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Regular Article TA |
a Cryobiology Research Institute, Herman B Wells Center for Pediatric Research, Department of Pediatrics, Indiana University School of Medicine, Indianapolis, Indiana 46202
b Department of Veterinary Clinical Sciences, School of Veterinary Medicine, Purdue University, West Lafayette, Indiana 47907
c Advanced Fertility Institute, Methodist Medical Plaza Carmel, Indianapolis, Indiana 46280
ABSTRACT
Current mammalian embryo cryopreservation protocols typically employ an interrupted slow freezing (ISF) procedure. In general, ISF consists of initial slow cooling, which raises the extracellular solute concentration, and results in cell dehydration. Permeating cryoprotective agents (CPAs), such as dimethyl sulfoxide (DMSO), are typically included in the medium to protect the cells against high solute concentrations. As this ISF procedure continues, slow cooling is terminated at an intermediate temperature (Tp), followed by plunging into liquid nitrogen (LN2). If the slow cooling step allowed a critical concentration ([CPA]c) of CPA to be reached within the cell, the CPA will interact with the remaining intracellular water during rapid cooling, resulting in the majority of the intracellular solution becoming vitrified and preventing damaging intracellular ice formation (IIF). This study presents a theoretical model to develop efficient ISF procedures, on the basis of previously developed data for the rat zygote. The model was used to select values of initial CPA concentrations and slow cooling rates (from initial estimated ranges of 0 to 4 molal DMSO and 0 to 2.5°C/min cooling rates) that would allow the intracellular solute concentration to exceed the critical concentration. The optimal combination was then determined from this range based on minimizing the duration of slow cooling.
assisted reproductive technology, embryo
INTRODUCTION
Current mammalian oocyte and embryo cryopreservation protocols have evolved from methods that have been successful with mouse and cattle embryos [1]. The basic strategy typically employed involves the use of a two-step, or interrupted slow freezing (ISF) procedure, which is now a common procedure to cryopreserve many different cell and tissue types [2]. In general, this procedure consists of an initial slow cooling period followed by rapid cooling as the sample is plunged into liquid nitrogen (LN2) for final storage. In the initial slow cooling step, extracellular ice is induced at a temperature just below the solution freezing point, and slow cooling is continued (at a given rate defined as B1) in the presence of this growing ice phase, which raises the extracellular solute concentration in the unfrozen fraction and results in water exiting the cell via exosmosmosis. (Table 1 contains a description of terms and abbreviations.) Permeating cryoprotective agents (CPAs) such as glycerol, dimethyl sulfoxide (DMSO), ethylene glycol (EG), or propylene glycol (PG) are typically included in the suspension medium to protect cells against injury from the high concentrations of electrolytes (so-called solution effects) that develop as water exits the solution as ice. These CPAs become increasingly concentrated intracellularly as a cell dehydrates.
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As this ISF procedure continues, the slow cooling step is terminated at an intermediate temperature at which plunging occurs (Tp), and is followed by a rapid cooling step (at a given rate defined as B2). If the initial cooling step is conducted in a way that allows formation of a critical concentration ([CPA]c) of intracellular solute ([S]i; in the present study defined as CPA + NaCl), then the CPA will interact with the remaining water in the cell, which results in the formation of a glass-like structure (vitrification) by the intracellular solution, and prevention of damaging intracellular ice formation (IIF). The concentration of CPA required to achieve intracellular vitrification during the subsequent rapid cooling (and to maintain the vitrified state during warming) is dependent upon the nature of the solute, the rate of rapid cooling during the plunge into LN2, and the rate of warming during subsequent thawing [3].
The Tp at which the [CPA]c is attained during the slow cooling phase of ISF depends on the initial concentration of CPA present in a cell (at the onset of the slow freezing step) and the initial slow cooling rate. For a given [CPA]c, this temperature could be theoretically determined for a specific initial concentration of CPA loaded into cells prior to freezing and a specific cooling rate. This suggests that theoretical determination of an ISF cryopreservation protocol is possible, if the [CPA]c is known.
