Vol.2, No.12, 1375-1385 (2010)
doi:10.4236/ns.2010.212168
Copyright © 2010 SciRes. Openly accessible at http:// www. scirp.org/journal/NS/
Natural Science
Modifed pennes' equation modelling bio-heat transfer in
living tissues: analytical and numerical analysis
Ahmed Lakhssassi1, Emmanuel Kengne1*, Hicham Semmaoui2
1Laboratoire d'Ingénierie des Microsystèmes Avancés, Département d'informatique et d'ingénierie, Université du Québec en Ou-
taouais, Gatineau, Canada; *Corresponding Autho r: kengem01@uqo.ca
2Electronic Engineering Department, University of Montreal 2900 Chemin de la tour Montreal Qc H3C3A7, Montreal, Canada
Received 20 September 2010; revised 25 October 2010; accepted 28 October 2010.
ABSTRACT
Based on modified version of the Pennes'
bio-heat transfer equation, a simplified one-
dimensional bio-heat transfer model of the liv-
ing tissues in the steady state has been applied
on whole body heat transfer studies, and by
using the Weierstrass' elliptic function, its cor-
responding analytic periodic and non-periodic
solutions have been derived in this paper. Using
the obtained analytic solutions, the effects of
the thermal diffusivity, the temperature-inde-
pendent perfusion component, and the tem-
perature-dependent perfusion component in
living tissues are analyzed numerically. The re-
sults show that the derived analytic solution is
useful to easily and accurately study the ther-
mal behavior of the biological system, and can
be extended to applications such as parameter
measurement, temperature field reconstruction
and clinical treatment.
Keywords: Bio-Heat Transfer Equation;
Pennes' Modified Equation; Weierstrass' Functions;
Jacobi Elliptic Functions
1. INTRODUCTION
The effects of blood flow on heat transfer in living
tissues have been examined for more than a century,
dating back to the experimental studies of Bernard in
1876 [1]. Since that time, mathematical modelling of the
complex thermal interaction between the vasculature and
tissue has been a topic of interest for numerous physi-
ologists, physicians, and engineers. It is a very difficult
task to establish an appropriate physical model for the
heat transport in the human body. The first quantitative
relationship that described heat transfer in human tissue
and included the effects of blood flow on tissue tem-
perature on a continuum basis was presented by Harry H.
Pennes, a researcher at the College of Physicians and
Surgeons of Columbia University [2]. His landmark pa-
per, which appeared in 1948, is cited in nearly all re-
search articles involving bio-heat transfer [3-10]. Ap-
propriately, the equation derived in this paper is often
referred to as the “traditional” or “classic” or “Pennes”
bio-heat equation. The general form of Pennes’ bio-heat
equation is [7]


=,
bb am
T
ckTwcTT
t
 
q (1)
where
, , and are the density (kg/m3), the spe-
cific heat ck
J
kg.K , and the tissue thermal conduc-
tivity
.KWm , respectively; b is the mass flow
rate of blood per unit volume of tissue
w

3
Kg s.m;
c is the blood specific heat; q is the metabolic heat
generation per unit volume
b m

; represents
the temperature of arterial blood
3
Wm a
T
K
; is the tem-
perature rise above th e ambient lev el; T
Tt
T
w
wc
is the rate
of temperature rise. Here, we need to know the arterial
blood temperature a
T. Pennes compressed all of the
perfusion information into the term bba . He
checked the valid ity of this approxi mation by comparing
temperatures predicted by his equation with experimen-
tally measured temperatures in the human forearm. In
his approach, the blood perfusion term b was adjusted
until the predicted temperatures agreed well with the
measured temperatures. To facilitate the solution of the
equation, the added blood perfusion term to account for
perfusion heat transfer is linear in the temperature.
T
Pennes’ primary premise was that energy exchange
between blood vessels and the surrounding tissue occurs
mainly across the wall of capillaries (blood vessels with
0.0054-0.015 mm in diameter) [7], where blood velocity
is very low. He assumed, therefore, that the thermal con-
tribution of blood can be modelled as if it enters an
imaginary pool (the capillary bed) at the temperature of
major supply vessels, , and immediately equilibrates
a
T
A. Lakhssassi et al. / Natural Science 2 (2010) 1375-1385
Copyright © 2010 SciRes. Openly accessible at http://www.scirp.org/journal/NS/
1376
(thermally) with the surrounding tissue. Thus it exits the
“pool” and enters the venous circulation at tissue tem-
perature, . He postulated therefore, that the total en-
ergy exchange by the flowing blood can be modelled as
a nondirectional heat source, whose magnitude is pro-
portional to the volumetric blood flow and the difference
between local tissue and major supply arterial tempera-
tures.
T
Boundary conditions are obtained from
out
kT nTT 
=h, where out is the environmental
temperature, and h is the heat transfer coefficient. The
main assumption of this model is that blood enters the
local tissue with the body core temperature and leaves it
with the local tissue temperature. Hence, convective heat
flow can be described by the temperature difference
and an empirical transfer coefficient [11].
T
bbb
The mathematical model for heat transfer in tissue
used to estimate tissue temperature profiles for each ex-
periment based on tissue temperature measurements just
before the animals were killed [12] was based on
one-dimensional bio-heat equation that describes heat
transfer in tissue [2] and was modified to account for
temperature-dependent variability in tissue perfusion
[13,15]. The model defines local tissue temperature
as
TTcw
,Txt
2
01
2
=,
aa
a
TT
TT TT
tT
x



