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Mathematical Biosciences 269 (2015) 1–9

Contents lists available at ScienceDirect

Mathematical Biosciences

journal homepage: www.elsevier.com/locate/mbs

Bioheat transfer problem for one-dimensional spherical

biological tissues

Emmanuel Kengne∗, Ahmed Lakhssassi

Département d’informatique et d’ingénierie, Université du Qué bec en Outaouais, 101 St-Jean-Bosco, Succursale Hull, Gatineau (PQ) J8Y 3G5, Canada

a r t i c l e i n f o

Article history:

Received 30 March 2015

Revised 6 July 2015

Accepted 20 August 2015

Available online 1 September 2015

Keywords:

Pennes bioheat transfer model

Point-heating source

Bioheat transfer problems

Tumors

Spherical living biological bodies

a b s t r a c t

Based on the Pennes bioheat transfer equation with constant blood perfusion, we set up a simplified one-

dimensionalbioheat transfer model of the spherical living biological tissues for application in bioheat transferproblems. Using the method of separation of variables, we present in a simple way the analytical solution of

the problem. The obtained exact solution is used to investigate the effects of tissue properties, the cooling

medium temperature, and the point-heating on the temperature distribution in living bodies. The obtained

analytical solution can be useful for investigating thermal behavior research of biological system, thermal

parameter measurements, temperature field reconstruction and clinical treatment.

© 2015 Elsevier Inc. All rights reserved.

1. Introduction

Spatiotemporal temperature distribution in living biological tis-

sues plays a vital role in many physiological processes. Investigationof bioheat transfer problems requires the evaluation of temporal and

spatial distributions of temperature. This class of problems has been

traditionally addressed using the Pennes bioheat equation [1]. Scien-

tific research in the bioheat transfer research field has paved a key

foundation in hyperthermia cancer therapy, thermal diagnosis, cryo-

genic surgery etc. [2–8]. The quantitative, qualitative, and accurate

analysis of bioheat transfer is to effectively understand and model

the heat transfer mechanism of the biological system.

Heat transfer analysis on thermal medical problems, such as the

thermal diagnostics [9] and thermal comfort analysis [10,11], thermal

parameter estimation [12–16], or burn injury evaluation [17], usually

has to simultaneously face the transient or spatial heating both on

skin surface and in interior of the biological bodies. The complexity

underlying in this class of medical problems remains not only for its

heterogeneity and anisotropy but also for conduction, convection,

and radiation heat flow, cell’s metabolism, and blood perfusion etc.

Therefore, to obtain a flexible solution, which is capable of solving

any one of the above thermal medical problems, is very desirable.

Indeed, analytical solutions reflect actual physical feature of the

models and can be used as standards to verify the corresponding

numerical results and as a proof to the reasonability of in-vitro mode

∗ Corresponding author. Tel.: +18196430402.

E-mail address: [email protected], [email protected] (E. Kengne).

analysis. Although people relied too much on numerical approaches

such as finite difference method (FDM), finite element method (FEM),

and boundary element method (BEM) for solving thermal medical

problems, the analytical solutions, if they can be obtained, areoften preferred [18,19]. Analytical solutions are very attractive since

their efficiency depends weakly on the dimensions of the problem,

in contrast to the numerical methods. Knowing analytical solution,

temperature at a desired point at a given time canbe performed inde-

pendentlyfrom that of the other points within thedomain, which can

be an asset when temperatures are needed at only some isolated sites

or times. It is also important to point out that analytical solutions

of bioheat transfer problems will save computational time greatly,

which is valuable in some hyperthermia practices. Motivated by the

importance of exact solutions of bioheat transfer problems in hy-

perthermia cancer therapy, thermal diagnosis, or cryogenic surgery,

we aimed in this paper to present analytical solutions to the Pennes’

bioheat model in one-dimensional spherical coordinate system with

relatively complex boundary or volumetric heating conditions [20].

