- Research
- Open Access
Mathematically optimized cryoprotectant equilibration procedures for cryopreservation of human oocytes
- Allyson Fry Davidson^{1},
- James D Benson^{2} and
- Adam Z Higgins^{1}Email author
https://doi.org/10.1186/1742-4682-11-13
© Davidson et al.; licensee BioMed Central Ltd. 2014
- Received: 16 July 2013
- Accepted: 19 February 2014
- Published: 20 March 2014
Abstract
Background
Simple and effective cryopreservation of human oocytes would have an enormous impact on the financial and ethical constraints of human assisted reproduction. Recently, studies have demonstrated the potential for cryopreservation in an ice-free glassy state by equilibrating oocytes with high concentrations of cryoprotectants (CPAs) and rapidly cooling to liquid nitrogen temperatures. A major difficulty with this approach is that the high concentrations required for the avoidance of crystal formation (vitrification) also increase the risk of osmotic and toxic damage. We recently described a mathematical optimization approach for designing CPA equilibration procedures that avoid osmotic damage and minimize toxicity, and we presented optimized procedures for human oocytes involving continuous changes in solution composition.
Methods
Here we adapt and refine our previous algorithm to predict piecewise-constant changes in extracellular solution concentrations in order to make the predicted procedures easier to implement. Importantly, we investigate the effects of using alternate equilibration endpoints on predicted protocol toxicity. Finally, we compare the resulting procedures to previously described experimental methods, as well as mathematically optimized procedures involving continuous changes in solution composition.
Results
For equilibration with CPA, our algorithm predicts an optimal first step consisting of exposure to a solution containing only water and CPA. This is predicted to cause the cells to initially shrink and then swell to the maximum cell volume limit. To reach the target intracellular CPA concentration, the cells are then induced to shrink to the minimum cell volume limit by exposure to a high CPA concentration. For post-thaw equilibration to remove CPA, the optimal procedures involve exposure to CPA-free solutions that are predicted to cause swelling to the maximum volume limit. The toxicity associated with these procedures is predicted to be much less than that of conventional procedures and comparable to that of the corresponding procedures with continuous changes in solution composition.
Conclusions
The piecewise-constant procedures described in this study are experimentally facile and are predicted to be less toxic than conventional procedures for human oocyte cryopreservation. Moreover, the mathematical optimization approach described here will facilitate the design of cryopreservation procedures for other cell types.
Keywords
- Optimization
- Toxicity
- Vitrification
- Cell membrane transport
- Permeability
Introduction
Cryopreservation theoretically allows nearly indefinite storage of viable biological material [1]. Conventional cryopreservation techniques are usually thought of as slow-cooling methods (~1°C/min) that utilize relatively low (1 to 2 mol/L) concentrations of cryoprotectants (CPAs) such as glycerol, ethylene glycol, or dimethyl sulfoxide. Although these conventional techniques are sufficient for many cell types, this approach is less successful for cells that have a reduced tolerance to sub-physiologic temperatures (e.g. oocytes [2, 3]) or are easily damaged by extracellular ice formation (e.g., three dimensional tissues [4, 5]). For these sensitive cell types, an alternative cryopreservation technique widely known as vitrification may be used that preserves cells in a glassy state devoid of ice crystals.
In order to completely avoid the liquid to crystal phase transition, these vitrification techniques require combinations of very high cooling and warming rates (typically >>100°C/min) with cryopreservation solutions that contain very high concentrations of CPA (typically > 5 mol/L). In addition to avoiding damage associated with ice formation, vitrification techniques are appealing because they require much less precise cooling rates compared to conventional methods, and as such can be implemented without costly or complicated controlled rate freezing devices.
