Exemplary Final Term Paper 3

Final Term Paper

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

Angela Reynolds et. al.

Introduction

The world is full of germs, known as pathogens. Pathogens are microbes or microorganisms such as viruses, bacterium, prions, or fungi that causes disease in an animal or plant host. In humans, pathogens are particularly concerning because they lead to an inflammatory response in our bodies . When the body is infected, it mounts an initial immune response, known as the acute inflammatory response, to rid itself of the pathogens and restore health. In a healthy response, the inflammatory response becomes activated, clears the pathogen (in the event of infection), begins a repair process and stops. Uncontrolled acute inflammation due to infection is defined clinically as sepsis and can culminate in organ failure and death. Death can occur in two possible ways: “septic death” or “aseptic death.” The aseptic death state corresponds to an outcome where pathogen has been eliminated but with high and persistent immune activation and damage. The septic death state corresponds to a state in which there is insufficient immune activation to clear pathogen.

Septic death occurs when the acute immune response generates uncontrolled inflammation, which leads to severe tissue damage, and the pathogen is not eliminated either. Suffering from uncontrolled inflammation due to a infection is called “sepsis”. Sepsis is a common condition, and can be caused by any sort of infection which is why it is so concerning. It is characterized by a systemic inflammatory response that can quickly become fatal and is seen in association with a large number of compromising diseases or infections. Severe sepsis occurs in more than 750 000 individuals in the United States each year, with a hospital mortality of about 30%. Current clinical treatments for sepsis are utilizing antibiotics, surgery to remove the source of the infection, and utilizing corticosteroids. Despite the use of these therapies, the mortality rates of patients with sepsis remains high. One reason for the lack of effective treatments may be that the complex nature of the inflammatory response renders the effect of targeting isolated components of inflammation difficult to predict. Another complication stems from how the initial progression of systemic inflammation can have different manifestations depending on how it is triggered, i.e., infection or trauma. Understanding the complexities of the inflammatory response is important for developing a treatment for sepsis.

Much research has been devoted to understanding the immune systems, in particular, the acute immune response. Advances in understanding the host immune response have fueled considerable interest finding a definitive solution to sepsis mortality. Day et al. completed an experiment which considers scenarios of repeated endotoxin administration to purposefully induce sepsis in animal models. Their results showed the the administration of the second dose of endotoxins was time dependent; at certain points in the immune response the body had recovered enough to fight the secondary wave of endotoxins. 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 death and the healthy state. Our reproduction of the model will sought to verify this hypothesis and show that the timing of the anti-inflammatory mediator is crucial.

We were able to accurately reproduce the model and found results which supported our hypothesis. Our results clearly showed that a dynamic, time-responsive anti-inflammation response creates a greater number of combinations between pathogen growth rate and initial pathogen which still result in a healthy outcome. The results of Reynold's paper illustrate that a dynamic anti-inflammatory response provides the best odds for a patient to find a healthy resolution to the infection. This paper by Reynolds, et al. provides new insight into the inflammatory response and offers possible sepsis treatments which may be effective due to the new knowledge of the acute inflammatory response. We sought to expand on the results of Reynolds paper by investigating the most effective method of administering an external anti-inflammatory mediator for patients who have been diagnosed with sepsis or septic shock. The use of corticosteroids as a means of therapy is a very controversial topic in literature concerning sepsis. It has been repeatedly reported that high, "stress" doses of corticosteroids have been proven to be harmful for patients suffering from sepsis, even increasing the mortality rates. However, two methods are utilized in clinical applications: applying additional corticosteroids (anti-inflammatory drugs)either as one continuous infusion at a lower dose, or as daily low doses. We investigated the effects of incorporating these two treatment methods into our model. We sought to determine which method was a more effective treatment of sepsis management.

Description

In this paper, we will be focusing on the reduced of the acute inflammatory response proposed by Reynolds, which consists of a system of four differential equations in which there are 4 distinct state variables to consider in regard to an acute inflammatory response. The variable $P$ represents the level of pathogen in the body, and indicates the level of infection. The variable $N^*$the number of activated phagocytes in the body. The amount of this variable can suggest the amount of immune response. The variable $D$ represents tissue damage due to inflammation. When phagocytes respond to an infection, their presence may also lead to collateral tissue damage, due to consuming healthy tissue. The variable $C_{A}$amount of anti-inflammatory mediators in the body. These mediators may be molecules such as cortisol and interleukin-10, and are excreted by the body to control the inflammatory response. Each of these variables are key to describe the interactions between the initiating event, inflammation, anti-inflammation, and damage.

The reduced model was originally developed by combining subsystems created by Reynold, et al. We will accept their methods of combination and derivation to produce the final reduced model, because our interest lies with the complexities offered by the combined model and we desire to expand upon their reduced model. The parameters used for the reduced model are provided in the table below. The parameters do not change, except where noted, and the values for the parameters are selected to remain within the given ranges and constraints found in the experimental literature. Parameters that could not be documented from existing data were estimated such that the subsystems behave in a biologically appropriate manner for plausible levels of the anti-inflammatory mediators.

