forward problem of electrocardiology forward problem of electrocardiology the forward problem of electrocardiology is a computational and mathematical approach to study the electrical activity of the heart through the body surface. the principal aim of this study is to computationally reproduce an electrocardiogram (ecg), which has important clinical relevance to define cardiac pathologies such as ischemia and infarction, or to test pharmaceutical intervention. given their important functionalities and the relative small invasiveness, the electrocardiography techniques are used quite often as clinical diagnostic tests. thus, it is natural to proceed to computationally reproduce an ecg, which means to mathematically model the cardiac behaviour inside the body. simulation of a realistic heart beat. schematic diagram of normal sinus rhythm for a human heart as seen on ecg. the three main parts of a forward model for the ecg are: a model for the cardiac electrical activity; a model for the diffusion of the electrical potential inside the torso, which represents the extracardiac region; some specific heart-torso coupling conditions. thus, to obtain an ecg, a mathematical electrical cardiac model must be considered, coupled with a diffusive model in a passive conductor that describes the electrical propagation inside the torso. the coupled model is usually a three-dimensional model expressed in terms of partial differential equations. such model is typically solved by means of finite element method for the solution's space evolution and semi-implicit numerical schemes involving finite differences for the solution's time evolution. however, the computational costs of such techniques, especially with three dimensional simulations, are quite high. thus, simplified models are often considered, solving for example the heart electrical activity independently from the problem on the torso. to provide realistic results, three dimensional anatomically-realistic models of the heart and the torso must be used. another possible simplification is a dynamical model made of three ordinary differential equations. heart tissue models the electrical activity of the heart is caused by the flow of ions across the cell membrane, between the intracellular and extracellular spaces, which determines a wave of excitation along the heart muscle that coordinates the cardiac contraction and, thus, the pumping action of the heart that enables it to push blood through the circulatory system. the modeling of cardiac electrical activity is thus related to the modelling of the flow of ions on a microscopic level, and on the propagation of the excitation wave along the muscle fibers on a macroscopic level. between the mathematical model on the macroscopic level, willem einthoven and augustus waller defined the ecg through the conceptual model of a dipole rotating around a fixed point, whose projection on the lead axis determined the lead recordings. then, a two-dimensional reconstruction of the heart activity in the frontal plane was possible using the einthoven's limbs leads i, ii and iii as theoretical basis. later on, the rotating cardiac dipole was considered inadequate and was substituted by multipolar sources moving inside a bounded torso domain. the main shortcoming of the methods used to quantify these sources is their lack of details, which are however very relevant to realistically simulate cardiac phenomena. on the other hand, microscopic models try to represent the behaviour of single cells and to connect them considering their electrical properties. these models present some challenges related to the different scales that need to be captured, in particular considering that, especially for large scale phenomena such as re-entry or body surface potential, the collective behaviour of the cells is more important than that of every single cell. the third option to model the electrical activity of the heart is to consider a so-called "middle-out approach", where the model incorporates both lower and higher level of details. this option considers the behaviour of a block of cells, called a continuum cell, thus avoiding scale and detail problems. the model obtained is called bidomain model, which is often replaced by its simplification, the monodomain model. bidomain model