Transcript 2-Presentations\Ottesen_HO

The Mathematical Microscope
by
Johnny T. Ottesen
Department of Science, Systems and Models
Roskilde University, Denmark
Copenhagen, IMFUFA, RUC
REx workshop, 2009
Methodology for developing
individual and patient
specific models
 Data, pre-knowledge and structure
 Canonical models based on physiology
 Parameter identification and estimation
 Validation and analysis of models
 Suggestion and identification of biomarkers
 Integration of models and their interactions
Data, pre-knowledge and structure
All reliable models in physiology are based on solid
knowledge and adequate data. Such knowledge and a
huge data material related to diabetes do exist.
Statistical methods such as approximated entropy
(regularity statistics known from nonlinear dynamics)
and generalized principal component analysis may
reveal further information, which forthcoming models
have to encompass.
Canonical models
based on physiology
Models should be developed so they incorporate the
responsible mechanisms for the modelled phenomena.
In order to identify and estimate patient specific
parameters in an effective and reliable way the number
of parameters has to be kept as low as possible, thus all
unimportant factors and elements should be excluded,
i.e. the so-called principle of parsimony have to be
obeyed.
The models Should be based on first principles
(conservation laws etc) whenever possible and the
parameters shall have a physiological interpretations.
Such models are denoted canonical models.
Parameter identification
and estimation
The parameters have to be estimated by statistically
founded algorithms (Extended Kalman filter, NelderMead algorithm combined with the simplex methods,
Multidirectional Search, Particle filter/Sequential
Monte Carlo (SMC) methods, generic algorithms, etc).
Not all the parameters will necessarily be identificable
due to limitation in available data. Thus the estimation
process has to be an iterative procedure coupled with
sensitivity analysis or generalised sensitivity analysis
combined with subset selection strategies for
instance.
Validation and analysis
of models
An important part of the validation process (i.e. lack of
falsification) is to compare model results with data
(ideally with data independent of the data set used to
estimate parameters).
Model reduction, analysis of variations of submechanisms, analysis of stability and bifurcation,
analysing possible limit cycle behaviour etc. are all
supplementary validation methods.
If a model fails to be validated it needs to be adjusted
which often gives rise to new insights into the
underlying physiology.
Suggestions and
identification of biomarkers
When well validated models with patient specific estimated
parameters exist the identification of potential biomarkers
become achievable. Different groups of patients, i.e.
pathological subjects versus non-pathological subjects
can be examined. Some of the parameters for two different
groups have to vary which suggest biomarkers.
To determine whether there is a ‘real’ difference between
values of the parameters (i.e. the biomarkers) within two
groups or whether suggested biomarkers can identify
variant causes of the illness (diagnosed by symptoms),
statistical tests has to be performed.
The biomarkers will for sure give rise to a classification of
variants of the illness because they are born to agree with
data from clinical diagnoses.
Integration of models and
their interactions
It must be analysed how systems of coupled and
integrated models behave compared to the isolated
canonical (sub-)models (e.g. an insulin-glucose model
coupled to a cardiovascular model or an exocytose
model coupled to a insulin-glucose model at system
level).
In cases where biomarkers have to be adjusted, we
expect that the adjustment is merely refinements of
the original validated biomarker.
Modeling point 1
Modeling may make the inaccessible accessible!
When William Harvey discovered the circulation of blood
in the cardiovascular system, he used mathematical
modeling as a tool and as the argument. He obtained a
contradiction (or a grotesque consequence), whereby he
falsified the existing ancient paradigm.
Harvey made the invisible capillary
visible by use of a model (46 years
before they became visible to the
human eye by help of the light
microscope).
Modeling point 2
Modeling is an outstanding, a superb and a
unrivalled way to obtain knowledge, insight
and understanding in science.
Parts which can’t be isolated experimentally
may be studied (separately) by modeling.
Change in heart rate for
a normal young subject
For fitting the
heart rate
curve one
need 7 pieces
of lines which
demand 14
parameters in
a maximum
likelihood
estimate (a
least square
formulation)
Muscle sympathetic stimulation, central command or vestibular effect
Conceptual Model
Impulse Function:
muscle sympathetic stimulation,
central command or vestibular effect
Sequential component model
- suitable for sub-model validations and effective calculations
Model compared to data
Heart rate model predictions (blue trace) plotted
against measured data (green trace). Left panel
shows results from a healthy young subject, middle
shows results from a healthy elderly subject, and
right shows results from a hypertensive elderly
subject.
