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Review
Modelling methodology in physiopathology
Jean-Pierre Boissela,b,c,, Benjamin Ribbab,d, Emmanuel Grenierb,e,
Guillemette Chapuisatb,f, Marie-Aimée Dronnea,b
a
CNRS, UMR5558, Lyon, France
Institute for Theoretical Medicine, UMR5558, Claude Bernard University Lyon 1, RTH, Laennec School of Medicine, rue Guillaume,
Paradin, 69008 Lyon, France
c
Léon Bérard Anticancer Centre, Lyon, France
d
EA, Lyon-Sud School of Medicine, Lyon, France
e
UMPA, UMR5669, ENS Lyon, 46 allée d’Italie, 69364 Lyon, France
f
CMLA, ENS Cachan, CNRS, Pres UniverSud, 94235 Cachan, France
b
Abstract
Diseases are complex systems. Modelling them, i.e. systems physiopathology, is a quite demanding, complicated,
multidimensional, multiscale process. As such, in order to achieve the goal of the model and further to optimise a rathertime and resource-consuming process, a relevant and easy to practice methodology is required. It includes guidance for
validation. Also, the model development should be managed as a complicated process, along a strategy which has been
elaborated in the beginning. It should be flexible enough to meet every case. A model is a representation of the available
knowledge. All available knowledge does not have the same level of evidence and, further, there is a large variability of the
values of all parameters (e.g. affinity constant or ionic current) across the literature. In addition, in a complex biological
system there are always values lacking for a few or sometimes many parameters. All these three aspects are sources of
uncertainty on the range of validity of the models and raise unsolved problems for designing a relevant model. Tools and
techniques for integrating the parameter range of experimental values, level of evidence and missing data are needed.
r 2007 Elsevier Ltd. All rights reserved.
Keywords: Systems physiopathology; Modelling; Methodology; Strategy
Contents
1.
2.
Beyond systems biology, systems physiopathology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1. Why should we pay attention to methodology? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2. Challenges and strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3. A few general principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Knowledge and uncertainty management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2
4
5
6
Corresponding author at: Institute for Theoretical Medicine, UMR5558, RTH Laennec School of Medicine, rue Guillaume, Paradin,
69008 Lyon, France.
E-mail address: [email protected] (J.-P. Boissel).
0079-6107/$ - see front matter r 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.pbiomolbio.2007.10.005
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2
3.
4.
5.
6.
7.
8.
9.
Sub-models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.1. Splitting the discursive model in sub-models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2. Chronology and organisational levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Perhaps the most challenging: parameter valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Phenomenological or mechanistic?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Top-down or bottom-up? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Validation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1. Beyond systems biology, systems physiopathology
1.1. Why should we pay attention to methodology?
Systems biology has been devised to tackle the barrier of complexity in life science. Beyond the complexity
issue, numerical modelling has been introduced in physiopathology and therapeutic research because of the
decreasing innovation efficiency in therapeutics and the growing difficulties for physicians to make optimal
treatment decision for a given patient through combining the many available modestly efficacious treatments.
Altogether, these two worrying developments result in a loss of chance for patients when compared to the
amount and quality of available knowledge.
The ultimate goal of ‘‘in silico’’ models in physiopathology is to improve health. Through designing
numerical models of a disease and running computer simulations, we expect a better understanding of the
links between its various components, weighting the respective influence of its various factors, deciphering
new and more relevant targets for innovative medicines and a quicker way of testing a new therapeutic
intervention at the very beginning of its development, before a sizeable amount of resources has been invested
(early proof of concept). Altogether, these developments should result in improving the way new therapies
are discovered and in providing doctors with tools predicting the outcome of combining therapies. Also,
following the introduction of patient parameter values into the numerical models of his/her disease and of all
the available therapies, a reasonable choice among all the alternatives and their combinations would be
facilitated.
