Local Coronary Flow and Stress Distribution

Abstract

The global flow analysis over the entire vascular tree presented in the previous chapters is not separable from the local flow patterns as the former determines the boundary conditions for the latter. The changes in geometry of blood vessels during branching can lead to significant flow disturbances (e.g., flow separation, secondary flow, stagnation point flow, reversed flow, and/or turbulence) due to convective inertia (Asakura & Karino, 1990). These disturbed flows affect various local hemodynamic parameters, such as wall shear stress (WSS), wall shear stress spatial gradient (WSSG), and oscillatory shear index (OSI). It has been found that spatial and temporal WSS and WSSG can locally induce abnormal biological response, such as dysfunction of endothelial cells, monocyte deposition, elevated wall permeability to macromolecules, particle migration into the vessel wall, smooth muscle cell proliferation, microemboli formation, and so on (Kleinstreuer et al., 2001; Traub & Berk, 1998), which can lead to atherogenesis and progression of atherosclerosis.

Appendices

Appendix 1: Hemodynamic Parameters (Huo et al., 2007)

The Reynolds (Re), Womersley (α), and Dean (D_n) numbers are defined, respectively, in main trunk of LAD epicardial arterial tree as follows:

$$ \mathit{\operatorname{Re}}=\frac{\rho V\cdot D}{\mu } $$

(8.1)

$$ \alpha =R\sqrt{\frac{\omega \rho}{\mu }} $$

(8.2)

$$ {D}_n={\left(2\frac{R}{R_{\mathrm{curve}}}\right)}^{1/2}\cdot 4\mathit{\operatorname{Re}} $$

(8.3)

where V = V_min, V_max, or V_mean, R and D, ω, ρ, and μ represent minimum, maximum, or time-averaged velocity at the inlet of LAD arterial tree, radius and diameter of LAD, angular frequency of beating hearts, blood mass density, and viscosity, respectively.

At any point of 3D finite element model (FEM), the stress can be represented as a nine-component tensor ($\overline{\overline{\tau}}$), which can be written as follows:

$$ \overline{\overline{\tau}}=\left[\begin{array}{lll}{\tau}_{11}& {\tau}_{12}& {\tau}_{13}\\ {}{\tau}_{21}& {\tau}_{22}& {\tau}_{23}\\ {}{\tau}_{31}& {\tau}_{32}& {\tau}_{33}\end{array}\right]=2\mu \overset{=}{D}=\mu \left[\begin{array}{ccc}2\frac{\partial u}{\partial x}& \frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}& \frac{\partial u}{\partial z}+\frac{\partial w}{\partial x}\\ {}\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}& 2\frac{\partial v}{\partial y}& \frac{\partial v}{\partial z}+\frac{\partial w}{\partial y}\\ {}\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x}& \frac{\partial v}{\partial z}+\frac{\partial w}{\partial y}& 2\frac{\partial w}{\partial z}\end{array}\right] $$

(8.4)

where $\overset{=}{D}=0.5\cdot \left[\left(\nabla \mathbf{v}\right)+{\left(\nabla \mathbf{v}\right)}^{\mathrm{T}}\right]$ is the shear rate tensor. The stress on the wall, its normal component, and its two tangential components can be written as, respectively:

$$ \overrightarrow{\tau}=\overline{\overline{\tau}}\cdot \mathbf{n},\kern1em {\tau}_n=\mathbf{n}\cdot \overline{\overline{\tau}}\cdot \mathbf{n},\kern1em {\tau}_{t_1}={\mathbf{t}}_1\cdot \overline{\overline{\tau}}\cdot \mathbf{n}\kern1em \mathrm{and}\kern1em {\tau}_{t_2}={\mathbf{t}}_2\cdot \overline{\overline{\tau}}\cdot \mathbf{n} $$

(8.5)

where n, t₁, and t₂ are the unit vectors in the normal and two tangential directions, respectively. The present time-averaged oscillatory shear index (OSI) can be written as follows:

$$ \mathrm{OSI}=\frac{1}{2}\left(1-\frac{\left|\frac{1}{T}{\int}_0^T\overrightarrow{\tau}\right|}{\frac{1}{T}{\int}_0^T\left|\overrightarrow{\tau}\right|}\right) $$

(8.6)

