Generalized dual-phase lag bioheat equations based on nonequilibrium heat transfer in living biological tissues

Based on a nonequilibrium heat transfer model in the living tissue obtained by performing volume average to the local instantaneous energy equations for blood and tissues, the dual-phase lag bioheat equations with blood or tissue temperature as sole unknown temperature are obtained by eliminating the tissue or blood temperature from the nonequilibrium model. The present dual-phase model successfully overcame the drawbacks of the existing dual-phase lag bioheat equation obtained by simply modifying the classical Pennes bioheat equation. Under the dual-phase model developed in this work, the phase lag times are expressed in terms of the properties of blood and tissue and the interphase convective heat transfer coefficient and blood perfusion rate. The phase lag times for heat flux and temperature gradient for the living tissue are estimated using the available properties from the literature. It is found that the phase lag times for heat flux and temperature gradient for the living tissue are very close to each other.     

Uncertainty analysis of thermal damage to living biological tissues by laser irradiation based on a generalized duel-phase lag model

The effects of uncertainties of laser exposure time, phase lag times, blood perfusion coefficient, scattering coefficient, and diffuse reflectance of light on the thermal damage of living biological tissue by laser irradiation are investigated using a sample-based stochastic model. The variabilities of input and output parameters are quantified using the coefficient of variance (COV) and interquartile range (IQR), respectively. The IQR analysis concluded that phase lag times for temperature gradient and heat flux, laser exposure time, and blood perfusion rate have more significant influences on the maximum temperature and maximum thermal damage of the living biological tissue induced by laser irradiation than the diffuse reflectance of light and scattering coefficient.     

Non‐Fourier Boundary Conditions Effects on the Skin Tissue Temperature Response

Thermal wave and dual phase lag bioheat transfer equations are solved analytically in the skin tissue exposed to oscillatory and constant surface heat flux. Comparison between the application of Fourier and non‐Fourier boundary conditions on the skin tissue temperature distributions is studied. The amplitude of temperature responses increases and also the phase shift between the temperature responses and heat flux decreases under the non‐Fourier boundary conditions for the case of an oscillatory surface heat flux. It is supposed the stable temperature cycles in order to estimate the blood perfusion rate via the existing phase shift between the surface heat fluxes and the temperature responses. It is shown that the higher rates of the blood perfusion correspond to lower phase shift between the surface temperature responses and the imposed heat flux.     

Analytical solution of the parabolic and hyperbolic heat transfer equations with constant and transient heat flux conditions on skin tissue

In this article, the parabolic (Pennes bioheat equation) and hyperbolic (thermal wave) bioheat transfer models for constant, periodic and pulse train heat flux boundary conditions are solved analytically by applying the Laplace transform method for skin as a semi-infinite and finite domain. The bioheat transfer analysis with transient heat flux on skin tissue has only been studied by Pennes equation for a semi-infinite domain. For modeling heat transfer in short duration of an initial transient, or when the propagation speed of the thermal wave is finite, there are major differences between the results of parabolic and hyperbolic heat transfer equations. The non-Fourier bioheat transfer equation describes the thermal behavior in the biological tissues better than Fourier equation. The outcome of transient heat flux condition shows that by penetrating into the depths beneath the skin subjected to heat, the amplitude of temperature response decreases significantly. The blood perfusion rate can be predicted using the phase shift between the surface temperature and transient surface heat flux. The thermal damage of the skin is studied by applying both the parabolic and hyperbolic bioheat transfer equations.     

Numerical Simulation of Thermal Damage to Living Biological Tissues Induced by Laser Irradiation Based on a Generalized Dual Phase Lag Model

A generalized dual phase lag (DPL) bioheat model based on the nonequilibrium 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, are carried out in this study. It is shown that the generalized DPL model could predict significantly different temperature and thermal damage from the classical DPL model and Pennes bioheat 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.     

