Bioheat transfer: application to therapies using laser irradiation

Simão Pedro de Sousa Nóbregasimao.nobrega@tecnico.ulisboa.pt

Instituto Superior Técnico, Lisboa, Portugal

Abstract

Bioheat transfer, and its sensitivity to parameters change, are analysed using three heat conductionmodels: Pennes, thermal wave and dual-phase lag models. Skin is modelled using a three layer structurecomposed of epidermis, dermis and subcutaneous tissue where it is assumed that there is a cancer inits superficial area. For cancer elimination, a laser is applied at its surface and the temperature andthermal damage distributions are obtained for the three models and for the two possible modellingapproaches for the laser effect: a) surface heat flux where is assumed that the tissue is highly absorbent;b) body heat source, using broad beam method, where it is assumed that the scattering and absorptioneffect have similar weight. The finite volume method is applied in order to solve the three partialdifferential equations. Central differences are applied in the special discretization and the implicitscheme is used for the temporal discretization. The temperature distributions obtained for the threemodels have significant differences. However, at the time the laser is turned off, the temperatures inthe domain had a similar distribution. It was concluded that the type of model has more influencethan the type of modelling used for the laser. As a final remark, it is possible to claim that, for theconditions used in this work, the use of hyperthermia treatment is not effective because the cancer isonly partially destroyed and there is thermal damage in the healthy tissue.Keywords: Pennes’s model, thermal wave model, dual-phase lag model, cancer, laser, finite volumemethod

1. Introduction

The use of heat for thermal treatments of cancergoes back to 2000 b. C. [1]. The thermal interactionbetween the highly complex vasculature and the tis-sue is hard to model and has been a topic of interestin the past decades. Fan and Wang [2] refer twopossibilities for the modelling of bioheat at macro-scale. The first approach, based on the mixture the-ory of continuum mechanics, considers only macro-scale quantities. The second approach, based onporous-media theory, considers the vessels’ presencewhich make possible to account for the thermal in-teraction between the vasculature and the tissue.A third approach is mentioned by Wren et al. [3],and is a hybrid model where the two previous ap-proaches are combined. In the present work, threemodels from the continuum theory are applied.

1.1. Models

In the mid-20th century, Pennes [4] developed abioheat equation based on an experimental inves-tigation to the human forearm. Pennes equationuses, as a basis, the heat conduction equation, thatapplies Fourier’s law, and adds two additional termsto take in account the metabolic effect and the ther-mal change between the blood and tissue. The last

term is given by the following equation:

Qb/t = wbρbcb (Ta − T ( #»r , t)) (1)

with wb (ml/s/ml) representing blood perfusion,ρb(kg/m3

)its volumetric mass, cb (J/[kg ·K]) its

specific heat, Ta(◦C) the temperature of the arterial

blood, T (◦C) the tissue temperature, t(s) the timeand #»r (m) the position vector. Equation (1) impliesthat the thermal equilibrium between the blood andtissue occurs at the capillary bed. In other words,Pennes assumed that the tissue and venous temper-ature are equal. Continuing the analysis of equation(1), it can be concluded that the perfusion effect ishomogeneous and isotropic, that is, the directionaleffect of the blood flow is not taken into account.As mentioned before, Pennes’ model uses Fourier’sheat conduction law:

#»q ′′( #»r , t) = −k #»∇T (~r, t) (2)

with #»q ′′(W/m2) representing the heat flux,

k(W/m ·K−1) the thermal conductivity and #»∇(/m)the gradient operator. An immediate consequenceof the equation (2) is that there is an infinite veloc-ity of heat propagation, i.e, a perturbation in onepoint of the domain is felt everywhere. This law, de-spite being applicable to many real cases, does not

1

provide good results in inhomogeneous materials asbiological tissues.

In order to eliminate the paradox of Fourier’s lawand to account for the thermal interaction in non-homogeneous systems, Cattaneo [5] and Vernotte[6] introduced the next relation:

#»q ′′( #»r , t+ τq) = −k#»∇T (~r, t) (3)

with τq(s) representing the relaxation time. Equa-tion (3) states that the heat flux arises τq secondsafter the thermal gradient at time t is imposed.Kaminski [7] suggests that this parameter is thetime required for thermal interaction between thestructural elements to take place.

