MATHEMATICAL MODELLING:
GENERAL CONCEPTS AND INTRODUCTION
Mario Grassi
(mariog@dicamp.univ.trieste.it)
Department of Chemical Engineering (DICAMP)
UINVERSITY OF TRIESTE
WHAT IS A MODEL?
[1]
AN ARTIFICIAL REPRESENTATTION OF SOMETHING
NEEDED FOR
VISUALIZE SOMETHING
THAT CAN NOT BE
DIRECTLY OBSERVED
(MOLECULE)
IMITATION OR
EMULATION OF
SOMETHING
CONSEQUENTLY, MODELLING
IS A COGNITIVE ACTIVITY AIMED TO THE
DESCRIPTION OF HOW PHENOMENA
OCCUR OR OBJECTS BEHAVE
KNOWLEDGE
MODELLING
CAN BE PERFORMED ACCORDING TO:
1
SKETCHES
2
PHYSICAL MODELS
3
DRAWINGS
4
MATHEMATICS
Leonardo
da Vinci
Y  f (X , Z)
C
  2 DC   K Cs  C 
t
MATHEMATICAL MODEL IS
A MATHEMATICAL METAPHOR OF SOME
ASPECTS OF REALITY
[2]
(DRUG RELEASE AND ABSORPTION)
“... niuna umana investigazione si può dimandare
vera scienzia, s’essa non passa per le matematiche
dimostrazioni”
Leonardo da Vinci,
Trattato della Pittura 1651
MANY DIFFERENT FIELDS CAN TAKE
ADVANTAGE OF MATHEMATICAL MODEL
1
4
PHARMACY
5
BIOLOGY
6
MEDICINE
7
PSYCHOLOGY
ENGINEERING
2
PHYSIC
3
CHEMISTRY
[3-5]
WHY NOT TYPICAL IN BLUE FIELDS (4-7)
CULTURAL
REASONS
PRACTICAL
REASONS
HUMAN MIND CATEGORISATION
OF KNOWLEDGE
Field 1
no
no
Field 2
no
no
Field 3
no
[1]
no
Impermeable barriers hindering
transversal tools diffusion
INTRINSIC COMLEXITY OF
PHARMACEUTICAL
BIOLOGICAL
MEDICAL
PSYCHOLOGICAL
PHENOMENA
Phenomenon
analysis
Physical
hypotheses
Reliable
Model
Identification of the
mechanisms ruling the
phenomenon
Mathematical
hypotheses
Definition of proper
experiments aimed to
check the
hypotheses:
NO
Mechanisms
mathematical form
and assembling to get
the MODEL
MODEL
comparison with
EXPERIMENTAL DATA:
FITTING
OK
MODEL
comparison with
EXPERIMENTAL DATA:
PREDICTION
OK
NO
MATHEMATICAL MODEL EXPRESSION
Y = f(Z1, Z2, ….Zn, A1, A2, ….Am)
dependent
variable
Independent
variables
EMPIRICAL
A1, A2, ..Am do not posses
a physical meaning
Y  A0  t * A1
Linear relation
Model parameters:
Z1, Z2, … Zn
independent
THEORETICAL
A1, A2, ..Am do posses a
physical meaning
SC0 D
A1 A2 A3
Y  Mt 
t
* Z1
h
A4
C0
donor
D receiver
C=0
h
MODEL FITTING PARAMETERS
SOME MODEL PARAMETERS ARE KNOWN …
A1, A2, A4 in the
previous example
Y  Mt 
SC0 D
AA A
t  1 2 3 * Z1
h
A4
C0
donor
D receiver
C=0
h
…THE REMAINING ARE UNKOWN AND ARE CALLED
FITTING PARAMETERS AND NEED TO BE DETERMINED
A3 in the previous
example
Y  Mt 
SC0 D
AA A
t  1 2 3 * Z1
h
A4
MODEL DATA FITTING
EXPERIMENTAL DATA
Cr(t)
DIFFERENCE
[Cr experim – Cr model]
MODEL
t
t1 t2
C
2
   
i 1 
i 6
t 3 t4
i
r exp erim
C
i
t5 t6
i
r mod el




Standard deviation
2
i 


2
1 N
xj  x

N  1 j 1
Statistic tells us that the most probable set of model
parameters (A1, A2, … Am) is that yielding the lowest 2
value
IS THE FITTING SATISFACTORY?
 y  y  σ   
2
N
MSR 
i 1
MSR
F
MSE
i
2
i
MSE 
M 1
2
N M
Mean Square Error
degrees of freedom n2 = N-M
Mean Square Regression
degrees of freedom n1 = M-1
N
y
2
y
σ
 i i
i 1
N
2