The final reduced model incorporates differential equations for amounts of pathogen($P$), activated phagocytes ($N^*$), tissue damage ($D$), and anti-inflammatory mediator ($C_{A}$). The production of the anti-inflammatory mediator is associated with the presence of activated phagocytes and elevated markers of tissue damage, so it is important for us to use the final reduced mode because it incorporates the $C_{A}$ term. The anti-inflammatory mediator ($C_{A}$) regulates the immune response by inhibiting the production and effects of activated phagocytes($P$) and tissue damage ($D$). More specifically, the presence of $C_{A}$ decreases the ability of activated phagocytes to react to other cell types, reducing their effectiveness against the pathogen, their induction of damage, and their production of additional $C_{A}$. The recruitment of $C_{A}$ by tissue damage ($D$) is similarly inhibited. It is clear that the term $C_{A}$ is critical to each of the differential equations created to model each of the state variables.

We will be investigating 4 key figures from the paper, figures 5, 6, 7 and 8. These figures depict the typical response of the 4 state variable at certain initial conditions and values of the pathogen growth rate $k_{pg}$, a bifurcation diagram of the system with regard to pathogen growth rate and activated phagocytes ($N^*$), and two boundary diagrams which depict the basins of attraction and the boundaries between life and death for a patient based on certain initial conditions and values of the pathogen growth rate $k_{pg}$. In order to solve the system, we will be utilizing the built-in Mathematica function of NDSolve. For the first figure, we are provided with all parameter values as well as initial conditions, which will allow us to easily replicate the diagram. In order to create the bifurcation diagram, we will be utilizing the built-in Mathematica function of FindRoot. We will "guess" at the equilibrium points of the system, and use the FindRoot function to establish the exact points. By applying a Jacobian matrix and finding the eigen values, we can also determine the stability of each of the equilibrium points and plot them accordingly in a color of our choosing. For the final 2 boundary diagrams, we will be binary search method to quickly determine the equilibrium point, with regard to various combinations of initial conditions. We are provided with those initial conditions which are held constant, and can vary the others accordingly to construct our figures. It should be noted that we will be showing the basin of attraction in one figure by also plotting the boundaries with fixed values for $C_{A}$.

In order to expand our model to include an external dose of corticosteroids (anti-inflammatory mediator), we will be creating functions to model the levels of additional $C_{A}$ in the system. We will combine the external and internal values of $C_{A}$, and use the total as the variable which influences the other 3 state variables. We will be using NDSolve extensively to accomplish this, as well as the binary search method so that we can observe each variable's behavior over time, as well as the influence of each treatment method on the life and death boundary which characterizes the system.

Pathogen

The differential equation for the pathogen variable is defined as:

$\frac{\text{dP}}{\text{dt}}=k_{\text{pg}}P(1-\frac{P}{p_{\infty}})-\frac{{k_{pm}}{s_{m}}P}{\mu_m+{k_{mp}}P}$

This equation is comprised of two terms. The first term, defined by $k_{\text{pg}}P(1-\frac{P}{p_{\infty}})$ is a logistic growth term which is used to account for the pathogen's ability to rapidly divide and incorporate the dynamics of the pathogen population into the model. The second term is derived from the following table which depicts the reactions with the non-specific local response ($M$) and pathogen levels ($P$).

Table 1: Reactions involving the local immune response $(M)$ and pathogen $(P)$

$M + P \xrightarrow{k_{pm}} M$	$P$ is destroyed at the rate $k_{pm}$ when it encounters $M$
$M + P \xrightarrow{k_{mp}} P$	$M$ is consumed at the rate $k_{mp}$ when it encounters $P$
$* \xrightarrow{s_{m}} M$	Source of $M$
$M \xrightarrow{\mu_{m}}$	Death of $M$

From the reactions in Table 1, and based on mass action kinetics, the following equations are derived:

$\frac{\text{dM}}{\text{dt}}=s_{m} - \mu_{m}M - k_{mp}MP$

$\frac{\text{dP}}{\text{dt}}= - k_{pm}MP$

Reynolds et al. assumed that the local response reaches a quasi-steady state and so $M = \frac{s_{m}}{\mu _{m}+k_{mp}P}$ was substituted into the pathogen equation derived from Table 1, which gives the second term of the differential equation used to define the pathogen amount in the final reduced model.

