Mathematical Programming for the Design and Analysis of a Biologics Facility - The use of mathematical programming methods for automated schedule generation. - BioPharm International

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Mathematical Programming for the Design and Analysis of a Biologics Facility
The use of mathematical programming methods for automated schedule generation.


BioPharm International
Volume 23, Issue 2

THE ENGINEERING PROBLEM AND BUSINESS OPPORTUNITY

Almost as soon as computers first appeared, engineers recognized their potential as tools for use in the simulation and analysis of processes. As early as the 1950s, chemical companies developed in-house sequential modular simulation systems. By the 1970s, use of commercial sequential modular simulators was widespread. These systems were well suited to predict and analyze the behavior of continuous single-product facilities that operated at or near steady-state conditions over long periods of time. Indeed, many would find it unthinkable to carry out process development, and the design and startup of a large-scale continuous chemical plant, without an accurate computer simulation. Yet, nearly a decade into the 21st century, we routinely design and construct large-scale biologics facilities based on bench- scale and pilot plant data, and using primitive tools such as spreadsheets, which can provide only limited insight into dynamic processes. Why do we see such a marked disparity in the level of sophistication applied to modeling these two very different types of processes? The answer is that biologics facilities are beset with a myriad of complicating factors that require dealing with dynamics to get sufficiently accurate results.

Manufacturing processes for biologics typically consist of a series of batch reactors followed by purification and concentration steps that use chromatography columns and ultrafiltration skids to isolate the protein of interest and concentrate it to manageable volumes. The batch nature of these reactions renders steady-state simulators unusable for detailed dynamic analysis. Although a steady-state simulator might approximate multiple batches with a single continuous rate, this approximation breaks down when the series of batches must be interrupted. Such interruptions occur routinely in biologics processes, e.g., for clean-in-place (CIP) operations or periodic repacking of chromatography columns. In addition, many processes require enforcing batch integrity and individual batch tracking. This means that when a vessel is used to store an intermediate material, all material from that batch must be emptied before any material from a subsequent batch may be added to the vessel. Typically, the vessel also must be cleaned before the introduction of new material. In the presence of such constraints, no modeling technology will be accurate unless it explicitly tracks the location of inventory from individual batches.

The biologic nature of these processes also induces stochastic variability in batch yields (titers) and cycle times. Successive batches of the same reaction produce different yields of the desired products, even when executed under identical conditions. Furthermore, the cycle time often is correlated to titers so that high titer batches may take more time. Batches with different titers will produce effluent with different concentrations of the protein of interest and thus different loadings on the chromatography columns. To describe and predict the true capacity of the facility, it is necessary to model both the expected uncertainty in titers and the effect that it has on the required frequency of column repacking. Bioreactor titers also affect the batch-processing duration in the columns. Other complicating factors may arise, such as failed batches. These variables alter the true capacity of the process and many of them interact in a nonlinear fashion with the availability of parallel equipment available at the various process stages. As a result, without high fidelity approaches that can produce detailed feasible schedules for the process over a sufficiently long time period to reflect stochastic behavior, and that satisfy all necessary constraints, it is not possible to predict how a process design will in fact behave in the real world with an accuracy commensurate with the capital investment.

One approach to this challenge has been to model such processes using discrete event simulation.1 This method is more appropriate than steady state simulation. However, discrete event simulation requires a significant investment of time to develop the logic necessary to accommodate complex processes. Discrete event simulators work by advancing the time variable in a monotonically increasing fashion with an attendant design of supporting data structures (e.g., event stacks). Therefore, if we consider events happening at time t, it is no longer possible to initiate events that can occur at an earlier time. This complicates the logic to such an extent that high fidelity simulations of large-scale biologics facilities present a significant technical challenge. In highly constrained applications, such as the one studied here, temporally restricted decision-making often encounters local infeasibilities. Discrete event simulation approaches address these infeasibilities with heuristics such as restarting the simulation sufficiently far in the past in an attempt to avoid problematic behavior. Practically, these simulation heuristics degrade as the models consider increasing levels of process detail. This happens because (1) the number of events becomes large and the sequential processing of events results in prohibitive algorithmic complexity, (2) the extension and maintenance of simulation heuristics becomes arduous and their behavior unpredictable as model complexity grows, and (3) complexity tends to grow in successive studies as the model evolves to address more detailed interactions and answer increasingly sophisticated questions of interest. These limitations, combined with the combinatorial burden of searching the enormous number of timelines needed for design and schedule optimization, argue for an alternate approach supporting implicit search and designed to scale well with increasing detail and model evolution.

A mathematical programming approach transcends both steady state and discrete event simulators. This approach allows the solver to range freely over the entire timeline of interest and insert or remove scheduling events and activities whenever it is useful to do so. The approach can not only mimic the left-to-right behavior of simulation to capture causal reasoning, but also can range over the timeline to enhance algorithm speed and facilitate global reasoning about constraints. In this article, we show how to handle stochastic process variability, disposal of batches, periodic repacking of chromatography columns, and other process constraints necessary to accurately predict the real world performance of a large-scale biologics facility during the design stage.

THE RESOURCE TASK NETWORK AND MATHEMATICAL PROGRAMMING APPROACH TO BIOLOGICS FACILITY MODELING

A resource task network (RTN) description is used to provide a structured description of process details.2 We developed a high fidelity model of an industrial biologics facility using the VirtECS version 7.1 software system, which consists of a core mathematical programming solver designed around a uniform discretization model (UDM) and a customized outer layer that is specifically tailored to address biologics process behavior.1,3,4


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