Compartment Modeling with Applications to Physiology AiS Challenge Summer Teacher Institute 2002 Richard Allen

Compartment Modeling Compartment systems provide a systematic way of modeling physical and biological processes. In the modeling process, a problem is broken up into a collection of connected “black boxes” or “pools”, called compartments. A compartment is defined by a charac- teristic material (chemical species, biological entity) occupying a given volume.

Compartment Modeling A compartment system is usually open; it changes material with its environment I1I1 k 01 k 02 k 21 k 12 q1q1 q2q2

Applications Water pollution Nuclear decay Chemical kinetics Population migration Pharmacokinetics Epidemiology Economics – water resource management Medicine Metabolism of iodine and other metabolites Potassium transport in heart muscle Insulin-glucose kinetics Lipoprotein kinetics

Discrete Model: time line Y 0 Y 1 Y 2 Y 3 … Y n | | | | |---> t 0 t 1 t 2 t 3 … t n t 0, t 1, t 2, … are equally spaced times at which the variable Y is determined: Δt = t 1 – t 0 = t 2 – t 1 = …. Y 0, Y 1, Y 2, … are values of the variable Y at times t 0, t 1, t 2, ….

SIS Epidemic Model {S j+1 - S j }= Δt{- *S j +  *I j } {I j+1 - I j } = Δt{+ * S j -  * I j } S 0 and I 0 given SI Infecteds Susceptibles 

SIS Epidemic model depends on N = S + I and p, the probability of transmitting the disease in contact. Susceptibles make c*S disease transmitting contacts in time Δt Rate of Susceptibles becoming infected is p*(c*S)*(I/N) = (p*c*I/N)*S and = (p*c*I/N) S Susceptible I  p*c*I/N Infected =

SIR Epidemic model   {S j+1 - S j } = Δt{U -  *S j - *S j } {I j+1 - I j } = Δt{ *S j -  *I j -  *I j } {R j+1 - R j } = Δt{  *I j -  *R j } S 0, I 0, and R 0 given SR Infecteds Susceptible I Recovered    U Infected

Drug Elimination Model   Vc{Cp j+1 -Cp j } = Δt{-k el *Cp j - k tp *Cp j + k pt *Ct j } Vt{Ct j+1 -Ct j } = Δt{-k pt *Ct j + k tp *Cp j } Central/Plasma Compartment Cp, Vp Rapidly Equilibrated Peripheral/Tissue Compartment Ct, Vt Slowly Equilibrated k el Drug Eliminated k k

Insulin/Glucose Dynamics Glucose Utilization Active Insulin Absorption Insulin Elimination Plasma Insulin Plasma Glucose Insulin Injection Insulin Sensitivity Glucose Input Carbohydrate Intake Endogenous Production

Injection Plasma Insulin I p Active Insulin I a Elimination kaka kpkp Glucose k el {Ip j+1 - Ip j } = Δt{(Iabs/Vi - k el *Ip j - k p *Ip j } {Ia j+1 - Ia j ] = Δt{-k a *Ia j + k p *Ip j } Ia 0 and Ip 0 given   Insulin Model

Dialysis Model   Ve{Ce j+1 - Ce j } = Δt{G - (k d +k r )*Ce j -k ic *Ce j +k ci *Ci j } Vi*{Ci j+1 - Ci j } = Δt{- k ci *Ci j + k ic *Ce j } Master tools: dialysis model Extracellular Space Ve Ce Intracellular Space Vi Ci Urea Removal k d + k r k ic k ci Mass Transfer Urea Production G

References “Computer simulation of plasma insulin and glucose dynamics after subcutaneous insulin injection, Berger and Rodbard, Diabetes Care, Vol. 12, No. 10, Nov.-Dec. 1989” rt/ rt/ rt/docjacquez/node1.html rt/docjacquez/node1.html