Analytical characterization of heat transport through biological media incorporating hyperthermia treatment

Modeling and understanding heat transport and temperature variations within biological tissues and body organs are key issues in medical thermal therapeutic applications, such as hyperthermia cancer treatment. The biological media can be treated as a blood saturated tissue represented by a porous matrix. A comprehensive analytical investigation of bioheat transport through the tissue/organ is carried out including thermal conduction in tissue and vascular system, blood–tissue convective heat exchange, metabolic heat generation and imposed heat flux. Utilizing local thermal non-equilibrium model in porous media theory, exact solutions for blood and tissue phase temperature profiles as well as overall heat exchange correlations are established for the first time, for two primary tissue/organ models representing isolated and uniform temperature conditions, while incorporating the pertinent effective parameters, such as volume fraction of the vascular space, ratio of the blood and the tissue matrix thermal conductivities, interfacial blood–tissue heat exchange, tissue/organ depth, arterial flow rate and temperature, body core temperature, imposed hyperthermia heat flux, metabolic heat generation, and blood physical properties. A simplified solution based on the local thermal equilibrium between the tissue and the blood is also presented.     

Analysis for the dual-phase-lag bio-heat transfer during magnetic hyperthermia treatment

Magnetic fluid hyperthermia is one of hyperthermia modalities for tumor treatment. The control of temperatures is necessary and important for treatment quality. Living tissue is highly non-homogenous, and the velocity of heat transfer in it should be limited. Thus, this work analyzes the temperature rise behaviors in biological tissues during hyperthermia treatment within the dual-phase-lag model, which accounts the effect of local non-equilibrium on the thermal behavior. A small tumor surrounded by the health tissue is considered as a solid sphere. The influences of lag times, metabolic heat generation rate, blood perfusion rate, and other physiological parameters on the thermal response in tissues are investigated. While the metabolic heat generation takes little percentage of heating source, its effect on the temperature rise can be ignored. The control of the blood perfusion rate is helpful to have an ideal hyperthermia treatment. The lag times, τ_q and τ_T, affect the bio-heat transfer at the early times of heating. The total effect of τ_q and τ_T on the bio-heat transfer may be different for the same τ_T/τ_q value.     

Temperature response in biological tissue by alternating heating and cooling modalities with sinusoidal temperature oscillation on the skin

This study investigates the transient temperature response in biological tissue immersed in warm and cold water alternately with a sinusoidal temperature oscillation on the skin surface to simulate the contrast therapy which uses alternatively heat and cold modalities in subacute and chronic conditions. In addition, this type of analysis is also applied to assessing the blood perfusion rate in the skin through noninvasive technology by imposing a periodic temperature load onto the skin surface and then measuring the phase shift of the resulting surface heat flux. Based on the Pennes bio-heat transfer equation, the Laplace transform is used to derive an exact solution to the temperature variation in the tissue from the initial oscillation to the final steady periodic oscillation. Furthermore, the solutions to special cases under no perfusion rate, constant temperature, and the combination of these two assumptions are demonstrated in this study. The results show that both a larger perfusion rate and a greater tissue depth decrease the amplitude of the sinusoidal temperature response. Meanwhile, a larger perfusion rate can reduce the phase angle related to the sinusoidal surface temperature oscillation and surface heat flux.     

Simulation of heat transfer of biological tissue during cryosurgery based on vascular trees

During the freezing process in cryosurgery, the blood flow and blood perfusion have great influence on the heat transfer of the biological tissue. The effect of blood vessels on the temperature distribution of biological tissue is studied in this paper. The blood vessels are assumed as vascular trees using fractal methods. The method is based on the calculation of the ‘fractal dimension’ of blood vessels, considering the parameters of blood vessels and blood flows. The biological tissue is assumed as porous media and a numerical model of phase change heat transfer in biological tissue is established. The temperature distribution in biological tissue considering the effect of blood flow is simulated. The effect of the geometry of the vascular tree on the phase change heat transfer in biological tissue during cryosurgery is also analyzed.     

