QR
andM
athematical
Modeling
LilianaIroni
IstitutodiA
nalisiNum
erica-
CN
Rvia
Ferrata
1,I–27100
Pavia,Italy
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odeling/1
Ackno
wledgm
ent��
Riccardo
Bellazzi
��
LizB
radley
��
Antonio
Capelo
��
Piero
ColliFranzone
��
Raffaella
Guglielm
ann
��
Stefania
Tentoni
��
Cristina
Bonferoni
��
Carla
Caram
ella
��
Bruno
Pirotti
��
Silvia
Rossi
��
Andrea
Nauti
��C
esareP
atrini
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odeling/2
Outline
1.Introduction
2.M
odellingfrom
Data:
System
Identification(S
I)
3.P
roblems
inS
I
4.W
hatcanQ
Rdo
forS
I?
5.B
riefoverviewofrelated
work
6.P
art1-
QR
forstructural(param
etric)S
I
-C
aseS
tudy:A
utomated
modeling
systemofvisco-elastic
materials
andits
applicationto
pharmacology
7.P
art2-
QR
forblack-box
(non-parametric)
SI
-C
aseS
tudy:K
ineticsofT
hiamine
(vitamin �� )
inthe
cellsofthe
intestinetissue
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odeling/3
IntroductionQ
uantitativem
athematical
model
Am
athematicaldescription
oftherelations
between
theinputs�
(causes)and
theoutputs�
(effects)ofa
system
output input
MO
DE
L
yx
Model:
am
apping���� �
Three
modeling
approaches:
��
white
box
��
greybox
��
blackbox
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odeling/4
White
box
modeling
output input
y... pV
=R
TF=
ma
V=
RI
...x
��
Physicallaw
sare
available
��
Typicalexamples:
mechanicaland
electricalsystems
��
The
boxis
“transparent”
MO
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odeling/5
Gre
ybo
xm
odeling
output input
yV
=R
I pV
=R
T
F=m
a
x
��
Physicallaw
sare
availablebut
thevalues
ofsom
eparam
etersare
unknown
��
theinternalstructure
ofthebox
isonly
partiallyknow
n
��
Idea:tune
theunknow
nparam
etersuntil
theoutputs
predictedby
them
odelmatch
theobserved
data
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odeling/6
Blac
kbo
xm
odeling
output input
yx
?
��
physicalknowledge
isnotavailable
��
physicalknowledge
isvery
incomplete
��
parameter
estimation
isnot
possibledue
tothe
lackof
adequateobserved
datasets
��
usefulforvery
complex
systems
��
Idea:collectdata
anduse
themto
findthe
linksbetw
eeninputs
andoutputs
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odeling/7
Prob
lems
inS
I��
Choice
ofthestructuralm
odeloridentifier
scheme
��
appropriateset-up
ofnumericalprocedures
(e.g.initialconditions,startguess
...)
��
Choice
ofadequatenum
ericalmethods
(e.g.curve
fitting,OD
Esolvers,...)
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odeling/8
What
canQ
Rdo
forS
I?
QR
helpsto:
��
findm
odelclassesconsistentw
ithprior
knowledge
��
findan
initialguessofparam
etervalues
��
chooseproper
numericalm
ethods
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odeling/9
Related
work
��K
ay[1996],K
ay,Rinner,K
uipers[2000];
semi-quantitative
SI
��
Bradley
[1994],Bradley,O
’Gallagher,J.R
ogers[1997];
M.E
asley,E.B
radley[1999]
quantitativestructural
SI
��
Capelo,Ironi,Tentoni[1996,1998];
quantitativestructural
SI
��
Bellazzi,G
uglielmann,Ironi[1997,1998,2000];
quantitative“black-box”
SI
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odeling/10
Part
1
QR
forstructural
SI
Case
study:A
utomated
modeling
ofvisco-elastic
material
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odeling/11
Structural
modeling
fromdata
��P
hysicalinsighthelps
definingthe
modelspace
(greyvs.