Actually, in terms of complexity, physiopathology is far beyond systems biology. Disease mechanisms are
much more complex than anything else. The main reason is that a disease develops throughout a diversity of
levels, from gene expression up to population, whereas systems biology limits itself at a couple of levels,
molecules and cell, sometimes a population of cells. The tissue level is seldom accounted for, and neither
physiology nor anatomy is paid attention to. Diseases encompass several piled up organisational levels of
complex phenomenon, from genes to population (Fig. 1). Time scales vary from nanoseconds to several
decades, with chemical interactions for the former scale and evolution to clinical events for the latter. In
chronic diseases, such as cancer or atherosclerosis, the sequence of events at the molecule, cell, tissue and
target organ levels takes decades to achieve in death or myocardial infarction. In addition, new knowledge is
emerging at a high rate: data from clinical research are coming in on top of evolving biological knowledge
(Fig. 2). This observation has three consequences: (i) the ever-extensive amount of knowledge one
should incorporate in a numerical model of disease; (ii) the need of a careful knowledge management tool, and
(iii) models should be conceived flexible enough in order to integrate any new relevant knowledge.
At first sight, it seems to be an easy task: just collect all the necessary pieces of information in the literature,
and put them on a computer to draw pictures. But everyone who tried this knows that many difficulties arise.
First we must find all the pieces, and even if only one is lacking, we may not achieve our objective well.
Second, we must be sure that all the mathematical solutions represent properly the real world we are interested
in. Third, we should solve all the methodological and technological hurdles that we faced along the process by
achieving a compromise between a too stringent reduction which would make the model irrelevant to its goal
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Integration axis
population
organism
tissue
System
physiopathology
subcellular
cellular
interactions between
organisational levels
}
molecular
System biology
Ph
en
om
en
on
ax
is
Time axis
interactions between
phenomenon (sub-systems)
Fig. 1. Complexity of systems physiopathology and systems biology.
Number of published original reports
16000
14000
12000
10000
8000
6000
4000
2000
0
1970
1975
1980
1985
1990
1995
2000
2005
2010
Years
Fig. 2. Number of clinical trial reports recorded in PubMed from 1970 as of today.
and a too big amount of details which will make the objective impossible to reach. Fourth, it takes time to
build the model and to make it run properly and in an efficient way.
In addition, beyond pure science, numerical modelling of diseases is science applied to improving care to
patients. As a consequence, the researcher should pay attention to both quality and transparency. For ethical
reason, the former is compulsory, whereas the latter is required by regulators. Actually, regulations are
already mandatory, and one can bet they will be even more stringent in the future in order to protect
consumers. Models and processes which they are part of should comply with the regulations.
Overall, modelling diseases develop along four axes: the phenomenon or sub-systems axis, the time axis and
the integration axis (Fig. 3). Clearly, there is no unique way to design the model which would help achieving
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Integration axis
Model
Modelaim(s)
aim(s)
• Mathematical solutions
• Embedding the description levels
xis • Integrating sub-models
a
l
ca
eri
m
Time axis
Ph
en
om
en
ol
og
ica
la
xi
s
Nu
Fig. 3. The four axes along with the numerical modeller in physiopathology plays.
Table 1
The challenges of systems physiopathology
Amount of data
Quality of data
Lack of data
Diversity of time scales
Number and diversity of complexity levels (from molecules to population)
Number and diversity of components and types of relations at each level
Availability of relevant mathematical tools
Computing time and precision
Model validation
the researcher aims. All this makes modelling in physiopathology different. It explains why an appropriate
strategy and relevant methodologies are needed.
1.2. Challenges and strategy
The challenges with numerical modelling in systems physiopathology are listed in Table 1. In order to take
them up, one needs to follow a research strategy. The first, inescapable step is to answer the question: what do
we want to model and why? A precise formulation of the objective(s) is required for a proper fixing of many
choices that arise during the process. This is not solely to obtain a pure numerical representation of what is
known in human physiology or to aim at a global understanding of the real system. It is not merely to build a
new tool for the researcher in human biology. It is a practical objective, such as the discovery of new therapies
for human diseases.
The next major step is the integration of the approach in the virtuous circle shown in Fig. 4. In practice, this
means bringing together a variety of expertise from mathematicians to clinicians, including computation
scientists, molecular biologists and pharmacologists, and to ensure living connections with other groups
working on the same theme. In short, be multidisciplinary.
