## Abstract

Compliance mismatch between the graft and the host artery of an end-to-side (ETS) arterial bypass graft anastomosis increases the intramural stress in the ETS graft–artery junction, and thus may compromise its long-term patency. The present study takes into account the effects of collagen fibers to demonstrate how their orientations alter the stresses. The stresses in an ETS bypass graft anastomosis, as a man-made bifurcation, are compared to those of its natural counterpart with different fiber orientations. Both of the ETS bypass graft anastomosis and its natural counterpart have identical geometric and material models and only their collagen fiber orientations are different. The results indicate that the fiber orientation mismatch between the graft and the host artery may increase the stresses at both the heel and toe regions of the ETS anastomosis (the maximum principal stress at the heel and toe regions increased by 72% and 12%, respectively). Our observations, thus, propose that the mismatch between the collagen fiber orientations of the graft and the host artery, independent of the effect of the suture line, may induce aberrant stresses to the anastomosis of the bypass graft.

## 1 Introduction

Development of intimal thickening (IT) considerably limits the longevity of arterial bypass grafts. In the end-to-side (ETS) anastomosis of a bypass graft, IT occurs in selected regions including the arterial floor, the heel and the toe of the anastomosis, and the suture line [13]. A fully controlled in vivo experimental study [4] investigated the effect of compliance mismatch on the end-to-end (ETE) graft patency. The results showed that the level of circumferential compliance mismatch can influence the graft patency. By conducting in vivo experiments on an iliac artery of a canine, it was found that compliance mismatch has a marginal effect on the development of neointimal hyperplasia in a canine model of ETE bypass graft anastomosis [5]. For an ETS arterial bypass graft anastomosis, at least two different types of IT in the anastomosis region have been reported [6]: (i) suture line IT, which is related to vascular healing and intramural stresses, and (ii) arterial floor IT, which develops in regions with abnormal flow conditions. Therefore, the disrupted flow is responsible for initiation and aggravation of the disease on the floor, while in the graft–artery junction, the mechanical response of the vessel wall as well as the injury induced by the surgical process is the major reason for triggering IT.

Effect of compliance mismatch as a predictive factor of distal anastomotic intimal hyperplasia (DAIH) formation has also been studied in vivo [7]. Considering the effect of suture line on the structural stresses of the ETE and ETS anastomoses, a computational study has revealed that although compliance mismatch between the graft and the host artery can elevate stresses around graft-artery junctions of both bypass models, in the ETE model, unlike the ETS, the maximum anastomotic stress is not a function of graft compliance mismatch [8]. Therefore, the increase of compliance mismatch may exacerbate the DAIH in ETS bypass graft anastomoses, while its influence on ETE DAIH is insignificant.

Blood vessels are complex fiber-reinforced composite structures and their mechanical behavior is heavily dominated by the distribution and concentration (architecture) of their various components including elastin and collagen. Elastic fibers are interlinked and create a rubber-like (network of) material, which is responsible for the isotropic behavior of blood vessels at low stresses. Collagen molecules adhere to each other and build collagen fibrils and fibers, and unlike elastin, the collagen fibers are responsible for the anisotropic behavior of blood vessels. Due to the arterial pressure, during the cardiac cycle, the blood vessels endure high levels of stresses, and the collagen fibers and their orientations are thus the major determinants of the load-carrying capability of blood vessels.

In all aforementioned studies, the vessel walls were treated as an isotropic material. However, although at lower stress values, the blood vessels behave (almost) isotropically [9], they exhibit a strongly anisotropic behavior at higher stress levels due to their collagen content. The structure of the ETS anastomosis, the residual stresses, and the anisotropic and nonlinear vessel behavior have been considered in a simulation of coronary artery bypass graft (CABG) surgery [10]. Yet, to the best of our knowledge, no systematic study has been devoted to focus on the collagen fiber orientations.

