Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 734020 9 pageshttpdxdoiorg1011552013734020

Research ArticleAn Inverse Problem of TemperatureOptimization in Hyperthermia by Controllingthe Overall Heat Transfer Coefficient

Seyed Ali Aghayan1 Dariush Sardari1 Seyed Rabii Mahdi Mahdavi2

and Mohammad Hasan Zahmatkesh3

1 Faculty of Engineering Science and Research Branch Islamic Azad University PO Box 14515-775 Tehran Iran2Medical Physics Tehran University of Medical Science Tehran Iran3Medical Physics Shahid Beheshti University Tehran Iran

Correspondence should be addressed to Seyed Ali Aghayan ali aghayanymailcom

Received 14 January 2013 Revised 1 July 2013 Accepted 8 July 2013

Academic Editor Tin-Tai Chow

Copyright copy 2013 Seyed Ali Aghayan et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A novel scheme to obtain the optimum tissue heating condition during hyperthermia treatment is proposed To do this theeffect of the controllable overall heat transfer coefficient of the cooling system is investigated An inverse problem by a conjugatedgradient with adjoint equation is used in our model We apply the finite difference time domain method to numerically solve thetissue temperature distribution using Pennes bioheat transfer equation In order to provide a quantitative measurement of errorsconvergence history of the method and root mean square of errors are also calculated The effects of heat convection coefficient ofwater and thermal conductivity of casing layer on the control parameter are also discussed separately

1 Introduction

Hyperthermia treatment is recognized as an adjunct cancertherapy technique following surgery chemotherapy andradiation techniques In hyperthermia the tumor cells will beoverheated typically 40ndash45∘C to damage or kill the cancercells [1 2] The determination of temperature distributionthroughout the biological tissue by solving the simple Pennesbioheat Equation (Eq) and increment of tumor temperatureto therapeutic value avoiding the overheating of the healthytissue are important issues of investigation in hyperthermia[3] Because of obvious reasons which are not going to bediscussed here the water (or any other dielectric liquid) bolusis known as a necessary element of all contact applicatorsused in clinics in order to provide local heating of malignanttumors by means of electromagnetic (EM) energy [4] Thereare numerous papers discussing the effect of the water cou-pling bolus on the temperature of surface and tissue specificabsorption rate (SAR) during hyperthermia treatment [5 6]Sherar et al [7] presented a new way to design the water

bolus for external microwave applicators which increased theavailable treatment field size by removing the central hotspotcaused by the increased power from the applicator in thisregion They also used beam shaping bolus with an arrayof saline-filled patches inside the coupling water bolus forsuperficial microwave hyperthermia [8] Ebrahimi-Ganjehand Attari [9] numerically computed and compared boththe SAR distributions and effective field size in the presenceand absence of water bolus Neuman et al [10] examined theeffect of the various thickness of water bolus coupling layerson the SAR patterns from dual concentric conductor basedon the conformal microwave array superficial hyperthermiaapplicators

One aspect of our investigation is optimal control theorywhich is defined as amathematical approach of manipulatingthe parameters affecting the behaviour of a system to producethe desired or optimal outcome A recent development in thisfield is the optimal control of systemswith distributed param-eters which especially include the biological tissue in courseof the hyperthermia treatment planning [11] Butkovasky [11]

2 Journal of Applied Mathematics

treated the fundamentals in the theory of optimal controlof systems with distributed parameters Wagter [12] studiedan optimization procedure to calculate the transient temper-ature profiles in a plane tissue by multiple EM applicatorsKuznetsov [13] used the minimum principle of Pontryaginto solve an optimum control problem for heating a layerof tissue in order to destroy a cancerous tumor Althoughthe Levenberg-Marquardt and simplexmethods are commontechniques for solving optimization problems we tend todevelop a methodology to numerically optimize hyperther-mia treatments based on an inverse heat conduction problemby using the conjugate gradient method with an adjointequation (CGMAE) One of the advantages of the CGMAEis its insensitivity to initial variables however it requiresa numerical method for each iteration to determine thetemperature fields which involves computing the gradientsof objective function In the last two decades CGMAE hasbeen widely used in the solution of the general inverseheat transfer problems (IHTPs) regarding the optimizationprocedure [14] especially in therapeutic applications [15ndash17]The most common approach for determining the optimalestimation of the model parameters functions or boundaryconditions is to minimize an objective function

In this study the objective function to be minimized isbased on the criterion ofminimal square root of the differencebetween the desired temperature and the computed one Wedemonstrate that optimum heating conditions inside the bio-logical tissue which is caused by an applied radiofrequency(RF) heat source can be numerically attained within thetotal time of the process by means of controlling the overallheat transfer coefficient associated with the cooling systemTo do this and to reach the appropriate configuration ofcooling system thermal properties of water bolus and casinglayer are considered as variable parameters Recently fewstudies have been conductedwith the aimof beam-forming inhyperthermia which are based on utilizing the saline patchesinserted in the cooling system or the solid absorbing blocksbetween the applicator and water bolus [7 8 18] Howeverbest to our knowledge none of them has used an inversemodel to control the thermal or electrical properties of theseadditional patches or blocks In the perspective of the modelwe believe that the desired temperature profile inside thetumor according to its shape and size can be achieved byinversely optimizing the shape thickness electrical and ther-mal properties of solid or liquid patches which are insertedin the cooling system Therefore the proposed algorithmdeveloped in this study could be effectively used by physiciansto design an optimal treatment plan in a large variety oftherapeutic treatments

2 Optimization Scheme

21 Governing Equations and Model Geometry Schematicdiagram of the studied control zone is shown in Figure 1In order to simplify demonstrating the methodology heattransfer model is assumed to be one-dimensional It shouldbe noted that although we could develop a model with anirregular shape of a tumor embedded in the healthy tissue

Water bolus

Radi

atin

g an

tenn

a

Casing layer

0 a X

I III

Figure 1 Schematic diagram of control zone Cooling systemconsists of water bolus embedded in a casing layer which is placedbetween the applicator and biomedia Regions I and II represent thehealthy tissue and tumor region respectively

a homogeneous slab of tissue including the tumor region isconsidered Temperature distribution inside the tissue couldbe obtained by solving the Pennes bioheat equation [19] asfollows

120588119905119888119905

120597119879119905(119909 119905)

120597119905= nabla (119896

119905nabla119879119905(119909 119905)) + 120596bl120588bl119888bl (119879119886 minus 119879

119905(119909 119905))

+ 119876119898+ 119876119903(119909 119905) 0 lt 119909 lt 119886

(1)

119879119905(119909 119905) = 119879

119888119909 = 0 (2)

minus119896119905

120597119879119905(119909 119905)

120597119909= 119880 (119879

infinminus 119879119905(119909 119905)) 119909 = 119886 (3)

119879119905(119909 119905) = 119879

1198940 lt 119909 lt 119886 119905 = 0 (4)

where119880 is the overall heat transfer coefficient which definesthe effectiveness of the cooling system Because the tempera-ture distribution is very sensitive to the heat transfer betweenthe cooling liquid and the so-called tissue its numerical valuemust be properly estimated It is shown that the expression of119880 is [20] as follows

119880 =1

1ℎ + l119896cl (5)

Here ℎ is the heat convection coefficient of water and 119897 and119896cl are the thickness and thermal conductivity of the casinglayer respectively

For the inverse problem proposed here the overall heattransfer coefficient of cooling system is considered as anunknown function while the other quantities within theformulation of the direct problem presented previously are

Journal of Applied Mathematics 3

assumed to be known precisely Moreover the temperaturedistribution inside the tissue region is considered available

The optimization procedure includes maximizing thetemperature inside the tumor while minimizing it in thenormal surrounding tissue via a proposed objective func-tion In other words we tend to achieve the desiredtemperature 119884

119905(119909119895 119905) at 119873 points whether in the tumor

region or in the healthy one by optimal control of 119880 withinthe total time 119905

119891of the treatment process Therefore the

following function must be minimized

119866 (119880) =

119873

sum

119895=1

int

119905119891

0

[119879119905(119909119895 119905 119880) minus 119884

119905(119909119895 119905)]2

119889119905 (6)

where 119879119905(119909119895 119905 119880) is the computed temperature In order to

minimize 119866(119880) under the constraints mentioned in (1)ndash(4)we introduce a new function called the ldquoLagrangian functionrdquo(119871) which associates (6) and (1) with the following equation

119871 (119879119905 Ψ119905 119880) = 119866 (119880)

+ int

119905119891

0

int

119886

0

Ψ119905(119909 119905) [119896

119905

1205972

119879119905(119909 119905)

1205971199092

minus 120588119905119888119905

120597119879119905(119909 119905)

120597119905

+ 120596bl120588bl119888bl (119879119886 minus 119879119905(119909 119905))

+119876119898+ 119876119903(119909 119905) ] 119889119909 119889119905

(7)

where Ψ119905(119909 119905) is the Lagrange multiplier obtained by solving

the adjoint problem which is defined by the following set ofequations

119896119905

1205972

Ψ119905(119909 119905)

1205971199092+ 120588119905119888119905

120597Ψ119905(119909 119905)

120597119905minus 120596bl120588bl119888blΨ119905 (119909 119905)

+ 2 [119879119905(119909119895 119905) minus 119884

119905(119909119895 119905)]

times 120575 (119909 minus 119909119895) = 0 0 lt 119909 lt 119886

Ψ119905(119909 119905) = 0 119909 = 0

119896119905

120597Ψ119905(119909 119905)

120597119909= Ψ119905(119909 119905) 119880 119909 = 119886

Ψ119905(119909 119905) = 0 0 lt 119909 lt 119886 119905 = 119905

119891

(8)

where 120575 is the Dirac function Adjoint problem can also besolved by the samemethod as the direct problemof (1) that isby using the finite difference time domain method (FDTD)It can be shown that the gradient of residual functional nabla119866could be expressed by the following analytical expression

nabla119866 (119880) = int

119905119891

0

Ψ119905(119886 119905) (119879

119905(119886 119905) minus 119879

infin) 119889119905 (9)

The iterative procedure of CGMAE for the optimizationof 119880 is given by

119880119897+1

= 119880119897

minus 120573119897

119889119897

119897 = 0 1 2 (10)

Here 120573119897 is the step size and 119889119897 is the descent direction of the

119897th iteration which is defined as follows

119889119897

= nabla119866119897

+ 120574119897

119889119897minus1

1198890

= 0 (11)

The coefficient of conjugate gradient 120574119897 is given by

120574119897

=

10038171003817100381710038171003817nabla119866 (119880

119897

)10038171003817100381710038171003817

2

1003817100381710038171003817nabla119866 (119880119897minus1)10038171003817100381710038172

1205740

= 0 (12)

Then the step size could be derived from the followingequation

120573119897

=

sum119873

119895=1

int119905119891

0

119879119905(119909119895 119905) minus 119884

119905(119909119895 119905) Δ119879

119905(119909119895 119905) 119889119905

sum119873

119895=1

int119905119891

0

Δ119879119905(119909119895 119905)2

119889119905

(13)

Here Δ119879119905(119909119895 119905) is the answer to the sensitivity problem As

mentioned in the appendix we present themethodology usedto analytically derive the sensitivity the adjoint problems andthe gradient equation The iteration procedure is repeateduntil it satisfies the following truncation criteria and thenleads to optimal control of the overall heat transfer coefficientas follows

119880119897+1

minus 119880119897

119880119897le 119890 (14)

where 119890 is a small number (10minus4 sim10minus5) The computationalalgorithm for the foregoing procedure is given by the follow-ing

(1) Give an initial guess for 119880(2) Solve the direct problem(3) Examine (14) and if convergence is satisfied stop

otherwise continue(4) Solve the adjoint problem(5) Compute the gradient equation

(6) Compute the conjugate coefficient 120574119897 and the direc-tion of descent 119889119897

(7) Solve the sensitivity problem

(8) Compute the search step size 120573119897(9) Increment the controlling parameter using (10) and

return to Step 1

22 Model Parameters

221 Electromagnetic Model The spatial and temporalbehaviors of the EM field inside the tissues and are governed

4 Journal of Applied Mathematics

by Maxwellrsquos equations In a homogeneous tissue theseequations are expressed by [21] the following

nabla times 119864 = 119894120596120583119867

nabla sdot 119864 = 0

nabla sdot 119867 = 0

nabla times 119867 = minus119894120596120576lowast

119864

(15)

where 119864 and 119867 are respectively complex electric andmagnetic fields 120596 is the angular frequency and 120576 120590 and 120583

are the electrical permittivity the electrical conductivity andthe magnetic permeability of tissue respectively 120576lowast is thecomplex permittivity which is defined by

120576lowast

= 120576 +119894120590

120596 (16)

The volumetric EM power deposition which averaged overtime is calculated by

119876119903=120590|119864|2

2 (17)

where |119864| is the root mean square magnitude [Vm] of theincident electric field in a point which could be calculatedthrough (15)

222 Cooling System and Tissue Properties Thermophysicalproperties of tissue and blood pertaining to the model fornumerical simulation are provided in Table 1 [3 22 23]The thickness of water is set at 10minus2m and the electricalpermittivity and electrical conductivity of water related to anRF applicator operating at 432MHz are 78 and 00300 Smrespectively [23] Moreover thicknesses of 003m and 15 times10minus4mare used respectively for tissue and casing layer whichare typically made of thin flexible Silicon layers Temperatureof the cooling system 119879

infin is set at 15∘C

3 Assumptions

In order to simplify demonstrating the methodology heattransfer is assumed to be one-dimensional with a constantrate of tissue metabolism and blood perfusion The targettemperatures after the elapse of allowed heating periodare set between 42 and 45∘C for tumoral region and 42∘Cor lower for normal tissues In addition we consider threedifferent nodes one in the tumor region with 119884(0007 119905

119891) =

43∘C and two others in the normal tissue with 119884(0002 119905

119891)

and 119884(0021 119905119891) = 39

∘C as the desired temperaturesAs the thickness of the casing layer is much smaller thanits wavelength it could be neglected in the EM analysisMoreover the thermal contact between the casing layer andtissue is assumed to be perfect so that the thermal contactresistance between these layers is negligible

4 Results and Discussions

In the present work FDTD method is used for numericalsolution of the direct sensitivity and adjoint problems A

Table 1 Biophysical characteristics of model

Component Parameter Value Units

Blood119888bl 4200 jkg ∘C120588bl 1000 kgm3

120596bl 00005 mLsmL

Tissue

119888119905

3770 jkg ∘C119896119905

05 Wm ∘C119876119898

33800 Wm3

119879119886

37 ∘C119879119888

37 ∘C119879119894

34 ∘C120576119905

55 mdash120588119905

998 kgm3

120590119905

11 Sm

uniform grid of 31 times 31 mesh in space and time is utilized forall the results presented hereafterTheoptimization algorithmpresented previously is programmed in Fortran 90 compiledthrough the software Compaq Visual Fortran ProfessionalEdition on a computer with an Intel Core i7 35 GHz and16Gb of RAM The final time 119905

119891 is chosen equal to 1800 s

Moreover the computational precision (119890) is set at 10minus5 In thefollowing simulation the desired temperature distribution isdetermined by precisely prescribing a known 119880

Figure 2 indicates the sensitivity of the temperature-time curve for variations in 119880 for three different valuesof 119880 Because of the large values of the sensitivity withrespect to 119880 obviously the proposed optimization model isldquowell-conditionedrdquo To illustrate the accuracy of the presentinverse analysis the initial approximations of 119880 were eitherincreased or decreased from its known value Figure 3 showsthe number of convergent iterations with different initialguesses for 119880 Changing the initial value for example by15 and 30 of its known value does not affect the solutionand the optimization object is still satisfied Moreover it isclear that the number of convergent iterations is reducedwhen the initial guess value varies within a 15 deviationfrom the known one In addition this figure shows thatthe minimization of the objective function is valid when 119880

varies between 125 and 200Wm2∘C In order to provide aquantitative measurement of errors the convergence historyof the presented model is also calculated Figure 4 comparesthe convergence history is of three different initial guessesConvergence history is measured by the relative error at eachiteration that is |119880retrieved minus 119880prescribed|119880prescribed

The computed temperature distribution is obtained whenthe minimization of the objective function is satisfied thatis when the optimal value of 119880 is achieved Figure 5 showsthe computed temperature distribution with initial guess of119880 = 200Wm2∘C as well as the desired one which isobtained from the application of a known 119880 at 119905 = 1000 sIt is observed that the recovered temperature distribution isin good agreement with the desired one In addition thecomputed RMS error is equal to 024 Figure 6 presents thedesired temperature distributions result from the solution of

Journal of Applied Mathematics 5

0 01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

06

07

ΔT

(∘C)

ttfU = 150Wm2 ∘CU = 175Wm2 ∘CU = 200Wm2 ∘C

Figure 2 Temperature sensitivity resulting from variations in 119880

2 4 6 8 10 12 14 16 18 20125

150

175

200

Number of iterations

U(W

m2∘ C)

Figure 3 Convergence curve for different initial guesses of 119880

the nonstationary thermal problem during the total heatingprocess It is found to be sufficient to elevate the temperatureinside the tumor to 42∘C within approximately 20 minuteswhile the temperature on the surface remains approximatelyconstant in this period of time

According to the foregoing discussion as (5) and inorder to enhance the performance of the cooling system thethermal conductivity of casing layer and the heat convectioncoefficient of water have been considered as variable param-eters Although the casing layer is very thin as shown inFigure 7 the influence of thermal conductivity of casing layeris remarkable It could be observed that a slight increase inthermal conductivity reduces significantly the temperature atthe skin surface In addition the position of the maximumtemperature is shifted slightly toward the skin The secondpossibility to control the effectiveness of the cooling systemis through the variation of the heat convection coefficient ofwater Figure 8 indicates the influence of two different valuesof ℎ on the temperature of tissue It could be observed thata 75 increase in ℎ results in a decrease of about 25∘C inthe skin and only 2∘C in the maximum The same resultscould be obtained by only 20 of the variation in the thermal

2 4 6 8 10 12 14 16 18 20Iteration

Rela

tive e

rror

10minus0

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

Figure 4 Comparison of convergence histories of CGM for differ-ent initial guessesmdash loz 125 ◻ 150 and I 200Wm2∘C for 119880

0 001 002 003Depth of tissue (m)

Prescribed temperature distributionRecovered temperature distribution

34

37

40

43 II II

Tem

pera

ture

(∘C)

Figure 5 A comparison between recovered and desired tempera-ture distribution at 119905 = 1000 s

0 001 002 003Depth of tissue (m)

34

37

40

43

31

t = 200 st = 500 st = 600 st = 1000 s

t = 1200 st = 1600 st = 1800 s

Tem

pera

ture

(∘C)

Figure 6 Transient spatial temperature distribution inside thetissue

6 Journal of Applied Mathematics

0 001 002 003Depth of tissue (m)

37

40

43

34

Tem

pera

ture

(∘C)

Figure 7 Temperature distribution at 119905 = 1000 s for two differentthermal conductivities of casing layer Solid line 119896cl = 014Wm∘C119880 = 175Wm2∘C dashed line 119896cl = 011Wm∘C119880 = 150Wm2∘C

0 001 002 003Depth of tissue (m)

34

40

43

31

37

Tem

pera

ture

(∘C)

Figure 8 Temperature distribution at 119905 = 1000 s for twodifferent heat convection coefficients of water Solid line ℎ =

200Wm2∘C119880 = 175Wm2∘C dashed line ℎ = 350Wm2∘C119880 = 200Wm2∘C

conductivity of casing layer However depending on thetechnical facilities one can make an optimal decision in thiscase

Although the thickness and the electrical conductivityof water can also be optimized in the treatment process wedid not consider them in this study and we mainly focusedon the influence of the overall heat transfer coefficient onthe temperature field inside the tissue In the perspective ofthe present model we believe that the desired temperatureprofile inside the tumor according to its shape and size canbe achieved by inversely optimizing the shape thicknesselectrical and thermal properties of solid or liquid patcheswhich are inserted in the cooling system Such an optimiza-tion problem should also incorporate the best configurationof the additional patches which reasonably must resemblethe tumor shape for example L-form patches for L-formtumors However the clinician will first scan the geometryof the tumor and measure the tumor size and depth usingan MRI or CT or other noninvasive imaging techniquesAlthough the present model is based on the 1D heat transferbut with some modifications in the governing formulations

such as 119880 = 119891(119910 119911) it has the potential to provide amore comprehensive understanding of the cooling systemability for the purposes of beam-forming in two or threedimensions This modified optimization algorithm is thenimplemented to the tumor and its surrounding tissue todetermine the optimal properties of cooling system

5 Conclusions

In this paper we demonstrated the validity of the CGMAE inthe optimization of the temperature distribution inside thetissue by optimal control of the overall heat transfer coef-ficient associated with the cooling system The temperaturedistribution inside the tissue based on the bioheat transferequation has also been numerically computed In order toevaluate the effectiveness of the cooling system the thermalconductivity of casing layer and the heat convection coef-ficient of water have been considered separately Accordingto results the presented model allows obtaining the bestproperties of water and casing layer in the light of opti-mization and therefore significantly improves the treatmentoutcome However this method shows great promise forfurther extensions in both model and clinical performancesfor any shaped targeted tumor in hyperthermia treatments

