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

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.

Data, pre-knowledge and structure

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.

Canonical models based on physiology

The parameters have to be estimated by statistically founded algorithms (Extended Kalman filter, Nelder-Mead 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.

Parameter identification and estimation

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 sub-mechanisms, 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.

Validation and analysis of models

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.

Suggestions and identification of biomarkers

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.

Integration of models and their interactions

Modeling point 1

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.

Modeling may make the inaccessible accessible!

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 2Modeling 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

Sequential component model

- suitable for sub-model validations and effective calculations

Impulse Function:muscle sympathetic stimulation,

central command or vestibular effect

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.

Loop small, indicates reduced dynamics; loop also not closed

Pressure Data (blue) and Mean Pressure (red) vs. Time

Nerve Firing vs. Pressure (hysteresis loop is wide and closed)

Loop very narrow with decreased slope; indicates reduced dynamics

Parasympathetic (blue) and sympathetic (red) tones vs. Time: (significant dynamics in both tones)

Tone value dynamics are greatly minimized, particularly sympathetic tone

Young Subject

Healthy Elderly Subject

Hypertensive Elderly Subject

Reduced pressure dynamics and lower resting state once regulated

Reduced pressure dynamics upon standing, longer timescale for regulation

Sympathetic tone response almost null aside from anticipation impulse

HR Data (blue) and HR Model (red) vs. Time

Smaller scale on HR dynamics, higher resting state, slower regulation

Slightly decreased dynamics, regulation on slower timescale than young subject

Notice that the model gives access to the sympathetic and the parasympathetic tones (nerve activities) as functions of time.

Loop small, indicates reduced dynamics; loop also not closed

Pressure Data (blue) and Mean Pressure (red) vs. Time

Nerve Firing vs. Pressure (hysteresis loop is wide and closed)

Loop very narrow with decreased slope; indicates reduced dynamics

Parasympathetic (blue) and sympathetic (red) tones vs. Time: (significant dynamics in both tones)

Tone value dynamics are greatly minimized, particularly sympathetic tone

Young Subject

Healthy Elderly Subject

Hypertensive Elderly Subject

Reduced pressure dynamics and lower resting state once regulated

Reduced pressure dynamics upon standing, longer timescale for regulation

Sympathetic tone response almost null aside from anticipation impulse

HR Data (blue) and HR Model (red) vs. Time

Smaller scale on HR dynamics, higher resting state, slower regulation

Slightly decreased dynamics, regulation on slower timescale than young subject

New concept

New measure

New clinical method

Modeling point 3Modeling is the only way to strictly define concepts well and to obtain values for measureable quantities (in combination with experiments)

)(1

)( tVC

tP

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 4Mathematics 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

Complex models with inaccessible parts and processes can be used for estimating quantities / parameters describing these inaccessible parts and processes

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

Individual / patient specific measurements are performed indirectly by help of models and biomarkers are obtained

A good subject In

sulin

/Glu

cose

[m

M]

Time [min]

Insulin

Glucose

Schematic representation of a compartmental delay model

Tgh is the net difference between glucose production and glucoseelimination; Tigmax is the maximal rate of second phase insulin release

Vg

Vi

Dg

Tgh

Vi

Kxi

Tigmax

Kxgl

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;

Thank you for your attention