Heat Transfer—Asian Research, 46 (1), 2017
Non-Fourier Boundary Conditions Effects on the Skin Tissue
Temperature Response
P. Forghani,1 H. Ahmadikia,2 and A. Karimipour1
of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Isfahan,
Iran
2 Department of Mechanical Engineering, University of Isfahan, Isfahan, Iran
1 Department
Thermal wave and dual phase lag bioheat transfer equations are solved analytically in the skin tissue exposed to oscillatory and constant surface heat flux. Comparison between the application of Fourier and non-Fourier boundary conditions on
the skin tissue temperature distributions is studied. The amplitude of temperature responses increases and also the phase shift between the temperature responses and
heat flux decreases under the non-Fourier boundary conditions for the case of an oscillatory surface heat flux. It is supposed the stable temperature cycles in order to
estimate the blood perfusion rate via the existing phase shift between the surface
heat fluxes and the temperature responses. It is shown that the higher rates of the
blood perfusion correspond to lower phase shift between the surface temperature responses and the imposed heat flux. C⃝ 2015 Wiley Periodicals, Inc. Heat Trans Asian
Res, 46(1): 29–48, 2017; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21196
Key words: non-Fourier boundary conditions, thermal wave, dual phase lag,
skin tissue, analytical solution
1. Introduction
Thermal behavior of the human body is characterized by the combination of heat generation
and heat transfer effects at various scales, viz., microscale (molecular or cellular level), meso-scale
(tissue and organ level), and macroscale (thermal interaction of the complete body with environment). It is well known that the skin is the largest single organ of the human body. Generally, skin is
made of three layers: epidermis, dermis, and hypodermis. Therapeutic applications include hyperthermia treatment, cryosurgery, laser surgery, and estimation of burn injury depends on the thermal
behavior of the skin tissue; therefore, the spatial and temporal distribution of the temperature in skin
tissue is of much significance [1–3].
The general bioheat transfer equation was first introduced in 1948 by Pennes. He assumed
that heat transfer is performed between tissue and blood in a capillary bed [4]. Later, several researchers as well as Wolf [5], Klinger [6], and Chen and Holmes [7] presented different models of
bioheat transfer equations; however, the simplicity of the Pennes model made it more practical than
C⃝
29
2015 Wiley Periodicals, Inc.
the other ones. A great number of researchers have examined the heat transfer in biological tissues
using the Pennes model [8–13].
The Pennes bioheat transfer equation is obtained based on the Fourier heat conduction law
which assumes an infinitely thermal wave (TW) propagation speed in each medium; however, this
fact is incompatible with physical reality. According to the Fourier heat conduction law, the heat flow
starts (stops) simultaneously with the appearance (disappearance) of a temperature gradient. This is
in contrast with the causality principle which states that two causally correlated events cannot happen at a same time; rather, the cause must precede the effect [14]. Fourier heat conduction law can
be applied in many industrial applications, but it has no usage yet in the cases of extremely short
periods of time, very high temperature gradient, and very low temperatures approaching absolute
zero. Cattaneo [15] and Vernotte [16] independently presented the TW equation in order to consider
a finite speed of heat propagation. They indicated that the heat flux vector and temperature gradient
must occur at different times in a material. In this way, they modified the Fourier heat conduction
law by considering a phase lag between heat flux and temperature gradient regarding the heat flux
relaxation time 𝜏 q . In homogenous materials, such as common metals, 𝜏 q lies between 10−8 s to
10−14 s [17, 18]. Biological tissues have much higher𝜏 q than the homogenous ones, due to their
nonhomogeneous structure [19]. The nonhomogeneous inner structure of biological tissues causes
the non-Fourier behaviors like the temperature oscillation and wave-like behavior which phenomena
cannot be described by the Pennes model. Temperature oscillation in biological tissues was first observed in 1950 by Richardson and colleagues [20] and then in 1985 by Roemer and colleagues [21].
Mitra and colleagues [22] conducted four different experiments on processed meat and observed
the sudden changes in the temperature distributions. They found the value of 𝜏 q for processed meat
is about 15.5 s. By the combination of Cattaneo–Vernotte TW equation with the general bioheat
transfer equation [23], Liu and colleagues [24] presented the TW model of bioheat transfer equation
(TWMBTE) to describe the wave-like behavior.
Liu [25] investigated the heat transfer phenomenon in biological tissues under the constant, sinusoidal, and step surface heat flux boundary conditions using the TWMBTE. Employing
TWMBTE, Liu and colleagues [26] numerically obtained the first and second-degree burn times for
a skin tissue exposed to transient heating. Thermal behavior of biological tissues exposed to microwaves was studied by Ozen and colleagues [27]. They found that temperature variations in the
skin tissue depend on different factors, such as blood perfusion rate, thermal conductivity, frequency
and power density of the microwave, and also exposure time. Their results imply that TWMBTE
predicts lower heat rise at the initial times in comparison with that of Pennes model due to the finite
speed of heat propagation. Several other research works can be referenced which concern the TW
model of bioheat transfer in biological tissues [28–34].
To consider the effects of microstructural interactions in the heat transfer process, Tzou [35]
proposed the dual phase lag (DPL) equation as follows:
q′′ (x, t) + 𝜏q
)
(
𝜕q′′ (x, t)
𝜕
= −k ∇T (x, t) + 𝜏T ∇T (x, t) ,
𝜕t
𝜕t
(1)
where 𝜏 T is the temperature gradient relaxation time created by microstructural interactions [35,
36]. In the TW model, the gradient of temperature precedes the heat flux, that is, the gradient of
30
temperature causes the heat flux. However, the temperature gradient can precede the heat flux or
vice versa in the DPL model. Moreover, the DPL model can be converted into the classical Fourier
heat conduction and TW models regarding different values of 𝜏 q and 𝜏 T [37–39].