The objective of this study is to investigate the effect of collagen fiber orientations on the intramural stresses of the ETS bypass graft anastomoses. To this end, two simulations were carried out. In both of the simulations, the same geometric and material models were used. It was assumed that the material of the graft is akin to that of the host artery. The only difference between these two simulations is the way in which their collagen fiber orientations were set. In the first simulation, these fiber orientations were set as they are in a native coronary artery bypass graft anastomosis, while in the second simulation, the fiber orientations were modified so that the related effect can be investigated. Microstructural characteristics, and fiber orientation mismatch in particular, should be contemplated as an important parameter for the bypass graft design. For the second simulation, by getting inspiration from the collagen fiber orientations of native bifurcations in the human circulatory system, the collagen fiber orientations of the bypass graft anastomosis were set. For this reason, the collagen fiber orientations of a natural coronary artery bifurcation was considered [11].

In Sec. 2.1, the geometrical model of the coronary artery bypass graft anastomosis and the related dimensions are described. In Sec. 2.2, the required governing equations and the utilized computational procedure including the convergence criteria, discretization method, and the approach used to solve these equations are explained. Sections 2.3.1 and 2.3.2 are devoted to describe the material model employed to simulate the mechanical behavior of the arterial wall, i.e., the graft and the host artery. The mathematical model of the blood and the initial and boundary conditions used in the computational study are provided in Secs. 2.3.3 and 2.4, respectively. The obtained results are presented in Sec. 3, while in Secs. 4 and 5, a critical discussion is added and the study is concluded.

## 2 Materials and Methods

### 2.1 Geometry of a Coronary Artery Bypass Graft.

Using itk-snap [12], a patient-specific geometry of the anastomosis of a CABG was reconstructed from a series of computed tomography angiography images (Fig. 1(a)). The data set was a courtesy of a retrospective study, extracted with the permission of research group with prior consent agreement of the patient. The length of the proximal section of the host artery, located between the stenosis and the anastomosis, and the length of the graft were 5.5 and 55 mm, respectively. Thickness of the blood vessels were assumed to be 0.9 mm considering the overall thicknesses of intima, media, and adventitia extracted from Ref. [13]. This patient-specific geometry of a CABG was employed as the geometrical model for both of the computational simulations carried out in this investigation.

Fig. 1
Fig. 1
Close modal

### 2.2 Governing Equations and Numerical Approach.

The blood flow was simulated by using the two-way fluid–structure interaction (FSI) method. The three-dimensional, time-dependent, blood flow was simulated employing the ansyscfx software (ANSYS, Inc., Natick, MA), which uses a hybrid finite element and finite volume approach to discretize the Navier–Stokes equations and to solve the continuity and momentum equations of the fluid. Applying Gauss's theorem and the Leibniz rule, the related governing equations become
$ddt∫V(t)ρdV+∫Sρ(Uj−Wj)dnj=0$
(1)
$ddt∫V(t)ρUidV+∫Sρ(Uj−Wj)Uidnj=−∫Spdni+∫Sμ(∂Ui∂xj+∂Uj∂xi)dnj$
(2)
which are the continuity and momentum equations, respectively. Herein, ρ denotes the blood density (assumed to be 1060 kg/m3), Uj are the components of the velocity vector, Wj are the components of the velocity vector of each control volume, dnj are the differential Cartesian components of the outward normal surface vector, p is the fluid pressure, and μ denotes the apparent fluid viscosity. For the structural part, the governing equation is
$[M][u¨(t)]+[K][u(t)]=[F(t)]$
(3)

where [F] is the load vector, and [M], [K], [u], and $[u¨]$ are the structural mass matrix, the stiffness matrix, the nodal displacement vector, and the acceleration vector, respectively.

The mesh generation process was performed using ansysmeshing. Three layers of higher order solid elements, which are well suited for problems involving hyperelasticity and large deflection, are used through the wall's thicknesses. Due to the complexity of the arterial geometry, the fluid domain was discretized using tetrahedral elements, and three layers of boundary elements were used near walls to assure the accuracy of the simulations. The computational domains for the structure and fluid are shown in Fig. 1(b). The ultimate number of elements selected for the computational domains of the fluid and the structure were 626,000 and 89,000, respectively, and the independence of numerical simulation from grid resolution was guaranteed.

