Modeling Systems and Processes
Anthony McGoron, PhD
Associate Professor
Department of Biomedical Engineering
Florida International University
Mathematical Modeling
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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
a11x1  a12 x2  ...a1n xn  b1
a21x1  a22 x2  ...a2 n xn  b2
an1 x1  an 2 x2  ...ann xn  bn
dy1
 f1 (t , y1 , y 2 ,..., y n ); y1 (t 0 )  y1, 0
dt
dy 2
 f 2 (t , y1 , y 2 ,..., y n ); y 2 (t 0 )  y 2, 0
dt
dy n
 f n (t , y1 , y 2 ,..., y n ); y n (t 0 )  y n , 0
dt
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 Example
Food 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 - Transport
mass, energy, momentum
Hemodialysis
Heart Lung
Bypass Machine
An Example: Drug Distribution
Mass Transport
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
Concentration of drug in the body as a function of time for
two types of drug dosage forms
(Rowland and Beckett, 1964)
Locus of
Action
“receptors”
Bound
Free
Tissue
Reservoirs
Bound
Free
Systemic
Circulation
Free Drug
Absorption
Bound Drug
Excretion
Metabolites
Biotransformation
Physiochemical factors in transfer of drugs across membranes:
absorption, distribution, biotransformation, and excretion of a drug
involve its passage across cell membranes.
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)
Simple Compartmental Model (lumped)
Absorption
R or k0
dA
1st order absorption: dt  k0 A
Solution: A(t )  A exp(k t )
0
dB
 k Ak B
dt
dE
kB
dt
0
1
1
0
kA
 [exp(k t )  exp(k t )]
k k
  1
E (t )  A  A(t )  B(t )  A 1  
 k k
B(t ) 
Elimination
k1
Body
0
0
0
1
1
0
0
0
1
IC’s:A(o)=A0
100
B(0)=0
E(o)=0
80


[k exp(k t )  k exp(k t ]


1
0
0
0
1
% of Dose
E
A
E
A
60
40
B
B
20
0
0
5
10
15
20
Time (hrs)
25
30
35
Simple Compartmental Model (lumped)
k1
P
k12
k21
T
Elimination
dA/dt=-ko*A
dP/dt=k0*A-k1*P-k12*P+k21*T
dT/dt=k12*P-k21T
dE/dt=k1P
20
Plasma
15
mg
Absorption
k0
10
Tissue
5
0
0
5
10
15
t (minutes)
20
25
30
Medical Application
Nuclear Medicine Imaging
Plasma time activity curve and Tissue time activity curve
© 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 Simulation
and optimization
© 1994-2000 Crump Institute for Molecular Imaging
UCLA School of Medicine
Model Simulation
and optimization
© 1994-2000 Crump Institute for Molecular Imaging
UCLA School of Medicine
Model Simulation
and optimization
© 1994-2000 Crump Institute for Molecular Imaging
UCLA School of Medicine