
Y-gra�Offset
% inflow to RPA
% IV
C flo
w to
RPA
0%10%20%30%40%50%60%70%80%90%
100%
0% 20% 40% 60% 80%
Y-gra�Offset
% inflow to RPA
% IV
C flo
w to
RPA
0%10%20%30%40%50%60%70%80%90%
100%
0% 20% 40% 60% 80% 100%
Y-gra� IIOffset II
% inflow to RPA%
IVC
flow
to R
PA
Figure 4.6: Hepatic flow distribution changes with variations in pulmonary flowsplit. Patients’ original pulmonary flow splits are marked by the arrows at the xaxis. The table shows the averaged deviations with respect to the original hepaticflow distribution for a 25% change in pulmonary flow split.
which identified a pressure shielding effect causing lower SVC pressure with the
Y-graft design due to the lack of a head-on flow collision in the junction.2
4.2.3 Power loss
Table 4.2 gives the power loss values for the Y-graft, T-junction and offset
designs for each patient. Energy loss increases with increasing exercise levels. The
T-junction designs result in the highest energy losses in most patients due to a
direct flow collision, agreeing with previous studies.
4.2.4 Wall Shear Stress
Mean WSS values on the IVC graft generally increase with increasing inflow
rates (Table 4.2). Exceptions to this are the T-junction and offset designs for
93
Table 4.2: Mean SVC pressure (mmHg), power loss (mW) and mean (in space)WSS magnitude (dynes/cm2) on the IVC graft for the Fontan models. Comparedto the best Y-graft design for the same patient, increases in power loss for theT-junction and offset designs are also shown.
patient modelSVC pressure power loss mean WSS
rest 2X 3X rest 2X 3X rest 2X 3X
AY-graft 15.3 19.4 22.9 5.1 14.5 24.9 2.3 4.6 7.2
T-junction 15.5 19.6 23.4 16% 14% 14% 4.5 5.9 7.9Offset 15.4 19.7 23.4 10% 15% 17% 1.9 4.5 7.4
B
Y-graft I 12.9 16.7 19.9 2.4 5.4 8.6 6.5 6.5 8.7Y-graft II 12.8 16.6 19.8 2.0 4.6 7.5 2.2 5.2 7.6Y-graft III 12.8 16.6 19.8 2.0 4.7 7.6 2.3 4.6 7.0T-junction 13.1 17.1 20.5 60% 70% 76% 6.1 6.8 9.0
Offset 13.0 17.0 20.4 40% 57% 64% 3.1 5.1 7.4
CY-graft 10.4 11.9 13.0 1.8 2.9 4.1 3.0 3.3 4.2
T-junction 10.4 11.9 13.0 0% 7% 5% 5.5 5.4 5.3Offset 10.4 11.9 13.1 0% 10% 15% 3.0 2.2 2.4
DY-graft 10.8 13.5 15.8 5.8 12.8 19.9 3.8 6.9 9.6
T-junction 10.9 13.7 16.1 10% 15% 17% 4.0 4.5 5.9Offset 10.8 13.6 16.0 5% 11% 16% 2.5 6.0 9.8
E
Y-graft I 22.9 27.7 31.7 5.14 10.8 17.1 3.9 5.7 8.0Y-graft II 22.8 27.6 31.7 4.7 9.7 15.6 2.9 4.7 7.15T-junction 22.9 27.8 31.8 14% 15% 15% 5.4 7.1 10.2Offset I 22.8 27.6 31.6 -1% -2% -5% 3.1 6.6 6.8Offset II 23.4 28.6 33.1 65% 86% 90% 2.7 4.6 8.1
94
A
T-junction OffsetY-graft
C
Figure 4.7: Contours of time-averaged WSS (dynes/cm2) at rest for patients Aand C.
patient C, in which the SVC jet enters the tube-shaped graft and increases the
WSS at a low IVC flow rate. The Y-graft and offset designs result in lower mean
WSS values on the graft than the T-junction design. This is consistent with the
fact that the T-junction design usually causes more energy dissipation. Time-
averaged WSS at rest for two representative patients is shown in Figure 4.7. The
other three patients exhibited similar behavior.
4.2.5 Averaged results
By averaging the hepatic distributions and power losses in the best per-
forming Y-graft, T-junction and offset designs over the five patients in the study
95
02468
101214161820
rest 2X 3X
Y-gra�T-junc�onOffset
Ave.
Pow
er lo
ss (m
W)
*
*
*
0%5%
10%15%20%25%30%35%40%45%50%
rest 2X 3X
Y-gra�T-junc�onOffset
Ave.
diff
. fro
m th
eor.
opt.
** *
Figure 4.8: Averaged differences from the theoretical optima and power losses overfive patients. The best performing of the Y-graft and offset designs for patients Band E are used. The differences between the Y-graft and T-junction designs arestatistically significant (∗P < 0.05).
(see Figure 4.8), we found that the Y-graft design has the lowest average difference
from the theoretical optima under rest and exercise conditions (i.e., the Y-graft
design results in more even hepatic flow distribution than other designs overall),
and that the distribution approaches the theoretical optima with increasing exer-
cise level. Significant differences (P < 0.05) are found in the mean hepatic flow
distribution and power loss between the Y-graft and T-junction designs.
4.3 Discussion
In this study, a multi-parameter approach was employed to evaluate three
Fontan designs in five patients, with particular focus on hepatic flow distribu-
tion. This multiple-patient series has demonstrated that the Y-graft design can
significantly improve hepatic flow distribution and moderately improve energy loss
and SVC pressure. However our results emphasize that no one-size-fits-all design
96
achieves satisfactory hepatic distributions in all patients. The hemodynamics af-
fecting flow distribution are non-intuitive in many cases, and small differences in
geometry can dramatically influence results. Because these subtle changes are not
easily elucidated with standard imaging modalities, simulations should be used to
determine the best candidates for a Y-graft preoperatively, and to refine the graft
design for each patient.
4.3.1 Hepatic flow distribution
Hepatic flow distribution in the Fontan is driven by multiple factors. The
SVC jet can prevent hepatic flow from reaching the LPA, and this was the leading
cause of uneven hepatic flow distribution in our study, agreeing with previous re-
sults of Dasi et al.53 Although previous studies in idealized geometries showed that
the T-junction design effectively mixes the IVC and SVC flows and distributes the
hepatic flow evenly,69 these phenomena were not observed uniformly in the patient
specific geometries in our study. The Y-graft design distributes hepatic flow more
evenly than the T-junction in most patients by avoiding a straight flow collision,
but the wrong choice of Y-graft may lead to unfavorable hemodynamics. The
LPA-offset design generally achieves satisfactory hepatic distribution in patients
with high LPA resistance, but an unfavorable distribution in patients with an equal
pulmonary flow split. This result is consistent with previous work of Bove et al.52
which shows that the IVC flow in the total cavopulmonary connection and tradi-
tional extracardiac Fontan models is skewed to the LPA with a pulmonary flow
97
split close to 50/50.
The non-intuitive result for patient D demonstrates that the geometry of
the SVC and relevant boundary conditions play an important role in distributing
the IVC and SVC flows. The hepatic flow distribution depends on the SVC flow
distribution (Equation 4.2) with an inverse relationship between the percentages of
hepatic and SVC flow going to the RPA. Thus the effect of the SVC-PA anastomosis
to the hepatic flow distribution should be carefully considered in surgical planning
for the Glenn procedure.
Although the maximum change in hepatic flow distribution from rest to
exercise was less than 20%, values generally approached their theoretical optima
during exercise. When the IVC flow rate is increased during exercise, the inter-
action between the two caval flows in the T-junction design generally enhances
mixing and distributes hepatic flow more evenly. Salim et al.140 showed that the
contribution of IVC flow to the cardiac output increases from 45% at 2.5-3 years
old to 65% (the adult value) at 6.6 years old. In our study, 45-48% of systemic
venous return is contributed by the IVC in two patients (mean 4.3 years) while
the IVC contribution is between 29-35% in three patients (mean 2.9 years). Thus,
simulations under exercise conditions may reveal some trends in the hepatic flow
distribution with increasing age, though changes in distribution of caval flow to
the PAs with age are still unknown.33
Our robustness test confirms that hepatic flow distribution depends strongly
on the pulmonary flow split. For some patients, the Y-graft design is relatively
98
more robust than the offset design for a wide range of flow conditions with less
chance to skew all IVC flow to one lung but this characteristic is not universal
for all Y-graft designs. Although the T-junction design skews hepatic flow with
patients’ original pulmonary flow splits, it performs better under the conditions
with a high RPA resistance due to a slight RPA-offset for the anastomosis. Thus,
the optimal working condition for the LPA-offset design is opposite. No design
emerged as the clear winner over a wide range of flow splits.
4.3.2 Power loss
The mechanism of energy dissipation in the Fontan has been well dis-
cussed.2,44 Compared to traditional designs, the Y-graft design reduces energy
dissipation by bifurcating the IVC to decrease flow competition. In our previous
work,2 the Y-graft design demonstrated reduced energy loss in a single patient
specific model. This study further compared the energy loss between the Y-graft
and traditional designs by examining multiple patients. The Y-graft design re-
duces energy loss at rest by 5-27% in four out of five patients, compared to the
offset design. The differences in energy loss are more pronounced during exercise,
agreeing with previous work.
4.3.3 SVC pressure
It is clinically well accepted that lower Fontan pressure generally correlates
with better outcomes. Overall, the Y-graft design offers moderate reductions in the
99
SVC pressure, compared to the T-junction and offset designs. In the T-junction
design, a higher SVC pressure is usually required to overcome competing flow from
the IVC. However the average SVC pressure differences are not as pronounced as
those reported in our previous work,2 likely because IVC flow rates were lower in
the younger patients in this study. This trend suggests that the significance of the
pressure shielding effect observed with the Y-graft may increase as patient’s age
and their relative IVC flow increases.
4.3.4 Wall shear stress
The Y-shaped grafts have lower mean WSS values than the tube-shaped
grafts. These results are qualitatively consistent with our previous work on ide-
alized Y-graft shape optimization,136 which shows a trade-off between energy effi-
ciency and areas of low WSS. While the WSS values in the Y-graft are generally
comparable to values in offset designs, further investigation into this issue is war-
ranted, and patients with known thrombotic tendency should likely be excluded as
candidates for the Y-graft procedure. The impingement of SVC flow results in a
relatively high WSS region in the intervening segment of the PA (Figure 4.7), which
does not suggest an increased likelihood of flow stasis and thrombus formation in
that region.
100
4.3.5 Ranking
Based on power loss and hepatic flow distribution, the proposed surgical
designs are ranked for each patient (Table 4.3) at rest and exercise. For energy loss,
the Y-graft design is superior to the others in four out of five patients regardless
of rest or exercise conditions, and Offset I wins by a narrow margin in patient
E. However, more variations emerge in the ranking based on the hepatic flow
distribution.
For patients A and C, the Y-graft design is clearly the final winner, as
it was uniformly ranked first in both energy loss and hepatic flow distribution.