The spatial derivatives of the stress can be obtained as follows:

$$ \nabla \overrightarrow{\tau}=\left[\begin{array}{lll}\frac{\partial {\tau}_n}{\partial n}& \frac{\partial {\tau}_n}{\partial {t}_1}& \frac{\partial {\tau}_n}{\partial {t}_2}\\ {}\frac{\partial {\tau}_{t_1}}{\partial n}& \frac{\partial {\tau}_{t_1}}{\partial {t}_1}& \frac{\partial {\tau}_{t_1}}{\partial {t}_2}\\ {}\frac{\partial {\tau}_{t_2}}{\partial n}& \frac{\partial {\tau}_{t_2}}{\partial {t}_1}& \frac{\partial {\tau}_{t_2}}{\partial {t}_2}\end{array}\right] $$

(8.7)

where n, t₁, and t₂ are the natural coordinates as shown. As defined by Lei et al. (1996), the diagonal components $\frac{\partial {\tau}_{t_1}}{\partial {t}_1}$ and $\frac{\partial {\tau}_{t_2}}{\partial {t}_2}$ generate intracellular tension, which causes widening and shrinking of the cellular gap. The diagonal component $\frac{\partial {\tau}_n}{\partial n}$ can cause endothelial cells rotation, however, which may damage the endothelium. Hence, the WSSG is defined as follows:

$$ \mathrm{WSSG}={\left[\left(\frac{\partial {\tau}_n}{\partial n}\right)+\left(\frac{\partial {\tau}_{t_1}}{\partial {t}_1}\right)+\left(\frac{\partial {\tau}_{t_2}}{\partial {t}_2}\right)\right]}^{\frac{1}{2}} $$

(8.8)

The time-averaged wall shear stress spatial gradient (WSSG) can be written:

$$ \mathrm{time}\hbox{-} \mathrm{averaged}\kern0.17em \mathrm{WSSG}=\frac{1}{T}{\int}_0^T\mathrm{WSSG}\cdot dt $$

(8.9)

In order to plot the shear stress in the entire computational domain, (wall shear stress) WSS is determined as the product of viscosity (μ) and wall shear rate ($\dot{\gamma}$), which is defined as:

$$ \mathrm{WSS}=\mu \dot{\gamma}=\mu \left[\begin{array}{l}2\left({\left(\frac{\partial u}{\partial x}\right)}^2+{\left(\frac{\partial v}{\partial y}\right)}^2+{\left(\frac{\partial w}{\partial z}\right)}^2\right)+\left({\left(\frac{\partial u}{\partial y}\right)}^2+{\left(\frac{\mathrm{\partial v}}{\partial x}\right)}^2\right)\\ {}+\left({\left(\frac{\mathrm{\partial v}}{\partial z}\right)}^2+{\left(\frac{\partial w}{\partial y}\right)}^2\right)+\left({\left(\frac{\partial w}{\partial x}\right)}^2+{\left(\frac{\partial u}{\partial z}\right)}^2\right)\end{array}\right] $$

(8.10)

The time-averaged WSS and OSI can be written as follows:

$$ \mathrm{time}\hbox{-} \mathrm{averaged}\;\mathrm{WSS}=\frac{1}{T}{\int}_0^T\mathrm{WSS}\cdot dt $$

(8.11)

Equations (8.6) and (8.8–8.11) are used to calculate the OSI, WSSG, and WSS in the FEM model. In the FORTRAN program, the shear stress, spatial gradient of shear stress, and OSI for each FEM node is calculated. The values for WSS, WSSG, and OSI, however, are only considered on the endothelial surface of vessels.

Appendix 2: Correlation Between Wall Shear Stress (WSS) and Oscillatory Shear Index (OSI) (Huo et al., 2007)

Table 8.1 The exponent (β) and correlation coefficient (R²) for the power law relation (OSI ~ WSS^β) lateral to the junction orifice and opposite to the flow divider near bifurcations, where high oscillatory shear index (OSI) coincides with low wall shear stress (WSS)

Appendix 3: Hemodynamic Parameters and Atherosclerotic-Prone Region (Huo et al., 2007)

Table 8.2 Relationship between hemodynamic parameters and atherosclerotic-prone region in main trunk and primary branches of left anterior descending (LAD) epicardial arterial tree

Appendix 4: Computational Fluid Dynamics in a Compliant Coronary Artery (Huo et al., 2009)

