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3 Fundamentals of the Formation of Biosignals

can be specified. The location of the current source is described by the coordinates

x, y, z, the location of the measurement outside the volume in which the current

sources are located by the coordinates x^󸀠, y^󸀠, z^󸀠.

According to Equation 3.18, the vector potential A propagates in a body due to the

e-function e^−kr like a harmonic wave whose propagation constant k describes the mag-

nitude of the wave vector or the local wavelength. To estimate the maximum value of

the product krmax in the exponent of the e-function, for a maximum distance between

the location of the current source and the measurement location of rmax = 100 cm, a

frequency of 1 kHz and an average conductivity κ of 4 mS/cm one obtains the result

krmax = 0.04 resp. e^−krmax = e^−0.04 = 0.96. Since this value is very close to 1, this

e-function in Equation 3.18 can be approximately neglected for the measurement in a

body and we obtain:

A ≈^μ0

4π ^∫

Ji

r ^{dv .}

(3.19)

If this result is substituted into Equation 3.15, it follows for the potential Φ:

Φ = ⁻¹

4πκ ^{∫Ji ⋅∇󸀠(1}

r ^{) dv .}

(3.20)

The Nabla operator with the apostrophe ∇^󸀠is meant to indicate that the local deriv-

atives are to be performed according to the coordinates x^󸀠, y^󸀠, z^󸀠of the measurement

location, to which Equation 3.15 refers, while in Equation 3.19 it is integrated over the

coordinates of the source locations x, y, z. However, because of ∇^󸀠( ¹

r ^{) = −∇( 1}

r ^{), the po-}

tential equation can also be described with a nabla operator related to the coordinates

of the measurement location:

Φ =

1

4πκ ^{∫Ji ⋅∇(1}

r ^{) dv .}

(3.21)

The result is also obtained by assuming that a flow of charge carriers in the body does

not cause self-induction and that it is an almost static problem which can be solved

by means of Poisson’s equation [18]. It also allows the interpretation that the current

density multiplied by a volume element Ji ⋅dv is a current dipole and summed over all

inner current dipoles with the weight ∇( ¹

r ^{). A current dipole is a current source with}

an associated current sink multiplied by the distance between the source and the sink.

Probably the strongest current sources in the body are generated by the action

potentials of the pacemaker cells in the heart and can be measured particularly well

as a potential on the thorax (see Figure 3.14).

To simplify the measurement of the electrical activity of the heart, all the current

dipoles of the heart in Equation 3.21 can be combined into a single current dipole by

vectorial summation and its effects on the body studied. If, for example, measure-

ments are made on the left and right arms and on the foot, the measured potential

differences can be interpreted as projectionsindexprojection of the heart vector onto

the respective individual sections between the arms and a foot (see Figure 3.15).

Using the potentials at the left arm (ΦL), right arm (ΦR) and foot (ΦF), the follow-

ing potential differences important for an ECG can be determined: