Hindawi Publishing CorporationThe Scientific World JournalVolume 2013 Article ID 398386 7 pageshttpdxdoiorg1011552013398386

Research ArticleA Study on the Effect of Metabolic Heat Generationon Biological Tissue Freezing

Sonalika Singh and Sushil Kumar

Department of Applied Mathematics and Humanities S V National Institute of Technology Surat Gujarat 395007 India

Correspondence should be addressed to Sonalika Singh singhsonalika01gmailcom

Received 7 August 2013 Accepted 20 September 2013

Academic Editors A Carotenuto G Tresset and A Zamyatnin

Copyright copy 2013 S Singh and S Kumar 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

The effect of metabolic heat generation on the freezing of biological tissue has been studied Quasi-steady approximation is usedto solve the Pennes bioheat equation in tissues Temperature profile and motion of freezing interfaces are obtained for differentvalues of metabolic heat generation It is observed that metabolism has a significant effect on freezing of biological tissues duringcryosurgery

1 Introduction

The effect of volumetric energy generation on phase changeproblems is important in several applications includingnuclear energy geophysics material processing vivo freezingof biological tissues and solar collectors Cryosurgery is oneof the examples that involves freezing of biological tissue invivo In tissues heat is generated by metabolism and bloodperfusion and the heat that is generated during metabolicprocesses such as growth and energy production of the livingsystem is defined as metabolic heat

Cryosurgery is one of the most important therapies fortumor treatment In cryosurgery extreme cold is used todestroy the tissue for therapeutic purpose It involves localfreezing of tissues for their controlled destruction or removalIn 1960 the concept of injecting liquid nitrogen throughcryoprobe into the target tissue to freeze them from withinwas introduced Several advantages of cryosurgery includethe low invasiveness of the procedure minimal blood flowlocalizing the site of surgery and reducing the recovery andhospitalization time for the patient The earliest model ofheat transfer in the biological tissue is discussed by Pennes[1] Cryosurgery destroys cells and tissues by a complexmechanism containing ice-related factors [2] Advantagesof cryosurgery have initiated interest among researchers toapply it to the field of skin breast prostate liver and lungcancers [3ndash11]

The aim of cryosurgery is to maximize the damage to theundesired tissues within the defined domain and minimizethe injury to the surrounding healthy tissues [9 12] Theparameters which influence the process of cryosurgery arethe coolest temperature in the tissue the duration of frozencycle the rate of freezing front propagation the thawing rateand the freezing-thawing cycles [13ndash21] The factors whichaffect necrosis such as the lowest temperature in the tissueor the rate of freezing front propagation depend on the bio-physical parameters that are present in a given cryosurgicalprocedure some of which may be selected and controlledby the surgeons These parameters include the temperatureand duration of freezing-thawing process the shape and sizeof cryoprobe the heat capacity and thermal conductivity ofthe tissue the rate of blood flow and metabolism in theinvolved tissue [22] Gage et al [16] have studied the effect ofvarying freezing rate duration of freezing and thawing cyclesto investigate the effect of these factors on cell destruction indog skin They suggested that features like fast cooling slowthawing and repetition of the freezethaw cycle should bemodified by maintaining the tissues in the frozen state forseveral minutes and slow thawing

Blood perfusion and metabolic heat generation also havean important effect on heat transfer in tissues [23ndash26] Thecoolest temperature in the tissues is one of the crucialpoints in the process of cryosurgery Moreover the duration

2 The Scientific World Journal

Frozen region

x = 0 Phase change interface (xi) x = L

Unfrozen region

Figure 1 Schematic representation of one-dimensional model

of frozen state also has much influence on the success ofcryosurgery [16 27ndash31] The tissue destruction is increasedwhen it is held in the frozen state in the temperature rangeat which recrystallization occurs [14] A common problem incryosurgery is the extent of post operative bleeding causedby parenchyma fractures and related to the thermal stressinside the target tissue [32] Shi et al [33] have described thelarge volumetric expansion having the primary contributor tolarge stress development during the freezing of biomaterialthrough ice-crystallization The thermal gradient and theeffect of volumetric expansion associated with freezing arethe twomost important factors that induce thermal stress [3334] Consequently the study of the thermal gradient insidethe tissue is also an important issue for the optimization ofcryosurgery

The temperature transients in tumour and normal tissueare useful to say whether the tumour is damaged or not andto minimize the injury to healthy tissues during cryosurgeryThere is a need for a simple analytical solution to evaluatethe effect of parameters like metabolic heat generation onice-crystallization A process of simplification is used insolving a variety of problems which can eliminate the needfor numerical solutions In this paper the method of quasi-steady approximation to study the effect of metabolic heatgeneration on one-dimensional ice-crystallization duringcryosurgery has been used Temperature profiles and motionof freezing interface are obtained for different values ofmetabolic heat generation

2 Mathematical Model

In the present study one-dimensional ice-crystallization inbiological tissue of length 119871 has been considered as shown inFigure 1 Cryoprobe with temperature119879

0= minus196

∘C is appliedat 119909 = 0 while at the other end 119909 = 119871 an adiabatic conditionis used In the frozen region blood perfusion and metabolicheat generation are zero [4 5 10 11 13]

The governing equations for one-dimensional ice-crystallization in biological tissue are as follows

In frozen region

120588119891119888119891

120597119879119891

120597119905= 119896119891

1205972119879119891

1205971199092for 0 le 119909 le 119909

119894 (1)

In unfrozen region

120588119906119888119906

120597119879119906

120597119905= 119896119906

1205972119879119906

1205971199092+ 119902119887+ 119902119898

for 119909119894le 119909 le 119871 (2)

Initial conditions119879119906(119909 0) = 119879

119868= 37∘C

119879119891(119909 0) = 119879

119868= 37∘C

(3)

Boundary conditions

119879119891(0 119905) = 119879

0= minus196

∘C

120597119879119906(119871 119905)

120597119909= 0

(4)

Conditions at phase change interface

119879119891(119909119894 119905) = 119879ph = 119879119906 (119909119894 119905)

119896119891

120597119879119891(119909119894 119905)

120597119909minus 119896119906

120597119879119906(119909119894 119905)

120597119909= 120588119906119897119889119909119894

119889119905

(5)

where 120588 is the density of tissue 119888 the specific heat 119896 thethermal conductivity 119909

119894the interface position of freezing

front 119879 the temperature 119909 the space coordinate 119905 the time119902119887the blood perfusion term 119897 the latent heat of fusion and

119902119898the metabolic heat generation in the tissue Subscripts 119906

and 119891 are for unfrozen and frozen state respectively andph and 119868 are for phase change and initial states respectivelyAssuming the negligible effect of blood perfusion and usingthe following dimensionless variables and constants

120572119891=

119896119891

120588119891119888119891

120572119906=119896119906

120588119906119888119906

120572lowast=120572119906

120572119891

119909lowast=119909

119871 119905

lowast=120572119906119905

1198712Ste 119870

lowast=119896119906

119896119891

119879lowast

119891=

119879119891minus 1198790

119879ph minus 1198790 119879

lowast

119906=119879119906minus 1198790

119879ph minus 1198790

119902lowast

119898=

1199021198981198712

119896119906(119879ph minus 1198790)

(6)

where Ste is the Stefan number defined as Ste = 119888119906(119879phminus1198790)119897

Equations (1) and (2) become

Ste120572lowast120597119879lowast

119891

120597119905lowast=

1205972119879lowast

119891

120597119909lowast2for 0 le 119909lowast le 119909lowast

119894

Ste120597119879lowast

119906

120597119905lowast=1205972119879lowast

119906

120597119909lowast2+ 119902lowast

119898for 119909lowast119894le 119909lowastle 1

(7)

The initial and boundary conditions (3)-(4) become

119879lowast

119891(119909lowast 0) =

119879119868minus 1198790

119879ph minus 1198790

119879lowast

119906(119909lowast 0) =

119879119868minus 1198790

119879ph minus 1198790

(8)

119879lowast

119891(0 119905lowast) = 0 (9)

120597119879lowast

119906(1 119905lowast)

120597119909= 0 (10)

The Scientific World Journal 3

Condition at phase change interface equations (5) is trans-formed to

119879lowast

119891(119909lowast

119894 119905lowast) = 1 = 119879

lowast

119906(119909lowast

119894 119905lowast) (11)

1

119870lowast

120597119879lowast

119891(119909lowast

119894 119905lowast)

120597119909lowastminus120597119879lowast

119906(119909lowast

119894 119905lowast)

120597119909lowast=119889119909lowast

119894

119889119905lowast (12)

3 Quasi-Steady Approximation

Due to nonlinearity of the interface energy equation thereare few exact solutions to the problems with phase changeThe condition at phase change equation is nonlinear becausethe interface velocity 119889119909

119894119889119905 depends on the temperature

gradients In this model the Stefan number is taken smallcompared to the unity A small Stefan number corresponds tothe sensible heat which is small compared to the latent heatThe interface moves slowly for a small Stefan number andthe temperature distribution at each instant corresponds tothat of steady-state Quasi-steady approximation is justifiedfor Ste lt 01 [35 36] Setting Ste = 0 in (7) and we get

1205972119879lowast

119891

120597119909lowast2= 0 (13)

1205972119879lowast

119906

120597119909lowast2+ 119902lowast

119898= 0 (14)

Integrating (13) and using boundary conditions given by (9)and (11) the temperature distribution in frozen region isobtained as

119879lowast

119891(119909lowast 119905lowast) =

119909lowast

119909lowast

119894

(15)

Integrating (14) with boundary conditions given by (10) and(11) the temperature distribution in the unfrozen region isgiven as

119879lowast

119906(119909lowast 119905lowast) =

119902lowast

119898

2(119909lowast2

119894minus 119909lowast2) + 119902lowast

119898(119909lowastminus 119909lowast

119894) + 1 (16)

Substituting the temperature of frozen and unfrozen regionsgiven by (15) and (16) into the condition at phase changeinterface given by (12) we obtain

119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 1

119870lowast119909lowast

119894

=119889119909lowast

119894

119889119905lowast (17)

Integrating (17) and utilizing the initial condition at 119909lowast119894

int

119905lowast

0

119889119905lowast= int

119909lowast

119894

0

119870lowast119909lowast

119894119889119909lowast

119894

119902lowast119898119870lowast119909lowast2

119894minus 119902lowast119898119870lowast119909lowast

119894+ 1 (18)

119905lowast=

1

2119902lowast119898

[ log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

+119902lowast

119898119870lowastint

119909lowast

119894

0

119889119909lowast

119894

119902lowast119898119870lowast119909lowast2

119894minus 119902lowast119898119870lowast119909lowast

119894+ 1]

(19)

In (19) the value of 119902lowast119898is unknown Due to the unknown

value of 119902lowast119898 there arise three possibilities Therefore we

evaluate the above integral considering the three cases whichare mentioned below

Case 1 (119902lowast119898119870lowastgt 4 119905

lowastge 0) From (19) we have

119905lowast=

1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

+119870lowast

2int

119909lowast

119894

0

119889119909lowast

119894

119902lowast119898119870lowast119909lowast2

119894minus 119902lowast119898119870lowast119909lowast

119894+ 1

=1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

+1

2119902lowast119898

int

119909lowast

119894

0

119889119909lowast

119894

119909lowast2

119894minus 119909lowast

119894+ (1119902lowast

119898119870lowast)

=1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816+

1

2119902lowast119898

times int

119909lowast

119894

0

(119889119909lowast

119894(119909lowast

119894minus1

2(1 + radic1 minus

4

119902lowast119898119870lowast)

times119909lowast

119894minus1

2(1 minus radic1 minus

4

119902lowast119898119870lowast))

minus1

)

=1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

+119870lowast

radic119902lowast2119898119870lowast2 minus 4119902lowast

119898119870lowast

times log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2119902lowast

119898119870lowast119909lowast

119894minus 119902lowast

119898119870lowastminus radic119902lowast2119898119870lowast2 minus 4119902lowast

119898119870lowast

2119902lowast119898119870lowast119909lowast

119894minus 119902lowast119898119870lowast + radic119902lowast2

119898119870lowast2 minus 4119902lowast

119898119870lowast

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus119870lowast

radic119902lowast2119898119870lowast2 minus 4119902lowast

119898119870lowast

times log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus119902lowast

119898119870lowastminus radic119902lowast2119898119870lowast2 minus 4119902lowast

119898119870lowast

minus119902lowast119898119870lowast + radic119902lowast2

119898119870lowast2 minus 4119902lowast

119898119870lowast

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(20)Case 2 (119902lowast

119898119870lowastlt 4 119879

lowastge 0) From (19) we have

119905lowast=

1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

+119870lowast

2int

119909lowast

119894

0

119889119909lowast

119894

119902lowast119898119870lowast119909lowast2

119894minus 119902lowast119898119870lowast119909lowast

119894+ 1

=1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

+1

2119902lowast119898

int

119909lowast

119894

0

119889119909lowast

119894

119909lowast2

119894minus 119909lowast

119894+ (1119902lowast

119898119870lowast)

4 The Scientific World Journal

Table 1 Thermal properties of tissues [13 23]

Parameter ValueDensity of unfrozen tissue (kgm3) 1050Density of frozen tissue (kgm3) 921Specific heat of unfrozen tissue (Jkg ∘C) 3770Specific heat of frozen tissue (Jkg ∘C) 1800Thermal conductivity of unfrozen tissue (Wm ∘C) 05Thermal conductivity of frozen tissue (Wm ∘C) 2Latent heat (KJkg) 250The phase change temperature (∘C) minus8Arterial blood temperature (∘C) 37Length of tissue (m) 004