1
σ
 i
i 1
 yi  y
   
i
i 1 
N
2
Mean
p
i



2
M = model fitting parameters
number
N = experimental data number
n1 =
n 
1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
40
60
120

161.4
18.51
10.13
7.71
6.61
5.99
5.59
5.32
5.12
4.96
4.84
4.75
4.67
4.60
4.54
4.49
4.45
4.41
4.38
4.35
4.32
4.30
4.28
4.26
4.24
4.23
4.21
4.20
4.18
4.17
4.08
4.00
3.92
3.84
2
3
4
5
6
7
8
9
10
12
15
20
24
30
40
60

120
Theoretical F values with n 1 (numerator) and n 2 (denominator) degrees of freedom fixing p = 1 –  = 0.95 (F[n 1,n 2,p])
199.5
19.00
9.55
6.94
5.79
5.14
4.74
4.46
4.26
4.10
3.98
3.89
3.81
3.74
3.68
3.63
3.59
3.55
3.52
3.49
3.47
3.44
3.42
3.40
3.39
3.37
3.35
3.34
3.33
3.32
3.23
3.15
3.07
3.00
215.7
19.16
9.28
6.59
5.41
4.76
4.35
4.07
3.86
3.71
3.59
3.49
3.41
3.34
3.29
3.24
3.20
3.16
3.13
3.10
3.07
3.05
3.03
3.01
2.99
2.98
2.96
2.95
2.93
2.92
2.84
2.76
2.68
2.60
224.6
19.25
9.12
6.39
5.19
4.53
4.12
3.84
3.63
3.38
3.36
3.26
3.18
3.11
3.06
3.01
2.96
2.93
2.90
2.87
2.84
2.82
2.80
2.78
2.76
2.74
2.73
2.71
2.70
2.69
2.61
2.53
2.45
2.37
230.2
19.30
9.01
6.26
5.05
4.39
3.97
3.69
3.48
3.33
3.20
3.11
3.03
2.96
2.90
2.85
2.81
2.77
2.74
2.71
2.68
2.66
2.64
2.62
2.60
2.59
2.57
2.56
2.55
2.53
2.45
2.37
2.29
2.21
234.0
19.33
8.94
6.16
4.95
4.28
3.87
3.58
3.37
3.22
3.09
3.00
2.92
2.85
2.79
2.74
2.70
2.66
2.63
2.60
2.57
2.55
2.53
2.51
2.49
2.47
2.56
2.45
2.43
2.42
2.34
2.25
2.17
2.10
236.8
19.35
8.89
6.09
4.88
4.21
3.79
3.50
3.29
3.14
3.01
2.91
2.83
2.76
2.71
2.66
2.61
2.58
2.54
2.51
2.49
2.46
2.44
2.42
2.40
2.39
2.37
2.36
2.35
2.33
2.25
2.17
2.09
2.01
238.9
19.37
8.85
6.04
4.82
4.15
3.73
3.44
3.23
3.07
2.95
2.85
2.77
2.70
2.64
2.59
2.55
2.51
2.48
2.45
2.42
2.40
2.37
2.36
2.34
2.32
2.31
2.29
2.28
2.27
2.18
2.10
2.02
1.94
240.5
19.38
8.81
6.00
4.77
4.10
3.68
3.39
3.18
3.02
2.90
2.80
2.71
2.65
2.59
2.54
2.49
2.46
2.42
2.39
