a fundamental solution-based finite element model...
TRANSCRIPT
A FUNDAMENTAL SOLUTION-BASED FINITE
ELEMENT MODEL FOR ANALYZING MULTI-LAYER
SKIN BURN INJURY
HUI WANG* and QING-HUA QIN†,‡
*Institute of Scienti¯c and Engineering ComputationHenan University of Technology
Zhengzhou 450052, PR China
†Research School of Engineering
Australian National University
Canberra, ACT 0200, Australia‡[email protected]
Received 11 October 2011
Revised 4 April 2012
Accepted 11 April 2012Published 28 July 2012
To understand the physiology of tissue burns for successful clinical treatment, it is important to
investigate the thermal behavior of human skin tissue subjected to heat injury. In this paper, afundamental solution-based hybrid ¯nite element formulation is proposed for numerically
simulating steady-state temperature distribution inside a multilayer human skin tissue during
burning. In the present approach, since only element boundary integrals are involved, the
computational dimension is reduced by one as the fundamental solutions used analyticallysatis¯es the bioheat governing equation. Further, inmulti-layer skinmodeling, the burn is applied
via a heating disk at constant temperature on a part of the epidermal surface of the skin tissue.
Numerical results from the proposed approach are ¯rstly veri¯ed by comparing them with exact
solutions of a simple single-layered model or the results from conventional ¯nite element method.Thereafter, a sensitivity analysis is carried out to reveal the e®ect of biological and environmental
parameters on temperature distribution inside the skin tissue subjected to heat injury.
Keywords: Bioheat transfer; hybrid ¯nite element; fundamental solutions; boundary integrals;
sensitivity analysis.
1. Introduction
Burns may occur when human skin is directly exposed to severe external environ-
ments such as a heat source, cold environment, chemicals, and electromagnetic ¯elds.
Henriques and Moritz1 concluded that the rate of the induced injury is higher than
that of recovery to the skin when its temperature is maintained at 44�C or above. In
this case, a ¯rst-degree burn may ensue. Therefore, investigations on temperature
distribution inside a biological system is vital for burn prediction.
‡Corresponding author.
Journal of Mechanics in Medicine and Biology
Vol. 12, No. 5 (2012) 1250027 (22 pages)
°c World Scienti¯c Publishing Company
DOI: 10.1142/S0219519412500273
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Numerical methods based on the Pennes bioheat transfer equation including
the e®ect of blood perfusion have been developed during the past two decades
and proven to be e®ective tools in predicting temperature changes within biological
systems subjected to various boundary conditions, since theoretical analysis is only
suitable for the solution of simple mathematical models with simple boundary,
loading, and geometry conditions. In particular, the ¯nite element method (FEM),
the ¯nite di®erence method (FDM), and the boundary element method (BEM) have
been widely used in bioheat analysis of biological tissues.2�10 In the study of bioheat
transfer in human skin tissues, for example, Diller et al. established a ¯nite element
model for analyzing transient bioheat transfer behavior and thermal ¯eld distribu-
tion in skin during the burning process;11 Ng and Chua compared one- and two-
dimensional models for predicting the state of skin burns using the FDM and FEM,
respectively;12 Jiang et al. developed a one-dimensional ¯nite di®erence formulation
to investigate the in°uence of thermal properties and geometrical dimensions of
multi-layer skin on burn injury13; Cao et al.14 used the method of fundamental
solution (MFS) coupling with interior collocation points to analyze thermal behavior
of tumor and burned skin tissues. Torvi and Dale15 developed a multi-layered FEM
for predicting skin temperature and the time for second-and third-degree burns to
occur under simulated °ash ¯re conditions. Ng et al.3,5 used the axisymmetric BEM
to investigate the temperature distribution in three-layered human skin tissue under
contact heating. The BEM distinguishes itself from the FEM and FDM by employing
only the boundary division of the domain of interest to achieve signi¯cant time-
saving in the creation and modi¯cation of the mesh,16,17 thus the BEM can reduce
the dimension of the problem by one to simplify data preparation. However, the
BEM is inconvenient in solving multi-material or heterogeneous problems such as
multi-layered skin tissues considered in the present study, because extra interface
conditions between adjacent subregions have to be satis¯ed, in addition to the
boundary integral equations along the boundary of each subregion.3,8 As a result, the
coe±cient matrix of the ¯nal equation system is usually non-symmetric.