By combining the general bioheat transfer equation [23] with Eq. (1), the DPL model of
bioheat transfer equation (DPLMBTE) can be obtained as follows:
𝜌c𝜏q
)
(
𝜕T
𝜕3T
𝜕2T
𝜕2T
𝜕T
+ 𝜌c
+ 𝜏T k
+ 𝜛b 𝜌b cb Tb − T + qmet + qext
+ 𝜏q 𝜛b 𝜌b cb
=k
2
2
2
𝜕t
𝜕t
𝜕t
𝜕x
𝜕t𝜕x
(
)
𝜕qmet 𝜕qext
+ 𝜏q
+
.
(2)
𝜕t
𝜕t
Antaki [40] examined heat transfer in the processed meat by applying the DPLMBTE. He
predicted the values of 𝜏 q and𝜏 T for the processed meat to be 14 s to 16 s and 0.043 s to 0.056
s, respectively. Liu and colleagues [41] studied the non-Fourier thermal behavior in the multilayer
skin with the DPLMBTE. They used a hybrid numerical scheme to solve the DPLMBTE. Moreover,
Zhou and colleagues [42] analyzed the DPL behavior of heat transfer in the biological tissue exposed
to laser heating. They simulated the biological tissue as a two-dimensional axisymmetric cylinder
and then numerically solved the DPLMBTE in this region. Different researchers have studied the
heat transfer in the biological tissue using DPLMBTE [43–49].
In the present article, TW and DPL bioheat transfer equations are solved analytically using
the Laplace transform method and inverse theorem under the non-Fourier boundary conditions. It
is worth to say that the analytical solution of bioheat transfer equations in skin tissue by utilizing
Laplace transform method and inverse theorem were presented by Ahmadikia and his co-authors
[50–55] and by Askarizadeh and Ahmadikia [56, 57] for the DPL heat transfer equations. However,
this study is conducted according to the bioheat transfer in skin tissue with non-Fourier oscillatory
boundary conditions which is rare in the previous works.
Nomenclature
c:
cb :
k:
L:
q′′ :
q0 ′′ :
qmet :
qext :
s:
t:
T:
Tb :
x:
Specific heat of tissue (J/kg K)
Specific heat of blood (J/kg K)
Tissue thermal conductivity (W/m K)
Skin thickness (m)
Heat flux (W/m2 )
Heat flux amplitude (W/m2 )
Metabolic heat generation (W/m3 )
External heat source (W/m3 )
Laplace domain parameter
Time (s)
Tissue temperature (K)
Blood temperature (K)
Coordinate variable (m)
31
Greek Symbols
𝛼:
𝛾:
Γ:
𝜂:
𝜃:
Λ:
𝜆:
𝜉:
𝜉L :
𝜌:
𝜌b :
𝜏q :
𝜏T :
𝜏1 :
𝜏2 :
𝜓:
𝜔:
𝜛b :
Tissue thermal diffusivity (m2 /s)
Dimensionless heat flux relaxation time for the case of a constant heat flux
Dimensionless temperature gradient relaxation time for the case of a constant heat flux
Dimensionless time
Dimensionless tissue temperature
Dimensionless blood perfusion rate
Eigenvalues
Dimensionless coordinate
Dimensionless skin thickness
Density of skin tissue (kg/m3 )
Density of blood (kg/m3 )
Heat flux relaxation time (s)
Temperature gradient relaxation time (s)
Dimensionless heat flux relaxation time for the case of an oscillatory heat flux
Dimensionless temperature gradient relaxation time for the case of oscillatory heat flux
Dimensionless metabolic heat generation
Heat flux frequency (s−1 )
Blood perfusion rate (s−1 )
2. Analytical Solution of the TW Bioheat Transfer Equation
The geometry of the skin tissue with the thickness of L and boundary conditions are demonstrated in Fig. 1.
Fig. 1. Geometry of the skin tissue.
32
The TWMBTE is solved for cosine and constant heat fluxon skin tissue with the following
initial and boundary conditions:
𝜕T (x, 0)
=0
𝜕t
′′
𝜕q (0, t)
𝜕T(0, t)
q′′ (0, t) + 𝜏q
= −k
𝜕t
𝜕x
𝜕T (L, t)
′′
q (L, t) = 0 ⇒
= 0.
𝜕x
T (x, 0) = Tb ,
(3)
(4)
(5)
2.1 Cosine heat flux
The applied heat flux on the skin surface is considered as cosine heat flux:
q′′ (0, t) = q0 ′′ cos (𝜔t) .
(6)
Thus, the non-Fourier boundary condition at the skin surface can be expressed as follows:
q0 ′′ cos (𝜔t) − 𝜏q q0 ′′ 𝜔 sin (𝜔t) = − k
𝜕T (0, t)
.
𝜕x
(7)
The governing equations beside the initial and boundary conditions are given in dimensionless forms using the√following variables:
√
T (x, t) − Tb 𝜔
𝜔
𝜉=
x, 𝜂 = 𝜔t, 𝜃(𝜉, 𝜂) = k
,
𝛼
q0 ′′
𝛼
√
√
W c
𝜔
k qmet
,
(8)
L𝜓 =
Λ = b b , 𝜏1 = 𝜔𝜏q , 𝜏2 = 𝜔𝜏T , 𝜉 L =
𝜌c𝜔
𝛼
𝜌c𝜔 q0 ′′
𝜏1
𝜕𝜃
𝜕2𝜃
𝜕2𝜃
+ (1 + Λ𝜏1 ) + Λ𝜃 =
+ 𝜓,
𝜕𝜂
𝜕𝜂 2
𝜕𝜉 2
𝜃 (𝜉, 0) = 0,
𝜕𝜃 (𝜉, 0)
= 0,
𝜕𝜂
𝜕𝜃 (0, 𝜂)
= − cos (𝜂) + 𝜏1 sin (𝜂) ,
𝜕𝜉
)
(
𝜕𝜃 𝜉 L , 𝜂
= 0,
𝜕𝜉
(9)
(10)
(11)
(12)
where Wb = 𝜌b 𝜛b . The Laplace transform of Eqs. (9) to (12) would lead to:
𝜓
d 2𝜃
− 𝛽𝜃 = − , 𝛽 = 𝜏1 s 2 + (1 + Λ𝜏1 )s + Λ,
s
d𝜉 2
𝜏 −s
d𝜃 (0, s)
,
= 1
2
d𝜉
(
)s + 1
d𝜃 𝜉 L , s
= 0.
