ORIGINAL ARTICLE An improved curvilinear gradient method for parameter optimization in complex biological models David Szekely • Jamie I. Vandenberg • Socrates Dokos • Adam P. Hill Received: 6 March 2010 / Accepted: 8 July 2010 / Published online: 30 July 2010 Ó International Federation for Medical and Biological Engineering 2010 Abstract Mathematical modeling is an often used approach in biological science which, given some under- standing of a system, is employed as a means of predicting future behavior and quantitative hypothesis testing. How- ever, as our understanding of processes becomes more in depth, the models we use to describe them become corre- spondingly more complex. There is a paucity of effective methods available for sampling the vast objective surfaces associated with complex multiparameter models while at the same time maintaining the accuracy needed for local evaluation of minima—all in a practical time period. We have developed a series of modifications to the curvilinear gradient method for parameter optimization. We demon- strate the power and efficiency of our routine through fit- ting of a 22 parameter Markov state model to an electrophysiological recording of a cardiac ion channel. Our method efficiently and accurately locates parameter minima which would not be easily identified using the currently available means. While the computational over- head involved in implementing the curvilinear gradient method may have contributed to resistance to adopting this technique, the performance improvements allowed by our modifications make this an extremely valuable tool in development of models of complex biological systems. Keywords Modeling  Curvilinear gradient method  Parameter optimization  Minimization  Biological models 1 Introduction The power of mathematical modeling has broad-reaching applications in many disciplines of biological and medical science [7]. It has, however, proved particularly useful in describing stochastic processes such as ion channel gating which underlies neuronal and cardiac electrical activity. Within this subject area, modeling has provided an enor- mous amount of insight into both physiological and path- ophysiological processes [6, 12, 21, 22]. Central to all modeling is the optimization of the parameters within the model to fit observable data. Least- squares parameter optimization of a simple model, for example, a two-state chemical equilibrium characterized by the equation: S 1  k 1 k 1 S 2 ; is straight forward since the objective surface is simple and contains only a single minimum (Fig. 1a). Fitting more complex models, how- ever, is often an onerous task since the size and complexity of the search space (and hence, computation time) grow exponentially with increases in the number of parameters and the breadth of the constraints for each parameter (Fig. 1b, c). In these cases, an exhaustive gridsearch is clearly not an efficient approach, so highlighting the need for algorithms which are able to efficiently and effectively sample the large parameter spaces characteristic of com- plex models. Electronic supplementary material The online version of this article (doi:10.1007/s11517-010-0667-1) contains supplementary material, which is available to authorized users. D. Szekely  J. I. Vandenberg  A. P. Hill (&) Mark Cowley Lidwill Program in Cardiac Electrophysiology, Victor Chang Cardiac Research Institute, 405 Liverpool Street, Darlinghurst, NSW 2010, Australia e-mail: a.hill@victorchang.edu.au D. Szekely  J. I. Vandenberg  A. P. Hill St Vincents Clinical School, University of New South Wales, Victoria Street, Darlinghurst, NSW 2010, Australia S. Dokos Graduate School of Biomedical Engineering, University of New South Wales, Sydney, NSW 2052, Australia 123 Med Biol Eng Comput (2011) 49:289–296 DOI 10.1007/s11517-010-0667-1