Computer-Aided Patient-Specific Coronary Artery Graft Design Improvements Using CFD Coupled Shape Optimizer
Abstract
This study aims to (i) demonstrate the efficacy of a new surgical planning framework for complex cardiovascular reconstructions, (ii) develop a computational fluid dynamics (CFD) coupled multi-dimensional shape optimization method to aid patient-specific coronary artery by-pass graft (CABG) design and, (iii) compare the hemodynamic efficiency of the sequential CABG, i.e., raising a daughter parallel branch from the parent CABG in patient-specific 3D settings. Hemodynamic efficiency of patient-specific complete revascularization scenarios for right coronary artery (RCA), left anterior descending artery (LAD), and left circumflex artery (LCX) bypasses were investigated in comparison to the stenosis condition. Multivariate 2D constraint optimization was applied on the left internal mammary artery (LIMA) graft, which was parameterized based on actual surgical settings extracted from 2D CT slices. The objective function was set to minimize the local variation of wall shear stress (WSS) and other hemodynamic indices (energy dissipation, flow deviation angle, average WSS, and vorticity) that correlate with performance of the graft and risk of re-stenosis at the anastomosis zone. Once the optimized 2D graft shape was obtained, it was translated to 3D using an in-house “sketch-based” interactive anatomical editing tool. The final graft design was evaluated using an experimentally validated second-order non-Newtonian CFD solver incorporating resistance based outlet boundary conditions. 3D patient-specific simulations for the healthy coronary anatomy produced realistic coronary flows. All revascularization techniques restored coronary perfusions to the healthy baseline. Multi-scale evaluation of the optimized LIMA graft enabled significant wall shear stress gradient (WSSG) relief (~34%). In comparison to original LIMA graft, sequential graft also lowered the WSSG by 15% proximal to LAD and diagonal bifurcation. The proposed sketch-based surgical planning paradigm evaluated the selected coronary bypass surgery procedures based on acute hemodynamic readjustments of aorta-CA flow. This methodology may provide a rational to aid surgical decision making in time-critical, patient-specific CA bypass operations before in vivo execution.
Keywords
Surgical planning Coronary artery Bypass graft CFD Hemodynamics Shape optimization Sequential graft WSS WSSG Surgical designIntroduction
Statistics from the American Heart Association identify coronary heart diseases (CHD) as the principal cause of morbidity and mortality in the western world.14,42 Major causes of CHD include atherosclerosis and complications related to congenital cardiac defects. Atherosclerosis involves the agglomeration of fatty substances, cholesterol, and other deposits on the inner lining of an artery together with transverse growth of smooth muscle cells (i.e., artheroma). This results in reduced blood flow and other pathological complications.42
Bypass conduits provide an alternative route around critically blocked arteries. Current surgical anastomosis techniques and the design of synthetic coronary artery bypass grafts (CABG) frequently lead to post-surgical complications such as intimal thickening, restenosis, and eventual long term graft failure. Failure presents in 5–20% of patients within 1–5 years, and approximately 50% of patients within 10 years after CABG surgery.5 Pathological hemodynamic states are usually precursors of intimal hyperplasia or platelet deposition and may result in graft occlusion.6,13,57 From a fluid mechanics perspective, abnormalities in coronary flow include recirculation zones, low/oscillating shear stresses, vortices, and areas of stagnation within the CABG. These parameters relate to the variation in strain rate within the conduit, which in turn are influenced strongly by the shape of the flow domain.28 Therefore, in order to improve the success of the surgery, the optimal anastomosis geometry and angle have been actively researched. Walsh et al.53 demonstrated that the use of cuffs (i.e., Miller cuff) and patches (i.e., Taylor patch) can significantly reduce abnormal wall shear stress (WSS) and wall shear stress gradient (WSSG) by up to 60% in patient-specific models, when compared to a conventional distal end-to-side anastomosis. Studies based on idealized femoral bypass geometries provided a better insight on the influence of various design parameters such as the advantage of an acute anastomosis angle, i.e., 10–20°,3,11 creating enlarged lumen sections around the toe region to reduce WSS parameters and enable smooth transition of the flow from graft to host artery,3,22,23 and influence of the proximal artery flow.3,20 Apart from these local design considerations, the bulk shape of the bypass conduits has received little attention. Based on the rapid variation of high and low wall shear stress along the sinusoidal shaped vessel geometries, ill-shaped grafts may also be prone to atherosclerosis development.
