Dual-phase-lag Bioheat Transfer Research Articles

Numerical analysis of DPL bioheat transfer model with nonlocal impact on skin tissue during hyperthermia

This article discussed a numerical study of dual-phase-lag (DPL) bioheat transfer model with nonlocal impact on skin tissue during hyperthermia therapy when Gaussian type heat device is applied to the skin's outer surface. With the aid of a sufficient value of the parameters η,Qro and rp for a heating device of the Gaussian type, the temperature distribution at the targeted location is controlled and maintained. These parameters are employed to destroy a significant number of cancer cells in the targeted location while protecting the surrounding healthy tissue. The temperature profile at the targeted location decreases as lagging time τq and τT increases, and increases as spatial lagging λq increases. The blood perfusion effect can be shown when the value of α increases or when blood temperature decreases then it is seen that the temperature profile decreases. The numerical results obtained by the Finite element Legendre wavelet Galerkin (FELWG) approach are compared with the analytical results obtained in the specific situation to evaluate the precision. The obtained numerical results logically relate to the analytical results when we utilized the operational matrix of order M−1 (where M=100). All impacts of problem parameters are graphically represented during hyperthermia treatment.

Numerical simulation of the skin tissue subjected to hyperthermia treatment using a nonlinear DPL model

This article deals with the study of simulation-based mathematical modeling of tissue using the non-linear dual-phase-lag bioheat transfer (DPLBHT) model for hyperthermia treatment of tumor or cancer cells with Gaussian distribution type heat source. The metabolic heat source as an exponential variation, blood perfusion rate as a linear variation and thermal conductivity as a linear variation are considered to be temperature-dependent, resulted in a more accurate prediction of the non-linear DPLBHT model. At initial time, τq is more effective than τt in the targeted region. When the infected or cancerous cells increases in width then for effective treatment the value of the antenna constant could be decreased and the value of probe region can be changed according to the tumor location in the skin tissue. When the value of γ, Wbo, a1 and a2 increases, the temperature profile in the target region decreases. The numerical solution is obtained by hybrid scheme using Runge Kutta (4,5) and finite difference technique. This technique is compared for accuracy with an exact analytical result in a particular case as shown in figures and table and found in good compliance. The effect of lagging, metabolic heat source, blood perfusion rate, coefficient of thermal conductivity, external heat source parameter and the other associated parameter are studied in detail and the results are presented graphically.

Analytical Solution of Non Linear DPL Bioheat Transfer Model for Temperature Dependent Metabolic Heat Source During Thermal Therapy

In this research paper, the simulation based modelling of heat transfer in tissue under periodic boundary condition has been considered. The dual phase lag bioheat transfer (DPLBHT) model is implemented for computation of the temperature based thermal therapy treatment. The elements of volumetric heat source such as perfusion of blood, metabolism heat source and external heat source are considered in nonlinear DPL model. In this paper we have taken three cases for metabolic heat source namely, constant, linear and exponential. The combined two numerical methods which are based on finite difference scheme and Runge Kutta (4,5) scheme are exerted to solve the non-linear problem. We compute the exact solution for particular case. Numerical conclusions which are computed by numerical hybrid method are equated with exact result. It has been found that non linear DPL model with exponential metabolic heat source is closed to exact solution. We also expressed the effect of different parameters such as relaxation time, perfusion rate, metabolic heat source parameter, associated blood perfusion heat, heat source because of heat flux and temperature gradient etc.

Numerical solution of nonlinear dual‐phase‐lag model for analyzing heat transfer in tissue during thermal therapy

This article deals with mathematical modeling and simulation of heat transfer in tissue under periodic boundary condition using nonlinear dual-phase-lag-bioheat-transfer (DPLBHT). We have taken the temperature dependent blood perfusion and metabolic heat source as exponent variation in nonlinear DPLBHT model, both are experimentally validated function of temperature. In this article we applied finite difference method and Runge–Kutta (4,5) scheme to solve nonlinear problem. In particular case the exact solution is obtained and compared with numerical scheme and both are in good agreement. Effect of different parameters are discussed in detail such as blood perfusion rate, dimensionless heat source parameters, relaxation, and thermalization time on dimensionless temperature. The whole article is analyzed in dimensionless form.

Thermoelastic behavior of skin tissue induced by laser irradiation based on the generalized dual-phase lag model.

