Abstract
Cryobiology is a multiscale and interdisciplinary field. The scope and scale of interactions limit the gains that can be made by one theory or experiment alone. Because of this, modeling has played a critical role in both explaining cryobiological phenomena and predicting improved protocols. Modeling facilitates understanding of the biophysical and some of the biochemical mechanisms of damage during all phases of cryopreservation including CPA equilibration and cooling and warming. Moreover, as a tool for optimization of cryopreservation protocols, modeling has yielded many successes. Modern cryobiological modeling includes very detailed descriptions of the physical phenomena that occur during freezing, including ice growth kinetics and spatial gradients that define heat and mass transport models. Here we reduce the complexity and approach only a small but classic subset of these problems. Namely, here we describe the process of building and using a mathematical model of a cell in suspension where spatial homogeneity is assumed for all quantities. We define the models that describe the critical cell quantities used to describe optimal and suboptimal protocols and then give an overview of classical methods of how to determine optimal protocols using these models. We include practical considerations of modeling in cryobiology, including fitting transport models to cell volume data, performing optimization with cell volume constraints, and a look at expanding cost functions to cooling regimes.
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References
Mazur P (1963) Kinetics of water loss from cells at subzero temperatures and the likelihood of intracellular freezing. J Gen Physiol 47:347–369
Mazur P, Leibo S, Chu E (1972) A two-factor hypothesis of freezing injury. Evidence from Chinese hamster tissue-culture cells. Exp Cell Res 71:345–355
Woelders H, Chaveiro A (2004) Theoretical prediction of ‘optimal’ freezing programmes. Cryobiology 49:258–271
Liu J, Woods EJ, Agca Y, Critser ES, Critser JK (2000) Cryobiology of rat embryos II: a theoretical model for the development of interrupted slow freezing procedures. Biol Reprod 63:1303–1312
Agca Y, Gilmore J, Byers M, Woods EJ, Liu J, Critser JK (2002) Osmotic characteristics of mouse spermatozoa in the presence of extenders and sugars. Biol Reprod 67:1493–1501
Morris CE, Homann U (2001) Cell surface area regulation and membrane tension. J Membr Biol 179:79–102. https://doi.org/10.1007/s002320010040
Gilmore JA, Liu J, Gao DY, Critser JK (1997) Determination of optimal cryoprotectants and procedures for their addition and removal from human spermatozoa. Hum Reprod 12:112–118
Davidson AF, Benson JD, Higgins AZ (2014) Mathematically optimized cryoprotectant equilibration procedures for cryopreservation of human oocytes. Theor Biol Med Model 11:13
Benson JD, Kearsley AJ, Higgins AZ (2012) Mathematical optimization of procedures for cryoprotectant equilibration using a toxicity cost function. Cryobiology 64:144–151
Benson JD, Chicone CC, Critser JK (2012) Analytical optimal controls for the state constrained addition and removal of cryoprotective agents. Bull Math Biol 74:1516–1530
Levin RL (1982) A generalized method for the minimization of cellular osmotic stresses and strains during the introduction and removal of permeable cryoprotectants. J Biomech Eng 104:81–86
Songsasen N, Leibo SP (1997) Cryopreservation of mouse spermatozoa. II. Relationship between survival after cryopreservation and osmotic tolerance of spermatozoa from three strains of mice. Cryobiology 35:255–269
Mullen SF, Li M, Li Y, Chen ZJ, Critser JK (2008) Human oocyte vitrification: the permeability of metaphase II oocytes to water and ethylene glycol and the appliance toward vitrification. Fertil Steril 89:1812–1825
Karlsson JO, Szurek EA, Higgins AZ, Lee SR, Eroglu A (2014) Optimization of cryoprotectant loading into murine and human oocytes. Cryobiology 68:18–28. http://dx.doi.org/10.1016/j.cryobiol.2013.11.002
