[Paper Review] Electrophysiology of living organs from first principles

Günter Scharf, Christoph Scharf|arXiv (Cornell University)|Jun 17, 2010

stochastic dynamics and bifurcation6 references3 citations

TL;DR

This paper derives the electrophysiology of living organs from first principles using spatial averaging of microscopic Maxwell's equations, showing that the primary source of measured bioelectric potentials is the polarization charge density at cell membranes, not conduction currents. It establishes that the macroscopic polarization (dipole) density is the key quantity for localizing cardiac arrhythmias, offering a physically rigorous alternative to models like the bidomain or volume conductor.

ABSTRACT

Based on the derivation of the macroscopic Maxwell's equations by spatial averaging of the microscopic equations, we discuss the electrophysiology of living organs. Other methods of averaging (or homogenization) like the bidomain model are not compatible with Maxwell's theory. We also point out that modeling the active cells by source currents is not a suitable description of the situation from first principles. Instead, it turns out that the main source of the measured electrical potentials is the polarization charge density which exists at the membranes of the active cells and adds up to a macroscopic polarization. The latter is the source term in the Laplace equation, the solution of which gives the measured far-field potential. As a consequence it is the polarization or dipole density which is best suited for localization of cardiac arrhythmia.

Motivation & Objective

• To provide a model-independent, first-principles description of bioelectric potentials in living organs based on macroscopic Maxwell's equations.
• To resolve inconsistencies in existing models like the bidomain and volume conductor approaches by deriving macroscopic electrodynamics from microscopic averaging.
• To identify the true physical source of measured electrical potentials in ECG, EEG, and EMG as the polarization charge density at cell membranes.
• To demonstrate that the dipole (polarization) density is the optimal quantity for localizing cardiac arrhythmia sources.

Proposed method

• Derives macroscopic Maxwell's equations via spatial averaging of microscopic Maxwell's equations, rigorously linking microscopic charge and current densities to macroscopic fields.
• Introduces the polarization vector field $\vec{P}$ as the key macroscopic quantity arising from spatial averaging, with $\varrho_{\rm pol} = -\nabla\cdot\vec{P}$ as the source of the electric potential.
• Uses the Laplace equation $\nabla^2 V = -\frac{1}{\varepsilon_0}(\varrho_{\rm pol} + \varrho_c)$ to describe the far-field potential, with $\varrho_{\rm pol}$ as the dominant source term.
• Models active cell membranes as a 2D surface $S$ with normal polarization $\vec{P}(\vec{x}) = \vec{n}d(\vec{x})\delta_S(\vec{x})$, leading to a surface integral for the potential via $V_{\rm pol}(\vec{x}) = \frac{1}{4\pi\varepsilon_0}\int_S d(\vec{y})\frac{\partial}{\partial n}\frac{1}{|\vec{x}-\vec{y}|}d\sigma_y$.
• Applies the limiting case of measurement at the membrane surface to show $V(\vec{x}_S) \approx -2\pi d(\vec{x}_S) + \text{background}$, enabling direct estimation of dipole density $d(\vec{x}_S)$.

Experimental results

Research questions

• RQ1 What is the true physical origin of the measured electrical potentials in ECG, EEG, and EMG from first principles?
• RQ2 Why are conventional models like the bidomain and volume conductor incompatible with Maxwell's theory?
• RQ3 How does spatial averaging of microscopic Maxwell's equations lead to a consistent macroscopic description of living tissue?
• RQ4 What is the role of conduction current versus polarization charge in generating the far-field potential?
• RQ5 Can the dipole density on the endocardial surface be directly inferred from measured potentials, and how accurately?

Key findings

• The main source of the measured electrical potential in living organs is the polarization charge density $\varrho_{\rm pol} = -\nabla\cdot\vec{P}$, not conduction currents.
• Conduction current $\vec{j}_c$ is a dissipative process and cannot be the primary source of the potential; it appears only as a correction in the macroscopic equations.
• The macroscopic polarization $\vec{P}$, arising from spatial averaging of microscopic charge distributions, is the fundamental physical quantity governing the potential field.
• The potential at the endocardial surface is directly proportional to the dipole density $d(\vec{x}_S)$, with $V(\vec{x}_S) \approx -2\pi d(\vec{x}_S)$, enabling direct localization of arrhythmia sources.
• The surface integral formulation $V_{\rm pol}(\vec{x}) = \frac{1}{4\pi\varepsilon_0}\int_S d(\vec{y})\frac{\partial}{\partial n}\frac{1}{|\vec{x}-\vec{y}|}d\sigma_y$ provides a mathematically rigorous and physically consistent model for ECG signal generation.