Biomedical Engineering Reference
In-Depth Information
mechanical properties of the supporting tissue, on the local quantity of nutrients and
oxygen, on the local concentrations of pro and anti-growth chemical factors, etc.
PDE models integrating spatial dynamics are thus better suited than ODE models
for the design of realistic models of tumour growth. Greenspan [
63
]wasthefirst
author to take into account the spatial dynamics of tumour cells and oxygen through
the simplifying hypothesis of a spherical symmetry of the diseased tissue (tumour
spheroid). Several models are based on his [
32
,
33
,
43
,
45
,
101
,
127
].
Reaction-diffusion PDEs are best suited to describe the space and time evolution
of the concentrations of chemical substances and of cell densities. Such equations
allow to take into account the interactions of a diffusing chemical molecule or of a
population of cells with its environment.
Thus Swanson et al. [
131
-
133
] used the classical KPP-Fisher model that is
frequently used to represent the spatial progression of so-called “travelling waves”
(see [
103
] for details), to develop a model of brain tumour growth that takes into
account tumour cell proliferation and diffusion
∂
p
t
=
∇
.
(
D
(
x
)
∇
p
)+
ρ
p
(
1
−
p
)
,
(6)
∂
where
p
is the tumour cell density (that depends on space and time),
D
(
x
)
the
diffusion rate (that depends on space),
ρ
the net proliferation rate. In fact, its
linearised form around the origin
∂
p
t
=
∇
.
(
D
(
x
)
∇
p
)+
ρ
p
(7)
∂
is sufficient to describe tumour progression (and it has an analytic solution if
D
is constant). The difficulty here resides in the identification of the diffusion
coefficient
D
(
x
)
, which is in fact far from constant, since it depends on the nature of
cerebral matter, grey or white, and the brain is not known to possess a simple spatial
structure.
In [
132
], Swanson et al. modelled the action of a chemotherapy by introducing
in Eq. (
6
) a linear death term that also depends on space and time. They investigated
drug delivery according to the tissue heterogeneities of the brain (white or grey
matter).
Competition between cells for the gain of space and nutrients influences tumour
growth. Thus, on the basis of a Lotka-Volterra type model [
109
,
110
], Gatenby et al.
[
59
,
60
] modelled competition between healthy and tumour cells to phenomeno-
logically represent the mutual negative influences of the populations on each other.
They highlighted the limitations of the clinical cytotoxic strategies that solely focus
on killing tumour cells and not on preserving healthy cell populations from toxic
side effects of the anticancer drugs.
Cell proliferation and cell death induce changes in the tumour volume. This
phenomenon has to be taken into account to model tumour cell growth in a more
(
x
)
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