Computational fluid dynamics (CFD) provide a viable tool for pre-surgical planning and device design, and for improving the design of surgeries and interventions used in cardiovascular medicine.33,35,40,49, 50, 51 Coupled with accurate reconstructions of anatomical data (via magnetic resonance imaging, angiograms, or computational tomography),12 CFD simulations provide the ability to quantify local hemodynamics and allow evaluating the performance of surgical design templates26,27,37 and candidate endovascular devices.58
Anatomical three-dimensional shape editing is one of the major challenges of the pre-surgical planning paradigm as cardiovascular geometries involve non-uniform vessel caliber and curvature, and conduits require complex multiple inlet–outlet geometries, which cannot be easily modified by combinations of mathematically simple binary operations or shape primitives offered by state-of-art CAD software.19 We introduced the first generation ‘interactive’ surgical planning tool, SURGEM,36 which incorporates a two-hand haptic interface to freely deform, bend and position 3D surgical baffles real-time. This shape-morphing tool has been used successfully in routine basis to aid the pre-surgical decision making process of congenital heart surgeries before the in vivo execution.37,49 Integration of a sketch-based 3D platform16, 17, 18 now expands the capability of ‘interactive’ anatomy-editing systems by replacing the expensive haptic user interface with an easy-to-access digital sketch-based modeling environment, i.e., tablet, that takes user strokes as input, and seamlessly converts them into precise three-dimensional (3D) geometric data. This allows surgeons to construct and edit anatomical structures directly in 3D precisely the way they envisage on a two dimensional (2D) image.
To date, CFD has been utilized primarily for identifying an optimal design, based on trial-and-error, among a small number of geometrical variations and intuitive design alternatives. More recently, several investigators have demonstrated the benefits of coupling computer simulations with numerical shape optimization to provide cost-effective methods for the design of the medical devices,1,15 surgical connections,25 and particularly CABGs.2,39,43 The challenges associated with CFD coupled shape optimization for clinical problems have been identified previously by Marsden et al.25
Current CABG design paradigms target improved hemodynamics to achieve reduced hyperplasia at the distal anastomosis region by modulating the anastomosis angle1 and vessel curvature39,43 in simplified 2D tubular conduits. Studies incorporating out-of-plane features reported notable variations in end-to-side anastomosis hemodynamics.31,47 Hence, although in-plane (2D) optimization is appropriate to identify the primary design features, an accurate assessment on the CABG hemodynamics requires patient-specific 3D anatomical information for reliable feedback.
One objective of this study was to demonstrate the efficacy of a novel sketch-based surgical planning framework for complex cardiovascular problems, and to develop a CFD coupled multi-dimensional shape optimization method to aid patient-specific CABG design. Hemodynamic efficiency of patient-specific complete revascularization scenarios for right coronary artery (RCA), left anterior descending artery (LAD) and left circumflex artery (LCX) bypasses were investigated in comparison to the stenosis condition. Single objective multivariate constraint optimization was applied to improve the patient-specific design of the end-to-side anastomosis of left internal mammary artery (LIMA) to the distal site of the stenosed LAD. The 2D optimization procedure was comprised of geometry creation, parameterization, mesh generation, finite element (FEA) solution, and design optimization. Multiple hemodynamic indices including the local gradient of WSS, space-averaged WSS, energy efficiency, and recirculation at the anastomosis zone were selected as cost functions for the optimization problem. The optimized 2D graft shape was re-evaluated in a patient-specific 3D setting to validate the efficacy of the 2D optimization methodology. In addition, hemodynamic efficiency of the sequential grafting strategy was analyzed in comparison to the standard single CABG configuration. Sequential grafting is a routine surgical method of raising multiple braches from a parent bypass graft when atherosclerotic occlusions involve more than one coronary artery (CA). Clinically, local hemodynamic adjustments after sequential grafts remain unclear, and there is no established preference for single or sequential graft bypass.41
Methods
Reconstruction of Patient Anatomy
To demonstrate the efficacy of our in-house anatomical editing tool, 3D CABGs (LAD, LCX, LIMA, and sequential grafts) were created virtually based on the post-op CT patient data. This interactive surgical planning platform is based on the robust shape-morphing principles introduced earlier,36 and incorporates B-spine curves for generating highly flexible tubular conduits. The sketch-based computer interface allows surgeon/operator to use pen strokes to modulate the in-plane design features (i.e., curvature, proximal, and distal anastomosis points) of the 3D tubular conduit, while preserving all out-of-plane features that reside on the remaining planes. Hence, rotating the view plane surgeon can switch between on multiple design planes and perform the desired 3D shape adjustments. This ‘iterative multi-plane shape-design’ approach allows generating realistic bypass grafts virtually under 10 min of user time.