Seki S, Jin B, Mazur P (2014) Extreme rapid warming yields high functional survivals of vitrified 8-cell mouse embryos even when suspended in a half-strength vitrification solution and cooled at moderate rates to -196 oC. Cryobiology 68:71–78
Anderson DM, Benson JD, Kearsley AJ (2014) Foundations of modeling in cryobiology—I: concentration, Gibbs energy, and chemical potential relationships. Cryobiology 69:349–360
Anderson DM, Benson JD, Kearsley AJ (2019) Foundations of modeling in cryobiology—II: heat and mass transport in bulk and at cell membrane and ice-liquid interfaces. Cryobiology. 91:3–17 https://doi.org/10.1016/j.cryobiol.2019.09.014
Anderson DM, Benson JD, Kearsley AJ (2019) Foundations of modeling in cryobiology—III: heat and mass transport in a ternary system. Cryobiology 92:34–46
Anderson DM, Benson JD, Kearsley AJ (2019) Numerical solution of inward solidification of a dilute ternary solution towards a semi-permeable spherical cell. Math Biosci 316:108240
Karlsson JOM, Cravalho EG, Rinkes IHMB, Tompkins RG, Yarmush ML, Toner M (1993) Nucleation and growth of ice crystals inside cultured-hepatocytes during freezing in the presence of dimethyl-sulfoxide. Biophys J 65:2524–2536
Yang G, Zhang A, Xu LX, He X (2009) Modeling the cell-type dependence of diffusion-limited intracellular ice nucleation and growth during both vitrification and slow freezing. J Appl Phys 105:114701
Chang A, Dantzig JA, Darr BT, Hubel A (2007) Modeling the interaction of biological cells with a solidifying interface. J Comput Phys 226:1808–1829
Liu Z, Wan R, Muldrew K, Sawchuk S, Rewcastle J (2004) A level set variational formulation for coupled phase change/mass transfer problems: application to freezing of biological systems. Finite Elem Anal Des 40:1641–1663
Zeng C, He L, Peng W, Ding L, Tang K, Fang D, Zhang Y (2014) Selection of optimal reference genes for quantitative RT-PCR studies of boar spermatozoa cryopreservation. Cryobiology 68:113–121. http://dx.doi.org/10.1016/j.cryobiol.2014.01.004
Kashuba Benson CM, Benson JD, Critser JK (2008) An improved cryopreservation method for a mouse embryonic stem cell line. Cryobiology 56:120–130
Kashuba CM, Benson JD, Critser JK (2014) Rationally optimized cryopreservation of multiple mouse embryonic stem cell lines: II—Mathematical prediction and experimental validation of optimal cryopreservation protocols. Cryobiology 68:176–184. http://dx.doi.org/10.1016/j.cryobiol.2013.12.003
Kashuba CM, Benson JD, Critser JK (2014) Rationally optimized cryopreservation of multiple mouse embryonic stem cell lines: I—comparative fundamental cryobiology of multiple mouse embryonic stem cell lines and the implications for embryonic stem cell cryopreservation protocols. Cryobiology 68:166–175. http://dx.doi.org/10.1016/j.cryobiol.2013.12.007
Agca Y, Liu J, Critser E, Critser J (2000) Fundamental cryobiology of rat immature and mature oocytes: hydraulic conductivity in the presence of Me(2)SO, Me(2)SO permeability, and their activation energies. J Exp Zool 286:523–533
Ridgway D, Broderick G, Lopez-Campistrous A, Ru’aini M, Winter P, Hamilton M, Boulanger P, Kovalenko A, Ellison MJ (2008) Coarse-grained molecular simulation of diffusion and reaction kinetics in a crowded virtual cytoplasm. Biophys J 94:3748–3759
Lacelle PL, Rothstein A (1966) The passive permeability of the red blood cell to cations. J Gen Physiol 50:171–188
Agca Y, Liu J, Mullen S, Johnson-Ward J, Gould K, Chan A, Critser J (2005) Chimpanzee (Pan troglodytes) spermatozoa osmotic tolerance and cryoprotectant permeability characteristics. J Androl 26:470–477
Newton H, Pegg DE, Barrass R, Gosden RG (1999) Osmotically inactive volume, hydraulic conductivity, and permeability to dimethyl sulphoxide of human mature oocytes. J Reprod Fertil 117:27–33
Gao DY, Chang Q, Liu C, Farris K, Harvey K, McGann LE, English D, Jansen J, Critser JK (1998) Fundamental cryobiology of human hematopoietic progenitor cells I: osmotic characteristics and volume distribution. Cryobiology 36:40–48. https://doi.org/10.1006/cryo.1997.2060