The other steps are tied together specifically according to every research topic. However, it is always
structured along the main steps that are shown in Table 2. Unlike what comes up from this table, the sequence
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Knowledge in
biology and
Mathematical and
medicine
computing tools
Observation and
Numerical models
experiments:
in vivo, ex vivo, in
vitro
Experiments:
in silico
Fig. 4. The virtuous circle.
Table 2
Major steps of a model-building strategy
Setting the objective
Making up and organising the multidisciplinary team
Organising connections with in vitro and in vivo expert groups
Selecting the knowledge management tool(s)
Collecting the data
Writing the discursive model
Sub-setting the discursive model
Choosing a phenomenological approach or a mechanistic one (or both)
Finding the mathematical solutions
Modelling the sub-sets of the discursive model (sub-modelling)
Arranging the computational tools
Writing the numerical solutions
Integrating the sub-models
Exploring the model robustness
Reducing the model
Validating the model
Using the model: ‘‘in silico’’ experiments
of steps is not fully linear. Step connection should not be looked at as a no-return process. Quite often, if not
always, it is necessary to look back from a step to a former one.
In the following we will consider some of these steps. Others, such as computing, are dealt with or illustrated
elsewhere in this issue of the journal (Descombes and Dumont, 2007; Grenier et al., 2007; Dronne et al.,
2007a).
1.3. A few general principles
The way the challenges are dealt with depends on the main objective of the research. However, there are a
few general principles that are quite helpful in finding appropriate ways to solve the problems: (i) modelling of
biological processes can be piecewise, clearly identified input and output with for each piece or sub-model and
biomarkers that could be used for the sake of validating the global model (this is the ‘‘Legos-like’’ principle);
(ii) each piece can be numerically solved at a different level of complexity from the others; (iii) at anytime in
the progress of the modelling process, a sub-model can be replaced by a more detailed one (the ‘‘plug-in’’
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principle). However, replacement by a more detailed sub-model should be prepared (consistency in input and
output) and may require new developments and re-calibration of the whole model.
2. Knowledge and uncertainty management
Systems physiopathology requires collection and analysis of all the available evidence and data, before
selecting those relevant for the model. Their uncertainty and strength of evidence should be weighted and
recorded. The knowledge needed to model diseases such as acute stroke or cancer is enormous and diverse.
Building a model in physiopathology relies on the basis of evidence that are considered as sound enough and
that look crucial for integration into the model. This is a kind of ‘‘basis of knowledge’’ which will be
represented in the discursive model. However, these data are embedded in uncertainty. They range from
in vitro experimental results in basic biology, including structural biology, to randomised clinical trials and
clinical imaging data. The first idea consists of scanning the whole literature, collecting all the possible
experimental data, together with the experimental conditions, the type of cells and species, then to compare all
these data and try to exclude the erroneous ones. This would give an up-to-date basis of knowledge. Because
of the diversity of experimental and observational settings, values of a given parameter may vary over a wide
range. Thus, it seems important to score the record by its variability and strength of evidence. In addition,
identifying erroneous values is a hard task because most of the time the experimental setting is not described in
such details that wrong measurements can be detected for sure. In addition, the measurements might well be
correct, but the experimental setting may not be representative of the situation of interest. Instead, we may
better take all the available knowledge. If moreover one takes into account the experimental errors and the
individual variations, one realises that it is impossible to sort out the ‘‘true data’’. Thus, we cannot view the
data extracted from the literature as definite values. An illustration of the challenge facing model builder,
because of uncertain and insufficient knowledge is given in other articles in this journal issue, in published
material and in Section 4.
Bearing this major hurdle in mind, all the data must be collected, reviewed and stored in formats and in
repositories that allow repeated access by all the team members. The development of databases or knowledge
bases is thus needed to house quantitative, qualitative and structural information with a scoring of their
strength of evidence (Ribba et al., 2006c). Having said that, the way of achieving such scoring remains an open
question for biochemical and genetic data. One of the solutions may be to address the problem by adopting a
statistical approach, i.e. meta-analysis, and a conceptualisation of the fundamentals of the empirical process
by which the parameter values are determined (Grenier et al., 2007; Dronne et al., 2007a; Dronne et al., 2006).