As an alternative to the FEM/FDM and BEM, a new hybrid ¯nite element
approach, which is based on fundamental solutions, has been developed recently18,19
and applied to heat transfer analysis in the human eye,20 plane elastic problems
in homogeneous21,22 and heterogeneous media,23 heat transfer analysis in ¯ber-
composites,24 and nonlinear Poisson-type problems.25 The hybrid ¯nite element
model mentioned above was referred to as HFS-FEM to distinguish it from other
numerical algorithms, due to the use of fundamental solutions. The HFS-FEM is
based on two independent ¯elds: the interior ¯eld de¯ned inside an element, which is
formed by the linear combination of fundamental solutions of the problem of interest,
and the frame ¯eld de¯ned over its element boundary, which is approximated by
conventional shape functions. Hence, it is versatile in generating multi-node elements
and special purposed elements.21,23,24 The satisfaction of the governing equations for
the interior ¯elds makes it possible to convert element domain integrals that appear
in the hybrid variational functional into element boundary integrals. In addition,
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as the material de¯nition can be prescribed over element level, it is easy to apply
the developed approach to multi-material problems such as the multi-layer
skin tissues used in this study.
To the best of the authors' knowledge, the application of a hybrid ¯nite element
formulation with fundamental solutions kernels to burns of the skin tissue has not
been reported in the literature, thus it is interesting to examine the feasibility of this
method in simulating temperature distributions within the human skin during
burning. The objective of this preliminary study is to extend the hybrid ¯nite ele-
ment technique presented in Refs. 18 and 20 to two-dimensional steady-state bioheat
transfer analysis in a system of a four-layer skin tissue when subjected to a constant
temperature heat source at the skin surface. The numerical solutions of temperature
distribution inside the skin tissue during burns are evaluated under various biological
conditions to e®ectively estimate the seriousness of di®erent burns.
The remainder of this paper is organized as follows: in Sec. 2, the bioheat for-
mulations of human skin are reviewed for introducing the mathematical model of a
four-layer skin system in rectangular coordinates, Sec. 3 presents the derivation of
the hybrid ¯nite element formulation, Sec. 4 discusses the numerical results obtained
using the present approach under various conditions, and our conclusions are pre-
sented in Sec. 5.
2. Mathematical Model of Multi-Layer Skin Tissue
2.1. Basic equations
In biomechanical engineering, the human skin tissue is usually modeled as a three-
layer material structure including the epidermis, the dermis, and the subcutaneous
fat layer. Further, an inner tissue which is in the region from the inner surface of the
subcutaneous fat layer to the core of the body is also introduced, as in Ref. 13. In a
four-layer biomechanical model proposed in this study shown in Fig. 1, each layer is
supposed to be ideally homogeneous, within which the blood perfusion, thermal
conductivity, and heat capacity are assumed to be constant, and layers are assumed
to be perfectly bonded each other to allow a continuous °ow of heat °ux across
interfaces.
The steady-state heat transfer in the biological tissue is governed by the well-
known Pennes bioheat equation26:
kr2T þ �bcb!bðTb � T Þ þQr þQm ¼ 0; ð1Þ
in which k is the thermal conductivity, T is the temperature change of the tissue, r2
is the Laplace operator, �b, cb, and !b are respectively the density, speci¯c heat, and
perfusion rate of blood, Tb is the temperature of arterial blood, Qm and Qr are
metabolic heat generation and heat deposition in the tissue caused by outer heating
factor such as laser and microwave, respectively.
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The bioheat transfer (Eq. (1)) is a statement of the law of conservation of energy.
The ¯rst term on the left-hand-side of Eq. (1) represents the heat conduction in the
tissue caused by the temperature gradient, and the second term stands for the heat
transport between the tissue and microcirculatory blood perfusion. The third and
last terms are two internal heat generations respectively due to tissue metabolism
and outer heating sources.
In this study, a contact heat source, i.e., a heating disk as displayed in Fig. 1,
represented the potential outer burning injury to investigate the induced tempera-
ture variation in the multi-layer skin tissue under di®erent heating temperatures.