Therefore, in the context of ISF protocol development, the goal is to determine the best selection of the initial CPA concentration ([CPA]0), B1, and Tp that will allow [S]i to reach [CPA]c. These determinations can be achieved by experimental and theoretical approaches. The procedures usually involve defining ranges of prospective selections of [CPA]0 and B1 for a given cell and CPA type, followed by evaluation of these ranges to determine which will yield optimal results as judged by experimental or established theoretical criteria. Many models have been proposed to integrate these aspects (to differing extents) in an attempt to quantitatively examine the biophysical events that occur during cryopreservation. Mazur [4, 5] first introduced a model that allowed examination of the kinetics of water loss from cells at subzero temperatures, relating it to the effects of cooling and warming velocity on cryosurvival. Liu et al. [6] established a theoretical model that incorporates the transmembrane movement of cryoprotectant at low temperatures and the DMSO/NaCl/water ternary phase diagram. Karlsson et al. [7] conducted a comprehensive study in which a coupled mechanistic model was used to design and optimize a two-step cryopreservation protocol for mouse oocytes. Briefly, their method consisted of fixing a given CPA concentration (1.5 M), then optimizing the cryopreservation protocol by 1) minimizing the time taken to reach the final temperature (to reduce injury by solution effects) and 2) avoiding IIF. The optimization process developed by Karlsson et al. [7] involved defining a cost-function equivalent to the duration of the freezing protocol. The protocol was then theoretically optimized by using a sequential simplex algorithm to minimize the cost function, subject to the constraint that the predicted incidence of IIF remains below 5% [7].
The objective of the present study was to theoretically optimize an ISF protocol to cryopreserve rat zygotes. In this study, DMSO was chosen as the permeable CPA because the most complete set of information exists for solutions of this permeating cryoprotectant; specifically, critical concentrations/cooling and warming rates necessary for vitrification and ternary solution phase diagram information [3, 8, 9]. The basic phenomenological strategy employed in the current model was similar to the studies by Karlsson et al. [7, 10] in that the consequences of the duration of slow cooling and the probability of IIF were considered; however, the current strategy employed a different theoretical method. Specifically, the current model assumed, first, that the cell membrane was permeable to CPA at any temperature (with permeability being calculated using the Arrhenius relationship); second, that the actual ternary solution phase diagram was used instead of the more general Clausius-Clapeyron equation; and third, that a simplified model was used to predict IIF. Essentially, development of the model followed three steps: 1) an initial range of DMSO concentrations from 0 to 4 molal, and a range of cooling rates from 0 to 2.5°C/min were evaluated theoretically to determine the selections of [CPA]0 and B1 that would allow the [S]i to reach the [CPA]c; 2) using Mazur's IIF model [11], the selections that could result in IIF were eliminated; and 3) the associated plunging temperatures for the combinations of [CPA]0 and B1 ranges were then calculated. The optimum set of conditions from the final range was then selected based on minimum duration of slow cooling.
MATERIALS AND METHODS
Theoretical Prediction of Intracellular Water Volume and Solute Mole Number at Varying Temperatures
During slow cooling, temperature is typically decreased linearly, causing [S]i to increase monotonically, as shown in Figure 1. For a specific cell and CPA, the change in intracellular CPA concentration is affected by cooling conditions (i.e. the B1 of the process and the [CPA]0 loaded into cells prior to the onset of freezing). As shown conceptually in Figure 1, for three different cooling conditions (initial CPA concentrations and cooling rates) a, b, and c, [S]i reaches [CPA]c at different temperatures, A, B and C, respectively. These temperature points are defined as Tp.
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The [S]i during slow cooling can be readily calculated when the cell water volume and the number of moles of CPA are known. The changes in intracellular water volume and intracellular CPA mole number during the temperature change of rate B1 were predicted as described by Liu et al. [6] using a model consisting of the following coupled equations:
where
The values of the hydraulic conductivity (Lp) and solute permeability (PCPA) parameters and their associated activation energies required for the calculations were obtained from Pfaff et al. (see accompanying article in this issue) and are listed in Table 2. The terms Vw and A indicate cell water volume and cell surface area, respectively; and 
is the number of moles of CPA inside the cell. Temperature and universal gas constant are indicated by T and R, respectively. C indicates the total extracellular concentration of CPA and NaCl in g/100 g, MW is molecular weight, and v10 is the molar volume of water. In the current study, Rt is defined as 
(W is the weight of a substance) instead of the standard notation, R, to distinguish it from the universal gas constant.
