o_19nf61j2o10399kdgfam962n4a.pdf

Int J Thermophys (2013) 34:139–159
DOI 10.1007/s10765-013-1396-0
Analytical Solution of Thermal Wave Models
on Skin Tissue Under Arbitrary Periodic
Boundary Conditions
R. Fazlali · H. Ahmadikia
Received: 10 August 2012 / Accepted: 9 January 2013 / Published online: 26 January 2013
© Springer Science+Business Media New York 2013
Abstract Modeling and understanding the heat transfer in biological tissues is important
in medical thermal therapeutic applications. The biothermomechanics of skin
involves interdisciplinary features, such as bioheat transfer, biomechanics, and burn
damage. The hyperbolic thermal wave model of bioheat transfer and the parabolic
Pennes bioheat transfer equations with blood perfusion and metabolic heat generation
are applied for the skin tissue as a finite and semi-infinite domain when the
skin surface temperature is suddenly exposed to a source of an arbitrary periodic
temperature. These equations are solved analytically by Laplace transform methods.
The thermal wave model results indicate that a non-Fourier model has predicted the
thermal behavior correctly, compared to that of previous experiments. The results of
the thermal wave model show that when the first thermal wave moves from the first
boundary, the temperature profiles for finite and semi-infinite domains of skin become
separated for these phenomena; the discrepancy between these profiles is negligible.
The accuracy of the obtained results is validated through comparisons with existing
numerical results. The results demonstrate that the non-Fourier model is significant in
describing the thermal behavior of skin tissue.
Keywords Laplace transforms · Non-Fourier · Pennes equation · Skin tissue ·
Thermal wave
R. Fazlali (B)
Young Researchers & Elites Club, Hamedan Branch, Islamic Azad University, Hamadan, Iran
e-mail: rz_fazlali@yahoo.com; rz_fazlali@iauh.ac.ir
H. Ahmadikia
Department of Mechanical Engineering, University of Isfahan, 81746-73441 Isfahan, Iran
e-mail: ahmadikia@eng.ui.ac.ir
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140 Int J Thermophys (2013) 34:139–159
List of Symbols
Variables
A Frequency factor (1/s)
C Thermal wave speed in the medium (m · s −1 )
c b Specific heat of blood (J · kg −1 · ◦C −1 )
c t Specific heat of tissue (J · kg −1 · ◦C −1 )
E a Activation energy (J · kmol −1 )
H Heaviside theta function
I 0 Zero rank-modified Bessel function
I 1 First rank-modified Bessel function
k Thermal conductivity of tissue (W · m −1 · ◦C −1 )
L Skin thickness (m)
q ext Heat generated (W · m −3 )
q met Metabolic heat generation (W · m −3 )
R Universal gas constant (J · mol −1 · K −1 )
T Tissue temperature ( ◦ C)
T a Blood temperature ( ◦ C)
T c Minimum temperature of skin surface boundary condition ( ◦ C)
T s Maximum temperature of skin surface boundary condition ( ◦ C)
t Time (s)
t 0 Period time (s)
x Distance along the surface (m)
U Unit step function
Greek Symbols
α Thermal diffusivity (m 2 · s −1 )
Δ Duration time of a period that the boundary condition of skin surface is
equal to T s (s)
δ Dirac delta function
ρ b Density of blood (kg · m −3 )
ρ t Density of skin tissue (kg · m −3 )
τ q Thermal relaxation time (s)
Ω Dimensionless thermal damage
ϖ b Blood perfusion rate (ml · ml −1 · s −1 )
Subscripts
b
P
t
Blood
Pennes equation
Tissue
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Int J Thermophys (2013) 34:139–159 141
1 Introduction
The skin is the most extensive living organ of the human body. The skin consists
of three layers: epidermis, dermis, and hypodermis (fat subcutaneous tissue). Its
contribution to the body is essential and includes sensory, thermoregulation, host
defense, etc. Advances in laser, microwave, and similar technologies have led to
recent developments in the thermal treatment of skin diseases and injured skin tissue
such as skin cancer and skin burn. Understanding heat transfer and related thermomechanical
properties in soft tissue such as skin is important in medical applications.
The thermal behavior of heat transfer in skin tissue is a kind of heat conduction
problem that is coupled with complicated physiological problems such as blood
circulation, metabolic heat generation, and other processes. Skin is affected by
many factors such as age, gender, etc. Furthermore, simple elastomeric constitutive
models are not suitable to describe the complicated mechanical behavior of the
skin [1].
Solving the heat transfer problem in soft organs such as skin has attracted the attention
of many researchers. The Fourier model (Pennes equation) introduced by Pennes
[2] and some non-Fourier models such as the thermal wave model of bioheat transfer
and the dual-phase-lag (DPL) model were applied in modeling bioheat transfer across
the tissue.
One of the most complicated problems in the heat transfer of skin tissue is blood
perfusion. Arkin et al. [3] have investigated the effect of blood perfusion in heat transfer
on the tissues. They argued that the Pennes interpretation of the vascular contribution
to heat transfer in perfuse tissues fails to account for the actual thermal equilibration
process between the flowing blood and the surrounding tissue. Lang et al. [4] used
the nonlinear three-dimensional heat transfer model based on temperature-dependent
blood perfusion in order to predict the temperature distribution. Xu et al. [5] used
the Pennes equation for modeling heat transfer in skin tissue. They applied the Green
function method for solving the Pennes equation.
