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Conceived and designed the experiments: SM EC UD. Performed the experiments: SM. Analyzed the data: SM EC. Wrote the paper: SM EC UAG UD.

The authors have declared that no competing interests exist.

High throughput drop-on-demand systems for separation and encapsulation of individual target cells from heterogeneous mixtures of multiple cell types is an emerging method in biotechnology that has broad applications in tissue engineering and regenerative medicine, genomics, and cryobiology. However, cell encapsulation in droplets is a random process that is hard to control. Statistical models can provide an understanding of the underlying processes and estimation of the relevant parameters, and enable reliable and repeatable control over the encapsulation of cells in droplets during the isolation process with high confidence level. We have modeled and experimentally verified a microdroplet-based cell encapsulation process for various combinations of cell loading and target cell concentrations. Here, we explain theoretically and validate experimentally a model to isolate and pattern single target cells from heterogeneous mixtures without using complex peripheral systems.

Cell encapsulation in nanoliter volume droplets and patterning has a broad range of applications including tissue engineering using biodegradable hydrogels

Challenges still remain to enable efficient extraction, isolation, and patterning of cells from heterogeneous cell suspensions and to keep them alive throughout the process. Although microfluidic approaches offer deterministic control over the cell encapsulation process, they require complex instrumentation involving hydrodynamic focusing and flow control for tracking multiple cells in these systems

Recently, microchip technologies have created multiple new avenues through experimental studies to isolate, capture, pattern cells in microscale fluidic volumes impacting a variety of fields. However, the emphasis has been on engineering, device modeling and medical applications, whereas the statistical analysis of such events has fallen short of focus. Among these microfluidic manipulation technologies, cell encapsulation processes within microscale droplet volumes have not been theoretically investigated from a statistical or stochastic point of view for droplet ejectors. In this paper, we statistically modeled and experimentally analyzed random cell encapsulation processes in microdroplets.

To model the encapsulation process, we assumed that droplets were generated from a heterogeneous cell suspension consisting of target and non-target cells. The cell encapsulation process can be described by using four random variables with respective probability distribution functions (PDFs). These random variables are: number of droplets containing cells, number of cells per droplet, number of target cells per droplet, and number of droplets containing only a single target cell. Employing different loading cell concentrations and target cell concentrations, each random process can be characterized by the rate of empty droplets, average number of cells per droplet, number of target cells in the printed droplets, and the rate of single target cell encapsulation, respectively. The PDF corresponding to each random variable can be used to estimate process characteristics based on experimental results. We followed four steps to analyze our single target cell encapsulation process: (i) we defined our system as a set of stochastic processes with random variables, (ii) estimated the minimum number of droplets at which the variables follow (approximately) a normal distribution, and determined whether the suggested processes are biased or un-biased, (iii) established statistical models and parameters for each random process, e.g., mean (μ), variance (σ), and Poisson distribution parameter (λ) for the cell encapsulation process, and (iv) evaluated overall system efficiency to find and encapsulate single target cells using central limit theorem (CLT) under conditions of simple random sampling (SRS).

Random variables for cell encapsulation process in droplets were schematically shown in _{d}, X_{c}, and X_{t}, respectively, and are defined at the sampling space from droplet array (n = 100 droplets forming a 10×10 array of patterned droplets). These random variables are proposed to have three probability distributions, Binomial, Poisson and a combined distribution, respectively as shown in

A heterogeneous solution of target and non-target cells is loaded to a droplet ejector. (_{c}, since the number of cells per droplet is the count of occurrence of a rare event (i.e., probability of the event is very low) in probability space with respect to the number of sampled droplets and droplet volume. (_{d} and X_{c}, are used for the overall PDF of the system. The parameter λ represents the Poisson coefficient, and μ, and σ represent mean, and variance of the underlying probability distributions, respectively. All distribution functions were interpolated to a continuous curve with colored bars on graph indicating the discrete values.

The cell encapsulation process in droplets can be classified into two discrete random processes: (i) random cell encapsulation in droplets from a homogeneous cell mixture, and (ii) encapsulation of a single target cell from a heterogeneous cell mixture. The single target cell encapsulation could be modeled resulting from an ensemble effect combined with a high probability of cell encapsulation using Binomial distribution, and a rare probability event process that follows Poisson distribution as shown in _{d}, which gives the number of successes in _{c}, is expected to follow a Poisson distribution, since the number of cells per droplet is a rare event in discrete probability space. The overall system PDF can be obtained by combining PDFs of the two independent random variables, X_{d} and X_{c} and each PDF is shown in _{s}, is a variable that depends on two other variables, namely, number of droplets containing cells, and number of target cells in a droplet. The parameters for each PDF can be estimated by using experimental results for independent random variables in discrete probability space, i.e., in the space of ejected cell encapsulating droplets.

