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1. Vol.:(0123456789)
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Journal of the Brazilian Society of Mechanical Sciences and Engineering (2020) 42:62
https://doi.org/10.1007/s40430-019-2132-x
TECHNICAL PAPER
Numerical simulation of bioheat transfer: a comparative study
on hyperbolic and parabolic heat conduction
Sri Charan Akula1
· Ranjith Maniyeri1
Received: 16 November 2018 / Accepted: 5 December 2019 / Published online: 19 December 2019
© The Brazilian Society of Mechanical Sciences and Engineering 2019
Abstract
Cancer is one of the most death-causing diseases across the world. Radiotherapy is an important modality with which it is
treated. In the present study, temperature distributions due to the application of radiation on skin are estimated using para-
bolic and hyperbolic models of heat transfer and are compared as to which one better estimates the practical temperature
distribution. From the temperature distributions obtained, it can be inferred that parabolic and hyperbolic models produce
similar results for smaller phase lag times, while they predict significantly different results when the lag times are large, sug-
gesting that hyperbolic models are suitable to be used for biological systems with very complex internal structure. Thermal
damage is also calculated for three different models of heat transfer. It is observed that thermal wave model predicts higher
damage, followed by dual phase lag model and Pennes model. Time required for tumor necrosis is more for higher phase
lag values, suggesting longer radiation exposure durations for treating complex biological systems. When phase lag times
for heat flux and temperature gradient are equal, both dual phase lag model and Pennes model have shown similar results.
Effect of Gaussian distribution of applied heat flux on time required for tumor necrosis is also studied.
Keywords Dual phase lag model · Gaussian heat flux distribution · Necrosis time · Pennes model · Thermal damage ·
Thermal wave model
1 Introduction
One of the largest causes for deaths globally is cancer. Inter-
national Agency for Research on Cancer has published in
2010 [1] that most of the developing countries have high
incidences of cancer cases, with the majority of 12.7 million
new cancer cases and 7.6 million deaths. This is a significant
portion of 63% of all the cancer-related deaths worldwide.
There are many modalities with which cancer is treated. Sur-
gery, chemotherapy and radiotherapy are to name some of
them. Delaney et al. [2] have estimated optimal radiotherapy
utilization rates from their review of evidence-based clinical
treatment guidelines to be 52.3%, which means this percent
of cancer patients will undergo radiotherapy in their course
of treatment. Of these, 60% of the people are treated with
curative intent [3]. Hence, radiotherapy can be understood
to be one of the most important cancer treatment modalities.
One of the challenges faced in radiotherapy is maximiz-
ing radiation dosage to cancer cells while minimizing dam-
age to the surrounding tissues. Generally, cancer cells can
be killed by increasing their temperature to about 40–45 °C
[4]. Thus, understanding the mechanism of heat transfer in
biological tissues has great therapeutic value in devising
treatment protocols and optimizing the radiation dosages
and time of exposure to radiation. The governing mecha-
nism of bioheat transfer can be understood by analyzing the
temperature variation with respect to space and time dur-
ing radiotherapy. Estimating spatial and temporal variations
is hampered by the pain tolerance of the subject and the
accuracy of the equipment used for measuring the tempera-
ture. Bioheat transfer consists of heat conduction by tissues,
metabolic heat generation by various cells and heat exchange
between tissues and blood due to blood perfusion. Having
a mathematical model or some other analytical tools would
be very helpful for easier treatment planning.
Technical Editor: Adriano Almeida Gonçalves Siqueira.
* Ranjith Maniyeri
mranji1@nitk.edu.in
1
Biophysics Laboratory, Department of Mechanical
Engineering, National Institute of Technology Karnataka,
Surathkal, Karnataka 575025, India

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Pennes [5] has proposed one of the earliest mathematical
models for governing qualitative relationship between tissue
and arterial temperature. According to this, arterial blood is
assumed to be constant at the core temperature and venous
temperature to be equal to that of the tissue temperature.
