Exemplary Model Description Draft 3

Term Paper Model Description

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

Angela Reynolds et. al.

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.

Equations

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.

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.

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

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

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

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 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.

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 $C_A$