Equation (3) implies that heat flux is a conse-quence of the temperature gradient (cause). Togive a more general model where a duality betweencause and consequence is possible to achieve, Tzoupresented a constitutive relation given by [8]:

#»q ′′( #»r , t+ τq) = −k#»∇T (~r, t+ τT ) (4)

with τT (s) representing the relaxation time for thetemperature gradient.

1.2. Thermal damageThe determination of the temperature distribu-

tion in the irradiated tissue is a necessary condi-tion, but not sufficient, to determine the effect of aheat source in the human tissue. The extension ofthe thermal damage depends on factors such as theintensity of the heat source, its duration and areaof application. Moritz and Henriques [9] were thefirst investigators to do research work in the field ofthermal damage and they proposed an expressionfor the denaturation process based on the first or-der approximation of the Arrhenius equation wherethere is an exponential relation between the denat-uration rate, K, and the tissue temperature:

K(T ) = A exp

(−EaRT

)(5)

with A(s−1) representing the frequency factor ofthe tissue, R(J/ [mole.K]) the universal gas con-stant and Ea(J/mole) the activation energy of thedenaturation reaction.

1.3. LaserThe effect of the laser can be modelled using dif-

ferent approaches. If it is considered that the radi-ation is highly absorbed in the tissue, such as ra-diation coming from a CO2 laser, it is possible tosimulate the laser effect using a Neumann boundarycondition. If the scattering effect as an intensitycompared to the absorption effect, it is necessaryto solve the transient radiative transfer equation inorder to determine the laser effect. However, there

are approximated methods that allow this calcu-lation such as the broad-beam irradiance method.This approach, introduced by Gardner et al. [10],uses the following equation to characterize the uni-dimensional fluence rate, φ(z)[W/m2]:

φ(z) =

φsup [C1 exp (−k1z/δ)− C2 exp (−k2z/δ)] (6)

where

δ =1√

3µa [µa + µs(1− g)](7)

with C1, C2, k1 and k2 representing the model con-stants, that are dependent on the diffuse reflectanceand determined by the Monte Carlo method, δ(m)the effective penetration depth, determined by thediffusion theory, µs(m

−1) the scattering coefficient,µa(m

−1) the absorption coefficient and g the scat-tering anisotropy factor.

1.4. Cancer and skinSkin is the biggest human organ and has several

functions such as the regulation of the body tem-perature and the defence against outdoor micro-organism. The skin layers change over the bodyposition and are formed by the epidermis and der-mis. Adjacent to dermis, and sometimes consideredas a layer from the skin, there is the subcutaneoustissue (s. t.). At the present work all three layerswill be considered and, at the surface of the skin,it will be assumed that a non-melanoma cancer ispresent: squamous cell carcinoma.

2. Governing equationsThe generic heat conduction equation is given by

the following expression [11]:

ρtct∂T

∂t= − #»∇ · #»q ′′ + Ṡ (8)

with subscript t indicating properties referring tothe tissue and Ṡ representing the source term. ForPennes, hyperbolic and dual models, the sourceterm is the same and is given by the following rela-tion:

Ṡ = wbρbcb (Ta − T ) +Qm +Qlaser (9)

with Qm(W/m3) representing the metabolic heat

generation and Qlaser(W/m3) the volumetric effect

of the laser. Combining equation (8) and (9), it isobtained equation (10) that will be used in the de-termination of the heat conduction equations thatcharacterize each model.

ρtct∂T

∂t=

− #»∇ · #»q ′′ + wbρbcb (Ta − T ) +Qm +Qlaser (10)

The difference among the models that will beused is given by the different constitutive relationsfor the heat flux that characterize each one.

2

2.1. Pennes’s modelCombining Fourier’s law, given by equation (2),

with equation (10), it is obtained Pennes’s bioheatequation:

ρtct∂T

∂t=

#»∇ · [k #»∇T ]+

wbρbcb (Ta − T ) +Qm +Qlaser (11)

2.2. Thermal wave modelThe first step in order to obtain the heat conduc-

tion equation that characterizes the thermal wavemodel is to make a first order Taylor expansion toequation (3) around time t:

#»q ′′( #»r , t) + τq∂ #»q ′′( #»r , t)

∂t≈ −k #»∇T (~r, t) (12)

Equation (12) isn’t ready to be introduced in equa-tion (10). To achieve this goal, it is first necessaryto do a time derivation to equation (10) and to mul-tiply it by τq:

τqρtct∂2T

∂t2= −τq

#»∇ · ∂#»q ′′

∂t+

τq

[∂Qm∂t− wbρbcb

∂T

∂t+∂Qlaser∂t

](13)