 


(2)
where is the local tissue temperature, is the time,
T t
x
is distance from the heated surface, a is the input
temperature of arterial blood, T
is the thermal diffu-
sivity, 0
is the temperature-independent (basal) perfu-
sion component, and 1
is the temperature-dependent
(vasodilation and angiogenesis) perfusion component.
Eq.2 is a nonlinear version of the Pennes bio-heat trans-
fer equation (temperature-dependent perfusion). Several
simulations reveal that the temperature at the steady state
is significantly higher with a temperature-dependent per-
fusion. Throughout this paper, Eq.2 will be referred to as
the “modified” Pennes bio-heat (MPBH) equation.
Generally, the complexity of the nonlinear bio-heat
transfer equations makes it difficult to obtain their ana-
lytic solutions. Many nonlinear equations can only be
solved by numerical methods. However, analytic solu-
tions of these equations, if attainable, are of important
significance in the study of bio-heat transfer because
they can accurately reflect not only the actual physical
feature of equations but also be used as standards to ver-
ify the corresponding results of numerical calculation. In
this paper, the derivation of analytic solution for one-
dimensional steady-state model of living tissue is con-
ducted by adopting the applicable nonlinear bio-heat
transfer equation. The rest of the paper is organized as
follows. In section II, we present analytic solutions of
the MPBH Eq.2 in the steady state. The effects of the
thermal diffusivity, the temperature-independent perfu-
sion component, and the temperature-dependent perfu-
sion component are discussed in section III via numeri-
cal computation. The main results are summarized in
section IV.
2. ANALYT IC SOLUTIONS
For analytic solutions, we suppose the input tempera-
ture a of arterial blood to be constant. The steady-
state condition is
T=0Tt
and the MPBH Eq.2 be-
comes

210
201
1
2
2
=.
a
a
T
dT TT
T
dx


 
 (3)
Clearly, Eq.3 is a second-order nonlinear ordinary
differential equation. If we multiply both sides of this
equation by dTdx and integrate the resulting equation,
we obtain

232
=46 4= ,
dT TTTRT
dx

 

 (4)
where

01 10
12
=,= ,=
66 2
a
a
T
T
 
 


,
(5)
being a constant of integration. The integration con-
stant
plays the role of a control parameter. Eq.4 with
the coefficients (5) is an elliptic ordinary differential
equation and admits various periodic and solitary wave
solutions, some of which can be found in Ref. [14,16].
As is well known [17,18] the so lution of
Eq.4
can be written as

Tx



0
0
23 0
=,
1
4;, 24
RT
Tx T
xggR T


(6)
where 0 is a simple nonnegative root of T
RT and
the prime denotes differentiation with respect to
T
. The
general solution
Tx of Eq.4 is [17,18]

 
 
 
23 0
0230
02
2300 0
;, 11
;,
22424
= ,
11
2;,2448
''' '''
'' ''''
dxgg RT
RTxg gRTRT RT
dx
Tx T
xg gRTRTRT

 



 


00
(7)
A. L
Copyright © 2010 SciRes. Openly accessible at http://www.scirp.org/journal/NS/
akhssassi et al. / Natural Science 2 (2010) 1375-1385
137
1377
where is a real constant, not necessarily a zero of
e invariants
0
T
. Th

