Modeling Microglia Activation and Inflammation-Based Neuroprotectant Strategies During Ischemic Stroke

Abstract

Neural inflammation immediately follows the onset of ischemic stroke. During this process, microglial cells can be activated into two different phenotypes: the M1 phenotype, which can worsen brain injury by producing pro-inflammatory cytokines; or the M2 phenotype, which can aid in long term recovery by producing anti-inflammatory cytokines. In this study, we formulate a nonlinear system of differential equations to model the activation of microglia post-ischemic stroke, which includes bidirectional switching between the microglia phenotypes, as well as the interactions between these cells and the cytokines that they produce. Further, we explore neuroprotectant-based modeling strategies to suppress the activation of the detrimental M1 phenotype, while promoting activation of the beneficial M2 phenotype. Through use of global sensitivity techniques, we analyze the effects of the model parameters on the ratio of M1 to M2 microglia and the total number of activated microglial cells in the system over time. Results demonstrate the significance of bidirectional microglia phenotype switching on the ratio of M1 to M2 microglia, in both the absence and presence of neuroprotectant terms. Simulations further suggest that early inhibition of M1 activation and support of M2 activation leads to a decreased minimum ratio of M1 to M2 microglia and allows for a larger number of M2 than M1 cells for a longer time period.

Appendices

Appendix A Local Parameter and Initial Condition Ranges Maintaining Observed Trends Without Treatment

We perform a local, one-at-a-time analysis to determine individual ranges for the parameters and initial conditions over which Model (7) maintains the observed trends without terms simulating treatment. More specifically, we perturb each parameter individually while holding the others constant (fixed to their nominal values in Table 1). Table 3 lists the resulting ranges for each parameter over which it may be individually varied while maintaining the qualitative trends observed in the model output, assuming baseline initial conditions of $M1(0)=100$ cells, $M2(0)=100$ cells, $P(0)=10 \ \frac{\mathrm{pg}}{\mathrm{ml}}$, and $A(0)=10 \ \frac{\mathrm{pg}}{\mathrm{ml}}$. The qualitative trends maintained over these parameter ranges are a clear increase in pro-inflammatory cytokines after stroke onset, with anti-inflammatory cytokines remaining similar to starting level; early dominance of M2 microglia, with eventual takeover of M1 microglia; and increased levels of both M1 and M2 microglia over three days post-ischemic stroke.

Table 3 Local ranges for the parameters in Model (7)

Table 4 Local ranges for the initial conditions in Model (7)

Similarly, Table 4 gives ranges for each initial condition over which it may be individually varied while maintaining the qualitative trends observed in the model output, assuming that the model parameters are fixed to their nominal values in Table 1 and the remaining initial conditions are fixed to their baseline values. Further, Fig. 6 shows how the ratio of M1 to M2 microglia is affected when individually varying the initial conditions of Model (7) over larger ranges, while holding the others fixed at their baseline values. Note that while not much difference is observed beyond the first few hours when varying the initial values of M1 and M2, the initial levels of pro-inflammatory and anti-inflammatory cytokines have a more clear effect: When P(0) is large, as with $P(0)=20 \ \frac{\mathrm{pg}}{\mathrm{ml}}$ in Fig. 6c, the ratio of M1 to M2 microglia does not go below 1, indicating there are always more detrimental M1 than beneficial M2 microglia in the system. Similar results hold when A(0) is too small, as seen in particular when $A(0)=1 \ \frac{\mathrm{pg}}{\mathrm{ml}}$ in Fig. 6d.

Fig. 6

Ratios of M1 to M2 microglial cells (M1 : M2) when varying each initial condition in Model (7) individually, while fixing the parameters to their nominal values in Table 1 and the remaining initial conditions to their baseline values of $M1(0)=100$ cells, $M2(0)=100$ cells, $P(0)=10 \ \frac{\mathrm{pg}}{\mathrm{ml}}$, and $A(0)=10 \ \frac{\mathrm{pg}}{\mathrm{ml}}$ (Color figure online)

Appendix B Effects of Microglia Phenotype Switching With and Without Treatment

The global sensitivity results presented in Figs. 3 and 5 highlight the significant influence of the parameters relating to bidirectional switching between microglia phenotypes on the corresponding model output, in particular when considering the ratio of M1 to M2 microglia. Here, we further analyze the effects of varying the values of these phenotype switching parameters (namely, $s_{M1 \rightarrow M2}$ and $s_{M2 \rightarrow M1}$) on Model (7) and Model (21), respectively, thereby simulating situations both with and without treatment. Figure 7 shows how the ratio of M1 to M2 cells changes when varying the phenotype switching parameters between 0 and 1 for each model, highlighting values at which the behavior of the ratio changes.