=1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

+1

2119902lowast119898

int

119909lowast

119894

0

119889119909lowast

119894

(119909lowast

119894minus (12))

2+ (radic(1119902lowast

119898119870lowast) minus (14))

2

=1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

+119870lowast

radic4119902lowast119898119870lowast minus 119902lowast2

119898119870lowast2

times tanminus1(2119902lowast

119898119870lowast119909lowast

119894minus 119902lowast

119898119870lowast

radic4119902lowast119898119870lowast minus 119902lowast2

119898119870lowast2

)

minus119870lowast

radic4119902lowast119898119870lowast minus 119902lowast2

119898119870lowast2

times tanminus1(minus119902lowast

119898119870lowast

radic4119902lowast119898119870lowast minus 119902lowast2

119898119870lowast2

)

(21)

Case 3 (119902lowast119898119870lowast= 4 119905

lowastge 0) From (19) we have

119905lowast=

1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

+119870lowast

2int

119909lowast

119894

0

119889119909lowast

119894

119902lowast119898119870lowast119909lowast2

119894minus 119902lowast119898119870lowast119909lowast

119894+ 1

=1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

+1

2119902lowast119898

int

119909lowast

119894

0

119889119909lowast

119894

119909lowast2

119894minus 119909lowast

119894+ (1119902lowast

119898119870lowast)

=1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

minus1

2119902lowast119898119909lowast

119894minus 119902lowast119898

minus1

119902lowast119898

(22)

Table 2 Penetration distance of interface and time taken fordifferent values of 119902lowast

119898

119902lowast

119898119902119898(Wm3) Interface penetration

distance 119909lowast119894

Time 119905lowast

13 763750 1 03596135 793125 1 0400914 822500 1 04571145 851875 1 0540215 881250 1 06807155 910625 1 1000816 940000 04 01494165 989375 04 0667317 998750 03 01554175 1028125 03 0174718 1057500 03 02036185 1086875 03 02561

0 02 04 06 08 1 12 140

01

02

03

04

05

06

07

08

09

1

qlowastm = 13 qlowastm = 16

qlowastm = 17

qlowastm = 18

qlowastm = 14

qlowastm = 15

qlowastm = 135 qlowastm = 165

qlowastm = 175

qlowastm = 185

qlowastm = 145

qlowastm = 155

Non

dim

ensio

nal i

nter

face

pos

ition

(xlowast i

)

Nondimensional time (tlowast)

Figure 2 Interface position with time

4 Results and Discussion

Thevalues of parameters used are given inTable 1 [13 23]Theposition of freezing interface with time for different values of119902lowast

119898is plotted in Figure 2 It is observed that when 119902lowast

119898lt 16

(ie 119902119898lt 940000Wm3) the freezing interface reaches to

the boundary 119909 = 119871 and the time require for solidification ofthe complete tissue increases with the increase in 119902

119898 When

119902lowast

119898ge 16 (ie 119902

119898ge 940000Wm3) the interface does not

reach to the boundary 119909 = 119871 this is because the equilibriumbetween cooling and heat generation is obtained before thefully freezing of tissue and hence freezing interface doesnot move forward Total penetration distance of freezing

The Scientific World Journal 5

0 01 02 03 04 05 06 07 08 09 10

1

2

3

4

5

6

7

8

tlowast = 10008

tlowast = 09715

tlowast = 09290

tlowast = 08588

tlowast = 07160

tlowast = 03886

tlowast = 00017

tlowast = 00094

tlowast = 00335

tlowast = 01132

Non

dim

ensio

nal t

empe

ratu

re (T

lowast)

Nondimensional distance (xlowast)

Figure 3 Temperature distribution at 119902lowast119898= 155

0 02 04 06 08 10

1

2

3

4

5

6

7

8

tlowast = 00017

tlowast = 00097

tlowast = 00365

tlowast = 01494

Non

dim

ensio

nal t

empe

ratu

re (T

lowast)

Nondimensional distance (xlowast)

Figure 4 Temperature distribution at 119902lowast119898= 16

interface and time taken as given in Table 2 show thatfreezing slows down with the increase in metabolic heatgeneration

The temperature profiles are required to optimize thedamage to diseased tissues Temperature profiles for differentvalues of 119902lowast

119898 that is 119902lowast

119898= 155 119902lowast

119898= 16 and 119902lowast

119898= 165

are plotted in Figures 3 4 and 5 respectively From Figure 3it is observed that temperature in tissue decreases with theincrease in time and at 119905lowast = 10008 it is in frozen stateWhile in case of 119902lowast

119898= 16 as shown in Figure 4 a steady

state is obtained at 119905lowast = 01494 and tissue is partially frozenSimilarly for 119902lowast

119898ge 16 the partially frozen state of tissue is

observed in steady state at 119905lowast = 06673 (Figure 5)

0 02 04 06 08 10

1

2

3

4

5

6

7

8

tlowast = 00175

tlowast = 00528

tlowast = 01413tlowast = 06673

Non

dim

ensio

nal t

empe

ratu

re (T

lowast)

Nondimensional distance (xlowast)

Figure 5 Temperature distribution at 119902lowast119898= 165

5 Conclusion

A quasi-steady approximation is used to get the temperatureprofile and position of freezing interface in the biologicaltissue during the freezing of biological tissues for differentvalues of metabolic heat generation It is observed that thefreezing process slows down with increase in metabolic heatgeneration Freezing of the entire tissue is even not possiblewhen the value of metabolic heat generation is extended toa higher value This shows that metabolism has a significanteffect on the freezing of biological tissues during cryosurgeryThe obtained information can be used to optimize thetreatment planning

Nomenclature

119909 Distance (0 le 119909 le 119871) (m)119871 Length of tissue (m)119909119894 Position of freezing interface (m)

119905 Time (s)119879 Temperature (∘C)120588 Density (kgm3)119888 Specific heat (Jkg ∘C)120572 Thermal diffusivity (m2sec)119896 Thermal conductivity (Wm ∘C)119897 Latent heat (KJkg)119902119898 Metabolic heat generation (Wm3)

119902119887 Blood perfusion term (Wm3 ∘C)

Subscripts

ph Phase change119891 Frozen state119906 Unfrozen state119868 Initial state (119905 = 0)119900 Cryoprobe

6 The Scientific World Journal

Superscript

lowast Dimensionless

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Sonalika Singh is thankful to S V National Institute ofTechnology Surat India for providing Senior ResearchFellowship during the preparation of this paper

References

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

[2] J G Baust A A Gage D Clarke J M Baust and RVan Buskirk ldquoCryosurgerymdasha putative approach to molecular-based optimizationrdquo Cryobiology vol 48 no 2 pp 190ndash2042004

[3] E G Kuflik andA A Gage ldquoThe five-year cure rate achieved bycryosurgery for skin cancerrdquo Journal of the American Academyof Dermatology vol 24 no 6 pp 1002ndash1004 1991

[4] A A Gage J A Caruana Jr andMMontes ldquoCritical tempera-ture for skin necrosis in experimental cryosurgeryrdquoCryobiologyvol 19 no 3 pp 273ndash282 1982

[5] S Kumar andV K Katiyar ldquoMathematical modeling of thawingproblem in skin and subcutaneous tissuerdquo in Proceedings of theIFMBE6thWorldCongress of Biomechanics (WCB rsquo10) C T Limand J C H Goh Eds pp 1611ndash1614 August 2010

[6] E D Staren M S Sabel L M Gianakakis et al ldquoCryosurgeryof breast cancerrdquo Archives of Surgery vol 132 no 1 pp 28ndash341997

[7] D K Bahn F Lee R Badalament A Kumar J Greski andM Chernick ldquoTargeted cryoablation of the prostate 7-yearoutcomes in the primary treatment of prostate cancerrdquoUrologyvol 60 no 2 pp 3ndash11 2002

[8] R Adam E Akpinar M Johann F Kunstlinger P Majno andH Bismuth ldquoPlace of cryosurgery in the treatment ofmalignantliver tumorsrdquo Annals of Surgery vol 225 no 1 pp 39ndash50 1997

[9] J C Bischof J Bastacky and B Rubinsky ldquoAn analytical studyof cryosurgery in the lungrdquo ASME Journal of BiomechanicalEngineering vol 114 no 4 pp 467ndash472 1992

[10] S Kumar and V K Katiyar ldquoNumerical study on phase changeheat transfer during combined hyperthermia and cryosurgicaltreatment of lung cancerrdquo Journal of Applied Mathematics andMechanics vol 3 no 3 pp 1ndash17 2007

[11] S Kumar S Ali andV K Katiyar ldquoA parametric study on phasechange heat transfer process during cryosurgery of lung tumorrdquoIndian Journal of Biomechanics no 7-8 pp 210ndash213 2009

[12] K J Chua S K Chou and J C Ho ldquoAn analytical study onthe thermal effects of cryosurgery on selective cell destructionrdquoJournal of Biomechanics vol 40 no 1 pp 100ndash116 2007

[13] S Kumar andV K Katiyar ldquoMathematical modeling of freezingand thawing process in tissues a porous media approachrdquoInternational Journal of AppliedMechanics vol 2 no 3 pp 617ndash633 2010

[14] A A Gage and J Baust ldquoMechanisms of tissue injury incryosurgeryrdquo Cryobiology vol 37 no 3 pp 171ndash186 1998

[15] J J Smith and J Fraser ldquoAn estimation of tissue damage andthermal history in the cryolesionrdquoCryobiology vol 11 no 2 pp139ndash147 1974

[16] A A Gage K Guest M Montes J A Caruana and D AWhalen Jr ldquoEffect of varying freezing and thawing rates inexperimental cryosurgeryrdquo Cryobiology vol 22 no 2 pp 175ndash182 1985

[17] T C Hua and H S Ren Cryobiomedical Techniques SciencePress Beijing China 1994

[18] R H Miller and P Mazur ldquoSurvival of frozen thawed humanred cells as a function of cooling and warming velocitiesrdquoCryobiology vol 13 no 4 pp 404ndash414 1976

[19] A A Gage J A Caruana Jr andMMontes ldquoCritical tempera-ture for skin necrosis in experimental cryosurgeryrdquoCryobiologyvol 19 no 3 pp 273ndash282 1982

[20] J N Tackenberg ldquoCryolumpectomy another option for breastcancerrdquo Nursing vol 20 no 5 pp 32Jndash40J 1990

[21] R W Rand R P Rand and F A Eggerding ldquoCryolumpectomyfor breast cancer an experimental studyrdquo Cryobiology vol 22no 4 pp 307ndash318 1985

[22] R I Andrushkiw ldquoMathematical modeling of freezing frontpropagation in biological tissuerdquo Mathematical and ComputerModelling vol 13 no 10 pp 1ndash9 1990

[23] Y Rabin and A Shitzer ldquoNumerical solution of multidi-mensional freezing problem during cryosurgeryrdquo Journal ofBiomechanical Engineering vol 120 no 1 pp 32ndash37 1998

[24] J C Rewcastle G A Sandison K Muldrew J C Salikenand B J Donnelly ldquoA model for the time dependent three-dimensional thermal distribution within iceballs surroundingmultiple cryoprobesrdquo Medical Physics vol 28 no 6 pp 1125ndash1137 2001

[25] Z-S Deng and J Liu ldquoNumerical simulation of 3-D freezingandheating problems for combined cryosurgery andhyperther-mia therapyrdquoNumerical Heat Transfer A vol 46 no 6 pp 587ndash611 2004

[26] G Zhao H-F Zhang X-J Guo D-W Luo and D-Y GaoldquoEffect of blood flow andmetabolism onmultidimensional heattransfer during cryosurgeryrdquo Medical Engineering and Physicsvol 29 no 2 pp 205ndash215 2007

[27] E D Staren M S Sabel L M Gianakakis et al ldquoCryosurgeryof breast cancerrdquo Archives of Surgery vol 132 no 1 pp 28ndash341997

[28] J C Bischof D Smith P V Pazhayannur C Manivel JHulbert and K P Roberts ldquoCryosurgery of dunning AT-1 ratprostate tumor thermal biophysical and viability response atthe cellular and tissue levelrdquo Cryobiology vol 34 no 1 pp 42ndash69 1997

[29] S M Burge J P Shepherd and R P R Dawber ldquoEffect offreezing the helix and the rim or edge of the human and pigearrdquo Journal of Dermatologic Surgery and Oncology vol 10 no10 pp 816ndash819 1984

[30] A A Gage ldquoExperimental cryogenic injury of the palateobservations pertinent to cryosurgical destruction of tumorsrdquoCryobiology vol 15 no 4 pp 415ndash425 1978

[31] G JacobM N Kurzer and B J Fuller ldquoAn assessment of tumorcell viability after in vitro freezingrdquo Cryobiology vol 22 no 5pp 417ndash426 1985

[32] X M He and J C Bischof ldquoAnalysis of thermal stress incryosurgery of kidneysrdquo Journal of Biomechanical Engineeringvol 127 no 4 pp 656ndash661 2005

The Scientific World Journal 7

[33] X Shi A K Datta and Y Mukherjee ldquoThermal stresses fromlarge volumetric expansion during freezing of biomaterialsrdquoASME Journal of Biomechanical Engineering vol 120 no 6 pp720ndash726 1998

