Biomedical Engineering Reference
In-Depth Information
the relaxation time constant (s), α is an empirical distribution
parameter that broadens the dispersion in each frequency range,
σ
i
is the static ionic conductivity, and ε
0
is the permittivity of
free space. Relaxation, permittivity, and distribution parameters
have been described in the literature for most common biologi-
cal tissues (Gabriel 1996b). The dielectric properties of tissue are
generally thought to be relatively isotropic, but this assumption
may not be valid in some situations in tissue with anisotropic
structure such as muscle (Epstein 1983).
The temperature dependencies of relative permittivity and
conductivity are not yet as fully characterized as frequency
dependence. Most of the reported data are clustered around
the ambient room temperature (~20°C) or normal body tem-
perature (~37°C). In the range of 10-60°C, temperature depen-
dence for both relative permittivity and conductivity have been
assumed to take a linear form (Duck 1990, Lazebnick 2006, Pop
2003, Bircan 2002, Chin 2001, Stauffer 2003). However, near
temperatures of water phase change (0°C or 100°C), substan-
tial deviations from the linear model have been reported (Brace
2008). At temperatures in excess of approximately 60°C, protein
and cellular structure changes can also affect tissue permittiv-
ity. Such effects have not been well characterized in the avail-
able literature on bulk tissue, but some data suggest that protein
denaturation may be detectable by a sudden slight change in the
temperature curve of relative permittivity or conductivity in
some tissues (Bircan 2002, Wall 1999).
Penetration of a microwave field into a tissue medium is
also dependent on the dielectric properties of the tissue. For a
plane wave in a homogenous isotropic medium, the penetration
depth,
δ (m), of an electromagnetic field is defined as the dis-
tance required for the electric field to attenuate to 1/
e
(~37%) of
its initial value:
One final point of consideration: much of the energy radiated
by microwave antennas in lossy media such as biological tissues
is absorbed in the near or Fresnel zones of the antenna and may
not propagate as a plane wave. In this case, the aforementioned
calculations of attenuation and penetration depth should be
viewed as approximations. Electromagnetic simulation and, in
particular, simulation of electromagnetic-thermal interactions
can provide more accurate estimates of temporal heating for
comparing devices and frequencies for microwave ablation.
9.4.2 Electromagnetic Interactions with tissue
When electromagnetic waves propagate through a lossy medium,
some of the energy is converted into heat. In biological tissues,
dielectric hysteresis losses dominate, so heat is primarily gener-
ated by the rotation of polar water molecules. The rapid oscillation
of these molecules effectively increases their kinetic energy and,
hence, temperature. Heat generation is proportional to the energy
applied (i.e., microwave power) and effective conductivity of the
tissue medium:
σ
2
2
Wm
−
3
(9.15)
Q= E
EM
|| (
),
where the square of the electric field intensity, |
E
|
2
, is linearly
proportional to the applied power. Comparing Equation 9.15 to
Equation 9.14 reveals that while penetration depth is inversely
proportional to the effective conductivity, the heat generation rate
is directly proportional to effective conductivity. This makes intu-
itive sense from a conservation of energy perspective: the primary
cause of field attenuation is the conversion of microwave energy
to heat. Note also that since the effective conductivity of the tis-
sue is related to frequency, temperature, water content, and other
factors, the heat generation rate is also dependent on these same
factors. Once heat is generated inside the tissue, heat transfer can
be modeled using the so-called bioheat equation (Pennes 1948):
1
δ=
( .
m
(9.13)
1/2
2
1
2
σ
ωε
−
ωµε
1
+
1
∂
∂
T
t
ρ
C
=kT+QQ
∇⋅ ∇
(
)
−
(
Wm
3
−
)
(9.16)
p
T
EM
perfusion
In the case of most tissues and microwave ablation frequen-
cies, the penetration depth from Equation 9.13 can be approxi-
mated by assuming the tissue is a good dielectric ([σ/ωε]2 << 1):
where ρ is density (kg m
−3
),
C
p
is the specific heat capacity at con-
stant pressure (J kg
−1
m
−3
),
T
is temperature (K),
t
is time (s),
k
T
is thermal conductivity (W/m K),
Q
EM
is the heat generation rate
illustrated in Equation 9.15, and
Q
p
is the rate of heat lost to blood
perfusion (W m
−3
). The term on the left-hand side of Equation
9.16 is the heating rate for a given volume of tissue, while the first
term on the right-hand side describes thermal conduction inside
the tissue. Heat lost to blood perfusion can be represented as a
convective process using the following equation:
2
ε
µ
δ=
σ
( .
m
(9.14)
It is important to note from Equation 9.14 that penetration depth
is inversely related to conductivity. Deeper plane-wave penetra-
tion occurs with lower frequencies, which may be more desirable
for non-focal heating applications such as regional hyperther-
mia (Fotopoulou 2010, van Rhoon 1998, Turner 1989). Higher
frequencies with less energy penetration may be desirable in
more superficial applications such as ablation of the endome-
trium (Feldberg 1998, Hodgson 1999).
Wm
3
−
Q=wCTT
perf
ρ
(
−
)
(
).
(9.17)
bl
bl
p,bl
0
This heat loss term encompasses microvascular blood perfu-
sion and blood flow through large vessels, which can both be
approximated as a convective heat transfer process (Pennes 1948).