Thermodynamic stability analysis of non-Fourier model-based bio-heat transfer in laser-irradiated cylindrical-shaped multilayered biological tissue

The present research focuses on examining the thermic response of living tissue in the form of a triple-layered cylindrical structure when subjected to laser light and the compatibility analysis of non-Fourier heat transfer with thermodynamics second law. The temperature field in the triple-layered cylindrical living tissue subjected to laser light is determined by numerically solving the transient radiative transfer equation in conjunction with the dual phase lag (DPL) based bioheat equation. Once the temperature field is known, the equilibrium and nonequilibrium entropy production rate (EPR) is calculated based on the hypothesis of classical irreversible thermodynamics and extended irreversible thermodynamics, respectively. The present results are verified against the data from the literature and found a good match between them. A comparative analysis of the Fourier and non-Fourier models is accomplished. The equilibrium and nonequilibrium EPR values for the Fourier model are found to be positive. While the equilibrium EPR is negative for non-Fourier heat conduction and does not satisfy the thermodynamics second law, nonequilibrium EPR is always a positive value for Fourier, DPL, and hyperbolic models and satisfies thermodynamics second law. It has been investigated how thermal relaxation times affect the temperature field and EPRs in tissue are subjected to laser light.     

Thermal action of a short light pulse on biological tissues

A simple model for optical and thermal properties of two-component biological tissues is proposed as applied to studies of thermal fields under external illumination. The model comprises a small number of varying input parameters to enable one to find all the optical characteristics required to compute light fields in tissue and to state the thermal source function. Thermal parameters of tissues determining heat transfer in a two-component medium are calculated with accounting for heat exchange conditions between the components and at the interface with various external media. A set of heat conduction equations is stated for the two-component medium simulating biological tissues. Its analytical solution is derived. Spatial distributions of the fluence rate and temperature over the tissue depth are investigated at varying time moments after the irradiation by a short light pulse. Localized absorption of light by blood vessels and its effect on optical parameters of the medium, more intense heating of blood as compared with its surrounding (basic) tissue and heat exchange between the blood and tissue, as well as heat transfer at the interface with different environments are taken into account. The solutions are derived via characteristic times of thermal processes to enable one to easy and vividly evaluate the features in tissue heating as well as the effects of optical and thermal parameters on temperature distributions of the components. The calculations are illustrated by examples.     

Numerical analysis of an equivalent heat transfer coefficient in a porous model for simulating a biological tissue in a hyperthermia therapy

An equivalent heat transfer coefficient between tissue and blood in a porous model is investigated, which is applied to the thermal analysis of a biological tissue in a hyperthermia therapy. This paper applies a finite difference method to solving the tissue temperature distribution using Pennes’ bio-heat transfer equation and a two-equation porous model, respectively, and then employs a conjugate gradient method to estimate the equivalent heat transfer coefficient in the two-equation porous model with a known perfusion rate in Pennes’ bio-heat transfer equation. The results indicate that the equivalent heat transfer coefficient is not a strong function of the perfusion rate, blood velocity and heating conditions, but is inversely related to the blood vessel diameter.     

生物组织热边界条件反演研究

对肿瘤热疗过程中生物组织表面热流及内部温度协同反演进行了研究。首先介绍了激光辐照下生物组织内部光热传输模型，并采用有限体积法和离散坐标法相结合求解生物组织内光热传输问题。然后介绍了模糊推理方法基本原理，并采用改进分散模糊推理方法同时反演了激光诱导肿瘤热疗过程中生物组织表面入射热流及内部温度场。最后分析了热流形式和测量误差对反演结果的影响。结果表明，改进分散模糊推理方法可以准确地同时反演组织表面热流及内部温度分布，并具有较强稳定性和抗误差干扰能力。     

Semi-analytical solutions for the dual phase lag heat conduction in multilayered media

