solution of an inverse heat conduction problem in a bi-layered spherical tissue

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Solution of an Inverse Heat ConductionProblem in a Bi-Layered Spherical TissueKuo-Chi Liu a & Chin-Tse Lin ba Department of Mechanical Engineering, Far East University, Tainan,Taiwan, Republic of Chinab Department of Computer Application Engineering, Far EastUniversity, Tainan, Taiwan, Republic of ChinaPublished online: 19 Nov 2010.

To cite this article: Kuo-Chi Liu & Chin-Tse Lin (2010) Solution of an Inverse Heat Conduction Problemin a Bi-Layered Spherical Tissue, Numerical Heat Transfer, Part A: Applications: An InternationalJournal of Computation and Methodology, 58:10, 802-818, DOI: 10.1080/10407782.2010.523329

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SOLUTION OF AN INVERSE HEAT CONDUCTIONPROBLEM IN A BI-LAYERED SPHERICAL TISSUE

Kuo-Chi Liu1 and Chin-Tse Lin21Department of Mechanical Engineering, Far East University, Tainan,Taiwan, Republic of China2Department of Computer Application Engineering, Far East University,Tainan, Taiwan, Republic of China

This work attempts to estimate the phase lag times of a tissue based on the dual-phase-lag

model from the experimental data. The inverse dual-phase-lag bioheat transfer problem in

the bilayered spherical tissue is studied. The difference between two layers in the thermo-

physical parameters, geometry effects, and measurement errors of the input data make it

hard to be solved. To solve the present problem, a hybrid scheme based on the Laplace

transform, change of variables, and the least-squares scheme is proposed. In order to evi-

dence the validity and accuracy of the estimated results, the comparison of the history of

temperature increase between the calculated results and the experimental data is made

for various measurement locations. The effect of measurement location on the estimated

results is also investigated.

INTRODUCTION

The use of heat to necrotize undesirable tissue for therapeutic purposes is notan innovative idea. It has been in many applications, such as cauterizing a wound,treating inflammation, and eliminating port wine stains. These days, the use ofhyperthermia has been developed as an important technique to destroy malignanttumors. In hyperthermia, the targeted tissue must be elevated to temperatures inthe range 42�C–46�C for a specified period of time. An ideal hyperthermia treatmentshould selectively destroy the target region without damaging the surroundinghealthy tissue. However, it is not easy to accurately determine the temperature fieldover the entire treatment region during clinical hyperthermia treatments, because thepain tolerance of patients makes the number of invasive temperature probes limited[1]. In order to further improve the thermal treatment methods, many researchershave aimed at hyperthermia treatments. The bioheat models are essential duringdevelopment of equipment, for pre-planning purposes, for online monitoring and

Received 10 February 2010; accepted 7 August 2010.

Support for this work by the National Science Counsel under grant no. NSC 97-2212-E-269-023 is

gratefully acknowledged.

Address correspondence to Kuo-Chi Liu, Department of Mechanical Engineering, Far East

University, 49 Chung Hua Rd., Hsin-Shih, Tainan 744, Taiwan, Republic of China. E-mail: kcliu@

cc.feu.edu.tw

Numerical Heat Transfer, Part A, 58: 802–818, 2010

Copyright # Taylor & Francis Group, LLC

ISSN: 1040-7782 print=1521-0634 online

DOI: 10.1080/10407782.2010.523329

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decision support, as well as for evaluation of the extent of thermal damage [2].Therefore, one of the ways current research seeks improvement is to develop a heattransfer model for the analysis and modeling of the underlying thermal mechanismsin the treated region.

Most analyses of temperature predictions for biological bodies are based on thewell-known Pennes’ bioheat equation. The Pennes’ bioheat equation was derivedwith the classical Fourier’s law that depicts an infinite velocity of thermal propa-gation. However, the contents of the literature [3–7] indicated that thermal behaviorin biological tissues requires a relaxation time to accumulate enough energy to trans-fer to the nearest element. The relevant researchers [8–11], thus employed the ther-mal wave model to remedy this physically unreasonable deficiency. The thermalwave model cannot capture the microstructural interaction effects [12], and intro-duce some unusual behaviors and physical solutions [13–15]. The thermal wavemodel becomes open to debate for its validity.

