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.     

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.     

Non-Fourier effects on transient temperature resulting from periodic on-off heat flux

The transient temperatures resulting from a periodic on-off heat flux boundary condition have many applications, including, among others, the sintering of catalysts frequently found during coke burnoff, and the use of laser pulses for annealing of semiconductors. In such situations, the duration of the pulses is so small (i.e. picosecond-nanosecond) that the classical heat diffusion phenomenon breaks down and the wave nature of energy propagation characterized by the hyperbolic heat conduction equation governs the temperature distribution in the medium. In this work, an explicit analytic solution is presented for a linear transient heat conduction problem in a semi-infinite medium subjected to a periodic on-off type heat flux at the boundary x = 0 by solving the hyperbolic heat conduction equation. The non-linear case allowing for the added effect of surface radiation into an external ambient is studied numerically.     

Numerical analysis of coupled effects of pulsatile blood flow and thermal relaxation time during thermal therapy

The main purpose of this study is to investigate the coupled effects of the pulsatile blood flow in thermally significant blood vessels and the thermal relaxation time in living tissues on temperature distributions during thermal treatments. Considering the fact that propagation speed of heat transfer in solid tissues is actually finite according to experiments, the traditional Pennes bioheat transfer equation (PBTE) was modified to a wave bioheat transfer equation (WBTE) that contains both wave transportation and diffusion competing with each other and characterized by the thermal relaxation time. The wave behavior will be more dominant when the relaxation time is large. WBTE together with a coupled energy transport equation for blood vessel flow was used to describe the temperature evolution of our current tumor–blood vessel system, and the equations were numerically solved by the highly accurate multi-block Chebyshev pseudospectral method. Numerical results showed that temperature evolution from WBTE was quite different from their counterparts from PBTE due to the dominant wave feature under large relaxation time. For example, larger relaxation time would preserve high temperature longer and this effect is even more pronounced when heating is fast. It further implies that heat is drained more slowly when relaxation time is large, and would make thermal lesion region cover the tumor tissue, the heating target, better. This phenomenon would therefore hint that the traditional PBTE simulations might under-estimate the thermal dose exerted on tumor. As to the pulsation frequency of blood flow from heart beat which was originally predicted to be important here, it turned out that the thermal behavior is quite insensitive to pulsation frequency in the current study.     

Integral equation solution for hyperbolic heat conduction with surface radiation

Temperature distributions for the hyperbolic heat conduction in a semi-infinite medium with surface radiation are found from the solutions of a nonlinear Volterra equation for the surface temperature. The integral equation is obtained by the Laplace transform. This method has the advantage that the temperature distributions do not involve numerical oscillations around the thermal wave front.     

Non-invasive estimation of size and location of a tumor in a human breast using a curve fitting technique

This article deals with the estimation of the size and location of a tumor in the breast. Estimation is based on the measurement of the skin surface temperature. With skin surface temperature known, the estimation is done using the newly proposed curve fitting technique. For the present study, justification is shown for the consideration of a 2-D geometry of the breast instead of its 3-D hemispherical shape. Heat transfer in a blood perfused tissue is analyzed using the Pennes bioheat equation. The steady-state temperature distribution in the tissue-tumor system is obtained by solving the bioheat equation using the finite volume method. The size and location of the tumor are accurately estimated. Computationally, the procedure is highly efficient.     

THERMALLY INDUCED VIBRATION IN A THIN PLATE UNDER THE WAVE HEAT CONDUCTION MODEL

Thermally induced vibration in a thin plate under a thermal excitation is investigated. The excitation is in the form of a suddenly applied laser pulse (thermal shock). The resulting transient variations of temperature are predicted using the wave heat conduction model (hyperbolic model), which accounts for the phase lag between the heat flux and the temperature gradient. The resulting heat conduction equation is solved semianalytically using the Laplace transformation and the Riemann sum approximation to calculate the temperature distribution within the plate. The equation of motion of the plate is solved numerically using the finite difference technique to calculate the transient variations in deflections.     

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.     

