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

28 The DuCa Model
Figure 5.3 To the left: dynamics of the Balb/c-mouse. To the right: dynamics of the NODmouse.
Based on parameters found in Marée et al. (2006), see table 5.1, and initial conditions
that correspond to a healthy rest state, i.e. (M, Ma, Ba, Bn, C) = (4.77 × 10 5 , 0, 0, 0, 0). The
apoptotic wave and the effect of it is seen as the peak at 9 days on the curves describing Ba-,
Bn- and C-concentrations. We see that inflammation persists in the NOD-mouse, whereas
nearly all apoptotic β-cells are removed before 21 days have transpired in Balb/c-mice.
5.5 Discussion of the Compartment Model
As we pointed out in the above section there are some discrepancies between equations
5.6-5.10 and figure 5.2, compared to how the equations in 5.5 corresponded to
the compartment system in figure 5.1. Not least the use of stippled arrows makes a
transformation from the compartment-system to a set of conventional reaction-kinetic
equations ambiguous.
Here we will make these matters more precise, devise some stringent rules for when it is
appropriate to use stippled arrows, or when the use of such arrows can be circumvented,
and when full-drawn arrows should be used. Ultimately we want to present a compartment
model that is in accordance with our rules and has a clearer correspondence with
the equations.
Firstly let us compare figure 5.2 to equations 5.6 to 5.10, and start with the most blatant
mismatches.
First of all the crowding terms are nowhere to be seen, secondly the apoptotic wave
is not shown, this could be because it is a short-lived event, but still it would at least
merit an explanatory remark why it is not there. Thirdly the stippled arrows identified
by f1 and f2 meet. While we do understand the meaning of this, and granted the
article is not written with compartment diagram-puritans in mind, it is still not a clear
way of setting up a compartment diagram – after the two arrows have merged it is
impossible to tell which of f1 and f2 we should assign to the arrow that continues after
the merging.
If we follow the aforementioned stippled arrows we get to the first of the two “clearance”
arrows. These arrows are not (directly) associated with any rate constant, as
they should be. Lastly there is the fully drawn arrow from the Ma compartment to
the cytokine compartment. This would normally imply that the active macrophages