## Introduction

As the monoclonal antibody (mAb) sector has matured, it has become critical to rapidly identify the most cost-effective purification processes that can handle increasing upstream productivities in a timely manner and overcome existing purification bottlenecks.[1-3] Chromatography operations are identified as critical steps in a mAb purification process and can represent a significant proportion of the purification material costs, particularly due to the use of expensive affinity matrices as well as the high amounts of buffer reagents required. Higher product titers allow meeting larger demands and decreasing the relative cost of upstream activities. However they increase the protein load on chromatography steps resulting in an increase in the number of cycles or further investment in larger columns and hence the relative cost of downstream increases.[3, 4] Although alternatives to traditional column chromatography platforms are emerging (e.g., non-chromatography operations, membrane adsorbers), industry practitioners are still reluctant to perform major process changes.[1-3] At the same time, it is important to determine how best to use existing installed production capacity for mAbs.[5, 6] In this context, continuous improvement of existing processes, particularly the optimization of chromatography operations, is a valuable approach to address the current challenges. The development of computer-based decisional tools for the bioprocess sector is an emerging area[7-11] and frameworks have been developed to assess different solutions for the design and operation of chromatography steps. Joseph et al.[12] present a simulation model to identify windows of operation for a chromatography step, using productivity and cost of goods (COG) as performance criteria. A model to find combinations of protein load and loading flow rate that meet yield and throughput constraints has been developed by Chhatre et al.[13] The discrete-event simulation framework proposed by Stonier et al.[14] allows the selection of optimal chromatography column sizes over a range of titers by brute force simulation. However, such an approach may not be feasible for very large decision spaces, and, particularly, when the variables have integer domain, as is the case in the problem addressed in the present article.

There are a large number of possible permutations and trade-offs related to running packed-bed chromatography operations such as opting for a smaller column run for several cycles so as to reduce resin costs vs. a large column run for fewer cycles so as to save time and labor costs. Decision makers usually have empirical approaches to come to a solution, mainly based on previous experience, and so may be missing good opportunities for improvement. The combinatorial optimization (CO) nature of the decision problem consists of selecting the most appropriate sizing strategy for the chromatography operation. In this article, the decisions are addressed using mixed-integer programming (MIP) techniques due to their widely recognized ability to handle CO problems.

Mixed-integer linear (MILP) and non-linear programming (MINLP) models have been developed to address capacity planning problems in the pharmaceutical[15, 16] and biopharmaceutical[17, 18] industries. At the process level, MIP models have focused on determining optimal purification sequences, using physicochemical data of protein mixtures and mathematical correlations of the separation techniques.[19-21] In some cases, the process synthesis optimization has also considered product loss by incorporating the decisions on the time of product collection and the start and finishing cut-points.[22, 23] More recently, efficient MILP models were developed using the discretization[24] and piecewise linearization approximation[25] to overcome the computational expense of MINLP models. These models use the number of chromatography steps, purity, and yield as performance metrics, but do not account for overall process costs.

Optimization of chromatography equipment sizing strategies for a sequence of chromatography steps on the basis of a global criterion, such as cost of goods per gram (COG/g), requires the use of either MINLP approaches or heuristic search methods such as evolutionary algorithms to handle the complex model dependencies. Meta-heuristic methods have been developed that integrate evolutionary algorithms with detailed process economics models to determine the most cost-effective purification sequences and chromatography sizing strategies that meet purity constraints.[26, 27] MINLP approaches have the advantage of providing exact solutions in the cases where commercial solvers or linearization techniques allow a feasible solution to be identified. However, an MINLP model for this problem domain does not exist in the literature. Hence, this article presents a novel mathematical programming model based on an MINLP formulation to determine the best chromatography equipment sizing strategies for the production of mAbs. The CO model addresses the challenge of optimizing the chromatography sizing strategy for a sequence of chromatography steps in a downstream purification train whilst considering several key decision variables for each step, including column bed height, column diameter, number of columns, and number of cycles. Furthermore, the model is used also to determine the optimal facility fit configuration for products with higher titers. A related problem has been previously addressed by Stonier et al.[28] using a stochastic simulation framework and multivariate analysis to identify root causes of facility mismatches.

The problem under study in this work—optimization of chromatography sizing strategies for facility design and facility fit—is formulated as an MINLP model, which can be solved to global optimality using commercially available global optimization solvers.