In Fig. 7a, we see that when $s_{M1 \rightarrow M2} = 0$ without treatment, there is an initial decrease of the ratio such that M2 is dominant for a short amount of time, after which the ratio grows in an exponential fashion. Similar behavior occurs for values of $s_{M1 \rightarrow M2}$ between 0 and 0.05. When $s_{M1 \rightarrow M2} = 0.05$ and up to 0.1, the ratio initially decreases and M2 dominates for a short amount of time, then begins to increase somewhat linearly. When $s_{M1 \rightarrow M2} = 0.1$, the ratio shows an initial decrease below 1 followed by an increase and eventual leveling off at a value above 1. Similar behavior occurs for values of $s_{M1 \rightarrow M2}$ between 0.1 and 0.25. When $s_{M1 \rightarrow M2} =0.25$, the ratio begins under 1 and decreases slightly before increasing and leveling off at a value below 1. This behavior holds for values of $s_{M1 \rightarrow M2}$ up to around 0.33. When $s_{M1 \rightarrow M2} = 0.33$, the ratio begins under 1 and decreases slightly with no increase before leveling off at a value below 1. For values of $s_{M1 \rightarrow M2}$ between 0.33 and 1, the ratio has an initial decrease and then gradually decreases for the full 72 h, with the ratio remaining below 1.

Figure 7b shows that when $s_{M2 \rightarrow M1} =0$ without treatment, the ratio of M1 to M2 cells starts below 1 and decreases for the full time period, ending at a value below 1. This behavior occurs for values of $s_{M2 \rightarrow M1}$ up until 0.12. At this value, the ratio has an initial decrease and then levels off around 15 h at a value below 1. For values of $s_{M2 \rightarrow M1}$ between 0.12 and 0.16, the ratio decreases initially and then increases, ending at a value below 1 after 72 h. Between 0.16 and 0.6, the ratio has an initial decrease and then increases gradually, ending at a value above 1. For values of $s_{M2 \rightarrow M1}$ larger than 0.6 and up until 0.8, the ratio begins at 1 and increases before moving to a more gradual linear-like increase. For $s_{M2 \rightarrow M1}$ between 0.8 and 1, the ratio begins at 1 and increases in a linear manner, with a more significant increase over time.

In Fig. 7c, we observe that when $s_{M1 \rightarrow M2} = 0$ with treatment applied, the ratio has a short decrease for about 1 h, followed by an increase that rises above 1 around 2.5 h. After this, there is another short decrease until around 10 h. The ratio then increases gradually until about the 45 h mark, at which point the increase becomes more significant. A similar behavior is observed for values of $s_{M1 \rightarrow M2}$ up until 0.01. When $s_{M1 \rightarrow M2} = 0.01$, we see a short initial decrease, followed by an increase; however, the ratio will not go above 1 in this increase. This is followed by a decrease until around 10 h. After this time, the ratio increases gradually and again begins to increase more significantly around 45 h. Similar behavior occurs for $s_{M1 \rightarrow M2}$ values up until 0.05. At this value, we observe a short decrease (about 1 h), followed by a small increase with the ratio not going above 1. A decrease in the ratio occurs around 5 h and persists until about 10 h. After this, the ratio increases gradually until around the 45 h mark, where it then begins to increase with a steeper incline. This trend continues for values of $s_{M1 \rightarrow M2}$ up until around 0.22. When $s_{M1 \rightarrow M2} = 0.22,$ the ratio decreases gradually and levels off until around 45 h, after which it increases but ends at a value below 1. Similar behavior occurs for values of this parameter up until 0.5. When $s_{M1 \rightarrow M2} = 0.5$, the ratio decreases until around the 45-h mark, at which point the ratio increases slightly until around 50 h where it then levels off at a value below 1. A similar trend occurs up until and at $s_{M1 \rightarrow M2} = 1$.

Figure 7d shows that when $s_{M2 \rightarrow M1}=0$ with treatment, the ratio of M1 to M2 cells begins below 1 and decreases until around 45 h. The ratio then increases slightly and levels off at a value under 1. Similar behavior occurs for values up until $s_{M2 \rightarrow M1} = 0.1$. When $s_{M2 \rightarrow M1} = 0.1$, the ratio decreases until around 45 h, after which it increases slightly for the rest of the 72 h and ends at a value under 1. This trend persists for values of $s_{M2 \rightarrow M1}$ up until 0.2. When $s_{M2 \rightarrow M1} = 0.2$, the ratio decreases gradually and then increases until around the 45-h mark, at which point it has a sharper increase and then increases more gradually, ending at a value above 1. This occurs for values up until $s_{M2 \rightarrow M1} = 0.3$. When $s_{M2 \rightarrow M1} = 0.3$ and up until 0.7, there is a short decrease in the ratio (for about 1 h), followed by an increase until around 4 h and another decrease until around 10 h. Then the ratio increases until the 45-h mark, at which point there is a sharper increase for the remaining time. When $s_{M2 \rightarrow M1} = 0.7$, the ratio does not go below 1 for the full 72 h. Instead it increases gradually until around 45 h, when we see a steeper incline. A similar trend occurs for values up until $s_{M2 \rightarrow M1} = 1$, with the ratio increasing more steeply at the end as $s_{M2 \rightarrow M1}$ approaches 1.

Fig. 7

Ratios of M1 to M2 microglial cells (M1 : M2) when varying each microglia phenotype switching parameter individually, while fixing the remaining model parameters to their nominal values in Table 1 and the initial conditions to their baseline values of $M1(0)=100$ cells, $M2(0)=100$ cells, $P(0)=10 \ \frac{\mathrm{pg}}{\mathrm{ml}}$, and $A(0)=10 \ \frac{\mathrm{pg}}{\mathrm{ml}}$ (Color figure online)