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IEEE TRANSACTIONS ON NANOBIOSCIENCE, VOL. 10, NO. 4, DECEMBER 2011
Numerical Study of Temperature Distribution in a
Spherical Tissue in Magnetic Fluid Hyperthermia
Using Lattice Boltzmann Method
Mansour Lahonian and Ali Akbar Golneshan
Abstract—This work applies a three-dimensional lattice Boltzmann method (LBM), to solve the Pennes bio-heat equation
(BHE), in order to predict the temperature distribution in a
spherical tissue, with blood perfusion, metabolism and magnetic nanoparticles (MNPs) heat sources, during magnetic fluid
hyperthermia (MFH). So, heat dissipation of MNPs under an
alternating magnetic field has been studied and effect of different
factors such as induction and frequency of magnetic field and
volume fraction of MNPs has been investigated. Then, effect of
MNPs dispersion on temperature distribution inside tumor and
its surrounding healthy tissue has been shown. Also, effect of
blood perfusion, thermal conductivity of tumor, frequency and
amplitude of magnetic field on temperature distribution has been
explained. Results show that the LBM has a good accuracy to
solve the bio-heat transfer problems.
Index Terms—Bio-heat equation, lattice Boltzmann method,
magnetic field, magnetic fluid hyperthermia, magnetic nanoparticle.
I. INTRODUCTION
M
FH IS A NOVEL method used for cancer therapy. In
MFH, the MNPs are delivered to the tumor. An alternating magnetic field is then applied to the target region, and
then MNPs generate heat as localized heat sources. The heat
generated increases the temperature of the tumor. The cancerous
cells are then destroyed by raising their temperature to about
whereas healthy cells will be safe at this temperature.
Moroz et al. [1] stated that MFH has the maximum potential for
such selective targeting. In order to kill cancer cells without injury to normal tissues, the ability to predict the temperature distribution inside as well as outside the target region as a function
of the exposure time, possesses a high degree of importance.
This helps to provide a level of therapeutic temperature and to
avoid overheating and dammaging of the surrounding healthy
tissue [2]–[4].
Two techniques are currently used to deliver MNPs to a
tumor. The first is to deliver particles to the tumor vasculature
through its supplying artery; however, this method is not effective for poorly perfused tumors. Furthermore, for a tumor with
an irregular shape, inadequate MNPs distribution may cause
under-dosage of heating in the tumor or overheating of the
Manuscript received June 01, 2011; accepted November 09, 2011. Date of
current version January 20, 2012. Asterisk indicates corresponding author.
*M. Lahonian is with the Mechanical Engineering Department, Kurdistan
University, Sanandaj, Iran (e-mail: lahonian@shirazu.ac.ir).
A. A. Golneshan is with the Mechanical Engineering Faculty, Shiraz University, Shiraz, Iran (e-mail: golnshan@shirazu.ac.ir).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TNB.2011.2177100
normal tissue. The second approach, is to directly inject MNPs
into the extracellular space in tumors. MNPs diffuse inside the
tissue after injection of ferrofluid. If the tumor has an irregular
shape, multisite injection can be exploited to cover the entire
target region [5].
Maenosono and Saita [6] investigated the feasibility of using
face centered cubic iron-platinum (fcc FePt) MNPs for magnetic hyperthermia, as fcc FePt present a high Curie temperature, high saturation magnetization, and high chemical stability.
To show the heating capability of these MNPs, Maenosono and
Saita [6] carried out theoretical assessment of fcc FePt MNPs as
heating elements for hyperthermia by combining the heat generation model and the BHE. Lin and Liu [4] developed a hybrid
numerical scheme for solving the transient BHE in spherical coordinates. They estimated the temperature rise in biological tissues for the heating effect of fcc FePt MNPs and claimed that
the results of Maenosono and Saita [6] are incorrect. Zhang et al.
[7] illustrated the temperature distribution within tumor-normal
composite tissue by establishing a multiregion finite difference
algorithm using low Curie temperature MNPs. Zablotskii et al.
[8] studied the heating effect of tunable arrays of nanoparticles in cancer therapy. Liangruksa et al. [9] studied heating in a
tumor with blood perfusion effect due to MFH. They modeled
the problem of a spherical tumor and its surrounding healthy
tissue, that are heated by exciting a homogeneous dispersion of
MNPs infused only into the tumor, under an alternating magnetic field.
Pennes BHE model has been widely used among different
bio-heat models [10], [11]. This model that is valid only for a region far from large blood vessels, shows the effect of blood flow
as a temperature-dependent heat sink term. On complication of
tissues and their complex geometry, exact solutions aren’t available for many cases. For many practical applications, numerical
models, such as finite element method [12]–[15], finite difference method [16], [17], and Monte Carlo method [18], [19], are
widely used nowadays.
