Differential Evolution for the Optimization of DMSO-Free Cryoprotectants: Influence of Control Parameters

Abstract

This study presents the influence of control parameters including population (NP) size, mutation factor (F), crossover (Cr), and four types of differential evolution (DE) algorithms including random, best, local-to-best, and local-to-best with self-adaptive (SA) modification for the purpose of optimizing the compositions of dimethylsufloxide (DMSO)-free cryoprotectants. Post-thaw recovery of Jurkat cells cryopreserved with two DMSO-free cryoprotectants at a cooling rate of 1 °C/min displayed a nonlinear, four-dimensional structure with multiple saddle nodes, which was a suitable training model to tune the control parameters and select the most appropriate type of differential evolution algorithm. Self-adaptive modification presented better performance in terms of optimization accuracy and sensitivity of mutation factor and crossover among the four different types of algorithms tested. Specifically, the classical type of differential evolution algorithm exhibited a wide acceptance to mutation factor and crossover. The optimization performance is more sensitive to mutation than crossover and the optimization accuracy is proportional to the population size. Increasing population size also reduces the sensitivity of the algorithm to the value of the mutation factor and crossover. The analysis of optimization accuracy and convergence speed suggests larger population size with F > 0.7 and Cr > 0.3 are well suited for use with cryopreservation optimization purposes. The tuned differential evolution algorithm is validated through finding global maximums of other two DMSO-free cryoprotectant formulation datasets. The results of these studies can be used to help more efficiently determine the optimal composition of multicomponent DMSO-free cryoprotectants in the future.

Introduction

In many biomedical applications, cells are collected in one location for use in a different location at a later time. Without appropriate storage conditions during transport, these cells degrade or die, greatly limiting their utility. Cryopreservation is a commonly used technique to reduce this degradation. Cryopreservation of cells is a critical supporting technology for a variety of fields including cell therapy, cell banking, and biotechnology [1–6].

Unfortunately, there is no universally optimal set of cryopreservation conditions, and thus conditions need to be reoptimized for each new cell type. A variety of factors influence the post-thaw recovery of a cell after cryopreservation: the composition of the freezing medium, the method of introduction and removal of the medium, the freezing and thawing rates and storage conditions [7]. Protocols are typically developed using empirical approaches: the composition, cooling rate, or other parameters are varied and the resulting post-thaw recovery measured. This approach is costly and time consuming [8,9].

An alternative to this inefficient empirical approach is to use an optimization scheme designed to significantly reduce the number of experiments necessary to arrive at the optimized cryopreservation protocol. Among optimization methods, differential evolution (DE) algorithm is a simple and powerful technique for multidimensional and global optimization [10]. Advantages of DE algorithm includes (1) simple and straightforward implementation in comparison to other optimization algorithms; (2) a small numbers of control parameters including mutation (F), crossover (Cr), and population (NP) in classical DE; and (3) low-space complexity of DE is low compared to most alternative real parameter optimization methods [11]. In addition, DE is also applicable to optimize discontinuous space [12,13], which is helpful in extending DE for handling large-scale and expensive optimization problems. Due to these advantages, DE algorithm and its variants have been applied in many fields such as bioinformatics, chemical engineering, molecular configuration, and urban energy management [11,14]. Recently, DE has been adopted in optimizing culture media formulations [15]. We have developed and applied DE to the development of dimethylsufloxide (DMSO)-free cryopreservation protocols in our lab [16], and we demonstrated the ability to optimize both solution composition and cooling rate using a relatively small number of experiments [17].

Robust application of the DE algorithm requires selection of the proper control parameters for the algorithm. Improper selection of control parameters may reduce the optimization efficiency or even result in stagnation. Reducing the number of experiments required to optimize the preservation of a given biological system is critical for applications in which the material is rare or hard to acquire (gametes from endangered species) or expensive (cell therapies are valued at hundreds of thousands of dollars). In this work, we use post-thaw recovery for a range of solution compositions spanning the parameter space to rationally select the control parameters settings for the DE algorithm to arrive at optimal cryopreservation conditions most efficiently. We then validate the performance of the optimized DE algorithm through optimizing other DMSO-free cryoprotectant conditions using other datasets. The knowledge obtained from this study is critical to expanding the use of this approach in future studies designed to optimize cryopreservation protocols to maximize high post-thaw recovery.

