Biomedical Engineering Reference
In-Depth Information
As will be shown later, the homeostatic balance reproduced by the model
depends primarily on the minimal fraction of SCs in the particular SC's immediate
neighborhood that would lead to initiating its differentiation. For simplicity, in the
first, general model this parameter (referred to as the QS parameter) was set to
1. The second condition guaranteeing homeostasis is a strictly positive time-delay
between a cell's “birth” and its differentiation (
). Since the latter condition exists
for all biological cells, it will not be discussed any further. The other parameters of
the model determine factors such as speed of cell production but do not influence
the ability of tissue cell populations to reach homeostasis. This demonstrates the
importance of the negative feedback, depicted in the model by rule (
2
), in which an
SC does not differentiate unless its immediate microenvironment is saturated with
SCs. This regulatory feedback has a crucial role in the homeostatic characteristics
described above.
Moreover, further analysis of the model shows that under certain assumptions,
the model guarantees stability in the proportion of SCs in the population [
45
].
Minimalistic and biologically plausible limitations on the cells' kinetic parameters,
and some constraints on the symmetry of the initial SC subset, enable derivation
of an expression for the fraction of SCs (and of DCs) in the population, averaged
over a period of
Ψ
3 time steps. During this time period, which is
the minimal time for an automaton cell to go through all states (proliferation,
differentiation, and death), the SC population size fluctuates. However, for a special
case of tube-like tissues, the size of the SC population is bounded from above and
from below. When cylindrical symmetry is imposed on the graph, by constructing it
as
h
Ψ
+
Θ
+
Φ
+
1 similar-sized layers, the numbers of all SCs and DCs at each time step do
not differ from the average value by more than
+
γ
%, where
400
(
Ψ
+
Θ
+
Φ
+
3
)
1
,
600
(
Φ
+
1
)
γ
=
<
(4)
h
+
1
h
(proof in [
45
]). Importantly, given such a cylindrical structure, it is possible to
calculate how many initial SCs are needed in the system in order to generate a stable
cell population. This is of interest for tissue engineering, where tube-like tissues are
constructed using SCs implemented in an artificial scaffold [
81
].
What can go wrong in tissue homeostasis? To examine the effect of deranged in-
tercellular communication in the microenvironment, Agur and colleagues modified
the model slightly [
3
]. They allowed the QS parameter to be less than unity, now
denoting it
K
i
, representing the intensity of a signal that reaches an SC from another
SC located at a distance of
i
on the connected graph. Rule (
2
) of the CA iterative
operator was generalized, such that an SC differentiates only if the overall signal
intensity it is exposed to (from all SCs in its proximity) is above a certain threshold.
The model was modified to have a cubic geometrical structure in order to simplify
quantification of this demand (see Fig. 1 in [
3
]).
Numerical simulations of this model were performed under various values of
K
i
and with
10
4
,and
different randomly chosen initial states. Most of the simulations resulted in one of
∼
possible triplets of values for cell kinetic parameters
Φ
,
Ψ
,
Θ
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