The model assumes Fourier heat transfer to be applicable
to tissue and that the effect of blood perfusion is uniformly
distributed in space thus not accounting for the effect of
directional blood flow. This model was quite successful
in interpreting temperatures in various cases. However, it
is not very convincing as the model permits infinite speed
of thermal disturbance propagation. Further, no means is
provided in accounting for non-homogeneity present in
biological systems. Catteneo [6] has introduced a new heat
transfer model which eliminates this paradox of instantane-
ous propagation by introducing wave-like relation between
heat flux and temperature gradient. This is a hyperbolic heat
transfer equation. Mitra et al. [7] have shown experimental
evidence of hyperbolic heat transfer in processed bologna
meat. They have used meats at different temperatures and
brought them into contact suddenly and used thermocouples
to measure the instantaneous temperature distributions. They
have estimated a thermal phase lag time value to be around
16 s. Kaminski [8] has also conducted some experiments
with material with non-homogeneous inner structure and has
found a phase lag time value of 20 s. Herwig and Beckert
[9] and Graumann and Peters [10] have conducted similar
experiments as above but found that Fourier heat transfer
model was sufficient to represent their observations, thus
asserting that Mitra et al. [7] and Kaminski [8] works to be
misinterpreted. However, nothing can be ruled out as there
is no means to know and compare the actual constitution of
the materials used by different researchers.
Further in Thermal wave model, there are abrupt jumps
in temperatures which are again unphysical. This was over-
come by dual phase lag (DPL) model consisting of two
phase lag time values τq and τT. Minkowycz et al. [11]
have considered local thermal non-equilibrium. In their
theoretical study, they have shown that local thermal equi-
librium conditions depend on mean pore size, interstitial
heat transfer coefficient and other thermo-physical proper-
ties. Hooshmand et al. [12] have considered similar non-
equilibrium model and have solved them using separation
of variables and Duhamel’s integral methods. Zhou et al.
[13] have used finite volume method to solve DPL model
for an axisymmetric domain. All three of them have shown
that DPL model reduces to Pennes’ model only when both
τq and τT are zero, unlike Liu and Wang [14] who have sug-
gested that DPL reduces to Pennes’ model even when they
are equal, not necessarily to zero. Antaki [15] interpreted
τq to be a measure of delay in conduction and τT to be a
measure of conduction along microscopic paths. Museux
et al. [16] have experimentally observed skin burns when
a porcine tissue was exposed to two different lasers of
different wavelengths. They have compared their results
with those predicted by Pennes’ model using finite element
method. Degree of burns was estimated using different
thermal damage models to find which matches with the
experimental results the best.
Jiang et al. [17] have used finite difference method to
solve Pennes’ bioheat transfer equation and Arrhenius
equations to evaluate thermal damage in a one-dimensional
multi-layer model and have studied the effects of thermo-
physical properties and physical dimensions of the domain
on the temperature and thermal damage function distribu-
tions. Bedin and Bazan [18] have considered a two-dimen-
sional bioheat model and obtained an explicit Fourier-
based solution. Further, they have used a pseudospectral
collocation method to construct highly accurate numerical
solutions for their problem. The model is subjected to con-
vective boundary conditions and space-dependent perfu-
sion coefficient. Tung et al. [19] have compared Pennes’
model and thermal wave model for their physical differ-
ences. Analytical results have been illustrated for radiofre-
quency heating and laser heating used for refractive error
correction in the cornea. Ahmadikia et al. [20] have used
Laplace transform method to obtain analytical solutions to
Pennes’ and Thermal wave models applied to a one-dimen-
sional skin domain. Transient temperature responses have
been studied with the skin surface subjected to a constant,
cosine and pulse train heat fluxes. Lakhssassi et al. [21]
have developed analytical solutions to Modified Pennes’
model which accounts for blood perfusion variation with
temperature on a one-dimensional domain. The obtained
analytical solutions were used to conduct various paramet-
ric studies like the effect on temperature distributions due
to variation in thermal diffusivity, temperature depend-
ent and independent blood perfusion. Deng and Liu [22]
have solved transient three-dimensional bioheat transfer
equations subject to convective, radiative and evaporative
boundary conditions with the Monte Carlo method. Using
statistical principles, they have proposed various thermal
criteria for disease diagnostic.
Motivated by the previous studies and noting down the
limitations like physically relevant values of phase lag
were not considered, we aim to analyze temperature distri-
butions using a developed computational model by incor-
porating them. This enables us to understand as to which
model is better at predicting temperatures in real-life situ-
ations and at a faster time and lesser computational cost.
Finite difference method is used for the analysis. Effect of
applying these models to a two-dimensional [2D] domain
is considered. Further, the effect of phase lag time is also
studied. Time required for temperature to reach 42.5 °C in
entire tumor region is estimated.

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2 Mathematical models
2.1 Heat transfer
In the present work, we will analyze three bioheat transfer mod-
els—Pennes’ model of bioheat transfer (PMBT), thermal wave
model of bioheat transfer (TWMBT) and dual phase lag (DPL)
model. Constitutive equation for heat transfer used in PMBT is
For TWMBT it is
And for DPL model it is
Here q is heat flux (W/m2
), k is the thermal conductiv-
ity (W/m2
) of the material under discussion, and T (°C) is
temperature of the material. τq (s) and τT (s) are the thermal
lag times for heat flux and temperature gradient, respectively.
