Biomedical Engineering Reference
In-Depth Information
where
~
r
~
q is the change in thermal flux. The equation of state for thermal energy is
U
¼
r
ð
17
:
70
Þ
CT
is an absolute temperature (K), r (kg/m
3
) is the density, and
where
T
C
(joules/kg K) is the
specific heat of the material.
At this point, as in the derivation of the photon phase, a phenomenological relation for
the thermal flux is needed. The commonly used relation is known as Fourier's law, and it
is written as
¼k r
T
~
q
ð
17
:
71
Þ
where
(W/m K) is the thermal conductivity of the material. Proceeding now by sub-
stituting Eqs. (17.71) and (17.70) into Eq. (17.69), the heat conduction equation is
obtained:
k
@ðr
CT
Þ=@
t
¼ rk r
T
þ
Q
L
þ
Q
o
ð
17
:
72
Þ
This equation, solved with application of proper initial and boundary conditions, can, in
theory, predict the temperature distribution in a material under laser irradiation.
In the case of biological tissue with other energy rates involved,
Q
o
can be written as
Q
o
¼
Q
b
þ
Q
m
ð
17
:
73
Þ
where
is the rate of metabolic heat
generation. These terms are not considered here. They are in general much smaller than the
laser heat source term
Q
b
is the heat rate removed by blood perfusion, and
Q
m
Q
L
except when lower laser powers are used for longer times, as in
hyperthermia treatments.
The material properties k and
have been measured for many biological tissues,
including human tissues. In general these properties may depend on temperature and
water content, which could cause a nonlinear effect in Eq. (17.72). Another source of nonlin-
earity may be due to temperature dependence of the optical properties that, when present,
has a more dominant role.
r
C
17.5.2 Thermal Coagulation and Necrosis: The Damage Model
The idea of quantifying the thermal denaturation process was first proposed
in the 1940s. Using a single constituent kinetic rate model, this early work grossly incor-
porates the irreversible biochemical processes of coagulation, denaturation, and necro-
sis associated with thermal injury to biological tissue in terms of a single function.
Defining an arbitrary nondimensional function, O, as an index for the severity of
damage, the model assumes that the rate of change of this function follows an Arrhenius
relation
d
O
dt
¼
A
E
RT
exp
ð
17
:
74
Þ
