Fig 1.
Schematic diagram of the model.
Diagram of processes in the endothelium and intima in early plaque formation with a flow-chart representation of the model interactions on the endothelium (top) and in the intima (bottom) between modLDL, monocytes/macrophages, chemoattactants, ES cytokines, foam cells and HDL. A plaque is initiated when the endothelium is injured and allows LDL to enter and causes it to become modified so that the LDL particles become oxidised or modified in other ways. These modLDL particles provoke an immune reaction that causes monocytes to enter the blood vessel wall from the blood stream and differentiate into macrophages which consume modLDL. The macrophages become filled with cholesterol and take on a foamy appearance under the microscope. If the cholesterol is not removed from the cells, these foam cells accumulate in the intima. HDL particles transport cholesterol out of foam cells and cause the plaque either to regress or grow more slowly. HDL also acts to reduce the modification of LDL and the excitation of the endothelium which reduces the rate that monocytes enter the plaque.
Table 1.
Parameter values used in the model.
It is difficult to obtain experimentally valid parameter values for all the parameters in the model and consequently these values are not available in the literature. In particular it is extremely hard to measure in vivo values, but most of the parameter that the model requires have not even been measured in vitro. Therefore, most values are order of magnitude estimates. Since the aim of this study is to produce qualitative results, having exact, experimentally determined parameter values is less critical than for a quantitative predictive model. The values below have been rescaled in space and time in order to normalise the intima width. We take an intima width of 40 μm [59]. We use a time scale of ∼ 7.7 × 106 s (approximately 89 days) to rescale the equations. Further details on the rescaling can be found in [58].
Fig 2.
Macrophage and foam cell density over time.
(a) Macrophage density in the intima and (b) foam cell density when the plaque grows unboundedly. Plots (c) and (d) show macrophage and foam cell density respectively where foam cell numbers settle to a fixed equilibrium. Note: all variables are scaled with respect to intima width and indicative time scale.
Fig 3.
Model predictions from bifurcation diagram.
(a) Bifurcation diagram showing the density of foam cells at equilibrium as a function of the rate of influx of HDL. The solid blue curve represents the attracting equilibrium and the dashed red curve the repellor. The black arrows and lines represent various plaques labeled A–E whose size changes with time; the dashed horizontal lines represent rapid changes in HDL influx rate; the vertical arrows represent changes with time due to intrinsic dynamics in the model tissue. (b) Plaques A–E plotted as a function of scaled time. As shown in the bifurcation diagram plaques B, C and E tend to the fixed equilibrium which has low foam cell density, but the density of foam cells in plaques A and D continue to grow. The scales on the axes are a qualitative indication only as there is limited information about the exact values of the input parameters in the model.
Fig 4.
Equilibrium foam cell density for changing physiological parameters.
Foam cell density at equilibrium (the attractor) and the repellor are plotted against HDL influx showing the effect of (a) increasing the cholesterol efflux capacity (CEC) of HDL particles; (b) increasing the influx of modLDL into the plaque and (c) increasing the proportion of macrophages that revert to M2 type after RCT. The attracting equilibria are indicated by the solid curves; the repellors by dashed curves. In (a), as cholesterol efflux capacity increases, the bifurcation point moves left so that the equilibrium exists for lower rates of HDL influx. In (b), as modLDL influx increases, the bifurcation point moves right so equilibrium only exists for higher rates of HDL influx. In (c), as the proportion of foam cells revert to M2 rather than M1 macrophages increases, the slope of the repellor curve increases so that it is more likely that a plaque will fall below the repellor curve if HDL influx is increased and consequently will regress. The blue curve in each plot is the same curve as in Fig 3(a).
Fig 5.
The effect of different HDL actions on equilibrium foam cell density.
