nr. 477 - 2011 - Institut for Natur, Systemer og Modeller (NSM)

4 Mathematical Modelling
The language of mathematics is widely used to describe a myriad of naturally occurring
phenomena. This is partly due to the fact that mathematical modelling along with
stability analysis allows for a qualitative description/understanding of systems in which
one or more of the components are not sufficiently or accurately determined, as it is
particularly the case in the medical sciences, where it is next to impossible to obtain data
for many in vivo parameters, e.g. rate constants. It can also help estimate parameters
based on which dynamical behavior is expected from a given system, or it can aide in
the understanding of which parameters are the most significant for the behavior – given
of course that the model is accurate enough, but more about this in the next paragraph.
The construction of mathematical models is not something that is based on a set of
well-defined rules or prescriptions, and one must always keep in mind what the purpose
of the model is; Do we (think we) know all the parameters and just want to make
long-term simulations? Are there stochastic processes involved, so we are content with
results within a confidence interval? Is the model made with the single purpose of
estimating parameters or do we just want to mimic a certain behavior regardless of
quantitative agreement with nature? To each purpose different a priori guidelines for
constructing a mathematical model comes to mind. In terms of mathematical modelling
in the biosciences we find these to be:
1. A reliable model should be based on an observable structure, by structure we
mean the underlying physiological/physical/biochemical etc. system which is of
interest.
2. The model should be rid of non-natural behavior, e.g. concentrations should not
be able to reach infinity or assume negative values.
3. The model must exhibit the measured/expected behavior within a given range of
known parameter-values.
The first guideline should secure that the model is not taken out of the thin air. The
second and third guidelines serve to validate the model’s foundation in reality. 1
In this study we will concern ourselves with modelling of type 1 diabetes. To the
uninitiated this may sound as a well-defined and isolated task, but the workings of the
pancreas like most physiological systems is astonishingly complex. Thus the mathematical
description of such systems becomes a difficult balancing act between including
relevant factors and not making the model impossible to work with. Including every
single mechanism involved in the onset of T1D would very likely obscure rather than
elucidate which are the important components in the system dynamics. Therefore it is
often advisable to seek a parsimonious model. Or as Murray (2002) puts it (Murray,
2002, p.175)
1 Notice that question i (cf. section 1.1) deals with the second guideline.
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