However, a lot of development is needed to achieve a relevant and valid array of tools capable of solving this
critical issue.
Besides data from literature, imaging and clinical data are often required (e.g. for building sham patients).
These data are available somewhere. The issue here is to know where there are stored and to assess their
quality. One needs an international network of such databases open to modellers.
3. Sub-models
3.1. Splitting the discursive model in sub-models
The discursive model is a text or a chart (or, better, both) that brings together all the components of the
disease and their interactions that are thought to be relevant enough to the objective of the modelling process.
The discursive model comes out from the knowledge review mentioned above. It is the basis for the later steps
(Table 2). Components and their connections will show up in the final numerical model. It is presented as a
text, summarised by a chart and several series of rules. An example of the chart is given in Fig. 4 (Dronne
et al., 2004). The discursive model comes in a variety of embedded forms: at the molecular level, at the cellular
level, etc. Use of a dedicated software may help in mastering this step. The final discursive model could be
quite intricate. Often, in physiopathology, it consists of thousands of pieces and even more functional
relations.
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In the numerical modelling steps, the challenge is therefore to integrate this large amount of heterogeneous
knowledge and data. A solution to this is splitting the discursive model by identifying independent subsystems. These are characterised by their studiability and the possibility of modelling them independently as
sub-models, while respecting the global dynamic system. For example, in modelling acute stroke, apoptosis is
described as an entire process by itself. Independence of a sub-system is defined on the basis of a couple of
rules: (i) the underlying biological phenomenon has a recognised specific functional status; (ii) there are wellcharacterised signals connecting with other sub-systems (input/output of the sub-system); and (iii) it
encompasses at least a biomarker which is measurable in vitro or in vivo (for the sake of validation of
simulation results). Clearly, apoptosis can be viewed as a sub-system with relatively simple input/output
signals, e.g. calcium concentration, energy stores as inputs, energy consumption and eventually cell death as
output.
3.2. Chronology and organisational levels
However, like the whole system, sub-systems have two other characteristics: one should pay attention to a
chronological component and an organisational level. The molecular level is the lower level, whereas the
population level is the higher level. As critical pieces of the whole discursive model, sub-systems should be
organised along two dimensions, the time and organisational axes. Both axes are descriptors of the sequence
of events, i.e. chronology and causal relationship. As an example, multiscale mathematical models of cancer
growth have been recently proposed (Ribba et al., 2005, 2006a, b).
In acute stroke, cells die first by necrosis, caused by a cellular oedema, resulting in abnormal ion exchanges
because of energy deprivation (Fig. 4). Then cells of the penumbra area may die through a completed
apoptotic process (Dronne et al., 2004). However, apoptosis occurs later, and can last several days. These
examples show that sub-systems can be linked by causal relations and can have different chronologies.
Cell and tissue mechanical properties are increasingly recognised as regulating factors of many biological
processes ranging from gene transcription to tissue re-modelling. Cell elasticity is a key parameter for
mechanical signal transduction (Huang et al., 2004; Janmey and Weitz, 2004), while extracellular matrix
stiffness regulates cell adhesion and migration (Gray et al., 2003). Environmental mechanical forces are known
to affect many cellular functions, such as cell growth, proliferation, protein synthesis and gene expression.
Thus, any numerical modelling at tissue level would have to point to different sub-systems at different
organisational levels that account for, e.g., cell proliferation, genetic expression and protein synthesis. It is
thus clear that one of the challenges is to allow the integration of a variety of organisational levels. Moreover,
computational power will soon allow the integration of cell proliferation regulated at molecular and genetic
levels, as well as macroscopic changes of tissue compression and deformation.
Thus, the connections between sub-models support causality and chronology also, as well as pure functional
relations. In Fig. 1, the phenomenon axis is arranged according to causality.