In order to explore another possibility, Antaki [16] used the dual-phase-lag(DPL) heat conduction model to interpret the thermal behavior in processed meats.The DPL model describes a macroscopic temperature with the microstructural effectby introducing the phase lag times of heat flux and temperature gradient. Followingin Antaki’s steps, there are a few papers that study the bioheat transfer problemswith the DPL model. Liu and Chen [17] studied temperature rise behavior in atwo-layer concentric spherical region during magnetic tumor hyperthermia treatment

NOMENCLATURE

A estimated parameter

c specific heat of tissue, J=kg �Kcb specific heat of blood, J=kg �Kdn correction of An

em deviation between hcalm and hmeam

f parameter defined in Eq. (20)

H new dependent variable,

H¼ r(T�T0)eHH Laplace transform of H

k thermal conductivity, W=m �KK parameter defined in Eq. (21)

‘ distance between two neighboring

nodes, m

M total number of nodes

N total number of estimated parameters

P power density, W=m3

qm metabolic heat generation, W=m3

qr spatial heating source, W=m3

r space coordinate, m

R radius of tumor, m

s Laplace transform parameter

t time, s

T temperature of tissue, K

Tb arterial temperature, K

T0 initial temperature of tissue, K

wb perfusion rate of blood, m3=s=m3

xmn parameter defined in Eq. (34)

dmn Kronecker delta

h temperature increase defined in

Eq. (27)

e standard deviation of the

measurements

k parameter defined in Eq. (19)

q density, kg=m3

w volume fraction of magnetic particles

sq phase lag of the heat flux, s

sT phase lag of the temperature gradient,

s

Superscripts

cal calculated value

mea measured data

Subscripts

g magnetic particle

i node number

j number of sub-space domain

k number of layer

m number of time node

n number of estimated parameter

t tumor tissue

INVERSE PROBLEM IN SPHERICAL TISSUE 803

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with the DPL model. Zhou et al. [18] recently proposed a two-dimensionalaxisymmetric DPL model to describe heat transfer in living biological tissues withnonhomogenous inner structures. Zhang [19] derived the general dual phase bio-heat equations from the nonequilibrium model for living biological tissues. How-ever, there is limited experimental evidence for showing the physical meanings ofthe DPL mode in the bioheat transfer.

This work attempts to estimate the phase lag times based on the DPLmodel from the experimental data given by Andra et al. [20]. The inverse problemis an ill-posed problem. Various methods [21–25] were developed for the inverseheat conduction problems. However, it is well-known that there are mathematicaldifficulties in dealing with the non-Fourier heat transfer problem, not to mentionthe inverse problem is an ill-posed problem. The literature about the estimationof the relaxation times in tissues are not numerous. Aside from references [4–7,16], the literature [26–30] mainly estimated the boundary conditions with the ana-lytical solution in conjunction with measurement errors. Due to the differencebetween two layers in the thermophysical parameters, geometry effects, andmeasurement errors of the input data, the present problem introduces complexityand causes more mathematical difficulties. To solve the present problem, a hybridnumerical scheme based on the Laplace transform, change of variables, and theleast-squares scheme is proposed. In order to evidence the validity and accuracyof the estimated results, the comparisons of the temperature increase historybetween the results calculated with the estimates and the experimental data aremade for various measurement locations. The solution of inverse problem willdepend on location [31]. Therefore, the investigation of the effect of locationon the estimated results is involved.

MATHEMATICS MODEL

To accommodate the microstructural effect, the DPL model suggested by Tzou[32] is applied. The linear version of the DPL model is expressed as

sqqqqt

þ q ¼ �kqTqr

� ksTq2Tqtqr

ð1Þ

where T is the temperature and q is the heat flux. sq means the phase lag of the heatflux and sT means the phase lag of the temperature gradient. In bioheat transfer,Antaki [16] interpreted sq as a delay time for contact resistance between tissue par-ticles. On the other hand, sT was interpreted as a measure of the conduction thatoccurs within tissues particles. The values of sq and sT may be different in tumorand normal tissue as well as the other physiological parameters. The heat flux pre-cedes the temperature gradient for sq< sT. The temperature gradient precedes theheat flux for sq> sT. The DPL model combines the wave features of hyperbolic con-duction with a diffusion-like feature of the evidence not captured by the hyperboliccase. As sT¼ 0, the DPL model reduces to the thermal wave model of heat transfer.By further letting sq¼ 0, it becomes the classical Fourier law.