The heating disk was assumed to distribute along the direction perpendicular to the
cross-section (x�y plane in the ¯gure) of the skin tissue, where the two-dimensional
model can be used to simplify the bioheat analysis. In our analysis, there was an
assumption that no interfacial resistance exists between the heating source and the
skin surface employed. Therefore, the temperature at the skin surface in contact with
the heating disk remained constant during heating. Moreover, the temperature
change caused by the heating disk was much greater than metabolic heat generation,
thus the metabolic heat generation is negligible here.3 Simultaneously, the internal
heat generation caused by outer heating source was also neglected. As a result, the
bioheat Eq. (1) reduces to:
kr2T þ �bcb!bðTb � T Þ ¼ 0: ð2ÞSpecially, when the blood perfusion rate is zero, no blood °ow exists in the
epidermis layer. Hence, the governing Eq. (2) reduces to the standard Laplace
equation:
kr2T ¼ 0: ð3Þ
Fig. 1. Schematic diagram of the multi-layer skin tissues (¯gure not to scale).
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In the bioheat transfer model under consideration, the boundary �1 represents the
bottom-most surface of the skin, thus we assume the temperature on it is equivalent
to the body core temperature Tc, that is:
T ¼ Tc at boundary �1 ð4ÞAt the upper and bottom surfaces, no heat °ow runs into the skin tissue along
these two edges with the assumption that the tissue far from the center area of the
solution domain is not a®ected by the imposed thermal disturbance at the center
domain,3,5,7,27,28 hence the adjacent condition is given by:
�k@T
@n¼ 0 at boundaries �2 and �3: ð5Þ
The part of the epidermal surface is directly exposed to the environmental °uid,
therefore the heat exchange occurs between the environmental °uid and skin via
convection and radiation. This is because in biological tissues, the e®ect of radiation
from the surrounding is very small in contrast to convection, thus radiation is neglected
here.20 Also, the cooling of the human skin by the evaporation of sweat should be
considered since the heat loss due to evaporation has been found to contribute
approximately 15% of the total heat loss from the skin surface.3,5 Thus, we have:
�k@T
@n¼ h1ðT � T1Þ þ Es at boundaries �4 and �5; ð6Þ
where h1 andT1 are respectively the ambient convection coe±cient and temperature,
and Es is the heat loss due to sweat evaporation on the skin surface.
Finally, on the boundary where the heating disk is applied, the temperature is
assumed to be equal to the temperature of heating disk Td, i.e.,
T ¼ Td at boundary �6: ð7Þ
2.2. Dimensionless form
Due to the signi¯cant scale di®erence of variables in Eq. (1), the dimensionless
variables de¯ned as follows are introduced:
X ¼ x
L0
; Y ¼ y
L0
; � ¼ ðT � TbÞk0Q0L
20
; K ¼ k
k0; ð8Þ
where L0 is a reference length of the biological body, k0, �0, c0, and Q0 are respec-
tively reference values of the thermal conductivity, density, speci¯c heat, and heat
source term.
Making use of the new variables de¯ned by Eq. (8), the Laplace operator in
Eq. (2) becomes:
@2T
@x2þ @2T
@y2¼ Q0L
20
k0
1
L20
@2�
@X2þ @2�
@Y 2
� �ð9Þ
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Equation (1) can then be rewritten as follows:
Kr2�� Sb�þ Qr þQm
Q0
¼ 0; ð10Þ
where
Sb ¼L2
0�bcb!b
k0: ð11Þ
At the same time, the corresponding boundary conditions reduce to:
� ¼ �c on �1
q ¼ �K@�
@n¼ 0 on �2 and �3
q ¼ �K@�
@n¼ H1ð�� �1Þ þ ~Es on �4 and �5
� ¼ �d on �6;
8>>>>>>>><>>>>>>>>:
; ð12Þ
where
H1 ¼ h1L0
k0; ~Es ¼
Es
Q0L0
: ð13Þ
3. Hybrid Finite Element Implementation
3.1. Variational functional
In the present hybrid ¯nite element formulation, a hybrid functional associated with
two independent ¯elds �, ~� de¯ned inside the element domain and over the element
boundary, respectively, is constructed based on the existing functional:29,30
� ¼ �1
2
Z�e
K@�
@X
� �2
þ @�
@Y
