A Review on the Importance of Volume Currents

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A Review on the Importance of Volume Currents

• John C. MosherBiological and Quantum Physics
  Group
• Los Alamos National Laboratory

2
Acknowledgements

• Cite as Mosher JC, A Review on the Importance
  of Volume Currents, (invited presentation), 14th
  International Conference on Biomagnetism, Boston,
  Massachusetts, August 2004, available as Los
  Alamos Technical Report LA-UR-04-6117.
• This work was supported by the National
  Institutes of Health under grant R01-EB002010,
  and by Los Alamos National Laboratory, operated
  by the University of California for the United
  States Department of Energy, under Contract
  W-7405-ENG-36.

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Abstract

• Given an elemental current dipole inside the
  brain, the forward problem is the calculation of
  the external scalp potential or the magnetic
  field by superposition, any more complicated
  distribution of primary current can be found by
  integration or summation of the basic solution.
  Although the solutions have been derived over the
  last four decades under a variety of situation,
  some users remain uncertain about the effects of
  the volume currents in the models. The confusion
  may arise in part because most forward models
  have been reworked to make the volume currents
  implicit, rather than explicit. We review the
  general development of the forward solution,
  including our discovery of an early 1971 paper
  missed by the MEG community that elegantly yields
  the general solution. We discuss the principal
  computational issues in boundary element methods
  (BEMs) in accurately accounting for these volume
  currents, and how it impacts both MEG and EEG
  models. We also review the myth of the silent
  radial dipole, reviewing classic and recent work
  that shows that radial dipoles are generally
  measurable in MEG data under realistic
  conditions.

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Outline

• Basic Assumptions, and the definition of Primary
  and Volume currents
• Simple MEG solution and the possible confusion
  about volume currents
• Historical review of the development of the
  general solutions for EEG and MEG
• The myth of the silent radial dipole

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Excellent Mathematical Reference

• Jukka Sarvas, 1987, Physics in Medicine and
  Biology.

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Sarvas References
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Basic Assumptions

• Source region is non-magnetic
• Currents are quasi-static
• Electric field is gradient of a scalar
• Curl of magnetic field is the current
• Divergence of magnetic field is zero
• Define static magnetic field as the curl of a
  vector potential A(r), yielding

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Magnetic Vector Potential

• Integrate the total current density flowing in
  the head, divided by its distance to the
  observation.

BrainStorm
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Biot-Savart Law
Vector potential
CURL yields magnetic field
But the Biot-Savart Law is expressed in total
current, we need the solution in terms of the
neural source generators.
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Primary vs. Secondary Currents

• Picture primary current as a small battery inside
  the brain.
• Secondary or volume currents are the gradient
  currents to complete the circuit.
• Boundaries shape the volume currents.

BrainStorm
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Primary Neural Sources

• Primary currents are produced by current flow in
  apical dendrites in cortical pyramidal neurons.
• Millions of EPSPs summed over ten milliseconds.
• Macrocellular vs. microcellular.

Ramon y Cajal 1888 from Hamalainen et al. 1993
Reviews of Modern Physics
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Two Types of Current

• Volume currents flow simply due to the presence
  of a macroscopic voltage gradient in conducting
  medium
• Simply define Primary Current as not volume

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Unbounded Solutions
Assume an unbounded homogeneous region
CURL Homogeneous magnetic field
DIVERGENCE Homogeneous electric potential
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Numerical Example

• Consider a dipole 7 cm up from an origin, and
  observation points arranged 12 cm from the origin.

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Primary Dipole Magnetic Fields
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Unbounded Regions

• Volume currents do not contribute to the
  potential or magnetic field in infinite
  homogeneous regions
• Volume currents only contribute when bounded
  regions are nearby

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Bounded Regions

• Given primary current, what is the magnetic
  field?
• MEG general solution includes the general
  solution of EEG surface potentials.

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Boundary Effects (cf. Sarvas)
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Fictitious Currents

• Standard vector identities allow us to delete the
  true 3-D volume currents and replace them with
  fictitious 2-D currents only on the boundaries,
  normally oriented

True physical currents
All surface current elements discontinuous
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Radial Field Outside of a Sphere

• In a sphere, then all fictitious currents are
  radial.
• If a sensing coil is oriented radially outside of
  the a perfect sphere, then none of the fictitious
  currents are visible

Radial fictitious currents are Unobservable by a
radial sensor
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Radial Field Strength
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The Simple MEG Solution

• Biot-Savart Law using All currents
• Spherical Case, Radial Direction
• Looks like the Biot-Savart Law, but only involves
  the primary current

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The Confusion

• We can also write the spherical solution for the
  radial measurement as
• But this often leads novices to the conclusion
  that the volume currents are unimportant. For
  non-radial measurements in the orientation o,
  they attempt

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Slightly Non-radial Measurements

• Six degrees from radial in the y-direction

Correct Primary and Volume Currents
Incorrect Primary Currents Only
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Non-radial Field Outside of a Sphere

• In non-radial directions, the fictitious currents
  are visible and must be included in the
  calculation

Non-radial orientation
Sensor and dipole in same orientation
Primary current does NOT contribute!
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All Sensors in the X-direction

• Sensors cannot see the primary current, only the
  volume currents

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General Solution Approach

• Specify an elemental primary current source,
  calculate the infinite homogeneous potential,
  then solve the Fredholm Integral of the Second
  Kind for all boundary potentials (cf Sarvas)
• Using this EEG solution, plug into Geselowitz
  (1970) equation for MEG

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Classical Solution Approach

• First Edition 1962

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Vector Spherical Harmonics
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Grynszpan and Geselowitz 1973

• Formalized the lead-field work of Baule and McFee
  (1965,1970), using vector spherical harmonics

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The Simple Acknowledgement

• 1975 Second Edition

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The Footnote

• The clue, quite overlooked, that a simple
  solution existed.