For patient B, the offset design provides 10% improvement in the hepatic flow
distribution at exercise, with better robustness to the pulmonary flow split, but
produces over 40% more power loss than Y-graft II. We therefore select Y-graft
II as the final candidate for patient B. For this patient, we note that Y-graft III
overcomes the disadvantage of Y-graft II in robustness, while keeping the power
loss almost unchanged. If we were to weight the robustness more heavily, then
Y-graft III would likely be the recommended choice. For patient D, the Y-graft
design is chosen because of lower energy losses and progressive improvements in the
hepatic flow distribution during exercise, but the T-junction would be preferred
if the robustness were weighted more heavily than the power loss. For patient
E, Y-graft II is chosen because it is well balanced between energy loss and the
hepatic flow distribution, and it is more robust to changes in the pulmonary flow
101
Table 4.3: Ranking of energy loss and hepatic flow distribution for each patient.The ranking of the hepatic flow distribution is based on the differences from thetheoretical optima. In patients C and D, there are two designs tied for the hepaticflow distribution.
patient modelpower loss hepatic flow
recommended designsrest 2X 3X rest 2X 3X
AY-graft 1 1 1 1 1 1 �
T-junction 3 2 2 3 3 3Offset 2 3 3 2 2 2
B
Y-graft I 3 3 3 5 5 5Y-graft II 1 1 1 1 2 2 �Y-graft III 2 2 2 3 3 3T-junction 5 5 5 5 5 5
Offset 4 4 4 2 1 1
CY-graft 1 1 1 1 1 1 �
T-junction 3 2 2 2 2 3Offset 2 3 3 3 2 2
DY-graft 1 1 1 2 1 1 �
T-junction 3 3 3 1 1 2Offset 2 2 2 3 3 3
E
Y-graft I 3 3 3 3 3 3Y-graft II 2 2 2 2 2 2 �T-junction 4 4 4 4 4 5Offset I 1 1 1 5 5 4Offset II 5 5 5 1 1 1
split compared to other designs. These surgical recommendations are limited to
the patients we studied. Generalization of these choices should be made carefully,
as metrics used in this study could change as additional clinical data is obtained
in future work. The reader should be cautioned that not all variants of the Y-graft
design that we tested in this study performed well. However, the potential of the
Y-graft design to improve hemodynamic performance is promising, and a variant
of the Y-graft design was our final recommendation for all patients.
102
4.4 Limitations
A main limitation in this study is the lack of postoperative data on patients’
resistances and caval flow rates. Although this information would not typically be
available in a clinical pre-surgical design study, the development of sophisticated
models that can predict changes and remodeling in inflows and outlet boundary
conditions will be crucial for future surgical design and management of patients.
The evolution of this dynamic process is still an open question, and beyond the
scope of this study. Validation studies of pre-surgical design, long term hemody-
namics and uncertainty analysis should be incorporated into future work.
In this study we assumed an optimal hepatic flow distribution of 50/50,
and evaluated designs according to how closely they met this criterion. The tar-
get values chosen in this study could be adjusted on an individual basis as our
understanding of the relationship between hepatic flow concentration and lung de-
velopment is improved. Because the theoretical optimum was not achieved for all
patients in the study, it is possible that further design optimization may improve
some underperforming Y-grafts. Future work using patient specific optimization
would likely lead to further design refinement in some cases, as this would allow for
more systematic exploration of the design space. In addition, Bove et al.52 show
lateral tunnel (LT) Fontan models constructed from hemi-Fontan models result in
even hepatic flow distribution and lower power loss because of better mixing in the
right atrium, so future studies on the Y-graft should include comparisons with the
103
LT Fontan as well. Finally, inevitable discrepancies will occur between the virtual
design and its actual surgical implementation. This introduces certain geomet-
rical uncertainties that could result in differences between computer simulations
and the actual conditions. In addition, the use of rigid walls and the Newtonian
assumption for blood may affect the results presented in this study.
4.5 Conclusions
Using five Glenn patient specific models, we performed virtual Fontan surg-
eries and compared the hemodynamic performance of the Y-graft, T-junction and
offset configurations. We have demonstrated that the geometry considerably in-
fluences the hepatic flow distribution, and the hepatic flow split is not necessarily
equal to the pulmonary flow split. Theoretical analysis showed that a 50/50 hepatic
flow split is not attainable for some patients. Overall, the Y-graft design results
in more even hepatic flow distribution and moderate improvements in energy loss
and SVC pressure. The offset design is able to achieve an even hepatic flow dis-
tribution for patients with highly unequal pulmonary flow splits, but is sensitive
to variations in pulmonary flow split. It is important to note that, while a Y-graft
design was the best choice for all patients in the study, not all Y-graft configura-
tions performed well. The results of this study indicate that graft designs should
be optimized for individual patients prior to surgery. In conclusion, the Y-graft
is a promising new design that warrants testing in clinical application and long
104
term clinical trials. With further validation, simulations should be used to identify
the best candidates for the Y-graft procedure, and to rule out those patients who
should continue to receive conventional treatments.
4.6 Acknowledgments
This work was supported by the American Heart Association, a Burroughs
Wellcome Fund Career Award at the Scientific Interface, a Leducq Foundation
Network of Excellence grant and INRIA associated team grant. We are grateful
to Shawn Shadden for sharing his expertise and codes in particle tracking as well
as Sethuraman Sankaran, Frandics Chan, Heidi Terwey, and Mary Hunt Martin
for their helpful discussions and expertise. We also wish to acknowledge the use
of Simvascular (simtk.org), as well as the numerical modeling expertise of Nathan
Wilson and Charles Taylor.
Chapter 4, in full, is a reprint of the material as it appears in Yang, W.,
Vignon-Clementel, I. E., Troianowski, G., Reddy, V. M., Feinstein, J. A. and
Marsden, A. L. Hepatic blood flow distribution and performance in traditional
and Y-graft Fontan Geometries: A Case Series Computational Fluid Dynamics
Study. J. Thorac. Cardiovasc. Surg. 2012;143: 1086-1097.
Chapter 5
Y-graft optimal design for
improved hepatic flow
distribution
In Chapter 4, we demonstrated improved hemodynamic performance of the
Y-graft in multiple patient models. However, the use of a non-optimized design
resulted in underperforming Y-graft designs in two out of five patient-specific mod-
els.3 Automated shape optimization was applied to the Y-graft design using an
idealized model to reduce energy loss in a study of Yang et al.136 However, formal
shape optimization has not previously been applied to improve HFD. Preliminary
results have shown that the Fontan connection geometry is a critical determinant
of HFD.3 Thus, the impact of Fontan geometry and flow conditions on HFD, and
methods to improve underperforming designs merit further exploration.
105
106
In this chapter, we couple shape optimization to HFD quantification to
systematically improve Y-graft performance. The goals of this study are: 1) to
evaluate a new Y-graft design with unequal branch sizes, 2) understand the influ-
ence of flow splits on choice of optimal Y-graft, 3) examine the effect of SVC flaring,
and 4) improve HFD in previously underperforming Y-grafts. To achieve this, we
couple Lagrangian particle tracking to a derivative-free optimization framework
for cardiovascular geometries introduced by Marsden and colleagues76, 136 to opti-
mize HFD in Y-graft models. Because patients often present clinically with uneven
pulmonary flow splits, due to differing pulmonary resistances and lung sizes, we
identify optimal designs for a range of flow conditions. While a 55/45 (RPA/LPA)
ratio is commonly accepted,2, 141 Seliem et al.31 observed that 35% of patients
immediately prior to Fontan have a moderate to severe uneven pulmonary flow
distribution ranging from 28% to 2% flow to one lung. In agreement with this,
moderately uneven pulmonary flow distribution was found in two out of five Glenn
patients in our previous studies.3, 5 To achieve optimal HFD in these patients, we
hypothesize that unequal Y-graft branch sizes may be needed. Simple conservation
of mass analysis3 demonstrates that a 50/50 hepatic flow split is not possible in pa-
tients with a highly uneven overall pulmonary flow split. We therefore identify the
best theoretical distribution for each pulmonary flow split scenario, and set that as
our target for optimization. In addition, the theoretical analysis shows that HFD
also depends on the IVC/SVC flow ratio. Thus, the impacts of IVC/SVC flow
ratio on the optimal shapes are studied in idealized and patient-specific models.
107
Since the SVC-to-PA anastomosis may play an important role in flow interaction,3
we also investigate effects of the SVC anastomosis on the choice of optimal graft
design. Finally, two patient-specific models with underperforming Y-grafts are
optimized to improve skewed HFD.
5.1 Methods
This study is divided into two parts. In the first part, we optimize an ide-
alized Fontan model to identify optimal shapes for even HFD with variations in
the pulmonary flow split, IVC/SVC flow ratio and SVC-to-PA anastomosis geom-
etry. In the second part, we present two patient-specific examples to demonstrate
that optimization-derived designs effectively improve the HFD in cases that were
previously underperforming.
5.1.1 Geometrical model construction
Idealized cases
The construction of the idealized models follows our previous work (Chapter
3) in which an automated script is used to generate the model given a set of
geometric input parameters.136 Since we have hypothesized that unequal-sized
branches may be necessary to improve HFD, we also allow branch diameters to vary
independently in the model during optimization (Figure 5.1 upper right), resulting
in 7 design parameters for this case. Design parameters used to define the Y-graft
108
design space include the branch diameters DR and DL, the distances between the
branch anastomosis and the SVC XR and XL, the branch curvatures ΔR and ΔL,
and the trunk length LIV C . In addition, there are six constant parameters used to
define the dimensions of the IVC trunk, PA and SVC.136
In the Glenn procedure, the SVC is connected to the PA in an end to side
fashion. The anastomosis may be flared to direct flow preferentially to one lung
or the other or, in preparation for the subsequent Fontan procedure.5, 142 To study
the influence of the SVC geometry on the choice of optimal graft design, we replace
the regular SVC-PA junction in selected models with two representative cases: 1.
LPA-flared junction (Figure 5.1 bottom left) and 2. curved junction (Figure 5.1
bottom right).
Patient-specific cases
Since previous patient-specific studies3, 51 have shown large variations in PA
topology, the idealized model (Figure 5.1) constructed for general analysis cannot
sufficiently represent a specific patient’s geometric and hemodynamic characteris-
tics. We therefore optimized a semi-idealized model that incorporates key features
of the patient specific model, and then use a patient-specific model for final verifi-
cation.
For each patient, we create a semi-idealized model (Figure 5.2) as follows.
The patient-specific centerline path of the PA with circular segmentations and the
patient’s SVC are incorporated into the semi-idealized Glenn model to preserve
109
−10 −8 −6 −4 −2 0 2 4 6 8 100
2
4
6
8
10
12
DLPADRPA C2
XL
C4
C0
LIVC
XR C3
C1
RR RL
Figure 5.1: Model parameterization and flared SVC anastomosis. Upper left: De-sign parameters and centerlines of an idealized Y-graft Fontan model. Upper right:A representative Y-graft model. Parameters DL and DR allow two branches tovary independently. Bottom left: An LPA-flared SVC anastomosis with a straightjunction for the RPA side. Bottom right: A curved-to-LPA SVC anastomosis.
each patient’s main geometric characteristics (Figure 5.2). The PA diameter is
set to an average measured from the patient-specific Glenn model. Then, a Y-
graft is implanted into the Glenn model to form a semi-idealized Fontan model for
optimization.
