Biomedical Engineering Reference
In-Depth Information
lar, the high negative value set for J
int
N;C
prevents cells from fragmenting (we
refer to [298, 352] for comments). J
ext
T;T
represents instead the adhesive strength
between the membranes of two nearby cells, a measure of the quantity of ac-
tive and exposed cadherins . Finally, J
ext
T;M
evaluates the heterophilic adhesive
bonds between the integrins on the cell surface and suitable ligands (such
as laminin and fibronectin) in the extracellular matrix . By setting constant
and homogeneous values for both J
ext
T;T
and J
ext
T;M
, we assume a uniform dis-
tribution of adhesion molecules on the cell surface and a homogenous density
of ligands in the external microenvironment. In particular, since malignant
cells have been demonstrated to have reduced cell{cell adhesiveness but in-
creased cell{ECM adhesiveness, due to a change in the relative expression of
the corresponding adhesive molecules [193], we set J
ext
T;T
2 J
ext
T;M
.
H
haptotaxis
reproduces the effect of cell preferential movement in the direc-
tion of zones with higher concentration of extracellular matrix proteins (such
as fibrin, vitronectin, and some of the collagen family) and is implemented
with a local linear-type relation of the form (1.9):
H
haptotaxis
=
hapt
[p
t
(x
target
;t) p
t
(x
source
;t)] ;
(8.2)
where x
source
and x
target
are, respectively, the source and the final lattice site
randomly selected during a trial update in a MCS, and
p
t
(x;t) = p(x;t) +
X
x
0
2
0
x
p(x
0
;t);
where x 2fx
source
; x
target
g; evaluates the local level of ECM proteins sensed
by the moving cell membrane site, as p(x;t) is their amount at site x (de-
fined in Equation (8.6)). Finally,
hapt
2R
+
represents the local strength of
haptotaxis, which is assumed constant for all malignant individuals.
Given the Hamiltonian, the transition probability of a spin flip has the
form of Equation (4.24). In particular, as usual in this topic, p(T
;
(t)) =
tanh(T
;
(t)). For any cell , T
;
, with (
) = N, is a low constant
positive value, T
;N
that mimics the passive motion of the nucleus of . If
(
) = C, T
;
= T
;C
represents instead, as seen, the intrinsic motility of
the overall cell, which is a measure of the agitation of its cytoskeleton (i.e.,
by the random actin polymerization/depolarization dynamics) and is in this
case assumed to be enhanced by the intracellular level of growth factors (i.e.,
motogen agents) in a dose-dependent manner. Adopting again the notation of
Equations (4.9) and (4.10), for any malignant individual and for (
) = C,
we have therefore that:
s
T
;
(x;t) = s
T;A
;
(x;t) = (n(x;t));
(8.3)
and
n
(t)
N
0
+ h (n
(t) N
0
)
T
;
(t) = T
;
(s
T;A
;
(x;t)) = f
T
(s
T;A
;
(x;t)) = T
0
;
(8.4)
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