Activated Phagocytes

The differential equation for the activated phagocytes variable is defined as: $\frac{\text{dN}^*}{\text{dt}}=\frac{s_{\text{nr}}R}{\mu _{\text{nr}}+R}-\mu _nN^*$

A key component of the acute immune response is the removal of the pathogen by immune cells, known as phagocytes. Resting phagocytes are activated by pathogen, and once activated, a phagocyte becomes efficient at eliminating pathogens. In order to properly capture the transition of phagocytes from the resting state to the activated state, the activated phagocyte equation was primarily derived from the following table:

Table 2: Reactions involving resting and activated phagocytes

$N_{R} \xrightarrow{k_{np}P + k_{nn}N^*} N^*$	Activation of the resting phagocytes $(N_{R})$ is induced by the presence of pathogen $(P)$ and by positive feedback from the activated phagocytes $(N^*)$ via pro-inflammatory cytokines
$* \xrightarrow{s_{nr}} N_{R}$	Source of $N_{R}$
$N_{R} \xrightarrow{\mu_{nr}} M$	Death of $N_{R}$
$N^* \xrightarrow{\mu_{n}}$	Death of $N^*$

From these series of reactions, the following equations can be derived:

$\frac{dN_{R}}{dt}=s_{nr} - \mu_{nr}N_{R} - R_{1}N_{R}$

$\frac{\text{dN}^*}{\text{dt}}=R_{1}N_{R} - \mu_{n}N^*$

and $R_{1}$ is defined as

$R_{1} = f\left(k_{\text{nn}}N^*+k_{\text{np}}\right.\text{P}$

Reynolds et al. assume that $N_{R}$ is in a quasi-steady state, and so the two differential equations can be reduced to a single equation which defines the activated phagocytes.

However, when activated phagocytes respond to an infection, their presence in the tissue not only kills pathogens, but may also lead to additional tissue damage. Damaged tissue releases pro-inflammatory cytokines, molecules which causes further phagocyte activation. This positive feedback interaction between phagocytes and damage was accounted for by modifying the function of $R_{1}$ to include a term accounting for the additional phagocytes activated by the tissue damage and the finalized function becomes $R$ and is defined as:

$R = f\left(k_{\text{nn}}N^*+k_{\text{np}}\right.\text{P+}\left.k_{\text{nd}}D\right)$

In normal individuals, the anti-inflammatory mediator inhibits the activation of phagocytes and reduce the ability of activated phagocytes to attack pathogen. We incorporate this inhibition into the final equation for $N^*$ by defining R with the function, $f$.

$f(V)=\frac{V}{\left(1+\left(\frac{C_A}{c_{\infty }}\right){}^2\right)}$

The parameter $c_{\infty}$ is set such that when the anti-inflammatory mediators reach their maximum level in response to an infection, their inhibitory effects are roughly equivalent to a 75% reduction in the amount of phagocytes, and thus the amount of inflammation.

Tissue Damage

The differential equation for the tissue damage variable is defined as: $\frac{\text{dD}}{\text{dt}}=k_{\text{dn}}f_s\left(f\left(N^*\right)\right)-\mu _dD$

The first term of $k_{\text{dn}}f_s\left(f\left(N^*\right)\right)$ accounts for the relationship between activated phagocytes and tissue damage. At low counts, activated phagocytes do not cause significant damage. However, as they accumulate in response to an infection, the activated phagocytes will cause tissue damage to accrue. Finally, once levels of activated phagocytes are sufficiently high, damage saturates, such that the activation of additional phagocytes has little impact on damage creation. This nonlinearity in the induction of damage by activated phagocytes is modeled via the Hill-type function, $f_{s}$. The saturation function is defined as:

$f_s(V)=\frac{V^6}{\left(x_{\text{dn}}^6+V^6\right)}$

Also, to complete the model and include the factor of the anti-inflammatory mediator which leads to inhibition of activated phagocytes, $N^*$ is replaced with $f (N^*)$. The ability of activated phagocytes to cause damage is also inhibited by the anti-inflammatory mediator, which is why the term of $f (N^*)$ is included in the saturation function, so we can be sure that the proper behavior is achieved in the overall function to define tissue damage $D$.

The second term in the overall differential equation for the tissue damage variable represents tissue repair, resolution, and regeneration.

Anti-Inflammatory Mediator

The differential equation for the anti-inflammatory variable is defined as: $\frac{\text{dC}_A}{\text{dt}}=s_c+\frac{k_{\text{cn}}f\left(N^*+k_{\text{cnd}}D\right)}{1+f\left(N^*+k_{\text{cnd}}D\right)}-\mu _cC_A$

This $(C_{A})$ variable is time dependent and the production of the anti-inflammatory mediator is associated with the presence of activated phagocytes and elevated levels of tissue damage. The anti-inflammatory mediator $(C_{A})$ regulates the immune response by preventing the production and effects of activated phagocytes and tissue damage. The presence of $(C_{A})$ decreases the ability of activated phagocytes to react pathogens, which reduces their effectiveness against the pathogen, the causation of tissue damage, and the production of additional $(C_{A})$. The recruitment of $(C_{A})$ by tissue damage $(D)$ is similarly inhibited. Also, $(C_{A})$ compromises all means of activation of resting phagocytes.

