The LDL-HDL profile determines the risk of atherosclerosis: a mathematical model

Abstract

Atherosclerosis, the leading death in the United State, is a disease in which a plaque builds up inside the arteries. As the plaque continues to grow, the shear force of the blood flow through the decreasing cross section of the lumen increases. This force may eventually cause rupture of the plaque, resulting in the formation of thrombus, and possibly heart attack. It has long been recognized that the formation of a plaque relates to the cholesterol concentration in the blood. For example, individuals with LDL above 190 mg/dL and HDL below 40 mg/dL are at high risk, while individuals with LDL below 100 mg/dL and HDL above 50 mg/dL are at no risk. In this paper, we developed a mathematical model of the formation of a plaque, which includes the following key variables: LDL and HDL, free radicals and oxidized LDL, MMP and TIMP, cytockines: MCP-1, IFN-γ, IL-12 and PDGF, and cells: macrophages, foam cells, T cells and smooth muscle cells. The model is given by a system of partial differential equations with in evolving plaque. Simulations of the model show how the combination of the concentrations of LDL and HDL in the blood determine whether a plaque will grow or disappear. More precisely, we create a map, showing the risk of plaque development for any pair of values (LDL,HDL).

Figure 1. Atherosclerosis schematics: the presence of ox-LDL in the intima causes monocytes to migrate from the lumen into the intima.

Monocytes differentiate into macrophages which endocytose ox-LDL and become foam cells. SMCs are attracted from the media into intima by chemotaxis and haptotaxis. Cytokines released by macrophages, foam cells and SMCs activate T cells. T cells enhance activation of macrophages. HDL helps prevent atherosclerosis.

Figure 2. Schematic network of atherosclerosis.

LDL and HDL are oxidized by free radicals, and become ox-LDL and ox-HDL respectively. Ox-LDL recruits macrophages to intima. By ingesting ox-LDL, macrophages are transformed to foam cells. SMCs are attracted into the intima by MCP-1 (secreted by endothelial cells) and PDGF (secreted by macrophages and foam cells). Macrophages, foam cells and SMCs secrete IL-12, which activates T cells. IFN-γ secreted by T cells enhance the activity of macrophages which contributes the plaque built-up.

Figure 3. Two 2D cross sections of a plaque.

Γ_M is the boundary of the intima in contact with the media, and Γ_I is the boundary of the intima in contact with the lumen. In (B) Γ_L and Γ_R are parts of the intima.

Figure 4. Simulations for the atherosclerosis model of 300 days after an initial plaque is formed with H ₀ = 40 mg/dL and L ₀ = 190 mg/dL.

(A: Cross sections of a blood vessel, B:Cross sections along the blood vessel).

Figure 5. Simulations for the atherosclerosis model of 300 days after an initial plaque is formed with H ₀ = 50 mg/dL and L ₀ = 130 mg/dL.

(A: Cross section of a blood vessel; B: Cross section along the blood vessel).

Figure 6. Simulations for the atherosclerosis model of 300 days after an initial plaque is formed with H ₀ = 60 mg/dL and L ₀ = 70 mg/dL.

(A: Cross section of a blood vessel; B: Cross section along the blood vessel).

Figure 7. Plaque weights for different levels of LDL and HDL.

The units of H ₀ and L ₀ are mg/dL.

Figure 8. Risk map for plaque development: Region I high risk; Region II low risk; Region III no risk.

The five points whose plaque's weight was simulated in Fig. 7 over a period of 300 days are indicated by “x”.

Figure 9. The PRCC of parameters for sensitivity analysis.

Figure 10. Drug treatment recommended for individuals A, B and C.