International Journal of Heat and Mass Transfer

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Thermal lagging in living biological tissue based on nonequilibrium heat

transfer between tissue, arterial and venous bloods

Nazia Afrin, Yuwen Zhang

, J.K. Chen

Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65201, USA

a r t i c l e

i n f o

Article history:

Received 3 September 2010

Received in revised form 11 January 2011 Accepted 11 January 2011

Available online 5 March 2011

Keywords: Bioheat transfer Laser irradiation Non-equilibrium Thermal damage Dual-phase lagging Protein denaturation

a b s t r a c t

Arterial, venous blood and solid tissue are the three energy carriers that contribute to heat transfer in the living biological tissues. A generalized dual-phase lag mode for living biological tissues based on nonequi-librium heat transfer between tissue, arterial and venous bloods is presented in this paper. The phase lag times for heat flux and temperature gradient only depend on properties of artery, vein and tissue, blood perfusion rate and convective heat transfer rate and are estimated using the available properties from the literature. It is found that the phase lag times for heat flux and temperature gradient are the identical for the case that the tissue and blood have the same properties. However, the phase lag times are different for the case that the properties of tissue and bloods are different. The phase lag times for brain and mus-cles are also discussed.

Ó2011 Elsevier Ltd. All rights reserved.

1. Introduction

The transport of thermal energy in living tissue is a complex process. It involves multiple phenomenological mechanisms including conduction in tissue, convection between blood and tis-sues, blood perfusion or advection and diffusion through microvas-cular beds, and metabolic heat generation. The bioheat transfer modeling is the basis of thermotherapy and the thermoregulation system in a human body. Variations of temperature and heat trans-fer in a human body depend on the arterial and venous blood flow rates, blood perfusion rate, and metabolic heat generation, heat conduction within the tissue, thermal properties of blood and tis-sue, and also on the human body geometry. The whole anatomical structure can be considered as a fluid saturated porous medium as tissue can be considered as a solid matrix and blood penetrate the pore space of the medium. Thus, heat transfer phenomenon can be considered as convection heat transfer in porous medium with internal heat generation.

Pennes[1]bioheat equation is the most widely applied model for temperature distribution in the living biological tissues. The ef-fect of arterial blood on the heat transfer in a living tissue is taken into account by a blood perfusion term, which is proportional to the volumetric rate of blood perfusion and the difference between the average arterial blood and tissue temperatures. Pennes bioheat model is valid only if when the venous blood flows from the capil-lary bed to the main supply vein, its temperature remains the same

as the tissue temperature regardless the size of vessel and the flow rate. To take metabolic heat generation within the tissue and local variation of the thermal properties of tissue into account, core and shell model[2]and four layer model[3]were developed for the thermoregulatory application, in which temperature changes of both arterial and venous blood flows were treated by the lumped parameter models.

The temperature variation in the axial direction is greater than that in the radial direction due to the blood perfusion through the tissue and the countercurrent effect between the arterial and ve-nous blood flows[4]. The axial heat transfer and temperature gra-dient are not negligible; this post additional challenge in analyzing bioheat transport in living biological tissue. Therefore, heat trans-fer analysis in living biological tissue should consider the effects of the directions of the blood flow. The complex vascular architec-ture is the fundamental problem in heat transfer process within the human body[5], including the variation of number, size and spacing of the vessels, the thermal interaction among arteries, veins and tissues, metabolic heat generation, convection and blood perfusion through the capillary beds and interaction with the envi-ronment in a complete model. In their series of papers, Weinbaum et al.[6–8]proposed a bioheat equation considering the variation of the number, density and size and flow velocity of the counter-current arterial-venous vessels. That model was applied for the sin-gle organ rather than the whole human body for thermoregulation. In Pennes bioheat equation, the heat conduction in biological tissue is modeled by using Fourier’s law; it assumes an infinite thermal energy propagation speed, i.e. any local temperature dis-turbance causes an instantaneous perturbation in the temperature

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⇑ Corresponding author.

E-mail address:zhangyu@missouri.edu(Y. Zhang).

Contents lists available atScienceDirect

International Journal of Heat and Mass Transfer

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j h m t

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at each point in the medium[9]. In fact, heat is always found to propagate at a finite speed. With inhomogeneous biological struc-tures, heat flux responds to the temperature gradient via a relaxa-tion behavior[10]. Cattaneo[11]and Vernotte[12]simultaneously suggested a modified heat flux model:

q00ðr;tÞ þ

s

@q00ðr;tÞ

@t ¼ k

r

Tðr;tÞ ð1Þ

Eq.(1)assumes that the heat flux and the temperature gradient oc-cur at different times. The delay between the heat flux and temper-ature gradient is defined as the thermal relaxation time;

s

. Kaminski

[13]suggested that the theoretical value of the thermal relaxation time

s

for biological tissue is in the range of 20–30 s while the experimental value was observed to be 16 s[14]. If Eq.(1)is used in replacement of the classical Fourier’s law of heat conduction in derivation of Pennes bioheat equation, the following bio heat equa-tion is obtained[15,16]:

