Bioreactors are used in a wide variety of processes and industries, from food processing to waste treatment. Typically, in these reactors, one or more organisms act on one or more substrates, creating more organisms together with desired chemical products. For the analysis, design and operation of bioreactor systems, one may need to consider their dynamics over a wide variety of conditions. Since performing physical experiments is likely to be impractical, numerical models are commonly used in the analysis of such systems. These models typically involve some degree of uncertainty, as some parameters and/or initial conditions may be known only to within some interval. In some cases, information on the probability distribution of the uncertain values may also be available. For analysis of the impact of such uncertainties on modeling results, a Monte Carlo simulation approach is widely used. However, since it is not possible to sample the complete space of the uncertain quantities in a finite number of simulations, the results of Monte Carlo analysis may fail to capture some system behaviors, especially in the case of nonlinear systems.

In this presentation, we demonstrate a method for the verified solution of nonlinear bioreactor models. This method computes rigorous bounds on the concentration profiles in the reactor over a given time horizon, based on specified ranges for uncertain parameters and/or initial conditions. The method is based on the general approach described by Lin and Stadtherr [1], which uses an interval Taylor series to represent dependence on time, and uses Taylor models to represent dependence on uncertain quantities. We also demonstrate an approach for the propagation of uncertain probability distributions in one or more model parameter and/or initial condition through the bioreactor model. Assuming an uncertain probability distribution for each parameter and/or initial condition of interest, we use a method, based on Taylor models and probability boxes (p-boxes) and recently described by Enszer et al. [2], that propagates these distributions through the dynamic model. As a result, we obtain a p-box describing the probability distribution for each state variable at any given time of interest. As opposed to the traditional Monte Carlo simulation approach, which may not properly bound all possible system outputs, these Taylor model methods for verified uncertainty analysis provide completely rigorous results that fully capture all possible system behaviors under the uncertain conditions. These methods are tested using a variety of nonlinear bioreactor models, with uncertainties in some parameters and/or initial conditions. Comparisons are made to the results of Monte Carlo simulations.

[1] Lin, Y., Stadtherr, M.A. Validated Solutions of Initial Value Problems for Parametric ODEs. Applied Numerical Mathematics, 57: pp. 1145-1162, 2007.

[2] Enszer, J.A., Lin, Y., Ferson, S., Corliss, G.F., Stadtherr, M.A. Propagating Uncertainties in Modeling Nonlinear Dynamic Systems. In Proceedings of the 3rd International Workshop on Reliable Engineering Computing, Georgia Institute of Technology, Savannah, GA: pp. 89-105, 2008.