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

106 Expanding and Modifying the DuCa model
II When the amount of β-cells do tend to 0, due to persistent inflammation or natural
apoptosis, after a while so must the concentrations of cytokines, apoptotic- and
necrotic β-cells as well as the active macrophages.
III The concentration of β-cells in NOD-mice must die out quicker than in Balb/cmice.
IV In the Balc/c-mice, the depletion of β-cells must be due to natural apoptosis only,
after the neonatal phase.
I is based on the fact that macrophages, cytokines and apoptotic-, necrotic- and healthy
β-cells are not the only cell/protein-types that are involved in the pathogenesis of T1D,
as T cells also play a significant part. In NOD-mice they enter the scene after about 4-5
weeks (Trudeau et al., 2003, p.219), and symptoms of T1D become apparent at about
30 weeks of age (Sreenan et al., 1999, p.989). II is a result of the causal relationship
between the components of the model – if there are no apoptotic β-cells then resting
macrophages cannot become activated, and no apoptotic β-cells can become necrotic –
there will be some residual apoptotic cells after the healthy β-cells are gone, so to be
very precise it is not until these cells have been phagocytized that all concentrations will
tend to zero (except for the resting macrophages). Thus once the deleterious process of
β-cells has been completed, the NOD-mice, too, will tend to what we called the HRS in
the bifurcation analysis, i.e. (M, Ma, Ba, Bn, C) = ( a/c, 0, 0, 0, 0), but in this case it will
obviously not be a sign of health, quite the contrary. III serves as a necessary, though
not sufficient, condition for the validity of the model. IV is explained below.
Because T cells become significant at 4-5 weeks of age, the expanded models, presented
below, become more or less unrealistic at the same time; they will not include T cells.
However, including several weeks more than 5 in our simulations (and in the adhering
figures) makes us more able to determine if the models comply with the criteria.
11.2 First approximation to a governing equation – Model A
At the neonatal stage the behavior of the β-cell mass in NOD-mice is a little more
complex than a simple depletion from day one. The pancreas is still undergoing remodeling
(Steer et al., 2006, p.262), e.g. the apoptotic wave. But even after these early
events and after the perpetual inflammation has begun, data exists that suggests that
the inflammation itself stimulates β-cell regeneration (Akirav et al., 2008, p.2883). This
could be because some of the cytokines produced in the inflammatory process are antiinflammatory
or benign growth factors (Souza et al. (2008)). This has been disputed by
others, whose data indicates a more linear decline in the β-cell population after insulitis
has been established (Akirav et al., 2008, p.2883). However a decrease in the β-cell
population is not reserved for those destined to become diabetic. Everybody would be
become diabetic at some point, given that they live long enough (Pociot (2009)), due
to the natural occurring apoptosis. This is why we have the fourth criterion.
One thing that all parts can agree on is that the β-cells are being removed, so as a first
approximation we will use a simple linearly decreasing model for the β-cell population.
The equation for the β-cell population, at this stage, is given by
dB
dt
= −x1
(11.1)