d𝜉
33
(13)
(14)
(15)
Equation (13) is solved regarding the boundary condition Eqs. (14) and (15), and consequently the dimensionless tissue temperature distribution in the Laplace domain is obtained as follows:
(√
)
⎛
⎞
cosh
𝛽(𝜉
−
𝜉
)
L
s − 𝜏1
⎜
⎟ 𝜓
𝜃 (𝜉, s) = (
) ⎟+ .
(√
)√ ⎜
2
𝛽 ⎜ sinh
s +1
⎟ s𝛽
𝛽𝜉 L
⎝
⎠
(16)
Applying the inverse theorem of the Laplace transform [58], Eq. (16) is inverted and𝜃(𝜉, 𝜂)
is obtained as:
)
)
(√
(√
(
)
(
)
e−𝜂i −i − 𝜏1 cosh
𝛽(i)(𝜉 − 𝜉 L )
𝛽(−i)(𝜉 − 𝜉 L )
e𝜂i i − 𝜏1 cos h
+
𝜃(𝜉, 𝜂) =
)
)
(√
(√
√
√
2i 𝛽(i) sin h
𝛽(i)𝜉 L
−2i 𝛽(−i) sin h
𝛽(−i)𝜉 L
) )
((
(
)
𝜉−𝜉 L
𝜂S1n S1 − 𝜏 cos
∞
𝜆n
2e
∑
n
1
𝜉L
+
)
(
)
(
2
n=1 S1n + 1 2𝜏1 S1n + Λ𝜏1 + 1 𝜉 L cos(𝜆n )
) )
((
(
)
𝜉−𝜉 L
𝜂S2n S2 − 𝜏 cos
∞
𝜆n
2e
∑
n
1
𝜉L
+
)
(
)
(
2
n=1 S2n + 1 2𝜏1 S2n + Λ𝜏1 + 1 𝜉 L cos(𝜆n )
)
(
)
−𝜂 (
−𝜂
e 𝜏1 −1 − 𝜏12 𝜏1
e−𝜂Λ Λ + 𝜏1
𝜓𝜏1 e 𝜏1
𝜓
𝜓e−𝜂Λ
+(
+ + (
,
+(
)(
)
)−
)
)(
Λ Λ Λ𝜏1 − 1
Λ𝜏1 − 1
Λ2 + 1 Λ𝜏1 − 1 𝜉 L
1 + 𝜏12 Λ𝜏1 − 1 𝜉 L
(17)
where
√
−(1 + Λ𝜏1 ) ±
(
( )2 )
𝜆
(1 + Λ𝜏1 ) − 4𝜏1 Λ + 𝜉 n
2
L
S1n , S2n =
𝜆n = n𝜋,
2𝜏1
n = 1, 2, 3, …
𝛽(i) = −𝜏1 + (1 + Λ𝜏1 )i + Λ,
𝛽(−i) = −𝜏1 − (1 + Λ𝜏1 )i + Λ.
(18)
(19)
2.2 Constant heat flux
The non-Fourier boundary condition at the skin surface can be written as follows by considering the constant heat flux through it:
q0 ′′ = − k
𝜕T(0, t)
.
𝜕x
(20)
The governing equation beside the initial and boundary conditions can be expressed in dimensionless forms by using the following dimensionless variables:
√
W c
W c
T (x, t) − Tb √
Wb cb
kWb cb , 𝛾 = b b 𝜏q ,
x, 𝜂 = b b t, 𝜃(𝜉, 𝜂) =
𝜉=
′′
k
𝜌c
𝜌c
q0
34
W c
Γ = b b 𝜏T ,
𝜌c
√
𝜓=
k qmet
,
Wb cb q0 ′′
√
𝜉L =
Wb cb
L.
K
(21)
Take the Laplace transform of the dimensionless TW equation and dimensionless boundary
conditions and then applying the inverse theorem of the Laplace transform leads to:
) )
((
𝜉−𝜉 L
𝜂
𝜂S1n cos
∞
−
𝜆n
2e
∑
cos h(𝜉 − 𝜉 L )
𝜉L
𝛾e 𝛾
e−𝜂
−
+
+
𝜃(𝜉, 𝜂) =
(
)
sin h(𝜉 L )
(𝛾 − 1)𝜉 L (𝛾 − 1)𝜉 L n=1 S1n 2𝛾 S1n + 𝛾 + 1 𝜉 L cos(𝜆n )
) )
((
𝜉−𝜉 L
∞
−𝜂
𝜆n
2e𝜂S2n cos
∑
𝜉L
𝜓e−𝜂 𝜓𝛾e 𝛾
+𝜓 +
−
,
(22)
+
(
)
𝛾 −1
𝛾 −1
n=1 S2n 2𝛾 S2n + 𝛾 + 1 𝜉 L cos(𝜆n )
where
√
(
( )2 )
𝜆
− (1 + 𝛾) ± (1 + 𝛾)2 − 4𝛾 1 + 𝜉 n
L
S1n , S2n =
𝜆n = n𝜋,
2𝛾
n = 1, 2, 3, …
(23)
The Fourier boundary conditions at the skin tissue exposed to constant and cosine heat fluxes
were solved by Ahmadikia and colleagues [50].
