Canonical modeling: review of concepts with emphasis on environmental health

The article reviews concepts of canonical modeling in the context of environmental health. Based on biochemical systems theory, the canonical approach was developed over the past thirty years and applied to complex systems primarily in biochemistry and the regulation of gene expression. Canonical modeling is based on nonlinear ordinary differential equations whose right-hand sides consist of products of power-law functions. This structure results from the linearization of complex processes in logarithmic space. The canonical structure has many intriguing features. First, almost any system of smooth functions or ordinary differential equations can be recast equivalently in a canonical model, which demonstrates that the model structure is rich enough to deal with all relevant nonlinearities. Second, a large body of successful applications suggests that canonical models are often valid and accurate representations of quite complex, real-world systems. Third, a set of guidelines supports the modeler in all phases of analysis. These guidelines address model design, algebraic and numerical analysis, and the interpretation of results. Fourth, the structure of canonical models, especially those in S-system form, facilitates algebraic and numerical analyses. Of particular importance is the derivation of steady-state solutions in an explicit symbolic or numerical form, which allows further assessments of stability and robustness. The homogeneous structure of canonical models has also led to the development of very efficient, customized computer algorithms for all steps of a typical analysis. Fifth, a surprising number of models currently used in environmental health research are special cases of canonical models. The traditional models are thus subsumed in one modeling framework, which offers new avenues of analysis and interpretation.