Biomedical Engineering Reference
In-Depth Information
This equation describes the temperature field as a function of the dimensionless space vari-
able
x
and dimensionless time
t
and is valid up to the onset of ablation when y
¼
1. The
Þ¼
Ð
1
z
symbol ierfc indicates the integral of the function erfc, ierfc
ð
z
erfc
ð
t
Þ
dt. This equation
can be implemented with relative ease using modern user-friendly computer tools
such as MATLAB
TM
or Mathematica
TM
. An example of the graph of this solution, which
is the progression in time of the temperature profile as function of depth, is shown in
Figure 17.16.
By letting y
0inEq. (17.78), the following transcendental alge-
braic equation is obtained, which can be solved numerically for the time for the onset of
ablation
¼
1 (i.e.,
T
¼
T
ab
)at
x ¼
t
ab
:
2
p
t
ab
p
t
ab
B
2
t
ab
erfc
B
l
¼
p
B
þ
e
½
B
1
ð
17
:
81
Þ
2
p
Note that for large values of
B
l
, the behavior is almost linear with a slope of
p
.
The present analysis did not consider the case of scattering tissue. An analytical solution
for
could be obtained using the diffusion approximation approach described in
Section 17.2. However, implementation of that solution into Eq. (17.72) to solve analytically
for temperature would be highly cumbersome at best. Therefore, a solution of temperature
field would normally require a numerical approach, such as the finite difference or the
finite element method.
Q
L
120
100
Onset of
Ablation
80
Increasing
time
60
40
20
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
Depth (mm)
FIGURE 17.16
Nondimensional temperature as a function of nondimensional depth at various times prior to
the onset of ablation.
