Biomedical Engineering Reference
In-Depth Information
@c
@t
@
@x
ch(c)
@u
@x
= k
:
(5)
where the term f(u) represents cell division and death; f(0) must clearly be
zero, and for simplicity Sherratt assumed that the cell density was rescaled
so that u = 1 is the equilibrium level within the tumor, implying f(1) = 0.
Random cell movement was assumed, and kinetics of extracellular matrix
are neglected, in keeping with expansive growth hypothesis, so that the
extracellular matrix density only changes because of convection with the
cells. This convection does not imply large-scale movement of intact matrix
by a cell; rather it is the net result of local matrix movement and remod-
eling during cell movement. This will increase with local matrix density
and is represented in the model as kc, and the function h(c) represent the
reduction in cell motility at high matrix density.
Another hypothesis, foreign body hypothesis is derived from the notion
that capsule formation is an attempt by the body localize the tumor and
assumes that, when stress, normal cells begin to secrete collagen or other
brous components of ECM. This view is essentially of an active process
where the body mounts a response akin to inammation to create a brous
barrier. Ewing's
25
work suggested that the encapsulated tumors may; thus
be shielded from cellular attack. Similarly, Enneking's
24
work suggests that
the hosts attempt to encapsulate and contain tumors. Barr et al
8
gave
a detail review of the mechanism of encapsulation and also suggested a
compromise hypothesis embodying both of the above mechanisms.
The major dierences between the foreign body hypothesis and the
expansive growth hypothesis is that the latter is a passive process, where the
former is an active response of the host. Since it is dicult to discriminate
the two hypotheses using experimental techniques, mathematical modeling
provides a natural approach for testing and comparing the assumptions
and the consequences associated with each of them. Jackson and Byrne
37
developed a mechanical approach to study both hypotheses.
9. Coupling and unifying dierent scales
9.1. Unication of model results of dierent scales
The problem of relationships between the various scales of description seems
to be the most important problems of the mathematical modeling of com-
plex systems, for example modeling of solid tumor growth. The following
strategy can be applied. One starts with the deterministic extracellular
scale model for which the identication of parameters by an experiment is
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