The numerical procedure carried out to achieve the solution with desired accuracy is as follows: During the FSI simulations, using the initial values, Eqs. (1) and (2) are solved by the flow solver at the first time-step to obtain the flow field. The pressure and wall stress are then transferred to the structural solver in the form of force vectors. Equation (3) is then solved to obtain the wall displacements, which are then returned to the fluid solver to update the force vectors. This process is recursively repeated till the specified convergence criteria (explained bellow) are met, and the solver then proceeds to the next time-step (see, e.g., Eladi et al. [14]).

The advection scheme was set to “high resolution,” which uses the second-order upwind scheme whenever possible (preventing numerical diffusion) and reverts to the first-order upwind scheme (or backward Euler) if a bounded solution is required. The transient scheme was set to the “second-order backward Euler,” which is second-order accurate in time.

The convergence criterion of the simulations was based on the residual mean square of principal variables targeted to be 10−5. The total time was set to 2.7 s (three cardiac cycles), and the maximum allowable time-step was 0.009 s. To obtain the results independent of temporal discretizations, each of the computational domains (fluid and structure) was separately evaluated, and then the smaller satisfactory time-step was used for the FSI simulation.

The independence of the result from the temporal step size and spatial discretization (sensitivity analysis) for the fluid simulation was assured plotting the velocity profiles in different cross section of the computational domain at different grid resolution and varying time steps. For the structural solver, same strategy was followed considering the distribution of the maximum principal stress contours.

### 2.3 Structural and Fluid Models.

We describe now the constitutive model for the vessel wall in addition to the constitutive parameters and how the element orientations of the ETS anastomosis are defined to help reveal their effects on its mechanical environment.

#### 2.3.1 Constitutive Model.

As mentioned above, in this study, it was assumed that the material of the graft and the host artery are the same. To model the structural behavior of the arterial walls, the following anisotropic strain-energy function Ψ was used [15]:
$Ψ=ΨJ+Ψg+∑i=1,2Ψfi$
(4)
where ΨJ is a penalty function, Ψg denotes an isotropic function related to the noncollagenous ground matrix, and Ψfi refers to the two embedded families of collagen fibers, which were calculated based on the following equations:
$ΨJ=KB2(J−1)2, Ψg=c2(I¯1−3)$
(5)
$Ψfi=k12k2{ exp {k2[κI¯1+(1−3κ)I¯4i−1]2}−1}, i=1,2$
(6)
and
$I¯1=trC¯, I¯4i=a0i⊗a0i:C¯, i=1,2$
(7)
Herein, $C¯=J−2/3C$ is the modified form of the right Cauchy–Green tensor C, with $detC¯≡1$, and J = (det C)1∕2 denotes the volume ratio. The quantities $I¯1$ and $I¯4i$, i =1, 2, are invariants [16], while KB = 25 MPa is the bulk modulus used here as a penalty parameter. In addition, c, k1, k2, and κ are additional model parameters, which are determined fitting the relations for the Cauchy stresses in the longitudinal (σzz) and circumferential (σθθ) directions to the experimental data (note that the dispersion parameter κ ∈ [0, 1/3] should be determined from images). These stresses were defined as
$σzz=λz∂Ψ∂λz, σθθ=λθ∂Ψ∂λθ$
(8)

where λz and λθ denote the stretches in the longitudinal and the circumferential direction, respectively, and Ψ is taken from Eq. (4).

The mean orientations of the two families of collagen fibers in the reference configuration within the material were described by the two unit vectors a0i, i =1, 2, and are assumed to be located in the tangential plane of the arterial wall [16]; hence, these vectors do not have a radial component. In a cylindrical polar coordinate system, the components of the direction vectors a01 and a02 have, in matrix notation, the forms
$[a01]=[0 cos γ sin γ], [a02]=[0 cos γ−sin γ]$
(9)

where γ is the angle between the collagen fibers (arranged in symmetrical spirals) and the circumferential direction, used here as a phenomenological (fitting) parameter.

