non-Fourier Bioheat Transfer Research Articles

Non-Fourier Bioheat Transfer Analysis in Brain Tissue During Interstitial Laser Ablation: Analysis of Multiple Influential Factors

This work presents the dual-phase lag-based non-Fourier bioheat transfer model of brain tissue subjected to interstitial laser ablation. The finite element method has been utilized to predict the brain tissue's temperature distributions and ablation volumes. A sensitivity analysis has been conducted to quantify the effect of variations in the input laser power, treatment time, laser fiber diameter, laser wavelength, and non-Fourier phase lags. Notably, in this work, the temperature-dependent thermal properties of brain tissue have been considered. The developed model has been validated by comparing the temperature obtained from the numerical and ex vivo brain tissue during interstitial laser ablation. The ex vivo brain model has been further extended to in vivo settings by incorporating the blood perfusion effects. The results of the systematic analysis highlight the importance of considering temperature-dependent thermal properties of the brain tissue, non-Fourier behavior, and microvascular perfusion effects in the computational models for accurate predictions of the treatment outcomes during interstitial laser ablation, thereby minimizing the damage to surrounding healthy tissue. The developed model and parametric analysis reported in this study would assist in a more accurate and precise prediction of the temperature distribution, thus allowing to optimize the thermal dosage during laser therapy in the brain.

Dynamic responses of wearable thermoelectric generators used for skin waste heat harvesting

A thermodynamic model for skin and wearable thermoelectric generators (WTEGs) system is developed based on dual-phase-lag (DPL) bioheat transfer. Human skin is regarded as a multi-layer structure consisted of subcutis, dermis and epidermis. Analytical solutions for temperature profile inside skin-WTEG system and energy conversion performance of WTEGs are obtained. Numerical results show that very small deviations for power output of WTEGs are caused by using the room-temperature physical properties of Bi2Te3-based thermoelectric semiconductors. However, the effect of heat convection by blood perfusion inside skin tissue should be considered to estimate energy conversion performance of WTEGs accurately. The influence of the contact thermal resistance between the skin and WTEG can be neglected when the thermal conductance ratio of the skin-WTEG interface to the flexible substrate is larger than 0.1. It takes more than 10 min to attain the steady-state again when a thermal perturbation is applied to the skin-WTEG system. The classic Fourier bioheat transfer model may also provide acceptable accuracy for the energy conversion analysis of WTEG compared with the non-Fourier bioheat transfer model if the response time exceeds 1 min. This paper provides a useful theoretical model for designing WTEG devices.

Three-Phase-Lag Bio-Heat Transfer Model of Cardiac Ablation

Significant research efforts have been devoted in the past decades to accurately modelling the complex heat transfer phenomena within biological tissues. These modeling efforts and analysis have assisted in a better understanding of the intricacies of associated biological phenomena and factors that affect the treatment outcomes of hyperthermic therapeutic procedures. In this contribution, we report a three-dimensional non-Fourier bio-heat transfer model of cardiac ablation that accounts for the three-phase-lags (TPL) in the heat propagation, viz., lags due to heat flux, temperature gradient, and thermal displacement gradient. Finite element-based COMSOL Multiphysics software has been utilized to predict the temperature distributions and ablation volumes. A comparative analysis has been conducted to report the variation in the treatment outcomes of cardiac ablation considering different bio-heat transfer models. The effect of variations in the magnitude of different phase lags has been systematically investigated. The fidelity and integrity of the developed model have been evaluated by comparing the results of the developed model with the analytical results of the recent studies available in the literature. This study demonstrates the importance of considering non-Fourier lags within biological tissue for predicting more accurately the characteristics important for the efficient application of thermal therapies.

