Biomedical Engineering Reference
In-Depth Information
We first analyze the behavior of the tumor and its vessels in the absence of
therapies or under continuous infusion therapies (CITs) of infinite temporal length,
i.e., for v
is small enough,
then the tumor can in fact be eradicated by anti-angiogenic action alone.
(
t
) ≡
V
≥
0, u
(
t
) ≡
U
≥
0and w
(
t
)=
W
≥
0. If F
( ∞ )
Lemma 6.1. If
ϕ
V
>
F
( ∞ )
, then, in the limit t
→ ∞
, the tumor is eradicated,
lim t → + ∞ p
(
t
)=
0 .
F − 1
The p -nullcline, p
=
0, is given by q
=
A
(
V
)
p where A
(
V
)=
( ϕ
V
)
and,
setting q
=
0, we obtain that
β − 1 I
(
p
)+ μ + γ
U
+ η
V
=
(
)=
.
q
Q
p
p
(20)
θ (
W
,
V
)
It is then straightforward to prove the following proposition:
Proposition 6.1. Under continuous infusion therapies, U
≥
0 and V
≥
0 ,if
θ (
W
,
V
) β (
A
(
V
)) > ( μ + γ
U
+ η
V
) ,
(21)
then there exists a unique, non-null, globally asymptotically stable equilibrium point
EQ
=(
p e (
U
,
V
,
W
) ,
q e (
U
,
V
,
W
))
that satisfies
(
,
,
)=
(
,
,
)
(
)
q e
U
V
W
p e
U
V
W
A
V
(22)
and
I − 1
(
,
,
)=
[ θ (
,
) β (
(
)) − ( μ + γ + η
)] .
p e
U
V
W
W
V
A
V
V
(23)
Moreover, the orbits of the system are bounded and the set
M
2
+
Ω ( U , V , W ) =
(
p
,
q
) ∈ R
: q
≤
M
=
max
p ∈ [ 0 , p e ]
Q
(
p
)
and 0
≤
p
≤
A
(
V
)
is positively invariant and attractive.
Thus, in case of infinitely long therapies, in principle it is possible to eradicate
the tumor under suitable constraints on the drug density in the blood. A first
condition to reach this target has been illustrated in Lemma 6.1 , but it is simply
the translation to the angiogenic setting of the eradication constraint R
)
from the chemotherapy setting. Here we are interested in results that genuinely
relate to the tumor-vessel interaction, and we also would like to show possible
synergies between chemotherapy and the anti-angiogenic therapies. This leads to
the following proposition:
(
0
) <
H
(
C
Proposition 6.2. Under continuous infusion therapy, if
θ (
W
,
V
) β (
A
(
V
)) ≤ μ + γ
U
+ η
V
(24)
 
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