In magnetic tumor hyperthermia, fine magnetic particles are localized at thetumor tissue. The literature [20, 33, 34] regarded the small tumor as a solid

804 K.-C. LIU AND C.-T. LIN

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sphere with radius R, and becomes a heat source of constant power density P inthe small tumor for excitation of alternating magnetic field. The heat source isassumed to be, surrounded by a medium of homogeneous heat conductivity.The heating material, e.g., magnetic particles injected into the tumor and the sur-rounding medium is characterized by the values of their heat conductivity k, theirspecific heat capacity c, and their mass density q. Because of the spherical sym-metry of the system and the homogeneous time-independent power density Pinside the sphere, the temperature distribution depends only on distance r fromthe center of the sphere and on time t.

The present work regards the temperature and the heat flux at the interface oftwo regions is continuous. Therefore, the boundary conditions are described as

qT1ð0; tÞqr

¼ 0 and T1ð0; tÞ is finite ð2Þ

T1ðR; tÞ ¼ T2ðR; tÞ ð3Þ

q1ðR; tÞ ¼ q2ðR; tÞ ð4Þ

T2ð1; tÞ ¼ T0 ð5Þ

and the initial conditions are

Tkðr; 0Þ ¼ T0;qTkðr; 0Þ

qt¼ 0; and qkðr; 0Þ ¼ k ¼ 1; 2 ð6Þ

The indices 1 and 2 mean the interior and exterior of the sphere r�R, respectively.In a local energy balance, the one-dimensional energy equation of the present

problem is given as

qcqTqt

¼ � qqqr

� 2

rqþ wbqbcbðTb � TÞ þ qm þ qr ð7Þ

where qb and cb, respectively, are the density and specific heat. The spatial heatingsource qr is defined as qr¼Pu(t), where u(t) is a step function. The present workomits the metabolic heat generation qm, and the perfusion rate of blood wb for thatthe experiment was not performed with living tissue.

Substituting Eq. (1) into the energy conservation Eq. (7) leads to the heat trans-port equations in the heating material, and the extended muscle tissue with constantphysiological parameters as the following.

k11

r2qqr

�r2�qT1

qrþ sT1

q2T1

qtqr

��¼

�1þ sq1

qqt

��q1c1

qT1

qt� P

�for 0 � r � R ð8Þ

k21

r2qqr

�r2�qT2

qrþ sT2

q2T2

qtqr

��¼

�1þ sq2

qqt

�q2c2

qT2

qtfor R � r � 1 ð9Þ


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ESTIMATION SCHEME

For convenience of analysis, a new dependent variable H is defined as

H ¼ rðT � T0Þ ð10Þ

Under the circumstances, Eqs. (8) and (9) in terms of H can be rewritten as

k1

�1þ sT1

qqt

�q2H1

qr2

¼�1þ sq1

qqt

��q1c1

qH1

qt� P r

�for 0 � r � R ð11Þ

k2

�1þ sT2

qqt

�q2H2

qr2

¼�1þ sq2

qqt

�q2c2

qH2

qtfor R � r � 1 ð12Þ

The boundary conditions and the initial conditions become

H1ð0; tÞ ¼ 0 ð13Þ

H1ðR; tÞ ¼ H2ðR; tÞ ð14Þ

q1ðR; tÞ ¼ q2ðR; tÞ ð15Þ

H2ð1; tÞ ¼ 0 ð16Þ

and

Hkðr; 0Þ ¼ 0;qHkðr; 0Þ

qt¼ 0; and qkðr; 0Þ ¼ 0 k ¼ 1; 2 ð17Þ

Subsequently, the Laplace transform technique is used to map the transientproblem into the steady one. The differential Eqs. (11) and (12) are transformedunder the initial conditions (17) as

d2 eHHk

dr2� k2k eHHk ¼ �fkr k ¼ 1; 2 ð18Þ

where s is the Laplace transform parameter of time t. k2k, f1, f2, and Kk are defined as

