In silico experiments of intimal hyperplasia development: disendothelization in an axisymmetric idealized artery

Abstract

We use in silico experiments to study the role of the hemodynamics and of the type of disendothelization on the physiopathology of intimal hyperplasia. We apply a multiscale bio-chemo-mechanical model of intimal hyperplasia on an idealized axisymmetric artery that suffers two kinds of disendothelizations. The model predicts the spatio-temporal evolution of the lesions development, initially localized at the site of damages, and after few days displaced downstream of the damaged zones, these two stages being observed whatever the kind of damage. Considering macroscopic quantities, the model sensitivity to pathology-protective and pathology-promoting zones is qualitatively consistent with experimental findings. The simulated pathological evolutions demonstrate the central role of two parameters: (a) the initial damage shape on the morphology of the incipient stenosis, and (b) the local wall shear stresses on the overall spatio-temporal dynamics of the lesion.

Appendices

A Reproducibility data, computational costs and sensitivity analysis

For the sake of further studies, we present in this section the reproducibility data in Tables 2 and 3 and the computational costs of our two disendothelization models simulations.

Numerical simulations were carried out in parallel on twelve processors of the PMCS2I of École Centrale de Lyon. The simulations lasted 1.17 days and 3.96 days, respectively, for the bump damage (BD) and the Gaussian damage (GD). The axisymmetric artery geometry is more CPU time consuming than the one in monodimensional artery. For information, the computational cost of 1D test-case in Jansen et al. (2022) is about 0.2 min.

Because of the prohibitive computational cost of the axisymmetric cases compared to the 1D test-case, we perform a sensitivity study of the parameters of our model only in the 1D configuration. In (Jansen 2021, section 6.2) we propose a qualitative and quantitative exploration of the parameter space of our model using a methodology that nests two Global Sensitivity Analysis (GSA) methods. The realization of such sensitivity analysis methods in axisymmetric geometry could not have been achieved because of its computational cost.

Table 1 OpenFOAM boundary conditions used at boundaries of the luminal axisymmetric domain $\Omega _{\textrm{l}}$ shown in Fig. 1

Table 2 Numerical schemes of OpenFOAM used for spatial and temporal discretization of equation (1)

Table 3 Definitions, values and units of the parameters of the model in a axisymmetric artery configuration

B Mesh convergence study

In this section, we present the convergence studies used to set relevant mesh parameters for the presented simulations. These parameters define the meshes of the luminal domain linked to CFD and of the parietal domain linked to the model of intimal hyperplasia. An illustration of the mesh parameters of domains is proposed in Fig. 14d. The convergence studies were carried out on the Gaussian damage (GD). The fixed and variable parameters of each study are presented in Table 4.

Table 4 Summary of the parameters of the luminal and parietal meshes in the three convergence studies proposed in Fig. 14

A first study, shown in Fig. 14a, impose conformed meshes between luminal and parietal meshes to choose the relevant discretization in the damaged zone of parietal mesh. This figure shows, as a function of the number of parietal mesh points in the central zone of the GD case, $N_{\textrm{TG}}^{M}$ (see Fig. 14d), the relative error of luminal radius at $t=60$ days, i.e., $R_{\textrm{l}}(z,t=60\textrm{d})$, based on a finer mesh which has $N_{\textrm{TG}}^{M}=1301$. From this study, we choose $N_{\textrm{TG}}^{M} = 1001$.

As the whole coupling between hemodynamics and tissue growth is based on the WSS distribution, and regarding the strong influence of WSS on vascular species dynamics (see Fig. 12), the most limiting factor for an accurate simulation of a spatio-temporal lesion evolution is the WSS computation. The more the hemodynamics will be finely resolved in the near endothelium region, the more precise the WSS will be obtained, within the feasible simulation limit.

In order to evaluate the influence of the numerical resolution of hemodynamics in axisymmetric configuration, two parameters of the luminal mesh were tested: discretization in the radial direction and discretization in the longitudinal direction.

Imposing a constant radial compression of the luminal mesh with a thickness of the near endothelium cell of $\Delta _y={1 \times 10^{-5}}$ m, we study the influence of the number of points $N_{y}$ and $N_{z}^M$, respectively, in Fig. 14b, c. In Fig. 14b, we observe a strong dependence of the number of points of mesh in the radial direction, with constant radial compression. From this study, we choose $N_{y} = 51$. Fixing $N_{\textrm{TG}}^{M}=1000$, Fig. 14c shows the influence of the increase in the longitudinal discretization of the luminal mesh on the relative errors of the luminal radius at $t=60$ days, based on the finer luminal mesh with $N_{z}^M/N_{\textrm{TG}}^{M}=8$. From this study, we impose the ratio $N_{z}^M/N_{\textrm{TG}}^{M}=7$.

Fig. 14

Means and maximums of the relative error based on luminal radius at $t=60$ days with respect to the finer mesh in the Gaussian damage. The fixed and variable parameters of each study are shown in Table 4. a Relative errors as a function of the number of points in the longitudinal direction in the central zone $N_{z}^M$. The finer mesh has $N_{z}^M=1301$ points. This study imposed conform meshes between luminal and parietal domains. b Relative errors as a function of the number of points in the radial direction on the luminal mesh $N_{y}$. The finer mesh has $N_{y}=51$ points. This study imposes constant radial compression of the luminal mesh. c Relative errors as a function of the ratio between the number of longitudinal points of luminal and parietal meshes for $N_{\textrm{TG}}^{M}=1000$. d Illustration in the plane of symmetry of the artery of the meshes in the luminal and parietal domains with the various mesh parameters considered. A hatched rectangle with red lines shows the initially damaged zone limited