Biomedical Engineering Reference
In-Depth Information
Usually the description of growth of tumor is carried out at extracellu-
lar scale. The mathematical structures are deterministic reaction diusion
equations 55;56;63 .
Following Chaplain and Anderson 20 , Lavichowicz considered a system of
deterministic reaction-diusion-chemotaxis equations that is able to model
the invasive spatial spread of solid tumors. The model is able to cap-
ture some aspects of solid tumor growth and invasion at the extracellular
scale (tissue level). The model is based on genetic solid tumor growth at
the avascular stage, and it describes the interactions between the tumor
and surrounding tissue (ECM). The variables in the model are tumor cell
density, 1 , ECM density, 2 , and MDE (certain factors produced by the
tumor cells and known as matrix degrading or degrading enzymes) concen-
tration, 3 . The model describes one key aspect of tissue invasion, namely
the ability of tumors cells to produce and secrete MDEs and their migra-
tory response. Chaplain and Anderson made assumptions that the tumor
cells produce MDEs which degrade the ECM locally; the ECM degradation
aids in tumor cells motility; movement of tumor cells up to a gradient of
ECM is referred as haptotaxis; tumor cell motion is driven only by ran-
dom motility and haptotaxis; the proliferation of tumor cells is not taken
into account.
With these assumptions the model (in dimensionless form) of Chaplain
and Anderson 20 yields
@ 1
@t
= d 1 r 2 1 r( 1 r 1 )
@ 2
@t
(6)
= 2 3
@ 3
@t
= d 3 r 2 3 + 1 3
where d 1 ; ; ; d 3 ; ; are given positive constants (the macroscopic pa-
rameters), d 1 r 2 1 , r( 1 r 1 ), 2 3 , d 3 r 2 3 , 1 , and 3 represent
random motility, haptoxis, degradation, diusion, production and decay re-
spectively. To improve the above model, one should construct cellular model
that correspond to the macroscopic model dened by equation (6).
9.2. Coupling of dierent scales
Most existing models focus on one scale. They dier considerably from each
other, according to the modeling scale (subcellular, cellular and extracellu-
lar) they focus on. While this may provide valuable insight into processes
occurring at that scale, it does not address fundamental problems of how
phenomena dierent scales are coupled. This is because one obstacle that
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