Adding equation (13) to equation (10), it is ob-tained equation (14) that allows the introductionof the thermal wave constitutive relation:

[ρtct + τqwbρbcb]∂T

∂t+ τqρtct

∂2T

∂t2=

− #»∇·[

#»q ′′ + τq∂ #»q ′′

∂t

]+τq

[∂Qm∂t

+∂Qlaser∂t

]+Qm+

wbρbcb [Ta − T ] +Qlaser (14)

Combining equation (12) and equation (14), it isobtained the heat conduction equation for the ther-mal wave model:

[ρtct + τqwbρbcb]∂T

∂t+ τqρtct

∂2T

∂t2=

#»∇ ·[k

#»∇T]

+ τq

[∂Qm∂t

+∂Qlaser∂t

]+Qm+

wbρbcb [Ta − T ] +Qlaser (15)

2.3. Dual-phase lag modelAs for the thermal wave model, the first step is

to make a first order Taylor expansion to equation(4) around time t:

#»q ′′( #»r , t) + τq∂ #»q ′′( #»r , t)

∂t≈

− k #»∇T ( #»r , t)− kτT∂

#»∇T ( #»r , t)∂t

(16)

Combining equations (14) and (16), it is obtainedthe heat conduction equation for the dual phase lag:

[ρtct + τqwbρbcb]∂T

∂t+τqρtct

∂2T

∂t2= Qlaser+Qm−

#»∇ ·

[−k #»∇T ( #»r , t)− kτT

∂#»∇T ( #»r , t)∂t

]+

τq

[∂Qm∂t

+∂Qlaser∂t

]+ wbρbcb [Ta − T ] (17)

3. Numerical methods and verificationEquations (11), (15) and (17) are three partial

differential equations that don’t have analytical so-lution for the most common engineering cases. Nu-merical methods overcome this problem allowingthe equations to have an approximated solution.In this work, finite volume method is going to beimplemented because of its advantage in conserv-ing energy in each control volume. For the spa-tial and temporal discretization, finite central dif-ferences and the implicit scheme, respectively, areused.

The solution of the algebraic equations are ob-tained with Thomas algorithm.

The verification of the computational code hasbeen done using exact solutions for the Pennesand thermal wave models. For the dual-phase lagmodel, the method of the manufactured solutionwas applied.

4. ResultsSkin is going to be modelled as a layered structure

with a cancer placed in the middle of the domain.In figure 1 there are the dimensions of the domain,layers and cancer and in table 2 the properties usedto get the results are presented.

The initial temperature distribution was obtainedimposing zero flux at all boundaries, except thebottom boundary where a convection process withh = 7 Wm2·K and T∞ = 20

oC was assumed, and inthe top boundary where it was imposed a fixed tem-perature of 37oC. For the dual-phase lag and ther-mal wave models, it is also necessary to assign avalue for ∂T∂t

∣∣t=0s

, which was assumed zero.For the boundary conditions, all boundaries had

zero flux, except the bottom face (tissue’s surface)where it was imposed for the surface heat flux mod-elling, at the cancer’s surface (area of figure 1 (b)represented by the symbol +), a heat flux witha value of 50000W/m2. In the area outside theone represented by the symbol +, still in the bot-tom boundary, a convection process was assumedwith the same characteristics used to determine theinitial temperature distribution. When body heatsource modelling was used, all the bottom surfacehad a convection process and the laser effect was in-troduced using the following equation: Qlaser(z) =

3

φ(z)× µa(z)[W/m3]. The optical properties neces-sary to define the problem are presented in table 1.For the mesh, a value of ∆x = ∆y = 1, 00× 10−4mand ∆z = 4, 15 × 10−5m were used. A time stepof ∆t = 0, 1 was implemented. In relation to thethermal damage, for T ≤ 55oC it was used a valueof Ea = 75000(J/mole) and A = 3, 1 × 1098s−1.For T > 55oC it was utilized a value of Ea =35406, 7(J/mole) and A = 5, 0× 1045s−1 [12].

Table 1: Optical properties [13].µa(mm

−1) µs(mm−1) g Rd

Cancer 2,5 1,70,7 0,05Dermis 2,2 2,1

S.t. 1,2 1,9

Cancer

0,09mm

1,47mm

7,44mmSubcutaneous Tissue

Epidermis

Dermis1mm

z

x

(a) yz or xz cut

XCancer=2mm

YCancer=2mm

XFlux=1,25mm YFlux=1,25m

m

XDomain=18mm

YD

omain=18m

m

x

y

(b) xy cut

Figure 1: Representation of skin layers and cancerposition.

