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CHAPTER 3 ANALYTICAL SOLUTION:
STEADY STATE
PENNES’BIO-HEAT
EQUATION
In this chapter, based on the Pennes Bio-heat Equation, a
simplified one dimensional bio-heat transfer model of cylindrical living
tissue in the steady state has been set up for application and by using the
Bessel’s equation, its corresponding analytical solution has been derived,
with the obtained analytical solution, the effects of the thermal
conductivity, the blood perfusion, the metabolic heat generation, and the
coefficient of heat transfer on the temperature distribution in living
tissues are analized. The derived analytical solution is useful to study the
thermal behaviour of the biological tissue accurately.
Before the derivation of analytical solution let us briefly
understand the topics concerned with the solution.
3.1 NON-DIMENSIONALIZATION
Nondimensionalization is one kind of asymptotic reduction based on the
idea that certain terms are small and can be neglected when models are
far too complicated to analized rigorously. In this way terms are
evaluated by means of dimensionless ratios.
If a model has a variable u, say then we nondimensionalize that variable
by writing *][ uuu where ][u is the chosen scale and
*u is
corresponding dimensionless variable. Similarly timescale can be written
34
as *][ ttt . The process of nondimensionalization will give a set of
equations, each of whose terms is dimensionless after division through by
the dimension of the equation and hence it is possible to compare terms in
a meanningful way. The asterisked terms are called dimensionless
parameters.
3.2 CONCEPT OF BESSEL’S FUNCTIONS
In mathematics, Bessel functions, first defined by
the mathematician Daniel Bernoulli and generalized by Friedrich Bessel,
are particular solutions )(xy of Bessel's differential equation:
0)( 222
22 yx
dxdyx
dxydx (3.2.1)
for an arbitrary real or complex number α (the order of the Bessel
function). The most common and important special case is where α is an
integer n. Although α and −α produce the same differential equation, it is
conventional to define different Bessel functions for these two orders
Since this is a second-order differential equation, there must be
two linearly independent solutions.
35
Now 0x is the regular singular point of the equation hence we
take its solution as km
kk xay
0
Then,
1
0)(
km
kk xakm
xdyd
2
0)1)((2
2
km
kk xakmkm
xdyd
Substituting these values in one we have,
0])[( 2
00
22
km
kk
km
kk xaxakm
Equating to zero the coefficient of mx (putting k = 0), we have, m
are the indicial roots.
Equating to zero the coefficient of 1mx (putting k = 1), we have, 01 a
Hence Equating to zero the coefficient of 2kmx (putting k = k+2), we
have,
2,1,0;)2()2(2
k
kmkm
aa k
k
Then,
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for ,...5,3,1k we have 0...753 aaa and
for ,...4,2,0k we have
,...)4()4(
,)2()2(
24
02
mm
aa
mm
aa and so on
respectively.
...
])4[(])2[()2(1 2222
4
22
20
mm
x
m
xxay m (3.2.2)
depending upon values of , we have different types of solutions.
Bessel functions of the first kind : Jα
Bessel functions of the first kind, denoted by )(xJ , are solutions
of Bessel's differential equation that are finite at the origin (x = 0) for
non-negative integer α, and diverge as x approaches zero for negative
non-integer α.
When 0 and is not an integer then we get two independent
solutions for m and m
When m , we have
m
m
xmm
xJm 2
0 2)1(!)1()( (3.2.3)
37
which is called Bessel’s equation of first kind of order , where Γ(z) is
the gamma function, a generalization of the factorial function to non-
integer values and substituting for , we get Bessel’s equation of
first kind of order .
For non-integer α, the functions )(xJ and )(xJ are linearly
independent, and are therefore the two solutions of the differential
equation. On the other hand, for integer order α, the following
relationship is valid (note that the Gamma function becomes infinite for
negative integer arguments):
)()1()( xJxJ nn
n (3.2.4)
This means that the two solutions are no longer linearly
independent. In this case, the second linearly independent solution is then
found to be the Bessel function of the second kind, as discussed below.
Bessel functions of the second kind : Yα
The Bessel functions of the second kind, denoted by )(xY , are
solutions of the Bessel differential equation. They have a singularity at
the origin (x =0). )(xY is sometimes also called the Neumann function,
and is occasionally denoted instead by Nα(x) and given by,
38
2)()()(
xJxdxxJxY
(3.2.5)
when α is an integer, )(xY is the second linearly independent solution
of Bessel's equation. Both )(xJ and )(xY are holomorphic functions
of x on the complex plane cut along the negative real axis. When α is an
integer, the Bessel functions J are entire functions of x. If x is kept fixed,
then the Bessel functions are entire functions of α.
Modified Bessel functions : Iα , Kα
The Bessel functions are valid even for complex arguments x, and
an important special case is that of a purely imaginary argument. In this
case, the solutions to the Bessel equation are called the modified Bessel
functions (or occasionally the hyperbolic Bessel functions) of the first
and second kind, and are defined by any of these equivalent alternatives:
)(2)sin(
)()(2
)(
2)1(!1)()(
)1(1
2
0
xiHixIxIxK
xmm
xijixIm
m
(3.2.6)
39
There exist many integral representations of these functions. The
series expansion for Iα(x) is thus similar to that for Jα(x), but without the
alternating (−1)m factor.