[34] B Rubinsky E G Cravalho and B Mikic ldquoThermal stresses infrozen organsrdquo Cryobiology vol 17 no 1 pp 66ndash73 1980

[35] L M Jiji Heat Conduction Springer Berlin Germany 3rdedition 2009

[36] J Crank The Mathematics of Diffusion Oxford Science Publi-cations 2nd edition 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

Hindawi Publishing Corporationhttpwwwhindawicom

Nucleic AcidsJournal of

Volume 2013

ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Biotechnology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

GenomicsInternational Journal of

Volume 2013

Evolutionary BiologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

Virolog y

ISRN Microbiology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Marine BiologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

BioMed Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Zoology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Signal TransductionJournal of

ISRN Cell Biology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

BioinformaticsAdvances in

PeptidesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Enzyme Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Biochemistry Research International

ISRN Molecular Biology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

2 The Scientific World Journal

Frozen region

x = 0 Phase change interface (xi) x = L

Unfrozen region

Figure 1 Schematic representation of one-dimensional model

of frozen state also has much influence on the success ofcryosurgery [16 27ndash31] The tissue destruction is increasedwhen it is held in the frozen state in the temperature rangeat which recrystallization occurs [14] A common problem incryosurgery is the extent of post operative bleeding causedby parenchyma fractures and related to the thermal stressinside the target tissue [32] Shi et al [33] have described thelarge volumetric expansion having the primary contributor tolarge stress development during the freezing of biomaterialthrough ice-crystallization The thermal gradient and theeffect of volumetric expansion associated with freezing arethe twomost important factors that induce thermal stress [3334] Consequently the study of the thermal gradient insidethe tissue is also an important issue for the optimization ofcryosurgery

The temperature transients in tumour and normal tissueare useful to say whether the tumour is damaged or not andto minimize the injury to healthy tissues during cryosurgeryThere is a need for a simple analytical solution to evaluatethe effect of parameters like metabolic heat generation onice-crystallization A process of simplification is used insolving a variety of problems which can eliminate the needfor numerical solutions In this paper the method of quasi-steady approximation to study the effect of metabolic heatgeneration on one-dimensional ice-crystallization duringcryosurgery has been used Temperature profiles and motionof freezing interface are obtained for different values ofmetabolic heat generation

2 Mathematical Model

In the present study one-dimensional ice-crystallization inbiological tissue of length 119871 has been considered as shown inFigure 1 Cryoprobe with temperature119879

0= minus196

∘C is appliedat 119909 = 0 while at the other end 119909 = 119871 an adiabatic conditionis used In the frozen region blood perfusion and metabolicheat generation are zero [4 5 10 11 13]

The governing equations for one-dimensional ice-crystallization in biological tissue are as follows

In frozen region

120588119891119888119891

120597119879119891

120597119905= 119896119891

1205972119879119891

1205971199092for 0 le 119909 le 119909

119894 (1)

In unfrozen region

120588119906119888119906

120597119879119906

120597119905= 119896119906

1205972119879119906

1205971199092+ 119902119887+ 119902119898

for 119909119894le 119909 le 119871 (2)

Initial conditions119879119906(119909 0) = 119879

119868= 37∘C

119879119891(119909 0) = 119879

119868= 37∘C

(3)

Boundary conditions

119879119891(0 119905) = 119879

0= minus196

∘C

120597119879119906(119871 119905)

120597119909= 0

(4)

Conditions at phase change interface

119879119891(119909119894 119905) = 119879ph = 119879119906 (119909119894 119905)

119896119891

120597119879119891(119909119894 119905)

120597119909minus 119896119906

120597119879119906(119909119894 119905)

120597119909= 120588119906119897119889119909119894

119889119905

(5)

where 120588 is the density of tissue 119888 the specific heat 119896 thethermal conductivity 119909

119894the interface position of freezing

front 119879 the temperature 119909 the space coordinate 119905 the time119902119887the blood perfusion term 119897 the latent heat of fusion and

119902119898the metabolic heat generation in the tissue Subscripts 119906

and 119891 are for unfrozen and frozen state respectively andph and 119868 are for phase change and initial states respectivelyAssuming the negligible effect of blood perfusion and usingthe following dimensionless variables and constants

120572119891=

119896119891

120588119891119888119891

120572119906=119896119906

120588119906119888119906

120572lowast=120572119906

120572119891

119909lowast=119909

119871 119905

lowast=120572119906119905

1198712Ste 119870

lowast=119896119906

119896119891

119879lowast

119891=

119879119891minus 1198790

119879ph minus 1198790 119879

lowast

119906=119879119906minus 1198790

119879ph minus 1198790

119902lowast

119898=

1199021198981198712

119896119906(119879ph minus 1198790)

(6)

where Ste is the Stefan number defined as Ste = 119888119906(119879phminus1198790)119897

Equations (1) and (2) become

Ste120572lowast120597119879lowast

119891

120597119905lowast=

1205972119879lowast

119891

120597119909lowast2for 0 le 119909lowast le 119909lowast

119894

Ste120597119879lowast

119906

120597119905lowast=1205972119879lowast

119906

120597119909lowast2+ 119902lowast

119898for 119909lowast119894le 119909lowastle 1

(7)

The initial and boundary conditions (3)-(4) become

119879lowast

119891(119909lowast 0) =

119879119868minus 1198790

119879ph minus 1198790

119879lowast

119906(119909lowast 0) =

119879119868minus 1198790

119879ph minus 1198790

(8)

119879lowast

119891(0 119905lowast) = 0 (9)

120597119879lowast

119906(1 119905lowast)

120597119909= 0 (10)

The Scientific World Journal 3

Condition at phase change interface equations (5) is trans-formed to

119879lowast

119891(119909lowast

119894 119905lowast) = 1 = 119879

lowast

119906(119909lowast

119894 119905lowast) (11)

1

119870lowast

120597119879lowast

119891(119909lowast

119894 119905lowast)

120597119909lowastminus120597119879lowast

119906(119909lowast

119894 119905lowast)

120597119909lowast=119889119909lowast

119894

119889119905lowast (12)

3 Quasi-Steady Approximation

Due to nonlinearity of the interface energy equation thereare few exact solutions to the problems with phase changeThe condition at phase change equation is nonlinear becausethe interface velocity 119889119909

119894119889119905 depends on the temperature

gradients In this model the Stefan number is taken smallcompared to the unity A small Stefan number corresponds tothe sensible heat which is small compared to the latent heatThe interface moves slowly for a small Stefan number andthe temperature distribution at each instant corresponds tothat of steady-state Quasi-steady approximation is justifiedfor Ste lt 01 [35 36] Setting Ste = 0 in (7) and we get

1205972119879lowast

119891

120597119909lowast2= 0 (13)

1205972119879lowast

119906

120597119909lowast2+ 119902lowast

119898= 0 (14)

Integrating (13) and using boundary conditions given by (9)and (11) the temperature distribution in frozen region isobtained as

119879lowast

119891(119909lowast 119905lowast) =

119909lowast

119909lowast

119894

(15)

Integrating (14) with boundary conditions given by (10) and(11) the temperature distribution in the unfrozen region isgiven as

119879lowast

119906(119909lowast 119905lowast) =

119902lowast

119898

2(119909lowast2

119894minus 119909lowast2) + 119902lowast

119898(119909lowastminus 119909lowast

119894) + 1 (16)

Substituting the temperature of frozen and unfrozen regionsgiven by (15) and (16) into the condition at phase changeinterface given by (12) we obtain

119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 1

119870lowast119909lowast

119894

=119889119909lowast

119894

119889119905lowast (17)

Integrating (17) and utilizing the initial condition at 119909lowast119894

int

119905lowast

0

119889119905lowast= int

119909lowast

119894

0

119870lowast119909lowast

119894119889119909lowast

119894

119902lowast119898119870lowast119909lowast2

119894minus 119902lowast119898119870lowast119909lowast

119894+ 1 (18)

119905lowast=

1

2119902lowast119898

[ log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

+119902lowast

119898119870lowastint

119909lowast

119894

0

119889119909lowast

119894

119902lowast119898119870lowast119909lowast2

119894minus 119902lowast119898119870lowast119909lowast

119894+ 1]

(19)

In (19) the value of 119902lowast119898is unknown Due to the unknown

value of 119902lowast119898 there arise three possibilities Therefore we

evaluate the above integral considering the three cases whichare mentioned below

Case 1 (119902lowast119898119870lowastgt 4 119905

lowastge 0) From (19) we have

119905lowast=

1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

+119870lowast

2int

119909lowast

119894

0

119889119909lowast

119894

119902lowast119898119870lowast119909lowast2

119894minus 119902lowast119898119870lowast119909lowast

119894+ 1

=1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

+1

2119902lowast119898

int

119909lowast

119894

0

119889119909lowast

119894

119909lowast2

119894minus 119909lowast

119894+ (1119902lowast

119898119870lowast)

=1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816+

1

2119902lowast119898

times int

119909lowast

119894

0

(119889119909lowast

119894(119909lowast

119894minus1

2(1 + radic1 minus

4

119902lowast119898119870lowast)

times119909lowast

119894minus1

2(1 minus radic1 minus

4

119902lowast119898119870lowast))

minus1

)

=1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

+119870lowast

radic119902lowast2119898119870lowast2 minus 4119902lowast

119898119870lowast

times log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2119902lowast

119898119870lowast119909lowast

119894minus 119902lowast

119898119870lowastminus radic119902lowast2119898119870lowast2 minus 4119902lowast

119898119870lowast

2119902lowast119898119870lowast119909lowast

119894minus 119902lowast119898119870lowast + radic119902lowast2

119898119870lowast2 minus 4119902lowast

119898119870lowast

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus119870lowast

radic119902lowast2119898119870lowast2 minus 4119902lowast

119898119870lowast

times log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus119902lowast

119898119870lowastminus radic119902lowast2119898119870lowast2 minus 4119902lowast

119898119870lowast

minus119902lowast119898119870lowast + radic119902lowast2

119898119870lowast2 minus 4119902lowast

119898119870lowast

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(20)Case 2 (119902lowast

119898119870lowastlt 4 119879

lowastge 0) From (19) we have

119905lowast=

1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

+119870lowast

2int

119909lowast

119894

0

119889119909lowast

119894

119902lowast119898119870lowast119909lowast2

119894minus 119902lowast119898119870lowast119909lowast

119894+ 1

=1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

+1

2119902lowast119898

int

119909lowast

119894

0

119889119909lowast

119894

119909lowast2

119894minus 119909lowast

119894+ (1119902lowast

119898119870lowast)

4 The Scientific World Journal

Table 1 Thermal properties of tissues [13 23]

Parameter ValueDensity of unfrozen tissue (kgm3) 1050Density of frozen tissue (kgm3) 921Specific heat of unfrozen tissue (Jkg ∘C) 3770Specific heat of frozen tissue (Jkg ∘C) 1800Thermal conductivity of unfrozen tissue (Wm ∘C) 05Thermal conductivity of frozen tissue (Wm ∘C) 2Latent heat (KJkg) 250The phase change temperature (∘C) minus8Arterial blood temperature (∘C) 37Length of tissue (m) 004

=1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

+1

2119902lowast119898

int

119909lowast

119894

0

119889119909lowast

119894

(119909lowast

119894minus (12))

2+ (radic(1119902lowast

119898119870lowast) minus (14))

2

=1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

+119870lowast

radic4119902lowast119898119870lowast minus 119902lowast2

119898119870lowast2

times tanminus1(2119902lowast

119898119870lowast119909lowast

119894minus 119902lowast

119898119870lowast

radic4119902lowast119898119870lowast minus 119902lowast2

119898119870lowast2

)

minus119870lowast

radic4119902lowast119898119870lowast minus 119902lowast2

119898119870lowast2

times tanminus1(minus119902lowast

119898119870lowast

radic4119902lowast119898119870lowast minus 119902lowast2

119898119870lowast2

)

(21)

Case 3 (119902lowast119898119870lowast= 4 119905

lowastge 0) From (19) we have

119905lowast=

1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

+119870lowast

2int

119909lowast

119894

0

119889119909lowast

119894

119902lowast119898119870lowast119909lowast2

119894minus 119902lowast119898119870lowast119909lowast

119894+ 1

=1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

+1

2119902lowast119898

int

119909lowast

119894

0

119889119909lowast

119894

119909lowast2

119894minus 119909lowast

119894+ (1119902lowast

119898119870lowast)

=1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

minus1

2119902lowast119898119909lowast

119894minus 119902lowast119898

minus1

119902lowast119898

(22)

Table 2 Penetration distance of interface and time taken fordifferent values of 119902lowast

119898

119902lowast

119898119902119898(Wm3) Interface penetration

distance 119909lowast119894

Time 119905lowast

13 763750 1 03596135 793125 1 0400914 822500 1 04571145 851875 1 0540215 881250 1 06807155 910625 1 1000816 940000 04 01494165 989375 04 0667317 998750 03 01554175 1028125 03 0174718 1057500 03 02036185 1086875 03 02561

0 02 04 06 08 1 12 140

01

02

03

04

05

06

07

08

09

1

qlowastm = 13 qlowastm = 16

qlowastm = 17

qlowastm = 18

qlowastm = 14

qlowastm = 15

qlowastm = 135 qlowastm = 165

qlowastm = 175

qlowastm = 185

qlowastm = 145

qlowastm = 155

Non

dim

ensio

nal i

nter

face

pos

ition

(xlowast i

)