This paper analyzes the thermoelastic responses of skin tissue during laser irradiation based on a generalized dual-phase-lag (DPL) model. The method of separation of variables is utilized to obtain the analytical solutions for thermal and mechanical responses. The influences of some crucial parameters on temperature, displacement and stress evolutions are discussed, including the phase lag of heat flux, the phase lag of temperature gradient and the phase lag of laser pulse, the coupling factor between tissue and blood, the porosity of tissue, the equivalent diameter of tissue and the diameter of blood vessels. The generalized DPL bio-heat transfer model predicts different results from those by the classical DPL model and Pennes model. The equivalent diameter of tissue affects the coupling factor between tissue and blood, while the diameter of blood vessels mainly affects the porosity of tissue.

Relationship Between the Nonlocal Effect and Lagging Behavior in Bioheat Transfer

Abstract Lots of generalized heat conduction models have been developed in recent decades, such as local thermal nonequilibrium model, phase lagging model, and nonlocal heat conduction model. But no attempt was made to prove which model is better (or worse) than others, or whether there is a certain relationship between these different models. With this inspiration, we establish the nonlocal bioheat transfer equations with lagging time, and the two and three-temperature bioheat transfer equations with considering all the carrier's heat conduction effect are also constructed. Comparing the two (or three)-temperature equation model with the nonlocal bioheat transfer models with lagging time, one may obtain: the lagging time of temperature gradient τtand the nonlocal characteristic length λq in the space derivative items of heat flux have the same effect on heat transfer; when the heat transport occur among N energy carriers with considering the conduction effects of all carries, the heat transfer processes are dependent upon the high-order effect of τqN-1, τtN-1 and λt(2N-1) in nonlocal dual phase lag bioheat transfer model. This phenomenon is very important for biological and medical systems where numerous carriers may exist on the cellular level.

A new boundary element algorithm for a general solution of nonlinear space-time fractional dual-phase-lag bio-heat transfer problems during electromagnetic radiation

The main aim of this paper is to propose a new boundary element method (BEM) formulation for solving the nonlinear space-time fractional dual-phase-lag bio-heat transfer problems during electromagnetic radiation. Due to the advantages of BEM, such as not requiring a discretization of the interior of the treated region and providing a low RAM and CPU time. BEM is therefore a flexible and efficient tool for modeling bio-heat transfer problems. The efficiency of our proposed methodology has been improved by applying the communication-avoiding versions of the Arnoldi (CA-Arnoldi) preconditioner for solving the resulting linear systems arising from the BEM to reduce the iterations number and CPU time. Numerical results are depicted graphically to show the effects of time-fractional derivative order and space-fractional derivative order on the nonlinear temperature distributions. The numerical results also show the significant differences between the nonlinear temperature distributions of the classical Fourier, single-phase-lag, and dual-phase-lag bio-heat conduction models. To demonstrate the validity and accuracy of the proposed BEM methodology, numerical solutions for two-dimensional (2D) special case of the nonlinear space-time fractional dual phase lag bio-heat transfer problems are obtained and compared to experimental, Legendre wavelet collocation method (LWCM) and Fractional order Legendre functions and Galerkin method (FOLFs-GM).

Theoretical analysis of thermal response in biological skin tissue subjected to multiple laser beams

Laser has become a well-accepted technique for surgical procedures. To fit the physically irradiated area to the clinically planned target area, a multifibre laser system can be used. This paper theoretically investigated the temperature distribution in a three-dimensional biological tissue when irradiated by multifibre lasers with the aids of the dual-phase-lag (DPL) bio-heat conduction model. First, four laser beams located on the corners of a square are used as the heat sources. The method of separation of variables is adopted to obtain the analytical expression of temperature. The characteristics of DPL bio-heat transfer model and the difference with Pennes model were checked graphically. It is found that the effect of multiple laser beams on temperature response differs from those of single laser beam distinctly. A second heating phenomenon occurs and the contour of a square with curved corners are obtained. Then the laser beams are located to form a triangle and a trapezoid and the corresponding temperature responses are studied. Different irradiated zone can be obtained and accomplished by changing the spot size, the arrangement layout and the interval distance of the laser beams.

A numerical solution of nonlinear DPL bioheat model in biological tissue due to laser irradiations

In this study, the numerical solutions of the thermal damages of biological tissue by nonlinear dual-phase-lag theory under different boundary conditions are constructed for the medicinal treatment of tumors cell. The volumetric heat source component in nonlinear dual phase lag bio-heat transfer model such as the metabolic heat generation is considered an experimentally temperature-dependent function. Due to the nonlinearity of the governing relations, the finite element schemes are used to get the solutions of these problems. The numerical outcomes for temperature are given graphically. Furthermore, the comparison between the linear and nonlinear results displays the present nonlinear mathematical models effective tools to estimate the bio-heat transfer in the living tissue.

