Human Reproduction vol.11 no.6 pp. 1296-1305, 1996

Fertilization and development of mouse oocytes cryopreserved using a theoretically optimized protocol

Jens O-M.Karlsson1, Ali Eroglu2, Thomas L.Toth2, Ernest G.Cravalho3 and Mehmet Toner1'4 'Center for Engineering in Medicine and Surgical Services, Massachusetts General Hospital and the Shriners Burns Institute, and Harvard Medical School, Obstetrics and Gynecology Service, Massachusetts General Hospital, Harvard Medical School, Boston, MA and 'Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA •^o whom correspondence should be addressed at: Shriners Research Center, One Kendall Square, 1400W, Cambridge, MA 02139, USA

Rational design of a cryopreservation protocol was demonstrated by using theoretical models of the cryopreservation process to develop an optimal freezing protocol for mouse oocytes. A coupled mechanistic model of the processes of freeze-induced cell dehydration and intracellular ice formation was developed, and cryomicroscopical measurements of intracellular ice formation kinetics were used to determine biophysical parameters required by the model, and to test model predictions of the freezing behaviour of mouse oocytes. A simple phenomenological model for oocyte damage resulting from exposure to concentrated electrolyte and cryoprotectant solutions during cryopreservation was obtained by defining a cost function equal to the duration of the freezing protocol. A two-step freezing protocol was theoretically optimized by using a sequential simplex algorithm to minimize the cost function, subject to the constraint that the predicted probability of intracellular ice formation remain below 5%, yielding a putative optimum at the cooling rate B = (L59°C/min, and plunge temperature Tp = - 6 7 ° C . By systematically varying B and Tp about these values in experiments with mouse oocytes cryopreserved in 1.5 M dimethyl sulphoxide, the maximal recovery of intact oocytes with a normal morphology (82%) was obtained for B = 0.5oC7min and Tp = -80°C. Further evaluation of the fertilizability and developmental capacity of oocytes cryopreserved using the optimized protocol yielded cleavage to the 2-cell stage in 65% of oocytes inseminated, and blastocyst formation in 50% of these 2-cell embryos. Key words: cryopreservation/mouse oocytes/optimization/ rational design/theoretical modelling

Introduction The ability to cryopreserve mammalian oocytes would represent an important advance in reproductive medicine, especially in the treatment of infertility. Although the frozen 1296

storage of surplus embryos obtained from in-vitro fertilization (IVF) in combination with ovarian stimulation regimens has become a widely used technique for increasing pregnancy rates, there are serious ethical and legal questions associated with the practice of freezing human embryos. A preferable approach would be the cryopreservation of unfertilized oocytes, which would additionally make possible the opportunity of future pregnancy to young women undergoing cancer therapy, and would allow the creation of frozen oocyte banks for oocyte donation programmes. Unfortunately, compared with the success rates obtained with embryos, die rates of in-vitro development of frozen—thawed oocytes are still very low (Van Steirteghem and Van den Abbeel, 1985; Trounson, 1986; Levran et al, 1990). To date, only five pregnancies have been reported in humans following IVF of >900 frozen oocytes (Chen, 1986; Friedler et al, 1988; Medical Research International, 1992). Current methods for freezing oocytes are typically adaptations of protocols that have been successful with embryos. The effect of various freezing parameters has been investigated experimentally in the mouse, in efforts to develop cryopreservation protocols with a maximal yield of viable, fertile oocytes after thawing (Trounson, 1986; Friedler et al., 1988). However, because there is a large number of protocol variables potentially affecting cell viability, an exhaustive experimental search for the optimal combination of these parameters would be prohibitively expensive in terms of time and resources. Thus, even though researchers have attempted to freeze mouse oocytes since the initial study of Sherman and Lin in 1958, only recently have high rates of survival and the successful fertilization of cryopreserved oocytes been reported (Carroll et al, 1990; Schroeder et al., 1990; Hunter et al, 1991). Blastocyst formation rates for frozen-thawed oocytes remain low, with only one group obtaining normal development to the blastocyst stage in >50% of all cryopreserved oocytes (Fuller and Bernard, 1984; Hunter et al, 1991). Moreover, a review of the literature reveals a high degree of variability in the success rates reported by different research groups, and a failure among investigators to reach consensus on the optimal freezing method for mouse oocytes. Here, we present a rational design approach to developing cryopreservation protocols, in which a theoretical model is used to efficiendy optimize a freezing protocol for mouse oocytes. We propose that such a model-based, dieoretical optimization approach can be more efficient than conventional experimental optimizations, and may thus be critical in developing successful cryopreservation methods for human oocytes by drastically reducing the number of oocytes and experiments required for optimization. In © European Society for Human Reproduction and Embryology

Oocyte cryopreservation by theoretical protocol optimization

addition, theoretical models can be the basis for increasing consistency in results between researchers by providing an improved understanding of the complex behaviour of cells during cryopreservation. The use of mathematical models to predict the effect of cryopreservation on cells was pioneered by Mazur (1963). Rational design of freezing protocols has become possible in the wake of recent advances in the theoretical modelling of intracellular ice formation, a major mechanism of freezing injury (Toner et al, 1990b; Pitt, 1992; Muldrew and McGann, 1994). Pitt (1992) has demonstrated the power of mathematical models in the optimization of non-linear freezing protocols, using a hypothetical cell type as an example. Muldrew (1993) predicted an optimal protocol for the cryopreservation of articular cartilage to — 80°C using the osmotic rupture model (Muldrew and McGann, 1994), but has not attempted to test this protocol experimentally. Toner et al. (1990b) have used their intracellular ice formation model to optimize a protocol for the rapid freezing of 1-cell mouse embryos to -45°C in the absence of cryoprotectants, obtaining good agreement between experimental results and model predictions (Toner et al., 1993). The model of Toner et al. (1990b) has recently been extended to include cryoprotectants, as well as the effects of intracellular crystal growth (Karlsson et al, 1994). Theoretical predictions by Karlsson et al. (1993a) have been shown to be consistent with the results of an experimental protocol optimization (Borel Rinkes et al, 1992), thus demonstrating the feasibility of using the model for the rational design of freezing protocols. In our present study, the intracellular ice formation model of Karlsson et al. (1993a, 1994) is used in an a priori theoretical optimization of a freezing protocol for unfertilized mouse oocytes in the presence of dimethyl sulphoxide (DMSO). Model predictions are then tested experimentally, resulting in the recovery of >80% morphologically normal oocytes after cryopreservation to — 196CC. The rates of fertilization and development of these oocytes were comparable with values obtained using current cryopreservation techniques for mouse oocytes.