Bio-thermo-viscoelastic behavior in multilayer skin tissue

A theoretical model for skin thermo-viscoelastic behavior is proposed based on the non-Fourier bioheat transfer theory in this article. Skin tissue is regarded as a multilayer structure composed of epidermis, dermis, and subcutis, the properties of each layer is assumed to be homogeneous and thermoviscoelasticity. Analytical solutions of temperature profile and stress distribution in skin tissue are obtained, and the influence of non-Fourier effect, viscoelastic properties of skin, blood perfusion, and time-dependent thermal boundary conditions at skin surface is considered. Numerical results show that the peak stress occurred at skin surface induced by thermal injury is much larger than mechanical threshold of pain perception. The temperature profile in skin tissue may be overestimated for the long-time hyperthermia treatment when the heat convection of blood perfusion in dermis and subcutis is neglected. The thermal stress calculated based on thermoviscoelastic model is significantly lower than that based on classical thermoelastic model due to the inherent rheological properties of skin tissue. This article may be helpful in improving the fundamental understanding of complicated thermomechanical behavior of human skin and designing of biomedical devices for hyperthermia treatment.

Effects of non-Fourier bioheat transfer on bone drilling temperature in orthopedic surgery: Theoretical and in vitro experimental investigation.

The mechanical drilling process is a typical step in treating bone fractures to fix broken parts with screws and plates. Drilling generates a significant amount of heat and elevates the temperature of the bone, which can cause thermal osteonecrosis and damage to the surrounding bone tissue and nerves. Thermal inertia between heat flux and temperature gradient in nonhomogeneous interior structural medium-like biological tissues is arguable. Therefore, this paper proposes an analytical model of heat propagation in bone drilling for orthopedic surgery based on the hyperbolic Pennes bioheat transfer equation (HPBTE). Drilling experiments in bovine cortical bone samples were also carried out using an infrared thermography approach to confirm the proposed analytical model. Around the drilled hole surface, thermal necrosis is spread out from 1 to 10 mm. Increased feed rate reduces necrosis penetration distance and increases intense bone necrosis. The HPBTE includes thermal relaxation time effect and internal convective function of tissue perfusion rate. As these factors are not considered in the parabolic heat transfer equation (PHTE), the results show that the HPBTE is more accurate in predicting temperature and thermal osteonecrosis than the PHTE. As a result, proposed analytical model is a handy tool for calculating temperature to avoid thermal damage while improving process efficiency. Furthermore, it has the capability of controlling the manual or robotic drilling procedure for minimally invasive operations.

Analytical Investigation of Non-Fourier Bioheat Transfer in the Axisymmetric Living Tissue Exposed to Pulsed Laser Heating Using Finite Integral Transform Technique

Abstract This article proposes the closed-form solution of the generalized non-Fourier model-based bioheat transfer equation (BHTE) in Cylindrical coordinates to understand the thermal behavior of living tissue heated by a pulsed laser. The axisymmetric living tissue exposed to the non-Gaussian temporal profile of laser heating has been considered to investigate the non-Fourier bioheat transfer phenomena. The closed-form solution of the generalized non-Fourier model-based BHTE with time-dependent thermal energy generation has been obtained through the finite integral transform (FIT) technique. The analytical solution was juxtaposed to the corresponding numerical solution in order to determine its reliability. The numerical solution of the aforementioned governing equation has been obtained by the finite volume method (FVM). The results of both analytical and numerical solutions have been verified using results given in published literature. Subsequently, the dual-phase-lag (DPL) model's findings were juxtaposed to those obtained using the hyperbolic and traditional Fourier models. The effect of different parameters like relaxation times corresponding to the temperature gradient and heat flux, metabolic energy generation, and blood perfusion on the resultant temperature distribution inside the axisymmetric living tissue exposed to pulsed laser heating has been discussed. The importance of this study might be found in various applications such as laser-based-photothermal therapy, melting of the surface of metal and alloys by laser heating.

Analysis of the time-space fractional bioheat transfer equation for biological tissues during laser irradiation

Laser technology has been widely used in biomedical therapies and external surgeries. To promote its applications, a good thermal model is required to analyze the temperature distribution within the living tissues. In this paper, we develop a time-space fractional hyperbolic bioheat transfer model to study the non-Fourier bioheat transfer process within the living biological tissues during laser irradiation. Based on the the L1 approximation for the Caputo time fractional derivative and the central difference scheme for the Riesz fractional derivative, the finite difference algorithm is developed for the laser irradiation problem. The efficiency and accuracy of this method have been verified by using three numerical examples. In view of the thermophysical properties of the biological tissues, the effect of time and space fractional parameters, phase lag time and blood perfusion rate on temperature distribution within living biological tissues have been analyzed and shown graphically.