k2k ¼1

Kkqkcks k ¼ 1; 2 ð19Þ

f1 ¼Pð1þ sq1sÞk1ð1þ sT1sÞs

ð20aÞ


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f2 ¼ 0 ð20bÞ

Kk ¼ kk1þ sTks1þ sqks

k ¼ 1; 2 ð21Þ

In accordance with Eq. (1), the boundary conditions (13)–(16) in the Laplace trans-form domain can be written as

eHH1ð0; sÞ ¼ 0 ð22Þ

eHH1ðR; sÞ ¼ eHH2ðR; sÞ ð23Þ

K1

�d eHH1ðR; sÞ

dr�

eHH1

R

�¼ K2

�d eHH2ðR; sÞ

dr�

eHH2

R

�ð24Þ

eHH2ð1; sÞ ¼ 0 ð25Þ

The analytical solution of the governing Eq. (18) is easily obtained as

eHHk ¼ Bk sinh kkrþ Ck cosh kkrþfk

k2kr ð26Þ

The coefficients Bk and Ck can be determined with the boundary conditions(22)–(25). The value of Hk in the physical domain can be determined with the appli-cation of the Gaussian elimination algorithm and the numerical inversion of theLaplce transform [35].

The temperature increase h is equal to

h ¼ T � T0 ¼ H=r ð27Þ

The value of H=r at r¼ 0 is indeterminate and must be replaced by its limit as r! 0.Thus, the value of the transient temperature increase at the center is evaluated byusing L’Hospital’s rule as

hð0; tÞ ¼ Tð0; tÞ � T0 ¼ limr!0

H

r¼ dH

drð28Þ

In order to estimate the target parameters from the experimental data, theleast-squares minimization technique is applied to minimize the sum of the squaresof the deviations between the calculated values and the experimental data at thespecified measurement location ri. The sum of the squares of the deviations betweenthe calculated values and the measurement values can be expressed as

EðA1;A2; . . . ;ANÞ ¼XNm¼1

ðhcalm � hmeam Þ2 ð29Þ

where hcalm and hmeam are the calculated temperature increase and the measurement

temperature increase at the mth time node, respectively. An, n¼ 1, 2, 3, . . . , N, are


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used to denote the estimated parameters. The estimated values of An are determinedwith that the value of E is minimum. The computational procedures are described asfollows.

First, the initial guesses of An are given. Afterwards, the calculated temperature

increase hcalm at the specified measurement location r¼ ri is taken from Eqs. (26–28).

Deviations between hcalm and hmeam are expressed as

em ¼ hcalm � hmeam for m ¼ 1; 2; 3; . . . ;N ð30Þ

The next calculated value hcal;�m can be expanded in a first-order Taylor series as

hcal;�m ¼ hcalm þXNn¼1

qhcalm

qAndAn ð31Þ

In order to obtain the derivative qhcalm =qAnin Eq. (31), the next guessed value of An,A�

n, is introduced as

A�n ¼ An þ dndmn for m; n ¼ 1; 2; 3; . . . ;N ð32Þ

where dn denotes the correction. The symbol dmn is Kronecker delta.

The next calculated value hcal;�m , similarly, with respect to A�n can be determined

from Eqs. (26)–(28). Deviations between hcal;�m and hmeam are written as

e�m ¼ hcal;�m � hmeam for m ¼ 1; 2; 3; . . . ;N ð33Þ

The derivative qhcalm =qAn can be expressed in the finite-difference form as

xmn ¼qhcalm

qAn¼ hcal;�m � hmea

m

A�n � An

for m; n ¼ 1; 2; 3; . . . ;N ð34Þ

Substituting Eqs. (30), (32), and (33) into Eq. (34) leads to

xmn ¼e�m � em

dnfor m; n ¼ 1; 2; 3; . . . ;N ð35Þ

The substitution of Eqs. (34) and (35) into Eq. (31) yields

hcal;�m ¼ hcalm þXNn¼1

xmnd�n for m ¼ 1; 2; 3; . . . ;N ð36Þ

where d�n ¼ dAn denotes the new correction of An.