4. Perhaps the most challenging: parameter valuation
A numerical model is a series of equations or a series of rules characterised by algebraic or logical functions
and parameters. The choice of functions depends on the connections or interactions between the system
components or entities. Parameter values determine the spatial and temporal behaviour of the model. Thus,
parameter selection and valuation are critical. Because of the uncertainty and inadequacy of available
knowledge, parameter valuation is a challenging issue.
This issue arises in two different settings: (i) the parameter values are drawn from the literature, i.e. the
experimental data they were derived from are not accessible and (ii) the parameters are adjusted on a set of
experiments to a statistical method (e.g. maximum likelihood) that allows fitting the model to the data. As the
appropriate experimental data are often not accessible, the former is the most frequently used approach. We
will limit this section to it. A practical example is given in Dronne et al. (2004).
To illustrate the challenge and ways to make it, let us take a case where ordinary differential equations are
the choice for the mathematical solution. The equation system is parameter sensitive. As in acute ischaemic
stroke, let us consider a series of ionic channels of sub-system 2 in Fig. 5. The ordinary differential equations
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ococclusion
c lu s ion
1
4
¯O2
¯O 2
¯cellular
¯ cellular
metabolismt
me abolism
Ri
¯ATP
¯A
TP
Ionic pumps,
pumps,
Ionic
channels,
channels,
exchangers
exchangers
5
spreading
spreading
depression
depression
glutamate
glutamate
uptake
uptake
reverse
reverse
glutamate
--glutamate
3
-[K+]e
- [K+]e
oedema
oedema
-[Na+]i
- [N a+ ]i
[Ca2+ ]i
--[Ca2+]i
NMDA
NM DA
receptor
receptor
2
necrosis
necrosis
rereperfusion
p e rfu s io n
AMPA-AMPA
Kainate
Kainate
eptor
receptor
9
8
apoptosis
apoptosis
-ATP
- ATP
-O2
- O2
NOS 22
NOS
(astrocyte)
(astrocyte)
cellular
cellular
metabolism
metabolism
10
NOS 11
NOS
(neuron)
(neuron)
NO
radicals
free radicals
--free
inflammation
inflammation
6
7
Fig. 5. Example of a chart representing a discursive model. Ten sub-systems are assumed to describe the whole process of an acute stroke.
system is so large and depends on so many parameters (three equations per channel and between five and ten
parameters per channel) that a small error on a parameter value may lead to a physiologically wrong
behaviour. Also, different parameter values can lead to alternative, although correct, physiological behaviour.
The same set of equations differing only by parameter values describes both action potential of neurons and
repetitive firing and pacemaker behaviour of cardiac cells. As said above, the parameter values are not known
precisely and even not known at all in many instances. Some data for glial cell ionic channels are lacking
(Dronne et al., 2007b). Available data come from species or types of neurons that are the ones we want to
model, and values differ across species (Hodgkin and Huxley, 1952). It is impossible to find the necessary data
for all the channels of a given type of cells. There are so many possibilities that not all situations will be
explored in vivo or even in vitro and most of the parameters of interest will remain unknown. Nevertheless, we
have to play with all the parameters, whose values are ‘‘partially’’ known or ‘‘totally’’ unknown, in order to
catch the physiologically correct behaviour.
In practice, one should begin with an experimental set of parameter values for every parameter in the model
and move on to a ‘‘reasonable’’ set. The experimental or observed set is directly drawn from the literature
scan. It includes all the values that have been observed in rather similar experimental settings, as close as
possible to the model the system is embedded in. The knowledge management tool mentioned above helps in
finding the whole range. It remains to define what ‘‘reasonable’’ values mean. We can state that parameters are
‘‘reasonable’’ if the model shows physiologically relevant responses to stimuli, correct rest state and if the
parameter values remain within plausible ranges. Note that this definition does not imply precise ‘‘real’’
values. It only assigns them in a range said plausible according to qualitative dynamic and resting properties of
the model. These properties arise from pre-specified rules included in the discursive model and drawn from the
knowledge basis. Hence the basis of knowledge must contain not only experimental data but also a qualitative
description of the global behaviour.
The reasonable set can be searched either using probabilistic methods or using deterministic methods.