4.1. Temperature and thermal damage distribu-tions

In figure 2 it is represented the temperature andthermal damage for Pennes’ model using surfaceheat flux. The maximum temperature obtainedwas 99,8oC, corresponding to the center of the do-main at the surface and for the instant of timethe laser was turned off. The thermal damage ob-tained at the cancer and skin were, respectively,

Table 2: Tissue and cancer properties.

Parameter Value Reference

Thickness Epid. 0, 09 [14](mm) Dermis 1, 47 [14]

S. t. 7, 44 Assumed

Volumetric Epid. 1190 [15]mass (kg/m3) Dermis 1116 [15]

S. t. 911 [16]Cancer 1050 [16]

Metabolism Epid. 368, 1 [17](W/m3) Dermis 368, 1 [17]

S. t. 464, 6 [16]Cancer 4000

Blood Epid. - [12]perf. (s−1) Dermis 0, 00187

S. t. 0, 0006 [16]Cancer 0, 007 [16]

Thermal cond. Epid. 0, 266 [18](W/m ·K−1) Dermis 0, 498 [18]

S. t. 0, 210 [16]Cancer 0, 510 [16]

Specific heat Epid. 3700 [18](J/kg ·K−1) Dermis 3200 [18]

S. t. 2348 [16]Cancer 3950 [16]

58, 34% and 6, 30%. For a better analysis of theresults from the three models, using surface heatflux, the temperature, (a), and thermal damage,(b), distributions along time are presented in fig-ure 3 for two points: P1(xmax/2, ymax/2, 0) andP2(xmax/2, ymax/2, zcancer).

The first remark, based in figure 3, is that thetemperature distribution for the three models isvery distinct. In point 2, is possible to see that forthe thermal wave model, despite having source/sinkterms in its modelling, there is a constant temper-ature level. This happens because the initial con-ditions were obtained until stationarity was veri-fied. At t = 5s, again for the thermal wave model,there is an increase of the temperature which canbe interpreted as the arrival of the thermal wave.For the dual-phase lag model the wave behaviouris not observed, allowing to conclude that the ex-tra term comparatively to the thermal wave model,#»∇ ·[kτT

∂#»∇T ( #»r ,t)

∂t

], destroys this behaviour.

From figure 3 (a) it can be seen that the ther-mal wave/dual model has the highest/lowest tem-perature when the laser power is turned off. How-

4

Table 3: Thermal damage and maximum temperature for the three models and the two modelling.Surface heat flux Body heat source

Tmax(oC) Ωc ≥ 1 Ωs ≥ 1 Tmax(oC) Ωc ≥ 1 Ωs ≥ 1

Pennes 99,8 58,34 0,102 78,7 56,17 0,059Thermal wave 100,0 64,05 0,169 78,9 60,96 0,110Dual 98,5 52,27 0,037 77,5 49,86 0,015

ever, the temperatures at t = 33, 3s are similar andthere is only a difference of 1, 5oC for all modelsand at any point of the domain. This is the reasonwhy only Pennes’s model bi-dimensional figures areshown .

From table 3, it is checked that the thermalwave/dual model forecast the highest/lowest ther-mal damage in the skin(s) and cancer(c).

In figure 4 it is represented the temperature andthermal damage for Pennes’s model using body heatsource to simulate the laser effect. Comparing fig-ure 2 and 4, two direct conclusions can be men-tioned: a) modelling using surface heat flux orig-inates, at the center of the domain, higher tem-peratures until half of the cancer thickness. Nearthe surface, the temperature difference can reach21, 1oC. After this section, z > 0, 0005m, the tem-perature obtained using body heat source is higherthan the one obtained from the surface heat fluxmodelling but the difference of temperatures are lesssignificant; b) thermal damage for z = zcancer andat the center of the domain is higher for body sourcemodelling. This result is in concordance with a).

From figure 5 it is possible to draw similar conclu-sions as those obtained for surface heat flux mod-elling. For brevity of exposure, it will only be em-phasized the difference between the two modellingused in the work. Body heat source modelling gotthe following differences:

1. Lower maximum temperatures - this fact iseasy to understand because in the surface heatflux modelling, all the energy is introduced atthe surface giving a larger increase in the tem-perature of this points.

