Biomedical Engineering Reference
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where
K
angio
is a positive function that depends on the drug concentration and on
the chemical concentration; in this case, Eq. (
9
) remains unchanged.
Thus, in [
73
], Jackson et al. considered a model with two kinds of cells
differing by their sensitivity to a cytotoxic treatment: one cell type was less
sensitive than the other one. They assumed that the tumour was a spheroid, thus
reducing the dimension from three to one, using radial symmetry. The drug fate
was modelled through the variations of its tissue concentration, via a term of blood-
to-tissue transfer (the drug concentration in blood being prescribed by the therapy
scheduling). The authors compared the tumour response to an equal amount of
drug administered either by bolus injection or by continuous infusion. Jackson
based herself on this work to develop a model of the action of an anti-cancer
agent (doxorubicin) on tumour growth [
72
]. This model is composed of a submodel
of tumour growth coupled to a three-compartment submodel of intratumour drug
concentration (extracellular space, intracellular fluid space, nucleus space) and to a
submodel of the plasma concentration of the drug. The intracellular action of the
drug on tumour cells is modelled through a Hill-type function. This model allows to
study the tumour response to repeated rounds of chemotherapy.
In [
112
], Norris et al. investigated the effects of different drug kinetics (linear
vs. Michaelis-Menten kinetics) and different drug schedules (single infusion vs.
repeated infusions) on tumour growth.
Frieboes et al. [
57
] developed a mathematical model of tumour drug response
that takes into account the local concentration of drug and nutrients. The authors
considered two cell phenotypes, viable and dead tumour cells, and supposed that
their mitosis and apoptosis rates depended on the nutrients and drug concentration.
This model was calibrated on in vitro cultures of breast cancer cells.
More mechanistic (i.e., more molecular than purely phenomenological) models
have been used to take into account details of the angiogenic process. Endothelial
cells that constitute the blood vessel wall migrate towards a gradient of a chemoat-
tractant substance secreted by quiescent (or hypoxic) tumour cells (this movement is
termed chemotaxis). Continuous models of angiogenesis usually take into account
the density of endothelial cells and the tissue concentration of the chemoattractant
substance. They are based on the following equations
∂
m
∂
t
=
∇
.
(
D
m
∇
m
)+
α
m
m
−
χ
m
∇
.
(
m
∇
w
)
−
δ
m
m
,
(13)
∂
w
t
=
∇
.
(
)+
α
(
)
−
δ
,
D
w
∇
w
q
w
w
w
(14)
w
∂
where
m
denotes the density of endothelial cells,
D
m
their diffusion rate,
α
m
their
proliferation rate,
δ
m
their death rate,
w
the concentration
of the chemoattractant substance,
D
w
its diffusion rate,
χ
m
their chemotaxis rate,
δ
w
its production rate that
depends on the density of quiescent tumour cells
q
,
δ
w
its degradation rate. Such
models can also include other kinds of cells or chemical substances, see for instance
[
34
] for a review of angiogenesis models.
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