Zhao et al. [6] solved the one-dimensional Pennes equation with a two-level finite
difference scheme. They compared the numerical and experimental results in order
to validate the new numerical scheme. Shih et al. [7] adopted the Laplace transform
method for solving the Pennes equation with an in-surface sinusoidal heating condition.
They observed that the temperature oscillation in the initial period caused
by the sinusoidal heating on the skin was unstable. Liu and Xu [8] performed the
blood perfusion estimation by the phase shift method in the temperature response
to sinusoidal heating for the Pennes equation. In their research, the sinusoidal heating
flux was considered constant, while this parameter was changeable in Ref. [7].
Erdmann et al. [9] studied the optimization of the temperature distribution for regional
hyperthermia based on the Pennes model by the finite element method. Ng et al.
[10,11] predicted the skin burn injury and thermal profiles within heated human
skin using the boundary element method. Durkee et al. [12] presented the exact
analytical solution to the multiregional time-dependent bioheat equation (Pennes
equation).
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142 Int J Thermophys (2013) 34:139–159
Fourier’s law assumes that any thermal distribution on the body is instantaneously
felt throughout the body. This means that the propagation speed of thermal distribution
is infinite. This assumption is reasonable in the majority of engineering applications.
This assumption is not true in the particular thermal condition, where heat conduction
shows a non-Fourier feature or thermal wave phenomena, or hyperbolic heat conduction.
The phase-lag behavior in the thermal wave may be justified if the layer has a
very small thickness [13] or the time scale of the problem is very short [14]. Hader
et al. [15] investigated the thermal behavior of a thin slab, under the effect of a fluctuating
volumetric thermal disturbance described by the hyperbolic and dual-phase-lag
heat conduction models. They found that the use of non-Fourier models is essential at
high frequencies of the volumetric disturbance. The validity of using the microscopic
hyperbolic heat conduction model under a harmonic fluctuating boundary heating
source was investigated by Naji et al. [16]. They found that the use of the microscopic
hyperbolic heat conduction model is essential when the value of the angular velocity
of the fluctuating temperature is greater than 1 × 10 9 rad · s −1 for most metallic layers
independent of their thickness.
The modification of Fourier’s law presented by Cattaneo [17] and Vernotte [18] is
a linear extension of the Fourier equation. The hyperbolic heat conduction model was
extended to describe the thermal behavior of an anisotropic material by Al-Nimr and
Naji [19]. In general, when materials (such as skin tissue) have a large relaxation time,
the thermal wave phenomenon is observed in heat transfer conduction. Liu et al. [20]
introduced a general form of the thermal wave model of bioheat transfer (TWMBT)
in living tissues for the first time. Liu and Lu [21] and Lu et al. [22,23] reported
that some thermal wave effects in bioheat transfer cannot be described by the Pennes
equation. Mitra et al. [24] performed different experiments on processed meat with
different boundary conditions. They observed wave-like phenomena in conduction
heat transfer.
Non-Fourier heat conduction of living tissue due to different kinds of heating methods
are studied by employing a thermal wave model of bioheat transfer, such as heating
by contact with hot material [25], laser heating [26–30], microwave heating [31,32],
radio-frequency heating [33], and the dual-phase-lag model [34].
In this study, the non-Fourier heat transport of skin biothermomechanics is solved
analytically, where the skin surface temperature is exposed to a suddenly hot source
of an arbitrary periodic temperature. The closed form of the temperature distribution
function is obtained analytically for both the Pennes equation (Fourier) and thermal
wave equation (non-Fourier) models for the skin as a finite and semi-infinite domain.
The Pennes and thermal wave equations are solved analytically by the Laplace transform
and the effect of metabolic heat. The thermal damage for skin for both Fourier
and non-Fourier models are studied here.
The finding of this study generates an exact analytical solution of the Pennes and
thermal wave models of bioheat transfer for skin tissue as a finite and semi-infinite
domain by considering an arbitrary periodic temperature for the skin surface boundary
condition. Studying the thermal behaviors of skin tissue with this boundary condition
by numerical solutions for Pennes, thermal wave, and dual-phase-lag bioheat transfer
models is future work of the authors.
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Int J Thermophys (2013) 34:139–159 143
2 Heat Transfer Models
2.1 Pennes Bioheat Transfer Equation (PBHTE)
For bioheat transfer, the Pennes equation is well established. This equation is based on
the classical Fourier law. The Pennes bioheat transfer equation (PBHTE) is expressed
as [2]
ρ t c t
∂T
∂t
+ ρ b ϖ b c b (T − T a ) = k ∂2 T
∂x 2 + q met + q ext (1)
Equation 1 is known as a parabolic bioheat equation. In this study, q ext is considered
as zero.
2.2 Thermal Wave Model of Bioheat Transfer (TWMBT)
Cattaneo [17] and Vernotte [18] reported a modified unsteady heat conduction equation
based on the concept of the finite heat propagation velocity as
∂q(x, t)
q(x, t) + τ q =−k∇T (x, t) (2)
∂t
BasedonEq.1, for a heat flux with the characteristic time τ q as well as the Pennes
equation, a general form of the thermal wave model of bioheat transfer (TWMBT) in
living tissues is expressed by Liu et al. [20]:
τ q ρ t c t
∂ 2 T
∂t 2
+ (ρ tc t + τ q ρ b ϖ b c b ) ∂T
∂t
+ ρ b ϖ b c b (T − T a )
)
(
= k ∂2 T
∂x 2 + ∂q met
∂q ext
q met + τ q + q ext + τ q
∂t
∂t
(3)
where τ q = α/C 2 is the thermal relaxation time [24,35].