s

q @2Ts @t2 þ 1þ wbcb Cs

s

q @T s @t ¼

a

s

r

2 Tsþ wbcc Cs ðTbTsÞ þwbcb

s

Cs @Tb @t þ SmþS Cs þ

s

q Cs @Sm @t þ @S @t ð2Þ

where

s

qis phase lag time for heat flux,wbis the blood perfusion

rate,Csis heat capacity of tissue,Smis metabolic heat generation

andSis heat source due to hyperthermia therapy. The second order derivative of temperature with respect to time appears, and for this reason Eq. (2)is referred to as hyperbolic bioheat equation[17]. In arrival to Eq. (2), it is assumed that the temperature gradient is established before heat flux, which is referred to as gradient-precedence type heat flow. Tzou [18] established a dual phase thermal lag (DPL) model that allows either the temperature gradient to precede heat flux vector or the heat flux vector precede temperature gradient. i.e.,

q00ðr;tþ

s

qÞ ¼ k

r

Tðr;tþ

s

TÞ ð3Þ

where

s

qis the phase lag for the heat flux vector, and

s

Tis the phase

lag for the temperature gradient. If the local heat flux vector results in the temperature gradient at the same location but an early time (

s

q>

s

T), the heat transfer is gradient-precedence type. On the other

hand, if the temperature gradient results in the heat flux at an early time (

s

q<

s

T), the heat flow is called flux-precedence type. The first

order approximation of Eq.(3)is

q00þ

s

q @q00 @t ¼ k

r

s

T @

r

T @t ð4Þ

If the classical Fourier’s law of conduction is replaced by Eq.(4), the bioheat equation becomes

s

q @2Ts @t2 þ 1þ wbcb Cs

s

q @T s @t ¼

a

s

r

2T

s

T @ @tð

r

2T sÞ þwbcb Cs ð TbTsÞ þ SmþS

q

scs þ

s

q

q

scs @Sm @t þ @S @t ð5Þ

Under the assumption of constant blood temperature (i.e.

@Tb=@t¼0) and the condition

s

s

T¼0, Eq.(5)reduces to the

classical bioheat equation. The DPL bioheat equation (5) is the modification of the Pennes bioheat equation by considering non-Fourier effect. Because of the lacking of appropriate theoretical model on estimation of the two phase lag model, DPL is still not widely accepted by the researchers in the field. Zhang[15] devel-oped a generalized DPL bioheat equation based on nonequilibrium between arterial blood and tissue. The phase lag times were ex-pressed in terms of properties of blood and tissue, interphase con-vection heat transfer coefficient, and blood perfusion coefficient. In a living biological tissue, both the arterial and venous blood flow through the vessels and disperse through the tortuous capillary beds. Therefore, the constituencies in the living tissue include arte-rial blood, venous blood and surrounding tissues. The DPL model proposed in Ref. [15] only considered nonequilibrium between the arterial blood and tissue while the venous blood was assumed to be in equilibrium with the surrounding tissue. In this paper, a new DPL model based on non-equilibrium heat transfer in arterial blood, venous blood and living tissue will be developed. The phase lag times for heat flux and temperature gradient under different conditions will be estimated based on the available properties in the literature.

2. Heat transfer in arteries, venous and solid tissue

The heat transfer in the whole biological tissue involves heat conduction in the tissue, convection heat transfer between tissue and blood in artery and vein, as well as blood perfusion. The tissue is treated as a solid matrix part of the saturated porous medium, and the blood permeate in the pore space of the porous medium

[19,20](seeFig. 1).

In the human body, blood flows in artery and vein through the compound matrix of tissue. The blood flow rate and direction are totally different in artery and vein because the vein is much narrower than the artery. Therefore, thermal equilibrium between

Nomenclature

a specific heat transfer area [m2/m3] c specific heat of artery [J/(kg K)]

db diameter of the blood vessel [m]

Ga coupling factor between artery and tissue [W/(m3K)]

h heat transfer coefficient [W/(m2K)] k thermal conductivity [W/(m K)]

Nu Nusselt number

q00 heat flux vector [W/m2] r position vector [m]

S heat source due to hyperthermia therapy [W/m3] Sm source terms due to metabolic heating [W/m3]

t time [s]

T average temperature [K]

V Intrinsic phase averaged velocity vector [m/s]

w blood perfusion rate [kg/s]

R vascular resistance

Greek symbols

a

thermal diffusivity [m2/s]

e

porosity

q

a artery blood mass density [kg/m3]

q

v venous blood mass density [kg/m3]

q

s tissue density [kg/m3]

s thermal relaxation time in hyperbolic model [s]

s

q phase lag time of the heat flux [s]

s

T phase lag of the temperature gradient [s]

