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

110 Expanding and Modifying the DuCa model
11.3 Modifying the governing equation – Model B
Based on the insights we obtained from Model A, we will now modify the apoptosisterm
to obtain a more realistic behavior.
Verchere (2009) and Pociot (2009) reports that a rule of thumb is that approximately
0.1%, or 1 in 1000, β-cells become apoptotic per day in adult mice; i.e. the number of
β-cells that become apoptotic at a given time, depends on how many are left. 4 Thus we
must adjust the governing equation accordingly. This is done simply by changing the
term that describes the natural apoptosis from x1 to x2B in equation 11.8 and 11.11.
The new constant, x2, has the dimension “’per day” (d −1 ) and a value of 0.001. We
will refer to this model as Model B.
In figure 11.2 we present a simulation of the Balb/c- mouse based on Model B, while
figure 11.3 shows the NOD-simulation.
Model B – Balb/c-simulation
We see that making the amount of β-cells, that enter natural apoptosis per day, dependent
on the concentration of β-cells has improved the Balb/c-behavior in that the
concentrations of apoptotic β-cells and activated macrophages tend toward zero as
t → ∞, and the concentration of β-cells also exhibits a more natural behavior. The
Balb/c-simulation is in agreement with criteria II and IV, and if we compare the plots
portraying the concentrations of healthy β-cells in figures 11.2 and 11.3 we see that the
third criterion is also fulfilled.
Figure 11.2 Behavior of the concentrations for Balb/c-mice with the inclusion of a compartment
of healthy β-cells; based on Model B. Left: the concentrations of M, Ma, Ba, Bn, C. Right: the
concentration of B. Initial values are (M, Ma, Ba, Bn, C, B) = (4.77 × 10 5 , 0, 0, 0, 0, 1 × 10 9 )
4 Teta et al. (2005) give an apoptosis rate of 0.0011 percent per day in one year old mice (Teta et al.,
2005, p.2562).