Young Subject
Pressure Data (blue) and Mean Pressure (red)
vs. Time
Nerve Firing vs. Pressure (hysteresis loop
is wide and closed)
Parasympathetic (blue) and sympathetic
(red) tones vs. Time: (significant dynamics
in both tones)
HR Data (blue) and HR Model (red) vs. Time
Notice that the model gives access to
the sympathetic and the
parasympathetic tones (nerve
activities) as functions of time.
ealthy Elderly Subject
Reduced pressure dynamics and lower resting
state once regulated
Loop very narrow with decreased slope;
indicates reduced dynamics
Tone value dynamics are greatly minimized,
particularly sympathetic tone
Smaller scale on HR dynamics, higher resting
state, slower regulation
Loop small, indicates reduced dynamics; loop
also not closed
Sympathetic tone response almost null aside
from anticipation impulse
Slightly decreased dynamics, regulation on slowe
timescale than young subject
Hypertensive Elderly
Subject
Reduced pressure dynamics upon standing,
longer timescale for regulation
Young Subject
Pressure Data (blue) and Mean Pressure (red)
vs. Time
Nerve Firing vs. Pressure (hysteresis loop
is wide and closed)
Parasympathetic (blue) and sympathetic
(red) tones vs. Time: (significant dynamics
in both tones)
HR Data (blue) and HR Model (red) vs. Time
New concept
lthy Elderly Subject
New measure
New clinical method
Reduced pressure dynamics and lower resting
state once regulated
Loop very narrow with decreased slope;
indicates reduced dynamics
Tone value dynamics are greatly minimized,
particularly sympathetic tone
Smaller scale on HR dynamics, higher resting
state, slower regulation
Loop small, indicates reduced dynamics; loop
also not closed
Sympathetic tone response almost null aside
from anticipation impulse
Slightly decreased dynamics, regulation on slower
timescale than young subject
ypertensive Elderly
Subject
Reduced pressure dynamics upon standing,
longer timescale for regulation
Modeling point 3
Modeling is the only way to strictly define concepts
well and to obtain values for measureable quantities
(in combination with experiments)
P (t ) 
1
V (t )
C
Modeling is and outstanding tool for suggesting new
experiments which were hardly possible without the model
(and leads to parameter estimations and model validation)
Modeling point 4
Mathematics is able to unfold the influence that each
of the processes has on the overall dynamical
behaviour of a complex system
Modern experimental science - especially
modern biology - is very good at
separating systems, into components
simple enough for their structures and
functions to be studied in isolation.
Mathematical modelling is the only
controlled way to put the pieces back
together, with equations that represent the
system's components and processes, as
well as the structures and interactions.
Modeling point 5
Modeling is an excellent tool for design purposes
Modelling is an invaluable tool
• for decision support in diagnostics
and therapy (theranostics)
• for the development of drugs
(models make it possible to target
the cause of a disease directly)
• for developing and constructing
industrial devises and equipments
Modeling (main) point 6
Patient specific parameter estimation is the future – it is
possible and it is an opportunity for pharmaceutical
industry and medical doctors to target causes instead
of treating symptoms
Complex models with inaccessible
parts and processes can be used for
estimating quantities / parameters
describing these inaccessible parts
and processes
Individual / patient specific measurements are performed
indirectly by help of models and biomarkers are obtained
A good subject
Insulin/Glucose [mM]
Insulin
Glucose
Time [min]
Schematic representation of a
compartmental delay model
Dg
Tgh
Vg
Kxgl
Vi
Vi
Kxi
Vg and Vi are the distribution
volumes for Glucose (G) and
Insulin (I).
Dg stands for the glucose bolus
administered;
KxgI is the second order net
elimination rate of glucose per unit
insulin concentration;
Kxi is the first order elimination rate
of insulin;
Tigmax
Tgh is the net difference between
glucose production and glucose
elimination;
Tigmax is the maximal rate of second
phase insulin release
Thank you for your attention