Exemplary Results Draft 3

Term Paper Model Results

A reduced mathematical model of the acute inflammatory response: I. Derivation of model and analysis of anti-inflammation.

Angela Reynolds et. al.

Results

A particularly notable paper, which provides further insight into the time-dependency of the acute immune system, is "A reduced mathematical model of the acute inflammatory response: Derivation of model and analysis of anti-inflammation" written by Reynolds et al. The authors constructed a model of the body's inflammatory response and included a time dependent anti-inflammatory response. The hypothesis proposed by Reynolds et. al. is that a time-dependent anti-inflammatory response results in a healthy immune response, compared to a static anti-inflammatory response, and the time-dependent anti-inflammatory response is characterized by a more stable equilibrium between aseptic and septic death which defines the healthy state. Our reproduction of the model will seek to verify this hypothesis and show that the timing of the anti-inflammatory mediator is crucial. The results of Reynold's paper illustrate that the point at which administration of an anti-inflammatory mediator occurs during the response to an infection may compromise outcome.

In the following sections, we have reproduced critical figures from the paper, utilizing the reduced model of the 4 variable system. By replicating these figures, we have verified the system proposed by Reynolds, et. al. This verification is crucial to our ability to provide an extension to the model, both by ensuring that the basis of our extension in sound, and by making sure that our extension investigates portions of the model which are overlooked by our replications.

Figure 5

Part A: Healthy Outcome

Below, I have included on the left, the original Figure 5, part A from the paper by Reynolds et. al. On the right is our replication of the figure.

Reynold's Results	Our Results

The conditions for creating this figure include:

• $k_{pg} = 0.3/h$
• $P[0] = 1$
• $N^*[0] = 0$
• $D[0] = 0$
• $C_{A}[0] = 0.125$

The original figure 5 part A depicts a health outcome, which Reynolds defines as a fixed point with $P = 0$, $N^* = 0$, $D = 0$, and $C_{A} = s_{c} / \mu_{c}$. Referring to the parameters table for the values of $s_{c}$ and $\mu_{c}$, we find that $C_{A} = s_{c} / \mu_{c} = 0.0125 / 0.1$.

We used NDSolve to solve the reduced model of the 4 differential equations, from 0 to 200 hours, and utilizing the initial conditions given in the original figure. From the solutions we calculated, we plotted each of the state variables over the time period of 0 to 200 hours to create each recreated plot to match the original Figure 5. We saw a perfect match of our recreated plots to the original, with no discrepancies.

With regard to the original hypothesis, this figure does not depict any influence of the time-dependent anti-inflammatory variable on the healthy outcome of the system. This figure mainly served as a way for us to identify what characterizes a healthy outcome to our system. From the figures, both the original and our recreation, we can say that a healthy outcome is characterized by a quick removal of the pathogen from the system, a initial rise and then decline of tissue damage, activated phagocytes and anti-inflammatory mediator.

Part B: Aseptic Death

Below, I have included on the left, the original Figure 5, part B from the paper by Reynolds et. al. On the right is our replication of the figure.

Reynold's Results	Our Results

The conditions for creating this figure include:

• $k_{pg} = 0.3/h$
• $P[0] = 1.5$
• $N^*[0] = 0$
• $D[0] = 0$
• $C_{A}[0] = 0.125$

The original figure 5 part B depicts aseptic death, which Reynolds defines as an outcome where pathogen has been eliminated but with high and persistent immune activation and damage, is a fixed point where $P = 0$, $N^*>0$, $D>0$ and $C_{A}>0$.

We used NDSolve to solve the reduced model of the 4 differential equations, from 0 to 200 hours, and utilizing the initial conditions given in the original figure. From the solutions we calculated, we plotted each of the state variables over the time period of 0 to 200 hours to create each recreated plot to match the original Figure 5. We saw a perfect match of our recreated plots to the original, with no discrepancies.

With regard to the original hypothesis, this figure does not depict any influence of the time-dependent anti-inflammatory variable on the aseptic death outcome of the system. This figure mainly served as a way for us to identify what characterizes the aseptic death outcome in our system. From the figures, both the original and our recreation, we can say that aseptic death is characterized by a quick removal of the pathogen from the system, and a rise of tissue damage, activated phagocytes and anti-inflammatory mediator with no decline or removal from the system.

Part C: Septic Death

Below, I have included on the left, the original Figure 5, part C from the paper by Reynolds et. al. On the right is our replication of the figure.

Reynold's Results	Our Results

The conditions for creating this figure include:

• $k_{pg} = 0.6/h$
• $P[0] = 1$
• $N^*[0] = 0$
• $D[0] = 0$
• $C_{A}[0] = 0.125$

The original figure 5 part C depicts septic death, which Reynolds defines as an outcome in which there is insufficient immune activation to clear the pathogen.

We used NDSolve to solve the reduced model of the 4 differential equations, from 0 to 200 hours, and utilizing the initial conditions given in the original figure. From the solutions we calculated, we plotted each of the state variables over the time period of 0 to 200 hours to create each recreated plot to match the original Figure 5. We saw a perfect match of our recreated plots to the original, with no discrepancies.

With regard to the original hypothesis, this figure does not depict any influence of the time-dependent anti-inflammatory variable on the septic death outcome of the system. This figure mainly served as a way for us to identify what characterizes the septic death outcome in our system. From the figures, both the original and our recreation, we can say that septic death is characterized by a steep rise and small decline of the pathogen in the system, and a rise of tissue damage, activated phagocytes and anti-inflammatory mediator with no decline or removal from the system.

