DUAL PHASE LAG BEHAVIOR OF LIVING BIOLOGICAL TISSUES IN LASER HYPERTHERMIA

Transcription

1 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

2 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

3 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. ii

4 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 is gratefully acknowledged. iii

5 TABLE OF CONTENTS ACKNOWLEDGEMENT ii LIST OF FIGURES.vii LIST OF TABLES...ix NOMENCLATURE...x ABSTRACT...xii CHAPTER 1: INTRODUCTION Background Heat Conduction Model Pennes Bioheat Equation Thermal Wave Model Dual Phase Lag Model 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 Tissue-laser interactions 9 iv

6 2.2 Laser parameters Classical DPL model Generalized DPL model Physical model Problem statement Calculation of the laser irradiance Calculation of thermal damage parameter Numerical Analysis Discretization scheme of space Discretization of governing equation Results and Discussions Conclusion 38 CHAPTER 3: THERMAL LAGGING IN LIVING BIOLOGICAL TISSUE BASED ON NONEQUILIBRIUM HEAT TRANSFER BETWEEN TISSUE, ARTERIAL AND VENOUS BLOODS Heat transfer in arteries, venous and solid tissue Governing equations Dual phase bioheat equation for three carriers system..42 v

7 3.4 Results and Discussion Conclusion.56 CHAPTER 4: SUMMERY AND CONCLUSIONS.59 REFERENCES..60 vi

8 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 τ T values 27 Fig. 7 Thermal damage at the irradiated surface of a scattering tissue for different τ T values..28 Fig. 8 Temperature distribution at the irradiated surface of a scattering tissue calculated by the non-equilibrium DPL model for different τ q values..29 Fig. 9 Thermal damage at the irradiated surface of a scattering tissue calculated by the nonequilibrium DPL model for different τ q 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. vii

9 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 viii

10 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 ix

11 NOMENCLATURES a specific heat transfer area [m 2 / m 3 ] c specific heat of artery [J/ (kg K)] G coupling factor between blood and tissue [W/ (m 3 K)] h heat transfer coefficient [W/(m 2 K)] k thermal conductivity [W/(m K)] q heat flux vector [W/m 2 ] r position vector [m] Q L heat source due to hyperthermia therapy [W/m 3 ] Q m source terms due to metabolic heating [W/m 3 ] t time [s] T average temperature [K] V intrinsic phase averaged velocity vector [m/s] w blood perfusion rate [m 3 /m 3 tissue] R d 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 d b diameter of the blood vessel [m] S heat source due to hyperthermia therapy [W/m 3 ] S m source terms due to metabolic heating [W/m 3 ] R vascular resistance Greek symbols α thermal diffusivity [m 2 /s] porosity ρ density [kg/m 3 ] ρ a artery blood mass density [kg/m 3 ] x

12 ρ v venous blood mass density [Kg/m 3 ] ρ s tissue density [kg/m 3 ] τ q τ T τ L φ in phase lag time of the heat flux [s] phase lag of the temperature gradient [s] laser exposure time incident laser irradiance µ a absorption coefficient [cm -1 ] µ s 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 xi

13 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 (τ T ) and heat flux vector (τ q ) 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. xii

14 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. xiii

15 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 1

16 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 2

17 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 3

18 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 Pennes Bioheat Equation: The general bioheat equation considering blood perfusion and metabolic heat generation is as follows [3]: T q ρc = + QL + Qm + wb ρbcb ( Tb T) t x (1) where q is heat flux; ρ and c are respectively density and specific heat of the tissue ; ρ b and c b are the density and specific heat of blood, w b is the blood perfusion rate; T b and T are the temperatures of blood and tissue; Q m and Q L 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 T b 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. 4

19 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 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. 5