Appendix

In order to find the minimum of the residual function bygradienttype method it is required to derive its gradient bythe following processes first computation of the variationproblem when the unknown has a small variation andsecond inspection of the conditions under which we obtain astationary point for the Lagrange functional To compute thevariation problem the unknown function 119880 is perturbed byΔ119880 that is

119880120576= 119880 + Δ119880 (A1)

Then the temperature undergoes the variation Δ119879119905(119909 119905) as

follows

119879119905120576(119909 119905 119880) = 119879

119905(119909 119905 119880) + Δ119879

119905(119909 119905 119880) (A2)

where 120576 refers to the perturbed variables By insertingthe a aforementioned equations in the direct problem andsubtracting the original (1)ndash(4) from the resulting expressionsand ignoring the second order terms the following directproblem in variation is obtained

120588119905119888119905

120597Δ119879119905(119909 119905)

120597119905= 119896119905

1205972

Δ119879119905(119909 119905)

1205971199092

minus 120596bl120588bl119888blΔ119879119905 (119909 119905) 0 lt 119909 lt 119886

(A3)

Δ119879119905(119909 119905) = 0 119909 = 0 (A4)

minus119896119905

120597Δ119879119905(119909 119905)

120597119909= Δ119880 (119879

infinminus 119879119905(119909 119905))

minus 119880Δ119879119905(119909 119905) 119909 = 119886

(A5)

Δ119879119905(119909 119905) = 0 0 lt 119909 lt 119886 119905 = 0 (A6)

Journal of Applied Mathematics 7

The second step is to calculate the variation of the objectivefunction Δ119866(119880) which is given by

Δ119866 (119880) = int

119905119891

0

int

119886

0

2 Δ119879119905(119909119895 119905) [119879

119905(119909119895 119905) minus119884

119905(119909119895 119905)]

times 120575 (119909 minus 119909119895) 119889119909 119889119905

(A7)

We obtain the variation of the augmented functionalΔ119871(119879Ψ

119905 119880) by considering the variation problem given by

(A3) and the variation of the objective function (A7) asfollows

Δ119871 = int

119905119891

0

int

119886

0

2 Δ119879119905(119909119895 119905) [119879

119905(119909119895 119905) minus 119884

119905(119909119895 119905)]

times 120575 (119909 minus 119909119895) 119889119909 119889119905

+ int

119905119891

0

int

119886

0

Ψ119905(119909 119905) [119896

119905

1205972

Δ119879119905(119909 119905)

1205971199092

minus 120588119905119888119905

120597Δ119879119905(119909 119905)

120597119905

minus120596bl120588bl119888blΔ119879119905 (119909 119905) ] 119889119909 119889119905

(A8)

Using integration by parts in (A8)

int

119905119891

0

int

119886

0

1205972

Δ119879119905(119909 119905)

1205971199092Ψ119905(119909 119905) 119889119909 119889119905

= int

119905119891

0

[120597Δ119879119905(119886 119905)

120597119909Ψ119905(119886 119905) minus

120597Ψ119905(119886 119905)

120597119909Δ119879119905(119886 119905)

minus120597Δ119879119905(0 119905)

120597119909Ψ119905(0 119905)

+120597Ψ119905(0 119905)

120597119909Δ119879119905(0 119905)] 119889119905

+ int

119905119891

0

int

119886

0

1205972

Ψ119905(119909 119905)

1205971199092Δ119879119905(119909 119905) 119889119909 119889119905

(A9)

int

119886

0

int

119905119891

0

120597Δ119879119905(119909 119905)

120597119905Ψ119905(119909 119905) 119889119905 119889119909

= int

119886

0

[Δ119879119905(119909 119905119891)Ψ119905(119909 119905) minus Δ119879

119905(119909 0) Ψ

119905(119909 0)] 119889119909

minus int

119886

0

int

119905119891

0

Δ119879119905(119909 119905)

120597Ψ119905(119909 119905)

120597119909119889119905 119889119909

(A10)

and by using the boundary conditions of the direct problemin variations (Equations (A3)ndash(A6) and the rearrangement

of the mathematical terms) the final form of (A8) can bewritten as follows

Δ119871 = int

119905119891

0

int

119886

0

[119896119905

1205972

Ψ119905(119909 119905)

1205971199092+ 120588119905119888119905

120597Ψ119905(119909 119905)

120597119905

minus 120596bl120588bl119888blΨ119905 (119909 119905)

+ 2 [119879119905(119909119895 119905) minus 119884

119905(119909119895 119905)]

times 120575 (119909 minus 119909119895) ]Δ119879

119905(119909 119905) 119889119909 119889119905

minus int

119905119891

0

119896119905Ψ119905(0 119905)

120597Δ119879119905(0 119905)

120597119909119889119905

+ int

119905119891

0

(Ψ119905(119886 119905) 119880 minus 119896

119905

120597Ψ119905(119886 119905)

120597119909)Δ119879119905(119886 119905) 119889119905

minus int

119886

0

120588119905119888119905Ψ119905(119909 119905119891) Δ119879119905(119909 119905119891) 119889119909

+ int

119905119891

0

Δ119880 Ψ119905(119886 119905) (119879

119905(119886 119905) minus 119879

infin) 119889119905

(A11)

To satisfy the optimality condition that isΔ119871(Δ119879Ψ

119905 Δ119880) = 0 it is required that all coefficients

of Δ119879119905(119909 119905) are to be vanished Finally the following term is

left

Δ119871 (119879Ψ119905 119880) = int

119905119891

0

Δ119880 Ψ119905(119886 119905) (119879

119905(119886 119905) minus 119879

infin) 119889119905

(A12)

Since we know that 119879119905(119909 119905) and Ψ

119905(119909 119905) are respectively the

direct and the adjoint problem solutions it follows that

119871 (119879Ψ119905 119880) = 119866 (119880) (A13)

and then

Δ119871 (119879Ψ119905 119880) = Δ119866 (119880) (A14)

Based on the definition the variation of the functional 119866(119880)can be written as follows

Δ119866 (119880) = ⟨nabla119866 Δ119880⟩ (A15)

where the associated inner scalar product ⟨ ⟩ of two functions119891(119905) and 119892(119905) in the Hilbert space of 119871

2(0 119905119891) is defined by

⟨119891 (119905) 119892 (119905)⟩1198712

= int

119889

119888

119891 (119905) 119892 (119905) 119889119905 (A16)

From the definition of the scalar product the variation of thefunctional 119866(119880) can be written as follows

Δ119866 (119880) = ⟨nabla119866 (119880) Δ119880⟩1198712

= nabla119866 (119880)Δ119880 (A17)

A comparison between (A12) (A14) and (A17) reveals thatthe gradient of the residual functional is given by (9) Moredetails about the derivation of the equations are mentionedin [24ndash26]

8 Journal of Applied Mathematics

Nomenclature

119888 Heat capacity119889 Vector of descent direction119890 Computational precision119864 Complex electric field119866(119880) Residual functionalΔ119866(119880) Residual functional variationnabla119866(119880) Gradient of residual functionalℎ Heat convection coefficient119867 Complex magnetic field119896 Thermal conductivity119897 Thickness of casing layer119871 Lagrangian functional119873 Number of nodes119876119898 Tissue metabolism

119876119903 Volumetric heat source

119879 Temperature119905 Time119880 Overall heat transfer coefficient119909 Space119884 Desired temperature

Greek Symbols

120573 Descent parameter120574 Descent direction parameter120575 Dirac function120576 Electrical permeability120588 Density120583 Magnetic permeability120590 Electrical conductivityΨ Lagrange multiplier120596 Angular frequency120596 Blood perfusion

Subscripts

119886 Arterialbl Blood119888 Corecl Casing layer119891 Final119894 Initial time119905 Tissueinfin Ambient

Superscripts

119897 Iteration number

References

[1] S R H Davidson and D F James ldquoDrilling in bone model-ing heat generation and temperature distributionrdquo Journal ofBiomechanical Engineering vol 125 no 3 pp 305ndash314 2003

[2] W G Sawyer M A Hamilton B J Fregly and S A BanksldquoTemperature modeling in a total knee joint replacement usingpatient-specific kinematicsrdquo Tribology Letters vol 15 no 4 pp343ndash351 2003

[3] R Dhar P Dhar and R Dhar ldquoProblem on optimal distributionof induced microwave by heating probe at the tumor site inhyperthermiardquo Advanced Modeling and Optimization vol 13no 1 pp 39ndash48 2011

[4] E A Gelvich and V N Mazokhin ldquoResonance effects in appli-cator water boluses and their influence on SAR distributionpatternsrdquo International Journal of Hyperthermia vol 16 no 2pp 113ndash128 2000

[5] M de Bruijne T Samaras J F Bakker and G C van RhoonldquoEffects of waterbolus size shape and configuration on the SARdistribution pattern of the Lucite cone applicatorrdquo InternationalJournal of Hyperthermia vol 22 no 1 pp 15ndash28 2006

[6] E R Lee D S Kapp AW Lohrbach and J L Sokol ldquoInfluenceof water bolus temperature onmeasured skin surface and intra-dermal temperaturesrdquo International Journal of Hyperthermiavol 10 no 1 pp 59ndash72 1994

[7] M D Sherar F F Liu D J Newcombe et al ldquoBeam shaping formicrowave waveguide hyperthermia applicatorsrdquo InternationalJournal of Radiation Oncology Biology Physics vol 25 no 5 pp849ndash857 1993

[8] J C Kumaradas and M D Sherar ldquoOptimization of a beamshaping bolus for superficial microwave hyperthermia waveg-uide applicators using a finite element methodrdquo Physics inMedicine and Biology vol 48 no 1 pp 1ndash18 2003

[9] M A Ebrahimi-Ganjeh and A R Attari ldquoStudy of water boluseffect on SAR penetration depth and effective field size for localhyperthermiardquo Progress in Electromagnetics Research B vol 4pp 273ndash283 2008

[10] D G Neuman P R Stauffer S Jacobsen and F RossettoldquoSAR pattern perturbations from resonance effects in waterbolus layers used with superficial microwave hyperthermiaapplicatorsrdquo International Journal of Hyperthermia vol 18 no3 pp 180ndash193 2002

[11] A G Butkovasky Distributed Control System American Else-vier New York NY USA 1969

[12] C De Wagter ldquoOptimization of simulated two-dimensionaltemperature distributions induced by multiple electromagneticapplicatorsrdquo IEEE Transactions on Microwave Theory and Tech-niques vol 34 no 5 pp 589ndash596 1986

[13] A V Kuznetsov ldquoOptimization problems for bioheat equationrdquoInternational Communications in Heat and Mass Transfer vol33 no 5 pp 537ndash543 2006

[14] C Huang and S Wang ldquoA three-dimensional inverse heatconduction problem in estimating surface heat flux by conju-gate gradient methodrdquo International Journal of Heat and MassTransfer vol 42 no 18 pp 3387ndash3403 1999

[15] P K Dhar and D K Sinha ldquoTemperature control of tissueby transient-induced-microwavesrdquo International Journal of Sys-tems Science vol 19 no 10 pp 2051ndash2055 1988

[16] T Loulou and E P Scott ldquoThermal dose optimization in hyper-thermia treatments by using the conjugate gradient methodrdquoNumerical Heat Transfer A vol 42 no 7 pp 661ndash683 2002

[17] T Loulou and E P Scott ldquoAn inverse heat conduction problemwith heat flux measurementsrdquo International Journal for Numer-ical Methods in Engineering vol 67 no 11 pp 1587ndash1616 2006

[18] H Kroeze M Van Vulpen A A C De Leeuw J B Van DeKamer and J J W Lagendijk ldquoThe use of absorbing structuresduring regional hyperthermia treatmentrdquo International Journalof Hyperthermia vol 17 no 3 pp 240ndash257 2001

[19] H H Pennes ldquoAnalysis of tissue and arterial blood tem-peratures in the resting human forearmrdquo Journal of AppliedPhysiology vol 1 pp 93ndash122 1948

Journal of Applied Mathematics 9

[20] J P HolmanHeat Transfer McGraw Hill New York NY USA9th edition 2002

[21] C Thiebaut and D Lemonnier ldquoThree-dimensional modellingand optimisation of thermal fields induced in a human bodyduring hyperthermiardquo International Journal of Thermal Sci-ences vol 41 no 6 pp 500ndash508 2002

[22] Z S Deng and J Liu ldquoAnalytical study on bioheat transferproblems with spatial or transient heating on skin surface orinside biological bodiesrdquo Journal of Biomechanical Engineeringvol 124 no 6 pp 638ndash649 2002

[23] N K Uzunoglu E A Angelikas and P A Cosmidis ldquoA432MHz local hyperthermia system using an indirectly cooledwater-loaded waveguide applicatorrdquo IEEE Transactions onMicrowave Theory and Techniques vol 35 no 2 pp 106ndash1111987

[24] OM Alifanov ldquoSolution of an inverse problem of heat conduc-tion by iteration methodsrdquo Journal of Engineering Physics vol26 no 4 pp 471ndash476 1974

[25] O M Alifanov Inverse Heat Transfer Problem Springer NewYork NY USA 1994

[26] M N Ozisik and H R B Orlande Inverse Heat TransferFundamentals and Applications Taylor amp Francis New YorkNY USA 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

treated the fundamentals in the theory of optimal controlof systems with distributed parameters Wagter [12] studiedan optimization procedure to calculate the transient temper-ature profiles in a plane tissue by multiple EM applicatorsKuznetsov [13] used the minimum principle of Pontryaginto solve an optimum control problem for heating a layerof tissue in order to destroy a cancerous tumor Althoughthe Levenberg-Marquardt and simplexmethods are commontechniques for solving optimization problems we tend todevelop a methodology to numerically optimize hyperther-mia treatments based on an inverse heat conduction problemby using the conjugate gradient method with an adjointequation (CGMAE) One of the advantages of the CGMAEis its insensitivity to initial variables however it requiresa numerical method for each iteration to determine thetemperature fields which involves computing the gradientsof objective function In the last two decades CGMAE hasbeen widely used in the solution of the general inverseheat transfer problems (IHTPs) regarding the optimizationprocedure [14] especially in therapeutic applications [15ndash17]The most common approach for determining the optimalestimation of the model parameters functions or boundaryconditions is to minimize an objective function

In this study the objective function to be minimized isbased on the criterion ofminimal square root of the differencebetween the desired temperature and the computed one Wedemonstrate that optimum heating conditions inside the bio-logical tissue which is caused by an applied radiofrequency(RF) heat source can be numerically attained within thetotal time of the process by means of controlling the overallheat transfer coefficient associated with the cooling systemTo do this and to reach the appropriate configuration ofcooling system thermal properties of water bolus and casinglayer are considered as variable parameters Recently fewstudies have been conductedwith the aimof beam-forming inhyperthermia which are based on utilizing the saline patchesinserted in the cooling system or the solid absorbing blocksbetween the applicator and water bolus [7 8 18] Howeverbest to our knowledge none of them has used an inversemodel to control the thermal or electrical properties of theseadditional patches or blocks In the perspective of the modelwe believe that the desired temperature profile inside thetumor according to its shape and size can be achieved byinversely optimizing the shape thickness electrical and ther-mal properties of solid or liquid patches which are insertedin the cooling system Therefore the proposed algorithmdeveloped in this study could be effectively used by physiciansto design an optimal treatment plan in a large variety oftherapeutic treatments

2 Optimization Scheme

21 Governing Equations and Model Geometry Schematicdiagram of the studied control zone is shown in Figure 1In order to simplify demonstrating the methodology heattransfer model is assumed to be one-dimensional It shouldbe noted that although we could develop a model with anirregular shape of a tumor embedded in the healthy tissue

Water bolus

Radi

atin

g an

tenn

a

Casing layer

0 a X

I III

Figure 1 Schematic diagram of control zone Cooling systemconsists of water bolus embedded in a casing layer which is placedbetween the applicator and biomedia Regions I and II represent thehealthy tissue and tumor region respectively

a homogeneous slab of tissue including the tumor region isconsidered Temperature distribution inside the tissue couldbe obtained by solving the Pennes bioheat equation [19] asfollows

120588119905119888119905

120597119879119905(119909 119905)

120597119905= nabla (119896

119905nabla119879119905(119909 119905)) + 120596bl120588bl119888bl (119879119886 minus 119879

119905(119909 119905))

+ 119876119898+ 119876119903(119909 119905) 0 lt 119909 lt 119886

(1)

119879119905(119909 119905) = 119879

119888119909 = 0 (2)

minus119896119905

120597119879119905(119909 119905)

120597119909= 119880 (119879

infinminus 119879119905(119909 119905)) 119909 = 119886 (3)

119879119905(119909 119905) = 119879

1198940 lt 119909 lt 119886 119905 = 0 (4)

where119880 is the overall heat transfer coefficient which definesthe effectiveness of the cooling system Because the tempera-ture distribution is very sensitive to the heat transfer betweenthe cooling liquid and the so-called tissue its numerical valuemust be properly estimated It is shown that the expression of119880 is [20] as follows

119880 =1

1ℎ + l119896cl (5)

Here ℎ is the heat convection coefficient of water and 119897 and119896cl are the thickness and thermal conductivity of the casinglayer respectively

For the inverse problem proposed here the overall heattransfer coefficient of cooling system is considered as anunknown function while the other quantities within theformulation of the direct problem presented previously are

Journal of Applied Mathematics 3

assumed to be known precisely Moreover the temperaturedistribution inside the tissue region is considered available

The optimization procedure includes maximizing thetemperature inside the tumor while minimizing it in thenormal surrounding tissue via a proposed objective func-tion In other words we tend to achieve the desiredtemperature 119884

119905(119909119895 119905) at 119873 points whether in the tumor

region or in the healthy one by optimal control of 119880 withinthe total time 119905

119891of the treatment process Therefore the

following function must be minimized

119866 (119880) =

119873

sum

119895=1

int

119905119891

0

[119879119905(119909119895 119905 119880) minus 119884

119905(119909119895 119905)]2

119889119905 (6)

where 119879119905(119909119895 119905 119880) is the computed temperature In order to

minimize 119866(119880) under the constraints mentioned in (1)ndash(4)we introduce a new function called the ldquoLagrangian functionrdquo(119871) which associates (6) and (1) with the following equation

119871 (119879119905 Ψ119905 119880) = 119866 (119880)

+ int

119905119891

0

int

119886

0

Ψ119905(119909 119905) [119896

119905

1205972

119879119905(119909 119905)

1205971199092

minus 120588119905119888119905

120597119879119905(119909 119905)

120597119905

+ 120596bl120588bl119888bl (119879119886 minus 119879119905(119909 119905))

+119876119898+ 119876119903(119909 119905) ] 119889119909 119889119905

(7)

where Ψ119905(119909 119905) is the Lagrange multiplier obtained by solving

the adjoint problem which is defined by the following set ofequations

119896119905

1205972

Ψ119905(119909 119905)

1205971199092+ 120588119905119888119905

120597Ψ119905(119909 119905)

120597119905minus 120596bl120588bl119888blΨ119905 (119909 119905)

+ 2 [119879119905(119909119895 119905) minus 119884

119905(119909119895 119905)]

times 120575 (119909 minus 119909119895) = 0 0 lt 119909 lt 119886

Ψ119905(119909 119905) = 0 119909 = 0

119896119905

120597Ψ119905(119909 119905)

120597119909= Ψ119905(119909 119905) 119880 119909 = 119886

Ψ119905(119909 119905) = 0 0 lt 119909 lt 119886 119905 = 119905

119891

(8)

where 120575 is the Dirac function Adjoint problem can also besolved by the samemethod as the direct problemof (1) that isby using the finite difference time domain method (FDTD)It can be shown that the gradient of residual functional nabla119866could be expressed by the following analytical expression

nabla119866 (119880) = int

119905119891

0

Ψ119905(119886 119905) (119879

119905(119886 119905) minus 119879

infin) 119889119905 (9)

The iterative procedure of CGMAE for the optimizationof 119880 is given by

119880119897+1

= 119880119897

minus 120573119897

119889119897

119897 = 0 1 2 (10)

Here 120573119897 is the step size and 119889119897 is the descent direction of the

119897th iteration which is defined as follows

119889119897

= nabla119866119897

+ 120574119897

119889119897minus1

1198890

= 0 (11)

The coefficient of conjugate gradient 120574119897 is given by

120574119897

=

10038171003817100381710038171003817nabla119866 (119880

119897

)10038171003817100381710038171003817

2

1003817100381710038171003817nabla119866 (119880119897minus1)10038171003817100381710038172

1205740

= 0 (12)

Then the step size could be derived from the followingequation

120573119897

=

sum119873

119895=1

int119905119891

0

119879119905(119909119895 119905) minus 119884

119905(119909119895 119905) Δ119879

119905(119909119895 119905) 119889119905

sum119873

119895=1

int119905119891

0

Δ119879119905(119909119895 119905)2

119889119905

(13)

