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

30 The DuCa Model
a
✓❄
✏
✛ M
✒ ✑
c
❄
✛
e1
❄
f1
W (t)
apoptosis
✻
Amax
kc+C
b
✻
f1
✓
✲ Ba
✒
✲
d
❄
✏
✑
✓ ✏
k
✲ Ma
✒ ✑e2
❅
❅❅❘ Figure 5.4 shows the non-necrotic part of our modified compartment model. M, Ma, and
Ba are the concentrations of macrophages, activated macrophages and apoptotic β-cells
respectively. An explanation of a, b, c, d, e1, e2, f1, f2, Amax, kc is given in table 5.1.
Besides standardizing the use of arrows we have also included the apoptotic wave, and
the crowding terms to make a compartment model that, albeit divided into two figures,
agrees with the governing equations.
Initially we wanted to rid the model of stippled arrows all together, in order to be able
to make a translation from the compartment model to a set of reaction equations, and
from there to the set of governing equations.
It is not impossible to get rid of the stippled arrows, but it requires extra compartments,
that contain intermediate combinations of, say, active macrophages that have engulfed
necrotic β-cells, and also we must introduce some extra parameters. In figure 5.6 we
have picked out a part of the original compartment model and added a compartment
containing active macrophages that have engulfed one or more necrotic β-cells. The
macrophages will either secrete a cytokine, at rate k4, or merely clear the necrotic
β-cell, at rate k5, without releasing any cytotoxic compounds – either way the active
macrophage will return to its compartment, at rate k3. If k1 and k2 are equal it means
that one macrophage engulfs one necrotic β-cell – if this engulfment is similar to the
engulfment of apoptotic β-cells, this need not be the case 7 (Marée et al. (2005)).
Upon engulfment the macrophage will either consume the β-cell quietly, at rate k5, or
it will secrete a cytokine, at rate k4. We will not concern ourselves with the fate of the
cytokines in this example.
When we decompose the compartment system in figure 5.6 into mass action kinetic
7 In Marée et al. (2005) they observed that an activated macrophage could contain up to 7 apoptotic
β-cells (that were at different levels of phagocytosis), and this did not appear to be an upper bound
(Marée et al., 2005, p.542).
f2
f2
❄✲