The optimization for these cases only included designs with equal-sized
branches, which reduced the number of design parameters to four (Figure 5.2) :
XL, XR, LIV C and Dbranch. For the PA path, we can define a parametric spline
110
Table 5.1: Bounds on the design parameters for the semi-idealized model. XR
and XL are measured from the SVC-PA junction to the right and left anastomosispoints, respectively. Since the PA path is a parametric spline S(t), the anastomosislocation can be changed by varying the spline parameter t. In our previous study,3
patient specific models employed a uniform 20-15-15 mm Y-graft. To optimize thegraft size, the branch diameter was allowed to vary between 12 and 16 mm.
Patient Dbranch mm LIV C mm XR mm XL mm
Patient A 12-16 10-20 0.7-15 0.5-15.2Patient B 12-16 10-25 0.5-20.5 0.6-22
S(t) =
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩
x = s1(t)
y = s2(t)
z = s3(t)
, (5.1)
such that the center of each PA segmentation lies on the curve. Thus, design pa-
rameters XL and XR are a function of parameter t. To determine the bounds of
XL and XR, the aortic arch and previous non-optimized patient specific models
constructed under the guidance of a surgeon were used as a reference. The param-
eter bounds are listed in Table 5.1. In this study, we use the terms “proximal” and
“distal” to describe the anastomosis location relative to the SVC . The anastomosis
points (marked with solid squares in Figure 5.2) are allowed to slide along the PA
path within the bounds. Similarly, the path of the Y-graft is defined by a Hermite
spline.
Based on surgical practice at our institution , the PA at the anastomosis
will be enlarged to match the graft size when a large graft is anastomosed to the
111
1 2 3
. .XLPA
Dbranch
LIVC
XRPA
Figure 5.2: 1. A patient-specific Glenn model. 2. In the semi-idealized Glennmodel, the PA is approximated by uniform circular segmentations and the pul-monary artery branches are neglected. The PA diameter is equal to the averageddiameter of the patient-specific PA. 3. A Y-graft is implanted forming a semi-idealized Fontan model for the same patient. The design parameters for the Y-graft are XL, XR, LIV C and Dbranch. When large branches are anastomosed, thesegmentation at the anastomosis is enlarged to the graft size. Then the rest ofthe PA segmentations are enlarged linearly according to the distance to the closestanastomosis.
PA and pressurized. For example, in measurements taken from images of a post-
operative Fontan patient who had a 20 mm extracardiac conduit placed, the size
of the PA at the anastomosis was about 19mm. To account for this change, the
circular segmentations along the PA path are adjusted automatically when the
branch size is larger than the PA size. First, the segmentation at the anastomosis
point is enlarged to the graft size. Then the rest of the PA segmentations are
enlarged linearly according to the distance to the closest anastomosis. The two
outlets of the PA are kept unchanged.
To improve initially underperforming Y-grafts, we first optimize the Y-graft
in the semi-idealized model. After an optimal Y-graft is identified, we implant it
into the patient-specific model to verify the hemodynamic performance.
112
Table 5.2: Mean pulsatile inflow rates, pulmonary flow splits and pressure. Arespiratory model4 was superimposed to the IVC flow acquired from PCMRI foreach patient following our previous work. No respiratory model was added tothe SVC input. The flow rates used for the idealized model were taken from atypical Fontan patient.4,5 We varied the RPA/LPA flow split in the idealized modelfor different conditions and set a Fontan pressure (central venous pressure) of 12mmHg. For patients A and B, pulmonary flow splits and pressure data were takenfrom MRI and catheterization prior to the Fontan procedure. Transpulmonarygradient (TPG) is the mean pressure difference between the SVC and the leftatrium.5
Model IVC (cc/s) SVC (cc/s) RPA/LPA flow split pressure (mmHg)
Idealized 17.9 14.1 varied 12Patient A 6.3 15.2 54/46 SVC:9, TPG:4Patient B 14.8 27.8 55/45 SVC:8, TPG:4
5.1.2 Flow simulation and boundary conditions
A custom stabilized finite element Navier-Stokes solver94 was employed to
simulate blood flow, assuming rigid walls and Newtonian flow with a density of
1.06g/cm3 and viscosity of 0.04 g/(cm s). Anisotropic mesh adaptation based on
the Hessian of the velocity field was performed for the patient-specific models, to
ensure mesh convergence of the solution.143
Pulsatile inflow conditions were employed for the IVC and SVC inlets with
a parabolic profile. A respiratory model was superimposed on the IVC flow wave-
form acquired from PCMRI for each patient following our previous work.4 No
respiratory model was added to the SVC input because little respiratory variation
is typically found in the SVC flow.4, 36 The mean flow rates for inlets are listed in
Table 5.2.
Resistance boundary conditions were employed for the idealized and semi-
113
idealized models. For a given pulmonary flow split, resistance values (R = P/Q)
were determined by setting 12 mmHg as the mean Fontan pressure, a typical clin-
ical value. For patient-specific simulations, a three element RCR circuit model96
was applied at each outlet. Total resistance was tuned to match the MRI-derived
pulmonary flow split and catheterization-derived transpulmonary gradient (TPG)
(Table 5.2), assuming that the pre-Fontan outlet boundary conditions are still valid
for immediate post-operative flow conditions.3, 5 The proximal and distal resis-
tances and capacitance for each outlet were determined based on a morphometry-
based pulmonary arterial tree and outlet areas.5 Resistances for each patient’s
semi-idealized model matched the total LPA and RPA resistances of the corre-
sponding patient-specific model.
Quantification of HFD
We assume that hepatic flow is well mixed in the IVC such that the hepatic
and IVC flow distributions are the same, and that the theoretical optimal HFD
is the closest value to 50/50. Based on Equation 4.2, the theoretical optimum for
the HFD can be obtained for any combination of the pulmonary flow split and
IVC-to-SVC flow ratio, as shown in Figure 5.3. The fraction of SVC flow going to
the RPA, SRPA, is allowed to change between 0 and 1 to achieve the best HFD.
A few conclusions can be drawn from Equation 4.2 and Figure 5.3: 1) A perfect
50/50 split is not always possible for some combinations of the pulmonary flow split
and IVC/SVC flow ratio without violating conservation of mass. For example, if
114
a patient has an overall flow split to LPA and RPA of 19/81, and an IVC/SVC
flow split ratio of 0.93 then the best hepatic distribution one can achieve is 0.63,
in the case that all the SVC flow goes to the RPA. 2) FRPA = 0.5 is the only value
for which the theoretical optimum is always 50/50, regardless of QIV C
QSVC. 3) For an
IVC/SVC ratio of 1, the theoretical optimum is 50/50 for a percentage inflow to
the RPA between 25% and 75%.
00.1
0.20.3
0.40.5
0.60.7
0.80.9
1
0
0.2
0.4
0.6
0.8
10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
QIVC
/(QIVC
+QSVC
)% inflow to RPA
best
% h
epat
ic fl
ow to
RPA
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 5.3: Optimal values for the HFD. Based on Equation (4.2), the theoreticaloptimum for the HFD, defined as the value closest to 50/50, is determined givenan inflow ratio QIV C
QSVCand a pulmonary flow split FRPA (% inflow to RPA).
To quantify the HFD, we used the same particle tracking introduced in
Chapter 4.
5.1.3 Optimization algorithm
The surrogate management framework (SMF)98 together with mesh adap-
tive direct search (MADS)86 is employed for optimizing HFD, following the work
of Dennis, Audet and Marsden.76, 102, 136 We consider the optimization problem,
115
minimize J(x),
subject to x ∈ Ω, (5.2)
where J : Rn → R is the cost function, Ω ⊂ Rn denotes the feasible region and x
is the vector of parameters.
The cost function J in this study is defined as
J = |HRPA − 0.5| , (5.3)
where HRPA is the fraction of IVC flow going to the RPA. For a given x, J is
obtained by performing a 3D simulation and particle tracking. We optimize the
HFD to achieve a target flow split of 50/50. For cases in which the theoretical
optimum is not 0.5, the best cost function value is therefore larger than zero.
5.2 Results
5.2.1 Idealized cases
We first examine the question of whether unequal sized branches may be
advantageous to improve HFD for cases of highly uneven pulmonary flow split.
To mimic an uneven pulmonary flow split, the LPA/RPA resistance ratio was set
116
to 4. In the first case, we allowed branch diameters to vary independently during
optimization to determine if unequal branches are needed to achieve optimal per-
formance. In subsequent tests, we restricted the number of design variables to use
equal-sized branches only and kept boundary conditions unchanged to determine
whether unequal-sized branches are the only way to achieve optimal HFD.
Figure 5.4 shows that shape optimization for a case with a highly uneven
pulmonary flow split 79/21 (RPA/LPA) caused a large difference in the optimal
branch diameters. The branch size for the RPA reduced to the lower bound,
increasing resistance for IVC flow streaming to the RPA. In contrast, optimization
results for the case of equal-sized branches achieved almost the same optimal HFD,
with a 9% smaller energy loss. In addition, outflow rates in both cases were almost
unchanged, confirming that the overall pulmonary flow split is largely determined
by the outlet boundary conditions, and not the local geometry.
These initial results suggested that unequal-sized branch diameters were
unnecessary for achieving optimal HFD. To further confirm this hypothesis, we
optimized the idealized model with equal-sized branches over a large range of pul-
monary flow splits. Figure 5.5 shows that the theoretical optima for the pulmonary
flow splits we tested are achieved, and the optimal geometry depends on the pul-
monary flow split. We observe that anastomosis locations were more distal on the
side of higher pulmonary resistance.
Shape optimization of the Y-graft was also performed for two different SVC
flaring configurations. Figure 5.6 shows the optimal Y-grafts for these two flared
117
1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00
0% 20% 40% 60% 80% 100%
unequal branchesequal branches
% IVC flow to RPA
Pow
er lo
ss (m
W)
Figure 5.4: A comparison of HFD and energy loss for optimal unequal and equal-sized branches. HFDs for the unequal and equal-sized branches are 63/37 and65/35 (IVC-RPA/IVC-LPA), respectively, but equal-sized branches perform betterin reducing energy loss. The pulmonary flow split is 79/21 (RPA/LPA).
0%10%20%30%40%50%60%70%80%90%
100%
0% 20% 40% 60% 80% 100%
resultsTheor. Opt.
% inflow to RPA
% IV
C flo
w to
RPA
1.6%
0%1%
0%
Figure 5.5: Optimal Y-grafts with equal-sized branches for a large range of pul-monary flow splits. Theoretical optima given by Equation 4.2 are achieved by usingoptimization. The difference from the theoretical value is shown at each point.
118
SVC anastomoses and a straight anastomosis under the same flow split conditions.
While the optimal geometry for the straight SVC connection is almost symmetric,
the optimal geometry for both flared configurations is asymmetric with the RPA
connection more proximal, and the LPA connection more distal. These differences
in geometry result from SVC streaming to the LPA side in case of a flared SVC
anastomosis.
straight LPA flared curved
Figure 5.6: Time-averaged flow fields of optimal Y-grafts for a straight SVC-PAjunction and two types of flared SVC anastomoses. The pulmonary flow split is55/45 (RPA/LPA). Compared to the model with a straight SVC-PA junction, theoptimal Y-grafts for two flared SVC anastomoses have a more distal anastomosisfor the LPA.
Since the HFD is also a function of the IVC/SVC flow ratio, we altered
the ratio without changing the cardiac output to examine the impact on different
models. Figure 5.7a shows that the idealized Y-graft model is insensitive to changes
in this ratio. However, the changes in the IVC/SVC flow ratio can significantly
influence the HFD in a patient-specific model due to a more complex flow field.
