טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentMarkovits Michal
SubjectModeling and Computational Simulation of Atherosclerosis
DepartmentDepartment of Mechanical Engineering
Supervisor Professor Emeritus Pinhas Bar-Yoseph
Full Thesis textFull thesis text - English Version


Abstract

Early stages of atherosclerosis are developed through highly complicated biochemical inflammatory processes that include many molecules and reactions and occur inside the intima layer of the artery wall. It is widely believed that accumulated modified low density protein (LDL) cholesterol, white blood cells (monocytes and macrophages) and fat-laden foam cells are the main components of the atherosclerotic plaque, which is separated from the blood flow by a thin fibrous cap. Researches pointed out that the rupture of a vulnerable plaque accompanied with thrombus, are the main clinical causes of cardiovascular and cerebrovascular diseases. The initiation, growth and rupture of an atherosclerotic plaque is a complex biomechanical time-dependent cascade that includes non-linear coupled processes, fluid-structure interactions in addition to a patient specific geometry. Hence, coupled hemodynamic, structural stress analyses and macromolecules’ reactions are pivotal to understand plaque initiation, growth and vulnerability to rupture.

In this work, through critically reviewing of the published mathematical models with various complexity degrees, we try to describe different aspects of the mechanisms that drive atherosclerosis, through mathematical modelling. We believe that simplified zeroth-order models, where the plaque geometrical complexity is not considered, may help to reveal new insights on the early stage of atherosclerosis. Non-spatial time dependent (zeroth order) model allows us to include many parameters in a simple and elegant way, perform stability analyses and investigate the effect of different components on the evolution of the plaque. The leading challenges are choosing the correct main components and their dependence one on each other (the equations’ terms).

We follow the dimensionless mathematical model of (Bulelzai and Dubbeldam, 2012), and study the different components and the nonlinear coupling between the different terms in the ODE system. In addition, we investigate the components which may play a major role. We conduct sensitivity analyses for the physical parameters and do rough calculations indicating how plaque constituents build up with time. Interestingly, we find out that the model has too many limitations, and it may even be not physiological at all. Understanding the limitations of zeroth-order modeling, we develop an original 2-D model in order to try simulate the atherosclerosis process and gain some qualitative insights. This model includes diffusion and advection of LDL within the blood flow, while the LDL is involved in a surface-based reaction. Unfortunately, the explicit method we use in order to solve the system of equations is unstable, due to a high convection coefficient of LDL in the blood. Moreover, the solution is too sensitive for the different parameters of the problem.

In a nutshell, reduced-order models are simple and elegant though they are more qualitative, rather than give a detailed and realistic look on the interesting and highly important biological phenomenon of atherosclerosis.