� �2� �þ Sb�
2
� �d�
�Z�qe
�q ~�d�þZ�e
q ~�� �� �
d�� 1
2
Z�ce
H1 ~�� �1� �2
d�; ð14Þ
where �qe and �ce are element boundaries with speci¯ed heat °ux and convec-
tion condition, respectively. �e represents the domain of element e with boundary
�e, as shown in Fig. 2. In addition, in this ¯gure, �te and �Ie respectively stand
for the elemental boundary on which the temperature is prescribed and the inter-
element boundary between element e and its adjacent elements, e.g., f and g. It is
obvious that:
�e ¼ �te þ �qe þ �ce þ �Ie: ð15Þ
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Particularly, for the bioheat model considered in the paper, we have:
�te ¼ �e \ �1 or �e \ �6
�qe ¼ �e \ �2 or �e \ �3
�ce ¼ �e \ �4 or �e \ �5 ð16ÞBy invoking the divergence theorem:
Z�
@f
@X
@g
@Xþ @f
@Y
@g
@Y
� �d� ¼
Z�
g@f
@nd��
Z�
gr2fd� ð17Þ
for any smooth functions f and g in the domain, the ¯rst-order variation of Eq. (14) is
written as:
�� ¼Z�e
ðKr2�� Sb�Þ��d�
þZ�te
q�~�d�þZ�qe
ðq� �qÞ�~�d�þZ�ce
½q�H1ð~�� �1Þ��~�d�
þZ�e
�qð~�� �Þd�; ð18Þ
from which it can be seen that the ¯rst, third, and fourth integrals are associated
with the governing Eq. (10), speci¯ed heat °ux condition, and convection condition
in Eq. (12), respectively. The second integral will disappear when ~� is assumed to
satisfy the speci¯ed temperature constraint on the boundary �te. The last integral
enforces the equality of � and ~� along the element frame �e.
Fig. 2. Illustration of a typical hybrid element.
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If the intra-element temperature � analytically satis¯es the governing Eq. (10),
then the hybrid functional Eq. (14) can be further simpli¯ed by again using the
divergence theorem, i.e.,
� ¼ � 1
2
Z�e
q�d�þZ�e
q~�d��Z�qe
�q ~�d��Z�ce
H12
ð~�� �1Þ2d�; ð19Þ
which includes only boundary integrals and can be used to derive the corresponding
element sti®ness equation.
3.2. Assumed ¯elds
As was done in the conventional FEM, the solution domain � is divided into a
number of elements in the present study. For a particular element, say element e,
occupying a sub-domain �e, with the element boundary �e, two groups of inde-
pendent ¯elds � and ~� are assumed as follows:
3.2.1. Non-conforming intra-element ¯elds
In the proposed fundamental solution-based hybrid ¯nite element formulation, in
order to construct the solution satisfying the governing Eq. (10) within the element
domain, the temperature approximation � at any given point P within the element
domain is expressed by a combination of fundamental solutions, as was done in the
MFS,31,32 for example,
� ¼Xns
i¼1
G�ðP ;QiÞci ¼ NeðPÞce; P 2 �e; Qi 62 �e; ð20Þ
where ci is undetermined coe±cients and ns is the number of virtual sources Qi
surrounding the element domain. G�ðP ;QiÞ denotes the free-space Green's function
(fundamental solutions) for the governing Eq. (10):
Kr2G�ðP ;QiÞ � SbG�ðP ;QiÞ þ �ðP ;QiÞ ¼ 0; ð21Þ
whose solution is given by:14
G�ðP ;QiÞ ¼ � 1
2�KK0ð�rÞ: ð22Þ
In Eqs. (21) and (22), � stands for the Dirac delta function, K0 is the modi¯ed
Bessel function of the second kind with order zero, r ¼ P �Qik k is the distance
between the ¯eld point P and source point Qi, and
� ¼ffiffiffiffiffiffiSb
K
r: ð23Þ
In the MFS, the coordinates of the source points Qi are prescribed and usually
they are located on a pseudo-boundary whose shape is similar to the element
boundary �e. Here, the locations of those source points were determined by means of
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element nodes using the following relation33,34:
XQi¼ Xi þ �ðXi �XcÞ
YQi¼ Yi þ �ðYi � YcÞ; ð24Þ
where (XQi;YQi), (Xi;Yi), and (Xc;Yc) are respectively the coordinates of source
pointQi, nodal point i, and element center. The dimensionless parameter � is used to
control the distance of source points to the element physical boundary.34 The source
points generated by Eq. (24) are located outside the elemental domain.
