Computational Biology, Part 18 Compartmental Analysis Robert F. Murphy Copyright  1996, All rights reserved.

Compartmental Systems Compartmental system 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 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 Inputs from/outputs to the environment are permitted If they occur, systems is open (otherwise closed) If they occur, systems is open (otherwise closed)

Problems in Compartmental Analysis Development of plausible models for particular biological systems Development of plausible models for particular biological systems Development of analytic theory for each class of compartmental systems Development of analytic theory for each class of compartmental systems Estimation of model parameters and determination of “best” model - so-called “inverse problem” 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) “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 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 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 X 0 and evaluate 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 X 0 and evaluate

Example: Lead Accumulation Yeargers, section 7.10 (pp ) Yeargers, section 7.10 (pp ) Three compartments: blood, soft tissues, bone Three compartments: blood, soft tissues, bone Open system (input from environment only into blood) Open system (input from environment only into blood) First-order compartment model First-order compartment model

Lead Accumulation Model Compartment 1 = blood, Compartment 2 = soft tissue, Compartment 3 = skeletal system, Compartment 0 = environment Compartment 1 = blood, Compartment 2 = soft tissue, Compartment 3 = skeletal system, Compartment 0 = environment x i for i=1..3 is amount of lead in compart. i x i for i=1..3 is amount of lead in compart. i a ij for i=0..3,j=1..3 is rate of transfer to compartment i from compartment j a ij for i=0..3,j=1..3 is rate of transfer to compartment i from compartment j I L (t) is the rate of intake into blood from environment I L (t) is the rate of intake into blood from environment

Lead Accumulation Model

(Maple sheet 1) (Maple sheet 1)

Pharmacokinetics Yeargers, section 7.11 (pp ) Yeargers, section 7.11 (pp ) x = amount of drug in GI tract x = amount of drug in GI tract y = amount of drug in blood y = amount of drug in blood D(t) is dosing function 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) (Maple sheet 2)