Woods EJ, Zieger MA, Lakey JR, Liu J, Critser JK (1997) Osmotic characteristics of isolated human and canine pancreatic islets. Cryobiology 35:106–113. https://doi.org/10.1006/cryo.1997.2029
Willoughby CE, Mazur P, Peter AT, Critser JK (1996) Osmotic tolerance limits and properties of murine spermatozoa. Biol Reprod 55:715–727
Du J, Tao J, Kleinhans FW, Peter AT, Critser JK (1994) Determination of boar spermatozoa water volume and osmotic response. Theriogenology 42:1183–1191
Du J, Tao J, Kleinhans FW, Mazur P, Critser JK (1994) Water volume and osmotic behaviour of mouse spermatozoa determined by electron paramagnetic resonance. J Reprod Fertil 101:37–42
Benson C, Liu C, Gao D, Critser E, Critser J (1993) Determination of the osmotic characteristics of hamster pancreatic islets and isolated pancreatic islet cells. Cell Transplant 2:461–465
Mazur P, Schneider U (1986) Osmotic responses of preimplantation mouse and bovine embryos and their cryobiological implications. Cell Biophys 8:259–285
Shapiro H (1948) The change in osmotically inactive fraction produced by cell activation. J Gen Physiol 32:34–51
Prickett RC, Elliott JAW, Hakda S, McGann LE (2008) A non-ideal replacement for the Boyle van’t Hoff equation. Cryobiology 57:130–136
Ponder E (1940) The red cell as an osmometer. In: Cold Spring Harbor Symposia on Quantitative Biology, vol 8. Cold Spring Harbor Laboratory Press, Cold Spring Harbor, pp 133–143
Katkov II (2011) On proper linearization, construction and analysis of the Boyle–van’t Hoff plots and correct calculation of the osmotically inactive volume. Cryobiology 62:232–241
Katkov II (2008) Challenge from the simple: some caveats in linearization of the Boyle-van’t Hoff and Arrhenius plots. Cryobiology 57:142–149
Benson JD (2012) Some comments on recent discussion of the Boyle van’t Hoff relationship. Cryobiology 64:118–120
Casula E, Traversari G, Fadda S, Klymenko OV, Kontoravdi C, Cincotti A (2019) Modelling the osmotic behaviour of human mesenchymal stem cells. Biochem Eng J 151:107296
Sun M, Northup N, Marga F, Huber T, Byfield FJ, Levitan I, Forgacs G (2007) The effect of cellular cholesterol on membrane-cytoskeleton adhesion. J Cell Sci. https://doi.org/10.1242/jcs.001370
Benson JD, Chicone CC, Critser JK (2011) A general model for the dynamics of cell volume, global stability and optimal control. J Math Biol 63:339–359
Moore WJ (1972) Physical chemistry, 4th edn. Prentice-Hall, Englewood Cliffs
Prickett RC, Elliott JAW, McGann LE (2011) Application of the multisolute osmotic virial equation to solutions containing electrolytes. J Phys Chem B 115:14531–14543
Benson JD, Bagchi A, Han X, Critser JK, Woods EJ (2010) Melting point equations for the ternary system water/sodium chloride/ethylene glycol revisited. Cryobiology 61:352–356
Elliott JAW, Prickett RC, Elmoazzen HY, Porter KR, McGann LE (2007) A multisolute osmotic virial equation for solutions of interest in biology. J Phys Chem B 111:1775–1785
Landau LD, Lifshitz EM (1980) Statistical physics, vol 5. Course of theoretical physics, 3rd edn. Pergamon Press, Oxford
Kleinhans FW, Mazur P (2007) Comparison of actual vs. synthesized ternary phase diagrams for solutes of cryobiological interest. Cryobiology 54:212–222
Benson JD (2011) Stability analysis of several non-dilute multiple solute transport equations. J Math Chem 49:859–869
Bird RB, Stewart WE, Lightfoot EN (2002) Transport phenomena, 2nd edn. Wiley, New York
Elliott JAW, Elmoazzen HY, McGann LE (2000) A method whereby Onsager coefficients may be evaluated. J Chem Phys 113:6573–6578
Kedem O, Katchalsky A (1958) Thermodynamic analysis of the permeability of biological membranes to non-electrolytes. Biochim Biophys Acta 27:229–246
Kleinhans FW (1998) Membrane permeability modeling: Kedem-Katchalsky vs. a two-parameter formalism. Cryobiology 37:271–289
Finkelstein A (1987) Water movement through lipid bilayers, pores, and plasma membranes: theory and reality. Wiley, New York
