DUAL PHASE LAG BEHAVIOR OF LIVING BIOLOGICAL TISSUES IN LASER

HYPERTHERMIA

_______________________________________

A Thesis

presented to

the Faculty of the Graduate School

at the University of Missouri-Columbia

_______________________________________________________

In Partial Fulfillment

of the Requirements for the Degree

Master of Science

_____________________________________________________

by

NAZIA AFRIN

Dr. Yuwen Zhang, Thesis Supervisor

MAY 2011

The undersigned appointed by the Dean of the Graduate Faculty, have examined a thesis entitled

DUAL PHASE LAG BEHAVIOR OF LIVING BIOLOGICAL TISSUES IN

LASER HYPERTHERMIA

presented by NAZIA AFRIN

a candidate for the degree of Master of Science in Mechanical and Aerospace Engineering,

and hereby certify that in their opinion it is worthy of acceptance

---------------------------------------------------------------------------------

Dr. Yuwen Zhang

-------------------------------------------------------------------------------

Dr. J. K. Chen

--------------------------------------------------------------------------------

Dr. Stephen Montgomery-Smith

ii

ACKNOWLEDGEMENTS

I am highly grateful to my supervisor Professor Dr. Yuwen Zhang, Department of

Mechanical and Aerospace Engineering for his encouragement, support, patience and guidance

throughout this research work also in daily life. This dissertation would not have been possible

without guidance and help of him. He contributed and extended his valuable assistant in the

preparation nd completion of this study. I would like to thank the members of my thesis

evaluation committee, Dr. J. K. Chen and Dr. Stephen Montgomery-Smith for giving the time to

provide valuable comments and criticism. Special thanks must be extended to Dr. Stephen

Montgomery-Smith for his assistance and courage of confidence.

I would like to express my sincere thanks to Professor Dr. Robert Tzou, Chairman of

Department of Mechanical and Aerospace Engineering for all the guidance, assistance and help

throughout this study. I am very grateful to him.

I would like to thanks Marilyn Estes and Melanie Carraher for all their helps on my

graduate paperwork.

I would like to thank my all coworkers, Jianhua Zhou, Sejoong Kim, Tao Jia, Yijin Mao,

Yunpeng Ren. It is really great time to work with them and I really enjoy their company in our

lab.

Also I would like to thank my friends, Rilya Rumbayan, Roxana Mtz C, Srisharan G.

Govindarajan and Weijun Huang. It is my pleasure to spend time with such wonderful friends

and I am wishing their successful life.

iii

I would like to express my gratitude to my parents, Shamsun Nahar Islam and Late F. K.

M Aminul Islam. My mother always gives me inspirations all the time about my study although

she is far away from me. Even though my father is not alive in this world, however, still I feel his

contribution on my every success in my life.

Support for this work by the University of Missouri Research Board and U.S. National

Science Foundation (NSF) under Grant No. CBET-0730143 is gratefully acknowledged.

iv

TABLE OF CONTENTS

ACKNOWLEDGEMENT ………………………………………………………………………ii

LIST OF FIGURES…………………………………………………………………………….vii

LIST OF TABLES ……………………………………………………………………………...ix

NOMENCLATURE ………………………………………………………………………….....x

ABSTRACT …………………………………………………………………………………....xii

CHAPTER 1: INTRODUCTION………………………………………………………………..1

1.1 Background………………………………………………………………………………1

1.2 Heat Conduction Model………………………………………………………………….4

1.2.1 Pennes Bioheat Equation………………………...…………………………………4

1.2.2 Thermal Wave Model……………………………………………………………...5

1.2.3 Dual Phase Lag Model……………………………………………………………..6

1.3 Thesis Objectives…………………………………………………………………………7

CHAPTER 2: NUMERICAL SIMULATION OF THERMAL DAMAGE TO LIVING

BIOLOGICAL TISSUES INDUCED BY LASER IRRADIATION BASED ON A

GENERALIZED DUAL PHASE LAG MODEL……………………………………………....9

2.1 Tissue-laser interactions…………………………………………………………………9

v

2.2 Laser parameters…………………………………………………………………….….10

2.3 Classical DPL model…………………………………………………………………...11

2.4 Generalized DPL model……………………………………………………….…….…12

2.5 Physical model……………………………………………………………………….…17

2.6 Problem statement……………………………………………………………………....17

2.7 Calculation of the laser irradiance………………………………………………………18

2.8 Calculation of thermal damage parameter………………………………………………19

2.9 Numerical Analysis……………………………………………………………………..20

2.9.1 Discretization scheme of space…………………………………………………...20

2.9.2 Discretization of governing equation…………………………………………......20

2.10 Results and Discussions…………………………………………………………………22

2.11 Conclusion………………………………………………………………………………38

CHAPTER 3: THERMAL LAGGING IN LIVING BIOLOGICAL TISSUE BASED ON

NONEQUILIBRIUM HEAT TRANSFER BETWEEN TISSUE, ARTERIAL AND VENOUS

BLOODS………………………………………………………………………………………..40

3.1 Heat transfer in arteries, venous and solid tissue……………………………………….40

3.2 Governing equations…………………………………………………………………….41

3.3 Dual phase bioheat equation for three carriers system…………………………………..42

vi

3.4 Results and Discussion……………………………………………………………….….46

3.5 Conclusion……………………………………………………………………………….56

CHAPTER 4: SUMMERY AND CONCLUSIONS…………………………………………….59

REFERENCES…………………………………………………………………………………..60

vii

LIST OF FIGURES

Figure Page

Fig. 1 Physical model and grid system ………………………………………………………….17

Fig. 2 Temperature evolution at the irradiated surface of a highly absorbing tissue …………...23

Fig. 3 Thermal damage at the irradiated surface of a highly absorbing tissue…………………..24

Fig. 4 Temperature evolution at the irradiated surface of a scattering tissue …………………...25

Fig. 5 Thermal damage at the irradiated surface of a scattering tissue…………………………..26

Fig. 6 Temperature distribution at the irradiated surface of a scattering tissue for different τT

values…………………………………………………………………………………………… 27

Fig. 7 Thermal damage at the irradiated surface of a scattering tissue for different τT values…..28

Fig. 8 Temperature distribution at the irradiated surface of a scattering tissue calculated by the

non-equilibrium DPL model for different τq values……………………………………………..29

Fig. 9 Thermal damage at the irradiated surface of a scattering tissue calculated by the

nonequilibrium DPL model for different τq values………………………………………………30

Fig. 10 Effects of laser irradiance on temperature at the irradiated surface of a scattering

tissue……………………………………………………………………………………………..31

Fig. 11 Effects of laser irradiance on damage parameter at the irradiated surface of a scattering

tissue……………………………………………………………………………………………..32

.

viii

Fig. 12 Effects of laser exposure time on temperature at the irradiated surface of a scattering

tissue……………………………………………………………………………………………33

Fig. 13 Effects of laser exposure time on damage parameter at the irradiated surface of a

scattering tissue…………………………………………………………………………………34

Fig. 14 Effects of coupling factor on temperature at the irradiated surface of a scattering

tissue…………………………………………………………………………………………….35

Fig. 15 Effects of coupling factor on damage parameter at the irradiated surface of a scattering

tissue…………………………………………………………………………………………….36

Fig. 16 Effects of blood perfusion rate on temperature at the irradiated surface of a scattering

tissue…………………………………………………………………………………………….37

Fig. 17 Effects of blood perfusion rate on thermal damage at the irradiated surface of a scattering

tissue………………………………………………………………………………………….....38

Fig. 18 Schematic view of artery and vein surrounding by tissue………………………………40

ix

LIST OF TABLES

Table Page

Table: 1 Structure and perfusion coefficient studied in Ref [27]……………………………..47

Table: 2 Coupling factors and phase lag times………………………………………………..49

Table: 3 Phase lag times for different thermo physical properties of tissue…………………..50

Table: 4 Phase lag times for same blood perfusion rate ……………………………………...52

Table: 5 Phase lag times for the same diameter of tissue……………………………………..53

Table: 6 Phase lag time for brain……………………………………………………………...55

Table: 7 Phase lag time for muscle…………………………………………………………....56

x

NOMENCLATURES

a specific heat transfer area [m2/ m3]

c specific heat of artery [J/ (kg K)]

G coupling factor between blood and tissue [W/ (m3 K)]

h heat transfer coefficient [W/(m2 K)]

k thermal conductivity [W/(m K)]

q heat flux vector [W/m2]

r position vector [m]

