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

40 The DuCa Model
the apoptotic wave, they all reach a (logarithmic) concentration of 0 within the same
amount of time as in figure 5.3. For the irreversible model it is another case. Here we
see the effects of setting k = 0. When we compare the behavior of the irreversible model
with figure 5.3 (the one showing Balb/c behavior) we see that it has a drastic influence
on the concentration of active macrophages. In the case of the irreversible model, there
is no natural efflux of the active macrophages, so it is the crowding term only that
provides a limitation to the growth of the concentration of active macrophages, which
as we can see in the righthand figure in figure 5.10 implies a much lengthier decline
in Ma concentration. This is in contrast with the behavior we should expect from the
concentration of active macrophages in Balb/c-mice (Marée et al., 2006, p.1279), where
it should tend to zero after the apoptotic wave has taken its toll. Thus if the DuCa
model is essentially correct, then the irreversible model cannot qualify as a model of
phagocytosis (statistically indistinguishable from the reversible model or not), and if the
irreversible model is to be taken as a good approximation of macrophage phagocytosis
in Balb/c-mice, then the DuCa model needs to be revised. Besides these observations
we must remember that the irreversible and the reversible model were statistically indistinguishable
– a conclusion that is based on measurements without any connection
to the DuCa model. Figure 5.11 shows how the different concentrations evolve in the
Figure 5.11 shows simulations with NOD phagocytosis parameters from table 5.3. On the left
we have the reversible model, i.e. the DuCa model. To the right is the irreversible model, i.e.
with k = 0. The two figures are qualitatively indistinguishable.
reversible and the irreversible model with NOD phagocytosis rates. Relative to the
differences we observed for Balb/c rates, there is not much to speak of for the NOD
rates, except that the concentrations of apoptotic and necrotic β-cells and cytokines
stabilize at a higher concentration in the reversible model, while the resting and active
macrophages stabilize at a lower concentration. In biological terms this means that the
β-cells will be depleted faster in this scenario.
When we compare figure 5.11 to the NOD behavior in figure 5.3, we see that qualitatively
there is no difference in how the concentrations evolve – following an initial
period of oscillations the course is set for the demise of the population of β-cells.
One very important thing to notice is that the phagocytosis rates presented in table 5.2
are based on in vitro experiments. This is naturally a much more artificial environment