dV/dt = (LA/v)exp{(EJR)[(l/To) R71n{(V - <}>VO)/[V + (Tlv - )V0]}),

where V is the cytosol volume (i.e. the difference between the total cell volume and the osmotically inactive volume); Vo is the initial value of V; / is time; L is the water permeability at To, the equilibrium melting point of water; E is the activation energy for water transport; A is the membrane surface area; R is the gas constant; T is temperature; H is the specific heat of fusion of water, v is the specific volume of water, and if and n are initial values of the volume fraction and osmolarity of intracellular solutes (NaCl and DMSO) respectively. Values of the water transport parameters (L and E) measured for oocytes in the absence of cryoprotectants by Toner et al (1990a) were used in this study. By numerically integrating Equation 1 using a fourth-order Runge-Kutta algorithm with adaptive step size control, the cell volume and the concentrations of DMSO and NaCl in the cytosol could be determined as functions of time, given a freezing protocol T\t). Intracellular ice nucleation The total rate of ice nucleation per cell, J, is calculated by adding together the contributions from all active nucleation mechanisms: J(t) = w(t)exp[-k(t)Wt) + Clvw(t)exp[-Kvk(t)]V(t) + Clsw(t)exp[-Ksk(t)]A(t), (Equation 2) where fly a n d ^ s a r e kinetic coefficients for the mechanisms of volume-catalysed nucleation (VCN) and surface-catalyzed nucleation (SCN) respectively (Toner et al, 1990b); and Kv and KS are thermodynamic coefficients for the VCN and SCN mechanisms respectively. Calculation of the functions w(t) and kit) is described in detail elsewhere (Karlsson et al, 1994). The probability of intracellular ice formation (PIF) is defined as the probability that a given cell will contain at least one ice nucleus. This probability can be determined from the calculated rate of intracellular ice nucleation using the approach of Toner et al. (1990b), which is based on earlier work by Carte (1959) and assumes a population of identical cells: PIF(f) = 1 - exp[-l'QJ (z)dx].

Theoretical background A theoretical model of intracellular ice crystallization, described in detail elsewhere (Karlsson et al, 1993a, 1994), was used here to predict the probability of intracellular ice formation and the volume of intracellular ice resulting from various freezing protocols. Briefly, the model integrates mathematical descriptions of three coupled physico-chemical phenomena: (i) freeze-induced dehydration of the cell; (ii) formation of intracellular ice nuclei; and (iii) growth of these nuclei into ice crystals. For convenience, the relevant equations are summarized below. Here, the theoretical model was used to optimize a freezing protocol for mouse oocytes. Water transport Water transport across the cell plasma membrane can be described using a two-compartment membrane-limited transport model as follows (Mazur, 1963; Younis et al, 19%):

- 1] (Equation 1)

(Equation 3)

The evaluation of Equation 3 requires knowledge of the coefficients Q s , Q w xs and Kv; these were measured experimentally as described below. For an analysis of the sensitivity of model predictions to uncertainties in model parameter measurements, the reader is referred to studies by Karlsson et al. (1993a) and Toner (1993). Freezing protocol optimization To maximize cell survival after a freeze-thaw cycle, rates of injury caused by all active mechanisms of damage must be minimized. The injury mechanisms acting during cryopreservation are typically grouped into two broad categories: (i) injury induced by intracellular ice formation (Karlsson et al., 1993b); and (ii) slow-cooling or 'solution effects' injury. The latter mode of damage is thought to be caused by high intracellular and extracellular solute concentrations, excessive cell shrinkage, mechanical deformation or other forces (Mazur, 1297

J.O.M.Karlsson et al 1984), but has not been characterized as well as intracellular ice formation. Whatever the exact mechanism underlying solution effects damage, the corresponding rate of injury typically decreases with decreasing temperature, and becomes negligible at cryogenic temperatures. Thus, to design optimized freezing protocols, two criteria were used in our study: (i) the time taken to reach the final temperature should be minimized to reduce injury by solution effects; and (ii) intracellular ice formation should be avoided. The first criterion was used to define the cost to be minimized, the second criterion specified a constraint to the solution. Thus, the cost associated with a freezing protocol 7X0 was defined as the duration of the protocol, described by the functional: ***{7X0}

=

^o

U

7

d/

[ ^ ~ Vl '

(Equation 4)

where U(x) is the Heaviside step function [U(x) = 1 for x ssfj; U(x) = 0 for x < 0], and 7> = -196°C (see Appendix for a discussion of assumptions in the definition of the cost functional). The 'optimal' protocol was defined as the function 7X0 which would minimize *P, subject to the constraint that the cumulative fraction of cells with intracellular ice remain at all times below 5%. This cumulative probability of intracellular ice formation during freezing was calculated using Equations 2 and 3 coupled with Equation 1. The freezing protocol optimized here was a piecewise linear cooling protocol consisting of two steps (Karlsson et al., 1995). First, the sample was cooled at a rate B from the seeding temperature 7"s to an intermediate temperature Tp, at which point the sample was immersed directly into liquid nitrogen, thus cooling the sample from Tp to the final storage temperature Tf, at an average rate Bp. The cooling rate of the initial step (B), as well as the plunge temperature (Tp), were the variable parameters in the optimization. Thus, the cost function for this particular problem can be written as: ¥ ( 0 , r p ) = (7; - Tp)/B + (Tp - Tf)/Bp.

(Equation 5)

The global minimum of the cost functional was determined using a variable-size simplex algorithm (Nelder and Mead, 1965) in conjunction with Monte Carlo techniques.

Materials and methods Source of oocytes and spermatozoa Freshly ovulated oocytes were obtained from a hybrid-inbred strain (C57B1/6XDBA) of virgin female BDF, mice, 4-10 weeks of age (Charles River Laboratories, Boston, MA, USA), maintained on a 12 h (07:00/19:00 h) light/dark cycle. Female mice were induced to superovulate by the i.p. injection of 7.5 IU pregnant mare's serum gonadotrophin (Sigma, St Louis, MO, USA) in the late afternoon. Approximately 49-50 h later, mice were injected i.p. with 7.5 IU human chorionic gonadotrophin (HCG; Sigma). These females were killed by cervical dislocation 12-15 h after HCG injection for the collection of oocytes. Oviducts were excised and placed in Petri dishes containing a large drop of Dulbecco's phosphate-buffered saline solution (PBS; Sigma) supplemented with 4 mg/ml bovine serum albumin (BSA; Sigma) at ambient temperature. The cumulus mass was released from each oviduct by puncturing the ampulla with a needle, and the cumulus was enzymatically removed by treatment in 120 IU/ml hyaluronidase (Sigma) in PBS for 5 min at ambient 1298