Methods

Post-thaw recovery as a function of solution composition was obtained from previous studies [18,19]. Jurkat cells (ATCC TIB-152) were cryopreserved in candidate cryoprotectant solutions in 96-well plates (Corning, NY) using a cooling rate of 1 °C/min and later thawed in a 37 °C water bath. Calcein acetoxymethyl (Calcein-AM, Life Technologies, Carlsbad, CA) and propidium iodine (PI, Life Technologies, Carlsbad, CA) were used to measure the numbers of live and dead cells in each well post-thaw. The post-thaw recovery was defined as the ratio of the number of live cells post-thaw to the number of live cells prefreeze.

Differential Evolution Algorithm

The DE algorithm was developed by Storn and Price [10] and utilizes stochastic direct search and independent perturbation of population vectors to identify a global maximum from within the parameter space. An initial random vector consisting of a given NP across the parameter space is selected $x_{i,G},\hspace{0.17em}i=1,\hspace{0.17em}2,\hspace{0.17em}3\dots \mathrm{NP}$ where $G$ denotes the generation. For the purpose of this investigation, the vector is composed of a given number of solution compositions and the number of candidate solution compositions tested in each generation remains constant during the optimization process. Cells are resuspended in a candidate solution of interest, frozen, and thawed and post-thaw recovery is measured. The DE algorithm utilizes the post-thaw recovery associated with a given population vector/generation to predict solutions that may result in more favorable live cell recovery. In classical DE algorithm, there are three factors: mutation (F), crossover (Cr), and population size (NP) that affect the manner by which a next-generation vector is chosen. These factors must be properly selected for proper function of the optimization process [20], and different applications of DE usually require different combinations of control parameters [21]. The general convention used in the DE community is DE/x/y/z, where DE means “differential evolution,” x stands for the base vector to be perturbed, y is the number of difference vectors considered for perturbation of x, and z is the type of crossover being used.

Mutation.

The mutation process involves a target vector from the current generation ( $x_{i,G}$), a donor vector ( $v_{i,G+1}$) which is a mutated vector and a trial vector ( $u_{i,G+1}$) formed from the combination of the target and donor vectors. Three common mutation types are used including “random,” “best,” and “local-to-best”. For each type, a donor vector $v_{i,G+1}$ is generated with different target vector as below.

DE/random/1/bin

$$v_{i,G+1}=x_{r1,G}+F\left(x_{r2,G}-x_{r3,G}\right)$$ (1)

DE/best/1/bin

$$v_{i,G+1}=x_{\mathrm{best},G}+F\left(x_{r2,G}-x_{r3,G}\right)$$ (2)

DE/local-to-best/1/bin

$$v_{i,G+1}=x_{i,G}+F\left(x_{\mathrm{best},G}-x_{i,G}\right)+F\left(x_{r2,G}-x_{r3,G}\right)$$ (3)

where $r1,\hspace{0.17em}r2,\hspace{0.17em}r3\hspace{0.17em}\in \left[1,\hspace{0.17em}\mathrm{NP}\right]$ are randomly chosen indices, $x_{\mathrm{best},G}$ is the vector with the highest performance in Gth generation, $F$ is a mutation factor ( $F\in \left[0,\hspace{0.17em}1\right]$) that controls the amplification of vector difference.

Crossover.

Crossover is used to enhance the diversity of the population after generating the donor vector through mutation. The donor vector exchanges its component with the target vector under crossover to form the trial vector. The crossover is performed on each of the D variables whenever a randomly generated number between 0 and 1 is less than or equal to the crossover value. The scheme is outlined as the following equation:

$$u_{ji,G+1}=\{\begin{array}{c}{v}_{ji,G+1}\\ x_{ji,G}\end{array}\hspace{0.17em}\begin{array}{c}\text{if}\hspace{0.17em}\hspace{0.17em}\text{rand}\left(j\right)\le \text{Cr}\\ \text{otherwise}\end{array}$$ (4)

For $j=1,\hspace{0.17em}2,\dots ,D$, $\mathrm{rand}\left(j\right)\in \left[0,\hspace{0.17em}1\right]$ is the jth evaluation of a uniform random number. $\mathrm{Cr}$ is the crossover constant $\mathrm{Cr}\in \left[0,\hspace{0.17em}1\right]$.

Selection.

