A study on thermal damage during hyperthermia treatment based on DPL model for multilayer tissues using finite element Legendre wavelet Galerkin approach

Hyperthermia is a process that uses heat from the spatial heat source to kill cancerous cells without damaging the surrounding healthy tissues. Efficacy of hyperthermia technique is related to achieve temperature at the infected cells during the treatment process. A mathematical model on heat transfer in multilayer tissues in finite domain is proposed to predict the control temperature profile at hyperthermia position. The treatment technique uses dual-phase-lag model of heat transfer in multilayer tissues with modified Gaussian distribution heat source subjected to the most generalized boundary condition and interface at the adjacent layers. The complete dual-phase-lag model of bioheat transfer is solved using finite element Legendre wavelet Galerkin approach. The present solution has been verified with exact solution in a specific case and provides a good accuracy. The effect of the variability of different parameters such as lagging times, external heat source, metabolic heat source and the most generalized boundary condition on temperature profile in multilayer tissues is analyzed and also discussed the effective approach of hyperthermia treatment. Furthermore, we studied the modified thermal damage model with regeneration of healthy tissues as well. For viewpoint of thermal damage, the least thermal damage has been observed in boundary condition of second kind. The article concludes with a discussion of better opportunities for future clinical application of hyperthermia treatment.     

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.     

A review on numerical modeling for magnetic nanoparticle hyperthermia: Progress and challenges

Recent progress in nanotechnology has advanced the development of magnetic nanoparticle (MNP) hyperthermia as a potential therapeutic platform for treating diseases. Due to the challenges in reliably predicting the spatiotemporal distribution of temperature in the living tissue during the therapy of MNP hyperthermia, critical for ensuring the safety as well as efficacy of the therapy, the development of effective and reliable numerical models is warranted. This article provides a comprehensive review on the various mathematical methods for determining specific loss power (SLP), a parameter used to quantify the heat generation capability of MNPs, as well as bio-heat models for predicting heat transfer phenomena and temperature distribution in living tissue upon the application of MNP hyperthermia. This article also discusses potential applications of the bio-heat models of MNP hyperthermia for therapeutic purposes, particularly for cancer treatment, along with their limitations that could be overcome.     

Triangular node for Transmission-Line Modeling (TLM) applied to bio-heat transfer

Transmission-Line Modeling (TLM) is a numerical method used to solve complex and time-domain bio-heat transfer problems. In TLM, rectangles are used to discretize two-dimensional problems. The drawback in using rectangular shapes is that instead of refining only the domain of interest, a large additional domain will also be refined in the x and y axes, which results in increased computational time and memory space. In this paper, we developed a triangular node for TLM applied to bio-heat transfer that does not have the drawback associated with the rectangular nodes. The model includes heat source, blood perfusion (advection), boundary conditions and initial conditions. The boundary conditions could be adiabatic, temperature, heat flux, or convection. A matrix equation for TLM, which simplifies the solution of time-domain problems or solves steady-state problems, was also developed. The predicted results were compared against results obtained from the solution of a simplified two-dimensional problem, and they agreed within 1% for a mesh length of triangular faces of 59 µm±9 µm (mean±standard deviation) and a time step of 1 ms.     

On the solution of the non-linear bio-heat equation

With reference to microwave localized hyperthermia, a non-linear model of the thermal behavior of living tissues, where local thermoregulating convective and conducting effects due to blood flow are taken into account, has been assumed. The non-linear operator equation for the space and time temperature distribution, which describes local energy balance (bio-heat equation), has been linearized and solved by using a variant of the Newton iterative method. Numerical calculations for plane stratified structures simulating living bodies, irradiated by plane electromagnetic waves, have been carried out.     

