Applied and Computational Mathematics Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899-8910, United States; Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, United States.
Applied and Computational Mathematics Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899-8910, United States; Department of Biology, University of Saskatchewan, Saskatoon, Saskatchewan, S7N 5E2, Canada.
Math Biosci. 2019 Oct;316:108240. doi: 10.1016/j.mbs.2019.108240. Epub 2019 Aug 27.
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.
模拟细胞对侵入冰的反应为冷冻保存协议的制定提供了四十年的指导。大量文献记录表明，了解细胞状态作为介质和冷却速率的函数有助于制定明智的冷冻保存协议，并解释损伤机制。然而，以前的研究忽视了溶质与侵入冰前沿之间的相互作用及其对细胞状态的影响。为了解决这个问题，我们在这里研究了一个球形对称的生物细胞模型的冷冻生物学相关环境，该模型与侵入的固-液界面通过三元流体混合物隔开。细胞和液体区域分别含有细胞膜不可渗透的胞内和胞外盐、一种在冷冻保存协议中常用的细胞膜可渗透的溶质（通常称为冷冻保护剂）以及作为细胞膜可渗透溶剂的水。随着冷却和凝固的进行，细胞外的化学环境发生变化，并导致跨细胞膜的质量传输。因此，凝固前沿和细胞膜都是自由边界，它们的动力学通过固、液和细胞区域的传输过程耦合在一起。我们描述了一种基于域变换和线方法的数值方法来解决这个耦合的自由边界问题。我们还研究了不同的外部边界冷却方案如何影响细胞内部的热和化学状态。最后，我们观察到，以前未被考虑的在前进的固-液界面处的部分溶质排斥作用增加了细胞外液体中的 CPA 和盐浓度，这是界面速度和分离系数的函数，这表明以前冷冻保存过程中细胞状态的模型预测是不准确的。
