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

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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.

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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)

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)( 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:

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)()(

)(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.

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