However, there is a high cost associated with these techniques: the equilibration of cells with and from high CPA concentrations (CPA addition and removal, respectively) dramatically increases the risk of damage due to osmotically driven cell volume changes and CPA induced cytotoxicity. Volumetric damage can be caused by rapid exposure to anisosmotic media, during which the differential permeability of water and CPA drives a biphasic volume response. This damage occurs when the cell either rapidly loses and then slowly regains its intracellular water in traditional CPA addition schemes, or vice versa with traditional removal schemes. These responses, if large enough, may drive the cell beyond critical volumes known as osmotic tolerance limits, outside of which irreversible cell damage occurs [6, 7]. Additionally, high CPA concentrations also increase the risk of cell damage or death due to chemical toxicity; it has been claimed that preventing toxicity is the biggest challenge in achieving successful vitrification [8].
Rational design approaches combine mathematical models and cell biophysical parameters to predict optimized CPA addition and removal procedures. Because the damage due to extending cell volumes beyond osmotic tolerance limits is relatively well understood, the most common rational design method has been to use membrane transport equations and osmotic tolerance limits to predict multi-step procedures that prevent osmotic damage [9–11]. With an argument that cytotoxicity due to CPA exposure is time-sensitive, rational design strategies have also been extended to reduce toxic damage by minimizing the duration of the CPA addition and removal procedures while still maintaining cell volumes between osmotic tolerance limits [12, 13].
While CPA cytotoxicity is time sensitive, it is also concentration sensitive [8, 14, 15]. Therefore, in order to account for this time and concentration dependence, we recently described mathematical methods that predict optimal procedures based on the minimization of a toxicity cost function, a term that describes the accrual of toxic damage [16]. However, our previous mathematical algorithm predicted procedures with continuous concentration changes for CPA and non-permeating solutes. These procedures are difficult to implement and would require specialized fluidic systems and computerized control. Moreover, because most previous rationally designed procedures used an isosmotic volume as the final state for CPA addition and removal [9, 10], our previous study used an isosmotic volume to define the target final cellular state. This final state may be less optimal than one where the cell is dehydrated to its osmotic tolerance limit at the end of CPA loading. In fact there has been discussion in the literature about the advantages of cooling in a pre-dehydrated state (see, e.g., [17]).
In the current study, we describe adaptations to our previous algorithm in order to make the predicted procedures easier to implement. The minimization of a toxicity cost function remains the basis of our algorithm. However, instead of predicting procedures with continuous concentration changes, the new algorithm predicts multi-step procedures with piecewise constant changes in the CPA and non-permeating solute concentrations. Also, rather than specifying an isotonic final cell volume, the new algorithm uses the intracellular CPA concentration to define the target final state, which allows exploration of alternate final cell volumes. We predict procedures for the addition and removal of vitrification solutions for human oocytes; a valuable, clinically relevant, and challenging to cryopreserve cell type. Our results demonstrate the potential to significantly reduce the toxicity of vitrification procedures with an experimentally and clinically facile CPA equilibration protocol.
Methods
where w is the intracellular water volume normalized to the water volume under isotonic conditions, s is the moles of intracellular CPA normalized to the moles of intracellular solute under isotonic conditions, τ is a dimensionless temporal variable, b is a dimensionless relative permeability constant, and m_{1} and m_{2} are the extracellular concentrations (in molal units) of non-permeating solute and CPA, respectively, normalized to the isotonic solute concentration (0.3 Osm/kg).
where γ is the product of the isotonic solute concentration and the partial molar volume of CPA. In the case of EG, γ = 0.0168.
where α = 1.6 is a constant describing the concentration dependence of the toxicity rate, and τ^{f} is the total duration of the procedure [16].
We used ϵ = 10^{−3} for CPA addition and ϵ = 10^{−1} for CPA removal, which was found to result in convergence near the goal state.
In order to identify optimal CPA addition and removal procedures it is first necessary to parameterize the procedural details. We assumed a constant temperature and only considered the solute concentrations m_{1}(τ) and m_{2}(τ) in the optimization scheme. In our previous study, we parameterized m_{1}(τ) and m_{2}(τ) using a piecewise linear approach [16]. The temporal domain between τ = 0 and τ^{f} was divided into 49 equally spaced segments and the concentrations m_{1} and m_{2} were assumed to vary linearly with time in each segment. This corresponds with 50 parameters for m_{1}, 50 parameters for m_{2} and one additional temporal parameter τ^{f}, resulting in a total of 101 parameters to be optimized.