Considering metabolic heat generation and blood perfusion,
bioheat transfer equation becomes [5]
where ρ is density (kg/m3
), c is mass specific heat (J/kgK) of
tissue, Qm is volumetric heat generation, wb is blood perfu-
sion ­(m3
/(m3
s)), ρb is density(kg/m3
), and Tb is temperature
(°C) of blood. When we use the aforementioned different
constitutive equations, we get different Bioheat transfer
models. It is assumed here that Tb represents arterial tem-
perature of blood and to be at some constant, generally equal
to body core temperature of around 37 °C and that the blood
leaves the tissue at a venous temperature equal to that of the
tissue. Using Eqs. (3) and (4), we get DPL model as
If τT is zero in the above equation, we get TWMBT and
when τq is also zero, we get PMBT. The above equation
considers all the physical properties to be constant.
(1)
q = −k∇T
(2)
q
(
t + 𝜏q
)
= −k∇T
(3)
q
(
t + 𝜏q
)
= −k∇T
(
t + 𝜏T
)
(4)
𝜌c(𝜕T∕𝜕t) = −∇q + Qm + wb𝜌bcb
(
Tb − T
)
(5)
𝜌c𝜏q
(
𝜕2
T∕𝜕t2
)
+ 𝜌c
(
1 +
(
wb𝜌bcb𝜏q
)
∕𝜌c
)
(𝜕T∕𝜕t)
= k∇2
T + k𝜏T
(
𝜕∇2
T∕𝜕t
)
+ Qm + wb𝜌bcb
(
Tb − T
)
2.2 Thermal damage model
Widely studied Arrhenius protein denaturation burn integral
equation, proposed by Henriques and Mortitz [23], will be
used here to estimate the thermal damage in the skin when
exposed to radiation. The equation is as follows
where Ω is the dimensionless burn parameter. Here, A is the
frequency factor of Arrhenius equation for rate of chemical
reaction. The value of the material parameter equivalent to
it is taken to be 3.1 × 1098
s−1 and the ratio of activation
energy of necrosis, Ea, to the universal gas constant, R, is
taken to be 75,000. In basal layer, thermal damage can be
estimated by performing the integration in Eq. (6). A first-
degree burn is considered to occur when 0.53Ω1 and
T44 °C. Similarly, a second-degree burn is considered to
occur when Ω = 1 and T 44 °C and a third-degree burn
when Ω = 10,000 and T 44 °C. French Society for Burn
Study and Treatment (SFETB) has classified burns and the
corresponding physical effects as in Table 1 [16].
3
Physical problem description
We have adopted the skin as the physical domain from
Verma et al. [24] for the present study as it is more realistic
in approach. It is a two-dimensional domain of size 2L ×L
with three layers. Figure 1 represents the line diagram show-
ing the skin layers with tumor.
Tumor is placed in the top of Dermis, representing the
common place where cells are generally located and hence
a probable site for melanoma occurrence. We will be ana-
lyzing only right half of the above full domain. This is to
reduce computational effort. Initial condition and boundary
conditions are shown in Table 2. Steady-state temperature,
Tsteady, can be solved from Eq. (5) with boundary conditions
(iii), (iv) and (v). Later, when t0, boundary condition (vi)
is applied at the tumor location until the entire tumor region
attains the temperature which ensures killing of all tumor
cells.
(6)
𝛺 =
t
∫
0
Ae−
Ea
RT dt
Table 1 Degree of Burns and their Physical effects [16]
Thermal damage factor Degree of burn Physical effects
0.53 𝛺 1 1st degree Superficial epidermal involvement
1 𝛺 10, 000 2nd degree Whole thickness epidermal involvement; Basal membrane disruption; Papillary dermis involvement
10,000Ω 3rd degree Full-thickness epidermal necrosis, including hair follicles; complete basal membrane necrosis; deep
dermis/hypodermis involvement

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4 Numerical method
Finite difference method (FDM) is used, with central differ-
ence scheme and implicit scheme for spatial and temporal
discretization of second-order derivatives. These expressions
are second-order accurate. Discretizing the second derivative
with time as that present in the first term of Eq. (5) is done
as in Eq. (7).