2. Lower thermal damage - this difference can bejustified due to the large difference of temper-atures in the top half of the cancer for surfaceheat flux modelling. This significant differencecauses a higher diffusion of heat to the periph-eral areas of the cancer yielding a larger ther-mal damage in the cancer. This last statementis also the reason for a skin thermal damagelower for body heat source modelling.

3. Thermal wave model without constant temper-ature level - from figure 5, it is observed forpoint 2 that there is no evidence of the ther-mal wave arrival, i.e, the temperature increases

when t→ 0s.

From table 3, it is possible to state that the typeof model has a larger influence in the thermal dam-age than the type of modelling. It is also possibleto claim that the effectiveness of the thermal treat-ment was not very prominent because the maximumcancer thermal damage had a value of 64, 05% and,even for this low value of destruction, there was apercentage of healthy tissue damaged.

4.2. Sensitivity analysisDue to inherent complexity of the human skin,

it is possible to claim that there is an uncertaintyregarding the values that characterize the cancerparameters. For the thermal conductivity and thespecific heat, values in a range of ±15% [19, 20],comparatively to the values used in the last sec-tion, were found . For the metabolic heat genera-tion, the value 40000(W/m3) was found in Zhang’swork [21]. Regarding the blood perfusion, Zhang[21] claimed a maximum value of 0, 01s−1 and aminimum value of 0, 00018s−1 was found in [22].With the aim of understanding the impact of thedifferent values found in the literature, a parametricstudy was done and the results are here presentedfor the dual-phase lag model.

In figure 6, the results of the parametric studyusing the dual-phase lag model and the surface heatflux modelling are presented. From figure 6 (a), it ispossible to see that the increase of the temperaturegradient relaxation time, τT , causes a decrease ofthe temperature for fixed value of τq. In contrastto the reports of Askarizadeh and Ahmadikia [23],equal ratios between the relaxations times do notproduce the same temperature distribution, i.e, thetemperature progress will depend on the absolutetimes. From table 4, it is also possible to claimthat the increase of τT needs a higher τq/τT ratioto compensate the decrease of the thermal damage.

From figure 6 (b) it is possible to verify that thedecrease of cancer’s thermal conductivity implies anincrease of the temperature, which is a result of thedecrease of the diffusion process. Analysing tables4 and 5, it is checked that the dual-phase lag modelas a greater relative variation, for the thermal con-ductivity, in the thermal damage when comparedwith the Pennes and thermal wave models. Thislast statement can be justified due to the introduc-tion of a new diffusion term in the dual-phase lag

5

x(m)

y(m

)

6 7 8 9 10 11 12

x 10−3

6

7

8

9

10

11

12x 10

−3

40

50

60

70

80

90

(a) Temperature(◦C) for z = 0m and t = 33, 3s

x(m)

y(m

)

6 7 8 9 10 11 12

x 10−3

6

7

8

9

10

11

12x 10

−3

0

2

4

6

8

10

12

(b) Thermal damage logarithm for z = 0m and t =43, 1s

x(m)

z(m

)

6 7 8 9 10 11 12

x 10−3

0.5

1

1.5

2

2.5

3

x 10−3

40

50

60

70

80

90

(c) Temperature(◦C) for y = ymax/2 and t = 33, 3s

x(m)

z(m

)

6 7 8 9 10 11 12

x 10−3

0.5

1

1.5

2

2.5

3

x 10−3

0

2

4

6

8

10

12

(d) Thermal damage logarithm for y = ymax/2 andt = 43, 1s

Figure 2: Temperature and thermal damage for Pennes’s model using surface heat flux.

0 10 20 30 40 50 6030

40

50

60

70

80

90

100

110

Time(s)

T(º

C)

O. t. 1Dual 1Pennes 1O. t. 2Dual 2Pennes 2

(a) Temperature (◦C)

0 10 20 30 40 50 600

1

2

3

4

5x 10

5

Poi

nt 1

− th

erm

al d

amag

e

Time(s)

0

0.5

1

1.5

2

Poi

nt 2

− th

erm

al d

amag

e

O. t. 1Dual 1Pennes 1O. t. 2Dual 2Pennes 2

(b) Thermal damage, Ω

Figure 3: Temperature and thermal damage as a function of time for point 1 and 2 assuming surface heatflux.

model.