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Substituting Eqs. (30) and (33) into Eq. (36) has

e�m ¼ em þXNn¼1

xmnd�n for m ¼ 1; 2; 3; . . . ;N ð37Þ

In accordance with Eqs. (29) and (33), the sum of the squares of the deviationsbetween the calculated values and the measurement values E(A1þDA1,A2þDA2, . . . , AnþDAn) can be expressed as

E ¼XNm¼1

ðe�mÞ2 ð38Þ

In order to yield the minimum value of E with respect to An, differentiating Ecorresponding to the new correction d�

n is performed. Thus, the correction equationscorresponding to An can be expressed as

XNl¼1

XNn¼1

xnlxmnd�l ¼ �

XNl¼1

xmlel for m ¼ 1; 2; 3; . . . ;N ð39Þ

Equation (39) is a set of four algebraic equations for the new correction d�n . The new

correction d�n are obtained from Eq. (39). Hence, the new values of An, An þ d�

n , canbe determined.

The above computation procedures were repeated until the value of

hcalm � hmeam

hmeam

�� < e for m ¼ 1; 2; 3; . . . ;N ð40Þ

where e is the standard deviation of the measurements.

RESULTS AND DISCCUSSION

For experimental study, Andra et al. [20] made the heating material of carra-geenan and a variable amount of magnetite with a mean grain size of 1 mm andembedded it in extended muscle tissue from cow. The spatial distribution of tem-perature increase as a function of exposure time was measured with thermocouples.Measured values of the increase of temperature for the various reduced distancesr=R¼ 1.10, 1.39, 1.68, and 1.98 are presented in Figure 1 with symbols. Thesymbols are discrete values of measured temperature that were extracted fromthe continuous thermocouple record, as presented in Figure 3 in reference [20].The experimental errors of both h and t are within the extent of the used symbols.In the experiment, the composite consists of 106mg magnetite and carrageenan gelwith the following parameters: k1¼ 0.778W=(K m), q1¼ 1.66 g=cm3, c1¼ 2.54 J=(g K),R¼ 3.15mm, and a power density of 6.15W=cm3. The corresponding parametersof the surrounding muscle tissue were taken as k2¼ 0.642W=(K m), q2¼ 1 g=cm3,3, and c2¼ 3.72 J=(g K).


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For theoretical study, Andra et al. [20] calculated the increase of temperaturewith exposure time t based on the classical bioheat transfer equation for the para-meters described above. It is observed from Figure 1 that there is an obvious differ-ence between the calculated values from the classical bioheat transfer equation andthe measured values indicated with symbols. This phenomenon implies that theclassical bioheat transfer equation does not completely describe the thermal behaviorcaptured in the experiment. The phase lag times for the composite of magnetite andcarrageenan gel and the surrounding muscle tissue, sq1, sT1, sq2, and sT2, are the tar-get estimated parameters in the present work, and the number of the estimated para-meters N is 4. The standard deviation of the measurements, e, was assumed to be aconstant 0.03. This work picks three sets of reference values at various measurementlocations, as shown in Tables 1–4. Each set is composed of four measured values. Toshow the rationality of the estimated results, the estimated values of sq1, sT1, sq2, and

Table 1. Reference values and estimated values based on the data measured at r=R¼ 1.1

Set number

Reference values

t(s) and h(K)

Estimated values of sq1, sT1,sq2, and sT2 (s)

1–1 t¼ 33, h¼ 7.35 t¼ 54, h¼ 10.675

t¼ 83, h¼ 13.3 t¼ 110, h¼ 14.875

sq1¼ 4.2679, sT1¼ 14.5383

sq2¼ 3.9135, sT2¼ 11.5800

1–2 t¼ 83, h¼ 13.3 t¼ 142, h¼ 16.45

t¼ 209, h¼ 18.375 t¼ 270, h¼ 19.775

sq1¼ 8.9464, sT1¼ 15.4121

sq2¼ 8.4322, sT2¼ 14.5465

1–3 t¼ 175, h¼ 17.5 t¼ 209, h¼ 18.375

t¼ 241, h¼ 19.25 t¼ 270, h¼ 19.775

sq1¼ 8.8539, sT1¼ 15.5921

sq2¼ 8.4299, sT2¼ 14.5515

Figure 1. Comparison between the calculated values from the classical bioheat transfer equation and the

experimental data.


