A 2D Mathematical Model of Blood Flow and its Interactions in an Atherosclerotic Artery

Abstract

A stenosis is the narrowing of the artery, this narrowing is usually the result of the formation of an atheromatous plaque infiltrating gradually the artery wall, forming a bump in the ductus arteriosus. This arterial lesion falls within the general context of atherosclerotic arterial disease that can affect the carotid arteries, but also the arteries of the heart (coronary), arteries of the legs (PAD), the renal arteries... It can cause a stroke (hemiplegia, transient paralysis of a limb, speech disorder, sailing before the eye). In this paper we study the bloodplaque and blood-wall interactions using a fluid-structure interaction model. We first propose a 2D analytical study of the generalized Navier-Stokes equations to prove the existence of a weak solution for incompressible non-Newtonian fluids with non standard boundary conditions. Then, coupled, based on the results of the theoretical study approach is given. And to form a realistic model, with high accuracy, additional conditions due to fluid-structure coupling are proposed on the border undergoing inetraction. This coupled model includes (a) a fluid model, where blood is modeled as an incompressible non-Newtonian viscous fluid, (b) a solid model, where the arterial wall and atherosclerotic plaque will be treated as non linear hyperelastic solids, and (c) a fluidstructure interaction (FSI) model where interactions between the fluid (blood) and structures (the arterial wall and atheromatous plaque) are conducted by an Arbitrary Lagrangian Eulerian (ALE) method that allows accurate fluid-structure coupling.

Numerical simulation of the fractional flow reserve (FFR)

“…Since we are interested in studying the flow in the coronary arteries, we decided to use a non-Newtonian flow model like in [3] and [2]. In fact, the coronary arteries have a small caliber (0.5 cm) compared to the aorta for example, where the use of non-Newtonian flow model is not really crucial.…”

“…In our case, the blood cannot be assimilated to a Newtonian fluid, since the coronary vessels caliber is very small (0.5 cm). We choose a non-Newtonian flow model, as in [3]. The mathematical model was studied in [3] and authors proved the existence of a solution to this type of problems.…”

“…We choose a non-Newtonian flow model, as in [3]. The mathematical model was studied in [3] and authors proved the existence of a solution to this type of problems. In this paper, we are more interested in giving a bi-dimensional based estimation of the VFFR on the one hand.…”

“…The simulation of blood flow in the presence of stenosis in a 2D geometry has been the subject of several works [3,9,10]. These works were based on the Navier-Stokes model and the arterial wall was considered to be rigid.…”

Numerical simulations of a 3D fluid-structure interaction model for blood flow in an atherosclerotic artery

“…One can find an attempt in [2], where, after a 2D analytical study to prove the existence of a weak solution for incompressible non-Newtonian fluids with non standard BCs, the authors tested the model numerically by coupling it with a nonlinear hyperelastic model for the arterial wall and the atheromatous plaque, with different values of Young modulus corresponding to each structure. They showed that considering the classical models of interaction between the blood flow (modeled as a Newtonian fluid), the vessel wall and the atherosclerotic plaque, the risk of rupture is less predictive because it underestimates the stress over the plaque comparing with the case where blood is modeled as a non-Newtonian fluid.…”