We have determined the required sample size that would render the normal approximation appropriate for our analysis. In this study, cell encapsulation was validated as a random process provided that it satisfies experimentally the conditions for the law of large numbers (LLN) for the normal approximation assumption to hold. As more details are provided in

The cell encapsulation process follows a sequential process based on Bernoulli trials to find a successful case. Binomial process represents how many times the sequential process will be a success (i.e., how many of the droplets contain cell(s)) regardless of the number of cells per droplet. Similarly, when the successful case is rare, the probability of encapsulating homogeneous target cells shows Poisson distribution as a special case of Bernoulli trials

PDF of binomial distribution is discussed in _{d}, number of droplets that contain cells, is associated with the occurrence of k successes in n trials _{n,p}(k), are μ = np and σ^{2} = npq. From the computational perspective, the fast growth of the factorial function is often handled by the Stirling approximation formula in the binomial probabilities _{n}), tends to zero as n goes to infinity. The convergence is rapid (e.g., n≥20), where the error of the approximation is below 0.5%.

Since high cell concentrations might have lower probability for single cell encapsulation compared to low cell concentrations, the low cell concentration and cell volume fraction can be represented with a Poisson distribution for encapsulation of “

In the case of encapsulating a single target cell in a droplet from a heterogeneous cell mixture, the probability to encapsulate a single target cell becomes lower than the probability for a homogenous cell mixture. So we attempt both a binomial distribution and a Poisson distribution model for different cell loading concentrations. Finally, overall process PDF for a single target cell encapsulation was modeled using two PDFs: the cell encapsulation process (modeled as binomial distribution) and single target cell encapsulation process (modeled as Poisson distribution). These PDFs were combined to make a single probability function describing the complete process. The PDF of the random variable X_{s} can be obtained by combining the PDFs of the two random variables, X_{d} and X_{t}, (see _{t} cannot be positive when X_{d} is not positive, in the sense that X_{d} = 0 forces X_{t} = 0. However, conditional on X_{d}>0, it is reasonable to assume that X_{d} and X_{t} are independent, and hence PDF of X_{s} can be written as follows:_{b} and k_{p} stand for the number of occurrences for the binomial and Poisson distribution, respectively. Note also that P(X_{s} = 0) = 1−P(X_{s} = 1).

Following CLT (as described in _{d}_{c}_{t}_{s}_{%}) shows same concentration as the reservoir concentration for 10.0% to 50.0% at 1.0×10^{5} cells/ml concentration (C_{opt}) under conditions of 90.0% confidence level and 15.0% tolerance.

For cell encapsulation model, 90% confidence level, (1−α), and 15% tolerance, ε, for lowest probability (p = 0.6) were used to determine the number of ejected droplets encapsulating cells. LLN was shown by investigating the PDF shape with 1.0×10^{5} cells/ml to 2.0×10^{5} cells/ml cell loading concentrations. As the sampling number of random variable, X_{d}, increases from n = 10 to 100, the shape of its PDF, which actually is a binomial PDF, gets closer to a normal distribution as indicated in _{d} = 1) = 58.3% and 87.3% for the cell loading concentrations of 1.0×10^{5} cells/ml to 2.0×10^{5} cells/ml, respectively. The PDF curve shows that the results follow normal PDF which has 58.3±3.8% for average and standard deviation with a 10% maximum probability of cell encapsulation at 1.0×10^{5} cells/ml concentration (^{5} cells/ml, also follows normal distribution (

Probabilities of cell encapsulation in droplets are P(X_{d} = 1) = 58.3% and 87.3%, respectively. As number of droplets increase, high cell concentration, 2×10^{5} cells/ml, also follows normal distribution.

The formula in (eq. 3.2) was utilized to calculate the fitted PDF curves using MATLAB® (Version R2010a, The MathWorks, Inc, MA). ^{5} cells/ml to 2.0×10^{5} cells/ml as shown in ^{5}, 1.0×10^{5}, 1.5×10^{5}, and 2.0×10^{5} cells/ml, the probabilities of cell encapsulation were 27.1%, 58.3%, 76.8%, and 87.3%, respectively. These probability values are estimated from the means of the empirical PDFs shown in

(^{5}, 1.0×10^{5}, 1.5×10^{5}, and 2.0×10^{5} cells/ml, respectively (_{test} = 100 droplets). Exponential regression curves fit the experimental results (coefficients of exponential regression: a = 131, b = 0.558, R^{2} = 0.995). (^{5} cells/ml, probability of cell encapsulation was 98.0% and the volume fraction was 1.7% (which represent the cell loading concentration and the minimum droplet volume to encapsulate a single target cell with the proposed mechanical valve system, respectively). In summary, 1.7% cell volume fraction is the optimal value to achieve a very high cell encapsulation probability.

