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

anti-inflammation

Neil ParikhBased on a paper by: G. Bard Ermentrout, Angela Reynolds, Jonathan Rubin, Gilles Clermont, Judy Day, and Yoram Yodocotz

Mentor: G. Bard Ermentrout

PurposeCreate a model that accurately depicts the anti-inflammatory response in relation to macrophage activity and pathogen levelsStudy the effects of a time-dependent anti-inflammatory response in an immune system while simultaneously monitoring pathogen levels and macrophage counts

Review of TermsPathogen: an agent that causes disease.Phagocytes: A cell, such as a white blood cell, that engulfs and absorbs waste material, harmful microorganisms, or other foreign bodies in the bloodstream and tissues.Septic Death: death caused by the presence of pathogenic organisms in the blood or tissue.Aseptic Death: death caused by excessive tissue damage, due to increased phagocyte levels.

Definitions obtained from dictionary.com

Important VariablesM: non-specific local responseP: initiating event (pathogen levels) N*: inflammation (# of phagocytes)NR: # of resting phagocytesD: collateral damage to tissueCA: anti-inflammation

InteractionsInitiating Event occurs and alerts the immune system (non-specific local response)Phagocytes lower pathogen levels but also cause inflammationInflammation runs in a positive feedback loopInflammation causes damage in tissueInflammation and damage in tissue both cause anti-inflammation levels to rise

ModelM/P SubsystemN*/P SubsystemN*/D SubsystemThree-Variable SubsystemFour-Variable Subsystem

M/P SubsystemThe M/P subsystem models the human immune system defending its body against foreign attack.

(1 )

m m mp

mp

pm mpg

m mp

dM s M k MPdt

dP k MPdt

k s PdP Pk Pdt P k MP

μ

μ∞

= − −

= −

= − −+

N*/P SubsystemAs pathogen levels increase, phagocytes are induced, and inflammation occurs as a result.Resting phagocytes are activated by active phagocytes.

*

*

** *

(1 )

( )

( )

pm mpg

m mp

Rnr nr R nn np R

nn np R n

k s PdP Pk P kN Pdt P k P

dN s N k N k P Ndt

dN k N k P N Ndt

μ

μ

μ

= − − −∞ +

= − − +

= + −

N*/D SubsystemActivated phagocytes induce collateral tissue damage.Damaged tissue releases pro-inflammatory cytokines, which causes further phagocyte activation.

6

6

***

*

*

6 *

( )( )

( )

nr nn ndn

nr nn nd

dn ddn

s k N k DdN Ndt k N k D

dD Nk Ddt x N

μμ

μ

+= −

+ +

= −+

Three-Variable Subsystem

6

6

***

*

*

6 *

( )( )

( )

nr nn np ndn

nr nn np nd

dn ddn

s k N k P k DdN Ndt k N k P k D

dD Nk Ddt x N

μμ

μ

+ += −

+ + +

= −+

*(1 ) pm mpg pn

m mp

k s PdP Pk P k N Pdt P k Pμ

= − − −∞ +

Three Variable Subsystem Cont’d

Four Variable Subsystem

*(1 ) ( )pm mpg pn

m mp

k s PdP Pk P k f N Pdt P k Pμ

= − − −∞ +

***

*

( )( )

nr nn np ndn

nr nn np nd

s f k N k P k DdN Ndt f k N k P k D

μμ

+ += −

+ + +* 6

6 * 6

( )( )( )dn d

dn

dD f Nk Ddt x f N

μ= −+

*

*

( )1 ( )

cn cndAc c A

cnd

k f N k DdC s Cdt f N k D

μ+= + −

+ +

2( )

1 ( )A

Vf V Cc∞

=+

P=1, N*=0, D=0, Ca=0.125, Kpq=0.3

P=1.5, N*=0, D=0, Ca=0.125, Kpq=0.3

P=1, N*=0, D=0, Ca=0.125, Kpq=0.6

ConclusionIt is advantageous to have dynamic anti-inflammatory levels.There is a specific range of N* and CA for optimal health.

LimitationsBecause of our oversimplified model, the biological aspects are not as accurate as we had hoped them to be. It is difficult to provide quantitative measurements for functions like “pro-inflammation”, “anti-inflammation”, and “damage”.

Future ResearchHow the various features of the inflammatory response interact to govern the outcome following multiple insults.Models that are more detailed in anti-inflammatory substances and analyze anti-inflammatory mediators as “therapeutic agents”.