Governing Equations and Boundary Conditions

The governing equations are formulated for an incompressible, Newtonian fluid. The equations of Continuity and Navier–Stokes can be written as:

$$ \nabla \cdot \mathbf{v}=0 $$

(8.12)

$$ \rho \frac{\partial \mathbf{v}}{\partial t}+\rho \mathbf{v}\cdot \nabla \mathbf{v}=-\nabla P+\nabla \cdot \mu \left(\nabla \mathbf{v}+{\left(\nabla \mathbf{v}\right)}^{\mathrm{T}}\right) $$

(8.13)

where v, P, ρ, and μ represent the velocity vector, pressure, blood density, and viscosity, respectively. Equations (8.12) and (8.13) are solved for velocity and pressure given appropriate boundary and initial conditions. Because the moving wall boundary due to vessel compliance is considered, the 3D mesh moving technique is used to solve the moving fluid boundaries. The mesh is updated by solving a Laplace problem which provides the displacement (s = x ⋅ e₁ + y ⋅ e₂ + z ⋅ e₃) of each point as follows:

$$ \nabla \cdot \left(\nabla \mathbf{s}\right)=0 $$

(8.14)

After Eq. (8.14) is solved, the nodal velocity (v_node) due to the transient changes of meshes, is calculated. The convective term in Eq. (8.13) is changed to ρ(v + v_node) ⋅  ∇ v, based on the ALE frameworks.

The solution of governing equations in the right coronary artery (RCA) is determined subject to the following boundary conditions:

$$ {\mathbf{v}}_{\mathbf{inlet}}={\mathbf{v}}_{\mathbf{measured}\kern0.17em \mathbf{flow}\kern0.17em \mathbf{velocity}\;\mathbf{at}\;\mathbf{the}\kern0.17em \mathbf{inlet}} $$

(8.15)

$$ {\mathbf{v}}_{\mathbf{wall}}={\mathbf{v}}_{\mathbf{measured}\kern0.17em \mathbf{wall}\kern0.17em \mathbf{velocity}} $$

(8.16)

$$ {\mathbf{v}}_{\mathbf{outlet}}={\mathbf{v}}_{\mathbf{estimated}\kern0.17em \mathbf{flow}\kern0.17em \mathbf{velocity}\;\mathbf{at}\;\mathbf{the}\kern0.17em \mathbf{outlet}} $$

(8.17)

Equations (8.15)–(8.17) are for inlet flow (v_inlet), moving wall (v_wall), and outlet flow (v_outlet) boundary conditions, respectively. The inlet flow and moving wall boundary conditions are obtained from experimental measurements. The outlet flow boundary conditions are estimated based on scaling laws (Huo & Kassab, 2009a; 2009b; Kassab, 2006).

Method of Solution

Equations (8.12) and (8.13) are solved using the finite element (FE) method described in our previous studies (Huo & Li, 2004, 2006). Briefly, the computational domain of blood flow is first divided into small elements. With each element, the dependent variables v and P are interpolated by the shape functions, ϕ(x) and ψ(x) (x is the coordinate), as:

$$ {\mathbf{v}}^m\left(\mathbf{x},t\right)={\phi}^{\mathrm{T}}\left(\mathbf{x}\right){\mathbf{V}}^m(t);\kern1em \mathbf{P}\left(\mathbf{x},t\right)={\psi}^{\mathrm{T}}\left(\mathbf{x}\right)\mathbf{P}(t) $$

(8.18)

where V^m(t) and P(t) (superscript m denotes the mth component of velocity vector) are the unknown column vectors at each element nodal point. The matrix form of the discretized equations may be written as follows:

$$ \left[\begin{array}{cc}\mathbf{M}& 0\\ {}0& 0\end{array}\right]\left[\begin{array}{l}\dot{\mathbf{V}}\\ {}\dot{\mathbf{P}}\end{array}\right]+\left[\begin{array}{cc}\mathbf{A}\left(\mathbf{V}\right)+\mathbf{K}& -\mathbf{C}\\ {}-\mathbf{C}& \mathbf{0}\end{array}\right]\left[\begin{array}{l}\mathbf{V}\\ {}\mathbf{P}\end{array}\right]=\left[\begin{array}{l}{\mathbf{F}}_{\mathbf{1}}\\ {}{\mathbf{F}}_{\mathbf{2}}\end{array}\right] $$