black-boxm
odels)
��
The
modelspace
definitionrequires
modeling
expertise��
difficulttask,noteasilym
adeautom
atic
System
Identification:given
them
odelspace,theprocess
ofderivinga
goodm
odelfor
thesystem
dynamics
fromthe
observations
��
SIgrey
modeling
mustnotreduce
toa
mere
numericalfitprocess
–adherence
tothe
observations
–m
inimalcom
plexity
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System
Identification
Plausible m
odelsubset
Tentative m
odelstructure
Quantitative m
odel
Physical assum
ptionsand law
sA
pplication task
� �� ���� ���� ���� �� �
No
Model evaluation
A priori know
ledge
OK
Increase model
complexity
Modeling expertise
IdentificationS
tructural
of system dynam
ics(O
DE
)
Model space
Observations
Quantitative m
odel
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Use
ofQ
Rin
SI
��
Intelligentdataanalysis
��
Structuralidentification
��
Param
eterestim
ation
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odeling/14
Autom
atedm
odelingof
visco-elasticm
aterials
Motivations:
assessmentofvisco-elastic
materials
fromdata
data from standard
rheological experiments
quantitative accuratem
odel of mechanical behavior
of tested material
��
derivingm
odelsby
handis
ahard
task
��
models
canbe
usedfor
simulations,
andprovide
adeeper
insightw
.r.to
am
ereexperim
entalstudy
Goal:to
formulate
theconstitutive
equation������ ��
(linearO
DE
)describing
them
echanicalbehaviorofa
materialunder
suitableassum
ptions
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odeling/15
Modeling
issues��
Modeling
approach:compositionalstrategy
(rheology)
The
modelspace
was
automatically
generated,andpartitioned
intoQ
B-hom
ogeneousclasses
(seeC
apelo,Ironi,Tentoni1998)
��
Experim
entaldata:S
tandardstatic
tests-
stepinputsignal
–C
reepexperim
ents:��� �����
–R
elaxationexperim
ents:��� �����M
ON
ET
Sum
mer
School/Liliana
IroniB
5.1-
Quantitative
Modeling/16
Model
spacecharacterization
(1)
The
mathem
aticalmodeldescribes
therelation
between� �
and� � :
� !#"$�&%�� �
' !#($' %'�
! "$� � !#($'
) *
Form
almodel(F
M):
symbolic
OD
Ew
iththe
same
OD
Estructure
and �+ �� �,
The
modelspace-.
canbe
partitionedas-. �
/10� 2� -.� ,and
eachclass
isassociated
with
itsow
nQ
B
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odeling/17
Model
spacecharacterization
(2)
345768:9;5< =?>
=@A 6BDC AE 6=@A 6B C AF< =G BHJI
K L576
(T,T,F);
(T,F,F)=
QB
(H)
34M 68 9;M< = >
=@A 6B C AE 6 =N 5@A 65 C AF< =G BH IK LM 6
(F,T,T);
(F,F,T)=
QB
(N)
34O 68 9;O< = >
=@A 6B C AE 6 =N 5@A 6B C AF< =G BH IK LO 6
(F,T,F);
34P 68:9;P< = > =N 5@A 6B C AE 6 =N 5@A 65 C AF< =G BHJIK LP 6
(T,T,T);
(T,F,T)=
QB
(H–N
)
Creep
test
t t
t t
01
01
eK
eH
N e
es
0 s
t t
Qualitative
strainresponse:� �
�Q R�S R�T
QB �
�UQ� US� UT
,whereUV �
TrueW �V+ ��
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odeling/18
Intelligent
Data
Analysis
��O
bservationsdrive
thew
holem
odelingprocess
data polish
measured
response
qualitativeresponse
SystemIdentification
expertiseencoded
(thresholds, filters..)
geometric
characterization ofbasic physical features
instrumentation
pre-processeddata
geometric shape
recognition
observed physicalfeature assesm
ent
system know
ledge-base
Qualitative R
esponse Abstraction
Data
pre-processing
��
removalofoutliers
��
filtering
��
evaluationofm
easureuncertainty
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odeling/19
Qualitative
Response
Abstraction
relevantgeom
etricpatterns
Behavioral description
geometric characterization
of basic physical properties
Data
Geom
etric shape recognition&
physical features assessment
of data
��
Geom
etricreasoning:shaperecognition
anddata
segmentation
τde 0τ
vτde 1
ooτ
τ1τ
0
t
e
��
Inferenceofobservedbehavior
fromthe
extractedgeom
etricfeatures
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odeling/20
Structural
Identification
Issue:select,within
them
odelspace,thesubsetofplausible
models
��m
oreefficientcom
putation(reduced
SIsearch
space)
��
ensuredphysicalaccuracy
How
:%XYX����������
QBZ
��-.[ Q
B\ �
QBZ
-.
