Elsevier

Journal of Biotechnology

Volume 117, Issue 4, 29 June 2005, Pages 407-419
Journal of Biotechnology

Dynamic optimization of bioprocesses: Efficient and robust numerical strategies

https://doi.org/10.1016/j.jbiotec.2005.02.013Get rights and content

Abstract

The dynamic optimization (open loop optimal control) of non-linear bioprocesses is considered in this contribution. These processes can be described by sets of non-linear differential and algebraic equations (DAEs), usually subject to constraints in the state and control variables. A review of the available solution techniques for this class of problems is presented, highlighting the numerical difficulties arising from the non-linear, constrained and often discontinuous nature of these systems.

In order to surmount these difficulties, we present several alternative stochastic and hybrid techniques based on the control vector parameterization (CVP) approach. The CVP approach is a direct method which transforms the original problem into a non-linear programming (NLP) problem, which must be solved by a suitable (efficient and robust) solver. In particular, a hybrid technique uses a first global optimization phase followed by a fast second phase based on a local deterministic method, so it can handle the nonconvexity of many of these NLPs.

The efficiency and robustness of these techniques is illustrated by solving several challenging case studies regarding the optimal control of fed-batch bioreactors and other bioprocesses. In order to fairly evaluate their advantages, a careful and critical comparison with several other direct approaches is provided. The results indicate that the two-phase hybrid approach presents the best compromise between robustness and efficiency.

Introduction

In recent years, many efforts have been devoted to the model-based optimization of processes in biotechnology and bioengineering. An example of a problem which has received major attention is the dynamic optimization (open loop optimal control) of fed-batch bioreactors, as reviewed by Johnson, 1987, Rani and Rao, 1999 and, more recently, by Banga et al. (2003). Dynamic optimization allows the computation of the optimal operating policies for these units, e.g. the best time-varying feed rate(s) which ensure the maximization of a pre-defined performance index (usually, a productivity, or an economical index derived from the operation profile and the final concentrations). Once computed in a reliable way, these operating policies can be implemented using different control strategies, such as adaptive control (Smets et al., 2004) or model predictive control (Mahadevan and Doyle, 2003).

Most bioprocesses have highly non-linear dynamics, and constraints are also frequently present on both the state and the control variables. Thus, efficient and robust dynamic optimization methods are needed in order to successfully obtain their optimal operating policies. In this work, the general problem of dynamic optimization of non-linear bioprocesses with unspecified final time is considered. Several solution strategies, both deterministic and stochastic, are compared based on their results for three challenging case studies: the optimal operation of two fed-batch bioreactors and the optimal drug scheduling for cancer chemotherapy. A hybrid (stochastic-deterministic) approach is also presented and evaluated, showing significant advantages over the other methods in terms of robustness and computational effort.

Section snippets

Problem statement

The general dynamic optimization (optimal control) problem of a bioprocess, considering a free terminal time, can be stated as finding the control vector u_{t} and the final time tf to minimize (or maximize) a performance index J[x_,u_]:J[x_,u_]=Θ[x_{tf}]+t0tfΦ[x_{t},u_{t},t]dtsubject to a set of ordinary differential equality constraints, Eq. (2):dx_dt=Ψ[x_{t},u_{t},t]where x is the vector of state variables, with initial conditions x{t0}=x_0, and also subject to sets of algebraic equality

Review of solution methods

The dynamic optimization of fed-batch bioreactors is a very challenging problem due to several reasons:

  • first, the control variable (e.g. feed rate) often appears linearly in the system differential equations, so the problem is singular, creating additional difficulties for its solution (especially using indirect methods, as discussed below). For this type of problems, the optimal operating policy will be either bang-bang, or singular, or a combination of both.

  • second, most bioprocesses have

Methods considered

This contribution has two main objectives. First, to present a careful comparison of available recent approaches for the dynamic optimization of non-linear bioprocesses. The purpose of this comparison is to present a critical review which can serve as a guideline for the selection of suitable solvers for other similar problems. Second, to illustrate how a suitable hybrid method presents the best efficiency and robustness for the solution of these problems. The following deterministic (local

Case studies

Three challenging case studies are examined here. For the sake of brevity, only a brief description of each case study is given. The detailed statements of the dynamic optimization problems are given in the cited references.

Results and discussion

In order to perform a proper comparative evaluation of the methods considered, quality of the solution and computational cost (i.e. performance index values and computation times) will be considered. Further, these results will also be compared with the best found in the literature when possible.

In order to ensure high accuracy and consistent results, in all the CVP-based methods the relative and absolute error tolerances for integrations of the system dynamics were set to tight values ranging

Conclusions

In this work, we have compared several direct numerical methods for the optimal control of non-linear bioprocesses, a problem of major importance in bioprocess engineering. The methods considered here essentially cover most of the currently known direct approaches, including control vector parameterization (CVP) and complete parameterization, and deterministic and stochastic solvers for the resulting non-linear programming problems.

Considering three challenging case studies, many of the

Acknowledgements

The authors thank the Spanish Ministry of Science and Technology (MCyT project AGL2001-2610-C02-02) and Xunta de Galicia (grant PGIDIT02PXIC40211PN) for financial support.

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