Nonsmooth Modeling for Simulation and Optimization of Continuous Pharmaceutical Manufacturing Processes

This chapter advocates a nonsmooth differential-algebraic equations (DAEs) modeling paradigm for dynamic simulation and optimization of continuous pharmaceutical process operations. Process intensification of pharmaceuticals production by continuous manufacturing is challenged by short campaigns, due to the low volume of some products. This calls for optimization of the entire dynamic campaign to achieve the highest overall yields, productivity and minimize waste. A variety of processes prevailing in the pharmaceutical industry are traditionally viewed as exhibiting hybrid continuous and discrete (i.e. discontinuous) behavior. In many cases such behavior is actually nonsmooth, rather than discontinuous. For such cases a distinction has not always been made, and these can be readily modeled by nonsmooth DAEs. A computationally relevant theory of nonsmooth DAEs (i.e. well-posedness and sensitivity analysis) has recently been established which is suitable for numerical implementations that scale efficiently for large-scale dynamic optimization problems. This chapter gives a brief overview of the nonsmooth DAEs framework. Such a formulation has been recently shown to admit a unique solution and computationally tractable sensitivity analysis, facilitating its embedding in optimization schemes to find optimal operational procedures. The mathematical modeling of selected processes using nonsmooth DAEs is outlined in detail. The nonsmooth dynamics of several units are illustrated. We then demonstrate how the overall campaign performance can be optimized in terms of on-specification productivity and yield, which are considered as objectives given a short time horizon for the production campaign. The optimal solutions give very valuable insights regarding the best start-up and shutdown procedures.