The second term in the right side of Eq. (5) can be dis-
cretized as follows
Spatial discretization is according to the standard central
difference schemes and hence not shown explicitly. After
(7)
𝜌c𝜏q𝜕2
T
𝜕t2
≈ 𝜌c𝜏q
(
Tt+Δt
P
− 2Tt
P
+ Tt−Δt
P
)
Δt2
(8)
k𝜏T𝜕∇2
T
𝜕t
≈ k𝜏T
(
∇2
Tt+Δt
− ∇2
Tt
)
Δt
spatial and temporal discretization is done, all the terms are
arranged into the standard form as below
Uniform grid is adopted, and hence the coefficients
become as follows
where
Successive over relaxation (SOR) method is used to solve
the discretized Eq. (9). Once the temperatures are obtained in
each time step, boundary temperatures are calculated using the
boundary conditions (iii), (iv) and (v). Then these temperature
(9)
aPTP = aETE + aWTW + aNTN + aSTS + b
(10)
aE = aW = aN = aS =
(
k
𝜌cΔx2
)
×
(
1 +
𝜏T
Δt
)
=
(
k
𝜌cΔy2
)
×
(
1 +
𝜏T
Δt
)
(11)
b = at
P
Tt
P
− at
E
Tt
E
− at
W
Tt
W
− at
N
Tt
N
− at
S
Tt
S
− at−Δt
P
Tt−Δt
P
+ SC
(12)
at
E
= at
N
= at
N
= at
S
=
k𝜏T
𝜌cΔtΔx2
(13)
at
P
=
2𝜏q
Δt2
+
(
1 +
wb𝜌bcb𝜏q
𝜌c
)
×
1
Δt
+ at
E
+ at
W
+ at
N
+ at
S
(14)
at−Δt
P =
𝜏q
Δt2
(15)
SC = −SPTb +
Qm
𝜌c
(16)
SP = −
wb𝜌bcb
𝜌c
(17)
aP = aE + aW + aN + aS +
𝜏q
Δt2
+
(
1 +
wb𝜌bcb𝜏q
𝜌c
)
×
1
Δt
− SP
Fig. 1 Schematic of the domain used in the study
Table 2 Initial Conditions and Boundary Conditions
Sl. no. Boundary condition Location and time of application
1. T = Tsteady At t=0
2. 𝜕T∕𝜕t = 0 At t=0
3. 𝜕T∕𝜕x = 0 At left and right boundaries of the domain and t≥0
4. T=37 °C At the bottom boundary and t≥0 as many of the large arteries are at this location.
5. 𝜕T∕𝜕y = h ×
(
T∞ − T
)
At entire top boundary when t=0. And at top boundary except the part straight
above the tumor when t0
6. 𝜕T∕𝜕y = q At the part of top boundary straight above the tumor when t0

5. Journal of the Brazilian Society of Mechanical Sciences and Engineering (2020) 42:62
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values are used to solve temperatures in the next time step. It
is to be noted with care that any random selection of time step
value Δt may result in unphysical results. To avoid this, time
step Δt and grid sizes Δx and Δy should be chosen in such a
way that [25]
For the case when thermal conductivity, k, changes with
space, the discretization is as follows. For the discussion, kX
means thermal conductivity at the node X and ̄
kX means aver-
age of thermal conductivities at node P and node X, i.e.,
Using the standard form and uniform grid system, the coef-
ficients change as follows
(18)
aE = aW = aN = aS at−Δt
P
(19)
̄
kX =
(
kX + kP
)
∕2
(20)
aE =
( ̄
kE
𝜌cΔx2
+
̄
kN
2𝜌cΔxΔy
−
̄
kS
2𝜌cΔxΔy
)
×
(
1 +
𝜏T
Δt
)
(21)
aW =
( ̄
kW
𝜌cΔx2
−
̄
kN
2𝜌cΔxΔy
+
̄
kS
2𝜌cΔxΔy
)
×
(
1 +
𝜏T
Δt
)
(22)
aN =
( ̄
kE
2𝜌cΔxΔy
−
̄
kW
2𝜌cΔxΔy
+
̄
kN
𝜌cΔy2
)
×
(
1 +
𝜏T
Δt
)
(23)
aS =
(
−
̄
kE
2𝜌cΔxΔy
+
̄
kW
2𝜌cΔxΔy
+
̄
kS
𝜌cΔy2
)
×
(
1 +
𝜏T
Δt
)
(24)
b = at
P
Tt
P
− at
E
Tt
E
− at
W
Tt
W
− at
N
Tt
N
− at
S
Tt
S
− at−Δt
P
Tt−Δt
P
+ SC
(25)
at
E
=
( ̄
kE
Δx
+
̄
kN
2Δy
−
̄
kS
2Δy
)
×
(
𝜏T
𝜌cΔtΔx
)
(26)
at
W
=
( ̄
kW
Δx
−
̄
kN
2Δy
+
̄
kS
2Δy
)
×
(
𝜏T
𝜌cΔtΔx
)
(27)
at
N
=
( ̄
kE
2Δx
−
̄
kW
2Δx
+
̄
kN
Δy
)
×
(
𝜏T
𝜌cΔtΔy
)
(28)
at
S
=
(
−
̄
kE
2Δx
+
̄
kW
2Δx
+
̄
kS
Δy
)
×
(
𝜏T
𝜌cΔtΔy
)
(29)
at
P
=
2𝜏q
Δt2
+
(
1 +
wb𝜌bcb𝜏q
𝜌c
)
×
1
Δt
+ at
E
+ at
W
+ at
N
+ at
S
It can be observed that when thermal conductivity is uni-
form, all the equations from Eq. (20) to (33) get reduced to
Eq. (10) to (17).
