Biomedical Engineering Reference
In-Depth Information
methods of their analysis and therapeutic control [
26
,
27
,
36
], in particular for
cancer chronotherapeutics, i.e., when the drug control is made 24 h-periodic to take
advantage of favourable circadian times.
Physiologically structured cell population dynamics models have been exten-
sively studied in the last 20 years, see Sect.
3.4
for some examples. We consider
here typically age-structured cell cycle models, in which the cell division cycle
is divided into
I
phases (classically 4:
G
1
,
S
,
G
2
and
M
), and the variables are the
densities
n
i
(
t
,
x
)
of cells having age
x
at time
t
in phase
i
. Equations read
⎧
⎨
∂
n
i
(
t
,
x
)
+
∂
n
i
(
t
,
x
)
x
+
d
i
(
t
,
x
)
n
i
(
t
,
x
)+
K
i
→
i
+
1
(
t
,
x
)
n
i
(
t
,
x
)=
0
,
∂
t
∂
∞
n
i
+
1
(
t
,
0
)=
K
i
→
i
+
1
(
t
,
x
)
n
i
(
t
,
x
)
d
x
,
(19)
⎩
0
2
∞
0
n
1
(
t
,
0
)=
K
I
→
1
(
t
,
x
)
n
I
(
t
,
x
)
d
x
(
(
=
,.
))
together with an initial condition
≤
i
≤
I
. This model was first introduced
in [
39
] and further studied in other publications, among which [
40
,
41
]. In this
model, in each phase
i
, cells are ageing with constant speed 1 (transport term),
they die with rate
d
i
or with rate
K
i
→
i
+
1
go to next phase, in which they start with
age 0. To represent the effect of circadian clocks on phase transitions [
91
], one may
consider time-periodic coefficients
d
i
and
K
i
→
i
+
1
, the period being in this case 24 h.
n
i
t
0
1
7.3
Basic Facts About Age-Structured Linear Models
One of the most important facts about linear models is the trend of their solutions
to exponential growth. The study of the growth exponent, first eigenvalue of the
system, is therefore crucial. Solutions to system (
19
) satisfy (if the coefficients are
time-periodic, or stationary)
n
i
C
0
N
i
e
λ
t
(
t
,
x
)
∼
(
t
,
x
)
[
119
], where
N
i
are defined by
⎧
⎨
N
i
(
(
,
)
(
,
)
∂
N
i
t
x
+
∂
N
i
t
x
x
+
λ
+
d
i
(
t
,
x
)+
K
i
→
i
+
1
(
t
,
x
)
t
,
x
)=
0
,
∂
t
∂
∞
N
i
+
1
(
t
,
0
)=
K
i
→
i
+
1
(
t
,
x
)
N
i
(
t
,
x
)
d
x
,
0
2
∞
0
(20)
⎩
N
1
(
t
,
0
)=
K
I
→
1
(
t
,
x
)
N
I
(
t
,
x
)
d
x
,
∞
T
∑
i
N
i
>
0
,
N
i
(
t
+
T
,.
)=
N
i
(
t
,.
)
,
N
i
(
t
,
x
)
d
x
d
t
=
1
0
0
with
T
−
periodic coefficients.
We
focus
now
on
the
case
of
stationary
phase
transition
coefficients
(
K
i
→
i
+
1
(
0). Note that if
one considers constant nonzero death rates, the problem does not change, only the
t
,
x
)=
K
i
→
i
+
1
(
x
)
) and we do not consider death rates (
d
i
=
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