The $C_{A}$ equation contains a source of $C_{A}$, denoted $s_{c}$, and a term modeling the production of anti-inflammatory mediator from damage and activated phagocytes, which takes the form $\frac{k_{cn}N^*+k_{cnd}D}{1+N^*+k_{cnd}D}$, before inhibition is incorporated. This expression defines how $k_{cnd}$ controls the effectiveness of damage, relative to activated phagocytes, in producing $C_{A}$. In order to include inhibition in this term, we utilize the function $f(V)=\frac{V}{\left(1+\left(\frac{C_A}{c_{\infty }}\right){}^2\right)}$ so that the final term becomes $\frac{k_{\text{cn}}f\left(N^*+k_{\text{cnd}}D\right)}{1+f\left(N^*+k_{\text{cnd}}D\right)}$ and factors in how the amount of $C_{A}$ will inhibit phagocytes and tissue damage. The anti-inflammatory mediator $(C_{A})$regulates the immune response by inhibiting the production and effects of activated phagocytes and damage.

Additional Functions

These functions do not define any of the state variables but their importance to the reduced system is explained in the appropriate sections.

$R = f\left(k_{\text{nn}}N^*+k_{\text{np}}\right.\text{P+}\left.k_{\text{nd}}D\right)$

$f(V)=\frac{V}{\left(1+\left(\frac{C_A}{c_{\infty }}\right){}^2\right)}$

$f_s(V)=\frac{V^6}{\left(x_{\text{dn}}^6+V^6\right)}$

Extension

In order to include an external treatment of corticosteroids to act as additional anti-inflammatory mediators in the acute immune response, we needed to create functions to model the two types of treatment: steady infusion, or daily doses. The steady infusion was modeled by an arbitrary unit step function which decayed exponentially at the end of the infusion duration (we selected a 7 day duration, based on scientific literature).

$C_{A, external} = \frac{1}{2}+\frac{1}{\pi }\text{ArcTan}[a*t]$

We chose a value of $a=10^4$, which causes the function to be a very close approximation of a unit step function.

The daily treatment method was modeled by a recurrent exponential function, with reapplication every 24 hours, and also continued for 7 days.

$C_{A, external} = \text{A}*\left(\frac{1}{2}+\frac{1}{\pi }\text{ArcTan}\left[a*t\right]\right)* e^{\left[-\frac{\text{Log}[2]* t}{t_{1/2}}\right]}$

Again, we chose a value of $a=10^4$, which causes the function to be a very close approximation of a unit step function. The new parameter of $A$ creates the amplitude of the function, which allows us to distinguish between a low dose and high dose. When the exponential term is added, the clearance of the drug becomes incorporated into the equation to model the treatment. The term $t_{1/2}$ is the half life of the drug, in other words, how many hours it takes for the drug to be reduced to half its original volume. We selected a half life of $t_{1/2}=10$ hours, which is a typical value for cortisone or hydrocortisone, which are two common corticosteroids used to treat sepsis. By manipulating the time term, we can reapply the function every 24 hours.

We also sought to verify that a high, "stress" dose of corticosteroids was detrimental to patients, and we modeled this by a single exponential decay function with a much greater amplitude value.

Assumptions

• The equations in Table 1 are assumed to be correct as they are the basis for the defining the differential equation for the pathogen variable.
• It is assumed that the local response reaches quasi-steady state and we can substitute $M = \frac{s_{m}}{\mu _{m}+k_{mp}P}$ into the pathogen equation, as stated in the pathogen section.
• It is assumed that the resting phagocyte variable $N_{R}$ is in quasi-steady state, so that the $\frac{N_{R}}{N^*}$ system derived from Table 2 can be reduced to a single equation defining $N^*$.
• It is assumed that all anti-inflammatory effects can be included in the variable $C_{A}$.
• It is assumed that $C_{A}$ provides uniform inhibition of phagocytes and tissue damage, though it would also be reasonable to consider different levels of inhibition by the anti-inflammatory mediator for each interaction.
• In our extension, we assume our values for the various treatment methods are appropriate for low doses and high doses.
• We also assume that the externally applied anti-inflammatory mediator is cleared from the body at a rate which can be modeled by an exponential decay function.