3. Analytical Solution of the DPL Bioheat Transfer Equation
The solution of the DPLMBTE under the non-Fourier boundary conditions is presented in
this section. The corresponding non-Fourier boundary conditions are expressed as follows:
(
)
𝜕q′′ (0, t)
𝜕T(0, t)
𝜕 𝜕T(0, t)
= −k
− k𝜏T
,
𝜕t
𝜕x
𝜕t
𝜕x
(
)
𝜕T (L, t)
𝜕 𝜕T (L, t)
′′
+ 𝜏T
= 0.
q (L, t) = 0 ⇒
𝜕x
𝜕t
𝜕x
q′′ (0, t) + 𝜏q
(24)
(25)
3.1 Cosine heat flux
The non-Fourier boundary condition at the skin surface is presented as follows for this case:
q0 ′′ cos (𝜔t) − 𝜏q q0 ′′ 𝜔 sin (𝜔t) = −k
𝜕T (0, t)
𝜕
− k𝜏T
𝜕x
𝜕t
(
𝜕T(0, t)
𝜕x
)
.
(26)
The dimensionless DPL equation and dimensionless non-Fourier boundary conditions can
be obtained as follows by applying Eq. (8):
𝜏1
𝜕𝜃
𝜕2𝜃
𝜕2 𝜃
𝜕3𝜃
+ (1 + Λ𝜏1 ) + Λ𝜃 =
+ 𝜏2
+ 𝜓,
𝜕𝜂
𝜕𝜂 2
𝜕𝜉 2
𝜕𝜂𝜕𝜉 2
35
(27)
𝜕𝜃 (0, 𝜂)
𝜕 2 𝜃 (0, 𝜂)
+ 𝜏2
= − cos (𝜂) + 𝜏1 sin (𝜂) ,
𝜕𝜉
𝜕𝜂𝜕𝜉
(28)
𝜕𝜃 (L, 𝜂)
𝜕 2 𝜃 (L, 𝜂)
+ 𝜏2
= 0.
𝜕𝜉
𝜕𝜂𝜕𝜉
(29)
Taking the Laplace transform of the dimensionless DPL equation and dimensionless boundary conditions and then applying the inverse theorem of the Laplace transform would result in:
)
)
(√
(√
(
)
(
)
𝛽(i)(𝜉 − 𝜉 L )
𝛽(−i)(𝜉 − 𝜉 L )
e−𝜂i −i − 𝜏1 cos h
e𝜂i i − 𝜏1 cos h
𝜃(𝜉, 𝜂) = (
)+
)
(√
(√
)√
(
)√
2i 1 + 𝜏2 i
𝛽(i) sin h
𝛽(i)𝜉 L
−2i 1 − 𝜏2 i
𝛽(−i) sin h
𝛽(−i)𝜉 L
) )
((
(
)(
)
𝜉−𝜉 L
𝜂S1n S1 − 𝜏
∞
𝜆n
2e
S1
1
+
𝜏
cos
∑
n
1
2
n
𝜉L
+
(
)
(
)
2
2
n=1 S1n + 1 𝜏1 𝜏2 S1n + 2𝜏1 S1n + Λ𝜏1 − Λ𝜏2 + 1 𝜉 L cos(𝜆n )
) )
((
(
)(
)
𝜉−𝜉 L
∞
𝜆n
2e𝜂S2n S2n − 𝜏1 1 + 𝜏2 S2n cos
∑
𝜉L
+
)
(
)
(
2
2
n=1 S2n + 1 𝜏1 𝜏2 S2n + 2𝜏1 S2n + Λ𝜏1 − Λ𝜏2 + 1 𝜉 L cos(𝜆n )
)
(
)
−𝜂 (
−𝜂
e 𝜏1 −1 − 𝜏12 𝜏1
e−𝜂Λ Λ + 𝜏1
𝜓𝜏1 e 𝜏1
𝜓
𝜓e−𝜂Λ
+(
+ + (
,
+(
)(
)
)−
)
)(
Λ Λ Λ𝜏1 − 1
Λ𝜏1 − 1
Λ2 + 1 Λ𝜏1 − 1 𝜉 L
1 + 𝜏12 Λ𝜏1 − 1 𝜉 L
(30)
where
S1n , S2n =
(
( )2 )
𝜆
− 1 + Λ𝜏1 + 𝜏2 𝜉 n
±
L
√
(
(
( )2 )2
( )2 )
𝜆
𝜆
− 4𝜏1 Λ + 𝜉 n
1 + Λ𝜏1 + 𝜏2 𝜉 n
L
L
2𝜏1
𝜆n = n𝜋,
n = 1, 2, 3, …
𝛽(i) =
−𝜏1 + (1 + Λ𝜏1 )i + Λ
,
1 + 𝜏2 i
(31)
𝛽(−i) =
−𝜏1 − (1 + Λ𝜏1 )i + Λ
.
1 − 𝜏2 i
(32)
The Solution of the DPLMBTE under the Fourier boundary conditions was reported by
Askarizadeh and Ahmadikia [53].
3.2 Constant heat flux
The non-Fourier boundary condition at skin surface would be expressed as follows if a constant heat flux was applied:
(
)
𝜕T(0, t)
𝜕 𝜕T(0, t)
q0 ′′ = −k
− k𝜏T
.
