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

112 Expanding and Modifying the DuCa model
Now let us see how well Model B models the Balb/c-mouse. The Balb/c-simulation of
Model B reveals that a B-concentration of 4 × 10 8 cells ml −1 is reached shortly before
500 days have transpired. This speaks against the soundness of Model B. B = 1 × 10 8
is reached some time after 1500 days, which must be regarded as a Methuselan age for
a mouse.
Thus on one hand Model B appears to fit the NOD-mouse. On the other the hand
the Balb/c-mouse also becomes diabetic, when we adapt the assumption that removing
60% of the β-cells implies diabetes. But before we start making any final conclusions
there is something we have left out in these first two models, namely the regeneration
and replication of healthy β-cells that is concurrent with apoptosis – in young rodents
at least. Adding a replication-term to the model may save the Balb/c-mouse from its
ill-fated lab-colleague. The next subsection will deal with this aspect.
11.4 Adding β-cell growth – Model C
As we hinted earlier, there is experimental evidence that suggests that the β-cells do
not simply die out linearly or according to an exponential law (Bouwens and Rooman
(2005), Steer et al. (2006), Akirav et al. (2008), Saisho et al. (2008)), after the apoptotic
wave has done its damage – even during the initial phase where the apoptotic
wave occurs the β-cell mass may actually expand. In this subsection we add a growthor
replication term to equation 11.11. We will also do yet another modification of the
apoptosis term.
Saisho et al. (2008) find the replication rate in islets of 1 month old rats to be 0.15%
per hour, which can be calculated to be approximately 3.6% per day or 0.036d −1 . In
order to obtain this estimate they use an assay where they add a growth-medium to
their Langerhanian islets, which stimulates an increase in replication. 5 Thus the value
of 0.036d −1 should not be taken at face-value, nonetheless it is the only estimate of the
replication rate in young rodents we have been able to find.
From the same article the apoptosis rate at 1 month of age can be calculated to approximately
0.018d −1 (Saisho et al., 2008, p.E90-E91). This differs from the apoptosis
rate we used in Model A and B. One reason for this discrepancy could be the same as
the reason why the replication rate estimated by Saisho et al. (2008) is inflated, i.e. we
speculate that the growth-medium may attenuate the apoptosis rate. Another reason
is that Saisho et al. (2008) use young rodents whereas the estimate given by Verchere
(2009) and Pociot (2009) is for adult rodents.
Besides providing an estimate for the replication – and apoptosis – rate of young rodents,
Saisho et al. (2008) also reports that in 10 week old rats replication is a rare
phenomenon in agreement with the findings of Teta et al. (2005) who estimate the
number of β-cells that undergo replication per day, in one year old mice of the so-called
c57BL/6×129Sv type, to be 1 in 1400, or ≈ 0.07% per day (Teta et al., 2005, p.2561).
Teta et al. (2005) further reports that the rate eventually drops to 0 (Teta et al., 2005,
p.2563), and provide an estimate for the replication rate of 8 month old male and female
Balb/c-mice, which they find to be ≈ 0.679% per day for the males and ≈ 0.242% per
5 The purpose of the article by Saisho et al. (2008) is not specifically to obtain a value for the
replication rate, so this does not constitute tampering with the results.