Various techniques have been performed to obtain exactsolutions

of bioheat transfer problems in one dimensional Cartesian coordinate

[8,21–27]. In our knowledge, a combination of the method of separa-

tion of variables with the Green’s function method has not yet been

applied for explicitly solving bioheat transfer problems for spherical

symmetry. In this paper, we combine the Fourier method (method

of separation of variables) with the Green’s function method to de-

rive the exact solution of one-dimensional model of the spherical liv-

ing tissue. Such a combination of the two methods has been used

by Durkee and Antich [28,29] when addressing the time-dependent

Pennes’ equation in 1-D multi-region Cartesian and spherical

http://dx.doi.org/10.1016/j.mbs.2015.08.012

0025-5564/© 2015 Elsevier Inc. All rights reserved.

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E. Kengne, A. Lakhssassi/ Mathematical Biosciences 269 (2015) 1–9 3

Fig. 2. Steady-state temperature fields prior to heating for tissue properties given in the text. (a) Effect of the blood perfusion on the steady-state temperature distribution. (b)

Effect of the tissue thermal conductivity on the steady-state temperature distribution.

third BC. Through using the following transformation:T (r , t ) = u(r , t )+ f (t ), (5)

Eq. (1) takes the following form:

ρc

k

∂u

∂t = 1

r 2∂

∂r

r 2∂u

∂r

− αu + F , (6)

where

F (r , t ) = α(T a − f (t ))+qm + Q

k − ρc

k

df

dt . (7)

The corresponding boundary and initial conditions are then ex-

pressed as

k∂u

∂r

r =0 =0, (8a)

k∂u

∂r

r =R1

+ h f u|r =R1 = 0 (8b)

u(r , 0) = T 0(r )− f (0). (8c)Solving problems (6) and (8a)–(8c) with the method of separation of

variables combined with the Green’s function method [8] and using

Eq. (5) yield

T (r , t ) = f (t )− f (0)+ exp−kαρc

t

T 0(r )

+k

ρc t

0 dτ R1

0 G(r , t ; ξ , τ)dξ , (9)where

G(r , t ; ξ ,τ) = exp

kα

ρc (τ − t )

+∞n=1

F (ξ , τ)

U n2sin [λnξ ] sin [λnr ]

ξ r ,

(10)

λn are positive roots of equation

tan [λR1] −λkR1

k − R1h f = 0, (11)

and

U n

2

=λnSi[2λnR1]

−

sin2

[λnR1]

R1.

3. Results and discussion

In this section, we discuss the effect of major thermal parameters

on thetemperature distribution of spherical living tissues. In oursim-

ulations,the typical tissueproperties are applied as given in [9,32,33].

The apparent heat convection coefficient due to natural convec-

tion and radiation is taken as ha = 10 W/(m2 ◦C), while the forcedconvection coefficient is applied as h f = 100 W/(m2 ◦C) and thesurrounding fluid temperature is chosen as T e = 25 ◦C. Further, asdemonstrated in many works [34,35], the interior tissue temperature

usually tends to a constant within a short distance such as 2–4 cm.

Therefore 0.02 m ≤ L ≤ 0.04 m will be used in our study. Curves of Fig. 2 depict the steady-state temperature distributions of biological

bodies associated with the above typical tissue properties. Fig. 2(a)

and(b) shows theeffect of theblood perfusion andthe tissue thermal

conductivity on the steady-state temperature distribution. Obviously,

the steady-state temperature (initial temperature) of the tissue de-

creases from the body core to the skin surface. It is seen from Fig. 2(a)

that the steady-state temperature of the tissue increases as the blood

perfusion decreases; moreover, Fig. 2(a) shows that the blood per-

fusion has significant effects on the temperature of the body core.

Curves of Fig. 2(b) show that the steady-state temperature near the

body core (near the skin surface) decreases as the tissue thermal con-

ductivity increases (increases with the tissue thermal conductivity).

In our investigations, we study the temperature distribution when

the biological body is subjected either to a point-heating source or to

the most typical heat source that the heat flux decays exponentially

with the distance from the skin surface.

