Lead Field Theory

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31650 Biomedical Engineering

• Lead Field Theory
• Kaj-Åge Henneberg

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Introduction

• One of the primary objectives of experimental and
  clinical electrophysiology is to detect changes
  in the electrical and morphological properties of
  excitable tissue. The underlying philosophy is
  that such changes will cause characteristic
  changes in the potential distribution arising
  from the spatial superposition of action
  potentials from all active cells and that these
  changes can be identified in recorded potential
  waveforms.
• Although the potential distribution in the tissue
  is a function both of time and space, the
  potentials recorded from a single pair of
  electrodes can only be displayed as functions of
  time on the oscilloscope screen. Ideally the
  investigator would like to be able to reconstruct
  the spatial information and to achieve this
  he/she must understand
• the principles of volume conduction which causes
  the spatial summation of the action potentials
  from individual cells and
• the manner in which the recording electrode
  weights potentials in its pick-up range.
• In this note we develop a mathematical model
  which can aid in understanding the recording
  properties of an arbitrary electrode
  configuration.

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Example Lead Field
Projection plane
Electrode wires
5 mm
10 mm
5 mm
Lead field projected onto plane
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Example Electrode Short Circuiting

• Isopotential surfaces in an anisotropic volume
  conductor due to two monopoles of opposite
  polarity.
• Non-insulated wire electrodes perturb the
  potential distribution adjacent to the wires
  located inside the hollow tubes protruding from
  the isosurfaces.
• Insulated wires with exposed ends minimize the
  region of perturbed potential distribution.

a)
b)
Point sources
Electrode wires
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Concept Diagram
Normal case
We will consider two electrodes placed in a
volume conductor containing bioelectric sources
(normal case). We let u denote the potential
distribution due to bioelectric sources in a
bounded volume conductor and ? denotes the
potential distribution in the reciprocal case
where bioelectric sources are absent and a unit
current is passed through the electrodes. Then we
construct the following relations using vector
identities
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Integral Relations
Subtracting Eq. 2 from Eq. 1 and integrating over
the volume conductor (excluding the volumes
occupied by the electrodes) yields
The total current at a given location is the sum
of current sources and the drift current due to
the electric field
Since there are no external sources or sinks
introducing or removing current from the system
in the normal case, the divergence of the total
current must vanish everywhere
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Elimination of tems
Assuming that there is no sources on the body
boundary S
In the normal case current cannot leave the
volume conductor via the electrodes due to an
infinite amplifier input impedance
In the reciprocal case there are no volume
sources
Inserting these three conditions in (3) yields
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Integration by parts
Integration of the right hand side of (4) by
parts yields
Assuming that there are no bioelectric sources on
any boundary S
Inserting this condition and (5) into (4) we
obtain
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Boundary Conditions
In the reciprocal case current cannot leave
through the boundaries, except on the electrode
surface
Since the electrodes are assumed to be perfect
conductors, their potentials are constants u1 and
u2, respectively
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Lead Field Definition
In the reciprocal case we will pass one unit of
current into the volume through S1 and remove the
same current through S2
Inserting this result in (8) we obtain the
measured voltage
The gradient field of the reciprocal case
describes the electrode Lead Field
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Interpretation
The measured voltage is the volume integral of
the source vectors projected onto the lead field
at the site of the source element
In the unique situation where the source is a
single dipole source
a) Perpendicular case
b) Parallel case
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Applications ECG
In cases where the bioelectric sources are
confined to a bounded region, such as the heart,
and the radius of this region is sufficiently
limited that the lead field can be considered
constant in the region, we have
The recorded ECG is then the time varying
projection of the heart vector onto the lead
vectors
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Example Lead Fields
Unipolar, point electrode
Bipolar, point electrodes
Bipolar, large electrodes
1
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Applications EMG
Electrode lead field orientation is parallel to
bioelectric sources
Electrode lead field orientation is perpendicular
to bioelectric sources
In situations with strong contractions where
active muscles are located under both the
positive and the negative electrode, the
electrode lead field will be perpendicular to the
source vectors in a large fraction of the pick-up
range. Since this condition gives small signal
amplitudes, the perpendicular orientation is
rarely used.
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Comb Filtering by Bipolar Electrodes
When measuring with the parallel configuration in
a differential mode, the recorded potential
resembles a differentiation of the volume
conducted potential. The waveshape is influenced
by electrode spacing.
The frequency spectrum of the recorded waveform
also is influenced by the electrode spacing
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References

• J. Malmivuo and R. Plonsey, Bioelectromagnetism.
  Oxford University Press,1995.