(33)
𝜕x
𝜕t
𝜕x
DPLMBTE, initial, and boundary conditions can be derived in dimensionless forms by using
Eq. (21). Moreover, take the Laplace transform of the dimensionless DPL equation and dimensionless boundary conditions and then applying the inverse theorem of the Laplace transform would lead
36
Table 1. Thermophysical properties of the skin tissue and blood [53, 59, 60]
Parameters
Units
Values
3
Density
kg/m
Specific heat
J/kgK
Thermal conductivity of skin
Skin
1000
Blood
1060
Skin
4187
Blood
3860
W/mK
0.628
m
0.006
Skin depth
3
Metabolic heat generation
W/m
368.1
to:
−𝜂
cos h(𝜉 − 𝜉 L )
𝛾e 𝛾
e−𝜂
𝜃(𝜉, 𝜂) =
−
+
sin h(𝜉 L )
(𝛾 − 1)𝜉 L (𝛾 − 1)𝜉 L
) )
((
𝜉−𝜉 L
∞
𝜆n
2e𝜂S1n (1 + ΓS1n ) cos
∑
𝜉L
+
2
n=1 S1n (𝛾ΓS1n + 2𝛾 S1n + 𝛾 − Γ + 1)𝜉 L cos(𝜆n )
) )
((
(
)
𝜉−𝜉 L
∞
−𝜂
𝜆n
2e𝜂S2n 1 + ΓS2n cos
∑
𝜉L
𝜓e−𝜂 𝜓𝛾e 𝛾
+
+𝜓 +
−
, (34)
(
)
2
𝛾 −1
𝛾 −1
n=1 S2n 𝛾ΓS2n + 2𝛾 S2n + 𝛾 − Γ + 1 𝜉 L cos(𝜆n )
where
S1n , S2n =
(
( )2 )
𝜆
− 1 + 𝛾 + Γ 𝜉n
±
𝜆n = n𝜋,
L
√
(
(
( )2 )
( )2 )2
𝜆n
𝜆
− 4𝛾 1 + 𝜉 n
1+𝛾 +Γ 𝜉
L
L
2𝛾
n = 1, 2, 3, …
(35)
4. Results and Discussion
The TW and DPL bioheat transfer equations for non-Fourier boundary conditions are studied
and the results are compared with those of the Fourier boundary conditions. Thermophysical properties of the biological tissue and blood are presented in Table 1. The blood perfusion rate is ϖb =
1.87 × 10−3 s−1 [53, 59] and the oscillatory and constant heat flux amplitude is q0 ′′ = 5000 W/m2
[50].
To validate the accuracy of the analytical solution of this study, first the boundary conditions
are considered as Fourier for the oscillatory heat flux and then the obtained results are compared with
those of Ahmadikia and colleagues [50] for the TW model (Fig. 2(a)) and with those of Askarizadeh
and Ahmadikia [53] for the DPL model (Fig. 2(b)).
37
Fig. 2. Temperature responses at the skin surface from this study versus (a) Ahmadikia and
colleagues [50] for TW model, (b) Askarizadeh and Ahmadikia [53] for DPL model, and (c)
Tzou [35]. Comparison between the analytical and numerical inversions when for the
dimensionless temperature response in a finite slab when 𝛾 = 𝜂 = 0.05. [Color figure can be
viewed in the online issue, which is available at wileyonlinelibrary.com/journal/htj.]
38
Fig. 3. Temperature responses at the basal layer and DF interface under the Fourier and
non-Fourier boundary conditions for (a) TW model and (b) DPL model. [Color figure can be
viewed in the online issue, which is available at wileyonlinelibrary.com/journal/htj.]
Moreover, the accuracy of the analytical inversion procedure is also checked by comparing
it with the numerical one which is used by Tzou [35]. This comparison is conducted when both
analytical and numerical inversion procedure are applied on the transformed temperature field in
the Tzou [35] study. The results of this process are shown in Fig. 2(c), where the consistency of
them approved conducted analytical inversion once more. It should be noted that developing an advanced experimental technique for the study of temporal and spatial interaction effects, that can
be described theoretically by the DPL model, is very difficult and requires heavy costs. Thus, an
efficient analytical study could be very useful and applicable in these situations. Figures 3(a) and
3(b) show the temperature responses for x = 0.0001 m and x = 0.0016 m (DF interface) based on
39
the TW and DPL models under the Fourier and non-Fourier boundary conditions at skin surface.
In these figures, the amplitude and frequency of the applied oscillatory heat flux at the skin surface are q0 ′′ = 5000 W/m2 and 𝜔 = 0.05 s−1 , respectively. The relaxation times are 𝜏 q = 16 s and
𝜏 T = 8 s. It is observed that the amplitude of temperature response is decreased for both models of
TW and DPL under the Fourier and non-Fourier boundary conditions through the more inner regions.
Furthermore, the amplitude of temperature response is smaller than that of the Fourier boundary conditions when the boundary conditions are supposed as non-Fourier for the initial unstable time. The
amplitude of temperature response in the non-Fourier boundary conditions increases with time in
spite of the Fourier boundary conditions.
The effects of 𝜏 q and𝜏 T on the skin temperature response under the non-Fourier boundary
conditions are demonstrated in Figs. 4(a) to 4(c) when the amplitude and frequency of the applied
oscillatory heat flux are q0 ′′ = 5000 W/m2 and 𝜔 = 0.05 s−1 , respectively. As observed in Fig. 4(a),
more 𝜏 q corresponds to the phase lag in the temperature responses for the TW model. The skin
temperature responses in the DPL model is presented in Fig. 4(b) for different values of 𝜏 q when
𝜏 T = 8 s. It is as previously discussed, the increasing of 𝜏 q leads to the phase lag of the temperature
responses and an increase of the temperature responses amplitude. It should be noted that the phase
lag in this model is lower and the amplitude is higher than that of the TW model. Figure 4(c) presents
the skin temperature responses for the DPL model with different values of 𝜏 T for 𝜏 q = 16 s. It is shown
that a higher value for 𝜏 T leads to the precedence of the temperature response and decrease of the
temperature response amplitude.