In the random process, the physical cell encapsulation process is affected by the ejection mechanism and cell loading concentrations. In ^{5} cells/ml, probability of cell encapsulation was as high as 98.0%. This result represented that small cell volume to droplet volume ratio was needed to encapsulate cells within a large media droplet volume. For 100% cell encapsulation probability, minimum droplet volume and required cell concentration was calculated as 30.8 picoliter (pl) and 32.5×10^{6} cells/ml by the following equation:^{6} cells/ml with a droplet volume of 7.7 nanoliter (nl). Exponential regression curves fit the experimental results (the adjusted R^{2} value is 0.995). The coefficients of exponential regression are a = 131 and b = 0.558 as shown in ^{5} cells/ml than 1.0×10^{5} cells/ml. However, the cell concentration of 1.0×10^{5} cells/ml showed higher probability of “^{5} cells/ml. Since only volume fraction of 1.7% is required for encapsulating a cell with over 98% probability, high cell concentrations might have lower probability for single cell encapsulation compared to low cell concentrations. The low cell concentration and cell volume fraction can be represented with a Poisson distribution for encapsulation of “

_{c} = k). In the case of droplets containing a single cell, the Poisson distribution matches the experimental results as the occurrence of “single cell encapsulation” becomes rare. That is, the number of droplets containing cells can be modeled as a binomial process, and the same holds for the number of droplets that have single target cells. However, the latter seems to be closer to a Poisson distribution, since the probability of success (i.e., probability of a droplet having a single target cell) is low and the number of droplets is large. Moreover, the results showed the same probability distribution regardless of target cell concentrations for 10% (_{test} = 100 droplets (0.76 µl total sub-sampling volume, single droplet volume is 7.6 nl). Experimental results and modeled values of Poisson distribution parameters match with ±2.0% error from 10.0% to 50.0% cell loading concentration at 1.0×10^{5} cells/ml.

In the case of droplets containing a single cell, the Poisson distribution agrees with the experimental results since the probability becomes small. Experimental results and modeled values as Poisson distribution for single cell encapsulation process agrees with ±2% error at 1.0×10^{5} cells/ml (_{test} = 100 droplets. The maximum probability and PDF is not affected by cell loading concentrations. The curves are generated using the Poisson distribution instead of the binomial distribution in continuous random variable space (λ = μ = ^{5} cells/ml cell concentration show higher cell encapsulation probability as shown in ^{5} cells/ml corresponding to 1.0% of volume fraction. The result is obtained from the peak points of each PDF, which gives highest probability of X_{c} = 1.

In ^{5} cells/ml, we observed Poisson distribution, since the probability for encapsulating cells in a droplet is very low. The other distributions for the concentrations are closer to the binomial distribution than Poisson distribution due to higher cell concentration and higher cell encapsulation probability. Following the experimental results and the statistical model as shown in _{experiment} and P_{Poisson model}, 1.0×10^{5} cells/ml was indicated as an optimal concentration to encapsulate a single cell in a droplet for our cell printing platform, since it has the highest single cell encapsulation probability. Even though, 1.5×10^{5} and 2.0×10^{5} cells/ml cell concentration show higher cell encapsulation probability as shown in ^{5} cells/ml corresponding to 1.0% of volume fraction. The highest probability estimates are obtained from the peak points in each PDF, which gives highest probability for X_{c} = 1.

As for the case of encapsulating a single target cell in a droplet from a heterogeneous cell mixture, _{t}), for heterogeneous cell mixture with different cell loading concentrations from 0.5×10^{5} cells/ml to 2.0×10^{5} cells/ml. The shape of Poisson distributions were determined based on experimental results. Binomial distributions and Poisson distribution for different cell loading concentrations were plotted in both the discrete space and continuous random variable space. As shown in

The PDFs are based on experimental results, Poisson distribution, and binomial distribution treating the variables as continuous (e.g., it is not possible to have 0.1 cells, but we estimate the cell encapsulation probability for this value), for cell loading concentration (^{5} cells/ml, (^{5} cells/ml, (^{5} cells/ml, (^{5} cells/ml.

