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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