Nondimensional time (tlowast)

Figure 2 Interface position with time

4 Results and Discussion

Thevalues of parameters used are given inTable 1 [13 23]Theposition of freezing interface with time for different values of119902lowast

119898is plotted in Figure 2 It is observed that when 119902lowast

119898lt 16

(ie 119902119898lt 940000Wm3) the freezing interface reaches to

the boundary 119909 = 119871 and the time require for solidification ofthe complete tissue increases with the increase in 119902

119898 When

119902lowast

119898ge 16 (ie 119902

119898ge 940000Wm3) the interface does not

reach to the boundary 119909 = 119871 this is because the equilibriumbetween cooling and heat generation is obtained before thefully freezing of tissue and hence freezing interface doesnot move forward Total penetration distance of freezing

The Scientific World Journal 5

0 01 02 03 04 05 06 07 08 09 10

1

2

3

4

5

6

7

8

tlowast = 10008

tlowast = 09715

tlowast = 09290

tlowast = 08588

tlowast = 07160

tlowast = 03886

tlowast = 00017

tlowast = 00094

tlowast = 00335

tlowast = 01132

Non

dim

ensio

nal t

empe

ratu

re (T

lowast)

Nondimensional distance (xlowast)

Figure 3 Temperature distribution at 119902lowast119898= 155

0 02 04 06 08 10

1

2

3

4

5

6

7

8

tlowast = 00017

tlowast = 00097

tlowast = 00365

tlowast = 01494

Non

dim

ensio

nal t

empe

ratu

re (T

lowast)

Nondimensional distance (xlowast)

Figure 4 Temperature distribution at 119902lowast119898= 16

interface and time taken as given in Table 2 show thatfreezing slows down with the increase in metabolic heatgeneration

The temperature profiles are required to optimize thedamage to diseased tissues Temperature profiles for differentvalues of 119902lowast

119898 that is 119902lowast

119898= 155 119902lowast

119898= 16 and 119902lowast

119898= 165

are plotted in Figures 3 4 and 5 respectively From Figure 3it is observed that temperature in tissue decreases with theincrease in time and at 119905lowast = 10008 it is in frozen stateWhile in case of 119902lowast

119898= 16 as shown in Figure 4 a steady

state is obtained at 119905lowast = 01494 and tissue is partially frozenSimilarly for 119902lowast

119898ge 16 the partially frozen state of tissue is

observed in steady state at 119905lowast = 06673 (Figure 5)

0 02 04 06 08 10

1

2

3

4

5

6

7

8

tlowast = 00175

tlowast = 00528

tlowast = 01413tlowast = 06673

Non

dim

ensio

nal t

empe

ratu

re (T

lowast)

Nondimensional distance (xlowast)

Figure 5 Temperature distribution at 119902lowast119898= 165

5 Conclusion

A quasi-steady approximation is used to get the temperatureprofile and position of freezing interface in the biologicaltissue during the freezing of biological tissues for differentvalues of metabolic heat generation It is observed that thefreezing process slows down with increase in metabolic heatgeneration Freezing of the entire tissue is even not possiblewhen the value of metabolic heat generation is extended toa higher value This shows that metabolism has a significanteffect on the freezing of biological tissues during cryosurgeryThe obtained information can be used to optimize thetreatment planning

Nomenclature

119909 Distance (0 le 119909 le 119871) (m)119871 Length of tissue (m)119909119894 Position of freezing interface (m)

119905 Time (s)119879 Temperature (∘C)120588 Density (kgm3)119888 Specific heat (Jkg ∘C)120572 Thermal diffusivity (m2sec)119896 Thermal conductivity (Wm ∘C)119897 Latent heat (KJkg)119902119898 Metabolic heat generation (Wm3)

119902119887 Blood perfusion term (Wm3 ∘C)

Subscripts

ph Phase change119891 Frozen state119906 Unfrozen state119868 Initial state (119905 = 0)119900 Cryoprobe

6 The Scientific World Journal

Superscript

lowast Dimensionless

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Sonalika Singh is thankful to S V National Institute ofTechnology Surat India for providing Senior ResearchFellowship during the preparation of this paper

References

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

[2] J G Baust A A Gage D Clarke J M Baust and RVan Buskirk ldquoCryosurgerymdasha putative approach to molecular-based optimizationrdquo Cryobiology vol 48 no 2 pp 190ndash2042004

[3] E G Kuflik andA A Gage ldquoThe five-year cure rate achieved bycryosurgery for skin cancerrdquo Journal of the American Academyof Dermatology vol 24 no 6 pp 1002ndash1004 1991

[4] A A Gage J A Caruana Jr andMMontes ldquoCritical tempera-ture for skin necrosis in experimental cryosurgeryrdquoCryobiologyvol 19 no 3 pp 273ndash282 1982

[5] S Kumar andV K Katiyar ldquoMathematical modeling of thawingproblem in skin and subcutaneous tissuerdquo in Proceedings of theIFMBE6thWorldCongress of Biomechanics (WCB rsquo10) C T Limand J C H Goh Eds pp 1611ndash1614 August 2010

[6] E D Staren M S Sabel L M Gianakakis et al ldquoCryosurgeryof breast cancerrdquo Archives of Surgery vol 132 no 1 pp 28ndash341997

[7] D K Bahn F Lee R Badalament A Kumar J Greski andM Chernick ldquoTargeted cryoablation of the prostate 7-yearoutcomes in the primary treatment of prostate cancerrdquoUrologyvol 60 no 2 pp 3ndash11 2002

[8] R Adam E Akpinar M Johann F Kunstlinger P Majno andH Bismuth ldquoPlace of cryosurgery in the treatment ofmalignantliver tumorsrdquo Annals of Surgery vol 225 no 1 pp 39ndash50 1997

[9] J C Bischof J Bastacky and B Rubinsky ldquoAn analytical studyof cryosurgery in the lungrdquo ASME Journal of BiomechanicalEngineering vol 114 no 4 pp 467ndash472 1992

[10] S Kumar and V K Katiyar ldquoNumerical study on phase changeheat transfer during combined hyperthermia and cryosurgicaltreatment of lung cancerrdquo Journal of Applied Mathematics andMechanics vol 3 no 3 pp 1ndash17 2007

[11] S Kumar S Ali andV K Katiyar ldquoA parametric study on phasechange heat transfer process during cryosurgery of lung tumorrdquoIndian Journal of Biomechanics no 7-8 pp 210ndash213 2009

[12] K J Chua S K Chou and J C Ho ldquoAn analytical study onthe thermal effects of cryosurgery on selective cell destructionrdquoJournal of Biomechanics vol 40 no 1 pp 100ndash116 2007

[13] S Kumar andV K Katiyar ldquoMathematical modeling of freezingand thawing process in tissues a porous media approachrdquoInternational Journal of AppliedMechanics vol 2 no 3 pp 617ndash633 2010

[14] A A Gage and J Baust ldquoMechanisms of tissue injury incryosurgeryrdquo Cryobiology vol 37 no 3 pp 171ndash186 1998

[15] J J Smith and J Fraser ldquoAn estimation of tissue damage andthermal history in the cryolesionrdquoCryobiology vol 11 no 2 pp139ndash147 1974

[16] A A Gage K Guest M Montes J A Caruana and D AWhalen Jr ldquoEffect of varying freezing and thawing rates inexperimental cryosurgeryrdquo Cryobiology vol 22 no 2 pp 175ndash182 1985

[17] T C Hua and H S Ren Cryobiomedical Techniques SciencePress Beijing China 1994

[18] R H Miller and P Mazur ldquoSurvival of frozen thawed humanred cells as a function of cooling and warming velocitiesrdquoCryobiology vol 13 no 4 pp 404ndash414 1976

[19] A A Gage J A Caruana Jr andMMontes ldquoCritical tempera-ture for skin necrosis in experimental cryosurgeryrdquoCryobiologyvol 19 no 3 pp 273ndash282 1982

[20] J N Tackenberg ldquoCryolumpectomy another option for breastcancerrdquo Nursing vol 20 no 5 pp 32Jndash40J 1990

[21] R W Rand R P Rand and F A Eggerding ldquoCryolumpectomyfor breast cancer an experimental studyrdquo Cryobiology vol 22no 4 pp 307ndash318 1985

[22] R I Andrushkiw ldquoMathematical modeling of freezing frontpropagation in biological tissuerdquo Mathematical and ComputerModelling vol 13 no 10 pp 1ndash9 1990

[23] Y Rabin and A Shitzer ldquoNumerical solution of multidi-mensional freezing problem during cryosurgeryrdquo Journal ofBiomechanical Engineering vol 120 no 1 pp 32ndash37 1998

[24] J C Rewcastle G A Sandison K Muldrew J C Salikenand B J Donnelly ldquoA model for the time dependent three-dimensional thermal distribution within iceballs surroundingmultiple cryoprobesrdquo Medical Physics vol 28 no 6 pp 1125ndash1137 2001

[25] Z-S Deng and J Liu ldquoNumerical simulation of 3-D freezingandheating problems for combined cryosurgery andhyperther-mia therapyrdquoNumerical Heat Transfer A vol 46 no 6 pp 587ndash611 2004

[26] G Zhao H-F Zhang X-J Guo D-W Luo and D-Y GaoldquoEffect of blood flow andmetabolism onmultidimensional heattransfer during cryosurgeryrdquo Medical Engineering and Physicsvol 29 no 2 pp 205ndash215 2007

[27] E D Staren M S Sabel L M Gianakakis et al ldquoCryosurgeryof breast cancerrdquo Archives of Surgery vol 132 no 1 pp 28ndash341997

[28] J C Bischof D Smith P V Pazhayannur C Manivel JHulbert and K P Roberts ldquoCryosurgery of dunning AT-1 ratprostate tumor thermal biophysical and viability response atthe cellular and tissue levelrdquo Cryobiology vol 34 no 1 pp 42ndash69 1997

[29] S M Burge J P Shepherd and R P R Dawber ldquoEffect offreezing the helix and the rim or edge of the human and pigearrdquo Journal of Dermatologic Surgery and Oncology vol 10 no10 pp 816ndash819 1984

[30] A A Gage ldquoExperimental cryogenic injury of the palateobservations pertinent to cryosurgical destruction of tumorsrdquoCryobiology vol 15 no 4 pp 415ndash425 1978

[31] G JacobM N Kurzer and B J Fuller ldquoAn assessment of tumorcell viability after in vitro freezingrdquo Cryobiology vol 22 no 5pp 417ndash426 1985

[32] X M He and J C Bischof ldquoAnalysis of thermal stress incryosurgery of kidneysrdquo Journal of Biomechanical Engineeringvol 127 no 4 pp 656ndash661 2005

The Scientific World Journal 7

[33] X Shi A K Datta and Y Mukherjee ldquoThermal stresses fromlarge volumetric expansion during freezing of biomaterialsrdquoASME Journal of Biomechanical Engineering vol 120 no 6 pp720ndash726 1998

[34] B Rubinsky E G Cravalho and B Mikic ldquoThermal stresses infrozen organsrdquo Cryobiology vol 17 no 1 pp 66ndash73 1980

[35] L M Jiji Heat Conduction Springer Berlin Germany 3rdedition 2009

[36] J Crank The Mathematics of Diffusion Oxford Science Publi-cations 2nd edition 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

Hindawi Publishing Corporationhttpwwwhindawicom

Nucleic AcidsJournal of

Volume 2013

ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Biotechnology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

GenomicsInternational Journal of

Volume 2013

Evolutionary BiologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

Virolog y

ISRN Microbiology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Marine BiologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

BioMed Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Zoology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Signal TransductionJournal of

ISRN Cell Biology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

BioinformaticsAdvances in

PeptidesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Enzyme Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Biochemistry Research International

ISRN Molecular Biology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

The Scientific World Journal 3

Condition at phase change interface equations (5) is trans-formed to

119879lowast

119891(119909lowast

119894 119905lowast) = 1 = 119879

lowast

119906(119909lowast

119894 119905lowast) (11)

1

119870lowast

120597119879lowast

119891(119909lowast

119894 119905lowast)

120597119909lowastminus120597119879lowast

119906(119909lowast

119894 119905lowast)

120597119909lowast=119889119909lowast

119894

119889119905lowast (12)

3 Quasi-Steady Approximation

Due to nonlinearity of the interface energy equation thereare few exact solutions to the problems with phase changeThe condition at phase change equation is nonlinear becausethe interface velocity 119889119909

119894119889119905 depends on the temperature

gradients In this model the Stefan number is taken smallcompared to the unity A small Stefan number corresponds tothe sensible heat which is small compared to the latent heatThe interface moves slowly for a small Stefan number andthe temperature distribution at each instant corresponds tothat of steady-state Quasi-steady approximation is justifiedfor Ste lt 01 [35 36] Setting Ste = 0 in (7) and we get

1205972119879lowast

119891

120597119909lowast2= 0 (13)

1205972119879lowast

119906

120597119909lowast2+ 119902lowast

119898= 0 (14)

Integrating (13) and using boundary conditions given by (9)and (11) the temperature distribution in frozen region isobtained as

119879lowast

119891(119909lowast 119905lowast) =

119909lowast

119909lowast

119894

(15)

