Permeation of dimethyl sulfoxide into articular cartilage at subzero temperatures

Determining Thermophysical Parameters of Cryopreserved Articular Cartilage Using Evolutionary Algorithms and Experimental Data

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.

Crystallisation Degree Analysis during Cryopreservation of Biological Tissue Applying Interval Arithmetic

This paper presents the numerical modelling of heat transfer and changes proceeding in the homogeneous sample, caused by the crystallisation phenomenon during cryopreservation by vitrification. Heat transfer was simulated in a microfluidic system in which the working fluid flowed in micro-channels. The analysed process included single-phase flow during warming, and two-phase flow during cooling. In the model under consideration, interval parameters were assumed. The base of the mathematical model is given by the Fourier equation, with a heat source including the degree of ice crystallisation. The formulated problem has been solved using the interval version of the finite difference method, with the rules of the directed interval arithmetic. The fourth order Runge-Kutta algorithm has been applied to determine the degree of crystallisation. In the final part of this paper, examples of numerical computations are presented.

Numerical Study of Heat and Mass Transfer during Cryopreservation Process with Application of Directed Interval Arithmetic

In the present paper, numerical modelling of heat and mass transfer proceeding in a two-dimensional axially symmetrical articular cartilage sample subjected to a cryopreservation process is presented. In the model under consideration, interval parameters were assumed. The heat transfer process is described using the Fourier interval equation, while the cryoprotectant transport (DMSO) across the cell membrane is analyzed using a two-parameter model taking into account the simulation of the water volume in the chondrocytes and the change in DMSO concentration over time. The liquidus tracking (LT) protocol introduced by Pegg et al. was used to model the cryopreservation process. This procedure divides the heating and cooling phases into eight and seven steps, respectively, allowing precise regulation of temperature and cryoprotectant (CPA) concentration of bathing solutions. This protocol protects chondrocytes from ice crystal, osmotic stress, and electrolyte damage. The obtained interval concentrations of cryoprotectant in chondrocytes were compared with previous simulations obtained using the deterministic model and they are mostly in agreement with the simulation data.

Numerical Modeling of Heat and Mass Transfer during Cryopreservation Using Interval Analysis

In the paper, the numerical analysis of heat and mass transfer proceeding in an axially symmetrical articular cartilage sample subjected to the cryopreservation process is presented. In particular, a two-dimensional (axially symmetrical) model with imprecisely defined parameters is considered. The base of the heat transfer model is given by the interval Fourier equation and supplemented by initial boundary conditions. The phenomenon of cryoprotectant transport (Me2SO) through the extracellular matrix is described by the interval mass transfer equation. The liquidus-tracking (LT) method is used to control the temperature, which avoids the formation of ice regardless of the cooling and warming rates. In the LT process, the temperature decreases/increases gradually during addition/removal of the cryoprotectant, and the articular cartilage remains on or above the liquidus line so that no ice forms, independent of the cooling/warming rate. The discussed problem is solved using the interval finite difference method with the rules of directed interval arithmetic. Examples of numerical computations are presented in the final part of the paper. The obtained results of the numerical simulation are compared with the experimental results, realized for deterministically defined parameters.

A model for predicting the permeation of dimethyl sulfoxide into articular cartilage, and its application to the liquidus-tracking method

Long-term storage of articular cartilage (AC) has excited great interest due to the practical surgical significance of this tissue. The liquidus-tracking (LT) method developed by Pegg et al. (2006) [29] for vitreous preservation of AC achieved reasonable survival of post-warming chondrocytes in situ, but the design of the entire procedure was more dependent on trial and error. Mathematical modeling would help to better understand the LT process, and thereby make possible improvements to attain higher cell survival. Mass transfer plays a dominant role in the LT process. In the present study, a diffusion model based on the free-volume theory and the Flory-Huggins thermodynamics theory was developed to predict the permeation of dimethyl sulfoxide (Me2SO) into AC. A comparison between the predicted mean concentration of Me2SO in the AC disc and the experimental data over wide temperature and concentration ranges [-30 to 37 °C, 10 to 64.5% (w/w)] shows that the developed model can accurately describe the permeation of Me2SO into AC [coefficient of determination (R(2)): 0.951-1.000, mean relative error (MRE): 0.8-12.8%]. With this model, the spatial and temporal distribution of Me2SO in the AC disc during a loading/unloading process can be obtained. Application of the model to Pegg et al.'s LT procedure revealed that the liquidus line is virtually not followed for the center part of the AC disc. The presently developed model will be a useful tool in the analysis and design of the LT method for vitreous preservation of AC.