Probabilistic methods consist of choosing at random a set of parameters and checking whether it leads to
correct macroscopic behaviour, i.e. whether it meet a series of rules that represent the qualitative knowledge.
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Table 3
Procedure for selecting the ‘‘reasonable’’ set
Choose the experimental/observed range of all parameter values
Choose at random a set of values for all parameters
Test numerically the values with the mathematical model according to the rules
If the rules are met, keep this set in the ‘‘reasonable’’ set
With the deterministic methods, we first build a ‘‘distance’’ which measures the difference between the
numerical result and the desired one, and we then try to minimise the distance. The former is easier to apply
(Table 3). Taking again the ionic channels case, the basis of rules consists of: (i) existence of a stable rest
equilibrium potential and (ii) a short (1 ms) strong enough imposed current leading to an action potential.
These two rules can be translated into precise mathematical statements and can be easily checked by an
automatic procedure. If the tested set meets the rules, it is stored. At the end of a run, a few thousands of
parameter sets have been tested, leading to a few dozen of suitable sets, which can then be explored in more
detail by adding more stringent criteria to the rules, e.g.: (i) an action potential is created for short-duration
external stimuli; below, there should be no action potential; above, the cell is depolarised; and (ii) for longlasting external stimuli, either multiple action potentials are created or repetitive firing occurs. Of course, the
more and more stringent rules we impose, the harder we get the reasonable set of parameters. In some cases it
appears impossible: the rules are contradictory. If the number of rules is great then random trials are no longer
efficient and must be helped by ‘‘local refinement’’ algorithms which locally increase the density of trials when
almost all criteria are satisfied.
As we see later in this article, there is a continuum between parameter valuation and exploration of the
model robustness.
5. Phenomenological or mechanistic?
A phenomenological numerical model is reduced to a representation of the envelope of the phenomenon of
interest. For instance, apoptosis in acute stroke could be modelled by any mathematical equation which
increases along time up to a maximum, then levels off and eventually goes down to baseline after a couple of
days (Chapuisat et al., 2007). In the same issue of the journal the approach of acute stroke by Chapuisat is an
illustration of a phenomenological model, whereas the approach by Dronne of the same disease is a
mechanistic model (Dronne et al., 2007a; Chapuisat et al., 2007). Another rather pure example of the
phenomenological approach is presented in the article by Grenier et al. (2007) in the same issue of the journal.
They assume migraine as a wave phenomenon and used a mathematic descriptor of a reaction–diffusion
elementary phenomenon to explore how migraine spreads in the brain.
Although parameters of phenomenological models may have a biological meaning, most of the time their
links with biology are somewhat indirect, and often they have no meaning at all. However, the main advantage
of phenomenological models is their simplicity. On the contrary, mechanistic models aim at bringing all
known details of the system. Details mean molecules—genes, proteins, ARNs, ions,y—with functional
capacities and relations, e.g. signalling. Parameters in mechanistic models have direct biological or
biochemical meaning, e.g. affinity constant for an enzyme, ionic conductance for an ionic channel,
and they can be measured in vitro or in vivo. Obviously, the difficulty with mechanistic models is the great
amount of information they should integrate, and, as discussed above, the imprecision and the lack of
parameter values.
The choice between the two alternatives depends on: (i) the objective of the modelling process; (ii) the
availability of information regarding the system; and (iii) the chosen strategy. However, other considerations
may play a role in the choice. The plug-in principle makes it possible to use a phenomenological model for a
sub-system while the other sub-systems are modelled mechanistically. If the need of detailing the sub-system
arises later, the phenomenological sub-model is replaced by a mechanistic one with the same entries and
outputs. In addition, it looks wise to build in parallel two global models, one phenomenological and one
mechanistic, or, perhaps better, to begin with the phenomenological approach in order to explore which
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sub-systems should be detailed. In such a case, parameters (and thus equations) should be chosen as direct or
almost direct representative of sub-systems. And, conversely, all sub-systems should be represented by at least
one parameter in the phenomenological model. Exploring the robustness of the global phenomenological
model will sort out the sensitive parameters, and hence the sub-systems that need more attention. This can
only be done if entries and output, hence the objective, are well defined.