Furthermore, in the absence of blood perfusion rate, the fundamental solutions
used for intra-element approximation is given by20:
G�ðP ;QiÞ ¼ � 1
2�KlnðrÞ: ð25Þ
Moreover, the heat °ux is approximated by:
q ¼ �K@�
@n¼ �K
Xns
i¼1
@G�ðP ;QiÞ@n
ci ¼ Qece; ð26Þ
where
@G�ðP ;QiÞ@n
¼ �
2�KK1ð�rÞ
@r
@nð27Þ
for the case of !b > 0, and
@G�ðP ;QiÞ@n
¼ � 1
2�Kr
@r
@nð28Þ
for the case of !b ¼ 0.
3.2.2. Conforming frame ¯elds de¯ned on element boundary
An independent frame temperature ¯eld de¯ned over the element boundary can be
approximated by the shape function interpolation widely used in the conventional
FEM and BEM:
~�ðP Þ ¼Xnd
i¼1
~NiðPÞdei ¼ ~NeðP Þde; P 2 �e; ð29Þ
where nd is the number of nodes in the element, de is the nodal temperature vector,
and ~Ni is the shape function.
The substitution of Eqs. (20), (26), and (29) into the functional Eq. (19) yields:
� ¼ � 1
2cTeHece � dT
e ge þ cTeGede �
1
2dT
eFede þ dTe f e � ae; ð30Þ
in which
He ¼Z�e
QTeNed�; Ge ¼
Z�e
QTe~Ned�; ge ¼
Z�qe
~NTe �qd�
Fe ¼Z�ce
H1 ~NTe~Ned�; f e ¼
Z�ce
H1�1 ~NTe d�; ae ¼
Z�ce
H1�212
d�
: ð31Þ
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By virtue of the stationary conditions:
@�
@cTe
¼ 0;@�
@dTe
¼ 0; ð32Þ
we obtain following sti®ness equation for determining nodal temperature vector de:
Kede ¼ ge � f e; ð33Þand the relationship of unknown coe±cient ce and de:
ce ¼ H�1e Gede: ð34Þ
In Eq. (33), the element sti®ness matrix Ke has the following form:
Ke ¼ GTeH
�1e Ge � Fe: ð35Þ
Assembling the element sti®ness matrix element by element, we can obtain the
global sti®ness matrix, which still remains the sparse and symmetrical features of the
conventional ¯nite element sti®ness matrix.
4. Numerical Assessments
Since no experimental study can be found in the literature for the problems that we
considered in this study, the numerical results from the proposed algorithm were
veri¯ed by comparing them with exact solutions of a benchmark example and
numerical results obtained using the FEM incorporating with the COMSOL, a ¯nite
element analysis, solver, and simulation software for various multi-physics and
engineering problems. The COMSOL can be used for analyzing the Pennes bioheat
transfer problems discussed in the paper. Subsequently, the temperature distribution
inside the solution domain was investigated with di®erent values of parameters
including the heating disk temperature, ambient temperature, ambient convection
coe±cient, evaporation rate of sweat, and blood perfusion rate and size of heating disk
to investigate the seriousness of burn. The purpose of such analysis, also called sensi-
tivity analysis, is to identify the e®ects of individual or a combination of parameters on
the temperature distribution within the skin tissue during outer burn process.
The typical material properties presented in Refs. 13 and 15 were employed
in the computation and are listed in Table 1. The height of the solution domain
Table 1. Thermal and physical properties used in the multi-layer skin
tissues.
Thickness Thermal conductivity Blood perfusion rate
(mm) (W/m/K) (ml/s/ml)
Epidermis 0.08 0.24 0
Dermis 2 0.45 0.0005
Subcutaneous 10 0.19 0.0005
Inner tissue 30 0.50 0.0005
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was taken as 80mm. The density and speci¯c heat of the blood were 1100 kg/m3 and
3300 J/kg/K, respectively. Additionally, the body core temperature was taken as
37�C, the ambient convection coe±cient was 7W/m2/K which corresponded to
natural convection, the ambient temperature was 20�C, the average temperature of
air at spring and autumn, the approximated sweat evaporation rate was chosen as
10W/m,2,3 and the heating disk temperature was set as 90�C in the calculation.
