Modeling Systems and Processes

Anthony McGoron, PhD

Associate Professor

Department of Biomedical Engineering

Florida International University

Mathematical Modeling

A model is any representation of a real system.– May deal with structure or function– May involve words, diagrams, mathematical notation,

physical structure– May have the same meaning as “hypothesis”– Must always involve simplification of the real system– A mathematical model may be as simple as a single equation

relating a single dependent variable (y) to another independent variable (x) such as: y = ax + b

– May be multi-component involving the interaction of many equations having several mutually dependent variables

nnnnnn

nn

nn

bxaxaxa

bxaxaxa

bxaxaxa

...

...

...

2211

22222121

11212111

0,021

0,2022122

0,1012111

)();,...,,,(

)();,...,,,(

)();,...,,,(

nnnnn

n

n

ytyyyytfdt

dy

ytyyyytfdt

dy

ytyyyytfdt

dy

Building Models

Stepwise replacement of a system component with a model equation.

1. Conceptual model of the real system. Without an understanding of the real system and the interaction of the system with its environment, no model can be developed.

2. Design experiments and collect “good” data that accurately represents the real system.

3. Examine the data to determine the parameter set that defines the system f(x,y,t,a,b,c…).

4. Define an equation based on the data (empirical) and/or based on the characteristics of the system (theory based). For example, y = ax + b. y and x are variables. a and b are parameters.

5. Find the optimal (most correct) values for the parameters a and b.

6. Implement the model to “experiment” with new concepts.

Building Models: An ExampleFood Chain/Ecosystem/Photosynthesis

Conceptual components of a hypothetical system are replaced by equations to form a multi-component model of a system (Keen and Spain, 1992)

The role of quantitative modeling and simulation within the process of research (Keen and Spain, 1992)

Modeling Application - Transportmass, energy, momentum

HemodialysisHeart LungBypass Machine

Pharmacology – The history, source, physical and chemical properties, biochemical and physiological effect, mechanisms of action, absorption, distribution, biotransformation and excretion, and therapeutic and other uses of drugs.

Pharmacokinetics – Absorption, Distribution, Metabolism (biotransformation) and Excretion of drugs (ADME).

Pharmacodynamics – Biochemical and physiological effects and their mechanisms of action

An Example: Drug DistributionMass Transport

Concentration of drug in the body as a function of time for two types of drug dosage forms

(Rowland and Beckett, 1964)

Physiochemical factors in transfer of drugs across membranes: absorption, distribution, biotransformation, and excretion of a drug involve its passage across cell membranes.

SystemicCirculation

Absorption Excretion

Metabolites

Biotransformation

Free Drug

Bound Drug

Locus ofAction

“receptors”Bound Free

TissueReservoirs

Bound Free

General compartment model for the human body (Bischoff and Brown, 1966)

Numerical details of a specific pharmacokinetic model of the body. There will be 36 equations (Bischoff and Brown, 1966).

Model for a local tissue region (Bischoff and brown, 1966)

BodyAbsorption Elimination

k1

Simple Compartmental Model (lumped)

1st order absorption:

R or k0

 

)exp()(00tkAtA

Bkdt

dE1

 

Akdt

dA0 BkAk

dt

dB10

EA

B

IC’s:A(o)=A0

B(0)=0 E(o)=0

Solution:

]exp()exp([1

1)()()(1001

01

00tkktkk

kkAtBtAAtE

)]exp()[exp()(10

01

00 tktkkk

AktB

0

20

40

60

80

100

0 5 10 15 20 25 30 35Time (hrs)

% o

f D

ose

EA

B

P

T

k0k1

k12 k21

Absorption

Simple Compartmental Model (lumped)

dA/dt=-ko*AdP/dt=k0*A-k1*P-k12*P+k21*TdT/dt=k12*P-k21TdE/dt=k1P

Elimination

0

5

10

15

20

0 5 10 15 20 25 30t (minutes)

mg

Plasma

Tissue

© 1994-2000 Crump Institute for Molecular Imaging UCLA School of Medicine

Plasma time activity curve and Tissue time activity curve

Medical ApplicationNuclear Medicine Imaging

© 1994-2000 Crump Institute for Molecular Imaging UCLA School of Medicine

Three compartment FDG model

© 1994-2000 Crump Institute for Molecular Imaging UCLA School of Medicine

Building the TTAC from the ROI

© 1994-2000 Crump Institute for Molecular Imaging UCLA School of Medicine

Building the TTAC from the ROI

© 1994-2000 Crump Institute for Molecular Imaging UCLA School of Medicine

Building the TTAC from the ROI

© 1994-2000 Crump Institute for Molecular Imaging UCLA School of Medicine

Model Simulationand optimization

© 1994-2000 Crump Institute for Molecular Imaging UCLA School of Medicine

Model Simulationand optimization

© 1994-2000 Crump Institute for Molecular Imaging UCLA School of Medicine

Model Simulationand optimization