QL heat source due to hyperthermia therapy [W/m3]

Qm 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 [m3/m3 tissue]

Rd diffuse reflectance of light

A frequency factor [s-1]

R universal gas constant [J/(mol K)]

E energy of activation of denaturation reaction [J/ mol]

Nu Nusselt number

db diameter of the blood vessel [m]

S heat source due to hyperthermia therapy [W/m3]

Sm source terms due to metabolic heating [W/m3]

R vascular resistance

Greek symbols

α thermal diffusivity [m2/s]

porosity

ρ density [kg/m3]

ρa artery blood mass density [kg/m3]

xi

ρv venous blood mass density [Kg/m3]

ρs tissue density [kg/m3]

τq phase lag time of the heat flux [s]

τT phase lag of the temperature gradient [s]

τL laser exposure time

φin incident laser irradiance

µa absorption coefficient [cm-1]

µs scattering coefficient [cm-1]

φ (x) local light irradiance

δ effective penetration depth

g scattering anisotropy

Ω damage parameter

(δx)w distance between W and P (two grid points)

(δx)e distance between P and E( two grid points)

Subscripts

s solid matrix (tissue)

b blood vessel

a arterial blood

v venous blood

eff effective

xii

ABSTRACT

A Generalized dual phase lag behavior for living biological tissues are investigated for blood and

tissues and also developed a generalized dual phase model for artery, vein and tissues in this

thesis. There are two parts of this thesis:

a) A Generalized dual phase lag (DPL) bioheat model based on the non equilibrium heat

transfer in living biological tissues is applied to investigate thermal damage induced by

laser irradiation. Comparisons of the temperature responses and thermal damages

between the generalized and classical DPL bioheat model, derived from the constitutive

DPL model and Pennes bioheat equation, and as well as Fourier heat conduction model

are carried out. It is shown that the generalized DPL model could predict significantly

different temperature and thermal damage from the classical DPL model and Fourier heat

conduction model. The generalized DPL equation can reduce to the classical Pennes heat

conduction equation only when the phase lag times of temperature gradient (τT) and heat

flux vector (τq) are both zero. The effects of laser parameters such as laser exposure time,

laser irradiance, and coupling factor on the thermal damage are also studied.

b) 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 nonequilibrium heat transfer between tissue, artery and

venous bloods is presented in this thesis. 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.

xiii

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 muscles are also discussed.

1

CHAPTER 1

INTRODUCTION

1.1 Background

The role of lasers in medical applications has increased dramatically over the past four

decades. Laser radiation possesses unique characteristics and has been extensively used in

clinical science for diagnostic and therapeutic applications. Most of the laser medical treatments

such as surgery, angioplasty, hyperthermia of tumors and laser tissue soldering are concern with

the thermal effects.

Welch described a three-step model for predicting laser induced thermal damage in

biological tissues [1]. The laser energy deposition was described based on the light propagation

in tissue first, followed by analyzing thermal response by solving a heat conduction equation.

Finally, the damage of the tissue was determined based on protein denaturation evaluated by a

chemical rate process equation. Many researchers have adopted this approach by applying

different methods to solve the problems involved in the process. In most cases, the bioheat

conduction equation based on Fourier’s law was used to investigate laser-induced damage in

biological tissue.

A real biological tissue can be treated as a non-homogeneous fluid saturated porous

medium. Heat transfer in living biological tissues involves multiple mechanisms including

conduction in tissue, convection between blood and tissues, blood perfusion or advection and

diffusion through micro vascular beds, and metabolic heat generation [2]. To date, Pennes

bioheat equation [3] has most widely been applied to obtain temperature distribution in living

biological tissues. It is assumed that when the venous blood flows from the capillary bed to the

main supply vein, its temperature remains the same as the tissue temperature regardless the size

2

of the vessel and the flow rate. The heat conduction in biological tissue is modeled by using

Fourier’s law, which assumes thermal disturbance propagates at an infinite velocity. An infinite

speed of heat propagation implies that a thermal disturbance applied at a certain location in a

medium can be sensed immediately anywhere in the medium. There are many situations where

the assumption of infinite speed of thermal propagation could be inadequate. Parallel to Fourier’s

law, in thermal wave model (Cattaneo Vernotte wave model [4-5]), the heat flux and the

temperature gradient across a material volume are assumed to occur at different instants of time.

Although allowing for a delayed response between the heat flux and temperature gradient, the

temperature gradient is always the cause for heat flux while the heat flux is always the effect [6].

Tzou [6-8] established a DPL model which introduces two different time delays between the

temperature gradient and the heat flux. The aim of this model was to remove the precedence

assumption that was made in the thermal wave model. It allows either the temperature gradient to

precede the heat flux or the heat flux to precede the temperature gradient in a transient process.

Recently the DPL model has attracted considerable interests in the field of engineering and

medical science [8]. It has been used to interpret the non-Fourier heat conduction phenomenon in

the processed meats [9].

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

tissues, blood perfusion or advection and diffusion through micro vascular 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 transfer 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

3

tissue, and also on the human body geometry [2]. 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 porous space of the medium. So, heat transfer phenomenon can be

considered as convection heat transfer in porous medium with internal heat generation.

Pennes [3] bioheat equation is the most widely applied model for temperature distribution in

the living biological tissues. The effect 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 capillary 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 [10] and four

layer model [11] 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

venous blood flows [12]. The axial heat transfer and temperature gradient are not negligible

which post additional challenge in analyzing bioheat transport in living biological tissue. That

means analyzing the heat transfer phenomenon in living biological tissue should consider the

effects of direction of the blood flow. The complex vascular architecture is the fundamental

problem in heat transfer process within the human body [13], 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

4

interaction with the environment in a complete model. In their series of papers, Weinbaum et al.

[14-16] proposed a bioheat equation considering the variation of the number, density and size

and flow velocity of the countercurrent arterial-venous vessels. That model was applied for the

single organ rather than the whole human body for thermoregulation.

1.2 Heat Conduction Models

1.2.1 Pennes Bioheat Equation:

The general bioheat equation considering blood perfusion and metabolic heat generation

is as follows [3]:

( )L m b b b b

Tc Q Q w c T T

t xρ ρ

∂ ∂= − + + + −

∂ ∂

q (1)

where q is heat flux; ρ and c are respectively density and specific heat of the tissue ; ρb and cb are

the density and specific heat of blood, wb is the blood perfusion rate; Tb and T are the

temperatures of blood and tissue; Qm and QL are the source term due to the metabolic heating and

hyperthermia therapy. Pennes bioheat equation was obtained by applying the classical Fourier’s

law of heat conduction in Eq. (1) and assuming the uniform blood temperature Tb throughout the

tissue. The vein temperature was assumed to be same to the tissue temperature. In addition, the

blood perfusion effect was assumed to be homogeneous and isentropic.

5

1.2.2 Thermal Wave model

With nonhomogeneous biological structures, heat flux responds to the temperature gradient

via a relaxation behavior [17]. Cattaneo [4] and Vernotte [5] simultaneously suggested a

modified heat flux model:

, , , (2)

Equation (2) assumes that the heat flux and the temperature gradient occur at different times. The

delay between the heat flux and temperature gradient is defined as the thermal relaxation time; τ.

Kaminski [18] suggested that the theoretical value of the thermal relaxation time τ for biological

tissue is in the range of 20-30 s while the experimental value was observed to be 16 s [19]. If Eq.

(2) is used in replacement of the classical Fourier’s law of heat conduction in derivation of

Pennes bioheat equation, the following bio heat equation is obtained [20-21]:

1

!"#!

!" !

(3)

where, is phase lag time for heat flux, $ is the blood perfusion rate, c is heat capacity of

tissue , %& is metabolic heat generation and % is heat source due to hyperthermia therapy. The

second order derivative of temperature with respect to time appears, and for this reason Eq. (3) is

referred to as hyperbolic bioheat equation [22]. In arrival to Eq. (3), it is assumed that the

temperature gradient is established before heat flux, which is referred to as gradient–precedence

type heat flow.