2.37
2.34
2.32
2.30
2.28
2.27
2.25
2.24
2.22
2.21
2.12
2.04
1.96
1.88
241.9
19.40
8.79
5.96
4.74
4.06
3.64
3.35
3.14
2.98
2.85
2.75
2.67
2.60
2.54
2.49
2.45
2.41
2.38
2.35
2.32
2.30
2.27
2.25
2.24
2.22
2.20
2.19
2.18
2.16
2.08
1.99
1.91
1.83
243.9
19.41
8.74
5.91
4.68
4.00
3.57
3.28
3.07
2.91
2.79
2.69
2.60
2.53
2.48
2.42
2.38
2.34
2.31
2.28
2.25
2.23
2.20
2.18
2.16
2.15
2.13
2.12
2.10
2.09
2.00
1.92
1.83
1.75
245.9
19.43
8.70
5.86
4.62
3.94
3.51
3.22
3.01
2.85
2.72
2.62
2.53
2.46
2.40
2.35
2.31
2.27
2.23
2.20
2.18
2.15
2.13
2.11
2.09
2.07
2.06
2.04
2.03
2.01
1.92
1.84
1.75
1.67
248.0
19.45
8.66
5.80
4.56
3.87
3.44
3.15
2.94
2.77
2.65
2.54
2.46
2.39
2.33
2.28
2.23
2.19
2.16
2.12
2.10
2.07
2.05
2.03
2.01
1.99
1.97
1.96
1.94
1.93
1.84
1.75
1.66
1.57
249.1
19.45
8.64
5.77
4.53
3.84
3.41
3.12
2.90
2.74
2.61
2.51
2.42
2.35
2.29
2.24
2.19
2.15
2.11
2.08
2.05
2.03
2.01
1.98
1.96
1.95
1.93
1.91
1.90
1.89
1.79
1.70
1.61
1.52
250.1
19.46
8.62
5.75
4.50
3.81
3.38
3.08
2.86
2.70
2.57
2.47
2.38
2.31
2.25
2.19
2.15
2.11
2.07
2.04
2.01
1.98
1.96
1.94
1.92
1.90
1.88
1.87
1.85
1.84
1.74
1.65
1.55
1.46
251.1
19.47
8.59
5.72
4.46
3.77
3.34
3.04
2.83
2.66
2.53
2.43
2.34
2.27
2.20
2.15
2.10
2.06
2.03
1.99
1.96
1.94
1.91
1.89
1.87
1.85
1.84
1.82
1.81
1.79
1.69
1.59
1.50
1.39
252.2
19.48
8.57
5.69
4.43
3.74
3.30
3.01
2.79
2.62
2.49
2.38
2.30
2.22
2.16
2.11
2.06
2.02
1.98
1.95
1.92
1.89
1.86
1.84
1.82
1.80
1.79
1.77
1.75
1.74
1.64
1.53
1.43
1.32
253.3
19.49
8.55
5.66
4.40
3.70
3.27
2.97
2.75
2.58
2.45
2.34
2.25
2.18
2.11
2.06
2.01
1.97
1.93
1.90
1.87
1.84
1.81
1.79
1.77
1.75
1.73
1.71
1.70
1.68
1.58
1.47
1.35
1.22
254.3
19.50
8.53
5.63
4.36
3.67
3.23
2.93
2.71
2.54
2.40
2.30
2.21
2.13
2.07
2.01
1.96
1.92
1.88
1.84
1.81
1.78
1.76
1.73
1.71
1.69
1.67
1.65
1.64
1.62
1.51
1.39
125
1.00
MODEL COMPARISON: AKAIKE CRITERION1
 χ2 