Figures 5(a) and 5(b) depict the temperature responses at the skin surface for different values
of the applied oscillatory heat flux frequencies on skin surface for the TW and DPL models. The
Fourier and non-Fourier boundary conditions are used while the applied oscillatory heat flux amplitude is chosen as q0 ′′ = 5000 W/m2 . Here, the amplitude and cyclic time of the skin temperature
response decrease for higher amounts of the frequency of the applied oscillatory heat flux. However,
at the state of considering the non-Fourier boundary conditions, the temperature response would
experience a smaller decrease with more heat flux frequency compared to Fourier boundary conditions. Hence, the amplitude decline of temperature responses in the DPL model is greater than that
of the TW model for the non-Fourier boundary conditions at higher values of applied oscillatory
heat flux frequency.
Figures 6(a) and 6(b) illustrate the dimensionless temperature responses at the skin surface for the TW and DPL models under the Fourier and non-Fourier boundary conditions at q0 ′′
= 5000 W/m2 and 𝜔 = 0.05 s−1 , respectively. In these figures, the relaxation times are taken in to account as 𝜏 q = 16 s and 𝜏 T = 8 s. It is observed that regarding the non-Fourier boundary conditions, the
phase shift between the heat flux and temperature response is lower than those of Fourier boundary
conditions. This phenomenon leads to generate the blood perfusion rate [60]. Under the Fourier
boundary conditions, the first cycle of temperature response, for instability, cannot be used to estimate the blood perfusion rate. Hence the second cycle is used. The second cycle of temperature
response for the TW model under the Fourier boundary conditions occurs in the dimensionless time
of 7.425 which equals to 148.5 s; in addition, the second cycle of temperature response for DPL
model under the Fourier boundary conditions occurs in the dimensionless time of 7.2 which equals
144 s. Under the non-Fourier boundary conditions, the first and second cycles of temperature response, due to instability, cannot be used to estimate blood perfusion rate, thus the third cycle is
40
Fig. 4. Temperature responses under the non-Fourier boundary conditions at different relaxation
times for (a) TW model, (b) DPL model with different 𝜏 q values, and (c) DPL model for different
𝜏 T values. [Color figure can be viewed in the online issue, which is available at
wileyonlinelibrary.com/journal/htj.]
41
Fig. 5. Temperature responses at the skin surface under the Fourier and non-Fourier boundary
conditions for different heat flux frequencies for (a) TW model and (b) DPL model. [Color figure
can be viewed in the online issue, which is available at wileyonlinelibrary.com/journal/htj.]
used. The third cycle of temperature response for the TW model under the non-Fourier boundary
conditions occurs in the dimensionless time of 13.035 which equals 260.7 s and the third cycle of
temperature response for DPL model under the non-Fourier boundary conditions occurs in the dimensionless time of 13.1850 which equals 263.7 s. It can be deduced that the non-Fourier boundary
conditions lead to generate instability through the temperature response in the first and second cycles
and it would become stable after the third cycle.
42
Fig. 6. Dimensionless temperature responses at the skin surface and heat flux variations under the
Fourier and non-Fourier boundary conditions for (a) TW model and (b) DPL model. [Color figure
can be viewed in the online issue, which is available at wileyonlinelibrary.com/journal/htj.]
The dimensionless temperature response at the skin surface under the non-Fourier boundary
conditions and dimensionless heat flux variations for both TW and DPL models for different values of
the blood perfusion rate are shown in Figs. 7(a) and 7(b). The amplitude and frequency of the applied
oscillatory heat flux on skin surface are q0 ′′ = 5000 W/m2 and 𝜔 = 0.05 s−1 , respectively, and also
the relaxation times are 𝜏 q = 16 s and 𝜏 T = 8 s. As observed, for small values of the dimensionless
blood perfusion rate (Λ = 0.01 and Λ = 0.1), the dimensionless temperature responses coincide
with each other and also the difference of phase shift between these two lines is very difficult to
distinguish. Thus, a highly precise device is required to recognize the phase shift between these cases.
43
Fig. 7. Dimensionless temperature responses at the skin surface and heat flux variations under the
Fourier and non-Fourier boundary conditions for different Λ values, (a) TW model and (b) DPL
model. [Color figure can be viewed in the online issue, which is available at
wileyonlinelibrary.com/journal/htj.]
Higher blood perfusion corresponds to more transferred heat in a short time so that the amplitude
of the temperature response and the phase shift between temperature response and heat flux would
decrease. This phenomenon is observed clearly at the dimensionless blood perfusion rate Λ = 1 in
Figs. 7(a) and (b). Here, the phase shift of the TW model is less than the DPL model. It is well
known that the blood perfusion rate in the human body is 2.86 kg/m3 s [61]. As a result, the proper
dimensionless blood perfusion rate in this study is considered as Λ = 0.01 and Λ = 0.1 which is
equal to the blood perfusion rate of Wb = 0.542 kg/m3 s and Wb = 5.424 kg/m3 s, respectively. The
44
Fig. 8. Temperature response of the DPL model for the Fourier and non-Fourier boundary
conditions at the basal layer for constant heat flux. [Color figure can be viewed in the online issue,
which is available at wileyonlinelibrary.com/journal/htj.]
blood perfusion rate decreases with more oscillatory heat flux on the skin surface. Moreover, and
based on the relation between the applied oscillatory heat flux and blood perfusion rate for different
values of the applied heat flux frequency, different intervals are obtained for the blood perfusion rate
which is widely used in medicine.
Figure 8 demonstrates the temperature response at a depth of x = 0.0001 m (basal layer) for
the DPL model under the Fourier and non-Fourier boundary conditions for a constant amount of heat
flux with the amplitude of q0 ′′ = 5000 W/m2 along the skin surface. In this figure the relaxation times
are chosen as 𝜏 q = 16 s and 𝜏 T = 8 s. It is seen that when the boundary conditions are considered as
non-Fourier, the temperature response is lower than that of the Fourier boundary conditions.