Integrating (14) with boundary conditions given by (10) and(11) the temperature distribution in the unfrozen region isgiven as

119879lowast

119906(119909lowast 119905lowast) =

119902lowast

119898

2(119909lowast2

119894minus 119909lowast2) + 119902lowast

119898(119909lowastminus 119909lowast

119894) + 1 (16)

Substituting the temperature of frozen and unfrozen regionsgiven by (15) and (16) into the condition at phase changeinterface given by (12) we obtain

119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 1

119870lowast119909lowast

119894

=119889119909lowast

119894

119889119905lowast (17)

Integrating (17) and utilizing the initial condition at 119909lowast119894

int

119905lowast

0

119889119905lowast= int

119909lowast

119894

0

119870lowast119909lowast

119894119889119909lowast

119894

119902lowast119898119870lowast119909lowast2

119894minus 119902lowast119898119870lowast119909lowast

119894+ 1 (18)

119905lowast=

1

2119902lowast119898

[ log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

+119902lowast

119898119870lowastint

119909lowast

119894

0

119889119909lowast

119894

119902lowast119898119870lowast119909lowast2

119894minus 119902lowast119898119870lowast119909lowast

119894+ 1]

(19)

In (19) the value of 119902lowast119898is unknown Due to the unknown

value of 119902lowast119898 there arise three possibilities Therefore we

evaluate the above integral considering the three cases whichare mentioned below

Case 1 (119902lowast119898119870lowastgt 4 119905

lowastge 0) From (19) we have

119905lowast=

1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

+119870lowast

2int

119909lowast

119894

0

119889119909lowast

119894

119902lowast119898119870lowast119909lowast2

119894minus 119902lowast119898119870lowast119909lowast

119894+ 1

=1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

+1

2119902lowast119898

int

119909lowast

119894

0

119889119909lowast

119894

119909lowast2

119894minus 119909lowast

119894+ (1119902lowast

119898119870lowast)

=1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816+

1

2119902lowast119898

times int

119909lowast

119894

0

(119889119909lowast

119894(119909lowast

119894minus1

2(1 + radic1 minus

4

119902lowast119898119870lowast)

times119909lowast

119894minus1

2(1 minus radic1 minus

4

119902lowast119898119870lowast))

minus1

)

=1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

+119870lowast

radic119902lowast2119898119870lowast2 minus 4119902lowast

119898119870lowast

times log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2119902lowast

119898119870lowast119909lowast

119894minus 119902lowast

119898119870lowastminus radic119902lowast2119898119870lowast2 minus 4119902lowast

119898119870lowast

2119902lowast119898119870lowast119909lowast

119894minus 119902lowast119898119870lowast + radic119902lowast2

119898119870lowast2 minus 4119902lowast

119898119870lowast

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus119870lowast

radic119902lowast2119898119870lowast2 minus 4119902lowast

119898119870lowast

times log

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus119902lowast

119898119870lowastminus radic119902lowast2119898119870lowast2 minus 4119902lowast

119898119870lowast

minus119902lowast119898119870lowast + radic119902lowast2

119898119870lowast2 minus 4119902lowast

119898119870lowast

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(20)Case 2 (119902lowast

119898119870lowastlt 4 119879

lowastge 0) From (19) we have

119905lowast=

1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

+119870lowast

2int

119909lowast

119894

0

119889119909lowast

119894

119902lowast119898119870lowast119909lowast2

119894minus 119902lowast119898119870lowast119909lowast

119894+ 1

=1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

+1

2119902lowast119898

int

119909lowast

119894

0

119889119909lowast

119894

119909lowast2

119894minus 119909lowast

119894+ (1119902lowast

119898119870lowast)

4 The Scientific World Journal

Table 1 Thermal properties of tissues [13 23]

Parameter ValueDensity of unfrozen tissue (kgm3) 1050Density of frozen tissue (kgm3) 921Specific heat of unfrozen tissue (Jkg ∘C) 3770Specific heat of frozen tissue (Jkg ∘C) 1800Thermal conductivity of unfrozen tissue (Wm ∘C) 05Thermal conductivity of frozen tissue (Wm ∘C) 2Latent heat (KJkg) 250The phase change temperature (∘C) minus8Arterial blood temperature (∘C) 37Length of tissue (m) 004

=1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

+1

2119902lowast119898

int

119909lowast

119894

0

119889119909lowast

119894

(119909lowast

119894minus (12))

2+ (radic(1119902lowast

119898119870lowast) minus (14))

2

=1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

+119870lowast

radic4119902lowast119898119870lowast minus 119902lowast2

119898119870lowast2

times tanminus1(2119902lowast

119898119870lowast119909lowast

119894minus 119902lowast

119898119870lowast

radic4119902lowast119898119870lowast minus 119902lowast2

119898119870lowast2

)

minus119870lowast

radic4119902lowast119898119870lowast minus 119902lowast2

119898119870lowast2

times tanminus1(minus119902lowast

119898119870lowast

radic4119902lowast119898119870lowast minus 119902lowast2

119898119870lowast2

)

(21)

Case 3 (119902lowast119898119870lowast= 4 119905

lowastge 0) From (19) we have

119905lowast=

1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

+119870lowast

2int

119909lowast

119894

0

119889119909lowast

119894

119902lowast119898119870lowast119909lowast2

119894minus 119902lowast119898119870lowast119909lowast

119894+ 1

=1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

+1

2119902lowast119898

int

119909lowast

119894

0

119889119909lowast

119894

119909lowast2

119894minus 119909lowast

119894+ (1119902lowast

119898119870lowast)

=1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

minus1

2119902lowast119898119909lowast

119894minus 119902lowast119898

minus1

119902lowast119898

(22)

Table 2 Penetration distance of interface and time taken fordifferent values of 119902lowast

119898

119902lowast

119898119902119898(Wm3) Interface penetration

distance 119909lowast119894

Time 119905lowast

13 763750 1 03596135 793125 1 0400914 822500 1 04571145 851875 1 0540215 881250 1 06807155 910625 1 1000816 940000 04 01494165 989375 04 0667317 998750 03 01554175 1028125 03 0174718 1057500 03 02036185 1086875 03 02561

0 02 04 06 08 1 12 140

01

02

03

04

05

06

07

08

09

1

qlowastm = 13 qlowastm = 16

qlowastm = 17

qlowastm = 18

qlowastm = 14

qlowastm = 15

qlowastm = 135 qlowastm = 165

qlowastm = 175

qlowastm = 185

qlowastm = 145

qlowastm = 155

Non

dim

ensio

nal i

nter

face

pos

ition

(xlowast i

)

Nondimensional time (tlowast)

Figure 2 Interface position with time

4 Results and Discussion

Thevalues of parameters used are given inTable 1 [13 23]Theposition of freezing interface with time for different values of119902lowast

119898is plotted in Figure 2 It is observed that when 119902lowast

119898lt 16

(ie 119902119898lt 940000Wm3) the freezing interface reaches to

the boundary 119909 = 119871 and the time require for solidification ofthe complete tissue increases with the increase in 119902

119898 When

119902lowast

119898ge 16 (ie 119902

119898ge 940000Wm3) the interface does not

reach to the boundary 119909 = 119871 this is because the equilibriumbetween cooling and heat generation is obtained before thefully freezing of tissue and hence freezing interface doesnot move forward Total penetration distance of freezing

The Scientific World Journal 5

0 01 02 03 04 05 06 07 08 09 10

1

2

3

4

5

6

7

8

tlowast = 10008

tlowast = 09715

tlowast = 09290

tlowast = 08588

tlowast = 07160

tlowast = 03886

tlowast = 00017

tlowast = 00094

tlowast = 00335

tlowast = 01132

Non

dim

ensio

nal t

empe

ratu

re (T

lowast)

Nondimensional distance (xlowast)

Figure 3 Temperature distribution at 119902lowast119898= 155

0 02 04 06 08 10

1

2

3

4

5

6

7

8

tlowast = 00017

tlowast = 00097

tlowast = 00365

tlowast = 01494

Non

dim

ensio

nal t

empe

ratu

re (T

lowast)

Nondimensional distance (xlowast)

Figure 4 Temperature distribution at 119902lowast119898= 16

interface and time taken as given in Table 2 show thatfreezing slows down with the increase in metabolic heatgeneration

The temperature profiles are required to optimize thedamage to diseased tissues Temperature profiles for differentvalues of 119902lowast

119898 that is 119902lowast

119898= 155 119902lowast

119898= 16 and 119902lowast

119898= 165

are plotted in Figures 3 4 and 5 respectively From Figure 3it is observed that temperature in tissue decreases with theincrease in time and at 119905lowast = 10008 it is in frozen stateWhile in case of 119902lowast

119898= 16 as shown in Figure 4 a steady

state is obtained at 119905lowast = 01494 and tissue is partially frozenSimilarly for 119902lowast

119898ge 16 the partially frozen state of tissue is

observed in steady state at 119905lowast = 06673 (Figure 5)

0 02 04 06 08 10

1

2

3

4

5

6

7

8

tlowast = 00175

tlowast = 00528

tlowast = 01413tlowast = 06673

Non

dim

ensio

nal t

empe

ratu

re (T

lowast)

Nondimensional distance (xlowast)

Figure 5 Temperature distribution at 119902lowast119898= 165

5 Conclusion

A quasi-steady approximation is used to get the temperatureprofile and position of freezing interface in the biologicaltissue during the freezing of biological tissues for differentvalues of metabolic heat generation It is observed that thefreezing process slows down with increase in metabolic heatgeneration Freezing of the entire tissue is even not possiblewhen the value of metabolic heat generation is extended toa higher value This shows that metabolism has a significanteffect on the freezing of biological tissues during cryosurgeryThe obtained information can be used to optimize thetreatment planning

Nomenclature

119909 Distance (0 le 119909 le 119871) (m)119871 Length of tissue (m)119909119894 Position of freezing interface (m)

119905 Time (s)119879 Temperature (∘C)120588 Density (kgm3)119888 Specific heat (Jkg ∘C)120572 Thermal diffusivity (m2sec)119896 Thermal conductivity (Wm ∘C)119897 Latent heat (KJkg)119902119898 Metabolic heat generation (Wm3)

119902119887 Blood perfusion term (Wm3 ∘C)

Subscripts

ph Phase change119891 Frozen state119906 Unfrozen state119868 Initial state (119905 = 0)119900 Cryoprobe

6 The Scientific World Journal

Superscript

lowast Dimensionless

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Sonalika Singh is thankful to S V National Institute ofTechnology Surat India for providing Senior ResearchFellowship during the preparation of this paper

References

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

[2] J G Baust A A Gage D Clarke J M Baust and RVan Buskirk ldquoCryosurgerymdasha putative approach to molecular-based optimizationrdquo Cryobiology vol 48 no 2 pp 190ndash2042004

[3] E G Kuflik andA A Gage ldquoThe five-year cure rate achieved bycryosurgery for skin cancerrdquo Journal of the American Academyof Dermatology vol 24 no 6 pp 1002ndash1004 1991

[4] A A Gage J A Caruana Jr andMMontes ldquoCritical tempera-ture for skin necrosis in experimental cryosurgeryrdquoCryobiologyvol 19 no 3 pp 273ndash282 1982

[5] S Kumar andV K Katiyar ldquoMathematical modeling of thawingproblem in skin and subcutaneous tissuerdquo in Proceedings of theIFMBE6thWorldCongress of Biomechanics (WCB rsquo10) C T Limand J C H Goh Eds pp 1611ndash1614 August 2010

[6] E D Staren M S Sabel L M Gianakakis et al ldquoCryosurgeryof breast cancerrdquo Archives of Surgery vol 132 no 1 pp 28ndash341997

[7] D K Bahn F Lee R Badalament A Kumar J Greski andM Chernick ldquoTargeted cryoablation of the prostate 7-yearoutcomes in the primary treatment of prostate cancerrdquoUrologyvol 60 no 2 pp 3ndash11 2002

[8] R Adam E Akpinar M Johann F Kunstlinger P Majno andH Bismuth ldquoPlace of cryosurgery in the treatment ofmalignantliver tumorsrdquo Annals of Surgery vol 225 no 1 pp 39ndash50 1997

[9] J C Bischof J Bastacky and B Rubinsky ldquoAn analytical studyof cryosurgery in the lungrdquo ASME Journal of BiomechanicalEngineering vol 114 no 4 pp 467ndash472 1992

[10] S Kumar and V K Katiyar ldquoNumerical study on phase changeheat transfer during combined hyperthermia and cryosurgicaltreatment of lung cancerrdquo Journal of Applied Mathematics andMechanics vol 3 no 3 pp 1ndash17 2007

[11] S Kumar S Ali andV K Katiyar ldquoA parametric study on phasechange heat transfer process during cryosurgery of lung tumorrdquoIndian Journal of Biomechanics no 7-8 pp 210ndash213 2009

[12] K J Chua S K Chou and J C Ho ldquoAn analytical study onthe thermal effects of cryosurgery on selective cell destructionrdquoJournal of Biomechanics vol 40 no 1 pp 100ndash116 2007

[13] S Kumar andV K Katiyar ldquoMathematical modeling of freezingand thawing process in tissues a porous media approachrdquoInternational Journal of AppliedMechanics vol 2 no 3 pp 617ndash633 2010

[14] A A Gage and J Baust ldquoMechanisms of tissue injury incryosurgeryrdquo Cryobiology vol 37 no 3 pp 171ndash186 1998