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The value of parameter Lp(T) and PCPA(T) in equations (1) and (2) can be calculated using the following Arrhenius relationship [12]:
where Para = Lp or PCPA and Ea(Lp, PCPA) is the activation energy for the process, expressed in kcal/mole. The subscript "o" represents the values at a reference temperature, To.
The extracellular solute concentration, C = C (T, Rt), at temperature T can be obtained by solving the following equation of the melting point for the ternary solution of DMSO/NaCl/water [9]:
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By solving these equations, it is possible to quantitatively calculate 1) the change in cell water volume vs. temperature during cooling, 2) the change in [S]i vs. temperature, and 3) the change in extracellular solute concentration vs. temperature.
Assumptions Pertaining to Intracellular Ice Formation
Several theories have been proposed to quantitatively study the probability of IIF [2, 11, 13–17]. These models, to some extent, require additional parameters to describe the kinetics of IIF. These parameters are usually cell-specific and are typically determined by experiments that incorporate relatively high cooling rates in order to observe IIF. The present method makes assumptions on the basis of Mazur's "three requirements" for IIF [11]: 1) the sample temperature has reached the ice nucleation zone, or, in other words, the temperature has become lower than the nucleation temperature; 2) the intracellular water content at that time is 
10% of its isotonic value; and 3) the intracellular water is 2°C or more supercooled. IIF will not occur if any one of these requirements is not met. Cell water content can be calculated as described above while the nucleation temperature must be experimentally determined. The extent of supercooling can be calculated by taking the difference between the melting points (determined by the NaCl/DMSO/H2O phase diagram, Equation 4 [9]) of intracellular and extracellular solutions. For any given [CPA]0 and B1, the total [S]i can be calculated at any temperature. This information enables determination of Tp for specific [CPA]0 and B1 combinations, if [CPA]c for a given CPA is known.
Source of Embryos for Ice Nucleation Experiments
Each experimental replicate consisted of 5 to 10 postpubertal, naturally cycling female rats (8–10 wk of age) from an outbred stock (HSD: Sprague-Dawley, SD, Harlan Sprague-Dawley, Inc., Indianapolis, IN) which were synchronized using a GnRH agonist (des-Gly10,[p-Ala6]-LH-RH ethylamide (Sigma Chemical Co., St. Louis, MO). The lyophilized powder was dissolved and reconstituted using physiological saline to yield a 200 µg/ml solution. Each female rat was injected (i.p.) with 0.2 ml (40 g) of GnRH agonist solution between 0900 and 1030 h and placed in a cage with a mature male rat in the afternoon of the fourth day after synchronization. The following morning, the female rats were screened for presence of vaginal plugs, and vaginal cytology was evaluated for the presence of spermatozoa and stage of estrous cycle [18]. For collection and recovery of zygotes, rats were injected (i.m.) with 0.1 ml of a 10:2 (v/v) mixture of 100 mg/ml ketamine (Ketaset, Fort Dodge Animal House, Fort Dodge, IA):100 mg/ml xylazine (Rompun, Bayer Co., Shawnee Mission, KS) for sedation followed by cervical dislocation. The oviducts were excised and placed into a Petri dish containing PBS (Life Technologies Inc., Grand Island, NY), supplemented with 1 mg/ml hyaluronidase (Sigma) and 4 mg/ml BSA (Sigma). The extended, translucent oviductal ampulla was dissected to release the clutch of zygotes into the solution in the Petri dish. The zygotes remained in the Petri dish with the hyaluronidase for approximately 5 min to enable dissociation and removal of the cumulus cells from the zygotes. A stereodissecting microscope (Nikon SMZ-2T, Fryer Co. Inc., Huntley, IL) was used to evaluate the zygotes for the presence of polar bodies, and pronuclei to verify fertilization. The cumulus-free zygotes were washed by transferring them through three drops of hyaluronidase-free PBS before being placed into 40-µl culture drops of Hams F10 under oil in Petri dishes, and maintained in an incubator in humidified air with 5% CO2 and 90% N2, at 37°C.