6. Top-down or bottom-up?
In a seminal article, Noble (2003) has spotted two opposite strategies to get into modelling a living system,
the bottom-up and the top-down approaches. The former starts at the lowest organisational level (e.g. genes)
with the detailed components of the system, their properties and their interactions. Higher levels are
constructed from this lowest level, by assembling its components and accounting for interactions between
levels. This approach assumes that, to understand the functioning of the system, it is first necessary to decipher
its lowest organisational levels. Behind it, we find the idea that living systems are built from a limited number
of standard parts, each amenable to standard formal model and that interactions between levels are encoded
in the lower levels. A step further, we get into an entirely different field, theoretical biology (Noble, 2002). As
Brenner stated, ‘‘Although I am convinced that Schrödinger’s equation underlies the behaviour of everything,
it does not provide a useful basis for the construction of motor cars or bridges’’ (Brenner, 1998). The bottomup strategy is somewhat similar to the mechanistic approach. It faces the same challenges, i.e. uncertainty and
lack of knowledge, and computational problems. Also, it neglects the observation that lower levels functioning
depends on higher levels (see Fig. 1).
According to the top-down strategy, one starts at the highest level on line with the modelling process
objective, and models functions rather than entities. Then, the modeller replaces each functional block with a
model of the mechanism which implements it. This strategy resembles the phenomenological approach.
However, the difference is that it is a strategy, addressing the various organisational levels from the top-down
to the lowest level, whereas the phenomenological approach can be used at any organisational level. However,
as a phenomenological approach, the top-down strategy has been proposed as a way to characterise the basic
building pieces that up a biological system.
Actually, neither strategy seems the one to be followed. A compromise or middle-out, as it is called by
Brenner, is more sensible (Brenner, 1998). Both the objective and the availability of data will determine where
the compromise is.
7. Validation
A model cannot be used to achieve the researcher’s aim without a convincing enough validation. Having
said that, we would be glad to give the reader precise and relevant methods aimed at validating a model.
Unfortunately, we can only propose a few general principles as the validation issue is almost entirely
dependent on the objectives of the research.
Assessing the validity of a model is a complex operation, based on several approaches: (i) checking the
internal consistency; (ii) checking the content of the model; (iii) checking the output. The methods
are different. However, they are complementary and are all prepared after the beginning of the process, at the
stage of building the discursive model.
Internal consistency means that the arrangement of the pieces in the model does not misrepresent the
content of the discursive model which is supposed to represent all the available knowledge about the system of
interest. It is neither a mathematical nor a logical operation. It consists in reviewing the details of the
numerical solutions and the parameter values to check whether they correctly translate the discursive model.
A detailed account of all the hypotheses (i.e., simplifying or bridging—filling lack of knowledge) made is
mandatory for this operation to be relevant. Finally, the model content should fit with a few general laws, such
as there is no life without energy.
The validity of content of the model is addressed by looking if all the sub-model markers behave as expected
from available evidence. This operation refers to data from in vitro and in vivo experiments. New such
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experiments might be requested. Examples are given in Dronne et al. (2006) with the currents of the ionic
channels involved in the model.
Eventually, the outputs of the model will support its validity if it is possible to compare them to available
data or to results of in vivo or in vitro experiments designed on purpose. Also, whatever they are, outputs
should comply with our general understanding of the living system and the particular system of interest.
However, this argument should be balanced by the expectation of entirely new knowledge arising from
simulations with the model.
8. Robustness
If we add a piece to a formal model that has been shown to work properly, we may completely destabilise it
and observe non-physiologic behaviours. In the example above, the addition of simply one channel may alter
the resting potential by a few millivolts. This may be sufficient to bifurcate into another type of behaviour, for
instance, an action potential type cell turns into a pacemaker type one.
