Cryopreservation is the process of freezing and storing biological cells and tissues with the purpose of preserving their essential physiological properties after re-warming. The process is applied primarily in medicine in the cryopreservation of cells and tissues, for example stem cells, or articular cartilage. The cryopreservation of articular cartilage has a crucial clinical application because that tissue can be used for reconstruction and repair of damaged joints. This article concerns the identification of the thermophysical parameters of cryopreserved articular cartilage. Initially, the direct problem was formulated in which heat and mass transfer were analyzed by applying the finite difference method. After that, at the stage of inverse problem investigations, an evolutionary algorithm coupled with the finite difference method was used. The identification of the thermophysical parameters was carried out on the basis of experimental data on the concentration of the cryoprotectant. In the last part, this article presents the results of numerical analysis for both the direct and inverse problems. Comparing the results for the direct problem, in which the thermophysical parameters are taken from the literature, with the experimental data, we obtained a relative error between 0.06% and 15.83%. After solving the inverse problem, modified values for the thermophysical parameters were proposed.

The article presents the one-dimensional numerical modelling of heat transfer in thin metal films irradiated by ultra-short laser pulses. In the mathematical description the relaxation times and the boundary conditions for phonons and electrons are given as interval numbers. The direct problem has been solved by means of the in-terval lattice Boltzmann method using the rules of directed interval arithmetic, while at the stage of the inverse problem solution, the evolutionary algorithm is applied. On the stage of inverse problems investigations, the evolutionary algorithm coupled with the interval lattice Boltzmann method is used.

In oncology, hyperthermia is understood as a planned, controlled technique of heating cancerous changes in order to destroy their cells or stop their growth. In clinical practice, hyperthermia is used in combination with radiotherapy, chemotherapy, or immunological therapy. During the hyperthermia, the tissue is typically exposed to a temperature in the range of 40–45 °C, the exception is thermoablation, during which the temperatures reach much higher values. Thermoablation is characterized by the use of high temperatures up to 90 °C. The electrode using the radiofrequency is inserted into the central area of the tumor. Interstitial thermoablation is used to treat, among others, breast and brain cancer. The therapy consists of inducing coagulation necrosis in an area that is heated to very high temperatures. Mathematical modeling is based on the use of a coupled thermo-electric model, in which the electric field is described by means of the Laplace equation, while the temperature field is based on the Pennes equation. Coupling occurs at the level of the additional source function in the Pennes equation. The temperature field obtained in this way makes it possible to calculate the Arrhenius integral as a determinant of the destruction of biological tissue. As a result of numerical calculations regarding the temperature field and the Arrhenius integral, it can be concluded that, with the help of numerical tools and mathematical modeling, one can simulate the process of destroying cancerous tissue.

The paper presents the mathematical modeling of the interstitial hyperthermia process caused by the action of internal applicator using electric field. As one knows, the increase of the tissue temperature causes the destruction of the cancer tissue. To estimate the degree of damage of the tumor tissue the Arrhenius scheme is used. From the point of view of the possibility of supporting thermotherapeutic techniques, it is important to examine the impact of individual parameters occurring in the mathematical model on the degree of tissue destruction. Therefore, it is necessary to estimate, based on the sensitivity analysis methods, which parameter has the greatest effect on the computations results. In the paper also the results of the inverse problem consisting in the simultaneous identification of the selected parameters controlling the process of tissue damage are also shown. The inverse problem is solved using the evolutionary algorithm. As a parameter of the fitness function for the evolutionary algorithm, the Arrhenius integral, which determines the degree of tissue destruction, is applied. The finite element method is used to solve the coupled problem related to the heating of biological tissues.

A cylindrical skin tissue domain subjected to an external heat flux is considered. Thermal processes in the domain considered are described by the Cattaneo-Vernotte equation supplemented by the appropriate boundary and initial conditions. The aim of considerations is the identification of external heat flux and relaxation time on the basis of ‘measured’ heating/cooling curves at the set of selected points located on the surface of the skin. The direct problem is solved using the implicit scheme of the Finite Difference Method (FDM), while at the stage of the inverse problem solution, the evolutionary algorithm is applied. In the final part of the paper the examples of computations are presented.