To the best of our knowledge, there are no means to reveal how different layers of the graft wall would be connected to those of the host artery after performing a CABG. Therefore, an equivalent single-layer model for the arterial wall was used here and the model parameters for this single-layer vessel wall are then calculated. Accordingly, two strips of blood vessel tissues were used, one in the longitudinal and the other in the circumferential direction, each consisting of three layers (intima, media, and adventitia). The thickness of the intimal, medial, and adventitial layers, and the length and width of the specimens, extracted from the literature [13], are 0.24, 0.32, 0.34, 7.21, and 2.81 mm, respectively.

For each layer, the model parameters were obtained by fitting Eq. (8) to experimental data [13], extracted from the stretch–stress responses of different layers of human left anterior descending coronary artery (specimen IX). The data and the utilized fitting method, a standard nonlinear Levenberg–Marquardt algorithm, are presented in details in Ref. [13]. The vessel wall was modeled as a nearly incompressible material. The model responses for each layer of the blood vessel are shown in Fig. 2, and the values of the model parameters are summarized in Table 1.

Fig. 2
Fig. 2
Close modal
Table 1

Values of the constitutive parameters of the anisotropic model (c, k1, k2, κ, γ) for the adventitial, medial, and intimal layers of the arterial wall

c15.794.9254.8133.15
k1996.692.875748.623180.0
k22453.298.503814.00999.80
κ0.3090.3120.2820.226
γ80.7212.8629.5842.19
c15.794.9254.8133.15
k1996.692.875748.623180.0
k22453.298.503814.00999.80
κ0.3090.3120.2820.226
γ80.7212.8629.5842.19

First column on the right presents the coefficients of the equivalent single-layer model.

c and k1 (kPa), k2 and κ (dimensionless), γ (deg).

The numerical uniaxial tensile tests of the strip specimens in the longitudinal and circumferential directions were simulated by using the ansys finite element solver. Two rigid plates were attached to the ends of the strips. Fixing a plate at one end, the applied force was gradually increased at the other. The Cauchy stress–stretch curves for an equivalent single-layer vessel wall in the longitudinal and circumferential directions were calculated as follows: the Cauchy stress is σ = (F/A)λ, where F is the applied force, A is the cross-sectional area of the undeformed specimen, and λ is the corresponding stretch of the specimen, i.e., (L + d)/L, where L is the specimen length and d is the plate displacement. Finally, the constitutive parameters of the equivalent single-layer wall model were calculated based on the data obtained by the virtual tensile test (the numerical uniaxial tensile test) (Table 1). The related curves are shown in Fig. 2.

The anisotropic model was incorporated in ansys by using the USERMAT routine. To verify the performance of the USERMAT routine, the validity of the calculated constitutive parameters for the intimal, medial, and adventitial layers of the vessel, and the equivalent single-layer wall, a numerical uniaxial tension test was performed for the strip specimens of each of these layers; then the results were compared to the analytical response obtained from Eq. (8). A good agreement between the finite element simulations and the experimental data is observed (Fig. 2).