20 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, τ q is the phase lag for the heat flux vector, and τ T 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 (τ q > τ 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 (τ q < τ T ), the heat flow is called fluxprecedence 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 T s wc b b Ts 2 2 wc b b τq + 1 T T 2 + τq = αs Ts + τt s + t Cs t t Cs Sm + S τq Sm S ρc ρc t t s s s s ( T ) ( b s ) (6) Under the assumption of constant blood temperature (i.e. w c τ b b q b C s T t =0) and the condition τ q = τ = 0, Eq. (6) reduces to the classical bioheat equation. The DPL bioheat equation (6) is T 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 6

21 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. 7

22 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. 8

23 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 9

24 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. 10

25 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): T wb ρbcb T T T QL Qm wb ρbcb τ q + (1 + τ ) ( ) 2 q = α + ατ 2 T T 2 b T t ρc t x t x ρc ρc ρc (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: 11

26 q q q q QL T T τ q α ατ 2 2 T α + = + + αw 2 bρbcb + αwb ρbcbτ T t t x t x x x t x (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 12

27 Tb ερbcb[ + V. Tb ] =.( kb, eff Tb ) + abhb ( Ts Tb ) + εql (9) t T (1 ε ) ρ c =.( k T ) + a h ( T T ) + (1 ε ) Q + (1 ε ) Q t s s s s, eff s b b b s m L (10) where a b is the specific heat transfer area, and h b is the heat transfer coefficient inside the blood vessel, the temperatures of blood and tissue are volume averaged values; k b,eff and k s,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, h b and the specific area on the blood vessel in the tissue, a b 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: Tb ερbcb[ + V. Tb ] =.( kb, eff Tb ) + abhb ( Ts Tb ) + wbcb ( Ts Tb ) + εql (11) t T (1 ε ) ρ c =.( k T ) + a h ( T T ) + w c ( T T ) + (1 ε ) Q + (1 ε ) Q t s s s s, eff s b b b s b b b s m L (12) 13

28 Considering not only interfacial convection heat transfer but also the blood perfusion, the two step model can be written in the following forms [20]: Tb 2 ερbcb[ + V. Tb ] = εkb Tb + G( Ts Tb ) + εql (13) t T t s 2 (1 ε ) ρscs = (1 ε ) ks Ts + G( Tb Ts ) + (1 ε ) Q + (1 ε ) Qm L (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: G = abhb + wbc b (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: ερ Tb Ts 2 2 bcb (1 ) scs bcb. Tb kb Tb (1 ) ks Ts (1 ) Qm Q L t ε ρ + + t ερ V = ε + ε + ε + (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: Tb ερbcb = G( Ts Tb ) t (17) 14

29 The rearrange form of this equation is T s ερ c = Tb + G T t b b b (18) Substituting Eq. (18) into Eq. (16), the following equation with the blood temperature as sole unknown is obtained: (1 ε ) Qm + Q L τ + + V. = α [ + τ ( )] + (19) t t ( c) t ( ρc) 2 Tb Tb ερbcb 2 2 q T 2 b eff Tb T Tb ρ eff eff where the phase lags for heat flux and temperature gradient are τ q ε (1 ε ) ρ c ρ c = G( ρc) b b s s eff (20) τ T ε (1 ε ) ρ b c b k = s Gk eff (21) where ( ρc) = ερ c + (1 ε ) ρ c (22) eff b b s s k = ε k + (1 ε ) k (23) eff b s α eff keff = (24) ( ρc) eff To obtained the equation with tissue temperature as sole unknown, Eq. (19) can be substitute on Eq. (18) and the final bioheat equation is 15

30 ερ ε ερ τ α τ ε 2 T s Ts bcb 2 2 (1 ) Qm QL bcb Qm q + + = [ ( )] [(1 ) 2 eff Ts + T Ts t t ( ρc) eff t ( ρc) eff G( ρc) eff t Q + t L ] (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] ερ c V. T G( T T ) (26) b b s b s Substituting Eq. (26) into Eq. (25), Zhang obtained the following equation with tissue temperature as sole unknown [20] 2 Ts Ts 2 2 G (1 ε ) Qm + QL q + = [ ( )] ( ) 2 eff Ts + T Ts + Tb Ts + t t t ( ρc) eff ( ρc) eff τ α τ ερbcb Qm QL + [(1 ε ) + ] G( ρc) t t eff (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: q q q q QL Gα s T Gα sτ T T τ q + = α + α τ α s 2 s T 2 s t t x t x x (1 ε ) x (1 ε ) t x (28) 16