A semi-analytical solution procedure for transient heat transfer in composite mediums consisting of multi-layers within the framework of the dual phase lag model is presented. The procedure is then used to derive solutions for the temperature-, temperature gradient-, and heat flux distributions in a two-layer composite planar slab, a bi-layered solid-cylinder and sphere. The solutions obtained are applicable to the classical Fourier heat diffusion, hyperbolic heat conduction, phonon–electron interaction, and phonon scattering models with perfect or imperfect contact and with layers of different materials. The interfacial contact resistance, the heat flux and temperature gradient phase lags, thermal diffusivities and conductivities, initial temperatures of the composite medium and a general time-dependent boundary heat flux enter the solutions as parameters, allowing the solutions obtained to be applicable to a wide range of arrangements including perfect and imperfect contact. Analysis of thermal wave propagation, transmission and reflection in planar, cylindrical and spherical geometries with imperfect interfaces are presented, and geometrical—as well as the temperature gradient phase lag—effects on the thermal lagging behavior in different layered media are discussed.     

An axisymmetric bioheat transfer analysis through a unified model

A unified model is developed for the analysis of heat transfer (radiation and non-Fourier conduction) in an axisymmetric participating medium. The proposed model includes three different variants of hyperbolic–parabolic heat conduction models, that is, the single phase lag model, dual phase lag model, and the Fourier (no phase lag) model. The radiating-conducting medium is radiatively absorbing, emitting, and isotropically scattering. Significance of all the above mentioned models on the heat transfer characteristics is investigated in a two-dimensional axisymmetric geometry. The equation of transfer and the coupled non-Fourier conduction-radiation equation are solved via finite volume method. A fully implicit scheme is used to resolve the transient terms in the energy equation. For spatial resolution of radiation information, the STEP scheme is applied. Tri-diagonal-matrix-algorithm is used to solve the resulting set of linear discrete equations. Effects of two important influencing parameters: the scattering albedo and the radiation- conduction parameter are studied on the temporal evolution of temperature field in the radiatively participating medium. The non-Fourier effect of heat transport captured well with the proposed unified model. A good agreement can be found between the proposed model predictions and those available in the literature. It is also found that when the phase lag of the temperature gradient and the heat flux are the same, it reduces to conventional Fourier conduction-radiation and the wave behavior diminishes. However, the reduction to this Fourier model fails in the presence of constant blood perfusion and metabolic heat generation.     

Numerical simulation of three-phase-lag bio-heat transfer model for generalized coordinate system amidst thermal ablation

Various heat transmits models exhibit a thermal response in the tissue during thermal therapies. In the present work, we proposed a three-phase-lag (TPL) bio-heat-transmitting model including an external laser heating term amidst thermal treatment for the generalized coordinate system. The TPL model is based on the law of heat conduction containing heat flux, temperature gradient, and thermal displacement gradient terms. The solution is determined by implementing a discretization scheme of finite difference in the company of the Runge–Kutta (4,5) technique to examine the temperature repercussion throughout the thermal treatment of the tumor. In addition to this, the impact of various parameters such as phase lag in heat flux, temperature gradient, thermal displacement gradient, the heat generated because of blood perfusion, metabolism, and external heat source term is studied. A comparison has been shown graphically during the thermal treatment. The temperature profile is studied for different coordinate systems, and results are discussed. Moreover, validation has been made of the TPL model used in this paper with the experimental results. Thus the outcomes achieved from this study are purposeful in the therapeutical field, especially for oncologists.     

Investigation for the dual phase lag behavior of bio-heat transfer

The success of hyperthermia treatment depends on the precise prediction and control of temperature distribution in the tissue. It was absolutely a necessity for hyperthermia treatment planning to understand the heat transport occurring in biological tissue. The tissue is highly non-homogenous, and non-Fourier thermal behavior in biological tissue has been experimentally observed. The dual phase lag model of heat conduction has been used to interpret the non-Fourier thermal behavior. This work attempts to be an extension study of Antaki [12] and explore whether the DPL thermal behavior exists in tissue. The inverse non-Fourier bio-heat transfer problem in the bi-layer spherical geometry is analyzed. In order to further address whether the dual phase lag model of bio-heat transfer merits additional study, the comparisons of the history of temperature increase among the present calculated results, the calculated values from the classical bio-heat transfer equation, and the experimental data are made for various measurement locations.     