Parameters

Table 3: Parameters

Parameter	Value	Description
$k_{pm}$	$0.6/M-units/h$	Rate at which the general local immune response eliminates pathogen
$k_{mp}$	$0.01/P-units/h$	Rate at which the general local immune response is exhausted by the pathogen $P$
$s_{m}$	$0.005 M-units/h$	Rate at which the the general local immune response is strengthened
$\mu_{m}$	$0.002/h$	Decay rate for the general local immune response
$k_{pg}$	$0.021-2.44/h$	Growth rate of the pathogen $P$; various values are used.
$p_{\infty}$	$20x10^{6} /cc$	Maximum pathogen population.
$k_{pn}$	$1.8/N^*-units/n$	Rate at which activated phagocytes $N^*$ consume pathogen
$k_{np}$	$0.1/P-units/h$	Activation of resting phagocytes by pathogen
$k_{nn}$	$0.01/N^*-units/h$	Activation of resting phagocytes by activated phagocytes $N^*$ and their cytokines
$s_{nr}$	$0.08 N^*-units/hr$	Rate at which resting phagocytes are created
$\mu_{nr}$	$0.12/h$	Decay rate of resting phagocytes
$\mu_{n}$	$0.05/h$	Decay rate of activated phagocytes
$k_{nd}$	$0.02/D-units/h$	Rate of activation of resting phagocytes by tissue damage $D$
$k_{dn}$	$0.35 D-units/h$	Maximum rate of tissue damage $D$ produced by activated phagocytes $N^*$ (including cytokines and free radicals)
$x_{dn}$	$0.06 N^*-units$	Level of activated phagocytes $N^*$ necessary to increase rate of damage production to half its maximum
$\mu_{d}$	$0.02/h$	Decay rate of tissue damage $D$
$c_{\infty}$	$0.28 C_{A}-units$	Strength of the anti-inflammatory mediator $C_A$
$s_{c}$	$0.0125 C_{A}-units/h$	Rate at which anti-inflammatory mediator is produced.
$k_{cn}$	$0.04 C_{A}-units/h$	Maximum rate of production of the anti-inflammatory mediator $C_A$
$k_{cnd}$	$48 N^*-units/D-units$	Ratio of the rate of anti-inflammatory mediator produced by activated phagocytes $N^*$ and tissue damage $D$
$\mu_{c}$	$0.1/h$	Decay rate of the anti-inflammatory mediator $C_A$

Results

The results of Reynold's paper illustrate that a time-dependent anti-inflammatory mediator is necessary to improve the possible outcomes of a patient suffering from an acute inflammatory response. Our reproduction of the model will seek to verify these results and show that the responsiveness of the anti-inflammatory mediator is crucial. 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.

When implementing our extension, we sought to compare the corticosteroid treatment methods that have been used by clinicians for patients suffering from sepsis. We compared the steady infusion method to a daily dose administration, with the hope of verifying which method was more effective in stabilizing the patient. We also investigated the effect of a single, high stress dose on the system, in order to verify that those doses were indeed ineffective, and potentially more harmful for a patient suffering from sepsis.

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 original figures from Reynold, et. al. which shows the behavior of each state variable over the course of 200 hours. The initial conditions were selected so that this figure would model a typical healthy outcome.	Our reproduced figures, which were an exact match to the original figures from Reynold's paper. We are able to characterize a healthy outcome as complete removal of pathogen, large decline of phagocytes and tissue damage, and a minimal decline in the anti-inflammatory mediator variable.

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 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 original figures from Reynold, et. al. which shows the behavior of each state variable over the course of 200 hours. The initial conditions were selected so that this figure would model a typical aseptic outcome.	Our reproduced figures, which were an exact match to the original figures from Reynold's paper. We are able to characterize a aseptic death as complete removal of pathogen, no decline of phagocytes and tissue damage, and a carrying capacity reached in the anti-inflammatory mediator variable.

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 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 original figures from Reynold, et. al. which shows the behavior of each state variable over the course of 200 hours. The initial conditions were selected so that this figure would model a typical septic outcome.	Our reproduced figures, which were an exact match to the original figures from Reynold's paper. We are able to characterize septic death as only slight removal of pathogen, no decline of phagocytes and tissue damage, and a carrying capacity reached in the anti-inflammatory mediator variable.

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 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
The original figure from Reynold, et. al. which shows the bifurcation diagram of the activated phagocyte variable versus the pathogen growth rate. This diagram illustrates the dependence of each of the state variables on the parameter of $k_{pg}$.	Our reproduced figure, was a very close match to the original, and includes an additional portion of bifurcation between phagocyte values of 0 and 0.6. We are also unable to determine the outcome at each of the branches, aseptic or septic death or health, without further analysis.

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
The original figure from Reynold, et. al. which shows the boundaries between health (below the line) and death (above the line), when comparing the initial pathogen amount to the growth rate of the pathogen.	Our reproduced figure, which was a close match to the original figures from Reynold's paper. We are able to concur with Reynold's conclusion that a dynamic, time-dependent anti-inflammatory response is better, and creates a greater possibility for healthy outcomes.

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
The original figure from Reynold, et. al. which shows the threshold between health and death with respect to initial pathogen and initial anti-inflammatory mediator amounts. Initial conditions on the left of each curve will lead to health, but initial conditions to the right of each curve lead to death.	Our reproduced figure, which was a very close match to Reynold's paper. We confirm the connection between the initial values of each variable, though our exact solutions differ slightly.

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 faster 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 right 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.

Extension

Our externally applied treatments were incorporated into the model once a trigger value of the activated phagocytes ($N^*$)is reached, because a change in the number of white blood cells is a means of diagnosing sepsis. We tested a trigger value for N * at various levels, in an attempt to correlate our results to the clinical standards for diagnosing sepsis: a white blood count of greater than 12,000/cu mm. If the trigger point was reached, then the various treatments were applied, and we observed the variables values over a period of $t = 400$ hours. We also modeled the new boundary between life and death when graphing the dynamic boundary between life and death for Figure 7.