A Study on Non-Linear DPL Model for Describing Heat Transfer in Skin Tissue during Hyperthermia Treatment.

The article studies the simulation-based mathematical modeling of bioheat transfer under the Dirichlet boundary condition. We used complex non-linear dual-phase-lag bioheat transfer (DPLBHT) for analyzing the temperature distribution in skin tissues during hyperthermia treatment of infected cells. The perfusion term, metabolic heat source, and external heat source were the three parts of the volumetric heat source that were used in the model. The non-linear DPLBHT model predicted a more accurate temperature within skin tissues. The finite element Runge–Kutta (4,5) (FERK (4,5)) method, which was based on two techniques, finite difference and Runge–Kutta (4,5), was applied for calculating the result in the case of our typical non-linear problem. The paper studies and presents the non-dimensional unit. Thermal damage of normal tissue was observed near zero during hyperthermia treatment. The effects of the non-dimensional time, non-dimensional space coordinate, location parameter, regional parameter, relaxation and thermalization time, metabolic heat source, associated metabolic heat source parameter, perfusion rate, associated perfusion heat source parameter, and external heat source coefficient on the dimensionless temperature profile were studied in detail during the hyperthermia treatment process.

Effects of Thermal Relaxation on Temperature Elevation in Ex Vivo Tissues During High Intensity Focused Ultrasound

The introduction of thermal relaxation makes non-Fourier heat transfer more effective than Pennes equation to reflect rapid bio-heat transferring during HIFU, however, the contribution of thermal relaxation in specific tissues needs to be further clarified. In this study, we first measured the thermal relaxation times of porcine muscle and porcine fat. Combining with experimental measurements, the effects of thermal relaxation on temperature elevation in the tissues were investigated by using Pennes equation, thermal wave model of bio-heat transfer (TWMBT) and dual phase-lag (DPL) bio-heat transfer. Results showed that: a) The thermal relaxation times of porcine muscle and porcine fat are experimentally determined as 5.71± 0.11 s and 5.02± 0.06 s, respectively. b) In the absence of cavitation, DPL bio-heat transfer is more accurate to predict the temperature elevation at the focus and 2 mm from the focus than Pennes equation and TWMBT. Particularly, comparing with the reported of bologna (16 s) used in most of the theoretical analysis, the utilization of the measured thermal relaxation time for a specific tissue in DPL bio-heat transfer is more effective in predicting the temperature elevation during HIFU. c) Acoustic cavitation and nonlinear propagation is easier to happen in fat, under which all bio-heat transfer models are failed to forecast the temperature elevation induced by HIFU. The results demonstrate that different tissues have different thermal relaxation, DPL bio-heat transfer with the measured thermal relaxation times of specific tissues can accurately predict the temperature elevation during HIFU without cavitation.

A unique technique for analytical solution of 2-D dual phase lag bio-heat transfer problem with generalized time-dependent boundary conditions

The eigenfunction-based solutions for various heat conduction models (Fourier and non-Fourier) exhibit mismatch at the boundaries due to the discrepancy between the boundary conditions (BCs) and the corresponding eigenfunction profiles. This mismatch leads to spurious oscillations and erroneous solutions near the boundaries. Recently developed finite integral transform (FIT) based analytical solution of the dual phase lag (DPL) bio-heat transfer problem with generalized time-dependent BCs in a 2-D planar skin tissue phantom also encounters the similar mismatch, especially for Dirichlet BCs. The present study aims at removing such mismatch from eigenfunction-based solutions, by the homogenization of generalized time-dependent BCs which is essentially carried out by subtracting an auxiliary function from the temperature of the domain of interest. This novel treatment yields a modified problem in terms of the modified temperature with homogeneous BCs and contains a modified source. For this, the above-mentioned auxiliary functions need to satisfy a set of conditions at the boundaries. However, the functions satisfying such conditions may not be unique. Therefore, an additional condition (pseudo-steady state condition) is imposed with a view to obtaining relatively simple, unique auxiliary functions. The current study also explores the homogenization process for 2-D bio-heat transfer problem for the tissue phantom in conjunction with the use of orthogonal eigenfunction expansion method (OEEM). The present developed solution approach is demonstrated using two test problems comprising of (a) constant surface temperature and (b) sinusoidal heat flux on the surface, respectively. In case of constant surface temperature, the present approach gets rid of large spurious oscillations near the boundary. In case of surface heat flux, temperature prediction by the present approach shows good agreement in the domain compare to that predicted by the FIT approach. Besides that, the flux distribution predicted on the boundary by present approach is a realistic non-zero value, whereas, the FIT approach produces zero heat flux for the same applied non-zero heat flux at the boundary. Moreover, for both the test problems, the developed novel approach facilitates significant reduction in number of terms in the series summation as compared to the corresponding FIT solutions, for the same level of accuracy.

