rivera2010

32nd Annual International Conference of the IEEE EMBS
Buenos Aires, Argentina, August 31 - September 4, 2010
Analytical Validation of COMSOL Multiphysics for Theoretical
Models of Radiofrequency Ablation Including the Hyperbolic
Bioheat Transfer Equation
Maria J. Rivera, Juan A. López Molina, Macarena Trujillo, Vicente Romero-García, and
Enrique J. Berjano
Abstract—In this paper we outline our main findings about
the differences between the use of the Bioheat Equation and the
Hyperbolic Bioheat Equation in theoretical models for
Radiofrequency (RF) ablation. At the moment, we have been
working on the analytical approach to solve both equations, but
more recently, we have considered numerical models based on
the Finite Element Method (FEM). As a first step to use FEM,
we conducted a comparative study between the temperature
profiles obtained from the analytical solutions and those
obtained from FEM. Regarding the differences between both
methods, we obtain agreement in less than 5% of relative
differences. Then FEM is a good alternative to model heating of
biological tissues using BE and HBE in, for example, more
complex and realistic geometries.
I. INTRODUCTION
R
adiofrequency (RF) heating of biological tissues is
currently employed in many surgical and therapeutic
procedures such as the elimination of cardiac
arrhythmias, the destruction of tumors, the treatment of
gastroesophageal reflux disease, and the heating of the
cornea for refractive surgery. In order to investigate and
develop new RF ablation techniques, besides understanding
the complex electrical and thermal phenomena involved in
the heating process, numerous theoretical models have been
employed [1]. To date, all these models have employed the
Bioheat Equation (BE) proposed by Pennes [2], in which the
heat conduction term is based on Fourier’s theory (i.e. they
have employed a parabolic heat transfer equation).
r
Therefore, it related to heat flux ( q ) in the following way:
r r
r r
q (r , t ) = −k∇T (r , t )
(1)
r
where k is the thermal conductivity (W/m⋅K) and T (r , t ) the
r
temperature at point r at time t. This approach assumes an
infinite thermal energy propagation speed, and although it
Manuscript received March 31, 2010. This work was supported in part by
the Spanish Government by means of the following Grants: TEC200801369/TEC, MTM2007-64222, MTM2010-14909 and MAT2009-09438,
and by Valencia Government by means of the Grant ACOMP/2010/008.
M. J. Rivera, J. A. López Molina and M. Trujillo are with the
Departamento de Matemática Aplicada, Instituto de Matemática Pura y
Aplicada, Universidad Politécnica de Valencia, Valencia, Spain (e-mails:
mjrivera@mat.upv.es, jalopez@mat.upv.es , matrugui@mat.upv.es).
V. García-Romero is with the Centro de Tecnologías Físicas: Acústica,
Universidad Politécnica de Valencia, Valencia, Spain (e-mail:
virogar1@upvnet.upv.es).
E. J. Berjano is with the Electronic Engineering Department,
Universidad Politécnica de Valencia, Valencia, Spain (e-mail:
eberjano@eln.upv.es).
978-1-4244-4124-2/10/$25.00 ©2010 IEEE
might be suitable for most RF ablation procedures, it has
been suggested that under certain conditions (such as very
short heating times), a non-Fourier model should be
considered by means of the Hyperbolic Bioheat Equation
(HBE), i.e. considering a nonzero thermal relaxation time (τ)
in the tissue [3]. It is known that heat is always found to
propagate at a finite speed [4], and in fact Cattaneo [5] and
Vernotte [6] simultaneously suggested a modified heat flux
model in the form:
r r
r r
(2)
q (r , t + τ ) = −k∇T (r , t )
where τ is the thermal relaxation time of the biological
tissue. Equation (2) assumes that the effect (heat flux) and
the cause (temperature gradient) occur at different times and
that the delay between heat flux and temperature gradient is
τ. The particular case of considering τ = 0 obviously
corresponds to the BE.
In order to study how the temperature profiles could be
altered when HBE is considered in place of BE, we have
conducted different theoretical studies based on onedimensional analytical models [7-9]. In these models, we
solved both BE and HBE under different circumstances.
Obviously, since the analytical approach does not allow
easily to consider complex geometries or to solve non-linear
equations, recently we are using a complementary approach
based on numerical techniques, specifically the Finite
Element Method (FEM). However the use of non standard
equations like the HBE and non standard functions as the
Heaviside function and the Dirac’s Delta function can add
new problems in the use of numerical methods. The aim of
this paper is the validation of COMSOL Multiphysics in the
solution of the HBE in models for RF ablation by the
analytical solution. For this reason the geometry we have to
consider must be simple enough so that the exact solution
can be attained.
II. ANALYTICAL APPROACH
Briefly, we considered a r0 radius spherical electrode
completely imbedded in the biological tissue (see Fig. 1),
which had an infinite dimension. Although some RF ablation
applications use wet and cooled electrodes, in our model we
considered the simplest case, i.e. a dry electrode.
