Bypass graft (BG) surgeries require surgical construction of a conduit or graft over a blocked blood vessel. Coronary artery bypass graft (CABG) surgery is usually performed when one or more coronary arteries, that supply blood to the heart muscle, are fully or partially blocked due to the build up of atherosclerotic plaque. Approximately 500,000 CABG surgeries are performed annually in the U.S. Coronary grafts are typically harvested from native tissue such as saphenous vein from the leg, left/right internal thoracic artery or radial artery from the forearm, or harvested from cadavers, or (rarely) composed of synthetic materials such as poly tetra-fluoro ethylene (PTFE). Percutaneous coronary intervention (PCI) with angioplasty (inserting and inflating a balloon in the coronary artery) and stenting (inserting and expanding a permanent mesh structure) are alternative treatments, which may be chosen when progression of disease is less severe, the patient is not a candidate for surgery, or when indicated by other clinical factors.

The operative mortality rate for CABG is approximately 3%. Following surgery, the patency and longevity of a CABG depends on multiple factors including the type of conduit used, the local hemodynamic environment, and structural properties of the walls. It is known that arterial grafts such as internal mammary artery (IMA) and radial artery grafts provide, in general, better patency than vein grafts. Approximately 15–30% of saphenous vein grafts occlude within the first year of surgery, with the rate increasing to over 50% after 10 years.^{32} Therefore, typically arterial grafts are preferred. However, because multiple vessels are bypassed in most CABG surgeries, vein grafts are usually used as well.

The dominant complications post-CABG are restenosis^{36} and graft occlusion with associated myocardial infarction,^{5} and are attributed primarily to deposition of atherosclerotic plaque,^{11,28} which may subsequently rupture leading to thrombosis, and associated hypo-perfusion. Graft incompatibility, and hemodynamic factors such as blood recirculation, wall shear stress (WSS), and wall shear stress gradients (WSSG) play an important role in the onset and development of plaque (atherogenesis).^{2,22,23,26,27,38,54} Because these factors are difficult to predict *a priori*, surgeons use intuition and geometric guidelines to make decisions about graft design, despite the well known links between hemodynamics and restenosis. While mean flow and pressure waveforms can be obtained from invasive coronary catheterization, local hemodynamic information cannot be obtained reliably using currently available methods. Hence, patient-specific modeling and accurate numerical simulations provide a means to obtain data on hemodynamics and WSS that is currently unavailable.^{15,34,50}

It has been shown experimentally (*in vivo*) that anastomosis angle effects flow fields^{47} and that flow patterns effect graft patency.^{19} Using three angles, Rittgers *et al.*^{39} did not observe a significant correlation between angle and IH on external iliac arteries for dogs. Using three *in vivo* patient-specific cases, Giordana *et al.*^{12} showed that 80° is better than 30° (since platelet activation is reduced with increased flow mixing) and the low angles studied are predisposed to occlude earlier. These *in vivo* do not present a standardized or holistic case study to recommend a specific anastomosis angle and further experiments are still needed.^{13} Earlier computational studies of the influence of CABG on mechanical environment have largely focussed on idealized geometrical models, simplifying physical assumptions, and/or a limited trial-and-error approach to identify BG shapes that improve outcome.^{1} In our previous work, we studied the influence of anastomosis angles and graft radius on flow in idealized BG models^{41} demonstrating the effect of geometrical parameters on local hemodynamics. Using regions of low WSS as a measure of undesirable (stagnant) flow patterns, we observed that these regions correlated with three known locations of BG failure (toe, heel, and arterial floor region) and that they could be minimized by changing the graft anastomosis angles.^{31,41} We also showed that uncertainty quantification^{42} influences the outcome of surgical optimization. Holzapfel *et al.*^{14} showed the interplay between BG shape, suture length, residual stresses and the structural stresses induced by CABG. They showed that small variations of the arterial incision have big effects on the size of the arterial opening and thereby, anastomosis angles and WSS.

Modeling flow in the coronary arteries is particularly challenging due to the influence of ventricular contraction, making traditional boundary conditions inappropriate. Sankaranarayanan *et al.*^{43} used constant pressure boundary conditions to study flow in an idealized CABG. This requires knowledge of patient-specific coronary pressure information, which is not usually available from standard clinical data. Pressure boundary conditions are also limited in their capability, since they cannot be used to predict a post-surgical outcome. Dur *et al.*^{8} investigated surgical planning for CABG through shape optimization, but their studies were limited to hemodynamic efficiency. Recently, Kim *et al.*^{20,21} developed an algorithm that models coronary circulation and microcirculation using a lumped parameter network (LPN). In this work, a set of ODEs describing the coronary LPN was solved analytically to prescribe the relation between flow and pressure at the coronary boundary faces. Inlet boundary conditions were switched from Neumann to Dirichlet when the aortic valve was closed, and the authors employed a penalty method assuming a parabolic flow profile to prevent divergence due to back flow. Here, we build on this previous work by employing an implicit numerical approach to solve the coupled PDE/ODE system. Using a finite element framework (FEM),^{51} we simultaneously solve the coupled boundary condition problem, modeling the effect of heart behavior and contraction in a closed loop network. This work was also motivated by the closed loop LPN models of Migliavacca and colleagues,^{18,24,33} which have been applied to model single ventricle physiology. This technique generalizes the computational framework for cases in which analytical solutions are not possible, and eliminates the need to switch between two different boundary conditions (i.e., Dirichlet and Neumann) for the same surface.

In this work, we propose a general cost function for CABG design, but point out that the choice of cost function can change with the acquisition of future experimental data. The use of a formal cost function is meant to set the stage for future work using optimization algorithms for shape design. A cost function can be chosen for BG design using WSS, oscillatory shear index (OSI), gradient oscillatory number (GON), aneurysmal flow index (AFI), or a combination of them. Our cost function is based on a simple linear combination of various risk factors for IH, and further experimental studies are needed to refine this choice. Though exact numerical values that correspond to diseased states are not fully understood, experimental studies can provide guidance on acceptable values.^{7,29} WSS values between 4 and 15 dynes/cm^{2} are, in general, acceptable. In coronary arteries, WSS between 10 and 25 dynes/cm^{2} are considered normal.^{40} WSSG values of 400 dynes/cm^{2} or higher are considered abnormal leading to excessive cell proliferation. Values of oscillation in WSS over 5 dynes/cm^{2} are undesirable, which roughly translates to values of OSI greater than 0.25. Acceptable ranges for other derived variables, GON and AFI, have not yet been experimentally determined. We also develop tools to automatically post-process hemodynamic quantities and test different graft designs for the CABG surgery.

The three goals of this paper are (1) to present a computational framework that could be used (with proper validation) in future work to predict coronary flow based solely on non-invasive clinical measurements, (2) to investigate the dependence of local hemodynamics on anastomosis angle, and (3) to compare the pre- and post-surgical flow conditions using multiscale modeling. The paper is organized as follows. In “Patient-Specific Simulations of Coronary Flow” section, we provide details of patient-specific geometric modeling. We then provide details of boundary condition implementation and post-processing. In “Parameterizing Surgical Geometry” section, we describe a spline-based method to parameterize surgical geometry. Finally, we present patient-specific modeling results.