Biomedical Engineering Reference
In-Depth Information
is Fourier number, where
α
i
=
k
i
/ρ
i
c
i
is the thermal diffusivity of the ice.
Moreover, the following dimensionless parameters are introduced:
Ste
i
=
c
i
(
T
i
−
T
p
)
: Stefan number
(1.97a)
h
sf
Sr
=
ρ
c
c
c
(
T
0
−
T
i
)
(1.97b)
ρ
i
h
sf
ω
∗
=
ω
eff
R
p
α
c
(1.97c)
Cr
=
k
i
(
T
i
−
T
p
)
k
c
(
T
0
−
(1.97d)
T
i
)
S
m
R
p
k
c
(
T
0
−
Met
=
(1.97e)
T
i
)
The foregoing ordinary differential equation (1.95) may readily be inte-
grated using any standard integration scheme such as Runge-Kutta-Gill, to
find the dimensionless time
t
∗
=
α
c
t/R
p
required for freezing the tumor of a
given dimensionless radius
R
i
=
R
i
/R
p
. Obviously, the quasi-steady assump-
tion is valid when
t
∗
Ste
i
/
(1 +
Sr
)
>
1, which roughly gives
t>
1 sec. Thus,
the assumption holds most part of the freezing process except its initial short
period.
It is interesting to note that there exists the limiting radius
R
lim
of the
tumor that one single cryoprobe can freeze at the maximum. Its dimension-
less value
R
lim
=
R
lim
/R
p
may be obtained setting
dR
lim
/dt
∗
= 0, for which
equation (1.95) yields
2
= 1
ln
R
lim
Cr
ω
∗
+6
Met
4(
ω
∗
+3
Met
)
3
(
ω
∗
+3
Met
)
1
/
2
R
lim
+
(1.98)
This implicit equation gives the dimensionless limiting radius
R
lim
for a given
set of the dimensionless values,
Met
,
Cr
, and
ω
∗
. Usually,
ω
∗
is much larger
than
Met
. For such cases, the following explicit expression based on Newton's
shooting method may be used to give a reasonably accurate value for
R
lim
:
3
2
ω
∗
Cr
+
R
0
R
0
+
3
32
ω
∗
R
0
(1
ln
R
0
)
−
R
lim
=
R
0
(1+ln
R
0
)+
3
(1.99)
32
ω
∗
where
1
ln
3
2
ω
∗
Cr
3
Cr
2
ω
∗
+
3
Cr
8
ω
∗
−
R
0
=
3
2
ω
∗
Cr
1+ln
3
2
ω
∗
Cr
+
3
(1.100)
32
ω
∗
For the present case of
Cr
= 15.4, equation (1.99) along with (1.100) gives
R
lim
= 29.9 and 12.9 for
ω
∗
= 0.031 (
ω
= 0.004/sec) and 0.310 (
ω
= 0.04/sec),
respectively.
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