119
0%10%20%30%40%50%60%70%80%90%
100%
0% 20% 40% 60% 80% 100%
idealizedpa�ent-specific
QIVC/Qinflow
% IV
C flo
w to
RPA
Figure 5.7: HFD vs. QIV C
Qinflowfor an idealized model and a patient-specific model
(patient B). Patient B’s original inflow ratio QIV C
Qinflowis marked by an arrow. Total
inflow is kept constant in this comparison. The idealized Y-graft is optimized foran IVC inflow-to-total inflow ratio of 45%. There is only 1% change in the Y-graft model when the ratio is altered. However, the patient specific model is moresensitive to the change of IVC inflow-to-total inflow ratio.
5.2.2 Patient-specific cases
To test the ability of optimization to improve hepatic flow in a patient
specific model, we now present the results for patient specific optimization using
a semi-idealized model. Figure 5.8 shows that the optimal Y-graft identified via
a semi-idealized model for patient A resulted in a consistent HFD after it was
implanted into the patient-specific Glenn model. Compared to the original non-
optimized design, the HFD is improved by 79% and the left branch anastomosis in
the optimal design is more proximal. With a more proximal anastomosis of the Y-
graft to the RPA, the SVC flow suppressed hepatic flow, reducing its concentration
120
in the RPA to a balanced level. An additional optimization case for patient A was
run in which a uniform PA diameter was used. When the resulting optimal Y-graft
was implanted into the patient specific model, there was no improvement in HFD.
In patient B, the right upper lobe (RUL) is adjacent to the SVC. In our
first optimization run, the RUL was not included in the semi-idealized model for
optimization and the target HFD was set to 50/50 as before. Figure 5.9 shows that
the target optimal Y-graft made no improvement in the patient-specific model even
though it achieved the target value in the semi-idealized model. By examining each
outflow rate, we found that the flow to the RUL accounted for 15% of the RPA
outflow. Since the RUL mainly received flow from the SVC, the concentration of
SVC flow in the RPA flow was sensitive to the presence of the RUL. To compensate
for the presence of the RUL in the semi-idealized model, the target hepatic flow
split value was changed to 0.65 in the second optimization run. Figure 5.10 shows
that a Y-graft that achieved a hepatic flow split of 65/35 in the idealized model
successfully distributed hepatic flow evenly when it was implanted to the patient-
specific model. In the resulting model, the right anastomosis is more distal, while
the left one is more proximal in the optimized model and the branches have less
curvature in the optimized design compared to the previous one. Table 5.3 lists
geometric parameters and power loss for patient specific models. A smaller branch
size resulted in more power loss in both patients.
121
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Model 1 Model 2 Model 3
IVC-RPAIVC-LPA
% IV
C flo
w
Model 1: optimal Y-graftwith semi-idealized model
Model 2: optimal Y-graftwith patient-specific model
Model 3: non-optimized Model 2
RPALPA
a)
b)
Figure 5.8: a) Time-averaged velocity vector fields in the semi-idealized andpatient-specific models for patient A. b) Particle snapshots taken at T=3s for thenon-optimized and optimal models. The bar chart shows the semi-idealized model(upper left) has a similar hepatic flow split to the patient-specific model (upperright) for the same optimal Y-graft, and that the optimized Y-graft improves theHFD by 79%, compared to the original non-optimized design (lower left). Theoptimal and non-optimized branch sizes are 12.9 and 15 mm, respectively.
122
Model 1: optimized for 50/50 Model 2: patient specific
RUL
0%10%20%30%40%50%60%70%80%90%
100%
Model 1 Model 2 Model 2 w/o RUL
IVC-RPAIVC-LPA
% IV
C flo
w
Figure 5.9: Time-averaged velocity vector fields in the semi-idealized and patient-specific models with and without the RUL for patient B. The Y-graft is optimizedfor a HFD of 50/50. Due to the effect of the RUL, the optimized Y-graft skewedthe hepatic flow by around 15% after it was implanted into the patient-specificmodel. When the RUL is excluded from the patient-specific model, the HFD isconsistent with the idealized model prediction.
Table 5.3: Geometric parameters and power loss for patient specific models. Inpatient A, the optimal XR was reduced resulting in a more proximal anastomosisfor the right branch. In patient B, the right anastomosis is more distal while theleft one is more proximal in the optimized model. In both cases, a smaller branchsize resulted in more power loss.
Patient Dbranch (mm) LIV C (mm) XR (mm) XL (mm) Power loss(mW)
A optimized 12.9 15.25 7.3 15.2 1.9A non-optimized 15.0 18.5 15.0 19.0 1.8B optimized 16.0 20.8 20.2 21.9 5.3
B non-optimized 15.0 13.5 17.0 23.0 5.8
123
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Model 1 Model 2 Model 3
IVC-RPAIVC-LPA
% IV
C flo
w
Model 1: optimized for 65/35 Model 2: optimal Y-graftwith patient-specific model
Model 3: non-optimized Model 2
RPA LPA
a)
b)
Figure 5.10: a) Time-averaged velocity vector fields and HFD for patient B. TheY-graft in the semi-idealized model (upper left) is optimized for a hepatic flowsplit of 65/35 (RPA/LPA) to account for the overestimation of the RPA hepaticflow in the semi-idealized model. b) Particle snapshots taken at T=3s for the non-optimized and optimal models. The bar chart shows the optimal Y-graft improvesthe performance by 94% achieving an even HFD in the patient-specific model(upper right), compared to the non-optimized design (lower left). The optimaland non-optimized branch sizes are 16 and 15 mm, respectively.
124
5.3 Discussion
By coupling particle tracking to an optimization framework, we optimized a
series of Y-graft designs to achieve optimal HFD in different scenarios. In idealized
cases, the unequal branch design was optimized and compared to the equal branch
design. Results showed that unequal branches are not necessary to achieve an even
HFD in all cases that we evaluated. We studied the impacts of the pulmonary
flow split, IVC/SVC flow ratio and SVC flaring on choice of optimal Y-graft.
Again, the Fontan connection geometry played an important role in distributing
the hepatic flow. Two previously underperforming patient-specific Y-graft designs
were improved by optimizing semi-idealized models with the patients’ SVC and
PA paths and then implanting the optimal design into the corresponding patient
specific model.
5.3.1 Idealized cases
We have previously observed that Y-graft designs with equal-sized branches
resulted in skewed hepatic flow in some patient-specific models. To have greater
control over hepatic distribution, we hypothesized that unequal branches might
improve distribution in select cases, particularly for patients with highly uneven
right/left pulmonary resistances. In this study, we tested this hypothesis using
an idealized model with a right/left pulmonary flow split close to 80/20 and com-
pared to results for equal-sized branches. Optimization with and without unequal
125
branches were both shown to achieve the target optimal hepatic distribution. How-
ever, designs with unequal branches dissipated more energy due to a higher pressure
drop in the smaller branch. Although reducing the branch size on one side can
increase the resistance to that side, our results showed that changes in the branch
geometry had little-to-no influence on the overall pulmonary flow split. More im-
portantly, the use of unequal-sized branches had limited impact on the HFD, while
the anastomosis locations had a significant impact. Thus, for models with a normal
pulmonary flow split around 45/55, equal-sized branches can easily achieve the tar-
get optimal flow distribution through a proper choice of anastomosis locations and
branch curvature. In addition, graft designs with unequal branch diameters may
increase difficulties in manufacturing compared to equal-sized branches. Thus, the
equal-sized branches should be suitable for most cases regardless of the pulmonary
flow split.
Since patients often come in with a range of relative blood flow distribution,
our intent was to explore this range of starting conditions for Y-graft implantation.
For patients with widely unequal pulmonary flow distribution, we identified a series
of optimal Y-graft designs in which the theoretical optimal HFD is achieved over
a large range of pulmonary flow splits. Results show that the optimal anastomosis
location depends on the pulmonary resistances for the corresponding side. For a
fixed IVC/SVC ratio, if the inflow received by the RPA is reduced, one needs to
reduce the SVC flow that goes to the RPA to keep an even HFD (Equation 4.2).
This can be achieved by modifying the Y-graft such that its implantation facilitates
126
flow from the IVC to the RPA. This thus leads to a more distal anastomosis
of the graft to the RPA, which streams IVC flow towards the RPA, the branch
with higher resistance. Therefore, the PA with lower resistance requires a more
proximal anastomosis, while the PA with higher resistance requires a more distal
anastomosis. In addition, HFD is more sensitive to the anastomosis location than
the branch size.
During the Glenn procedure, some surgeons flare the SVC towards the
LPA intending to channel more SVC flow to the LPA which usually has a higher
resistance than the RPA.5, 142 However the pulmonary flow split is dictated by
the downstream resistance, and thus the distribution of SVC flow is equal to the
pulmonary flow split in the Glenn. After Fontan completion, SVC geometry may
play an important role in determining HFD. We performed Y-graft optimization
for two types of flared SVC anastomoses. For both flared geometries, the resulting
optimal designs had a more distal branch anastomosis on the LPA side compared
to the straight SVC case. This can be explained by the fact that flaring to the LPA
increases SVC flow to the LPA. Hence, according to Equation 4.2, to maintain the
optimum HFD for given flow split and IVC/SVC flow ratio, SVC flow to the RPA
needs to be facilitated. This can be achieved either by impeding IVC flow to the
RPA (leading to a more proximal anastomosis on that side) or by facilitating IVC
flow to the LPA (leading to a more distal anastomosis on that side). The latter
case lead here to the optimum solution. Compared to the flared SVC case, the
curved SVC anastomosis channeled more SVC flow to the LPA for most Y-grafts
127
evaluated during the optimization and required about 200% more evaluations to
achieve the theoretical optimum. The relative pulmonary blood flow in the Glenn
patient plays a signficant role in subsequent Fontan blood flow distribution and,
as such, consideration should be given to measuring this routinely in pre-operative
Fontan patients. In addition, surgeons should carefully consider the specifics of the
anastomosis for the SVC-PA junction in the Glenn procedure with an eye towards
the impact on the HFD in the Fontan procedure. For example, if a patient has an
RPA/LPA flow split of 80/20, it would be preferential to have most of the SVC
flow going to the RPA when the IVC is connected to the PA in order to optimize
hepatic flow split. If the Glenn, however, had the SVC-PA anastomosis flared
towards the LPA, a Y-graft that achieves the target HFD would be difficult.
We also studied the effect of changing the IVC/SVC flow ratio on HFD
in both idealized and patient-specific models. In the idealized case, for a Y-graft
optimized for a certain IVC/SVC flow ratio, the HFD remains almost unchanged
when the IVC/SVC ratio is altered. However, a patient specific model is more
sensitive to changes in IVC/SVC flow. A possible reason is that the Y-graft in
this patient-specific model has proximal anastomoses. The trend is qualitatively
consistent with the change from rest to exercise which was observed in our previous
study,3 in which increasing the IVC flow momentum improved the HFD. Thus a
significant change in IVC flow momentum directly influences the SVC-IVC flow
interaction. These findings indicate that the idealized model is not sufficient for
capturing this sensitivity.