This model presented radial symmetry and a onedimensional approach was possible. Regarding the electrical
problem, we always modeled a constant-power protocol, i.e.
the source term Qs(r,t) in W/m3 for the BE and HBE (i.e. the
Joule heat produced per unit volume of tissue) was always:
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P ⋅ r0
(3)
H (t )
4 ⋅π ⋅ r4
where P is the total applied power (W), r0 the electrode
radius (m), and H(t) is the Heaviside function. Although this
temporal function have not been included in the previous
study by Erez and Shitzer [10], later it was crucial to study
the pulsed protocol in RF ablation for the case HBE [11].
Qs (r , t ) =
Additionally, we consider the source term related with the
blood perfusion which can be expressed as:
Q p ( r , t ) = − ρ b cbωb (T ( r , t ) − T0 )
(4)
where ρb is the blood density (kg/m3), cb (J/kg⋅K) the blood
specific heat, ωb is the perfusion blood flux (s-1), and the T0
is the blood temperature (ºC). Both terms (Qp and Qs) are
source terms and therefore the total source term is:
∂ 2T ( r , t ) 2 ∂T (r , t )
∂T (r , t )
+
) +ζ
∂r 2
r ∂r
∂t
(9)
∂T (r , t ) Pα r0
+τ
=
( H (t ) + τ δ (t )) − B(T ( r , t ) − T0 )
∂t
4π k r 4
where δ(t) is Dirac’s function, B = αρ bcbωb , and ζ = 1 + τ B .
k
To set the boundary condition in r = r0, we adopted a
simplification assuming the thermal conductivity of the
electrode to be much larger than that of the tissue (i.e.
assuming that the boundary condition at the interface
between electrode and tissue is mainly governed by the
thermal inertia of the electrode). This obviously modeled a
dry electrode. Other thermal boundary conditions should be
considered for the case of internally cooled electrodes [13].
−α (
III. NUMERICAL APPROACH
Q ( r , t ) = Qs ( r , t ) + Q p ( r , t )
(5)
Fig. 1. Schematic diagram of the model geometry. A spherical
electrode (white circle) of radius r0 is completely imbedded and
in close contact with the biological tissue, which has an infinite
dimension. As a result, the model presented a radial symmetry,
and a one-dimensional approach is possible (dimensional variable
is r).
The HBE was obtained by combining the energy equation:
− ∇q ( r , t ) + Q ( r , t ) = ρ c
∂T ( r , t )
∂t
(6)
where ρ is the density (kg/m3) and c (J/kg⋅K) the specific
heat, with the heat transfer model derived from Equation (2)
and proposed by Özişik and Tzou [12]:
q(r , t ) + τ
∂q(r , t )
= −k∇T (r , t )
∂t
(7)
The result was:
1 ∂T (r , t )
∂ 2T ( r , t )
(
+τ
)=
α
∂t
∂ 2t
(8)
1
∂Q (r , t )
(Q(r , t ) + τ
)
k
∂t
where α is the thermal diffusivity (m2/s). Finally, we
− ∆T (r , t ) +
combined (5) and (8) to obtain the HBE:
The majority of heat transfer problems of real situations
involve complex geometries, are non-linear problems or their
initial and boundary conditions lead us to use numerical
methods to solve them. This is absolutely true in RF ablation.
Some widespread numerical methods to solve this kind of
problems are the Finite Element Method (FEM) and the
Finite Differences Method. There is abundant available
software for building models, solving them by the mentioned
methods and post-processing the results. We have chosen
COMSOL Multiphysics (Burlington, MA, USA), which has
been broadly employed in the study of the RF ablation of
biological tissues. However, all of those previous studies
considered the BE [14-16]. In this respect, our recent
objective has focused on the validation of COMSOL
Multiphysics for using the HBE in obtaining the temperature
distribution during RF ablation.
This issue is especially important by taking into account
the cuspidal-type singularities found in the analytical
solutions of the HBE, which are materialized as a
temperature peak traveling through the medium at a finite
speed [7]. In other words, it is necessary to know if this
behavior will be accurately modeled by numerical methods
in general, and by COMSOL in particular.
For this reason, we build with COMSOL the same onedimensional model previously solved by analytical methods,
and then we obtained the numerical solution. Our idea was to
validate COMSOL by comparing the numerical and
analytical solution.
We used COMSOL Multiphysics software version 3.2b,
which can virtually model and solve any physical
phenomenon, which can be described with Partial
Differential Equations (PDE) using the FEM. COMSOL
presents several models to solve a wide range of PDEs. We
have chosen a one-dimensional problem.
We used the automatic mesh generated by COMSOL, and
for this reason we conducted a sensibility analysis to check
that a more refinished meshes do not produce results closer
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to the analytical ones. The control parameter used to conduct
this sensibility analysis was the temperature reached at the
interface electrode-tissue (r=r0), r=2r0, and r=3r0, after 60 s.
These three different points were checked due to the
different thermal behavior at each location, especially in the
case of HBE.
