Statistical modeling of single target cell encapsulation

Abstract

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.

Conflict of interest statement

Competing Interests: The authors have declared that no competing interests exist.

Figures

Figure 1. Schematic notation of random variables and probability distribution functions for statistical analysis.

A heterogeneous solution of target and non-target cells is loaded to a droplet ejector. (a) Random cell encapsulation process. (b) Three random variables and one dependent variable were mapped to a patterned array of cell encapsulating droplets which represents number of droplets that contain cells, number of cells per droplet, number of target cells, and droplets encapsulating a single target cell, respectively. The probability of the process can be described as (c) binomial distribution, which represents success and failure corresponding to cell containing and empty droplets, respectively. (d) Poisson distribution is used for the random variable, Xc, 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. (e) Overall system random process becomes the combined function of suggested PDFs. The PDFs for the random variables, Xd and Xc, 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.

Figure 2. LLN (Law of large numbers) for different sampling numbers from n = 10 to 100 was shown according to cell loading concentrations (a) 1.0×105 cells/ml and (b) 2.0×105 cells/ml.

Probabilities of cell encapsulation in droplets are P(Xd = 1) = 58.3% and 87.3%, respectively. As number of droplets increase, high cell concentration, 2×105 cells/ml, also follows normal distribution.

Figure 3. Probability distribution functions of Bernoulli's random variable, Xd.

(a) Binomial distribution functions (n Bernoulli trials for discrete random space) are shown with fitted PDF curves. The mean values of modeled binomial distribution (b) 27.1%, 58.3%, 76.8%, and 87.3% for probability of cell encapsulation at cell loading concentrations of 0.5×105, 1.0×105, 1.5×105, and 2.0×105 cells/ml, respectively (ntest = 100 droplets). Exponential regression curves fit the experimental results (coefficients of exponential regression: a = 131, b = 0.558, R2 = 0.995). (c) Cell encapsulation probability and volume fraction (which is the ratio of cell volume divided by droplet volume, 7.7 nl) are shown as a function of cell concentrations. At cell concentration 2.5×105 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.

Figure 4. Probability distribution functions for the number of cells per droplet, Xc.

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 1.0×105 cells/ml (np has a moderate size, λ = 1.0 for n = 10 cells in a droplet) for (a) 10% and (b) 50% target cell mixture, ntest = 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 = np, σ = npq = λ(1−λ/n)). In spite of the fact that 1.5 and 2.0×105 cells/ml cell concentration show higher cell encapsulation probability as shown in Figure 3, highest probability for single cell encapsulation is achieved at the specific cell concentration of 1.0×105 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 Xc = 1.

Figure 5. Probability distribution functions for encapsulation of target cells from (a–d) 0.5×105 cells/ml to 2.0×105 cells/ml cell concentrations, P(Xt).

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 (a) 0.5×105 cells/ml, (b) 1.0×105 cells/ml, (c) 1.5×105 cells/ml, (d) 2.0×105 cells/ml.

Figure 6. The plot of single cell encapsulation probability versus number of target cells per droplet (a), and percentage of target cells in a reservoir (b) PDFs for a single target cell encapsulation, P(Xs) were shown with combined PDFs for selected cases: (a) Poisson distribution for 1.0×105 cells/ml cell concentration for four different target cell concentrations, and (b) cell encapsulation probability compared with experimental results from 10% to 50% target cell mixture.

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