128
5.3.2 Patient-specific cases
In patient A, we demonstrated that the patient’s PA path and size should
be included in the semi-idealized model to achieve consistent predictions when
the graft design is implanted in the patient specific model. Since optimization
involves systematically evaluating a series of designs, the use of idealized models
can significantly reduce overall computational cost. Simulations for a patient spe-
cific model are 15 times more expensive than for a semi-idealized model. However,
without proper consideration, an oversimplified model may be lacking important
local information and result in a failed prediction. Direct patient-specific shape
optimization requires a methodology solution to parameterize and automatically
manipulate the graft anastomosis. At the present time, it remains a challenge to
parameterize patient-specific models, due to the need for manipulating a complex
surface while maintaining its integrity using relatively few design variables. In the
future, a fully parameterized patient specific model could serve as a high fidelity
model and a multi-fidelity optimization could be performed with a low-fidelity
semi-idealized model.144
In patient B, the importance of including the RUL when computing the
HFD was revealed. In the model for patient A, the RUL flow was lumped together
with the RPA when constructing the semi-idealized model. However, the same
approach did not prove adequate for patient B because the RUL attachment point
was on the SVC. Results showed flow streaming from the SVC to the RUL with very
129
little mixing and less than 1% of hepatic flow delivered to the RUL. Therefore the
presence of the RUL directly influenced the concentration of SVC flow in the RPA
outflow. In contrast, including or excluding the RUL will not produce significant
differences in the HFD if the RUL is distal to the SVC. Thus, for patients with
a direct RUL-SVC connection, the semi-idealized model must compensate for the
RUL flow contribution, as done in patient B. The extra hepatic flow going to
the RPA in the semi-idealized model would be offset when the same Y-graft was
implanted into the patient-specific model with the RUL.
In patient A, without optimization, HFD is too skewed towards the RPA.
According to the semi-idealized simulations, the Y graft can achieve better HFD
either with a more proximal anastomosis on the RPA side (to reduce IVC flow to the
RPA) or a more distal anastomosis on the LPA side (to favor IVC flow towards the
LPA). Indeed, the optimized geometry lead to a more proximal anastomosis on the
RPA side. On the other side, due to geometric constraints, the anastomosis could
not be placed more distally. To compensate, the diameter was decreased to favor
hepatic flow to the LPA. By contrast for patient B, optimization needed to increase
IVC flow to the RPA. Here, the optimum was found by using a more proximal
anastomosis on the LPA side (impeding IVC flow to that side) and increasing
the graft diameter (facilitating flow to the RPA). The complex interplay of the 3D
geometry and flow is highlighted in these patient specific cases, where identification
of optimum configurations requires computer simulations using the full model.
No consistent choice for optimal branch size was observed. Patient A’s
130
optimal branch size (12.9 mm) was smaller than the non-optimized size (15 mm)
while a larger diameter (16 mm) was identified for patient B. This can be attributed
to the existence of multiple local minima. A larger branch size can slightly reduce
power loss, however, this was shown in a recent multi-scale modeling study to have
negligible effect on cardiac work load.54 The trunk length had a variable influence
on HFD with no particular trend for the values considered. The anastomosis
locations for the Y-graft play a more important role than the trunk length in
regulating HFD. However, we believe that the branch size and trunk length may
have a pronounced impact on wall shear stress (WSS) levels in the graft. Future
work should consider thrombotic risk in the surgical design, and thus a further
study with WSS and residence time constraints is warranted.
5.3.3 Technical considerations for Y-graft implantation
Space constraints are the main concern for surgical implantation of the Y-
graft. Since 20 mm grafts with an offset are routinely implanted in our institution,
our surgeons maintain there should be enough space for a Y-graft with 12 or 14
mm branches. The space available, however, can vary dramatically from patient
to patient, making some patients better candidates than others. The Y-graft may
also not be applicable to patients with abnormal anatomy and/or a history of
clotting disorder. Special care should be taken for these patients to identify an
optimal graft prior to surgery. Technical feasibility must be confirmed by a clinical
study.
131
5.4 Limitations
A HFD of 50/50 is assumed to be optimal in this study, and all Y-graft de-
signs are evaluated according to this criterion. Although the exact value required
for the prevention of PAVMs is still unknown, the goal of distributing hepatic flow
evenly is supported by clinical practice. As a step towards a better understand-
ing of the relationship between Fontan geometry and HFD, we chose 50/50 as a
reasonable target value which could be adjusted in future studies.
In this study, a single objective optimization with a simple bound con-
straint was performed. However it is clear that surgical design is multi-factorial
and physiological states can change over time. In addition, discrepancies from
surgical implementation may result in a large deviation in post-operative perfor-
mance. Multiple objectives, constraints and robustness should be addressed in
future studies.135
For the idealized model, the Y-graft is restricted to lie in a plane. Additional
variables for future study could include out-of-plane anastomosis angles and trunk
size to increase design flexibility though as mentioned previously, space within the
chest cavity is limited. The geometry of the SVC-PA junction should be exam-
ined in future studies to determine if the Glenn procedure should be performed
differently in patients with a planned Y-graft procedure.
Our assumption that the PA is enlarged to accommodate a large graft
anastomosis is based on common surgical practice. The work of Dobrin et al.145
132
showed that the maximum size of the anastomosis is determined by the Gore-tex
graft because the arteries are much more compliant than the Gore-tex material.
Although we do not model the process of vessel enlargement in this study, models
are constructed under the guidance of a surgeon to replicate realistic anastomoses.
Future work will focus on incorporating finite element modeling into the model
construction such that the process of anastomosing the graft in patient-specific
models can be implemented automatically.
The HFD is quantified numerically. Although the flow solver that provides
velocity fields for particle tracking has been validated against experiments and
theoretical solutions, predictions of patients’ HFD still need to be validated in vivo
and in vitro. Patients’ lung perfusion data will be used to compare with numerical
simulations in our future work, and comparisons to in vitro models should also be
made.
In this study, the HFD was divided into two parts (IVC-RPA and IVC-
LPA). However, we found that little hepatic flow was delivered to the RUL in
patient B though the overall hepatic flow split is close to 50/50. It is still unknown
whether an overall even hepatic flow split with a localized lack of hepatic flow in
a lobe can cause PAVMs. Future work would look at the incidence of PAVMs in
the RUL and consider controlling the HFD for each lobe.
A rigid wall assumption is used in this study. Our recent work has demon-
strated that HFD is insensitive to the use of rigid vs. deformable walls, which
increased confidence in our HFD predictions.57 However future predictions of
133
thrombotic risk linked to wall shear stress will likely require fluid structure in-
teraction, as there are large differences in WSS between rigid and flexible wall
simulations.
For the patient specific cases, flow measurements were taken from preop-
erative PCMRI data. Although we accounted for changes in IVC inflow patterns
after the Fontan procedure by scaling the IVC waveform, a multi-scale closed loop
model54 should be incorporated in future studies.
5.5 Conclusion
In this study, we coupled Lagrangian particle tracking to an optimization
framework to investigate the effect of geometry on HFD in a Y-graft Fontan con-
figuration. Y-graft models with unequal-sized branches were compared to models
with equal-sized branches. Optimized models with equal-sized branches are able
to distribute hepatic flow equally well as unequal-sized branches with lower en-
ergy loss under highly uneven pulmonary flow split conditions. In addition, the
theoretical optima are achieved using equal-sized branches over a large range of
pulmonary flow splits. Thus we do not recommend unequal-sized branches for fu-
ture Y-graft designs. A flared SVC anastomosis impacts optimal geometry of the
Y-graft by resulting in a more distal anastomosis for the branch on the flared side
and a more proximal one on the other side compared to the non-flared case. In
idealized models, the Y-graft design is more robust to changes in the IVC/SVC
134
flow ratio than the offset design. However, a patient-specific test did not support
this finding.
Two underperforming Y-grafts have been successfully optimized for patient-
specific cases by using semi-idealized Glenn models that incorporated key patient-
specific attributes, namely a patient-specific SVC and a curved PA path. Com-
pared to the original designs, these optimized Y-grafts for patients A and B im-
proved HFD by 79% and 94%, respectively. The strategy of using semi-idealized
models for optimization avoids a costly trial-and-error design process, requiring
laborious manual model revisions. We also found that ignoring the effect the right
upper lobe when it is adjacent to the SVC may result in failure to improve HFD in
patient-specific models. This study emphasizes that an optimization plan should
be tailored for each patient within the context of the overall framework we have
presented.
5.6 Acknowledgments
This work was supported by the American Heart Association, a Burroughs
Wellcome Fund Career Award at the Scientific Interface, a Leducq Foundation
Network of Excellence grant and an INRIA associated team project grant. The
authors wish to thank Charles Audet, Sebastien Le Digabel and Mohan V. Reddy
for sharing their expertise on optimization and pediatric cardiac surgery , and
Frandics Chan for image-data acquisition and expertise.
135
Chapter 5, in full, is a reprint of the material as it appears in Yang, W.,
Feinstein, J. A., Shadden, S. C., Vignon-Clementel, I. E. and Marsden, A. L.
Optimization of a Y-graft Design for Improved Hepatic Flow Distribution in the
Fontan Circulation. J. Biomech. Engrg., accepted.
Chapter 6
Simulations and validation for the
first cohort of Y-graft Fontan
patients
Simulations on idealized and patient specific models have shown that overall
the Y-graft design improves energy loss, SVC pressure and hepatic flow distribu-
tion (HFD) though no one-size-fits-all Y-graft design exists.2, 3, 136 Since Optiflo
introduces extra synthetic materials and technical difficulties, the dual bifurcated
design has not been studied further. Based on the previous simulation results, the
Y-graft design has been translated into clinical use in two institutions.146 In a pi-
lot study at Lucile Packard Children’s hospital at Stanford University, six patients
underwent a Y-graft EC Fontan surgery between June 2010 and March 2011. The
technical success demonstrated the feasibility of the Y-graft, which was a major
136
137
concern due to limited space in the chest cavity.
Previous hemodynamic evaluations of the Y-graft were based on virtual
surgeries.2,3 Although CFD for blood flow modeling has been validated against
theoretical solutions, in-vitro and in-vivo experiments,94, 117, 147 there is a lack of
in-vivo validation for Fontan patients in the literature. The study in this chapter
has two main goals: 1) to evaluate post-operative hemodynamic performance in
the first cohort of Y-graft Fontan patients, and 2) validate simulation predictions
of hepatic flow distribution against in-vivo clinical data. Based on the importance
of HFD, simulation-derived HFD and patients’ lung perfusion data were chosen to
validate the credibility of flow simulations for surgical design.
6.1 Methods
6.1.1 Surgical technique and clinical data acquisition
The Y-graft implanted in six patients (YF1 to YF6) was custom made for
each patient by the surgeon with an 18 mm trunk and 12 mm branches. Patients
received standard post-operative care, with no change in international normal-
ized ratio (INR) target. Although the six surgeries were technically successfully,
thrombus was found in the left branch in patient YF5 3 months after the Fontan
procedure.
Catheterization was routinely performed immediately prior to the Fontan
procedure. PAVMs were evident in the RPA of patient YF5. Early diffuse PAVMs
138
were found in the right upper lobe of patient YF4.