From figure 6 (c), it is possible to observe that adecrease in blood perfusion causes an increase of the

temperature. This result agrees with the modellingused for the blood perfusion where it is assumedthat this term is a heat sink for temperatures higher

6

x(m)

y(m

)

6 7 8 9 10 11 12

x 10−3

6

7

8

9

10

11

12x 10

−3

40

45

50

55

60

65

70

75

(a) Temperature(◦C) for z = 0m and t = 33, 3s

x(m)

y(m

)

6 7 8 9 10 11 12

x 10−3

6

7

8

9

10

11

12x 10

−3

0

1

2

3

4

5

6

7

(b) Thermal damage logarithm for z = 0m and t =42, 8s

x(m)

z(m

)

6 7 8 9 10 11 12

x 10−3

0.5

1

1.5

2

2.5

3

x 10−3

40

45

50

55

60

65

70

75

(c) Temperature(◦C) for y = ymax/2 and t = 33, 3s

x(m)

z(m

)

6 7 8 9 10 11 12

x 10−3

0.5

1

1.5

2

2.5

3

x 10−3

0

1

2

3

4

5

6

7

(d) Thermal damage logarithm for y = ymax/2 andt = 42, 8s

Figure 4: Temperature and thermal damage for Pennes’ model using body heat source.

0 10 20 30 40 50 6030

40

50

60

70

80

Time(s)

T(º

C)

O. t. 1Dual 1Pennes 1O. t. 2Dual 2Pennes 2

(a) Temperature (◦C)

0 10 20 30 40 50 600

500

1000

1500

Poi

nt 1

− th

erm

al d

amag

e

Time(s)

0

0.5

1

1.5

2

2.5

3

Poi

nt 2

− th

erm

al d

amag

e

O. t. 1Dual 1Pennes 1O. t. 2Dual 2Pennes 2

(b) Thermal damage, Ω

Figure 5: Temperature and thermal damage as a function of time for point 1 and 2 assuming body heatsource.

than 37oC.

Figure 6 (d) has the parametric study for the spe-cific heat. For Pennes’ model, and unlike the othertwo models where to opposite contributions exist,

the specific heat as only influence in the coefficientof the temporal derivative of the temperature. Inother words, it is expected that the relative varia-tion of this parameter for the Pennes’s model as a

7

0 10 20 30 40 50 60 7030

40

50

60

70

80

90

100

Time(s)

T(º

C)

τ

q1=5 e τ

T1=0

τq1

=5 e τT1

=5

τq1

=5 e τT1

=10

τq1

=10 e τT1

=10

τq1

=5 e τT1

=0

τq1

=5 e τT1

=5

τq1

=5 e τT1

=10

τq1

=10 e τT1

=10

(a) Relaxation time (s)

0 10 20 30 40 50

40

50

60

70

80

90

100

110

Time(s)

T(º

C)

k

1=0,434

k1=0,510

k1=0,587

k2=0,434

k2=0,510

k2=0,587

(b) Thermal conductivity (W/m ·K−1)

0 10 20 30 40 50

40

50

60

70

80

90

100

Time(s)

T(º

C)

ω

1=0,00018

ω1=0,007

ω1=0,01

ω2=0,00018

ω2=0,007

ω2=0,01

(c) Blood perfusion (s−1)

0 10 20 30 40 50

40

50

60

70

80

90

100

Time(s)

T(º

C)

c

1=3357,5

c1=3950

c1=4542,5

c2=3357,5

c2=3950

c2=4542,5

(d) Specific heat (J/kg ·K−1)

Figure 6: Parametric study for dual-phase lag model using surface heat flux.

Table 4: Thermal damage variation for the dual-phase lag model using the two modelling. The referencevalues are in table 3.

Surface heat flux Body heat sourceMagnitude Ωc ≥ 1 Ωs ≥ 1 Ωc ≥ 1 Ωs ≥ 1

τq(s)/τT (s)

5/05/55/1010/10

64,05(+11,78%)52,27%

42,30(-9,97%)42,34(-9,93%)

0,169(+0,132%)0,037%

0,000(-0,037%)0,000(-0,037%)

60,96(+11,10%)49,86%

39,04(-10,82%)38,78(-11,08%)

0,111(+0,096%)0,015%

0,000(-0,015%)0,000(-0,015%)

k(W/[m ·K])

0,43350,5865

56,12(+3,85%)48,49(-3,78%)

0,057(+0,020%)0,011(-0,026%)

53,92(+4,06%)45,48(-4,38%)

0,033(+0,018%)0,004(-0,011%)

ω(s−1)

0,000180,01

54,74(+2,47%)51,43(-0,84%)

0,057(+0,020%)0,026(-0,011%)

52,31(+2,45%)48,70(-1,16%)

0,027(+0,012%)0,006(-0,009%)

c(J/Kg ·K)

3357,54542,5

53,61(+1,34%)50,85(-1,42%)

0,046(+0,009%)0,017(-0,020%)

51,11(+1,25%)48,23(-1,63%)

0,019(+0,004%)0,006(-0,009%)

larger absolute value. From tables 4 and 5, this laststatement can be claimed.