FM ]�
����. �^`_a � ) * T! a$b^ �c ��d ���fee� gc
_a ih
� ! "$� %�� �
' !#($' %'�
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odeling/21
Quantitative
Identification
The
plausiblem
odelset. �^`_a c
ishierarchical
(j :m
odelcomplexity
index,k :m
odelparameters)
Problem
:F
inddV� V
suchthat:
�� V�
argm
inl TC�����m ���
n�
o
��
andrcond -qp
,e� �r ,
-
:inform
ationm
atrix
��dV�
argm
ina Xst d ,Xst
:A
kaikeInform
ationC
riterion
Properties
of _avu V :
��
numericaland
statisticalreliability
��
minim
alcomplexity
��
reasonablygood
datafitting
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odeling/22
Prob
lems
fw , �
Agood
startingguess
x
mustbe
provided
fw y �
Initialconditionsz{}|~ �{ ~
mustbe
given
(De
vectorofthe
time
derivativesof�
)
�
OD
Es_
a
may
bestiff
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odeling/23
Prob
lem
���
Agood
guess
x
isneededto
ensureconvergenceto
thetrue
(ratherthan
toa
local)minim
um.
But x
hasno
explicitphysicalm
eaning,and
extractinginform
ationfrom
datais
notastraightforw
ardtask.
�
QR
-drivencurve
fitting:
� �m QBZ������ �
� US�� �� 2� �� , �exp ����� R
� UT����� � �R� UQ����� o
(exploitsa
prioriknowledge
andqualitative
datainterpretation)
+least-squares
OD
Ecollocation: x
l.s.solution
of
� !#"$� %�� �a �
' !#($' %'� �a � d �,�fee�f�� e
MO
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odeling/24
Prob
lem
���
��Initial
conditions{ ~
mustbe
given.����
couldbe
treatedas
furtherparam
etersto
beidentified
asw
ell,but
thisw
ouldentaila
highercom
putationaleffort.
�
{ ~is
definedby:{ ~h �
z� �x
� �_a m
aybe
stiff,accordingto
theelasticcom
ponentsoftheresponse
ExplicitA
dams
orR
unge-Kutta
methods
may
beunstable
�Im
plicit,backw
arddifference
schemes
(BD
F,N
DF
)are
preferred(less
accuratebutstable)
Rem
ark:S
tiffsystems
arefrequentin
many
applicationdom
ains:chem
icalkinetics,chemistry
ofpolymers,m
echanics...A
“stiff”system
ischaracterized
bytim
econstants
widely
varyingin
magnitude.
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Rem
arks
TraditionalstructuralSIdoes
benefitfromthe
integrationw
ithQ
R
Integratedfram
eworks:
��
allowus
todealautom
aticallyw
ithm
odelingproblem
sdifficultto
behandled
byhand
��
providem
ethodologiesand
toolsfor
adeeper,m
orerobustand
eco-nom
icinvestigation
ofphysical
domains
traditionallystudied
ata
mere
experimentallevel
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odeling/26
Application
toP
harmacology
Motiv
ations
Polym
ericdrug
deliveryresearch
within
thedesign
ofDrug
Delivery
System
s(D
DS
’s)
vehiculantm
aterial dosage form
DD
Sactive
ingredient
Aim
:ensuring
optimaldrug
bioavailability(fasttargeting
+m
osteffective
deliverym
ode)
The
developmentofa
newD
DS
requiresassessm
entofthosephysicochem
icalpropertiesofcarrier
materials
which
affectbioavailability
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Mucoadhesion
Mechanism
whereby
apolym
ericcarrier
adheresto
am
ucosaltissue
�
Abetter
mucoadhesive
performance
would
improve
drug
bioavailability
Traditionalapproachis
entirelyexperim
ental:
-tim
econsum
ingand
costly
-ithardly
providesinfo
onthe
structuralrequirements
foradhesion
Am
odelbasedapproach
would
providea
deeper
comprehension
ofthepolym
er-mucus
interaction
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RH
EO
LO’s
architecture
IDE
AL
BE
HA
VIO
RS
MA
TH
EM
AT
ICA
LM
OD
ELS
PLA
US
IBLE
MO
DE
LSQ
UA
LITA
TIV
EO
BS
ER
VE
D
QU
AN
TIT
AT
IVE
MO
DE
LC
RE
EP
DA
TA
BE
HA
VIO
R
QU
ALIT
AT
IVE
LIBR
AR
YM
OD
EL
QR
A
RH
EO
LOG
ICA
LF
OR
MU
LAE
Iden
tification
System
Output:
OD
Em
odelcom
pliancem
odel
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odeling/29
Resum
e
Variables:
�������perturbation
onthe
system(input)
�������
elicitedsystem
response(output)
Data:��
Standard
creeptest:���������#���������
Models:
��
OD
Em
odel
��
Com
pliancem
odel
Structuralidentification �
classand
orderofthe
model
Param
eterestim
ation
valuesofthe
parameters
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odeling/30
Com
pliancem
odelE
xplicitlyrelated
tothe
rheologicalstructureofthe
material:
¡� ����¡� ¢
£¤ ¥¦ ¡¤ § �¨ ©ª �
�«¤¢
�¬
¡®���� ¯
(instantaneouselasticity)¡�
Prom
ptelasticstretching
ofbondsbetw
eenthe
primary
structuralunits
¡°� �� ¯
(retardedelasticity)± ¡¤² «¤³¤ ¥¦´´£
Bonds
breakand
reform,producing
aslow
er,stillrecoverable,deformation.
