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From Heart to Body Surface: What is the Mathematical Model Behind ECG?
In today's medical community, electrocardiogram (ECG) is an important tool for assessing heart health, but the mathematical model behind it is often unknown. Exploring the electrical activity of the heart and how to obtain electrical signals through the body surface are forward-looking questions in electrocardiology. An important computational goal is to reconstruct a clinically meaningful ECG to identify various cardiac pathologies such as ischemia or heart attack.
This modeling process mainly combines the electrical activity model of the heart, the potential diffusion model in the torso, and the coupling conditions between them. To obtain an ECG, a mathematical electrical heart model must be considered, placed within a diffusion model in a passive conductor. This coupled model is usually formulated in the form of partial differential equations and is often solved using finite element methods in order to track the evolution of the solution in three dimensions and in time with a semi-implicit numerical scheme.
However, such computational costs are quite high, especially in the case of three-dimensional simulations. Therefore, some simplified models are often considered that address the electrical activity of the heart and the torso independently.
Heart tissue model
The heart's electrical activity is caused by the flow of ions across cell membranes, which creates a wave of excitement in the heart muscle that coordinates the heart's contractions, allowing the heart to push blood into the circulatory system. Modeling of the electrical activity of the heart is therefore linked to modeling the flow of ions at the microscopic level and to the propagation of excitation waves at the macroscopic level.
The early conceptual model of electrocardiogram (ECG) was mainly proposed by Willem Einthoven and Augustus Waller. It was based on a dipole model rotating around a fixed point, and records were obtained through the projection of the lead axis. As research progressed, the model of the rotating cardiac dipole was replaced by the more complex multipole source model.
Dual domain model
The basic assumption of the dual-domain model is that cardiac tissue can be divided into two continuous ohmic conductive media - intracellular and extracellular regions. These regions represent cellular tissue and the spaces between cells, respectively. Although there are some theoretical imperfections in this model, it still reasonably captures realistic physiological phenomena related to transmembrane potential.
The standard formula of this model summarizes the behavior of the heart's transmembrane potential when an external current is applied. Mathematically, it shows the complexity of electrical current delivery in the heart in clinical practice.
Single domain model
The single-domain model is a simplified form of the dual-domain model that focuses on the transmission of transmembrane potential and ignores some unnecessary physiological assumptions in order to remain reasonable when explaining major physiological phenomena. Although this model is simplified, it can still provide some feasible explanations in electrophysiology.
Trunk tissue model
In the forward problem of electrocardiography, the torso is usually regarded as a passive conductor, the model of which can be derived from Maxwell's equations under quasi-static assumptions. The core of this model is to use the generalized Laplace equation to describe the potential distribution of the trunk, which in turn affects the reading results of the electrocardiogram.
For a healthy heart, many factors, such as the conductivity of the conductor, can affect the final ECG result, highlighting the importance of model accuracy.
Between these various models, whether it is from microscopic cell behavior to description of overall cardiac performance, the development of mathematical models provides necessary data and cognition for clinical practice, helping doctors make more accurate diagnoses. However, the use of simplified models may lead to deviations from the actual situation. In this case, are there other more effective ways to balance computational efficiency and model accuracy?