31 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 0 is considered. A flat-top laser beam is applied normally to the left surface of the slab at time t = 0+ (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 larger than the thickness of the thermally effected zone for the time period of interest. The right boundary surface is assumed to be thermally insulated (q = 0) while the boundary condition at the left surface depends on the laser light absorption and scattering of tissues. Fig. 1 Physical model and grid system 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 L, is zero and the boundary conditions are given by [24]: q = φin (1 R d ) for x = 0 when 0 < t < τ L (29) 17

32 q = 0 for x = L when 0 < t < τ L (30) where τ L is the laser exposure time, φ in is the incident laser irradiance and R d is the diffuse reflectance of light at the irradiated surface. For strongly scattering tissues, laser heating is considered as a body heat source (Q L 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 < τ L (31) q = 0 for x = L when 0 < t < τ L (32) The initial conditions for both cases are: q q = 0 and = 0 t for 0 < x < L, t = 0 (33) 2.7 Calculation of the laser irradiance (Q L ) 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: Q ( x) = µ φ( x) (34) L a 18

33 where µ is the absorption coefficient, and φ ( x) is the local light irradiance varying with depth a 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: φ( x) = φ [ C exp( k z / δ ) C exp( k z / δ )] (35) in where δ is the effective penetration depth; C 1, C 2, k 1 and k 2 are determined by Monte Carlo solutions, depending on the diffuse reflectance, R d ; the effective penetration depth δ can be obtained from the diffusion theory as δ = 1 3 µ [ µ + µ (1 )] a a s g (36) where µ s is the scattering coefficient and g is the scattering anisotropy. Equation (36) is valid when 0.1 µ s and 0.7 g 0.9 [25]. ( µ + µ ) a s 2.8 Calculation of thermal damage parameter (Ω) The damage parameter is evaluated according to the Arrenius equation [1, 31]: t f E Ω = A exp( ) dt (37) t0 RT where A is the frequency factor, s -1 [1]; E is the energy of activation of denaturation reaction, J/mol [1]; R is the universal gas constant, J/ (mol. K); T is the absolute 19

34 temperature of the tissue at the location where thermal damage is evaluated; t 0 is the time at onset of laser exposure; and t f 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 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 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: q q q q Q G T t t x t x x x e t+ t 2 e t+ t 2 3 L αs ( τ q + ) dtdx = ( α + α τ α s s T s (1 ε ) w t w t 2 + Gα sτ T T ) dtdx (1 ε ) 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): a q = a q + a q + b (39) t + t + t + P P E E W W 20

35 where τ x q ap = aw + ae + + x t (40) a E α s t α sτ T = + (41) ( δ x) e ( δ x) e a W αs t αsτ T = + (42) ( δ x) ( δ x) w w 2τ q x αsτ T αsτ T αsτ T αsτ τ T q x b = [ + x + + ] q q q q t ( δ x) ( δ x) ( δ x) ( δ x) t t t t t t P e w P e w e w α sµ a x t[ φin{ C1 Exp[ k1x / δ ] C2 Exp[ k2x / δ ]}] x x t t t+ t t+ t t t Gα s TE TW Gα sτ T TE TW TE TW + t + [ ] (1 ε ) 2 (1 ε ) 2 2 P (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 t+ t t+ t t+ t t t qe qw G t TP = TP + [ + µ aϕp + Qm + ( Tb TP )] ρ c 2 x (1 ε) s s (44) 21

