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.
[1] 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.
Thus, simplified models are often considered, solving for example the heart electrical activity independently from the problem on the torso.
[1][4] 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.
[5] 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.
[4] On the other hand, microscopic models try to represent the behaviour of single cells and to connect them considering their electrical properties.
[6][7][8] 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.
[4] The basic assumption of the bidomain model is that the heart tissue can be divided in two ohmic conducting continuous media, connected but separated through the cell membrane.
[2] The boundary conditions for this version of the bidomain model are obtained through the assumption that there is no flow of intracellular potential outside of the heart, which means that
In the forward problem of electrocardiography, the torso is seen as a passive conductor and its model can be derived starting from the Maxwell's equations under quasi-static assumption.
[1][2] The standard formulation consists of a partial differential equation with one unknown scalar field, the torso potential
First of all, since the electrical and magnetic activity inside the body is generated at low level, a quasi-static assumption can be considered.
Thus, the body can be viewed as a passive conductor, which means that its capacitive, inductive and propagative effect can be ignored.
[1] Finally, since aside from the heart there is no current source inside the torso, the current per unit volume can be set to zero, giving the generalized Laplace equation, which represents the standard formulation of the diffusive problem inside the torso[1]
The boundary conditions accounts for the properties of the media surrounding the torso, i.e. of the air around the body.
A simple example of conductivity parameters in a torso that considers the bones and the lungs is reported in the following table.
[2] Boundary conditions that represent a perfect electrical coupling between the heart and the torso are the most used and the classical ones.
However, between the heart and the torso there is the pericardium, a sac with a double wall that contains a serous fluid which has a specific effect on the electrical transmission.
Defining the electrodes positions on the torso, it is possible to find the time evolution of the potential on such points.
[2] The heart-torso models are expressed in terms of partial differential equations whose unknowns are function of both space and time.
They are in turn coupled with an ionic model which is usually expressed in terms of a system of ordinary differential equations.
This means that the bidomain and monodomain models can be solved for example with a backward differentiation formula for the time discretization, while the problems to compute the extracellular potential and torso potential can be easily solved by applying only the finite element method because they are time independent.
[2][11] To simulate and electrocardiogram using the fully coupled or uncoupled models, a three-dimensional reconstruction of the human torso is needed.
For example, the Visible Human Data[13] is a useful dataset to create a three-dimensional torso model detailed with internal organs including the skeletal structure and muscles.
[3] The quasi-periodicity of the heart beat is reproduced by a three-dimensional trajectory around an attracting limit cycle in the
The equations can be easily solved with classical numerical algorithms like Runge-Kutta methods for ODEs.