Steady Infusion Dose

In these figures, we used a unit step function which ends in an exponential decay to model a corticosteroid treatment which is delivered intravenously for 7 days. We utilized trigger values of $N^* = 0.25, 0.5, 0.75$.

$N^* = 0.25$

Steady Infusion Treatment at $N^* = 0.25$

The behaviors of the state variables over the course of 400 hours, with treatment being applied when $N^* = 0.25$.

We observe a few interesting results from applying a steady infusion of low dose drug, once the phagocyte level reaches a value of 0.25. Firstly, we see that the infusion seems to cause the pathogen to increase drastically, while also suppressing tissue damage. When the infusion ends, however, tissue damage increases drastically, which may suggest that a higher dose of anti-inflammatory is needed, or that no treatment should be implemented at a phagocyte level of 0.25.

A comparison between the boundary between life and death when $C_{A}$ is defined by the dynamic differential equation, and when $C_{A}$ also includes a steady infusion treatment of additional anti-inflammatory mediator.

We observe that the new dividing boundary between life and death, with the external mediator, has surpassed the original dynamic line which is indicative of a time-dependent immune response.

$N^* = 0.5$

The behaviors of the state variables over the course of 400 hours, with treatment being applied when $N^* = 0.5$.

We see similar results to the previous figure. However, the external dose is now applied at a later time.

A comparison between the boundary between life and death when $C_{A}$ is defined by the dynamic differential equation, and when $C_{A}$ also includes a steady infusion treatment of additional anti-inflammatory mediator.

In this figure, applying the dose at a later time had even further increased the area which leads to a healthy resolution for the patient.

$N^* = 0.75$

The behaviors of the state variables over the course of 400 hours, with treatment being applied when $N^* = 0.75$.

With a higher trigger point for the activated phagocytes, the shape of some of the curves has changed slightly, though they are very similar to each of the previous figures at lower trigger values for $N^*$.

A comparison between the boundary between life and death when $C_{A}$ is defined by the dynamic differential equation, and when $C_{A}$ also includes a steady infusion treatment of additional anti-inflammatory mediator.

In this figure, we observe that there no increase of the life and death boundary, so we can conclude that there is a maximum value of activated phagocytes at which there is no additional benefit to administering an external corticosteroid treatment by steady infusion.

Daily Dose

In these figures, we used a repeated exponential decay function to model a corticosteroid treatment which is periodically for 7 days. We utilized trigger values of $N^* = 0.25, 0.5, 0.75$.

$N^* = 0.25$

In these figures, we can clearly see the relationship between the variables, as all are affected by the changing value of the external dosage of anti-inflammatory mediator. During dosage, there does appear to be some temporary suppression of tissue damage and the activated phagocytes, but immediately upon removal of the drug, they increase largely.

A comparison between the boundary between life and death when $C_{A}$ is defined by the dynamic differential equation, and when $C_{A}$ also includes a daily treatment of additional anti-inflammatory mediator.

In this figure, we once again see an increase in the area under the curve which means a greater likelihood of a healthy resolution for a patient.

$N^* = 0.5$

In these figures, we see very little change in shape from the previous results. It does appear that waiting for a larger amount of activated phagocytes, will suppress the tissue damage longer, but again, with removal of the drug, we see large increases in the tissue damage and activated phagocyte variables.

A comparison between the boundary between life and death when $C_{A}$ is defined by the dynamic differential equation, and when $C_{A}$ also includes a daily treatment of additional anti-inflammatory mediator.

In this figure, we once again see an increase in the area under the curve which means a greater likelihood of a healthy resolution for a patient.

$N^* = 0.75$

In these figures, we see little to no change from the previous sections, save for the delay of the introduction of the external $C_{A}$ function, due to a larger value of $N^*$ being utilized.

A comparison between the boundary between life and death when $C_{A}$ is defined by the dynamic differential equation, and when $C_{A}$ also includes a daily treatment of additional anti-inflammatory mediator.

In this figure, we observe that there no increase of the life and death boundary, so we can conclude that there is a maximum value of activated phagocytes at which there is no additional benefit to administering an external corticosteroid treatment by daily doses.

High Stress Dose

In these figures, we used a a single, high amplitude, exponential decay to model a corticosteroid treatment which is such a high dose that it 'stresses' the system. We utilized trigger values of $N^* = 0.25, 0.5, 0.75$.

$N^* = 0.25$

Many of the figures exhibit similar forms to previous instances. We observe that when the stress dose is applied, causes the suppression of activated phagocytes and tissue damage, and as soon as the drug clears from the body, these variables continue to increase drastically.

A comparison between the boundary between life and death when $C_{A}$ is defined by the dynamic differential equation, and when $C_{A}$ also includes a high dose treatment of additional anti-inflammatory mediator.

Interestingly enough, we do see an increase of the life and death boundary of this figure. It is possible, that at a lower value of $N^*$, presumably earlier in the infection process, that any additional anti-inflammatory mediator will be beneficial to some degree.

