Probabilistically Robust Uncertainty Analysis and Optimal Control of Continuous Lyophilization via Polynomial Chaos Theory

TL;DR

The paper addresses probabilistic uncertainty in continuous lyophilization of suspended vials during the primary and secondary drying steps. It develops a mechanistic 1D model for the suspended-vial lyophilizer and couples it with a non-intrusive polynomial chaos expansion (PCE) to propagate uncertainty through outputs such as temperature, sublimation-front position, and bound-water concentration, formalized as a PCE $Y = \sum_{i=0}^{L-1} y_i \psi_i(\boldsymbol{\Theta})$ with $L = \frac{(N_\theta+N_P)!}{N_\theta! N_P!}$. The framework enables uncertainty quantification and enables stochastic design and optimal control, benchmarking PCE against Monte Carlo (MC) and achieving similar accuracy with orders-of-magnitude faster computation (e.g., cases A1/A2: $N_{P}=2$, 50 PCE points vs 2000 MC points). Key findings identify the cake resistance $R_p$ and heat-transfer coefficient $h$ as critical in primary drying and desorption kinetics $k_d$, initial moisture $c_{w,0}$, and $h$ as influential in secondary drying, guiding robust process design. The work demonstrates probabilistic design under uncertainty, with Case B1/B2 showing how shelf temperature and optimal control can meet probabilistic constraints (e.g., final $c_w$ below 0.01 wt/wt with $P=0.95$) and reports substantial computational speedups, enabling real-time or near-real-time decision support for continuous lyophilization.

Abstract

Lyophilization, aka freeze drying, is a process commonly used to increase the stability of various drug products in biotherapeutics manufacturing, e.g., mRNA vaccines, allowing for higher storage temperature. While the current trends in the industry are moving towards continuous manufacturing, the majority of industrial lyophilization processes are still being operated in a batch mode. This article presents a framework that accounts for the probabilistic uncertainty during the primary and secondary drying steps in continuous lyophilization. The probabilistic uncertainty is incorporated into the mechanistic model via polynomial chaos theory (PCT). The resulting PCT-based model is able to accurately and efficiently quantify the effects of uncertainty on several critical process variables, including the temperature, sublimation front, and concentration of bound water. The integration of the PCT-based model into stochastic optimization and control is demonstrated. The proposed framework and case studies can be used to guide the design and control of continuous lyophilization while accounting for probabilistic uncertainty.

Figures (7)

• Figure 1: Continuous lyophilization of suspended vials. Vials are suspended and move continuously along the lyophilizer.
• Figure 2: Schematic diagram showing the mechanistic modeling of suspended vials for (A) primary drying and (B) secondary drying.
• Figure 3: Uncertainty quantification of the primary drying step. (A) Mean and 95% confidence interval of the product temperature simulated via PCE. (B) Probability distribution of the temperature at the final time computed by PCE and MC. (C) Mean and 95% confidence interval of the sublimation front simulated via PCE. (D) Probability distribution of the sublimation front at the final time computed by PCE and MC.
• Figure 4: Uncertainty quantification of the secondary drying step. (A) Mean and 95% confidence interval of the product temperature simulated via PCE. (B) Probability distribution of the temperature at the final time computed by PCE and MC. (C) Mean and 95% confidence interval of the concentration of bound water simulated via PCE. (D) Probability distribution of the concentration of bound water at the final time computed by PCE and MC.
• Figure 5: Evolution of the concentration of bound water for the uncertain (A) initial concentration, (B) desorption kinetics, and (C) heat transfer coefficient.