Figure 6

Below, I have included on the left, the original Figure 6 from the paper by Reynolds et. al. On the right is our replication of the figure.

Reynold's Results	Our Results

This figure is the bifurcation diagram for the four-variable reduced model. In the paper, Reynolds et. al. describes the bifurcation diagram. They state, septic death comes into existence via a saddle-node bifurcation at $k_{pg}= 0.5137/h$. Health and aseptic death lose stability by transcritical bifurcations at $k_{pg}= 1.5$ and $1.755$, respectively. For $k_{pg}<0.5137$, the model is bistable between health and aseptic death. The model has all three states stable for $0.5137<k_{pg}<1.5$. There is bistability between aseptic and septic death for $1.5<k_{pg}<1.755$. Finally, above $k_{pg}= 1.755$, the only stable state is septic death.

We used a program to establish guesses of the equilibrium points and used FindRoot to solve for the equilibrium points. We then determined the stability of each point and plotted the $N^*$ values against the $k_{pg}$ values with the shade of grey determining the stability of the point.

With regard to the original hypothesis, this figure does not depict any influence of the time-dependent anti-inflammatory variable on the reduced system. This system only serves to depict the qualitative dependence of the existence of stable states on the pathogen growth rate $k_{pg}$.

Our recreation of Figure 6 does differ slightly from that seen in the original. We have shown an additional portion of the diagram was between the $N^*$ values of 0 and 0.6. This portion of the diagram corresponds to a non-physiological state, according to Reynolds, and is not included in the original Figure 6. Further investigation will be required for us to corroborate that the values from the bifurcation diagram do indeed correlate with a non-physiological state.

Figure 7

Below, I have included on the left, the original Figure 7 from the paper by Reynolds et. al. On the right is our replication of the figure.

Reynold's Results	Our Results

In the paper, Reynolds et. al. describes the diagram. The basin of attraction for the health state depends on $C_{A}$. For each constant $C_{A}$level shown, the three-variable subsystem was used to determine the level of initial pathogen that is the threshold between health and death (aseptic or septic), over a range of kpg. Using the reduced model, with initial conditions $N* = 0$, $D =0$, and $C_{A}=0.125$ and with dynamic $C_{A}$, the same was done, giving rise to the curve labeled ‘‘Dynamic’’. The dotted portion of the $C_{A}=0.7$ curve (black) represents a range of $k_{pg}$ where health is the only stable outcome.

In Figure 7, the outcomes associated with a dynamic $C_{A}$ response with those found with a variety of constant levels of $C_{A}$. At different $k_{pg}$ values, we determine the level of initial pathogen that is the threshold between health and death (aseptic or septic). The curve associated with $C_{A}=0$ lies below all other curves; the presence of the anti-inflammatory mediator, whether dynamic or constant, allows a larger initial pathogen load or growth rate to be tolerated over all values of the pathogen growth rate, $k_{pg}$. A notable result of the diagram is that a dynamic anti-inflammatory mediator is almost always more effective than constant levels of the anti-inflammatory mediator at producing a healthy outcome following infection, because the threshold is higher than all other curves at fixed $C_{A}$ values.

This figure is very important because it corroborates our hypothesis. We are investigating the claim that a time-dependent anti-inflammatory response results in a healthier immune response, compared to a static anti-inflammatory response. Our diagram supports this because the boundary between the healthy and unhealthy states is much higher when the anti-inflammatory mediator variable is dynamic (i.e. time-dependent). A higher boundary means that the system can tolerate more initial pathogen amounts over all possible $k_{pg}$ values, allowing for a greater possibility of a healthy immune response.

Our figure deviates a little bit in the shape of the curves from the original figure, but I believe this is only due to our method of defining the equilibrium points. The integrity of the figure is maintained because the same conclusions can be drawn.

Figure 8

Below, I have included on the left, the original Figure 8 from the paper by Reynolds et. al. On the right is our replication of the figure.

Reynold's Results	Our Results

In the paper, Reynolds et. al. describes the diagram. This figure depicts the impact of baseline anti-inflammatory mediator levels in the system on the response to infection. The threshold between health and death depends on the initial anti-inflammatory mediator and pathogen levels in the reduced model. At each $k_{pg}$ value indicated we find the initial $C_{A}$ level that is the threshold between health and death outcomes, given that $N^*$ and $D$ are initially at their baseline (zero) levels. Initial conditions to the left of each curve lead to health while those to the right give rise to either septic or aseptic death. The baseline level of $C_{A}$, corresponding to health in the reduced model, is indicated by the blue dashed line.

We observe that slower rates of pathogen growth, $k_{pg}$ give a greater area of healthy resolutions possible for various combinations of initial pathogen and anti-inflammatory mediator values. When $k_{pg}$ approaches 1, there area to the left of the curves (i.e. the area which leads to a healthy resoltion) decreases drastically. A deviation of our reproduction is the smooth nature of the lines. I suspect that this is due to the method of recreation we chose. We may not have had the time or resources available to get the fine detail as shown in the original model. As a result of meshing our results, we lose some of the local variation of each boundary line. I do not believe this detracts from the significance of the figure, nor the conclusions we can draw from the figure.

This figure does not specifically support the hypothesis because we are only considering the initial value of the anti-inflammatory mediator, with no time factor considered. However, this figure does allow us to observe basins of attraction of the healthy state which will be important as we consider our extension of adding a term to the reduced model to mimic the response of adding an externally applied anti-inflammatory mediator to the body to assist with achieving a healthy resolution.