6

1.2.3 Dual Phase Lag Model

Tzou [6] 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.,

′′, ', 4

where, τq is the phase lag for the heat flux vector, and τT is 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 (τq > τT ) , the heat transfer is gradient-precedence type. On the other hand, if the

temperature gradient results in the heat flux at an early time (τq < τT), the heat flow is called flux-

precedence type. The first order approximation of the Eq. (4) is

′′ ′′

(5)

If the classical Fourier’s law of conduction is replaced by Eq. (5), the bioheat equation

becomes

( ) ( )2

2 2

b s21 T T

s b b s b bq q s s T s

s s

qm m

s s s s

T w c T w cT T

t C t t C

S S S S

c c t t

τ τ α τ

τ

ρ ρ

∂ ∂ ∂ + + = ∇ + ∇ + − ∂ ∂ ∂

+ ∂ ∂ + + +

∂ ∂

(6)

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

s

w c T

C t

τ ∂

∂=0) and the condition

0q T

τ τ= = , Eq. (6) reduces to the classical bioheat equation. The DPL bioheat equation (6) 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

7

still not widely accepted by the researchers in the field. Zhang [20] developed a generalized DPL

bioheat equation based on nonequilibrium between arterial blood and tissue. The phase lag times

were expressed in terms of properties of blood and tissue, interphase convection 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 arterial blood, venous blood and

surrounding tissues. The DPL model proposed in Ref. [20] 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 thesis, 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.

1.3 Thesis Objectives

The main objectives of this thesis are:

1) A Generalized dual phase lag (DPL) bioheat model based on the non equilibrium heat

transfer in living biological tissues is applied to investigate thermal damage induced by

laser irradiation.

2) Comparisons of the temperature responses and thermal damages between the generalized

and classical DPL bioheat model, derived from the constitutive DPL model and Pennes

bioheat equation, are carried out.

3) The effects of laser parameters such as laser exposure time, laser irradiance, and coupling

factor on the thermal damage are also studied.

8

4) A generalized dual-phase lag model for living biological tissues based on nonequilibrium

heat transfer between tissue, artery and venous bloods is obtained.

5) The phase lag times for temperature gradient and phase lag times for heat flux vector are

calculated for different properties and also for brain and muscles.

9

CHAPTER 2

Numerical Simulation of Thermal Damage to Living Biological Tissues

Induced by Laser Irradiation based on a Generalized Dual Phase Lag Model

In this chapter, a generalized DPL model obtained by the performing volume average to

the local instantaneous energy equation for the blood and the tissue is used to investigate the

temperature response and thermal damage induced by laser irradiation. Comparisons of the

thermal responses and thermal damages between a generalized DPL, classical DPL model and

Fourier bioheat model are carried out also.

To study the evolution of temperature and thermal damage due to the laser irradiation, the

most important thing is the fundamental understanding of laser tissue interactions.

2.1 Tissue-laser interactions

Laser can interact with the tissue in four key ways: transmission, reflection, scattering

and absorption [23]. Transmission refers to the passing of laser through tissue without giving any

effect on that tissue or even the properties of the light. Reflection refers to the repelling of light

off the surface of the tissue without entering to the tissue. Approximately 4% to 7% of light is

reflected of skin. The amount of light reflection is proportional to the angle of incidence with the

least reflection occurring when the laser beam directed perpendicular to the tissue. The scattering

of light occurs after light has entered in the tissue. Scattering occurs is due to the heterogeneous

structure of tissue with the variations in particles size and the index of refraction between

10

different parts of tissue. Scattering spreads out the beam of light within the tissue which results n

radiation of area and then anticipated. Scattering depends on the depth of penetration because it

can be occur forward as well as backward. The amount of scattering is inversely proportional to

the wave length of the laser. Longer wave length laser thus penetrate tissue more deeply. And

laser light absorption by specific tissue targets is the fundamental goal of clinical lasers. The

absorption of the photons of light is reasonable for its effects on the tissue. The components of

the tissue that absorbed the photons depend on wavelength. These light absorbing tissue

components are known as chromophores. Absorption of energy by a chromophores results in

conversion of energy to thermal heat.

2.2 Laser parameters

1. Beam characteristics

An important feature of the light produced by a laser is how the intensity is distributed

across the beam diameter [23]. Most cutaneous lasers produce a beam with a Gaussian

profile in which the intensity peaks at the center of the beam and attenuates at the

periphery.

2. Spot Size

The spot size of a laser is equivalent to the laser beam cross section. The spot size

directly affects the fluence and the irradiance of a laser beam. Fluence and irradiance are

inversely proportional to the square of the radius of the spot size. A small spot size allows

more scattering both backwards and sideways than a larger spot size.

11

3. Pulse duration

Laser light can be delivered in a continuous wave or a pulse wave. Continuous wave

lasers emit a constant beam of light that may result in nonselective tissue injury. Pulsed

delivery of laser light allows for more selective tissue damage. The duration of time of

exposure of a laser beam determines the rate at which the laser energy is delivered. The

thermal relaxation time is generally proportional to the size of the target structure.

2.3 Classical DPL model

To capture the thermal lagging behaviors in biological tissues composing of

nonhomogeneous inner structures, the two lagging times will be include in the bioheat

conduction equation. Zhou et al. [24] proposed a DPL bioheat conduction model, together with a

broad beam irradiation method [25] and the rate process equation to investigate thermal damage

in laser-irradiated biological tissues. The temperature and damage parameter of the tissue was

compared with those obtained from classical Fourier and the hyperbolic bioheat conduction

model. It was found that the DPL heat conduction model predicted significantly different

temperature and thermal damage in tissue from hyperbolic and Fourier’s heat conduction model.

Combining Eqs. (1) and (5), while eliminating the heat flux q leads to the following DPL

bioheat equation for tissue temperature T (x, t):

2 2 3

2 2 2(1 ) ( )b b b m b b bL

q q T b

w c Q w cQT T T TT T

t c t x t x c c c

ρ ρτ τ α ατ

ρ ρ ρ ρ

∂ ∂ ∂ ∂+ + = + + + + −

∂ ∂ ∂ ∂ ∂ (7)

Alternatively, Zhou et al. [24] obtained the following bioheat conduction equation with

heat flux as unknown to simulate the DPL heat conduction in tissue:

12

2 2 3 2

2 2 2

Lq T b b b b b b T

Q T Tw c w c

t t x t x x x t xτ α ατ α α ρ α ρ τ

∂∂ ∂ ∂ ∂ ∂ ∂+ = + − + +

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

q q q q (8)

where α is thermal diffusivity of tissue. The transport of thermal energy in living tissue involves

conduction in tissue, convection between blood and tissues, blood perfusion or advection and

diffusion through micro vascular beds and metabolic heat generation.

2.4 Generalized DPL model

The DPL bioheat equation obtained by simply modification of the fundamental Pennes

bioheat equation is not very convincing approach. The main foundation of dual phase lag

phenomena in the living biological tissue is nonequilibrium between the blood and the

surrounding tissue. Zhang [20] derived a generalized DPL model based on nonequilibrium heat

transfer [26] in living biological tissue. It was demonstrated that, the phase lag times depended

on intrinsic properties of blood and tissue, blood perfusion rate and convection heat transfer. The

values of phase lag times might vary from place to place in human body.

For heat transfer in living biological tissues, the temperatures of blood and tissue are

different and the equilibrium assumption is invalid. Although Pennes bioheat equation assumed

nonequilibrium assumption but the blood temperature is assumed to be a constant. In reality, the

convective heat transfer between blood and tissue and blood perfusion results the change of

temperatures. Xuan and Roetzel [12] obtained a two-temperature model by performing volume

average to the local instantaneous governing equation for blood and tissues. With the presence of

internal heat source by hyperthermia therapy, the following energy equations for blood and tissue

are

13

,[ . ] .( ) ( )bb b b b eff b b b s b L

Tc T k T a h T T Q

tερ ε

∂+ ∇ = ∇ ∇ + − +

∂V (9)

,(1 ) .( ) ( ) (1 ) (1 )L

ss s s eff s b b b s m

Tc k T a h T T Q Q

tε ρ ε ε

∂− = ∇ ∇ + − + − + −

∂ (10)

where ab is the specific heat transfer area, and hb is the heat transfer coefficient inside the blood

vessel, the temperatures of blood and tissue are volume averaged values; kb,eff and ks,eff are

effective thermal conductivity of blood and solid matrix tissue, respectively. Those energy

equations include significant effects from the blood flow and direction, thermal diffusion and

local nonequilibrium between blood and the surrounding tissues. The convective heat transfer

coefficient, hb and the specific area on the blood vessel in the tissue, ab accounts the effects of

vascular geometry and size of the blood vessel.

