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

(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 (n _test = 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.