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

5.8 Discussion of Parameters 41
than an in vivo environment. This means that a lot of processes that occur in vivo have
been removed from the equation, so to speak. It may be that the only way of obtaining
good estimates of the phagocytosis rates is via such in vitro studies, but it implies that
we cannot readily assume that the in vivo phagocytosis rates are exactly the same.
Another thing that is interesting when we are speaking about phagocytosis rates, is the
fact that there is a major difference in the percentage of NOD-mice that develop T1D,
when one looks at female versus male specimens.
If Marée et al. (2006) are correct, i.e. if the difference between developing T1D or not
is predominantly a matter of macrophage phagocytosis rates, then this must mean that
there is a difference in these rates between male and female NOD-mice. Though we
are not capable of inferring any direct relation between the phagocytic differences in
male and female mice and their bodily attributes (such as weight) or other things that
may have an impact on their immune system or pancreas, such as living circumstances
(e.g. the female mice give birth which may have an impact on their immune system),
it would be very interesting if anyone had the expertise to make such an inference, as
this could take the model to a new level. This would mean that the model could be
used as a specimen-specific model, i.e. by adjusting the rate constants according to
some rule that relates phagocytosis rates to the attributes of the specimen at hand the
model should yield a behavior that is specific for the specimen at hand.
The DuCa model with turnover of Ma
In the figures we have presented above we have blindly used the DuCa equations,
though we have set k = 0 for the simulations of the irreversible model, and changed
the phagocytosis rates. Now we would like to present some simulations we have done
with a seemingly minor alteration that addresses the fact that active macrophages in
reality also succumb to turnover (Blasio et al., 1999, p.1680), with a turnover rate that
is equal to the turnover/efflux rate of resting macrophages (Wigginton and Kirschner,
2001, p.1963). This means implementing a term of −cMa in equation 5.7.
When we use the original model parameters, given in table 5.1, this alteration does not
change the behavior of the NOD nor the Balb/c concentrations. But when we introduce
it in the reversible as well as the irreversible model with matching parameters something
interesting happens to the behavior of the Balb/c concentrations. In figure 5.12 we have
the log-concentration versus time plot on the left and a phase space plot of M versus
Ma on the right. Introducing the turnover of active macrophages in the irreversible
model entails the occurrence of a stable spiral. Obviously this is at variance with the
dynamics we would expect from Balb/c-mice.
For the reversible model the change in behavior is less interesting (there is no spiral),
but as we can see in figure 5.13 the behavior differs essentially from the behavior we
saw in figure 5.3, in the key aspect that once again inflammation persists, though the
concentration of cytokines is removed. As we mentioned earlier the original DuCa model
with parameters given in table 5.1 retained its behavior after we added the turnover of
active macrophages. However as we learned that the reversible model with matching
Balb/c phagocytosis parameters also exhibited persistent apoptosis of β-cells (when
we added the turnover of active macrophages), we found it interesting to investigate
how much the Balb/c phagocytosis rates should deviate from their original values,
f1 = 2 × 10 −5 , f2 = 5 × 10 −5 ml cell −1 d −1 , before the DuCa model with Ma-efflux