5 Results and discussion
5.1 Validation
For the current study, the thermo-physical properties and
the geometry used are presented in Table 3 [24]. Blood
is considered to have a temperature Tb = 37 °C, density
ρb = 1052 kg/m3
, mass specific heat = 3800 J/kgK, heat
transfer coefficient of the ambient atmosphere h = 20 W/
m2
K, and the ambient temperature T∞ =20 °C. A length of
L=6 cm is considered as shown in Fig. 1. Numerical models
developed for the heterogeneous medium are validated with
analytical results for both PMBT [26] and TWMBT [14] as
shown in Figs. 2 and 3, respectively. It may be noted that the
analytical results considered for validation were developed
for homogeneous medium. However, there is no loss of gen-
erality as the codes developed for heterogeneous medium
can be used for homogeneous medium by just assigning
same thermo-physical parameters to various regions in the
computational domain considered for the current study.
Temperature distributions along the depth of the skin at
various instances are shown in Fig. 2. Figure 2a shows tem-
perature distribution variation with time when skin is irradi-
ated with constant heat flux of 250 W/m2
[26], and Fig. 2b
shows temperature distribution variation with time when a
(30)
at−Δt
P =
𝜏q
Δt2
(31)
SC = −SPTb +
Qm
𝜌c
(32)
SP = −
wb𝜌bcb
𝜌c
(33)
aP = aE + aW + aN + aS +
𝜏q
Δt2
+
(
1 +
wb𝜌bcb𝜏q
𝜌c
)
×
1
Δt
− SP
Table 3 Parameters of different tissues [24]
Type of tissue Epidermis Dermis Subcutaneous Tumor
Thickness (m) 0.0001 0.0015 0.0044 0.001
𝜌cp
(
J/m3
K
)
4.2 × 106
4.2 × 106
4.2 × 106
4.2 × 106
k (W/mK) 0.21 0.30 0.21 0.59
wb ­(m3
/m3
s) 0 1.63 × 10−3
1.0 × 10−3
5.0 × 10−3
Qm
(
W/m3
)
400 400 400 4000

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sinusoidal heat flux of 250+200 cos(0.02t) W/m2
is used.
In these figures, lines represent the temperature distributions
obtained using the present code and the discrete symbols are
those obtained from the analytical study of Deng and Liu
[26]. It can be seen that there is a slight difference between
present results with that of [26]. In Deng and Liu [26] work,
the analytical solution is derived based on Green’s function
method. Also, it can be noticed in their work that they have
omitted zero-order terms in the Green function series in the
analytical solution. We believe that the difference between
the two results may be due to the above reason. But there is
a qualitative agreement between both results.
For validation with [26], the thermo-physical prop-
erties used are density of blood and skin as 1000 kg/m3
,
mass specific heat of blood and skin as 4200 J/kg °C, body
core temperature as 37 °C, thermal conductivity of skin as
0.5 W/m °C, blood perfusion rate as 0.0005 ml/s/ml and
metabolic heat generation rate as 33,800 W/m3
. Tempera-
ture of the surrounding fluid is taken as 25 °C and natural
convective heat transfer coefficient as 10 W/m2
°C.
Figure 3a shows the comparison between present results
with that of [14], and Fig. 3b represents comparison between
present results with that of [14] for the PMBT, TWMBT and
DPL models. For this validation case, the heat flux is 2 W/
m2
; phase lag times used are τq =16 s and τT =0.05 s. From
Fig. 2 Validation with [26]
Fig. 3 Validation with [14]

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both the figures, it is clear that a good matching is obtained
between our results with that of [14] which confirms the
validity of the developed model. Further, it can be observed
that in Fig. 3a at the time of 5 s, there are unphysical tem-
perature oscillations. This is due to the condition explained
in Eq. (18) [25]. The condition in Eq. (18) can be met either
by reducing the spatial grid size or by increasing time step
size. It is recommended to reduce the grid size as the effect
of τT which is generally in the order of 0.01–10 s [27] will
not be captured if time step size is large.
