3D current–voltage–time surfaces unveil critical repolarization differences underlying similar cardiac action potentials: A model study M. Zaniboni ⇑ Dipartimento di Biologia Evolutiva e Funzionale – Sezione Fisiologia, Università degli Studi di Parma, V.le G.P. Usberti 11 A, 43124 Parma, Italy article info Article history: Received 12 February 2011 Received in revised form 18 June 2011 Accepted 27 June 2011 Available online 12 July 2011 Keywords: Cardiac action potential Ventricular repolarization Cardiac mathematical models Electrotonic interactions Models parameter choice abstract The number of mathematical models of cardiac cellular excitability is rapidly growing, and compact graphical representations of their properties can make new acquisitions available for a broader range of scientists in cardiac field. Particularly, the intrinsic over-determination of the model equations systems when fitted only to action potential (AP) waveform and the fact that they are frequently tuned on data covering only a relatively narrow range of dynamic conditions, often lead modellers to compare very sim- ilar AP profiles, which underlie though quite different excitable properties. In this study I discuss a novel compact 3D representation of the cardiac cellular AP, where the third dimension represents the instan- taneous current–voltage profile of the membrane, measured as repolarization proceeds. Measurements of this type have been used previously for in vivo experiments, and are adopted here iteratively at a very high time, voltage, current-resolution on (i) the same human ventricular model, endowed with two dif- ferent parameters sets which generate the same AP waveform, and on (ii) three different models of the same human ventricular cell type. In these 3D representations, the AP waveforms lie at the intersection between instantaneous time–voltage–current surfaces and the zero-current plane. Different surfaces can share the same intersection and therefore the same AP; in these cases, the morphology of the current sur- face provides a compact view of important differences within corresponding repolarization dynamics. Refractory period, supernormal excitability window, and extent of repolarization reserve can be visu- alized at once. Two pivotal dynamical properties can be precisely assessed, i.e. all-or-nothing repolariza- tion window and membrane resistance during recovery. I discuss differences in these properties among the membranes under study, and show relevant implications for cardiac cellular repolarization. Ó 2011 Elsevier Inc. All rights reserved. 1. Introduction Mathematical modelling of cardiac membrane excitability has paralleled over the years, as modelling does in any other branch of modern science, our rational knowledge of the mechanisms gov- erning the phenomenon. The field has been originally established by Van der Pol and Van der Mark in 1920s [1], and later extended by Fitzhugh to a simplified Hodgkin Huxley equations system [2]. However, it has only been in 1960 that Noble based his mathemat- ical model of cardiac action potential (AP) on actual experimental data [3], and in 1977 that Beeler and Reuter were able to recon- struct a ventricular myocardial AP [4]. Since then, many key con- cepts, like threshold potential, refractoriness, and all-or-nothing- repolarization (AONR), have been clarified and carefully described for cardiac membrane excitability (e.g. [5,6]). Given the rather complex formalism involved and the rapidly growing number of available models (e.g. [7–9]) though, not always physiologists, pharmacologists, or cardiologist, can easily take all these features into account in their studies, which often require screening only one or few parameters and discriminating between physiological and pathological states. An inherent problem with mathematical modelling is the generality of the model, i.e. how good the equa- tions system can predict different dynamical conditions of the ob- ject under study. This is always challenging, particularly when (1) the complexity of description increases, (2) the mathematical ap- proaches to modelling are not unique, and (3) models are tuned to fit only one or few experimental variables among the many that are involved. A clear example of this can be found in the attempt of modelling the electrophysiology of a very specific cardiac cell type; at least three different models have been recently developed in or- der to reproduce the human ventricular epicardial AP: the Priebe and Beuckelmann model [10], the Ten Tusscher et al. model [11], and the Iyer et al. model [12]. Although these three models repro- duce rather similar AP waveforms and meet several in vivo ob- served dynamical properties, nevertheless they have already been shown to differ substantially, for example, in the role of in- wardly rectifying potassium currents to repolarization [13]. Another example of this type has recently been described by Cherry and Fenton in the case of dog canine ventricular AP [7], 0025-5564/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.mbs.2011.06.008 ⇑ Tel.: +39 0521 905623; fax: +39 0521 905673. E-mail address: massimiliano.zaniboni@unipr.it Mathematical Biosciences 233 (2011) 98–110 Contents lists available at ScienceDirect Mathematical Biosciences journal homepage: www.elsevier.com/locate/mbs