One of the goals of the present study was to modify the optimization approach to yield procedures that are easier to implement experimentally. Thus we examined procedures consisting of piecewise constant concentration profiles for m_{1}(τ) and m_{2}(τ). We considered both two-step and three-step procedures—procedures with either two or three step-changes in the extracellular concentration. In the case of two-step procedures, the parameters to be optimized consist of the duration of the first step, the concentrations m_{1} and m_{2} in the first step, the duration of second step and the concentrations m_{1} and m_{2} in the second step, resulting in a total of 6 parameters. A total of 9 parameters are required for parameterization of three-step procedures. Unless otherwise noted, the concentration parameters to be optimized were bounded between a lower limit of m = 0 and an upper limit of m = 80. This corresponds with a maximal EG concentration of 60% w/w, or about 10.3 mol/L.
allowing the calculation and optimization to occur completely in the time transform space with the attendant exact solutions. The analytical solutions for w and s in terms of the variable x are provided in the Appendix.
To mathematically optimize piecewise constant procedures, the built-in constrained minimizer “fmincon” was used in MATLAB (MathWorks, Inc., Natick, MA) to implement the interior point algorithm [23–25]. This algorithm was used to minimize the value of the cost function (Eq. 4) subject to the constraints in Eq. 2, and a grid search approach was used with a wide range of initial parameter guesses to increase the potential for finding a global minimum. In practice, we found that several parameter combinations yielded nearly identical cost function values, an observation that is consistent with previous attempts to optimize piecewise constant CPA addition and removal procedures [12]. Consequently the “optimal” procedures reported here probably do not represent true global optimums, but rather procedures in the vicinity of the global optimum. Finally, to compare our new approach to non-piecewise constant controls, we solved the continuous control problem as before [16] but without the w^{f} + γs^{f} = 1 condition; i.e., we simply replaced the previous end point penalty cost function J_{ϵ} (Eq. 5) with its new expression (Eq. 6).
Results
Comparison of mathematically optimized methods for equilibration of human oocytes with 6 mol/L EG
Procedure | Concentration parameterization | Goal cell volume | Goal EG Conc. (s^{f}/w^{f}) | Toxicity (J_{α}) |
---|---|---|---|---|
Addition | Piecewise linear | Isotonic | 30 | 396 |
Addition | Piecewise linear | Not specified | 30 | 32.3 |
Addition | 2-step piecewise constant | Not specified | 30 | 49.5 |
Addition | 3-step piecewise constant | Not specified | 30 | 42.7 |
Removal | Piecewise linear | Isotonic | 0 | 38.4 |
Removal | Piecewise linear | Not specified | 0 | 12.1 |
Removal | 2-step piecewise constant | Not specified | 0 | 12.8 |
Removal | 3-step piecewise constant | Not specified | 0 | 12.4 |
Effects of parameter constraints on optimized piecewise constant procedures for equilibration of human oocytes with EG
Procedure | Step | Non-permeating solute, M_{1}(Osm/kg) | EG, M_{2}(Osm/kg) | Time, t(min) | Toxicity (J_{α}) |
---|---|---|---|---|---|
Constraints: | 0 ≤ M_{ 1 } ≤ 24 | 0 ≤ M_{ 2 } ≤ 24 | t ≥ 0 | ||
Addition | 1 | 0 | 1.4 | 20 | 130 |
2 | 0 | 2.4 | 16 | ||
3 | 0 | 24 | 0.094 | ||
Removal | 1 | 1.8 | 0 | 3.6 | 22 |
