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

11 Expanding and Modifying the DuCa
model
Thus far we have given a thorough description, analysis and critique of the DuCa model.
In this next section is we will expand the DuCa model in two steps. First by including
a compartment of healthy β-cell, and second, by incorporating some recent findings.
Instead of just presenting the last model as though it was the first we came up with,
we have chosen to include them all as this constitutes a more accurate reflection of the
modelling-process.
11.1 Including a Compartment of Healthy β-cells
A natural extension of the DuCa model would be a compartment constituting healthy
β-cells, with according changes in the system of differential equations. Including such a
compartment also addresses one of the major discrepancies between the progression of
diabetes in NOD-mice and how the DuCa model behaves. Remember that in the DuCa
model, the reservoir for apoptotic β-cells is de facto infinite, but in the NOD-mice there
will eventually be no more β-cells left, if no prophylactic measures are taken – the same
is in essence true for Balb/c-mice given that they could live long enough.
An equation describing the change in concentration in β-cells should naturally reflect
the “real-life” behavior associated with the population of β-cells, so before we try to
construct such an equation, let us first define some criteria an expanded model must
fulfill and next discuss how we should go about putting together the additional equation
based on these criteria.
Model criteria
A mathematical model’s ability to fit data is a necessary but not sufficient gauge for the
model’s applicability and soundness. Thus we need a bit more to validate the model(s)
to come.
As stated in chapter 4 there must be a correspondence between the physiological system
we aim to model and the mathematical model we construct. In our case this correspondence
consists of four key-features that our expanded model(s) must live up to (besides
the three guidelines stated in chapter 4)
I In the NOD-mice the β-cells should not be removed at the time where T cells come
into play – there must be room for T cells to play an active part in the depletion
of β-cells though this will not be included in this thesis.
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