Mathematical and numerical modeling of early atherosclerotic lesions

Atherosclerosis is an inflammatory disease that starts with intima alterations. It is now well accepted that the early stage of the process is the result of interactions between plasma Low Density Lipoproteins (LDL) that filtrate through the endothelium into the intima, cellular components (monocytes/macrophages, endothelial cells and smooth muscle cells) and the extracellular matrix of the arterial wall. The filtration phenomenon is closely linked to local hemodynamic factors through the action of the wall shear stress exerted by the lumen blood flow on the endothelium. This work is devoted to the construction of a mathematical model describing the early formation of atherosclerotic lesions. First, we model the biochemical process that takes place into the intima by means of a system of nonlinear reaction-diffusion partial differential equations. This is a slow process that may give rise to an atheromatous plaque over a period of months or years. For modeling the plaque growth, we use the fact that, by absorbing oxidized LDL, macrophages transform into foam cells that are large cells. By restricting to foam cells ($F$) and biomass ($W$), since other constituents (macrophages and oxidized LDL denoted by $M$ and $Ox$ respectively) are much smaller, the incompressibility assumption used by many authors \cite{Preziosi} in tumor growth modeling reads $$F+W=A \, , $$ where $A$ denotes the total number of cells per unit volume assumed to be invariant along time. Foam cell formation induces rearrangement and global domain deformation with velocity $\mathbf{V}$ solution of the following partial differential equation: $$ \div\, \mathbf{V} = k \frac{ Ox \cdot M }{A}\, .$$ Then, relying strongly on previous works by several authors, see \cite{Zunino2002, QuarteroniZunino2002}, we couple the inflammatory and the lesion growth models with the blood flow by means of a mass transfer model. To do so, we consider a fluid-wall model inspired by \cite{Zunino2002, QuarteroniZunino2002} with a single layer. This layer models the intima considered as a porous medium. In the lumen, we make the following assumptions: The blood is an incompressible Newtonian fluid, the arterial wall is rigid and the blood flow is stationary. As in \cite{Zunino2002, QuarteroniZunino2002}, the blood filtration in the wall is described by the Darcy model. The evolution of the plasma and the intima LDL concentrations are given by convection-diffusion equations. The fluid dynamics and the solute dynamics problems are coupled by the interface conditions involving the fluxes $J_v$ and $J_s$ that are obtained from Kedem-Katchalsky equations. Those equations prescribe the fluid velocity $J_v$ and the flux of chemicals $J_s$ through a semi-permeable membrane that separates two solutions where solute concentrations and fluid pressures are nonequal. Finally, using the finite element solver FreeFem++, we perform numerical simulations on a two-dimensional geometry (see Figure 1) inspired from \textit{Cheng et al} experiment \cite{Cheng2005, Cheng2006}. In this experiment conducted on mice, the carotidian vessel is partially obstructed with a cast. This modifies the blood flow, and in particular the WSS patterns. From this experiment and others, it appears that the growth of atheromatous plaques is correlated with low WSS (that occurs here right before the cast and after the cast). With our model, we aim at obtaining numerical results that fit with the experiments