(413d) Optimal Control of a Stefan Problem By Differential Algebraic Equation Reformulation: A Cell Thawing Case Study

A Stefan problem (aka moving boundary problem, moving interface problem) describes the evolution of a moving interface during phase change, e.g., freezing and melting (Carslaw and Jaeger, 1959; Bird et al., 2002). Different Stefan problem formulations were applied to study various industrial and natural systems, including polymorphous materials (Tao, 1979), steel casting (Hill and Wu, 1994; Petrus et al., 2010), biological tissue (Rabin and Shitzer, 1995, 1997), glaciation (Mikova et al., 2017), phase change materials (Brezina et al., 2018), alloy formation (Brosa Planella et al., 2019), cryopreservation (Dalwadi et al., 2020), and cell thawing (Srisuma et al., 2023). Several numerical techniques for implementing and simulating the Stefan problems are also well established (see examples in Srisuma et al., 2023).

Optimal control of Stefan problems was extensively investigated and proven useful for various industrial applications over the past few decades, with many different objective functions, constraints, and controls (manipulated variables) considered. Some examples are the control of the heating process to satisfy the heating speed and thermoelastic stress constraints by manipulating the furnace temperature (Roubíček, 1986), the stabilization of the moving boundary and temperature/concentration fields by varying the heat flux (Pawlow, 1987), the maximization of the amount of melted solid in the melting process via controlling the heat flux (Neto and White, 1994), the control of the moving boundary to follow the desired path in the solidification process by manipulating the wall temperature (Hinze and Ziegenbalg, 2007), the control of the water level in the drainage basin by varying the discharge velocity (Miyaoka and Kawahara, 2008), and the minimization of the metallurgical length (ML) deviation in steel casting via controlling the boundary heat flux (Chen et al., 2019).

All the aforementioned studies rely on some optimization algorithms/solvers to obtain the optimal control trajectories. This optimization-based approach is a very common technique for solving optimal control problems; the general steps are to discretize the PDE and ODE constraints, parameterize the time-varying control vector, and numerically solve the resulting optimization problem (Kishida et al., 2013; Nolasco et al., 2021). Numerous optimal control algorithms have been developed and improved for efficient solutions to facilitate more advanced applications such as model predictive control (see detailed discussion in Rodrigues et al. (2014); Nolasco et al., 2021). Alternatively, Berliner et al. (2022) showed the transformation of an optimal control problem into an equivalent system of index-1 differential algebraic equations (DAEs), where the optimal control vector can be obtained via a DAE solver, without any optimization solver, resulting in a more computationally efficient solution to the optimal control problem.

Although optimal control with Stefan problems has been explored for a wide range of processes, applications to cell thawing have not been available. Previous studies showed that accurate prediction, control, and optimization of cryopreservation and cell thawing can improve the viability and quality of the resulting cells, which directly benefits cell therapy (Hunt, 2019; Cottle et al., 2022; Uhrig et al., 2022). In addition, various cell thawing experiments can be formulated as optimal control problems (Seki and Mazur, 2008; Jang et al., 2017; Bojic et al., 2021). These benefits and case studies therefore motivate the development of an efficient Stefan problem-based optimal control algorithm for cell thawing.

This article presents a novel technique for optimal control of a Stefan problem with application to cell thawing. The proposed simulation-based technique reformulates an optimal control problem to a system of differential algebraic equations (DAEs), and the resulting equations are solved by a DAE solver. With this approach, the optimal control trajectory is determined by the selected DAE solver, eliminating the need for any optimization solver. The technique is demonstrated via cell thawing case studies, which includes both index-1 and high-index DAE systems.

Simulation results show that the DAE-based technique introduced in this work is highly accurate and computationally efficient. From all case studies, the computation times can be reduced by 70%–80% with the DAE-based approach, whereas the obtained solutions from the DAE-solver are as accurate as the optimization-based solutions. This finding suggests that the DAE-based approach is a reliable, efficient framework for solving optimal control problems, and hence can be readily extended to more advanced control strategies which require fast and efficient optimal control solvers such as model predictive control.

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