RT 2
g
and 3
g
of Weierstrass’ el-
tion liptic func

23
;,
x
gg
e rela
by thtiare related to the coeffi-
ons [
2
cients of

RT 19]
23
23
=43,=2.gg
 
 (8)
The discrimi nant (of
and [19]) R
32
23
=27,
g
g (9)
is suitable to classavior of

Tx. The condi-
tions ify the beh
lead s nditions
[2
solut
d in real, bounded, and nonnegative solutions
. Based on form (6) for the solution
2.1. Case
23
0or =0,>0,>0gg  (10)
to periodic solution[20], whereas the co
1]
23
=0, 0,gg0 (11)
are associated with solitary wave-like ions. We are
intereste

Tx

Tx, we list
in what follows some interesting particular solutions of
Eq.4 that cannot be found in Ref. [14].
>0
If >0, Weierstrass’ function
can be expressed
as [21]


13
23 3213
;, =,
sn ,
ee
xg geeexm

(12)
where

,
s
nx n
tion with modul
m deotes the Jacobi elliptic sine func-
us [14,22],
m

23 13
=mee ee
an h
0. (13)
Substituting Eq.12 into Eq.6 yield s [23] (14
For the boundedness of solution (14) we must have
1
.e
d 3
e being the roots of te equation
12
ee
323
4=sgsg
).
either
30 3
22 0or2<eTe 0
2 <2T

 )
It is easily seen that if conditions (15) are s
(15
atisfied
then



13
, 2>0mee
2
0 13
22 sne Teex

 
for every 3
x
, so that the bounded so
noative,
i.e Eqs.15 and 16 are boundedness and nonnega-
tivity conditions, resp ectively, of solution (14). The
the mathematical conditions. For the clinical condition,
we only need the boundedness and nonnegativity of
lution (14) will be
nnegative if, and on ly if, its numerator is nonneg
., if either (15).
Thus, se are
Tx on some segment
0
0,
X
. Indeed, if 0
X
is the
al distance fromurface, maxim the heated si.e., 0
x
X
in Eq.2, and if
Tx is nonnegative
ent and bounded on
the segm
0
0,
X
, then we say that the clinical con-
is vaditionlid.
Note: It should be noted that, in the case where
>0
, the polynomial

323
=4Pssgs g cannot
ha double root. Hence solution (14) contains neither
a trigonometric nor a hyperbolic so lution.
For the plot of thissubsection we employed the ther-
mal properties of tsue and blood based on Ref. [24].
The density of e is 1050 kg/m3, both specific heat
ca
ve a
is
tissu °
th ue is 0 W/m.C. In
°
pacity of tissue and blood are 3770 J/kg.C, and the
ermal conductivity of tiss.4°addition,
the arterial temperature is 37C. With these data, we
compute
and 0
. The temperature-dependent per-
fusion component 1
with be chosen properly.
Figure 1 shows the temperature profile along the dis-
tance from the heat source. For the curv es of this figu re,
we used the periodic solution (14) with 1=6.449
3
10
1
s
. With this value of the temperature-dependent
perfusion component, polynomial

RT admits three
different roots: 0=38T, 0= 35.634T, and 0= 33.268T.
These values of Tare used for plot (a), (b), and (c),
res 0
pectively. Here, the free parameter
has been taken
as
23 2
0
2
641
6
a
aa T
a


, with

373 1
2
a
the plots of Figure 1 show, the temperature response is a
steady periodic oscillation. The reponse temperature
oscillates around th temperature 0
T. We then set up the
following question: It is true that the response tempera-
ture (14) will be always maintained almost
. As
s
e
at the input
perature of arterial blood if is a positiv
t of polynomial
temC
a
Te
roo
0
T
RT so that 00?
t
a
TT
As we can see from Figure 2, the answer to the above
question is negative! Figure 2 is obtained with the same
parameters as in Figure 1, bu with 10
=
and
0
=456.33

. With these values of parameters, we
solved the equatio n
=0RT and as p ositive roots, we
obtain ed 0= 36.99CT and CT. Plot (a)
corresponds to 0= C,T whilained
us
0= 37.058
e plot (36.99 b) is obt
ing 0=37.058C.T These two plots show that al-
though 0
T is almost equal to the input temperature of






22 0
13
sn
2
e
e ee
030 0
2
30
4252
=
22 sn
TeT T
Tx e T




13 13
13
,2.
,
eexm eT
exm
 
(14)

2
030 0130
42<2 .T eeT
 
(15)
2
030 00
42520and0or0<TeT TT

 25eT T
 
A. L
Copyright © 2010 SciRes. Openly accessible at http://www.scirp.org/journal/NS/
akhssassi et al. / Natural Science 2 (2010) 1375-1385 1378
(a)
(b)
(c)
Figure 1. Temperature profiles along the distance
x
(a)
(b)
Figure 2. Temperature profiles along the distance x (in mm)
from the heat source when using the same parameters as in
Figure 1, but with 10
=
and 0
= 456.33.