Here Δ119879119905(119909119895 119905) is the answer to the sensitivity problem As

mentioned in the appendix we present themethodology usedto analytically derive the sensitivity the adjoint problems andthe gradient equation The iteration procedure is repeateduntil it satisfies the following truncation criteria and thenleads to optimal control of the overall heat transfer coefficientas follows

119880119897+1

minus 119880119897

119880119897le 119890 (14)

where 119890 is a small number (10minus4 sim10minus5) The computationalalgorithm for the foregoing procedure is given by the follow-ing

(1) Give an initial guess for 119880(2) Solve the direct problem(3) Examine (14) and if convergence is satisfied stop

otherwise continue(4) Solve the adjoint problem(5) Compute the gradient equation

(6) Compute the conjugate coefficient 120574119897 and the direc-tion of descent 119889119897

(7) Solve the sensitivity problem

(8) Compute the search step size 120573119897(9) Increment the controlling parameter using (10) and

return to Step 1

22 Model Parameters

221 Electromagnetic Model The spatial and temporalbehaviors of the EM field inside the tissues and are governed

4 Journal of Applied Mathematics

by Maxwellrsquos equations In a homogeneous tissue theseequations are expressed by [21] the following

nabla times 119864 = 119894120596120583119867

nabla sdot 119864 = 0

nabla sdot 119867 = 0

nabla times 119867 = minus119894120596120576lowast

119864

(15)

where 119864 and 119867 are respectively complex electric andmagnetic fields 120596 is the angular frequency and 120576 120590 and 120583

are the electrical permittivity the electrical conductivity andthe magnetic permeability of tissue respectively 120576lowast is thecomplex permittivity which is defined by

120576lowast

= 120576 +119894120590

120596 (16)

The volumetric EM power deposition which averaged overtime is calculated by

119876119903=120590|119864|2

2 (17)

where |119864| is the root mean square magnitude [Vm] of theincident electric field in a point which could be calculatedthrough (15)

222 Cooling System and Tissue Properties Thermophysicalproperties of tissue and blood pertaining to the model fornumerical simulation are provided in Table 1 [3 22 23]The thickness of water is set at 10minus2m and the electricalpermittivity and electrical conductivity of water related to anRF applicator operating at 432MHz are 78 and 00300 Smrespectively [23] Moreover thicknesses of 003m and 15 times10minus4mare used respectively for tissue and casing layer whichare typically made of thin flexible Silicon layers Temperatureof the cooling system 119879

infin is set at 15∘C

3 Assumptions

In order to simplify demonstrating the methodology heattransfer is assumed to be one-dimensional with a constantrate of tissue metabolism and blood perfusion The targettemperatures after the elapse of allowed heating periodare set between 42 and 45∘C for tumoral region and 42∘Cor lower for normal tissues In addition we consider threedifferent nodes one in the tumor region with 119884(0007 119905

119891) =

43∘C and two others in the normal tissue with 119884(0002 119905

119891)

and 119884(0021 119905119891) = 39

∘C as the desired temperaturesAs the thickness of the casing layer is much smaller thanits wavelength it could be neglected in the EM analysisMoreover the thermal contact between the casing layer andtissue is assumed to be perfect so that the thermal contactresistance between these layers is negligible

4 Results and Discussions

In the present work FDTD method is used for numericalsolution of the direct sensitivity and adjoint problems A

Table 1 Biophysical characteristics of model

Component Parameter Value Units

Blood119888bl 4200 jkg ∘C120588bl 1000 kgm3

120596bl 00005 mLsmL

Tissue

119888119905

3770 jkg ∘C119896119905

05 Wm ∘C119876119898

33800 Wm3

119879119886

37 ∘C119879119888

37 ∘C119879119894

34 ∘C120576119905

55 mdash120588119905

998 kgm3

120590119905

11 Sm

uniform grid of 31 times 31 mesh in space and time is utilized forall the results presented hereafterTheoptimization algorithmpresented previously is programmed in Fortran 90 compiledthrough the software Compaq Visual Fortran ProfessionalEdition on a computer with an Intel Core i7 35 GHz and16Gb of RAM The final time 119905

119891 is chosen equal to 1800 s

Moreover the computational precision (119890) is set at 10minus5 In thefollowing simulation the desired temperature distribution isdetermined by precisely prescribing a known 119880

Figure 2 indicates the sensitivity of the temperature-time curve for variations in 119880 for three different valuesof 119880 Because of the large values of the sensitivity withrespect to 119880 obviously the proposed optimization model isldquowell-conditionedrdquo To illustrate the accuracy of the presentinverse analysis the initial approximations of 119880 were eitherincreased or decreased from its known value Figure 3 showsthe number of convergent iterations with different initialguesses for 119880 Changing the initial value for example by15 and 30 of its known value does not affect the solutionand the optimization object is still satisfied Moreover it isclear that the number of convergent iterations is reducedwhen the initial guess value varies within a 15 deviationfrom the known one In addition this figure shows thatthe minimization of the objective function is valid when 119880

varies between 125 and 200Wm2∘C In order to provide aquantitative measurement of errors the convergence historyof the presented model is also calculated Figure 4 comparesthe convergence history is of three different initial guessesConvergence history is measured by the relative error at eachiteration that is |119880retrieved minus 119880prescribed|119880prescribed

The computed temperature distribution is obtained whenthe minimization of the objective function is satisfied thatis when the optimal value of 119880 is achieved Figure 5 showsthe computed temperature distribution with initial guess of119880 = 200Wm2∘C as well as the desired one which isobtained from the application of a known 119880 at 119905 = 1000 sIt is observed that the recovered temperature distribution isin good agreement with the desired one In addition thecomputed RMS error is equal to 024 Figure 6 presents thedesired temperature distributions result from the solution of

Journal of Applied Mathematics 5

0 01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

06

07

ΔT

(∘C)

ttfU = 150Wm2 ∘CU = 175Wm2 ∘CU = 200Wm2 ∘C

Figure 2 Temperature sensitivity resulting from variations in 119880

2 4 6 8 10 12 14 16 18 20125

150

175

200

Number of iterations

U(W

m2∘ C)

Figure 3 Convergence curve for different initial guesses of 119880

the nonstationary thermal problem during the total heatingprocess It is found to be sufficient to elevate the temperatureinside the tumor to 42∘C within approximately 20 minuteswhile the temperature on the surface remains approximatelyconstant in this period of time

According to the foregoing discussion as (5) and inorder to enhance the performance of the cooling system thethermal conductivity of casing layer and the heat convectioncoefficient of water have been considered as variable param-eters Although the casing layer is very thin as shown inFigure 7 the influence of thermal conductivity of casing layeris remarkable It could be observed that a slight increase inthermal conductivity reduces significantly the temperature atthe skin surface In addition the position of the maximumtemperature is shifted slightly toward the skin The secondpossibility to control the effectiveness of the cooling systemis through the variation of the heat convection coefficient ofwater Figure 8 indicates the influence of two different valuesof ℎ on the temperature of tissue It could be observed thata 75 increase in ℎ results in a decrease of about 25∘C inthe skin and only 2∘C in the maximum The same resultscould be obtained by only 20 of the variation in the thermal

2 4 6 8 10 12 14 16 18 20Iteration

Rela

tive e

rror

10minus0

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

Figure 4 Comparison of convergence histories of CGM for differ-ent initial guessesmdash loz 125 ◻ 150 and I 200Wm2∘C for 119880

0 001 002 003Depth of tissue (m)

Prescribed temperature distributionRecovered temperature distribution

34

37

40

43 II II

Tem

pera

ture

(∘C)

Figure 5 A comparison between recovered and desired tempera-ture distribution at 119905 = 1000 s

0 001 002 003Depth of tissue (m)

34

37

40

43

31

t = 200 st = 500 st = 600 st = 1000 s

t = 1200 st = 1600 st = 1800 s

Tem

pera

ture

(∘C)

Figure 6 Transient spatial temperature distribution inside thetissue

6 Journal of Applied Mathematics

0 001 002 003Depth of tissue (m)

37

40

43

34

Tem

pera

ture

(∘C)

Figure 7 Temperature distribution at 119905 = 1000 s for two differentthermal conductivities of casing layer Solid line 119896cl = 014Wm∘C119880 = 175Wm2∘C dashed line 119896cl = 011Wm∘C119880 = 150Wm2∘C

0 001 002 003Depth of tissue (m)

34

40

43

31

37

Tem

pera

ture

(∘C)

Figure 8 Temperature distribution at 119905 = 1000 s for twodifferent heat convection coefficients of water Solid line ℎ =

200Wm2∘C119880 = 175Wm2∘C dashed line ℎ = 350Wm2∘C119880 = 200Wm2∘C

conductivity of casing layer However depending on thetechnical facilities one can make an optimal decision in thiscase

Although the thickness and the electrical conductivityof water can also be optimized in the treatment process wedid not consider them in this study and we mainly focusedon the influence of the overall heat transfer coefficient onthe temperature field inside the tissue In the perspective ofthe present model we believe that the desired temperatureprofile inside the tumor according to its shape and size canbe achieved by inversely optimizing the shape thicknesselectrical and thermal properties of solid or liquid patcheswhich are inserted in the cooling system Such an optimiza-tion problem should also incorporate the best configurationof the additional patches which reasonably must resemblethe tumor shape for example L-form patches for L-formtumors However the clinician will first scan the geometryof the tumor and measure the tumor size and depth usingan MRI or CT or other noninvasive imaging techniquesAlthough the present model is based on the 1D heat transferbut with some modifications in the governing formulations

such as 119880 = 119891(119910 119911) it has the potential to provide amore comprehensive understanding of the cooling systemability for the purposes of beam-forming in two or threedimensions This modified optimization algorithm is thenimplemented to the tumor and its surrounding tissue todetermine the optimal properties of cooling system

5 Conclusions

In this paper we demonstrated the validity of the CGMAE inthe optimization of the temperature distribution inside thetissue by optimal control of the overall heat transfer coef-ficient associated with the cooling system The temperaturedistribution inside the tissue based on the bioheat transferequation has also been numerically computed In order toevaluate the effectiveness of the cooling system the thermalconductivity of casing layer and the heat convection coef-ficient of water have been considered separately Accordingto results the presented model allows obtaining the bestproperties of water and casing layer in the light of opti-mization and therefore significantly improves the treatmentoutcome However this method shows great promise forfurther extensions in both model and clinical performancesfor any shaped targeted tumor in hyperthermia treatments

Appendix

In order to find the minimum of the residual function bygradienttype method it is required to derive its gradient bythe following processes first computation of the variationproblem when the unknown has a small variation andsecond inspection of the conditions under which we obtain astationary point for the Lagrange functional To compute thevariation problem the unknown function 119880 is perturbed byΔ119880 that is

119880120576= 119880 + Δ119880 (A1)

Then the temperature undergoes the variation Δ119879119905(119909 119905) as

follows

119879119905120576(119909 119905 119880) = 119879

119905(119909 119905 119880) + Δ119879

119905(119909 119905 119880) (A2)

where 120576 refers to the perturbed variables By insertingthe a aforementioned equations in the direct problem andsubtracting the original (1)ndash(4) from the resulting expressionsand ignoring the second order terms the following directproblem in variation is obtained

120588119905119888119905

120597Δ119879119905(119909 119905)

120597119905= 119896119905

1205972

Δ119879119905(119909 119905)

1205971199092

minus 120596bl120588bl119888blΔ119879119905 (119909 119905) 0 lt 119909 lt 119886

(A3)

Δ119879119905(119909 119905) = 0 119909 = 0 (A4)

minus119896119905

120597Δ119879119905(119909 119905)

120597119909= Δ119880 (119879

infinminus 119879119905(119909 119905))

minus 119880Δ119879119905(119909 119905) 119909 = 119886

(A5)

Δ119879119905(119909 119905) = 0 0 lt 119909 lt 119886 119905 = 0 (A6)

Journal of Applied Mathematics 7

The second step is to calculate the variation of the objectivefunction Δ119866(119880) which is given by

Δ119866 (119880) = int

119905119891

0

int

119886

0

2 Δ119879119905(119909119895 119905) [119879

119905(119909119895 119905) minus119884

119905(119909119895 119905)]

times 120575 (119909 minus 119909119895) 119889119909 119889119905

(A7)

We obtain the variation of the augmented functionalΔ119871(119879Ψ

119905 119880) by considering the variation problem given by

(A3) and the variation of the objective function (A7) asfollows

Δ119871 = int

119905119891

0

int

119886

0

2 Δ119879119905(119909119895 119905) [119879

119905(119909119895 119905) minus 119884

119905(119909119895 119905)]

times 120575 (119909 minus 119909119895) 119889119909 119889119905

+ int

119905119891

0

int

119886

0

Ψ119905(119909 119905) [119896

119905

1205972

Δ119879119905(119909 119905)

1205971199092

minus 120588119905119888119905

120597Δ119879119905(119909 119905)

120597119905

minus120596bl120588bl119888blΔ119879119905 (119909 119905) ] 119889119909 119889119905

(A8)

Using integration by parts in (A8)

int

119905119891

0

int

119886

0

1205972

Δ119879119905(119909 119905)

1205971199092Ψ119905(119909 119905) 119889119909 119889119905

= int

119905119891

0

[120597Δ119879119905(119886 119905)

120597119909Ψ119905(119886 119905) minus

120597Ψ119905(119886 119905)

120597119909Δ119879119905(119886 119905)

minus120597Δ119879119905(0 119905)

120597119909Ψ119905(0 119905)

+120597Ψ119905(0 119905)

120597119909Δ119879119905(0 119905)] 119889119905

+ int

119905119891

0

int

119886

0

1205972

Ψ119905(119909 119905)

1205971199092Δ119879119905(119909 119905) 119889119909 119889119905

(A9)

int

119886

0

int

119905119891

0

120597Δ119879119905(119909 119905)

120597119905Ψ119905(119909 119905) 119889119905 119889119909

= int

119886

0

[Δ119879119905(119909 119905119891)Ψ119905(119909 119905) minus Δ119879

119905(119909 0) Ψ

119905(119909 0)] 119889119909

minus int

119886

0

int

119905119891

0

Δ119879119905(119909 119905)

120597Ψ119905(119909 119905)

120597119909119889119905 119889119909

(A10)

and by using the boundary conditions of the direct problemin variations (Equations (A3)ndash(A6) and the rearrangement

of the mathematical terms) the final form of (A8) can bewritten as follows

Δ119871 = int

119905119891

0

int

119886

0

[119896119905

1205972

Ψ119905(119909 119905)

1205971199092+ 120588119905119888119905

120597Ψ119905(119909 119905)

120597119905

minus 120596bl120588bl119888blΨ119905 (119909 119905)

+ 2 [119879119905(119909119895 119905) minus 119884

119905(119909119895 119905)]

times 120575 (119909 minus 119909119895) ]Δ119879

119905(119909 119905) 119889119909 119889119905

minus int

119905119891

0

119896119905Ψ119905(0 119905)

120597Δ119879119905(0 119905)

120597119909119889119905

+ int

119905119891

0

(Ψ119905(119886 119905) 119880 minus 119896

119905

120597Ψ119905(119886 119905)

120597119909)Δ119879119905(119886 119905) 119889119905

minus int

119886

0

120588119905119888119905Ψ119905(119909 119905119891) Δ119879119905(119909 119905119891) 119889119909

+ int

119905119891

0

Δ119880 Ψ119905(119886 119905) (119879

119905(119886 119905) minus 119879

infin) 119889119905

(A11)

To satisfy the optimality condition that isΔ119871(Δ119879Ψ

119905 Δ119880) = 0 it is required that all coefficients

of Δ119879119905(119909 119905) are to be vanished Finally the following term is

left

Δ119871 (119879Ψ119905 119880) = int

119905119891

0

Δ119880 Ψ119905(119886 119905) (119879

119905(119886 119905) minus 119879

infin) 119889119905

(A12)

Since we know that 119879119905(119909 119905) and Ψ

119905(119909 119905) are respectively the

direct and the adjoint problem solutions it follows that

119871 (119879Ψ119905 119880) = 119866 (119880) (A13)

and then

Δ119871 (119879Ψ119905 119880) = Δ119866 (119880) (A14)

Based on the definition the variation of the functional 119866(119880)can be written as follows

Δ119866 (119880) = ⟨nabla119866 Δ119880⟩ (A15)

where the associated inner scalar product ⟨ ⟩ of two functions119891(119905) and 119892(119905) in the Hilbert space of 119871

2(0 119905119891) is defined by

⟨119891 (119905) 119892 (119905)⟩1198712

= int

119889

119888

119891 (119905) 119892 (119905) 119889119905 (A16)

From the definition of the scalar product the variation of thefunctional 119866(119880) can be written as follows

Δ119866 (119880) = ⟨nabla119866 (119880) Δ119880⟩1198712

= nabla119866 (119880)Δ119880 (A17)

A comparison between (A12) (A14) and (A17) reveals thatthe gradient of the residual functional is given by (9) Moredetails about the derivation of the equations are mentionedin [24ndash26]

8 Journal of Applied Mathematics

Nomenclature

119888 Heat capacity119889 Vector of descent direction119890 Computational precision119864 Complex electric field119866(119880) Residual functionalΔ119866(119880) Residual functional variationnabla119866(119880) Gradient of residual functionalℎ Heat convection coefficient119867 Complex magnetic field119896 Thermal conductivity119897 Thickness of casing layer119871 Lagrangian functional119873 Number of nodes119876119898 Tissue metabolism

119876119903 Volumetric heat source

119879 Temperature119905 Time119880 Overall heat transfer coefficient119909 Space119884 Desired temperature

Greek Symbols

120573 Descent parameter120574 Descent direction parameter120575 Dirac function120576 Electrical permeability120588 Density120583 Magnetic permeability120590 Electrical conductivityΨ Lagrange multiplier120596 Angular frequency120596 Blood perfusion

Subscripts

119886 Arterialbl Blood119888 Corecl Casing layer119891 Final119894 Initial time119905 Tissueinfin Ambient

Superscripts

119897 Iteration number

References

[1] S R H Davidson and D F James ldquoDrilling in bone model-ing heat generation and temperature distributionrdquo Journal ofBiomechanical Engineering vol 125 no 3 pp 305ndash314 2003

[2] W G Sawyer M A Hamilton B J Fregly and S A BanksldquoTemperature modeling in a total knee joint replacement usingpatient-specific kinematicsrdquo Tribology Letters vol 15 no 4 pp343ndash351 2003

[3] R Dhar P Dhar and R Dhar ldquoProblem on optimal distributionof induced microwave by heating probe at the tumor site inhyperthermiardquo Advanced Modeling and Optimization vol 13no 1 pp 39ndash48 2011

[4] E A Gelvich and V N Mazokhin ldquoResonance effects in appli-cator water boluses and their influence on SAR distributionpatternsrdquo International Journal of Hyperthermia vol 16 no 2pp 113ndash128 2000

[5] M de Bruijne T Samaras J F Bakker and G C van RhoonldquoEffects of waterbolus size shape and configuration on the SARdistribution pattern of the Lucite cone applicatorrdquo InternationalJournal of Hyperthermia vol 22 no 1 pp 15ndash28 2006

[6] E R Lee D S Kapp AW Lohrbach and J L Sokol ldquoInfluenceof water bolus temperature onmeasured skin surface and intra-dermal temperaturesrdquo International Journal of Hyperthermiavol 10 no 1 pp 59ndash72 1994

[7] M D Sherar F F Liu D J Newcombe et al ldquoBeam shaping formicrowave waveguide hyperthermia applicatorsrdquo InternationalJournal of Radiation Oncology Biology Physics vol 25 no 5 pp849ndash857 1993

[8] J C Kumaradas and M D Sherar ldquoOptimization of a beamshaping bolus for superficial microwave hyperthermia waveg-uide applicators using a finite element methodrdquo Physics inMedicine and Biology vol 48 no 1 pp 1ndash18 2003

[9] M A Ebrahimi-Ganjeh and A R Attari ldquoStudy of water boluseffect on SAR penetration depth and effective field size for localhyperthermiardquo Progress in Electromagnetics Research B vol 4pp 273ndash283 2008

[10] D G Neuman P R Stauffer S Jacobsen and F RossettoldquoSAR pattern perturbations from resonance effects in waterbolus layers used with superficial microwave hyperthermiaapplicatorsrdquo International Journal of Hyperthermia vol 18 no3 pp 180ndash193 2002

[11] A G Butkovasky Distributed Control System American Else-vier New York NY USA 1969

[12] C De Wagter ldquoOptimization of simulated two-dimensionaltemperature distributions induced by multiple electromagneticapplicatorsrdquo IEEE Transactions on Microwave Theory and Tech-niques vol 34 no 5 pp 589ndash596 1986

[13] A V Kuznetsov ldquoOptimization problems for bioheat equationrdquoInternational Communications in Heat and Mass Transfer vol33 no 5 pp 537ndash543 2006

