Biomedical Engineering Reference
In-Depth Information
where
is the total light intensity. The total
intensity at a point is the sum of radiative fluxes received at that point. In the case presented
here, using purely absorbing tissue, a single radiative flux is used, and thus,
m
a
is defined as the absorption coefficient and
I
I
¼
q
ð
rad
Þ
z
ð
17
:
28
Þ
Proceeding by replacing Eqs. (17.27) and (17.28) into Eq. (17.26), the following differential
equation is obtained
dI
dz
¼m
a
I
ð
17
:
29
Þ
which has the simple solution
I
ð
z
Þ¼
I
0
exp
ðm
a
z
Þ
ð
17
:
30
Þ
where
I
0
denotes the intensity at the surface,
z
¼
0. This equation is the well-known
Beer-
Lambert
law of absorption, as given in Eq. (17.17) for a purely absorbing medium, which
was mentioned earlier in the chapter.
The laser heat source term can now be written (Eqs. (17.27) and (17.30))as
Q
L
¼ m
a
I
0
exp
ðm
a
z
Þ:
ð
17
:
31
Þ
This equation can be generalized to an axisymmetric three-dimensional case to include
the effect of the radial beam profile of a laser light incident orthogonally on a slab by
writing it as
Q
L
ð
r
,
z
Þ¼m
a
I
0
exp
ðm
a
z
Þ
f
ð
r
Þ
,
ð
17
:
32
Þ
where
) is the radial profile of an axisymmetric laser beam. For a Gaussian beam profile,
which is a common mode of laser irradiation,
f
(
r
2
o
2
o
2
r
f
ð
r
Þ¼
exp
ð
17
:
33
Þ
2
radius” of the beam, since at
1/e
2
.
where o
o
is known as the “1/
e
r
¼
o
o
,
f
(
r
)
¼
EXAMPLE PROBLEM 17.4
For a Gaussian laser beam irradiating at a wavelength of 2.1
m, the absorption coefficient
is estimated to be 25 cm
1
and scattering is negligible. The radial profile of light intensity and
the rate of heat generation at the tissue surface,
m
z
¼
0, and its axial profile along the center
axis of the beam,
r
¼
0, can be found using Eqs. (17.32) and (17.33). Graph
I
(
r
, 0) for
r
ranging from
-2o
o
to
þ
2o
o
and
I
(0,
z
) for
z
¼
0to
z
¼
5/
m
a
, as well as graph
Q
L
(
r
, 0) and
Q
L
(0,
z
). For simplicity,
1 W/cm
2
can be used.
I
o
¼
Solution
Plot of the radial and axial profile of light intensity, I, and volumetric rate of absorption,
Q
L
,in
a purely absorbing tissue.
Continued