Analytical solutions of thermal damage in living tissues due to laser irradiation

ABSTRACT In this article, the analytical solutions of dual-phase-lag (DPL) bioheat equation with various boundary conditions have been constructed for curative therapy of cancerous cells. The volumetric thermal source components in the model of DPL bioheat transfer, such as the metabolic heat generation and the rate of blood perfusion, are considered. The analytical solutions in the Laplace domain are obtained. The thermal injuries to the tissue are assessed by the denatured protein range, using the formulation of Arrhenius. Numerical outcomes for temperatures are graphically introduced. A parametric analysis is devoted to the identification of an appropriate procedure for selecting important design variables in order to predict more accurate to reach effective heating in hyperthermia treatment.

A new LRBFCM-GBEM modeling algorithm for general solution of time fractional-order dual phase lag bioheat transfer problems in functionally graded tissues

The main objective of this article is to propose a new hybrid modeling algorithm based on combining local radial basis function collocation method (LRBFCM) and general boundary element method (GBEM) for solving time fractional-order dual phase lag bioheat transfer problems in functionally graded tissues. The LRBFCM was developed and implemented using an implicit time-stepping technique and Caputo time fractional derivative for solving the fractional-order governing equation without dual phase lags. Due to suitability of the GBEM for modeling of bioheat transfer in functionally graded tissues. Therefore, GBEM is applied for solving the dual phase lags governing equation without fractional-order derivative. The numerical results are depicted graphical forms to show the effects of functionally graded parameter, fractional-order parameter and anisotropy on the nonlinear temperature distribution. Also, these numerical results demonstrate the validity and accuracy of the proposed algorithm and technique.

Verified non-linear DPL model with experimental data for analyzing heat transfer in tissue during thermal therapy

In this paper, mathematical modeling and simulation of heat transfer in tissue using non-linear dual-phase-lag bioheat transfer (DPLBHT) model under Dirichlet boundary condition has been studied for therapeutic treatment of cancerous or tumorous cells. The components of volumetric heat source in non-linear DPLBHT model such as blood perfusion and metabolism are assumed experimentally validated temperature-dependent function which results in non-linear DPLBHT model in order to predict more accurate. A hybrid numerical method which is based on finite difference and Runge-Kutta (4, 5) schemes, is used to solve the present non-linear problem. The exact solution has been obtained in a particular case and compared with the present numerical scheme, and we found that those are in good agreement. A comparison of models has been made when the variation of blood perfusion rate and metabolism is realistic function of temperature, is of non-linear DPLBHT model, non-linear single-phase-lag bioheat transfer (SPLBHT) model and non-linear Pennes bioheat transfer (PBHT) model with experimental data in same situation and it has been found that non-linear DPL model is closest to the experimental data. The whole paper is analyzed and presented in dimensionless form. The effect of coefficient of blood perfusion rate, dimensionless heating source parameters, relaxation and thermalisation time on dimensionless temperature distribution has been analyzed in treatment process.

Solving the dual-phase lag bioheat transfer equation by the generalized finite difference method

The modeling of bioheat transfer process described by the dual-phase lag equation is considered. The basic equation is supplemented by the appropriate boundary-initial conditions. In the central part of the cylindrical domain the heated sub-domain is located. In this region the additional component determining the capacity of an internal heat source is taken into account. At the stage of numerical computations the generalized finite difference method (GFDM) is used. The GFDM nodes distribution is generated in a random way (with some limitations). The examples of computations for different nodes distribution and comparison with the classical finite difference method are presented. In the final part of the paper the conclusions are formulated.

Numerical solution of non-linear dual-phase-lag bioheat transfer equation within skin tissues

This paper deals with numerical modeling and simulation of heat transfer in skin tissues using non-linear dual-phase-lag (DPL) bioheat transfer model under periodic heat flux boundary condition. The blood perfusion is assumed temperature-dependent which results in non-linear DPL bioheat transfer model in order to predict more accurate results. A numerical method of line which is based on finite difference and Runge–Kutta (4,5) schemes, is used to solve the present non-linear problem. Under specific case, the exact solution has been obtained and compared with the present numerical scheme, and we found that those are in good agreement. A comparison based on model selection criterion (AIC) has been made among non-linear DPL models when the variation of blood perfusion rate with temperature is of constant, linear and exponential type with the experimental data and it has been found that non-linear DPL model with exponential variation of blood perfusion rate is closest to the experimental data. In addition, it is found that due to absence of phase-lag phenomena in Pennes bioheat transfer model, it achieves steady state more quickly and always predict higher temperature than thermal and DPL non-linear models. The effect of coefficient of blood perfusion rate, dimensionless heating frequency and Kirchoff number on dimensionless temperature distribution has also been analyzed. The whole analysis is presented in dimensionless form.