Magnetic fluid hyperthermia simulations in evaluation of SAR calculation methods

PurposeThe purpose of this study is to employ magnetic fluid hyperthermia simulations in the precise computation of Specific Absorption Rate functions -SAR(T)-, and in the evaluation of the predictive capacity of different SAR calculation methods.MethodsMagnetic fluid hyperthermia experiments were carried out using magnetite-based nanofluids. The respective SAR values were estimated through four different calculation methods including the initial slope method, the Box-Lucas method, the corrected slope method and the incremental analysis method (INCAM). A novel numerical model combining the heat transfer equations and the Navier-Stokes equations was developed to reproduce the experimental heating process. To address variations in heating efficiency with temperature, the expression of the power dissipation as a Gaussian function of temperature was introduced and the Levenberg-Marquardt optimization algorithm was employed to compute the function parameters and determine the function’s effective branch within each measurement’s temperature range. The power dissipation function was then reduced to the respective SAR function.ResultsThe INCAM exhibited the lowest relative errors ranging between 0.62 and 15.03% with respect to the simulations. SAR(T) functions exhibited significant variations, up to 45%, within the MFH-relevant temperature range.ConclusionsThe examined calculation methods are not suitable to accurately quantify the heating efficiency of a magnetic fluid. Numerical models can be exploited to effectively compute SAR(T) and contribute to the development of robust hyperthermia treatment planning applications.     

Variational finite element approach to study heat transfer in the biological tissues of premature infants

The body temperature of newborn preterm infants depends on the heat transfer between the infant and the external environment. Factors that influence the heat exchange include the temperature and humidity of the air and the temperature of surfaces in contact with and around the infant. Neonatal thermoregulation has a different pattern as they have an immature thermoregulatory system. For this purpose, mathematical models can provide detailed insights for the heat transfer processes and its applications for clinical purposes. A new multi-compartment mathematical model of the neonatal thermoregulatory system is presented. The formulation of the model is based on the Pennes’ bio-heat equation with suitable boundary and initial conditions. The variational finite element method has been employed to determine heat transfer and exchange in the biological tissues of premature infants. The results obtained in this paper have shown that premature infants are unable to maintain a constant core temperature and resemble the empirically obtained results, proving the validity and feasibility of our model.AMS (2010): Subject classification92BXX, 92CXX, 92C35, 92C50, 46N60.     

A direct substitution, equation error technique for solving the thermographic tomography problem

A new technique for solving the combined state and parameter estimation problem in thermographic tomography is presented. The technique involves the direct substitution of known skin temperatures into the finite difference form of the bio-heat transfer equation as formulated for solving an initial value problem with a convection boundary condition at the skin surface. These equations are then used to solve the inverse bio-heat transfer problem for the unknown subcutaneous tissue temperatures and physiological parameters. For a small number of nodal points, closed form algebraic solutions are obtained. For larger sets of equations, a hybrid technique is used in which the problem is initially posed as an unconstrained optimization problem in which the model equation error is minimized using the conjugate gradient descent technique to get close to a solution. Then a generalized Newton-Raphson technique was used to solve the equations. A numerical simulation of a one-dimensional problem is investigated both with and without noise superimposed on the input (transient) skin temperature data. The results show that the technique gives very accurate results if the skin temperature data contains little noise. It is also shown that if the physical properties of the tissue and the metabolism are known, that a given set of proper transient skin temperature inputs yields a unique solution for the unknown internal temperatures and blood perfusion rates. However, the similar problem with known blood perfusion rates and unknown metabolisms does not yield a unique solution for the internal temperatures and metabolisms.     

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.     

Uncertainty analysis for temperature prediction of biological bodies subject to randomly spatial heating

An analytical solution to the Pennes bioheat transfer equation in three-dimensional geometry with practical hyperthermia boundary conditions and random heating was obtained in this paper. Uncertainties for the predicted temperatures of tissues due to approximate parameters were studied based on analyzing one-dimensional heat transfer in the biological bodies subject to a spatially decay heating. Contributions from each of the thermal parameters such as heat conductivity, blood perfusion rate, and metabolic rate of the tissues, the scattering coefficient and the surface power flux of the heating apparatus were compared and the uncertainty limit for temperature distribution in this case was estimated. The results are useful in a variety of clinical hyperthermia and biological thermal parameter measurement.     

Temperature and burn injury prediction of human skin exposed to microwaves: a model analysis

A one-dimensional multi-layer model is presented to characterize skin temperature rises and burn processes resulting from skin exposure to microwaves. Temperature variations and damage function analyses in the skin tissue exposed to microwaves were predicted depending on blood perfusion rate, thermal conductivity, power density, and exposure time. Thermal wave model was applied and the bio-heat transfer equation was solved using the finite difference time domain method. The thermal wave model of bio-heat transfer predicts a lower temperature rise than a model that uses Pennes’ equation. When approaching steady state, the solutions overlaps with that obtained using the Pennes’ equation. The results obtained may help to analyze the consequences of short-time high-power MW exposures in biological tissues.     