4.1. Veri¯cation of the algorithm
To verify the proposed approach, a homogeneous biological tissue with rectangular
domain was considered (Fig. 3). The thermal conductivity of the tissue was 0.5W/
m/K, and the thickness and width of the tissue were 30 and 80mm, respectively, as
used in the model by Liu et al.35 If the outer surface of the tissue is subjected to
convection condition, the analytical solution can be written as36:
T ðxÞ ¼ Aþ ðTc �AÞ½� coshð�xÞ þB sinhð�xÞ��coshð�LÞ þB sinhð�LÞ þ BðT1 �AÞ sinh½�ðL� xÞ�
� coshð�LÞ þB sinhð�LÞ ; ð36Þ
where
A ¼ Tb; B ¼ h1k
; � ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi�bcb!b
k
r: ð37Þ
Due to the symmetry, only half of the domain was modeled (see the shaded region
in Fig. 3) and the adiabatic condition was imposed along the symmetry edge. A total
of 20 eight-node quadrilateral hybrid elements with 79 nodes were employed to
discretize the computing domain (see Fig. 3). To show the numerical accuracy and
Fig. 3. Boundary de¯nition and mesh division in the computing domain.
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stability of the present approach, di®erent blood perfusion rates whose values
changed from 0.00001 to 0.001ml/s/ml were employed and the corresponding
parameter � changed from 8.521 to 85.206.
In addition, to determine the locations of source points which were outside the
element domain, the e®ect of the dimensionless parameter � on numerical results was
studied for di®erent blood perfusion rates. The percentage relative error of temper-
ature at the point of (0,0) is displayed in Fig. 4, from which it can be seen that
numerical results were stable for a wide range value of the parameter �. The small
value of � indicates the small distance from source points to the element boundary. It
is evident from Fig. 4 that the singular disturbance of fundamental solutions
increases rapidly when � approaches zero. Inversely, if the value of � is too large, the
numerical accuracy of the inversion of matrix H also decreases because the attenu-
ation characteristic of fundamental solutions theoretically makes the entries of H
close to zero for the case of a larger �. Thus, in the present work, the parameter � was
chosen to be 10. Here, the percentage relative errors in the temperature at the sample
point (0,0) are 0.008%, 0.04%, and 0.285%, corresponding to blood perfusion rates
given by 10�5, 10�4, and 10�3 ml/s/ml, respectively.
On the other hand, di®erent values of the blood perfusion rate were studied to
reveal their e®ect on temperature distribution. Figure 5 shows the temperature
distribution along the thickness of the tissue for various values of blood perfusion
rates. The results predicted by the proposed HFS-FEM are in good agreement with
the exact solutions obtained from Eq. (36). Besides, it can be seen from Fig. 5 that
the larger the blood perfusion rate, the steeper the temperature curve. A larger blood
Fig. 4. Percentage relative error at the temperature at point (0,0) versus � .
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perfusion is found to produce a higher skin temperature which is evident in the
thermoregulatory response of the biological tissue.9 For an environment with a
temperature (i.e., 20�C) lower than the body temperature (37�C), the larger perfu-
sion rate allows for better heat transfer into the body from the blood vessels to heat
the body, resulting in a hotter surface of the tissue. Further, the results for zero blood
perfusion rate are included in Fig. 5 for the purpose of comparison, from which the
e®ects of blood perfusion can be seen clearly. The existence of blood perfusion causes
the temperature distribution in the body to be nonlinear.
4.2. Sensitivity analysis
It has long been revealed that the body surface temperature is controlled by factors
such as the blood circulation underneath the skin, heat exchange between the skin
and its environment, and so on. Changes in any of these parameters can induce
variations of temperature at the skin surface. As the surface temperature can be
easily measured in clinical diagnosis as non-invasive thermometry, it is possible to
detect the temperature change re°ecting the physiological state of the human body
and predict the potential severity of burns and thermal injury in°icted on the skin.
In this subsection, the sensitivity analysis10 of control parameters investigated
under a contact heating through a disk at the local surface of the skin is discussed.
It may bene¯t e®ective clinical applications if we can properly adjust the
control parameters to achieve a reasonable temperature variation associated
with the tissue burn. The e®ects of heating temperature and heating area, and
Fig. 5. Temperature distribution along the thickness of the tissue for various blood perfusion rates.
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environmental parameters including the convective coe±cient and sink temperature
and the perfusion rate on skin temperature are assessed here. Due to the symmetry,
half of a domain was computed, which modeled via 1,280 quadratic elements with
3,985 nodal degrees of freedom, as shown in Fig. 6. The height of the heating disk was
initially taken to be 30mm.