Ebertz S, McGann L (2002) Osmotic parameters of cells from a bioengineered human corneal equivalent and consequences for cryopreservation. Cryobiology 45:109–117
Mazur P, Koshimoto C (2002) Is intracellular ice formation the cause of death of mouse sperm frozen at high cooling rates? Biol Reprod 66:1485–1490
Fedorow C, McGann L, Korbutt G, Rayat G, Rajotte R, Lakey J (2001) Osmotic and cryoprotectant permeation characteristics of islet cells isolated from the newborn pig pancreas. Cell Transplant 10:651–659
Benson CT, Liu C, Gao DY, Critser ES, Benson J, Critser J (1998) Hydraulic conductivity (Lp) and its activation energy (Ea), cryoprotectant agent permeability (Ps) and its Ea, and reflection coefficients (sigma) for golden hamster individual pancreatic islet cell membranes. Cryobiology 37:290–299. https://doi.org/10.1006/cryo.1998.2124
Liu J, Zieger MAJ, Lakey JRT, Woods EJ, Critser JK (1997) The determination of membrane permeability coefficients of canine pancreatic islet cells and their application to islet cryopreservation. Cryobiology 35:1–13
Benson CT, Liu C, Gao DY, Critser ES, Benson JD, Critser JK (1996) Hydraulic conductivity (Lp) and its activation energy (Ea), cryoprotectant agent permeability (Ps) and its Ea, and reflection coefficients (ς) for golden hamster individual pancreatic islet cell membranes. Cryobiology 37(4):290–299
Liu C, Benson CT, Gao DY, Haag BW, Mcgann LE, Critser JK (1995) Water permeability and its activation-energy for individual hamster pancreatic islet cells. Cryobiology 32:493–502. https://doi.org/10.1006/cryo.1995.1049
Rule GS, Law P, Kruuv J, Lepock JR (1980) Water permeability of mammalian cells as a function of temperature in the presence of dimethylsulfoxide: correlation with the state of the membrane lipids. J Cell Physiol 103:407–416
Devireddy RV, Fahrig B, Godke RA, Leibo SP (2004) Subzero water transport characteristics of boar spermatozoa confirm observed optimal cooling rates. Mol Reprod Dev 67:446–457
Devireddy RV, Smith DJ, Bischof JC (1999) Mass transfer during freezing in rat prostate tumor tissue. AIChE J 43(3):639–654
Devireddy RV, Raha D, Bischof JC (1998) Measurement of water transport during freezing in cell suspensions using a differential scanning calorimeter. Cryobiology 36:124–155
Drobnis E, Crowe L, Berger T, Anchordoguy T, Overstreet J, Crowe J (1993) Cold shock damage is due to lipid phase transitions in cell membranes: a demonstration using sperm as a model. J Exp Zool 265:432–437
Mcnaught AD, Wilkinson A (1997) IUPAC. Compendium of chemical terminology (the “Gold Book”), 2nd edn. Wiley Blackwell, Oxford. ISBN 978-0865426849
Katkov I (2000) A two-parameter model of cell membrane permeability for multisolute systems. Cryobiology 40:64–83
Lusianti RE, Benson JD, Acker JP, Higgins AZ (2013) Rapid removal of glycerol from frozen-thawed red blood cells. Biotechnol Prog 69:609–620
Kreyszig E (2006) Advanced engineering mathematics, 9th edn. Wiley, New York
Benson JD, Chicone CC, Critser JK (2005) Exact solutions of a two parameter flux model and cryobiological applications. Cryobiology 50:308–316
Boyce W, DiPrima R (1992) Elementary differential equations and boundary value problems, 6th edn. Wiley, New York
Benson JD, Higgins AZ, Desai K, Eroglu A (2018) A toxicity cost function approach to optimal CPA equilibration in tissues. Cryobiology 80:144–155
Katkov I (2002) The point of maximum cell water volume excursion in case of presence of an impermeable solute. Cryobiology 44:193–203
Zhang S, Chen G (2002) Analytical solution for the extremums of cell water volume and cell volume using a two-parameter model. Cryobiology 44:204–209
Elmoazzen HY, Elliott JAW, McGann LE (2009) Osmotic transport across cell membranes in nondilute solutions: a new nondilute solute transport equation. Biophys J 96:2559–2571
Prickett RC, Elliott JAW, McGann LE (2010) Application of the osmotic virial equation in cryobiology. Cryobiology 60:30–42