Comparing Eq. (10) and with Eq. (1) it can shown that the blood perfusion term is simply

replaced by the convective heat transfer. But the interfacial convective heat transfer and blood

perfusion are totally different processes [27]. In the presence of blood perfusion, due to the

temperature difference between blood and tissue, the convective heat transfer occurs. And on the

other hand, blood perfusion is the process of delivery the nutrition of the arterial blood to the

capillary bed in the biological tissue. Nakayama and Kuwahara [27] presented a developed

mathematical model based on volume averaging theory and the following governing equations

for bioheat transfer in tissue:

,[ . ] .( ) ( ) ( )bb b b b eff b b b s b b b s b L

Tc T k T a h T T w c T T Q

tερ ε

∂+ ∇ = ∇ ∇ + − + − +

∂V (11)

,(1 ) .( ) ( ) ( ) (1 ) (1 )L

ss s s eff s b b b s b b b s m

Tc k T a h T T w c T T Q Q

tε ρ ε ε

∂− = ∇ ∇ + − + − + − + −

∂ (12)

14

Considering not only interfacial convection heat transfer but also the blood perfusion, the

two step model can be written in the following forms [20]:

2[ . ] ( )bb b b b b s b L

Tc T k T G T T Q

tερ ε ε

∂+ ∇ = ∇ + − +

∂V (13)

2(1 ) (1 ) ( ) (1 ) (1 )L

ss s s s b s m

Tc k T G T T Q Q

tε ρ ε ε ε

∂− = − ∇ + − + − + −

∂ (14)

where ε is a proportional rate, subscript s is referred to tissue, and G is coupling factor between

blood and tissue and can be expressed as follows:

b b b bG a h w c= + (15)

It is evident from Eq. (15) that the coupling factor depends upon convection heat transfer

and blood perfusion rate.

Dual phase lag bioheat equation can be obtained by eliminating either tissue or blood

temperature from the two temperature model. Adding Eqs. (13) and (14) the following equation

can be obtained:

2 2(1 ) . (1 ) (1 )L

b sb b s s b b b b b s s m

T Tc c c T k T k T Q Q

t tερ ε ρ ερ ε ε ε

∂ ∂+ − + ∇ = ∇ + − ∇ + − +

∂ ∂V (16)

Minkowycz at al [28] assumed the assumption that before onset of equilibrium, the

temperature of blood undergoes a transient process. This assumption can be written by this

following equation:

( )bb b s b

Tc G T T

tερ

∂= −

∂ (17)

15

The rearrange form of this equation is

b b bs b

c TT T

G t

ερ ∂= +

∂ (18)

Substituting Eq. (18) into Eq. (16), the following equation with the blood temperature as

sole unknown is obtained:

22 2

2

(1 ). [ ( )]

( ) ( )L

mb b b bq b eff b T b

eff eff

Q QT T cT T T

t t c t c

εερτ α τ

ρ ρ

− +∂ ∂ ∂+ + ∇ = ∇ + ∇ +

∂ ∂ ∂V (19)

where the phase lags for heat flux and temperature gradient are

(1 )

( )

b b s sq

eff

c c

G c

ε ε ρ ρτ

ρ

−= (20)

(1 ) b b sT

eff

c k

Gk

ε ε ρτ

−= (21)

where

( ) (1 )eff b b s s

c c cρ ερ ε ρ= + − (22)

(1 )eff b s

k k kε ε= + − (23)

( )

eff

eff

eff

k

cα

ρ= (24)

To obtained the equation with tissue temperature as sole unknown, Eq. (19) can be

substitute on Eq. (18) and the final bioheat equation is

16

22 2

2

(1 )[ ( )] [(1 )

( ) ( ) ( )

]

s s b b m L b b mq eff s T s

eff eff eff

L

T T c Q Q c QT T

t t c t c G c t

Q

t

ερ ε ερτ α τ ε

ρ ρ ρ

∂ ∂ − + ∂∂+ + = ∇ + ∇ + + −

∂ ∂ ∂ ∂

∂+

∂

(25)

The contribution of blood flow on the temperature distribution is represented by the third

term on the left –hand side of Eq. (25) or the second term on the right –hand side of Eq. (14).

These two terms represent same physical phenomenon, one can write [29]

. ( )b b s b sc T G T Tερ ∇ ≈ −V (26)

Substituting Eq. (26) into Eq. (25), Zhang obtained the following equation with tissue

temperature as sole unknown [20]

22 2

2

(1 )[ ( )] ( )

( ) ( )

[(1 ) ]( )

s s m Lq eff s T s b s

eff eff

b b m L

eff

T T Q QGT T T T

t t t c c

c Q Q

G c t t

ετ α τ

ρ ρ

ερε

ρ

∂ ∂ − +∂+ = ∇ + ∇ + − +

∂ ∂ ∂

∂ ∂+ − +

∂ ∂

(27)

This is the generalized dual phase lag (DPL) bioheat equation based on the non

equilibrium heat transfer in living biological tissues.

The objective of this paper is to investigate temperature response and thermal damage

induced by laser irradiation using the generalized dual phase lag (DPL) bioheat model based on

the non equilibrium heat transfer in living biological tissues. A generalized DPL model in terms

of heat flux can be obtained from the two step model and the Dual phase lag model as follows:

2 2 3 2

2 2 2 (1 ) (1 )

α α ττ α α τ α

ε ε

∂∂ ∂ ∂ ∂ ∂ ∂+ = + − + +

∂ ∂ ∂ ∂ ∂ ∂ − ∂ − ∂ ∂s s TL

q s s T s

G GQq q q q T T

t t x t x x x t x (28)

Equation (28) will be used as the governing equation in this study.

2.5 Physical Model

A finite slab of a biological tissue with a thickness ‘L’ and initial temperature T

considered. A flat-top laser beam is applied normally to the left

(Fig. 1). A 1-D model will be sufficient to analyzing the thermal response of the heated medium

when the spot size of the broad beam laser is much

effected zone for the time peri

thermally insulated (q = 0) while the boundary condition at the left

light absorption and scattering of

Fig. 1

2.6 Problem Statements

For highly absorbed tissues, the laser heating is approximated as boundary condition of

second kind. The laser volumetric heat source or laser irradiance, Q

conditions are given by [24]:

(1 )φ= −in dq R for x

17

Equation (28) will be used as the governing equation in this study.

finite slab of a biological tissue with a thickness ‘L’ and initial temperature T

laser beam is applied normally to the left surface of the slab at time t

D model will be sufficient to analyzing the thermal response of the heated medium

when the spot size of the broad beam laser is much larger than the thickness of the thermally

effected zone for the time period of interest. The right boundary surface is assumed

while the boundary condition at the left surface depends on the laser

of tissues.

Fig. 1 Physical model and grid system

Problem Statements

For highly absorbed tissues, the laser heating is approximated as boundary condition of

he laser volumetric heat source or laser irradiance, QL, is zero and the boundary

or x = 0 when 0 < t < Lτ

finite slab of a biological tissue with a thickness ‘L’ and initial temperature T0 is

surface of the slab at time t = 0+

D model will be sufficient to analyzing the thermal response of the heated medium

than the thickness of the thermally

is assumed to be

surface depends on the laser

For highly absorbed tissues, the laser heating is approximated as boundary condition of

is zero and the boundary

(29)

18

0=q for x = L when 0 < t < Lτ (30)

where τL is the laser exposure time, φin is the incident laser irradiance and Rd is the diffuse

reflectance of light at the irradiated surface.

For strongly scattering tissues, laser heating is considered as a body heat source (QL ≠ 0)

but the irradiated surface is thermally insulated. The boundary conditions in this case can be

represented as:

q = 0 for x = 0 when 0 < t < Lτ

(31)

q = 0 for x = L when 0 < t < Lτ (32)

The initial conditions for both cases are:

q = 0 and 0∂

=∂

q

t for 0 < x < L, t = 0 (33)

2.7 Calculation of the laser irradiance (QL)

When the laser light irradiation is absorbed within a very small depth of tissue (~1 µm),

the laser heating can be predicated by considering the laser irradiation as a surface heat flux on

the irradiated surface (see Eq. (29)). When the scattering is considerable over the visible and near

infrared wavelength [30], heat flux boundary condition is not enough to describe the laser

deposition into a tissue. Rather, the laser light attenuation depends on the properties of laser light

and its propagation. The absorbed laser is considered as a body heat source. The laser volumetric

heat source can be determined as follows:

( ) ( )L aQ x xµ φ= (34)

19

where aµ is the absorption coefficient, and ( )xφ is the local light irradiance varying with depth

of the tissue.