3 Analytical Solution of the Problem
3.1 Arbitrary Periodic Surface Temperature Heating on a Finite-Domain Skin
The skin surface temperature is exposed to a suddenly hot source with a temperature
of T s (t). The skin tissue is considered as a perfect and infinitely wide and long layer.
The adiabatic thermal condition is selected for the right boundary of the skin tissue.
The geometrical problem and boundary conditions are shown in Fig. 1a.
3.1.1 Solution of Thermal Wave Bioheat Transfer Equation (TWMBT)
The bioheat transfer equation of a thermal wave as given in Eq. 3 is applied to describe
the wave-like heat transfer process through the skin tissue. For simplicity, the new
variable is defined as
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144 Int J Thermophys (2013) 34:139–159
T(0,t)=T s
Skin
T x (L,t)=0
T(0,t)=T s
Skin
T x (∞,t)=0
x=L
x
x
(a)
skin as a finite domain
(b)
skin as a semi-infinite domain
Fig. 1 Geometrical problem
θ = T − T a (4)
Equation 3, with respect to the constant q met and q ext = 0, can be rewritten in terms
of a new variable and it becomes
where
τ q γ 1
∂ 2 θ
∂t 2 + (γ 1 + τ q γ 2 ) ∂θ
∂t + γ 2θ = ∂2 θ
∂x 2 + q m (5)
γ 1 = ρ tc t
k ,
γ 2 = ρ bϖ b c b
, q m = q met
k
k
(6)
The initial and boundary conditions can be written as
θ(x, 0) = 0, θ t (x, 0) = 0 (7)
θ(0, t) = T s (t) − T a = θ s (t), θ x (L, t) = 0 (8)
If the boundary condition at the skin surface is periodic with an arbitrary function,
we can write it in the Fourier series form. By considering the step function form at the
skin surface that is shown in Fig. 2, the Fourier series form of the surface boundary
condition can be written as
123
∞∑
∞∑
θ s (t) = T s (t) − T a = a 0 + a n cos(ω n t) + b n sin(ω n t) (9)
n=1
n=1

Int J Thermophys (2013) 34:139–159 145
Fig. 2 Surface boundary
condition
where ω n = 2nπ/t 0 and the coefficients are obtained as
a 0 =
( )
(Tb −T a )+(T c −T a )(t 0 −)
t 0
a n = 2 t 0
[
(Tb −T a ) sin(ω n )
ω n
b n =− 2 t 0
[
(Tb −T a )(cos(ω n )−1)
ω n
]
+ (T c−T a )(sin(ω n t 0 )−sin(ω n ))
ω n
]
+ (T c−T a )(cos(ω n t 0 )−cos(ω n ))
ω n
(10)
By using the superposition theorem, this problem is divided into three sub-problems
with the following boundary conditions:
θ 1 (0, t) = a 0 , θ x (L, t) = 0 (11)
θ 2 (0, t) = cos (ω n t) , θ x (L, t) = 0 (12)
θ 3 (0, t) = sin (ω n t) , θ x (L, t) = 0 (13)
By taking the Laplace transform of Eq. 5 and applying the initial conditions, the
partial differential equation is changed into the following ordinary differential equation:
d 2 θ
dx 2 − P2 θ =− q m
s ,
P = √
τ q γ 1 s 2 + (γ 1 + τ q γ 2 )s + γ 2 (14)
The constant boundary condition problem (Eq. 11) is solved with q m and the other
boundary conditions are solved with considering q m = 0, because Eq. 5 is nonhomogeneous
due to term q m .
The solution of Eq. 14 and applying the Laplace transform of boundary conditions
(Eq. 11), can be obtained through
¯θ 1 (x, s) = a 0 cosh (P(x − L))
s cosh (PL)
− q m cosh (P(x − L))
sP 2 cosh(PL)
+ q m
sP 2 (15)
The inverse theorem is applied for the inverse Laplace transform. For complementary
information about the inverse theorem, refer to [36]. The first term of Eq. 15 has a pole
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146 Int J Thermophys (2013) 34:139–159
at s = 0 and two other simple poles at s = S m + , and s = S− m , these poles are obtained
as follows:
⎡
S m ± = 1
( ) ]
⎣−(γ 1 + τ q γ 2 ) ± √ (γ1 + τ q γ 2 )