2 | 0.66 | 0 | 15 | ||
Constraints: | 0 ≤ M_{ 1 } ≤ 24 | 0 ≤ M_{ 2 } ≤ 24 | t ≥ 1 | ||
Addition | 1 | 0 | 1.3 | 19 | 240 |
2 | 0 | 2.4 | 16 | ||
3 | 0 | 17 | 1.0 | ||
Removal | 1 | 2.0 | 0 | 3.6 | 25 |
2 | 0.65 | 0 | 15 | ||
Constraints: | 0 ≤ M_{ 1 } ≤ 24 | 0 ≤ M_{ 2 } ≤ 16 | t ≥ 1 | ||
Addition | 1 | 0 | 1.4 | 20 | 250 |
2 | 0 | 2.5 | 16 | ||
3 | 1.2 | 16 | 1.0 | ||
Removal | 1 | 1.8 | 0 | 3.5 | 22 |
2 | 0.68 | 0 | 13 | ||
Constraints: | 0.05 ≤ M_{ 1 } ≤ 24 | 0 ≤ M_{ 2 } ≤ 16 | t ≥ 1 | ||
Addition* | 1 | 0.050 | 1.4 | 24 | |
2 | 0.050 | 2.4 | 20 | 280 | |
3 | 1.2 | 16 | 1.0 | ||
Removal* | 1 | 1.8 | 0 | 3.6 | 23 |
2 | 0.66 | 0 | 14 |
Table 2 shows the effects of these practical constraints on procedures for addition and removal of EG. We designed procedures using a goal concentration of 8.5 mol/L because this EG concentration is expected to allow vitrification of the sample at the cooling and warming rates that are achievable using 1/4 mL freezing straws. When the step duration was limited to a minimum of 1 min, the only essential difference was an increase in the duration of the final addition step and a corresponding increase in the predicted toxicity cost by nearly two-fold. On the other hand, constraining the EG concentration to a maximum of 8.5 mol/L (i.e., m_{2} = 53.7) had very little effect on the toxicity cost. The main difference is that the resulting procedure calls for a non-zero concentration of non-permeating solute in the final addition step. Limiting the non-permeating solute concentration to a minimum of 0.05 Osm/kg resulted in longer equilibration times in steps one and two, and a corresponding modest (< 15%) increase in the toxicity cost. All of the parameter constraints considered in Table 2 resulted in nearly identical procedures for EG removal.
Discussion
CPA induced cytotoxicity has been identified as a principal impediment to achieving successful vitrification [8]. However, the conventional approach for rational design of CPA equilibration procedures focuses only on avoidance of osmotic damage and does not consider mitigation of toxicity [9–11]. To address this deficiency, rational design approaches have recently been developed for minimizing protocol duration [12, 13]; while these approaches would be expected to reduce toxicity compared with conventional methods, they do not account for the concentration dependence of toxicity. In our previous study [16] we described a new strategy for designing minimally-toxic CPA equilibration procedures using a concentration-dependent toxicity cost function. The resulting procedures are predicted to be less toxic than conventional methods for CPA equilibration as well as procedures with minimized duration. In this study we address two drawbacks of our previously reported mathematical optimization approach [16]. Our previous study relied on the concatenation of many linear changes in CPA and non-permeating solute concentrations which are difficult to achieve experimentally. Therefore, the primary objective of this study was to develop a method for designing multi-step CPA addition and removal procedures that are similar to conventional procedures with abrupt changes in CPA and non-permeating solute concentrations [7]. In addition, our previously reported optimization algorithm required cells to reach an isotonic final volume, potentially a suboptimal equilibration endpoint. Thus, an additional objective of this study was to evaluate alternate equilibration endpoints.