Determination of Ice Nucleation Temperature
A cryomicroscope system (Hoxan Corporation, Japan) was used to determine the ice nucleation temperature. Rat zygotes were placed into a customized glass cell with a temperature probe, which was mounted on the cooling chamber of the system. The chamber was fixed onto the stage of an inverted microscope (Nikon). Cooling was controlled by the system with a microprocessor. The cooling process was videotaped, and images were captured by a computer for further analysis. IIF can be detected when a sudden refractive index change occurs in the cytoplasm, causing it to become dark or opaque when viewed with phase-contrast optics. Temperature-dependence of the probability of IIF was determined by counting the number of cells in serial images that had IIF at different temperatures.
Determination of Critical Concentration
For the purposes of this study, [CPA]c was assumed to be the greater value between the concentration that was necessary to induce vitrification during cooling (
) and that required to preclude devitrification during warming (
) at rates that could be attained in the given cell containers. This concentration is defined as 
, and was the value used in the optimization calculation. Using the 
ensures a conservative estimate of [CPA]c for determination of [CPA]0, B1, and Tp. For the plunging step, the following formula introduced by Sutton [8] was used for calculating the critical cooling rate (Bc) for a given concentration of solute (CPA and NaCl) in solution with H2O:
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was determined. For the warming process, data published by Ren et al. [3] were used to estimate the critical concentration (). Measurement of Cooling and Warming Rates Inside Straws
For the present study, standard 0.25-ml plastic straws were chosen as the freezing container. For measurement of B2, the straws were placed in LN2 vapor. The distance from the straws to the LN2 surface was adjusted until the temperature stabilized at approximately -30°C, then they were plunged into the liquid phase. Data were recorded at the rate of 20 data points per second using a computer-interfaced temperature measurement board (Omega, Stamford, CT). Cooling rates were calculated as B2 = 
T/t, where T is the temperature difference between two adjacent points, which are apart by t in time (in this case, t = 0.05 sec). For determining B3, the straws were transferred to a 37°C water bath directly from LN2. Data were recorded and analyzed as described earlier.
Prediction of the Theoretically Optimized Cryopreservation Protocol
The procedure for theoretical optimization involved three steps. First, the was determined based on measurement of B2 and B3 inside the straws. Next, the initial investigation ranges for [CPA]0 and B1 were determined and these ranges were divided into appropriate intervals for subsequent numerical calculation of [S]i and degree of supercooling. In the final step of model development, combinations of [CPA]0 and B1 values that could be used were determined and Tp values for these combinations were calculated. To be more specific, these combinations are determined by using the criterion that intracellular dehydration must reach to restrict the selections of [CPA]0 and B1 values, then using Mazur's three requirements IIF model [11] to eliminate the selections that could result in IIF. The optimum set of conditions can then be selected by the minimum duration of slow cooling, which minimizes solution effects.
In this study, [CPA]0 values ranging from 0 to 4 molal (4 molal 
3.1 molar) were chosen in conjunction with B1 values ranging from 0 to -2.5°/min. These ranges were then divided into 50 equal intervals. The calculations cover a temperature range (Tseed - Tend), where Tseed is the seeding temperature (which was set equal to the melting point) and Tend is the end point of the calculation. Because Tp is unknown prior to the calculation, a lower temperature (-80°C in the current study) was set to ensure the calculated Tp would be included in the range. For each of these 2500 (50 x 50) combinations of [CPA]0 and B1, the intracellular CPA concentration, intracellular water volume, and extent of supercooling at each temperature point were calculated. The temperature points ranged from the Tseed to the Tend at the interval of (Tseed - Tend)/100.
The intracellular concentration for each combination of [CPA]0 and B1 was calculated at each temperature point starting from Tseed to Tend, and the value was compared to . When the predicted value became greater than the , the calculation was terminated for this combination of [CPA]0 and B1 and the temperature point was recorded. The extent of supercooling and intracellular water volume were then calculated for the temperature points that fell below the nucleation temperature to determine if IIF would occur based on Mazur's three requirements [11]. Any combination that would likely result in IIF was then ruled out. If the [S]i could not reach the over the entire temperature range, this combination of [CPA]0 values and B1 values were also ruled out. The remaining points were considered the optimum range for plunging in LN2 for that particular combination of [CPA]0 and B1 values.