µ¶
number
ofbondtypes
·¸ ¶
intensityofeach
bondtype
¹¸ ¶
times
atwhich
thegreater
partofeachbond
typeestablishes
¡���� ¯
(viscousflow
)¬
Irreversiblerupture
ofbonds.Inparticular:
1.the
#retardation
times
(model
order),relatedto
theestablishm
entof
newtypes
ofbonds,characterizesthe
materialcom
plexity
2.the
compliance
valuesexpress
thestrength
ofthestructuralunits
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The
applicationprob
lem��
Materials
NaC
MC
:solutionsofpolym
eratthree
viscositygrades
(LV,MV,H
V),
eachone
atthreeconcentration
levels(low
,medium
,high).P
olymer+
mucin:
mixtures
ofeachpolym
erw
ithm
ucinatthree
differentconcentrations
��
Aim
Model-based
investigationofpolym
erm
ucoadhesiveperform
ance,to
getadeeper
knowledge
onthe
polymer-m
ucininteraction
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odeling/32
Method
1.Q
uantitativecharacterization
ofrheologicalpropertiesof
eachm
aterialbym
eansofm
odelorderand
parameters
2.H
ighlightstructuralconditionsatw
hichpolym
er-mucin
synergyis
higher(bestm
ucoadhesiveperform
ance)
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1-
Results
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odeling/34
1-
Results
Modelevaluation:
Akaike
indexesand
conditionnum
bers
Polym
er(H
.V.NaC
MC
1.6%)
+8%
mucin
µrcond
º» µ¼
01.00
e+00
1005.51
4.00e-04
780.4
µ¾½¿2
9.33e-03
591.43
2.19e-08
423.6
Optim
almodelorder
andparam
eterestim
ates(95%
confidenceintervals)
Polym
er(H
.V.NaC
MC
1.6%)
+8%
mucin
µ ½
2
À ½Á
1.568e+
2[1.562
e+2,1.574
e+2]
Pa Â
s
À ½Ã
2.880e+
3[2.879
e+3,2.882
e+3]
Pa Â
s Ã
À ½Ä
8.128e+
2[8.123
e+2,8.132
e+2]
Pa Â
s ÄM
ON
ET
Sum
mer
School/Liliana
IroniB
5.1-
Quantitative
Modeling/35
2-
Results
k*
L.V.NaC
MC
low0
Mixture
with
mucin
3L.V.N
aCM
Cm
edium0
Mixture
with
mucin
2H
.V.NaC
MC
low1
Mixture
with
mucin
2H
.V.NaC
MC
medium
2M
ixturew
ithm
ucin2
LV-N
aCM
Cand
HV
-NaC
MC
atdifferent
concentrations,and
theirm
ixturew
ithm
ucinat8%
concentration:
optimalm
odelorder(ÅÇÆ
)
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odeling/36
2-
Results
The
additionofm
ucincauses
anincrease
inthe
elasticproperties,
bythe
establishments
ofnewbonds:
��
increasein
modelorder¯
betterinteraction
between
polymer
andm
ucinchains
��
increasein
thecom
pliancevalues¯
furherstrengtheningofthe
mu-
coadhesiveinterface
The
polymer-m
ucininteraction
ishighestw
henLV
-NaC
MC
isused
atthelow
estconcentration(deeper
interpenetration)
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odeling/37
Conc
lusiverem
arks��
RH
EO
LOhas
favoureda
modelbased
approachto
theinvestigation
ofphysicochem
icalproperties
relevantin
DD
S’s
design(e.g.
mu-
coadhesion)
��
The
proposedapproach
canbe
usedto
investigatephenom
enain-
volvingvariations
inthe
material
structurerevealed
bychanges
inthe
rheologicalbehavior
��
The
modelbased
approachhas
provided
-deep
insightintothe
polymer-m
ucininteractions
-cheaper
andm
oreeffective
evaluationof
polymer
mucoadhe-
siveperform
ancesthrough
model
parameters
andcom
plexity(rheologicalproperties)
New
application:H
emodynam
ics:study
ofblood
rheologicalproperties
fordiagnostic
andtherapeutic
purposes
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odeling/38
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