Subscripts

a arterial blood

v venous blood

eff effective

(3)

different carriers do not exist. In addition, Pennes bioheat equation assumes the constant blood temperature but in anatomical human structure, convective heat transfer between the blood and the tis-sue causes a constant change of the blood temperature. The gov-erning equations for three carriers in a human body[4]are:

e

aCa @Ta @t þVa

r

Ta ¼

e

aka

r

2TaþGaðTsTaÞ þ

e

aS ð6Þ

e

vCv @Tv @t Vv:

r

Tv ¼

e

vkv

r

2TvþGvðTsTvÞ þ

e

vS ð7Þ ð1

e

ÞCs @Ts @t ¼ ð1

e

Þks

r

2 TsþGaðTaTsÞ þGvðTvTsÞ þ ð1

e

ÞSmþ ð1

e

ÞS ð8Þ

where the blood (arterial and venous) and tissue temperature are volume averaged values; the second term of the right hand side of Eqs.(6) and (7)and the second and the third terms of Eq.(8) rep-resent the contribution of blood perfusion on the energy balance in blood and tissue, andGis referred to as a coupling factor between the blood (arterial and venous) and the tissue; kaea, kvev and

ks(1

e

) are the effective thermal conductivities of arterial blood,

venous blood and tissue, respectively. Those three equations in-clude significant effects of the directional blood flow, thermal diffu-sivity and local thermal nonequilibrium between the blood and the surrounding tissue. The two coupling factors are a measure of com-bined convection and perfusion[15]. The porosity of the porous media is equal to summation of volume fractions of arterial and ve-nous blood, i.e.,

e

¼

e

e

v ð9Þ

If the sufficient information about the thermal and anatomical properties are available and also if the blood velocities and direc-tions are known, Eqs. (6)–(8) can be used to determine average temperature distributions. The second term on the left hand side of Eqs(6) and (7) expresses the counter current effect between the arterial and venous blood flows. In Eq.(8), the effect of metab-olites heat generation is also taken into account. The main way to control and regulate the temperature of the human body is via heat exchange as well as the metabolites heat generation between the blood and the solid matrix.

3. Dual phase bioheat equation

The dual phase lag bioheat equation can be obtained by elimi-nate either blood (arterial or venous) or tissue temperature from the multi-temperature model equation. In this paper operator method is used to obtain one equation with tissue temperature as sole unknown. The total energy equation can be established by adding individual energy equations for artery, vein and tissue. Adding Eqs. (6)–(8), the following energy equation can be obtained:

e

aCa @Ta @t þ

e

vCv @Tv @t þ ð1

e

ÞCs @Ts @t þ

e

aCaVa:

r

Ta

e

vCvVv:

r

Tv ¼

e

aka

r

2Taþ

e

vkv

r

2Tvþ ð1

e

Þks

r

2Tsþ

e

aS þ

e

vSþ ð1

e

ÞSmþ ð1

e

ÞS ð10Þ

Under a rapid heating condition, the tissue and blood (both ar-tery and venous) are not at the same temperature at a local level. Following the assumption by Minkowycz et al.[21], it is hypothe-sized that before the onset of equilibrium, the blood temperatures for both artery and vein undergo transient processes can be ob-tained by:

e

aCa

@Ta

@t ¼GaðTsTaÞ ð11Þ

e

vCv@@Ttv¼GvðTsTvÞ ð12Þ

Substituting Eqs.(11) and (12)into Eq.(10), the bioheat equa-tion with tissue temperature as a sole unknown can be expressed as:

s

q @2 @t2Tsþ @ @tTsþ

aCa Ceff Va

vCv Ceff Vv

r

Tsþð1

e

Þ

e

aCs

e

vCaCv@ 3 @t3Ts ¼

a

eff½

r

2Tsþ

s

T @ @tð

r

2 TsÞþð

e

vCvGv

e

aCaVv

e

aCaGa

e

vCvVaÞ GaGvCeff @ @t

r

Ts þ

e

aCa

e

vCv @2S @t2þð1

e

Þ

e

aCa

e

vCv @2Sm @t2 þð1

e

Þks

e

a

e

vCaCv GaGvceff @2 @t2

r

2T sþ½ Sþð1

e

ÞSm Ceff þðGv

e

aCaþGa

e

vCvÞ@@StþðGv

e

aCaþGa

e

vCvÞð1

e

Þ@@Stm ð13Þ

(4)

where the phase lags for heat flux and temperature gradient are gi-ven by:

s

e

aCaGa

e

vCvþ

e

vCvGv

e

aCaþ ð1

e

ÞCsGv

e

aCaþ ð1

e

ÞCs

e

vGaCv GaGvCeff ð14Þ

s

e

aKaGa

e

vCvþ

e

vKvGv

e

aCaþ ð1

e

ÞKsGv

e

aCaþ ð1

e

Þ

e

vGaCvKs GaGvKeff ð15Þ

with the effective properties being defined as,

Ceff¼

e

aCaþ

e

vCvþ ð1

e

ÞCs ð16Þ keff¼

e

akaþ

e

vkvþ ð1

e

Þks ð17Þ

a

eff¼ keff Ceff ð 18Þ

Equation(13)represents the DPL bioheat equation with average tis-sue temperature as a sole unknown and the phase lags times for heat flux and temperature are functions of the properties of artery, vein and solid tissue and the coupling factors between the three carriers. In Eq.(13), the third order time derivative term indicates the effect of three carriers (artery, vein and tissue) system. Although Eq.(13)is more complex than Eq.(5), it is more accurate because it is based on nonequilibrium between different energy carriers. In addition, Eq.(13)has only average tissue temperature as a sole un-known and no arterial and venous temperatures are involved. It ac-counted for the conduction and convection (blood perfusion) effects in the arterial and venous blood. The directions of the blood flow in arteries and veins can be accurately accounted by the convection terms on the left hand side of the equation. Eq.(13)is distinct from Eq.(5)by another fact that the phase lag times can be easily ob-tained from Eqs.(14) and (15).

While Eq.(13)is in the form that can be directly used to obtain the tissue temperature if the arterial and venous blood velocities are known, it would be helpful if it can be casted in the form that is similar to the Pennes bioheat equation. The contribution of blood flow on the temperature distribution is represented by the third term on the left-hand side of Eq.(13)or the second and third terms on the right-hand side of Eq.(8). Since both of these two terms rep-resents the same physical phenomenon, one can expect that[22]:

ð

e

aCaVa

e

vCvVvÞ

r

TsGaðTaTsÞ þGvðTvTsÞ ð19Þ

which converts the effect of blood flow on the tissue temperature to coupling between blood temperatures to tissue temperature. Obvi-ously, the information about the effect of directions of blood flow on the tissue temperature has been, in theory, included in the coupling factorGaandGv.

Substituting Eq. (19)into Eq. (13), the following DPL bioheat equation is obtained

s

q @2 @t2Tsþ @ @tTsþ ð1

e

Þ

e

aCs

e

vCaCv GaGvCeff @3 @t3Ts ¼

a

eff½

r

2 Tsþ

s

T @ @tð

r

2 TsÞ þð

e

vCvGv

e

aCaVv

e

aCaGa

e

vCvVaÞ GaGvCeff @ @t

r

Ts þð1

e

Þks

e

a

e

vCaCv GaGvceff @2 @t2

r

2T s þ½GaðTaTsÞ þGvðTvTsÞ Ceff þ ½Sþ ð1

e

ÞSm Ceff þ

e

aCa

e

vCv@ 2S @t2þ ð1

e

Þ

e

aCa

e

vCv @2Sm @t2 þ ðGv

e

aCa þGa

e

vCvÞ @S @tþ ðGv

e

aCaþGa

e

vCvÞð1

e

Þ @Sm @t ð20Þ

The difference between the present DPL bioheat equation(20)and the classical DPL bioheat equation(5)is that the latter considers heat conduction in tissue only but Eq.(20)considers the contribu-tions to conduction by both tissue and blood.

4. Results and discussion

Eq. (13) or (20) convey one of the most important advantages of the present DPL bioheat equation over the classical DPL bioheat equation(5). The phase lag times for heat flux and temperature gradient can be obtained as functions of known quantities such as the properties of blood and tissue, interphase convection heat transfer coefficient and blood perfusion rate. The present DPL mod-el also reveals that the root of dual phase lag is the nonequilibrium thermal transport between blood and tissue. It is evidence from Eqs.(14) and (15)that the phase lag times are governed by the coupling factor (G), porosity of the medium, and heat capacities of blood and tissues. The coupling factor (G) describes the energy exchange between the (arterial and venous) bloods and their sur-rounding tissues. It is an important property for analyzing the bio-logical system. The coupling factor depends upon convection heat transfer and blood perfusion rate:

Ga¼aahaþwaca ð21Þ

Gv¼avhvþwvcv ð22Þ

For the bundle of vascular artery and veins with diametersda

anddv, the respective coupling factors are[15]:

Table 1

Structure and perfusion coefficient studied in Ref.[24].

Case ds(mm) da(mm) dv(mm) ea ev wa(kg/m3s) wv(kg/m3s) 1 17.83 1.14 1.254 0.004 0.005 1 1.43 2 12.85 1.14 1.254 0.008 0.01 2 2.86 3 10.75 1.14 1.254 0.01 0.014 3 4.29 4 9.7 1.14 1.254 0.014 0.017 4 5.71 5 8.65 1.14 1.254 0.017 0.021 5 7.14 6 19.82 2.28 2.508 0.013 0.02 1 1.43 7 14.42 2.28 2.508 0.025 0.03 2 2..86 8 12.06 2.28 2.508 0.036 0.043 3 4.29 9 10.48 2.28 2.508 0.05 0.06 4 5.71 10 9.92 2.28 2.508 0.053 0.064 5 7.14 11 20.98 4.56 5.016 0.05 0.057 1 1.43 12 15.73 4.56 5.016 0.164 0.102 2 2.86 13 13.58 4.56 5.016 0.11 0.14 3 4.29 14 12.06 4.56 5.016 0.143 0.173 4 5.71 15 11.27 4.56 5.016 0.164 0.2 5 7.14