Early (< 1 month) and six-month postoperative magnetic resonance imag-
ing (MRI) scans were performed on patients YF1, YF2 and YF3. Pulmonary and
vena cava flow was measured by phase contrast MRI (PC-MRI). For patients YF4
and YF6, only CT images were collected. To quantify the HFD in vivo, a lung
perfusion scan was performed on these three patients in the early post-operative
stage. Authors who performed flow simulations were blinded to lung perfusion
data until all corresponding simulation results were reported. Patient YF5 under-
went pre-Fontan and 3 month post-operative MRI scans. MRI and lung perfusion
scans were performed with an institutional review board approved protocol.
6.1.2 Model construction
Following our previous work, patient specific models were constructed us-
ing a custom version of the open source Simvascular software package.2 From the
acquired image data, centerline paths were created in the Y-graft, SVC and PAs,
and segmentations of the vessel lumen were created in all vessels. Finally, a 3D
solid model was created by lofting all segmentations. Since patient YF5 developed
thrombus in the left branch of the Y-graft, an additional virtual unblocked Y-graft
was constructed for comparison. Patient specific models and MRI/CT images
are shown in Figure 6.1. To examine the influence of graft size and anastomosis
location on wall shear stress (WSS) and HFD, two modified Y-graft designs con-
structed for patient YF5. First, the right and left branch sizes were changed to 14
139
YF1
YF4
YF3YF2
YF5 YF6
Figure 6.1: Post-operative MRI/CT images and models. Since patients YF5 de-veloped thrombus in the left branch, an unblocked Y-graft was reconstructed forstudy.
mm and 10 mm, respectively (referred to as R14-L10). Then, model R12-L12 was
created to have a distal anastomosis and a proximal anastomosis for the right and
left branches respectively without changing the branch size.
140
6.1.3 Flow simulation and boundary conditions
To simulate blood flow, a 3D finite element Navier-Stokes solver was em-
ployed with a Newtonian approximation for the viscosity and a rigid wall assump-
tion.94 Pulsatile flow boundary conditions were applied to the IVC and SVC inlets.
To account for the respiratory effects in IVC flow, MRI-derived pulsatile inflow
data were superimposed with a respiratory model following our previous work.4
RCR boundary conditions were tuned to achieve the target Fontan pressure and
pulmonary flow split.3,5 Post-operative catheterization derived Fontan pressure of
11 mmHg was used as the target for patient 5, and 12 mmHg was assumed for
all other patients since no catheterization data was available. For patients YF4
and YF6, three pulmonary flow splits, 35/65, 55/45 and 75/25, were applied since
these patients were imaged with CT. The IVC and SVC inflow conditions for these
patients were taken from patient YF5 and scaled according to BSA. In the fol-
lowing paragraphs, we use the format, RPA/LPA, to present pulmonary flow split
and HFD. Table 2 lists all patients’ flow data and pulmonary flow splits used in
simulations.
To investigate the potential factors that led to the formation of thrombus in
patient YF5, three scenarios were studied. We first simulated the flow environment
at the time of the 3 month post-operative MRI, after formation of the thrombus.
A Y-graft model with a blocked left branch was constructed directly from the
post-operative MRI data and boundary conditions from the PCMRI acquired dur-
141
Table 6.1: Patients’ flow conditions used in simulations. The vena cava flow andpulmonary flow split were measured by PC-MRI except for patients YF4 and YF6,who had CT imaging. We use the format, RPA/LPA, to present pulmonary flowsplit. For patients YF1, YF2 and YF3, “early” and “6 month” denote measure-ments taken in the early (< 1 month) and 6 month post-operative stages, respec-tively. For patient YF5, pre-operative and 3 month post-operative measurementswere performed.
Data acquisition Patient IVC ml/s SVC ml/s Pulmonary flow split
Measured
YF1early: 6 early: 14 early: 81/19
6 month: 10.7 6 month: 17.8 6 month: 70/30
YF2early: 11.8 early: 15 early: 60/406 month: 8.2 6 month: 16.8 6 month: 66/34
YF3early: 9.3 early: 9.5 early: 59/41
6 month: 21.5 6 month: 15.3 6 month: 53/47
YF5pre-Fontan: 11.5 pre-Fontan: 14 pre-Fontan: 81/193 month: 11.7 3 month: 12.2 3 month: 54/46
ScaledYF4 11.4 11.9 35/65, 55/45, 75/25YF6 11.2 11.7 35/65, 55/45, 75/25
ing this scan were used. Second, a complete Y-graft was virtually reconstructed
by removing the branch blockage, and the same post-operative boundary condi-
tions were used. Third, we simulated the presumed flow conditions in the early
post-operative state using an unblocked Y-graft model with the pre-operative pul-
monary flow split, assuming that pulmonary remodeling progresses gradually and
the pulmonary flow split does not change significantly within a short period (less
than one month) after Fontan completion.
Lagrangian particle tracking was used to quantify the HFD.2, 139 Approxi-
mately 10,000 particles were released at the IVC inlet every 1/100 cycle for a cycle.
HFD was derived by computing the particle flux for particles traveling to the RPA
and LPA, respectively.3 In addition, previous work has shown that simulation
142
predictions of HFD are insensitive to the use of rigid vs. deformable walls.58
6.2 Results
Results are divided into three parts. We first present validation and lon-
gitudinal results for patients YF1, YF2 and YF3. Second, HFD under a range
of pulmonary flow splits for patients YF4 and YF6 is shown. Third, the issue of
thrombosis in patient YF5 is explored in detail to identify possible causes. For
patient YF5 three different scenarios were simulated and compared in order to
identify possible factors that triggered thrombus formation.
6.2.1 Simulation vs. lung perfusion
Figure 6.2a shows a comparison between simulation- and lung perfusion-
derived HFD. In the early post-operative stage, simulations agree within 10% of
with in-vivo measurements for all three patients. In patients YF1 and YF3, the
SVC flow blocked hepatic flow in the right and left branch respectively due to
a more proximal anastomosis. In patient YF3, a distal anastomosis for the left
branch which bypassed the stenosis in the LPA and a medial anastomosis for the
right branch resulted in over 60% of hepatic flow streaming to the LPA despite the
fact that the RPA received 60% of the total systemic venous flow.
143
YF3YF2YF1
a)
b)
0%10%20%30%40%50%60%70%80%90%
100%
Simula�on Lung perfusion
IVC-LPAIVC-RPA
PFS: 81/19
0%10%20%30%40%50%60%70%80%90%
100%
0 mon. post-op 6 mon. post op
IVC-LPAIVC-RPA
PFS: 81/19 PFS: 70/30
0%10%20%30%40%50%60%70%80%90%
100%
Simula�on Lung perfusion
IVC-LPAIVC-RPA
PFS: 60/40
0%10%20%30%40%50%60%70%80%90%
100%
Simula�on Lung perfusion
IVC-LPAIVC-RPA
PFS: 59/41
0%10%20%30%40%50%60%70%80%90%
100%
0 mon. post-op 6 mon. post-op
IVC-LPAIVC-RPA
PFS: 59/41 PFS: 53/47
0%10%20%30%40%50%60%70%80%90%
100%
0 mon. post-op 6 mon. post-op
IVC-LPAIVC-RPA
PFS: 60/40 PFS: 66/34
Figure 6.2: a) Comparison between early post-operative simulation-derived HFDand lung perfusion data for patients YF1, YF2 and YF3. HFD in the early post-operative stage was quantified by simulation and lung perfusion. b) Changes inHFD from the early to six-month post-operative stages derived from simulations.
6.2.2 Longitudinal HFD
HFD was predicted by particle tracking at the < one month and six month
time points, and pulmonary flow split was measured by MRI (Figure 6.2b). Lon-
gitudinally, HFD in patients YF1 and YF3 became more even as pulmonary flow
split approached 50/50 and IVC flow increased. In contrast, an opposite trend was
found in patient 2, but the changes in pulmonary flow split and HFD were minor.
The average change in HFD for these three patients is an 11% improvement and
in pulmonary flow split is 7.7%.
144
6.2.3 HFD estimation without in vivo flow conditions
Since PCMRI flow rates were not available for patients YF4 and YF6, HFD
was quantified under three assumed pulmonary flow splits of 35/65, 55/45 and
75/25. In patient YF4, this resulted in percentages of hepatic flow streaming to
the RPA of 55%, 69% and 88%, respectively. In patient YF6, results are 61%, 85%
and 95%, respectively. RPA-skewed HFD was attributed to the blockage effect of
SVC flow on the proximal anastomosis for the left branch.
6.2.4 Thrombus investigation
Because patient YF5 had an occurrence of thrombosis in the left branch,
this patient is examined in more detail for possible causes. The patient’s HFD
under different flow conditions is summarized in Table 6.2. In the early post-
operative period, hepatic flow in patient YF5 was highly skewed to the RPA due
to low resistance. Patient YF5’s HFD was still skewed due to the thrombus in the
left branch even though the patient’s pulmonary flow split was found to be 54/46
at the three month post-operative time point. If the left branch remained patent,
the HFD would be improved achieving 63/37 at the three month post-operative
time point.
To examine whether patient YF5 is different from other Y-graft patients
in WSS, Table 6.2 and Figure 6.3 show the time-averaged WSS magnitude for
patients YF1, YF2, YF3 and YF5. For patient YF5, there is a persistent region
145
Table 6.2: Mean WSS magnitude for Y-graft branches and HFD. Compared toother patients, patient YF5 had low WSS in the left branch in the early postoperative stage but the WSS in the left branch increased in the 3 month post-operative stage in which the pulmonary flow split changed from 81/19 to 54/46.The mean WSS for patient YF3 is low due to a lower cardiac output. The modifiedY-grafts for patient YF5 increased mean WSS in the left branch in the early post-operative stage compared to the original Y-graft. All Y-graft designs for patientYF5 skewed hepatic flow to the RPA with PAVMs in the early post-operative stagebut the HFD was improved in the 3 month post-operative stage.
Type Patient (flow cond.) Left branch Right branch HFD(dynes/cm2) (dynes/cm2)
Initial
YF1 (early) 5.0 10.8 63/37YF2 (early) 8.4 11.3 83/17YF3 (early) 4.2 5.4 35/65YF5 (early) 4.6 25 88/12
YF5 (3 mon.)9.1 22 63/37 (unclotted)N.A. 19.7 82/18 (clotted)
Modified
YF5 R14-L10 (early) 9.4 19.5 95/5YF5 R14-L10 (3 mon.) 12.7 13.8 72/28YF5 R12-L12 (early) 11.7 13.2 99/1YF5 R12-L12 (3 mon.) 9.9 8.1 85/15
146
of low WSS for the left branch for the early post-operative case when we have
a measured pulmonary flow split of 81/19. Overall, mean WSS is higher in the
right branch than the left branch. The WSS in the left branch increased in our
simulation of the later time point when pulmonary flow split is changed to 54/46.
To quantify the low WSS area which we hypothesize is related to thrombus
formation, we computed the branch surface area in which the WSS value is lower
than a threshold value. While it is known that low WSS is conducive to thrombosis
formation, the exact threshold value remains unknown, and is certain to vary
among patients.148 Figure 6.4 shows the percentage area of low WSS with different
threshold values for each branch in patients YF1, YF2, YF3 and YF5. Compared
to the early post-operative results for patients YF1, YF2 and YF3, YF5’s left
branch had a larger portion of low WSS for all threshold values used, though the
mean WSS magnitude for the left branch in patient YF3 was also low due to a
lower cardiac output. However, the percentage area of low WSS was significantly
reduced in the three month post-operative stage. For the right branch, patient
YF5 had the smallest low WSS area in both post-operative stages at all threshold
levels.
