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Computational Biology, Part 19
Compartmental Analysis
Robert F. Murphy
Copyright  1996, 1999, 2000, 2001.
All rights reserved.
Compartmental Systems

Compartmental system
 made
up of a finite number of macroscopic
subsystems, called compartments, each of
which is homogeneous and well-mixed
 interactions between compartments consist of
exchanging material
Compartmental Systems
All interactions between compartments are
transfers of material in which some type of
mass conservation condition holds
 Inputs from/outputs to the environment are
permitted
 If they occur, systems is open (otherwise
closed)

Problems in Compartmental
Analysis
Development of plausible models for
particular biological systems
 Development of analytic theory for each
class of compartmental systems
 Estimation of model parameters and
determination of “best” model - so-called
“inverse problem”

Definition of Compartment

“A compartment is an amount of a
material that acts kinetically like a distinct,
homogeneous, well-mixed amount of the
material.” (Jacquez)
 Not
a physical volume or space
First-order Compartment Models

A common, important category of
compartment models is that set of models in
which the rates of all transfers between
compartments are given by first-order rate
constants
Handling first-order
compartment models
Don’t need to solve (e.g. dsolve) the model
from the differentials, since the general
form of the solution is known
 Just need to enter the rate constants for the
allowed transfers into the matrix A, the
environmental transfers into vector f, the
initial concentrations into vector X0 and
evaluate

Xe
At
X
1
0

1
A f  A f
Example: Lead Accumulation
Yeargers, section 7.10 (pp. 220-224)
 Three compartments: blood, soft tissues,
bone
 Open system (input from environment only
into blood)
 First-order compartment model

Lead Accumulation Model
Compartment 1 = blood, Compartment 2 =
soft tissue, Compartment 3 = skeletal system,
Compartment 0 = environment
 xi for i=1..3 is amount of lead in compart. i
 aij for i=0..3,j=1..3 is rate of transfer to
compartment i from compartment j
 IL(t) is the rate of intake into blood from
environment

Lead Accumulation Model

(Maple sheet 1)
Pharmacokinetics
Yeargers, section 7.11 (pp. 226-229)
 x = amount of drug in GI tract
 y = amount of drug in blood
 D(t) is dosing function

 drug
taken every six hours and dissolves within
one half-hour
 a = half-life of drug in GI tract
 b = half-life of drug in blood
Pharmacokinetics

(Maple sheet 2)
Reading for next class

Yeargers, Chapter 8