Biomedical Engineering Reference
In-Depth Information
is applied simultaneously at each time step on all vertices in
V
to define the state of
the system at any time
t
. The operator definition is as follows:
(
N
,
0
)
if
τ
=
Φ
,
x
t
x
t
+
1
(
v
)=(
D
,
τ
)
−→
(
v
)=
(1)
(
D
,
τ
+
1
)
otherwise;
⎧
⎨
(
D
,
0
)
if
τ
=
Ψ
and each
v
's neighbor
is a stem cell
,
x
t
+
1
x
t
(
v
)=(
S
,
τ
)
−→
(
v
)=
(2)
(
S
,
τ
)
if
τ
=
Ψ
and
v
has a non-stem
neighbor
⎩
(
S
,
τ
+
1
)
otherwise;
⎧
⎨
(
N
,
0
)
if
v
has no stem neighbor
,
(
S
,
0
)
if
v
has a stem neighbor
and
x
t
x
t
+
1
(
v
)=(
N
,
τ
)
−→
(
v
)=
(3)
⎩
τ
=
Θ
,
(
N
,
τ
+
1
)
otherwise;
where a vertex is defined as a neighbor of
v
if the distance between the two vertices
in the shortest-path metric induced by
G
is equal to 1.
5.2
Tissue Homeostasis
In order to prove that this simple description of fate-decision regulation is sufficient
to reproduce tissue homeostasis, Agur and colleagues conducted a mathematical
analysis of the model [
4
,
45
]. This resulted in a set of propositions, analytically
proven, that together show that the model retains the basic properties essential
for maintaining tissue homeostasis, reaching stable SC and DC populations. These
theorems are nonquantitative and are robust for any potential refinements involving
more elaborate rules. In other words, the model represents a family of cellular
automata, and it can be modified to describe more specifically the cell-population
control of specific cell types in different tissues. For example, imposing limitations
on the kinetic parameters
or imposing a certain geometrical structure
will not affect the system's homeostatic properties, since the theorems that follow
directly from the basic model assumptions will stay valid.
It was proven that, after some limited initial number of time steps, the tissue
model sustains a minimal density of SCs at any time point. A constant supply of
mature cells is also assured, owing to the existence of a lower bound for the rate
of production of DCs. (The proofs are detailed in [
4
].) The authors also analyzed
the dynamics leading to a state in which the system dies out, i.e., when all vertices
are in the state of N. They proved that the system never dies out, regardless of the
initial SC population size, except under specific extreme conditions. This feature of
the model reflects the tissue's ability to recover after SC depletion.
Φ
,
Ψ
,and
Θ
Search WWH ::
Custom Search