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
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

5
Discrete Model time line

• Y0 Y1 Y2 Y3
  Yn
• --------------------------
  --------------------gt
• t0 t1 t2 t3
  tn
• t0, t1, t2, are equally spaced times at which
  the variable Y is determined
  ?t t1 t0 t2 t1 .
• Y0, Y1, Y2, are values of the variable Y at
  times t0, t1, t2, .

6
SIS Epidemic Model

• Sj1 - Sj ?t- ?Sj ?Ij
• Ij1 - Ij ?t? Sj - ? Ij
• S0 and I0 given

7
SIS Epidemic model
?
I
S
pcI/N
Susceptible
Infected
m
m

• ? depends on N S I and p, the probability of
  transmitting the disease in contact.
• Susceptibles make cS disease transmitting
  contacts in time ?t
• Rate of Susceptibles becoming infected is
  p(cS)(I/N) (pcI/N)S and ? (pcI/N)

8
SIR Epidemic model ?
U
S
R
I
g
l
Recovered
Susceptible
Infected
Infecteds
m
m
m

• Sj1 - Sj ?tU - ? Sj - ? Sj
• Ij1 - Ij ?t?Sj - ? Ij - ? Ij
• Rj1 - Rj ?t ?Ij - ?Rj
• S0, I0, and R0 given

9
Drug Elimination Model ?

• VcCpj1-Cpj ?t-kelCpj - ktpCpj
  kptCtj
• VtCtj1-Ctj ?t-kptCtj ktpCpj

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
Ipj1 - Ipj ?t(Iabs/Vi - kelIpj -
kpIpj Iaj1 - Iaj ?t-kaIaj kpIpj Ia0
and Ip0 given ?
12
Dialysis Model ?
kic
Extracellular Space Ve Ce
Intracellular Space Vi Ci
Urea Production G
Mass
Transfer
kci

• VeCej1 - Cej ?tG - (kd kr)Cej -kicCej
  kciCij
• ViCij1 - Cij ?t- kciCij kicCej
• 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/docja
  cquez/node1.html