To calculate light distribution in scattering tissue, a broad beam laser method [25] is

adopted and the light distribution can be determined by the following relation:

1 1 2 2( ) [ exp( / ) exp( / )]inx C k z C k zφ φ δ δ= − − − (35)

where δ is the effective penetration depth; C1, C2, k1 and k2 are determined by Monte Carlo

solutions, depending on the diffuse reflectance, Rd; the effective penetration depth δ can be

obtained from the diffusion theory as

1

3 [ (1 )]a a s

gδ

µ µ µ=

+ − (36)

where µs is the scattering coefficient and g is the scattering anisotropy. Equation (36) is valid

when 0.1≤ ( )

s

a s

µµ µ+

≤ 0.999 and 0.7 ≤ g ≤ 0.9 [25].

2.8 Calculation of thermal damage parameter (Ω)

The damage parameter is evaluated according to the Arrenius equation [1, 31]:

0

exp( )ft

t

EA dt

RTΩ = −∫ (37)

where A is the frequency factor, 3.1×1098 s-1 [1]; E is the energy of activation of denaturation

reaction, 6.28×105 J/mol [1]; R is the universal gas constant, 8.314 J/ (mol. K); T is the absolute

20

temperature of the tissue at the location where thermal damage is evaluated; t0 is the time at

onset of laser exposure; and tf is the time of thermal damage evaluation. When Ω = 1.0, the tissue

is assumed irreversibly damaged which causes the denaturation of 63% of the molecules.

2.9 Numerical Analysis

2.9.1 Discretization scheme of space

The total thickness L is divided into (L1-1) equal width control volumes (Fig. 1). The

grid points are located at the geometric center of each control volume. w and e denote the faces

of the control volume where P is located, and W and E are the adjacent grid points. ∆x is the

width of each control volume. (δx)w and (δx)e are the distance of the two adjacent grid points

measured from the point P, respectively.

2.9.2 Discretization of governing equation

The finite volume method [32] is employed to discretize the governing equation (28) and

the boundary conditions. Performing integration of Eq. (28) over the control volume of grid point

P (Fig. 1) and over the time step from t to t+∆t leads to:

2 2 3

2 2 2

2

( ) ((1 )

)(1 )

ατ α α τ α

ε

α τ

ε

+∆ +∆∂∂ ∂ ∂ ∂ ∂

+ = + − +∂ ∂ ∂ ∂ ∂ ∂ − ∂

∂+

− ∂ ∂

∫ ∫ ∫ ∫e t t e t t

sLq s s T s

w t w t

s T

GQq q q q Tdtdx

t t x t x x x

G Tdtdx

t x

(38)

Applying backward difference in time and piecewise-linear profile in space, the

following algebraic equation for heat flux can be obtained from Eq. (38):

+∆ +∆ +∆= + +t t t t t t

P P E E W Wa q a q a q b (39)

21

where

q

P w e

xa a a x

t

τ ∆= + + + ∆

∆ (40)

( ) ( )

s s TE

e

ta

x e x

α α τ

δ δ

∆= + (41)

( ) ( )

s s TW

w w

ta

x x

α α τ

δ δ

∆= + (42)

1 1 2 2

2[ ]

( ) ( ) ( ) ( )

[ [ / ] [ / ]]

[ ](1 ) 2 (1 ) 2 2

q qt t t t ts T s T s T s T

P e w P

e w e w

s a in

P

t t t t t t t t

s E W s T E W E W

x xb x q q q q

t x x x x t

x t C Exp k x C Exp k xx x

G T T G T T T Tt

τ τα τ α τ α τ α τ

δ δ δ δ

α µ φ δ δ

α α τ

ε ε

−∆

+∆ +∆

∆ ∆= + ∆ + + − − −

∆ ∆

∂ ∂− ∆ ∆ − − −

∂ ∂

− − −+ ∆ + −

− −

(43)

Following the general procedures are described in Ref. [32], the discretization equation

for the boundary grid points can also be obtained. This Discretization of the present bioheat

transfer model involves three time instants, i.e. t-∆t, t and t+∆t. The current time at which the

heat flux needs to be solved is t+∆t. After replacing the values of the temperature-involved terms

into Eq. (43) for the source term b, the discretization Eq. (39) becomes a linear system of

algebraic equations and can be solved by TDMA (Tri-diagonal matrix algorithm). Once the heat

flux at the grid point P is determined, the temperature can be computed from the discretization

form of the bioheat transfer as below

[ ( )]2 (1 )

t t t t

t t t tE w

P P a P m b P

s s

q qt GT T Q T T

c xµ ϕ

ρ ε

+∆ +∆+∆ − −∆

= + + + + −∆ −

(44)

22

Equation (43) involves the value of T at the current time t t+ ∆ , so an iterative solution

between Eqs. (39) and (44) is required in each time step until convergence of the value of T is

met.

2.10 Results and Discussion

The following properties of a living biological tissue are used for this analysis.

Thermophysical properties of tissues [33]: ρ =1000 kg/m3, k = 0.628 W/ (m K), c = 4187 J/(kg

K); thermo-physical properties of blood vessel: bρ = 1060 kg/m3, cb = 3860 J/(kg K), wb =

1.87)10-3 m3/ (m3 tissue s): optical properties [34]: sµ = 120.0 cm-1, a

µ = 0.4 cm-1, g = 0.9 ;

blood temperature: Tb = 37oC; metabolic heat generation: Qm = 1.19×103 W/m3 [33]. The

thickness of the slab of tissue is L = 5 cm, and the initial temperature is T0 = 37˚C. The diffuse

reflectance Rd = 0.05 is used for the laser light distribution of scattering tissue. Two laser

irradiances are considered, φin= 2 W/cm2 and 30 W/cm2. The laser duration time τL is 5s. After

the model convergence test, a total of 120 grid points and a time step (∆t) of 0.001s are

employed. Three different values of the coupling factor are taken based on the blood perfusion

rate. According to the blood perfusion rate wb= 1.87×10-3 m3/ (m3 tissue s), the values of ε are

0.0079, 0.025 and 0.0845 [20] and the coupling factors are 67435, 55078 and 47488 W/m3K [20,

35].

The first case studied is that the laser light is highly absorbed by the tissue. As stated

earlier, the heat flux boundary condition Eq. (29) is applied at the laser irradiated surface. Figure

2 compares the temperature responses at the irradiated surface obtained from the Fourier heat

conduction, constitutive DPL model and generalized DPL model, and Fig. 3 displays the change

of the resulting thermal damage parameters. The laser irradiance is taken as 2W/cm2 for all the

23

three cases. The lag times used in this computation are τq =16 s and τT = 0.05 s for the

generalized DPL model and also the constitutive DPL model [36-38], and τq = τT = 0 for the

Fourier heat conduction model. As shown in Fig. 2, the generalized DPL model predicts lower

temperature response compared to the classical DPL model, especially after laser pulse is off.

Fig. 2 Temperature evolution at the irradiated surface of a highly absorbing tissue

The reason is that the coupling factor between blood and the tissue in the generalized

DPL model includes not only blood perfusion but also convection heat transfer in the tissue,

whereas the constitutive DPL model allows only the blood perfusion effects. On the other hand,

the classical Fourier’s heat conduction model predicts the lowest temperature rise. This is

because the Fourier heat conduction model predicts the infinite propagation speed of heat. When

the laser light impinges onto tissue surface, heat is transferred into deeper part of the tissue

without any delay.

24

Fig. 3 Thermal damage at the irradiated surface of a highly absorbing tissue

For those case of laser light highly absorbed by the tissue, the predicted thermal damage

response at the irradiated surface shown in Fig. 3 indicates that the constitutive DPL model

results in the highest irreversible tissue damage compared to the generalized DPL model. On the

other hand, Fourier heat conduction shows the mildest thermal damage. When the two phase lags

are present, the generalized DPL model can be used for photothermal reaction for laser irradiated

biological tissue.

Figure 4 illustrates the temperature response of a scattering tissue at the irradiated

surface. In a scattering tissue, the laser irradiation is considered as a volumetric heat source that

is determined by the light propagation. The phase lags times used are the same as those in the

previous case (Figs. 2 and 3); but, the laser irradiance is increased to 30 W/cm2. The results

obtained from the generalized DPL model are significantly different from the constitutive DPL

25

model and the Fourier heat conduction model for the later time. After the laser is turned off, the

temperature dropped more significantly in generalized DPL model than others.