Benson J, Haidekker M, Benson C, Critser J (2005) Mercury free operation of the Coulter counter MultiSizer II sampling stand. Cryobiology 51:344–347
Higgins A, Karlsson J (2008) Curve fitting approach for measurement of cellular osmotic properties by the electrical sensing zone method. I. osmotically inactive volume. Cryobiology 57:223–233
Gao DY, Benson CT, Liu C, McGrath JJ, Critser ES, Critser JK (1996) Development of a novel microperfusion chamber for determination of cell membrane transport properties. Biophys J 71:443–450
Mullen SF, Rosenbaum M, Critser JK (2007) The effect of osmotic stress on the cell volume, metaphase ii spindle and developmental potential of in vitro matured porcine oocytes. Cryobiology 54:281–289
Zhao G, Zhang Z, Zhang Y, Chen Z, Niu D, Cao Y, He X (2017) A microfluidic perfusion approach for on-chip characterization of the transport properties of human oocytes. Lab Chip 17:1297–1305
Mbogba MK, Haider Z, Hossain SM, Huang D, Memon K, Panhwar F, Lei Z, Zhao G (2018) The application of convolution neural network based cell segmentation during cryopreservation. Cryobiology 85:95–104. https://doi.org/10.1016/j.cryobiol.2018.09.003
Chaveiro A, Liu J, Engel B, Critser JK, Woelders H (2006) Significant variability among bulls in the sperm membrane permeability for water and glycerol: possible implications for semen freezing protocols for individual males. Cryobiology 53:349–359
Fry AK, Higgins AZ (2012) Measurement of cryoprotectant permeability in adherent endothelial cells and applications to cryopreservation. Cell Mol Bioeng 5:287–298
Zhurova M, Olivieri A, Holt A, Acker JP (2014) A method to measure permeability of red blood cell membrane to water and solutes using intrinsic fluorescence. Clin Chim Acta Int J Clin Chem 431C:103–110. https://doi.org/10.1016/j.cca.2014.01.045
Toner M, Cravalho EG, Karel M (1993) Cellular-response of mouse oocytes to freezing stress - prediction of intracellular ice formation. J Biomech Eng Trans ASME 115:169–174
Karlsson JOM, Cravalho EG, Toner M (1994) A model of diffusion–limited ice growth inside biological cells during freezing. J Appl Phy 75:4442–4455
Karlsson JOM (2010) Effects of solution composition on the theoretical prediction of ice nucleation kinetics and thermodynamics. Cryobiology 60:43–51
Glazar AI, Mullen SF, Liu J, Benson JD, Critser JK, Squires EL, Graham JK (2009) Osmotic tolerance limits and membrane permeability characteristics of stallion spermatozoa treated with cholesterol. Cryobiology 59:201–206
Yoshimori T, Takamatsu H (2009) 3-D measurement of osmotic dehydration of isolated and adhered PC-3 cells. Cryobiology 58:52–61. http://dx.doi.org/10.1016/j.cryobiol.2008.10.128
Blanco JM, Long JA, Gee G, Donoghue AM, Wildt DE (2008) Osmotic tolerance of avian spermatozoa: influence of time, temperature, cryoprotectant and membrane ion pump function on sperm viability. Cryobiology 56:8–14
Salinas-Flores L, Adams SL, Lim MH (2008) Determination of the membrane permeability characteristics of pacific oyster, crassostrea gigas, oocytes and development of optimized methods to add and remove ethylene glycol. Cryobiology 56:43–52
Si W, Benson J, Men H, Critser J (2006) Osmotic tolerance limits and effects of cryoprotectants on the motility, plasma membrane integrity and acrosomal integrity of rat sperm. Cryobiology 53:336–348
Agca Y, Mullen S, Liu J, Johnson-Ward J, Gould K, Chan A, Critser J (2005) Osmotic tolerance and membrane permeability characteristics of rhesus monkey (Macaca mulatta) spermatozoa. Cryobiology 50:1–14
Walters EM, Men H, Agca Y, Mullen SF, Critser ES, Critser JK (2005) Osmotic tolerance of mouse spermatozoa from various genetic backgrounds: acrosome integrity, membrane integrity, and maintenance of motility. Cryobiology 50:193–205
Hunt C, Armitage S, Pegg D (2003) Cryopreservation of umbilical cord blood: 2. Tolerance of CD34(+) cells to multimolar dimethyl sulphoxide and the effect of cooling rate on recovery after freezing and thawing. Cryobiology 46:76–87