CFD Coupled 2D Shape Optimization
The optimization procedure was applied to the LIMA graft, which was parameterized based on actual surgical settings extracted from static 2D CT images as shown in Fig. 1. The CABG was simplified as a 2D cylindrical tube for the in-plane optimization. The scaffold of the coronary vessel was based on a third-order Bezier curve whose shape was dictated by four design parameters (s): the proximal (θ_{p}), and distal (θ_{d}) anastomosis angles and proximal (v_{p}) and distal curvature vectors (v_{d}). The latter two parameters defined the curvature of the conduit away from the anastomosis regions. The final 2D axisymmetric tube was formed by extruding the vessel walls in the normal directions along the centerline. In order to consider the hemodynamics of the proximal anastomosis zone, the LAD artery was also included in the optimization model as a static 2D cylindrical tube with corresponding patient-specific vessel caliber and orientation based on the CT image. Second-order Bezier curves were used to smooth geometric perturbations and sharp-corners at the anastomosis zone.
The 2D geometry of the anastomosis region was discretized using unstructured tetragonal mesh that enabled high mesh quality for dynamic shape alterations during optimization. Grid independence was ensured by comparing solutions at six refinement levels. The results presented in this manuscript were based on 9k triangular elements. In addition, local grid sensitivity checks were performed in order to guarantee grid quality proximal to the anastomosis zone. A steady state solver using the Petrov–Galerkin finite-element formulation was incorporated to solve the governing Navier–Stokes (N–S) equations for each geometric configuration.
A parabolic velocity profile \( \left( {V\left( s \right) = V_{\max } \cdot s\left( {1 - s} \right)} \right) \) where V_{max} is the maximum velocity in the core region and s is the arch length along the inlet boundary, was prescribed at the inlet boundary of the graft to yield the typical coronary flow velocity (0.1 m/s).10 No slip boundary condition was enforced on the vessel walls and the proximal end of the LAD, which was assumed to be completely (100%) stenosed. No traction boundary condition was assigned at the distal end of LAD with a constant myocardial pressure of 10 mmHg. In order to consider the non-Newtonian behavior of coronary blood in low shear regions, two blood rheology models (Ballyk and Carreau48) were implemented. Blood density was specified as 1060 kg/m^{3}. Simulations were performed on a dual-core T7200 computer with 2 GB of memory. Approximately 5 min were required to complete one converged steady state solution.
Optimization Problem and Cost Functions
Summary of hemodynamic cost functions evaluated for CFD-coupled 2D CABG optimization
Cost function | Formulation |
---|---|
Velocity gradient based energy dissipation | \( E_{\text{loss}} = \frac{1}{V}\int {\phi dv,} \quad \phi = \frac{1}{2}\mu \left( {{\frac{{\partial u_{i} }}{{\partial X_{j} }}} + {\frac{{\partial u_{j} }}{{\partial X_{i} }}}} \right) \) |
Severity parameter | \( {\text{SP}} = \frac{1}{A}\int {{\text{WSSG}}\,ds,} \) \( {\text{WSSG}} = \sqrt {\left( {{\frac{{\partial {\text{WSS}}}}{\partial x}}} \right)^{2} \left( {{\frac{{\partial {\text{WSS}}}}{\partial y}}} \right)^{2} \left( {{\frac{{\partial {\text{WSS}}}}{\partial z}}} \right)^{2} } \) |
Pressure drop | \( \Updelta P = P_{\text{in}} - P_{\text{out}} \) |
Mean wall shear stress | \({\text{WSS}}_{\text{m}} = \frac{1}{A}\int {{\text{WSS}}\,ds,\quad {\text{WSS}} = \mu ( {{ {\Updelta u + (\Updelta u}) ^{\text{T}}}}} ) \) |
Flow deviation angle | \( \phi = \frac{1}{V}\int {|{\text{dev}}|dV,\quad {\text{dev}} = a\tan \left( {{{u_{\text{N-S}} } \mathord{\left/ {\vphantom {{u_{\text{N-S}} } {u_{\text{S}} }}} \right. \kern-\nulldelimiterspace} {u_{\text{S}} }}} \right)} \) |
Vorticity | \( \zeta ^2 = \frac{1}{V}\int {|{\omega}^{2} |dV,\quad \omega = \overrightarrow {\nabla } \times \overrightarrow {u} } \) |
3D Hemodynamic Evaluation
Results
Shape Optimization of the 2D LIMA Graft
Variation of performance parameters between original and optimal LIMA graft configurations based on multivariate single objective, i.e., based on severity parameter (SP), CFD-coupled 2D shape optimization