[14] C Huang and S Wang ldquoA three-dimensional inverse heatconduction problem in estimating surface heat flux by conju-gate gradient methodrdquo International Journal of Heat and MassTransfer vol 42 no 18 pp 3387ndash3403 1999

[15] P K Dhar and D K Sinha ldquoTemperature control of tissueby transient-induced-microwavesrdquo International Journal of Sys-tems Science vol 19 no 10 pp 2051ndash2055 1988

[16] T Loulou and E P Scott ldquoThermal dose optimization in hyper-thermia treatments by using the conjugate gradient methodrdquoNumerical Heat Transfer A vol 42 no 7 pp 661ndash683 2002

[17] T Loulou and E P Scott ldquoAn inverse heat conduction problemwith heat flux measurementsrdquo International Journal for Numer-ical Methods in Engineering vol 67 no 11 pp 1587ndash1616 2006

[18] H Kroeze M Van Vulpen A A C De Leeuw J B Van DeKamer and J J W Lagendijk ldquoThe use of absorbing structuresduring regional hyperthermia treatmentrdquo International Journalof Hyperthermia vol 17 no 3 pp 240ndash257 2001

[19] H H Pennes ldquoAnalysis of tissue and arterial blood tem-peratures in the resting human forearmrdquo Journal of AppliedPhysiology vol 1 pp 93ndash122 1948

Journal of Applied Mathematics 9

[20] J P HolmanHeat Transfer McGraw Hill New York NY USA9th edition 2002

[21] C Thiebaut and D Lemonnier ldquoThree-dimensional modellingand optimisation of thermal fields induced in a human bodyduring hyperthermiardquo International Journal of Thermal Sci-ences vol 41 no 6 pp 500ndash508 2002

[22] Z S Deng and J Liu ldquoAnalytical study on bioheat transferproblems with spatial or transient heating on skin surface orinside biological bodiesrdquo Journal of Biomechanical Engineeringvol 124 no 6 pp 638ndash649 2002

[23] N K Uzunoglu E A Angelikas and P A Cosmidis ldquoA432MHz local hyperthermia system using an indirectly cooledwater-loaded waveguide applicatorrdquo IEEE Transactions onMicrowave Theory and Techniques vol 35 no 2 pp 106ndash1111987

[24] OM Alifanov ldquoSolution of an inverse problem of heat conduc-tion by iteration methodsrdquo Journal of Engineering Physics vol26 no 4 pp 471ndash476 1974

[25] O M Alifanov Inverse Heat Transfer Problem Springer NewYork NY USA 1994

[26] M N Ozisik and H R B Orlande Inverse Heat TransferFundamentals and Applications Taylor amp Francis New YorkNY USA 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

assumed to be known precisely Moreover the temperaturedistribution inside the tissue region is considered available

The optimization procedure includes maximizing thetemperature inside the tumor while minimizing it in thenormal surrounding tissue via a proposed objective func-tion In other words we tend to achieve the desiredtemperature 119884

119905(119909119895 119905) at 119873 points whether in the tumor

region or in the healthy one by optimal control of 119880 withinthe total time 119905

119891of the treatment process Therefore the

following function must be minimized

119866 (119880) =

119873

sum

119895=1

int

119905119891

0

[119879119905(119909119895 119905 119880) minus 119884

119905(119909119895 119905)]2

119889119905 (6)

where 119879119905(119909119895 119905 119880) is the computed temperature In order to

minimize 119866(119880) under the constraints mentioned in (1)ndash(4)we introduce a new function called the ldquoLagrangian functionrdquo(119871) which associates (6) and (1) with the following equation

119871 (119879119905 Ψ119905 119880) = 119866 (119880)

+ int

119905119891

0

int

119886

0

Ψ119905(119909 119905) [119896

119905

1205972

119879119905(119909 119905)

1205971199092

minus 120588119905119888119905

120597119879119905(119909 119905)

120597119905

+ 120596bl120588bl119888bl (119879119886 minus 119879119905(119909 119905))

+119876119898+ 119876119903(119909 119905) ] 119889119909 119889119905

(7)

where Ψ119905(119909 119905) is the Lagrange multiplier obtained by solving

the adjoint problem which is defined by the following set ofequations

119896119905

1205972

Ψ119905(119909 119905)

1205971199092+ 120588119905119888119905

120597Ψ119905(119909 119905)

120597119905minus 120596bl120588bl119888blΨ119905 (119909 119905)

+ 2 [119879119905(119909119895 119905) minus 119884

119905(119909119895 119905)]

times 120575 (119909 minus 119909119895) = 0 0 lt 119909 lt 119886

Ψ119905(119909 119905) = 0 119909 = 0

119896119905

120597Ψ119905(119909 119905)

120597119909= Ψ119905(119909 119905) 119880 119909 = 119886

Ψ119905(119909 119905) = 0 0 lt 119909 lt 119886 119905 = 119905

119891

(8)

where 120575 is the Dirac function Adjoint problem can also besolved by the samemethod as the direct problemof (1) that isby using the finite difference time domain method (FDTD)It can be shown that the gradient of residual functional nabla119866could be expressed by the following analytical expression

nabla119866 (119880) = int

119905119891

0

Ψ119905(119886 119905) (119879

119905(119886 119905) minus 119879

infin) 119889119905 (9)

The iterative procedure of CGMAE for the optimizationof 119880 is given by

119880119897+1

= 119880119897

minus 120573119897

119889119897

119897 = 0 1 2 (10)

Here 120573119897 is the step size and 119889119897 is the descent direction of the

119897th iteration which is defined as follows

119889119897

= nabla119866119897

+ 120574119897

119889119897minus1

1198890

= 0 (11)

The coefficient of conjugate gradient 120574119897 is given by

120574119897

=

10038171003817100381710038171003817nabla119866 (119880

119897

)10038171003817100381710038171003817

2

1003817100381710038171003817nabla119866 (119880119897minus1)10038171003817100381710038172

1205740

= 0 (12)

Then the step size could be derived from the followingequation

120573119897

=

sum119873

119895=1

int119905119891

0

119879119905(119909119895 119905) minus 119884

119905(119909119895 119905) Δ119879

119905(119909119895 119905) 119889119905

sum119873

119895=1

int119905119891

0

Δ119879119905(119909119895 119905)2

119889119905

(13)

Here Δ119879119905(119909119895 119905) is the answer to the sensitivity problem As

mentioned in the appendix we present themethodology usedto analytically derive the sensitivity the adjoint problems andthe gradient equation The iteration procedure is repeateduntil it satisfies the following truncation criteria and thenleads to optimal control of the overall heat transfer coefficientas follows

119880119897+1

minus 119880119897

119880119897le 119890 (14)

where 119890 is a small number (10minus4 sim10minus5) The computationalalgorithm for the foregoing procedure is given by the follow-ing

(1) Give an initial guess for 119880(2) Solve the direct problem(3) Examine (14) and if convergence is satisfied stop

otherwise continue(4) Solve the adjoint problem(5) Compute the gradient equation

(6) Compute the conjugate coefficient 120574119897 and the direc-tion of descent 119889119897

(7) Solve the sensitivity problem

(8) Compute the search step size 120573119897(9) Increment the controlling parameter using (10) and

return to Step 1

22 Model Parameters

221 Electromagnetic Model The spatial and temporalbehaviors of the EM field inside the tissues and are governed

4 Journal of Applied Mathematics

by Maxwellrsquos equations In a homogeneous tissue theseequations are expressed by [21] the following

nabla times 119864 = 119894120596120583119867

nabla sdot 119864 = 0

nabla sdot 119867 = 0

nabla times 119867 = minus119894120596120576lowast

119864

(15)

where 119864 and 119867 are respectively complex electric andmagnetic fields 120596 is the angular frequency and 120576 120590 and 120583

are the electrical permittivity the electrical conductivity andthe magnetic permeability of tissue respectively 120576lowast is thecomplex permittivity which is defined by

120576lowast

= 120576 +119894120590

120596 (16)

The volumetric EM power deposition which averaged overtime is calculated by

119876119903=120590|119864|2

2 (17)

where |119864| is the root mean square magnitude [Vm] of theincident electric field in a point which could be calculatedthrough (15)

222 Cooling System and Tissue Properties Thermophysicalproperties of tissue and blood pertaining to the model fornumerical simulation are provided in Table 1 [3 22 23]The thickness of water is set at 10minus2m and the electricalpermittivity and electrical conductivity of water related to anRF applicator operating at 432MHz are 78 and 00300 Smrespectively [23] Moreover thicknesses of 003m and 15 times10minus4mare used respectively for tissue and casing layer whichare typically made of thin flexible Silicon layers Temperatureof the cooling system 119879

infin is set at 15∘C

3 Assumptions

In order to simplify demonstrating the methodology heattransfer is assumed to be one-dimensional with a constantrate of tissue metabolism and blood perfusion The targettemperatures after the elapse of allowed heating periodare set between 42 and 45∘C for tumoral region and 42∘Cor lower for normal tissues In addition we consider threedifferent nodes one in the tumor region with 119884(0007 119905

119891) =

43∘C and two others in the normal tissue with 119884(0002 119905

119891)

and 119884(0021 119905119891) = 39

∘C as the desired temperaturesAs the thickness of the casing layer is much smaller thanits wavelength it could be neglected in the EM analysisMoreover the thermal contact between the casing layer andtissue is assumed to be perfect so that the thermal contactresistance between these layers is negligible

4 Results and Discussions

In the present work FDTD method is used for numericalsolution of the direct sensitivity and adjoint problems A

Table 1 Biophysical characteristics of model

Component Parameter Value Units

Blood119888bl 4200 jkg ∘C120588bl 1000 kgm3

120596bl 00005 mLsmL

Tissue

119888119905

3770 jkg ∘C119896119905

05 Wm ∘C119876119898

33800 Wm3

119879119886

37 ∘C119879119888

37 ∘C119879119894

34 ∘C120576119905

55 mdash120588119905

998 kgm3

120590119905

11 Sm

uniform grid of 31 times 31 mesh in space and time is utilized forall the results presented hereafterTheoptimization algorithmpresented previously is programmed in Fortran 90 compiledthrough the software Compaq Visual Fortran ProfessionalEdition on a computer with an Intel Core i7 35 GHz and16Gb of RAM The final time 119905

119891 is chosen equal to 1800 s

Moreover the computational precision (119890) is set at 10minus5 In thefollowing simulation the desired temperature distribution isdetermined by precisely prescribing a known 119880

Figure 2 indicates the sensitivity of the temperature-time curve for variations in 119880 for three different valuesof 119880 Because of the large values of the sensitivity withrespect to 119880 obviously the proposed optimization model isldquowell-conditionedrdquo To illustrate the accuracy of the presentinverse analysis the initial approximations of 119880 were eitherincreased or decreased from its known value Figure 3 showsthe number of convergent iterations with different initialguesses for 119880 Changing the initial value for example by15 and 30 of its known value does not affect the solutionand the optimization object is still satisfied Moreover it isclear that the number of convergent iterations is reducedwhen the initial guess value varies within a 15 deviationfrom the known one In addition this figure shows thatthe minimization of the objective function is valid when 119880

varies between 125 and 200Wm2∘C In order to provide aquantitative measurement of errors the convergence historyof the presented model is also calculated Figure 4 comparesthe convergence history is of three different initial guessesConvergence history is measured by the relative error at eachiteration that is |119880retrieved minus 119880prescribed|119880prescribed

The computed temperature distribution is obtained whenthe minimization of the objective function is satisfied thatis when the optimal value of 119880 is achieved Figure 5 showsthe computed temperature distribution with initial guess of119880 = 200Wm2∘C as well as the desired one which isobtained from the application of a known 119880 at 119905 = 1000 sIt is observed that the recovered temperature distribution isin good agreement with the desired one In addition thecomputed RMS error is equal to 024 Figure 6 presents thedesired temperature distributions result from the solution of

Journal of Applied Mathematics 5

0 01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

06

07

ΔT

(∘C)

ttfU = 150Wm2 ∘CU = 175Wm2 ∘CU = 200Wm2 ∘C

Figure 2 Temperature sensitivity resulting from variations in 119880

2 4 6 8 10 12 14 16 18 20125

150

175

200

Number of iterations

U(W

m2∘ C)

Figure 3 Convergence curve for different initial guesses of 119880

the nonstationary thermal problem during the total heatingprocess It is found to be sufficient to elevate the temperatureinside the tumor to 42∘C within approximately 20 minuteswhile the temperature on the surface remains approximatelyconstant in this period of time

According to the foregoing discussion as (5) and inorder to enhance the performance of the cooling system thethermal conductivity of casing layer and the heat convectioncoefficient of water have been considered as variable param-eters Although the casing layer is very thin as shown inFigure 7 the influence of thermal conductivity of casing layeris remarkable It could be observed that a slight increase inthermal conductivity reduces significantly the temperature atthe skin surface In addition the position of the maximumtemperature is shifted slightly toward the skin The secondpossibility to control the effectiveness of the cooling systemis through the variation of the heat convection coefficient ofwater Figure 8 indicates the influence of two different valuesof ℎ on the temperature of tissue It could be observed thata 75 increase in ℎ results in a decrease of about 25∘C inthe skin and only 2∘C in the maximum The same resultscould be obtained by only 20 of the variation in the thermal

2 4 6 8 10 12 14 16 18 20Iteration

Rela

tive e

rror

10minus0

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

Figure 4 Comparison of convergence histories of CGM for differ-ent initial guessesmdash loz 125 ◻ 150 and I 200Wm2∘C for 119880

0 001 002 003Depth of tissue (m)

Prescribed temperature distributionRecovered temperature distribution

34

37

40

43 II II

Tem

pera

ture

(∘C)

Figure 5 A comparison between recovered and desired tempera-ture distribution at 119905 = 1000 s

0 001 002 003Depth of tissue (m)

34

37

40

43

31

t = 200 st = 500 st = 600 st = 1000 s

t = 1200 st = 1600 st = 1800 s

Tem

pera

ture

(∘C)

Figure 6 Transient spatial temperature distribution inside thetissue

6 Journal of Applied Mathematics

0 001 002 003Depth of tissue (m)

37

40

43

34

Tem

pera

ture

(∘C)

Figure 7 Temperature distribution at 119905 = 1000 s for two differentthermal conductivities of casing layer Solid line 119896cl = 014Wm∘C119880 = 175Wm2∘C dashed line 119896cl = 011Wm∘C119880 = 150Wm2∘C

0 001 002 003Depth of tissue (m)

34

40

43

31

37

Tem

pera

ture

(∘C)

Figure 8 Temperature distribution at 119905 = 1000 s for twodifferent heat convection coefficients of water Solid line ℎ =

200Wm2∘C119880 = 175Wm2∘C dashed line ℎ = 350Wm2∘C119880 = 200Wm2∘C

conductivity of casing layer However depending on thetechnical facilities one can make an optimal decision in thiscase

Although the thickness and the electrical conductivityof water can also be optimized in the treatment process wedid not consider them in this study and we mainly focusedon the influence of the overall heat transfer coefficient onthe temperature field inside the tissue In the perspective ofthe present model we believe that the desired temperatureprofile inside the tumor according to its shape and size canbe achieved by inversely optimizing the shape thicknesselectrical and thermal properties of solid or liquid patcheswhich are inserted in the cooling system Such an optimiza-tion problem should also incorporate the best configurationof the additional patches which reasonably must resemblethe tumor shape for example L-form patches for L-formtumors However the clinician will first scan the geometryof the tumor and measure the tumor size and depth usingan MRI or CT or other noninvasive imaging techniquesAlthough the present model is based on the 1D heat transferbut with some modifications in the governing formulations

such as 119880 = 119891(119910 119911) it has the potential to provide amore comprehensive understanding of the cooling systemability for the purposes of beam-forming in two or threedimensions This modified optimization algorithm is thenimplemented to the tumor and its surrounding tissue todetermine the optimal properties of cooling system

5 Conclusions

In this paper we demonstrated the validity of the CGMAE inthe optimization of the temperature distribution inside thetissue by optimal control of the overall heat transfer coef-ficient associated with the cooling system The temperaturedistribution inside the tissue based on the bioheat transferequation has also been numerically computed In order toevaluate the effectiveness of the cooling system the thermalconductivity of casing layer and the heat convection coef-ficient of water have been considered separately Accordingto results the presented model allows obtaining the bestproperties of water and casing layer in the light of opti-mization and therefore significantly improves the treatmentoutcome However this method shows great promise forfurther extensions in both model and clinical performancesfor any shaped targeted tumor in hyperthermia treatments

Appendix

In order to find the minimum of the residual function bygradienttype method it is required to derive its gradient bythe following processes first computation of the variationproblem when the unknown has a small variation andsecond inspection of the conditions under which we obtain astationary point for the Lagrange functional To compute thevariation problem the unknown function 119880 is perturbed byΔ119880 that is

119880120576= 119880 + Δ119880 (A1)

Then the temperature undergoes the variation Δ119879119905(119909 119905) as

follows

119879119905120576(119909 119905 119880) = 119879

119905(119909 119905 119880) + Δ119879

119905(119909 119905 119880) (A2)

where 120576 refers to the perturbed variables By insertingthe a aforementioned equations in the direct problem andsubtracting the original (1)ndash(4) from the resulting expressionsand ignoring the second order terms the following directproblem in variation is obtained

120588119905119888119905

120597Δ119879119905(119909 119905)

120597119905= 119896119905

1205972

Δ119879119905(119909 119905)

1205971199092

minus 120596bl120588bl119888blΔ119879119905 (119909 119905) 0 lt 119909 lt 119886

(A3)

Δ119879119905(119909 119905) = 0 119909 = 0 (A4)

minus119896119905

120597Δ119879119905(119909 119905)

120597119909= Δ119880 (119879

infinminus 119879119905(119909 119905))

minus 119880Δ119879119905(119909 119905) 119909 = 119886

(A5)

Δ119879119905(119909 119905) = 0 0 lt 119909 lt 119886 119905 = 0 (A6)

Journal of Applied Mathematics 7

The second step is to calculate the variation of the objectivefunction Δ119866(119880) which is given by

Δ119866 (119880) = int

119905119891

0

int

119886

0

2 Δ119879119905(119909119895 119905) [119879

119905(119909119895 119905) minus119884

119905(119909119895 119905)]

times 120575 (119909 minus 119909119895) 119889119909 119889119905

(A7)

We obtain the variation of the augmented functionalΔ119871(119879Ψ

119905 119880) by considering the variation problem given by

(A3) and the variation of the objective function (A7) asfollows

Δ119871 = int

119905119891

0

int

119886

0

2 Δ119879119905(119909119895 119905) [119879

119905(119909119895 119905) minus 119884

119905(119909119895 119905)]

times 120575 (119909 minus 119909119895) 119889119909 119889119905

+ int

119905119891

0

int

119886

0

Ψ119905(119909 119905) [119896

119905

1205972

Δ119879119905(119909 119905)

1205971199092

minus 120588119905119888119905

120597Δ119879119905(119909 119905)

120597119905

minus120596bl120588bl119888blΔ119879119905 (119909 119905) ] 119889119909 119889119905

(A8)

Using integration by parts in (A8)

int

119905119891

0

int

119886

0

1205972

Δ119879119905(119909 119905)

1205971199092Ψ119905(119909 119905) 119889119909 119889119905

= int

119905119891

0

[120597Δ119879119905(119886 119905)

120597119909Ψ119905(119886 119905) minus

120597Ψ119905(119886 119905)

120597119909Δ119879119905(119886 119905)

minus120597Δ119879119905(0 119905)

120597119909Ψ119905(0 119905)

+120597Ψ119905(0 119905)

120597119909Δ119879119905(0 119905)] 119889119905

+ int

119905119891

0

int

119886

0

1205972

Ψ119905(119909 119905)

1205971199092Δ119879119905(119909 119905) 119889119909 119889119905

(A9)

int

119886

0

int

119905119891

0

120597Δ119879119905(119909 119905)

120597119905Ψ119905(119909 119905) 119889119905 119889119909

= int

119886

0

[Δ119879119905(119909 119905119891)Ψ119905(119909 119905) minus Δ119879

119905(119909 0) Ψ

119905(119909 0)] 119889119909

minus int

119886

0

int

119905119891

0

Δ119879119905(119909 119905)

120597Ψ119905(119909 119905)

120597119909119889119905 119889119909

(A10)

and by using the boundary conditions of the direct problemin variations (Equations (A3)ndash(A6) and the rearrangement

of the mathematical terms) the final form of (A8) can bewritten as follows

Δ119871 = int

119905119891

0

int

119886

0

[119896119905

1205972

Ψ119905(119909 119905)

1205971199092+ 120588119905119888119905

120597Ψ119905(119909 119905)

120597119905

minus 120596bl120588bl119888blΨ119905 (119909 119905)

+ 2 [119879119905(119909119895 119905) minus 119884

119905(119909119895 119905)]

times 120575 (119909 minus 119909119895) ]Δ119879

119905(119909 119905) 119889119909 119889119905

minus int

119905119891

0

119896119905Ψ119905(0 119905)