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sT2 would bring into Eqs. (26)–(28) to calculate the increase of temperature withexposure time t at various reduced distances. Figures 2–5 show the comparison ofthe calculated results with the experimental data.

Table 1 shows the three sets of reference values chosen from the data measuredat r=R¼ 1.10 and the estimated results. It is found that the estimated values of sq1,sT1, sq2, and sT2 for set 1–2 almost agree with those for set 1–3, but are obviouslydifferent with those for set 1–1. However, the calculated results of the increase oftemperature with exposure time t for these three sets are very similar and approachthe experimental data at various reduced distances, as shown in Figure 2. The timenodes are 33 s, 54 s, 83 s, and 110 s for the reference values of set 1–1. The history oftemperature increase with sq1¼ 4.2679s, sT1¼ 14.5383 s, sq2¼ 3.9135 s, andsT2¼ 11.5800 s coincides with the experimental data at the reduced distancer=R¼ 1.10 during 0 s� t� 150 s, but does not coincide at the reduced distancesr=R¼ 1.39, 1.68, and 1.98. After t¼ 150 s, it gradually splits from the experimental


Set number

Reference values

t(s) and h(K)

Estimated values

of sq1, sT1, sq2, and sT2, (s)

3–1 t¼ 32, h¼ 1.225 t¼ 48, h¼ 2.625

t¼ 64, h¼ 3.675 t¼ 109, h¼ 6.125

sq1¼ 12.9945, sT1¼ 5.3008

sq2¼ 12.2865, sT2¼ 18.9981

3–2 t¼ 109, h¼ 6.125 t¼ 129, h¼ 7.175

t¼ 157, h¼ 8.225 t¼ 212, h¼ 9.45

sq1¼ 6.6475, sT1¼ 20.6194

sq2¼ 6.3299, sT2¼ 17.0494

3–3 t¼ 129, h¼ 7.175 t¼ 157, h¼ 11.2

t¼ 212, h¼ 12.775 t¼ 274, h¼ 14

sq1¼ 7.5586, sT1¼ 21.9667

sq2¼ 7.401, sT2¼ 17.8216


Set number

Reference values

t(s) and h(K)

Estimated values of sq1, sT1,sq2, and sT2 (s)

4–1 t¼ 49, h¼ 0.875 t¼ 64, h¼ 1.925

t¼ 86, h¼ 3.15 t¼ 109, h¼ 4.025

——–

4–2 t¼ 109, h¼ 4.025 t¼ 157, h¼ 5.6

t¼ 212, h¼ 6.65 t¼ 274, h¼ 7.875

sq1¼ 8.1768, sT1¼ 22.5628

sq2¼ 7.7672, sT2¼ 21.3071

4–3 t¼ 129, h¼ 4.725 t¼ 157, h¼ 5.6

t¼ 212, h¼ 6.65 t¼ 274, h¼ 7.875

sq1¼ 6.1462, sT1¼ 28.7482

sq2¼ 5.7370, sT2¼ 22.5038


Set number

Reference values

t(s) and h(K)

Estimated values of

sq1, sT1, sq2, and sT2 (s)

2–1 t¼ 32, h¼ 2.8 t¼ 48, h¼ 4.9

t¼ 64, h¼ 6.3 t¼ 86, h¼ 8.05

——–

2–2 t¼ 109, h¼ 9.1 t¼ 157, h¼ 11.2

t¼ 212, h¼ 12.775 t¼ 274, h¼ 14

sq1¼ 7.6140, sT1¼ 20.1088

sq2¼ 7.3629, sT2¼ 18.7825

2–3 t¼ 157, h¼ 11.2 t¼ 184, h¼ 12.25

t¼ 240, h¼ 13.475 t¼ 274, h¼ 14

sq1¼ 8.3776, sT1¼ 18.6416

sq2¼ 8.0022, sT2¼ 17.7723


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Figure 2. History of temperature increase with the estimated values of sq1, sT1, sq2, and sT2 based on the

experimental data at r=R¼ 1.1.




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data. These phenomena do not exist for set 1–2 and set 1–3. In the meantime, thecurves of the history of temperature increase gradually approach consistent withthe reduced distance increasing for sets 1–1, 1–2, and 1–3.