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Ga¼ 4

e

aKa d2a Nuþwaca ð23Þ Gv¼4

e

vKv d2v Nuþwvcv ð24Þ

where the Nusselt number is approximatelyNu= 4.93[23,24]. With the assumptions of uniform distribution of blood vessel in the tissue [25], different diameters of blood vessels, porosity

ð

e

a¼ ðda=dsÞ2Þ, and blood perfusion rate are investigated and listed

inTable 1. The thermophysical properties of blood and tissue are assumed to be identical: ka=km=kz= 0.5 W/m K,

q

a=

q

m=

q

s= 1020 kg/m3andca=cm=cs= 3770 J/kg K. The perfusion rate is

defined as, the mean pressure difference between artery and vein divided by the vascular resistance[26], i.e.,

w¼PaPv

R ð25Þ

According to Poiseuille–Hagen formula [26], the relation be-tween the volumetric flow rate in a long narrow tube, the viscosity of the fluid and the radius of the tube is expressed mathematically as follows:

F¼ ðPaPvÞ

p

81

g

r

4

L ð26Þ

whereF,

g

; randLare the volumetric flow rate, viscosity, radius and length of the tube. Since the volumetric flow rate is pressure difference divided by resistance, the vascular resistance is ex-pressed as

R¼8

g

L

p

r4 ð27Þ

From the above three Eqs. (25)–(27), it can be seen that the arterial and venous blood perfusion rate are only functions of diameters. The approximate values of lumen diameter and the wall thickness for artery are 4 mm and 1 mm, respectively. The approx-imate values of lumen diameter and wall thickness of vein are 5 mm and 0.5 mm, respectively[26]. Using these approximate val-ues, a relationship established between the diameter for artery and vein is that the diameter of vein is 1.1 times the diameter of artery. And the relation between the two blood perfusion rates is

wv¼ 1:43wa. Those two relations are used to calculate different

properties of venous blood such as porosity, diameter and blood perfusion rate that are summarized inTable 1. The contributions of convection heat transfer and blood perfusion rate along with the above properties in the coupling factor, the phase lag times are listed inTable 2. For all cases inTable 2, the phase lag times for heat flux and temperature gradient are exactly the same

becauseksa¼Csa andksv¼Csv where,ksa¼ks=ka andCsa¼Cs=Ca.

Comparison between the phase lag times shows that the DPL phe-nomena is more pronounced when the blood vessel diameter is large because the phase lag times significantly increase when the diameter of the blood vessel increases.

The thermophysical properties of tissue depend on the type and location of the tissue in the body. Therefore, the assumption of the same thermophysical properties for blood and tissue is not always valid. The DPL phenomena are then investigated for the case that the properties of tissues and blood differ:

q

s= 1000 kg/m3, ks=

0.628 W/m K, cs= 4187 J/kg K,

q

a=

q

v= 1060 kg/m3 and ca= cv=

3860 J/kg K. The thermal conductivity of blood used by Yuan[25]

waskb= 0.5 W/m K and this value agrees with other sources[15].

The thermal conductivities of artery and vein are taken as the same value, i.e.,ka= kv= 0.5 W/m K.Table 3shows the

contribu-tions of convection heat transfer and blood perfusion, in the coupling factors and phase lag times. The phase lag times for heat flux and temperature gradient shown inTable 3are not exactly the same, but the differences are small from 0.07% to 3.0%. With the same properties, phase lag time for heat flux is less than the phase lag time for temperature gradient. When the effect of different thermo physical properties are considered, the phase lag times for the temperature gradient are increased by 3–12% compared to the values inTable 2with the same porosity and blood perfusion rate. The difference between the results inTable 2 and 3is more significant compared to the similar study by Zhang[15]because in this case the venous blood is accounted as an additional carrier. When the diameter of arterial blood vessel increases from 1.14 mm to 4.56 mm, the highest difference from its previous value for

s

qis

over 11% (Case 12). Although the thermal conductivity, specific heats of arterial and venous blood are different from those of the tissue inTable 3, the phase lag times for heat flux and temperature gradient are approximately close to each other although the maximum difference is 7%. Since the fluctuating behavior of phase lag times with the same diameter are shown in this case,Table 4

is generated to show the continuous decreasing phase lag times. It can be seen fromTable 4that with the same blood perfusion rate, the phase lag times are identical to each other and gradually decreasing when the arterial diameter is kept the same (da=

1.14 mm) and tissue diameter varies.

With the same diameter of tissue (ds= 17.83 mm), the effect of

blood perfusion rate on the phase lag times are shown inTable 5. It is seen that the phase lag times for heat flux and temperature gra-dient are identical for all cases and gradually increase from its pre-vious value with the increasing blood perfusion rate.