A Non-Fourier Bioheat Transfer Model for Cryosurgery of Tumor Tissue with Minimum Collateral Damage

Background and objectivesIncorporation of non-Fourier heat conduction while studying heat transfer phenomena in biological materials has emerged has an important approach as it predicts better and more realistic results than Fourier based models. In this article we have proposed a non-Fourier computational model and applied the same to simulate cryosurgery of lung tumor and attempted minimization of freezing damage of healthy lung tissue using pulsed laser irradiation. MethodsA non-Fourier bioheat transfer model for phase change in biological tissues is solved via a Fourier heat conduction based solution approach. A unified model is proposed combining all variants of bioheat models: Fourier's heat conduction based Pennes’ bioheat model, hyperbolic heat conduction model and dual phase lag model. The proposed model takes into account the different thermophysical properties of frozen and unfrozen regions. In order to mimic the actual biotransport process, the blood perfusion and metabolic heat generation are switched off in the frozen region. Implicit source based enthalpy method is used to model phase change process. A new iterative enthalpy update equation is developed for capturing evolution of freezing front implicitly. Finite Volume based numerical discretization technique is used to discretize the governing PDE. The resulting discrete algebraic equation set is solved implicitly by Tri-diagonal Matrix Algorithm. The proposed model is verified with existing results from the literature. ResultsFor Fourier heat conduction, freezing time of 99.99% of tumor is 1247s, which increases to 1267s for τq= 5s (τT= 0s) and again reduces to 1255s for τq= 5s and τT= 3s. τq and τT are phase lag parameters for non-Fourier heat conduction. For τq= 5s and τT= 0.05s, the freezing damage of healthy tissue decreases by 23.76% when pulsed laser irradiation (Io = 106 W/m2) is used to warm the neighboring healthy tissue. ConclusionsSo non-Fourier bioheat transport models are better and more accurate in predicting temperature history, freezing time and freezing front propagation as compared to Fourier based models. Pulsed laser irradiation can prove to be a very efficient technique in minimizing collateral damage during cryosurgery.

Convergence of finite element methods for hyperbolic heat conduction model with an interface

The paper concerns numerical study of non-Fourier bio heat transfer model in multi-layered media. Specifically, we employ the Maxwell–Cattaneo equation on the physical media with discontinuous coefficients. A fitted finite element method is proposed and analyzed for a hyperbolic heat conduction model with discontinuous coefficients. Typical semidiscrete and fully discrete schemes are presented for a fitted finite element discretization with straight interface triangles. The fully discrete space–time finite element discretizations are based on second order in time Newmark scheme. Optimal a priori error estimates for both semidiscrete and fully discrete schemes are proved in L∞(L2) norm. Numerical experiments are reported for several test cases to confirm our theoretical convergence rate. Finite element algorithm presented here can be used to solve a wide variety of hyperbolic heat conduction models for non-homogeneous inner structures.

Finite element methods for non-fourier thermal wave model of bio heat transfer with an interface

We propose a fitted finite element method for non-Fourier bio heat transfer model in multi-layered media. Specifically, we employ the Maxwell–Cattaneo equation on the physical media that have a heterogeneous conductivity. Well-posedness of the model interface problem is established. A continuous piecewise linear finite element space is employed for the spatially semidiscrete approximation and the temporal discretization is based on backward scheme. Optimal order error estimates for both semidiscrete and fully discrete schemes are proved in $$L^{\infty }(H^1)$$ norm. Finally, we give numerical examples to verify our theoretical results. The new results and finite element schemes can be applied in the fields of engineering, medicine, and biotechnology.