The two-step and three-step CPA equilibration procedures described in this study would be much easier to implement experimentally than the procedures described in our previous study [16]. Moreover, it is simpler and faster to predict optimal two-step and three-step procedures because there are fewer parameters to optimize and because an analytical solution to the membrane transport model is available for piecewise constant changes in solution composition [21]. However, it is important to evaluate the potential increase in toxicity associated with restricting the optimization to two-step and three-step piecewise constant concentration changes. Compared with the corresponding piecewise linear EG addition procedure, the two-step and three-step procedures had toxicity costs that were 50% and 30% higher, respectively (Table 1). Thus, it may be worthwhile to use continuous changes in concentration during CPA addition. However, CPA removal using two-step and three-step procedures is predicted to yield a toxicity cost that is nearly identical to that obtained using the corresponding piecewise linear CPA removal procedure, which indicates that the increased complexity of the piecewise linear procedure would probably not be worth the effort in this case. To fully evaluate the tradeoffs between experimental expediency and toxicity, it will be necessary to more precisely define the relationship between oocyte viability and the predicted toxicity cost.
The goal state defined in our previous study required that cells achieved an isotonic volume at the end of CPA addition. However, it is a common strategy to intentionally induce shrinkage in the final CPA addition step and to vitrify the sample while the cells are in the shrunken state [17, 26–28]. For instance, multi-step vitrification procedures for oocytes commonly involve loading of CPA at relatively low concentrations followed by exposure to the final vitrification solution for a brief period of time directly before cooling [29–31]. In other words, with these procedures, the cooling process is initiated while the cells are in the shrunken state. The rationale behind this strategy is that water loss concentrates intracellular solutes, allowing a vitrifiable cytoplasm composition to be reached with a shorter exposure to the final vitrification solution [17]. Another advantage of vitrification in the shrunken state is that it facilitates removal of intracellular CPA after warming [17]. This is because the cell contains less total CPA in the shrunken state, and also has more capacity for swelling during the first removal step. The mathematically optimized procedures we describe in this study are consistent with this vitrification strategy in that the final CPA addition step comprises exposure to a concentrated CPA solution, which induces shrinkage to the minimum tolerable volume. Thus, our results provide a theoretical basis for the common practice of exposing cells to the final vitrification solution for a short time, and then initiating cooling while the cells are in the shrunken state.
The most unique aspect of our optimized procedures is that cells are loaded with CPA by inducing swelling to the maximum volume limit using a solution lacking non-permeating solutes (e.g., salts). In comparison, typical CPA loading solutions contain an isotonic concentration of non-permeating solutes and consequently do not induce swelling. Swelling is advantageous because it allows a given amount of CPA to be loaded into the cells using a relatively low CPA concentration. This is because the amount of intracellular CPA is equal to the product of the intracellular concentration and the cell volume. To our knowledge, loading CPA intracellularly while forcing cells to be in a swollen state is a novel result of our toxicity minimization strategy. While this approach is promising, it may be damaging to expose oocytes to solutions lacking salts because of potential perturbations in ion homeostasis. Studies with red blood cells show that complete lack of salts in the extracellular medium causes the cell membrane to become leaky, resulting in substantial loss of intracellular ions over a period of hours [32, 33]. However, the presence of even a small amount of salt in the extracellular medium dramatically slows the rate of ion leakage [32, 33]. This suggests that it may be possible avoid problems with ion leakage by including some minimal concentration of salts in the CPA loading solution. Recently, Karlsson and colleagues showed that mouse oocytes are not damaged by exposure to a CPA solution containing only 0.05 Osm/kg salts [34]. Therefore, we also optimized a CPA loading procedure using 0.05 Osm/kg as a minimum constraint on the non-permeating solute concentration (Table 2 and Figure 5). The resulting procedure still takes advantage of swelling, and hence would be expected to be much less toxic than conventional CPA loading methods, and is only marginally more toxic than our optimized protocols without the minimal salt constraint (Table 2).