3.1. Temperature distribution under a point-heating source

In our analysis, we first discuss the temperature response subject

to a point-heating. Practical examples for such heating can be found

in clinics where heat is depositedthough insertinga conducting heat-

ing probein the deep tumor site. Here, the point-heating source to be

used is [8]

Q (r , t ) = P 1(t )δ(r − r 0), (12)

where P 1(t ) is thepoint-heating power, δ(r − r 0) is theDiracfunction,and r 0 is theposition of thepoint-heating source. FollowingAnderson

and Burnside [36], a sinusoidal point-heating power

P 1(t ) = q0 + qw cos [ω0t ] (13)


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4 E. Kengne, A. Lakhssassi / Mathematical Biosciences 269 (2015) 1–9

will be used. Here, q0 and qw are the constant term and the oscilla-

tion amplitude of sinusoidal point-heating power, respectively, and

ω0 is heating frequency. It is important to point out that the spatialsinusoidal heating was also proposed [8] to measure the blood per-

fusion where the heat was deposited to the biological body using ul-

trasound, and the temperature response was monitored at the skin

surface. As the time-dependent temperature of the cooling medium,

we will consider a sinusoidal temperature as follows:

f (t ) = q0 f + qw f cosω0 f t

,

where q0 f and qwf are theconstantterm andthe oscillation amplitude,

respectively, of sinusoidal temperature of the cooling medium, and

ω0 f is the temperature frequency.Under the above considerations, Eq. (10) becomes

G(r , t ; ξ , τ) = exp

kα

ρc (τ − t )

+∞n=1

1

U n2

P 1(τ)

k

sin [λnr 0]

r 0

+ sin [λnξ ]ξ

αT a − α f (τ)+

qmk − ρc

k

df

dt (τ)

sin [λnr ]

r . (14)

Therefore, the temperature response is obtained from Eqs. (9) and

(14):

T (r , t ) = f (t )− f (0)+ exp−kαρc

t

T 0(r )+

+∞n=1

g I n(t )

U n2sin [λnr ]

r ,

(15)

where

g I n(t ) = R1qw sin [λnr 0]r 0

k2α2 +ω20ρ2c 2kα cos [ω0t ] +ω0ρc sin [ω0t ]

− kα exp−kαρc

t

+

R1q0 sin [λnr 0]

αkr 0+ Si[λnR1]

×αkT a + qm − kαq0 f αk

1 − exp

−kαρc

t

+qw f Si[λnR1]

exp

−kαρc

t

− cos

ω0 f t

.

Fig. 3 depicts temporal temperature distributions in the tissues

under a surface point-heating with the sinusoidal heating power

P 1(t ) = 2500+ 2450 cos[0.02t ] at r 0 0. The sinusoidal temperature

Fig. 3. Temperature distribution at three positions when a surface point-heating with the sinusoidal point-heating power and a sinusoidal temperature of the cooling medium are

adopted.

Fig. 4. Temperature distribution at different times when a surface point-heating with the sinusoidal point-heating power P 1(t ) = 1500 + 1450 cos [0.02t ] W/m3 is applied. A

sinusoidal temperature of the cooling medium f (t ) = 25+ cos [0.02t ] is used.


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Fig. 5. Effect of some biological parameter on the temperature distribution for typical tissue parameters shown in Table 1. (a) Influence of the tissue density ρ . (b) Influence of the

tissue thermal conductivity k.

Table 1

Typical material properties of the tissue.

Paramete rs Val ue and uni t

ρ, ρb 1000 [kg/m3]

c, c b 4200 [J/(kg°

C)]T a 37 [°C]

k 0.5 [W/(m °C)]

ωb 0.0005 [ml/(s ml)]

qm 33,800 [W/m3]

of the cooling medium f (t ) = 25 + cos [0.02t ] °C is used. Due to thesurface cooling by the flowing medium, the lowest tissue tempera-

ture occurs at the skin surface ( x = L = 0.03) while the highest tissuetemperature reaches at the body core.

Fig. 4 depicts the temperature distributions at time t = 100 s of biological bodies subject to a surface sinusoidal point-heating and

sinusoidal temperature of the cooling medium. The surface point-

heating is applied at r 0 = 0.01. Due to the surface cooling by the flow-ing medium, the tissue temperature decreases as both the time and

the depth increase. At a given time, the position for the local high-

est temperature is just stayed at the site of the point source. This is

very beneficial for hyperthermia therapy since one can then selec-

tively heat the deep regional tumor.

