Page 110 - Cardiac Electrophysiology | A Modeling and Imaging Approach

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4. THE ELECTRIC FIELD OF THE HEART

The electrical activity of cardiac muscle cells generates an electric field in the extracellular

space. This field depends on time and location; on the body surface it is recorded as the electro-
cardiogram (ECG). In this section, a formulation based on electromagnetic theory is used to relate
the electrocardiographic field potentials to the biophysical process of cardiac excitation described
in the previous sections of this monograph.

4.1 The Field Generated by a Single Cardiac Cell

The ionic mechanisms that underlie the cardiac action potential (AP) are discussed
extensively in the previous sections. The AP is the transmembrane potential during excitation,
defined as V = ϕ – ϕ where ϕ and ϕ are the intracellular and extracellular potentials,
m i 0 i 0
respectively, measured adjacent to the cell membrane. Because the membrane is very thin

(~75 Angtrom), it can be considered as a surface of zero thickness. This surface is the boundary
between the intracellular and extracellular domains (Figure 4.1), at which the following boundary
conditions hold:

φ −φ = V m ≠ 0 (4.1)
o
i

and

∂φ ∂φ
σ i = σ o
i
n ∂ ° n ∂ (4.2)

The first condition states that the potential is discontinuous across the cell membrane and
the discontinuity equals the transmembrane potential. During the AP this potential difference
across the membrane varies in time. The second condition is a conservation of charge statement.

It implies that the current density normal to the membrane surface is continuous, i.e. no charge
can be gained or lost in crossing the membrane.
Define a scalar function ψ: 244
ψ =σϕ (4.3)

where ϕ is the potential and σ is a piecewise constant conductivity function, σ = σ inside the cell
i
and σ = σ in the extracellular space. The potential function ϕ satisfies Laplace’s equation 2 ∆ ϕ = 0
o
in the intracellular and extracellular domains, which do not contain electrical sources (the sources

are confined to the cell membrane). Consequently, ψ as defined in (4.3) also satisfies Laplace’s
equation 2 ψ = 0, which identifies it as a potential function. Expressing the boundary conditions
∆