Many studies are available describing the vitrification of human oocytes. In particular, the study by Kuwayama et al. [29] resulted in 7 healthy babies and 3 ongoing pregnancies at the time of publication. This study was also the first to use the Cryotop cooling device, a minimal volume device that offers an alternative to vitrification in freezing straws by taking advantage of the higher cooling and warming rates achieved with smaller sample volumes. The procedure for EG loading described by Kuwayama et al. results in a calculated toxicity of J_{α} = 60.6. Using the same goal state (an EG concentration of 5 mol/L, or s^{f}/w^{f} = 23), our toxicity-minimization strategy predicts a procedure with a twofold lower toxicity of J_{α} = 29.3. It is important to note that the procedure described by Kuwayama et al. is predicted to yield oocyte volumes that exceed the osmotic tolerance limits that we used for designing our mathematically optimized method. Since the procedure reported by Kuwayama et al. has been successful, this may indicate that the osmotic tolerance limits that we used in this study were too restrictive. Broadening the osmotic tolerance limits would be expected to lead to even further reductions in the toxicity cost or increases in maximally achievable CPA concentration at the same cost.
While successful, the disadvantage of the Cryotop method employed by Kuwayama et al. [29] is that it is a potentially nonsterile system, where cells are directly exposed to liquid nitrogen. This open system is a requirement due to the ultrahigh cooling rates needed to avoid crystallization at such low CPA concentrations. However, if we assume that the calculated toxicity from their protocol, J_{α} = 60.6, is acceptable, then we can use our optimization approach to determine the maximal EG concentration that would result in the same level of toxicity. In this case we would be able to achieve a much increased goal concentration of approximately 6.6 mol/L (s^{f}/w^{f} = 35) using two-step or three-step toxicity minimized procedures. This approach is useful because with higher goal concentrations, it is possible to achieve vitrification using less extreme cooling and warming rates. Thus, the ability to reach higher goal concentrations without significant cytotoxicity would enable the use of other devices that offer more sterility but have a greater thermal mass, such as freezing straws, and would offer considerably more margin for error in cooling and warming rates under the present Cryotop protocol.
Therefore, instead of minimizing toxicity under current cooling regimes such as the Cryotop method, we may use our optimization approach to calculate the anticipated added cost of achieving a concentration that would facilitate vitrification under more sterile conditions. In particular, Baudot and Odagescu [35] determined that a 50% w/w EG solution required a cooling rate of 11°C/min to achieve vitrification and a warming rate of 853°C/min to prevent devitrification. Cooling rates up to 2000°C/min can be achieved by directly immersing 1/4 mL freezing straws into liquid nitrogen, and warming rates up to 3000°C/min can be achieved by immersing straws into a 25°C water bath [36]. Thus, 50% w/w EG should conservatively enable vitrification at the cooling and warming rates achievable using freezing straws. An EG concentration of 50% w/w corresponds with a goal state of s^{f}/w^{f} = 53.7. Using our toxicity-minimized procedures, achieving a goal state of s^{f}/w^{f} = 53.7 would result in a toxicity of J_{α} = 130 (Table 2). This is larger than the predicted toxicity cost associated with the procedure reported by Kuwayama et al. [29] which has been proven successful. However, greater toxic damage may be an acceptable tradeoff for increased sterility and improved stability of the glassy state during storage. The clinical application of this approach will require a more precise understanding of the cost function J_{α}, and the determination of acceptable values of this cost in the context of reproductive medicine.
Our results show that to minimize toxicity during CPA addition, the final step should induce shrinkage to the minimum volume limit and last only long enough for this minimum volume to be achieved. For instance, the two-step and three-step CPA addition procedures shown in Figure 1 had final steps with durations of about 5 seconds. However, it may not be practical to perform a 5 second equilibration with sufficient accuracy and repeatability for clinical application. A distinct advantage of our approach is that it allows the determination of optimal protocols even after the addition of practical design constraints to the problem. Most previously reported CPA equilibration procedures for vitrification of human oocytes involve exposure to the final vitrification solution for at least 30 seconds [29–31]. Thus, we assumed that a one-minute final step would be feasible and determined optimal two-step and three-step procedures with this constraint (see Table 2). Interestingly, when such procedures were designed using a maximum concentration constraint equal to the goal concentration, the final addition step called for the presence of non-permeating solute at a concentration of approximately 1 Osm/kg. This is consistent with the common practice of including 0.5-1 mol/L sucrose in the final vitrification solution for human oocytes [29–31]. The presence of non-permeating solute in the final vitrification solution is potentially advantageous because it results in equilibration of the cells in a shrunken state. For example, exposure to the final solution compositions shown in Table 2 is predicted to cause rapid shrinkage and subsequent equilibration at the minimum volume limit in less than 20 seconds, as shown in Figure 5. These procedures would be expected to be relatively robust to variations in the exposure time in the final CPA addition step, since equilibrium is achieved quickly.