5.2 Temperature distributions
For the present study, τq =16 s and τT =0.5 s are used. Fig-
ure 4 shows the temperature distributions predicted by three
different bioheat transfer models at the end of 7 s. It can be
observed that at this time, temperature estimated by PMBT
at the lower right corner of the tumor region is higher than
that of those predicted by other models. On the other hand,
temperatures at the surface are predicted to be very high
by TWMBT and DPL models when compared to PMBT.
This is because energy easily diffuses through the domain
according to PMBT, while according to the other two, it
takes more time to travel to deeper parts of the skin. Very
high temperatures at the surface predicted by the hyper-
bolic models can be explained as follows. Same amount of
energy, 11,000 W/m2
, is being applied to the domain for all
the models. However, according to PMBT, energy diffuses
to a larger domain and hence temperature rise is small over
a larger domain, while hyperbolic models suggest energy
distribution in smaller domain and hence large temperature
rise over small domain. This is in agreement with energy
conservation.
In Fig. 4b, it can be observed that TWMBT predicts sharp
temperature gradients typical to wave type of propagation.
The effect of τT can be observed from the third figure in
Fig. 4c that it has a diffusing effect at the wave front pre-
dicted by the TWMBT. This also explains the slight lower
surface temperature predicted by DPL model. Such behavior
was observed in experiment I in Mitra et al. [7]. Figure 5
shows the temperature distributions predicted by three dif-
ferent bioheat transfer models at the end of 15 s of expo-
sure. In Fig. 5a PMBT predicts that the effect of applied
heat flux is observed in the entire part of the domain shown.
Temperature at the corner of the tumor is between 38 and
40 °C already. Figure 5b, c shows the temperature distribu-
tion according to TWMBT and DPL models, respectively.
It is found that maximum temperature rises by about 10 °C
according to PMBT, while it is only about 5 °C according
to the hyperbolic models for same duration of exposure to
radiation. This is due to the wave type of propagation that
the temperature rise by hyperbolic models is not as high as
that suggested by PMBT.
Figure 6 shows temperature distributions at 7 s of expo-
sure to radiation. However, this time, τq and τT are taken to
be 22 s and 7 s, respectively, indicative of more complex
tissues like those with greater vasculature [27]. Compar-
ing Figs. 4b and 6b, we can observe that the increase in τq
from 15 to 22 s retards the energy penetration further, thus
modeling the effect of contact resistances due to greater
complexity. Here again, there is greater temperature of
90 °C predicted at the surface due to greater retardation
and hence greater energy concentration while it is only
85 °C in the case of Fig. 4b. Higher value of τT has a
greater diffusing effect on the thermal wave of TWMBT
as seen in Fig. 6c. It can be observed in Fig. 7 that for the
case of more complex tissues, for a given time of radiation
exposure, the order of temperature rise from 7 s of expo-
sure to 15 s of exposure is same for all the models. This is
because the thermal wave is not yet completely formed by
this time for the considered values of thermal phase lag
time. As thermal phase lag time for temperature gradient
Fig. 4 Temperature distributions at 7 s of exposure with 11,000 W/m2
heat flux with τq =16 s and τT =0.5 s

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62 Page 8 of 13
τT increases, the heat transfer mechanism becomes more
diffusive, as seen in Fig. 7c. Though τT has only diffusing
effect at the wave front only, as its magnitude increases,
its effect spreads into a larger domain. The higher the τq,
the steeper the temperature gradient at the wave front, and
the higher the τT, the smoother the wave front becomes.