In the paper the soft tissue freezing process is considered. The tissue sub-domain is subjected to the action of cylindrical cryoprobe. Thermal processes proceeding in the domain considered are described using the dual-phase lag equation (DPLE) supplemented by the appropriate boundary and initial conditions. DPLE results from the generalization of the Fourier law in which two lag times are introduced (relaxation and thermalization times). The aim of research is the identification of these parameters on the basis of measured cooling curves at the set of points selected from the tissue domain. To solve the problem the evolutionary algorithms are used. The paper contains the mathematical model of the tissue freezing process, the very short information concerning the numerical solution of the basic problem, the description of the inverse problem solution and the results of computations.

The article presents the mathematical modeling of the artificial hyperthermia process caused by the action of electric field. The increase in the tissue temperature causes, as one knows, the destruction of the cancer tissue. To estimate the degree of damage of the tumor tissue the Arrhenius scheme has been used. In the paper the inverse problem consisting in the simultaneous identification of the exposure time of the electric field and the parameters controlling the process of tissue damage is solved using the evolutionary algorithm. As a parameter of the fitness function for the evolutionary algorithm, the Arrhenius integral which determines the degree of tissue destruction is applied. The electric field is induced by an internal applicator placed inside the biological tissue attacked by the cancer. The mathematical description is based on the Pennes equations (for the temperature field) and the Laplace equations (for the electric field). Electro-thermal coupling is taken into account by means of an additional internal heat source occurring in the bioheat transfer equation. On the stage of numerical computations, the evolutionary algorithm coupled with the finite element method is used.

The article presents the mathematical modeling of the phenomenon of artificial hyperthermia which is caused by the interaction of an electric field. The electric field is induced by the applicator positioned within the biological tissue with cancer. In addition, in order to estimate the degree of tumor destruction under the influence of high temperature an Arrhenius integral has been used. The distribution of electric potential in the domain considered is described by the Laplace system of equations, while the temperature field is described by the Pennes system of equations. These problems are coupled by source function being the additional component in the Pennes equation and resulting from the electric field action. The boundary element method is applied to solve the coupled problem connected with the heating of biological tissues.

Purpose - The purpose of this study is to show that the methods of the numerical simulation can be a very effective tool for a proper choice of control parameters of artificial hyperthermia. An electromagnetic field induced by two external electrodes and a temperature field resulting from electrodes action in a 3D domain of biological tissue is considered. An important problem is the appropriate directing of heat in the region of tumor, so as to avoid damaging healthy cells surrounding the tumor. Recently, to concentrate the heat on the tumor, magnetic nanoparticles, which are introduced into the tumor, were used. The nanoparticles should be made of material that ensures appropriate magnetic properties and has a high biocompatibility with the biological tissue. External electric field causes the heat generation in the tissue domain.Design/methodology/approach - The distribution of electric potential in the domain considered is described by the Laplace system of equations, while the temperature field is described by the Pennes' system of equations. These problems are coupled by source function being the additional component in the Pennes' equation and resulting from the electric field action. The boundary element method is applied to solve the coupled problem connected with the heating of biological tissues.Findings - The aim of investigations is to determine an electric potential of external electrodes and the number of nanoparticles introduced to a tumor region to obtain the artificial hyperthermia state. The tests performed showed that the proposed tool to solve the inverse problem provides correct results.Research limitations/implications - In the paper the steady state bioheat transfer problem is considered, so the thermal damage is a function of the temperature only. Therefore, the solution can be considered as the maximum ablation zone of cancer. Additionally, the choice of appropriate parameters will be affected on the position and shape of the tumor and the electrodes.Originality/value - In the paper the inverse problem has been solved using the evolutionary algorithm, gradient method and hybrid algorithm which is a combination of the two previous.

The paper presents numerical modeling of RF (Radiofrequency) hyperthermia caused by the introduction of internal electrode to the tumor. The main purpose of the publication is to analyze the relationship between electrode voltage and the duration of hyperthermia treatment necessary to achieve adequate temperature in the tumor region. Mathematical modeling based on the coupling of two problems: electrical-to generate additional heat and thermal-to estimate the change and rise of the temperature in the tumor. The distribution of electric potential in domain considered is described by the Laplace equation, while the temperature field is described by the Pennes equation. At the stage of numerical simulation the Boundary Element Method (BEM) is used. In the final part of the paper the results of numerical computations are shown.

