Optimization of Organ Freezing Protocols With Specified Allowable Thermal Stress Levels (original) (raw)
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Approximate analytical solution for one-dimensional tissue freezing around cylindrical cryoprobes
International Journal of Thermal Sciences, 2009
An approximate analytical solution for the temperature distribution and interface motion is determined for the freezing of blood-perfused tissue around a cylindrical cryoprobe. The solution is based on an improved quasi-steady model in which assumed temperature profiles in the frozen and unfrozen tissue are used to determine the interface motion. The approximate solution satisfies all temperature boundary conditions as well as the transient heat equations at the interface. Due to blood perfusion in the unfrozen tissue, a steady state is reached where the interface becomes stationary. The solution converges to the exact steady state interface location. Improvement over the quasi-steady solution and the accuracy of the present theory are verified by comparison with numerical solutions for the limiting case of zero blood perfusion and metabolic heat production. Results show that a typical quasi-steady error of 73% is reduced to 8% using the present theory. Parametric charts are presented to evaluate the effect of the governing parameters on interface location. (L.M. Jiji), ganatos@ccny.cuny.edu (P. Ganatos).
On One Mathematical Model of Cooling Living Biological Tissue
Mathematics and Statistics, 2021
When cooling living biological tissue (active, non-inert medium), cryomedicine uses cryo-instruments with various forms of cooling surface. Cryoinstruments are located on the surface of biological tissue or completely penetrate into it. With a decrease in the temperature of the cooling surface, an unsteady temperature field appears in the tissue, which in the general case depends on three spatial coordinates and time. To date, there are a large number of scientific publications that consider mathematical models of cryodestruction of biological tissue. However, in the overwhelming majority of them, the Pennes equation (or some of its modifications) is taken as the basis of the mathematical model, from which the linear nature of the dependence of heat sources of biological tissue on the desired temperature field is visible. This character of the dependence does not allow one to describe the actually observed spatial localization of heat. In addition, Pennes' model does not take into account the fact that the freezing of the intercellular fluid occurs much earlier than the freezing of the intracellular fluid and the heat corresponding to these two processes is released at different times. In the proposed work, a new mathematical model of cooling and freezing of living biological tissue are built with a flat rectangular applicator located on its surface. The model takes into account the above features and is a three-dimensional boundary-value problem of the Stefan type with nonlinear heat sources of a special type and has applications in cryosurgery. A method is proposed for the numerical study of the problem posed, based on the use of locally one-dimensional difference schemes without explicitly separating the boundary of the influence of cold and the boundaries of the phase transition. The method was previously successfully tested by the author in solving other two-dimensional problems arising in cryomedicine.
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.
International Journal of Heat and Mass Transfer, 2017
In the paper, the problem of biological tissue freezing is discussed. In contrast to the previously presented models based on the Pennes equation, the thermal interactions between the cryoprobe tip and soft tissue are described using the dual-phase lag model (DPLM). This model contains two delay times (the relaxation and thermalization times) and in this way, the finite velocity of the thermal wave is considered. The model of the freezing process is based on the introduction of a parameter called 'substitute thermal capacity' to the dual-phase lag equation. At the stage of numerical computations, the explicit scheme of the finite difference method is used. In the final part of the paper the examples of the computation are shown. A comparison with the solution resulting from the adoption of the Pennes equation and also the model verification with the experimental data are presented in the final part of the paper.
Computer Methods and Programs in Biomedicine, 2007
As a part of ongoing efforts to develop computerized planning tools for cryosurgery, the current study focuses on developing an efficient numerical technique for bioheat transfer simulations. Our long-term goal is to develop a planning tool for cryosurgery that takes a 3D reconstruction of a target region, and suggests the best cryoprobe layout. Toward that goal, a planning algorithm, termed "force-field analogy," has been recently presented, based on a sequence of bioheat transfer simulations, which are by far the most computationally expensive part of the planning method. The objective in the current study is to develop a finite difference numerical scheme for bioheat transfer simulations, which reduces the overall run time of computerized planning, thereby making it clinically relevant. While the general concept of variable grid size and time intervals is not new, its application to the phase change problem of cryosurgery is the unique contribution of the current study.
The Energy Equation for Freezing of Biological Tissue
Journal of Heat Transfer, 1989
In the past, the process of freezing in biological tissue was modeled using the regular energy equation for a homogeneous compound. New experimental evidence shows that in tissue the water freezes separately in the vascular system and in the cells. The freezing process is affected by the water transport between the cells and the blood vessels. A new equation was developed to model the experimental results. In this work the general equation for freezing of biological tissue will be presented together with a linearized version of the new equation. A perturbation solution is obtained for the linearized equation to illustrate the effect of the water transport on the freezing process in tissue.
Heat Transfer In Biological Tissue Subjected ToThe Action Of Cylindrical Cryoprobe
WIT transactions on modelling and simulation, 1970
The application of the boundary element method for numerical solution of the non-steady state problem in domain oriented in cylindrical coordinate system is presented. In particular, the thermal processes proceeding in biological tissue subjected to the action of a disk-shaped cryoprobe are analyzed. The problem is treated as the axially symmetrical one. The non-steady temperature field in domain considered is determined by nonlinear bio-heat transfer equation. The boundary condition on the contact surface between cryoprobe and tissue results from the temporary surface temperature (e.g. the constant cooling rate can be assumed). On the remaining parts of the boundary, the adiabatic condition can be accepted. The initial condition results from the natural tissue temperature. In order to use the boundary element method, the additional numerical algorithm called the artificial source method is applied. In the final part of the paper the results of numerical computations are presented.
A Study on the Effect of Metabolic Heat Generation on Biological Tissue Freezing
The effect of metabolic heat generation on the freezing of biological tissue has been studied. Quasi-steady approximation is used to solve the Pennes bioheat equation in tissues. Temperature profile and motion of freezing interfaces are obtained for different values of metabolic heat generation. It is observed that metabolism has a significant effect on freezing of biological tissues during cryosurgery.