This question appears to be crucial in formal modelling. To be confident in the results of the global model we
need to study its dependence with respect to the parameter values of its own components. If the results do not
change when the parameters take ‘‘reasonable’’ values, then the qualitative results of the global model may
be considered as convenient. If they change, this may be an indication that some threshold is hidden within the
combination of the sub-models. Further investigations are then necessary to precise the overall behaviour. The
threshold may figure out a real ‘‘in vivo’’ threshold. More often, it simply points to a weakness of the model,
possibly due to a lack of our knowledge. In such case, one must encourage further in silico or in vivo investigations.
Actually, exploring the robustness of a model is not entirely different from searching for the ‘‘reasonable’’ set of
parameter values. An example is given with the ionic model of acute stroke by Dronne et al. (2006). In this model,
28 ionic conductances had to be specified. Three rules were pre-specified: (i) steady state at t ¼ 0; (ii) steady state
reached at equilibrium; and (iii) cell depolarised when pumps do not operate. One thousand sets of conductance
values were randomly drawn from the plausible ranges. One hundred and eighty of the 1000 random sets met the
three rules, defining the ‘‘reasonable’’ set. Robustness was explored by simulating model behaviour in
experimental situations that were used to validate the model, i.e. their results could be compared to in vivo or
in vitro data or were consistent with the knowledge content of the model. All 180 sets gave qualitative as well as
quantitative results that were very close to the expected results. Hence, the model was said to be robust.
As we are interested in a particular behaviour of a sub-model it is rather easy to find a set of parameter
values which is physiologically reasonable and which gives the desired qualitative properties. However, several
different sub-models would have to be integrated in the global model—for instance, models of ionic
exchanges, of apoptosis, of vascular system, of energy management (Dronne et al., 2004). Now, the question is
how far does the behaviour of the global system depend on the details of each of its pieces?
Sensitivity of the model behaviour to a parameter value points to this parameter. However, interpreting this
finding deserves some caution. Rather than meaning that the parameter is pivotal in the system under study, it
could reflect a constructed parameter, i.e. a parameter which does not exist biologically speaking and which in
the model is hiding several more real parameters. In any case, sensitivity is a marker of something that should
be explored more deeply.
9. Conclusion
Methodology of modelling in physiopathology is a mix of strategy, tasks, work packages and validating
steps structured by and organised around the biological entity that is to be modelled and the objective of the
research. With an entity of a few molecules, the methodology is rather straightforward. It is almost limited to
the choice of the biomarker(s) for validation, the mathematical solution, the parameter values, the simulation
procedure and a few other things. With an organ or, say, a disease, appropriate strategy and methodology can
make the difference between a successful model and one which does not achieve the researcher’s objective(s).
Hence, deep thoughts prior to start are required. At this stage, no time spent to think is wasted.
Other major initial steps in elaborating on a model-building strategy are to define the objective, then
to collect and analyse the available knowledge and data to identify those that will be integrated in the model.
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The uncertainty and the strength of evidence of these data should be accounted for. Then, the researcher has
to identify the pieces, that is the biological processes and entities that can be put together and separated out
from the others to constitute a sub-system which will be worked out independently. This is particularly needed
in the case of a disease where knowledge is scattered across different organisational levels, such as genetic,
clinical, or epidemiological.
Eventually, methodology in this setting has to do with epistemology. Obviously, investigating the
management of unknown parameter values, which leads to a reasonable model behaviour, rises the issue of the
boundaries of real world as well as the place of in silico simulation on deciphering life. Thus, one important
concluding remark is the need for more research on methodology and epistemology of numerical modelling in
physiopathology. Unfortunately, there is no standard recipe which would ensure a successful achievement. Art
is not a fashionable buzz-word in science. However, it could well be appropriate here, beyond expertise.
Acknowledgements
We thank Michel (i) Cucherat, who kindly passed on Fig. 2 to the authors, and (ii) all the juniors and
seniors of the Institute of Theoretical Medicine, and particularly those, besides the authors of this article, who
are involved in the elaboration of the textbook on methodology of numerical modelling in physiopathology:
Amine Achaachi, Frédérique Billy, Loubna El Zein, Catherine Genty, Ivanny Marchant, Patrice Nony,
Nancie Raymond and Diéré Sonko.
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