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The main objective of this paper is to investigate the response of human skin to an intense temperature drop at the surface. In addition, this paper aims to evaluate the efficiency of finite difference and finite volume methods in solving the highly nonlinear form of Pennes’ bioheat equation.

One-dimensional linear and nonlinear forms of Pennes’ bioheat equation with uniform grids were used to study the behavior of human skin. The specific heat capacity, thermal conductivity and blood perfusion rate were assumed to be linear functions of temperature. The nonlinear form of the bioheat equation was solved using the Newton linearization method for the finite difference method and the Picard linearization method for the finite volume method. The algorithms were validated by comparing the results from both methods.

The study demonstrated the capacity of both finite difference and finite volume methods to solve the one-dimensional and highly nonlinear form of the bioheat equation. The investigation of human skin’s thermal behavior indicated that thermal conductivity and blood perfusion rate are the most effective properties in mitigating a surface temperature drop, while specific heat capacity has a lesser impact and can be considered constant.

This paper modeled the transient heat distribution within human skin in a one-dimensional manner, using temperate-dependent physical properties. The nonlinear equation was solved with two numerical methods to ensure the validity of the results, despite the complexity of the formulation. The findings of this study can help in understanding the behavior of human skin under extreme temperature conditions, which can be beneficial in various fields, including medical and engineering.

The purpose of this paper is to provide a method for determining the numerical solutions of the thermal damage of cylindrical living tissues using hyperbolic bioheat model. Due to the complex governing equation, the finite element approach has been adopted to solve these problems. To approve the accuracy of the numerical solution, the numerical outcomes obtained by the finite element approach are compared with the existing experimental study. In addition, the comparisons between the numerical outcomes and the existing experimental data displays that the present mathematical models are efficient tools to evaluate the bioheat transfer in the cylindrical living tissue. Numerical computations for temperatures and thermal damage are presented graphically.

In this section, the complex equation of bioheat transfer based upon one relaxation time in cylindrical living tissue is summarized by using the finite element method. This method has been used here to get the solution of equation (8) with initial conditions (9) and boundary conditions (10). The finite element technique is a strong method originally advanced for numerical solutions of complex problems in many fields, and it is the approach of choice for complex systems. Another advantage of this method is that it makes it possible to visualize and quantify the physical effects independently of the experimental limits. Abbas and his colleagues [26-34] have solved several problems under generalized thermoelastic theories.

In this study, the different values of blood perfusion and thermal relaxation time of the dermal part of cylindrical living tissue are used. To verify the accuracy of the numerical solutions, the numerical outcomes obtained by the finite element procedure and the existing experimental study have been compared. This comparison displays that the present mathematical model is an effective tool to evaluate the bioheat transfer in the living tissue.

The validation of the obtained results by using experimental data the numerical solution of hyperbolic bioheat equation is presented. Due to the nonlinearity of the basic equation, the finite element approach is adopted. The effects of thermal relaxation times on the thermal damage and temperature are studied.

The purpose of this paper is to present the computational model of the neonate’s brain cooling process. The main aim of the analysis is to tune the developed computational model, make it convergent and representing the hypothermia therapy reasonably. To find the appropriate model parameters the trial of an inverse analysis, based on the standard least-square method, is performed. Having partially validated model the number of numerical simulations are carried out to compare their results with measurements made during real therapy.

The geometrical model of the newborn’s body is built using MRI and CT scans utilizing Mimics software and the Design Modeler while Ansys Fluent with its User Defined Function capability was used to implement the whole model and to carry out simulations. To model the bioheat transfer the Pennes bioheat equation is applied. In the mathematical model blood perfusion rate, metabolic heat generation rate as well as the arterial blood temperature are dependent on the tissue temperature. In order to determine the proper values of model parameters of bioheat transport in neonate’s body the attempt to inverse analysis is also performed.

The performed inverse analysis resulted in the values of model parameters (metabolic heat sources, blood perfusions etc.). Tuned model was then applied to simulate brain cooling process with reasonable accuracy. Obtained model parameters were also compared to the data obtained from neonatologists.

The presented numerical model still requires tests and simulations. The results from the inverse analysis based on the real measurements can be very valuable.

The determination of the proper parameters of the bioheat transfer in the neonatal body can finally be used to control the numerical simulations of the brain cooling process. The simulation of the re-warming process after hypothermic therapy can be improved considerably.

The performance of the numerical simulations of the brain cooling process in the proper way can finally helps protect newborns’ health and life.

In the paper 3-D real geometrical model of the newborn’s body includes head, torso and limbs and different types of tissues are distinguished in the model. The considered bioheat transfer problem is also fully 3-D. This model is then utilised together with inverse analysis in order to determine the model parameters for the newborn’s body.

Second-generation biofuel is regarded as a sustainable alternatives to fossil energy in transportation where electricity is not feasible. The main purpose of this study is to analyze how large-scale second-generation biofuel based on wood may affect the competitiveness of more mature bioenergy technologies such as bioheat through competition in the biomass market. The impacts on forest industries are also included.