Fig. 4 Temperature evolution at the irradiated surface of a scattering tissue

Figure 5 illustrates the thermal damage transient in the scattering tissue. It is shown from

Fig. 5 that the generalized DPL model predicts mildest thermal damage compared to classical

DPL model and Fourier heat conduction.

26

Fig. 5 Thermal damage at the irradiated surface of a scattering tissue

Figure 6 shows the temperature response for different phase lag times (τT) for

temperature gradient while the phase lag time for heat flux (τq) is kept constant, 16 s. The

incident irradiance of laser beam is set as before, 30W/cm2. It can be seen from Fig. 6 that the

temperature variations during the laser irradiation time period (5s) are almost same for different

τT values, although the temperature becomes diverse after the laser is turned off.

27

Fig. 6 Temperature distribution at the irradiated surface of a scattering tissue for different

τT values

As expected, this predicts a different damage progress and in turn, results in different

final damage extents in the biological tissues (Fig. 7). The longer the phase lag τT, the larger the

saturated value of damage parameter. The saturated damage parameter predicted with τT = 32s is

about 2.6 times that predicted by the Fourier’s law of heat conduction.

28

Fig. 7 Thermal damage at the irradiated surface of a scattering tissue for different τT values

To further investigate the condition under which the DPL results approach the prediction

by Fourier’s law, the simulations are performed for constant τT (0.05 s) and different values of τq

(32 to 0.05s) and the results are shown in Figs. 8 and 9. It tends to induce more thermal effects as

τq increased. Figures 8 and 9 illustrate the effect of τq on the evolution of irradiated surface

temperature and thermal damage. It can be observed that the longer the delay time τq, the higher

the temperature rise and the larger the saturated values of damage parameter. As τq decrease to

0.05s, the curve almost overlaps with that obtained from the Fourier’s law.

29

Fig. 8 Temperature distribution at the irradiated surface of a scattering tissue calculated by the

nonequilibrium DPL model for different τq values

30

Fig. 9 Thermal damage at the irradiated surface of a scattering tissue calculated by the

nonequilibrium DPL model for different τq values

Comparing the temperature responses Figs. 6 and 8 shows that for laser irradiated

biological tissues τq has more impact on the temperature in the early time while τT has more

impact on the temperature in the later time. The generalized DPL model will be close to classical

Fourier’s heat conduction when the phase lags are very small. Otherwise, the heat conduction

described by the generalized DPL bioheat transfer model would differ from the classical

Fourier’s heat conduction even if τT = τq = 0.

The next investigation of this study is to illustrate the effects of laser parameters and the

coupling factor on the thermal damage in the tissue using the generalized DPL bioheat model.

31

Figures 10 and 11 show the effects of laser irradiance on temperature and damage

parameter. As expected, the higher the laser irradiance, the higher the temperature and the

earlier, steeper and greater the damage parameter. The higher denaturation process resulting from

the higher irradiance prolongs due to the fact that it takes longer time to cool down the tissue.

For the laser exposure time 5 s, the minimum irradiance that causes the irreversible thermal

damage is found to be in between 20-25 W/cm2.

Fig. 10 Effects of laser irradiance on temperature at the irradiated surface of a scattering

tissue

32

Fig. 11 Effects of laser irradiance on damage parameter at the irradiated surface of a scattering

tissue

Figures 12 and 13 present the effects of the laser exposure time on the temperature and

the resulting thermal damage. It is shows that the effects of the laser exposure time are smaller to

those of the laser irradiance. The longer the laser exposure time is, the higher the temperature

rises and more the thermal damage is induced in the irradiated surface of the tissue. From Fig.

13, it can be observed that the tissue would be more irreversibly damaged (Ω≥1.0) when the

laser exposure time is more than 4 s.

33

Fig. 12 Effects of laser exposure time on temperature at the irradiated surface of a scattering

tissue

34

Fig. 13 Effects of laser exposure time on damage parameter at the irradiated surface of a

scattering tissue

The effects of the coupling factor on temperature and thermal damage is shown in

Figs.14 and 15. The coupling factor indicates the energy exchange between the blood and the

tissues. Both the blood perfusion and the convective heat transfer have effects on the coupling

factor. In this study, the blood perfusion rate is assumed to be constant 1.87 ×10-3 m3/ (m3s

tissue). Thus, the coupling factor change only depends upon the change of the blood vessel [20]

diameter. As can be seen from Figs. 14 and 15, the higher the coupling factors, the more the

temperature decrease. The thermal damage is increased with the decrease of blood tissue

coupling factor.

35

Fig. 14 Effects of coupling factor on temperature at the irradiated surface of a scattering tissue

36

Fig. 15 Effects of coupling factor on damage parameter at the irradiated surface of a scattering

tissue

Blood perfusion rate depends on the location of the tissue. Convection cooling effect of

the blood flow plays a significant role in an optimized treatment procedure in laser-induced

thermotherapy. Figure 16 shows the effects of blood perfusion rate on the temperature. The

higher the blood perfusion rate the stronger the convection heat loss due to the faster blood flow.

It is shown in Fig. 17 that as the blood perfusion increase, the less extent of thermal damage is

caused with the consequence of decrease temperature.

37

Fig. 16 Effects of blood perfusion rate on temperature at the irradiated surface of a scattering

tissue

38

Fig. 17 Effects of blood perfusion rate on thermal damage at the irradiated surface of a

scattering tissue

2.11 Conclusion

The treatment efficiency as well as safety is a primary concern in the laser medical

applications. Therefore, the most important issue is to understand and accurately assess the laser

induced thermal damage in the biological tissue. DPL model obtained by performing volume

average to the local instantaneous energy equations for the blood and the tissue is used to

investigate the thermal response of the laser irradiated biological tissues in this thesis. The broad

beam laser irradiation method based on Monte Carlo simulation is used to determine the laser

39

light propagation in the biological tissue. The generalized DPL bioheat model based on

nonequilibrium heat transfer and the classical DPL model are compared with Fourier’s heat

conduction model. It is shown that the generalized DPL model predicts significantly different

temperature and thermal damage in the irradiated surface of tissue. It is also found that for the

laser irradiated biological tissues the phase lag time of heat flux (τq) has more impact on the

temperature in the early time while the phase lag time of temperature gradient (τT) has more

impact on the temperature in the later time. The generalized DPL model reduces to the Fourier’s

heat conduction model only when τq = τT = 0. The influences of laser exposure time and

irradiance, blood perfusion, and the coupling factor on temperature and thermal damage are also

studied. The result shows that the overall effects of the laser parameters on the temperature and

damage parameter are similar to those of the time delay τT.

40

CHAPTER 3

Thermal Lagging in Living Biological Tissue based on Nonequilibrium Heat

Transfer between Tissue, Arterial and Venous Blood

3.1 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 [39, 27] (see Fig. 18).

Fig. 18 Schematic view of artery and vein surrounding by tissue.

41

3.2 Governing Equations

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 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 tissue causes a constant

change of the blood temperature. The governing equations for three carriers in a human body

[26] are:

*+* ,-*- .* . *0 *** 1* * *% 45

3+3 ,-3- .3 . 30 333 13 3 3% 46

1 + -- 1 1** 133 1 %& 1 % 47

where the blood (arterial and venous) and tissue temperature are volume averaged values; the

second term of the right hand side of the equation (45) and (46) and second and third term of the

eq. (47) represents the contribution of blood perfusion on the energy balance in blood and tissue

and G is referred to as a coupling factor between the blood (arterial and venous) and the tissue;

ka *, kv 3 and ks(1-ε) are the effective thermal conductivities of arterial blood, venous blood

and tissue, respectively. Those three equations include significant effects of the blood flow and

direction, thermal diffusivity and local thermal nonequilibrium between the blood and the

42

surrounding tissue. The two coupling factors are a measure of combined convection and

perfusion [20]. The porosity of the porous media is equal to summation of volume fractions of

arterial and venous blood, i.e.,

a vε ε ε= + (48)

If the sufficient information about the thermal and anatomical properties are available and

also if the blood velocities and directions are known, Eqs. (45) - (47) can be used to determine

average temperature distributions. The second term on the left hand side of Eqs (45) and (46)

expresses the counter current effect between the arterial and venous blood flows. In Eq. (47), the

effect of metabolites 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.