M  1N  2
AIC  N ln    2
N M 2
N
N = number of experimental data
M = number of fitting parameters
The most probable model is that characterised by the smallest AIC
The probability of being right is defined by:
e 0.5
pAIC 
;
0.5 
1 e
  AICom  AICsmallest
EXAMPLE: IF  = 2 =====> pAIC = 0.73
MODEL EVALUATION: PREDICTION
FITTING
UNKNOWN MODEL
PARAMETERS (FITTING P.)
Y = f(Z1, Z2, ….Zn, A1, A2, ….Am)
Cr
MODEL
PREDICTION
EXPERIMENTAL DATA
t
IS MODEL PREDICTION SATISFACTORY?
YES
E.D. STD. DEV.
MODEL
PREDICTION
Cr
EXP. DATA
MODEL STD. DEV.
t
MODEL STD DEV IS CALCULATED ASSUMING FITTING PARAMETERS
SETS THAT MAXIMISE AND MINIMISE PREDICTION: Ai = value ± S.D.
ROBUST FITTING
EXP. DATA
Traditional Fitting
d4
Cr(t)
MODEL
EXP. DATA
d2
MODEL
d6
d5
d3
d1
t1 t2
C
2
   
i 1 
i 6
t3 t 4
i
r exp erim
t5 t6
C
i
i
r mod el




2
t1
t2
t3
t4
 di
   
i 1   i
i 6
2
t5



2
t6
EXAMPLE 1: SLIDING SLAB
Y
FRICTION
(FR = b*M*v)
FM
FM
X
FR
t=0
HEAT due to friction
(WRINKLED SURFACE)
MODEL: NEWTON EQUATION
dV
Ma  M
 FM  b M V
dt
SLAB MOTION CAN BE SIMULATED
POSITION, VELOCITY, ACCELERATION,
TEMPERATURE
0.7
25
0.6
POSITION
0.5
15
v(m/s)
x(m)
20
10
0.4
VELOCITY
0.3
0.2
5
0.1
0
0
10
20
30
0
40
0
10
t(s)
30
40
t(s)
0.12
0.01
0.1
ACCELERATION
TEMPERATURE
PROFILE
0.008
y(m)
0.08
a(m/s2)
20
0.06
0.006
0.004
0.04
0.002
0.02
0
0
0
10
20
t(s)
30
40
25
25.5
26
T(°C)
26.5
27
Energy(J)
25
Supplied
Lost
Kinetic
20
15
10
5
0
0
5
10
15
20
25
30
Power(w)
t(s)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Supplied
Lost
Kinetic
0
5
10
15
t(s)
20
25
30
EXAMPLE 2: HUMAN MEMORY [6]
Stimulus
vp
Performance
Sensory
buffer
kr(d)
kj+1(d)
kj(d)
kj-1(d)
k1=d
pi  pPi  pRi
STS
r
j+1
j
j-1
B
u
ff
e
r
1
pPi  1  e
I t
retrieval
pRi 
t*/n
I
LTS
  d 1 

  i



i
 1  k 
r
r 1 j
j 1
mij
f(r,d,t,q)
Model fitting parameters
mij
FREE VERBAL RECALL TEST [7]
48 young subjects (19-28 years old) are
asked to remember a list containing
twelve items (word) each (8 repetitions)
Vp = 0.5 item/s
exp
i
p
# Corrected remembrance position " i"

# Corrected remembrance position "12"
exp
12
p
1
1.2
1
0.8
p
0.6
0.4
0.2
experimental
model best fitting
simulation
simulation
simulation
0.25 item/s
0.5 item/s
1 item/s
2 item/s
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13
item presentation order
r = 3;  = 1.75; t = 0.76; d = 0.27
Fitting parameters
REFERENCES
1. Grassi, M., Grassi, G., Lapasin, R., Colombo, I. Understanding drug
release and absorption mechanisms: a physical and mathematical
approach. CRC, Boca Raton, Florida, USA, 2006.
2. Israel, G. Balthazar van der Pol e il primo modello matematico del battito
cardiaco, in Modelli matematici nelle scienze biologiche, Freguglia, P.
Ed., Edizioni Quattro Venti, 1998.
3. Dym, C. L. Principles of Mathematical Modeling, 2nd edition, Elsevier
Academic Press, San Diego, London 2004.
4. Bourne, D. W. A. Mathematical Modeling of Pharmacokinetic Data,
Technomic publishing CO., INC., Lancaster, Basel, 1995.
5. Cartensen, J. T. Modeling and data treatment in the pharmaceutical
sciences, Technomic publishing CO., INC., Lancaster, Basel, 1996.
6. Atkinson, R. C. and Shiffrin, R. M., Human memory: A proposed system
and its control processes, in The Psychology of Learning and
Motivation: Advances in Research and Theory, Spence, K. W. and
Spence J.T. Eds., Academic Press, 1968
7. Peirano, F. Rievocazione libera e interpretazione della curva di posizione
seriale negli anziani, Tesi di Laurea, Università di Padova, Dipartimento
di Psicologia, Giugno 2000.
8. Grassi, M., Voinovich, D., Moneghini, M., Franceschinis, Perissutti, B.,
Filipovic-Grcic, J. Preparation and evaluation of a melt pelletised
paracetamol stearic acid sustained release delivery system. Journal of
Controlled Release 88 (2003) 381–391.