An economic model for the energy and forest sectors based on partial equilibrium modeling is used to quantify the impacts of four different locations of biofuel production in Norway.

The results show that there are regional variations in biomass price effects depending on local raw material availability and costs of transport and import. Technologies allowing for a larger variety of wood biomass qualities will face lower biomass prices than technologies using only one species as raw material, causing less reduction in the production of bioheat and forest industrial products. For Norway specifically, the paper concludes that even if there is a potential for both increased bioheat generation and large-scale biofuel production, the production of second-generation biofuels based on domestic wood resources will cause a 5-20 percent reduction in bioheat generation depending on the scale of biofuel production.

This study demonstrates how impacts on biomass markets from establishment of biofuel production vary quite substantially with location, production level and choice of feedstock. One main finding is the quite large biomass cost impact that is seen in the model runs when introducing large-scale biofuel production. Increased biomass costs reduce the profitability and this must be taken into account when establishing a biofuel installation.

The originality of the paper is the analyses of biofuel impacts with a detailed model for biomass supply as the bioenergy and forest sectors.

This paper aims to test the inverse analysis, based on the standard least-square method, which will finally lead to find the appropriate parameters of modelling of the brain cooling process.

To test the presented in this paper method of inverse analysis the numerical simulations of the bioheat transfer process in the neonatal body were performed. To model the bioheat transfer the Pennes bioheat equation and the modified Fiala model were applied.

The performed tests of the inverse analysis proved that it is possible to estimate the proper parameters of the process using this tool, but always with the small mistake.

The presented method still requires a lot of tests. The test with the data from real measurements can be very valuable.

The determination of the proper parameters of the bioheat transfer in the neonatal body can finally be used to perform the numerical simulations of the brain cooling process.

The performance of the numerical simulations of the brain cooling process in the proper way can finally helps protect newborns’ health and life.

In the paper the attempt of the inverse analysis in order to determine the parameters of bioheat transfer in the newborn's body is made.

The purpose of this paper is to focus on applications related to the hyperthermia treatment of cancer, with heating imposed either by a laser in the near-infrared range or by radiofrequency waves. The particle filter algorithms are compared in terms of computational time and solution accuracy.

The authors extend the analyses performed in their previous works to compare three different algorithms of the particle filter, as applied to the hyperthermia treatment of cancer. The particle filters examined here are the sampling importance resampling (SIR) algorithm, the auxiliary sampling importance resampling (ASIR) algorithm and Liu & West’s algorithm.

Liu & West’s algorithm resulted in the largest computational times. On the other hand, this filter was shown to be capable of dealing with very large uncertainties. In fact, besides the uncertainties in the model parameters, Gaussian noises, similar to those used for the SIR and ASIR filters, were added to the evolution models for the application of Liu & West’s filter. For the three filters, the estimated temperatures were in excellent agreement with the exact ones.

This work may help medical doctors in the future to prescribe treatment protocols and also opens the possibility of devising control strategies for the hyperthermia treatment of cancer.

The natural solution to couple the uncertain results from numerical simulations with the measurements that contain uncertainties, aiming at the better prediction of the temperature field of the tissues inside the body, is to formulate the problem in terms of state estimation, as performed in this work.

The purpose of this paper is to present an inverse analysis procedure based on a coupled numerical formulation through which the coefficients describing non‐linear thermal properties of blood perfusion may be identified.

The coupled numerical technique involves a combination of the dual reciprocity boundary element method (DRBEM) and a genetic algorithm (GA) for the solution of the Pennes bioheat equation. Both linear and quadratic temperature‐dependent variations are considered for the blood perfusion.

The proposed DRBEM formulation requires no internal discretisation and, in this case, no internal nodes either, apart from those defining the interface tissue/tumour. It is seen that the skin temperature variation changes as the blood perfusion increases, and in certain cases flat or nearly flat curves are produced. The proposed algorithm has difficulty to identify the perfusion parameters in these cases, although a more advanced genetic algorithm may provide improved results.

The coupled technique allows accurate inverse solutions of the Pennes bioheat equation for quantitative diagnostics on the physiological conditions of biological bodies and for optimisation of hyperthermia for cancer therapy.

The proposed technique can be used to guide hyperthermia cancer treatment, which normally involves heating tissue to 42‐43°C. When heated up to this range of temperatures, the blood flow in normal tissues, e.g. skin and muscle, increases significantly, while blood flow in the tumour zone decreases. Therefore, the consideration of temperature‐dependent blood perfusion in this case is not only essential for the correct modelling of the problem, but also should provide larger skin temperature variations, making the identification problem easier.

The purpose of this paper is to apply the Steady State Kalman Filter for temperature measurements of tissues via magnetic resonance thermometry. Instead of using classical direct inversion, a methodology is proposed that couples the magnetic resonance thermometry with the bioheat transfer problem and the local temperatures can be identified through the solution of a state estimation problem.

