Biomedical Engineering Reference
In-Depth Information
T
Frozen tumor
Tumor
T
0
T
i
R
m
T
p
R
p
R
i
(t)
0
r
FIGURE 1.13
Temperature profile around a cryoprobe.
The freezing front moves so slowly that quasi-steady approximation may
be valid. Thus, assuming that the temperature profile within the frozen region
follows that obtained at the steady state, namely,
T
−
T
p
ln(
r
/
R
p
)
ln (
R
i
/
R
p
)
:
R
p
≤
T
p
=
r
≤
R
i
(1.87)
T
i
−
where we may estimate the first term on the right-hand side as
r
=
R
i
∂T
∂r
T
p
ln (
R
i
/
R
p
)
T
i
−
R
i
k
i
=
k
i
(1.88)
To estimate the second term on the right-hand side (representing the heat
flux from the unfrozen tumor to the interface), we write the bioheat equation
(1.79) for the unfrozen tumor region using the cylindrical coordinate system,
which, under the quasi-steady approximation, may be integrated to give
r
=
R
i
−
ρ
c
c
c
ω
eff
R
m
R
i
T
0
)
dr
+
S
m
R
m
R
i
dT
dr
−
k
c
R
i
r
(
T
−
rdr
= 0
(1.89)
Let us assume that the temperature in this unfrozen region follows:
=
1
2
T
−
T
0
T
i
−
r − R
i
R
m
−
−
:
R
i
≤
r
≤
R
m
(1.90)
T
0
R
i
The equation satisfies
T
=
T
i
at
r
=
R
i
and
T
=
T
0
and
∂T /∂r
=0at
r
=
R
m
such that the boundary condition given by (1.85b) is satisfied in an
approximate sense. Then substituting this temperature profile into equation
(1.89), we have
T
0
)
1
R
i
)
2
T
0
R
m
−
T
i
−
1
12
(
R
m
−
2
k
c
R
i
R
i
−
ρ
c
c
c
ω
eff
(
T
i
−
3
(
R
m
−
R
i
)
R
i
+
+
S
m
R
2
m
−
R
i
= 0
(1.91)
2
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