Biomedical Engineering Reference
In-Depth Information
z
=(
1
+
α
s
)
w
,
z
(
0
)=
0
,
(41)
s
=
−
σ
s
+
w
,
s
(
0
)=
0
,
(42)
over all Lebesgue measurable functions
u
:
[
0
,
T
]
→
[
0
,
u
max
]
and
w
:
[
0
,
T
]
→
[
0
,
w
max
]
for which the corresponding trajectory satisfies the end-point constraints
y
(
T
)
≤
y
max
and
z
(
T
)
≤
z
max
.
The coefficients are assumed constant and generally are positive;
u
max
,
w
max
represent maximum dose rates at which the agents can be administered and
y
max
and
z
max
limit the total amounts of the respective agents to be given. A medically
reasonable selection for all parameter values is given in Table 1 in [
8
]. The main
difference between model
[AR]
and the model in [
8
] is that we dropped the so-
called early-tissue constraint which prevents an overestimation of the damage done
to the early tissue. Also, rather than distinguishing between separate variables
r
p
and
r
q
that model the damage done to
p
and
q
, here, for simplicity, we only use one
variable
r
as the numerical values given in [
8
] for the coefficients in these equations
agree. Furthermore, we consider a continuous time version of radiotherapy that is
not necessarily given in fractionated doses. We shall comment below on how such a
treatment protocol can be derived from our version.
The addition of the radio-therapy terms has no structural effects on the anti-
angiogenic treatment and if the control
u
is singular, then regardless of the form of
the radio-therapy schedule, we have the following direct extension of Proposition
8.2
for model
[AR]
:
Proposition 9.1 ([
29
]).
If the optimal anti-angiogenic dose rate u follows a singu-
lar control on an open interval I and if the radiotherapy dose rate is given by w,
then we have the following relation between the controls u and w:
γ
u
sin
(
t
)+[(
η
+
δ
r
)
−
(
ϕ
+
β
r
)]
w
(
t
)=
Ψ
(
p
(
t
)
,
q
(
t
))
(43)
with
the function defined in Eq. (
34
) for the optimal anti-angiogenic monotherapy.
Given w, this determines the anti-angiogenic dose rate.
Ψ
Like for combinations with chemotherapy, also in this case there is an immediate
and mathematically simple extension of the formula that determines the optimal
singular anti-angiogenic dose rate to the more structured and more complicated
mathematical model that describes the combination treatment with radiotherapy.
However, now the structure of the second control is very different. In fact, it typically
(if the bounds on the dose rates permit) is singular as well. The controls
u
and
w
are
said to be
totally singular
on an open interval
I
if they are singular simultaneously.
This is not optimal for the combination therapy model
[AC]
with chemotherapy,
but it is the defining structure for the combination of anti-angiogenic therapy with
radiotherapy. For this we need a second equation that links
u
sin
with
w
sin
.
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