Selection is used to determine whether the target or the trial vector survives to the next generation. A selection scheme is used as the following equation:

$$x_{i,G+1}=\{\begin{array}{c}{u}_{i,G}\\ x_{i,G}\end{array}\hspace{0.17em}\begin{array}{c}\text{if}\hspace{0.17em}f\left(u_{i,G+1}\right)>f\left(x_{i,G}\right)\hspace{0.17em}\text{for}\hspace{0.17em}\text{maximization}\\ \text{otherwise}\end{array}$$ (5)

If and only if the trial vector $u_{i,G+1}$ yields a better result than $x_{i,G}$, then $x_{i,G+1}$ is set to $u_{i,G}$; otherwise, the old value is reused. According to Storn and Price [10], the efficiency of DE is sensitive to these controlled parameters. The common choices are: $F\in \left[0.5,\hspace{0.17em}1\right]$, $\mathrm{Cr}\in \left[0.8,\hspace{0.17em}1\right]$, and $\mathrm{NP}=10D$.

Self-Adaptive Differential Evolution.

Trial-and-error method used for tuning parameters requires several runs. According to Brest et al. [21], self-adaptive (SA) DE was proposed to adjust both Cr and mutation factor (F) in each generation for eliminating the manual tuning of control parameters. The self-adaptive strategy can be described using the following equations:

$$F_{i,G+1}=\{\begin{array}{c}{F}_l+{\text{rand}}_1\cdot F_u\\ F_{i,G}\end{array}\hspace{0.17em}\begin{array}{c}{\text{if}\hspace{0.17em}\hspace{0.17em}\text{rand}}_2<{\tau }_1\\ \text{otherwise}\end{array}$$ (6)

$${\text{Cr}}_{i,G+1}=\{\begin{array}{c}{\text{rand}}_3\\ {\text{Cr}}_{i,G}\end{array}\hspace{0.17em}\begin{array}{c}{\text{if}\hspace{0.17em}\hspace{0.17em}\text{rand}}_4<{\tau }_2\\ \text{otherwise}\end{array}$$ (7)

where ${\mathrm{rand}}_j,\hspace{0.17em}j\in \left\{1,\hspace{0.17em}2,\hspace{0.17em}3,\hspace{0.17em}4\right\}$, are uniformly distributed random values between 0 and 1; ${\tau }_1$ and ${\tau }_2$ are constants, 0.1, which represent the probabilities that update both mutation and crossover; $F_l$ and $F_u$ are constants values, 0.1 and 0.9, respectively. The new F and new Cr take a value from $\left[0.5,\hspace{0.17em}1\right]$ and $\left[0,\hspace{0.17em}1\right]$, respectively.

The differential evolution algorithm and its variants were coded with Python 3.4.0 in this work. Each cryoprotectant formulation was taken as a vector in the software. Concentration level was used instead of absolute concentration for convenience [18,19]. The code searched the corresponding post-thaw recovery automatically and suggested the composition of cryoprotectant to test in the following iteration.

Evaluation Criteria.

Four experimental datasets were used to find optimal control parameters and most effective type base for the two indices to evaluate the performance of each type: (1) accuracy, which is the ratio of the post-thaw recovery of converged solution and optimal post-thaw recovery, and (2) convergence speed, which is defined as the generations to convergence without a change of accuracy. Convergence can be measured by observing an increase in cumulative best formulation, a decrease in the number of improved solutions within the emergent population after each generation, or by the average of each generation. For the purpose of this investigation, the convergence was defined when the same best member was observed in three consecutive generations and a decreasing number of improved solutions is also observed.

The accuracy was defined as the following equation:

$$\text{Accuracy}=\frac{\text{localized}\hspace{0.17em}\hspace{0.17em}\text{optimal}\hspace{0.17em}\hspace{0.17em}\text{recovery}}{\text{real}\hspace{0.17em}\hspace{0.17em}\text{optimal}\hspace{0.17em}\hspace{0.17em}\text{recovery}}\times 100\%$$ (8)

Each experiment was repeated for 50 runs.

Results

Post-Thaw Recovery for Multicomponent Solutions.

Previous studies have established that multicomponent osmolyte solutions are effective in cryopreserving cells [17–19,22]. The post-thaw recovery of Jurkat cells cryopreserved with two combinations of osmolytes, sucrose–glycerol–creatine (SGC), and sucrose–glycerol–isoleucine (SGI), were screened over the entire parameter space [18,19] as shown in Figs. 1 and 2, respectively. The concentration space of each component was discretized to six levels with equal scale, with the highest level being determined by either toxicity limits or solubility limits. A total of 216 formulations were tested across the parameter space for one DMSO-free cryoprotectant. As described previously [18,19], the SGC optimal formulation composition was 438  mM sucrose, 10% glycerol, and 10 mM creation with 80% post-thaw recovery (Fig. 1(d)). The SGI optimal formulation composition was 146 mM sucrose, 10% glycerol, and 43 mM isoleucine with 84% post-thaw recovery (Fig. 2(b)).

