A Quantitative Model of Early Atherosclerotic Plaques Parameterized Using In Vitro Experiments

Abstract

There are a growing number of studies that model immunological processes in the artery wall that lead to the development of atherosclerotic plaques. However, few of these models use parameters that are obtained from experimental data even though data-driven models are vital if mathematical models are to become clinically relevant. We present the development and analysis of a quantitative mathematical model for the coupled inflammatory, lipid and macrophage dynamics in early atherosclerotic plaques. Our modeling approach is similar to the biologists’ experimental approach where the bigger picture of atherosclerosis is put together from many smaller observations and findings from in vitro experiments. We first develop a series of three simpler submodels which are least-squares fitted to various in vitro experimental results from the literature. Subsequently, we use these three submodels to construct a quantitative model of the development of early atherosclerotic plaques. We perform a local sensitivity analysis of the model with respect to its parameters that identifies critical parameters and processes. Further, we present a systematic analysis of the long-term outcome of the model which produces a characterization of the stability of model plaques based on the rates of recruitment of low-density lipoproteins, high-density lipoproteins and macrophages. The analysis of the model suggests that further experimental work quantifying the different fates of macrophages as a function of cholesterol load and the balance between free cholesterol and cholesterol ester inside macrophages may give valuable insight into long-term atherosclerotic plaque outcomes. This model is an important step toward models applicable in a clinical setting.

Appendices

Appendix 1: Experiment-Specific Parameters of Submodels

The following Tables 5, 6 and 7 contain the experiment-specific parameters of the mathematical submodels of in vitro systems in analogy to the experimental setups in Henriksen et al. (1983), Leake and Rankin (1990), Mackness et al. (1993), Brown et al. (1979) and Brown et al. (1980).

Table 5 Experiment-specific parameters of the mathematical submodel of LDL modification and ingestion (submodel 1) in analogy to the experimental setups in Henriksen et al. (1983) and Leake and Rankin (1990)

Table 6 Experiment-specific parameters of the mathematical submodel of HDL protection against LDL modification (submodel 2) in analogy to the experimental setups in Mackness et al. (1993)

Table 7 Experiment-specific parameters of the mathematical submodel of cholesterol cycle and reverse cholesterol transport (submodel 3) in analogy to the experimental setups in Brown et al. (1979) and Brown et al. (1980)

Appendix 2: Least-Squares Fits of Submodels

The following Figs. 5, 6, 7, 8, 9 and 10 contain all least-squares fits of the mathematical submodels of in vitro systems to experimental results in Henriksen et al. (1983), Leake and Rankin (1990), Mackness et al. (1993), Brown et al. (1979), Brown et al. (1980) and Yao and Tabas (2000). Appendix 1: Experiment-specific parameters of submodels and Table 4 give the experiment-specific and least-squares fitted parameters of the submodels, respectively.

Fig. 5

Comparison of results of the mathematical submodel of LDL modification and ingestion (submodel 1) with results from various experimental setups in Henriksen et al. (1983). Least-squares fits of the simulated ingestion of native and modified LDL per macrophage $\frac{a_{\mathrm {Ing}}(T_\mathrm {Ing})}{m_\mathrm {Ing}}$ to experimental results a in Henriksen et al. (1983), Fig. 1 for varying ingestion time periods $T_\mathrm {Ing}$, b in Henriksen et al. (1983), Fig. 2 for varying modification time periods $T_\mathrm {Mod}$ and c in Henriksen et al. (1983), Fig. 5 for varying initial LDL ingestion concentrations ${\ell }_{\mathrm {Ing},0}$ (Color figure online)

Fig. 6

Comparison of results of the mathematical submodel of LDL modification and ingestion (submodel 1) with results from various experimental setups in Leake and Rankin (1990). Least-squares fits of the simulated ingestion of native and modified LDL per macrophage $\frac{a_{\mathrm {Ing}}(T_\mathrm {Ing})}{m_\mathrm {Ing}}$ to experimental results (a in Leake and Rankin (1990), Fig. 1a for varying modification time periods $T_\mathrm {Mod}$ and b in Leake and Rankin (1990), Fig. 4 for varying initial LDL ingestion concentrations ${\ell }_{\mathrm {Ing},0}$ (Color figure online)

Fig. 7

Comparison of results of the mathematical submodel of the HDL protection against LDL modification (submodel 2) with results from various experimental setups in Mackness et al. (1993). Least-squares fits of the simulated lipid peroxide content per lipoprotein $\frac{N_{\tilde{\ell }}{\tilde{\ell }}(T_\mathrm {Mod})}{{\ell }_{0}}$ and the HDL protection to experimental results a in Mackness et al. (1993), Fig. 4 for varying modification time periods $T_\mathrm {Mod}$ and b in Mackness et al. (1993), Fig. 5 for varying initial HDL concentrations $h_0$ (Color figure online)