Parametric analysis of effective tissue thermal conductivity,thermal wave characteristic,and pulsatile blood flow on temperature distribution during thermal therapy

This study examines the coupled effects of pulsatile blood flow in a thermally significant blood vessel, the effective thermal conductivity of tumor tissue, and the thermal relaxation time in solid tissues on the temperature distributions during thermal treatments. Due to the cyclic nature of blood flow as a result of the heartbeat, the blood pressure gradient along a blood vessel was modeled as a sinusoidal change to imitate a pulsatile blood flow. Considering the enhancement in the thermal conductivity of living tissues due to blood perfusion, the effective tissue thermal conductivity was investigated. Based on the finite propagation speed of heat transfer in solid tissues, a modified wave bio-heat transfer transport equation in cylindrical coordinates was used. The numerical results show that a larger relaxation time results in a higher peak temperature. In the rapid heating case I (i.e., heating power density of 100 W cm^− 3 and heating duration of 1 s) and a heartbeat frequency of 1 Hz, the maximum temperatures were 62.587 and 63.107 °C for thermal relaxation times of 0.464 and 6.825 s, respectively. In contrast, the same total heated energy density of 100 J cm^− 3 in a slow heating case (i.e., heating power density of 5 W cm^− 3 and heating duration of 20 s) revealed maximum temperatures of 57.724 and 61.233 °C for thermal relaxation times of 0.464 and 6.825 s, respectively. In rapid heating cases, the occurrence of the peak temperature exhibits a time lag due to the influence of the thermal relaxation time. In contrast, in slow heating cases, the peak temperature may occur prior to the end of the heating period. Moreover, the frequency of the pulsatile blood flow does not appear to affect the maximum temperature in solid tumor tissues.     

Performance of shape-stabilized phase change material wallboard with periodical outside heat flux waves

Numerical simulations were carried out to investigate the performance of shape-stabilized phase change material (SSPCM) wallboard with sinusoidal heat flux waves on the outer surface and compared with traditional building materials – brick, foam concrete and expanded polystyrene (EPS). One-dimensional enthalpy equation was solved using control volume-based implicit finite-difference scheme. Time lag (φ), decrement factor (f) and phase transition keeping time (ψ) of inner surface were applied to analyze the effects of PCM thermo-physical properties, inner surface convective heat transfer coefficient and thickness of SSPCM wallboard. The results showed that for SSPCM, there exist two flat segments within one wave length period of inner surface heat flux lines and it has larger time lag and lower decrement factor than those three ordinary building materials. It was found that melting temperature and thermal conductivity of SSPCM have little effects on φ, f and ψ, which is different from the case of temperature waves; for a certain outside heat flux wave, there exist critical values of latent heat of fusion and thickness of SSPCM above which the heat flux wave amplitude can be diminished to zero; inner surface convective heat transfer coefficient is one important factor which significantly influences the decrement factor; and the phase transition zone leads to small fluctuations of the original flat segments of inner surface heat flux line.     

Electromagnetic field-induced thermal management of biological materials

The study of temperature profiles and heat transport within the human body when subjected to electromagnetic waves is crucial for development and improvement of radiofrequency cardiac ablation treatments (radio frequency ablation). The present study provides an analytical solution for computing the temperature profiles for blood and tissue for various biological media along with heat transfer behavior during various ablation processes. The local thermal nonequilibrium model is used to characterize the bioheat transport through the biological medium. The two energy equation model for tissue and blood phase is considered. To understand the effects induced by imposed electromagnetic field, the specific absorption rate of body tissues is also studied. The results obtained have been validated against the pertinent numerical results in the literature. This study provides benchmark analytical solutions for heat transport through biological media, thereby helping in understanding the thermophysiologic response of human body toward imposed electromagnetic radiation.