The usage of the LBM to analyze problems in science and
engineering has increased significantly in recent years. As a different approach from the conventional computational fluid dynamics (CFD), the LBM has been demonstrated to be successful
in simulation of fluid flow and heat transfer problems and other
types of complex physical systems [20]–[22]. In comparison to
conventional numerical methods, the LBM advantages include a
simple calculation procedure, efficient implementation for parallel computation, and robust handling of complex geometries
with regard to numerical stability and accuracy.
1536-1241/$26.00 © 2011 IEEE
LAHONIAN AND GOLNESHAN: NUMERICAL STUDY OF TEMPERATURE DISTRIBUTION
While the applicability of the LBM in fluid mechanics is well
established, its application to thermal energy transport is also
gaining momentum recently [23]–[28].
Zhang [29] developed the LBM as a potential solver for the
bio-heat problems using Pennes BHE. The results showed that
the LBM can give a precise prediction of the temperature distribution, and it is efficient to deal with the space- and time-dependent heat source, which are often encountered in the treatment
planning of tumor hyperthermia. To our knowledge, Zhang’s
work was the first attempt to use LBM to solve bio-heat problems. Huang et al. [30], introduced a thermal curved boundary
condition for the doubled-population thermal lattice Boltzmann
equation model, for uniform regular lattices.
In this work we use LBM to solve the Pennes BHE with
blood perfusion, metabolism, MNPs heat sources, and Neumann
curved boundary condition (spherical tissue). Accordingly, heat
dissipation of MNPs under an alternating magnetic field is
studied and effect of different parameters such as induction and
frequency of magnetic field and volume fraction of MNPs is
investigated. The temperature rise in biological tissues is predicted for the heating effect of MNPs dispersion. Also, effect
of the blood perfusion, thermal conductivity of the tumor, and
frequency and amplitude of the magnetic field on temperature
distribution is depicted.
263
Fig. 1. Schematic plot of the D3Q15 lattice.
where the distribution functions is a set of populations representing the probability of finding a particle at position at
time moving along the direction identified by the propagation
velocity , the subscript
the direction of the
thermal population (see Fig. 1),
the time step, the dimensionless relaxation time, and
the equilibrium distribution of
the evolution population
(6)
is the weight factor and equal to
, and
propagation velocity is defined as
,
and the
where
II. BIO-HEAT EQUATION
for
for
for
In a generalized form, the Pennes BHE reads
(1)
where , , and are the density, the specific heat, and the
thermal conductivity of the tissue respectively, is the temperature, the time,
, ,
, and
are the perfusion, the
density, the specific heat, and the temperature of the blood,
and
are the metabolic heat generation of the tissue and the
distributed volumetric heat source due to spatial heating.
Equation (1) can be written in a simple form as
(2)
to 7
to 15
(7)
is the lattice velocity, and
where
are the discrete lattice sizes.
The dimensionless relaxation time is defined as
(8)
where
is the thermal diffusivity.
The macroscopic physical quantities can be obtained from the
distribution function. For the temperature and heat flux, they are
[31], [32]
(9)
where
(3)
(10)
and
where
is the dimensionless blood perfusion defined as
(4)
(11)
III. THERMAL LATTICE BOLTZMANN METHOD
The internal energy evolution equation of the three-dimensional fifteen-speed (D3Q15) LBM is as below [20]:
(5)
IV. BOUNDARY CONDITION
In the curved boundary condition (see Fig. 2), to evaluate
internal energy density distribution functions, the postcollision
distribution function obtained by [30]
(12)
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IEEE TRANSACTIONS ON NANOBIOSCIENCE, VOL. 10, NO. 4, DECEMBER 2011
Fig. 2. Curved boundary and lattice nodes (open large circle is computational
boundary node, open small circle is media node, filled circle is the physical
boundary node in the link of media node and computational boundary).
Fig. 3. Schematic plot of tissue and tumor.
where
is the postcollision distribution function,
and
are the equilibrium and nonequilibrium part of
and
.
can be obtained from (6). To determine the
, extrapolation method is used.
is evaluated as
(13)
is the fraction of the intersected link in the media.
where
According to Fig. 2, for calculating the unknown internal energy
density distribution functions such as
, can be obtained
as
(14)
In the Neumann boundary condition, the temperature at the
computational boundary nodes (see Fig. 2) can be calculated as
below:
Fig. 4. 2-D projection of discretized domain and computational boundary
nodes (filled circles) for a spherical boundary of radius 7.5 lattice units.