Guthrie H, Liu J, Critser J (2002) Osmotic tolerance limits and effects of cryoprotectants on motility of bovine spermatozoa. Biol Reprod 67. https://doi.org/10.1095/biolreprod67.6.1811
Koshimoto C, Mazur P (2002) The effect of the osmolality of sugar-containing media, the type of sugar, and the mass and molar concentration of sugar on the survival of frozen-thawed mouse sperm. Cryobiology 45:80–90
Liu J, Christian J, Critser J (2002) Canine RBC osmotic tolerance and membrane permeability. Cryobiology 44:258–268
Koshimoto C, Gamliel E, Mazur P (2000) Effect of osmolality and oxygen tension on the survival of mouse sperm frozen to various temperatures in various concentrations of glycerol and raffinose. Cryobiology 41:204–231
Gao DY, Liu J, Liu C, McGann LE, Watson PF, Kleinhans FW, Mazur P, Critser ES, Critser JK (1995) Prevention of osmotic injury to human spermatozoa during addition and removal of glycerol. Hum Reprod 10:1109–1122
Gao DY, Ashworth E, Watson PF, Kleinhans FW, Mazur P, Critser JK (1993) Hyperosmotic tolerance of human spermatozoa: separate effects of glycerol, sodium chloride, and sucrose on spermolysis. Biol Reprod 49:112–123
Fahy G, Wowk B, Wu J, Paynter S (2004) Improved vitrification solutions based on the predictability of vitrification solution toxicity. Cryobiology 48:22–35
Elmoazzen HY, Poovadan A, Law GK, Elliott JAW, McGann LE, Jomha NM (2007) Dimethyl sulfoxide toxicity kinetics in intact articular cartilage. Cell Tissue Bank 8:125–133
Wang L, Liu J, Zhou GB, Hou YP, Li JJ, Zhu SE (2011) Quantitative investigations on the effects of exposure durations to the combined cryoprotective agents on mouse oocyte vitrification procedures. Biol Reprod 85:884–894
Benson JD (2009) Mathematical problems from cryobiology. Ph.D. thesis, University of Missouri
Karlsson JO, Younis AI, Chan AW, Gould KG, Eroglu A (2009) Permeability of the rhesus monkey oocyte membrane to water and common cryoprotectants. Mol Reprod Dev 76:321–333
Lee E, Markus L (1968) Foundations of optimal control theory. The SIAM series in applied mathematics. Wiley, New York.
Royden HL (1988) Real analysis, 3rd edn. Prentice-Hall, Englewood Cliffs
Sch attler H, Ledzewicz U (2012) Geometric optimal control: theory, methods and examples. Springer, Berlin
Benson JD (2013) Cost functional dependence of optimal CPA equilibration trajectories. Cryobiology 67:404
Mazur P (1977) The role of intracellular freezing in the death of cells cooled at supraoptimal rates. Cryobiology 14:251–272
Muldrew K, Acker JP, Elliott JA, McGann LE (2004) The water to ice transition: implications for living cells. In: Fuller BJ, Lane N, Benson EE (eds) Life in the Frozen State. CRC Press, London, pp 93–134
Mazur P, Miller R (1976) Permeability of the human erythrocyte to glycerol in 1 and 2 M solutions at 0 or 20 ∘C. Cryobiology 13:507–522
Morris G, Acton E, Avery S (1999) A novel approach to sperm cryopreservation. Hum Reprod 14:1013–1021
Karlsson JOM, Eroglu A, Toth TL, Cravalho EG, Toner M (1996) Fertilization and development of mouse oocytes cryopreserved using a theoretically optimized protocol. Hum Reprod 11:1296–1305
Wowk B (2010) Thermodynamic aspects of vitrification. Cryobiology 60:11–22
Seki S, Mazur P (2009) The dominance of warming rate over cooling rate in the survival of mouse oocytes subjected to a vitrification procedure. Cryobiology 59:75–82
Barry PH, Diamond JM (1984) Effects of unstirred layers on membrane phenomena. Physiol Rev 64:763–872
Prickett RC (2010) The application of the multisolute osmotic virial equation to cryobiology. Ph.D. thesis, University of Alberta, Edmonton, Alberta
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Benson, J.D. (2021). Mathematical Modeling and Optimization of Cryopreservation in Single Cells. In: Wolkers, W.F., Oldenhof, H. (eds) Cryopreservation and Freeze-Drying Protocols. Methods in Molecular Biology, vol 2180. Humana, New York, NY. https://doi.org/10.1007/978-1-0716-0783-1_4
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