temperature. The oocytes were then washed three times in PBS before use in the experiments. Any degenerate oocytes (typically 0-10% of the total yield) were discarded. Spermatozoa for FVF were obtained from the cauda epididymides of mature (~3 months old) male BDF| mice (Charles River Laboratories) by puncturing the epididymides with a needle and allowing the spermatozoa to disperse for 15-20 min into a small drop of human tubal fluid (HTF) medium (Irvine Scientific, Santa Ana, CA, USA) at 37°C. The sperm suspension was then diluted in HTF medium supplemented with 5 mg/ml BSA, to yield a final concentration of 1-2X106 spermatozoa/ml. The spermatozoa were capacitated by incubation at 37°C for 1-2 h prior to IVF. Cryomicroscopy and the determination ofnucleation rate parameters To measure the biophysical parameters required by the theoretical model and to test model accuracy, a cryomicroscopy system, described in detail elsewhere (Cosman et al., 1989), was used. Our study employed a Thermascope (Interface Techniques, Cambridge, MA, USA) programmable thermal microscope stage in conjunction with a video microscopy system. For experiments to measure intracellular ice formation kinetics, 10-20 oocytes were equilibrated in a solution of 1.5 M DMSO (Aldrich Chemical Co., Milwaukee, WI, USA) and then placed on the cryomicroscope stage under a coverslip sealed with silicone grease. The initiation of sample freezing was achieved by slightly supercooling the solution and manually triggering ice growth by contacting the edge of the sample with the tip of a chilled forceps. Controlled cooling, at a rate of 120°C/min, was initiated as soon as the growing ice front had engulfed all oocytes in the field of view. Depending on the distribution of the oocytes on the freezing stage, 10-20 oocytes could be observed at a time with a X4 objective. Under bright-field illumination, intracellular ice formation was manifested by a sudden darkening of the cytoplasm, believed to be caused by light scattering due to microscopic ice crystals and/or air bubbles in the cells. Examination of the video recordings of the freezing experiments permitted the determination of the intracellular ice formation temperature for each cell. At the end of each freezing experiment, the thermal gradients across the sample window were determined by recording the location of the ice front at several temperatures close to the melting point of the solution. Using these data, the instrument temperature reading could be corrected to estimate the actual temperature at the location of each individual' oocyte. The nucleation rate parameters fij. flv. K s a n d K v were determined for mouse oocytes in 1.5 M DMSO by measuring intracellular ice formation kinetics during cooling to — 60°C at a rate of 120°C/min. A rapid cooling rate was chosen to minimize water efflux, thus reducing the effect of water transport on intracellular ice nucleation. The cumulative fraction of cells with intracellular ice at a given temperature provided a measure of the probability of intracellular ice formation for cooling to that temperature. The nucleation rate parameters were then determined from the experimental data and Equations 1-3 by fitting Equation 3 to the measured probability of intracellular ice formation using previously described techniques (Toner et al., 1990b; Karlsson et aL, 1993a; Toner, 1993; Younis et al, 1996). Briefly, a simplex algorithm (Nelder and Mead, 1965) was used to determine the combination of values for the nucleation rate coefficients which minimized the sample variance (s2) (Bevington, 1969) between the experimental and the predicted probability of intracellular ice formation data. Uncertainties in the estimated coefficients were determined from the parameter covariance matrix (Bevington, 1969).

Oocyte cryopreservation by theoretical protocol optimization

Oocyte cryopreservation Oocytes to be cryoprcserved were loaded with cryoprotectant by transfer into a 1 ml drop of 1.5 M DMSO in PBS solution at 4°C. They were equilibrated for 15 min at this temperature (manipulations were performed in a 4°C cold room using pre-chilled solutions). Next, oocytes were loaded into 0.25 ml plastic straws (TS Scientific, Perkasie, PA, USA) by aspiration with a syringe. Each straw was loaded with 15 mm cryoprotectant solution, 5 mm air, 20 mm cryoprotectant solution, five to 15 oocytes and an additional 20 mm of cryoprotectant solution, so that the oocytes were approximately in the centre of a contiguous 40 mm length of cryoprotectant solution. After successive aspirations were completed, the straws were sealed using Crito-Seal putty (TS Scientific) and placed horizontally in a Planer Kryo 10 Series II programmable freezer (TS Scientific) which had been pre-cooled to 0°C. The oocytes were exposed to 1.5 M DMSO at 4°C for -30 min before cryopreservation commenced. Straws were cooled to —5°C and extracellular ice was seeded by touching the straws with a metal forceps pre-chilled in liquid nitrogen. At 15 min after seeding, straws were cooled at a rate of between 0.05 and 3.00°C/min, to a given plunge temperature of between —30 and — 150°C; upon reaching its designated plunge temperature, each straw was immediately removed from the freezer and plunged directly into a Dewar flask containing liquid nitrogen. Straws were kept in liquid nitrogen for at least 5 min before thawing. Oocytes were thawed by reintroducing the straws into the freezer preset to a temperature of -80°C, and then wanning to 4°C at 8°C/min. At 4°C, straws were removed from the freezer, and their contents expelled into 1 ml drops of 4°C 1.5 M DMSO. DMSO was then removed from the oocytes by a three-step dilution, transferring cells sequentially through 1 ml drops of 1.0 M DMSO (equilibrating for 10 min at 4°C), 0.5 M DMSO (10 min at 4°C) and PBS (10 min at room temperature, 22-25°C). In pilot experiments which did not use fertilization as an end-point, cryoprotectant was introduced and removed at room temperature, using two-step loading/dilution procedures. Before an evaluation of viability and function, oocytes were washed twice in PBS and incubated for 1 h in bicarbonate-buffered HTF medium supplemented with 5 mg/ml BSA and overlaid with silicone oil (Aldrich Chemical Co.) under a humidified gas atmosphere of 5% CO 2 in air at 37°C. Assays for oocyte viability and function In pilot experiments to test and refine the theoretically predicted cryopreservation protocol, the viability of frozen-thawed oocytes was evaluated by assessing membrane integrity and morphological appearance at 1 and 24 h after thawing and cryoprotectant removal. The final optimized freezing protocol (fi = 0.5°C/min, Tp = -80°C; see Results below) was further evaluated by determining the ability of cryopreserved oocytes to undergo IVF and development to the blastocyst stage. Membrane integrity was assessed using two fluorescent dyes, calcein AM (Molecular Probes, Eugene, OR, USA), which stains the cytoplasm of cells with intact membranes, and ethidium homodimer 1 (Molecular Probes), which stains DNA in cells with damaged membranes. Oocytes were counted and examined for fluorescent staining and normal morphological appearance under a microscope with epifluorescence and phase-contrast optics respectively. Oocytes were considered morphologically abnormal if they exhibited a broken zona pellucida, a ruptured vitelline membrane, an unusual polar body, parthenogenic activation or a shrunken, expanded or granular vitellus. For IVF, morphologically normal oocytes were transferred into insemination dishes containing capacitated spermatozoa, and incubated at 37°C under 5% CO2 in air for 5-6 h. Oocytes were then washed three times and cultured in bicarbonate-buffered HTF medium