3.3 Dual Phase Bioheat Equation for Three Carriers System

The dual phase lag bioheat equation can be obtained by eliminate either blood (arterial or

venous) or tissue temperature from the multi-temperature model equation. In this thesis, 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. (45)- (47), the following energy equation can be obtained:

43

*+* -*- 3+3 -3- 1 + -- *+*6*. * 3+363. 3 *** 333 1 *% 3% 1 %& 1 % 49

Under a rapid heating condition, the tissue and blood (both artery and venous) are not at

the same temperature at a local level. With the assumption taken by Minkowycz et al. [28] that it

is hypothesized that before the onset of equilibrium, the blood temperatures for both artery and

vein undergo transient processes can be obtained by:

*+* -*- 1* * 50

3+3 -3- 13 3 51

Substituting Eqs. (50) and (51) into Eq. (49), the bioheat equation with tissue temperature as

a sole unknown can be expressed as:

9 --

-- :;* +*+<== .* ;3+3+<== .3 > 1 *+3+*+3 -?

-?

<== @ -- A 3 +313*+* .3 *+*1*3+3.*1*13+<==--

*+*3+3 -%

- 1 *+*3+3 -%& - 1 *3+*+31*13B<==

--

C% 1 %&D+<== 13*+* 1*3+3 -%- 13*+* 1*3+31 -%& - 52

where the phase lags for heat flux and temperature gradient are given by:

44

9 *+*1*3+3 3+313*+* 1 +13*+* 1 +31*+31*13+<== 53

*G*1*3+3 3G313*+* 1 G13*+* 1 3 1*+3G1*13G<== 54

with the effective properties being defined as,

+<== *+* 3+3 1 + 55 <== ** 33 1 56

<== <==+<== 57 Equation (52) represents the DPL bioheat equation with average tissue 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. (52), the third order time derivative term indicates the effect of three carriers (artery, vein

and tissue) system. Although Eq. (52) is more complex than Eq. (6), it is more accurate because

it is based on nonequilibrium between different energy carriers. In addition, Eq. (52) has only

average tissue temperature as a sole unknown and no arterial and venous temperatures are

involved. It accounted 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. Equation (52) is distinct

from Eq. (6) by another fact that the phase lag times can be easily obtained from Eqs. (53) and

(54).

45

While Eq. (52) 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. (52) or the second and

third terms on the right-hand side of Eq. (47). Since both of these two terms represents the same

physical phenomenon, one can expect that [29]:

;* +*6* ;3+363 H 1** 133 58

which converts the effect of blood flow on the tissue temperature to coupling between blood

temperature to the tissue temperature. Obviously, the information about the effect of directions

of blood flow on the tissue temperature has been, in theory, included in the coupling factor Ga

and Gv. This approximation is only valid when the blood temperature is not equal to the tissue

temperature and the direction of the blood flow is taken into account in consideration in bioheat

equation.

Substituting Eq. (58) into Eq. (52), the following DPL bioheat equation is obtained

9 --

--

1 *+3+*+31*13+<==-?-? <== @ -- A

3 +313*+* .3 *+*1*3+3.*1*13+<==--

1 *3+*+31*13B<==--

C1* * 133 D +<== C% 1 %&D+<== *+*3+3 -%

-

1 *+*3+3 -%& - 13*+* 1*3+3 -%-

13*+* 1*3+31 -%& - 59

46

The difference between the present DPL bioheat equation (59) and the classical DPL

bioheat equation (6) is that the latter considers heat conduction in tissue only but Eq. (59)

considers the contributions to conduction by both tissue and blood.

3.4 Results and Discussion

Equation (52) or (59) conveys one of the most important advantages of the present DPL

bioheat equation over the classical DPL bioheat equation (6). 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 model also reveals that the root of dual phase lag is the nonequilibrium thermal

transport between blood and tissue. It is evidence from Eqs. (53) and (54) 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 surrounding tissues. It is an important property for analyzing the

biological system. The coupling factor depends upon convection heat transfer and blood

perfusion rate:

1* J*K* $*B* (60)

13 J3K3 $3B3 (61)

For the bundle of vascular artery and veins with diameters da and dv, the respective

coupling factors are [20]:

47

1* 4*G*L*

MN $*B* 62

13 43G3L3

MN $3B3 63 where the Nusselt number is approximately Nu = 4.93 [40, 41].

With the assumptions of uniform distribution of blood vessel in the tissue [42], different

diameters of blood vessels, porosity ( 2( / )ε =a a sd d ), and blood perfusion rate are investigated

and listed in Table 1. The thermo physical properties of blood and tissue are assumed to be

identical: * 3 0.5 W/m K, O* O3 O 1050 kg/m3 and B* B3 B 3770

J/kg K.

Table 1: Structure and perfusion coefficient studied in Ref [41]

Case ds

(mm)

da

(mm)

dv

(mm) * 3

wa

(kg/m3

sec)

wv

(kg/m3

sec)

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

48

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

The perfusion rate is defined as, the mean pressure difference between artery and vein

divided by the vascular resistance [35], i.e.,

$ P* P3Q 64

According to Poiseuille-Hagen formula [35], the relation between 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:

R P* P3 S81TUVW 65

where, F, T, U and W are 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

expressed as

Q 8TWSUV 66

From the above three equations (64) - (66), 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 1mm, respectively. The approximate values of

lumen diameter and wall thickness of vein are 5 mm and 0.5 mm, respectively [35]. Using these

49

approximate values, 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 rate is 1.43v aw w= − . Those two relations are used to calculate different properties of

venous blood such as porosity, diameter and blood perfusion rate that are summarized in Table 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 in Table 2. For all cases in

Table 2, the phase lag times for heat flux and temperature gradient are exactly the same because

=sa sak C and =sv svk C where, /=sa s ak k k and /=sa s aC C C . Comparison between the phase lag

times shows that the DPL phenomena 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.

Table 2: Coupling factors and phase lag times (* 3 0.5 W/m K, O* O3 O=

1050 Kg/m3 and B* B3 B 3770 J/Kg K)

Case

aaha

(W/m3

K)

avhv

(W/m3

K)

cawa

(W/m3

K)

cvwv

(W/m3

K)

Ga

(W/m3 K)

Gv

(W/m3 K)

9

(sec)

(sec)

1 30,348 31,351 3,770 -5,391 34,118 25,960 1.221 1.221

2 60,696 62,702 7,540 -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 3,770 -5,391 28,428 25,960 4.793 4.793

7 47,418 47,027 7,540 -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

50

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 3,770 -5,391 27,479 16,947 19.449 19.449

12 39,832 39,973 7,540 -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

The thermo-physical properties of tissue depend on the type and location of the tissue in

the body. Therefore, the assumption of the same thermo-physical 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: ρs = 1000 kg/m3, ks = 0.628 W/m K, cs = 4187 J/ kg K, ρa= ρv =1060

kg/m3 and ca = cv = 3860 J/kg K. The thermal conductivity of blood used by Yuan [42] was kb=

0.5W/m K and this value agrees with other sources [20]. The thermal conductivities of artery and

vein are taken as the same value, i.e., ka= kv = 0.5 W/m K. Table 3 shows the contributions of

convection heat transfer and blood perfusion, in the coupling factors and phase lag times.

Table 3: Phase lag times for different thermo physical properties of tissue (ρs=1000 kg/ m3,

ka = 0.5 W/m K, ks= 0.628 W/m K, cs= 4187 J/ kg K, ρa= ρv= 1060 kg/m3, ca= cv = 3860 J/ kg K)

Case

aaha

(W/m3

K)

avhv

(W/m3

K)

cawa

(W/m3

K)

cvwv

(W/m3

K)

Ga

(W/m3

K)

Gv

(W/m3

K)

9

(sec)

(sec)

1 30,348 31,351 3,860 -5,520 34,208 25,831 1.265 1.267

2 60,696 62,702 7,720 -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

51

5 128,978 131,674 19,300 -27,560 148,278 104,114 1.269 1.274

6 24,657 31,351 3,860 -5,520 28,518 25,831 4.948 4.963

7 47,418 47,027 7,720 -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 3,860 -5,520 27,569 16,818 20.153 20.345

12 39,832 39,973 7,720 -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

The phase lag times for heat flux and temperature gradient shown in Table 3 are 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% to 12% compared to the values in Table 2 with the

same porosity and blood perfusion rate. The difference between the results in Table 2 and 3 is

more significant compared to the similar study by Zhang [20] 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 τq is over 11%

(Case 12). Although the thermal conductivity, specific heats of arterial and venous blood are

different from those of the tissue in Table 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 from Table 4

52

that 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.

