# Numerical Modeling of Heat and Mass Transfer during Cryopreservation Using Interval Analysis
## 1 Introduction
The models present a multi-physical field's weak coupling problem, including the modelling of processes such as heat transfer, crystallization, moving-boundary problem, mass transport at the macroscale, and mass transfer across the cell membrane (with cell dehydration) [2]. Numerical modeling is an opportunity for the further development of cryopreservation. Models representing cryopreservation by slow freezing [23] and vitrification [12; 24; 25] have been created. Different biological tissues and cells such as stem cells [23; 26; 27] and articular cartilage [28; 29; 30; 31] have been analyzed. During the cryopreservation for heat transport modelling, the Fourier equation is used [12; 23; 24; 28; 29]. The heat transfer in cryopreservation organisms or during a cryosurgery can be also described by the following models: the Pennes parabolic equation (1948) [32; 33; 34; 35], the Catatneo-Vernotte hyperbolic model (1958) [36; 37; 38; 39], and the dual-phase lag model [40; 41; 42]. The heat proceeding during cryopreservation should be supplemented by parameters related to the phase-change phenomena [33; 34; 35; 38; 39; 41; 42; 43]. The mathematical model of the cryopreservation process should also consider the phenomenon of crystallization and the moving-boundary problem. Crystallization is associated with the nucleation of ice crystals and their growth in intracellular and extracellular areas. This happens when the liquid's temperature approaches the crystallization temperature [2; 23; 24; 12]. The problem of the moving boundary during the cryopreservation process is related to the selection of a numerical method for calculating the moving boundary between solid and liquid phases [2; 43; 44]. The mass transport modeling during cryopreservation is divided into: (1) macroscale flow and (2) cell-level transfer. To model the mass transfer phenomenon on a macroscale, [PERSON]'s second law is used, which determines the local concentration over time caused by diffusion [12; 28; 29; 30; 31; 45]. Sometimes, [PERSON]'s second law is complemented by a velocity vector, which is defined by, e.g., the Navier-Stokes equation [11; 17; 34; 46]. All cells have membranes that separate the extracellular environment from the intracellular solution. Hence, in the cryopreservation process model, the mass transfer across the cell membrane, which contains the simultaneous transport of water and permeable solutes, should also be included. It can be described by the following models: the [PERSON] and Steward model (1932) [47; 48], two-parameter (2-P) formalism [11; 13; 14; 29; 45; 47; 48; 49; 50],the [PERSON] and Katchalsky model (1958) [13; 27; 48; 50; 51; 52], and the [PERSON] model (1963) [53] or its later-modified model by [PERSON] et al. (1973) [54; 55; 10; 56]. Heat and mass transfer problems are usually solved by assuming that the equations appearing in the mathematical model and all parameters in those equations are deterministic. Such assumption does not produce an exact image of the thermal and mass processes met in the cryopreservation process. It seems more reasonable to consider uncertainties related, for example, to the material parameters. Such analysis is particularly important in biomechanics because thermophysical parameters can change in a wide range (thermal conductivity, volumetric specific heat). This stems from the parameters depending on numerous individual traits such as age, sex, occupation, etc. [57; 58]. In the present paper, a numerical analysis of the cryopreservation process, including heat and mass transfer proceeding, is presented. In the model, the LT method protocol for a cryopreserved articular cartilage sample was simulated. Furthermore, some thermophysical parameters cannot be determined deterministically; therefore, the values were approximated experimentally. This often causes the parameters to be inaccurate. Because of that, the interval arithmetic is used to properly interpret uncertain thermophysical parameters [57; 58].