Biomedical Engineering Reference
In-Depth Information
We briefly extend the growth model considered in the previous section to include
chemotherapy. In this model we assumed that
M
(
0
)
<
Π
(
0
)
since a negative net
proliferation rate
R
0 implies the self-extinction of the neoplasm, a case only
of interest for immunogenic tumors. But a negative net proliferation rate is exactly
what chemotherapeutic agents aim at and thus this has relevance in the theoretical
analysis of chemotherapy where the agents either reduce the proliferation rate or
increase the death rate of the neoplastic cells. When a drug is delivered to a human or
an animal host, two different types of processes take place called
pharmacokinetics
(PK)
and
pharmacodynamics (PD)
: pharmacokinetics determines the density of the
drug in the blood, i.e.,
what the body does to the drug
, and pharmacodynamics
models the effects the drugs have,
what the drug does to the body
. If we administer
a drug whose density profile in the blood is
c
(
p
)
<
, in many cases it is considered
realistic [
68
] to assume that the number of cells killed per time unit is proportional
to
c
(
t
)
, i.e., the pharmacodynamic model is linear in both the concentration
c
and
p
. This hypothesis is called the
linear log-kill hypothesis
, and it modifies the
basic growth model to become
(
t
)
p
(
t
)
p
=
pR
(
p
)
−
ϕ
cp
(7)
with
a positive parameter. For example, a simple model of chemotherapy
assuming a Gompertz law for unperturbed growth can be written as
ϕ
ln
p
)
p
∞
(
t
(
)=
−
ξ
(
)
−
ϕ
(
)
(
)
,
p
t
p
t
c
t
p
t
(8)
where
p
∞
denotes the (constant) carrying capacity. In a more general setting, one
can assume that the pharmacodynamics still is linear in
p
, but nonlinear in
c
,say
p
=
pR
(
p
)
−
H
(
c
)
p
.
(9)
If the therapy is delivered by constant continuous infusion therapy, after some initial
transient, it is reasonable to assume that
c
(
t
)
≈
C
, and in this case the tumor can be
eradicated if
R
(
0
)
<
H
(
C
)
. In the important case of periodic therapy,
c
(
t
+
T
)=
c
(
t
)
,
this eradication condition becomes
R
(
0
)
<
H
(
c
)
,where
f
denotes the average
of a periodic function
f
over one period.
4
Modeling Radiotherapeutic Treatments
The log-kill model used in Eq. (
8
) can also be considered a rudimentary approx-
imation for including effects of radiotherapy when only first-order killing effects
are considered. While such a linear model is reasonable for a cytotoxic agent, it
is, however, only a crude approximation for the effects of radiotherapy. It is more
realistic to assume that the damage to DNA made by the effects of ionization
radiation consists of a linear component that corresponds to a simultaneous break
Search WWH ::
Custom Search