Biomedical Engineering Reference
In-Depth Information
surface ablation temperature so thermal energy can be provided to the surface for the abla-
tion process to proceed. This is in accordance with one of the corollaries of the second law
of thermodynamics, stating that heat can be transported only from hotter points to colder
points. In fact, the higher than ablation threshold temperature puts the subsurface tissue
in a “metastable” equilibrium condition, which may be perturbed by internal tissue condi-
tions and result in nucleation and vaporization initiating subsurface and manifesting itself
by an “explosion” and mechanical tearing of the tissue surface. It should be noted at this
point that initiation of the ablation process involves nonequilibrium nucleation processes
that are not considered by the present approach.
17.5.4 Analytical Solution of Light Heating in a Purely Absorbing Medium
The governing equations for light irradiation with ablation were described in
Section 17.5.3. Here, the governing equations will be repeated for a one-dimensional,
semi-infinite, and purely absorbing medium. The governing equation for the preablation
heating stage can be written as
2
T
@r
cT
¼
@
z
2
þ m
a
Ie
m
a
z
ð
17
:
77
Þ
@
t
@
where
m
a
is the absorption coefficient. This equation is valid up to the onset of ablation,
which is assumed to occur when the surface temperature reaches the ablation threshold
temperature,
. The details of the analysis of ablation are beyond the scope of this chapter
and will not be considered.
In the late 1970s, a nondimensionalization of the heat conduction equation for an axisym-
metric three-dimensional case of preablation laser heating of tissue by a Gaussian beam in
an absorbing medium was solved. The following relations will transform the governing
equations into a dimensionless moving frame:
T
ab
¼
T
T
0
T
ab
T
y
x
¼ð
I
c
=
kL
Þ
z
t
¼ð
I
c
=
kL
Þð
I
=
r
L
Þ
t
ð
17
:
78
Þ
0
0
0
0
The variables introduced are, respectively, dimensionless temperature y, dimensionless
coordinate
x
in the moving frame with origin at the ablation front, and dimensionless time
t
. A dimensionless absorption parameter,
B
, and a dimensionless heating parameter
l
are
also defined as
B
¼ð
kL
=
I
c
Þ
a
0
ð
17
:
79
Þ
l
¼
c
ð
T
ab
T
0
Þ=
L
Analytical solutions of the nondimensional form of the governing equations can be
found by Laplace transformation of the space variable
x
. The solution is as follows:
<
:
=
;
2
4
3
5
0
@
2
4
3
5
þ
2
4
3
5
1
A
p
ierfc
p
p
1
Bl
2B
x
2
1
e
Bx
erfc B
x
2
e
Bx
erfc B
x
2
2
e
B
2
t
e
Bx
y
ð
x, t
Þ¼
p
þ
p
þ
p
ð
17
:
80
Þ
