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

5.4 The DuCa Model – Compartment Model and Equation System 27
As a macrophage engulfs an apoptotic β-cell it becomes activated, moves from the compartment
of resting macrophages, and into the compartment of activated macrophages,
yielding a positive contribution to this compartment. This can be seen in equation
5.6 and 5.7 where f1MBa is an efflux in equation 5.6 and reappears as the positive
input in equation 5.7 – remember that the macrophage engulfs the apoptotic β-cell i.e.
“it carries it with it” to the active macrophages compartment. We must remark that
in reality the macrophage does not move into a specific part of the tissue once it has
engulfed a β-cell, but rather into another state.
Now that we have introduced the compartment model, the equations and the para-
Parameter Meaning Balb/c NOD Units
a Normal macrophage influx 5 – ×10 4 cells ml −1 d −1
b Recruitment rate of M by Ma 0.09 – d −1
c Macrophage egress rate 0.1 – d −1
d Ba non-specific decay rate 0.5 – d −1
k Ma deactivation rate 0.4 – d −1
l Ba apoptosis induced per Ma 0.41 – d −1
f1 Basal phagocytosis rate per M 2 1 ×10 −5 ml cell −1 d −1
f2 Activated phagocytosis rate per Ma 5 1 ×10 −5 ml cell −1 d −1
e1 = e2 Anti-crowding rates 1 – ×10 −8 ml cell −1 d −1
Amax Maximal cytokine-induced β-cell apoptosis rate 2 – ×10 7 cells ml −1 d −1
kc Cytokine concentration for half-maximal apoptosis rate 1.0 – nM
α Cytokine secretion rate by Ma due to Bn 5 – ×10 −9 nM cell −2 d −1
δ Cytokine turnover rate 25 – d −1
kb
δ/αkc 5 – ×10 10 cell 2
Table 5.1 summarizes the model-parameters and their values – d −1 is days to the power of
minus one, or “per day”. If not stated otherwise the parameters for NOD-mice are the same
as for Balb/c-mice. The rate constant l is not included in the DuCa model but appears in the
intermediate model in section 6. kb is not in the DuCa model either, but it appears in section
7 in equation 8.5. The table is a combination of the table on p. 1271 and p. 1278 in Marée
et al. (2006)
meters it is time to take a look at how the two systems (NOD and Balb/c) evolve over
time.
In figure 5.3 matlab-simulations of the systems with parameters as given in table 5.1
are shown. The concentrations in figure 5.3 are logarithmic.
On the left we see that the Balb/c-mouse is rid of everything but resting macrophages
after 50 days. 6 This is not the case for the NOD-mouse. Here all concentrations become
constant, and greater than 0. This is in itself not a problem as long as they settle at a
very low concentration (save for the resting macrophages). The concentration of apoptotic
β-cells stabilizes at approximately 12 after 50 days. Thus after day 50 e 12 β-cells
per ml become apoptotic. This is not a huge number, but if we add to this the number
of β-cells that were depleted before the first 50 days it starts adding up. We must also
remember that around 4-5 weeks of age, T cells will start infiltrating the islets, and add
to the destruction. The most important consequence of the constant concentration is
that the β-cells will keep on becoming apoptotic.
6 Remember that the concentrations are logarithmic, so a concentration of 0 is not really a concentration
of 0, rather it is a concentration of 1.