The
curves of this plot nd to solution (14) with
(a) and (b).
arterial blood , the response temperature corre-
sponding to so14) oscillates far from
An obviousis of Eq.14 allows usake the
following conclusion: In order that the response tem-
perature corresponding to the periodic solution (14)
should be maintained at almost the input temperature of
arterial blood , it is necessary and sufficient that
correspo
0= 37.058T
0= 36.99CT
C
C
a
T
lution (
analysa
T.
to m
C
a
T


0
2
0010
1and
23 322
a
TT
TTeT
 
 
1.
(17)
For example, if we use the same parameters as in Fig.
1 but with 1= 0.12898
s
-1 and , we com-
pute the roo270.86a
ts 0
T of
RT
36.81
0= 3
With
(in )
from the heat source when mm
72
/J/m, =1.2631W
10
41
0= 4.761910,
s

=37C,
a
T and 1= 61031
..449
s
with
(c).
espond to solu
and 0= 33.268T
The curves of this p
0=38CT (a), 0=T
lot corr
35.63 C
(b), tion (14)
C
and
find 0
T
C. Only one
813 C,
satisf
0= 36.813C
=37.05C
,
of these
ies both
we have
0= 36.932T
values of
conditions (1
C
, and 0=T
0
T, namely T
5) and (16).
3
6.
T
A. Lakhssassi et al. / Natural Science 2 (2010) 1375-1385
Copyright © 2010 SciRes. Openly accessible at http://www.scirp.org/journal/NS/
137
1379

0
2
00
10
=36.81337= 0.187
23 3
and= 0.23838,
22
a
TT
TT
eT
 




and conditions (17) are satisfied. We then conclude
that for the response temperature will
be maine input temperature of arterial
blood irmed by Figur e 3.
satisfy the clini-
ure 3(a) and 3(c)
ed
h and
co read
0= 36.813C,T
tained at almost th
C.
a
This is conf
37.05 C
and T
T
=
0
T
cal condition as on
where we us
respectively. For t
nditions (17)
0= 36.932C
e can see from Fig
= 37.05C
and
0=37.05CT
0
T
ese
0= 36.932C,T
0= 36.932CT

0=37.05= 0.05
a
TT
2
00
10
23 3
and= 0.23503
22
TT
eT
 



and
37

0=36.93237=0.068
a
TT
2
00
23 3TT
 

Thus, satisfies conditions (17), but
satisfy this condition.hus for
temperature is matained at
erature of arterial blood (see
r the rem-
taine inpupera-
gure 3(Figure
is noicient
e to be mtained
rial blood
2.2. Case
10
and =46.462.
22eT


0=37.05T
32 does not
the response
e input temp
a)), while fo
re is not main
terial blood
ows that cond
for the resp
st the input tem
0=36.9T
0=37.05T
almost th
3(
peratu
ture of ar
3(b) sh
condition
at almo
T
in C
a
T
sponse te
t tem
).
t a suff
ain
a
T
Figure 0=36.932T,
ed at almost th
C
a
T (see Fi
n 0a
TT
se temperatur
erature of arte
itio
on
p
c)
C.
<0
In the present case, we express Weierstrass’ function
as [21]



23 2
1cn2 ,
;, =1cn2, ,
H
xm
xg geH
H
xm

(18)
where 2
=12 34meH
, 22
22
=3 4
H
eg, and
,cnx m
us m. If is the Jacobi elliptic cosine function with modul
we insert expression (18) into Eq.6 we obtain
 



2
20 0
02
2524 2cn
=222cn
2
0 20
20
2 ,24252.
2,222
H
eTT
Tx TH e



 


HxmT HeT
HxmHeT


 
  (19)
For the mathematical boundedness of solution (19) it
is necessary and sufficient that either

02
222222>0HeTT He
 

 