A revised approach for an exact analytical solution for thermal response in biological tissues significant in therapeutic treatments

The genesis of the present research paper is to develop a revised exact analytical solution of thermal profile of 1-D Pennes’ bioheat equation (PBHE) for living tissues influenced in thermal therapeutic treatments. In order to illustrate the temperature distribution in living tissue both Fourier and non-Fourier model of 1-D PBHE has been solved by ‘Separation of variables’ technique. Till date most of the research works have been carried out with the constant initial steady temperature of tissue which is not at all relevant for the biological body due to its nonhomogeneous living cells. There should be a temperature variation in the body before the therapeutic treatment. Therefore, a coupled heat transfer in skin surface before therapeutic heating must be taken account for establishment of exact temperature propagation. This approach has not yet been considered in any research work. In this work, an initial condition for solving governing differential equation of heat conduction in biological tissues has been represented as a function of spatial coordinate. In a few research work, initial temperature distribution with PBHE has been coupled in such a way that it eliminates metabolic heat generation. The study has been devoted to establish the comparison of thermal profile between present approach and published theoretical approach for particular initial and boundary conditions inflicted in this investigation. It has been studied that maximum temperature difference of existing approach for Fourier temperature distribution is 19.6% while in case of non-Fourier, it is 52.8%. We have validated our present analysis with experimental results and it has been observed that the temperature response based on the spatial dependent variable initial condition matches more accurately than other approaches.     

Analytical prediction of sub–surface thermal history in translucent tissue phantoms during plasmonic photo–thermotherapy (PPTT)

Knowledge of thermal history and/or distribution in biological tissues during laser based hyperthermia is essential to achieve necrosis of tumour/carcinoma cells. A semi–analytical model to predict sub–surface thermal distribution in translucent, soft, tissue mimics has been proposed. The model can accurately predict the spatio–temporal temperature variations along depth and the anomalous thermal behaviour in such media, viz. occurrence of sub-surface temperature peaks. Based on optical and thermal properties, the augmented temperature and shift of the peak positions in case of gold nanostructure mediated tissue phantom hyperthermia can be predicted. Employing inverse approach, the absorption coefficient of nano-graphene infused tissue mimics is determined from the peak temperature and found to provide appreciably accurate predictions along depth. Furthermore, a simplistic, dimensionally consistent correlation to theoretically determine the position of the peak in such media is proposed and found to be consistent with experiments and computations. The model shows promise in predicting thermal distribution induced by lasers in tissues and deduction of therapeutic hyperthermia parameters, thereby assisting clinical procedures by providing a priori estimates.     

Simulation and investigation of quantum dot effects as internal heat-generator source in breast tumor site

The Pennes bio-heat transfer equation, which introduces the exchange magnitude of heat transfer between tissue and blood, is often used to solve the temperature distribution for thermal imaging and sensing. Near-infrared light has the ability to be used as a non-invasive means of diagnostic imaging within the woman's breast. Due to the diffusive nature of light in different tissue, computational model-based methods are required for functional imaging within the breast. In this article, the time-dependent bio-heat transfer is solved by a numerical method. In our model, the heat generation source (intrinsic and extrinsic) involves laser, metabolism, and quantum dot that the metabolism and heat generated by QDs are considered as intrinsic. We supposed the injected quantum dots would target the tumor site by a passive targeting process and then by interaction of laser radiation and quantum dot, the photoluminescence of quantum dot is converted to heat in the tumor site. The extra generated heat can impact on the extracted heat profile. One of the important applications of this research has led to a sensitivity improvement of the imaging system, which is potentially useful in the diagnosis and detection of breast cancer.     

An Efficient Wavelet-Based Approximation Method to Gene Propagation Model Arising in Population Biology

In this paper, we have applied an efficient wavelet-based approximation method for solving the Fisher’s type and the fractional Fisher’s type equations arising in biological sciences. To the best of our knowledge, until now there is no rigorous wavelet solution has been addressed for the Fisher’s and fractional Fisher’s equations. The highest derivative in the differential equation is expanded into Legendre series; this approximation is integrated while the boundary conditions are applied using integration constants. With the help of Legendre wavelets operational matrices, the Fisher’s equation and the fractional Fisher’s equation are converted into a system of algebraic equations. Block-pulse functions are used to investigate the Legendre wavelets coefficient vectors of nonlinear terms. The convergence of the proposed methods is proved. Finally, we have given some numerical examples to demonstrate the validity and applicability of the method.     