Velocity fields confirmed that an area of flow stagnation is evident in the
left branch of patient YF5 under a pulmonary flow split of 81/19. In models YF4
and YF6 with HFD similar to YF5, less flow stagnation region is observed due to
the impingement of the SVC jet.
Table 6.2 shows that the mean WSS increased with decreasing graft size,
147
YF3YF2YF1
YF5 YF5 R14-L10 YF5 R12-L12
Figure 6.3: Time-averaged WSS magnitude for patients YF1, YF2 YF3 and YF5in the early post-operative stage. YF5 R14-L10 and YF5 R12-L12 are two modifiedY-graft designs for patient YF5. In the baseline model for patient YF5, a distalanastomosis for the left branch and a highly skewed pulmonary flow split resultedin larger low WSS area in the left branch. In model R12-L12, the WSS in theleft branch was enhanced due to a proximal anastomosis that allowed SVC flow towash the left branch.
00.10.20.30.40.50.60.70.80.9
1
0.5 1 2 4
YF1
YF2
YF3
YF5 early post-opYF5 3 mon. post-op
cri�
cal a
rea
le� branch
τ0
0.10.20.30.40.50.60.70.80.9
1
0.5 1 2 4
YF1
YF2
YF3
YF5 early post-op.YF5 3 mon. post-op
τ
cri�
cal a
rea
right branch
Figure 6.4: Percentage of low WSS region for two branches in patients YF1, YF2,YF3 and YF5. For each threshold value τ , the low WSS area relative to eachbranch surface was computed.
148
00.10.20.30.40.50.60.70.80.9
1
0.5 1 2 4
R14-L10 le� branchR12-L12 le� branch
cri�
cal a
rea
τ
pulmonary flow split: 81/19
00.10.20.30.40.50.60.70.80.9
1
0.5 1 2 4
R14-L10 le� branchR12-L12 le� branch
cri�
cal a
rea
τ
3 mon post-op.pulmonary flow split: 54/46
early post-op.
Figure 6.5: Percentage of low WSS region for patient YF5’s modified Y-grafts. Inmodel R12-L12, the low WSS area in the left branch can be effectively minimizedby using a proximal anastomosis in which the SVC jet impinged the wall and theimpact of the SVC jet on the WSS was reduced with increasing LPA flow. Com-pared to the baseline model (Figure 6.4), model R14-L10 has a similar percentageof low WSS area for the left branch in the early post-operative stage for thresholdvalues below 2 dynes/cm2.
and that the proximal anastomosis resulted in less low WSS area in the left branch
due to the effect of the SVC jet. In Figure 6.5, model R12-L12 effectively minimized
the low WSS area in the left branch by using a proximal anastomosis. Compared to
the baseline model (see Figure 6.4), model R14-L10 achieved a similar percentage
of low WSS area for the left branch in the early post-operative stage for threshold
values below 2 dynes/cm2.
6.3 Discussion
In this study, we evaluated the hemodynamic performance for the first co-
hort of Y-graft Fontan patients. For the first time, we obtained MRI measurements
at two time points, which provided a direct comparison between the early and six-
month post-operative stages for Y-graft patients. Although only short-term data
149
are available, measurements at two post-operative time points show that the ra-
tios of caval flow and pulmonary flow split are not constant in time. Within six
months after Fontan completion, a uniform increase of 33% was observed in the
SVC flow. The IVC flow increased significantly by 105% in two patients, YF1
and YF3, but decreased by 31% in patient YF2. Although the systemic venous
flow increased in all patients, the IVC flow was still less than the SVC flow in all
patients except YF3 at the six-month post-operative time point. Similarly, Fogel
et al.33 and Houlind et al.34 previously reported that the SVC contributed more
flow to the venous return in lateral tunnel Fontan patients. This indicates that
Y-graft patients likely follow the same trend in systemic venous flow distribution
as other Fontan patients. At least four out of six patients in this study had a
RPA-predominant split, consistent with the measurements of Houlind et al.34
In validating simulations against the lung perfusion data, good agreement
was obtained in the three patients studied. The largest difference (10%) between
the simulation-derived HFD and lung perfusion in patient YF2 is still accept-
able, considering potential sources of uncertainty in both simulation and clinical
measurement. Although patient specific data were incorporated into previous sim-
ulation studies, simulation-derived results had not been directly validated against
clinical data in a blinded fashion in prior studies. The significance of this study
is a validation that flow simulations correctly quantify patients’ HFD, which is
an important parameter for Fontan surgical design. This provides a basis for fu-
ture predictive studies, albeit more patients should be used to achieve statistical
150
significance.
HFD is largely influenced by the anastomosis locations and amount of SVC
flow. In the first cohort of Y-graft patients, the Y-graft was anastomosed to the
PAs proximally relative to the SVC. For an uneven pulmonary flow split, a proximal
anastomosis on the PA side with lower resistance is beneficial for reducing skewed
hepatic flow. In patient YF1, the hepatic flow was relatively even despite a severely
skewed pulmonary flow split, since the SVC jet blocked a part of hepatic flow
streaming to the RPA. However, the same effect caused a highly uneven HFD in
patient YF2, who had a mildly uneven pulmonary flow split. These in-vivo results
were consistent with our previous virtual studies.3 Limited space in the chest cavity
is a major concern for implanting a Y-graft. Although the Y-graft in this study was
anastomosed proximally, it does not imply that the branch cannot be anastomosed
more distally. With more surgical experience, more distal anastomoses could be
made in some patients to improve HFD. For example, anastomosing the left branch
more distally could potentially avoid the SVC flow impingement in the left branch
in patients YF2, YF4 and YF6.
Unknown flow boundary conditions for patients YF4 and YF6 made simula-
tion results uncertain for these patients. Since there was good agreement between
simulation and lung perfusion for patients with MRI scans, the boundary condi-
tions are the major source of uncertainty for patients YF4 and YF6. Under a
range of pulmonary flow splits, patients YF4 and YF6 showed a consistent trend
that the hepatic flow streaming to the RPA increased with increasing RPA flow.
151
Since patient YF4 had PAVMs in the right upper lobe, we infer that HFD might
have changed from severely RPA-skewed to moderate RPA-skewed with PAVM
regression because PAVMs usually result in lower pulmonary resistance.13, 23
Since caval flow ratios and pulmonary resistances change with age, HFD is
unlikely to remain constant over time. We found that HFD values moved changed
towards 50/50 over time in patients who had increased IVC flow and a less skewed
pulmonary flow split. This agrees with our prior exercise simulation findings in,3
such that increasing IVC flow could mitigate the adverse effects of high SVC flow
and uneven pulmonary resistances on HFD.
The pre and post operative measurements in patient YF5 provide some in-
teresting insight into pulmonary remodeling in patients with PAVMs. Patient YF5
had a pre-operative pulmonary flow split of 81/19. It is well known that the lung
with PAVMs typically has a low pulmonary resistance due to a precapillary con-
nection between the systemic and pulmonary venous return.13, 23 At the 3 month
post-operative examination, the pulmonary flow distribution was nearly even and
PAVMs in the RPA had regressed. Based on the simulation results for patient
YF5, we can infer that an uneven pulmonary flow split for patients with PAVMs
prior to the Fontan procedure would likely change dramatically post surgery, and
that streaming hepatic flow to the malformed lung reversed the PAVMs in this
case, resulting in a less skewed pulmonary flow split.
For patients without PAVMs, the question of how pulmonary flow split
changes over time remains unclear and long term follow up is needed. Our data
152
suggest that the pulmonary flow split in patients without PAVMs is relatively
stable within six months. Recently, Yin et al.149 reported a five year follow-up
study on the pulmonary perfusion of 43 Fontan patients . Their results showed
that there are small differences (3% and 2%, respectively) in the pulmonary and
IVC flow perfusion between the early post-operative and follow-up groups, though
pulmonary vascular resistance and relative perfusion for upper and lower lobe
lung showed statistically significant differences after 5 years.149 However perfusion
data prior to the Fontan surgery were not available in their study. If there were
significant changes in pulmonary flow split, they likely occurred during the early
post-operative stage because a small difference was found between the early and
5 year post-operative time points. If these trends were universal, HFD would
remain relatively stable with possible larger changes in the relative distribution for
the upper and lower lobes. Relative hepatic and pulmonary flow distribution in
each lung should be examined in future clinical studies.
The risk of thrombosis is another important issue which should be consid-
ered in addition to HFD. In patient YF5, thrombus was found in the left branch,
even though PAVMs in the right lung had regressed. Although the choice of crit-
ical WSS value for thrombosis is still unclear, calculations with different critical
values consistently showed that a low WSS region was evident in the left branch
for patient YF5 in the early post-operative stage. These results indicate that low
WSS and flow stasis in the left branch increased thrombotic risk and consequently
led to complete occlusion of the left branch. Moreover, simulations with an un-
153
blocked Y-graft and a pulmonary flow split of 54/46 suggested that the thrombus
likely formed before the PAVM regression, which caused the pulmonary flow split
to drop from 81/19 to 54/46. Had PAVM regression occurred earlier, the increased
WSS level in the left branch may have prevented thrombus formation. Without
further clinical evidence and data to support this timing, these findings, however,
remain speculative.
Although the presumed early development of thrombus in the left branch
in patient YF5 may be driven by multiple factors, our findings suggest that low
WSS and flow stasis are likely to be important causes. Thus, for patient YF5, the
immediate surgical design goal should have been to distribute the hepatic flow to
the lung with PAVMs without introducing a large low WSS area, while at the same
time keeping the long-term goal to achieve even HFD after PAVMs are regressed.
Compared to the original Y-graft for patient YF5, an 18-12-12 Y-graft
with a proximal anastomose for the left branch (YF5 R12-L12) minimized the
low WSS area in the simulated early post-operative stage, though HFD was still
skewed at the early time point. However, HFD is later improved with increasing
IVC flow.3 The unequal sized Y-graft (YF5 R14-L10) did not show substantial
improvement in minimizing the lowWSS area. Previous optimal designs considered
a single objective solely. The formation of thrombus in patient YF5 showed that
a single objective design without considering the dynamic process of pulmonary
remodeling and thrombotic risk is inadequate. Future studies should incorporate
multi-objective constrained optimization to balance the need for even HFD with
154
thrombotic risk. Our results suggest that the the first three months post Fontan
may be a critical time period to prevent thrombosis in the Y-graft for patients
with PAVMs, and that the risk likely drops after PAVM regression. As pointed
out in our previous studies, the superiority of the Y-graft cannot be guaranteed
without a customized optimal design for each patient. In addition to optimizing
the Y-graft geometry to minimize low WSS area, more aggressive anticoagulation
therapy may be needed for patients during the initial high risk period. MRI and
CFD could be used in these cases to monitor the WSS level in Y-graft Fontan
patients, providing extra information for their treatment.
