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

12 Discussion of the Expanded Model
Just as Marée et al. (2006) modified the qualitative Copenhagen model to obtain a
model that was more quantitatively correct so we have modified the DuCa model to
make it do better in terms of realistic properties, i.e. we changed it so that there is
no longer an infinite supply of healthy β-cells. 1 In short we expanded the DuCa model
based on the fact that the β-cell population is not infinite. This last step was taken
in chapter 11 where we started out by stating some criteria that an expanded model
should live up to. Then we made some basic assumptions about what influenced the
concentration of β-cells based on recent literature, incorporated this into a differential
equation that described the change in β-cell concentration, and altered the equation
describing the change in concentration of apoptotic β-cells accordingly. Matlab simulations
were then done, and we analyzed the figures that were the product of these
simulations. Based on the figures we were able to determine how well the different
models complied with our criteria. In the following we will compare the model to the
data given below, and compare Model B and Model C to each other.
Perhaps the most important step towards validating one’s model(s) is to compare it to
data. The only readily useable data we have is
i The age at which symptoms of T1D becomes overt in the NOD-mouse – 30 weeks
ii The number of β-cells that are left when symptoms of T1D present themselves (in
NOD-mice) – 10-40 % of the initial β-cell mass
iii The time at which T cells become significant in the depletion of β-cells in NOD-mice
– 4-5 weeks
iv The median lifespan of a laboratory mouse – 30 months
which is not an overwhelming amount of data. It is, however, some of the most important
data that we could have hoped for, since it allows us to determine if our addition
to the DuCa model behaves realistically, i.e. lets us answer questions ii and iii of the
thesis statement; cf. section 1.1. Question i will be answered based on the simulations
presented in the previous chapter.
Based on the simulations of the first expanded model, Model A, we found that it did
not agree with model criteria I and II nor with the second guideline from chapter 4 (it
exhibited nonphysiological behavior), and almost by implication it is at variance with
the third guideline as well. We concluded that we needed to formulate an addendum,
regarding what we would consider healthy behavior, to our criteria. This addendum
had not previously occurred to us even though, looking back, it seems obvious.
A little wiser we proceeded to alter Model A slightly, so that we would avoid negative
1 This should not be interpreted as though the DuCa model is irrelevant or wrong. It was not
constructed to mimic Langerhanian Islets over long periods of time, only to analyze the hypothesis of
Marée et al. (2006).
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