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

there is an exponential relation between f1 and the concentration of active macrophages
that must be exceeded to induce a non-inflammatory state. We propose
For h(f1) we require
A suitable candidate is
M + a (f1) ≈ Ma,h + (M + a,0 − Ma,h) × exp(−h(f1)) (9.3)
lim h(f1) = ∞, lim
f1→f1,h
f1→0 h(f1) = 0 (9.4)
M + a (f1) ≈ Ma,h + (M + a,0 − Ma,h) × exp
pf1,h
−
− p
f1,h − f1
99
(9.5)
where p ∈ R + \0 and f1 ∈ [0; f1,h[. The slope of the M + a (f1)-curve is determined by
which values we use for p. 2 Based on equation 9.5 we can estimate the concentration that
must be added to an NOD-mouse to overcome chronic inflammation; at least in theory.
We find the concentration to be ∆Ma := M + a (f1)−M(f1). Looking at figure 8.9 we find
that at f1 = 1 × 10−5 we have M + a ≈ 10 × 106cells ml−1 and Ma ≈ 5 × 106cells ml−1 , so
at the estimated NOD-value of f1 we need to add at least 5 × 106cells ml−1 , or double
the concentration of activated macrophages, to stop the inflammation. However if we
were able to turn the value of f1 up to, say, 1.5×10−5 , then we find ∆Ma ≈ 1×106cells ml−1 ; i.e. a fifth of what was needed at the estimated f1-value for NOD-mice.
There are two problems with the expression given in equation 9.5; . First have assumed
that M + a,0
is finite, secondly, even though this may be the case, there is still the matter
of fitting all the cells into the volume available. If the former is the case, then our
mathematical relation between M + a and f1 is wrong, and the M + a -curve would in this
case better be approximated by a hyperbola. However in reality NOD-mice are never
born with a phagocytosis rate of 0, so we can more or less disregard this problem,
and accept equation 9.5 as a reasonable approximation for the cases where f1 is not
unreasonably small. This allows us to address the second problem. From Poulter and
Turk (1975) we obtain an estimate of the volume of a macrophage to be 1450µm 3
(Poulter and Turk, 1975, p.198) which converts to 1.45 × 10 −9 ml. Hence based on
the result of Poulter and Turk (1975) there is room for 689655172 macrophages within
1 ml. So, depending on the volume of the other cells, we will not rule out adding
supplementary activated macrophages as a viable mean of inducing health in NODmice
with chronic islet inflammation; especially if it can be combined with some way of
enhancing the phagocytic ability of the macrophages. However it is up to readers with
a medical background to check it experimentally.
Please be aware that we under no circumstances claim that the expression for M + a (f1)
given in equation 9.5 is canonical. We could just as well have chosen a function of the
form
M + a (f1) = Ma,h + (M + a,0 − Ma,h) × exp(−kf1) (9.6)
where k is a constant much greater than 1.
This much we learned from the NOD-bifurcation diagrams. Now let us look at what
information can be extracted from the Balb/c-diagrams.
2 By using matlab to better approximate the UUB-curve at several values of f1 we could fit equation
9.5 to these data points to obtain to a value for p; i.e. be more meticulous with the method we used
to estimate the UUB in the first place.