100 -r

-20

-30

-40

•50

Temperature [°C]

Figure 1. Non-linear curve fit to determine the heterogeneous nucleation rate parameters for mouse oocytes in 1.5 M dimethylsulphoxide. The cumulative incidence of intracellular ice formation was measured at a cooling rate of 120°C/min. Open symbols indicate intracellular ice formation by the surface-catalysed nucleation (SCN) mechanism; closed symbols represent volumecatalysed nucleation (VCN). The dashed curve shows model predictions for SCN only; the solid line is the theoretical prediction including both SCN and VCN. supplemented with 5 mg/ml BSA at 37°C under 5% CO2 in air. Fertilization was assessed after 40 h in culture by counting oocytes that had cleaved to the 2-cell stage or beyond. Developmental capacity was assessed by determining the number of embryos that had reached the blastocyst stage after 5 days of culture. Some of the frozenthawed oocytes were cultured without insemination to determine the parthenogenic activation rate. Results Ice nucleation parameters The nucleation rate parameters (Q s , Qv. — 34CC only were used (open symbols in Figure 1), obtaining the best fit between model predictions and experimental data for £i s = 1299

J.O.M.Karlsson et al -100

100-90 -

Lethal Intracellular ice formation region

T -20

-50 -

- 4 0 - 6 0 - 8 0

-100

-120

-140

Plunge temperature [°C] -40

0.0

0.2 0.4 0.6 0.8 Dehydration cooling rate [°C/mln]

1.0

Figure 2. Shape of the cost surface in protocol parameter space. Thin lines are contours of constant cost (protocol duration); the heavy line is a predicted contour of constant probability of intracellular ice formation (=5%), representing the constraint on the solution space for optimization. (•) The theoretical cost optimum; (O) the experimental optimum, see text for details. 3.8X 1(T 4 7 °° ± ° 0 5 m and KS = (2.4 ± 0.3) X 10"3 [equivalent to ftoSCN = 3.9 X 106 m - 2 s - ' and KoSCN = 3.6 X 109 K5 respectively, when converted to the units used by Toner et al. (1990b), for purposes of comparison], with a sample variance s2 = 1.5 X 10~4. The VCN parameters were then determined using intracellular ice formation data at temperatures < - 3 4 ° C (closed symbols in Figure 1), and subtracting the predicted contribution from the SCN mechanism to the cumulative incidence of intracellular ice formation at lower temperatures (dashed line in Figure 1). The resulting best fit parameters wereQ v = 2.2 X 1 0 " 2 7 0 - ° 8 a n d K V = 0.29 ± 0.02 (equivalent to OoVCN = 2.3 X 1026 m" 3 s" 1 and KoVCN = 4.4 X 10" K5 respectively, using the units of Toner et al. (1990b)), with a sample variance s2 = 2.0 X 10~3 The heterogeneous nucleation parameters determined from this experiment were used in all subsequent model simulations. Model performance using the measured nucleation rate parameters was evaluated in an independent set of cryomicroscopy experiments. Experimentally measured intracellular ice formation kinetics were found to be in good agreement with model predictions under a wide variety of physico-chemical conditions (data not shown). The model was then used for theoretical optimization of a cryopreservation protocol for mouse oocytes. Theoretical optimization of the cryopreservation protocol The cryopreservation strategy employed here was to use a two-step protocol, consisting of a dehydration step (B) to achieve an intracellular DMSO concentration sufficient to prevent intracellular ice formation, followed by rapid cooling from Tp to T{ at the highest experimentally attainable rate (flp). The plunge cooling rate for direct immersion into liquid nitrogen was in the order of lO^lO^C/min. For the purposes of numerical simulation, Bp was assumed to be constant at 300°C/min (model predictions were not sensitive to this assumption; see Karlsson etaL, 1995). Thus, using the mathem1300

00.0

r 1.0 1.5 Cooling rate [°C/min]

2.0

2.5

Figure 3. (A) Effect of plunge temperature on the viability of mouse oocytes frozen at a dehydration cooling rate of 0.5°C/min. (B) Effect of dehydration cooling rate on the viability of mouse oocytes plunged into liquid nitrogen at — 80°C. Experimental data ( ) are morphological survival rates 1 h post-thaw, mean and SE from two or three experiments, using 20-60 oocytes. Theoretical data (—) are fractions of cells with no intracellular ice. atical model, the amount of intracellular ice formation resulting from any combination of the parameters B and Tp could be predicted. To determine the optimal two-step freezing protocol, the criterion that the final fraction of cells with intracellular ice remain below a 5% threshold level defined the domain of permissible combinations of B and Tp. In Figure 2, showing the protocol parameter space (B, Tp), this domain is demarcated by the curve labelled 'PIF = 5%'. Within the permissible solution space, the parameter combination which minimized the cost function (Equation 5) was determined using a simplex algorithm (Nelder and Mead, 1965). Figure 2 shows contours of constant cost Q¥ = 3, 4, 5 and 15 h are marked), indicating the shape of the three-dimensional cost surface in the protocol parameter space. The optimal combination of freezing protocol parameters for mouse oocytes frozen in the presence of 1.5 M DMSO was predicted to be a dehydration cooling rate of 0.59°C/min with a plunge temperature of — 67CC, and this point is also marked ( • ) in Figure 2. Experimental testing of the theoretical optimization Cell survival was experimentally measured in the neighbourhood of this predicted optimum to verify model predictions and to establish the sensitivity of the freezing outcome to the

