Modeling fibrous cap formation in atherosclerotic plaque development: stability and oscillatory behavior

Atherosclerosis usually occurs within the large arteries. It is characterized by the inflammation of the intima, which involves dynamic interactions between the plasma molecules; namely, LDL (low density lipoproteins), monocytes or macrophages, cellular components and the extracellular matrix of the arterial wall. This process is referred to as plaque formation. If the accumulation of LDL cholesterol progresses unchecked, atherosclerotic plaques will form as a result of increased number of proliferating smooth muscle cells (SMCs) and extracellular lipid. This can thicken the artery wall and interfere further with blood flow. The growth of the plaques can become thrombotic and unstable, ending in rupture which gives rise to many life threatening illnesses, such as coronary heart disease, cardiovascular diseases, myocardial infarction, and stroke. A mathematical model of the essential chemical processes associated with atherosclerotic plaque development is analyzed, considering the concentrations of LDLs, oxidized LDLs, foam cells, oxidized LDL-derived chemoattractant and macrophage-derived chemoattractant, the density of macrophages, smooth muscle cells (SMCs), and extracellular matrix (ECM). The positive invariant set is found and local stability is established. Oscillatory behavior of the model solutions is also investigated. Numerical solutions show various dynamic behaviors that can occur under suitable conditions on the system parameters.