LecturesPart19
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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
Xe
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