(8.19)

where the global matrices M, A(V), C, and K are assembled from the following element matrices

$$ {\mathbf{M}}^e={\int}_{\Omega^e}{\rho \phi \phi}^{\mathrm{T}} dV;\kern1em {\mathbf{A}}^e={\int}_{\Omega^e}\rho \phi \left(\mathbf{v}+{\mathbf{v}}_{\mathbf{node}}\right)\cdotp \nabla {\phi}^{\mathrm{T}} dV;\kern1em {\left({\mathbf{C}}^e\right)}_j={\int}_{\Omega^e}{e}_j\cdotp \nabla {\phi \psi}^{\mathrm{T}} dV $$

and

$$ {\left({\mathbf{K}}^{\mathbf{e}}\right)}_{ij}=\left({\int}_{\Omega^e}\mu \nabla \phi \cdot \nabla {\phi}^{\mathrm{T}} dV\right){\delta}_{ij}+{\int}_{\Omega^e}\mu \left({\mathbf{e}}_j\cdotp \nabla \phi \right)\left({\mathbf{e}}_j\cdotp \nabla {\phi}^{\mathrm{T}}\right) dV $$

where Ω^e is the computational domain of an element, e_i (i = 1, 2, 3) is a unit vector in the ith direction. The right-hand side of Eq. (8.19) (F₁ and F₂) is obtained from the inlet and outlet conditions, Equations (8.15) and (8.17), respectively, which is iteratively updated at every time step in the transient computation.

With each element, the displacement variable s is interpolated by the shape functions, s(x) = φ(x)S. The matrix form of the discretized Eq. (8.14) may be written as follows:

$$ \left({\delta}_{ij}{\int}_{\Omega^e}\mu \nabla \varphi \cdot \nabla {\varphi}^{\mathrm{T}} dV\right)\cdot \mathbf{S}={\mathbf{F}}_{\mathbf{3}} $$

(8.20)

where F₃ is determined by Eq. (8.16). The nodal velocity (v_node) due to the transient changes of meshes is calculated in each node, which is incorporated in the convective term A(V). The assembled global matrix equations are stored and solved using the LU decomposition with partial pivoting and triangular system solvers through forward and backward substitution (Superlud_dist, which is implemented in ANSI C, and MPI for communications) in the BigRed Cluster of Indiana University, which is a 1024-node (4096-processors) distributed shared-memory cluster, designed around IBM’s BladeCenter JS21.

The fluid calculation with moving wall is solved iteratively using the backward method. A new mesh is generated from Eq. (8.20) at the next time and the velocity v_node is calculated at each nodal point and Eq. (8.19) is then solved for the fluid flow field. The iteration continues until all the variables converge within a preset tolerance (relative error of flow velocity <1 × 10⁻⁴).

Appendix 5: Transmural Stress Distribution in Pseudo-Elastic Model (Zhang et al., 2004)

The behavior of the arterial wall is assumed to obey an orthotropic, pseudo-elastic, incompressible, exponential model proposed by Fung (Chap. 4). The pseudo-strain energy density $\overline{W}$ can be written as:

$$ \overline{W}=\frac{c_0}{2}\exp \left\{{c}_1{\overline{E}}_r^2+{c}_2{\overline{E}}_z^2+{c}_3{\overline{E}}_{\theta}^2+2\left({c}_4{\overline{E}}_r{\overline{E}}_z+{c}_5{\overline{E}}_z{\overline{E}}_{\theta }+{c}_6{\overline{E}}_{\theta }{\overline{E}}_r\right)\right\} $$

(8.21)

where ${\overline{E}}_r$, ${\overline{E}}_z$, and ${\overline{E}}_{\theta }$ represent the normalized Green strains in the radial, longitudinal and circumferential directions, respectively. These strains are related to the normalized stretch ratios by

$$ {\overline{E}}_i=\frac{1}{2}\left({\overline{\lambda_i}}^2-1\right)\kern1em \left(i=r,z,\theta \right) $$

(8.22)

The normalized stretch ratios are defined as the corresponding principal stretch ratios (λ_r, λ_z, λ_θ), divided by the cubic root of the third invariant (J = λ_rλ_zλ_θ) of the deformation gradient as:

$$ {\overline{\lambda}}_r={\lambda}_r{J}^{-1/3},\kern1em {\overline{\lambda}}_z={\lambda}_z{J}^{-1/3},\kern1em {\overline{\lambda}}_{\theta }={\lambda}_{\theta }{J}^{-1/3} $$

(8.23)