Fourier’s law can be considered as a special case of the dual phase heat conduction model only when the phase lag times for

Table 2

Coupling factors and phase lag times (ka=sv=ks= 0.5 W/m K,qa=qv=qs= 1050 kg/m3

andca=cv=cs= 3770 J/kg K). Case aaha(W/m3K) avhv(W/m3K) cawa(W/m3K) cvwv(W/m3K) Ga(W/m3K) Gv(W/m3K) sa(s) sT(s) 1 30,348 31,351 3770 5391 34,118 25,960 1.221 1.221 2 60,696 62,702 7540 10,782 68,236 51,920 1.216 1.216 3 83,456 87,783 11,310 16,173 94766 71,610 1.218 1.218 4 106,217 106,594 15,080 21,527 121,297 85,065 1.229 1.229 5 128,978 131,674 18,850 26,918 147,828 104,757 1.223 1.223 6 24,657 31,351 3770 5391 28,428 25,960 4.793 4.793 7 47,418 47,027 7540 10,782 54,959 36,244 4.932 4.932 8 68,283 67,405 11,310 16,173 79,593 51,231 4.906 4.906 9 94,837 94,053 15,080 21,527 109,917 72,526 4.815 4.815 10 100,527 100,323 18,850 26,918 119,377 73,406 4.896 4.896 11 23,709 22,338 3770 5391 27,479 16,947 19.449 19.449 12 39,832 39,973 7540 10,782 47,372 29,190 18.856 18.856 13 52,160 54,864 11,310 16,173 63,470 38,691 18.439 18.439 14 67,808 67,797 15,080 21,527 82,889 46,270 18.094 18.094 15 77,766 78,378 18,850 26,918 96,616 51,460 17.964 17.964

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temperature gradient and heat flux are zero. Under the assumption of Fourier’s law, there is no time lag between the heat flux and temperature gradient. In this paper, DPL bioheat equation with a sole unknown average tissue temperature is obtained for the three

carrier systems. By solving the continuity and momentum equa-tions in porous medium, the intrinsic averaged velocity vector for artery and vein in the solid matrix can be obtained, which is needed to solve Eq. (13) or (20). The intrinsic averaged velocity is

Table 3

Phase lag times for different thermophysical properties of tissue (qs= 1000 kg/m3

,ka= 0.5 W/m K,ks= 0.628 W/m K,cs= 4187 J/kg K,qa=qv= 1060 kg/m3,ca=cv= 3860 J/kg K). Case aaha(W/m3K) avhv(W/m3K) cawa(W/m3K) cvwv(W/m3K) Ga(W/m3K) Gv(W/m3K) sa(s) sT(s) 1 30,348 31,351 3860 5520 34,208 25,831 1.265 1.267 2 60,696 62,702 7720 11,040 68,416 51,662 1.259 1.261 3 75,870 87,783 11,580 16,559 95,036 71,223 1.262 1.265 4 106,217 106,594 15,440 22,041 121,257 84,553 1.273 1.277 5 128,978 131,674 19,300 27,560 148,278 104,114 1.269 1.274 6 24,657 31,351 3860 5520 28,518 25,831 4.948 4.963 7 47,418 47,027 7720 11,040 55,138 35,987 5.121 5.146 8 68,283 67,405 11,580 16,559 79,863 50,845 5.094 5.131 9 94,837 94,053 15,440 22,041 110,277 72,013 4.975 5.023 10 100,527 100,323 19,300 27,560 119,827 72,763 5.089 5.143 11 23,709 22,338 3860 5520 27,569 16,818 20.153 20.345 12 39,832 39,973 7720 11,040 39,831 28,933 20.899 21.236 13 52,160 54,864 11,580 16,559 63,740 38,305 19.197 19.609 14 67,808 67,797 15,440 22,041 83,248 45,756 18.875 19.366 15 77,766 78,378 19,300 27,560 97,066 50,817 18.735 19.280 Table 4

