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

5.4 The DuCa Model – Compartment Model and Equation System 25
with initial conditions (M, Ma, Ba, Bn, C) = (4.77 × 10 5 , 0, 0, 0, 0). 5 The equations
should be interpreted thusly:
Equation 5.6: the rate of change in the concentration of macrophages, M, is made up
of a gain from the influx, a, the deactivation rate of the active macrophages times the
concentration of active macrophages, kMa, and the recruitment rate times the concentration
of active macrophages, bMa, and a loss constituted by the efflux, c, times
M, the activation rate of resting macrophages, f1, times M times the concentration of
apoptotic β-cells, Ba, and lastly the crowding rate, e1, times M(M + Ma).
Equation 5.7: the rate of change of the concentration of active macrophages, Ma, is
made up of a gain from f1 times MBa and a loss or efflux due to the deactivation rate,
k times Ma, and the crowding rate, e2, times Ma(M + Ma).
Equation 5.8: the rate of change in the concentration of apoptotic β-cells, Ba, is governed
by the phagocytosis rates, f1 and f2, times MBa and MaBa respectively and the
nonspecific decay rate, d, times Ba. These make up the negative contributions while
the apoptotic wave (not shown in the figure), W (t), and the Michaelis-Menten saturation
function of cytokines, (AmaxC)/(kc+C), where Amax is the maximal rate of apoptotsis
that the cytokines can induce, and kc is the Michaelis constant, constitute the positive
contributions.
Equation 5.9: the rate of change of the necrotic β-cells, Bn, is influenced by a positive
contribution from dBa, and negative contributions from f1 and f2 times MBn and
MaBn respectively.
Equation 5.10: the rate of change in the cytokine concentration, C, depends on the
decay rate, δ, times C and the secretion rate by active macrophages, α times MaBn.
The meaning and units of the different parameters are given in table 5.1. At a first
glance the equations almost seem to be in accordance with what we learned when we
looked at generic compartments in subsection 5.3. But soon we see that something is
amiss. We will return to these matters in section 5.5. For now let us return to the
DuCa model, or rather the equations that comprise it.
To unfold equations 5.6 to 5.10 a little more and to give ourselves the opportunity to
think thoroughly about every term on the right hand side in said equations we present
a comprehensive list of how we understand each term:
• a is daily inflow of resting macrophages from the surrounding tissue
• kMa is the deactivation rate of activated macrophages times the concentration
of activated macrophages, i.e. the dimensions are cells ml −1 d −1 , where “d” in
this case is days, not to be confused with the parameter d. This term represents
the flow of deactivated macrophages from the Ma-compartment into the
M-compartment
• bMa is a measure of the extra influx of resting macrophages due to signalling to
the surrounding tissue by active macrophages.
• cM is daily efflux of resting macrophages, i.e. the outflow of resting macrophages
to the surrounding tissue.
• f1MBa in equation 5.6 represents the outflow of resting macrophages that have
become activated upon clearance of an apoptotic β-cell, hence Ba is included in
5 For a statement of the existence and uniqueness theorem, and its application to the DuCa model cf.
appendix A.1.