Recently, Haggerty et al.150 reported post-operative simulation results for
five Y-graft Fontan patients at Children’s Healthcare of Atlanta showing that com-
pared to virtual T-junction and offset designs, the Y-graft design resulted in im-
proved HFD but little difference in the connection resistance. These findings are
likely due to the use of a smaller branch size (9mm and 10mm) used for the Y-graft
and more idealized graft geometry for the virtual tube-shaped models. Instead of
custom Y-grafts, commercial bifurcated aorto-illiac grafts were used in the Y-graft
Fontan surgery in their study.150 Although use of an “off the shelf” graft is consis-
tent and well controlled in terms of fabrication, it is less flexible than the custom
Y-graft in the choice of geometry such as trunk-branch size and bifurcation angle.
Since Fontan patients show a variety of anatomic patterns, a custom graft may
be more suitable for customized Fontan design. In addition, our findings indicate
that anastomosis location is likely equally or more important than the graft branch
155
diameter in determining WSS and HFD values.
6.4 Limitations
The follow up time in this study was limited and the number of study
patients was relatively small. It is still too early to answer questions such as how
pulmonary and hepatic flow distributions change over a long period and whether
the Y-graft results in better mid-term or long term outcomes than traditional
designs. Longitudinal and serial data collection including the pre-Fontan stage
should be performed on multiple patients in future studies.
Owing to a lack of PC-MRI measurements in the early post-operative stage
for patients YF4, YF5 and YF6, flow boundary conditions were approximated.
Since patient YF5 underwent pre-operative and three month post-operative MRI
examinations, results for the early post-operative stage were less uncertain com-
pared to patients YF4 and YF6 who had CT images only.
The prediction of post-operative pulmonary flow split from pre-operative
data should be addressed systematically by multi-scale modeling in future studies.
Rigid wall and Newtonian flow assumptions were employed in this study. Long et
al.58 showed that differences in HFD and energy efficiency due to wall compliance
are small but that WSS was over-predicted by up to 17% in rest conditions by rigid
wall simulations.57 Future accurate predictions of thrombotic risk may therefore
necessitate use of fluid structure interaction, however the relative values used in
156
this study may be adequate for risk assessment.
6.5 Conclusions
By comparing in-vivo lung perfusion data, the accuracy of simulation-
derived HFD was validated. Although the technical success of the Y-graft Fontan
surgery demonstrated the feasibility of the Y-graft concept, simulations for the
first six patients show that a proximal anastomosis of the left branch resulted in
uneven HFD due to the SVC blockage effect in some patients. Overall, hepatic
flow was skewed to the RPA in the early post-operative stage and HFD was im-
proved as the IVC flow increased and pulmonary flow split became less unequal.
It was confirmed that the overall pulmonary flow split changes over time. A pa-
tient with unilateral PAVMs showed a significant change in pulmonary flow split
after Fontan completion due to the regression of PAVMs while variations in other
patients were less pronounced. Simulations can provide insight in patients with
adverse events. Although sufficient hepatic flow was channeled to the malformed
right lung to resolve PAVMs, thrombus likely developed in one branch with stag-
nant flow. Compared to other patients, YF5’s left branch had a larger region of
low WSS in the early post-operative period. Thus a plausible explanation is that
the thrombus in the left branch developed due to flow stasis soon after Fontan
completion, and that PAVM regression occurred after this. Therefore, particular
attention should be paid to WSS values and flow stasis in the early post-operative
157
period in order to reduce the thrombotic risk in the Y-graft. This should be ex-
amined during the pre-operative surgical planning phase, particularly for patients
with existing PAVMs.
6.6 Acknowledgments
This study was supported by the American Heart Association, a Burroughs
Wellcome Fund Carreer Award at the Scientific Interface, a Leducq Foudantion
Network of Excellent Grant, a NSF CAREER Award and a UCSD Kaplan Fel-
lowship. We thank Shawn Shadden, Irene Vignon-Clementel, Mahdi Esmaily
Moghadam, and John Lamberti for their expertise in numerical simulations and
pediatric cardiac surgery, as well as Christina Ngo for model construction. We also
wish to acknowledge the use of Simvascular (simtk.org, www.osmsc.com).
Chapter 6, in full, is a reprint of the material as it appears in Yang, W.,
Chan, F. P., Feinstein, Reddy, V. M., Marsden, A. L., and Feinstein, J. A. Flow
Simulations and Validation for the First Cohort of Y-graft Fontan Patients., in
preparation.
Chapter 7
Conclusions and future work
7.1 Conclusions
In this dissertation, we focused on hemodynamics of a novel Y-graft de-
signed for the Fontan procedure using numerical simulations. This work extended
the preliminary results obtained by Marsden and colleagues.2, 4, 76 We applied a
derivative-free optimization framework to evaluate and improve upon designs for
the Fontan surgery. Simulations in idealized Y-graft models showed that a 24mm
branch size (maximum size) with a large bifurcation angle achieved the highest en-
ergy efficiency at rest and the graft size decreased with increasing IVC flow under
exercise conditions. Optimal designs for the rest condition were significantly influ-
enced by the wall shear stress (WSS) constraint showing a trade-off relationship
between the energy efficiency and low WSS area. However this impact was less
significant under exercise conditions because a smaller graft size resulted in less
158
159
energy loss. Although this is the first study that applied formal optimal design
to the Fontan procedure, the use of idealized models resulted in low energy loss
values that are not realistic in the in-vivo Fontan circulation. Later surgical prac-
tice suggested that a Y-graft with 24 mm branches may not be feasible in 2-4 year
old Fontan patients. In addition, a recent multiscale modeling study by Baretta et
al.54 showed that differences in energy loss with different Fontan geometries had
negligible effects on the ventricular pressure-volume loop. Thus, the hypothesis
that lower hydrodynamic energy loss improves clinical outcomes is not supported
by recent studies, although further investigations are required to solidify these
findings. Based on our current results, the importance of energy efficiency in the
Fontan geometry is secondary to other factors.
Studies in multiple patient specific models showed that the hemodynamic
performance of the Y-graft design was patient specific. We found that the pul-
monary resistance, anastomosis location, flow condition and stenosis treatment
play important roles in local hemodynamic performance. Instead of energy loss,
more attention was paid to the pulmonary flow distribution (HFD). The SVC flow
was found to be a major reason causing skewed HFD because the downward SVC
jet blocked the IVC flow, resulting in unilateral streaming with as much as 97%.
Compared to the traditional T-junction and offset designs, the Y-graft design dis-
tributed hepatic flow to two lungs more evenly with moderate improvement in
energy loss and SVC pressure. However, the cases of underperforming Y-graft de-
signs demonstrated the importance and needs for customizing surgical designs for
160
individual patients.
The findings in the patient specific study motivated us to investigate the
necessity of the use of unequal branches for patients with highly uneven pulmonary
flow splits. We demonstrated that optimized Y-graft with equal sized branches can
achieve the same target HFD with less energy loss for a large range of pulmonary
flow splits compared to the design with unequal sized branches. Anastomosis
locations for the Y-graft are more important in determining HFD than the graft
size. Although the SVC jet blockage effect was attributed to skewed HFD, it
can be utilized to optimize HFD by anastomosing the branch more proximally
(towards the SVC) for the PA with a smaller resistance value. We effectively
improved previously underperforming Y-grafts in two patients by optimizing semi-
idealized models. Comparisons between the semi-idealized and patient specific
models showed that the semi-idealized model can approximate the patient specific
model for HFD prediction with treatments for the PA and right upper lobe adjacent
to the SVC.
Based on these simulation results, the Y-graft design has been translated
into clinical use. The technical success demonstrated the feasibility of the Y-graft
design. Simulation derived HFD showed excellent agreement with in vivo lung per-
fusion data for three patients in the immediate post-operative stage. However non-
optimized Y-grafts did not achieve a 50/50 split for hepatic flow, channeling more
hepatic flow to the right lung in most patients. The six month post-operative simu-
lations and MRI measurements showed that increased IVC flow rate and more even
161
pulmonary flow split improved HFD, compared to the immediate post-operative
data. Investigations on thrombus formation demonstrated that skewing hepatic
flow to the malformed lung might facilitate the regression of PAVMs but resulted
in larger areas of low WSS and increased flow stagnation, which might be a pos-
sible factor causing thrombus. This case suggested that future surgical design
should carefully balance the need of hepatic flow for the lung with PAVMs and
the risk of thrombosis due to low WSS, together with considerations of pulmonary
remodeling.
7.2 Future work
7.2.1 Pre-operative prediction and assessment
Post-operative validation with lung perfusion data demonstrated the capa-
bility of numerical simulations to correctly calculate physiologically relevant pa-
rameters with MRI-derived boundary conditions. Current computational tools are
able to evaluate individual Fontan patient’s hemodynamic performance with rea-
sonable accuracy. However clinicians also need a tool capable of predicting post-
operative performance based on pre-operative data in order to choose the best
surgical plan. Therefore our future studies should focus on predictive simulation
capabilities for the Fontan procedure.
Although similar issues have been involved in the comparison study using
multiple patient specific models in Chapter 4, the assumption that patients’ pul-
162
monary flow split and cardiac output remain unchanged limits the scope of the
study within the immediate post-operative period. Our post-operative follow up
showed that there are nonnegligible changes in the pulmonary flow split and caval
flow within six month after Fontan completion. Therefore future predictive simula-
tions have to account for these changes in the boundary conditions. The multiscale
modeling methods54 that couple closed-loop lumped parameter networks to a 3D
flow solver offer a promising means to systematically model the effects of physio-
logic changes due to the surgery.
We still know little about Fontan patients’ hemodynamic evolution. The
study on the first cohort of Y-Fontan patients has not yet provided enough inputs
to predict the outcomes of the Y-graft design due to a small sample size and a
short follow up period. Therefore, launching a long term follow up study with
a sufficient sample size is crucial to determine the actual outcomes and provide
first-hand data for predictive modeling.
7.2.2 Patient specific optimal design
As we showed in this work, the variability of the Fontan geometry lead to
dramatically different flow fields, and idealized models are unable to fully char-
acterize individual Fontan patient’s hemodynamics. Therefore, patient specific
designs are necessary. A few attempts have been made in patient specific design
optimization.151,152 The major limitations included the use of a trial-and-error ap-
proach and optimized Y-graft derived from the semi-idealized models. Although
163
some treatments were applied to the semi-idealized model in order to approximate
the corresponding patient specific model, discrepancies were inevitable and manual
interventions were needed. In addition, the assumption about the size of the end-
to-side anastomosis is empirical and qualitative. In future studies, a physics-based
model manipulation method for the end-to-side anastomosis will be developed such
that one can manipulate the Y-graft design in patient specific models without la-
borious manual interventions. One possible way is to model the surface of the
PA and graft as an elastic structure. The geometry of the graft could then be
determined by applying forces to deform the graft and the end-to-side anastomosis
could be modeled by pressurizing the elastic structure.
The semi-idealized model can be used as a low fidelity model which is less
expensive to evaluate compared to the patient specific model. Recent optimization
studies took advantages of low fidelity models to enhance the efficiency of expensive
simulation based optimization.144
7.2.3 Validation against 4D MRI
Although simulation-derived HFD has been validated against to lung per-
fusion data, the in-vivo velocity field and WSS have not been compared. With
the advent of new imaging technology, 4D MRI allows one to measure in-vivo
time-dependent 3D flow. Although current spatial resolution limits our ability to
quantify WSS and highly dynamic flow structures,39 4D MRI provides a valuable
tool to comprehensively compare with CFD results and we can expect to be able to
164
perform direct validations for WSS calculations and turbulence modeling in blood
flow in the near future.
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