#### 2.3.2 Fiber Directions.

To assess the effect of the collagen fiber orientations on the wall stresses of the bypass graft anastomosis, the fiber orientations of the man-made bifurcation were set in two different ways. First, they were set as they are oriented in a real bypass graft anastomosis and in the second setup, their orientations were modified with the inspiration from a natural coronary artery bifurcation. To set the solid elements' orientations, the following procedure was followed: first, a surface guide was defined to provide a surface normal direction (N-vector) at a location of the selected surface, which is closest to each element's centroid and is aligned to the z-axis. Then a curve guide was defined to obtain tangential directions (T-vector) of each element. Subsequently, using the cross product of the T-vector and the N-vector, the direction of the second axis (the y-axis) was calculated. Finally, the third axis (x-axis) was obtained by the cross product of the second axis (y-axis) and the N-vector (z-axis). To set the element orientations of the bypass graft anastomosis, its geometry was divided into two parts: the bypass graft and the host artery (Fig. 3(a)). As each of these parts is a segment of a straight vessel, their centerlines and inner surfaces were defined as their curve and surface guides, respectively. Coronary artery bifurcation was selected as the representative example of a natural bifurcation. The element orientations of a typical natural bifurcation were extracted from several published studies [11,17,18]. To define the element orientations based on the collagen fiber orientations of the natural coronary artery bifurcation, the related geometry was divided into three parts, as can be seen in Fig. 3(b). The surfaces of each of these three parts and their centerlines were selected as their surface and curve guides. The element orientations for the bypass graft anastomosis before and after modification in the longitudinal direction are shown in Figs. 3(c) and 3(d), respectively. Based on the aforementioned instructions, it is obvious that the circumferential directions of the elements orientations are tangential to their surface guides and perpendicular to their longitudinal directions that are shown in Figs. 3(c) and 3(d), and hence for the sake of clarity, they are not depicted in these figures. Note that the material model used in this study depends on the angle between the two families of collagen fibers, and hence any change in this angle will change the material behavior. However, here, the collagen fiber orientation was modified wherever needed, while the mentioned angle was fixed.

Fig. 3
Fig. 3
Close modal

#### 2.3.3 Fluid Model.

Blood was assumed to be a homogenous, incompressible, and non-Newtonian fluid. The shear-thinning behavior of blood was modeled using the Carreau–Yasuda equation [19], i.e.,
$μ=μ∞+(μ0−μ∞)[1+(λ|γ˙|)a]n−1a$
(10)
where the viscosities at infinite and zero shear rates are μ = 2.2 × 10−3  Pa·s, μ0 = 22 × 10−3  Pa·s, respectively, the relaxation time, here λ, is 0.11 s, and two power indexes are a =0.644 and n =0.392, while the shear rate $γ˙$ involves the second invariant of the rate of deformation tensor defined as (see also Gijsen et al. [20])
$γ˙=γ˙ijγ˙ij/2$
(11)

### 2.4 Boundary and Initial Conditions.

A typical velocity profile of the CABG (with fully occluded host artery) with a time period of 0.9 s was applied at the entrance of the computational domains. The entrance was sufficiently extended far away from the anastomosis to attain a fully developed velocity profile and minimize the erroneous effects of boundary (Fig. 4). The outlet boundary condition for the fluid domain was set to “entrainment,” which is equivalent to the traction-free outflow boundary condition. This type of the outflow boundary condition has been shown to be optimal for the case that the flow direction is unknown, similar to the one investigated in this study. Hereby, the flow direction through the boundary was obtained by enforcing a zero velocity gradient perpendicular to the boundary. For the FSI simulations, a “fixed support” constraint was applied to the three ends of the model, including the graft inlet, the host artery outlet, and the proximal section of the host artery to restrict rotation, displacement, and deformation at boundaries.

Fig. 4
Fig. 4
Close modal

In the FSI simulations, it is very important to properly define the initial conditions, especially when the material is nonlinear. Hence, the steady-state FSI solution was employed here as a suitable initial condition for the transient simulations. The transmural pressure increased linearly from 0 to 100 mmHg to prestress the vessel walls.

## 3 Results

The effect of a variation in the collagen fiber orientations of the bypass graft anastomosis on its structural response and flow pattern has been investigated. It was found that although the fiber orientations can considerably affect the structural behavior of the bypass graft anastomosis inducing local stress concentration, their effect on the flow pattern and the wall shear stress are negligible. In this section, the wall stress distributions and the deformations were compared in the coronary artery bypass graft (ETS anastomosis) before modifying the collagen fiber orientations and also after that, during a cardiac cycle.

### 3.1 Wall Stress.

Contours of maximum principal stresses of the vessel walls for the bypass graft anastomosis (ETS anastomosis with real collagen fiber orientation) and the natural bifurcation (ETS anastomosis with modified collagen fiber orientation) cases at t =0.4 s are shown in Fig. 5. More specifically, Figs. 5(a) and 5(b), 5(c) and 5(d), and 5(e) and 5(f) show the distributions of the maximum principal stress at the heel, toe, and side walls, respectively, while the left column refers to the bypass graft anastomosis, the right one refers to the corresponding regions of the natural bifurcation.