Figure 8 shows temperatures along a line passing through
the tumor at a quarter width line, that is x = 0.0005 m,
Fig. 5 Temperature distributions at 15 s of exposure with 11,000 W/m2
heat flux with τq =16 s and τT =0.5 s
Fig. 6 Temperature distributions at 7 s of exposure with 11,000 W/m2
heat flux with τq =22 s and τT =7 s
Fig. 7 Temperature distributions at 15 s of exposure with 11,000 W/m2
heat flux with τq =22 s and τT =7 s

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parallel to y-axis at 7 s of radiation exposure. The thermal
wave propagation is clearly found in the figure. It may be
observed that PMBT overestimates temperatures in deeper
parts of the domain; however, the temperature variation is
very smooth. On the other hand, the temperatures predicted
by hyperbolic models exhibit abrupt changes. From Fig. 8b,
we see that as τT increases, the temperature distribution pre-
dicted by DPL model moves away from that predicted by
TWMBT toward the one predicted by PMBT. Effect of the
applied heat flux is experienced at a depth of y=0.0057 m
when τq =16 s, while it is experienced only till a depth of
y=0.0058 m when τq =22 s at same 7 s of exposure. From
Fig. 9a, it may be seen that temperature predicted by all the
heat transfer models is about to reach the tumor killing value
of 42.5 °C throughout the tumor region at a close time inter-
val. Though temperature according to PMBT is supposed
to be higher than that according to hyperbolic models, it
is still below the threshold tumor killing temperature. Fur-
ther, for the present case, the thermal wave happens to have
reached entire depth of tumor, which is y=0.0049 m from
the surface. Hence, for the case of τq =16 s and τT =0.5 s,
the predicted time of exposure to ensure killing of entire
tumor is almost same by all the models as presented in
Table 4. In Fig. 9b, such behavior is not observed. There
Fig. 8 Temperature distributions at 7 s of exposure with 11,000 W/m2
heat flux with a τq =16 s and τT =0.5 s and b τq =22 s and τT =7 s near
quarter tumor along a line parallel to y-axis
Fig. 9 Temperature distributions at 15 s of exposure with 11,000 W/m2
heat flux with a τq =16 s and τT =0.5 s and b τq =22 s and τT =7 s near
quarter tumor along a line parallel to y-axis

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is a considerable difference in the depth and temperatures
predicted by the three models. Time taken for necrosis of
cancer cells to occur for this case of τq = 22 s and τT = 7 s
is presented in Table 4. It can be seen that it takes around
21.4 s for cancer killing according to PMBT, while it takes
about 50.2 s and 56.0 s according to TWMBT and DPL
models, respectively.
5.3 Thermal damage
Figure 10 shows the thermal damage at 7 s of radiation
exposure for the case of τq =16 s and τT =0.5 s. It may be
observed that PMBT predicts much lower level of thermal
damage than the hyperbolic models. At this instant, only a
second-degree burn is seen as per PMBT and over a small
area of the tissue. As can be seen in the case of TWMBT in
Fig. 10b, a third-degree burn is seen over a large part of the
tissue above the tumor. The diffusing effect of phase lag time
for temperature gradient τT is evident from a slight decrease
in the area of burn as seen in Fig. 10c. Thermal damage in
the tumor region is set to zero as thermal damage of skin is
only studied. Figure 11 shows the thermal damage at 15 s
of radiation exposure for the case of τq =16 s and τT =0.5 s.
In the case of PMBT, a small region of third-degree burn is
seen in Fig. 11a. In the case of TWMBT and DPL models,
the region of third-degree burn is almost unchanged when
compared to that for 7 s of exposure, while the region of
second and first-degree burns has increased to similar extent
in both the models. It can be seen that the effect of having
small τT is not very significant. Further, PMBT does not
predict thermal damage beyond the region of radiation expo-
sure, i.e., beyond the region immediately above the tumor.
However, the hyperbolic models do predict such behavior
because the concentration of energy has been explained ear-
lier in Sect. 5.2.
Thermal damage at 7 s of exposure to radiation with
τq = 22 s and τT = 7 s is shown in Fig. 12. From Fig. 12b,
it can be observed that the extent of third-degree damage
is more in this case as compared to that present in the case
of Fig. 10b. Thermal damage has reduced in this case for
DPL model than that present in Fig. 10c. For 15 s of radia-
tion exposure, thermal damage is shown in Fig. 13. From
Fig. 13b, the damage is seen not to have increased a lot
from that in Fig. 11b. However, thermal damage is decreased
considerably in the case of DPL model as is observed when
Figs. 11c and 13c are compared. Increasing the value of
τq increases the thermal damage while increasing the value
of τT reduces the thermal damage. From the heat transfer
models, a DPL model would predict same results as that of
PMBT only when both τq and τT are zeroes. In [14] it was
Table 4 Time taken for killing of tumor cells
Values of phase lag time Model Estimated time for necrosis
of cancer cells to occur (s)
τq =16 s and τT =0.5 s PMBT 21.4
TWMBT 21
DPL 21.4
τq =22 s and τT =7 PMBT 21.4
TWMBT 50.2
DPL 56.0
Fig. 10 Thermal damage at 7 s of exposure with11, 000 W
m2
heat flux with 𝜏q = 16s and 𝜏T = 0.5 s
Fig. 11 Thermal damage at 15 s of exposure with11, 000 W
m2
heat flux with 𝜏q = 16s and 𝜏T = 0.5 s

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proposed that even when they are equal to each other, not
necessarily to zero, DPL will predict same results as that
of PMBT. However, in [12] and [13] it can be found that
when both τq and τT are equal but not to zero, there are dis-
crepancies between parabolic, i.e., PMBT, and hyperbolic,
i.e., TWMBT and DPL, models. In our study, we have ana-
lyzed this case of both phase lag time values being equal to
each other, for a particular value of τq =7 s and τT =7 s and
found that there is not much difference in the temperature
fields predicted by both PMBT and DPL at 15 s of exposure
with 11,000 W/m2
heat flux near quarter tumor line paral-
lel to y-axis as shown in Fig. 14. The small oscillations at
y=0.0016 m can be attributed to a very small deviation from
condition depicted in Eq. (18).