5. Conclusion
An analytical solution of the TW and DPL bioheat transfer equations on skin tissue exposed
to constant and oscillatory heat flux under the non-Fourier boundary conditions was presented. To do
this, the Laplace transform method and inverse theorem were applied. The results of the oscillatory
heat flux revealed that non-Fourier boundary conditions led to decrease the temperature response
amplitude at the initial unstable period of time and it is increased in the stable time interval. The
non-Fourier boundary conditions reduced the phase shift between temperature response and heat
flux.
The amplitude and cyclic time of the skin temperature response decreased at higher values
of applied oscillatory heat flux frequency along the skin surface. Moreover, with an increase in the
frequency of the applied heat flux, the temperature response amplitude experienced less reduction for
the state of non-Four boundary conditions compared to the Fourier one. The non-Fourier boundary
conditions led to generate instability through the temperature response in the first and second cycles.
However, the temperature response became stable after the third cycle. Hence, the first and second
45
cycles could not be used to estimate the blood perfusion rate and the third cycle had to be applied.
Furthermore, the phase shift between temperature response and the heat flux, decreased with more
blood perfusion rate.
Literature Cited
1. Whitton JT, Everall JD. The thickness of the epidermis. British J Dermatol 1973;89:467–476.
2. Bhowmik A, Singh R, Repaka R, Mishra SC. Conventional and newly developed bioheat transport models in vascularized tissues: A review. J Thermal Biol 2013;38:107–125.
3. Fan J, Wang L. A general bioheat model at macroscale. Int J Heat Mass Transfer 2011;54:722–
726.
4. Pennes HH. Analysis of tissue and arterial blood temperature in the resting human forearm. J
Appl Physiol 1948;1:93–122.
5. Wulff W. The energy conservation equation for living tissue. IEEE Trans Biomed Eng
1974;21:494–495.
6. Klinger HG. Heat transfer in perfused biological tissue. II. The ‘macroscopic’ temperature distribution in tissue. Bull Math Biol 1978;40:183–199.
7. Chen MM, Holmes KR. Microvascular contributions in tissue heat transfer. Ann N Y Acad Sci
1980;335:137–150.
8. Liu J, Xu LX. Estimation of blood perfusion using phase shift in temperature response to sinusoidal heating the skin surface. IEEE Trans Biomed Eng 1999;46:1037–1042.
9. Ng EYK, Chua LT. Comparison of one-and two-dimensional programmes for predicting the
state of skin burns. Burns 2002;28:27–34.
10. Chan CL. Boundary element method analysis for the bioheat transfer equation. J Biomech Eng
1992;114:358–365.
11. Chatterjee I, Adams RE. Finite element thermal modelling of the human body under hyperthermia treatment for cancer. Int J Computer Appl Technol 1994;7:151–159.
12. Zhao JJ, Zhang J, Kang N, Yang F. A two level finite difference scheme for one dimensional
pennes bioheat equation. Appl Math Comput 2005;171:320–331.
13. Karaa S, Zhang J, Yang F. A numerical study of a 3D bioheat transfer problem with different
spatial heating. Math Computers Simul 2005;68:375–388.
14. Cimmelli VA. Boundary conditions for diffusive hyperbolic systems in non-equilibrium thermodynamics. Technische Mechanik 2002, 181, Band 22, Heft 3.
15. Cattaneo C. A form of heat conduction equation which eliminates the paradox of instantaneous
propagation. Compte Rendus 1958;247:431–433.
16. Vernotte P. Les paradoxes de la theorie continue de l’equation de la chaleur. Compte Rendus
1958;246:3154–3155.
17. Luikov AV. Analytical heat diffusion theory. New York: Academic Press; 1968.
18. Braznikov AM, Karpychev VA, Luikova AV. One engineering method of calculating heat conduction process. Inzhenerno-Fizicheskii Zhurnal 1975;28:677–680.
19. Kaminski W. Hyperbolic heat conduction equation for materials with a non-homogeneous inner
structure. J Heat Transf 1990;112:555–560.
20. Richardson AW, Imig CG, Feucht BL, Hines HM. Relationship between deep tissue temperature and blood flow during electromagnetic irradiation. Archives Phys Med Rehab 1950;31:
19–25.
21. Roemer RB, Oleson JR, Cetas TC. Oscillatory temperature response to constant power applied
to canine muscle. Am J Physiol 1985;249:153–158.
22. Mitra K, Kumar S, Vedavarz A, Moallemi MK. Experimental evidence of hyperbolic heat conduction in processed meat. J Heat Transf1995;117:568–573.
46
23. Liu J, Ren Z, Wang C. Interpretation of living tissue’s temperature oscillations by thermal wave
theory. Chinese Sci Bull 1995;40:1493–1495.
24. Zhu L. Standard handbook of biomedical engineering and design. McGraw-Hill; 2004.
25. Liu KC. Thermal propagation analysis for living tissue with surface heating. Int J Thermal Sci
2008;47:507–513.
26. Liu J, Chen X, Xu LX. New thermal wave aspects on burn evaluation of skin subjected to instantaneous heating. IEEE Trans Biomed Eng 1999;46(4):420–428.
27. Özen Ş, Helhel S, Çerezci O. Heat analysis of biological tissue exposed to microwave by using
thermal wave model of bio-heat transfer. Burns 2008;34:45–49.
28. Lu WQ, Liu J, Zeng Y. Simulation of the thermal wave propagation in biological tissues by the
dual reciprocity boundary element method. Eng Analysis Boundary Elements 1998;22:167–
174.
29. Zhou J, Zhang Y, Chen JK. Non-Fourier heat conduction effect on laser-induced thermal damage in biological tissues. Numerical Heat Transf Part A, 2008;54:1–19.
