INTRODUCTION
Hyperthermia uses the physical methods to heat certain organ or tissue to the
temperatures in the range of 4044Â°C with treatment times over 30 min. In
general, cancer cells have a higher chance of dying when the temperature is
above 42.5Â°C and the rate of death drastically increases with increasing
temperature (Moroz et al., 2002). An ideal hyperthermia
treatment should selectively destroy the tumor cells without damaging the surrounding
healthy tissue. Magnetic fluid hyperthermia is one of hyperthermia modalities
for tumor treatment. It is absolutely a necessity to understand the temperature
rise behavior occurring in biological tissues during treatment. Especially,
the temperature distribution inside as well as outside the target region must
be known as function of the exposure time in order to provide a level of therapeutic
temperature and on the other hand, to avoid overheating and damaging of the
surrounding healthy tissue.
AndrAAO et al. (1999) , Bagaria
and Johnson (2005) and Maenosono and Saita (2006) studying
magnetic fluid hyperthermia used the Pennes` equation to describe the behavior
of heat transfer in biological tissues. AndrAƒAO et
al. (1999) modeled small breast carcinomas surrounded by extended health
tissue as a solid sphere with constant heat generation and gave an elementary
solution of the original heat conduction problem without the effects of blood
perfusion and metabolism. Bagaria and Johnson (2005)
considered the tissue model as two finite concentric spherical regions with
the blood perfusion effect. Analytical and numerical solutions to the model
with the mixed boundary conditions were calculated by separation of variables
and an explicit finite differencing technique, respectively. Maenosono
and Saita (2006) carried out theoretical assessment of FePt magnetic nanoparticles
as heating elements for hyperthermia using the heat generation model and the
bioheat transfer equation. To estimate the temperature rise behavior in vivo,
the Pennes bioheat equation with the Neuman boundary conditions in spherical
coordinates was solved with the same thermal properties in the diseased and
healthy tissues. Durkee et al. (1990) also gave
the exact solutions to the Pennes bioheat equation in onedimensional multilayer
spherical geometry. Tsuda et al. (1996) developed
an inverse method to optimize the heating conditions during a hyperthermia treatment.
The present study models the tissues within magnetic fluid hyperthermia as
an infinite concentric spherical domain. Several methods for solving the heat
conduction problems in multilayer spherical structures have also been developed
by Virseda and Pinazo (1998), Chen
et al. (2003) and Shirmohammadi (2008). This
study would develop a hybrid numerical scheme based on the Laplace transform,
change of variables and the modified discretization technique in conjunction
with the hyperbolic shape functions for solving the present problem. In reality,
there exists the difference in metabolic heat generation rate, blood perfusion
rate and other physiological parameters between tumor and normal tissues (Hu
et al., 2004; Gonzalez, 2007). For complete
analysis, this study explores the effects of those differences on the thermal
response.
MATHEMATICAL FORMULATION
In magnetic fluid hyperthermia, fine magnetic particles are localized at the
tumor tissue. An alternating magnetic field is then applied to the target region,
which heats the magnetic particles by magnetic hysteresis losses. These particles
might act as localized heat sources. A small tumor surrounded by the normal
tissue was modeled as a sphere of the radius R (AndrAƒAO
et al., 1999; Bagaria and Johnson, 2005;
Maenosono and Saita, 2006). As magnetic particles are injected into and
homogenously distributed in the tumor, a spherical heat source of constant power
density P is excited by an alternating magnetic field. Afterward, heat symmetrically
transfers in the radius direction. The temperature distribution in the tumor
and normal tissues is the function of distance r from the center of the sphere
and time t. This system can be mathematically described in the temperature domain
as (AndrAƒAO et al., 1999):
with the boundary conditions:
and the initial conditions:
Here, Ï?, c, k and T denote density, specific heat, thermal conductivity
and temperature in two regions. Ï?_{b}, c_{b} and
w_{b} are, respectively, the density, specific heat and perfusion
rate of blood. q_{m} is the metabolic heat generation. T_{b}
is the arterial temperature and was specified as 37Â°C. The initial
temperature T_{0} is regarded as the arterial temperature.