[15] J J Smith and J Fraser ldquoAn estimation of tissue damage andthermal history in the cryolesionrdquoCryobiology vol 11 no 2 pp139ndash147 1974

[16] A A Gage K Guest M Montes J A Caruana and D AWhalen Jr ldquoEffect of varying freezing and thawing rates inexperimental cryosurgeryrdquo Cryobiology vol 22 no 2 pp 175ndash182 1985

[17] T C Hua and H S Ren Cryobiomedical Techniques SciencePress Beijing China 1994

[18] R H Miller and P Mazur ldquoSurvival of frozen thawed humanred cells as a function of cooling and warming velocitiesrdquoCryobiology vol 13 no 4 pp 404ndash414 1976

[19] A A Gage J A Caruana Jr andMMontes ldquoCritical tempera-ture for skin necrosis in experimental cryosurgeryrdquoCryobiologyvol 19 no 3 pp 273ndash282 1982

[20] J N Tackenberg ldquoCryolumpectomy another option for breastcancerrdquo Nursing vol 20 no 5 pp 32Jndash40J 1990

[21] R W Rand R P Rand and F A Eggerding ldquoCryolumpectomyfor breast cancer an experimental studyrdquo Cryobiology vol 22no 4 pp 307ndash318 1985

[22] R I Andrushkiw ldquoMathematical modeling of freezing frontpropagation in biological tissuerdquo Mathematical and ComputerModelling vol 13 no 10 pp 1ndash9 1990

[23] Y Rabin and A Shitzer ldquoNumerical solution of multidi-mensional freezing problem during cryosurgeryrdquo Journal ofBiomechanical Engineering vol 120 no 1 pp 32ndash37 1998

[24] J C Rewcastle G A Sandison K Muldrew J C Salikenand B J Donnelly ldquoA model for the time dependent three-dimensional thermal distribution within iceballs surroundingmultiple cryoprobesrdquo Medical Physics vol 28 no 6 pp 1125ndash1137 2001

[25] Z-S Deng and J Liu ldquoNumerical simulation of 3-D freezingandheating problems for combined cryosurgery andhyperther-mia therapyrdquoNumerical Heat Transfer A vol 46 no 6 pp 587ndash611 2004

[26] G Zhao H-F Zhang X-J Guo D-W Luo and D-Y GaoldquoEffect of blood flow andmetabolism onmultidimensional heattransfer during cryosurgeryrdquo Medical Engineering and Physicsvol 29 no 2 pp 205ndash215 2007

[27] E D Staren M S Sabel L M Gianakakis et al ldquoCryosurgeryof breast cancerrdquo Archives of Surgery vol 132 no 1 pp 28ndash341997

[28] J C Bischof D Smith P V Pazhayannur C Manivel JHulbert and K P Roberts ldquoCryosurgery of dunning AT-1 ratprostate tumor thermal biophysical and viability response atthe cellular and tissue levelrdquo Cryobiology vol 34 no 1 pp 42ndash69 1997

[29] S M Burge J P Shepherd and R P R Dawber ldquoEffect offreezing the helix and the rim or edge of the human and pigearrdquo Journal of Dermatologic Surgery and Oncology vol 10 no10 pp 816ndash819 1984

[30] A A Gage ldquoExperimental cryogenic injury of the palateobservations pertinent to cryosurgical destruction of tumorsrdquoCryobiology vol 15 no 4 pp 415ndash425 1978

[31] G JacobM N Kurzer and B J Fuller ldquoAn assessment of tumorcell viability after in vitro freezingrdquo Cryobiology vol 22 no 5pp 417ndash426 1985

[32] X M He and J C Bischof ldquoAnalysis of thermal stress incryosurgery of kidneysrdquo Journal of Biomechanical Engineeringvol 127 no 4 pp 656ndash661 2005

The Scientific World Journal 7

[33] X Shi A K Datta and Y Mukherjee ldquoThermal stresses fromlarge volumetric expansion during freezing of biomaterialsrdquoASME Journal of Biomechanical Engineering vol 120 no 6 pp720ndash726 1998

[34] B Rubinsky E G Cravalho and B Mikic ldquoThermal stresses infrozen organsrdquo Cryobiology vol 17 no 1 pp 66ndash73 1980

[35] L M Jiji Heat Conduction Springer Berlin Germany 3rdedition 2009

[36] J Crank The Mathematics of Diffusion Oxford Science Publi-cations 2nd edition 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

Hindawi Publishing Corporationhttpwwwhindawicom

Nucleic AcidsJournal of

Volume 2013

ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Biotechnology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

GenomicsInternational Journal of

Volume 2013

Evolutionary BiologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

Virolog y

ISRN Microbiology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Marine BiologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

BioMed Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Zoology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Signal TransductionJournal of

ISRN Cell Biology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

BioinformaticsAdvances in

PeptidesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Enzyme Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Biochemistry Research International

ISRN Molecular Biology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

4 The Scientific World Journal

Table 1 Thermal properties of tissues [13 23]

Parameter ValueDensity of unfrozen tissue (kgm3) 1050Density of frozen tissue (kgm3) 921Specific heat of unfrozen tissue (Jkg ∘C) 3770Specific heat of frozen tissue (Jkg ∘C) 1800Thermal conductivity of unfrozen tissue (Wm ∘C) 05Thermal conductivity of frozen tissue (Wm ∘C) 2Latent heat (KJkg) 250The phase change temperature (∘C) minus8Arterial blood temperature (∘C) 37Length of tissue (m) 004

=1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

+1

2119902lowast119898

int

119909lowast

119894

0

119889119909lowast

119894

(119909lowast

119894minus (12))

2+ (radic(1119902lowast

119898119870lowast) minus (14))

2

=1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

+119870lowast

radic4119902lowast119898119870lowast minus 119902lowast2

119898119870lowast2

times tanminus1(2119902lowast

119898119870lowast119909lowast

119894minus 119902lowast

119898119870lowast

radic4119902lowast119898119870lowast minus 119902lowast2

119898119870lowast2

)

minus119870lowast

radic4119902lowast119898119870lowast minus 119902lowast2

119898119870lowast2

times tanminus1(minus119902lowast

119898119870lowast

radic4119902lowast119898119870lowast minus 119902lowast2

119898119870lowast2

)

(21)

Case 3 (119902lowast119898119870lowast= 4 119905

lowastge 0) From (19) we have

119905lowast=

1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

+119870lowast

2int

119909lowast

119894

0

119889119909lowast

119894

119902lowast119898119870lowast119909lowast2

119894minus 119902lowast119898119870lowast119909lowast

119894+ 1

=1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

+1

2119902lowast119898

int

119909lowast

119894

0

119889119909lowast

119894

119909lowast2

119894minus 119909lowast

119894+ (1119902lowast

119898119870lowast)

=1

2119902lowast119898

log 10038161003816100381610038161003816119902lowast

119898119870lowast119909lowast2

119894minus 119902lowast

119898119870lowast119909lowast

119894+ 110038161003816100381610038161003816

minus1

2119902lowast119898119909lowast

119894minus 119902lowast119898

minus1

119902lowast119898

(22)

Table 2 Penetration distance of interface and time taken fordifferent values of 119902lowast

119898

119902lowast

119898119902119898(Wm3) Interface penetration

distance 119909lowast119894

Time 119905lowast

13 763750 1 03596135 793125 1 0400914 822500 1 04571145 851875 1 0540215 881250 1 06807155 910625 1 1000816 940000 04 01494165 989375 04 0667317 998750 03 01554175 1028125 03 0174718 1057500 03 02036185 1086875 03 02561

0 02 04 06 08 1 12 140

01

02

03

04

05

06

07

08

09

1

qlowastm = 13 qlowastm = 16

qlowastm = 17

qlowastm = 18

qlowastm = 14

qlowastm = 15

qlowastm = 135 qlowastm = 165

qlowastm = 175

qlowastm = 185

qlowastm = 145

qlowastm = 155

Non

dim

ensio

nal i

nter

face

pos

ition

(xlowast i

)

Nondimensional time (tlowast)

Figure 2 Interface position with time

4 Results and Discussion

Thevalues of parameters used are given inTable 1 [13 23]Theposition of freezing interface with time for different values of119902lowast

119898is plotted in Figure 2 It is observed that when 119902lowast

119898lt 16

(ie 119902119898lt 940000Wm3) the freezing interface reaches to

the boundary 119909 = 119871 and the time require for solidification ofthe complete tissue increases with the increase in 119902

119898 When

119902lowast

119898ge 16 (ie 119902

119898ge 940000Wm3) the interface does not

reach to the boundary 119909 = 119871 this is because the equilibriumbetween cooling and heat generation is obtained before thefully freezing of tissue and hence freezing interface doesnot move forward Total penetration distance of freezing

The Scientific World Journal 5

0 01 02 03 04 05 06 07 08 09 10

1

2

3

4

5

6

7

8

tlowast = 10008

tlowast = 09715

tlowast = 09290

tlowast = 08588

tlowast = 07160

tlowast = 03886

tlowast = 00017

tlowast = 00094

tlowast = 00335

tlowast = 01132

Non

dim

ensio

nal t

empe

ratu

re (T

lowast)

Nondimensional distance (xlowast)

Figure 3 Temperature distribution at 119902lowast119898= 155

0 02 04 06 08 10

1

2

3

4

5

6

7

8

tlowast = 00017

tlowast = 00097

tlowast = 00365

tlowast = 01494

Non

dim

ensio

nal t

empe

ratu

re (T

lowast)

Nondimensional distance (xlowast)

Figure 4 Temperature distribution at 119902lowast119898= 16

interface and time taken as given in Table 2 show thatfreezing slows down with the increase in metabolic heatgeneration

The temperature profiles are required to optimize thedamage to diseased tissues Temperature profiles for differentvalues of 119902lowast

119898 that is 119902lowast

119898= 155 119902lowast

119898= 16 and 119902lowast

119898= 165

are plotted in Figures 3 4 and 5 respectively From Figure 3it is observed that temperature in tissue decreases with theincrease in time and at 119905lowast = 10008 it is in frozen stateWhile in case of 119902lowast

119898= 16 as shown in Figure 4 a steady

state is obtained at 119905lowast = 01494 and tissue is partially frozenSimilarly for 119902lowast

119898ge 16 the partially frozen state of tissue is

observed in steady state at 119905lowast = 06673 (Figure 5)

0 02 04 06 08 10

1

2

3

4

5

6

7

8

tlowast = 00175

tlowast = 00528

tlowast = 01413tlowast = 06673

Non

dim

ensio

nal t

empe

ratu

re (T

lowast)

Nondimensional distance (xlowast)

Figure 5 Temperature distribution at 119902lowast119898= 165

5 Conclusion

A quasi-steady approximation is used to get the temperatureprofile and position of freezing interface in the biologicaltissue during the freezing of biological tissues for differentvalues of metabolic heat generation It is observed that thefreezing process slows down with increase in metabolic heatgeneration Freezing of the entire tissue is even not possiblewhen the value of metabolic heat generation is extended toa higher value This shows that metabolism has a significanteffect on the freezing of biological tissues during cryosurgeryThe obtained information can be used to optimize thetreatment planning

Nomenclature

119909 Distance (0 le 119909 le 119871) (m)119871 Length of tissue (m)119909119894 Position of freezing interface (m)

119905 Time (s)119879 Temperature (∘C)120588 Density (kgm3)119888 Specific heat (Jkg ∘C)120572 Thermal diffusivity (m2sec)119896 Thermal conductivity (Wm ∘C)119897 Latent heat (KJkg)119902119898 Metabolic heat generation (Wm3)

119902119887 Blood perfusion term (Wm3 ∘C)

Subscripts

ph Phase change119891 Frozen state119906 Unfrozen state119868 Initial state (119905 = 0)119900 Cryoprobe

6 The Scientific World Journal

Superscript

lowast Dimensionless

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Sonalika Singh is thankful to S V National Institute ofTechnology Surat India for providing Senior ResearchFellowship during the preparation of this paper

References

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

[2] J G Baust A A Gage D Clarke J M Baust and RVan Buskirk ldquoCryosurgerymdasha putative approach to molecular-based optimizationrdquo Cryobiology vol 48 no 2 pp 190ndash2042004

[3] E G Kuflik andA A Gage ldquoThe five-year cure rate achieved bycryosurgery for skin cancerrdquo Journal of the American Academyof Dermatology vol 24 no 6 pp 1002ndash1004 1991

[4] A A Gage J A Caruana Jr andMMontes ldquoCritical tempera-ture for skin necrosis in experimental cryosurgeryrdquoCryobiologyvol 19 no 3 pp 273ndash282 1982

[5] S Kumar andV K Katiyar ldquoMathematical modeling of thawingproblem in skin and subcutaneous tissuerdquo in Proceedings of theIFMBE6thWorldCongress of Biomechanics (WCB rsquo10) C T Limand J C H Goh Eds pp 1611ndash1614 August 2010

[6] E D Staren M S Sabel L M Gianakakis et al ldquoCryosurgeryof breast cancerrdquo Archives of Surgery vol 132 no 1 pp 28ndash341997

[7] D K Bahn F Lee R Badalament A Kumar J Greski andM Chernick ldquoTargeted cryoablation of the prostate 7-yearoutcomes in the primary treatment of prostate cancerrdquoUrologyvol 60 no 2 pp 3ndash11 2002