Original LIMA | %Variation | Optimal LIMA |
---|---|---|
E_{diss} = 35.3 W/m^{3} | −0.5 | E_{diss} = 35.1 W/m^{3} |
SP = 703 N/m^{3} | −58 | SP = 300 N/m^{3} |
∆P = 11 mmHg | −12 | ∆P = 9.6 mmHg |
WSS_{m} = 7.2 dyn/cm^{2} | −1.4 | WSS_{m} = 7.1 |
|Φ| = 7.4° | −4 | |Φ| = 7.1° |
ζ^{2} = 9448 s^{−2} | 1 | ζ^{2} = 9551 s^{−2} |
L = 18.2 cm | −11 | L = 16.2 cm |
Evaluation of 3D Patient-Specific CABG Configurations
Coronary flow (mL) within the first level branches of coronary tree for the healthy and stenosed coronary arteries, the complete revascularization with single and sequential grafting configurations obtained from 3D CFD simulations
CA branches | Healthy CA | Stenosed CA | Single graft revascularization^{a} | Sequential graft LIMA-OM |
---|---|---|---|---|
Inno. A | 554 | 479 (−13) | 477 (−14) | 475 (−14) |
LCCA | 125 | 129 (4) | 128 (3) | 128 (3) |
VA | 62 | 71 (14) | 70 (13) | 70 (13) |
LSA | 93 | 107 (15) | 106 (14) | 106 (14) |
Total H–N | 833 | 786 (−6) | 782 (−6) | 779 (−7) |
s1 | 13 | 12 (−3) | 12 (−4) | 12 (−4) |
LAD_main | 32 | 28 (−13) | 55 (69) | 33 (3) |
Diagonal | 20 | 17 (−12) | 17 (−13) | 21 (4) |
Total LAD | 65 | 58 (−11) | 84 (30) | 66 (2) |
RM | 27 | 29 (5) | 28 (3) | 28 (4) |
LCX_main | 36 | 31 (−14) | 37 (2) | 37 (2) |
OM | 31 | 27 (−14) | 32 (2) | 32 (2) |
m1 | 21 | 18 (−15) | 21 (1) | 21 (1) |
Total LCX | 88 | 76 (−14) | 90 (2) | 90 (2) |
RCA | 77 | 59 (−24) | 79 (3) | 80 (3) |
DAo | 3955 | 4039 (2) | 3984 (1) | 4004 (1) |
Discussion
For the fixed distal and proximal anastomosis sites, i.e., not necessarily the anastomosis angles, multi-scale CFD coupled shape optimization was used to investigate the optimal in vivo shape of the CABGs. Preceding the 2D formal shape optimization, 3D patient-specific CFD simulations validated the utility of the proposed 2D CABG optimization method and evaluated the hemodynamic performance of the selected CABG configurations. Starting the surgical design process with reduced order 2D optimization allowed a narrower the search space and a computationally efficient framework, which provided fast evaluation of initial design alternatives. Optimization results for the LIMA graft indicated that the WSSG at the anastomosis zone can be reduced by ~30% between suboptimal and optimal configurations. Clinically, the lower WSSG may translate to improved hemodynamic performance; in turn, it will provide higher patency rates to prevent postoperative graft restenosis. We demonstrated that hemodynamics efficiency of the LIMA graft depended not only on the anastomosis angle but also on the vessel curvature for the fixed anastomosis angle. Therefore, for the first time in literature, this study identified the importance of the bulk shape of the CABG, which has been overlooked previously. In addition, high WSSG identified proximal to CA bifurcations may provide a guideline for selecting the distal anastomosis site to prevent postoperative graft restenosis.
According to our patient-specific 3D CFD analysis, sequential grafting improved the local hemodynamics proximal to LAD and diagonal bifurcation by lowering the local WSSG and WSS in comparison to original single LIMA grafting. These results promise higher potency for the sequential grafting method and agree with previous results based on idealized models45 and the long term follow up studies favoring the performance of composite grafting over single grafts.7,8,52 As opposed to the clinical data30 and results based on circuit-analog lumped models of coronary circulation,38 sequential grafting failed to increase proximal graft (LIMA) flow. For the present 3D model, the distal end of the sequential graft was anastomosed approximately at a similar peripheral distance from the coronary sinus in comparison to the distal anastomosis site of LIMA graft. Hence, the downstream pressure at the distal end of the sequential graft, which affects the graft flow and patency,29 was similar to downstream pressure at the distal end of the LIMA graft. Future clinical and numerical studies should investigate the effect of peripheral anastomosis of the distal end of the sequential graft to improve the proximal graft flow.