Fig. 5 illustrates the temperature distribution at time

t = 100 s for the tissues subjected to point source P 1(t ) =2500+ 2450 cos[0.02t ] W/m3, r 0 = 0.015 m under the sinusoidalcooling medium temperature f (t ) = 25 + cos [0.02t ] °C for the

typical biological parameters shown in Table 1. The effect of the

tissue density ρ and the effect of the tissue thermal conductivity k onthe temperature distribution are illustrated in Fig. 5(a) and Fig. 5(b),

respectively. For each value of either the tissue density ρ or the tissuethermal conductivity k, the position for the local highest temperature

is just stayed at the site of the point source. Plots of Fig. 5(a) and (b)show that the position for the local highest temperature depends

both on the tissue density and the tissue thermal conductivity. It

is seen from Fig. 5(a) that the higher the tissue density, the higher

the temperature near the skin surface. The effect of the tissue

thermal conductivity on the temperature distribution is illustrated

in Fig. 5(b); this figure indicates that the thermal conductivity has

insignificant effect on the temperature distribution at the body core

as it varied between 0.48 W/(m °C) and 0.54W/(m °C). The curves

of Fig. 5(b) indicate that the higher the tissue thermal conductivity,

the lower the temperature near the body core and the higher the

temperature near the skin surface. Such a phenomenon in thermal

distribution of biological body produces a distinguishable elevation

in either the skin surface or the body core temperature and is a tool

for analyzing benign stage tumour.Fig. 6 depicts the influence of tissue specific heat c on the tem-

perature distribution at time t = 100 s when tissue was subjectedto point-heating source Q (r , t ) = P 1(t )δ(r − 0.015), and the tempera-ture of cooling medium is f (t ) = 25 ◦C. It shows that, the larger tissuespecific heat, the higher temperature increases.

Fig. 7 shows the temperature distributions at time t = 100 s forthe tissues subjected to constant point source P 1(t ) = 2500 W/m3,

Fig. 6. Effect of the tissue specific heat c on the temperature distribution in biological tissues subject to a constant point-heating under a constant temperature of the cooling

medium.


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Fig. 7. Influence of cooling medium temperature to tissue temperature distribution.

Fig. 8. Influence of heating power on tissue temperature distribution.

r 0 = 0.018 m and under different cooling medium temperatures. Aswe can see from plots of this figure, the larger the temperature of the

cooling medium, the lower the body core temperature decreases. It is

also seen from this figure that themagnitude and position of thelocal

highest temperature are changeless on the whole. This means that

an optimum heating can be obtained through regulating the surface

cooling.

We show in Fig. 8 the influence of heating power on the temper-

ature distribution when tissue was subjected to a constant point-

heating source Q (r , t )=

P 1(t )δ(r −

0.018), and the temperature of

cooling medium is f (t ) = 25 ◦C. It shows that although the effectof the heating power on the temperature distribution is negligible,

the larger the heating power, the higher the temperature increases.

Moreover, the position for the local highest temperature is fixed for

all these heatings.

Fig.9 gives outthe effects of thecooling medium temperature (top

plot) and the effects of the heating power (bottom plot) on the tem-

perature transients at skin surface. Curves of the top panel are ob-

tained when the biological tissue was subjected to a constant point-

heating source Q (r , t ) = 2500δ(r − 0.018). To generate the plots of the bottom panel, we used the constant cooling medium temper-

ature f (t ) = 25 ◦C. As it is seen from the curves of the top panel,the skin surface temperature decreases when the cooling medium

temperature increases. The bottom panel of Fig. 9 indicates that, the

larger the heating power, the higher the skin surface temperature

increases.

