Modeling a cell's response to encroaching ice has informed the development of cryopreservation protocols for four decades. It has been well documented that knowledge of the cellular state as a function of media and cooling rate faciliate informed cryopreservation protocol design and explain mechanisms of damage. However, previous efforts have neglected the interaction between solutes and the encroaching ice front and their effects on the cell state. To address this, here we examine the cryobiologically relevant setting of a spherically-symmetric model of a biological cell separated by a ternary fluid mixture from an encroaching solid-liquid interface. The cell and liquid regions contain cell membrane impermeable intracellular and extracellular salts, respectively, a cell membrane permeable solute commonly used in cryopreservation protocols known as a cryoprotective agent (CPA), and water as a membrane permeable solvent. As cooling and solidification proceed the extracellular chemical environment evolves and leads to mass transport across the cell membrane. Consequently, both the solidification front and the cell membrane are free boundaries whose dynamics are coupled through transport processes in the solid, liquid and cell regions. We describe a numerical procedure to solve this coupled free-boundary problem based on a domain transformation and method of lines approach. We also investigate how the thermal and chemical states inside the cell are influenced by different cooling protocols at the external boundary. Finally, we observe that the previously unaccounted-for partial solute rejection at the advancing solid-liquid interface increases the CPA and salt concentrations in the extracellular liquid as a function of the interface speed and segregation coefficients, suggesting that previous model predictions of the cell state during cryopreservation were inaccurate.

中文翻译：

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稀三元溶液向半渗透性球状细胞向内凝固的数值解。

对细胞对冰侵袭的反应进行建模已经为低温保存协议的发展提供了四十年的经验。已有充分的文献记载，关于细胞状态与介质和冷却速率的函数关系的知识有助于进行低温保存规程的设计并解释损伤的机理。然而，先前的努力已经忽略了溶质与侵蚀冰面之间的相互作用以及它们对细胞状态的影响。为了解决这个问题，我们在这里研究了一种生物细胞的球形对称模型的冷冻生物学相关设置，该模型由三元流体混合物从侵入的固液界面中分离出来。细胞和液体区域分别包含细胞膜不可渗透的细胞内和细胞外盐，通常在冷冻保存规程中用作防冻剂​​（CPA）的细胞膜可渗透溶质，而水作为膜可渗透溶剂。随着冷却和固化的进行，细胞外化学环境不断演变，并导致整个细胞膜的质量迁移。因此，凝固前沿和细胞膜都是自由边界，其动力学通过固体，液体和细胞区域中的传输过程耦合。我们描述了一种基于域变换和线法的数值程序来解决这种耦合的自由边界问题。我们还研究了电池内部的热和化学状态如何受到外部边界处不同冷却协议的影响。最后，