Because B1 and Tp values for the optimum region are known, the duration of slow cooling can readily be calculated as:
where 
i is the time required to cool the cells from seeding temperature to plunging temperature, and Tpi, Tseedi, and B1i are the associated Tp, Tseed, and B1 for each individual combination, respectively. Each i can be determined for each of the combinations of initial concentration and cooling rate.
RESULTS
Cooling/Warming Rates in 0.25-ml Plastic Straws
Experimental measurements of temperature during plunging of four straws (0.25 ml) are presented in Figure 2A and during warming in Figure 2B. During plunging, the cooling rate changed from 0 to a maximum of approximately 1 x 104°C/min, then decreased to 0 as the sample temperature equilibrated with LN2 at -196°C. During warming (in a 37°C water bath), the warming rate changed from 0 to a maximum of approximately 2 x 104°C /min, then decreased to 0 when the sample temperature equilibrated with the water bath.
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Estimation of Critical Concentration
To precisely calculate the from measured B2 and B3 is beyond the scope of the current study. The B2 for 40% (w/w) DMSO and NaCl concentration was calculated using Sutton's formula [8] and was determined to be 3.5 x 102°C/min, which is much lower than the actual cooling rate inside the straw during plunging. The B3 for 45% and 47.5% (w/w) binary solutions of DMSO (e.g., DMSO without NaCl) have been determined to be 2.38 x 103°C/min and 1.86 x 103°C/min, respectively [3], which is also much less than the actual warming rate. Considering both the cooling and warming rate requirements, the critical concentration was estimated as 40% w/w for the experimental conditions.
Experimental Determination of Intracellular Nucleation Temperature
Images captured from the recorded cryomicroscopy experiments were examined to determine the intracellular nucleation temperature. When rat zygotes were cooled at a rate of 10°C/min in 1.5 M DMSO, IIF reached 5% when the temperature reached -36°C and all zygotes had IIF at -46°C (no zygotes had IIF at -34°C; Fig. 3). Based upon these measurements, the onset of nucleation was estimated to be -35°C for rat zygotes.
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Estimation of Optimal Plunging Temperature
A hypothetical rat zygote was considered suspended in 50 equally spaced [CPA]0 values ranging from 0.08 to 4 molal and cooled from Tseed to Tend at 1 of 50 equally spaced B1 values ranging from 0.05 to 2.5°C/min. For each of the 2500 combinations of [CPA]0 and B1 values, the model calculated [S]i, water volume, and supercooling at each temperature point (100 equally spaced points from Tseed to Tend). The criteria of exceeding 40% (w/w) and no predicted IIF were used to divide the plane of [CPA]0 vs. B1 into three regions (Fig. 4) as follows:
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Region I
In this region, no combination of [CPA]0 and B1 allows the intracellular solute concentration to reach the at any temperature point. This region contains the high cooling rate zone and low initial CPA concentration zone. The high cooling rates in this region (bottom of Fig. 4) would not allow the cell to dehydrate sufficiently, and the low concentrations (left part of Fig. 4) are simply not high enough for the intracellular solute concentration (CPA and NaCl) to reach the . The cooling rates and concentrations in this region were rejected in the developing optimized ISF procedure.
Region II
The combinations of [CPA]0 and B1 allow the [S]i to reach at certain temperature points, but under these conditions there is a high probability of IIF during slow cooling. If a cell is cooled under these conditions, the cell would be supercooled more than 2°C when the temperature reaches the nucleation zone (-35°C in this case) and intracellular water content would be greater than 10% of its isotonic volume. These conditions were also rejected in developing the ISF procedure.
Region III
The combinations of [CPA]0 and B1 allow the [S]i to reach the , and no IIF is predicted during slow cooling. Figure 5 shows the Tp for each [CPA]0 and B1, indicated by the points in a three-dimensional plot (panel A) and a two-dimensional plot (panel B). These points range from approximately -25°C (for points around the high concentration/low cooling rate corner) to approximately -35°C (for points around the low concentration/high cooling rate corner).