Similarly, there are three sets of reference values chosen from the measureddata of r=R¼ 1.39, as shown in Table 2. However, the estimated values of sq1,sT1, sq2, and sT2 based on the reference values of set 2–1 can not let the calculatedvalues of the temperature increase up to the standard deviation of the measurements.The calculated results of the variation of temperature increase for sets 2–2 and 2–3are compared with the experimental data, as shown in Figure 3. They approach theexperimental, data and almost agree with each other. However, the estimated valuesof sq1, sT1, sq2, and sT2 are sq1¼ 7.6140 s, sT1¼ 20.1088 s, sq2¼ 7.3629 s, andsT2¼ 17.0494 s for set 2–2 and are sq1¼ 8.3776 s, sT1¼ 18.6416 s, sq2¼ 8.0022 s,and sT2¼ 17.7723 s for set 2–3. It is in character with the ill-posed problem. Underthe circumstances, how to judge which one is more correct has became the importantissue for analysis of the inverse heat transfer problem.

The estimated values of sq1, sT1, sq2, and sT2 with the reference values extractedfrom the measured data of r=R¼ 1.68 are presented in Table 3. Figure 4 displays thecalculated results of the variation of temperature increase with them. The estimatedresults in Tables 1 and 2 show that the ratio of sT to sq is greater than 1. It is foundfrom Table 3 that the value of sq1 is greater than the value of sT1 for set 3–1. In themeantime, the estimated values of sq1, sT1, sq2, and sT2 for set 3–1 are sq1¼ 12.9945 s,sT1¼ 5.3008 s, sq2¼ 12.2865 s, and sT2¼ 18.9981 s. They are quite different withthose for sets 3–2 and 3–3, but the curves of the history of temperature increase alsoapproach the experimental data. This phenomenon further shows the fact that thesolution of the inverse heat transfer problem is not unique. Maybe the distanceinduces the uncertainty in magnitude, the difference gradually becomes obviousbetween the calculated results of the variation of temperature increase and theexperimental data at the reduced distance r=R¼ 1.1.

The response time of measurement instruments, probably causes the uncer-tainty of the experimental data. Table 4 also shows the difficulty to estimate thevalues of sq1, sT1, sq2, and sT2 based on the reference values at the following times:49 s, 64 s, 86 s, and 109 s. The estimated values of sq1, sT1, sq2, and sT2 aresq1¼ 8.1768 s, sT1¼ 22.5628 s, sq2¼ 7.7672 s, and sT2¼ 21.3071 s for set 4–2 andare sq1¼ 6.1462 s, sT1¼ 28.7482 s, sq2¼ 5.7370 s, and sT2¼ 22.5038 s for set 4–3.The difference is large in the value of sT1 between sets 4–2 and set 4–3. Comparingthe reference values of set 4–2 with those of set 4–3 finds only the reference valueh¼ 4.025 at t¼ 109 is replaced with h¼ 4.725 at t¼ 129. From this, the sensitivityof the solution of the inverse heat transfer problem to the reference value is observed.However, the effect of the difference in the estimated results on the calculated resultsof the variation of temperature increase is slight in this case. The comparison inFigure 5 shows that the difference between the calculated results of the variationof temperature increase and the experimental data is more obvious than that shownin Figure 4 at the reduced distance r=R¼ 1.1.

The curves of the history of temperature increase, which are presented inFigures 2–5, approach consistent with the reduced distance increasing and the differ-ence is more obvious with the experimental data at early times. In accordance withthe statement of Rabin [36], the measurement uncertainties would propagate into the


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mathematical solution of heat transfer problem. Rabin [36] further indicated that thetemperature analysis based on a larger region is expected to be significantly moreaccurate and the effects of the measurement uncertainties on the prediction of tem-perature are more obvious at the early times. As a result, the present results arereasonable and the quality of the experimental data is important for the presentproblem. On the other hand, no matter what location the reference values are pickedfrom, the present calculated results of thermal response are close to the experimentaldata at various measurement locations. This phenomenon shows the efficiency ofthe present method for such a problem. In the meantime, the longer the distanceof the measurement location from the center is, the later the beginning of thermalresponse is. The behavior of finite propagation of thermal signal is graduallyreplaced by the diffusion behavior with the penetration distance of thermal signalincreasing. The above results enhance the features of the non-Fourier thermal beha-vior in the experimental data.

