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

46 The Intermediated Model
The differential equations that correspond to the compartment system in figure 6.1 are
dM
dt = a + (b + k)Ma − cM − f1MA (6.1)
dMa
= f1MA − kMa (6.2)
dt
dA
dt = lMa − mA (6.3)
The purpose of the CPH model was not to give a quantitative prediction of how the
onset of T1D comes about, rather it was to illustrate that T1D is caused by multiple
factors coming together in a synergetic way, and to show that the difference between
onset and not onset depends on a threshold that incorporates these factors, i.e. the
onset of T1D implies crossing a “threshold surface” that is spanned in the parameter
space, of the model. This threshold was found through stability analysis and defined
as f0 (Blasio et al., 1999, p.1684)
f0 ≡ f1l
(6.4)
ck
Where the system is guaranteed to be stable for f0 < 1. Marée et al. (2006) wants to
illustrate that chronic inflammation can not occur for biologically reasonable parameter
values in the CPH model, and thus the CPH model needs revision if it is to be used
as a quantitative model. Therefore the DuCa model can be seen as an extension of the
CPH model that has been modified to investigate a different hypothesis.
In this section we are going to take a look at what we have dubbed the Intermediated
Model (IM). This model bridges the gap between the CPH model and the DuCa model.
The IM can be seen as a slightly modified CPH model and as a rudimentary version of
the DuCa model. Marée et al. (2006) does not, however, tell the story of how to revise
the IM model to transform it into the DuCa model. This implies that even though the
DuCa model originates from the IM there is no straight line between the imperfections
of the CPH model/IM model and the modifications done on the IM to reach the DuCa
model. However the extension of the CPH model is quite straightforward since Marée
et al. (2006) aims to investigate whether the effectiveness in phagocytosis can make
the difference between a diabetic and a non-diabetic mice. As stated in section 5.1
this hypothesis is rooted in an observed difference in phagocytosis rates for Balb/c and
NOD-mice, seen in the basal phagocytosis rate of the resting macrophages, but also in
their ability to undergo an activation step, when the first apoptotic cell is engulfed. To
investigate the effect of these differences they implement the clearance of apoptotic βcells
done by the resting and activated macrophages into the CPH model, by including
the terms f1MBa and f2MaBa. This modification addresses another peculiar aspect
of the CPH model since there was no clearance of the antigens due to macrophage
engulfment, but only a nonspecific decay. This seems odd since one should expect
that upon engulfment of a protein, there is one less to activate the remaining resting
macrophages. This leads us to the next modification, albeit of a more conceptual
matter. Instead of monitoring the antigens – i.e. small protein fractions of the β-cells
that underwent apoptosis, they look at the clearance of an apoptotic β-cell as a whole.
These changes lead to the system of differential equations seen in 6.5 - 6.7, and the
corresponding compartment can be seen in figure 6.2.