A comparison of hyperbolic and parabolic models of phase change of a pure metal

The solidification history of individual thermal spray particles has been the subject of many experimental and theoretical studies. Yet it is customary to assume that solidification occurs at the equilibrium temperature, and that heat propagates according to Fourier’s Law. To account for a finite thermal diffusion speed, a hyperbolic heat conduction equation is usually adopted to analyze heat transfer. However, under certain circumstances, this equation can violate the second law of thermodynamics, and so others have modified the original hyperbolic equation via theories of extended irreversible thermodynamics. In this work, we study non-equilibrium effects of rapid solidification of a pure metal particle, and compare the so-called parabolic, hyperbolic and modified hyperbolic equations for heat transfer, to predict the interface undercooling due to thermal effects and velocity as a function of time, for different relaxation times. Results indicate that differences are limited to the early part of the solidification process, when undercooling is most significant, the interface velocity is highest, and non-equilibrium effects are most evident. As solidification progresses, the non-equilibrium effects wane and solidification can then be properly modeled as an equilibrium process.     

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.     

General formulation and analysis of hyperbolic heat conduction in composite media

Some recent experimental results show the existence of reflections of thermal waves at the interface of dissimilar materials in superfluid helium. In light of these results, a theoretical investigation of thermal waves in composite is provided to give a theoretical foundation to the observed phenomenon. A general one-dimensional temperature and heat flux formulation for hyperbolic heat conduction in a composite medium is presented. Also, the general solution, based on the flux formulation, is developed for the standard three orthogonal coordinate systems. Unlike classical parabolic heat conduction, heat conduction based on the modified Fourier's law produces non-separable field equations for both the temperature and flux and therefore standard analytical techniques cannot be applied in these situations. In order to alleviate this difficulty, a generalized finite integral transform technique is proposed in the flux domain and a general solution is developed for the standard three orthogonal coordinate systems. The general solution is applied to the case of a two-region slab with a pulsed volumetric source and insulated exterior surfaces which displays the unusual and controversial nature associated with heat conduction based on the modified Fourier's law in composite regions.     

Difference Scheme for Hyperbolic Heat Conduction Equation with Pulsed Heating Boundary

horotctionIt is well known that the basic law of heatconduction is the Fourier law. It has the formq = --k' vT, and one-dimensional heat conduchondifferenhal equation is. The aboveequations are derived from the hypothesis that thevelocity to establish the thermal balance is infinitelygreat. In the modem heat conduchon theory heattransacts in materials in a licited velocity. The factorsto affect the velocity are the thermal propelles of thematerials. In order tO describe this Problem, the sch…     

A general bioheat transfer model based on the theory of porous media

A volume averaging theory (VAT) established in the field of fluid-saturated porous media has been successfully exploited to derive a general set of bioheat transfer equations for blood flows and its surrounding biological tissue. A closed set of macroscopic governing equations for both velocity and temperature fields in intra- and extravascular phases has been established, for the first time, using the theory of anisotropic porous media. Firstly, two individual macroscopic energy equations are derived for the blood flow and its surrounding tissue under the thermal non-equilibrium condition. The blood perfusion term is identified and modeled in consideration of the transvascular flow in the extravascular region, while the dispersion and interfacial heat transfer terms are modeled according to conventional porous media treatments. It is shown that the resulting two-energy equation model reduces to Pennes model, Wulff model and their modifications, under appropriate conditions. Subsequently, the two-energy equation model has been extended to the three-energy equation version, in order to account for the countercurrent heat transfer between closely spaced arteries and veins in the circulatory system and its effect on the peripheral heat transfer. This general form of three-energy equation model naturally reduces to the energy equations for the tissue, proposed by Chato, Keller and Seiler. Controversial issues on blood perfusion, dispersion and interfacial heat transfer coefficient are discussed in a rigorous mathematical manner.     

SPECIALLY TAILORED TRANS FINITE-ELEMENT FORMULATIONS FOR HYPERBOLIC HEAT CONDUCTION INVOLVING NON-FOURIER EFFECTS

The phenomenon of hyperbolic heat conduction in contrast to the classical (parabolic) form of Fourier heat conduction involves thermal energy transport that propagates only at finite speeds as opposed to an infinite speed of thermal energy transport. To accommodate the finite speed of thermal wave propagation, a more precise form of heat flux law is involved, thereby modifying the heat flux originally postulated in the classical theory of heat conduction. As a consequence, for hyperbolic heat conduction problems, the thermal energy propagates with very sharp discontinuities at the wave front. The primary purpose of the present paper is to provide accurate solutions to a class of one-dimensional hyperbolic heat conduction problems involving non-Fourier effects that can precisely help understand the true response and furthermore can be used effectively for representative benchmark tests and for validating alternate schemes. As a consequence, the present paper purposely describes modeling/analysis formulations via specially tailored hybrid computations for accurately modeling the sharp discontinuities of the propagating thermal wave front. Comparative numerical test models are presented for various hyperbolic heat conduction models involving non-Fourier effects to demonstrate the present formulations.     