Fourier and non-Fourier bio-heat transfer models to predict ex vivo temperature response to focused ultrasound heating

Although predicting temperature variation is important for designing treatment plans for thermal therapies, research in this area is yet to investigate the applicability of prevalent thermal conduction models, such as the Pennes equation, the thermal wave model of bio-heat transfer, and the dual phase lag (DPL) model. To address this shortcoming, we heated a tissue phantom and ex vivo bovine liver tissues with focused ultrasound (FU), measured the temperature response, and compared the results with those predicted by these models. The findings show that, for a homogeneous-tissue phantom, the initial temperature increase is accurately predicted by the Pennes equation at the onset of FU irradiation, although the prediction deviates from the measured temperature with increasing FU irradiation time. For heterogeneous liver tissues, the predicted response is closer to the measured temperature for the non-Fourier models, especially the DPL model. Furthermore, the DPL model accurately predicts the temperature response in biological tissues because it increases the phase lag, which characterizes microstructural thermal interactions. These findings should help to establish more precise clinical treatment plans for thermal therapies.

HEATING ANALYSIS OF SOFT TISSUE AT FINITE DEFORMATION DURING THERMAL ABLATION

Heating analysis of soft tissue at finite deformation is of importance during treatment planning process for thermal ablation. This paper presents a method by combining non-Fourier bio-heat transfer and nonlinear biomaterial model to simulate thermal and mechanical responses of soft tissues under combined thermal and mechanical loads. At the same time, anisotropic feature of biological tissue is also concerned by including the stored energy of collagen fiber bundles into the traditional neo-Hookean free energy function. Simulation results demonstrate that in spite of relatively small thermal-induced displacement, the resultant thermal strain and stress contribution is nonnegligible and can be used as theoretical basis to measure tissue pain during thermal ablation process.

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.

Non-Fourier based thermal-mechanical tissue damage prediction for thermal ablation

ABSTRACTPrediction of tissue damage under thermal loads plays important role for thermal ablation planning. A new methodology is presented in this paper by combing non-Fourier bio-heat transfer, constitutive elastic mechanics as well as non-rigid motion of dynamics to predict and analyze thermal distribution, thermal-induced mechanical deformation and thermal-mechanical damage of soft tissues under thermal loads. Simulations and comparison analysis demonstrate that the proposed methodology based on the non-Fourier bio-heat transfer can account for the thermal-induced mechanical behaviors of soft tissues and predict tissue thermal damage more accurately than classical Fourier bio-heat transfer based model.

Heat treatment modelling using strongly continuous semigroups

In this paper, mathematical simulation of bioheat transfer phenomenon within the living tissue is studied using the thermal wave model. Three different sources that have therapeutic applications in laser surgery, cornea laser heating and cancer hyperthermia are used. Spatial and transient heating source, on the skin surface and inside biological body, are considered by using step heating, sinusoidal and constant heating. Mathematical simulations describe a non-Fourier process. Exact solution for the corresponding non-Fourier bioheat transfer model that has time lag in its heat flux is proposed using strongly continuous semigroup theory in conjunction with variational methods. The abstract differential equation, infinitesimal generator and corresponding strongly continuous semigroup are proposed. It is proved that related semigroup is a contraction semigroup and is exponentially stable. Mathematical simulations are done for skin burning and thermal therapy in 10 different models and the related solutions are depicted. Unlike numerical solutions, which suffer from uncertain physical results, proposed analytical solutions do not have unwanted numerical oscillations.

Numerical Study on non-Fourier Bio heat Transfer During Thermal Ablation

In this paper, a mathematical model describing the process of non-Fourier heat transfer in biological tissues for generalized coordinate system during thermal ablation by electromagnetic radiation with Gaussian external heat source is studied. The boundary value problem governing this process has been solved using hybrid numerical method taking Legendre wavelet basis function. The result obtained from finite element Legendre wavelet Galerkin method (FELWGM) is compared with exact analytical solution and shows a good agreement. The effect of variability of time, generalized coordinate system, location of tumor or cancer, relaxation time and external heat source coefficient on temperature distribution is discussed in detail. It has been observed that treatment of tumor or cancerous cell is independent of the generalized coordinate system at the thermal ablation position.