3D in silico study of magnetic fluid hyperthermia of breast tumor using Fe3O4 magnetic nanoparticles

Modeling and simulation of the temperature distribution, the mass concentration, and the heat transfer in the breast tissue are hot issues in magnetic fluid hyperthermia treatment of cancer. The breast tissue can be visualized as a porous matrix with saturated blood. In this paper, 3D in silico study of breast cancer hyperthermia using magnetic nanoparticles (MNPs) is conducted. The 3D FEM models are incorporated to investigate the infusion and backflow of nanofluid in the breast tumor, the diffusion of nanofluid, temperature distribution during the treatment, and prediction of the fraction of tumor necrosis while dealing with the thermal therapy. All the hyperthermia procedures are simulated and analyzed on COMSOL Multiphysics. The sensitivity of frequency and amplitude of the applied magnetic field (AMF) is investigated on the heating effect of the tumor. The mesh dependent solution of Penne's bioheat model is also analyzed. The simulated results demonstrate successful breast cancer treatment using MNPs with minimum side effects. Validation of current simulations results with experimental studies existing in literature advocates the success of our therapy. The increase in the amplitude and frequency of the AMF increases of the temperature in the tumor. The variation of mesh from coarser to finer increased the temperature through small fractions. We have also simulated the magnetic induction problem where the magnetic field is generated by current-carrying coil conductors induce heat in nearby breast tumors due to excitation of MNPs by magnetic flux. This research will aid treatment protocols and real-time clinical breast cancer treatments.     

Numerical investigation of thermal response of laser-irradiated biological tissue phantoms embedded with gold nanoshells

The work presented in this paper focuses on numerically investigating the thermal response of gold nanoshells-embedded biological tissue phantoms with potential applications into photo-thermal therapy wherein the interest is in destroying the cancerous cells with minimum damage to the surrounding healthy cells. The tissue phantom has been irradiated with a pico-second laser. Radiative transfer equation (RTE) has been employed to model the light-tissue interaction using discrete ordinate method (DOM). For determining the temperature distribution inside the tissue phantom, the RTE has been solved in combination with a generalized non-Fourier heat conduction model namely the dual phase lag bio-heat transfer model. The numerical code comprising the coupled RTE-bio-heat transfer equation, developed as a part of the current work, has been benchmarked against the experimental as well as the numerical results available in the literature. It has been demonstrated that the temperature of the optical inhomogeneity inside the biological tissue phantom embedded with gold nanoshells is relatively higher than that of the baseline case (no nanoshells) for the same laser power and operation time. The study clearly underlines the impact of nanoshell concentration and its size on the thermal response of the biological tissue sample. The comparative study concerned with the size and concentration of nanoshells showed that 60 nm nanoshells with concentration of 5×10¹⁵ mm⁻³ result into the temperature levels that are optimum for the irreversible destruction of cancer infected cells in the context of photo-thermal therapy. To the best of the knowledge of the authors, the present study is one of the first attempts to quantify the influence of gold nanoshells on the temperature distributions inside the biological tissue phantoms upon laser irradiation using the dual phase lag heat conduction model.     

Inverse investigation of non-Fourier heat conduction in tissue

This paper attempts to describe the heat conduction in tissue using the dual-phase-lag mode. Evaluating the thermo-physical parameters is one of the ways to certify the thermal behavior. As a result, the paper simultaneously and inversely estimates the values of τ_q, τ_T and α for bologna based on the dual-phase-lag mode with the measurement data in the literature. The inconsistence in theory discovered in the literatures is eliminated. The calculated results of temperature variation with the estimated values of τ_q, τ_T and α at the measurement location are very close to the experimental data and address the rationality of the present results.     

内热源影响下的均质组织表面温度分布研究

现有的医用红外热像技术几乎都采自体表,为了提高诊断准确性,必须进一步确定体表温度分布与体内热源间的关系。基于人体内部传热机制和人体与环境的传热机制,推导出稳态下均质（各向同性）组织的表面温度分布与内部球状热源之间的关系,通过模拟计算可验证结论的准确性。