In order to compare analytical and numerical solution we
plotted the progress of temperature from each solution. In the
case of the analytical solution we used the software
Mathematica 7.0 software (Wolfram Research, Champaign,
IL, USA). To make graphics of the numerical solution we
used the post-processing option of COMSOL.
In order to plot the results we particularized the solutions
for a specific case. We chose an electrode radius of r0=1.5
mm. As biological tissue we chose the liver with the
following characteristics: density ρ of 1060 kg/m3, specific
heat c of 3600 J/kg⋅K and thermal conductivity k of 0.502
W/m⋅K.
The electrode characteristics were the density ρ of 21500
kg/m3 and the specific heat c of 132 J/kg⋅K. This
corresponds with a platinum-iridium electrode such as used
in RF ablation. The initial temperature of tissue was 37ºC.
The applied power was of P = 1 W. This values was chosen
in order to obtain tissue temperatures capable of producing
thermal lesion (>50ºC) and below 80ºC (where non linear
phenomena occur, such as vaporization and desiccation).l
Moreover, we included the term of blood perfusion both in
BE and HBE. Finally, these numerical solutions were
compared to those obtained analytically in order to validate
the FEM tool.
IV. RESULTS AND DISCUSSION
Regarding the analytical solutions, we found, from a
mathematical point of view, that the HBE solution shows
cuspidal-type singularities in the form of a temperature peak
traveling through the medium at finite speed (see Fig. 2).
This peak arises at the electrode surface, and clearly reflects
the wave nature of the thermal problem. In [11] we tried to
provide an explanation about this behavior which is based on
the interaction of forward and reverse thermal waves.
At the beginning of heating (i.e. when the considered time
was comparable to or shorter than the thermal relaxation
time), HBE provided temperature values lower than those
provided by BE. In general, and for points far from the
electrode surface, the speed of temperature change at the
beginning of the heating in the case of HBE was slightly
slower than BE. This can be explained due to the fact that
when using HBE a period of time is needed for heat to travel
to a particular location inside the tissue.
When these conclusions were particularized for specific
tissues, once more the differences between BE and HBE
temperature profiles were greater for lower times and shorter
distances. For this reason, our results suggested that the HBE
should be considered in the case of RF heating of the cornea
(heating time 0.6 s), and for short time ablation in cardiac
tissue (less than 30 s) [8]. Although in this study we used the
characteristics of the liver, these values are very similar to
those found in the cornea [8], and for this reason, the
temperatures obtained will be almost equal.
Regarding the numerical results, Figure 3 shows the
temperature progress for HBE and for two values of thermal
relaxation time (1 and 16 s) and for two blood perfusion
conditions (without perfusion ω=0, and perfusion ω=0.01
1/s).
To obtain the numerical solution, we have meshed both
the numerical and the temporal domains. On one hand,
spatial domain has been meshed in 16 nodes using Lagrangequadratic elements, whereas the temporal domain has been
analyzed form 0 s to 160 s in steps of 0.1 s. The mesh in both
cases is equi-distributed, that means, every node present the
same size. The solution has been obtained by the
unsymmetric multifrontal sparse LU factorization package
(UMFPACK), which is a set of routines for solving
unsymmetric sparse linear systems using the Unsymmetric
MultiFrontal method.
The results obtained from analytical approach, both using
BE and HBE, and from COMSOL were almost coincident.
We have analyzed the relative differences between the
numerical and analytical temperature profiles, using the next
expression:
ε (%) =
TN − TA
× 100
TA
(10)
Where, TN is the temperature profile obtained by FEM and
TA is the analytical temperature profile. For all the cases
analyzed in this work, the differences have been less 5%.
These results suggest that COMSOL can be a suitable tool to
model the heating of biological tissues using BE and HBE.
Now, future work will be conducted to implement theoretical
models based on FEM (COMSOL) with more realistic
geometries.
Fig. 2. Temperature progress obtained from analytical solution
during 60 s of RF ablation using the HBE and for two values of
thermal relaxation time (1 and 16 s) and for two blood perfusion
conditions: without perfusion ωb=0 (solid line), and perfusion ωb
=0.01 1/s (dashed line). The plots correspond with a location r =
2r0.
V. CONCLUSION
In this paper, we have outlined our main findings about
the differences between the BE and HBE models for RF
ablation. These differences encourage the use of the HBE
approach for processes in which great amounts of heat are
transferred to any material in very short times, e.g. RF
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heating in the cornea. At the moment, we have been working
on the analytical modeling of the HBE, but more recently,
we have considered numerical models based on FEM. As the
first step to use FEM should be the validation, we have
conducted a comparative study between the temperature
profiles obtained from the analytical solutions and those
obtained from FEM.
Fig 3. Temperature progress obtained from COMSOL (top) and
analytical solution (bottom) during 60 s of RF ablation using the
HBE and for two values of thermal relaxation time (1 and 16 s)
and for two blood perfusion conditions: without perfusion ω=0
(solid line), and perfusion ω=0.01 1/s (dashed line). The plots
correspond with a location r = 2r0.
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