120597Δ119879119905(0 119905)

120597119909119889119905

+ int

119905119891

0

(Ψ119905(119886 119905) 119880 minus 119896

119905

120597Ψ119905(119886 119905)

120597119909)Δ119879119905(119886 119905) 119889119905

minus int

119886

0

120588119905119888119905Ψ119905(119909 119905119891) Δ119879119905(119909 119905119891) 119889119909

+ int

119905119891

0

Δ119880 Ψ119905(119886 119905) (119879

119905(119886 119905) minus 119879

infin) 119889119905

(A11)

To satisfy the optimality condition that isΔ119871(Δ119879Ψ

119905 Δ119880) = 0 it is required that all coefficients

of Δ119879119905(119909 119905) are to be vanished Finally the following term is

left

Δ119871 (119879Ψ119905 119880) = int

119905119891

0

Δ119880 Ψ119905(119886 119905) (119879

119905(119886 119905) minus 119879

infin) 119889119905

(A12)

Since we know that 119879119905(119909 119905) and Ψ

119905(119909 119905) are respectively the

direct and the adjoint problem solutions it follows that

119871 (119879Ψ119905 119880) = 119866 (119880) (A13)

and then

Δ119871 (119879Ψ119905 119880) = Δ119866 (119880) (A14)

Based on the definition the variation of the functional 119866(119880)can be written as follows

Δ119866 (119880) = ⟨nabla119866 Δ119880⟩ (A15)

where the associated inner scalar product ⟨ ⟩ of two functions119891(119905) and 119892(119905) in the Hilbert space of 119871

2(0 119905119891) is defined by

⟨119891 (119905) 119892 (119905)⟩1198712

= int

119889

119888

119891 (119905) 119892 (119905) 119889119905 (A16)

From the definition of the scalar product the variation of thefunctional 119866(119880) can be written as follows

Δ119866 (119880) = ⟨nabla119866 (119880) Δ119880⟩1198712

= nabla119866 (119880)Δ119880 (A17)

A comparison between (A12) (A14) and (A17) reveals thatthe gradient of the residual functional is given by (9) Moredetails about the derivation of the equations are mentionedin [24ndash26]

8 Journal of Applied Mathematics

Nomenclature

119888 Heat capacity119889 Vector of descent direction119890 Computational precision119864 Complex electric field119866(119880) Residual functionalΔ119866(119880) Residual functional variationnabla119866(119880) Gradient of residual functionalℎ Heat convection coefficient119867 Complex magnetic field119896 Thermal conductivity119897 Thickness of casing layer119871 Lagrangian functional119873 Number of nodes119876119898 Tissue metabolism

119876119903 Volumetric heat source

119879 Temperature119905 Time119880 Overall heat transfer coefficient119909 Space119884 Desired temperature

Greek Symbols

120573 Descent parameter120574 Descent direction parameter120575 Dirac function120576 Electrical permeability120588 Density120583 Magnetic permeability120590 Electrical conductivityΨ Lagrange multiplier120596 Angular frequency120596 Blood perfusion

Subscripts

119886 Arterialbl Blood119888 Corecl Casing layer119891 Final119894 Initial time119905 Tissueinfin Ambient

Superscripts

119897 Iteration number

References

[1] S R H Davidson and D F James ldquoDrilling in bone model-ing heat generation and temperature distributionrdquo Journal ofBiomechanical Engineering vol 125 no 3 pp 305ndash314 2003

[2] W G Sawyer M A Hamilton B J Fregly and S A BanksldquoTemperature modeling in a total knee joint replacement usingpatient-specific kinematicsrdquo Tribology Letters vol 15 no 4 pp343ndash351 2003

[3] R Dhar P Dhar and R Dhar ldquoProblem on optimal distributionof induced microwave by heating probe at the tumor site inhyperthermiardquo Advanced Modeling and Optimization vol 13no 1 pp 39ndash48 2011

[4] E A Gelvich and V N Mazokhin ldquoResonance effects in appli-cator water boluses and their influence on SAR distributionpatternsrdquo International Journal of Hyperthermia vol 16 no 2pp 113ndash128 2000

[5] M de Bruijne T Samaras J F Bakker and G C van RhoonldquoEffects of waterbolus size shape and configuration on the SARdistribution pattern of the Lucite cone applicatorrdquo InternationalJournal of Hyperthermia vol 22 no 1 pp 15ndash28 2006

[6] E R Lee D S Kapp AW Lohrbach and J L Sokol ldquoInfluenceof water bolus temperature onmeasured skin surface and intra-dermal temperaturesrdquo International Journal of Hyperthermiavol 10 no 1 pp 59ndash72 1994

[7] M D Sherar F F Liu D J Newcombe et al ldquoBeam shaping formicrowave waveguide hyperthermia applicatorsrdquo InternationalJournal of Radiation Oncology Biology Physics vol 25 no 5 pp849ndash857 1993

[8] J C Kumaradas and M D Sherar ldquoOptimization of a beamshaping bolus for superficial microwave hyperthermia waveg-uide applicators using a finite element methodrdquo Physics inMedicine and Biology vol 48 no 1 pp 1ndash18 2003

[9] M A Ebrahimi-Ganjeh and A R Attari ldquoStudy of water boluseffect on SAR penetration depth and effective field size for localhyperthermiardquo Progress in Electromagnetics Research B vol 4pp 273ndash283 2008

[10] D G Neuman P R Stauffer S Jacobsen and F RossettoldquoSAR pattern perturbations from resonance effects in waterbolus layers used with superficial microwave hyperthermiaapplicatorsrdquo International Journal of Hyperthermia vol 18 no3 pp 180ndash193 2002

[11] A G Butkovasky Distributed Control System American Else-vier New York NY USA 1969

[12] C De Wagter ldquoOptimization of simulated two-dimensionaltemperature distributions induced by multiple electromagneticapplicatorsrdquo IEEE Transactions on Microwave Theory and Tech-niques vol 34 no 5 pp 589ndash596 1986

[13] A V Kuznetsov ldquoOptimization problems for bioheat equationrdquoInternational Communications in Heat and Mass Transfer vol33 no 5 pp 537ndash543 2006

[14] C Huang and S Wang ldquoA three-dimensional inverse heatconduction problem in estimating surface heat flux by conju-gate gradient methodrdquo International Journal of Heat and MassTransfer vol 42 no 18 pp 3387ndash3403 1999

[15] P K Dhar and D K Sinha ldquoTemperature control of tissueby transient-induced-microwavesrdquo International Journal of Sys-tems Science vol 19 no 10 pp 2051ndash2055 1988

[16] T Loulou and E P Scott ldquoThermal dose optimization in hyper-thermia treatments by using the conjugate gradient methodrdquoNumerical Heat Transfer A vol 42 no 7 pp 661ndash683 2002

[17] T Loulou and E P Scott ldquoAn inverse heat conduction problemwith heat flux measurementsrdquo International Journal for Numer-ical Methods in Engineering vol 67 no 11 pp 1587ndash1616 2006

[18] H Kroeze M Van Vulpen A A C De Leeuw J B Van DeKamer and J J W Lagendijk ldquoThe use of absorbing structuresduring regional hyperthermia treatmentrdquo International Journalof Hyperthermia vol 17 no 3 pp 240ndash257 2001

[19] H H Pennes ldquoAnalysis of tissue and arterial blood tem-peratures in the resting human forearmrdquo Journal of AppliedPhysiology vol 1 pp 93ndash122 1948

Journal of Applied Mathematics 9

[20] J P HolmanHeat Transfer McGraw Hill New York NY USA9th edition 2002

[21] C Thiebaut and D Lemonnier ldquoThree-dimensional modellingand optimisation of thermal fields induced in a human bodyduring hyperthermiardquo International Journal of Thermal Sci-ences vol 41 no 6 pp 500ndash508 2002

[22] Z S Deng and J Liu ldquoAnalytical study on bioheat transferproblems with spatial or transient heating on skin surface orinside biological bodiesrdquo Journal of Biomechanical Engineeringvol 124 no 6 pp 638ndash649 2002

[23] N K Uzunoglu E A Angelikas and P A Cosmidis ldquoA432MHz local hyperthermia system using an indirectly cooledwater-loaded waveguide applicatorrdquo IEEE Transactions onMicrowave Theory and Techniques vol 35 no 2 pp 106ndash1111987

[24] OM Alifanov ldquoSolution of an inverse problem of heat conduc-tion by iteration methodsrdquo Journal of Engineering Physics vol26 no 4 pp 471ndash476 1974

[25] O M Alifanov Inverse Heat Transfer Problem Springer NewYork NY USA 1994

[26] M N Ozisik and H R B Orlande Inverse Heat TransferFundamentals and Applications Taylor amp Francis New YorkNY USA 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

by Maxwellrsquos equations In a homogeneous tissue theseequations are expressed by [21] the following

nabla times 119864 = 119894120596120583119867

nabla sdot 119864 = 0

nabla sdot 119867 = 0

nabla times 119867 = minus119894120596120576lowast

119864

(15)

where 119864 and 119867 are respectively complex electric andmagnetic fields 120596 is the angular frequency and 120576 120590 and 120583

are the electrical permittivity the electrical conductivity andthe magnetic permeability of tissue respectively 120576lowast is thecomplex permittivity which is defined by

120576lowast

= 120576 +119894120590

120596 (16)

The volumetric EM power deposition which averaged overtime is calculated by

119876119903=120590|119864|2

2 (17)

where |119864| is the root mean square magnitude [Vm] of theincident electric field in a point which could be calculatedthrough (15)

222 Cooling System and Tissue Properties Thermophysicalproperties of tissue and blood pertaining to the model fornumerical simulation are provided in Table 1 [3 22 23]The thickness of water is set at 10minus2m and the electricalpermittivity and electrical conductivity of water related to anRF applicator operating at 432MHz are 78 and 00300 Smrespectively [23] Moreover thicknesses of 003m and 15 times10minus4mare used respectively for tissue and casing layer whichare typically made of thin flexible Silicon layers Temperatureof the cooling system 119879

infin is set at 15∘C

3 Assumptions

In order to simplify demonstrating the methodology heattransfer is assumed to be one-dimensional with a constantrate of tissue metabolism and blood perfusion The targettemperatures after the elapse of allowed heating periodare set between 42 and 45∘C for tumoral region and 42∘Cor lower for normal tissues In addition we consider threedifferent nodes one in the tumor region with 119884(0007 119905

119891) =

43∘C and two others in the normal tissue with 119884(0002 119905

119891)

and 119884(0021 119905119891) = 39

∘C as the desired temperaturesAs the thickness of the casing layer is much smaller thanits wavelength it could be neglected in the EM analysisMoreover the thermal contact between the casing layer andtissue is assumed to be perfect so that the thermal contactresistance between these layers is negligible

4 Results and Discussions

In the present work FDTD method is used for numericalsolution of the direct sensitivity and adjoint problems A

Table 1 Biophysical characteristics of model

Component Parameter Value Units

Blood119888bl 4200 jkg ∘C120588bl 1000 kgm3

120596bl 00005 mLsmL

Tissue

119888119905

3770 jkg ∘C119896119905

05 Wm ∘C119876119898

33800 Wm3

119879119886

37 ∘C119879119888

37 ∘C119879119894

34 ∘C120576119905

55 mdash120588119905

998 kgm3

120590119905

11 Sm

uniform grid of 31 times 31 mesh in space and time is utilized forall the results presented hereafterTheoptimization algorithmpresented previously is programmed in Fortran 90 compiledthrough the software Compaq Visual Fortran ProfessionalEdition on a computer with an Intel Core i7 35 GHz and16Gb of RAM The final time 119905

119891 is chosen equal to 1800 s

Moreover the computational precision (119890) is set at 10minus5 In thefollowing simulation the desired temperature distribution isdetermined by precisely prescribing a known 119880

Figure 2 indicates the sensitivity of the temperature-time curve for variations in 119880 for three different valuesof 119880 Because of the large values of the sensitivity withrespect to 119880 obviously the proposed optimization model isldquowell-conditionedrdquo To illustrate the accuracy of the presentinverse analysis the initial approximations of 119880 were eitherincreased or decreased from its known value Figure 3 showsthe number of convergent iterations with different initialguesses for 119880 Changing the initial value for example by15 and 30 of its known value does not affect the solutionand the optimization object is still satisfied Moreover it isclear that the number of convergent iterations is reducedwhen the initial guess value varies within a 15 deviationfrom the known one In addition this figure shows thatthe minimization of the objective function is valid when 119880

varies between 125 and 200Wm2∘C In order to provide aquantitative measurement of errors the convergence historyof the presented model is also calculated Figure 4 comparesthe convergence history is of three different initial guessesConvergence history is measured by the relative error at eachiteration that is |119880retrieved minus 119880prescribed|119880prescribed

The computed temperature distribution is obtained whenthe minimization of the objective function is satisfied thatis when the optimal value of 119880 is achieved Figure 5 showsthe computed temperature distribution with initial guess of119880 = 200Wm2∘C as well as the desired one which isobtained from the application of a known 119880 at 119905 = 1000 sIt is observed that the recovered temperature distribution isin good agreement with the desired one In addition thecomputed RMS error is equal to 024 Figure 6 presents thedesired temperature distributions result from the solution of

Journal of Applied Mathematics 5

0 01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

06

07

ΔT

(∘C)

ttfU = 150Wm2 ∘CU = 175Wm2 ∘CU = 200Wm2 ∘C

Figure 2 Temperature sensitivity resulting from variations in 119880

2 4 6 8 10 12 14 16 18 20125

150

175

200

Number of iterations

U(W

m2∘ C)

Figure 3 Convergence curve for different initial guesses of 119880

the nonstationary thermal problem during the total heatingprocess It is found to be sufficient to elevate the temperatureinside the tumor to 42∘C within approximately 20 minuteswhile the temperature on the surface remains approximatelyconstant in this period of time

According to the foregoing discussion as (5) and inorder to enhance the performance of the cooling system thethermal conductivity of casing layer and the heat convectioncoefficient of water have been considered as variable param-eters Although the casing layer is very thin as shown inFigure 7 the influence of thermal conductivity of casing layeris remarkable It could be observed that a slight increase inthermal conductivity reduces significantly the temperature atthe skin surface In addition the position of the maximumtemperature is shifted slightly toward the skin The secondpossibility to control the effectiveness of the cooling systemis through the variation of the heat convection coefficient ofwater Figure 8 indicates the influence of two different valuesof ℎ on the temperature of tissue It could be observed thata 75 increase in ℎ results in a decrease of about 25∘C inthe skin and only 2∘C in the maximum The same resultscould be obtained by only 20 of the variation in the thermal

2 4 6 8 10 12 14 16 18 20Iteration

Rela

tive e

rror

10minus0

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

Figure 4 Comparison of convergence histories of CGM for differ-ent initial guessesmdash loz 125 ◻ 150 and I 200Wm2∘C for 119880

0 001 002 003Depth of tissue (m)

Prescribed temperature distributionRecovered temperature distribution

34

37

40

43 II II

Tem

pera

ture

(∘C)

Figure 5 A comparison between recovered and desired tempera-ture distribution at 119905 = 1000 s

0 001 002 003Depth of tissue (m)

34

37

40

43

31

t = 200 st = 500 st = 600 st = 1000 s

t = 1200 st = 1600 st = 1800 s

Tem

pera

ture

(∘C)

Figure 6 Transient spatial temperature distribution inside thetissue

6 Journal of Applied Mathematics

0 001 002 003Depth of tissue (m)

37

40

43

34

Tem

pera

ture

(∘C)

Figure 7 Temperature distribution at 119905 = 1000 s for two differentthermal conductivities of casing layer Solid line 119896cl = 014Wm∘C119880 = 175Wm2∘C dashed line 119896cl = 011Wm∘C119880 = 150Wm2∘C

0 001 002 003Depth of tissue (m)

34

40

43

31

37

Tem

pera

ture

(∘C)

Figure 8 Temperature distribution at 119905 = 1000 s for twodifferent heat convection coefficients of water Solid line ℎ =

200Wm2∘C119880 = 175Wm2∘C dashed line ℎ = 350Wm2∘C119880 = 200Wm2∘C

conductivity of casing layer However depending on thetechnical facilities one can make an optimal decision in thiscase

Although the thickness and the electrical conductivityof water can also be optimized in the treatment process wedid not consider them in this study and we mainly focusedon the influence of the overall heat transfer coefficient onthe temperature field inside the tissue In the perspective ofthe present model we believe that the desired temperatureprofile inside the tumor according to its shape and size canbe achieved by inversely optimizing the shape thicknesselectrical and thermal properties of solid or liquid patcheswhich are inserted in the cooling system Such an optimiza-tion problem should also incorporate the best configurationof the additional patches which reasonably must resemblethe tumor shape for example L-form patches for L-formtumors However the clinician will first scan the geometryof the tumor and measure the tumor size and depth usingan MRI or CT or other noninvasive imaging techniquesAlthough the present model is based on the 1D heat transferbut with some modifications in the governing formulations

such as 119880 = 119891(119910 119911) it has the potential to provide amore comprehensive understanding of the cooling systemability for the purposes of beam-forming in two or threedimensions This modified optimization algorithm is thenimplemented to the tumor and its surrounding tissue todetermine the optimal properties of cooling system

5 Conclusions

In this paper we demonstrated the validity of the CGMAE inthe optimization of the temperature distribution inside thetissue by optimal control of the overall heat transfer coef-ficient associated with the cooling system The temperaturedistribution inside the tissue based on the bioheat transferequation has also been numerically computed In order toevaluate the effectiveness of the cooling system the thermalconductivity of casing layer and the heat convection coef-ficient of water have been considered separately Accordingto results the presented model allows obtaining the bestproperties of water and casing layer in the light of opti-mization and therefore significantly improves the treatmentoutcome However this method shows great promise forfurther extensions in both model and clinical performancesfor any shaped targeted tumor in hyperthermia treatments

Appendix

In order to find the minimum of the residual function bygradienttype method it is required to derive its gradient bythe following processes first computation of the variationproblem when the unknown has a small variation andsecond inspection of the conditions under which we obtain astationary point for the Lagrange functional To compute thevariation problem the unknown function 119880 is perturbed byΔ119880 that is

119880120576= 119880 + Δ119880 (A1)

Then the temperature undergoes the variation Δ119879119905(119909 119905) as

follows

119879119905120576(119909 119905 119880) = 119879

119905(119909 119905 119880) + Δ119879

119905(119909 119905 119880) (A2)

where 120576 refers to the perturbed variables By insertingthe a aforementioned equations in the direct problem andsubtracting the original (1)ndash(4) from the resulting expressionsand ignoring the second order terms the following directproblem in variation is obtained

120588119905119888119905

120597Δ119879119905(119909 119905)

120597119905= 119896119905

1205972

Δ119879119905(119909 119905)

1205971199092

minus 120596bl120588bl119888blΔ119879119905 (119909 119905) 0 lt 119909 lt 119886

(A3)

Δ119879119905(119909 119905) = 0 119909 = 0 (A4)

minus119896119905

120597Δ119879119905(119909 119905)

120597119909= Δ119880 (119879

infinminus 119879119905(119909 119905))

minus 119880Δ119879119905(119909 119905) 119909 = 119886

(A5)

Δ119879119905(119909 119905) = 0 0 lt 119909 lt 119886 119905 = 0 (A6)

Journal of Applied Mathematics 7

The second step is to calculate the variation of the objectivefunction Δ119866(119880) which is given by

Δ119866 (119880) = int

119905119891

0

int

119886

0

2 Δ119879119905(119909119895 119905) [119879

119905(119909119895 119905) minus119884

119905(119909119895 119905)]

times 120575 (119909 minus 119909119895) 119889119909 119889119905

(A7)

We obtain the variation of the augmented functionalΔ119871(119879Ψ

119905 119880) by considering the variation problem given by

(A3) and the variation of the objective function (A7) asfollows

Δ119871 = int

119905119891

0

int

119886

0

2 Δ119879119905(119909119895 119905) [119879

119905(119909119895 119905) minus 119884

119905(119909119895 119905)]

times 120575 (119909 minus 119909119895) 119889119909 119889119905

+ int

119905119891

0

int

119886

0

Ψ119905(119909 119905) [119896

119905

1205972

Δ119879119905(119909 119905)

1205971199092

minus 120588119905119888119905

120597Δ119879119905(119909 119905)

120597119905

minus120596bl120588bl119888blΔ119879119905 (119909 119905) ] 119889119909 119889119905

(A8)

Using integration by parts in (A8)

int

119905119891

0

int

119886

0

1205972

Δ119879119905(119909 119905)

1205971199092Ψ119905(119909 119905) 119889119909 119889119905

= int

119905119891

0

[120597Δ119879119905(119886 119905)

120597119909Ψ119905(119886 119905) minus

120597Ψ119905(119886 119905)

120597119909Δ119879119905(119886 119905)

minus120597Δ119879119905(0 119905)

120597119909Ψ119905(0 119905)

+120597Ψ119905(0 119905)