3.2. Temperature distribution under the most typical heating source

In this section, we study the temperature distribution when the

tissue is subjected to the typical heating source for which the heat

flux decays exponentially with the distance from the skin surface. In

other words, we work with the spatial heating [37]

Q (r , t ) = ηP 0(t )exp [−ηr ], (16)

where P 0(t ) is the time-dependent heating power on skin surface

and η is the scattering coefficient. We focus on the constant heat-ing power (P 0(t ) = q p = constant), reflecting the situation where thehuman skin was heated by a laser.

In the present situation, the temperature response is obtained

from Eqs. (9) and (14):

T (r , t ) = f (t )− f (0)+ exp−kαρc

t

T 0(r )+

+∞n=1

g II n(t )

U n2sin [λnr ]

r ,

(17)


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Fig. 9. Influence of cooling medium temperature (top) and influence of heating power (bottom) on the skin surface temperature response.

where

g II n(t ) =

kαT a + qm − kαq0 f

Si[λnR1]

αk

1 − exp

−kαρc

t

− qw f Si[λnR1]

cosω0 f t

− exp

−kαρc

t

+ q pηαk

1 − exp

−kαρc

t

R10

exp [−ηξ ] sin [λnξ ]ξ

dξ .

Fig. 10 shows the effect of the heating power P 0(t ) (top) and theeffect of the scattering coefficient η to the skin surface temperatureresponse. To generate the curves of this figure, we use the constant

cooling medium temperature f (t ) = 25 ◦C and the tissue propertiesshown in Table 1. The toppanelof Fig. 10 shows the transient temper-

atures of tissues subject to four constant heating with η = 200 m−1[37]. Obviously, the larger the heating power, the higher the tem-

perature increases. Such information is valuable for thermal comfort

evaluation. The bottom panel of Fig. 10 gives out the effects of the

scattering coefficient η on the temperature transients at skin surfacefor P 0(t ) = 250 W/m2. The plots of this figure show that the largerthe coefficient, the higher the temperature decreases. Although the

scattering coefficient η seems to have insignificant effect on thetemperature distribution on the skin surface, the bottom panel of

Fig. 10 reveals that the tissue temperature increases as the scattering

coefficient η decreases. Because different heating apparatus such aslaser or microwave may have different power P 0(t ) and scattering

coefficient η, we conclude that the above results are expected tobe useful for the heating-dose planning during the hyperthermia

treatment or parameter estimation.

Fig. 11 is the transient temperature at three positions of biological

bodies subject to a constant spatial heating with P 0(t ) = 250 W/m2and η = 200 m−1 [37]. Curves, A, B and C are the transient tempera-tures of tissues at positions r = 0.01 m, r = 0.015 m, and r = 0.025 m.Plots of Fig. 11 indicate that the tissue temperature decreases as the

tissue depth increases; this means that the highest temperature oc-curs at the body core, while the lowest temperature occurs at the skin

surface. This situation is due to the surface cooling by the flowing

medium. It is also seen from this figure that the temperature at any

given depth decreases as a function of time.

4. Conclusion

In this work, the bioheat transfer problems for the spherical liv-

ing biological tissues have been investigated based on 1-D Pennes

model, and the analytical solution of the corresponding equation

has been found with the help of the method of separation of vari-

ables. The obtained exact solution of the problem has been used to

study the tissue temperature distribution in radial direction. Our re-

sults show that the scattering coefficient η and the heating powers


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Fig. 10. Effects of the heating power (top) and scattering coefficient (bottom) on temperature response at skin surface for f (t ) = 25 °C. Top: η = 200 m−1; Bottom: P 0(t ) =250 W/m2.

Fig. 11. Transient temperatures at three positions whena constant spatial heating with P 0(t ) = 250W/m2 and η = 200m−1 was applied. The cooling medium temperature is takento be f (t ) = 25 ◦C.


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P 0(t ) and P 1(t ) of the spatial heating are neglected for further anal-

ysis of the temperature distribution, though the tissue density, the

tissue specific heat and the cooling medium temperature are to be

take into account when investigating temperature distribution in the

biological bodies. The solution of the bioheat problem presented in

this paper is found to be very useful for a variety of bio-thermal

studies.

Acknowledgment

This work was supported by the Natural Sciences and Engineering

Research Council of Canada.

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