The optimized procedures for EG removal presented here call for exposure to solutions containing non-permeating solutes, but lacking EG. However, some residual EG would be present in practice, regardless of the method for changing the extracellular composition. To examine the potential effects of residual EG, minimum constraints can be imposed on the EG concentration during each removal step. If the EG concentration is constrained to a 20-fold dilution in each step, the toxicity cost associated with the resulting procedure is about 40% higher than that obtained when the EG concentration is zero in each step. A 100-fold dilution in each step is only associated with a 6% increase in toxicity cost. Overall, these increases in toxicity would not be expected to substantially effect of the outcome of the cryopreservation process, since EG removal is still be predicted to be much less toxic than EG addition.
Although we used our optimization algorithm for human oocytes in this study, our approach is applicable to any cell type given the necessary biophysical parameters (i.e., the membrane permeability values and osmotic tolerance limits). Moreover, our general approach of minimizing a toxicity cost function provides a framework for optimizing other important aspects of the CPA equilibration process. For instance, it is generally recognized that CPA toxicity is reduced at lower temperatures (e.g., 4°C), but CPA loading also takes longer at low temperatures because the cell membrane permeability is lower. Thus, selection of the optimal temperature for CPA loading is not trivial and arguments have been presented for CPA equilibration at both low temperatures [27] and high temperatures [37]. Our optimization approach also provides a framework for rational comparison of different CPA types in terms of their toxicity. To extend our approach to optimization of factors such as temperature and CPA type will require an improved understanding of the effect of these factors on the rate of damage due to toxicity, and formulation of a toxicity cost function that accounts for these factors.
Another advantage of our approach is that the toxicity cost function provides a quantitative indicator of cell damage after cryopreservation, facilitating rational evaluation of feasibility. If the expected cost under the optimal protocol is unacceptable, exceeding a limit that indicates a significant level of damage, a completely new approach must be tried that mitigates this cost. For example, it may be possible to reduce toxicity by using a different combination of CPAs or by carrying out the procedure at a different temperature. Importantly, the model results can be used to direct the research focus to the source of damage. This aspect is unique to our approach and has the potential to save time by identifying non-feasible approaches without the need for fruitless experiments. To realize these benefits, it will be necessary to clarify the factors affecting the toxicity cost function, as well as the relationship between the cost function and cell viability for the cell type of interest.
Conclusions
In this study we have presented an adaptation of our toxicity-minimization strategy for predicting CPA addition and removal procedures. In particular, we have modified our previous strategy which relied on continuous concentration changes and instead predict procedures based on piecewise constant concentration changes. These new procedures are not only similar to conventional procedures but are also much simpler to implement experimentally. The mathematical algorithm is based on the minimization of a toxicity cost function, which describes the effect of CPA concentration on cytotoxicity. Although these procedures still require experimental validation, we have provided theoretical evidence suggesting that our procedures would reduce toxic damage relative to procedures that are currently in use. The employment of this cost function allows for rational comparison of potential experimental designs and facilitates the generation of cell damage hypotheses in the context of cryopreservation protocols. Finally, our strategy also provides a structure for incorporating other factors into the model-based design of toxicity-minimized vitrification procedures, including the effects of temperature on CPA toxicity.
Appendix
This analytical solution was used in our optimization algorithm to predict changes in cell volume and intracellular CPA concentration, thus allowing evaluation of the cell volume constraints as well as the toxicity cost function. We restricted m_{1} > 10^{− 4}.
Declarations
Acknowledgements
This work was supported by a National Science Foundation grant (#1150861) to Adam Higgins.
Authors’ Affiliations
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