Phase lag times for same blood perfusion rate (qs= 1000 kg/m3,k

s= 0.628 W/m K,cs= 4187 J/kg K,qa=qv= 1060 kg/m3,ca=cv= 3860 J/kg K). wa wv ea ev aaha avhv Ga Gv sq sT 1 1.42857 0.0041 0.005 31,015 31,015 34,785 25,629 1.224 1.224 1 1.42857 0.008 0.010 59,713 59,713 63,483 54,328 1.174 1.174 1 1.42857 0.011 0.014 85,322 85,322 89,092 79,936 1.159 1.159 1 1.42857 0.014 0.017 104,793 104,793 108,563 99,408 1.151 1.151 1 1.42857 0.017 0.021 131,779 131,779 135,549 126,393 1.143 1.143 2 2.8571 0.004 0.005 31,015 31,015 38,555 20,244 1.380 1.380 2 2.8571 0.008 0.010 59,713 59,713 67,253 48,942 1.223 1.223 2 2.8571 0.011 0.014 85,322 85,322 92,862 74,550 1.187 1.187 2 2.8571 0.014 0.017 104,793 104,793 112,333 94,022 1.172 1.172 2 2.8571 0.017 0.021 131,779 131,779 139,319 121,007 1.158 1.158 3 4.2857 0.004 0.005 31,015 31,015 42,325 14,858 1.692 1.692 3 4.2857 0.008 0.010 59,713 59,713 71,023 43,556 1.292 1.292 3 4.2857 0.011 0.014 85,322 85,322 96,632 69,165 1.224 1.224 3 4.2857 0.014 0.017 104,793 104,793 116,103 88,636 1.198 1.198 3 4.2857 0.017 0.021 131,779 131,779 143,089 115,621 1.177 1.177 4 5.7143 0.004 0.005 31,015 31,015 46,095 9472 2.407 2.407 4 5.7143 0.008 0.010 59,713 59,713 74,793 38,170 1.391 1.391 4 5.7143 0.011 0.014 85,322 85,322 100,402 63,779 1.271 1.271 4 5.7143 0.014 0.017 104,793 104,793 119,873 83,250 1.231 1.231 4 5.7143 0.017 0.021 131,779 131,779 146,859 110,236 1.199 1.199 5 7.1429 0.004 0.005 31,015 31,015 49,865 4087 5.091 5.091 5 7.1429 0.008 0.010 59,713 59,713 78,563 32,785 1.532 1.532 5 7.1429 0.011 0.014 85,322 85,322 104,172 58,393 1.332 1.332 5 7.1429 0.014 0.017 104,793 104,793 123,643 77,865 1.272 1.272 5 7.1429 0.017 0.021 131,779 131,779 150,629 104,850 1.225 1.225 Table 5

Phase lag times for the same diameter of tissue (qs= 1000 kg/m3

,ks= 0.628 W/m K,cs= 4187 J/kg K,qa=qv= 1060 kg/m3 ,ca=cv= 3860 J/kg K). wa wv ea ev aaha avhv cawa cvwv Ga Gv sq sT 1 1.429 0.0041 0.005 31,015 31,015 3770 5386 34,785 25,630 1.224 1.224 2 2.857 0.0041 0.005 31,015 31,015 7540 10,771 38,555 20,244 1.381 1.381 3 4.286 0.0041 0.005 31,015 31,015 11,310 16,157 42,325 14,858 1.692 1.692 4 5.714 0.0041 0.005 31,015 31,015 15,080 21,543 46,095 9472 2.407 2.407 5 7.143 0.0041 0.005 31,015 31,015 18,850 26,929 49,865 4087 5.091 5.091 1 1.429 0.0041 0.005 31,015 31,015 3770 5386 34,785 25,630 1.224 1.224 2 2.857 0.0041 0.005 31,015 31,015 7540 10,771 38,555 20,244 1.380 1.380 3 4.286 0.0041 0.005 31,015 31,015 11,310 16,157 42,325 14,858 1.692 1.692 4 5.714 0.0041 0.005 31,015 31,015 15,080 21,543 46,095 9472 2.407 2.407 5 7.143 0.0041 0.005 31,015 31,015 18,850 26,929 49,865 4047 5.091 5.091 1 1.429 0.0041 0.005 31,015 31,015 3770 5386 34,785 25,630 1.224 1.224 2 2.857 0.0041 0.005 31,015 31,015 7540 10,771 38,555 20,243 1.380 1.380 3 4.286 0.0041 0.005 31,015 31,015 11,310 16,157 42,325 14,858 1.692 1.692 4 5.714 0.0041 0.005 31,015 31,015 15,080 21,543 46,095 9472 2.407 2.407 5 7.143 0.0041 0.005 31,015 31,015 18,850 26,929 49,865 4086 5.091 5.091

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much higher in artery than vein, i.e.,Va>Vv. The only way that the

DPL bioheat equation can be simplified to the Pennes’ bioheat equation is only if the there are no lagging in the medium (i.e.,

s

q=

s

T=0). This is different from the case when no internal or

external source is present in the energy equation in which the DPL model is reduced to the classical parabolic energy equation when the phase lag times for heat flux and temperature gradient are equal to each other, even if they are not zero.

The same procedure is done to investigate the convection heat transfer, blood perfusion, coupling factors and phase lag times for brain and muscle and the results are listed inTables 6 and 7. In the case of brain, the phase lag time for heat flux is always less than the phase lag time for temperature gradient as previous. The maxi-mum difference between the phase lag time for heat flux and phase lag time for temperature gradient is 3.011 s (Case 12). In the case of muscle, the same pattern of phase lag times has been observed as the brain. In this case, the maximum difference of two phase lag times is 5.011 s. The values of the phase lag times for brain (Table 6) and muscles (Table 7) are much higher than the values with same thermophysical properties (Table 2) as well as with different ther-mophysical properties for blood and tissue (Table 3). The values of phase lag times for heat flux and temperature gradient with the ef-fect of vein (shown inTables 3 and 4) are more than those without the effect of vein in the biological tissue[15].