The Cauchy stress tensor and the Hencky (logarithmic) strain tensor are the proper measures of true stress and strain for large deformations (Atluri, 1984). ANSYS uses these mechanical measures and also provides them in the user subroutine USERMAT through which we implemented Fung’s pseudo-strain energy function as described below. The Cauchy stress tensor is conjugate to the Hencky strain tensor for isotropic materials. For anisotropic materials, however, the stress–strain relation should be properly established between the second Piola–Kirchhoff (P-K) stress and the Green-Lagrange strain. The second P-K stresses are given by the derivatives of strain energy function. The principal logarithmic strains are defined as:

$$ {\varepsilon}_i=\ln {\lambda}_i\kern1em \left(i=r,z,\theta \right) $$

(8.24)

The Cauchy stresses, with the incompressibility of blood vessels being taken into account by the method of a Lagrangian multiplier, can be expressed by:

$$ {\sigma}_{ij}=\frac{\rho }{\rho_0}\frac{\partial {x}_j}{\partial {X}_{\alpha }}\frac{\partial {x}_i}{\partial {X}_{\beta }}\frac{\partial \overline{W}}{\partial {E}_{\beta \alpha}}+\eta \frac{\partial \varphi }{\partial {\varepsilon}_{ij}}\kern1em \left(i,j,\alpha, \beta =r,z,\theta \right) $$

(8.25)

where x_i and X_α denote coordinates, and ρ and ρ₀ represent densities, in the deformed and reference states, respectively. Here, φ(J) = (J − 1)² is selected as the penalty function to impose the volume-conserving constraint (Atluri & Reissner, 1989). The parameter η reflects the incompressibility of the material.

For ideally incompressible materials, J = 1 and ρ = ρ₀, and the above equations correspond to those presented by Chuong and Fung (1986). The material constants summarized in Chap. 4 are selected. The constant c₀ scales all the stresses at certain strains. In other words, c₀ represents the overall rigidity of the vessel wall. The constants c₁, c_2, and c₃ reflect the relative elasticity in the principal directions, i.e., the mechanical anisotropy of the vessel wall. The constants c₄, c_5, and c₆ reflect the Poisson’s ratio of the vessel wall. A relatively large penalty parameter η = 1000 kPa is adopted. The density of the blood vessel wall is assumed to be ρ = 1100 kg/m³.

Rachev et al. (1996) showed that under the assumption of incompressibility, and based on the finite deformation theory, the principal stretch ratios can be expressed as:

$$ {\lambda}_r=\frac{R}{\lambda_z\chi r},\kern1em {\lambda}_z=\frac{z}{Z},\kern1em {\lambda}_{\theta }=\frac{\chi r}{R} $$

(8.26)

where R denotes the radius of an arbitrary point at zero-stress state (an open sector), r is the radial coordinate in the deformed configuration, χ = π/(π − Φ) is a factor depending on the opening angle Φ defined as the angle subtended by two radii connecting the midpoint of the inner wall. The relationship between the undeformed coordinate and the deformed (no-load) coordinate can be written as:

$$ R=\sqrt{\chi \left({r}^2-{r}_i^2\right)+{R}_i^2} $$

(8.27)

Given Eqs. (8.24), (8.26), and (8.27), the distributions of residual strains at the no-load state are computed. The constitutive model is coded and linked to ANSYS through user subroutine USERMAT as outlined below.

Using Equations (8.22)–(8.24) and ${E}_i=\left({\lambda}_i^2-1\right)/2$, the principal Cauchy stresses can be expressed in terms of normalized Green strains as:

$$ {\sigma}_r={\overline{\lambda_r}}^2\frac{\partial \overline{W}}{\partial {\overline{E}}_r}+\eta \xi J={c}_0\left(1+2{\overline{E}}_r\right)\left({c}_1{\overline{E}}_r+{c}_4{\overline{E}}_Z+{c}_6{\overline{E}}_{\theta}\right){e}^{\overline{Q}}+\eta \xi J $$

(8.28)

$$ {\sigma}_z={\overline{\lambda}}_z^2\frac{\partial \overline{W}}{\partial {\overline{E}}_z}+\eta \xi J={c}_0\left(1+2{\overline{E}}_z\right)\left({c}_2{\overline{E}}_z+{c}_4{\overline{E}}_r+{c}_5{\overline{E}}_{\theta}\right){e}^{\overline{Q}}+\eta \xi J $$