The related boundary conditions are
(18)
(19)
(15)
(20)
From (6) and (13) equilibrium and nonequlilibrium parts of
are obtained to fulfill the collision step.
(21)
and initial conditions are
V. RESULTS AND DISCUSSION
Two concentric spherical regions were chosen as the domain
of the analysis (Fig. 3). The inner sphere represents the diseased
tissue. MNPs are present in the tumor with
radius that act as
sources of heat generation when an alternating magnetic field is
applied. Fig. 4 shows the 2-D projection of discretized domain
and computational boundary nodes (filled circles) for a spherical
boundary of radius 7.5 lattice units.
The outer sphere represents the healthy tissue. According to
Pennes BHE, (16) and (17) define the transient heat transport in
the diseased and healthy tissues, respectively [4].
(16)
(17)
(22)
(23)
,
, , , , , ,
,
, ,
In the above equations
, ,
, are tumor radius, healthy tissue radius, radius, specific heat capacity, tissue thermal conductivity, blood thermal
conductivity, density, metabolism heat generation, energy dissipation of MNPs in an alternating magnetic field, time, temperature, initial temperature, blood perfusion, and subscripts
are diseased tissue, healthy tissue, and blood respectively.
In this study the properties of the tissue are taken as
,
,
,
,
where
is volume fraction,
,
for fcc
FePt MNPs,
,
for magnetite MNPs,
,
,
LAHONIAN AND GOLNESHAN: NUMERICAL STUDY OF TEMPERATURE DISTRIBUTION
265
Power dissipation of MNPs in an alternating magnetic field
is expressed as [35], [36]
(24)
where
is the permeability of free space,
is the equilibrium susceptibility,
and are the amplitude and
frequency of the alternating magnetic field, and is the effective
relaxation time, given by
(25)
Fig. 5. History of temperature at the center of the tumor for the cases of 9-nm
and 19-nm magnetite
fcc FePt
MNPs.
and
are the Néel relaxation and the Brownian rewhere
laxation time, respectively.
and
are written as (26)
(26)
(27)
Fig. 6. Temperature distributions in the tissue at
for the cases of
and 19-nm magnetite
9-nm fcc FePt
MNPs.
,
,
[4], [6].
,
,
,
For the LBM simulation, we use
lattices and
the lattice size is
. Therefore
.
In a magnetic field with fixed amplitude and frequency at
50 mT and 300 kHz and a volume fraction
,
9-nm fcc FePt and 19-nm magnetite MNPs can dissipate
, respectively [4], [6].
Fig. 5 shows the history of temperature at the center of the
tumor for the cases of 9-nm fcc FePt
and 19-nm magnetite
MNPs. In accordance with the results shown in Fig. 5, the temperature distribution has become steady at
. Also, Fig. 6 shows
the temperature distributions in the tissue at
for the
cases of 9-nm fcc FePt
and 19-nm
magnetite
MNPs. It is observed that
the results of the present work, match well with those given in
the literature [4].
The temperature that can be achieved in the tissue strongly
depends on the power dissipation of the MNPs, thermophysical
properties of the blood and the tissue, the blood perfusion in the
tissue and the duration of application of the magnetic field. Also,
the power dissipation of the MNPs depends on the properties of
the magnetic material used, the frequency and the strength of the
applied magnetic field. Therefore, carefully chosen parameters
and the means to monitor the real-time temperature are required
[3], [5], [33], [34].
where the shorter time constant tends to dominate in determining the effective relaxation time for any given size of
particle.
is the average relaxation time in response to a
thermal fluctuation, is the viscosity of medium,
is the
hydrodynamic volume of MNP,
is the Boltzmann constant,
, and
is the temperature. Here,
where is the magnetocrystalline anisotropy
constantn and
is the volume of MNPs. The MNPs volume
and the hydrodynamic volume including the ligand layer
are written as
(28)
(29)
where is the diameter of MNP and is the ligand layer thickness.
The equilibrium susceptibility
is assumed to be the chord
susceptibility corresponding to the Langevin equation
, and expressed as
(30)
where
,
,
,
and is the volume fraction of MNPs. Here,
and
are the
domain and saturation magnetization, respectively. The initial
susceptibility is given by
(31)
Based on the above-mentioned theory, we calculated the
power dissipation for aqueous dispersion of mono-dispersed
equiatomic fcc FePt MNPs varying the diameter of MNP in
adiabatic condition. In Table I, physical properties of fcc FePt
and magnetite
are shown.