4.2.1. E®ect of heating temperature of disk
In this study, the burn was applied via a heating disk at the skin surface; the
temperature change of the heating disk was thus expected to have signi¯cant
in°uence to the temperature distribution in the tissue. In the sensitivity analysis of
the disk's heating temperature, the environmental temperature was assumed to be
20�C and the ambient convective coe±cient was 7W/m2/K. Here, the convection
behavior corresponded to the natural convection between the skin surface and the
ambient air. The blood perfusion rates were assumed to be zero in the epidermis
layer, and 0.0005ml/s/ml in the remaining layers (an average blood perfusion
value for the human skin), respectively. The sweat evaporation rate was taken to be
10W/m2. The values of the control parameter, the heating disk temperature Td, were
separately taken to be 60�C, 90�C, and 120�C, to investigate the e®ects of the heatingtemperature on the isotherm of the skin tissues under constant burn heating.
Fig. 6. Mesh con¯guration for multi-layer skin tissue.
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Figure 7 shows the temperature distribution along the horizontal lines y ¼ 0m
and y ¼ 0:04m when the heating disk was kept at 60�C. It was seen that there was a
larger temperature gradient along the line y ¼ 0m (from 60�C to 37�C) than that
along the line y ¼ 0:04m. This might be attributed to the surface heating source
being just on the line in the central region near y ¼ 0m, while along the line
y ¼ 0:04m, and cooling took place at the surface of the skin exposed to the ambient
air. Figure 8 shows isotherms in the solution domain of the human skin tissue. We
found that the lowest temperature occurred at the core region of the skin tissue and
the hottest surface was on the surface where the heating disk was applied. It was also
found that the temperature at the convection surface had a higher average value
(approximately 51�C) than the core temperature 37�C, since heat supplied from the
heating disk was basically transferred to the environment via convection and sweat
evaporation, while the average temperature was lower than the heating temperature
of disk (60�C), because the cooling e®ect took place at the convection surface exposed
to the ambient °uid. Further, there was no observable change of temperature in the
epidermis layer because it was too thin (only 0.08mm). In Figs. 7 and 8, the ¯nite
element numerical results implemented with COMSOL (total number of 25,332
triangular elements with 37,998 nodal degrees of freedom) are also presented to
further verify the present HFS-FEM, where good agreement was observed. From
these results, we concluded that the present HFS-FEM can be e®ectively used to
simulate the bioheat e®ect in the complicated multi-layer skin tissues.
Next, the e®ect of heating disk with various burning temperatures (60�C, 90�C,and 120�C) was investigated to reveal how these values can a®ect the isotherms in
Fig. 7. Temperature variations along two horizontal directions for the case of disk temperature of 60�C.
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the skin tissues. The corresponding results are displayed in Fig. 9, from which we can
see that the values of temperature in the dermis and subcutaneous fat layers all
exceeded 44�C, which is the threshold value of the ¯rst degree burn. Meanwhile, it
was found that the skin temperature at the layer of inner tissue did not vary sig-
ni¯cantly when the temperature of the disk was dramatically changed. In addition
the temperature ¯eld displayed a rapid decrease for the case of the heating disk with
(a)
(b)
Fig. 8. Temperature distribution in the multi-layer skin model when heated by the disk with temperature
of 60�C. (a) Results from the HFS-FEM (b) Results from the FEM using COMSOL.
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a higher temperature, especially in the dermis and subcutaneous fat layers. It is not
surprising, given that more heat energy transport occurs between the tissue and
microcirculatory blood perfusion when the temperature di®erence is relatively larger
between the heating disk and tissue.
4.2.2. E®ect of environmental temperature
The environmental parameters related to convection condition imposed at the skin
surface include the ambient temperature T1 and ambient convective coe±cient h1.
First, let us consider the e®ect of ambient temperature and its three values at 0�C,20�C, and 40�C, chosen to simulate its e®ect. In the analysis, the ambient convection
coe±cient was assumed to be 7W/m2/K, which corresponded to natural convection
at the skin surface. The heating temperature was assumed to be 90�C. The results
displayed in Fig. 10 shows that the e®ect of ambient temperature on the temperature
distribution inside the tissue can be neglected. This implies that reducing the
ambient temperature may not be e®ective to avoid the risk of thermal damage of
the skin that has su®ered burns.
Subsequently, the e®ect of ambient convection coe±cient was studied. Its value
changed from 7W/m2/K (natural convection) to 25W/m2/K (onset from natural
convection to forced convection), and then to 50W/m2/K (forced convection). The
temperature of ambient °uid was assumed to be 20�C. Results of temperature
obtained from the proposed formulation are given in Fig. 11. It is found that a change
Fig. 9. Temperature variation along the horizontal coordinate axis due to di®erent heating disk
temperatures.
