Mathematical Modeling and Simulation of the Evolution of Plaques in Blood Vessels

The formation of plaques is one of the main causes for the blockage of arteries. This can lead to ischaemic brain or myocardial infarctions as well as other cardiovascular diseases. Possible biochemical and biomechanical processes contribute to the development of plaque growth and rupture. The main biochemical processes are the penetration of monocytes and the accumulation of foam cells in the vessel wall, leading to the formation and growth of plaques. The biomechanical forces can be measured by observing stresses in the blood flow and the vessel wall, which may lead to the rupture of plaques. In this thesis, we formulate an appropriate model to describe the evolution of plaques. The model consists of both the interaction between the blood flow and the vessel wall, and the growth of plaques due to the penetration of monocytes from the blood flow into the vessel wall. The Navier-Stokes equations and the elastic structure equations are used to describe the dynamics of fluid (blood flow) and the mechanics of structure (vessel wall). The motion of monocytes is described by the convection-diffusion-reaction equation, coupled with an equation for the accumulation of foam cells. Finally the metric of growth is introduced to accurately determine the stress tensor, and its evolution equation is derived. The variational formulation of the model is transformed into the ALE (Arbitrary Lagrangian-Eulerian) formulation, and all the equations are rewritten in the fixed domain. Temporal discretization is achieved with finite differences and spatial discretization is based on the Galerkin finite element method. The nonlinear systems are linearized and solved by the Newton method. Based on the model and the numerical methods above, numerical simulations are performed by using the software Gascoigne. The obtained numerical results make an agreement with the observation, and support the assumption that the penetration of monocytes and the accumulation of foam cells lead to the formation and growth of plaques, and that the evolution of plaques induces the increase of stresses in the vessel wall, which is an indicator of plaque rupture