A lipid-structured mathematical model of atherosclerosis with macrophage proliferation

Abstract

We extend the lipid-structured model for atherosclerotic plaque development of Ford et al (2019) to account for macrophage proliferation. Proliferation is modelled as a non-local decrease in the lipid structural variable that is similar to the treatment of cell division in size-structured models (e.g. Efendiev et al ( 2018)). Steady state analysis indicates that proliferation assists in reducing eventual necrotic core size and acts to spread the lipid load of the macrophage population amongst the cells. The relative contribution of plaque macrophages by proliferation and recruitment from the bloodstream is also examined. The model suggests that a more proliferative plaque differs from an equivalent (same lipid content and cell count) recruitment-dominant plaque only in the way lipid is distributed amongst the macrophages.

A New Lipid-Structured Model to Investigate the Opposing Effects of Ldl and Hdl on Atherosclerotic Plaque Macrophages

“…However, experiments show that, although proliferation is more significant in the later stages of plaque development, it is still detectable in early plaques (Robbins et al [34]). The effect of proliferation on lipid accumulation in atherosclerosis is the focus of the model by Chambers et al [18]. Based upon their results, we anticipate that incorporating proliferation into our model will decrease the average lipid burden of the macrophages, opposing the effects of lipid-dependent cell death and emigration.…”

“…It is not straightforward to analyse the lipid distribution 𝑚 𝑗 (𝑡), 𝑗 = 0, 1, … , 𝐽 by directly considering Eqs. ( 16)- (18). In Appendix B we show that closed-form solutions for the steady state values, 𝑚 ⋆ 𝑗 , 𝑗 = 0, 1, … 𝐽 , can be obtained via the method of generating functions.…”

“…A more natural approach of representing lipid accumulation in plaque macrophages is to use a structured population model. Lipid-structured models for atherosclerosis have been developed by several authors [17][18][19][20]. The models of Ford et al [17], Chambers et al [18] and Watson et al [19] assume, for simplicity, that the rates of lipid uptake and offloading are independent of lipid load.…”

“…Lipid-structured models for atherosclerosis have been developed by several authors [17][18][19][20]. The models of Ford et al [17], Chambers et al [18] and Watson et al [19] assume, for simplicity, that the rates of lipid uptake and offloading are independent of lipid load. The model of Meunier and Muller [20] accounts for an arbitrary lipid-dependent uptake rate of LDL, but does not account for uptake of cellular debris nor lipid offloading.…”

Bifurcation for a free boundary model of an atherosclerotic plaque formation in the presence of reverse cholesterol transport with Holling type III functional response

“…Over the last few decades, many researchers have developed a number of mathematical models of atherosclerosis that can be used to characterize the various stages of atherosclerosis (see [2,6,9,13,20,22,21,26,27,28,31,33] and the references therein), such as inflammation, plaque growth, fibrous cap formation, and interactions between plaque and blood. Almost all of these models recognize the "bad" cholesterol: LDL, which promotes plaque formation, whereas "good" cholesterol: HDL, which inhibits plaque formation during the RCT process.…”