Compartment Modeling with Applications to Physiology

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

2. 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.

3. Compartment Modeling • A compartment system is usually open; it changes material with its environment I1 k21 q1 q2 k12 k01 k02

4. 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 Applications

5. Discrete Model: time line Y0 Y1 Y2 Y3 … Yn |---------|----------|------- --|---------------|---> t0 t1 t2 t3 … tn • t0, t1, t2, … are equally spaced times at which the variable Y is determined: Δt = t1 – t0 = t2 – t1 = … . • Y0, Y1, Y2, … are values of the variable Y at times t0, t1, t2, … .

6. l S I Infecteds Susceptibles d SIS Epidemic Model {Sj+1 - Sj}= Δt{- *Sj + *Ij} {Ij+1 - Ij} = Δt{+* Sj - * Ij} S0 and I0 given

7. SIS Epidemic model  = I S p*c*I/N Susceptible Infected m m •  depends on N = S + Iand p, theprobability of transmitting the disease in contact. • Susceptibles make c*S disease transmitting contacts in time Δt • Rate of Susceptibles becoming infected isp*(c*S)*(I/N) = (p*c*I/N)*S and  = (p*c*I/N)

8. SIR Epidemic model  U S I R g l Recovered Susceptible Infected Infecteds {Sj+1 - Sj} = Δt{U -  *Sj - *Sj} {Ij+1 - Ij} = Δt{*Sj -  *Ij -  *Ij} {Rj+1 - Rj} = Δt{*Ij -*Rj} S0, I0, and R0 given m m m

9. k k Central/Plasma Compartment Cp, Vp Rapidly Equilibrated Peripheral/Tissue Compartment Ct, Vt Slowly Equilibrated kel Drug Eliminated Drug Elimination Model  Vc{Cpj+1-Cpj} = Δt{-kel*Cpj - ktp*Cpj + kpt*Ctj} Vt{Ctj+1-Ctj} = Δt{-kpt*Ctj + ktp*Cpj}

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

11. Insulin Model Injection Plasma Insulin Ip Active Insulin Ia kp Glucose kel ka Elimination {Ipj+1 - Ipj} = Δt{(Iabs/Vi - kel*Ipj - kp*Ipj} {Iaj+1 - Iaj] = Δt{-ka*Iaj + kp*Ipj} Ia0 and Ip0 given 

12. kd +kr Urea Removal Dialysis Model kic Extracellular Space Ve Ce Intracellular Space Vi Ci Urea Production G Mass Transfer kci Ve{Cej+1 - Cej} = Δt{G - (kd +kr)*Cej -kic*Cej +kci*Cij} Vi*{Cij+1 - Cij} = Δt{- kci*Cij + kic*Cej} Master tools: dialysis model

13. 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” • http://www.sph.umich.edu/geomed/mods/compart/ • http://www.shodor.org/aida/ • http://www.boomer.org/c/p1/Ch22/Ch2211.html • http://www.sph.umich.edu/geomed/mods/compart/docjacquez/node1.html