[8] R Adam E Akpinar M Johann F Kunstlinger P Majno andH Bismuth ldquoPlace of cryosurgery in the treatment ofmalignantliver tumorsrdquo Annals of Surgery vol 225 no 1 pp 39ndash50 1997

[9] J C Bischof J Bastacky and B Rubinsky ldquoAn analytical studyof cryosurgery in the lungrdquo ASME Journal of BiomechanicalEngineering vol 114 no 4 pp 467ndash472 1992

[10] S Kumar and V K Katiyar ldquoNumerical study on phase changeheat transfer during combined hyperthermia and cryosurgicaltreatment of lung cancerrdquo Journal of Applied Mathematics andMechanics vol 3 no 3 pp 1ndash17 2007

[11] S Kumar S Ali andV K Katiyar ldquoA parametric study on phasechange heat transfer process during cryosurgery of lung tumorrdquoIndian Journal of Biomechanics no 7-8 pp 210ndash213 2009

[12] K J Chua S K Chou and J C Ho ldquoAn analytical study onthe thermal effects of cryosurgery on selective cell destructionrdquoJournal of Biomechanics vol 40 no 1 pp 100ndash116 2007

[13] S Kumar andV K Katiyar ldquoMathematical modeling of freezingand thawing process in tissues a porous media approachrdquoInternational Journal of AppliedMechanics vol 2 no 3 pp 617ndash633 2010

[14] A A Gage and J Baust ldquoMechanisms of tissue injury incryosurgeryrdquo Cryobiology vol 37 no 3 pp 171ndash186 1998

[15] J J Smith and J Fraser ldquoAn estimation of tissue damage andthermal history in the cryolesionrdquoCryobiology vol 11 no 2 pp139ndash147 1974

[16] A A Gage K Guest M Montes J A Caruana and D AWhalen Jr ldquoEffect of varying freezing and thawing rates inexperimental cryosurgeryrdquo Cryobiology vol 22 no 2 pp 175ndash182 1985

[17] T C Hua and H S Ren Cryobiomedical Techniques SciencePress Beijing China 1994

[18] R H Miller and P Mazur ldquoSurvival of frozen thawed humanred cells as a function of cooling and warming velocitiesrdquoCryobiology vol 13 no 4 pp 404ndash414 1976

[19] A A Gage J A Caruana Jr andMMontes ldquoCritical tempera-ture for skin necrosis in experimental cryosurgeryrdquoCryobiologyvol 19 no 3 pp 273ndash282 1982

[20] J N Tackenberg ldquoCryolumpectomy another option for breastcancerrdquo Nursing vol 20 no 5 pp 32Jndash40J 1990

[21] R W Rand R P Rand and F A Eggerding ldquoCryolumpectomyfor breast cancer an experimental studyrdquo Cryobiology vol 22no 4 pp 307ndash318 1985

[22] R I Andrushkiw ldquoMathematical modeling of freezing frontpropagation in biological tissuerdquo Mathematical and ComputerModelling vol 13 no 10 pp 1ndash9 1990

[23] Y Rabin and A Shitzer ldquoNumerical solution of multidi-mensional freezing problem during cryosurgeryrdquo Journal ofBiomechanical Engineering vol 120 no 1 pp 32ndash37 1998

[24] J C Rewcastle G A Sandison K Muldrew J C Salikenand B J Donnelly ldquoA model for the time dependent three-dimensional thermal distribution within iceballs surroundingmultiple cryoprobesrdquo Medical Physics vol 28 no 6 pp 1125ndash1137 2001

[25] Z-S Deng and J Liu ldquoNumerical simulation of 3-D freezingandheating problems for combined cryosurgery andhyperther-mia therapyrdquoNumerical Heat Transfer A vol 46 no 6 pp 587ndash611 2004

[26] G Zhao H-F Zhang X-J Guo D-W Luo and D-Y GaoldquoEffect of blood flow andmetabolism onmultidimensional heattransfer during cryosurgeryrdquo Medical Engineering and Physicsvol 29 no 2 pp 205ndash215 2007

[27] E D Staren M S Sabel L M Gianakakis et al ldquoCryosurgeryof breast cancerrdquo Archives of Surgery vol 132 no 1 pp 28ndash341997

[28] J C Bischof D Smith P V Pazhayannur C Manivel JHulbert and K P Roberts ldquoCryosurgery of dunning AT-1 ratprostate tumor thermal biophysical and viability response atthe cellular and tissue levelrdquo Cryobiology vol 34 no 1 pp 42ndash69 1997

[29] S M Burge J P Shepherd and R P R Dawber ldquoEffect offreezing the helix and the rim or edge of the human and pigearrdquo Journal of Dermatologic Surgery and Oncology vol 10 no10 pp 816ndash819 1984

[30] A A Gage ldquoExperimental cryogenic injury of the palateobservations pertinent to cryosurgical destruction of tumorsrdquoCryobiology vol 15 no 4 pp 415ndash425 1978

[31] G JacobM N Kurzer and B J Fuller ldquoAn assessment of tumorcell viability after in vitro freezingrdquo Cryobiology vol 22 no 5pp 417ndash426 1985

[32] X M He and J C Bischof ldquoAnalysis of thermal stress incryosurgery of kidneysrdquo Journal of Biomechanical Engineeringvol 127 no 4 pp 656ndash661 2005

The Scientific World Journal 7

[33] X Shi A K Datta and Y Mukherjee ldquoThermal stresses fromlarge volumetric expansion during freezing of biomaterialsrdquoASME Journal of Biomechanical Engineering vol 120 no 6 pp720ndash726 1998

[34] B Rubinsky E G Cravalho and B Mikic ldquoThermal stresses infrozen organsrdquo Cryobiology vol 17 no 1 pp 66ndash73 1980

[35] L M Jiji Heat Conduction Springer Berlin Germany 3rdedition 2009

[36] J Crank The Mathematics of Diffusion Oxford Science Publi-cations 2nd edition 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

Hindawi Publishing Corporationhttpwwwhindawicom

Nucleic AcidsJournal of

Volume 2013

ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Biotechnology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

GenomicsInternational Journal of

Volume 2013

Evolutionary BiologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

Virolog y

ISRN Microbiology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Marine BiologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

BioMed Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Zoology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Signal TransductionJournal of

ISRN Cell Biology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

BioinformaticsAdvances in

PeptidesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Enzyme Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Biochemistry Research International

ISRN Molecular Biology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

The Scientific World Journal 5

0 01 02 03 04 05 06 07 08 09 10

1

2

3

4

5

6

7

8

tlowast = 10008

tlowast = 09715

tlowast = 09290

tlowast = 08588

tlowast = 07160

tlowast = 03886

tlowast = 00017

tlowast = 00094

tlowast = 00335

tlowast = 01132

Non

dim

ensio

nal t

empe

ratu

re (T

lowast)

Nondimensional distance (xlowast)

Figure 3 Temperature distribution at 119902lowast119898= 155

0 02 04 06 08 10

1

2

3

4

5

6

7

8

tlowast = 00017

tlowast = 00097

tlowast = 00365

tlowast = 01494

Non

dim

ensio

nal t

empe

ratu

re (T

lowast)

Nondimensional distance (xlowast)

Figure 4 Temperature distribution at 119902lowast119898= 16

interface and time taken as given in Table 2 show thatfreezing slows down with the increase in metabolic heatgeneration

The temperature profiles are required to optimize thedamage to diseased tissues Temperature profiles for differentvalues of 119902lowast

119898 that is 119902lowast

119898= 155 119902lowast

119898= 16 and 119902lowast

119898= 165

are plotted in Figures 3 4 and 5 respectively From Figure 3it is observed that temperature in tissue decreases with theincrease in time and at 119905lowast = 10008 it is in frozen stateWhile in case of 119902lowast

119898= 16 as shown in Figure 4 a steady

state is obtained at 119905lowast = 01494 and tissue is partially frozenSimilarly for 119902lowast

119898ge 16 the partially frozen state of tissue is

observed in steady state at 119905lowast = 06673 (Figure 5)

0 02 04 06 08 10

1

2

3

4

5

6

7

8

tlowast = 00175

tlowast = 00528

tlowast = 01413tlowast = 06673

Non

dim

ensio

nal t

empe

ratu

re (T

lowast)

Nondimensional distance (xlowast)

Figure 5 Temperature distribution at 119902lowast119898= 165

5 Conclusion

A quasi-steady approximation is used to get the temperatureprofile and position of freezing interface in the biologicaltissue during the freezing of biological tissues for differentvalues of metabolic heat generation It is observed that thefreezing process slows down with increase in metabolic heatgeneration Freezing of the entire tissue is even not possiblewhen the value of metabolic heat generation is extended toa higher value This shows that metabolism has a significanteffect on the freezing of biological tissues during cryosurgeryThe obtained information can be used to optimize thetreatment planning

Nomenclature

119909 Distance (0 le 119909 le 119871) (m)119871 Length of tissue (m)119909119894 Position of freezing interface (m)

119905 Time (s)119879 Temperature (∘C)120588 Density (kgm3)119888 Specific heat (Jkg ∘C)120572 Thermal diffusivity (m2sec)119896 Thermal conductivity (Wm ∘C)119897 Latent heat (KJkg)119902119898 Metabolic heat generation (Wm3)

119902119887 Blood perfusion term (Wm3 ∘C)

Subscripts

ph Phase change119891 Frozen state119906 Unfrozen state119868 Initial state (119905 = 0)119900 Cryoprobe

6 The Scientific World Journal

Superscript

lowast Dimensionless

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Sonalika Singh is thankful to S V National Institute ofTechnology Surat India for providing Senior ResearchFellowship during the preparation of this paper

References

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

[2] J G Baust A A Gage D Clarke J M Baust and RVan Buskirk ldquoCryosurgerymdasha putative approach to molecular-based optimizationrdquo Cryobiology vol 48 no 2 pp 190ndash2042004

[3] E G Kuflik andA A Gage ldquoThe five-year cure rate achieved bycryosurgery for skin cancerrdquo Journal of the American Academyof Dermatology vol 24 no 6 pp 1002ndash1004 1991

[4] A A Gage J A Caruana Jr andMMontes ldquoCritical tempera-ture for skin necrosis in experimental cryosurgeryrdquoCryobiologyvol 19 no 3 pp 273ndash282 1982

[5] S Kumar andV K Katiyar ldquoMathematical modeling of thawingproblem in skin and subcutaneous tissuerdquo in Proceedings of theIFMBE6thWorldCongress of Biomechanics (WCB rsquo10) C T Limand J C H Goh Eds pp 1611ndash1614 August 2010

[6] E D Staren M S Sabel L M Gianakakis et al ldquoCryosurgeryof breast cancerrdquo Archives of Surgery vol 132 no 1 pp 28ndash341997

[7] D K Bahn F Lee R Badalament A Kumar J Greski andM Chernick ldquoTargeted cryoablation of the prostate 7-yearoutcomes in the primary treatment of prostate cancerrdquoUrologyvol 60 no 2 pp 3ndash11 2002

[8] R Adam E Akpinar M Johann F Kunstlinger P Majno andH Bismuth ldquoPlace of cryosurgery in the treatment ofmalignantliver tumorsrdquo Annals of Surgery vol 225 no 1 pp 39ndash50 1997

[9] J C Bischof J Bastacky and B Rubinsky ldquoAn analytical studyof cryosurgery in the lungrdquo ASME Journal of BiomechanicalEngineering vol 114 no 4 pp 467ndash472 1992

[10] S Kumar and V K Katiyar ldquoNumerical study on phase changeheat transfer during combined hyperthermia and cryosurgicaltreatment of lung cancerrdquo Journal of Applied Mathematics andMechanics vol 3 no 3 pp 1ndash17 2007

[11] S Kumar S Ali andV K Katiyar ldquoA parametric study on phasechange heat transfer process during cryosurgery of lung tumorrdquoIndian Journal of Biomechanics no 7-8 pp 210ndash213 2009

[12] K J Chua S K Chou and J C Ho ldquoAn analytical study onthe thermal effects of cryosurgery on selective cell destructionrdquoJournal of Biomechanics vol 40 no 1 pp 100ndash116 2007

[13] S Kumar andV K Katiyar ldquoMathematical modeling of freezingand thawing process in tissues a porous media approachrdquoInternational Journal of AppliedMechanics vol 2 no 3 pp 617ndash633 2010

[14] A A Gage and J Baust ldquoMechanisms of tissue injury incryosurgeryrdquo Cryobiology vol 37 no 3 pp 171ndash186 1998

[15] J J Smith and J Fraser ldquoAn estimation of tissue damage andthermal history in the cryolesionrdquoCryobiology vol 11 no 2 pp139ndash147 1974

[16] A A Gage K Guest M Montes J A Caruana and D AWhalen Jr ldquoEffect of varying freezing and thawing rates inexperimental cryosurgeryrdquo Cryobiology vol 22 no 2 pp 175ndash182 1985

[17] T C Hua and H S Ren Cryobiomedical Techniques SciencePress Beijing China 1994

[18] R H Miller and P Mazur ldquoSurvival of frozen thawed humanred cells as a function of cooling and warming velocitiesrdquoCryobiology vol 13 no 4 pp 404ndash414 1976

[19] A A Gage J A Caruana Jr andMMontes ldquoCritical tempera-ture for skin necrosis in experimental cryosurgeryrdquoCryobiologyvol 19 no 3 pp 273ndash282 1982