30. Tung MM, Trujillo M, Molina JAL, Rivera MJ, Berjano EJ. Modeling the heating of biological
tissue based on the hyperbolic heat. Math Computer Model 2009;50:665–672.
31. Taitel Y. On the parabolic, hyperbolic and discrete formulation of the sheat conduction equation.
Int J Heat Mass Transfer 1972;15:369–371.
32. Bai C, Lavine AS. On the hyperbolic heat conduction and the second law of thermodynamics.
J Heat Transf 1995;117:256–263.
33. Al-Nimr M, Naji M. On the phase-lag effect on the non-equilibrium entropy production. Microscale Thermophys Eng 2000;4(4):231–243.
34. Ahmadikia H, Rismanian M. Analytical solution of non-Fourier heat conduction problem on a
fin under periodic boundary conditions. J Mech Sci Technol 2011;25(11):2919–2926.
35. Tzou DY. A unified field approach for heat conduction from macro- to micro-scales. J Heat
Transf 1995;117:8–16.
36. Tzou DY. Macro-to micro-scale heat transfer: the lagging behavior. Washington, DC: Taylor
and Francis; 1997.
37. Qiu TQ, Tien CL. Short-pulse laser heating on metals. Int J Heat Mass Transf 1992;35:719–726.
38. Qiu TQ, Tien CL. Heat transfer mechanisms during short-pulse laser heating of metals. J Heat
Transf 1993;115:835–841.
39. Guyer RA, Krumhansl JA. Solution of the linearized Boltzmann equation. Phys Rev
1966;148:766–778.
40. Antaki PJ. New interpretation of non-Fourier heat conduction in processed meat. J Heat Transf
2005;127:189–193.
41. Liu KC, Cheng PJ, Wang YN. Analysis of non-Fourier thermal behavior for multi-layer skin
model. Thermal Sci 2011;15:61–67.
42. Zhou J, Zhang Y, Chen JK. An axisymmetric dual-phase-lag bioheat model for laser heating of
living tissues. Int J Thermal Sci 2009;48:1477–1485.
43. Liu KC, Chen HT. Analysis for the dual-phase-lag bio-heat transfer during magnetic hyperthermia treatment. Int J Heat Mass Transf 2009;52:1185–1192.
44. Liu KC, Chen HT. Investigation for the dual phase lag behavior of bio-heat transfer. Int J Thermal Sci 2010;49:1138–1146.
45. Liu KC. Nonlinear behavior of thermal lagging in concentric living tissues with Gaussian distribution source. Int J Heat Mass Transf 2011;54:2829–2836.
46. Liu KC., Wang Y, Chen Y. Investigation on the bio-heat transfer with the dual phase-lag effect.
Int J Thermal Sci 2012;58:29–35.
47. Narasimhan A, Sadasivam S. Non-Fourier bio heat transfer modelling of thermal damage during
retinal laser irradiation. Int J Heat Mass Transf 2013;60:591–597.
47
48. Sahoo N, Ghosh S, Narasimhan A, Das SK. Investigation of non-Fourier effects in bio-tissues
during laser assisted photothermal therapy. Int J Thermal Sci 2014;76:208–220.
49. Poor HZ, Moosavi H, Moradi A. Analysis of the DPL bio-heat transfer equation
with constant and time-dependent heat flux conditions on skin surface. Thermal Sci
doi:10.2298/TSCI140128057Z.
50. Ahmadikia H, Fazlali R, Moradi A. Analytical solution of the parabolic and hyperbolic heat
transfer equations with constant and transient heat flux conditions on skin tissue. Int Commun
Heat Mass Transf 2012;39:121–130.
51. Ahmadikia H, Moradi A, Fazlali R, Parsa AB. Analytical solution of non-Fourier and
Fourier bioheat transfer analysis during laser irradiation of skin tissue. J Mech Sci Technol
2012;26(6):1937–1947.
52. Fazlali R, Ahmadikia H. Analytical solution of thermal wave models on skin tissue under arbitrary periodic boundary conditions. Int J Thermophys 2013;34:139–159.
53. Askarizadeh H, Ahmadikia H. Analytical analysis of the dual-phase-lag model of bioheat transfer equation during transient heating of skin tissue. Heat Mass Transf 2014;50(12):1673–1684.
54. Askarizadeh H, Ahmadikia H. Analytical study on the transient heating of a two-dimensional
skin tissue using parabolic and hyperbolic bioheat transfer equations. Appl Math Model
2014;39(13):3704–3720.
55. Askarizadeh H, Ahmadikia H. Nonequilibrium dual-phase-lag heat transport through biological
tissues. J Porous Media 2015;18(1):57–69.
56. Askarizadeh H, Ahmadikia H. Analytical analysis of the dual-phase-lag heat transfer equation
in a finite slab with periodic surface heat flux. Int J Eng 2013;27(6):971–978.
57. Askarizadeh H, Ahmadikia H. Periodic heat transfer in convective fins based on dual-phase-lag
theory. J Thermophys Heat Transf 2015; (doi: 10.2514/1.T4602, Ahead of print).
58. Arpaci VC. Conduction heat transfer. Addisson Wesley Publication; 1966.
59. Afrin N, Zhou J, Zhang Y, Tzou DY, Chen JK. Numerical simulation of thermal damage to living
biological tissues induced by laser irradiation based on a generalized dual phase lag model.
Numer Heat Transf 2012;61:483–501.
60. Xu F, Seffen KA, Lu TJ. Non-Fourier analysis of skin biothermomechanics. Int J Heat Mass
Transf 2008;51:2237–2259.
61. Shih TC, Yuan P, Lin WL, Kou HS. Analytical analysis of the Pennes bioheat transfer equation
with sinusoidal heat flux condition on skin surface. Med Eng Phys 2007;29:946–953.
□▴
▾ ▴
▾ ▴
▾□
48