Biomedical Engineering Reference
In-Depth Information
Fig. 2
Schematic description
of the general tissue model.
Three cell types—stem (S),
differentiated (D), and null
(N) cells—are represented.
The colored areas show QS
regulation on the SC fate
decision
Migration/
Death
Maturation
N
D
Proliferation
Differentiation
S
N
S
Sense
neighbors
Sense
neighbors
Mathematically,
this
system
is
represented
by
dynamics
on
a
connected
undirected graph
G
,where
V
and
E
are sets of vertices and edges,
respectively. Each vertex is a cell, and the edges connect each vertex with its closest
neighbors. The distance between each two vertices joined by an edge is defined as
1. Each vertex is equipped with an internal counter
=(
V
,
E
)
, measuring the cell's progress
towards replication or differentiation, if it is an SC, or progress of maturation in the
case of DCs. Note that the connected graph formulation compels no restrictions on
the geometrical structure or dimensionality of the cellular automaton.
The state
x
of a vertex
v
at any time
t
(denoted
x
t
τ
) is a two-component variable,
the first dimension denoting the cell's “type” (either S, D, or N), while the second
is a nonnegative integer that denotes its internal counter status. Agur et al. assumed
that at each time step, the cell state can be changed due to differentiation (from
S to D), proliferation (from N to S), or cell death (from D to N). These changes
happen according to the following rules, depending on three nonnegative integer
parameters, namely
(
v
)
:
A DC increases its lifetime counter at each time step from
Φ
,
Ψ
,and
Θ
τ
to
τ
+
1, until when
τ
=
Φ
represents DC maturation time.
An SC increases its internal counter in the same way, until
it dies, and its state becomes
(
N
,
0
)
.
Φ
Ψ
represents the duration of SC differentiation time. Then, if all of the SC's closest
neighbors are SCs, the cell differentiates (its state becoming
τ
=
Ψ
,where
(
D
,
0
)
). However, if an
SC has a non-stem neighbor when
, it does not differentiate but remains in
the same state. This stipulation corresponds to the QS hypothesis of an SC receiving
negative feedback signals from the other SCs in its microenvironment.
An N cell does not change its state, unless it has a stem neighbor, which provides
the N cell with the potential to become occupied by the SC's daughter cell following
the SC's replication. If the N cell has a stem neighbor, it increases its internal
counter over time, until
τ
=
Ψ
represents the cell-cycle time period for
SC proliferation. Then the N cell is replaced with a new SC (i.e., its state becomes
(
τ
=
Θ
,where
Θ
).
These rules are described by an iterative operator, which defines what happens
to a single vertex during the transition between time
t
and time
t
S
,
0
)
+
1. This operator
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