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Fig. 11. Temperature variations along the horizontal direction with di®erent ambient convection
coe±cients.
Fig. 10. Temperature variation along the horizontal direction subjected to di®erent environmental
temperatures.
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of convection coe±cient had little e®ect on the temperature distribution along the
horizontal coordinate axis (y ¼ 0m) inside the skin tissue. However, at the convec-
tion surface of the skin, the temperature appeared to be signi¯cantly reduced as the
ambient convection coe±cient increases. Hence, it is important to convert natural
convection into forced convection to alleviate the thermal damage in clinical practice.
4.2.3. E®ect of blood perfusion rate
To estimate the e®ect of blood perfusion rate on temperature distribution in the skin
tissue, the values of perfusion rate were chosen to be from 0 to 0.00125ml/s/ml, the
maximum of which was 2.5 times of the average perfusion rate (0.0005ml/s/ml) to
distinguish conventional heat transfer and bioheat transfer, and simultaneously, to
understand the role of blood °ow in the thermoregulation of the biological tissue
when it su®ers burns.
The e®ects of blood perfusion rate on the temperature distribution inside the skin
tissue are shown in Fig. 12. As expected, a larger blood perfusion rate was found to
produce a lower skin temperature. It is reasonable that when the tissue temperature
increases, the immediate e®ect is for the body to cause the blood vessels to expand
and then to increase the blood °ow. As a result, the heat energy accumulated in the
body is dissipated by the heat exchange between the body and the vessels.
4.2.4. E®ect of size of heating disk
The e®ect of the heating disk's size on bioheat transfer here was studied using
the proposed method. In the sensitivity analysis, the environmental temperature was
Fig. 12. Temperature variations along the horizontal directions due to di®erent blood perfusion rates.
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assumed to be 20�C, which may induce natural convection, and the ambient con-
vective coe±cient was 7W/m2/K. The blood perfusion rates were taken to be zero in
the epidermal layer and 0.0005ml/s/ml in the remaining layers. The sweat evapo-
ration rate was 10W/m2. The heating temperature of the disk was set to be 90�C.Heating disks with lengths of 10, 20, and 30mm are, respectively, considered to
investigate how the size of heating disk can a®ect temperature distribution in the
tissue under constant burn heating. Figure 13 displays the temperature distributions
for three di®erent sizes of the heating disk. It can be seen that the temperature
distribution along the horizontal axis is not sensitive to the size of the heating disk.
However, a rapid temperature increase of 19:1�C (from 34:5�C to 53:6�C) was
detected at the surface point (0, 0.04m) as the size of the heating disk increase from
10 to 30mm, as expected. Meanwhile, a minor increase in temperature (about 4:3�C)was found when the size of the heating disk increased from 10 to 20mm.
5. Conclusion
The fundamental solution-based hybrid ¯nite element model is established in the
paper for solving the steady-state bioheat transfer problem in multi-layer skin
tissues subjected to contact surface heating via a disk to simulate its burning injury
process. The e®ectiveness and accuracy of the present formulation were assessed
through examples whose analytical solutions are known or numerical solutions
obtained using the conventional ¯nite element method. Thereafter, the sensitivity
analysis was performed to investigate the e®ects of several control parameters in-
cluding the heating temperature, the size of heating disk, the temperature of ambient
Fig. 13. Temperature variations along the horizontal directions with the size of heating disk.
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°uid, the ambient convection coe±cient, and blood perfusion rate. Results obtained
from the proposed model show that the temperature of the heating disk plays a
signi¯cant role in altering the temperature distribution inside the body and the
severity of the thermal damage in°icted in the skin tissue. In addition, the ambient
convection coe±cient and blood perfusion rate are also e®ective in reducing the
highest temperature.
It should be mentioned that the proposed approach cannot be directly used for
transient or nonlinear bioheat analysis in multi-layered biological tissues because
time discretization or nonlinear iteration will induce domain integrals. The work on
how to e®ectively handle the induced domain integrals is still underway.
Acknowledgments
The research in this paper is partially supported by the Australian Endeavour
Awards 2011 and Foundation for University Key Teacher by the Henan Province,
China, under the grant no. 2011GGJS-083.
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