 
20 02
2 0
and 222222> 0eTTHe
 
  (20)
or

20 02
02
222222<0
and 4222<0.
HeTTHe
HTH e
 

 
 (21)
For the mathematical positivity of solution (19) it is
necessary and sufficient that one of the following condi-
tions should be satisfied:
1) Condition (20) is satisfied, moreover,



22
020
2
2452 >0
eT
 
 
002
2
20 020 0
248252 >0,
2524 22524 2
THTT e
H eTTH eTT
 
 
 
  (22)
and

2
20 0
0
22TH
2) Conditions (21) and (23) are satisfied and
2
25242
>0,
2
HeTT
e

 
 (23)
 
2
0
24 2TH

 2
0
2
20 0
24 2eTT
 
3) Condition (23) is en eith
(20) and (24) are satisfieons (21
satisfied.
02
20 0
52 4
<0 <0,
252524 2
eT HT
H HeTT
 
  (24)
violated wher conditions
d or conditi) and (22) are
Figure 4 shows the plot of the response temperature
along distance
x
from the heated surface. Here we
used the same parameters as in Figure 1, but with dif-
ferent values of the temperature-dependent perfusion
component s
-1 for plot (1),
4
11
:= 7.11910

4
10
1= 6.5086
s-1 for plot (2), and 4
1=6.3492 10
s
we used =0
-1
for plot (3). For all these plots,
and
A. Lakhssassi et al. / Natural Science 2 (2010) 1375-1385
Copyright © 2010 SciRes. Openly accessible at http://www.scirp.org/journal/NS/
1380
(a)
(b)
(c)
Figure 3. Plot of the response temperature along the distance
x
(in ) from the heat source with the same parameter
if Figure 1 but, with
mm
as values1= 0.12898
s
-1 and
. The roots , and
have beenc),
270.86
36.813
ively.
a
0
T
res
0= 37.05CT
used for , 0= 36.932T
curves (a), (b),C
and (=
pect C
Figure 4. Plot of the response temperature along distance x (in
mm) from the heated surface for the same parameter value
Figure 1, but with different values of s as in
41
0
11
:= 7.1191
s

(1), 41
1= 6.508610
s
(2), and 1= 6.34921041
s
(3).
0=0T in Eq.19. The curves of this figure show that for
the same value of =0T0, the maximal va
onse temperature increases as the tem
lue of the re-
perature-de-
perfusion cnt
sp
pendentompone 1
decreases.
2.3. Case =0
Solving equation =0
with respect to
, we ob-
tain
3222
336 T


01
0
1
3332
1
=.
108 54 36
a




Inserting this expression for
in the expression for
3
g
, we find that
3
0
33
=,
213
g
which is always positive. Hence in the case where
=0
, Weierstrass’ function

23
;,
x
gg
ions [21] and (can be ex-
by trigonometric funct6) reads pressed



2
01
00
10
2
11
3
sin 2
=,
24 3
12 sin
62
RT ex
Tx TeRT
eex
if0,

(25)
and


0
00
2
0
6
=,if=0,and
24 a
RT
TxTT T
RTx

0
<,
(26)
here w3
13
=2eg in Eq.2 nd is a simple root
5, a0
T
of
RT .
A. Lakhssassi et al. / Natural Science 2 (2010) 1375-1385
Copyright © 2010 SciRes. Openly accessible at http://www.scirp.org/journal/NS/
138
1381
In Figure 5 we plot the response temperature
Tx
e same
ues of
along the distance from the heated surface for th
parameters as in Figure 1 but with different val
1
:2
1=1.7619 10
s
-1 for plot (a), 5
9 10
1=4.761
plot (c).
37.7 C,
Plots (a)
orre-
s-1
For
0
T
and
for plot (b), and
plots (a), (b),
= 38.85C
, and 0
T
(b) correspond to s
3 s
-1 for
d 0=T
respectively.
hile plot (c) c
1=1.666810
and (c) we use
=36.5C,
olution (25), w
sponds to solution (26) .
3. NUMERICAL COMPUTATION
For the numerical computation of the nonlinear heat
transfer Eq.2 we use the the Crank-Nicolson method, as
it is stable and accurate tcond order in space and
me [25]. The accuracy of erical computations
s checked by testing different time and space steps. The
variables t and x are measured in units of time and space,
respectively. In this study, we consider the input tem-
perature of arterial blood as an x-dependent quantity.
This inpuerature is chosen among the steady-state
solutions ained in the previous section. Any other
coefficient i Eq.2 is considered to be constant. We con-
sider the initial time is and distance
x from
the heateurface is compbetween 0 and
o se
our numti
i
a
T
t temp
obt
n
at th
d s0=0t
rised 0
X
so
that thest x distance furthermwe
designate by
e small is 0
x=0; ore
and h the time step and the
x
step,
respectively thatented b, so t will be incremy
and
x
bynote h. We de by n
j
T the nu