Fig. 5
Fig. 5
Close modal

The maximum principal stress for the bypass graft anastomosis at the heel region is 72% higher than the natural bifurcation (2.59 × 10+5  Pa versus 1.51 × 10+5  Pa, Figs. 5(a) and 5(b)). The maximum principal stress for the bypass graft anastomosis at the toe region is also 12% higher than the natural bifurcation (1.2 × 10+5  Pa versus 1.07 × 10+5  Pa, Figs. 5(c) and 5(d)). Majority of regions with high stress are located near the suture line in the bypass graft anastomosis. No significant difference is observed between the maximum principal stress for the side walls of the bypass graft anastomosis and the natural bifurcation (Figs. 5(e) and 5(f)).

The time history of the maximum principal stress variations for the bypass graft anastomosis and the natural bifurcation during a cardiac cycle are shown in Fig. 6. The highest value of the principal stress for both the bypass graft anastomosis and the natural bifurcation occurs at t =0.4 s, which corresponds to a time at which the pressure difference between the inlet and outlet of the fluid domain possesses its maximum amount, while the minimum value happens at end diastole (t =0.85 s). The occurrence of maximum and minimum pressure differences at these time points is consistent with published literature [21]. The magnitude of the maximum principal stress for the bypass graft anastomosis varies between 2.5 × 10+5 and 2.6 × 10+5  Pa, while the values of this stress at the walls for the natural bifurcation changes between 1.46 × 10+5 and 1.51 × 10+5  Pa. Hence, the amplitude of variation in the values of the maximum principal stress for the bypass graft anastomosis, calculated based on the difference of maximum stresses normalized by the lower limit, is about 100% higher than that of the natural bifurcation (10 kPa versus 5 kPa).

Fig. 6
Fig. 6
Close modal

### 3.2 Wall Deformation and Strain.

Total wall deformations were also compared for the bypass graft anastomosis (ETS anastomosis with real collagen fiber orientation) and the natural bifurcation cases (Figs. 7(a) and 7(b)). The side wall of the bypass graft anastomosis experiences a relatively higher deformation when compared with the natural bifurcation. Variations in the maximum total wall deformation of the bypass graft anastomosis and those of the natural bifurcation during a cardiac cycle are shown in Fig. 7(c). As can be seen similar to maximum principal stresses, the maximum and minimum values of the total wall deformation occur at t =0.4 and 0.85 s, respectively. Moreover, the values of the total wall deformation of the bypass graft during the whole cardiac cycle are slightly higher than those of the natural bifurcation 5%. In the present study, the coronary artery bypass graft anastomosis experiences a maximum diameter variation of 4.4% at the time of the maximum flow rate. In a published study [22], the maximum diameter variation of human coronary arteries during a cardiac cycle has been reported to be about 2%. In addition, diameter variation of about 4% for a conventional ETS bypass graft anastomosis at the peak internal pressure has also been reported over a cardiac cycle [21]. Therefore, the obtained maximum diameter variation in the present study is consistent with the published literature.

Fig. 7
Fig. 7
Close modal

Contours of the maximum principal strain for the bypass graft anastomosis and the natural bifurcation (Fig. 8) show that, around the suture line of the bypass graft anastomosis, there is a region with high strain values. While, in the corresponding region of natural bifurcation, the value of strain is higher than the neighboring regions, it shows a more uniform distribution than that of the bypass graft anastomosis. Based on Figs. 8(c) and 8(d), the region with high strain values in the natural bifurcation exclusively resides in one surface guide which is labeled as “surface guide 2,” while in the bypass graft anastomosis, it occurs in both surface guides of the graft and the host artery. In the toe of the bypass graft anastomosis, there are regions with relatively higher strain values than the adjacent areas, which partially coincide with the location of suture line, and hence take place in two different surface guides, i.e., those of the graft and the host artery. In the corresponding region of the natural bifurcation, the magnitude of the strain gets higher than that of the bypass graft anastomosis; but, akin to the toe situation, here the whole region with high strain magnitude resides only in a different single surface guide, “surface guide 1” (Figs. 8(e) and 8(f)).