2τ q γ 2 − 4τ q γ 1
[γ ⎤ 2 λm
2 + ⎦ ,
1 L
( ) 2m − 1
λ m =
π, m = 1, 2, ... (16)
2
By calculating the residues at s = 0, s = S m + , and s = S− m , the inverse Laplace
transform of the first term of Eq. 15 is obtained by
θ 11 (x, t) = a 0 cosh ( √
γ2 (x −L) ) ⎛
( )
∞
cosh ( √
γ2 L ) + ∑ 2a 0 e S+ m t λ m i cosh λm i
L
⎝
(x −L) ⎞
⎠
L
m=1
2 S m + (2τ q γ 1 S m + +γ 1 +τ q γ 2 ) sinh (λ m i)
⎛
( )
∞∑ 2a 0 e S− m t λ m i cosh λm i
L
+ ⎝
(x − L) ⎞
⎠
L 2 Sm − (2τ q γ 1 Sm − (17)
+ γ 1 + τ q γ 2 ) sinh (λ m i)
m=1
The poles of the second term of Eq. 15 are derived as follows:
s = 0, s =− 1 τ q
, s =− γ 2
γ 1
, s = S + m , s = S− m (18)
By calculating the residues based on the above poles, the inverse Laplace transform
of the second term of Eq. 15 is obtained as
θ 12 (x, t) =− q m cosh ( √
γ2 (x − L) )
γ 2 cosh ( √
γ2 L ) + q m τ q
( )e − t q
τq m γ 1
+ ( ) e − γ 2
γ1 t
τq γ 2 − γ 1 γ1 − τ q γ 2 γ2
(
∞∑ 2q m e S+ m t cosh ( λ m i ( x
−
L
− 1 )) )
S + m=1 m λ m i(2τ q γ 1 S m + + γ 1 + τ q γ 2 ) sinh (λ m i)
(
∞∑ 2q m e S− m t cosh ( λ m i ( x
−
L
− 1 )) )
Sm − λ m i(2τ q γ 1 Sm − (19)
+ γ 1 + τ q γ 2 ) sinh (λ m i)
m=1
The inverse Laplace transform of the third term of Eq. 15 is obtained by
θ 13 (x, t) = q m
⎛
⎝ 1 γ 2
−
τ qe − t
τq
τ q γ 2 − γ 1
+
γ 1 e − γ 2 t
γ 1
( )
γ 2 τq γ 2 − γ 1
⎞
⎠ (20)
Finally, by adding Eqs. 17, 19, and 20, the closed form of the function θ 1 (x, t) is
obtained by
123
θ 1 (x, t) = θ 11 (x, t) + θ 12 (x, t) + θ 13 (x, t) (21)

Int J Thermophys (2013) 34:139–159 147
In this section, q m is assumed to be zero in Eq. 14. With a similar method, the inverse
Laplace transforms of the solutions of this equation can be obtained by applying the
boundary conditions (Eqs. 12, 13) in the Laplace domain.
(√
)
cosh −τ q γ 1 ωn 2 + (γ 1 + τ q γ 2 )i ⌢ ω n + γ 2 (x − L) e iω nt
θ 2 (x, t) =
(√
)
2 cosh −τ q γ 1 ωn 2 + (γ 1 + τ q γ 2 )i ⌢ ω n + γ 2 L
(√
)
cosh −τ q γ 1 ωn 2 + (γ 1 + τ q γ 2 )i ⌢ ω n + γ 2 (x − L) e −iω nt
+
(√
)
2 cosh −τ q γ 1 ωn 2 − (γ 1 + τ q γ 2 )i ⌢ ω n + γ 2 L
⎡
∞∑ 2S
+ ⎣
m + λ mi cosh ( λ m i ( x
L
− 1 )) ⎤
e S+ m t
(
m=1 L 2 S m +2 (2τq
+ ωn) 2 γ 1 S m + )
⎦
+ γ 1 + τ q γ 2 sinh (λm i)
⎡
∞∑ 2S
+ ⎣
m −λ mi cosh ( λ m i ( x
L
− 1 )) ⎤
e S− m t
(
m=1 L 2 Sm −2 (2τq
+ ω n
2)
γ 1 Sm − )
⎦ (22)
+ γ 1 + τ q γ 2 sinh (λm i)
(√
)
cosh −τ q γ 1 ωn 2 + (γ 1 + τ q γ 2 )iω n + γ 2 (x − L) e iω nt
θ 3 (x, t) =
(√
)
2i cosh −τ q γ 1 ωn 2 + (γ 1 + τ q γ 2 )iω n + γ 2 L
(√
)
cosh −τ q γ 1 ωn 2 − (γ 1 + τ q γ 2 )iω n + γ 2 (x − L) e −iω nt
−
(√
)
2i cosh −τ q γ 1 ωn 2 − (γ 1 + τ q γ 2 )iω n + γ 2 L
⎡
∞∑
2ω
+ ⎣
n λ m i cosh ( λ m i ( x
L
− 1 )) ⎤
e S+ m t
(
m=1 L 2 S m +2 (2τq
+ ω n
2)
γ 1 S m + )
⎦
+ γ 1 + τ q γ 2 sinh (λm i)
⎡
∞∑ 2ω
+ ⎣
n λ m i cosh ( λ m i ( x
L
− 1 )) ⎤
e S− m t
( )
⎦ (23)
m=1 L 2 Sm
−2 + ωn
2 (2τ q γ 1 Sm − + γ 1 + τ q γ 2 ) sinh (λ m i)
Finally, the closed form of function θ(x, t) is obtained by
∞∑
∞∑
θ(x, t) = θ 1 (x, t) + a n θ 2 (x, t) + b n θ 3 (x, t) (24)
n=1
n=1
3.1.2 Solution of the Pennes Bioheat Transfer Equation (PBHTE)
The Pennes bioheat transfer equation as given in Eq. 1 is applied here to describe the
parabolic heat transfer process through the skin tissue. Substituting Eq. 4 by Eq. 1, it
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148 Int J Thermophys (2013) 34:139–159
gives
γ 1
∂T
∂t
+ γ 2 θ = ∂2 θ
∂x 2 + q m (25)
The boundary and initial conditions are similar to that of the thermal wave model.