Oocyte cryopreservation by theoretical protocol optimization

protocol parameters cooling rate and plunge temperature. Because of limitations of the experimental system, the set of freezing parameters closest to the predicted optimum that could be reliably achieved was a cooling rate of 0.5°C/min arid a plunge temperature of -70°C. Of 30 oocytes frozen under those conditions, all were recovered after thawing and cryoprotectant removal, and 47 ± 20% were intact and morphologically normal after incubation for 1 h. After incubation for 24 h, 43 ± 17% were morphologically normal. To test the model further in the vicinity of this set of protocol parameters, Tp and B were varied independently and the resulting cell viabilities were compared with predictions of cell damage caused by intracellular ice formation. The effect of varying the plunge temperature with a constant dehydration cooling rate of 0.5°C/min is shown in Figure 3A. Cell survival (morphological survival rate 1 h after thawing; represented by a dashed line in Figure 3) was found to increase with decreasing plunge temperature, and the maximum survival rate (82% of the total number of oocytes frozen) was achieved with a plunge temperature of -80 c C. When straws were plunged before reaching -60°C, no intact cells were recovered. There was little or no variation between the morphological and membrane integrity assays for cell survival, or between evaluations at 1 and 24 h post-thaw (data not shown). Also shown in Figure 3A (represented by a solid line) are theoretical predictions for the fraction of cells containing no ice nuclei, i.e. 1 — PIF (Equation 3). The model predicted a sharp increase in the fraction of ice-free cells (and hence, presumably, survival) as the plunge temperature is lowered from —50 to —60°C, and then a 'plateau' region in which the predicted fraction of cells without intracellular ice remained above 90%, decreasing only slightly with decreasing plunge temperature; as the plunge temperature is lowered beyond —110°C, a sharp drop-off in survival was predicted because of an increase in PIF. Although the predicted transition region at high plunge temperatures was observed experimentally at a somewhat lower temperature (—70°C) than that predicted by the model (—55 C C), the overall model predictions were in good qualitative agreement with the corresponding experimental data. The effect of varying the dehydration cooling rate with the plunge temperature fixed at the experimental optimum value (Tp = -80°C) was investigated next As shown in Figure 3B, maximum survival (86% morphologically normal 1 h postthaw) was obtained with a cooling rate of 0.3°C/min, although the difference between this value and the 82% morphological survival rate at 0.5°C/min was not statistically significant (P > 0.70). At higher dehydration cooling rates, cell viability decreased; no cells survived when cooled at a rate of 2°C/ min. In addition, a reduction in the 1 h post-thaw morphological survival rate was observed when the dehydration cooling rate was decreased from 0.3 to 0.1°C/min. The predicted fraction of oocytes without intracellular ice, shown in Figure 3B, remained above 90% for cooling rates <~0.8°C/min, decreasing only slightly with increasing cooling rates. However, at a dehydration cooling rate of ~l°C/min, 50% of all oocytes were predicted to contain intracellular ice nuclei, and at ~1.3°C/ min, the model predicted that no cells were ice-free. This predicted transition corresponded well with the experi-

100-r

Recovery

Morphology Fertilization Development

Figure 4. Viability and function of mouse oocytes cryopreserved using the optimized protocol (shaded) and unfrozen control oocytes (unshaded). 'Recovery' was the fraction of frozen cells recovered after thawing and cryoprotectant dilution. 'Morphology' was the fraction of recovered cells that were morphologically normal. 'Fertilization' was the fraction of inseminated cells which cleaved at the 2-cell stage. 'Development' was the fraction of 2-ceIl embryos which developed to the blastocyst stage. Error bars are SE from four experiments, each using 20-50 oocytes. The total number of oocytes was 165 in the cryopreservation group and 91 in the control group. mentally observed drop-off in survival at high cooling rates, even though the cooling rates at which intracellular ice formation becomes significant were slightly overestimated by the theoretical model. The experimentally observed reduction in survival at 0.1°C/min was not predicted by the intracellular ice formation model, because cell injury at this cooling rate was presumably caused by solution effects and not intracellular crystallization. The above experiments to test protocols in the vicinity of the theoretically predicted optimum established that a freezing procedure with B = 0.5°C/min and Tp = -80°C (Figure 2, represented by O) yielded the optimal recovery of intact and morphologically normal cells. This protocol was then evaluated further by determining the capacity of frozen-thawed oocytes to undergo IVF and development to the blastocyst stage. Fertilization and embryonic development after optimal cryopreservation As shown in Figure 4, of 165 oocytes frozen using the optimized protocol, 160 (97%) were recovered after thawing and cryoprotectant removal; of these, 124 (78%) appeared morphologically normal after incubation for 1 h. After insemination of 110 of these oocytes, 72 (65%) fertilized and cleaved to the 2-cell stage. Following embryo culture for 5 days, 36 (50%) of the fertilized oocytes had reached the blastocyst stage. The corresponding fertilization and blastocyst formation rates for the 91 unfrozen control oocytes were 88 and 81% respectively, representing a statistically significant difference (P < 0.001) compared with the cryopreserved eggs. No parthenogenic activation was observed after a 1 day culture of the 14 morphologically normal frozen-thawed oocytes not inseminated. 1301

J.O-M.Karlsson et aL

Table I. Review of literature on the conventional cryopreservation of mouse metaphase U oocytes Reference

Year

TfCC)

Sherman and Lin Parkening et al. Tsunoda et al. Parkening and Chang Whittingham Leibo et aL Fuller and Bernard Chen Glenister et al. Ko and Threlfall Carroll el aL Trounson and Kirby Carroll et al. Schroeder et al. Hunter et al. Carroll et aL Van Blerkom and Davis

1958 1976 1976 1977 1977 1978 1984 1986 1987 1988 1989 1989 1990 1990 1991 1993 1994

-20 -75 -1% -75 -196 -196 -196 -196 -196 -196 -196 -1% -196 -196 -196 -196 -1%

Protocol duration (min) 14

154 210 154 195 129 100 58 148 140 125 146 125 148 145 121 209

•

Normal morphology*

Fertilization rate1

Blastocyst formation*

90 72 12 66 58 65 _ 80 63 57 61 5 64 84 89 95 55

46 4 38 36 _ 30 11 28 1 31 74 53 78 -

_ 17 13 54 _ 21 23 50 47 -

•Expressed as percentages of the total number of cells frozen.

