Biomedical Engineering Reference
In-Depth Information
let the upper-case letters
E
(East),
W
(West),
N
(North), and
S
(South)
denote its neighboring nodes. Furthermore, we let the lowercase versions of
the same letters
e
,
w
,
n,
and
s
denote four faces of the control volume, and
(
δx
)
e
,(
δx
)
w
,(
δy
)
n
, and (
δy
)
s
denote the distances between the nodes. Then,
the discretized version of the bioheat equation may be written as follows (see
e.g., Nakayama [1995] for details):
a
P
T
P
=
a
E
T
E
+
a
W
T
W
+
a
N
T
N
+
a
S
T
S
+
b
(1.82)
where
a
E
=
∆
y
δx
e
1
T
E
−
T
E
k
(
T
)
dT
(1.83a)
T
P
T
P
a
W
=
∆
y
δx
w
1
T
P
−
T
P
k
(
T
)
dT
(1.83b)
T
W
T
W
a
N
=
∆
x
δy
n
1
T
N
−
T
N
k
(
T
)
dT
(1.83c)
T
P
T
P
a
S
=
∆
x
δy
s
1
T
P
−
T
P
k
(
T
)
dT
(1.83d)
T
S
T
S
a
P
=
∆
x
∆
y
∆
t
ρc
+
a
E
+
a
W
+
a
N
+
a
S
+
ρ
f
c
p
f
ω
eff
∆
x
∆
y
(1.83e)
b
=
∆
x
∆
y
∆
t
ρcT
P
+
ρ
f
c
p
f
ω
eff
T
f
+
S
m
∆
x
∆
y
(1.83f)
T
P
1
T
P
−
ρc
=
ρc
(
T
)
dT
(1.83g)
T
P
T
P
The superscript
o
indicates the value at the
old
time
t
, whereas no super-
script is assigned for the value at the new time
t
+∆
t
.
The present computer code is capable of dealing with arbitrary two-
dimensional shapes of cryoprobe and tumor as illustrated in Figure 1.12. The
initial and boundary conditions for the freezing process using the cryoprobe
of outer radius
R
p
are given as follows:
t
=0:
T
=
T
f
(everywhere)
(1.84)
t>
0:
T
|
x
2
+
y
2
=
R
P
=
T
p
(cryoprobe outer surface)
(1.85a)
T
|
x
2
+
y
2
→∞
=
T
f
(deep tissue region)
(1.85b)
Computations were carried out using highly nonunform grid systems,
namely, (250
×
500) to cover the right-half domain 30 mm
×
60 mm for
the case of the longitudinal tumor of 20 mm
×
27 mm, and (350
×
700) to
cover the right-half domain 160 mm
160 mm for the case of determining the
limiting radius. The results associated with the limiting radius are found to
be independent of any additional expansion of the calculation domain. Grid
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