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

108 Expanding and Modifying the DuCa model
as substituting W0(t)B and A′
max C/(kc+C)B we get
dM
dt = a + (k + b)Ma − cM − f1MBa − e1M(M + Ma) (11.6)
dMa
dt
= f1MBa − kMa − e2Ma(M + Ma) (11.7)
dBa
dt
= x1 + W0(t)B + A′ maxC
kc + C B − f1MBa − f2MaBa − dBa
(11.8)
dBn
dt
= dBa − f1MBn − f2MaBn (11.9)
dC
dt = αBnMa − δC (11.10)
dB
dt = −x1 − W0(t)B − A′ maxC
B
kc + C
(11.11)
with initial conditions (M(0), Ma(0), Ba(0), Bn(0), C(0), B(0)) = (4.77×10 5 , 0, 0, 0, 0, 1×
10 9 ). For future reference we shall call this version of the expanded DuCa model, simply,
Model A. Figure 11.1 shows simulations done in matlab of Model A – please notice
Figure 11.1 Simulation of Balb/c- and NOD-mice with the inclusion of a compartment of
healthy β-cells, i.e. equations 11.6-11.11, where the natural apoptosis has been assumed
to be constant. Left: the behavior of the concentrations in Balb/c-mice. Right: the
behavior of the concentrations in NOD-mice. Initial values are (M, Ma, Ba, Bn, C, B) =
(4.77 × 10 5 , 0, 0, 0, 0, 1 × 10 9 ).
the difference in scale on the y-axis. On the left we see how the concentrations behave
in Balb/c-mice, while on the right we see the concentrations in NOD-mice. The first
feature we should take notice of is how many days it takes before the healthy β-cells
have all died out. In the Balb/c-mice the β-cells are gone after approximately 800 days,
while it only takes about 200 days in an NOD-mouse, after which the concentration
becomes negative. This is already a hint that Model A is not the most optimal candidate
– the β-cells should ceteris paribus not be depleted at 800 days in a Balb/c-mouse,
and there should still be some β-cells left in the NOD-mouse at the time when T1D
becomes apparent. In addition to this Model A fails to live up to the second guideline