Mathematical models in physiology

Computational modelling of biological processes and systems has witnessed a remarkable development in recent years. The search-term (modelling OR modeling) yields over 58000 entries in PubMed, with more than 34000 since the year 2000: thus, almost two-thirds of papers appeared in the last 5–6 years, compared to only about one-third in the preceding 5–6 decades.\ud \ud The development is fuelled both by the continuously improving tools and techniques available for bio-mathematical modelling and by the increasing demand in quantitative assessment of element inter-relations in complex biological systems. This has given rise to a worldwide public domain effort to build a computational framework that provides a comprehensive theoretical representation of integrated biological function—the Physiome.\ud \ud The current and next issues of this journal are devoted to a small sub-set of this initiative and address biocomputation and modelling in physiology, illustrating the breadth and depth of experimental data-based model development in biological research from sub-cellular events to whole organ simulations

Approximate Bisimulations for Sodium Channel Dynamics

Abstract. This paper shows that, in the context of the Iyer et al. 67-variable cardiac myocycte model (IMW), it is possible to replace the detailed 13-state continuous-time MDP model of the sodium-channel dy-namics, with a much simpler Hodgkin-Huxley (HH)-like two-state sodium-channel model, while only incurring a bounded approximation error. The technical basis for this result is the construction of an approximate bisim-ulation between the HH and IMW channel models, both of which are input-controlled (voltage in this case) continuous-time Markov chains. The construction of the appropriate approximate bisimulation, as well as the overall result regarding the behavior of this modified IMW model, in-volves: (1) The identification of the voltage-dependent parameters of the m and h gates in the HH-type channel, based on the observations of the IMW channel. (2) Proving that the distance between observations of the two channels never exceeds a given error. (3) Exploring the sensitivity of the overall IMW model to the HH-type sodium-channel approximation. Our extensive simulation results experimentally validate our findings, for varying IMW-type input stimuli

Numerical solution for optimal control of the reaction-diffusion equations in cardiac electrophysiology

Bidomain model, Reaction-diffusion equations, Ionic model, Optimal control with PDE constraints, Existence and uniqueness, FEM, Rosenbrock type methods, NCG method,