5.4 Effect of realistic heat flux distribution
In all the previous studies, a uniform heat flux was consid-
ered over the tumor. However, this is not realistic and the
fact that the intensity of the projected radiation usually fades
out toward the end of the laser irradiation area can reduce
the temperature attained in the domain and thereby compel
one to increase the time of duration of radiation exposure to
ensure necrosis of the cancer cells. For studying this condi-
tion, we assumed Gaussian distribution shown in Eq. (34) of
heat flux to be a good approximation of the actual heat flux
distribution, as shown in Fig. 15. Here, qmax is the maximum
intensity applied, and xt is the half of the width of the tumor,
shown in Fig. 1 along x-axis.
(34)
𝐪 = 𝐪𝐦𝐚𝐱
(
𝐞
− 𝐱2
𝐱2
𝐭
)
We observed significant changes in the time required for
the cancer cell necrosis as shown in Table 5, only for the
cases of PMBT and DPL but not in the case of TWMBT
when τq and τT are considered to be 22 s and 7 s, respec-
tively. For the case of PMBT, it has increased by about 8 s,
while it has increased by about 5 s for the case of DPL model
when we go from uniform to Gaussian distribution. This is
a substantial change and, hence, must be taken care when
simulations are carried out to estimate the time required for
the treatment.
Fig. 12 Thermal damage at 7 s of exposure with110, 00 W/m2
heat flux with 𝜏q = 22 s and 𝜏T = 7 s
Fig. 13 Thermal damage at 15 s of exposure with11, 000 W/m2
heat flux with 𝜏q = 22 s and 𝜏T = 7 s
Fig. 14 Effect of thermal phase lag time being equal to each other
and non-zero value

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6 Conclusion
A two-dimensional computational model based on finite dif-
ference method is developed to understand temperature dis-
tributions in biological tissue. Three bioheat transfer models
are compared and analyzed. Finite difference method is used
for discretization. The developed computational model is
validated with analytical results of previous researchers.
Later, temperature distribution in the domain is determined
using different bioheat transfer models. It is observed that
heat reaches quickly to deeper layers of skin according to
PMBT. However, higher temperatures and hence suggest-
ing energy concentration at surface layers are predicted by
TWMBT and DPL models. The effect of τT is understood to
have a diffusing effect on heat at the wave front predicted by
TWMBT model. Time of exposure of heat source for killing
tumor cells is estimated according to different models. It is
observed that time of treatment is not always increased due
to the phase lag times as the thermal wave will have reached
the corner of the tumor before the temperature predicted by
PMBT is already reached the threshold cancer cell killing
temperature. The higher the value of τq, the greater is the
resistance to the penetration of heat; hence, only a small
part of the domain is affected by the applied heat flux. For
complex tissues which generally have very large values of
phase lag times, the time required for cancer cell necro-
sis is highest according to DPL followed by TWMBT and
PMBT. TWMBT predicts highest thermal damage among
all the three models. When a realistic heat flux distribution,
like that of a Gaussian distribution, is considered, a signifi-
cant rise in the estimated time for cancer cell necrosis is
observed. It is recommended that time for necrosis to occur
should be determined by all the three models and whatever
is higher should be used for designing the treatment protocol
so as not to leave a chance for some cancer cells being left
behind, which may lead to resurrection of tumor.
Acknowledgement This research was supported by National Institute
of Technology Karnataka, Surathkal. We extend our thanks to Prof.
Prasenjith Rath, IIT Bhuvaneshwar, for his valuable comments.
Compliance with ethical standards
Conflict of interest We declare that we do not have any commercial or
associative interest that represents a conflict of interest in connection
with the work submitted.
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