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There is a minimum [CPA]0 boundary below which no B1 (in the range of 0.05 to 2.5°C/min) allows the [S]i to reach the . There is also a maximum B1, above which any B1 causes the same effect. Roughly, these two limits construct a box of [CPA]0 and B1 values that are candidates for the optimized protocol. The optimum set of conditions was selected by minimizing i from equation 6 and determined to be a [CPA]0 of 1.2 M DMSO, a B1 of 0.95°C/min, and Tp of -35°C.
DISCUSSION
In one form or another, ISF methods have been used and have provided varying levels of success for a diverse group of cell and tissue types, including red blood cells [19], pancreatic islets [20], and embryos [21]. Historically, mammalian embryo ISF procedures have further been divided into two categories on the basis of optimal plunging temperature and warming conditions [22]. One category (slow warming method) utilizes slow cooling (<1°C/min) to temperatures below -60°C, followed by transfer into LN2 (-196°C) and subsequent slow warming at rates of 
25°C/min or less [23, 24]. The second procedure (rapid warming method) utilizes slow cooling to an intermediate temperature, usually between -25 and -40°C, followed by transfer into LN2 and subsequent rapid warming at rates above 300°C/min [25, 26].
The current study proposes a new method by which rapid warming ISF procedures may be enhanced through modeling, taking into account typical cryobiological practices (i.e., the use of 0.25-ml plastic straws, 0 to 4 molal DMSO concentrations, and cooling rates between 0 to 2.5°C/min). To apply this approach, specific fundamental biophysical and physical-chemical parameters must be known, including cell membrane permeability coefficients and their activation energies, ternary solution (NaCl/CPA/H2O) phase diagram information, and the solution's vitrification properties. This study presents a model designed to develop ISF procedures on the basis of two criteria: 1) the [S]i must reach a , at which the intracellular solution can be readily vitrified during plunging, and 2) lethal IIF must be avoided. The model was applied to the rat zygote using DMSO as the permeating CPA, and from the predicted optimal [CPA]0 range, B1 range, and associated Tp values, a protocol consisting of a [CPA]0 of 1.2 M DMSO, a B1 of 0.95°C/min, and a Tp of -35°C was predicted by calculating which set of conditions resulted in the minimum duration of slow cooling.
Several other models have been developed to attempt to quantitatively examine the biophysical events that occur during these procedures, the most recent being those developed by Karlsson et al. [7, 10, 17], who proposed a theoretical model of optimization and applied it to the mouse oocyte considering glycerol as the permeating CPA. In contrast to Karlsson's study, in the model described here, the changes of intracellular water volume and [S]i during cooling and warming were predicted as described by Liu et al. [6] considering that the cell remains permeable to CPA at low temperatures. In addition, a fixed initial CPA concentration was not a necessary constraint in the current model. This allows more flexibility in determining the set of conditions for optimum ISF protocols without needing to repeat the simulations for other fixed CPA concentrations. It also allows the possibility of considering other biological concerns that are equally crucial in cryopreservation. For example, the potential toxic effects of CPAs on cells usually become severe when CPA concentrations are high and the exposure time is long. In this regard, when the duration of slow cooling does not change greatly over a wide range of CPA concentrations, a lower CPA concentration should be chosen. This is possible using the current model, and from this point of view, the upper-right corner of Figure 5B (high [CPA]0, low B1) on the initial concentration/cooling rate plane could be readily eliminated.
Another issue involved in ISF procedure development is IIF estimation. Lacking the necessary parameters for the rat zygote, the most current IIF models could not be used (i.e., Karlsson et al. [7, 10, 17]); however, Mazur's model [11] agreed closely with experimental data (Fig. 3) and was therefore considered adequate. The use of different IIF models will affect the shape of zones II and III. However, the use of different IIF models would not fundamentally change our results because 1) the boundary between zones I and II (Fig. 4) is independent of the choice of slow cooling IIF model (it is based on the [CPA]c required for vitrification during the rapid cooling step), with the boundary between zones II and III (Fig. 4) constrained to stay within zone I; and 2) IIF curves from the experimental data agree closely with predictions made using Mazur's model [11] as well as those from Karlsson's predictions for mouse oocytes [7] (Fig. 3).