,:
nmerical valthe
p poiue of
tem erature atnt
,
j
nj jn
x
tT Txt, where
=tn
n
and =
n
x
jh . Using
a
Tx as the input tem-
perature, at =0t,
,0Tx is set the same as
a
Tx.
According to the Crank-Nicolson method, we use
1
111
1111
2
=;
22
=
2
n
jj
nn
nn nn
jjjj jj
TT
T
TTT TT
x


 
(27)
2
22
1
n
t
T
T
hh
Inserting Eq.27 into Eq.2, we obtain
.


111
11
2222
1
10
2
101
2
1=
222
12
.
2
nnnn
1
j
jj j
nn
jj
a
n
ja
TTTT
hhhh
TT
T
h
TT
h










 



As x-dependent input temperature of arterial blood,
we used either



25
25
327.41sn0.92823 10,0.5654.83
=,
8.6162sn0.92823 10,0.517.232
a
x
Tx x


(28)
(a)
(b)
(c)
distanceFigure 5. Plot of the response temperature along
x
(in mm) from the heated surface for the same parameter val-
ues as in Figure 1, but with different values of
21
11
:=1.7619 10 s

(a), 51
1=4.7619610

(b), and
31
1=1.66686 10
s
(c).
A. Lakhssassi et al. / Natural Science 2 (2010) 1375-1385
Copyright © 2010 SciRes. http://www.scirp.org/journal/NS/
1382
obtained fr om Eq.14, or

Openly accessible at
2
1070
= 36.5,
24 713.3
Tx
x
(29)
obtained from Eq.26. For the numerical simulation, we
used Mathematica.
3.1. Effect of the Thermal Diffusivity
For the numerical study of the effect of thermal diffu-
sivity, the following temperature-independent perfusion
component and temperature-dependent perfusion com-
ponent are used: 4
0=4.7619 10
s
-1 and
3
1= 6.44910
s
-1, reeters,
together with =1.2631
spectively. These param
ve been used for
plot (a) of Figu
Figure 6 shows the numerical solution for dit
2
/WJ.m ha
fferen
alues of
re 1.
v
. The first, second, and third rows of this
figure copond to , rres
11
82
=1.26311 10/WJ.m
42
=1.263 10/WJ.m
, and =1.26311 10/WJ.
22
m
s of the first, second, and th
erature at time =50t s, =8t
ectively. An
,
ird
0
respectiv The plot
columns sthe temp
s, and t s, resp
ely.
how
=100
x
-dependent inpu
ood (28) is used. The plots o
ponse temperature decrea
t
f
ses
temperatuof arterial bl
Figure 6ow that the res
as the thl diffusivity
re
sh
erma
increases.
3.2. Effect of the Temperature-Independent
Perfusion Component 0
To show the effect of the teerature-independent
perfusion componentmp
0
, we plotted Figures 7(a-f
with the parameter valu and
)
es 72
.26311 10/WJ.m
=1
(a) (b) (c)
(d) (e) (f)
(g)
Figure 6. Temperature plotted against distance x (in ) and at givan
(h) (i)
en time t for 0=4
d 31
1= 6.449
mm 41
.761910 s

10 s .
Each row is for a different thermal diffusivity :
/WJ
82
=1
22
.m f
, and
.26311 10/WJ
.m
or pts (g
(h), and =t
for plots
for plots (d), (e), and (f), and
for plots (a), (d), and (g),
(
ile each column is for
and (i).
a), (b), and (c), 42
=1.26311 10/WJ.m
), (h), and (i), wh different time: =50ts
100s for plots (c), (f),
=1.26311 10
8, for plots (b), (e)lo
=t0s
A. Lakhssassi et al. / Natural Science 2 (2010) 1375-1385
Copyright © 2010 SciRes. Openly accessible at http://www.scirp.org/journal/NS/
138
1383
(a) (b) (c)
(d) (e) (f)
Figure 7. Plot of the temperature response along the distance
x
(in ) from the heated surface at time for
72 and mm =100st
=1.26311 10/WJ.m
41
11
== 6.44910s