Taking the Laplace transform of Eq. 25 and considering the initial condition, we have
the following:
d 2 θ
dx 2 − (γ 1s + γ 2 ) θ =− q m
s
(26)
Substituting the boundary conditions of Eq. 11 for the general solution of Eq. 26,the
function ¯θ P1 (x, s) on the Laplace domain is expressed as
¯θ P1 (x, s) =
[ ] [√
a0
s − q m cosh γ1 s + γ 2 (x − L) ]
s (γ 1 s + γ 2 ) cosh ( √
γ1 s + γ 2 L ) + q m
s (γ 1 s + γ 2 )
By employing the inverse theorem, the inverse Laplace transform of Eq. 27 is obtained
by
(
θ P1 (x, t) = a 0 − q ) [√
m cosh γ2 (x − L) ]
γ 2 cosh ( √
γ2 L ) + q m
γ 2
(( [√ ∞∑ 2 cosh γ1 S P + γ 2 (x − L) ] )(
))
e S P t
+
S P Lγ 1 sinh (√ γ 1 S P + γ 2 L ) √ q m
a 0 γ1 S P + γ 2 − √
γ1 S p + γ 2
n=1
(27)
(28)
where
S P =− γ 2
− 1 ( ) 2 λm
, λ m =
γ 1 γ 1 L
( 2m − 1
2
)
π, m = 1, 2, ... (29)
In this section, q m is assumed to be zero in Eq. 26. With a similar method, the inverse
Laplace transform of the solutions of this equation by applying the boundary conditions
(Eqs. 12, 13), can be obtained as
θ P2 (x, t) =
(√
)
cosh γ 1 i ⌢ ω n + γ 2 (x − L) e iωnt
(√
) +
2cosh γ 1 i ⌢ ω n + γ 2 L
+
(√
)
cosh −γ 1 i ⌢ ω n + γ 2 (x − L)
)
2cosh( √−γ1
i ⌢ ω n + γ 2 L
[ √ ∞∑ 2SP γ1 S P + γ 2 cosh (√ γ 1 S P + γ 2 (x − L) ) ]
e S P t
(
S
2
P
+ ωn
2 )
Lγ1 sinh (√ γ 1 S P + γ 2 L )
m=1
e −iωnt
(30)
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Int J Thermophys (2013) 34:139–159 149
θ P3 (x, t) =
(√
)
cosh γ 1 i ⌢ ω n + γ 2 (x − L) e iωnt
2i cosh (√ γ 1 iω n + γ 2 L )
+
m=1
(√
)
cosh −γ 1 i ⌢ ω n + γ 2 (x − L) e −iωnt
− 2i cosh (√ −γ 1 iω n + γ 2 L )
[ √ ∞∑ 2ωn γ1 S P + γ 2 cosh (√ γ 1 S P + γ 2 (x − L) ) ]
e S P t
(
S
2
P
+ ωn
2 )
Lγ1 sinh (√ γ 1 S P + γ 2 L )
(31)
Finally, the closed form of the function θ P (x, t) is obtained by
∞∑
∞∑
θ P (x, t) = θ P1 (x, t) + a n θ P2 (x, t) + b n θ P3 (x, t) (32)
n=1
n=1
3.2 Arbitrary Periodic Surface Temperature Heating on a Semi-infinite Domain Skin
In this case, the skin is assumed as a semi-infinite domain. Thus, the new boundary
condition in the Laplace domain is defined as
¯θ x (∞, s) = 0 (33)
The geometrical problem and boundary conditions are shown in Fig. 1b. The boundary
conditions at the skin surface are similar to that of Eqs. 11–13.
3.2.1 Solution of Thermal Wave Bioheat Transfer Equation (TWMBT)
By using the Laplace transform of the boundary conditions of Eq. 11 at the skin surface
and replacing the boundary condition at x = L with Eq. 33, the solution of Eq. 14 is
obtained as
¯θ 1 (x, s) =
The inverse Laplace transform of Eq. 34 is calculated by
( a0
s − q m
sP 2 )
e −Px + q m
sP 2 (34)
θ 1 (x, t) = £ −1 [ ¯θ 1 (x, s) ] =
∫ t
0
G 1 (t − v) Q (x,v) dv + £ −1 ( q m
sP 2 )
(35)
where
G 1 (t) = £ −1 ( a 0
s − q m
sP 2 )
, Q (x, t) = £ −1 ( e −Px) (36)
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150 Int J Thermophys (2013) 34:139–159
The function Q (x, t) is simplified as follows:
( (
£ −1 e −Px) =£
[exp
−1 −x
√τ q γ 1
(s+ 1 )(
s+ γ ) )]
2
τ q γ 1
⎡ ⎛
√
(
=exp − γ )
√√√√ ⎛
⎞⎞⎤
2
£ −1 ⎢ ⎜
⎣exp ⎝−x √ ( )
τq γ 1
1
γ 1 − τ q γ 2 s ⎝s+ ( ) ⎠⎟⎥
γ 1 γ 1 − τ q γ 2 τq
⎠⎦ ,
γ 1
γ 1 −τ q γ 2
γ 1
>τ q (37)
γ 2
In order to obtain the inversion of Eq. 37, the following equation is applied [37]:
(
£
[exp
−1 −x √ )] ⎡ ⎛ √
κs 2 + s
√ = £ −1 ⎣exp ⎝ −x κs ( s + 1 ) ⎞⎤
κ
√ ⎠⎦
a a
= δ (t − xb) s 1 (x, t) + H (t − xb) s 2 (x, t) (38)
where
( )
s 1 (x, t) = e − 2κ t 1
I0
√t
2κ
2 −x 2 b 2 ,
xbe − 2κ
t
s 2 (x, t) =
2κ √ t 2 − b 2 x I 2 1
( )
1 √t
2κ
2 − x 2 b 2
(39)
where b = √ κ/a. δ and H represent the Dirac delta and Heaviside theta functions,
respectively. With a similar method and using the Laplace transform of the boundary
conditions of Eqs. 12 and 13 at the skin surface and assuming the skin as a semi-infinite