Numerical scheme: The present problem is transformed into a problem
in the rectangular coordinate system with change of variable. A new dependent
variable Î¸ is defined as:
Therefore, Eq. 1 and 2 can be written
in terms of Î¸, respectively, as:
The boundary condition at r = 0 can be stated from the definition of
Î¸ (r,t) shown in Eq. 8 as:
The other boundary conditions become to:
The initial conditions are also rewritten as:
And then, using the Laplace transform technique simplifies the transient
problem into the steady one. The differential Eq. 9
and 10 and the boundary conditions Eq.
1114 of the present problem are rewritten as:
and
Where:
and s is the Laplace transform parameter for time t.
It is found from Eq. 16 that the governing equations
of the present problem have become ordinary differential equations. Their
analytical solutions in the subspace domain k, [r_{i}, r_{i+1}],
with the boundary conditions:
are easily obtained and can be written as:
Similarly, Eq. 24 in the subspace domain k1, [r_{i1},
r_{i}], can be written as:
where, l denotes the length of subspace domain or the distance between
two neighboring nodes. The value of l can be different in the different
layer. The subscript i is the number of node. N is the total number of node.
A modified discretization technique based on Eq. 24 and 25
is developed for the governing algebraic equations in the present study. The
earlier studies have used the similar approximation functions to discretize
the hyperbolic and duallagphase diffusion equations (Liu,
2007a, b, 2008).
For continuities of temperature and heat flux within the whole space
domain, the following conditions can be required:
Substituting Eq. 24, 25 and 26
into Eq. 27 and then evaluating the resulting derivative
can lead to the discretized form for the interior nodes in layer j as
following:
The discretized form for the node at the interface of the tumor and normal
tissues, r = R, can be obtained from the boundary condition Eq.
19 and is written as:
Equation 28 and 29 in conjunction
with the discretized forms of the boundary conditions can be rearranged
as the following matrix equation:
where, [B] is a matrix with complex numbers,
is a column vector in the Laplace transform domain and {F} is a column vector
representing the forcing term. Thereafter, the value of H in the physical domain
can be determined with the application of the Gaussian elimination algorithm
and the numerical inversion of the Laplce transform (Honig
and Hirdes, 1984). At the same time, the temperature difference, TT_{0},
is equal to Î¸/r. However, the value of Î¸/r at r = 0 is indeterminate and
must be replaced by its limit as râ ’0. Thus, the value of the transient
temperature at the center, T (0, t), can be evaluated by using L`HÃ´spital`s
rule.
RESULTS AND DISCUSSION
AndrAƒAO et al. (1999) have numerically
simulated the thermal behavior of localized magnetic hyperthermia applied to
a breast carcinoma and estimated the minimum amount of iron oxide for getting
predetermined temperatures. They considered a small spherical tumor of radius
R = 0.00315 m with a constant power density of 6.15x10^{6} Wm^{3}
embedded in extended muscle tissue. The thermal parameters were taken to be
k_{1} = 0.778, W K^{1} m, Ï?_{1} = 1660 kg m^{3},
c_{1} = 2540 J kg^{1} K, k_{2} = 0.642, W K^{1}
m, Ï?_{2} = 1000 kg m^{3} and c_{2} = 3720 J kg^{1}
K. For low blood perfusion in breast, AndrAƒAO et
al. (1999) calculated the temperature distribution with neglecting the
effects of blood perfusion and metabolism. However, in accordance with the contents
of the literatures (Hu et al., 2004; Gonzalez,
2007), there exists an obvious difference in metabolic heat generation rate
and blood perfusion rate between tumor and normal tissue. This difference may
significantly affect the temperature rise during a hyperthermia treatment. Thus,
it is explored in the present study. All the computations are performed with
the uniform space size l = R/100.