Dual phase lag bio-heat transfer during cryosurgery of lung cancer: Comparison of three heat transfer models

The Pennes bio-heat model is based on Fourier's law of heat conduction, which assumed that a thermal signal propagate with infinite speed. This gives contradiction in physical situation. Also, the hyperbolic bio-heat model considers the micro scale response in time, but it does not explain the micro scale response in space. Therefore, to consider the thermal behaviour which is not captured by the Fourier's law and to take into account the microstructural effect in space, a dual phase lag (DPL) bio-heat conduction model would be advantageous. In this paper, a two dimensional DPL model is proposed to study the phase change heat transfer process during cryosurgery of lung cancer. The governing equations are solved numerically by enthalpy based finite difference method. The non-ideal behaviour of tissue and heat source terms, metabolism and blood perfusion are also considered. This study is made to examine the effects of phase lags in heat flux and temperature gradient on interface positions and temperature distribution during freezing process. A comparative study of DPL, parabolic and hyperbolic conduction models is thoroughly investigated. It is found that the phase lags of temperature gradient and heat flux have significant effect on interface positions and temperature distribution.

Finite integral transform-based analytical solutions of dual phase lag bio-heat transfer equation

• Analytical solution of the non-Fourier bio heat transfer equation. • Finite integral transform. • Time independent/dependent boundary conditions. • Coupling of the transient RTE and non-Fourier heat conduction models. • Thermal analysis of laser-irradiated biological tissue phantoms. A finite integral transform (FIT)-based analytical solution to the dual phase lag (DPL) bio-heat transfer equation has been developed. One of the potential applications of this analytical approach is in the field of photo-thermal therapy, wherein the interest lies in determining the thermal response of laser-irradiated biological samples. In order to demonstrate the applicability of the generalized analytical solutions, three problems have been formulated: (1) time independent boundary conditions (constant surface temperature heating), (2) time dependent boundary conditions (medium subjected to sinusoidal surface heating), and (3) biological tissue phantoms subjected to short-pulse laser irradiation. In the context of the case study involving biological tissue phantoms, the FIT-based analytical solutions of Fourier, as well as non-Fourier, heat conduction equations have been coupled with a numerical solution of the transient form of the radiative transfer equation (RTE) to determine the resultant temperature distribution. Performance of the FIT-based approach has been assessed by comparing the results of the present study with those reported in the literature. A comparison of DPL-based analytical solutions with those obtained using the conventional Fourier and hyperbolic heat conduction models has been presented. The relative influence of relaxation times associated with the temperature gradients ( τ T ) and heat flux ( τ q ) on the resultant thermal profiles has also been discussed. To the best of the knowledge of the authors, the present study is the first successful attempt at developing complete FIT-based analytical solution(s) of non-Fourier heat conduction equation(s), which have subsequently been coupled with numerical solutions of the transient form of the RTE. The work finds its importance in a range of areas such as material processing, photo-thermal therapy, etc.

Numerical simulation of time fractional dual-phase-lag model of heat transfer within skin tissue during thermal therapy

In this paper, we investigated the thermal behavior in living biological tissues using time fractional dual-phase-lag bioheat transfer (DPLBHT) model subjected to Dirichelt boundary condition in presence of metabolic and electromagnetic heat sources during thermal therapy. We solved this bioheat transfer model using finite element Legendre wavelet Galerkin method (FELWGM) with help of block pulse function in sense of Caputo fractional order derivative. We compared the obtained results from FELWGM and exact method in a specific case, and found a high accuracy. Results are interpreted in the form of standard and anomalous cases for taking different order of time fractional DPLBHT model. The time to achieve hyperthermia position is discussed in both cases as standard and time fractional order derivative. The success of thermal therapy in the treatment of metastatic cancerous cell depends on time fractional order derivative to precise prediction and control of temperature. The effect of variability of parameters such as time fractional derivative, lagging times, blood perfusion coefficient, metabolic heat source and transmitted power on dimensionless temperature distribution in skin tissue is discussed in detail. The physiological parameters has been estimated, corresponding to the value of fractional order derivative for hyperthermia treatment therapy.