Discussion This study represents a comprehensive attempt to use mechanistic models to design and optimize a cryopreservation protocol for reversibly freezing mouse oocytes to — 196CC. The technique allowed a rapid identification of the optimal protocol using a small number of experiments, a significant improvement in efficiency over the experimental optimization methods currently used. Furthermore, the theoretical optimization approach used here should be readily applicable to human oocytes or any other cell type for which the relevant biophysical parameters have been measured. Rational design of an optimal cryopreservation protocol for mouse oocytes first required that the biophysical parameters describing membrane permeability (L and E) and intracellular ice nucleation (£2S, *s> ^ v and *v) ^ known- Whereas nucleation rate coefficients for mouse oocytes have been measured previously only in the absence of cryoprotectants (Toner et al., 1990b), it was necessary to remeasure the nucleation rate parameters in the presence of DMSO. Compared with the parameter values obtained by Toner et aL (1990b) in the absence of DMSO, the thermodynamic coefficient (K) was only slightly affected by the presence of 1.5 M DMSO, while the kinetic coefficient (£2) was reduced by two orders of magnitude when the cryoprotectant was used. Nucleation rate parameters for VCN were also determined by Toner et aL (1990b) by freezing oocytes in a hypertonic (1035 mOsm) salt solution with no cryoprotectant Compared with the values measured here for oocytes in 1.5 M DMSO, the kinetic coefficient was significantly reduced (by a factor of 1024), while the thermodynamic coefficient was reduced by a factor of about two in the presence of DMSO. In addition to these effects on the nucleation rate reference coefficients, the presence of DMSO alters the viscosity and equilibrium melting point of the solution, which in turn affects the rate of intracellular ice nucleation (Karlsson et al, 1994). The overall effect of the presence of 1.5 M DMSO is a switch in the dominant nucleation mechanism from SCN (for no DMSO) to 1302

VCN (in the presence of DMSO), with a concomitant depression of the median intracellular ice formation temperature by ~27°C. Model predictions using the nucleation parameters obtained here were tested in an independent set of experiments, confirming adequate performance of the theoretical model under a variety of physico-chemical conditions. Having measured the necessary biophysical parameters and tested the mathematical model of intracellular ice formation, a simple two-step freezing protocol for mouse oocytes was theoretically optimized using a sequential simplex algorithm. Graphically, this process involved determining the minimum of three-dimensional cost surface in the protocol parameter space (Figure 2). To take into account both intracellular ice formation and solution effect injury, the parameter space was first divided into regions of lethal (>5%) and permissible (<5%) levels of intracellular ice formation, resulting in a nose-shaped demarcation. Thus, if plunge temperatures are too high, cells do not have time to dehydrate, and intracellular ice formation occurs as cells are ultra-rapidly cooled during the plunge into liquid nitrogen. On the other hand, if too low a plunge temperature is used, intracellular ice forms during the initial cooling to the plunge temperature because slow cooling rates afford sufficient time for water molecules to aggregate into stable ice nuclei (Karlsson et aL, 1994). The range of plunge temperatures for which levels of intracellular ice formation remain sublethal is a function of the dehydration cooling rate, and can be determined from Figure 2. The freezing process was optimized by determining, from the permissible combinations of cooling rate and plunge tempeTature, which protocol had the minimal duration (i.e. cost). Using this theoretical optimization as a guide for experiments, it was possible to rapidly find a combination of protocol parameters which resulted in recovering >80% morphologically normal oocytes post-thaw, with 49% of cryopreserved oocytes able to undergo FVF and cleavage to the 2-cell stage, and 24% of frozen—thawed eggs developing to the blastocyst stage. These results are comparable with the survival, fertiliza-

Oocyte cryopreservation by theoretical protocol optimization

tion and blastocyst formation rates reported for mouse oocytes in the literature (Table I). Direct comparison of the success of freezing protocols from different studies is often difficult or impossible because of confounding variables such as differing culture techniques, oocyte source and isolation techniques, freezing apparatus, pre- and post-preservation processing, and viability assays. Often, oocytes are frozen as cumulusoophorus complexes; inasmuch as it is usually not possible to establish the number of oocytes in such cumulus masses, the yield of cells (i.e. the percentage of cells frozen which were recovered) post-thaw cannot be determined, and all measured survival rates must be expressed as fractions of the number of cells recovered rather than the total number of cells frozen. Table I is a literature review of conventional freezing protocols for metaphase II oocytes, including only studies which report the total number of cells frozen, and thus permit absolute survival rates to be calculated. As can be seen in Table I, there is considerable variability in the success rates for oocyte cryopreservation. However, high rates of survival and function have been obtained recently by several groups. Schroeder et al. (1990) froze oocytes in 1 M DMSO at 0.5°C/ min, plunging at - 80°C, and recovered 84% morphologically normal oocytes, of which 88% were fertilizable; 68% of the fertilized eggs developed to the blastocyst stage. Although the freezing method used by Schroeder et al. (1990) was comparable with the optimized protocol developed here, a higher concentration of DMSO (1.5 M) was used in our study. Hunter et al. (1991) obtained 89% morphologically normal oocytes post-thaw, with 60% of these oocytes fertilizable and 89% of the resulting 2-cell embryos forming blastocysts. Hunter et al. (1991) cryopreserved their oocytes in 1.5 M DMSO, using an initial cooling rate of 0.4°C/min and plunging into liquid nitrogen at —65CC, i.e. a protocol very similar to that predicted by theoretical optimization here. Carroll et al (1993) reported morphological survival rates of 80-95%, with IVF rates as high as 82%, for oocytes frozen in 1.5 M DMSO using a three-step protocol: cells were first cooled to -40°C at a rate of 0.3°C/min, then at a rate of 10°C/min to -150°C, before being plunged into liquid nitrogen. Although it is expected that increasing the number of freezing steps will afford greater control over the cryopreservation process, thus potentially increasing cell survival (Pitt, 1992), a two-step freezing procedure was used in our study based on an initial theoretical analysis of the freezing behaviour of mouse oocytes (Karlsson et al., 1995). Also worth noting is the study by Trounson and Kirby (1989), because the DMSO concentration, dehydration cooling rate and plunge temperature used by that group were ostensibly the same as those determined to be optimal here (i.e. 1.5 M DMSO, B = 0.5°C/min, Tp = -80°C). However, in the experiments by Trounson and Kirby (1989), only 5% of all oocytes frozen were recovered intact (although when oocytes were frozen with their cumulus, 72% of the number of cells recovered were morphologically normal), and 20% of these were fertilizable. These numbers are significantly lower than the corresponding morphological survival and IVF rates obtained in our study (75 and 65% respectively), illustrating that survival rates are sensitive to factors other than the freezing protocol parameters. This conclusion is also relevant

