Estimation of the stable frozen zone volume and the extent of contrast for a therapeutic substance

BACKGROUND: In biomedical science and clinical practice, an estimation of the stable frozen zone volume and distribution of concentration fields of injected diagnostic and healing solutions in the tissues of living organisms is of great importance and does not currently have any mathematical solution aimed at its precise evaluation. OBJECTIVE: The aim of this research is the estimation of the stable frozen zone volume at ultra-low temperatures as well as the distribution of temperature areas and concentration fields of injected diagnostic and healing substances in vitro. The results can improve our understanding of the stable frozen zone volume and the extent of contrast for a therapeutic substance. MATERIALS AND METHODS: A cryogenic zone (ice ball) was generated at -180°C using liquid nitrogen without any difficulties in vitro. The effects of freeze-thaw processes using ultra-low temperature and the cryogenic response of a 1.5% gelatin solution in water (%g/v) kept at a constant temperature of 20°C and continuously stirred were mathematically analyzed. The stable frozen zone volume was illustrated in vitro and measured in terms of its length, depth and cryogenic margin using a standard medical ruler and Vernier caliper after a freezing period at -180°C, using liquid nitrogen to provide cooling and freezing of a small portion of this solution in the vessel at room temperature (20°C). Round-shaped cryoprobes with diameters of 15 mm and 50 mm were applied to create a frozen zone volume in vitro. A single cryoprobe was used per procedure. The sample exposure time was 3 min. After this time, the volume of the frozen region remains unchanged, which indicates that the equilibrium stationary state has been reached. The experimental design, cryogenic procedure and freeze-thaw processes of the hemisphere were described and illustrated in vitro item by item. The statistical analysis manifested significant differences that were found between the 50 mm and 15 mm cryoprobes with regards to the freezing diameter, depth, and cryogenic margin (P < 0.001). RESULTS: An illustrated analytical mathematical solution of equations determined the stable frozen zone volume and the radius of the sphere of the frozen medium in the equilibrium stationary state. The resulting assessment provided the basis for the creation of mini- and micro-cryoprobes as well as cryoneedles for local tissue freezing in living biological structures. A solution to the equations was obtained under the boundary conditions with a set stable temperature value on the boundary surface of the cryoprobe as well as at the surface well-away from it, where the temperature is equal to the stable temperature of the environment. For example, this solution gives that in the case of a hemispherical cryoprobe radius of 1 mm, the frozen zone volume was more than three orders of magnitude greater than the volume of the cryoprobe itself and was equal to approximately 4 cm(3). The determination of the fractal dimension can consider the individual characteristics of the spread of the contrast medium or therapeutic substance(s) in living tissue. Based on fractal theory, our innovative mathematical formulas allow for the assessment of the effective distribution of contrast medium in living biological structures, specifically for tissues assessed for diagnostic purposes, and they enable the selection of an optimal treatment strategy in medical practice. CONCLUSION: A simple mathematical approach to solving the problems of assessing the stable frozen zone volume and distribution of temperature areas and concentration fields of injected diagnostic and healing substances in living biological structures, particularly living tissue in vitro, is presented in this study. The expressed quantitative mathematical formulas determine the stable stationary frozen zone volume and provide the basis for the creation of mini- and micro-cryoprobes. The application of fractal theory is proposed for assessing the distribution efficiency of contrast medium and therapeutic substance(s) in living biological structures for diagnostic purposes and for selecting a compassionate treatment strategy in medical professional practice.