Heat transfer in the tissues is given by Pennes’ bioheat transfer model, while the Proton Resonance Frequency (PRF)-Shift technique is used for the magnetic resonance thermometry. The problem of measuring the transient temperature field of tissues is recast as a state estimation problem and is solved through the Steady-State Kalman filter. Noisy synthetic measurements are used for testing the proposed methodology.

The proposed approach is more accurate for recovering the local transient temperatures from the noisy PRF-Shift measurements than the direct data inversion. The methodology used here can be applied in real time due to the reduced computational cost. Idealized test cases are examined that include the actual geometry of a forearm.

The solution of the state estimation problem recovers the temperature variations in the region more accurately than the direct inversion. Besides that, the estimation of the temperature field in the region was possible with the solution of the state estimation problem via the Steady-State Kalman filter, but not with the direct inversion.

The recursive equations of the Steady-State Kalman filter can be calculated in computational times smaller than the supposed physical times, thus demonstrating that the present approach can be used for real-time applications, such as in control of the heating source in the hyperthermia treatment of cancer.

The original and novel contributions of the manuscript include: formulation of the PRF-Shift thermometry as a state estimation problem, which results in reduced uncertainties of the temperature variation as compared to the classical direct inversion; estimation of the actual temperature in the region with the solution of the state estimation problem, which is not possible with the direct inversion that is limited to the identification of the temperature variation; solution of the state estimation problem with the Steady-State Kalman filter, which allows for fast computations and real-time calculations.

This paper aims to develop and describe an improved process for determining the rate of heat generation in living tissue.

Previous work by the authors on solving the bioheat equation has been updated to include a new localized meshless method which will create a more robust and computationally efficient technique. Inclusion of this technique will allow for the solution of more complex and realistic geometries, which are typical of living tissue. Additionally, the unknown heat generation rates are found through genetic algorithm optimization.

The localized technique showed superior accuracy and significant savings in memory and processor time. The computational efficiency of the newly proposed meshless solver allows the optimization process to be carried to a higher level, leading to more accurate solutions for the inverse technique. Several example cases are presented to demonstrate these conclusions.

This work includes only 2D development of the approach, while any realistic modeling for patient‐specific cases would be inherently 3D. The extension to 3D, as well as studies to improve the technique by decreasing the sensitivity to measurement noise and to incorporate non‐invasive measurement positioning, are under way.

As medical imaging continuously improves, such techniques may prove useful in patient diagonosis, as heat generation can be correlated to the presence of tumors, infections, or other conditions.

This paper describes a new application of meshless methods. Such methods are becoming attractive due to their decreased pre‐processing requirements, especially for problems involving complex geometries (such as patient specific tissues), as well as optimization problems, where geometries may be constantly changing.

The purpose of this paper is to derive the Trefftz functions (T-functions) for the Pennes’ equation and for the single-phase-lag (SPL) model (hyperbolic equation) with perfusion and then comparing field of temperature in a flat slab made of skin in the case when perfusion is taken into account, with the situation when a Fourier model is considered. When considering the process of heat conduction in the skin, one needs to take into account the average values of its thermal properties. When in biological bodies relaxation time is of the order of 20 s, the thermal wave propagation appears. The initial-boundary problems for Pennes’ model and SPL with perfusion model are considered to investigate the effect of the finite velocity of heat in the skin, perfusion and thickness of the slab on the rate of the thermal wave attenuation. As a reference model, the solution of the classic Fourier heat transfer equation for the considered problems is calculated. A heat flux has direction perpendicular to the surface of skin, considered as a flat slab. Therefore, the equations depend only on time and one spatial variable.

First of all the T-functions for the Pennes’ equation and for the SPL model with perfusion are derived. Then, an approximate solutions of the problems are expressed in the form of a linear combination of the T-functions. The T-functions satisfy the equation modeling the problem under consideration. Therefore, approximating a solution of a problem with a linear combination of n T-functions one obtains a function that satisfies the equation. The unknown coefficients of the linear combination are obtained as a result of minimization of the functional that describes an inaccuracy of satisfying the initial and boundary conditions in a mean-square sense.

The sets of T-functions for the Pennes’ equation and for the SPL model with perfusion are derived. An infinite set of these functions is a complete set of functions and stands for a base functions layout for the space of solutions for the equation used to generate them. Then, an approximate solutions of the initial-boundary problem have been found and compared to find out the effect of finite velocity of heat in the skin, perfusion and thickness of the slab on the rate of the thermal wave attenuation.

The methods used in the literature to find an approximate solution of any bioheat transfer problems are more complicated than the one used in the presented paper. However, it should be pointed out that there is some limitation concerning the T-function method, namely, the greater number of T-function is used, the greater condition number becomes. This limitation usually can be overcome using symbolic calculations or conducting calculations with a large number of significant digits.

The T-functions for the Pennes’ equation and for the SPL equation with perfusion have been reported in this paper for the first time. In the literature, the T-functions are known for other linear partial differential equations (e.g. harmonic functions for Laplace equation), but for the first time they have been derived for the two aforementioned equations. The results are discussed with respect to practical applications.