120597119909Δ119879119905(0 119905)] 119889119905

+ int

119905119891

0

int

119886

0

1205972

Ψ119905(119909 119905)

1205971199092Δ119879119905(119909 119905) 119889119909 119889119905

(A9)

int

119886

0

int

119905119891

0

120597Δ119879119905(119909 119905)

120597119905Ψ119905(119909 119905) 119889119905 119889119909

= int

119886

0

[Δ119879119905(119909 119905119891)Ψ119905(119909 119905) minus Δ119879

119905(119909 0) Ψ

119905(119909 0)] 119889119909

minus int

119886

0

int

119905119891

0

Δ119879119905(119909 119905)

120597Ψ119905(119909 119905)

120597119909119889119905 119889119909

(A10)

and by using the boundary conditions of the direct problemin variations (Equations (A3)ndash(A6) and the rearrangement

of the mathematical terms) the final form of (A8) can bewritten as follows

Δ119871 = int

119905119891

0

int

119886

0

[119896119905

1205972

Ψ119905(119909 119905)

1205971199092+ 120588119905119888119905

120597Ψ119905(119909 119905)

120597119905

minus 120596bl120588bl119888blΨ119905 (119909 119905)

+ 2 [119879119905(119909119895 119905) minus 119884

119905(119909119895 119905)]

times 120575 (119909 minus 119909119895) ]Δ119879

119905(119909 119905) 119889119909 119889119905

minus int

119905119891

0

119896119905Ψ119905(0 119905)

120597Δ119879119905(0 119905)

120597119909119889119905

+ int

119905119891

0

(Ψ119905(119886 119905) 119880 minus 119896

119905

120597Ψ119905(119886 119905)

120597119909)Δ119879119905(119886 119905) 119889119905

minus int

119886

0

120588119905119888119905Ψ119905(119909 119905119891) Δ119879119905(119909 119905119891) 119889119909

+ int

119905119891

0

Δ119880 Ψ119905(119886 119905) (119879

119905(119886 119905) minus 119879

infin) 119889119905

(A11)

To satisfy the optimality condition that isΔ119871(Δ119879Ψ

119905 Δ119880) = 0 it is required that all coefficients

of Δ119879119905(119909 119905) are to be vanished Finally the following term is

left

Δ119871 (119879Ψ119905 119880) = int

119905119891

0

Δ119880 Ψ119905(119886 119905) (119879

119905(119886 119905) minus 119879

infin) 119889119905

(A12)

Since we know that 119879119905(119909 119905) and Ψ

119905(119909 119905) are respectively the

direct and the adjoint problem solutions it follows that

119871 (119879Ψ119905 119880) = 119866 (119880) (A13)

and then

Δ119871 (119879Ψ119905 119880) = Δ119866 (119880) (A14)

Based on the definition the variation of the functional 119866(119880)can be written as follows

Δ119866 (119880) = ⟨nabla119866 Δ119880⟩ (A15)

where the associated inner scalar product ⟨ ⟩ of two functions119891(119905) and 119892(119905) in the Hilbert space of 119871

2(0 119905119891) is defined by

⟨119891 (119905) 119892 (119905)⟩1198712

= int

119889

119888

119891 (119905) 119892 (119905) 119889119905 (A16)

From the definition of the scalar product the variation of thefunctional 119866(119880) can be written as follows

Δ119866 (119880) = ⟨nabla119866 (119880) Δ119880⟩1198712

= nabla119866 (119880)Δ119880 (A17)

A comparison between (A12) (A14) and (A17) reveals thatthe gradient of the residual functional is given by (9) Moredetails about the derivation of the equations are mentionedin [24ndash26]

8 Journal of Applied Mathematics

Nomenclature

119888 Heat capacity119889 Vector of descent direction119890 Computational precision119864 Complex electric field119866(119880) Residual functionalΔ119866(119880) Residual functional variationnabla119866(119880) Gradient of residual functionalℎ Heat convection coefficient119867 Complex magnetic field119896 Thermal conductivity119897 Thickness of casing layer119871 Lagrangian functional119873 Number of nodes119876119898 Tissue metabolism

119876119903 Volumetric heat source

119879 Temperature119905 Time119880 Overall heat transfer coefficient119909 Space119884 Desired temperature

Greek Symbols

120573 Descent parameter120574 Descent direction parameter120575 Dirac function120576 Electrical permeability120588 Density120583 Magnetic permeability120590 Electrical conductivityΨ Lagrange multiplier120596 Angular frequency120596 Blood perfusion

Subscripts

119886 Arterialbl Blood119888 Corecl Casing layer119891 Final119894 Initial time119905 Tissueinfin Ambient

Superscripts

119897 Iteration number

References

[1] S R H Davidson and D F James ldquoDrilling in bone model-ing heat generation and temperature distributionrdquo Journal ofBiomechanical Engineering vol 125 no 3 pp 305ndash314 2003

[2] W G Sawyer M A Hamilton B J Fregly and S A BanksldquoTemperature modeling in a total knee joint replacement usingpatient-specific kinematicsrdquo Tribology Letters vol 15 no 4 pp343ndash351 2003

[3] R Dhar P Dhar and R Dhar ldquoProblem on optimal distributionof induced microwave by heating probe at the tumor site inhyperthermiardquo Advanced Modeling and Optimization vol 13no 1 pp 39ndash48 2011

[4] E A Gelvich and V N Mazokhin ldquoResonance effects in appli-cator water boluses and their influence on SAR distributionpatternsrdquo International Journal of Hyperthermia vol 16 no 2pp 113ndash128 2000

[5] M de Bruijne T Samaras J F Bakker and G C van RhoonldquoEffects of waterbolus size shape and configuration on the SARdistribution pattern of the Lucite cone applicatorrdquo InternationalJournal of Hyperthermia vol 22 no 1 pp 15ndash28 2006

[6] E R Lee D S Kapp AW Lohrbach and J L Sokol ldquoInfluenceof water bolus temperature onmeasured skin surface and intra-dermal temperaturesrdquo International Journal of Hyperthermiavol 10 no 1 pp 59ndash72 1994

[7] M D Sherar F F Liu D J Newcombe et al ldquoBeam shaping formicrowave waveguide hyperthermia applicatorsrdquo InternationalJournal of Radiation Oncology Biology Physics vol 25 no 5 pp849ndash857 1993

[8] J C Kumaradas and M D Sherar ldquoOptimization of a beamshaping bolus for superficial microwave hyperthermia waveg-uide applicators using a finite element methodrdquo Physics inMedicine and Biology vol 48 no 1 pp 1ndash18 2003

[9] M A Ebrahimi-Ganjeh and A R Attari ldquoStudy of water boluseffect on SAR penetration depth and effective field size for localhyperthermiardquo Progress in Electromagnetics Research B vol 4pp 273ndash283 2008

[10] D G Neuman P R Stauffer S Jacobsen and F RossettoldquoSAR pattern perturbations from resonance effects in waterbolus layers used with superficial microwave hyperthermiaapplicatorsrdquo International Journal of Hyperthermia vol 18 no3 pp 180ndash193 2002

[11] A G Butkovasky Distributed Control System American Else-vier New York NY USA 1969

[12] C De Wagter ldquoOptimization of simulated two-dimensionaltemperature distributions induced by multiple electromagneticapplicatorsrdquo IEEE Transactions on Microwave Theory and Tech-niques vol 34 no 5 pp 589ndash596 1986

[13] A V Kuznetsov ldquoOptimization problems for bioheat equationrdquoInternational Communications in Heat and Mass Transfer vol33 no 5 pp 537ndash543 2006

[14] C Huang and S Wang ldquoA three-dimensional inverse heatconduction problem in estimating surface heat flux by conju-gate gradient methodrdquo International Journal of Heat and MassTransfer vol 42 no 18 pp 3387ndash3403 1999

[15] P K Dhar and D K Sinha ldquoTemperature control of tissueby transient-induced-microwavesrdquo International Journal of Sys-tems Science vol 19 no 10 pp 2051ndash2055 1988

[16] T Loulou and E P Scott ldquoThermal dose optimization in hyper-thermia treatments by using the conjugate gradient methodrdquoNumerical Heat Transfer A vol 42 no 7 pp 661ndash683 2002

[17] T Loulou and E P Scott ldquoAn inverse heat conduction problemwith heat flux measurementsrdquo International Journal for Numer-ical Methods in Engineering vol 67 no 11 pp 1587ndash1616 2006

[18] H Kroeze M Van Vulpen A A C De Leeuw J B Van DeKamer and J J W Lagendijk ldquoThe use of absorbing structuresduring regional hyperthermia treatmentrdquo International Journalof Hyperthermia vol 17 no 3 pp 240ndash257 2001

[19] H H Pennes ldquoAnalysis of tissue and arterial blood tem-peratures in the resting human forearmrdquo Journal of AppliedPhysiology vol 1 pp 93ndash122 1948

Journal of Applied Mathematics 9

[20] J P HolmanHeat Transfer McGraw Hill New York NY USA9th edition 2002

[21] C Thiebaut and D Lemonnier ldquoThree-dimensional modellingand optimisation of thermal fields induced in a human bodyduring hyperthermiardquo International Journal of Thermal Sci-ences vol 41 no 6 pp 500ndash508 2002

[22] Z S Deng and J Liu ldquoAnalytical study on bioheat transferproblems with spatial or transient heating on skin surface orinside biological bodiesrdquo Journal of Biomechanical Engineeringvol 124 no 6 pp 638ndash649 2002

[23] N K Uzunoglu E A Angelikas and P A Cosmidis ldquoA432MHz local hyperthermia system using an indirectly cooledwater-loaded waveguide applicatorrdquo IEEE Transactions onMicrowave Theory and Techniques vol 35 no 2 pp 106ndash1111987

[24] OM Alifanov ldquoSolution of an inverse problem of heat conduc-tion by iteration methodsrdquo Journal of Engineering Physics vol26 no 4 pp 471ndash476 1974

[25] O M Alifanov Inverse Heat Transfer Problem Springer NewYork NY USA 1994

[26] M N Ozisik and H R B Orlande Inverse Heat TransferFundamentals and Applications Taylor amp Francis New YorkNY USA 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

0 01 02 03 04 05 06 07 08 09 10

01

02

03

04

05

06

07

ΔT

(∘C)

ttfU = 150Wm2 ∘CU = 175Wm2 ∘CU = 200Wm2 ∘C

Figure 2 Temperature sensitivity resulting from variations in 119880

2 4 6 8 10 12 14 16 18 20125

150

175

200

Number of iterations

U(W

m2∘ C)

Figure 3 Convergence curve for different initial guesses of 119880

the nonstationary thermal problem during the total heatingprocess It is found to be sufficient to elevate the temperatureinside the tumor to 42∘C within approximately 20 minuteswhile the temperature on the surface remains approximatelyconstant in this period of time

According to the foregoing discussion as (5) and inorder to enhance the performance of the cooling system thethermal conductivity of casing layer and the heat convectioncoefficient of water have been considered as variable param-eters Although the casing layer is very thin as shown inFigure 7 the influence of thermal conductivity of casing layeris remarkable It could be observed that a slight increase inthermal conductivity reduces significantly the temperature atthe skin surface In addition the position of the maximumtemperature is shifted slightly toward the skin The secondpossibility to control the effectiveness of the cooling systemis through the variation of the heat convection coefficient ofwater Figure 8 indicates the influence of two different valuesof ℎ on the temperature of tissue It could be observed thata 75 increase in ℎ results in a decrease of about 25∘C inthe skin and only 2∘C in the maximum The same resultscould be obtained by only 20 of the variation in the thermal

2 4 6 8 10 12 14 16 18 20Iteration

Rela

tive e

rror

10minus0

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

Figure 4 Comparison of convergence histories of CGM for differ-ent initial guessesmdash loz 125 ◻ 150 and I 200Wm2∘C for 119880

0 001 002 003Depth of tissue (m)

Prescribed temperature distributionRecovered temperature distribution

34

37

40

43 II II

Tem

pera

ture

(∘C)

Figure 5 A comparison between recovered and desired tempera-ture distribution at 119905 = 1000 s

0 001 002 003Depth of tissue (m)

34

37

40

43

31

t = 200 st = 500 st = 600 st = 1000 s

t = 1200 st = 1600 st = 1800 s

Tem

pera

ture

(∘C)

Figure 6 Transient spatial temperature distribution inside thetissue

6 Journal of Applied Mathematics

0 001 002 003Depth of tissue (m)

37

40

43

34

Tem

pera

ture

(∘C)

Figure 7 Temperature distribution at 119905 = 1000 s for two differentthermal conductivities of casing layer Solid line 119896cl = 014Wm∘C119880 = 175Wm2∘C dashed line 119896cl = 011Wm∘C119880 = 150Wm2∘C

0 001 002 003Depth of tissue (m)

34

40

43

31

37

Tem

pera

ture

(∘C)

Figure 8 Temperature distribution at 119905 = 1000 s for twodifferent heat convection coefficients of water Solid line ℎ =

200Wm2∘C119880 = 175Wm2∘C dashed line ℎ = 350Wm2∘C119880 = 200Wm2∘C

conductivity of casing layer However depending on thetechnical facilities one can make an optimal decision in thiscase

Although the thickness and the electrical conductivityof water can also be optimized in the treatment process wedid not consider them in this study and we mainly focusedon the influence of the overall heat transfer coefficient onthe temperature field inside the tissue In the perspective ofthe present model we believe that the desired temperatureprofile inside the tumor according to its shape and size canbe achieved by inversely optimizing the shape thicknesselectrical and thermal properties of solid or liquid patcheswhich are inserted in the cooling system Such an optimiza-tion problem should also incorporate the best configurationof the additional patches which reasonably must resemblethe tumor shape for example L-form patches for L-formtumors However the clinician will first scan the geometryof the tumor and measure the tumor size and depth usingan MRI or CT or other noninvasive imaging techniquesAlthough the present model is based on the 1D heat transferbut with some modifications in the governing formulations

such as 119880 = 119891(119910 119911) it has the potential to provide amore comprehensive understanding of the cooling systemability for the purposes of beam-forming in two or threedimensions This modified optimization algorithm is thenimplemented to the tumor and its surrounding tissue todetermine the optimal properties of cooling system

5 Conclusions

In this paper we demonstrated the validity of the CGMAE inthe optimization of the temperature distribution inside thetissue by optimal control of the overall heat transfer coef-ficient associated with the cooling system The temperaturedistribution inside the tissue based on the bioheat transferequation has also been numerically computed In order toevaluate the effectiveness of the cooling system the thermalconductivity of casing layer and the heat convection coef-ficient of water have been considered separately Accordingto results the presented model allows obtaining the bestproperties of water and casing layer in the light of opti-mization and therefore significantly improves the treatmentoutcome However this method shows great promise forfurther extensions in both model and clinical performancesfor any shaped targeted tumor in hyperthermia treatments

Appendix

In order to find the minimum of the residual function bygradienttype method it is required to derive its gradient bythe following processes first computation of the variationproblem when the unknown has a small variation andsecond inspection of the conditions under which we obtain astationary point for the Lagrange functional To compute thevariation problem the unknown function 119880 is perturbed byΔ119880 that is

119880120576= 119880 + Δ119880 (A1)

Then the temperature undergoes the variation Δ119879119905(119909 119905) as

follows

119879119905120576(119909 119905 119880) = 119879

119905(119909 119905 119880) + Δ119879

119905(119909 119905 119880) (A2)

where 120576 refers to the perturbed variables By insertingthe a aforementioned equations in the direct problem andsubtracting the original (1)ndash(4) from the resulting expressionsand ignoring the second order terms the following directproblem in variation is obtained

120588119905119888119905

120597Δ119879119905(119909 119905)

120597119905= 119896119905

1205972

Δ119879119905(119909 119905)

1205971199092

minus 120596bl120588bl119888blΔ119879119905 (119909 119905) 0 lt 119909 lt 119886

(A3)

Δ119879119905(119909 119905) = 0 119909 = 0 (A4)

minus119896119905

120597Δ119879119905(119909 119905)

120597119909= Δ119880 (119879

infinminus 119879119905(119909 119905))

minus 119880Δ119879119905(119909 119905) 119909 = 119886

(A5)

Δ119879119905(119909 119905) = 0 0 lt 119909 lt 119886 119905 = 0 (A6)

Journal of Applied Mathematics 7

The second step is to calculate the variation of the objectivefunction Δ119866(119880) which is given by

Δ119866 (119880) = int

119905119891

0

int

119886

0

2 Δ119879119905(119909119895 119905) [119879

119905(119909119895 119905) minus119884

119905(119909119895 119905)]

times 120575 (119909 minus 119909119895) 119889119909 119889119905

(A7)

We obtain the variation of the augmented functionalΔ119871(119879Ψ

119905 119880) by considering the variation problem given by

(A3) and the variation of the objective function (A7) asfollows

Δ119871 = int

119905119891

0

int

119886

0

2 Δ119879119905(119909119895 119905) [119879

119905(119909119895 119905) minus 119884

119905(119909119895 119905)]

times 120575 (119909 minus 119909119895) 119889119909 119889119905

+ int

119905119891

0

int

119886

0

Ψ119905(119909 119905) [119896

119905

1205972

Δ119879119905(119909 119905)

1205971199092

minus 120588119905119888119905

120597Δ119879119905(119909 119905)

120597119905

minus120596bl120588bl119888blΔ119879119905 (119909 119905) ] 119889119909 119889119905

(A8)

Using integration by parts in (A8)

int

119905119891

0

int

119886

0

1205972

Δ119879119905(119909 119905)

1205971199092Ψ119905(119909 119905) 119889119909 119889119905

= int

119905119891

0

[120597Δ119879119905(119886 119905)

120597119909Ψ119905(119886 119905) minus

120597Ψ119905(119886 119905)

120597119909Δ119879119905(119886 119905)

minus120597Δ119879119905(0 119905)

120597119909Ψ119905(0 119905)

+120597Ψ119905(0 119905)

120597119909Δ119879119905(0 119905)] 119889119905

+ int

119905119891

0

int

119886

0

1205972

Ψ119905(119909 119905)

1205971199092Δ119879119905(119909 119905) 119889119909 119889119905

(A9)

int

119886

0

int

119905119891

0

120597Δ119879119905(119909 119905)

120597119905Ψ119905(119909 119905) 119889119905 119889119909

= int

119886

0

[Δ119879119905(119909 119905119891)Ψ119905(119909 119905) minus Δ119879

119905(119909 0) Ψ

119905(119909 0)] 119889119909

minus int

119886

0

int

119905119891

0

Δ119879119905(119909 119905)

120597Ψ119905(119909 119905)

120597119909119889119905 119889119909

(A10)

and by using the boundary conditions of the direct problemin variations (Equations (A3)ndash(A6) and the rearrangement

of the mathematical terms) the final form of (A8) can bewritten as follows

Δ119871 = int

119905119891

0

int

119886

0

[119896119905

1205972

Ψ119905(119909 119905)

1205971199092+ 120588119905119888119905

120597Ψ119905(119909 119905)

120597119905

minus 120596bl120588bl119888blΨ119905 (119909 119905)

+ 2 [119879119905(119909119895 119905) minus 119884

119905(119909119895 119905)]

times 120575 (119909 minus 119909119895) ]Δ119879

119905(119909 119905) 119889119909 119889119905

minus int

119905119891

0

119896119905Ψ119905(0 119905)

120597Δ119879119905(0 119905)

120597119909119889119905

+ int

119905119891

0

(Ψ119905(119886 119905) 119880 minus 119896

119905

120597Ψ119905(119886 119905)

120597119909)Δ119879119905(119886 119905) 119889119905

minus int

119886

0

120588119905119888119905Ψ119905(119909 119905119891) Δ119879119905(119909 119905119891) 119889119909

+ int

119905119891

0

Δ119880 Ψ119905(119886 119905) (119879

119905(119886 119905) minus 119879

infin) 119889119905

(A11)

To satisfy the optimality condition that isΔ119871(Δ119879Ψ

119905 Δ119880) = 0 it is required that all coefficients

of Δ119879119905(119909 119905) are to be vanished Finally the following term is

left

Δ119871 (119879Ψ119905 119880) = int

119905119891

0

Δ119880 Ψ119905(119886 119905) (119879

119905(119886 119905) minus 119879

infin) 119889119905

(A12)

Since we know that 119879119905(119909 119905) and Ψ

119905(119909 119905) are respectively the

direct and the adjoint problem solutions it follows that

119871 (119879Ψ119905 119880) = 119866 (119880) (A13)

and then

Δ119871 (119879Ψ119905 119880) = Δ119866 (119880) (A14)

Based on the definition the variation of the functional 119866(119880)can be written as follows

Δ119866 (119880) = ⟨nabla119866 Δ119880⟩ (A15)

where the associated inner scalar product ⟨ ⟩ of two functions119891(119905) and 119892(119905) in the Hilbert space of 119871

2(0 119905119891) is defined by

⟨119891 (119905) 119892 (119905)⟩1198712

= int

119889

119888

119891 (119905) 119892 (119905) 119889119905 (A16)

