(377a) Design Space Via Symbolic Computation

Design space (DS) is a very important concept in the pharmaceutical industry. The importance of DS definition lies in the assurance of quality, a key goal of Quality by Design (QbD), and in the broadening from an acceptable operating setpoint to a collection of tolerable operating regions [1]. Defining proper limits for the design space is vital to provide manufacturing flexibility and quality assurance. The design space is determined by the uncertainty in the model parameters along with the allowable set of conditions in the process parameter space [2]. Flexibility index is an important measure to evaluate the operability of a design model with uncertainties [3], and it also provides a way to describe the design space; for example, the design space can be approximately regarded as a largest inscribed hyperrectangle or hyperelliptic. In addition, if the nominal conditions are not given, the flexibility index can be extended to the design centering problem [4], which is another important problem of DS definition, where it might be desirable to obtain optimal nominal conditions that maximize the feasible region of operation. The ultimate goal of flexibility analysis is to accurately and explicitly describe the design space, regardless of the convex and nonconvex nature of the design space.

Design space description, flexibility index, and design centering are viewed as three different issues in the DS field, and they are commonly formulated as multi-level optimization models. In general, these issues can be handled by numerical computation methods; for example, through the KKT conditions and complementarity conditions, the multi-level optimization models can be transformed into MILP/MINLP models [5]. However, it is well known that it is difficult to find the global solutions of an MINLP model. Moreover, the numerical methods can only estimate the outer envelope of the design space, rather than providing an accurate description, especially for the nonconvex cases. Mathematically, the expressions with variables can be obtained through the symbolic computation methods, which is very different from the numerical methods.

As we will show in this presentation, the original design model can be obtained directly in the space of the uncertain parameters through the proposed symbolic computation method; thus, it is unnecessary to solve numerical optimization problems. In order to provide an accurate description of the design space, we propose to reformulate the traditional flexibility analysis problem as an existential quantifier model. Then, we introduce a symbolic computation method, cylindrical algebraic decomposition (CAD) [6, 7] to eliminate the quantifiers and transform the flexibility analysis formulation into a series of explicitly triangular formulas. Each of these formulas represents a subspace of the design space, and the logical combination of them constitutes the complete symbolic representation of the design space projected in the space of the uncertain parameters. We show that with this symbolic computation method, the nonconvex design space can also be derived effectively.

Symbolic computation provides a novel way to deal with the characterization of a design space. Moreover, the proposed unified symbolic framework can integrate the three approaches described above for design space definition. These three problems have a similar symbolic model, and the only difference is that the dimension is increased successively, and the corresponding computational complexity is also increased successively. For the design space description problem, through the CAD method, the original high-dimensional feasible space can be projected on the space of the uncertain parameters. For the flexibility index problem, the feasible space can be projected onto the one-dimensional space of flexibility index, and all of the candidate values of flexibility index can be generated at the same time. For the design centering problem, the feasible space can be projected onto the space of the flexibility index and the uncertain parameters, and a rule to locate the final nominal point is developed, which only requires a series of simple numerical operations. In summary, the proposed method for design space has four properties, (1) it is a unified symbolic framework; (2) it can avoid solving numerical optimization problems; (3) it can guarantee finding the optimal flexibility index or nominal points; (4) it can provide explicit algebraic expressions of the design space. A design space may be constructed for single unit operation, multiple unit operations, or for the entire process. However, as the scale of the design model increases, the complexity is greatly increased. For reducing the complexity of symbolic computation, we propose to use the adaptive sampling method to construct the surrogate model and eliminate the equalities of the model. Then, the CAD method can be adapted to obtain the explicit expressions of the design space, and the boundary of the design space can be validated and revised iteratively to satisfy a specified accuracy. Using several examples from the pharmaceutical industry, we illustrate the application of the proposed symbolic computation method to show its novelty and value.

Keywords: Design space, flexibility analysis, design centering, cylindrical algebraic decomposition, symbolic computation.

Reference:

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