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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.

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 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.

This study deals with the computational simulation and inverse analysis of the cooling treatment of the hypoxic-ischemic encephalopathy in neonates. A reduced-order model is implemented for real-time monitoring of the internal body temperatures. The purpose of this study is to sequentially estimate the transient temperatures of the brain and other body regions with reduced uncertainties.

Pennes’ model was applied in each body element, and Fiala’s blood pool concept was used for the solution of the forward bioheat transfer problem. A state estimation problem was solved with the Sampling Importance Resampling (SIR) algorithm of the particle filter method.

The particle filter method was stable and accurate for the estimation of the internal body temperatures, even in situations involving large modeling and measurement uncertainties.

The proposed reduced-order model was verified with the results of a high-fidelity model available in the literature. Validation of the proposed model and of the solution of the state estimation problem shall be pursued in the future.

The solution of the state estimation problem with the reduced-order model presented in this paper has great potential to perform as an observer of the brain temperature of neonates, for the analysis and control of the systemic cooling treatment of neonatal hypoxic-ischemic encephalopathy.

The main treatment for hypoxic-ischemic encephalopathy in neonates is the cooling of affected regions. Accurate and fast models might allow the development of individualized protocols, as well as control strategies for the cooling treatment.

This paper presents the application of the SIR algorithm for the solution of a state problem during the systemic cooling of a neonate for the treatment of the hypoxic-ischemic encephalopathy.

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.

The system of equations for fractional thermo-viscoelasticity is used to investigate two-dimensional bioheat transfer and heat-induced mechanical response in human skin tissue with rheological properties.

Laplace and Fourier’s transformations are used. The resulting formulation is applied to human skin tissue subjected to regional hyperthermia therapy for cancer treatment. The inversion process for Fourier and Laplace transforms is carried out using a numerical method based on Fourier series expansions.

Comparisons are made with the results anticipated through the coupled and generalized theories. The influences of volume materials properties and fractional order parameters for all the regarded fields are examined. The results indicate that volume relaxation parameters, as well as fractional order parameters, play a major role in all considered distributions.

Bio-thermo-mechanics includes bioheat transfer, biomechanics, burn injury and physiology. In clinical applications, knowledge of bio-thermo-mechanics in living tissues is very important. One can infer from the numerical results that, with a finite distance, the thermo-mechanical waves spread to skin tissue, removing the unrealistic predictions of the Pennes’ model.

In the present paper, the new concept of “memory dependent derivative” in the Pennes’ bioheat transfer and heat-induced mechanical response in human living tissue with variable thermal conductivity and rheological properties of the volume is considered.

A problem of cancerous layered with arbitrary thickness is considered and solved analytically by Kirchhoff and Laplace transformation. The analytical expressions for temperature, displacement and stress are obtained in the Laplace transform domain. The inversion technique for Laplace transforms is carried out using a numerical technique based on Fourier series expansions.

Comparisons are made with the results anticipated through the coupled and generalized theories. The influence of variable thermal, volume materials properties and time-delay parameters for all the regarded fields for different forms of kernel functions is examined.

The results indicate that the thermal conductivity and volume relaxation parameters and MDD parameter play a major role in all considered distributions. This dissertation is an attempt to provide a theoretical thermo-viscoelastic structure to help researchers understand the complex thermo-mechanical processes present in thermal therapies.

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 a numerical analysis of the laser surgery of port wine stain (PWS) with cryogen spray cooling to compare the treatment effect between pulse dye laser and Nd:YAG laser, explain the incomplete clear of the lesion and optimize the laser parameter.

The complex structure of skin and PWS is simplified to a multi-layer skin model that consists of top epidermal layer and underneath dermis layer embedded with discrete blood vessels. The cooling effect of cryogen spray before laser firing is quantified by a general correlation obtained recently from the experimental data. The light distribution is modeled by the Monte Carlo method. The heat transfer in skin tissue is calculated by Pennes bioheat transfer model. The thermal damage of blood vessel is quantified by the Arrhenius damage integral.

For the vessel size studied (10-120 µm), pulse duration is recommended shorter than 6 ms. Large and deeply buried vessels, which may survive from 595 nm laser irradiation, can be coagulated by 1,064 nm laser due to its deep light penetration depth in skin. Furthermore, a desired uniform heating within the large vessel lumen can be achieved by 1,064 nm laser whereas 595 nm laser produce non-uniform heating.

The possible reason for the poor responding and incomplete clearance lesions is clarified. Laser wavelength and pulse duration are suggested to improve the clinical results.

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.