Fig. 8

Comparison of results of the mathematical submodel of cholesterol cycle and reverse cholesterol transport (submodel 3) with results from various experimental setups in Brown et al. (1979). Least-squares fits of the simulated concentration of intracellular free cholesterol $\frac{f(T_\text {Cho})}{m}$ to the experimental results a in Brown et al. (1979), Fig. 1a and b in Brown et al. (1979), Fig. 1b for varying experimental time periods $T_\text {Cho}$ (Color figure online)

Fig. 9

Comparison of results of the mathematical submodel of cholesterol cycle and reverse cholesterol transport (submodel 3) with results from various experimental setups in Brown et al. (1980). Least-squares fits of the simulated concentration of intracellular free cholesterol $\frac{f(T_\text {Cho})}{m}$, intracellular cholesterol ester $\frac{b(T_\text {Cho})}{m}$ and excreted cholesterol $\frac{r(T_\text {Cho})}{m}$ per macrophage to the experimental results a in Brown et al. (1980), Fig. 1a for varying experimental time periods $T_\text {Cho}$, b in Brown et al. (1980), Fig. 1b for varying experimental time periods $T_\text {Cho}$, c in Brown et al. (1980), Fig. 1c for varying experimental time periods $T_\text {Cho}$, d in Brown et al. (1980), Fig. 2a for varying additions of HDL $h_0$, e in Brown et al. (1980), Fig. 4 for varying additions of HDL $h_0$ and f in Brown et al. (1980), Fig. 7b for varying experimental time periods $T_\text {Cho}$ (Color figure online)

Fig. 10

Comparison of least-squares fit of simulated macrophage apoptosis to experimental results in Yao and Tabas (2000), Fig. 4a (Color figure online)

Appendix 3: Proof of Proposition 1

In the following we prove Proposition 1. To prove the positivity of ${\ell }$, i.e., ${\ell }(t)\ge 0 \; \forall t\ge 0$ it is sufficient to note that ${\ell }(0)=0$ and that ${\ell }(t)=0$ implies

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}{\ell }(t) \overset{(6)}{=} \frac{r_{\ell }}{H} > 0. \end{aligned}$$

(11)

due to the strict positivity of the parameters. The positivity of $h, m$ and ${\tilde{\ell }}$ can be proved in an analogue fashion. Since $\frac{f(0)}{m(0)}=f_0 \ge f_\mathrm {Min}$ and $\frac{f(t)}{m(t)}=f_\mathrm {Min}$ implies

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}\left( \frac{f(t)}{m(t)}\right) = \frac{\frac{\mathrm {d}}{\mathrm {d}t}f(t) -\frac{f(t)}{m(t)}\frac{\mathrm {d}}{\mathrm {d}t}m(t)}{m(t)} \overset{(6)}{\ge } \frac{\frac{r_{m}}{H} f_\mathrm {In} -f_\mathrm {Min}\frac{r_{m}}{H}}{m(t)} {\ge } 0, \end{aligned}$$

(12)

it holds $\frac{f(t)}{m(t)} \ge f_\mathrm {Min} \; \forall t \ge 0$. This also implies the positivity of $f$. Given that $\frac{f(0)}{m(0)}=f_0 < f_\mathrm {Max}$ and that $\frac{f(t)}{m(t)}\rightarrow f_\mathrm {Max}$ implies

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}\left( \frac{f(t)}{m(t)}\right) = \frac{\frac{\mathrm {d}}{\mathrm {d}t}f(t) -\frac{f(t)}{m(t)}\frac{\mathrm {d}}{\mathrm {d}t}m(t)}{m(t)} {\longrightarrow } -\infty , \end{aligned}$$

(13)

it follows $\frac{f(t)}{m(t)} \le f_\mathrm {Max} \; \forall t \ge 0$. Hence, we conclude that

$$\begin{aligned} \frac{f(t)}{m(t)} \in \left[ f_\mathrm {Min} , f_\mathrm {Max} \right] \; \forall t \ge 0. \end{aligned}$$

(14)

The positivity of $b$ follows since $b(0)=0$ and $b(t)=0$ implies

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}b(t) \overset{(6)}{=} k_{f}{ \frac{{\left( {f(t)}-f_\text {Min}{m}\right) }^2}{f_\text {Max}{m}-{f(t)}} } \overset{(14)}{\ge } 0 \end{aligned}$$

(15)

which finishes the proof of 1. (It also follows that the time-dependent solution $({\ell }(t),{\tilde{\ell }}(t),h(t),f(t),b(t),m(t)), t\ge 0$ of the initial value problem is unique and smooth because the smoothness of the right-hand side of ordinary differential equation (6) is now straight-forward to show.)

