(316a) Nonsmooth Dynamic Optimization for Bioreactors with Flux Balance Models Embedded

A nonsmooth dynamic optimization algorithm for fed-batch biochemical reactor models based on dynamic flux balance analysis (DFBA) is presented.

Large-scale production of renewable liquid fuels through biochemical conversion is a growing market. The development of optimal process control strategies for this and similar processes is an important research problem. Optimal model-based fed-batch control strategies rely on the predictive capabilities of the process model. Dynamic flux balance analysis is a promising modeling framework that has these desired capabilities. DFBA assumes that the intracellular species are at equilibrium with the extracellular environment. The resulting underdetermined stoichiometric model is solved under the assumption of a cellular objective such as growth rate maximization. The cell model is coupled with the dynamic mass balance equations of the extracellular environment via expressions for the rates of substrate uptake and product excretion, which imposes additional constraints on the linear program (LP) defined by growth rate maximization of the cell. The linear program is embedded in the dynamic model of the bioreactor, giving an accurate model of the substrate consumption and biomass production during operation. The overall DFBA model is a system of ordinary differential equations for which the evaluation of the right-hand side requires not only function evaluations, but also the solution of one or more linear programs. The resulting dynamic optimization problem is a bilevel optimization problem, where the upper level aims to maximize the final desired product yield of the batch, while the lower level, representing the cell model, is optimizing the biomass generation.

Optimal fed-batch control for DFBA models faces two challenges: First, numerical integration of DFBA models requires a fast, scalable and reliable algorithm that takes into account the special structure of the system, since the naive approach of solving the LP at each time step is inaccurate, due to numerical noise from the LP solver, and inefficient, due to the computational cost of solving the LP, especially for large problems. Second, due to the nonsmoothness property the sensitivities with respect to the control inputs are possibly set-valued. This has to be accounted for during integration and optimization.

The proposed optimization algorithm overcomes both challenges. A novel algorithm, which efficiently integrates DFBA models, called DSL48LPR, is considered here. Using KKT theory, the DFBA model is rewritten as a hybrid differential-algebraic equation (DAE) system such that the embedded LP only has to be solved once at events where the hybrid mode switches. The hybrid modes can be characterized as regimes in which the intracellular kinetics are limited by different combinations of uptake constraints, e.g., aerobic and anaerobic growth phase. 

Using the concept of the generalized Jacobian, a set-valued map, a nonsmooth bundle solver in combination with the DSL48LPR yields an optimal solution. Due to the semismoothness property of the generalized Jacobian, we can compute an element of the generalized Jacobian to be passed to the nonsmooth bundle solver.