# Numerical Modeling of Heat and Mass Transfer during Cryopreservation Using Interval Analysis
## 1 Introduction  
Cryopreservation is a process which aims to preserve tissues or cells at low temperatures (between \(-80\) and \(-196\)\({}^{\circ}\)C). In this process, the biological activity of tissues is reduced or completely stopped and then the physiological temperature is restored again. Successful cryopreservation does not significantly impact the basic functions of cryopreserved biological tissues or cells, for example, mechanical properties [1, 2]. Cryopreservation of cells or tissues has many medical applications, such as cryobanking human cells or organ transplantation. It allows the transport of organs between different medical centers on time. The long-term storage of stem cells, for example, is the initial step toward tissue engineering, which will enable the treatment of incurable diseases and the regeneration of soft tissues [3]. Cryopreservation of articular cartilage can be helpful in its complicated treatment, considering the damage to the articular cartilage, which can lead to osteoarthritis [4, 5, 6, 7, 8, 9, 10]. Thermophysical phenomena occurring in cells or tissues during temperature decrease are regulated by chemical adjuvants, e.g., cryoprotective agents (CPA) or antifreeze proteins [1, 2]. Examples of CPA are glycerol, DMSO, trehalose, and propylene glycol [3]. In cryopreservation, the process of loading/unloading of cryoprotectants (CPA) is often time-consuming and can cause osmotic damage to cells or biological tissues [11]. This is why it is important to complete this step properly. A few methods of CPA transport have been developed, such as droplet-based methods [12] and channel-based methods [2]. Recently, membrane microfluids have been proposed, which have great potential to solveproblems with the cryoprotectant loading/unloading process [1; 13; 14; 15; 16; 17]. [PERSON] et al. [11] presented a new approach to achieve safe and efficient loading/unloading of cryoprotectants, namely microfiltration-based sequential perfusion (MSP). Generally, cryopreservation involves the following steps: (1) CPA loading, (2) freezing, (3) thawing, and (4) CPA unloading. However, there are two methods of cryopreservation: slow freezing and vitrification. The main differences between these two techniques are the cooling rate used and the CPA concentration. Slow-freezing protocols with a cooling rate of about 1 \({}^{\circ}\)C/min use a low concentration of CPA (approx. 1-2 M), so the tissues are not exposed to damage caused by the osmotic shock of cells and the chemical toxicity of cryoprotectants. This can be the reason for ice crystals forming, which causes dehydration and tissue destruction. In contrast, vitrification (cooling rate of about 100 \({}^{\circ}\)C/min) is the process during which the transition from the aqueous phase to a glass state occurs without the ice crystals' formation. In the case of vitrification, the problems are toxicity and osmotic shock associated with a high CPA concentration (approx. 4-8 M) [2; 3]. An alternative to the techniques is the liquidus-tracking (LT) method. It was invented by [PERSON] in 1962 [18] and developed by [PERSON] et al. in 1972 [19]. The basic idea of this technique is to control the temperature as well as the concentration of CPA in the cell/tissue so that the temperature/concentration condition of the sample is on the liquidus (melting point) line or above it. This approach prevents ice crystals formation and the occurrence of a high concentration of cryoprotectants [20]. Recently, the LT method has been applied to the cryopreservation of articular cartilage by [PERSON] et al. [21] and [PERSON] et al. [22]. The use of the standard methods of cryopreservation articular cartilage is problematic because of the ice crystallisation problem in chondrocytes [20]. Lately, mathematical models describing phenomena that occur during the cryopreservation process have been developed.