Plots showing where equilibrium plaques with a fixed density of foam cells exist as a function of σh the rate of influx of HDL and (a) σm which governs the rate of influx of monocytes and may be thought of as the availability of monocytes in the blood stream; (b) which governs the rate that LDL is modified and hence the rate that modLDL enters the model plaque; and (c)
which governs the excitability of the endothelium in response to modLDL and its propensity to express adhesion molecules and thereby recruit macrophages to enter the lesion. In each plot, the curve is the locus of the bifurcation point.
Fig 6.
The existence of equilibrium plaque as a function of cholesterol efflux capacity and HDL influx.
This plot shows where equilibrium plaques with fixed density of foam cells exist as a function of σh the rate of influx of HDL and νN which specifies the cholesterol efflux capacity of HDL and governs the rate of reverse cholesterol transport from foam cells. The solid curve is the locus of the bifurcation point; that is, of Σh as νN changes. This curve separates the region where an equilibrium plaque exists from the region where there is no equilibrium. As the cholesterol efflux capacity of the HDL particles decreases, the bifurcation point occurs at increasingly greater values of σh than when cholesterol efflux capacity is high. If cholesterol transport rates are very low then very high rates of HDL influx are required for a plaque to exist in equilibrium. On the locus of the bifurcation point, as νN → 0 then σh → ∞ which suggests that there it is impossible to have an equilibrium plaque when there is no reverse cholesterol transport.
Table 2.
Model results compared with experiments.
This table shows the results of different treatments by Feig et al [29] and the corresponding prediction of the model.
Fig 7.
Diagrammatic representation of model predictions described in Table 1.
Sketch of changes predicted by the model in plaque size after transplant into recipient mice: (a) ApoE-/-; (b) hAI/ApoE-/-; (c) ApoA-I-/-; (d) Wild Type. The curves in each plot represent the attractors and repellors. The solid curves represent the attractors or fixed equilibrium solutions and the dashed curves represent the repellors. Where there are two sets of curves, the heavier set of curves corresponds to the plaque after transplant and the lighter curve corresponds to the plaque before transplant. The solid arrow represents plaque growth before transplant and the dashed arrow represents post-transplant changes.
Fig 8.
Diagram illustrating of the effect of treatment by raising HDL influx into plaques, showing schematically the effect of a period of infusion of HDL that leads to increased HDL influx into the plaque, both with and without a simultaneous lessening of LDL influx.
(a) HDL influx into the plaque as a function of time where there is a period of regular infusions, preceded and followed by a period of no infusions. (b) Sketch of the bifurcation diagram showing the solution for the plaque under the HDL influx regime illustrated in (a) when LDL influx doesn’t change. The solution evolves over time along the black curve in the direction of the arrows. When the infusions are completed the plaque returns to a state where it cannot reach the attractor but continues to grow; (c) Sketch of plaque size as indicated by foam cell density as a function of time under the HDL influx regime illustrated in (a) when LDL influx doesn’t change. This plot corresponds to the progress of the plaque shown in (b) but here foam cell density is plotted explicitly as a function of time; (d) sketch of the changes in the bifurcation diagram when LDL influx is decreased before HDL infusion begins. (This might model the introduction of statin therapy for example.) The black arrow represents plaque growth before LDL influx decreases. We assume that the LDL influx rate is changed after the plaque has developed sufficiently so that the solution lies above the repellor on the new bifurcation curve. (e) Sketch of the bifurcation diagram showing the solution for the plaque under the HDL influx regime illustrated in (a) when the LDL influx is changed before HDL infusion is started. The grey arrow represents plaque growth prior to LDL increase and HDL infusion. Since the untreated level of HDL influx now lies to the right of the bifurcation point, the plaque will settle to an equilibrium once infusions stop since HDL infusion has reduced the size of the plaque so that the solution can approach the attractor and reduced LDL levels ensures that the attractor is now accessible when there is no HDL infusion. This new equilibrium is stable as long as LDL influx into the plaque remains low.; (f) Sketch of plaque size as indicated by foam cell density as a function of time under under reduced LDL influx and the HDL influx regime illustrated in (a). This plot corresponds to the progress of the plaque shown in (e) but here foam cell density is plotted explicitly as a function of time.