domain, the closed form of functions θ(x, t) is obtained by
∫ t
θ(x, t) = D 1 +
123
+
+
∞∑
n=1
∞∑
n=1
0
a n
⎛
⎝
b n
⎛
⎝
∫ t
(G 1 (t − v) D 2 ) dv + H (t − xb)
∫ t
0
∫ t
0
xb
(G 2 (t − v) D 2 ) dv + H (t − xb)
(G 3 (t − v) D 2 ) dv + H (t − xb)
(G 1 (t − v) D 3 ) dv
∫ t
xb
∫ t
xb
⎞
(G 2 (t − v) D 3 ) dv⎠
⎞
(G 3 (t − v) D 3 ) dv⎠
(40)

Int J Thermophys (2013) 34:139–159 151
where
⎛
⎞
D 1 = q m
⎝ 1 −
τ qe − τq
t
γ 1 e − γ 2 t
γ 1
+ ( ) ⎠ (41)
γ 2 τ q γ 2 − γ 1 γ 2 τq γ 2 − γ 1
⎛
D 2 =δ (v − xb) exp ⎝−
(
v γ 1 − τ q γ 2 + γ 2
2τ q γ 1
γ 1
)
⎞
(( )
γ1 − τ
⎠ q γ 2 √ )
I 0 v
2τ q γ 2 − x 2 b 2 (42)
1
xb ( ) (
) (( )
γ 1 − τ q γ 2 exp − v(γ 1−τ q γ 2 +1) γ1 −τ
2τ q γ 1
I q γ 2
√ )
1 2τ q γ 1 v 2 − x 2 b 2
D 3 =
√ (43)
2τ q γ 1 v 2 − x 2 b 2
⎛
( )
G 1 (t − v) = a 0 − q m
⎝ 1 − τ qe − t−v
⎞
τq
+
γ 1e − γ 2 (t−v)
γ 1
( ) ⎠ (44)
γ 2 τ q γ 2 − γ 1 γ 2 τq γ 2 − γ 1
G 2 (t − v) = cos (ω n (t − v)) , G 3 (t − v) = sin (ω n (t − v)) , b = √ τ q γ 1 (45)
3.2.2 Solution of the Pennes Bioheat Transfer Equation (PBHTE)
By substituting the boundary conditions of Eq. 11 at the skin surface and Eq. 33 for
Eq. 26, the function ¯θ P1 (x, s) is obtained as
¯θ P1 (x, s) =
(
a0
s −
)
q m
e −√ γ 1 s+γ 2 x +
s (γ 1 s + γ 2 )
q m
s (γ 1 s + γ 2 )
(46)
The inverse Laplace transform of Eq. 46 is calculated by
θ P1 (x, t) = 1 2
√
1
(
a 0 − q )
m
γ 2
γ
( ) (
qm
− e −γ t £ −1
γ 2
((
) )
e −γ t £ −1 1
1
(√ √ ) − (√ √ ) e −√ √
γ 1 sx
s− γ s+ γ
e −√ γ 1
√ sx
s
)
( ) ( )
qm 1
+ £ −1
γ 2 s − 1
(s + γ )
(47)
where γ = γ 2 /γ 1 . In order to obtain the inversion of Eq. 47, the following equations
are applied [37]:
£ −1 (
£ −1 (
)
e −z√ s
√ s + a
e −z√ s
s
)
= √ 1
(
e −z2 /4t − ae za e a2t erfc a √ t +
πt
( ) z
= erfc
2 √ t
z
2 √ t
)
z ≥ 0 (48)
(49)
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152 Int J Thermophys (2013) 34:139–159
Finally, the closed form of functionθ P1 (x, t)is obtained by
θ P1 (x, t) = 1 (
2 √ a 0 − q ) (
m √γ
e −γ t e
− √ γ √ (
γ 1 x e γ t erfc − √ √ )
γ1 x
γ t +
γ γ 2 2 √ t
+ √ γ e √ γ √ ( √ ))
γ 1 √γ x e γ t γ1 x
erfc t +
2 √ t
( ) (√ ) ( )
qm
+ e −γ t γ1 x
erfc
γ 2 2 √ qm (1
+ − e
−γ t ) (50)
t γ 2
By substituting the boundary conditions of Eq. 12 at the skin surface and Eq. 33 for
Eq. 26 and taking q m = 0, the function ¯θ P2 (x, s) is obtained as
(
s
¯θ P2 (x, t) =
s 2 + ωn
2
)
e −√ γ 1 s+γ 2 x
(51)
The inverse Laplace transform of Eq. 51 is calculated as follows:
( ) ∫t
θ P2 (x, t) = e −γ t 1
2 √ (θ P21 (t − v) θ P22 (x, t)) dv (52)
γ + iω n
where
0
θ P21 (t − v) =−iω n e (t−v)(γ −iωn) + δ (t − v) (53)
θ P22 (x, t) = √ (
γ + iω n e −√ √
γ +iω n γ1 x e (γ +iωn)t erfc − √ √ )
√ γ1 x
γ + iω n t +
2 √ t
+ √ γ + iω n e √ (
√
γ +iω n γ1 √γ
√ )
x e (γ +iωn)t √ γ1 x
erfc + iωn t +
2 √ t
(54)
By substituting the boundary conditions of Eq. 13 at the skin surface and Eq. 33 for
Eq. 26 and taking q m = 0, the function ¯θ P3 (x, s) is
( )
ωn
¯θ P3 (x, t) =
s 2 + ωn
2 e −√ γ 1 s+γ 2 x
(55)
The inverse Laplace transform of Eq. 55 is obtained by
123
θ P3 (x, t) = e−γ t
4i
(( ) ( ) )
1
1
√ Y 1 (x, t) − √ Y 2 (x, t)
γ + iωn γ − iωn
(56)

Int J Thermophys (2013) 34:139–159 153
where
(
Y 1 (x, t)=a 1 e −za 1e a2 1 t erfc
Y 2 (x, t)=a 2 e −za 2e a2 2 t erfc
√
−a 1 t +
z
2 √ t
( √
−a 2 t +
z
2 √ t
)
(
+a 1 e za 1e a2 1 t erfc
)
+a 2 e za 2e a2 2 t erfc
)
√
a 1 t +
z
2 √ t
( √
a 2 t +
z
2 √ t
)
(57)
(58)
and
a 1 = √ γ + iω n , a 2 = √ γ − iω n , z = √ γ 1 x (59)
Finally, the closed form of the equation θ P (x, t) is obtained by
∞∑
∞∑
θ P (x, t) = θ P1 (x, t) + a n θ P2 (x, t) + b n θ P3 (x, t) (60)
n=1
n=1
3.3 Thermal Damage
The burn evaluation is one of the most essential characteristics in the bioengineering
science of skin tissue. The thermal damage begins when the basal layer temperature
