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


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

Model Description of the Biologics Process

Figure 1. Biologics process schematic
The process and model descriptions in this paper reflect the actual industrial process for which this work was performed. Diagrams, figures, and results are taken from a sanitized model that is similar in nature and complexity. A diagram of the process is shown in Figure 1. In the inoculum stage, a working cell bank vial is thawed and expanded in a series of flasks and Wave bioreactors. The resulting material is then fed through a series of three bioreactors of increasing capacity and then sent to a large production bioreactor. There are three complete parallel trains for scaling up the vial thaw and five parallel production bioreactors. The cell culture growth in the production bioreactor requires 9 to 11 days. When cell growth is completed in the bioreactor train, the lot is harvested using centrifugation and filtration. Purification follows through a series of chromatographic columns and filters.

Figure 2. Gantt chart for a portion of a typical line
The model we developed considers each step including preparation time, processing time, and the time required to clean the equipment following processing. An important consideration in the model is the handling and storage of intermediate materials. Unlimited intermediate storage can be available for certain types of materials, e.g., frozen cell cultures that are sufficiently compact that storage space is never a problem. But for most materials in this process, storage is limited and must be modeled explicitly if the resulting model is to have real world validity and for confident engineering decisions to be made. This limited storage availability may be classified into two types: dedicated and process. Dedicated storage describes the case where dedicated tanks are available to hold an intermediate material. Both the capacity and identity of these tanks must be modeled explicitly because lots cannot be mixed. Process vessel storage takes place when the material produced by a given process step can be stored only in the vessel where it was produced. When process vessel storage is used, no activity can occur in that processing vessel until all of the stored material has been removed and fed to the downstream stage. Thus, even the preparation or cleanout activities must be delayed until the vessel is finished being used as a storage tank for the previous batch. In this process, there are many nontrivial material transfers that tie up both the feeding and receiving tanks, e.g., charging cycles to a column from an eluate tank. These were modeled in detail because they can force delays in processing the subsequent lot. The resulting model consists of 22 mainline manufacturing steps, 151 pieces of equipment, 479 materials, 353 activities (tasks), and 186 resources (e.g., operators, etc). For a representative instance, the final solution to mathematical programming formulation contained approximately 87,500 nonzero continuous variables, approximately 19,250 nonzero binary variables, and approximately 300,000 active nontrivial constraints. All of the instances reported below had comparable final solution statistics.

Figure 3. Material storage necessitated by a column repack
The manufacturing process begins with the thaw of a working cell bank that is fed to the flasks in the inoculum stage. Through subsequent scale-up processing, a new lot becomes available approximately every 3.5 days. By using parallel scale-up equipment, lots can be produced at a variety of rates. From an operational perspective, it is desirable to schedule bioreactor starts at regular intervals (cadence). This cadence is one of the main operational parameters investigated during the design study. Faster cadence produces more batches, but increases the likelihood of batch failure because the chance of holding a batch longer than the permitted hold time increases. By exploring cadences ranging from one lot every two days to one lot every four, we studied the effect of cadence on realizable plant capacity. Because of the biological nature of the process, the production bioreactors experience variability in run length, volume, and titers. As described above, the processing time varies from 9 to 11 days. The run length of the production bioreactor also affects the volume and titers of the resulting product. The volume and titer variables in turn affect the batch size and processing time of the columns. We modeled this behavior by specifying a series of time-volume-titer tuples that represent possible outcomes from bioreactor harvests—the Monte Carlo versions of the stochastic tasks described above. Each tuple has a specific probability of occurrence. We modeled product demands so as to require a single bioreactor lot for each demand. In this way, we are able to select among the literal time-volume-titer tuples for each lot/demand with a random number generator used by the solution algorithm. Then, during the scheduling of this particular lot, the algorithm only authorizes steps whose processing times and batch sizes correspond to the tuple that was randomly assigned to this demand. In addition to variability in bioreactor performance, we modeled a 1% random failure rate on bioreactor batches to account for the anticipated likelihood of both primary and backup sterility failure. The purification system begins with a series of four chromatography columns. The columns are not large enough to consume an entire bioreactor lot in one pass, so each lot is fed into the columns in a series of cycles. Here the behavior of the process is affected by the stochastic variability inherent in the bioreactors. The columns require periodic repacking after a given number of cycles. During repacking, the column is out of service for two days. This requirement introduces irregularities into the schedules, making it very difficult to analyze plant capacity using simplified methods like spreadsheets. Because lots with higher volume and titer load the column more heavily, we weigh the cycles of these lots more heavily, meaning that they will cause the column repacks to occur with a greater frequency. This accurately represents the true variability expected in actual plant operation.

Most of the equipment in this process requires a CIP procedure to be performed after each batch. This is an example of implementing process-specific constraints in the custom logic layer. The solution algorithm schedules CIP tasks as soon as possible after each piece of equipment completes a batch and assumes that equipment can be held in a clean state until its next use. This assumption was validated by inspection of early Gantt charts and expectation of clean-hold times. Because CIP requires a CIP skid, and a limited number of skids are available, manufacturing can be delayed because of this activity. This allows us to determine the minimum number of skids required if the plant is to avoid losing capacity because of CIP, and to assess the capacity reduction expected if fewer skids are provided.

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