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

98 Discussion of the Bifurcation Analysis
the inflammation is irreversible. Besides these results the experiments that were used
to obtain the phagocytosis rates were in vitro experiments, how this exactly affects the
macrophages we do not know. It would be interesting to obtain f1- and f2-values from
male NOD-mice to see if their mean value is above the Hopf-bifurcation value.
Hence when we compare the phagocytosis rates to the values at which the Hopfbifurcations
occur, we find that based on the parameter values from (female) NOD-mice
the DuCa-model is sound in the sense that no bifurcations occur within a physiologically
reasonable range.
So far we have used the NOD-bifurcation diagrams to discuss if bifurcations occur in f1and
f2-intervals that are close to the estimated f1- and f2-values, and found that this is
not the case. Now, let us see what extra information we can gather from the diagrams.
For instance, from a medical point of view, it would be interesting to know which of the
phagocytosis rates it is most opportune to manipulate with stopping the inflammation
in mind. Comparing figure 8.1 to 8.12 we conclude that, if we can only adjust one of
the phagocytosis rates in the NOD bio-model, enhancing f1 is most efficient. By this
we mean that f1 needs “only” be changed to little more than 2.57 × 10 −5 to arrest and
reverse any inflammation, whereas f2 must exceed a staggering 8.544 × 10 −5 to induce
the same effect. Furthermore, due to the irreversibility, we would only need to enhance
the phagocytic ability of the resting macrophages until they had phagocytized enough
apoptotic β-cells, as to make the concentration of activated macrophages drop beneath
the NODf1 -LUB.
While we are in the area of inferring treatment strategies from the bifurcation diagram,
we saw in figure 8.9 that there was more to figure 8.1 than had revealed itself from our
method of producing bifurcation-diagrams. What did not reveal itself was the so-called
upper unstable branch, or the UUB. The existence of the UUB (be it an unstable fixed
point or a separatrix) opens up for a completely novel approach to curing T1D (in
NOD-mice). 1 The mathematical analysis tells us, that if the inflammation has reached
a chronic/stable stage, then by adding an extra amount of activated macrophages to
the inflamed islets we can exceed the UUB thus making the flow tend to the HRS. The
specific amount will depend on f1. Thus if we are not able to make f1 exceed the Hopfbifurcation
value, we can (hypothetically) add active macrophages, to achieve a positive
outcome, where by positive we mean that the chronic inflammation seizes. On this basis
we would like to establish an approximative relation between f1 and the UUB. Let M + a
denote the UUB-curve in figure 8.9, f1,h be the f1-value at which the Hopf-bifurcation
occurs and Ma,h be the Ma-concentration at the Hopf-bifurcation, then we find that
Furthermore we hypothesize that
M + a (f1,h) ≈ Ma,h
M + a (0) = M + a,0
(9.1)
(9.2)
where M + a,0 is the Ma-value that is needed to make the flow tend to the HRS if we set
f1 = 0. We remind the reader that based on calculations done with matlab, we found
that the UUB decreases approximately exponentially for f1 < f1,h. Thus we guess that
1 The approach is novel in as much as we have not come across it in our literature studies.