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Modeling Systems and Processes

Anthony McGoron, PhD

Associate ProfessorDepartment of Biomedical Engineering

Florida International University

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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 anotherindependent variable (x) such as:y = ax + b

May be multi-component involving the interaction of manyequations 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

!!

!!

!!

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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 accuratelyrepresents the real system.

3. Examine the data to determine the parameter set that defines the

systemf(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 andx 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.

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Building Models: An ExampleFood Chain/Ecosystem/Photosynthesis

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Conceptual components of a hypothetical system are replaced by

equations to form a multi-component model of a system (Keen andSpain, 1992)

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The role of quantitative modeling and simulation within the process

of research (Keen and Spain, 1992)

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Modeling Application - Transportmass, energy, momentum

HemodialysisHeart LungBypassMachine

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PharmacologyThe history, source, physical and chemicalproperties, 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 (ADM

E).

Pharmacodynamics Biochemical and physiological effectsand their mechanisms of action

An Example: Drug DistributionMass Transport

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Concentration of drug in the body as a function of time for

two types of drug dosage forms

(Rowland and Beckett, 1964)

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Physiochemical factors in transfer of drugs across membranes:

absorption, distribution, biotransformation, and excretion of a drug

involve its passage across cell membranes.

Systemic

Circulation

Absorption Excretion

Metabolites

Biotransformation

Free Drug

Bound Drug

Locus of

Action

receptors

Bound Free

Tissue

Reservoirs

Bound Free

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General compartment model for the human body (Bischoff and

Brown, 1966)

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Numerical details of a specific pharmacokinetic model of the

body. There will be 36 equations (Bischoff and Brown, 1966).

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

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BodyAbsorption Elimination

k1

Simple Compartmental Model (lumped)

1st order absorption:

R or k0

)exp()(00tkAtA !

Bkdt

dE1

!Akdt

dA0! BkAkdt

dB10

!

EA

B

ICs:A(o)=A0

B(0)=0

E(o)=0

Solution:

!!]exp()exp([11)()()( 1001

01

00 tkktkkkk

AtBtAAtE

)]exp()[exp()(10

01

00 tktkkk

AktB !

!

0

20

40

60

80

100

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

%o

fDose

E

A

B

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P

T

k0k1

k12 k21

Absorption

Simple Compartmental Model (lumped)

dA/dt=-ko*A

dP/dt=k0*A-k1*P-k12*P+k21*T

dT

/dt=k12*P

-k21T

dE/dt=k1P

Elimination

0

5

10

15

20

0 5 10 15 20 25 30t (minutes)

mg

Plasma

Tissue

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1994-2000 Crump Institute for Molecular Imaging

UCLA School of Medicine

Plasma time activity curve and Tissue time activity curve

Medical ApplicationNuclear Medicine Imaging

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1994-2000 Crump Institute for Molecular Imaging

UCLA School of Medicine

Three compartment FDG model

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1994-2000 Crump Institute for Molecular Imaging

UCLA School of Medicine

Building the TTAC from

the ROI

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1994-2000 Crump Institute for Molecular Imaging

UCLA School of Medicine

Building the TTAC from

the ROI

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1994-2000 Crump Institute for Molecular Imaging

UCLA School of Medicine

Building the TTAC from

the ROI

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1994-2000 Crump Institute for Molecular ImagingUCLA School of Medicine

Model Simulation

and optimization

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1994-2000 Crump Institute for Molecular ImagingUCLA School of Medicine

Model Simulation

and optimization

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1994-2000 Crump Institute for Molecular ImagingUCLA School of Medicine

Model Simulation

and optimization