Fig. 1.

Post-thaw recoveries of Jurkat cells cryopreserved in varying concentrations of glycerol and creatine for a given concentration of sucrose (SGC) at a cooling rate of 1 °C/min for sucrose concentration (a) 0 mM, (b) 146 mM, (c) 292 mM, (d) 438 mM, (e) 584 mM, and (f) 730 mM

Fig. 2.

Post-thaw recoveries of Jurkat cells cryopreserved in varying concentrations of glycerol and isoleucine for a given concentration of sucrose (SGI) at a cooling rate of 1 °C/min for sucrose concentration (a) 0 mM, (b) 146 mM, (c) 292 mM, (d) 438 mM, (e) 584 mM, and (f) 730 mM

These studies determined that there is a narrow range of optimum composition for the three components (sugar, sugar alcohol, and amino acid) associated with maximum post-thaw recovery. This type of behavior provides an excellent opportunity to test DE algorithm and tune the selection of the mutation (F), crossover (Cr), and population size (NP) for DE optimization of cryopreservation protocols.

The 3D plots of post-thaw recovery for different combinations of osmolytes were presented as a function of composition and is shown in Figs. 1 and 2. The topology of SGI was complex and nonlinear with a sharp peak around the optimal formulation. The topology of SGC was smoother than SGI with broader regions of high post-thaw recovery. Statistical analysis of the data suggests that the overall post-thaw recovery was proportional to the concentration of glycerol for both SGC and SGI [19]. Post-thaw recovery increased then decreased as sugar concentration increased for both SGC and SGI. These experimental data were utilized to test different types of DE algorithm and to adjust the control parameters in order to achieve optimal accuracy and convergence speed.

Mutation and Crossover.

The next phase of the investigation involved using the experimental datasets described above to determine the influence of mutation factor (F) and crossover (Cr) on different DE types. Four different DE types including random, best, local-to-best, and local-to-best with SA were examined as well in order to investigate the influence of mutation strategies.

The accuracy maps of all combinations of mutation and crossover with 0.1 increment under same population size for all experimental data are shown in Figs. 3 and 4. The threshold of accuracy was set to 95% in order to filter combinations of mutation and crossover as gray area. These figures indicated that SA was tolerant to combinations of mutation and crossover for SGC but were sensitive for SGI. DE/best/1/bin showed the smallest acceptable range of mutation and crossover for SGC. The DE algorithm operated more efficiently for larger values of crossover but accepted a narrow range of mutation factors. For all DE types, the selection of mutation factor and crossover was critical for SGI. It is noteworthy that DE/rand/1/bin, the classical DE, was tolerant to mutation factor and crossover for SGI in comparison other DE types (Fig. 4).

Fig. 3.

Accuracy of post-thaw recovery for (a) DE/rand/1/bin, (b) DE/best/1/bin, (c) DE/local-to-best/1/bin, and (d) DE/1/local-to-best/1/bin with self-adaption for the SGC dataset with NP = 9 and 0.1 increment of mutation and crossover. Gray area represents the region with accuracy higher than 95%.

Fig. 4.

Accuracy of post-thaw recovery for (a) DE/rand/1/bin, (b) DE/best/1/bin, (c) DE/local-to-best/1/bin, and (d) DE/1/local-to-best/1/bin with self-adaption for the SGI dataset with NP = 9 and 0.1 increment of mutation and crossover. Gray area represents the region with accuracy higher than 95%.

Population Size.

The accuracy was proportional to the population size for all experimental datasets analyzed as shown in Fig. 5. The accuracy for the SGC dataset was proportional to population size and was below 95% for NP < 17 for F = 0.5 and Cr = 0.9 as shown in Fig. 5(a) but the accuracies were all above 95% for all NP and no significant differences, whereas NP was larger than 13 for F = 0.9 and Cr = 0.5 as shown in Fig. 5(b).

Fig. 5.

