Measurement and simulation of water transport during freezing in mammalian liver tissue

Optimization of cryosurgical procedure.^ on deep tissues such as liver requires an increased understanding of the fundamental mechanisms of ice formation and water transport in tissues during freezing. In order to further investigate and quantify the amount of water transport that occurs during freezing in tissue, this study reports quantitative and dynamic experimental data and theoretical modeling of rat liver freezing under controlled conditions. The rat liver was frozen by one of four methods of cooling: Method 1 —ultrarapid “slam cooling” (≥ 1000°; C/min) for control samples; Method 2-equilibrium freezing achieved by equilibrating tissue at different subzero temperatures (-4, -6, -8, -10°C); Method 3-two-step freezing, which involves cooling at 5°C/min. to -4, -6, -8, -10 or - 20°C followed immediately by slam cooling; or Method 4-constant and controlled freezing at rates from 5- 400°C/min. on a directional cooling stage. After freezing, the tissue was freeze substituted, embedded in resin, sectioned, stained, and imaged under a light microscope fitted with a digitizing system. Image analysis techniques were then used to determine the relative cellular to extracellular volumes of the tissue. The osmotically inactive cell volume was determined to be 0.35 by constructing a Boyle van't Hoff plot using cellular volumes from Method 2. The dynamic volume of the rat liver cells during cooling was obtained using cellular volumes from Method 3 (two-step freezing at 5°C/min). A nonlinear regression fit of a Krogh cylinder model to the volumetric shrinkage data in Method 3 yielded the biophysical parameters of water transport in rat liver tissue of: L_pg = 3.1 X 10^-13 m³/Ns (1.86 μ/min-atm) and E_ip = 290 kJ/mole (69.3 kcal/mole), with chi-squared variance of 0.00124. These parameters were then incorporated into the Krogh cylinder model and used to simulate water transport in rat liver tissue during constant cooling at rates between 5-100°C/min. Reasonable agreement between these simulations and the constant cooling rate freezing experiments in Method 4 were obtained. The model predicts that the water transport ceases at a relatively high subzero temperature (- 10°C), such that the amount of intracellular ice forming in the tissue cells rises from almost none (=extensive dehydration and vascular expansion) at ≥5°C/min to over 88 percent of the original cellular water at ≥50°C/min. The theoretical simulations based on these experimental methods may be of use in visualizing and predicting freezing response, and thus can assist in the planning and implementing of cryosurgical protocols.