To evidence the accuracy of the present results, the comparison of the present
results with those given by AndrAƒAO et al. (1999)
is made, as shown in Fig. 1 and 2. It is
found from these two figures that the present results agree well with the results
presented in the literature (AndrAƒAO et al.,
1999). This phenomenon demonstrates the efficiency of the present numerical
scheme for solving such a problem. Due to continuous heating, temperature at
different reduced distance r/R from the center increases with time, as plotted
in Fig. 1. On the other hand, heat energy continuously diffuse
away in the infinite domain, so the increasing rate of temperature at a specified
location decays with time increasing. Figure 2 presents the
distributions of temperature increase TT_{0} for various times. It
is observed that the temperatures in the domain occupied by the heat source
get higher and the affected domain increases with time. For an ideal hyperthermia
treatment, that should selectively destroy the tumor cells without damaging
the surrounding healthy tissue, it is well known the control of exposure time
is a necessity.

Fig. 1: 
History of temperature at different reduced distance
r/R from the center for the problem in infinite domain 
The metabolic heat generation rates and the blood perfusion rates of tumor
and normal tissue in female breast were estimated. Accordingly, Hu et al.
(2004) took the metabolic heat generation rates of tumor and normal tissue to
be q_{m1} = 29,000 W m^{3} and q_{m2} = 450 W m^{3}
and the corresponding perfusion rates were w_{b1} = 0.009 m^{3}/s/m^{3}
and w_{b2} = 0.0018 m^{3}/s/m^{3} . The product of density
and specific heat capacity of blood, Ï?_{b}c_{b} = 4.18x106
J/m^{3}/K (Maenosono and Saita, 2006) and the
above data are used to investigate the effects of the metabolic heat generation
rate and the blood perfusion rate on the temperature rise in the present study.
Figure 3 displays the transient temperature rises with and
without the effects of the metabolic heat generation rate and the blood perfusion
rate at the locations x = 0 and x = R. It can be observed in this case that
the effects of the metabolic heat generation rate and the blood perfusion rate
make the temperature rise up. However, the blood perfusion plays the cooling
role for that the temperature of blood is specified as a constant value, 37Â°C.
In other words, it is the metabolic heat generation to cause the temperature
increase in tissues. Judging from this, only using constant power density is
not easy to have a constant predetermined tumor temperature during a hyperthermia
treatment.

Fig. 2: 
Temperature distributions in the infinite domain for
various times 
Figure 4 shows the temperature distributions with metabolism
and with blood perfusion and metabolism for various times. The cooling
function of blood perfusion is obviously displayed in Fig.
4. Before t = 6 sec the difference between the tumor temperature and
the blood temperature is not large enough, so the amount of heat transfer
through the blood perfusion is few and the cooling function does not develop
yet. With time passing, the accumulation of heat in tissues increases
and the tissue temperatures rise up. As the difference between the tissue
and blood temperatures is enlarged, the heat loss through the blood perfusion
increases. Then the rate of temperature rise tends to be gradual. Probably,
the thermal balance will appear for the cooling effect of blood perfusion
and the temperature distribution will be steady in the tissues.

Fig. 3: 
Variation of temperature rise with and without the effects
of the metabolic heat generation rate and the blood perfusion rate
at the locations x = 0 and x = R 

Fig. 4: 
Effects of blood perfusion on the temperature distributions
at various times 
CONCLUSION
A hybrid numerical scheme based on the Laplace transform, change of variables
and the modified discretization technique in conjunction with the hyperbolic
shape functions is developed to solve the Pennes bioheat equation in onedimensional
multilayer spherical geometry and explores the effects of the difference in
metabolic heat generation rate, blood perfusion rate and other physiological
parameters between tumor and normal tissues on the thermal response. The accuracy
of the numerical scheme is evidenced by comparing with the analytical solution
and the results in the literature (AndrAƒAO et al.,
1999). The metabolic heat generation rate and the blood perfusion rate practically
affect the temperature rise behavior in vivo during hyperthermia treatment.
The temperature distribution in tissues can reach the equilibrium state for
the cooling effect of blood perfusion.