Table 4: Phase lag times for same blood perfusion rate (ρs= 1000 kg/ m3, ks= 0.628 W/m K,

cs=4187 J/ kg K, ρa= ρv = 1060 kg/ m3, ca= cv = 3860 J/ kg K)

wa wv εa εv aaha avhv Ga Gv τq τT

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 9,472 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 4,087 5.091 5.091

53

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

With the same diameter of tissue (ds = 17.83 mm), the effect of blood perfusion rate on the

phase lag times are shown in Table 5. It is seen that the phase lag times for heat flux and

temperature gradient are identical for all cases and gradually increase from its previous value

with the increasing blood perfusion rate.

Table 5: Phase lag times for the same diameter of tissue (ρs=1000 kg/ m3, ks=0.628 W/m K,

cs=4187 J/ kg K, ρa= ρv= 1060 kg/m3, ca= cv=3860 J/ kg K)

wa wv εa εv aaha avhv cawa cvwv Ga Gv τq τT

1 -1.429 0.0041 0.005 31,015 31,015 3,770 -5,386 34,785 25,630 1.224 1.224

2 -2.857 0.0041 0.005 31,015 31,015 7,540 -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 9,472 2.407 2.407

5 -7.143 0.0041 0.005 31,015 31,015 18,850 -26,929 49,865 4,087 5.091 5.091

1 -1.429 0.0041 0.005 31,015 31,015 3,770 -5,386 34,785 25,630 1.224 1.224

2 -2.857 0.0041 0.005 31,015 31,015 7,540 -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 9,472 2.407 2.407

5 -7.143 0.0041 0.005 31,015 31,015 18,850 -26,929 49,865 4,047 5.091 5.091

1 -1.429 0.0041 0.005 31,015 31,015 3,770 --5,386 34,785 25,630 1.224 1.224

2 -2.857 0.0041 0.005 31,015 31,015 7,540 -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 9,472 2.407 2.407

5 -7.143 0.0041 0.005 31,015 31,015 18,850 -26,929 49,865 4,086 5.091 5.091

54

Fourier’s law can be considered as a special case of the dual phase heat conduction model

only when the phase lag times for 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 thesis, DPL bioheat equation with a sole unknown average tissue temperature is obtained

for the three carrier systems. By solving the continuity and momentum equations in porous

medium, the intrinsic averaged velocity vector for artery and vein in the solid matrix can be

obtained, which is needed to solve the Eq. (52) or (59). The intrinsic averaged velocity is 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.,

τq= τT = 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 in Table 6 and

Table 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 maximum difference between the phase lag time

for heat flux and phase lag time for temperature gradient is 3.011 sec (Case 12).

55

Table 6: Phase lag time for brain: (ρs=1030 kg/m3, ks=0.5395 W/K m, $s=9.33 kg/m3 sec,

cs=3680 J/kg K, ρa=ρv=1060 kg/m3,ka=kv=0.5 W/m K and ca=cv=3860J/kg K)

Case aaha

(W/m3 K)

avhv

(W/m3 K)

cawa

(W/m3 K)

cvwv

(W/m3 K)

Ga

(W/m3 K)

Gv

(W/m3 K)

9

(sec)

(sec)

1 30,348 31,351 3,860 -5,520 34,208 25,831 1.290 1.472

2 60,696 62,702 7,720 -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 3,860 -5,520 28,518 25,831 5.040 5.756

7 47,418 47,027 7,720 -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 3,860 -5,520 27,569 16,818 20.438 23.35

12 39,832 39,973 7,720 -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

In the case of muscle (Table 7), 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 sec.

56

Table 7: Phase lag time for muscle: (ρs=1040 kg/m3, ks= 0.4935 W/K m, $s= 0.63 kg/m3 sec, cs=

3720 J/kg K, ρa= ρv= 1060 kg/m3, ka= kv = 0.5 W/m K and ca= cv= 3860J/kg K)

Case aaha

(W/m3 K)

avhv

(W/m3 K)

cawa

(W/m3 K)

cvwv

(W/m3 K)

Ga

(W/m3 K)

Gv

(W/m3 K)

9

(sec)

(sec)

1 30,348 31,351 3,860 -5,520 34,208 25,831 1.264 1.627

2 60,696 62,702 7,720 -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 3,860 -5,520 28,518 25,831 4.941 6.222

7 47,418 47,027 7,720 -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 3,860 -5,520 27,569 16,818 20.070 25.110

12 39,832 39,973 7,720 -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

The values of the phase lag times for brain (Table 6) and muscles (Table 7) are much

higher than the values with same thermo physical properties (Table 2) as well as different thermo

physical properties (Table 3). Also for same diameter, the difference of the values of phase lag

times is much higher for brain and muscles than the different properties of tissue and blood. The

values of phase lag times for heat flux and temperature gradient with the effect of vein (shown in

Table 3 and 4) are more than those without the effect of vein in the biological tissue [20].

3.5 Conclusions

For the three-carrier system (artery, venous and tissue) in a living biological system, a

dual phase lag bioheat equation with tissue temperature as the sole unknown can be obtained by

57

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 physical properties of blood and surrounding tissue, interphase convection

heat transfer, and the blood perfusion rate in biological system. The novelty of this paper is that

DPL model is derived from nonequilibrium between arterial blood, venous blood, which was not

accounted in classical Pennes bioheat equation, is taken into account. If the densities, specific

heats and thermal conductivities of arterial and venous blood are similar to the tissue, the phase

lag times for heat flux and temperature gradient are identical. When the densities are different

and the specific heat and thermal conductivities of blood and tissues are considered, the two

phase lag times are still similar. The phase lag times studied in this thesis range from 1 to 21 sec.

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.

58

CHAPTER 4 SUMMERY AND CONCLUSIONS

To ensure the treatment efficiency and personal safety in laser application in medical

sector, one of the most important issues is to understand and accurately access laser-induced

thermal damage to the living biological tissue. In this thesis, in chapter 2, a generalized DPL

model for blood and surrounding tissues is applied to investigate the temperature response and

thermal damage of laser irradiated biological tissue. The laser light propagation in tissues is

determined by Monte Carlo simulation. The thermally induced damage in tissue is evaluated by

using the rate process equation for protein denaturation process. For comparison, the results are

also computed with the classical DPL model and Fourier heat conduction model. It is shown that

the present approach predicts significantly different temperature and thermal damage in tissue

than the classical DPL model and Fourier heat conduction model. It is also found that for the

laser irradiated biological tissues the phase lag time of heat flux (τq) has more impact on the

temperature in the early time while the phase lag time of temperature gradient (τT) has more

impact on the temperature in the later time. The generalized DPL model reduces to the Fourier’s

heat conduction model only when τq = τT = 0. The influences of laser exposure time and

irradiance, blood perfusion, and the coupling factor on temperature and thermal damage are also

studied. The result shows that the overall effects of the laser parameters on the temperature and

damage parameter are similar to those of the time delay τT.

For the three-carrier system (artery, venous and tissue) in a living biological system, a

generalized 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

59

arterial and venous blood. The phase lags times are expressed as functions of the thermo-

physical properties of arterial and venous blood and surrounding tissues, interphase convective

heat transfer, and the blood perfusion rate in the biological system. In addition, the convection of

blood is taken into account to derive this model where it was not accounted in classical Pennes

bioheat equation. From this result, it is shown that if the densities, specific heat and thermal

conductivities of artery and venous blood are similar to the tissue, the phase lag times are

identical to each other. When the densities are different and specific heat and thermal

conductivity are considered then the phase lag times are also identical to each others. The phase

lag times for brain and muscle are also calculated in this chapter. The range of the phase lag time

for heat flux vector and phase lag time for temperature gradient studied in this chapter from 1 to

21 s. From results, it is shown that due to presence of blood perfusion in surrounding tissue, the

generalized DPL model can reduce to Fourier heat conduction model only when both phase lag

times are equal to zero.

60

References

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non-equilibrium heat transfer in a three-carrier system, International Journal of Heat and

Mass Transfer, Vol 54 (2011) 2419-2426.

[3] H.H. Pennes, Analysis of tissue and arterial blood temperatures in the resting forearm, J.

Appl. Physiol. 1 (1948) 93–122.

[4] C. Catteneo, A form of heat conduction equation which eliminates the paradox 503 of

instantaneous propagation, Compes Rendu 247 (1958) 431–433.

[5] P. Vernotte, Les paradoxes de la théorie continue de l´équation de la chaleur, 505 Comptes

Rendus 246 (1958) 3154–3155.

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