rises to 44 ◦ C[38]. The basal layer is located between the epidermis and dermis layers
of the skin. A quantitative analysis of thermal damage was first proposed by Moritz
and Henriques [39,40] based on the fact that the tissue damage could be represented
as an integral of a chemical process rate:
=
∫ t
0
A exp (−E a /RT) dt (61)
where A is a material parameter equivalent to a frequency factor, E a is the activation
energy, and R is the universal gas constant. The constantsA and E a are obtained experimentally.
By fitting these experimental data, a linear relation is observed betweenE a
and ln(A), expressed as [41]
E a = 21149.324 + 2688.367ln(A) (62)
4 Results and Discussion
In this study, the arterial blood and maximum temperature of the skin surface boundary
condition are considered as T a = 376 ◦ C and T s = 100 ◦ C, respectively. The blood
perfusion rate is considered as W b = ρ b ϖ b = 0.5kg · m −3 · s −1 here [7]. The
other blood properties and one-layer skin properties are given in Table 1. Itisworth
mentioning that the blood and tissue properties in all the results are based on data
given in Table 1.
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154 Int J Thermophys (2013) 34:139–159
Table 1 Thermophysical properties of blood and skin tissue [42,43]
Parameters
Skin density
Skin specific heat
Thermal conductivity of skin
Metabolic heat generation
Skin depth
Blood density
Blood specific heat
Value
1190 kg · m −3
3600 J · kg −1 · K −1
0.235 W · m −1 · K −1
368.1 W · m −3
0.006 m
1060 kg · m −3
3770 J · kg −1 · K −1
Fig. 3 Temperature distribution at x = 0.0001 m for both the Pennes and thermal wave models of bioheat
transfer
The accuracy of the obtained analytical solutions is validated through comparison
with the existing numerical results. Therefore, the temperature profiles for both the
Pennes and thermal wave models for skin as a finite domain are obtained by applying
the parameters introduced by Liu et al. [34], shown in Fig. 3. They assumed that the
skin surface is exposed to a sudden heat source of constant temperature of 100 ◦ C and
after contacting for 15 s, the heat source is removed and skin is cooled by a coolant
of 0 ◦ C for 30 s. In this case, T c = 0 ◦ C, Δ = 15 s, t 0 = 45 s, and τ q = 10 s are
considered. In fact, the results of the first period are compared with the numerical
results of Liu et al. [34]inFig.3. By comparing the obtained results here with that of
Liu et al. [34]atx = 0.0001 m, good agreement is observed. Of course, the results of
this study have a little difference with the numerical results of Liu et al. [34], especially,
at the locations where the thermal shocks (instantaneous jumping) have occurred. In
the numerical solution of the thermal wave model due to the thermal shock, a large
123

Int J Thermophys (2013) 34:139–159 155
Fig. 4 Comparison between temperature response for the skin as a finite and semi-infinite domain at t =
100 s and 250 s for Pennes bioheat transfer equation
amount of oscillations was observed. The oscillations were decreased by increasing
the accuracy of the numerical solution. In the thermal wave bioheat transfer model,
the thermal wave propagation speed is finite, so instantaneous jumping is observed
in the skin tissue temperature (see Fig. 3). The magnitude of the relaxation time τ q
is an important characteristic of the thermal modeling of biomedical tissue and has
an important effect on the temperature prediction. Mitra et al. [24] observed similar
experimental wave-like results. These results show that the Pennes bioheat transfer
model could not predict the instantaneous jump in the skin temperature, while the
thermal wave model could predict this phenomenon correctly.