Fig. 7 shows the dependence of power dissipation on induction of applied magnetic field, for fixed
. Note
that
is varied as 30, 50, and 80 mT. Increasing
earns a
raise for power dissipation. Also, Fig. 8 shows the dependence
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IEEE TRANSACTIONS ON NANOBIOSCIENCE, VOL. 10, NO. 4, DECEMBER 2011
TABLE I
PHYSICAL PROPERTIES OF FCC FEPT AND MAGNETITE MNPS [6]
Fig. 10. Steady state temperature distribution for 19-nm magnetite MNPs in
an alternating magnetic field.
Fig. 7. Dependence of power dissipation on
.
the human tissue [1]. The super-paramagnetic particles (10–40
nm) are recommended in clinical application as they are able
to generate substantial heat within low magnetic field strength
and frequency [37].
The typical magnetite dosage is
magnetite MNPs
per gram of tumor that has been reported in clinical studies [38],
[39]. Therefore, homogenous volume fraction is
(32)
Here, we consider three cases for the distribution of a constant
amount of magnetite MNPs inside the tumor. For each case, the
volume fraction
as a function of distance from the center
of the tumor can be modeled as follows:
(33)
Fig. 8. Dependence of power dissipation on .
(34)
(35)
Fig. 9. Dependence of power dissipation on .
of power dissipation on when
is fixed in 50 mT. Note that
is varied as 100, 200, and 300 kHz. Increasing earns a raise
and a gradual decrease, respectively, in the power dissipation
and
.
Fig. 9 shows the dependence of power dissipation on the
volume fraction,
. The power dissipation increases with increasing . No change in
is observed.
Iron oxides magnetite
nanoparticles are the most
studied to date due to their biocompatibility, when injected in
Now the effect of diffusion of MNPs on temperature distribution inside the tissue will be studied after applying an alternating
magnetic field. Generally, the practical range of frequency and
amplitudes are often described as 50–1200 kHz and 0–15 kA/m
[40].
Fig. 10 shows the temperature distribution for 19-nm magnetite MNPs in an alternating magnetic field with amplitude of
10 kA/m and frequency of 100 kHz for constant , linear and
parabolic distributions of MNPs inside the tumor. As shown in
Fig. 10, the temperature in the healthy tissue are the same for
three cases but the temperature in the tumor is the highest for
linear distribution and the lowest for homogenous distribution.
As a result, 19-nm magnetite MNPs are able to raise the temperature of the tumor above
in the magnetic field of amplitude of 10 kA/m and frequency 100 kHz with the dose of 10 mg
magnetite MNPs per gram of tumor.
Fig. 11 shows the effect of different factors such as the blood
perfusion, thermal conductivity of the tumor, frequency and amplitude of the magnetic field on temperature distribution inside
LAHONIAN AND GOLNESHAN: NUMERICAL STUDY OF TEMPERATURE DISTRIBUTION
267
Fig. 11. Temperature distribution inside the tumor and its surrounding healthy tissue under different conditions.
the tumor and its surrounding healthy tissue, for linear distribution of MNPs case. Results show increasing the blood perfusion decreases the temperature inside the tumor and its surrounding healthy tissue [Fig. 11(a)]. Also, increasing thermal
conductivity of the tumor decreases the temperature only in the
tumor area [Fig. 11(b)]. Furthermore, increasing amplitude and
frequency of the magnetic field increases the temperature of
both the tumor and the healthy tissue [Figs. 11(c) and 11(d)].
that increasing of the blood perfusion decreases the temperature inside the tumor and its surrounding healthy tissue. Also,
increasing thermal conductivity of the tumor only decreases the
temperature inside the tumor. In addition, increasing of amplitude and frequency of the magnetic field increases the temperature all over the tissue.
Results show that the LBM has good accuracy to solve the
bio-heat transfer problems.
REFERENCES
VI. CONCLUSION
We have applied a three-dimensional LBM for solving the
Pennes BHE to predict the temperature distribution in spherical
tissue in MFH treatment. The effects of the blood perfusion,
the metabolism and the magnetic nanoparticles (MNPs) heat
sources were introduced. To validation, the results were compared with those reported in the literature [4]. To show the effect
of MNPs dispersion on temperature distribution, firstly, the effect of different parameters such as induction and frequency of
the magnetic field and the MNPs volume fraction was studied.
Then, effect of MNPs dispersion for homogenous, parabolic and
linear distribution was investigated. As a result, 19-nm magnetite MNPs are able to rise the temperature of the tumor above
in the magnetic field of amplitude of 10 kA/m and frequency 100 kHz with the dose of 10 mg magnetite MNPs per
gram of tumor. Also, effect of the blood perfusion, thermal conductivity of the tumor, frequency and amplitude of the magnetic
field on temperature distribution was explained. Results show
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Authors’ photographs and biographies not available at the time of publication.