Coupled Simulation of Hemodynamics and Vascular Growth and Remodeling in a Subject-Specific Geometry

Abstract

A computational framework to couple vascular growth and remodeling (G&R) with blood flow simulation in a 3D patient-specific geometry is presented. Hyperelastic and anisotropic properties are considered for the vessel wall material and a constrained mixture model is used to represent multiple constituents in the vessel wall, which was modeled as a membrane. The coupled simulation is divided into two time scales—a longer time scale for G&R and a shorter time scale for fluid dynamics simulation. G&R is simulated to evolve the boundary of the fluid domain, and fluid simulation is in turn used to generate wall shear stress and transmural pressure data that regulates G&R. To minimize required computation cost, the fluid dynamics are only simulated when G&R causes significant vascular geometric change. For demonstration, this coupled model was used to study the influence of stress-mediated growth parameters, and blood flow mechanics, on the behavior of the vascular tissue growth in a model of the infrarenal aorta derived from medical image data.

Appendix

Algorithm for coupled simulation of G&R and hemodynamics

• (1) Generate initial homeostatic state through accelerated G&R.

• (2)

  • (a) Introduce initial mass loss to the homeostatic state.

  • (b) Simulate blood flow in the initial geometry and obtain initial WSS field $\tau _w(\mathbf{x}, t)$.

  • (c) Set initial values for mass production rate $m_i \rightarrow m^k(t)$.

• (3) Formulate the weak form for displacement $\mathbf{u}=(u, v, w)$ from the virtual work principle,

  $$\begin{aligned} \delta I=\int _S\delta w dA-\int _s P\mathbf{n}\cdot \delta \mathbf{x}da=0 \;, \end{aligned}$$

  and solve using a nonlinear FEM solver.

• (4)

  • (a) Generate stress measure field $\sigma ^k(\mathbf{x}, t)$ based on deformation and constitutive relations

  • (b) Update mass production rate:

    $$\begin{aligned} m^k(t)=\frac{M(t)}{M(0)}\left( K_{\sigma }\left( \sigma ^k(t)-\sigma ^h \right) -K_{\tau }\left( \tau _w(t)- \tau ^h_w \right) + \tilde{f_h}\right) \end{aligned}$$

• (5) If $\left\| m^k(t)-m_i \right\| /\left\| m_i\right\| <\text{ tolerance }$ (in this work, tolerance$=0.001$),

  • (a) Set $m_i \leftarrow m^k(t)$

  • (b) Set $t \leftarrow t+\Delta t$

  • (c) Update values for remaining fractions $Q^k(\cdot )$ and $q^k(\cdot )$

  • (d) Go to Step (6)

  Else,

  • (a) Set $m_i \leftarrow m^k(t)$

  • (b) Go to Step (3)

• (6) If the geometric change is significant for fluid domain,

  • (a) Solve blood flow in the new geometry

  • (b) Update WSS field $\tau _w(\mathbf{x}, t)$ and pressure $P(\mathbf{x}, t)$

  • (c) Go to Step (3)

  Else, go to Step (3)