In the thermal wave model of bioheat transfer, the thermal relaxation time generally
is considered as τ q = 16 s for the skin tissue in previous research [24]; thus, τ q = 16 s is
used here. The temperature profiles along the skin depth for both skins, having finite
and semi-infinite domains at two different times for the Pennes and thermal wave
models of bioheat transfer are shown in Figs. 4 and 5, respectively. Here, T c = 37 ◦ C,
Δ = 45 s, and t 0 = 90 s are applied. The Pennes equation is based on an infinite speed
of thermal wave propagation and due to this assumption, the type of bottom surface
boundary condition gains an important effect on the temperature distribution even
during the initial stages of heating. Figure 4 illustrates that the discrepancy between
the results of the finite and semi-infinite skin domains is increased with increasing
time. As shown is Fig. 5, in the results of the thermal wave model, there is good
agreement in the finite and semi-infinite skin domains up to about 100 s, but after
this time reaches about t = 102.47 s, the temperature profiles related to the skin as
finite and semi-infinite domains become separated since the first thermal wave reaches
x = L. This discrepancy is related to the right surface boundary condition. As for the
finite domain, at the thermal wave reaches x = L, the adiabatic boundary condition
123

156 Int J Thermophys (2013) 34:139–159
Fig. 5 Comparison between temperature response for the skin as a finite and semi-infinite domain at t =
100 s and 250 s for thermal wave model of bioheat transfer
Fig. 6 Temperature response for the skin as a finite domain at x = 0.0001 m for both the Pennes and
thermal wave models of bioheat transfer
at x = L is not a suitable assumption because it means that heat cannot transfer into
the depth more than x = L. Therefore, the results of Figs. 4 and 5 are not reality at
large times and only illustrate that the boundary condition of the end of the skin is
not important in the results of the thermal wave model while the first thermal wave is
reached at x = L.
123

Int J Thermophys (2013) 34:139–159 157
Fig. 7 Temperature response for the skin as a finite domain at x = 0.0016 m for both the Pennes and
thermal wave models of bioheat transfer
The temperature response of the Pennes and thermal wave models for two selected
blood perfusion rates at two different spatial locations, x = 0.0001 m and x =
0.0016 m, are shown in Figs. 6 and 7, respectively. Here, T c =37 ◦ C, Δ = 5 s, and t 0 =
10 s are applied. In these figures, for the temperature profile of the thermal wave model,
a wave-like behavior is observed. As shown in Figs. 6 and 7, the first instantaneous
temperature jump profile at the locations of x = 0.0001 m and 0.0016 m occurred at
about t = 1.7 s and t = 27.3 s, respectively. The effect of the blood perfusion rate
in tissue heat transfer is investigated. In general, the skin temperature decreases with
an increase in the blood perfusion rate, where large amounts of heat can be carried
away through the rate of blood perfusion. Figure 6 shows that the blood perfusion rate
has little effect on the temperature distribution near the skin surface (x = 0.0001 m).
On the other hand, Fig. 7 shows that the blood perfusion rate has a relatively large
influence on heat transfer at the skin depth (x = 0.0016 m).
The variation of thermal damage with time for both the Pennes and thermal wave
models with τ q = 16 s, T c = 37 ◦ C,Δ = 5 s, and t 0 = 10 s at the locations of
x = 0.0001 m and 0.0016 m is shown in Fig. 8. In this study, the thermal damage
of skin is calculated by using the burn integration of Eq. 61, with the frequency
factor A = 3.1 × 10 98 and the ratio of activation energy to universal gas constant
E a /R = 75 000 [40]. A wave-like behavior can also be observed in the results of
the thermal damage of the thermal wave model. The results in Fig. 8 illustrate that,
in general, the thermal damage predicted by the Pennes and thermal wave models
are different. This difference is due to the lack of consideration with respect to the
non-Fourier effects on the materials that have the great thermal relaxation times when
exposed to sudden temperature variations. Thus, it can be deduced that on neglecting
these conditions, large deviations are observed in the predictions of the temperature
distribution, thermal damage, and thermal stress.
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158 Int J Thermophys (2013) 34:139–159
Fig. 8 Thermal damage profile for the skin as a finite domain at x = 0.0001 m and 0.0016 m for both the
thermal wave model and Pennes equation
5 Conclusions
The thermal behavior of skin tissue was investigated here. The thermal wave and
Pennes bioheat transfer models were applied for the energy equation in skin as finite
and semi-infinite domains. The governing equations were solved analytically by means
of the Laplace transform. The inverse theorem is applied to calculate the inverse
Laplace transform. The thermal damage was studied at x = 0.0001 m and 0.0016 m
locations. The obtained results were compared with those obtained by Liu et al. [34]
to validate the accuracy of the analytical solution here. This comparison shows good
agreement between both results. The results show that wave-like behavior is observed
for the thermal wave model and the thermal relaxation time τ q has an important effect
on the temperature distribution. The results indicate that the thermal wave model
could predict the instantaneous jump in the temperature profile, consistent with the
experimental results obtained by Mitra et al. [24], while the Pennes equation does not
show a prediction of this kind of jump in the temperature profile.
The results of the Pennes equation for skin as either a finite or semi-infinite domain
are compared. There is a discrepancy between the temperature responses from the
finite and semi-infinite domains even during the initial stage heating. On the other
hand, the relaxation time of skin is great and the speed of the thermal wave is finite.
The results of the thermal wave model show that two graphs of finite and semi-infinite
models of skin tissue are quite identical before the first thermal wave is propagated
at the right boundary of the tissue. The temperature profiles become separated. This
phenomenon show that the boundary condition at the end of the skin is not important
in the results of the thermal wave model when the first thermal wave reached x = L.
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Int J Thermophys (2013) 34:139–159 159
The thermal damage results show that the obtained burn times for the Pennes and
thermal wave models are different in general. The thermal wave behavior is observed
in the thermal damage profile.
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