$N^* = 0.5$

Table 11: Stress Dosage with $N^* = 0.5$

The figures are similar to previous instances. When the stress dose is applied, there is suppression of activated phagocytes and tissue damage, and as soon as the drug clears from the body, these variables continue to increase drastically.

A comparison between the boundary between life and death when $C_{A}$ is defined by the dynamic differential equation, and when $C_{A}$ also includes a high dose treatment of additional anti-inflammatory mediator.

Unexpectedly, we see an even larger increase of the boundary between life and death in this figure. This means a greater likelihood of a healthy resolution for a patient, despite many studies which dispute the medical benefits of a stress dose of corticosteroids as a treatment for sepsis or septic shock.

$N^* = 0.75$

Table 12: Stress Dosage with $N^* = 0.75$

We see no significant changes from previous figures.

A comparison between the boundary between life and death when $C_{A}$ is defined by the dynamic differential equation, and when $C_{A}$ also includes a high dose treatment of additional anti-inflammatory mediator.

We achieve our expected result, that there is no additional benefit of a stress dose of corticosteroids in the treatment of sepsis. However, our model also does not show that there is any detrimental effect (i.e. we see no lowering of the boundary line). This may suggest that those studies which observed detrimental effects from a stress dose could have been based only on the circumstances of the patient.

Discussion

Our model is a reduced model of the acute inflammatory response of the body to a pathogen and by Reynold's own admission, this model as been immensely simplified, and therefore is considered reduced. We observed the 4 variables of $P$ (the amount of pathogen), $N^*$ (the amount of phagocytes), $D$ (the amount of tissue damage), and $C_{A}$ (the amount of anti-inflammatory mediator). Recall that our hypothesis was 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 the death and the healthy state. In two of our figures, we observed that our model did indeed support this hypothesis.

In Figure 7, we observed that the model of a dynamic anti-inflammatory mediator ($C_{A}$) which is responsive to time and the fluctuations of the variables for activated phagocytes ($N^*$) and tissue damage ($D$), showed a higher boundary between life and death. When compared to constant values of the $C_{A}$, the maximum amount of initial pathogen tolerated in the body, and still allowing a healthy resolution, was greatest when $C_{A}$ was able to change and fluctuate. This evidence strongly supports our hypothesis, perhaps not in regard to the stability of the health and death boundary, but it does promote the importance of modeling the acute immune response with a time-dependent anti-inflammatory response. It is also important to note that Figure 7 depicts the boundary between life and death over a range of pathogen growth rate ($k_{pg}$) values and at the smallest values of $k_{pg}$, the boundaries are undefined because the patient's health cannot be perturbed to critical levels, i.e. death, and thus health is the only resolution.

In Figure 8, we depicted the relationship between the initial amount of anti-inflammatory mediator ($C_{A}$) and the initial amount of pathogen ($P$) and found the equilibrium points which defines the boundary between life and death. At smaller growth rates $k_{pg}$, of the pathogen, there is a greater chance of a healthy resolution to the infection. The faster the growth rate of the pathogen, the smaller chance of survival for the patient. This figure was not specifically supportive of our hypothesis, except to show that at lower growth rates, the initial value of $C_{A}$ is critically linked to the intital value of pathogen. In the figure, we see that if $k_{pg}$ is at a low level, by increasing the initial value of $C_{A}$, we can allow the system to handle a larger initial value of $P$.

As our results and the original figures had minimal discrepancies, we draw similar conclusions to those of Reynolds et. al., though they discuss some additional observations. First, they claim that the anti-inflammation expands the basin of attraction of health compared to that present in models lacking anti-inflammation, which is a desirable feature, because we wish to avoid sepsis or septic shock in patients. Second, Reynolds et. al. assert that the figures demonstrate an advantage, in terms of healthy resolution of infection, conferred by the dynamic nature of the anti-inflammatory response, in comparison to a tonic response. This advantage holds in all situations except for the mildest of infections, which, in any case, do not present a vital threat. This is illustrated in current clinical practice, where distressing symptoms associated with mild infections are alleviated by the co-administration of antibiotics and anti-inflammatory mediators. The reduced model also underlines the importance of the different response rates of substances promoting inflammation, represented in the model by N*, and of the anti-inflammatory mediator that limit this response. They suggest that these rates are fairly well tuned to optimize healthy outcomes to pathogenic infection.

We have found the limitations of this reduced model to be difficult when attempting to translate our results into applicable conclusions. The lack of unit for each of the state variables is particularly frustrating because we cannot correlate any of the figures to a defined situation that may be experienced in a clinical setting. If the initial amount of pathogen and anti-inflammatory mediator could be determined in a patient and figure 8 could be recreated with pertinent units, this figure could serve as a basis for determining how aggressive the treatment methods should be.

