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Human ventricular myocyte models

Introduction

Mathematical modelling of cells began over 50 years ago with the work of Hodgkin and Nobel Prize winner Huxley (Hodgkin-Huxley, 1952). The first computer model of cardiac tissue was developed by Noble in 1962 (Noble, 1962) from the University of Oxford. During the early years the main restraint of the models including those commonly assessed as accurate like Luo-Rudy II (Luo-Rudy, 1994) was the source of experimental data as they came only from the animal studies. In 1998 Priebe and Beuckelmann published new model based on the Luo-Rudy but they replaced animal data (guinea pigs) for the ionic currents (IKr , IKs , ICa , Ito, IK1) with information derived from the studies on human miocytes and accordingly fitted the equations (Priebe-Beuckelmann, 1998). Priebe and Beuckelmann used their model to compare the electrophysiological properties of healthy and failing ventricular myocytes. The model was accurate, but its complexity and large number of variables make it computationally inefficient. In addition, as a , made up of ordinary differential equations describing the time dependence of membrane potential, gating variables and ion concentrations, the Priebe- Beuckelmann model was unstable. During the following years models development process was focused on the computational efficiency and stability improvement (mathematical side) and what is even more important on the animal to human data replacement (physiological side). Most commonly used and cited models were developed by Ten Tusscher (Ten Tusscher, 2004), Bernus (Bernus, 2002), Iyer (Iyer, 2004), Fink (Fink, 2008), Bueno-Orovio (Bueno-Orovio, 2008). The main scientific aims for their use were heart physiology and pathophysiology in term of the cardiac arrhythmias mechanism investigation but nowadays new, specialized models have been created which aim primarily in the influence of various ionic currents on cells action potential investigation (i.e. hERG potassium channels - Peitersen, 2008; Fink, 2008).

Mathematical background

As it was stated before almost all mathematical models describing heart cells electric activity are based on the primary Hodgkin-Huxley model assumptions. It includes one of the most commonly cited Ten Tusscher model (Ten Tusscher, 2004) where the cell membrane is treated as a capacitor connected with resistances and batteries representing ionic currents. Voltage values changes of a single cell (V) in time t (t) is derived from the cell capacitance per unit surface area (Cm), and sum of all transmembrane ionic currents (Iion) and externally applied stimulus current (Istim).

Assuming lack of discrete character of microscopic cardiac cell structure, a two-dimensional surface of cardiac cell is modelled with use of the partial differential equations. In such case additional parameters are taken under consideration including cellular resistivity and surface-to-volume ratio. It is possible to tracks not only action potential (AP) changes in time but also any other parameter.

Usage

Above described models can be mathematically implemented in any accessible computational environment, where ordinary and partial differential equations solvers could be execute including Matlab, R, Octave and other. Nevertheless due to popularity and problem importance there are specialized systems enabling implementation, saving, sharing and group work on the models. Good example of such activity is project lead by the Faculty of Physiology, Anatomy and Genetics Oxford University system called COR (Cellular Open Resource) based on the CellML environment. CellML is a language created and maintained by University of Auckland scientists and the core of this language is based on the XML and MathML standard. The main purpose of the project is to facilitate the models sharing (no necessarily biologically driven) between various scientific centres. Basic elements of every single model stored in the internet repository (www.cellml.org) include: model structure, equations, metadata and documentation. To use the downloaded model in a user friendly mode one can use OpenCell system. All described above elements are freely available as an OpenSource and are components of larger project – Physiome Project which is a worldwide public domain effort to provide a computational framework for understanding human and other eukaryotic physiology. It aims to develop integrative models at all levels of biological organisation, from genes to the whole organism via gene regulatory networks, protein pathways, integrative cell function, and tissue and whole organ structure/function relations (www.physiome.org.nz). Such activities stays in agreement with the general policy described as 3R (Reduce, Reuse, Recycle) what in the drug development area can be directly translated into the quantity of laboratory animals used during the research. One of the arguments which support such ideas is lack of possibilities to observe inter-individual variability based on results from animal studies what in reality exclude the chance of the extreme observations finding. They can be observed during the clinical trials and careful in-depth investigation is always needed to find the mechanism of their presence. Computational modelling and simulation systems use for automatic in vitro – in vivo extrapolation have been recognize as a valuable tools in some situations – as described above – superior in comparison with animal models as they enable to assess the variability in drug behaviour and human organism reaction. It’s been seen in the FDA policy which officially and consequently supports and effectively uses them on the daily basis.