Results based on two common blood rheology models indicated that shear-thinning behavior of the blood flow was significant along the modeled coronary artery tree (i.e., from the sinus to the end of the first generation branches). Hence, our findings suggest that analysis and optimization of the coronary flow conduits requires incorporating non-Newtonian blood rheology.
Various cost functions employed in this study highlighted the necessity for reliable case-specific cost functions for hemodynamics optimization problems confirming the earlier cardiovascular CFD-coupled optimal shape studies.25 Shape optimization based on energy efficiency indices and Φ resulted in small curvature topology, indicating that tortuosity of the CABG effects the rotational (dissipative) characteristics of blood flow. We suspect that the influence of Φ will be more prevalent in the presence of transient flow regimes and the retrograde flow borne flow structures. Future optimization studies under pulsatile coronary flow settings should investigate the importance of oscillating shear index and flow recirculation at the anastomosis site. Oscillating shear index has been correlated strongly with vasoregulation and disease states.24 The current optimization paradigm incorporated the anatomical constraints (pulmonary vasculature, location of aorta, and heart) based on a single 2D CT slice. Under transient flow conditions, an optimization paradigm incorporating a large design space, to account for the optimal CABG design based on 3D anatomical constraints is needed, but requires high performance computing resources.32
It is worthwhile to note that results presented in this study were based solely on a single clinical case with healthy aorta-coronary tree architecture. Therefore, the proposed methodology needs to be expanded on a larger patient cohort in order to address the complexity which could arise in a broader spectrum of CABG geometries, i.e., abnormal coronary tree, congenital coronary defects, flow competition between neighboring CABGs in comparison to the hemodynamics performance of a single graft. Likewise, the stenosis severity on each main coronary artery was modeled at a fixed clinically moderate level irrespective of the actual pathology of the patient’s coronary tree. Future studies will identify the effect of stenosis severity on the performance and optimal shape of the graft in order to validate the proposed pre-surgical planning paradigm. In addition, the post-op in vivo CABG exhibits dynamic conformational variations due to cardiac contractions. Therefore, the results presented in this manuscript may also demonstrate the extent of vascular resistance variation under physiological conditions. Shape optimization under dynamic myocardial loading conditions requires further investigation.
The CFD-coupled shape optimization framework evaluated in this study allowed fast convergence due to the efficient FEMLAB 2D solver and robust communication between the optimization toolbox and FEA package, both built in MATLAB. Particularly, the third-order Bezier curves allowed easy geometry modulation and reproduced the 2D graft shape accurately under the consideration of in vivo geometrical constrains. This shows promise for extending this methodology to optimize complex connections used in reconstructive surgeries for congenital heart defects and for the future surgical planning challenges. The design paradigm developed here could be expanded to other surgical connections such as the shunts used in reconstructive surgeries for single ventricles (i.e., Norwood Procedure) or femoral arteries (femoro-poptiteal bypass), etc. Recently, we embarked upon translating the CFD-guided optimal surgical design concept to the growth and remodeling of embryonic aortic arches, which represent one of the most complex components in the cardiovascular system.55 These studies highlight the prospective use of the CFD-coupled shape optimization approach in cardiovascular research and point to future directions for improving understanding of optimal surgical design and cardiovascular function.
Conclusions
An automated framework for coupling optimal shape design to non-Newtonian blood flow simulation in multi-scale patient-specific cardiovascular geometries is demonstrated. The proposed sketch-based surgical planning paradigm evaluated the selected coronary bypass surgery procedures based on local hemodynamics and acute hemodynamic readjustments of aorta-CA flow. Our results indicated lower local WSSG for both the optimized LIMA graft and the LAD–LCX sequential graft in comparison to the original LIMA graft. This procedure may provide a rational to aid surgical decision-making process in time-critical, patient-specific CA bypass operations before the in vivo execution. We showed that the SQP optimization method realizes robust convergence provided the proper cost function and initial condition are selected. Optimal shape design requires evaluating the relation between the cost functions and flow regions, which would cause CABG failure. It is also critical to understand the behavior of each cost function as it pertains to design quality. Future studies should benefit from multi-objective optimization to minimize local flow disturbances (WSSG) and flow rotationality (Φ), and to maximize the conduit energy efficiency, concurrently. Specifically, it is required to optimize the anastomosis geometry as well as the graft length and transitional curvature to achieve hemodynamic characteristics that promote failure-free bypass conduits.
Notes
Acknowledgments
The study was partially supported through NSF CAREER 0954465 and Pennsylvania Infrastructure Technology Alliance (PITA). The computational resources provided in part by Pittsburgh Supercomputing Center grant CCR080013. The authors would like to thank Gunay Orbay, MS for his valuable contributions in implementing the sketch-based anatomical shape editing progress.
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