From the definition of the scalar product the variation of thefunctional 119866(119880) can be written as follows

Δ119866 (119880) = ⟨nabla119866 (119880) Δ119880⟩1198712

= nabla119866 (119880)Δ119880 (A17)

A comparison between (A12) (A14) and (A17) reveals thatthe gradient of the residual functional is given by (9) Moredetails about the derivation of the equations are mentionedin [24ndash26]

8 Journal of Applied Mathematics

Nomenclature

119888 Heat capacity119889 Vector of descent direction119890 Computational precision119864 Complex electric field119866(119880) Residual functionalΔ119866(119880) Residual functional variationnabla119866(119880) Gradient of residual functionalℎ Heat convection coefficient119867 Complex magnetic field119896 Thermal conductivity119897 Thickness of casing layer119871 Lagrangian functional119873 Number of nodes119876119898 Tissue metabolism

119876119903 Volumetric heat source

119879 Temperature119905 Time119880 Overall heat transfer coefficient119909 Space119884 Desired temperature

Greek Symbols

120573 Descent parameter120574 Descent direction parameter120575 Dirac function120576 Electrical permeability120588 Density120583 Magnetic permeability120590 Electrical conductivityΨ Lagrange multiplier120596 Angular frequency120596 Blood perfusion

Subscripts

119886 Arterialbl Blood119888 Corecl Casing layer119891 Final119894 Initial time119905 Tissueinfin Ambient

Superscripts

119897 Iteration number

References

[1] S R H Davidson and D F James ldquoDrilling in bone model-ing heat generation and temperature distributionrdquo Journal ofBiomechanical Engineering vol 125 no 3 pp 305ndash314 2003

[2] W G Sawyer M A Hamilton B J Fregly and S A BanksldquoTemperature modeling in a total knee joint replacement usingpatient-specific kinematicsrdquo Tribology Letters vol 15 no 4 pp343ndash351 2003

[3] R Dhar P Dhar and R Dhar ldquoProblem on optimal distributionof induced microwave by heating probe at the tumor site inhyperthermiardquo Advanced Modeling and Optimization vol 13no 1 pp 39ndash48 2011

[4] E A Gelvich and V N Mazokhin ldquoResonance effects in appli-cator water boluses and their influence on SAR distributionpatternsrdquo International Journal of Hyperthermia vol 16 no 2pp 113ndash128 2000

[5] M de Bruijne T Samaras J F Bakker and G C van RhoonldquoEffects of waterbolus size shape and configuration on the SARdistribution pattern of the Lucite cone applicatorrdquo InternationalJournal of Hyperthermia vol 22 no 1 pp 15ndash28 2006

[6] E R Lee D S Kapp AW Lohrbach and J L Sokol ldquoInfluenceof water bolus temperature onmeasured skin surface and intra-dermal temperaturesrdquo International Journal of Hyperthermiavol 10 no 1 pp 59ndash72 1994

[7] M D Sherar F F Liu D J Newcombe et al ldquoBeam shaping formicrowave waveguide hyperthermia applicatorsrdquo InternationalJournal of Radiation Oncology Biology Physics vol 25 no 5 pp849ndash857 1993

[8] J C Kumaradas and M D Sherar ldquoOptimization of a beamshaping bolus for superficial microwave hyperthermia waveg-uide applicators using a finite element methodrdquo Physics inMedicine and Biology vol 48 no 1 pp 1ndash18 2003

[9] M A Ebrahimi-Ganjeh and A R Attari ldquoStudy of water boluseffect on SAR penetration depth and effective field size for localhyperthermiardquo Progress in Electromagnetics Research B vol 4pp 273ndash283 2008

[10] D G Neuman P R Stauffer S Jacobsen and F RossettoldquoSAR pattern perturbations from resonance effects in waterbolus layers used with superficial microwave hyperthermiaapplicatorsrdquo International Journal of Hyperthermia vol 18 no3 pp 180ndash193 2002

[11] A G Butkovasky Distributed Control System American Else-vier New York NY USA 1969

[12] C De Wagter ldquoOptimization of simulated two-dimensionaltemperature distributions induced by multiple electromagneticapplicatorsrdquo IEEE Transactions on Microwave Theory and Tech-niques vol 34 no 5 pp 589ndash596 1986

[13] A V Kuznetsov ldquoOptimization problems for bioheat equationrdquoInternational Communications in Heat and Mass Transfer vol33 no 5 pp 537ndash543 2006

[14] C Huang and S Wang ldquoA three-dimensional inverse heatconduction problem in estimating surface heat flux by conju-gate gradient methodrdquo International Journal of Heat and MassTransfer vol 42 no 18 pp 3387ndash3403 1999

[15] P K Dhar and D K Sinha ldquoTemperature control of tissueby transient-induced-microwavesrdquo International Journal of Sys-tems Science vol 19 no 10 pp 2051ndash2055 1988

[16] T Loulou and E P Scott ldquoThermal dose optimization in hyper-thermia treatments by using the conjugate gradient methodrdquoNumerical Heat Transfer A vol 42 no 7 pp 661ndash683 2002

[17] T Loulou and E P Scott ldquoAn inverse heat conduction problemwith heat flux measurementsrdquo International Journal for Numer-ical Methods in Engineering vol 67 no 11 pp 1587ndash1616 2006

[18] H Kroeze M Van Vulpen A A C De Leeuw J B Van DeKamer and J J W Lagendijk ldquoThe use of absorbing structuresduring regional hyperthermia treatmentrdquo International Journalof Hyperthermia vol 17 no 3 pp 240ndash257 2001

[19] H H Pennes ldquoAnalysis of tissue and arterial blood tem-peratures in the resting human forearmrdquo Journal of AppliedPhysiology vol 1 pp 93ndash122 1948

Journal of Applied Mathematics 9

[20] J P HolmanHeat Transfer McGraw Hill New York NY USA9th edition 2002

[21] C Thiebaut and D Lemonnier ldquoThree-dimensional modellingand optimisation of thermal fields induced in a human bodyduring hyperthermiardquo International Journal of Thermal Sci-ences vol 41 no 6 pp 500ndash508 2002

[22] Z S Deng and J Liu ldquoAnalytical study on bioheat transferproblems with spatial or transient heating on skin surface orinside biological bodiesrdquo Journal of Biomechanical Engineeringvol 124 no 6 pp 638ndash649 2002

[23] N K Uzunoglu E A Angelikas and P A Cosmidis ldquoA432MHz local hyperthermia system using an indirectly cooledwater-loaded waveguide applicatorrdquo IEEE Transactions onMicrowave Theory and Techniques vol 35 no 2 pp 106ndash1111987

[24] OM Alifanov ldquoSolution of an inverse problem of heat conduc-tion by iteration methodsrdquo Journal of Engineering Physics vol26 no 4 pp 471ndash476 1974

[25] O M Alifanov Inverse Heat Transfer Problem Springer NewYork NY USA 1994

[26] M N Ozisik and H R B Orlande Inverse Heat TransferFundamentals and Applications Taylor amp Francis New YorkNY USA 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

0 001 002 003Depth of tissue (m)

37

40

43

34

Tem

pera

ture

(∘C)

Figure 7 Temperature distribution at 119905 = 1000 s for two differentthermal conductivities of casing layer Solid line 119896cl = 014Wm∘C119880 = 175Wm2∘C dashed line 119896cl = 011Wm∘C119880 = 150Wm2∘C

0 001 002 003Depth of tissue (m)

34

40

43

31

37

Tem

pera

ture

(∘C)

Figure 8 Temperature distribution at 119905 = 1000 s for twodifferent heat convection coefficients of water Solid line ℎ =

200Wm2∘C119880 = 175Wm2∘C dashed line ℎ = 350Wm2∘C119880 = 200Wm2∘C

conductivity of casing layer However depending on thetechnical facilities one can make an optimal decision in thiscase

Although the thickness and the electrical conductivityof water can also be optimized in the treatment process wedid not consider them in this study and we mainly focusedon the influence of the overall heat transfer coefficient onthe temperature field inside the tissue In the perspective ofthe present model we believe that the desired temperatureprofile inside the tumor according to its shape and size canbe achieved by inversely optimizing the shape thicknesselectrical and thermal properties of solid or liquid patcheswhich are inserted in the cooling system Such an optimiza-tion problem should also incorporate the best configurationof the additional patches which reasonably must resemblethe tumor shape for example L-form patches for L-formtumors However the clinician will first scan the geometryof the tumor and measure the tumor size and depth usingan MRI or CT or other noninvasive imaging techniquesAlthough the present model is based on the 1D heat transferbut with some modifications in the governing formulations

such as 119880 = 119891(119910 119911) it has the potential to provide amore comprehensive understanding of the cooling systemability for the purposes of beam-forming in two or threedimensions This modified optimization algorithm is thenimplemented to the tumor and its surrounding tissue todetermine the optimal properties of cooling system

5 Conclusions

In this paper we demonstrated the validity of the CGMAE inthe optimization of the temperature distribution inside thetissue by optimal control of the overall heat transfer coef-ficient associated with the cooling system The temperaturedistribution inside the tissue based on the bioheat transferequation has also been numerically computed In order toevaluate the effectiveness of the cooling system the thermalconductivity of casing layer and the heat convection coef-ficient of water have been considered separately Accordingto results the presented model allows obtaining the bestproperties of water and casing layer in the light of opti-mization and therefore significantly improves the treatmentoutcome However this method shows great promise forfurther extensions in both model and clinical performancesfor any shaped targeted tumor in hyperthermia treatments

Appendix

In order to find the minimum of the residual function bygradienttype method it is required to derive its gradient bythe following processes first computation of the variationproblem when the unknown has a small variation andsecond inspection of the conditions under which we obtain astationary point for the Lagrange functional To compute thevariation problem the unknown function 119880 is perturbed byΔ119880 that is

119880120576= 119880 + Δ119880 (A1)

Then the temperature undergoes the variation Δ119879119905(119909 119905) as

follows

119879119905120576(119909 119905 119880) = 119879

119905(119909 119905 119880) + Δ119879

119905(119909 119905 119880) (A2)

where 120576 refers to the perturbed variables By insertingthe a aforementioned equations in the direct problem andsubtracting the original (1)ndash(4) from the resulting expressionsand ignoring the second order terms the following directproblem in variation is obtained

120588119905119888119905

120597Δ119879119905(119909 119905)

120597119905= 119896119905

1205972

Δ119879119905(119909 119905)

1205971199092

minus 120596bl120588bl119888blΔ119879119905 (119909 119905) 0 lt 119909 lt 119886

(A3)

Δ119879119905(119909 119905) = 0 119909 = 0 (A4)

minus119896119905

120597Δ119879119905(119909 119905)

120597119909= Δ119880 (119879

infinminus 119879119905(119909 119905))

minus 119880Δ119879119905(119909 119905) 119909 = 119886

(A5)

Δ119879119905(119909 119905) = 0 0 lt 119909 lt 119886 119905 = 0 (A6)

Journal of Applied Mathematics 7

The second step is to calculate the variation of the objectivefunction Δ119866(119880) which is given by

Δ119866 (119880) = int

119905119891

0

int

119886

0

2 Δ119879119905(119909119895 119905) [119879

119905(119909119895 119905) minus119884

119905(119909119895 119905)]

times 120575 (119909 minus 119909119895) 119889119909 119889119905

(A7)

We obtain the variation of the augmented functionalΔ119871(119879Ψ

119905 119880) by considering the variation problem given by

(A3) and the variation of the objective function (A7) asfollows

Δ119871 = int

119905119891

0

int

119886

0

2 Δ119879119905(119909119895 119905) [119879

119905(119909119895 119905) minus 119884

119905(119909119895 119905)]

times 120575 (119909 minus 119909119895) 119889119909 119889119905

+ int

119905119891

0

int

119886

0

Ψ119905(119909 119905) [119896

119905

1205972

Δ119879119905(119909 119905)

1205971199092

minus 120588119905119888119905

120597Δ119879119905(119909 119905)

120597119905

minus120596bl120588bl119888blΔ119879119905 (119909 119905) ] 119889119909 119889119905

(A8)

Using integration by parts in (A8)

int

119905119891

0

int

119886

0

1205972

Δ119879119905(119909 119905)

1205971199092Ψ119905(119909 119905) 119889119909 119889119905

= int

119905119891

0

[120597Δ119879119905(119886 119905)

120597119909Ψ119905(119886 119905) minus

120597Ψ119905(119886 119905)

120597119909Δ119879119905(119886 119905)

minus120597Δ119879119905(0 119905)

120597119909Ψ119905(0 119905)

+120597Ψ119905(0 119905)

120597119909Δ119879119905(0 119905)] 119889119905

+ int

119905119891

0

int

119886

0

1205972

Ψ119905(119909 119905)

1205971199092Δ119879119905(119909 119905) 119889119909 119889119905

(A9)

int

119886

0

int

119905119891

0

120597Δ119879119905(119909 119905)

120597119905Ψ119905(119909 119905) 119889119905 119889119909

= int

119886

0

[Δ119879119905(119909 119905119891)Ψ119905(119909 119905) minus Δ119879

119905(119909 0) Ψ

119905(119909 0)] 119889119909

minus int

119886

0

int

119905119891

0

Δ119879119905(119909 119905)

120597Ψ119905(119909 119905)

120597119909119889119905 119889119909

(A10)

and by using the boundary conditions of the direct problemin variations (Equations (A3)ndash(A6) and the rearrangement

of the mathematical terms) the final form of (A8) can bewritten as follows

Δ119871 = int

119905119891

0

int

119886

0

[119896119905

1205972

Ψ119905(119909 119905)

1205971199092+ 120588119905119888119905

120597Ψ119905(119909 119905)

120597119905

minus 120596bl120588bl119888blΨ119905 (119909 119905)

+ 2 [119879119905(119909119895 119905) minus 119884

119905(119909119895 119905)]

times 120575 (119909 minus 119909119895) ]Δ119879

119905(119909 119905) 119889119909 119889119905

minus int

119905119891

0

119896119905Ψ119905(0 119905)

120597Δ119879119905(0 119905)

120597119909119889119905

+ int

119905119891

0

(Ψ119905(119886 119905) 119880 minus 119896

119905

120597Ψ119905(119886 119905)

120597119909)Δ119879119905(119886 119905) 119889119905

minus int

119886

0

120588119905119888119905Ψ119905(119909 119905119891) Δ119879119905(119909 119905119891) 119889119909

+ int

119905119891

0

Δ119880 Ψ119905(119886 119905) (119879

119905(119886 119905) minus 119879

infin) 119889119905

(A11)

To satisfy the optimality condition that isΔ119871(Δ119879Ψ

119905 Δ119880) = 0 it is required that all coefficients

of Δ119879119905(119909 119905) are to be vanished Finally the following term is

left

Δ119871 (119879Ψ119905 119880) = int

119905119891

0

Δ119880 Ψ119905(119886 119905) (119879

119905(119886 119905) minus 119879

infin) 119889119905

(A12)

Since we know that 119879119905(119909 119905) and Ψ

119905(119909 119905) are respectively the

direct and the adjoint problem solutions it follows that

119871 (119879Ψ119905 119880) = 119866 (119880) (A13)

and then

Δ119871 (119879Ψ119905 119880) = Δ119866 (119880) (A14)

Based on the definition the variation of the functional 119866(119880)can be written as follows

Δ119866 (119880) = ⟨nabla119866 Δ119880⟩ (A15)

where the associated inner scalar product ⟨ ⟩ of two functions119891(119905) and 119892(119905) in the Hilbert space of 119871

2(0 119905119891) is defined by

⟨119891 (119905) 119892 (119905)⟩1198712

= int

119889

119888

119891 (119905) 119892 (119905) 119889119905 (A16)

From the definition of the scalar product the variation of thefunctional 119866(119880) can be written as follows

Δ119866 (119880) = ⟨nabla119866 (119880) Δ119880⟩1198712

= nabla119866 (119880)Δ119880 (A17)

A comparison between (A12) (A14) and (A17) reveals thatthe gradient of the residual functional is given by (9) Moredetails about the derivation of the equations are mentionedin [24ndash26]

8 Journal of Applied Mathematics

Nomenclature

119888 Heat capacity119889 Vector of descent direction119890 Computational precision119864 Complex electric field119866(119880) Residual functionalΔ119866(119880) Residual functional variationnabla119866(119880) Gradient of residual functionalℎ Heat convection coefficient119867 Complex magnetic field119896 Thermal conductivity119897 Thickness of casing layer119871 Lagrangian functional119873 Number of nodes119876119898 Tissue metabolism

119876119903 Volumetric heat source

119879 Temperature119905 Time119880 Overall heat transfer coefficient119909 Space119884 Desired temperature

Greek Symbols

120573 Descent parameter120574 Descent direction parameter120575 Dirac function120576 Electrical permeability120588 Density120583 Magnetic permeability120590 Electrical conductivityΨ Lagrange multiplier120596 Angular frequency120596 Blood perfusion

Subscripts

119886 Arterialbl Blood119888 Corecl Casing layer119891 Final119894 Initial time119905 Tissueinfin Ambient

Superscripts

119897 Iteration number

References

[1] S R H Davidson and D F James ldquoDrilling in bone model-ing heat generation and temperature distributionrdquo Journal ofBiomechanical Engineering vol 125 no 3 pp 305ndash314 2003

[2] W G Sawyer M A Hamilton B J Fregly and S A BanksldquoTemperature modeling in a total knee joint replacement usingpatient-specific kinematicsrdquo Tribology Letters vol 15 no 4 pp343ndash351 2003

[3] R Dhar P Dhar and R Dhar ldquoProblem on optimal distributionof induced microwave by heating probe at the tumor site inhyperthermiardquo Advanced Modeling and Optimization vol 13no 1 pp 39ndash48 2011

[4] E A Gelvich and V N Mazokhin ldquoResonance effects in appli-cator water boluses and their influence on SAR distributionpatternsrdquo International Journal of Hyperthermia vol 16 no 2pp 113ndash128 2000

[5] M de Bruijne T Samaras J F Bakker and G C van RhoonldquoEffects of waterbolus size shape and configuration on the SARdistribution pattern of the Lucite cone applicatorrdquo InternationalJournal of Hyperthermia vol 22 no 1 pp 15ndash28 2006

[6] E R Lee D S Kapp AW Lohrbach and J L Sokol ldquoInfluenceof water bolus temperature onmeasured skin surface and intra-dermal temperaturesrdquo International Journal of Hyperthermiavol 10 no 1 pp 59ndash72 1994

[7] M D Sherar F F Liu D J Newcombe et al ldquoBeam shaping formicrowave waveguide hyperthermia applicatorsrdquo InternationalJournal of Radiation Oncology Biology Physics vol 25 no 5 pp849ndash857 1993

[8] J C Kumaradas and M D Sherar ldquoOptimization of a beamshaping bolus for superficial microwave hyperthermia waveg-uide applicators using a finite element methodrdquo Physics inMedicine and Biology vol 48 no 1 pp 1ndash18 2003

[9] M A Ebrahimi-Ganjeh and A R Attari ldquoStudy of water boluseffect on SAR penetration depth and effective field size for localhyperthermiardquo Progress in Electromagnetics Research B vol 4pp 273ndash283 2008

[10] D G Neuman P R Stauffer S Jacobsen and F RossettoldquoSAR pattern perturbations from resonance effects in waterbolus layers used with superficial microwave hyperthermiaapplicatorsrdquo International Journal of Hyperthermia vol 18 no3 pp 180ndash193 2002

[11] A G Butkovasky Distributed Control System American Else-vier New York NY USA 1969

[12] C De Wagter ldquoOptimization of simulated two-dimensionaltemperature distributions induced by multiple electromagneticapplicatorsrdquo IEEE Transactions on Microwave Theory and Tech-niques vol 34 no 5 pp 589ndash596 1986

[13] A V Kuznetsov ldquoOptimization problems for bioheat equationrdquoInternational Communications in Heat and Mass Transfer vol33 no 5 pp 537ndash543 2006

[14] C Huang and S Wang ldquoA three-dimensional inverse heatconduction problem in estimating surface heat flux by conju-gate gradient methodrdquo International Journal of Heat and MassTransfer vol 42 no 18 pp 3387ndash3403 1999

[15] P K Dhar and D K Sinha ldquoTemperature control of tissueby transient-induced-microwavesrdquo International Journal of Sys-tems Science vol 19 no 10 pp 2051ndash2055 1988

[16] T Loulou and E P Scott ldquoThermal dose optimization in hyper-thermia treatments by using the conjugate gradient methodrdquoNumerical Heat Transfer A vol 42 no 7 pp 661ndash683 2002

[17] T Loulou and E P Scott ldquoAn inverse heat conduction problemwith heat flux measurementsrdquo International Journal for Numer-ical Methods in Engineering vol 67 no 11 pp 1587ndash1616 2006

[18] H Kroeze M Van Vulpen A A C De Leeuw J B Van DeKamer and J J W Lagendijk ldquoThe use of absorbing structuresduring regional hyperthermia treatmentrdquo International Journalof Hyperthermia vol 17 no 3 pp 240ndash257 2001