In the chapter, problems connected with the numerical modeling of bioheat transfer processes are presented. In particular the non-homogeneous system of a burn wound and healthy tissue is considered. The heat exchange between sub-domains and environment is described by a system of partial differential equations (the Pennes equations) supplemented by adequate boundary conditions. The first goal of the research is the estimation of the changes of temperature fields due to perturbations in thermal parameters using the direct method of sensitivity analysis. Both the basic problem and additional ones concerning the sensitivity with respect to selected parameters are solved using the boundary element method. The second goal is the problem of burn wound shape identification. The additional information necessary to solve such a task results from the knowledge of temperature distribution on the external surface of skin tissue. At the stage of solving the inverse problem, a gradient method has been used. In the final part of the chapter the results of computations are shown.

Purpose: The paper presents numerical modeling of the artificial hyperthermia induced by the electric field in order to destroy the abnormal tissue. In particular, the possibility of process control in order to increase the temperature of only the tumor tissue was discussed. Due to the fact that the external electrodes which generate the additional heat, heat not only the area of the tumor, but also healthy tissue which surrounds the tumor, increasing the temperature inside the cancer is possible by introducing the paramagnetic nanoparticles into the interior. Additionally, the proper selection of voltage on the electrodes and the number of nanoparticles will enable optimal effect of hyperthermia treatment to be achieved.Methods: The multiple reciprocity BEM is applied to solve the coupled problem connected with the biological tissue heating. In order to determine the appropriate values of the parameters the inverse problem has been formulated, connected with simultaneous identification of the voltage of the electrodes and the number of nanoparticles, which is solved using the evolutionary algorithm.Results: The changes of the voltage of electrodes cause the changes of temperature in the entire domain considered, but the possibilities of temperature field control (e.g., concentration of maximum temperature at the central point of tumor) are rather unrealizable, because the maximum temperature we could observe in the neighbourhood of the electrodes.Conclusions: The idea consisting in the introduction of nanoparticles to the tumor region (for the concentrated energy deposition at the target tissue) is very effective. We obtain the maximum temperature exactly in the tumor domain.

The Cattaneo-Vernotte equation describing the heat conduction process in domain of solid body results from the generalization of the well - known Fourier law, in which the 'delay time' (relaxation time tau(q)) is introduced. The Cattaneo-Vernotte equation should be, among others, used in a case of microscale heat transfer analysis when the thermal processes are characterized by the extremely short duration (e.g. ultrafast laser pulse), the considerable temperature gradients and the very small dimensions (e.g. thin metal film). In the paper the problem of relaxation time identification is considered. In particular, the heat conduction process proceeding in domain of thin metal film subjected to a laser pulse is analyzed. The inverse problem solution is obtained using the evolutionary algorithms. The information concerning the time-dependent temperature distribution on the surface of metal film is assumed to be known. At the stage of numerical computations the finite difference method (FDM) is applied. In the final part of the paper the example of computations is shown.

The energy equation corresponding to the dual phase lag model (DPLM) results from the generalized form of the Fourier law, in which the two 'delay times' (relaxation and thermalization time) are introduced. The DPLM should be used in the case of microscale heat transfer analysis, in particular when thermal processes are characterized by extremely short duration (e.g. ultrafast laser pulse), considerable temperature gradients and very small dimensions (e.g. thin metal film). In this paper, the problem of relaxation and thermalization time identification is discussed, at the same time the heat transfer processes proceeding in the domain of a thin metal film subjected to a laser beam are analyzed. The solution presented bases on the application of evolutionary algorithms. The additional information concerning the transient temperature distribution on a metal film surface is assumed to be known. At the stage of numerical realization, the finite difference method (FDM) is used. In the final part of the paper, an example of computations is presented.