This model could provide a way to diagnose the possible resolution of a patient's infection, if certain parameters could be measured. Currently, when a doctor is diagnosing a patient with sepsis or septic shock, it is not based on the measurement of pathogen, tissue damage or the anti-inflammatory mediator explicitly. Below are several indicators, according to the American College of Chest Physicians, which should be used to diagnose sepsis :

1. a body temperature greater than 38C or less than 36C
2. a heart rate greater than 90 beats per minute
3. tachypnea, manifested by a respiratory rate greater than 20 breaths per minute, or hyperventilation, as indicated by a PaCO2 of less than 32 mm Hg
4. an alteration in the white blood cell count, such as a count greater than 12,000/cu mm, a count less than 4,000/cu mm, or the presence of more than 10 percent immature neutrophils (“bands”)

In the list, it can be observed that only the amount of phagocytes in the body may lead the clinician to a diagnosis of sepsis or septic shock. Suspicion of infection is involved in the diagnosis, but with no need to quantify the amount of pathogen. The inclusion of the relationship between the 4 state variables involved in Reynolds model to the diagnostic variables of patient temperature, heart rate, and respiratory rate would allow this model to become a diagnostic tool.

We were able to reproduce figure 5 perfectly, with no observable deviations for the original figure. We were also able to reproduce figure 6 with slight formatting difference from the original model. We saw an extra branch to the bifurcation diagram, which was not included by Reynold's because the branch represented non-physiological conditions.

In Figure 7, 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.

We saw some discrepancies between our recreation of Figure 8 and the original figure from Reynold's paper. The specific curvature of each boundary curve is not observed in our recreation. 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.

In their model, Reynolds et. al. continue their analysis with a figure 9, which depicts the effect that manipulating the anti-inflammatory variable can have. In many studies, it has been observed that a large dose of anti-inflammation drug can be disastrous for patients who have proceeded far enough into sepsis . The high amount of anti-inflammatory mediator does not halt the progression of the infection, but rather drives it to result in further damage, and typically death. The complications associated with the use of corticosteroids (anti-inflammatory mediators) are dependent on the dose, the dosing strategy, and the duration of therapy. Much research, including Reynold's, discusses the potential applications for a low dose of anti-inflammation drug applied repeatedly, though the results remain controversial. In 1995, two analyses found no benefit for high dose corticosteroids in sepsis and septic shock and in 2004 another two analyses found benefit for long courses of low dose corticosteroids. This redundant application method is purported to be better for stabilizing the patient and slowly reduce the infection. Shocking the system with a high dose of corticosteroids can be ineffective and even harmful, and there are no studies documenting that stress doses of steroids improve the outcome of sepsis in the absence of shock unless the patient requires stress dose replacement due to a prior history of steroid therapy or adrenal dysfunction.

We observed that the steady infusion of a corticosteroid does increase the potential for healthy outcomes, particularly at low pathogen growth rates and with higher tolerance for large initial pathogen amounts. We also observed an equal increase in potential healthy outcomes when applying a daily repeated dose of corticosteroid. A single large stressful dose of corticosteroid also showed an increase in possible healthy outcomes, which was an unexpected result. We did not see any detrimental effects, so it is possible that a single stressful dose simply does not affect the immune response in a positive way. We observed the effects of each of the treatment methods at various trigger levels. We used these trigger values of activated phagocytes to mimic how a clinician would diagnose sepsis or septic shock, but without tangible variables provided by Reynolds et. al., we could only estimate that a value of $N^* =0.5$ is an approximate point at which a doctor may begin treatment. By acknowledging that any of our treatments, which are varying with time, we confirm that a time-dependent response of the anti-inflammatory mediator is necessary, both in a clinical setting and to produce an accurate model.

Future Work

Our true desire was to create a method for doctors to determine the likely outcomes for their patients, if they knew the patients conditions. We attempted to incorporate clinical practices into our model, but the lack of appropriate units was very frustrating. In future work, it would be very useful to determine a model which accurately reflect units, such as the amount of white blood cells per mL of blood. As we sought to find practical applications for our model, we realized the incongruities between Reynold's model and how sepsis diagnosis occurs in hospitals today. Augmenting this reduced model to find the correlation between core body temperature or heart rate would help to apply the results of the model to practical applications. It may be necessary to construct a completely new model to seek the connections between pathogen, anti-inflammatory mediator, and the indicator variables of temperature, white blood cell count and heart rate.

We also recommend further investigation of the various treatment dosage methods, utilizing an animal model. We struggle with the ethical complications of carrying out such a study in the human population, and would like to recommend using such a study as a last resort. Much research has already been conducted concerning sepsis in animal models. An animal model could easily help provide data to corroborate or dispel theories about which of the 3 treatment methods (steady infusion, daily dosage, or single stress dose) are most effective. Simultaneously, the animal model could also provide preliminary data for research investigating the correlation between sepsis and infection to the physical indicators of temperature, heart rate, and even white blood cell count. Clearly, much more research should be conducted in order to create results which directly correlate to patient conditions, and aid doctors in their diagnoses.

Acknowledgements

I would like to thank my partner for all his hard work on this project, in particular his code expertise! I am also grateful for the guidance of Dr. Chiel, Kendrick Shaw, Jeff McManus, and Jeff Gill.

Appendix A: Mathematica Code

[link to Mathematica code here]