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

114 Expanding and Modifying the DuCa model
where t ′ 1 is given by equation 11.14.
Because we have found different values for the apoptosis rate in young and adult animals,
we hypothesize that the (natural) apoptosis rate might also decline with age. Again we
choose a logistic function to describe the apoptosis rate as a function of time, but this
time we do not want the rate to tend to zero as the animal ages. Instead we want it to
reach the constant value given by Verchere (2009); 0.001d −1 . Thus we find
x3(t) =
x3,0
1 + exp(p2(t − t ′ + x2
(11.16)
2 ))
As before we can relate p2 and x3,0 as well as t ′ 2 and x3,0, as seen in equations 11.17
and 11.18.
t ′ 2
=
p2 =
x3,0−0.018
x3,0−0.001
0.018 − 30 ln 0.001
0.001(x3,0−0.018)
ln 0.018(x3,0−0.001)
x3,0−0.018
0.018
365 ln
ln
30 − t ′ 2
(11.17)
(11.18)
t ′ 2 in equation 11.18 is given by 11.17.
By now we have altered the governing equations for the healthy and apoptotic β-cells
so many times that we find it appropriate to restate them
dBa
dt
= x3(t)B + W0(t)B + A′ maxC
kc + C B − f1MBa − f2MaBa − dBa (11.19)
dB
dt = −x3(t)B − W0(t)B − A′ maxC
B + r(t)B
kc + C
(11.20)
The other equations remain as given in equation 11.6, 11.7, 11.9 and 11.10.
We could avoid the use of the non-autonomous apoptosis and replication terms by
adding two additional differential equations to the system of equations (one describing
the change in the apoptosis-rate and one describing the change in the replication-rate),
and assume that x3,0 ≈ x3(30) and r(0) ≈ r(30).
If we estimate r0 = 0.040d −1 we obtain t ′ 1 ≈ 148d and p1 ≈ 0.019. While by estimating
x3,0 = 0.022d −1 we get t ′ 2 ≈ 141d and p2 ≈ 0.014. Based on these estimates, and
x2 = 0.001d −1 , the simulations in figure 11.4 and 11.5 were produced. 6 As with Model
B we will compare Model C to the data that is available to us. However we will save
the discussion of Model C for the next chapter.
Model C – Balb/c-simulation
Figure 11.4 shows that, after the β-cell population has endured the initial remodelling,
it grows to a concentration of nearly 3.5 × 10 9 before it starts to decline. After 1500
days the concentration is still above 3 × 10 8 , and at 30 months the concentration is
above what would be considered diabetic. The elevated number of β-cells, compared to
6 In appendix C.3 we have included the code that was used to do the simulations.