The accuracy of post-thaw recovery for DE/best/1/bin with NP = 9, 13, 17, 21 and 25 for the SGC dataset of (a) F = 0.5, Cr = 0.9, (b) F = 0.9, Cr = 0.5, and for the SGI dataset of (c) F = 0.5, Cr = 0.9, and (d) F = 0.9, Cr = 0.5

For the SGI dataset, the accuracies were proportional to population size and all below 90% independent of population size using F = 0.5 and Cr = 0.9 as shown in Fig. 5(c). The accuracy achieved over 95% while F = 0.9, Cr = 0.5, and NP > 17 as shown in Fig. 5(d). The searching converged after five generations for F = 0.5 and Cr = 0.9, which exhibited stagnation as shown in Figs. 5(a) and 5(c). In other words, the searching process became stuck at the local optimum. Adjusting the mutation and crossover to F = 0.9 and Cr = 0.5 resulted in improved accuracy for both SGI and SGC datasets for the same population size. The accuracy map for four DE types for both SGC and SGI datasets showed that the acceptable combinations of mutation factor and crossover were proportional to the population. When applied to datasets with a smooth topology, F > 0.7 and the full range of Cr tested achieved >95% accuracy. For datasets with a sharp topology such as SGI, F > 0.7 and Cr > 0.3 to DE/rand/1/bin and F > 0.7 and Cr > 0.3 to others achieved >95%.

Validation.

Based on the studies described above, the best control parameters are F > 0.7 and Cr > 0.3 in order to optimize the compositions of DMSO-free cryoprotectants using multiple osmolytes. The next phase of the investigation involved evaluating the universality and consistency of these control parameters using two new DMSO-free cryoprotectant formulations. Post-thaw recovery for two candidate solutions (trehalose–glycerol–creatine (TGC) and trehalose–glycerol–isoleucine (TGI)) was optimized using the DE algorithm and the control parameters described above. The outcomes of this optimization are shown in Figs. 6 and 7, respectively. The solution composition associated with the highest post-thaw recovery emerged in a relatively small number of generations. The optimum solution formulation associated with optimal post-thaw recovery converged after generation 6 (seven freezing experiments). Figures 6(a) and 6(d) and 7(a) and 7(d) demonstrate that the optimal cryopreservation solution composition occurred at either generation 2 or 3 and persisted until the convergence criteria was fulfilled. Figures 6(b) and 6(e) and 7(b) and 7(e) show the recovery associated with the best member solution increased and plateaued as the algorithm converged. Figures 6(c) and 6(f) and 7(c) and 7(f) present the number of improved solutions in each generation, which decreased as the algorithm converged. The population size influenced the rate at which the number of improved solutions declined. For NP = 18, the number of improved solutions per generation rapidly declined for generations > 1. For NP = 9, the number of improved solutions declined after generations > 3. The TGC optimal formulation composition of 61 mM trehalose, 10% glycerol, and 7 mM creatine with 83% post-thaw recovery. The TGI optimal formulation composition was 61 mM trehalose, 10% glycerol, and 43 mM isoleucine with 84% post-thaw recovery. Experiments spanning the entire parameter space confirmed that the best member identified in the algorithm represented the overall global maximum. It is noteworthy that the DE algorithm determined the optimum using 60 formulations and spanning the parameter space requires at least 216 formulations.

Fig. 6.

TGC optimized DE/local-to-best/1/bin with self-adaptive using initial F = 0.9, Cr = 0.5 for Jurkat cells. For NP = 9, the best formulations are presented in black (a) post-thaw recoveries of all formulations in every generation. (b) Post-thaw recovery of the best member per generation. (c) Number of improved formulations per generation. For NP = 18, (d) post-thaw recoveries of all formulations in every generation. (e) Post-thaw recovery of the best member per generation. (f) Number of improved formulations per generation.

Fig. 7.

TGI-optimized DE/local-to-best/1/bin with self-adaptive using initial F = 0.9, Cr = 0.5 for Jurkat cells. For NP = 9, the best formulations are presented in black (a) Post-thaw recoveries of all formulations in every generation. (b) Post-thaw recovery of the best member per generation. (c) Number of improved formulations per generation. For NP = 18, (d) post-thaw recoveries of all formulations in every generation. (e) Post-thaw recovery of the best member per generation. (f) Number of improved formulations per generation.