[19] H H Pennes ldquoAnalysis of tissue and arterial blood tem-peratures in the resting human forearmrdquo Journal of AppliedPhysiology vol 1 pp 93ndash122 1948

Journal of Applied Mathematics 9

[20] J P HolmanHeat Transfer McGraw Hill New York NY USA9th edition 2002

[21] C Thiebaut and D Lemonnier ldquoThree-dimensional modellingand optimisation of thermal fields induced in a human bodyduring hyperthermiardquo International Journal of Thermal Sci-ences vol 41 no 6 pp 500ndash508 2002

[22] Z S Deng and J Liu ldquoAnalytical study on bioheat transferproblems with spatial or transient heating on skin surface orinside biological bodiesrdquo Journal of Biomechanical Engineeringvol 124 no 6 pp 638ndash649 2002

[23] N K Uzunoglu E A Angelikas and P A Cosmidis ldquoA432MHz local hyperthermia system using an indirectly cooledwater-loaded waveguide applicatorrdquo IEEE Transactions onMicrowave Theory and Techniques vol 35 no 2 pp 106ndash1111987

[24] OM Alifanov ldquoSolution of an inverse problem of heat conduc-tion by iteration methodsrdquo Journal of Engineering Physics vol26 no 4 pp 471ndash476 1974

[25] O M Alifanov Inverse Heat Transfer Problem Springer NewYork NY USA 1994

[26] M N Ozisik and H R B Orlande Inverse Heat TransferFundamentals and Applications Taylor amp Francis New YorkNY USA 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 7

The second step is to calculate the variation of the objectivefunction Δ119866(119880) which is given by

Δ119866 (119880) = int

119905119891

0

int

119886

0

2 Δ119879119905(119909119895 119905) [119879

119905(119909119895 119905) minus119884

119905(119909119895 119905)]

times 120575 (119909 minus 119909119895) 119889119909 119889119905

(A7)

We obtain the variation of the augmented functionalΔ119871(119879Ψ

119905 119880) by considering the variation problem given by

(A3) and the variation of the objective function (A7) asfollows

Δ119871 = int

119905119891

0

int

119886

0

2 Δ119879119905(119909119895 119905) [119879

119905(119909119895 119905) minus 119884

119905(119909119895 119905)]

times 120575 (119909 minus 119909119895) 119889119909 119889119905

+ int

119905119891

0

int

119886

0

Ψ119905(119909 119905) [119896

119905

1205972

Δ119879119905(119909 119905)

1205971199092

minus 120588119905119888119905

120597Δ119879119905(119909 119905)

120597119905

minus120596bl120588bl119888blΔ119879119905 (119909 119905) ] 119889119909 119889119905

(A8)

Using integration by parts in (A8)

int

119905119891

0

int

119886

0

1205972

Δ119879119905(119909 119905)

1205971199092Ψ119905(119909 119905) 119889119909 119889119905

= int

119905119891

0

[120597Δ119879119905(119886 119905)

120597119909Ψ119905(119886 119905) minus

120597Ψ119905(119886 119905)

120597119909Δ119879119905(119886 119905)

minus120597Δ119879119905(0 119905)

120597119909Ψ119905(0 119905)

+120597Ψ119905(0 119905)

120597119909Δ119879119905(0 119905)] 119889119905

+ int

119905119891

0

int

119886

0

1205972

Ψ119905(119909 119905)

1205971199092Δ119879119905(119909 119905) 119889119909 119889119905

(A9)

int

119886

0

int

119905119891

0

120597Δ119879119905(119909 119905)

120597119905Ψ119905(119909 119905) 119889119905 119889119909

= int

119886

0

[Δ119879119905(119909 119905119891)Ψ119905(119909 119905) minus Δ119879

119905(119909 0) Ψ

119905(119909 0)] 119889119909

minus int

119886

0

int

119905119891

0

Δ119879119905(119909 119905)

120597Ψ119905(119909 119905)

120597119909119889119905 119889119909

(A10)

and by using the boundary conditions of the direct problemin variations (Equations (A3)ndash(A6) and the rearrangement

of the mathematical terms) the final form of (A8) can bewritten as follows

Δ119871 = int

119905119891

0

int

119886

0

[119896119905

1205972

Ψ119905(119909 119905)

1205971199092+ 120588119905119888119905

120597Ψ119905(119909 119905)

120597119905

minus 120596bl120588bl119888blΨ119905 (119909 119905)

+ 2 [119879119905(119909119895 119905) minus 119884

119905(119909119895 119905)]

times 120575 (119909 minus 119909119895) ]Δ119879

119905(119909 119905) 119889119909 119889119905

minus int

119905119891

0

119896119905Ψ119905(0 119905)

120597Δ119879119905(0 119905)

120597119909119889119905

+ int

119905119891

0

(Ψ119905(119886 119905) 119880 minus 119896

119905

120597Ψ119905(119886 119905)

120597119909)Δ119879119905(119886 119905) 119889119905

minus int

119886

0

120588119905119888119905Ψ119905(119909 119905119891) Δ119879119905(119909 119905119891) 119889119909

+ int

119905119891

0

Δ119880 Ψ119905(119886 119905) (119879

119905(119886 119905) minus 119879

infin) 119889119905

(A11)

To satisfy the optimality condition that isΔ119871(Δ119879Ψ

119905 Δ119880) = 0 it is required that all coefficients

of Δ119879119905(119909 119905) are to be vanished Finally the following term is

left

Δ119871 (119879Ψ119905 119880) = int

119905119891

0

Δ119880 Ψ119905(119886 119905) (119879

119905(119886 119905) minus 119879

infin) 119889119905

(A12)

Since we know that 119879119905(119909 119905) and Ψ

119905(119909 119905) are respectively the

direct and the adjoint problem solutions it follows that

119871 (119879Ψ119905 119880) = 119866 (119880) (A13)

and then

Δ119871 (119879Ψ119905 119880) = Δ119866 (119880) (A14)

Based on the definition the variation of the functional 119866(119880)can be written as follows

Δ119866 (119880) = ⟨nabla119866 Δ119880⟩ (A15)

where the associated inner scalar product ⟨ ⟩ of two functions119891(119905) and 119892(119905) in the Hilbert space of 119871

2(0 119905119891) is defined by

⟨119891 (119905) 119892 (119905)⟩1198712

= int

119889

119888

119891 (119905) 119892 (119905) 119889119905 (A16)

From the definition of the scalar product the variation of thefunctional 119866(119880) can be written as follows

Δ119866 (119880) = ⟨nabla119866 (119880) Δ119880⟩1198712

= nabla119866 (119880)Δ119880 (A17)

A comparison between (A12) (A14) and (A17) reveals thatthe gradient of the residual functional is given by (9) Moredetails about the derivation of the equations are mentionedin [24ndash26]

8 Journal of Applied Mathematics

Nomenclature

119888 Heat capacity119889 Vector of descent direction119890 Computational precision119864 Complex electric field119866(119880) Residual functionalΔ119866(119880) Residual functional variationnabla119866(119880) Gradient of residual functionalℎ Heat convection coefficient119867 Complex magnetic field119896 Thermal conductivity119897 Thickness of casing layer119871 Lagrangian functional119873 Number of nodes119876119898 Tissue metabolism

119876119903 Volumetric heat source

119879 Temperature119905 Time119880 Overall heat transfer coefficient119909 Space119884 Desired temperature

Greek Symbols

120573 Descent parameter120574 Descent direction parameter120575 Dirac function120576 Electrical permeability120588 Density120583 Magnetic permeability120590 Electrical conductivityΨ Lagrange multiplier120596 Angular frequency120596 Blood perfusion

Subscripts

119886 Arterialbl Blood119888 Corecl Casing layer119891 Final119894 Initial time119905 Tissueinfin Ambient

Superscripts

119897 Iteration number

References

[1] S R H Davidson and D F James ldquoDrilling in bone model-ing heat generation and temperature distributionrdquo Journal ofBiomechanical Engineering vol 125 no 3 pp 305ndash314 2003

[2] W G Sawyer M A Hamilton B J Fregly and S A BanksldquoTemperature modeling in a total knee joint replacement usingpatient-specific kinematicsrdquo Tribology Letters vol 15 no 4 pp343ndash351 2003

[3] R Dhar P Dhar and R Dhar ldquoProblem on optimal distributionof induced microwave by heating probe at the tumor site inhyperthermiardquo Advanced Modeling and Optimization vol 13no 1 pp 39ndash48 2011

[4] E A Gelvich and V N Mazokhin ldquoResonance effects in appli-cator water boluses and their influence on SAR distributionpatternsrdquo International Journal of Hyperthermia vol 16 no 2pp 113ndash128 2000

[5] M de Bruijne T Samaras J F Bakker and G C van RhoonldquoEffects of waterbolus size shape and configuration on the SARdistribution pattern of the Lucite cone applicatorrdquo InternationalJournal of Hyperthermia vol 22 no 1 pp 15ndash28 2006

[6] E R Lee D S Kapp AW Lohrbach and J L Sokol ldquoInfluenceof water bolus temperature onmeasured skin surface and intra-dermal temperaturesrdquo International Journal of Hyperthermiavol 10 no 1 pp 59ndash72 1994

[7] M D Sherar F F Liu D J Newcombe et al ldquoBeam shaping formicrowave waveguide hyperthermia applicatorsrdquo InternationalJournal of Radiation Oncology Biology Physics vol 25 no 5 pp849ndash857 1993

[8] J C Kumaradas and M D Sherar ldquoOptimization of a beamshaping bolus for superficial microwave hyperthermia waveg-uide applicators using a finite element methodrdquo Physics inMedicine and Biology vol 48 no 1 pp 1ndash18 2003

[9] M A Ebrahimi-Ganjeh and A R Attari ldquoStudy of water boluseffect on SAR penetration depth and effective field size for localhyperthermiardquo Progress in Electromagnetics Research B vol 4pp 273ndash283 2008

[10] D G Neuman P R Stauffer S Jacobsen and F RossettoldquoSAR pattern perturbations from resonance effects in waterbolus layers used with superficial microwave hyperthermiaapplicatorsrdquo International Journal of Hyperthermia vol 18 no3 pp 180ndash193 2002

[11] A G Butkovasky Distributed Control System American Else-vier New York NY USA 1969

[12] C De Wagter ldquoOptimization of simulated two-dimensionaltemperature distributions induced by multiple electromagneticapplicatorsrdquo IEEE Transactions on Microwave Theory and Tech-niques vol 34 no 5 pp 589ndash596 1986

[13] A V Kuznetsov ldquoOptimization problems for bioheat equationrdquoInternational Communications in Heat and Mass Transfer vol33 no 5 pp 537ndash543 2006

[14] C Huang and S Wang ldquoA three-dimensional inverse heatconduction problem in estimating surface heat flux by conju-gate gradient methodrdquo International Journal of Heat and MassTransfer vol 42 no 18 pp 3387ndash3403 1999

[15] P K Dhar and D K Sinha ldquoTemperature control of tissueby transient-induced-microwavesrdquo International Journal of Sys-tems Science vol 19 no 10 pp 2051ndash2055 1988

[16] T Loulou and E P Scott ldquoThermal dose optimization in hyper-thermia treatments by using the conjugate gradient methodrdquoNumerical Heat Transfer A vol 42 no 7 pp 661ndash683 2002

[17] T Loulou and E P Scott ldquoAn inverse heat conduction problemwith heat flux measurementsrdquo International Journal for Numer-ical Methods in Engineering vol 67 no 11 pp 1587ndash1616 2006

[18] H Kroeze M Van Vulpen A A C De Leeuw J B Van DeKamer and J J W Lagendijk ldquoThe use of absorbing structuresduring regional hyperthermia treatmentrdquo International Journalof Hyperthermia vol 17 no 3 pp 240ndash257 2001

[19] H H Pennes ldquoAnalysis of tissue and arterial blood tem-peratures in the resting human forearmrdquo Journal of AppliedPhysiology vol 1 pp 93ndash122 1948

Journal of Applied Mathematics 9

[20] J P HolmanHeat Transfer McGraw Hill New York NY USA9th edition 2002

[21] C Thiebaut and D Lemonnier ldquoThree-dimensional modellingand optimisation of thermal fields induced in a human bodyduring hyperthermiardquo International Journal of Thermal Sci-ences vol 41 no 6 pp 500ndash508 2002

[22] Z S Deng and J Liu ldquoAnalytical study on bioheat transferproblems with spatial or transient heating on skin surface orinside biological bodiesrdquo Journal of Biomechanical Engineeringvol 124 no 6 pp 638ndash649 2002

[23] N K Uzunoglu E A Angelikas and P A Cosmidis ldquoA432MHz local hyperthermia system using an indirectly cooledwater-loaded waveguide applicatorrdquo IEEE Transactions onMicrowave Theory and Techniques vol 35 no 2 pp 106ndash1111987

[24] OM Alifanov ldquoSolution of an inverse problem of heat conduc-tion by iteration methodsrdquo Journal of Engineering Physics vol26 no 4 pp 471ndash476 1974

[25] O M Alifanov Inverse Heat Transfer Problem Springer NewYork NY USA 1994

[26] M N Ozisik and H R B Orlande Inverse Heat TransferFundamentals and Applications Taylor amp Francis New YorkNY USA 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Applied Mathematics

Nomenclature

119888 Heat capacity119889 Vector of descent direction119890 Computational precision119864 Complex electric field119866(119880) Residual functionalΔ119866(119880) Residual functional variationnabla119866(119880) Gradient of residual functionalℎ Heat convection coefficient119867 Complex magnetic field119896 Thermal conductivity119897 Thickness of casing layer119871 Lagrangian functional119873 Number of nodes119876119898 Tissue metabolism

119876119903 Volumetric heat source

119879 Temperature119905 Time119880 Overall heat transfer coefficient119909 Space119884 Desired temperature

Greek Symbols

120573 Descent parameter120574 Descent direction parameter120575 Dirac function120576 Electrical permeability120588 Density120583 Magnetic permeability120590 Electrical conductivityΨ Lagrange multiplier120596 Angular frequency120596 Blood perfusion

Subscripts

119886 Arterialbl Blood119888 Corecl Casing layer119891 Final119894 Initial time119905 Tissueinfin Ambient

Superscripts

119897 Iteration number

References

[1] S R H Davidson and D F James ldquoDrilling in bone model-ing heat generation and temperature distributionrdquo Journal ofBiomechanical Engineering vol 125 no 3 pp 305ndash314 2003

[2] W G Sawyer M A Hamilton B J Fregly and S A BanksldquoTemperature modeling in a total knee joint replacement usingpatient-specific kinematicsrdquo Tribology Letters vol 15 no 4 pp343ndash351 2003

[3] R Dhar P Dhar and R Dhar ldquoProblem on optimal distributionof induced microwave by heating probe at the tumor site inhyperthermiardquo Advanced Modeling and Optimization vol 13no 1 pp 39ndash48 2011

[4] E A Gelvich and V N Mazokhin ldquoResonance effects in appli-cator water boluses and their influence on SAR distributionpatternsrdquo International Journal of Hyperthermia vol 16 no 2pp 113ndash128 2000

[5] M de Bruijne T Samaras J F Bakker and G C van RhoonldquoEffects of waterbolus size shape and configuration on the SARdistribution pattern of the Lucite cone applicatorrdquo InternationalJournal of Hyperthermia vol 22 no 1 pp 15ndash28 2006

[6] E R Lee D S Kapp AW Lohrbach and J L Sokol ldquoInfluenceof water bolus temperature onmeasured skin surface and intra-dermal temperaturesrdquo International Journal of Hyperthermiavol 10 no 1 pp 59ndash72 1994

[7] M D Sherar F F Liu D J Newcombe et al ldquoBeam shaping formicrowave waveguide hyperthermia applicatorsrdquo InternationalJournal of Radiation Oncology Biology Physics vol 25 no 5 pp849ndash857 1993

[8] J C Kumaradas and M D Sherar ldquoOptimization of a beamshaping bolus for superficial microwave hyperthermia waveg-uide applicators using a finite element methodrdquo Physics inMedicine and Biology vol 48 no 1 pp 1ndash18 2003

[9] M A Ebrahimi-Ganjeh and A R Attari ldquoStudy of water boluseffect on SAR penetration depth and effective field size for localhyperthermiardquo Progress in Electromagnetics Research B vol 4pp 273ndash283 2008

[10] D G Neuman P R Stauffer S Jacobsen and F RossettoldquoSAR pattern perturbations from resonance effects in waterbolus layers used with superficial microwave hyperthermiaapplicatorsrdquo International Journal of Hyperthermia vol 18 no3 pp 180ndash193 2002

[11] A G Butkovasky Distributed Control System American Else-vier New York NY USA 1969

[12] C De Wagter ldquoOptimization of simulated two-dimensionaltemperature distributions induced by multiple electromagneticapplicatorsrdquo IEEE Transactions on Microwave Theory and Tech-niques vol 34 no 5 pp 589ndash596 1986

[13] A V Kuznetsov ldquoOptimization problems for bioheat equationrdquoInternational Communications in Heat and Mass Transfer vol33 no 5 pp 537ndash543 2006

[14] C Huang and S Wang ldquoA three-dimensional inverse heatconduction problem in estimating surface heat flux by conju-gate gradient methodrdquo International Journal of Heat and MassTransfer vol 42 no 18 pp 3387ndash3403 1999

[15] P K Dhar and D K Sinha ldquoTemperature control of tissueby transient-induced-microwavesrdquo International Journal of Sys-tems Science vol 19 no 10 pp 2051ndash2055 1988

[16] T Loulou and E P Scott ldquoThermal dose optimization in hyper-thermia treatments by using the conjugate gradient methodrdquoNumerical Heat Transfer A vol 42 no 7 pp 661ndash683 2002

[17] T Loulou and E P Scott ldquoAn inverse heat conduction problemwith heat flux measurementsrdquo International Journal for Numer-ical Methods in Engineering vol 67 no 11 pp 1587ndash1616 2006

[18] H Kroeze M Van Vulpen A A C De Leeuw J B Van DeKamer and J J W Lagendijk ldquoThe use of absorbing structuresduring regional hyperthermia treatmentrdquo International Journalof Hyperthermia vol 17 no 3 pp 240ndash257 2001

[19] H H Pennes ldquoAnalysis of tissue and arterial blood tem-peratures in the resting human forearmrdquo Journal of AppliedPhysiology vol 1 pp 93ndash122 1948

Journal of Applied Mathematics 9

[20] J P HolmanHeat Transfer McGraw Hill New York NY USA9th edition 2002

[21] C Thiebaut and D Lemonnier ldquoThree-dimensional modellingand optimisation of thermal fields induced in a human bodyduring hyperthermiardquo International Journal of Thermal Sci-ences vol 41 no 6 pp 500ndash508 2002

[22] Z S Deng and J Liu ldquoAnalytical study on bioheat transferproblems with spatial or transient heating on skin surface orinside biological bodiesrdquo Journal of Biomechanical Engineeringvol 124 no 6 pp 638ndash649 2002

[23] N K Uzunoglu E A Angelikas and P A Cosmidis ldquoA432MHz local hyperthermia system using an indirectly cooledwater-loaded waveguide applicatorrdquo IEEE Transactions onMicrowave Theory and Techniques vol 35 no 2 pp 106ndash1111987

[24] OM Alifanov ldquoSolution of an inverse problem of heat conduc-tion by iteration methodsrdquo Journal of Engineering Physics vol26 no 4 pp 471ndash476 1974

[25] O M Alifanov Inverse Heat Transfer Problem Springer NewYork NY USA 1994

[26] M N Ozisik and H R B Orlande Inverse Heat TransferFundamentals and Applications Taylor amp Francis New YorkNY USA 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 9

[20] J P HolmanHeat Transfer McGraw Hill New York NY USA9th edition 2002

[21] C Thiebaut and D Lemonnier ldquoThree-dimensional modellingand optimisation of thermal fields induced in a human bodyduring hyperthermiardquo International Journal of Thermal Sci-ences vol 41 no 6 pp 500ndash508 2002

[22] Z S Deng and J Liu ldquoAnalytical study on bioheat transferproblems with spatial or transient heating on skin surface orinside biological bodiesrdquo Journal of Biomechanical Engineeringvol 124 no 6 pp 638ndash649 2002

[23] N K Uzunoglu E A Angelikas and P A Cosmidis ldquoA432MHz local hyperthermia system using an indirectly cooledwater-loaded waveguide applicatorrdquo IEEE Transactions onMicrowave Theory and Techniques vol 35 no 2 pp 106ndash1111987

[24] OM Alifanov ldquoSolution of an inverse problem of heat conduc-tion by iteration methodsrdquo Journal of Engineering Physics vol26 no 4 pp 471ndash476 1974

[25] O M Alifanov Inverse Heat Transfer Problem Springer NewYork NY USA 1994

[26] M N Ozisik and H R B Orlande Inverse Heat TransferFundamentals and Applications Taylor amp Francis New YorkNY USA 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of