For the overall process PDF for a single target cell encapsulation as in (eq. 3.6), _{s}, and conditional PDF for selected cases of 1.0×10^{5} cells/ml (

Modeled PDFs showed 5% error compared to the experimental results using specific parameters, μ, σ, and

Based on the suggested model on (eq. 3.6), the number of homogeneous droplets was modeled using the Poisson distribution. The single target cell encapsulation process in a droplet can be modeled following two control parameters: cell loading concentration and percent mixture of target cell. Average number of cells, λ, for four different cell loading concentrations and target cell concentrations were determined as shown in _{max} = 0.95 and λ_{min} = 0.03. Based on these experimental and analysis results, statistical models can be determined based on λ values, e.g., λ = 0.05 for 1.0×10^{5} cells/ml with 10% target cell mixture.

For certain applications, for instance in tissue engineering and/or high throughput testing

When performing cell encapsulation studies, commonly observed experimental conditions in cell culture needs to be taken into account, such as the aggregation and settling of cells in the suspension form. In our studies, cell encapsulating droplet generation took place within minutes after the cell suspension was prepared. Therefore, cell settling or aggregation of the cells in the reservoir was not considered to have a significant effect on the results. Various mixing methods, such as magnetic stirring, can be used to prevent cell aggregation and settling for experiments lasting longer durations. On the other hand, in our earlier work, we have shown that different fluid types with different rheological properties can be used to generate droplets and encapsulate cells

Encapsulation of a few to many cells in micro-scale droplets has been investigated for applications in tissue engineering, in which cell-encapsulating hydrogels can be used as building blocks for generating organized tissue structures. In these studies, the control over the number of cells in hydrogel building blocks (e.g., microscale droplets) is essential, where the number of cells per building block determines the overall cell density in the resulting tissues, and hence the structural and functional outcome

Cell clusters (e.g., pancreatic islets) are currently an important research area, which has the potential to offer alternative treatments for diseases such as, Type-1 diabetes

The presented drop-on-demand approach for single cell sorting has a trade-off from a deterministic cell encapsulation aspect compared to the microfluidic cell encapsulation approaches. For single cell encapsulation, microfluidic method provides a more deterministic method and better control over cell encapsulation. However, as the heterogeneity of sample increases, the types of cells that need to be tracked in the sample also increases leading to incline in complexity of microfluidic systems due to more complex peripheral setups and high-end computerized controls

We investigated the cell encapsulation process in microdroplets. We modeled the encapsulation process of a mechanical valve system that randomly encapsulates target cells from a heterogeneous cell suspension. Using four random variables and corresponding probability distribution functions (PDFs), the cell encapsulation process was described and used to estimate process characteristics such as mean (μ), variance (σ), and Poisson distribution parameter (λ) for different cell concentrations and target cell mixtures. These models exhibited Poisson distributions with 16 different values of a parameter as shown in

In conclusion, statistical and stochastic modeling proves to be a powerful and promising tool to determine the conditions for single target cell encapsulation. In this article, we have theoretically analyzed the encapsulation of a single target cell in microdroplets from heterogeneous cell mixtures and supported our theoretical results with experimental data. This analysis explains the statistical dependence of encapsulation of single cells in droplets.

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_{d}, _{c}, and _{t}) were defined in discrete independent domain and one dependent variable (i.e., _{s}) was defined by a combination of independent variables for overall process efficiency. Three variables were used to represent percentage of empty droplets, effect of number of cells in droplets as a function of loading cell concentrations, and target cell concentrations, respectively.

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_{s}). The number of homogeneous droplets was modeled using Poisson distribution in a random variable space, i.e., number of target cells. The model was verified using a coefficient, λ, and experimental results. Average number of cells, λ, for four different cell loading concentrations and target cell concentrations were determined. As cell loading density increases, target cell concentration, λ values increase, from λ_{min} = 0.03 to λ_{max} = 0.95. Based on these experimental and analysis results, statistical models can be determined based on λ values, e.g., λ = 0.10- for 1.0×10^{5} cells/ml with 10% target cell mixture.

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^{(**)}. Following the law of large numbers (LLN), minimum sample number was determined as 100 droplets for 90.0% confidence level and 15.0% tolerance. This sampling volume of a droplet represented (0.76 µl = 10×10×7.6 nl) 0.76% of the total volume of the ejection reservoir (0.1 mL). (_{d}_{c}_{t}_{s}^{5} cells/ml concentration (C_{opt}) under conditions of 90.0% confidence level and 15.0% tolerance.

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This work was performed at the Demirci Bio-Acoustic MEMS in Medicine (BAMM) Laboratories at Harvard-Massachusetts Institute of Technology Health Sciences and Technology, Brigham & Women's Hospital Center for Bioengineering at Harvard Medical School.