Analytical study on the transient heating of a two-dimensional skin tissue using parabolic and hyperbolic bioheat transfer equations

In this study, exact analytical analysis of two-dimensional Fourier and non-Fourier bioheat transfer equations in skin tissue exposed to an instantaneous heating condition is presented. The effects of blood perfusion and metabolic heat generation on the tissue thermal behavior are considered. Corresponding analytical approach is developed through Laplace transform (LT) technique in conjunction with the separation of variables method and inversion theorem. The dual-phase-lag (DPL), thermal wave (TW) and Pennes models of bioheat transfer equation are studied by utilizing a generalized model. The reliability of the presented results has been evidenced through enforcing appropriate circumstances on the DPL model and comparing the outcome results with that of predicted by the Pennes and TW models. It is proved that the DPL bioheat transfer equation with the effects of blood perfusion and metabolic heat generation can be reduced to the Pennes bioheat transfer equation when τq=τT. The effects of local non-equilibrium on the tissue thermal behavior are examined and discussed by comparing the gradient precedence (GP) and flux precedence (FP) heat flow regimes of the DPL model. The first- and second-degree burn times of a 2D skin tissue for different bioheat models are introduced and compared with the 1D case.

Analysis for high-order effects in thermal lagging to thermal responses in biological tissue

Abstract For effective medical treatment processes, to understand the evolution of thermal responses in the biological tissues is important. Therefore, a non-Fourier bio-heat transfer equation, which considers the second-order effects in thermal lagging, was developed and used to investigate the variation of temperature in a laser-irradiated biological tissue. And then, thermal damage in the tissue was assessed with the rate process equation. Because the non-Fourier bio-heat transfer equation involves the mixed-derivative terms and the high-order derivatives of temperature with respect to time, there are mathematical difficulties for solving the present problem. A hybrid numerical scheme based on Laplace transform was employed. For an advance understand, the results with the non-linear effects of phase lag times are compared with the results with the linear effects. The non-linear effects of phase lag times would enhance the behavior of non-Fourier heat transfer. The diffusion-like feature of heat transfer would assist heat energy expanding and reduce thermal damage.

Analysis of Thermal Damage in a Laser-Irradiated Based on the Non-Fourier Model

A modified non-Fourier equation of bio-heat transfer based on the second-order Taylor expansion of dual-phase-lag model is proposed to estimating thermal damage in a laser-irradiated biological tissue. This non-Fourier bio-heat transfer equation involves the mixed-derivative terms and the high-order derivatives of temperature with respect to time. The thermal damage in the tissue is assessed with the rate process equation. The effects of blood perfusion and metabolic heat generation on thermal response and thermal damage are explored. There are mathematical difficulties in dealing with such a problem. A hybrid numerical scheme is extended to solve the present problem. The deviations of the results from the bio-heat transfer equation based on the linear form of the dual-phase-lag model are presented and discussed.

Non-Fourier bio heat transfer modelling of thermal damage during retinal laser irradiation

Study of non-Fourier energy transport is important in various biomedical applications, in the understanding of heat transfer in biological tissues. One such biomedical application that involves non-Fourier heat conduction is laser irradiation of the retina, where a high heat flux is typically applied for a short period of time. In the present study, retinal laser irradiation is analysed using the dual phase lag model for heat conduction. A simplified one-dimensional model of the human eye with seven layers is used in the simulation. The laser heating is modelled as a volumetric heat source and the respective magnitudes are calculated based on the absorptivities of the various layers. Apart from the temperature distribution, the damage distribution is also computed using the Arrhenius damage integral approach. The effect of the two phase lags τq and τT on the temperature and damage distributions are analysed. Small τT/τq ratios (⩽0.01) indicate a thermal wave like behaviour, which diminishes as the ratio increases (i.e. for τT/τq⩾0.1). The results are also compared with corresponding results from the Fourier model. Choroidal blood perfusion rate is found to have no significant effect on the unsteady temperature and damage distributions.