Biomedical Engineering Reference
In-Depth Information
FIGURE 8.2: Schematic diagram of the key processes included in the mathe-
matical model. The extracellular environment provides to tumor cells soluble
growth factors, whose uptake enhances cell intrinsic motility and prolifera-
tion. Tumor cells, in turn, secrete matrix metalloproteinases (MMPs), which
degrade ECM proteins.
If
is such that (
) 2fN;Cg, s
;
(x;t) = (n(x;t)) 2R
+
, where
n(x;t) is the local intracellular concentration of growth factors, ex-
pressed in units of M.
The system Hamiltonian is given by:
H(t) = H
shape
(t) + H
adhesion
(t) + H
haptotaxis
(t):
(8.1)
H
shape
includes the constraints that regulate cell volume and surface (which, in
two dimensions, correspond to the cell area and perimeter), written as nondi-
mensional relative deformations in the quadratic form of Equation (4.6). The
cell target dimensions correspond to their initial measures and are given in the
Appendix. Assuming that the cells do not significantly grow during invasion,
the fluctuations of their volumes are kept negligible with high constant values
of
surface
;
:(
)2fN;Cg
. Moreover, because tumor cells are typically deformable,
as they are able to significantly remodel to invade their surroundings more
eciently, for any and for
such that (
) = C,
perimeter
;
=
perimeter
;C
needs to be suciently low. The relative rigidity of the nucleus (with respect
to the cytoplasm) is instead modeled, as usual, by a relatively higher value of
perimeter
;
=
perimeter
;N
when (
) = N.
H
adhesion
is straightforwardly differentiated into the internal and external
contact energy contributions, cf. Equations (4.1), (4.2), and (4.3). In particu-
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