Fig. 8
Fig. 8
Close modal

## 4 Discussion

Higher intramural stresses in the vicinity of the suture line promote intimal hyperplasia (IH) formation in these regions, and ultimately lead to arterial bypass grafts failure [6,23]. There have been a number of investigations studying the effect of stent oversizing on resultant arterial wall stress concentration [24]. As a result, it was found that high intramural stresses experienced by the arterial wall after the stent implantation may cause neointimal hyperplasia.

Results of a computational study conducted to investigate the effect of stent mis-sizing on the distribution of hemodynamic parameters and the intramural stresses of coronary arteries showed that intramural wall stress concentration is correlated with the IH development [25]. Therefore, high intramural stresses were proved to lead to IH formation, and as a result, the fiber orientation mismatch, which, in the present study, is shown to increase the stress concentration on the heel and the toe of the bypass graft anastomosis, may also cause IH development in these areas, ultimately leading to the arterial bypass grafts failure.

Herein, the effects of collagen fiber orientations on the wall stresses and deformations of CABG (ETS anastomosis) were comprehensively investigated using numerical simulation. To reconstruct the model, in the anisotropic model introduced by Gasser et al. [15], a new set of constitutive parameters related to the intima, media, and the adventitia layers of human coronary arteries were calculated (Fig. 2 and Table 1). Furthermore, for the first time, the values of these parameters for an equivalent single-layer artery that mechanically is able to mimic the three layer coronary artery are introduced. This equivalent single-layer model not only can model the mechanical behavior of a physiologically realistic artery but also can reduce the computational costs effectively. Therefore, it is recommended to simulate the three-dimensional large-scale problems of the blood flow in compliant coronary artery networks using the equivalent single-layer arterial wall model.

The presence and orientation of such fibers in the structure of the blood vessels cause them to exhibit anisotropic behavior under physiological loading conditions. The behavior was simulated using a fiber dispersion model [15], which considers family of fibers that are distributed with rotational symmetry about a mean (preferred) reference direction. Hence, this family contributes a transversely isotropic character to the overall material response. However, according to Niestrawska et al. [26], we now know that the collagen fiber dispersion in human arterial layers in the tangential plane is more significant than that out of plane; hence, a rotationally symmetric dispersion model is not able to capture this distinction. The model by Holzapfel et al. [27] provides a nonsymmetric dispersion model using bivariate von Mises distributions. Such a model would be able to capture more realistically the fiber dispersion in human blood vessels. In the present study, the fiber orientations within the vessel walls of the bypass graft anastomosis were defined in two different ways to assess the effect of their variations on the mechanical response of the ETS bypass graft anastomosis. In the first way, according to the fact that bypass graft anastomoses are formed suturing two straight blood vessels together, the fiber orientations were defined to mimic the collagen fibers' orientations of a real bypass graft anastomosis. In the second, considering the fiber orientations of the coronary artery bifurcation (see Refs. [11,17,18]), as a natural bifurcation, the fiber orientations within the vessel walls of the anastomosis were defined so that from the fiber orientations' point of view, it can resemble a natural arterial branching. It should be noted that in both models, the geometries and the materials were identical and only the fiber orientations were altered to isolate their effects on structural performance. It is worth mentioning that in this study the fibers' orientations were modified wherever needed while fixing the angle between the two families of fibers, and thus the mechanical properties of the material of the tissues did not change before and after modifying their collagen fibers. Moreover, with the aim of conducting physiologically realistic simulations, the blood flow inside both the bypass graft anastomosis and the natural bifurcation was modeled using the two-way FSI approach, by which the dispensability and deformation of the arterial wall over the cardiac cycle were included.