36 Equation (43) involves the value of T at the current timet + 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 Results and Discussion The following properties of a living biological tissue are used for this analysis. Thermophysical properties of tissues [33]: ρ =1000 kg/m 3, k = W/ (m K), c = 4187 J/(kg K); thermo-physical properties of blood vessel: ρ b = 1060 kg/m 3, c b = 3860 J/(kg K), w b = m 3 / (m 3 tissue s): optical properties [34]: µ s = cm -1, µ a = 0.4 cm -1, g = 0.9 ; blood temperature: T b = 37 o C; metabolic heat generation: Q m = W/m 3 [33]. The thickness of the slab of tissue is L = 5 cm, and the initial temperature is T 0 = 37 C. The diffuse reflectance R d = 0.05 is used for the laser light distribution of scattering tissue. Two laser irradiances are considered, φ in = 2 W/cm 2 and 30 W/cm 2. The laser duration time τ L 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 w b = m 3 / (m 3 tissue s), the values of ε are , and [20] and the coupling factors are 67435, and W/m 3 K [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/cm 2 for all the 22

37 three cases. The lag times used in this computation are τ q =16 s and τ T = 0.05 s for the generalized DPL model and also the constitutive DPL model [36-38], and τ q = τ T = 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. 23

38 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/cm 2. The results obtained from the generalized DPL model are significantly different from the constitutive DPL 24

39 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. 25

40 Fig. 5 Thermal damage at the irradiated surface of a scattering tissue Figure 6 shows the temperature response for different phase lag times (τ T ) for temperature gradient while the phase lag time for heat flux (τ q ) is kept constant, 16 s. The incident irradiance of laser beam is set as before, 30W/cm 2. It can be seen from Fig. 6 that the temperature variations during the laser irradiation time period (5s) are almost same for different τ T values, although the temperature becomes diverse after the laser is turned off. 26

41 τ T values Fig. 6 Temperature distribution at the irradiated surface of a scattering tissue for different 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 τ T, the larger the saturated value of damage parameter. The saturated damage parameter predicted with τ T = 32s is about 2.6 times that predicted by the Fourier s law of heat conduction. 27

42 Fig. 7 Thermal damage at the irradiated surface of a scattering tissue for different τ T values To further investigate the condition under which the DPL results approach the prediction by Fourier s law, the simulations are performed for constant τ T (0.05 s) and different values of τ q (32 to 0.05s) and the results are shown in Figs. 8 and 9. It tends to induce more thermal effects as τ q increased. Figures 8 and 9 illustrate the effect of τ q on the evolution of irradiated surface temperature and thermal damage. It can be observed that the longer the delay time τ q, the higher the temperature rise and the larger the saturated values of damage parameter. As τ q decrease to 0.05s, the curve almost overlaps with that obtained from the Fourier s law. 28

43 Fig. 8 Temperature distribution at the irradiated surface of a scattering tissue calculated by the nonequilibrium DPL model for different τ q values 29

44 Fig. 9 Thermal damage at the irradiated surface of a scattering tissue calculated by the nonequilibrium DPL model for different τ q values Comparing the temperature responses Figs. 6 and 8 shows that for laser irradiated biological tissues τ q has more impact on the temperature in the early time while τ T 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 τ T = τ q = 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. 30

45 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 W/cm 2. tissue Fig. 10 Effects of laser irradiance on temperature at the irradiated surface of a scattering 31

46 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. 32

47 Fig. 12 Effects of laser exposure time on temperature at the irradiated surface of a scattering tissue 33

48 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 m 3 / (m 3 s 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. 34

49 Fig. 14 Effects of coupling factor on temperature at the irradiated surface of a scattering tissue 35

50 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. 36

51 Fig. 16 Effects of blood perfusion rate on temperature at the irradiated surface of a scattering tissue 37

52 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 38

53 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 (τ q ) has more impact on the temperature in the early time while the phase lag time of temperature gradient (τ T ) has more impact on the temperature in the later time. The generalized DPL model reduces to the Fourier s heat conduction model only when τ q = τ T = 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 τ T. 39

54 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. 40