Simulation of intimal thickening induced by hemodynamical shear stresses
Date
2023
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Publisher
University of Delaware
Abstract
Atherosclerosis is an inflammatory disease in medium-sized and large arteries caused by accumulation of lipids within the arterial wall. It is a complex process that involves lipid infiltration through the sites of endothelial damage, formation of foam cells, and the creation of plaques. These plaques can rupture, releasing clotting agents into the bloodstream, causing a complete blockage of blood vessels, resulting in heart attacks and strokes. The complex process of atherosclerosis starts with a benign thickening of the intima. Thus, intimal thickening is a precursor to atherosclerosis. ☐ The goal of this work is to investigate intimal growth in arteries, induced by hemodynamical shear stress, through finite element simulation using the FEniCS computational environment. In this thesis, we develop a mathematical model that establishes a relationship between growth and hemodynamics, which should be viewed as providing a framework for coupling hemodynamic simulations to mathematical descriptions of atherosclerosis, both of which have been modeled separately in great detail. In our model, the growth of the intima depends on cross section geometry and features of hemodynamics (shear stress). In this work, the arterial wall is modeled as three distinct layers, the intima, the media, and the adventitia, each with different material properties. Since the average mechanical properties of an arterial wall are more or less uniform along the axial direction, we introduce a 2D model of an arterial cross section in which all variables of interest are invariant with respect to the axial direction. Blood flow through vessels is approximately non-turbulent and unidirectional in regions of the vessel away from vessel arches and bifurcations. In addition, in our model, the timescale for vessel growth and deformation is very large compared to the timescale of the blood velocity. Thus, taking time average of the velocity of blood flow with respect to the timescale of our simulations, the blood flow can be approximated to be steady. Further, the the dissipative forces near the boundary (the endothelium) becomes large so the tangential component of the velocity vanishes. Hence, we assume that the blood flow is steady, non-turbulent, and unidirectional. The blood flow is modeled with a Poisson equation (derived from time-independent and unidirectional Navier-Stokes equation in absence of external forces) with a zero-flow Dirichlet boundary condition at the endothelium. We calculate the shear stress from the solution to the Poisson equation. Damages to endothelial cells induce platelets from blood to attach to the sites of injury that release platelet-derived growth factor (PDGF). In addition, endothelial cells themselves release PDGF in response to shear stress. In this work, since we are interested in the release of PDGF by uninjured endothelial cells, we assume that the endothelial cells release PDGF in a shear-dependent manner. Although PDGF causes cell proliferation both in the intima and the media, but since we are only interested in the growth of intima, growth is assumed to occur only in the intima and is modeled using morphoelasticity theory. The PDGF transport is modeled as a steady-state diffusion-degradation equation since the time scale of diffusion and degradation of PDGF happens at a time scale much much smaller compared to timescale of the intimal growth. We solve the diffusion-degradation equation in all three layers subject to a Neumann boundary condition on the endothelium that registers the flux of PDGF through the endothelium into the intima and a Dirichlet boundary condition on the outer boundary of the adventitia. Since the endothelium is far away from the outer boundary of the adventitia and considering the diffusion and degradation of PDGF across the arterial wall, the PDGF concentration at the outermost boundary of the arterial wall can be considered to be negligible. Therefore, we impose a zero PDGF Dirichlet boundary condition on the outermost boundary of the arterial wall. Finally, the optimal displacement fields for the arterial wall and the lumen are obtained by minimizing strain energy functionals for the arterial wall and the lumen. ☐ We simulate intimal growth for three distinct arterial cross section geometries. We show that the rate of intimal thickening varies depending on different cross section geometries. For cross section geometries that are annular, the growth of the intima is uniform in the angular direction, the endothelium stays a disk as the intima grows. For non-annular cross section geometries, thicker intimas grow more compared to thinner ones, rate of intimal thickening depends on the distance to the center of flow (greater distance, less growth), shear stress is negatively correlated with distance from the flow center (where the flow velocity is maximal), the maxima and minima of curvature increase and decrease, respectively, with time, the PDGF concentration increase with time, the shape of the lumen becomes polygonal over time for non-annular cross sections. We also observe multiple remodelling phases of the lumen (the lumen dilates, followed by a contraction, followed by a dilation). This multiple remodelling feature is completely new and hasn't been addressed in any literature yet. Each layer of the artery has transversely helical collagen fiber fields which when stretched offer resistance to outward dilation, thus contributing to the total energy of the system. While there are several interpolation and probabilistic techniques available to model the fiber distributions, we implemented the technique of numerical conformal maps to model the fiber field distribution in each of the layers. ☐ The mathematical model established in this thesis should be viewed as providing a framework for coupling hemodynamics simulations to mathematical descriptions of atherosclerosis, both of which have been modeled separately in great detail. This work can be extended to include the process of atherosclerosis and its progression that will shed light on the association among hemodynamics, intimal thickening and atherosclerosis, which is associated with life-threatening diseases. With the advancement of medical technologies, our model, coupled with a model for atherosclerosis, can help predict cardiovascular diseases in advance.
Description
Keywords
Hemodynamics, Hyperelasticity, Intimal thickening, Morphoelasticity theory