Electromagnetic field induced by two external electrodes and temperature field resulting from electrode action in the domain of biological tissue being a composition of healthy region and a tumour is considered. To warrant the optimum conditions of tumour destruction, the magnetic nanoparticles are embedded in this region. It is assumed that the temperature which assures an effect of destruction should be higher than 42°C. In this article, the problems relating to the electrodes’ electric potential identification and the simultaneous identification of potential and number of nanoparticles introduced to the tumour region are discussed. Additional information necessary to solve the task results from the postulated temperature inside the tumour region assuring its destruction. The problem has been solved using both the gradient method and evolutionary algorithm. The boundary element method is applied to solve the coupled problem connected with the heating of biological tissues.

Generalization of Fourier law, in particular the introduction of two ‘delay times’ (relaxation time τ q and thermalization time τ T ) leads to the new form of energy equation called the dual-phase-lag model (DPLM). This equation should be applied in a case of microscale heat transfer modeling. In particular, DPLM constitutes a good approximation of thermal processes which are characterized by extremely short duration (e.g. ultrafast laser pulse), extreme temperature gradients and geometrical features of domain considered (e.g. thin metal film). The aim of considerations presented in this paper is the identification of two above mentioned positive constants τ q , τ T . They correspond to the relaxation time, which is the mean time for electrons to change their energy states and the thermalization time, which is the mean time required for electrons and lattice to reach equilibrium. In this paper the DPLM equation is applied for analysis of thermal processes proceeding in a thin metal film subjected to a laser beam. At the stage of computations connected with the identification problem solution the evolutionary algorithms are used. To solve the problem the additional information concerning the transient temperature distribution on a metal film surface is assumed to be known.

The dual phase lag model (DPLM) based on the generalized form of Fourier law, in particular the introduction of two 'delay times' (relaxation time τq and thermalization time τT) leads to the considered form of energy equation. This equation should be applied in the case of microscale heat transfer modeling. In p articular, DPLM constitutes a good ap- proximation of thermal processes which are characte rized by extremely short duration (e.g. ultrafast laser pulse), extreme temperature gradien ts and geometrical features of the domain considered (e.g. thin metal film). In this paper, t he identification problem of two of the above mentioned positive constants τq, τT is discussed and the thermal processes proceeding in the domain of thin metal film subjected to a la ser beam are analyzed. At the stage of computations connected with the identification prob lem solution, evolutionary algorithms are used. To solve the problem, additional informat ion concerning the transient temperature distribution on a metal film surface is assumed to be known.

Electromagnetic field induced by two external electrodes and temperature field resulting from electrodes action in 3D domain of biological tissue is considered. External electric field causes the heat generation in tissue domain. The distribution of electric potential in domain considered is described by the Laplace equation, while the temperature field is described by the Pennes equation. These problems are coupled by source function being the additional component in Pennes equation and resulting from the electric field action. The boundary element method is applied to solve the coupled problem connected with the biological tissue heating. In the final part of the paper the examples of computations are shown. 1. Governing equations In Figure 1 a typical radio frequency (RF) hyperthermia system is shown [1]. The mathematical model of the process analyzed consists of two parts [1-3]. The electric part concerns the Laplace equation to obtain the electric field distribution. The thermal part is connected with the bioheat transfer equation to obtain the temperature distribution. In the bioheat transfer equation the additional source term associated with the heat generation caused by electric field distribution appears. The potential inside the tissue is described by the Laplace equation ( ) ( ) ( ) , , : e , , φ , , 0 x y z x y z x y z ∈Ω ∇  ∇  =   (1) where e(x, y, z) [C /(Nm )] is the dielectric permittivity of tissue. On the external surface of tissue being in a contact with the electrodes the following condition is given ( ) ( ) ( ) ( ) 1 2 , , : φ , , , , : φ , , x y z x y z U x y z x y z U ∈Γ = ∈Γ = − (2) where U [V] is the electric potential of the electrode relative to the ground. Please cite this article as: Ewa Majchrzak, Marek Paruch, Numerical modelling of tissue heating by means of the electromagnetic field, Scientific Research of the Institute of Mathematics and Computer Science, 2010, Volume 9, Issue 1, pages 89-97. The website: http://www.amcm.pcz.pl/