Discussion

Protocols for cryopreservation have typically been determined empirically by exhaustively varying composition and cooling rates. We have previously developed DMSO-free conditions for cryopreserving cells that used combinations of osmolytes including sugars, sugar alcohols, and amino acids [18,19,22]. However, the performance (i.e., post-thaw recovery) of these conditions highly depended on the composition of the osmolytes and had to be redetermined for each new cell type. Finding the optimal formulation with traditional trial-and-error methodologies proved to be time consuming and costly. We previously applied a differential evolution algorithm to help optimize the composition of cryoprotectants [16], but three control parameters including mutation (F) and crossover (Cr) and population size (NP) may significantly influence the performance of DE. In this study, we examined the manner by which varying the control parameters affected the performance of the DE algorithm.

One approach for testing a new DE variant usually involved measuring the accuracy and convergence speed of the algorithm with several mathematic test functions [23,24]. This approach allows testing of a large population size in order to select mutation and crossover. In contrast, this study used actual datasets for tuning selection of the crossover and mutation.

Topology of Data.

For experimental datasets with smooth, broad peaks, a larger range of acceptable combinations of control parameters can be used. For example, the DE/local-to-best/1/bin with self-adaptive modifications can achieve 95% accuracy with most initial combination of mutation and crossover, F > 0.1 and Cr  >  0.2, with larger population size as shown in Fig. 3(d), and other DE strategies presented similar acceptable combinations, F > 0.7 and Cr > 0.3 as shown in Figs. 3(a)–3(c). However, when presented with a sharp maximum as seen in the SGI dataset, the number of acceptable combinations of mutation and crossover were F > 0.7 and Cr > 0.3 for DE/rand/1/bin only (Fig. 4(a)), which were significantly lower. Other strategies cannot search the global optimum with 95% accuracy as shown in Figs. 4(b)–4(d). Operating outside of that range of mutation and crossover values would result in the algorithm getting stuck in local optimum or passing over the global optimum.

Mutation, Crossover, and Population.

In comparison to suggested control parameters for bioprocess, biomedical, and bioinformatics [10,11,25], using DE algorithm to optimize cryoprotectants requires narrow range of mutation factor and wide range of crossover. Specifically, the accuracy of all DE types is greater for mutation factors ranging from 0.7 to 1.0 and crossovers ranging from 0.3 to 1.0. For SGI, three DE strategies presented performance below 95% as shown in Figs. 4(b)–4(d) but showed acceptable performance for combinations for F > 0.7 and Cr > 0.3 as shown in Fig. 4(a). Increasing the population size increased the acceptable combinations of mutation and crossover. For example, F = 0.5, Cr = 0.9, and NP = 9 resulted in low accuracy as shown in Figs. 5(a) and 5(c), but NP = 27 improved accuracy 10% and 20% to SGC and SGI, respectively. The improvement observed with increasing population size was distinct as shown in Fig. 5(c) and 5(d).

Selecting population size is a tradeoff between the convergence speed and cost. In addition, when biological experiments need to be performed, practical considerations are also relevant. These include availability of the system being preserved, duration and budget and the availability and capacity of labor and instrumentation. For example, in our experiments, NP = 18 was at the limit of testable formulations when factors such as the capacity of the controlled rate freezer and the time necessary for liquid handling and sample preparations were taken into account.

Future Work.

The current approach to optimizing cryoprotectant composition assumes a single output (viability) for a homogeneous population. Preservation of a heterogeneous cell population may be desired and future work could extend the application of DE to heterogeneous cell populations using multiobjective differential evolution [26–29]. Alternatively, it may also be desirable to optimize several post-thaw metrics for a single cell type (e.g., viability, surface markers, and functionality).

Conclusion

This work investigated the effects of control parameter selection on performance of a differential evolution algorithm for the purpose of optimizing the composition of DMSO-free cryoprotectant. The accuracy of classical DE algorithm depended on the combinations of mutation factor and crossover. Screening results showed typical mutation factor and crossover should be between [0.7, 1] and [0.3, 1], respectively. The self-adaption modification reduced the effects of control parameter selections but classical DE with classical mutation method (DE/rand/1/bin) presented a better accuracy in the discontinuous space. The topology of experimental data was also another critical issue to the optimization process. If the global optimum was a sharp peak, the DE algorithm might pass over that during mutation, crossover, or even numerical truncation and eventually get stuck in the local optimum. This work will help with more efficiently determining the optimal concentrations of multicomponent cryopreservation solutions using a differential evolution algorithm approach.

Funding Data

• National Institutes of Health (NIH) (Grant No. R01EB023880; Funder ID: 10.13039/100000002).