Using (14)

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}m(t) \overset{(6),(14)}{\ge } - \underbrace{\mu _{m}\frac{{(f_\mathrm {Max})}^{n_{m}}}{{(\xi _{m})}^{n_{m}}+{(f_\mathrm {Max})}^{n_{m}}}}_{=:z_\mathrm {Max}} m+\frac{r_{m}}{H} \end{aligned}$$

(16)

holds, and by solving this ordinary differential inequality with associated initial condition $m(0)=m_0$ it follows

$$\begin{aligned} \begin{aligned} m(t) \overset{(16)}{\ge }&m_0\exp \left( - z_\mathrm {Max} t \right) + \frac{r_{m}}{H z_\mathrm {Max}} \left( 1 - \exp \left( - z_\mathrm {Max} t \right) \right) \\&\ge \min \left( m_0 , \frac{r_{m}}{H z_\mathrm {Max}} \right) \overset{(7),f_\mathrm {Max}> f_0}{=} \frac{r_{m}}{H z_\mathrm {Max}} \\&= \frac{r_{m}}{H} \frac{{(\xi _{m})}^{n_{m}}+{(f_\mathrm {Max})}^{n_{m}}}{\mu _{m}{(f_\mathrm {Max})}^{n_{m}}} \; \forall t \ge 0. \end{aligned} \end{aligned}$$

(17)

In an analogue way, the upper bound for $m(t)$ can be found, leading to

$$\begin{aligned} m(t) \in \bigg [ \underbrace{\frac{r_{m}}{H} \frac{{(\xi _{m})}^{n_{m}}+{(f_\mathrm {Max})}^{n_{m}}}{\mu _{m}{(f_\mathrm {Max})}^{n_{m}}}}_{=: m_\mathrm {Min}} , \underbrace{\frac{r_{m}}{H} \frac{{(\xi _{m})}^{n_{m}}+{(f_\mathrm {Min})}^{n_{m}}}{\mu _{m}{(f_\mathrm {Min})}^{n_{m}}}}_{=: m_\mathrm {Max}} \bigg ] \; \forall t\ge 0 \end{aligned}$$

(18)

which finishes the proof of 2.

The boundedness of ${\ell }(t)$ is given by

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}{\ell }(t) \overset{(6),(18)}{\le } -\left( q_{{\ell },{m}} m_\mathrm {Min} + \frac{q_{{\ell },{e}}}{H} \right) {\ell }(t) + \frac{r_{\ell }}{H} \end{aligned}$$

(19)

since the solution of the ordinary differential inequality (with associated initial condition ${\ell }(0)=0$) is bounded by

$$\begin{aligned} \begin{aligned} {\ell }(t)&\overset{(19)}{\le } \frac{r_{\ell }}{H q_{{\ell },{m}} m_\mathrm {Min} + q_{{\ell },{e}}} \left( 1 - \exp \left( - \left( q_{{\ell },{m}} m_\mathrm {Min} + \frac{q_{{\ell },{e}}}{H} \right) t \right) \right) \\&\le \underbrace{\frac{r_{\ell }}{H q_{{\ell },{m}} m_\mathrm {Min} + q_{{\ell },{e}}}}_{{\ell }_\mathrm {Max}}. \end{aligned} \end{aligned}$$

(20)

In an analogue way, the boundedness of $h(t)$ is proved. We show boundedness of ${\tilde{\ell }}(t)$ under the condition $\frac{r_{\ell }}{H}< \mu _{\tilde{\ell }}m_\mathrm {Min}$ by a proof by contradiction. Hence, let ${\tilde{\ell }}(t)$ be unbounded, i.e., ${\tilde{\ell }}(t) \rightarrow \infty$ as $t\rightarrow \infty$ and $\frac{r_{\ell }}{H}< \mu _{\tilde{\ell }}m_\mathrm {Min}$. It follows

$$\begin{aligned} \begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}{\tilde{\ell }}(t)&\overset{(6),(20)}{\le } \underbrace{\left( - \mu _{{\tilde{\ell }}} + q_{{\ell },{m}} {\ell }_\mathrm {Max} \right) }_{< 0, \; \mathrm { since }\; \frac{r_{\ell }}{H}< \mu _{\tilde{\ell }}m_\mathrm {Min}} m(t) + \frac{q_{{\ell },{e}}}{H} {\ell }_\mathrm {Max} \\&\overset{(18)}{\le } - \mu _{{\tilde{\ell }}} m_\mathrm {Min} + q_{{\ell },{m}} {\ell }_\mathrm {Max} m_\mathrm {Min} + \frac{q_{{\ell },{e}}}{H} {\ell }_\mathrm {Max} =- \mu _{\tilde{\ell }}m_\mathrm {Min} + \frac{r_{\ell }}{H} < 0 \end{aligned} \end{aligned}$$

(21)

which is in contradiction to the assumed unboundedness of ${\tilde{\ell }}(t)$. Hence, this finishes the proof of Proposition 1. $\square$