5. Conclusions

For the three-carrier system (artery, venous and tissue) in a liv-ing biological system, a dual phase lag bioheat equation with tissue temperature as the sole unknown can be obtained by analyzing

non-equilibrium heat transfer and by eliminating the temperature of arterial and venous blood. The phase lag times for heat flux and temperature gradient are expressed as functions of thermo physi-cal properties of blood and surrounding tissue, interphase convec-tion heat transfer, and the blood perfusion rate in biological system. The novelty of this paper is that the DPL model is derived from nonequilibrium between arterial blood, venous blood and porous media tissue. In addition, the convection of blood, which was not accounted in classical Pennes bioheat equation, is taken into account. If the densities, specific heats and thermal conductiv-ities of arterial and venous blood are similar to the tissue, the phase lag times for heat flux and temperature gradient are identi-cal. When the densities are different and the specific heat and ther-mal conductivities of blood and tissues are considered, the two phase lag times are still similar. The phase lag times studied in this paper range from 1 to 21 s. The non-Fourier thermal behavior (DPL effect) allows us to study the microstructure interactions with heat transport. Due to the presence of blood perfusion in living tissue, the DPL bioheat equation can reduce to Pennes bioheat equation only when both phase lag times are equal to zero.

Acknowledgements

Support for this work by the University of Missouri Research Board and US National Science Foundation (NSF) under Grant No. CBET-0730143 is gratefully acknowledged.

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Table 6

Phase lag time for brain: (qs= 1030 kg/m3

,ks= 0.5395 W/K m,ws= 9.33 kg/m3s,cs= 3680 J/kg K,qa=qv= 1060 kg/m3,ka=kv= 0.5 W/m K andca=cv= 3860 J/kg K). Case aaha(W/m3K) avhv(W/m3K) cawa(W/m3K) cvwv(W/m3K) Ga(W/m3K) Gv(W/m3K) sa(s) sT(s) 1 30,348 31,351 3860 5520 34,208 25,831 1.290 1.472 2 60,696 62,702 7720 11,040 68,416 51,662 1.284 1.464 3 75,870 87,783 11,580 16,559 95,036 71,223 1.286 1.467 4 106,217 106,594 15,440 22,041 121,257 84,553 1.297 1.480 5 128,978 131,674 19,300 27,560 148,278 104,114 1.293 1.476 6 24,657 31,351 3860 5520 28,518 25,831 5.040 5.756 7 47,418 47,027 7720 11,040 55,138 35,987 5.209 5.752 8 68,283 67,405 11,580 16,559 79,863 50,845 5.174 5.947 9 94,837 94,053 15,440 22,041 110,277 72,013 5.044 5.763 10 100,527 100,323 19,300 27,560 119,827 72,763 5.158 5.894 11 23,709 22,338 3860 5520 27,569 16,818 20.438 23.35 12 39,832 39,973 7720 11,040 39,831 28,933 21.100 24.111 13 52,160 54,864 11,580 16,559 63,740 38,305 19.315 22.068 14 67,808 67,797 15,440 22,041 83,248 45,756 18.933 21.590 15 77,766 78,378 19,300 27,560 97,066 50,817 18.752 21.349 Table 7

Phase lag time for muscle: (qs= 1040 kg/m3,k

s= 0.4935 W/K m,ws= 0.63 kg/m3s,cs= 3720 J/kg K,qa=qv= 1060 kg/m3,ka=kv= 0.5 W/m K andca=cv= 3860J/kg K). Case aaha(W/m3K) avhv(W/m3K) cawa(W/m3K) cvwv(W/m3K) Ga(W/m3K) Gv(W/m3K) sa(s) sT(s) 1 30,348 31,351 3860 5520 34,208 25,831 1.264 1.627 2 60,696 62,702 7720 11,040 68,416 51,662 1.258 1.586 3 75,870 87,783 11,580 16,559 95,036 71,223 1.260 1.588 4 106,217 106,594 15,440 22,041 121,257 84,553 1.272 1.602 5 128,978 131,674 19,300 27,560 148,278 104,114 1.268 1.596 6 24,657 31,351 3860 5520 28,518 25,831 4.941 6.222 7 47,418 47,027 7720 11,040 55,138 35,987 5.109 6.422 8 68,283 67,405 11,580 16,559 79,863 50,845 5.078 6.370 9 94,837 94,053 15,440 22,041 110,277 72,013 4.953 6.196 10 100,527 100,323 19,300 27,560 119,827 72,763 5.066 6.333 11 23,709 22,338 3860 5520 27,569 16,818 20.070 25.110 12 39,832 39,973 7720 11,040 39,831 28,933 20.756 25.767 13 52,160 54,864 11,580 16,559 63,740 38,305 19.025 23.464 14 67,808 67,797 15,440 22,041 83,248 45,756 18.675 22.835 15 77,766 78,378 19,300 27,560 97,066 50,817 18.515 22.495

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Figure

Fig. 1. Schematic view of artery and vein surrounding by tissue.
Fig. 1. Schematic view of artery and vein surrounding by tissue. p.3

References