Study of an unsteady magnetohydrodynamic-free convective flow under parabolic ramped temperature and concentration in the presence of Soret and heat sink

The purpose of this paper is to investigate the effects of Soret, thermal radiation, and chemical reaction on an unsteady magnetohydrodynamic free convective flow past an impulsively initiated semi-infinite vertical plate with heat sink under parabolic ramped temperature and parabolic ramped concentration. Using some nondimensional parameters, the flow boundary equations in this case are first converted to dimensionless equations. The closed-form Laplace transform technique is employed here to solve the partial differential equations and get the solutions for fluid velocity, temperature, and concentration. The velocity, temperature, and concentration of the fluid tend to vary with the effect of various flow factors. These changes are graphically represented and analyzed. Differences in skin friction, Nusselt number, and Sherwood number for the different relevant parameters are also recorded. The Soret number hikes the fluid velocity and concentration. The rate of heat transfer, mass transfer, and momentum transfer improves due to the application of parabolic ramped conditions.     

Hyperbolic axial dispersion model for heat exchangers

Flow maldistribution in heat exchangers for steady-state and transient processes can be described by dispersion models. The traditional parabolic model and the proposed hyperbolic model which includes the parabolic model as a special case can be used for dispersive flux formulation. Instead of using the heuristic approach of parabolic or hyperbolic formulation, these models can be quantitatively derived from the axial temperature profiles of heat exchangers. In this paper both the models are derived for a shell-and-tube heat exchanger with pure maldistribution (without back mixing) in tube side flow and the plug flow on the shell side. The Mach number and the boundary condition which plays a key role in the hyperbolic dispersion have been derived and compared with previous investigation. It is observed that the hyperbolic model is the best suited one as it compares well with the actual calculations. This establishes the hyperbolic model and its boundary conditions.     

A generalized relation between the local values of temperature and the corresponding heat flux in a one-dimensional semi-infinite domain with the moving boundary

The paper presents generalized relation between the local values of temperature and the corresponding heat flux in a one-dimensional semi-infinite domain with the moving boundary. The generalized relation between the local values of temperature and the corresponding heat flux has been achieved by the use of a novel technique that involves generalized derivatives (in particular, derivatives of non-integer orders). Confluent hyper-geometric functions, known as Whittaker’s functions, appear in the course of the solution procedure, upon applying the Laplace transform to the original transport equation. The relation is written in the integral form and provides a relationship between the local values of the temperature and heat flux.     

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.     

LATTICE BOLTZMANN SCHEME FOR HYPERBOLIC HEAT CONDUCTION EQUATION

An extended lattice Boltzmann (LB) equation, the lattice Boltzmann equation with a source term, is developed for the system of equations governing the hyperbolic heat conduction equation. Mathematical consistence between the proposed extended LB equation and the governing equations are accomplished by the Chapman-Enskog expansion. Four illustrative examples, with both finite and semi-infinite computational domains and subjected to linear and nonlinear boundary conditions, are simulated. All numerical predications agree very well with the existing solutions in the literature. It is also demonstrated that the present scheme is stable and free of numerical oscillations especially around the wave front, where sharp change in temperature occurs.     

Convection heat and mass transfer in a hydromagnetic flow of a second grade fluid in the presence of thermal radiation and thermal diffusion

The convection heat and mass transfer in a hydromagnetic flow of a second grade fluid past a semi-infinite stretching sheet in the presence of thermal radiation and thermal diffusion are considered. The governing coupled non-linear partial differential equations describing the flow problem are transformed into non-linear ordinary differential equations by method of similarity transformation. The resulting similarity equations are solved numerically using Runge-Kutta shooting method. The results are presented as velocity, temperature and concentration fields for different values of parameters entering into the problem. The skin friction, rate of heat transfer and mass transfer are presented numerically in tabular form. In addition, the results obtained showed that these parameters have significant influence on the flow, heat and mass transfer.