to the work of Carroll et al. (1993), who obtained their highest survival rates by processing oocytes with polyvinyl alcohol and fetal calf serum. The discrepancy between the current results and those obtained by Trounson and Kirby (1989), as well as the generally high degree of variability in the success of freezing protocols reported in the literature (Table I), may be partly explained by the sensitivity of cell survival to the exact values of B and Tp for near-optimal freezing protocols. This is demonstrated in Figure 3A and B, in which the sharp transitions in the predicted intracellular ice formation levels and the corresponding measured survival rates indicate a high sensitivity to the protocol parameters in the neighbourhood of the optimum. Thus, slightly different protocol parameter values, or nominally identical protocols implemented on different experimental systems, may result in significantly different viability rates for cryopreserved cells. Furthermore, the high sensitivity of cell survival on the protocol parameter values underscores the need to narrow down any experimental search for optimal cryopreservation protocols to as narrow a domain as possible. Cell survival drops off rapidly in the vicinity of the optimum, and outside of the immediate neighbourhood of the optimal protocol, cell survival rates are uniformly low, making it difficult or impossible to identify improved protocols based on experimental viability data alone. Thus, even though the putative optimal protocol predicted by theoretical modelling yielded only 47% viable oocytes, the model identified the neighbourhood of the optimum in parameter space, allowing a rapid experimental determination of freezing protocols, yielding oocytes with rates of viability and function comparable with those reported in the recent literature. In summary, our work has demonstrated the use of mathematical models in the rational design of cryopreservation protocols, using mouse oocytes as a model system. We have shown that physico-chemical models can adequately predict water transport and intracellular ice formation behaviour, that the protocol duration is a reasonable phenomenological measure of slow-freezing damage, and that the method of designing cryopreservation protocols a priori using theoretical models is feasible. Model-based rational design can be significantly more efficient than conventional experimental approaches to protocol optimization, especially as freezing protocols become more complex and the number of interacting parameters to be optimized increases. While the results obtained here compare favourably with current techniques for oocyte cryopreservation, rates of IVF and blastocyst formation of frozen-thawed oocytes are still low compared with unfrozen controls, indicating that some oocytes sustain damage as a result of cryopreservation. To attain improved function after cryopreservation, the relevant mechanisms of damage must be elucidated and subsequently incorporated into the theoretical model of oocyte freezing. In particular, the greatest challenge facing the continued success of theoretical approaches to freezing protocol design is the development of effective mechanistic models of oocyte injury caused by solution effects. Acknowledgements This research was supported in part by the Whitaker Foundation, American Cancer Society, TAP Pharmaceuticals and the Shriners Hospitals for Crippled Children. 1303

J.O.M.Karlsson et at

References Bevington, P.R. (1969) Data Reduction and Error Analysis for the Physical Sciences. McGraw-Hill, Inc., New York, NY, USA. Borel Rinkes, I.H.M., Toner, M., Sheehan, SJ. et al. (1992) Long-term functional recovery of hepatocytes after cryopreservation in a threedimensional culture configuration. Cell Transplant., 1, 281-292. Carroll, J., Wames, G.M. and Matthews, C D . (1989) Increase in digyny explains polyploidy after in-vitro fertilization of frozen-thawed mouse oocytes. J. Reprod. Fertit., 85, 489-494. Carroll, J., Depypere, H. and Matthews, C D . (1990) Freeze-thaw-induced changes of the zona pellucida explain decreased rates of fertilization in frozen-thawed mouse oocytes. J. Reprod. Fertil., 90, 547-553. Carroll, J., Wood, MJ. and Whittingham, D.G. (1993) Normal fertilization and development of frozen-thawed mouse oocytes: protective action of certain macromolecules. Biol. Reprod., 48, 606-612. Carte, A.E. (1959) Probability of freezing. Proc. Phys. Soc., 73, 324. Chen, C. (1986) Pregnancy after human oocyte cryopreservation. Lancet, i, 884-886. Cosman, M.D., Toner, M., Kandel, J. and Cravalho, E.G. (1989) An integrated cryomicroscopy system. Cryo-Lett., 10, 17-38. Friedler, S., Giudice, L.C. and Lamb, EJ. (1988) Cryopreservation of embryos and ova. Fertil. Steril., 49, 743-764. Fuller, B J . and Bernard, A. (1984) Successful in vitro fertilization of mouse oocytes after cryopreservation using glycerol. Cryo-Lett., 5, 307-312. Glenister, P.H., Wood, MJ., Kirby, C. and Whittingham, D.G. (1987) Incidence of chromosome anomalies in first-cleavage mouse embryos obtained from frozen-thawed oocytes fertilized in vitro. Gamete Res., 16, 205-216. Hunter, J.E., Bernard, A., Fuller, B. et aL (1991) Fertilization and development of the human oocyte following exposure to cryoprotectants, low temperatures and cryopreservau'on: a comparison of two techniques. Hum. Reprod., 6, 1460-1465. Karlsson, J.O.M., Cravalho, E.G., Borel Rinkes, I.H.M. et al. (1993a) Nucleation and growth of ice crystals inside cultured hepatocytes during freezing in the presence of dimethyl sulfoxide. Biophys. J., 65, 2524-2536. Karlsson, J.O.M., Cravalho, E.G. and Toner, M. (1993b) Intracellular ice formation: causes and consequences. Cryo-Lett., 14, 323-336. Karlsson, J.O.M., Cravalho, E.G. and Toner, M. (1994) A model of diffusionlimited ice growth inside biological cells during freezing. J. Appl. Phys., 75, 4442^455. Karlsson, J.O.M., Eroglu, A., Toth, T.L. et al. (1995) Rational design and theoretical optimization of a cryopreservation protocol. In Advances in Heat and Mass Transfer in Biological Systems. American Society of Mechanical Engineers, New York, NY, USA, Vol. HTD-322, pp. 85-90. Ko, Y. and Threlfall, W.R. (1988) The effect of 1,2-propanediol as a cryoprotectant on the freezing of mouse oocytes. Theriogcnology, 29, 987-995. Leibo, S.P., McGrath, J J . and Cravalho, E.G. (1978) Microscopic observation of intracellular ice formation in unfertilized mouse ova as a function of cooling rate. Cryobiology, 15, 257-271. Levran, D., Dor, J., Rudak, E. et aL (1990) Pregnancy potential of human oocytes: the effect of cryopreservation. N. Eng. J. Med., 323, 1153-1156. 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. (1984) Freezing of living cells: mechanisms and implications. Am. J. PhysioL, 143, C125-C142. Medical Research International, The American Fertility Society (1992) In vitro fertilization-embryo transfer (IVF-ET) in the United States: 1990 results from the IVF-ET registry. FertiL Steril., 57, 15-24. Muldrew, K.B. (1993) The osmotic rupture hypothesis and itj application to the cryopreservation of articular cartilage. PhD thesis. Department of Laboratory Medicine and Pathology, University of Alberta, Canada. Muldrew, K. and McGann, L.E. (1994) The osmotic rupture hypothesis of intracellular freezing injury. Biophys. J., 66, 532-541. Myers, S.P., Pitt, R.E., Lynch, D.V. and Steponkus, P i - (1989) Characterization of intracellular ice formation in Drosophila melanogaster embryos. Cryobiology, 26, 472-484. Nelder, J.A. and Mead, R. (1965) A simplex method for function minimization. Comput. J., 7, 308-313. Parkening, T.A. and Chang, M.C. (1977) Effects of cooling rates and maturity of the animal on the recovery and fertilization of frozen-thawed rodent eggs. Biol. Reprod., 17, 527-531. Parkening, T.A., Tsunoda, Y. and Chang, M.C. (1976) Effects of various low temperatures, cryoprotective agents and cooling rates on the survival, fertilizability and development of frozen-ihawed mouse eggs. / . Exp. Zool., 197, 369-374. Pitt, R.E. (1992) Thermodynamics and intracellular ice formation. In