[20] J N Tackenberg ldquoCryolumpectomy another option for breastcancerrdquo Nursing vol 20 no 5 pp 32Jndash40J 1990

[21] R W Rand R P Rand and F A Eggerding ldquoCryolumpectomyfor breast cancer an experimental studyrdquo Cryobiology vol 22no 4 pp 307ndash318 1985

[22] R I Andrushkiw ldquoMathematical modeling of freezing frontpropagation in biological tissuerdquo Mathematical and ComputerModelling vol 13 no 10 pp 1ndash9 1990

[23] Y Rabin and A Shitzer ldquoNumerical solution of multidi-mensional freezing problem during cryosurgeryrdquo Journal ofBiomechanical Engineering vol 120 no 1 pp 32ndash37 1998

[24] J C Rewcastle G A Sandison K Muldrew J C Salikenand B J Donnelly ldquoA model for the time dependent three-dimensional thermal distribution within iceballs surroundingmultiple cryoprobesrdquo Medical Physics vol 28 no 6 pp 1125ndash1137 2001

[25] Z-S Deng and J Liu ldquoNumerical simulation of 3-D freezingandheating problems for combined cryosurgery andhyperther-mia therapyrdquoNumerical Heat Transfer A vol 46 no 6 pp 587ndash611 2004

[26] G Zhao H-F Zhang X-J Guo D-W Luo and D-Y GaoldquoEffect of blood flow andmetabolism onmultidimensional heattransfer during cryosurgeryrdquo Medical Engineering and Physicsvol 29 no 2 pp 205ndash215 2007

[27] E D Staren M S Sabel L M Gianakakis et al ldquoCryosurgeryof breast cancerrdquo Archives of Surgery vol 132 no 1 pp 28ndash341997

[28] J C Bischof D Smith P V Pazhayannur C Manivel JHulbert and K P Roberts ldquoCryosurgery of dunning AT-1 ratprostate tumor thermal biophysical and viability response atthe cellular and tissue levelrdquo Cryobiology vol 34 no 1 pp 42ndash69 1997

[29] S M Burge J P Shepherd and R P R Dawber ldquoEffect offreezing the helix and the rim or edge of the human and pigearrdquo Journal of Dermatologic Surgery and Oncology vol 10 no10 pp 816ndash819 1984

[30] A A Gage ldquoExperimental cryogenic injury of the palateobservations pertinent to cryosurgical destruction of tumorsrdquoCryobiology vol 15 no 4 pp 415ndash425 1978

[31] G JacobM N Kurzer and B J Fuller ldquoAn assessment of tumorcell viability after in vitro freezingrdquo Cryobiology vol 22 no 5pp 417ndash426 1985

[32] X M He and J C Bischof ldquoAnalysis of thermal stress incryosurgery of kidneysrdquo Journal of Biomechanical Engineeringvol 127 no 4 pp 656ndash661 2005

The Scientific World Journal 7

[33] X Shi A K Datta and Y Mukherjee ldquoThermal stresses fromlarge volumetric expansion during freezing of biomaterialsrdquoASME Journal of Biomechanical Engineering vol 120 no 6 pp720ndash726 1998

[34] B Rubinsky E G Cravalho and B Mikic ldquoThermal stresses infrozen organsrdquo Cryobiology vol 17 no 1 pp 66ndash73 1980

[35] L M Jiji Heat Conduction Springer Berlin Germany 3rdedition 2009

[36] J Crank The Mathematics of Diffusion Oxford Science Publi-cations 2nd edition 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

Hindawi Publishing Corporationhttpwwwhindawicom

Nucleic AcidsJournal of

Volume 2013

ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Biotechnology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

GenomicsInternational Journal of

Volume 2013

Evolutionary BiologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

Virolog y

ISRN Microbiology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Marine BiologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

BioMed Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Zoology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Signal TransductionJournal of

ISRN Cell Biology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

BioinformaticsAdvances in

PeptidesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Enzyme Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Biochemistry Research International

ISRN Molecular Biology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

6 The Scientific World Journal

Superscript

lowast Dimensionless

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Sonalika Singh is thankful to S V National Institute ofTechnology Surat India for providing Senior ResearchFellowship during the preparation of this paper

References

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

[2] J G Baust A A Gage D Clarke J M Baust and RVan Buskirk ldquoCryosurgerymdasha putative approach to molecular-based optimizationrdquo Cryobiology vol 48 no 2 pp 190ndash2042004

[3] E G Kuflik andA A Gage ldquoThe five-year cure rate achieved bycryosurgery for skin cancerrdquo Journal of the American Academyof Dermatology vol 24 no 6 pp 1002ndash1004 1991

[4] A A Gage J A Caruana Jr andMMontes ldquoCritical tempera-ture for skin necrosis in experimental cryosurgeryrdquoCryobiologyvol 19 no 3 pp 273ndash282 1982

[5] S Kumar andV K Katiyar ldquoMathematical modeling of thawingproblem in skin and subcutaneous tissuerdquo in Proceedings of theIFMBE6thWorldCongress of Biomechanics (WCB rsquo10) C T Limand J C H Goh Eds pp 1611ndash1614 August 2010

[6] E D Staren M S Sabel L M Gianakakis et al ldquoCryosurgeryof breast cancerrdquo Archives of Surgery vol 132 no 1 pp 28ndash341997

[7] D K Bahn F Lee R Badalament A Kumar J Greski andM Chernick ldquoTargeted cryoablation of the prostate 7-yearoutcomes in the primary treatment of prostate cancerrdquoUrologyvol 60 no 2 pp 3ndash11 2002

[8] R Adam E Akpinar M Johann F Kunstlinger P Majno andH Bismuth ldquoPlace of cryosurgery in the treatment ofmalignantliver tumorsrdquo Annals of Surgery vol 225 no 1 pp 39ndash50 1997

[9] J C Bischof J Bastacky and B Rubinsky ldquoAn analytical studyof cryosurgery in the lungrdquo ASME Journal of BiomechanicalEngineering vol 114 no 4 pp 467ndash472 1992

[10] S Kumar and V K Katiyar ldquoNumerical study on phase changeheat transfer during combined hyperthermia and cryosurgicaltreatment of lung cancerrdquo Journal of Applied Mathematics andMechanics vol 3 no 3 pp 1ndash17 2007

[11] S Kumar S Ali andV K Katiyar ldquoA parametric study on phasechange heat transfer process during cryosurgery of lung tumorrdquoIndian Journal of Biomechanics no 7-8 pp 210ndash213 2009

[12] K J Chua S K Chou and J C Ho ldquoAn analytical study onthe thermal effects of cryosurgery on selective cell destructionrdquoJournal of Biomechanics vol 40 no 1 pp 100ndash116 2007

[13] S Kumar andV K Katiyar ldquoMathematical modeling of freezingand thawing process in tissues a porous media approachrdquoInternational Journal of AppliedMechanics vol 2 no 3 pp 617ndash633 2010

[14] A A Gage and J Baust ldquoMechanisms of tissue injury incryosurgeryrdquo Cryobiology vol 37 no 3 pp 171ndash186 1998

[15] J J Smith and J Fraser ldquoAn estimation of tissue damage andthermal history in the cryolesionrdquoCryobiology vol 11 no 2 pp139ndash147 1974

[16] A A Gage K Guest M Montes J A Caruana and D AWhalen Jr ldquoEffect of varying freezing and thawing rates inexperimental cryosurgeryrdquo Cryobiology vol 22 no 2 pp 175ndash182 1985

[17] T C Hua and H S Ren Cryobiomedical Techniques SciencePress Beijing China 1994

[18] R H Miller and P Mazur ldquoSurvival of frozen thawed humanred cells as a function of cooling and warming velocitiesrdquoCryobiology vol 13 no 4 pp 404ndash414 1976

[19] A A Gage J A Caruana Jr andMMontes ldquoCritical tempera-ture for skin necrosis in experimental cryosurgeryrdquoCryobiologyvol 19 no 3 pp 273ndash282 1982

[20] J N Tackenberg ldquoCryolumpectomy another option for breastcancerrdquo Nursing vol 20 no 5 pp 32Jndash40J 1990

[21] R W Rand R P Rand and F A Eggerding ldquoCryolumpectomyfor breast cancer an experimental studyrdquo Cryobiology vol 22no 4 pp 307ndash318 1985

[22] R I Andrushkiw ldquoMathematical modeling of freezing frontpropagation in biological tissuerdquo Mathematical and ComputerModelling vol 13 no 10 pp 1ndash9 1990

[23] Y Rabin and A Shitzer ldquoNumerical solution of multidi-mensional freezing problem during cryosurgeryrdquo Journal ofBiomechanical Engineering vol 120 no 1 pp 32ndash37 1998

[24] J C Rewcastle G A Sandison K Muldrew J C Salikenand B J Donnelly ldquoA model for the time dependent three-dimensional thermal distribution within iceballs surroundingmultiple cryoprobesrdquo Medical Physics vol 28 no 6 pp 1125ndash1137 2001

[25] Z-S Deng and J Liu ldquoNumerical simulation of 3-D freezingandheating problems for combined cryosurgery andhyperther-mia therapyrdquoNumerical Heat Transfer A vol 46 no 6 pp 587ndash611 2004

[26] G Zhao H-F Zhang X-J Guo D-W Luo and D-Y GaoldquoEffect of blood flow andmetabolism onmultidimensional heattransfer during cryosurgeryrdquo Medical Engineering and Physicsvol 29 no 2 pp 205ndash215 2007

[27] E D Staren M S Sabel L M Gianakakis et al ldquoCryosurgeryof breast cancerrdquo Archives of Surgery vol 132 no 1 pp 28ndash341997

[28] J C Bischof D Smith P V Pazhayannur C Manivel JHulbert and K P Roberts ldquoCryosurgery of dunning AT-1 ratprostate tumor thermal biophysical and viability response atthe cellular and tissue levelrdquo Cryobiology vol 34 no 1 pp 42ndash69 1997

[29] S M Burge J P Shepherd and R P R Dawber ldquoEffect offreezing the helix and the rim or edge of the human and pigearrdquo Journal of Dermatologic Surgery and Oncology vol 10 no10 pp 816ndash819 1984

[30] A A Gage ldquoExperimental cryogenic injury of the palateobservations pertinent to cryosurgical destruction of tumorsrdquoCryobiology vol 15 no 4 pp 415ndash425 1978

[31] G JacobM N Kurzer and B J Fuller ldquoAn assessment of tumorcell viability after in vitro freezingrdquo Cryobiology vol 22 no 5pp 417ndash426 1985

[32] X M He and J C Bischof ldquoAnalysis of thermal stress incryosurgery of kidneysrdquo Journal of Biomechanical Engineeringvol 127 no 4 pp 656ndash661 2005

The Scientific World Journal 7

[33] X Shi A K Datta and Y Mukherjee ldquoThermal stresses fromlarge volumetric expansion during freezing of biomaterialsrdquoASME Journal of Biomechanical Engineering vol 120 no 6 pp720ndash726 1998

[34] B Rubinsky E G Cravalho and B Mikic ldquoThermal stresses infrozen organsrdquo Cryobiology vol 17 no 1 pp 66ndash73 1980

[35] L M Jiji Heat Conduction Springer Berlin Germany 3rdedition 2009

[36] J Crank The Mathematics of Diffusion Oxford Science Publi-cations 2nd edition 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

Hindawi Publishing Corporationhttpwwwhindawicom

Nucleic AcidsJournal of

Volume 2013

ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Biotechnology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

GenomicsInternational Journal of

Volume 2013

Evolutionary BiologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

Virolog y

ISRN Microbiology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Marine BiologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

BioMed Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Zoology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Signal TransductionJournal of

ISRN Cell Biology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

BioinformaticsAdvances in

PeptidesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Enzyme Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Biochemistry Research International

ISRN Molecular Biology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

The Scientific World Journal 7

[33] X Shi A K Datta and Y Mukherjee ldquoThermal stresses fromlarge volumetric expansion during freezing of biomaterialsrdquoASME Journal of Biomechanical Engineering vol 120 no 6 pp720ndash726 1998

[34] B Rubinsky E G Cravalho and B Mikic ldquoThermal stresses infrozen organsrdquo Cryobiology vol 17 no 1 pp 66ndash73 1980

[35] L M Jiji Heat Conduction Springer Berlin Germany 3rdedition 2009

[36] J Crank The Mathematics of Diffusion Oxford Science Publi-cations 2nd edition 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

Hindawi Publishing Corporationhttpwwwhindawicom

Nucleic AcidsJournal of

Volume 2013

ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Biotechnology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

GenomicsInternational Journal of

Volume 2013

Evolutionary BiologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

Virolog y

ISRN Microbiology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Marine BiologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

BioMed Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Zoology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Signal TransductionJournal of

ISRN Cell Biology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

BioinformaticsAdvances in

PeptidesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Enzyme Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Biochemistry Research International

ISRN Molecular Biology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

Hindawi Publishing Corporationhttpwwwhindawicom

Nucleic AcidsJournal of

Volume 2013

ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Biotechnology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

GenomicsInternational Journal of

Volume 2013

Evolutionary BiologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

Virolog y

ISRN Microbiology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Marine BiologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

BioMed Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Zoology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Signal TransductionJournal of

ISRN Cell Biology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

BioinformaticsAdvances in

PeptidesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Enzyme Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Biochemistry Research International

ISRN Molecular Biology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013