for plots (a) and (d), 0= 4
and for d of temperature-independent perfuponent
for plots (b) and (e), and c)
These plots show the tem-
distance x from the heated
or plots (a), (b), and (c), we
ure of arterial blood (28),
of arterial blood (29) is used
ots (a) and (d), (b) and (e),
with
iff
s erent valuession com
for plots (
41
00
:= 4.761910s


and (f).
3
.7619 10
21
0= 4.761910s

4
11
== 6.44910

perature response along t
surface at time =100t
have used the input te
while the input temp
for plots (d), (e), and (
and (c) and (f) are obtai
1
s.
he
s. F
mperat
erature
f). Pl
ned 4
0=4.7619 10
2
0= 4.761910
s
-1,
3
0= 4.761910
s
-1 and
can conclude that
ndent erfusion com
s
-1.
we the
ep po-
From the plots of Figur
effect of the temperature-
0
e 7
ind e p
nent
on the temperature response can be neglected.
3.3. Effect of the Temperature-Dependent
Perfusion Component 1
One of the effects of the temperature-dependent per-
fusion component 1
he te
10
can be obtained from Figure 8
where we plotted tmperature response as function of
the distance x from the heated surface at time s
and for =100t
72
=1.26311 /WJ.m
and 4
0= 4.761910
input tem
ratu
), and (
4
10
the -
d c)
4.7619
s-1 For pe used
peratu lots (a),and (c), w
re of arteri
ith
(b),
al blood (28), while plots (d), (e), and
(f) are obtained with the use of th e input tempere of
arterial blood (29). Plots (a) and (d), (b) an(e
and (f) are obtained w1=
s
-1,
3
1= 4.761910
s
-1 2
1= 4.761910
s
-1. It and
follows from the plots of Figure 8 that the temperature
response increases with the temperature-dependent per-
fusion component 1
.
he o
el of
on
e co
b
ction
e tem
ance from
been use
y, th
nd t
on th
lu
ent of t
4. CONCLUSION
In this paper, tne-dimensional steady-state bio-
heat transfer mod the living tissues has been taken
into account based the modified (nonlinear) Pennes
equation, and trresponding equation has been
he analytic
solution expressedy the Jacobi elliptic functions,
trigonometric funs and rational functions are de-
rived to obtain thperature changes with the varia-
tion of the dist the heated surface. The analytic
solutions have d to investigate the effects of the
thermal diffusivite temperature-independent perfu-
sion component, ahe temperature-dependent perfu-
sion component e temperature distribution. The
obtained analytic sotion can provide a good knowledge
are valuable
remhermal parameters, the recon-
structio and the thermal diag
an im
plant
h
solved both analytically and numerically. T
of thermal behavior of living tissues, which
for the measu
n of the temperature field -
nosis and treatment, the dosimetry, and the hum-
ation smart devices.
A. Lakhssassi et al. / Natural Science 2 (2010) 1375-1385
Copyright © 2010 SciRes. Openly accessible at http://www.scirp.org/journal/NS/
1384
(a) (b) (c)
(d) (e) (f)
Figure 8. Temperature
response along the distance x (in mm) from the heated surface at time =100 st for =1.2631
72
1 10/WJ.m
and 41
0= 4.761910s

and for different values of 4
11
:= 6.4410s

1
9
f)
for plots (a) and (d), 31
1= 6.44910
plots
21
9 10
.
5. ACKdevelopments in modeling heat trerfused
s
for
(b) andfor plots (c) and (
NOWLEDGEMENTS
The authors thank NSERC for
n the history of hea
ass transfer in. Journal of Biomec
lie
[3] Davalosa R.V. and Rubinsky B. (2008) Temperature con-
a-
s
[4]
ultrasou
rmotherapy combining nopagation
[7] Arkin, H., Xuolmes, K.R. (1994)
ansfer in blood p
tissues. IEEE Transactions on Biomedical Engineering,
erent thermal models. Bioelectromagnet-
ics, 30, 52-58. Solutions of the bio-heat transfer
dicine and Biology, 33, 785-792.
[1
sing bioheat transfer equation of
(e), and 1= 6.44
s
their financial support, CMC Micro-
systems and LIMA “Laboratoire d'Ingénierie des Microsystèmes
Avancés” at UQO (Université du Québec en Outaouais) for providing
design tools, support and associated technologies.
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