Biomedical Engineering Reference
In-Depth Information
evolution of a system comprising a large number of cells, then the system
of ordinary dierential equations (one for each cell) can be replaced by
a kinetic equation on the statistical distribution of the state of all cells.
The application of methods of mathematical kinetic theory to model the
competition between tumor and immune cells was initiated by Bellomo and
Forni 9 .
Cellular models 39;51 deal with interaction between cells, which is of
course strongly related to what happens at the subcellular level. Cellular
Automata models that treat cells as single points on a lattice, for exam-
ple, the LGCA model of Alarcon et al. 2 , Dormann and Deutsch 23 . They
adopt local rules specifying adhesion, pressure (cells are pushed towards
regions of low cell density) and couple the LGCA to a continuum chemical
dynamics. Their two-dimensional simulations produce a layered structure
that resembles a cross-section of an MTS.
6.1. Cellular Potts model
The cellular Potts model is a more sophisticated Cellular Automata model,
which describes individual cells as extended objects of variable shapes. The
cellular Potts model can be applied to model tumor growth. Any cellular
scale model of tumor growth must consider cell-cell adhesion, chemotaxis,
cell dynamics including cell growth, cell division and cell mutation, as well
as the reaction-diusion of chemicals: nutrients and waste products, and
eventually, angiogenesis factors and hormones. In additional to dierential
adhesion and chemotaxis, cellular models can include the reaction-diusion
dynamics for relevant chemicals:
@C 0
@t
= D 0 r 2 C 0 a(x);
(1)
@C n
@t
= D n r 2 C n b(x);
(2)
@C w
@t
= D w r 2 C w + c(x);
(3)
where C 0 ; C n ; and C w are concentrations of oxygen, nutrients (glucose),
and metabolic wastes (lactate), their initial values are a 0 ; b 0 and c 0 re-
spectively. D 0 ; D n ; and D w are their respective diusion constants; are
metabolic rates of the cell located at x; and c is the coecient of the
metabolic waste production.
C 0 C 0
C 0 C 0
a = a 0
;
Search WWH ::




Custom Search