Biomedical Engineering Reference
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optimised by determining drug infusion patterns that should maximise tumour cell
death under the constraint of minimising healthy cell death. Barbolosi and Iliadis
[
13
,
70
] coupled a Gompertz model of tumour growth, perturbed by a cytotoxic
efficacy term, to a two-compartment model of the chemotherapy PK (plasmatic
and active drug concentration). They investigated optimal drug delivery schedules
under constraints of maximal allowed drug (single doses and cumulative dose) and
leukopenia.
In an attempt to design a more realistic model of tumour growth under angiogenic
stimulator and inhibitor control, Hahnfeldt et al. [
67
] proposed a two-variable model
derived from the Gompertz model. It is based on observations made on experimental
data from lung tumours in mice treated by an antiangiogenic drug. Hahnfeldt et al.
introduced as a variable the carrying capacity of the environment,
K
:
N
ln
K
N
d
N
d
t
=
λ
,
(1)
d
K
d
t
=
dN
2
/
3
−
(
μ
+
)
−
η
(
)
,
bN
K
g
t
K
(2)
dN
2
/
3
μ
+
where
b
is the rate of the tumour-induced vasculature formation,
(
)
≥
represents the rate of spontaneous and tumour-induced vasculature loss,
g
t
0
represents the antiangiogenic drug concentration.
This model enables to take into account the vasculature that provides nutrients
and oxygen to tumour cells, and thus to study the effects of several anti-angiogenic
factors on tumour growth. D'Onofrio et al., based themselves on this model,
proposed different expressions for
K
, modelling for instance endothelial cell
proliferation or delayed death, to investigate the action on tumour growth of anti-
angiogenic therapies [
46
-
48
] and of combined therapies [
49
]. In the same way,
Ledzewicz et al. [
84
], basing themselves on [
55
], considered the following model
N
ln
N
K
d
N
d
t
=
−
λ
−
ϕ
vN
,
(3)
d
K
d
t
=
bK
2
/
3
dK
4
/
3
−
−
(
μ
+
γ
u
−
η
v
)
K
,
(4)
where
u
and
v
represent the doses of an anti-angiogenic drug and of a cytotoxic
drug, respectively, and
their effects on tumour cells and on vasculature. The
authors introduced an optimisation problem to minimise the tumour cell mass under
constraints on the quantity of drug to be delivered.
Most cancer chemotherapies are cell cycle phase-specific, which means that
they act only on cells that are in a specific phase of the cell division cycle, for
instance in
S
or in
M
. To take into account such specificities, models describing
the cell cycle have been designed. Without always entering into the details of cell
cycle phases, several models distinguish between proliferating and nonproliferating
cells [
114
,
115
,
139
,
140
]. They take into account two kinds of cell populations,
ϕ
,
γ
,
η
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