The results for the metabolic heat generationwere not presented in this section because the in-crease in 900% of its value, in relation to the stan-dard value, did not affect the temperature distribu-tion.

In figure 7, the results of the parametric studyusing the dual-phase lag model and the body heat

source modelling are presented. From this figure, itis possible to conclude that the destruction of thethermal wave implies a similar behaviour for thesurface heat flux and body heat source modelling.

5. Conclusions

The development of this work had, as the maingoal, the analysis of the destruction potential of acancer using a hyperthermia treatment executed by

8

0 10 20 30 40 50 60 7035

40

45

50

55

60

65

70

75

80

Time(s)

T(º

C)

τ

q1=5 e τ

T1=0

τq1

=5 e τT1

=5

τq1

=5 e τT1

=10

τq1

=10 e τT1

=10

τq1

=5 e τT1

=0

τq1

=5 e τT1

=5

τq1

=5 e τT1

=10

τq1

=10 e τT1

=10

(a) Relaxation time (s)

0 10 20 30 40 50

40

50

60

70

80

Time(s)

T(º

C)

k

1=0,434

k1=0,510

k1=0,587

k2=0,434

k2=0,510

k2=0,587

(b) Thermal conductivity (W/m ·K−1)

0 10 20 30 40 50 6035

40

45

50

55

60

65

70

75

80

Time(s)

T(º

C)

ω

1=0,00018

ω1=0,007

ω1=0,01

ω2=0,00018

ω2=0,007

ω2=0,01

(c) Blood perfusion (s−1)

0 10 20 30 40 5035

40

45

50

55

60

65

70

75

80

Time(s)

T(º

C)

c

1=3357,5

c1=3950

c1=4542,5

c2=3357,5

c2=3950

c2=4542,5

(d) Specific heat (J/kg ·K−1)

Figure 7: Parametric study for dual-phase lag model using body heat source.

Table 5: Thermal damage variation for the Pennes and thermal wave models using the two modelling.The reference values are in table 3.

Surface heat flux Body heat sourcePennes Thermal wave Pennes Thermal wave

Magnitude Ωc ≥ 1 Ωc ≥ 1 Ωc ≥ 1 Ωc ≥ 1k

(W/[m ·K])0,43350,5865

61,24(+2,90%)55,14(-3,20%)

59,96(+3,79%)52,33(-3,84%)

67,13(+3,08%)60,85(-3,19%)

64,05(+3,09%)56,97(-3,99%)

ω(s−1)

0,000180,01

60,12(+1,78%)57,32(-1,02%)

58,19(+2,02%)55,16(-1,01%)

66,22(+2,17%)63,07(-0,98%)

63,03(+2,07%)59,44(-1,52%)

c(J/Kg ·K)

3357,54542,5

59,90(+1,56%)56,82(-1,52%)

57,62(+1,45%)54,59(-1,58%)

64,73(+0,68%)62,81(-1,24%)

61,98(+1,02%)59,29(-1,67%)

a laser application. As a second objective, a para-metric study was done in order to understand theeffects of changing cancer parameters.

For the first part, it was concluded that the typeof model had a greater influence in the thermaldamage results than the type of modelling. Forthe parameters chosen, the hyperthermia treatmentwas not very effective. The maximum temperatureswere found for surface heat flux modelling and, theconstant temperature level at points far from thesurface at initial times, is only found for the ther-

mal wave model when surface heat flux is used.

Regarding the second study, it is possible to claimthat the increase of τq and τT implies, respectively,an increase and decrease of the thermal damage ob-tained. Thermal conductivity is the thermophysicalparameter which has the greater impact in the ther-mal damage. The dual-phase lag model forecasts ahigher change regarding the thermal conductivityand for the specific heat, the greater influence isobserved when the Pennes’ model is used.

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