Iα(x) and Kα(x) are the two linearly independent solutions to the modified
Bessel's equation:
0)( 222
22 yx
dxdyx
dxydx
Unlike the ordinary Bessel functions, which are oscillating
as functions of a real argument, Iα and Kα are exponentially
growing and decaying functions, respectively. Like the ordinary Bessel
function Jα, the function Iα goes to zero at x = 0 for α > 0 and is finite
at x = 0 for α = 0. Analogously, Kα diverges at x = 0.
3.3 ANALYTICAL SOLUTION Based on Pennes Equation, the one dimensional mathematical
model to describe the heat transfer of the cylindrical living tissues in the
steady state is shown as below where the governing equation is:
01
kQTT
kE
dr
dTr
drd
rm
Abbt
(3.3.1)
with boundary conditions given by:
40
0,0 dr
dTr t and
)(, TThdr
dTKRr A
t
where R is the radius of concerned tissue; Ah is the coefficient of heat
transfer which accounts for the effects of both convection and radiation
on the surface of the tissue; T is the ambient temperature.
Performing nondimensionalization of (3.3.1) by introducing
TT
TTT
Rrr
A
tt
** ; (3.3.2)
Subsituting (3.3.2) in (3.3.1) we have,
0)()()()(
1 **
***
K
QTTKETTTT
RrddRr
Rrdd
Rrm
Abb
At
0))(()(11 **
**
**2
K
QTTTTTKETT
drdTr
drd
rRm
AtAbb
At
0)]([)(1 2*
2
*
**
**
K
RQTTTTTK
RETTdrdTr
drd
rm
AtAbb
At
41
0)(
]1[1 2*2
*
**
**
TTKRQTR
KE
drdTr
drd
r A
mt
bbt (3.3.3)
Taking dimensionless parameters as ,
K
Rhh
TTK
RQQ
K
RE AA
A
mm
bbb
222* ** ;
)(;
The equation (3.3.1) along with boundary conditions can be rewritten as,
0][1 *****
**
**
mbtb
t QTdrdTr
drd
r (3.3.4)
Now,
0r 0* Rr
0* r BC(1)
0drdTt 0])([
)(*
* TTTTRrd
dAt
0*
*
drdTt BC(2)
Rr RRr *
1* r BC(3)
42
)( TThdrdTK A
t
])([])([)(
*** TTTTThTTTTRrd
dK AtAAt
**
*
tAt Th
drdT
RK
***
*
tAt Th
drdT
BC(4)
Assuming
MQmb ** ; Nb * ; *NTM
we have
0][1 *****
**
**
mbtb
t QTdrdTr
drd
r
01 **
**
**
MNT
drdTr
drd
r tt
01*
**
**
drdTr
drd
rt
43
0**
**
*
r
drdTr
drd t
01*
*
*2*
*2
drdT
rdr
Td tt
01**2*
2
NM
drd
rNM
drd
011**2*
2
drd
Nrdrd
N
01**2*
2
N
drd
rdrd
(3.3.5)
which is a zero order modified Bessel’s differential equation,whose
general solution can be expressed as,
)()()( 21 zKczIcz vv (3.3.6)
where vI and vK are the modified Bessel functions of first and second
kind respectively.
In order to determine if the analytical solution can be expressed by
Bessel’s functions, comparing (3.3.5) with generalized Bessel’s equation
as follows:
44
02212
)222()12(222222
2
Rxap
dxdR
xm
dxRd
x
vpmxmp
(3.3.7)
which has solution of the type
)()( 21
pv
pv
axm axYcaxJcexR (3.3.7(a))
where vJ and vY are modified Bessel’s functions of the first and second
kind respectively.
Now comparison of (3.3.5) and (3.3.7) gives
Napvm 2;1;0;0;0 (3.3.7(b))
since (3.3.5) is zero order modified bessel’s function (3.3.6) can be
rewritten using 3.3.7(a) and 3.3.7(b) as:
)()()( *2
*1 00 rNKcrNIcz
Hence
MQmb ** ; Nb * ; *tNTM
*tNTM )()( **
2**
1 00 rKcrIc bb
45
*tT
*
*
*
b
mb Q
)()( ***
2***
1
00 rKcrIcb
bb
b
Now when 0z we have 0)0(1 I and )0(1K
Considering given boundary conditions, we have
)(;0 **1*
1*
*
2 rIc
drdT
c bb
t
Hence
)(
)(**
1
*
**
*
****rI
drdT
QrTb
tb
b
mbt
)()(
)(1)1()(
****
******
10
0
b
A
bb
b
b
mt
Ih
I
rIQrT
So analytical solution for T is given by:
46
)()(
)(1)1)(()(
****
*****
10
0
b
A
bb
b
b
mAt
Ih
I
rIQTTTrT
(3.3.8)
which is the analytical solution of steady state Pennes Bioheat equation
without considering spatial heating.
If we consider spatial or Transient heating on skin surface then its
analytical solution can be obtained by Green’s function method which is
the concerned solution method of our study.
47
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