The current model could also be used to optimize a situation in which the cell type is biologically sensitive to supercooling and not to CPA exposure time. The high [CPA]0, low B1 region of Figure 5 indicates that the Tp values are much higher in this section (approximately 10 degrees higher). Although the cells are exposed to relatively high CPA concentrations and subzero temperatures much longer (due to lower B1 values), the absolute temperature values are much higher and supercooling is much lower. According to Pitt et al. [15], two effects dominate IIF: 1) supercooling and 2) exposure time. Therefore, it would be predicted that these cells should be cooled at a lower B1 value with a higher [CPA]0 value.
The model's prediction, indicated by Figure 5, that a higher [CPA]0 allows for a lower B1 value is entirely consistent with previous theory [27] and experimental data as well [28]. A contour map of maximum [S]i plotted for different [CPA]0 and B1 values is presented in Figure 6. For discussion purposes, the section "AZ " of the line "40%" was used as the trace of optimal combinations of [CPA]0 and B1 values. It is clear that the optimal B1 decreases as the [CPA]0 increases. Two distinguishing aspects are important to note. First, as explained previously by Mazur [28], the introduction of CPA makes the cells susceptible to IIF at even lower cooling rates. This statement supports the current prediction because the IIF region in Figure 4 has the same trend as the line of "40%" in Figure 6. The cause of this phenomenon is most likely that the higher [CPA]0 values tend to trap more water inside the cell and thereby increase the probability of IIF [17, 29]. For a situation of the same cooling rate but different [CPA]i, a cell with lower [CPA]i has to dehydrate more than a cell with higher [CPA]i to make its intracellular concentration equilibrate with extracellular concentration, which increases during slow cooling in the first step. For this scenario, the IIF occurring during slow cooling could be avoided by using a lower B1. Second, as illustrated in Figure 6, for a given B1, higher [CPA]0 values tend to have a lower Cmax during slow cooling (again, most likely due to the higher intracellular CPA concentration tending to trap more water inside the cell) [17, 29]. For this scenario, the issue is the [CPA], and a lower [CPA]0 must be chosen to ensure that the [S]i can become greater than the [CPA].
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The calculation results for 
, and predicted IIF zone for mouse oocytes first presented by Karlsson et al. [17] in 1994 are in close agreement with the results predicted in the current study for rat zygotes using DMSO. Their plot is reconstructed along with the current predictions (Fig. 7), and the results indicate close agreement until low [CPA]0 values (less than 0.6 molal) are considered. The discrepancy in this range may be attributed to the current model's prediction that the Cmax in this situation could not reach the required [CPA] if these [CPA]0 values were used.
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In the most recent study by Karlsson et al. [7], a cost function was used to optimize the duration time of the freezing protocol, and this may be accomplished in a simplified manner in the current model by calculating the duration of slow cooling using equation 6. From Figure 8A it is obvious that higher B1 values decrease i dramatically, even though the corresponding Tp values are lower than those using the lower B1 values. Figure 8B indicates that there is an optimal [CPA]0 for a given B1, which minimizes the duration of the slow cooling step. For the rat zygote, this optimal [CPA]0 changes slightly for different B1 values. When the B1 increases from 0.1°C/min to 0.95°C/min, the optimal [CPA]0 decreases from 1.7 molal to 1.2 molal. Similar to the predictions made by Karlsson's model [7], these optimal conditions occur right on the border of regions at which unacceptable cell damage is predicted. Therefore, although theoretically optimal, this set of conditions may actually be risky in practice because any procedural error (e.g., fluctuation in cooling rate, error in solution preparation) or error in solution properties (e.g., error in [CPA] estimation) could result in unexpected negative consequences. Indeed, the determination of zone III is very sensitive to the value of [CPA]. The estimated [CPA] values used in the model were based on published data for binary solutions (e.g., CPA and H2O only). If this approximation resulted in an underestimation of [CPA], it could be detrimental because, in practice, the [S]i would not be high enough at Tp to ensure vitrification during plunging. By selecting a point more toward the center of zone III (e.g., a [CPA]0 of 1.5 molal DMSO, a B1 of 0.5°C/min, and a Tp of -30°C) a more conservative estimation can be achieved.