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Bronzan, Am. Jrnl. Phys. 1971
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Bronzans Solution (1971)

• Magnetic scalar, given in terms of the total
  current. In the spherical boundaries case, only
  primary currents can contribute
• Magnetic field is simply the gradient.

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Further MEG Development

• Bronzans solution is still today not widely
  cited, and has only recently been cited in MEG
  literature (Jerbi et al. 2003 PMB).
• As the MEG community began experimental
  measurements in the 1980s, a solution was sought
  for the non-radial field outside of a sphere.

1984 Biomag in Vancouver
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Ambiguity of MEG Data

• Consider again the physical case. A
  radially-oriented sensor cannot distinguish
  between the following cases

Volume currents are invisible
Radial line currents are invisible
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General Solution for the Sphere

• Sum the magnetic field for the two radial lines
  and the tangential cross elements. Take the limit
  as the tangential element is made small,
  yielding
• Unfortunately, while correct, the formula
  contained a singularity 0/0 condition at
  certain observation points.

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Correct Insight

• Their insight was correct, however, that we did
  NOT need to calculate the volume currents
  explicitly

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Sarvas Solution

• 1987, Jukka Sarvas correctly exploited the
  observations of Ilmoniemi et al, yielding a
  concise closed-form linear algebra solution
  (independent of Bronzans development)

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Bronzan-Sarvas Model

• 1971 Bronzan solved the general magnetic scalar
  solution. No specialization to primary currents
  and spherical geometry.
• 1987 Sarvas independently addressed the specific
  spherical MEG case and provided the explicit
  gradient solution for the magnetic field.

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Full Magnetic Field
Primary plus Volume Currents
Primary Current only
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The Elegant Phantom

• Returning to the Ilmoniemi et al solution. Their
  insight led them to develop a phascinating
  phantom. These triangular

magnetic dipoles are experimentally
indistinguishable from a current dipole in a
perfect sphere of conducting solution.
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Spatial Ambiguity

• This phantom experimentally emphasizes
• (1) the importance of the volume currents, which
  are now embodied in the radial lines, and
• (2) the complete spatial ambiguity between two
  different current configurations.

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The Silent Radial Dipole

• Return to the fictitious currents model

No measurable signal, primary and fictitious are
all radial.
Indeed, there is NO external magnetic field in
any direction. The field from the volume currents
has exactly cancelled the field from the primary
current.
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Big Big Equals Zero

• Radially-oriented current dipole generates itself
  a substantial homogeneous field.
• Volume currents exactly negate this field
  everywhere.

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The Myth of the Radial Dipole

• Two critical conditions
• The dipole must be perfectly radial.
• The head must be perfectly spherical.
• Re Radial Hillebrand and Barnes (Neuroimage
  2002) recently found less the 5 of the cortical
  surface is within 15 degrees radial.

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Nearly Radial Dipole in a Sphere

• Dipole six degrees in x-direction from radial,
  sensors radially oriented.

Strength already 10 of that of the tangential
dipole. At 15 degrees, strength is 25 of that
of the tangential dipole.
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The Myth of the Spherical Head

• A much older and often overlooked result of
  Grynszpan and Geselowitz (1973) examines a slight
  perturbation of the spherical head.
• Let the minor axis of a prolate spheroid be 99.5
  that of the major axis.
• The maximum external magnetic field for a radial
  source is now 10 of that for a tangential
  source.
• Effectively, the radial dipole in a perfect
  sphere has been rotated to 6 degrees.

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Summary

• Primary current source generates volume currents.
• If no primary current, then no volume current.
• But can have closed loop primary currents that
  generate no volume current.
• The volume currents create differences in
  potentials on the scalp surface -gt EEG.
• Silent EEG sources include those with no volume
  currents.
• In general, both the primary current and the
  volume currents contribute to the magnetic field.
• Must first solve the EEG forward model before
  solving the MEG forward model.

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Summary continued

• In the special case of spherical geometry and
  radial MEG measurements, then volume currents do
  not contribute to the measurement.
• Only occurs in simulation, even a six degree
  offset from radial causes appreciable volume
  current signals in the sensor.
• In the special case of spherical geometry and a
  radial source, then volume currents exactly
  cancel the primary signal, such that NO external
  magnetic field exists.
• Only occurs in simulation, since head must be
  perfectly spherical.

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Summary continued

• In the special case of spherical geometry, then
  the tangential magnetic field components may be
  calculated directly from the radial magnetic
  field components.
• Therefore do not need explicitly to solve EEG
  problem first.
• First solved in general by Bronzan 1971 using
  magnetic scalars, but result remains obscure.
• Investigated by Ilmoniemi et. al in 1984,
  resulting in elegant dry phantom.
• Solved explicitly by Sarvas 1987 for the MEG case.