Steep gradients of the maximum principal stress at the heel and toe regions of the bypass graft anastomosis with real collagen fibers' orientation were observed, while such changes are not noticeable in the corresponding regions of the natural bifurcation. As the mere difference between the two models was the collagen fiber orientations, the noticeable differences between the maximum principal stress distributions in the bypass graft anastomosis and the natural bifurcation cases could be attributed to the mismatch between the orientations of collagen fibers of the graft and the host artery. The variations in the values of maximum principal stress are not only steep in gradients, they are also more pronounced in the magnitude where up to twofold of increase in stress values are observed in the anastomosis case compared to the control case.

Although there is a considerable mismatch between the collagen fiber orientations on the side wall of the bypass graft anastomosis, where the surface guide of the graft is connected to that of the host artery (Fig. 3(c)), and also on the anastomosis side wall of the natural bifurcation (Fig. 3(d)), no considerable changes in the distribution of stress are visible in these regions. At a first glance, this may cast serious doubt over the observed stress rise at the heel and toe of the bypass graft anastomosis we attributed to the fiber orientation mismatch. However, the distribution of the maximum principal strain (Fig. 8) provides a compelling justification for this phenomenon. The tensile response of soft tissues is highly nonlinear so that at low stretch values, it is nearly linear and there is no difference between their response in the circumferential and longitudinal directions (Fig. 2). However, elements of arterial tissue exposed to higher stretch values exhibit a nonlinear behavior where their response in the longitudinal direction differs considerably from that of the circumferential direction. Therefore, only when experiencing sufficiently high strain values, the fiber orientation mismatch affects the distribution of stresses. The side wall of the natural bifurcation and that of the bypass graft anastomosis both experience considerably lower amount of strains compared to the heel and the toe of the ETS anastomosis and this is why no considerable change is observed in the distribution of principal stress on the side wall of the natural bifurcation and that of the bypass graft anastomosis.

It should be mentioned that although the side wall of bypass graft anastomosis undergoes a higher deformation compared to that of the natural bifurcation (Fig. 7(a)), it is not a consequence of aberrant stress distribution. This increase in wall deformation is mainly caused by the transitional displacement of wall elements rather than stress-induced deformation as they do not overlap with high values of strains in the side wall of the bypass graft anastomosis (Fig. 8(c)). It is also noteworthy that although the heel and toe regions of the natural bifurcation experience a maximum principal strain sufficiently high to affect the stress distribution, each region with high strain value resides in one surface guide, where there is no collagen fiber orientation mismatch. While in the bypass graft anastomosis, the regions with relatively high strain values co-locate with the location of the suture line where two different conterminous surface guides with two distinct curve guides are connected to each other and this creates mismatch between the collagen fiber orientation in this region. Therefore, contrary to the bypass graft anastomosis, in the mentioned regions of the natural bifurcation, there is no fiber orientation mismatch. Thus, the values of stress in these regions of the natural bifurcation do not increase as high as the bypass graft anastomosis.

## 5 Conclusion

Several factors may limit the longevity of coronary artery bypass grafts including aberrant flow conditions, compliance mismatch between the graft and the host artery, and a surgical injury. Although, majority of the literature accredited the abnormal flow conditions as the principal contributing factor to clinical events and graft failure, there are few pieces of evidence to underline the importance of factors concerning the wall stress. There is a rising interest in research community to dedicate further research to understand the mechanical behavior of bypass grafts anastomoses under physiological loading condition which may potentially help surgeons update their interventional guidelines and hence increase the CABG lifespan. In this study, the structural effect of collagen fiber orientations manifested in bypass graft surgeries was investigated using a two-way FSI method. The mismatch between the collagen fiber orientations in a bypass graft anastomosis not only increases the values of the maximum principal stress at the heel and toe, but it can also create regions with steep gradients of wall stress within those areas. This can also diminish the graft patency through modulating the vascular healing process after CABG and consequently, disturb the blood flow conditions. Therefore, in addition to the surgical injury, compliance mismatch, and abnormal flow conditions, we offer a mismatch of the collagen fiber orientations between the graft and the host artery as an important factor contributing to the failure of ETS anastomoses.

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