Mathematical models for atherosclerotic plaque evolution

Atherosclerosis is a disease in which low density lipoproteins (LDL) accumulate in the arterial wall due to an inflammatory response, which is triggered by the oxidation of LDL molecules that are already present in the arterial wall. Progression of atherosclerotic plaques involves many components which includes, macrophages, monocytes, LDL, oxidized-LDL, cytokines etc. Clinical data is scarce for the long term evolution of plaques, therefore, mathematical models provide an alternative to enhance our knowledge in predicting and analyzing the complex processes in the advancement of atherosclerosis. In this thesis, we study different simplified mathematical models and analyse the long term behavior of evolution of plaques. One model we studied, consisted of ordinary differential equations that could be coupled and decoupled to the blood flow. Another model was also developed, which consisted of partial differential equations and that treats the artery wall as a moving boundary problem. The ordinary differential equations (ODE) model consisted of nonlinear coupled equations for oxidized-LDL, monocytes, macrophages, foam cells and radius of the artery (which can also be translated into an equation for endothelial shear stress). The model is distinguished by model A which is decoupled to the blood flow and model B, which is coupled to blood flow. The model B we call in thesis to be self-consistent system. Both models are time dependent and do not take account of geometry of the artery. Both these models have many parameters, which affect the growth of concentrations in time. A bifurcation analysis was therefore presented in this thesis to understand the impact of varying parameters on the system. For this purpose, a complete codimension two bifurcation analysis was carried out and singularities in the system were discussed. We found that the system, model A, can also have a codimension three bifurcation point (the degenerate Bogdanov-takens point). In model A, bistability and multistability was found to depend on the value of the parameter e. The parameter e is the ingestion rate of oxidized-LDL particles by macrophages in the equation for oxidized-LDL. The regions of stable and unstable equilibria and limit cycles were depicted with the two parameter diagrams. These regions were presented for cases with ingestion rate vs. LDL intake and shear stress vs. LDL intake. A threshold value dcrit was found for the LDL particles beyond which occlusion of the plaque seems inevitable. Two other critical parameter values for ingestion rates were also found. These values bcrit and ecrit were presented along with the Hopf curve as a function of parameters (d;b) and (s;d), which constitute a critical curve in the system. In model B, the multistability was disappeared and the attachment of blood flow profile stabilized the dynamics of the artery. For understanding the long term response of the suspended system in model B, we have carried out a slow-fast system analysis. This shows that the stable healthy state of (0;M0;0;R0) may be perturbed for short amount of times, but ultimately the suspended system always comes to a stationary point. This all depends on the values of parameters d;b;e. The reduction in radius was found to be proportional to cholesterol intake parameter. For longer times, the radius goes quasi-static suggesting the results to be physically viable. A complex time dependent moving endothelium model was also presented in an axi-symmetric setting in order to understand the role played by the geometry in the problem. To this end, the Navier-Stokes equations for blood flow were coupled with the convection-diffusion equation for LDL in lumen and a diffusion equation for oxidized-LDL in intima. These equations were coupled to a common boundary (the endothelium) and solved to study the long term effects of recirculation on LDL. The flux of low-density lipoproteins across the endothelium was shown to depend rather strongly on wall shear stress. For the limiting case of Poiseuille flow additional numerical simulations were performed. LDL concentrations and plaque volume were compared with the full model results. It was shown that recirculation leads to larger plaque volumes and is hence detrimental to plaque evolution. Our computations suggest that recirculation leads predominantly to a broader plaque. Comparison with Poiseuille flows also revealed that assuming a Poiseuille flow renders results that compare well with solutions of the full Navier-Stokes model. This implies that a good estimate of plaque progression can be obtained at less expense. It also validates results obtained in ODE model in which Poiseuille flow was implicitly assumed. For a better mechanical understanding of remodeling of the artery, the stress strain analysis may also be carried out. For this purpose, fluid-structure interaction (FSI) technique should be applied to the axisymmetric model we proposed. For a quantitative and realistic model the clinical data is vital to understand the growth of plaques. Such data will remain indispensable to construct increasingly realistic models and to obtain precise values of the model parameters, such that qualitatively predictive models can be realized in near future. Future close collaboration between clinical researchers and applied mathematicians will therefore be essential for progress in the this field.