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Steponkus, P. (ed.), Advances in Low-Temperature Biology. JAI Press Ltd, London, UK, Vol. 1, pp. 63-99. Schroeder, A.C, Champlin, A.K., Mobraaten, L.E. and Eppig, J J . (1990) Developmental capacity of mouse oocytes cryopreserved before and after maturation in vitro. J. Reprod. Fertil., 89, 43-50. Sherman, J.K. and Lin, T.P. (1958) Survival of unfertilized mouse eggs during freezing and thawing. Proc. Soc. Exp. Biol. Med., 98, 902-905. Toner, M. (1993) Nucleation of ice crystals inside biological cells. In Steponkus, P. (ed.). Advances in Low-Temperature Biology. JAI Press Ltd, London, UK, Vol. 2, pp. 1-51. Toner, M., Cravalho, E.G. and Armant, D.R. (1990a) Water transport and estimated transmembrane potential during freezing of mouse oocytes. J. Membr. Biol., 115, 261-272. Toner, M., Cravalho, E.G. and Karel, M. (1990b) Thermodynamics and kinetics of intracellular ice formation during freezing of biological cells. J. Appl. Phys., 67, 1582-1592 (erratum: 1991, J. AppL Phys., 70, 4653). Toner, M., Cravalho, E.G., Karel, M. and Armant, D.R. (1991) Cryomicroscopic analysis of intracellular ice formation during freezing of mouse oocytes without cryoadditives. Cryobiology, 28, 55—71. Toner, M., Cravalho, E.G., Stachecki, J. et al. (1993) Nonequilibrium freezing of one-cell mouse embryos. Biophys. J., 64, 1908-1921. Trounson, A.O. (1986) Preservation of human eggs and embryos. Fertil. Steril., 46, 1-12. Trounson, A. and Kirby, C. (1989) Problems in the cryopreservation of unfertilized eggs by slow cooling in dimethyl sulfoxide. Fertil. Steril., 52, 778-786. Tsunoda, Y., Parkening, T.A. and Chang, M.C. (1976) In vitro fertilization of mouse and hamster eggs after freezing and thawing. Experientia, 32, 223-224. Van Blerkom, J. and Davis, P.W. (1994) Cytogenetic, cellular, and developmental consequences of cryopreservation of immature and mature mouse and human oocytes. Microsc. Res. Tech., 27, 165—193. Van Steirteghem, A.C. and Van den Abbeel, E. (1985) Survey on cryopreservation. In SeppBIS, M. and Edwards, R.G. (eds). In Vitro Fertilization and Embryo Transfer, Ann. N.Y. Acad. Sci., 442, 571-574. Whittingham, D.G. (1977) Fertilization in vitro and development to term of unfertilized mouse oocytes previously stored at -196"C. J. Reprod. Fertil., 49, 89-94. Younis, A.I., Toner, M., Albertini, D.F. and Biggers, J.D. (1996) Cryobiology of non-human primate oocytes. Hum. Reprod., 11, 156-165. Received on January 22, 1996; accepted on April 9, 1996

Appendix To theoretically optimize a cryopreservation protocol, one must define a suitable cost function which is strongly associated with the actual probability of cell damage during cryopreservation. In principle, the cost associated with freezing is simply the probability of irreversible cell injury, which can be written as the combined probability of damage caused by intracellular ice formation and solution effects: «/ = p.

J.

p

p.p

(Equation 6)

where P, is the probability of cell injury caused by intracellular ice, which to a good approximation is estimated by PIF (Toner, 1993), and P s is the probability of damage by solution effects. However, because the mechanism of solution effects damage is unknown, and no effective phenomenological models for Ps are available, it is not possible to evaluate Equation 6. Instead, we have taken the approach of minimi/ing solution effects damage (PJ subject to the constraint that damage caused by intracellular ice remains below a fixed threshold (Pj < 5%). This formulation allows the use of qualitative knowledge about the nature of solution effects damage, without requiring an explicit calculation of (/"*). Specifically, the simple strategy of minimizing solution effects injury by minimizing protocol duration was employed in our study. Two assumptions about the nature of solution effects are implicit to this approach: (i)

Oocyte cryopreservation by theoretical protocol optimization 100-

00

200

400

600

800

1000 1200 1400

Protocol Duration [rnln]

Figure 5. Relationship between measured oocyte viability (morphological survival rate, 1 h post-thaw) and duration of freezing protocol. Open symbols represent experiments in which the predicted cumulative incidence of intracellular ice formation was >5%; closed symbols represent experiments in which the predicted cumulative incidence of intracellular ice formation was

solution effects damage is a stochastic process, i.e. a random occurrence with an underlying average rate of injury, implying that the probability of damage under constant conditions is directly proportional to the time of exposure to those conditions; and (ii) the rate of injury is negbgible at the cryogenic storage temperature. In fact, inspection of Equation 4 reveals that the rate of injury is assumed to be constant for all temperatures above T{, and zero at or below Tf. Although this is not an entirely realistic assumption, the approximation is reasonable and justified considering the lack of relevant experimental data. To test whether the putative measure of solution effects injury (protocol duration) used in our study was adequate, the experimental viability data for each freezing protocol reported here have been plotted in Figure 5 as a function of the duration of the corresponding protocol. As can be seen, a strong negative association between measured cell viability and protocol duration was observed for protocols longer than ~200 min; for these protocols, < 5 % intracellular ice formation was predicted to occur (Figure 5, • ) , thus suggesting that cell injury in this regime was caused by solution effects. For shorter freezing protocols (duration <~200 min), there was a drop in survival, which is expected because of the increased likelihood of intracellular ice formation in this regime (Figure 5, O). The strong association observed between protocol duration and reduced cell viability indicates that under the freezing conditions used here, protocol duration is a reasonable predictor of solution effects injury, and thus an adequate cost function for protocol optimization.

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