Living organisms can not be understood by analyzing individual components but analyzing the interactions among those components
[1, 2]. In this regard, many efforts are being devoted to formulate mathematical models that enable the possibility of developing and testing new hypotheses about biological systems.
In recent years the use of optimization techniques for the purpose of modeling has attracted substantial attention. In particular, mathematical optimization is the underlying hypothesis for model development in, for example, flux balance analysis
, or the activation of metabolic pathways
[4–6] and is at the core of model identification, including parameter estimation and optimal experimental design
Despite the success of modeling efforts in systems biology, the truth is that only in few occasions those models have been used to design or to optimize desired biological behaviors. This may be explained by the difficulty on formulating and solving those problems but also in the limited number of software tools that may be used for that purpose
. In this regard, the recently developed toolbox DOTcvpSB
 can handle the dynamic optimization of lumped systems (described in terms of ordinary differential equations), such as those related to biochemical processes (see the reviews by Banga et al.
[7, 8, 10] and the works cited therein), or to biomedical systems
It should be noted, however, that many biological systems of interest are being modelled by sets of partial differential equations (PDE). This is particularly the case of reaction diffusion waves in biology (see the recent review by
) or spatial organization in cell signaling
. The scarce works related to the optimization of this type of systems
[19, 20] reveal that the problem presents significant computational and conceptual challenges due mainly to the presence of suboptimal solutions and to the computational cost associated to the simulation and, thus, the optimization.
The use of global optimization techniques provides guarantees, at least in a probabilistic sense, of arriving to the global solution. Unfortunately the price to pay is the number of cost function evaluations and the associated computational cost, which increase exponentially with the number of decision variables. This aspect is particularly critical for PDE systems as they are usually solved with spatial discretization techniques (e.g. finite element or the finite differences methods) and the result is a large scale dynamic system whose simulation may take from several seconds to hours.
In this concern, the use of surrogate models has been proposed as the alternative to reduce total computation times. The most promising techniques based on kriging or radial basis functions have been incorporated to global optimization solvers
[21–23]. However these methodologies do not integrate any knowledge about the system being optimized, i.e. models are treated as blackboxes. Alternatives for PDE systems rely on the application of reduced order modeling techniques which take into account the phenomena of interest. In particular the use of the proper orthogonal decomposition (POD) approach has demonstrated to be an excellent candidate for simulation, optimization and control
This work presents the application of hybrid optimization techniques for the solution of complex dynamic optimization problems related to biological applications. Particular emphasis is paid to the efficiency and robustness of the proposed methodologies. In this regard, the use of a hybrid global-local methodology together with a control refining technique is proposed. In addition, the POD technique is used to reduce the dimensionality (and thus the computational effort) of the original distributed (full scale) models.
To illustrate the usage and advantages of the proposed techniques two challenging case studies will be considered. The first is related to bacterial chemotaxis and considers the achievement of two different objectives as formulated in
. In addition, a second dynamic optimization problem related to the FitzHugh-Nagumo (FHN) model
[27, 28], which describes a number of important physiological processes, such as the heart behavior, is formulated and solved.