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Consistent with the current model, Hirabayashi [30] recently reported cryopreservation of 2-cell rat embryos using an ISF method consisting of a 0.5°C/min B1, a [CPA]0 of 10%(v/v) DMSO (1.4 5M), and a Tp of -30°C. Using late morula or early blastocysts obtained from naturally ovulated and mated Wistar female rats, Utsumi et al. [31] investigated the effects of [CPA]0 on percent survival (at a fixed B1) and found a 0% and 20% survival rate for 0.15 M and 0.3 M DMSO, respectively. Survival increased to 60% and 59% for 1.0 M and 1.5 M. This same study also reported the effects of B1 and Tp (at a fixed [CPA]0) on embryo survival. Utsumi et al. [31] concluded that survival decreased dramatically when a 5°C/min B1 was used; while, for the same Tp, the percent survival tended to be higher if a lower B1 was used. All of these experimental results correspond well with the current model predictions (region III in Fig. 4). However, it is important to note that direct comparisons cannot be made because embryo developmental stages and specific experimental conditions were different from those used to develop the model in the current study.
Figure 6 suggests an alternative way to optimize ISF procedures. If a discrete isoconcentration line is roughly used as the guideline for selecting [CPA]0 and B1, the turning point (e.g., A and B in Fig. 6 at which the curve has the smallest radius) could be considered as the optimal point for corresponding isoconcentrations because B1 has the maximum value while [CPA]0 is close to the minimum value. If a comparison is made between the turning points of 40% and 45% isoconcentration lines, we can conclude that the optimal B1 is much higher for 40% than 45% and the [CPA]0 is relatively lower. The former condition, higher B1 and lower [CPA]0, would appear to be optimal because the higher B1 shortens the duration of slow cooling (therefore potentially reducing solution effects injury), and the lower [CPA]0 lowers the osmotic stress and lessens the potential CPA toxicity.
The is constrained by the B2 during plunging and B3 during thawing for a given solution. It is important to note that if these rates could be increased by using a thinner container, or material with higher heat transfer properties, then may be reduced. This would result in situations in which equilibrium cooling could be performed much more efficiently because higher B2 values and lower [CPA]0 values could be applied. The positive results reported by investigators who use thermal-pulled, thin straws may be due to these reasons [32].
Conclusion
Theoretically, these calculations may be conducted for any cell type and CPA if the appropriate information on the fundamental membrane permeability parameters and phase diagram solution characteristics are known. While the procedures described here focused on rat zygotes, the cryobiological issues apply directly to other species and other cell types, including human oocytes and embryos. In this regard, optimization of ISF methods for human oocytes and preimplantation embryos would address current concerns in assisted reproduction technology (ART) programs related to the increasing number of available oocytes and embryos per cycle; combined with the growing concern for avoiding multiple gestation rates. Approaches like the ones described in this work can be used to aid in the development of improved cryopreservation methods that allow high survival rates of human oocytes and early embryos. In this way, ART laboratories will have a wider range of options available to manage the growing need to preserve, rather than discard or transfer large cohorts of oocytes and embryos. However, to use these types of optimization procedures, basic knowledge of the fundamental cryobiology of these cells is required because oocytes and embryos of different species have different cryobiological characteristics (e.g., membrane permeability coefficients). Therefore, there is a critical need to develop noninvasive techniques to make such measurements while maintaining the clinical usefulness of the cells. Such techniques are currently being developed and could then be used in clinical programs.
ACKNOWLEDGMENTS
The authors acknowledge the helpful discussion and thoughtful criticism provided by Drs. Peter Mazur, Michael Zieger, Jens Karlsson, and Locksley McGann during the development of this manuscript.
FOOTNOTES
First decision: 23 March 2000.
1 Supported by The Cryobiology Research Institute, grants R01-AA10722 and R24RR13195 from the National Institutes of Health, and Harlan Sprague Dawley, Inc. 
2 Correspondence: John K. Critser, Indiana University School of Medicine, Cancer Research Building, Wells Center for Pediatric Research, 1044 West Walnut St., Room 454, Indianapolis, IN 46202. FAX: 317 274 8679; jcritser{at}iupui.edu
Accepted: June 6, 2000.
Received: February 9, 2000.
REFERENCES
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