The solution of the present problem depends on the location is ensured withthe estimated results shown in Tables 1–4. It is in an agreement with the statementof Ozisik [31]. This work attempts to do a further estimation without the effect oflocation. Therefore, the present work picks a reference value at each measurementlocation at the similar time node, as shown in Table 5. The estimated values ofsq1, sT1, sq2, and sT2 cannot reach the convergence situation with the reference valuespicked at the time node t¼ 33 s for the location r=R¼ 1.1 and at the time nodet¼ 32 s for the other measurement locations. This result is in coordination withthe above discussions and presents that the measurement uncertainties would causethe mathematical difficulties for solving the inverse heat transfer problem. For sets5–2, 5–3, and 5–4 the estimated values of sq1, sT1, sq2, and sT2 are around sq1¼ 12.5 s,sT1¼ 8 s, sq2¼ 12 s, and sT2¼ 17.24 s. In other words, reducing the effect of locationcan lead the estimated results to be more stable in magnitude. It is a pity that Andra

Table 5. Reference values and estimated values based on the data measured at various locations

Set number

Reference values

r(R), t(s) and h(K)

Estimated values of sq1,sT1, sq2, and sT2 (s)

5–1 r=R¼ 1.1, t¼ 33, h¼ 7.35

r=R¼ 1.39, t¼ 32, h¼ 2.8

r=R¼ 1.68, t¼ 32, h¼ 1.225

r=R¼ 1.98, t¼ 32, h¼ 0.875

———–

5–2 r=R¼ 1.1, t¼ 83, h¼ 13.3

r=R¼ 1.39, t¼ 86, h¼ 8.05

r=R¼ 1.68, t¼ 86, h¼ 5.425

r=R¼ 1.98, t¼ 86, h¼ 3.15

sq1¼ 12.3795, sT1¼ 8.0381

sq2¼ 12.0518, sT2¼ 17.2462

5–3 r=R¼ 1.1, t¼ 175, h¼ 17.5

r=R¼ 1.39, t¼ 184, h¼ 12.25

r=R¼ 1.68, t¼ 184, h¼ 8.75

r=R¼ 1.98, t¼ 184, h¼ 6.125

sq1¼ 12.4509, sT1¼ 8.2692

sq2¼ 11.7638, sT2¼ 17.2460

5–4 r=R¼ 1.1, t¼ 270, h¼ 19.775

r=R¼ 1.39, t¼ 274, h¼ 14.0

r=R¼ 1.68, t¼ 274, h¼ 10.5

r=R¼ 1.98, t¼ 274, h¼ 7.875

sq1¼ 12.9278, sT1¼ 7.5094

sq2¼ 12.5131, sT2¼ 17.2431


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et al. [20] did not give the experimental data for the location r=R¼ 1.1 at the time,nodes that were the same with those for the other locations. Otherwise, the estimatedvalues of sq1, sT1, sq2, and sT2 will be better. The corresponding curves of the vari-ation of temperature increase are shown in Figure 6. Similar to that shown inFigures 2–5, the curves are separated at the reduced distance r=R¼ 1.1 and graduallyclose to each other with the reduced distance increasing. Essentially, the non-Fouriereffect is more obvious in small-scales and small times. The temperature variationnearby the heating source would be sensible to the values of sq and sT. In the presentwork, r=R¼ 1.10 is the measurement location nearest the heating source. Therefore,there is a more obvious difference among the curves for r=R¼ 1.10.

CONCLUSION

This work studies the inverse bioheat transfer problem in the bilayered spheri-cal tissue based on the dual-phase-lag model of heat conduction. The phase lag timesare estimated with the experimental data. Due to the difference of the thermophysi-cal parameters between two layers, geometry effects, and measurement uncertainties,solving the present problem introduces complexity and causes the mathematical dif-ficulties. The fact that the calculated results of the history of temperature increaseagree with the experimental data shows the efficiency of the proposed method forsuch a problem. The solution of the inverse non-Fourier heat transfer problem isvery sensitive to the reference values. The measurement uncertainties and themeasurement locations strongly affect the estimated results.


experimental data at various locations.


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solution of an inverse heat conduction problem in a bi-layered spherical tissue

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