(8.29)

$$ {\sigma}_{\theta }={\overline{\lambda}}_{\theta}^2\frac{\partial \overline{W}}{\partial {\overline{E}}_{\theta }}+\eta \xi J={c}_0\left(1+2{\overline{E}}_{\theta}\right)\left({c}_3{\overline{E}}_{\theta }+{c}_5{\overline{E}}_z+{c}_6{\overline{E}}_r\right){e}^{\overline{Q}}+\eta \xi J $$

(8.30)

where $\xi ={\varphi}^{\prime }(J)=2\left(J-1\right)=2\left[\sqrt{\left(1+2{E}_r\right)\left(1+2{E}_Z\right)\left(1+2{E}_{\theta}\right)}-1\right]$. The shear stress is calculated as follows:

$$ {\sigma}_{rz}=\frac{\lambda_r{\lambda}_z}{2}\frac{\partial W}{\partial {E}_{rz}}={c}_0{c}_7\sqrt{\left(1+2{E}_r\right)\left(1+2{E}_z\right)}{E}_{rz}{e}^Q $$

(8.31)

The shear strain is assumed to be small in Eq. (8.31). Finally, the relationships ηξJ = 2η(J² − J) and J =  exp (ε_r + ε_z + ε_θ) can be expressed as:

$$ \frac{\partial \left(\eta \xi J\right)}{\partial {\varepsilon}_i}=2\eta \frac{\partial \left({J}^2-J\right)}{\partial {\varepsilon}_i}=2\eta \left(2J-1\right)\frac{\partial J}{\partial {\varepsilon}_i}=2\eta \left(2{J}^2-J\right)=\kappa $$

(8.32)

where κ is a scalar and i = r, z, or θ.

In the user subroutine USERMAT provided by ANSYS, the Jacobian matrix is defined between Cauchy stress and logarithmic strain. Considering the relationship between logarithmic strain and Green strain, as well as the relation between normalized and non-normalized Green strains, we can finally write the Jacobian matrix as:

$$ \left[\begin{array}{cccc}\left(1+2{\overline{E}}_r\right)\frac{\partial {\sigma}_r}{\partial {\overline{E}}_r}+\kappa & \left(1+2{\overline{E}}_z\right)\frac{\partial {\sigma}_r}{\partial {\overline{E}}_z}+\kappa & \left(1+2{\overline{E}}_{\theta}\right)\frac{\partial {\sigma}_r}{\partial {\overline{E}}_{\theta }}+\kappa & 0\\ {}\left(1+2{\overline{E}}_r\right)\frac{\partial {\sigma}_z}{\partial {\overline{E}}_r}+\kappa & \left(1+2{\overline{E}}_z\right)\frac{\partial {\sigma}_z}{\partial {\overline{E}}_z}+\kappa & \left(1+2{\overline{E}}_{\theta}\right)\frac{\partial {\sigma}_z}{\partial {\overline{E}}_{\theta }}+\kappa & 0\\ {}\left(1+2{\overline{E}}_r\right)\frac{\partial {\sigma}_{\theta }}{\partial {\overline{E}}_r}+\kappa & \left(1+2{\overline{E}}_z\right)\frac{\partial {\sigma}_{\theta }}{\partial {\overline{E}}_z}+\kappa & \left(1+2{\overline{E}}_{\theta}\right)\frac{\partial {\sigma}_{\theta }}{\partial {\overline{E}}_{\theta }}+\kappa & 0\\ {}0& 0& 0& \frac{\partial {\sigma}_{rz}}{\partial {\overline{E}}_{rz}}\end{array}\right] $$

(8.33)

where $\frac{\partial {\sigma}_{rz}}{\partial {\overline{E}}_{rz}}\approx \frac{\partial {\sigma}_{rz}}{\partial {\varepsilon}_{rz}}\approx \frac{\partial {\sigma}_{rz}}{\partial {E}_{rz}}\approx {c}_0{c}_7\sqrt{\left(1+2{E}_r\right)\left(1+2{E}_z\right)}{e}^Q$. The shear strain is verified to be small. Hence, the above formulation is not applicable to problems involving large rotation or shear deformation.

In the user subroutine USERMAT, ANSYS provides the current logarithmic strains. The strains are used to calculate the stresses and the Jacobian matrix and are passed back to ANSYS for computation of new strains. The USERMAT is compiled and linked with ANSYS using a batch command provided in the ANSYS package to create a new executable. In the input file, the material model for the vessel wall is defined as a user material (to call user subroutine USERMAT).