Source localization in electromyography: The Inverse Potential - - PDF document

Source localization in electromyography: The Inverse Potential Problem

Kees van den Doel

• Dept. of Computer Science,

University of British Columbia, Vancouver, Canada with Uri Ascher, Dinesh Pai, Benjamin Gilles, Armin Curt, John Steeves

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Forward Potential Problem

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Inverse Potential Problem

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Non-uniqueness

u u q1 q2 Two charge distributions create same u(x) outside, if total charge is the same =

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Applications

• Geophysics (u is gravitational pot.)
• EEG, locate seizure focus in brain
• Cardiology (find current sources in heart)
• CMG, locate muscle activity

– Novel work with D. Pai and U. Ascher

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Tikhonov regularization

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Computed Electromyography

• Measure nu(x) on skin
• Reconstruct sources q(x)
• First need a forward model:

– Geometry

• skin, fat, muscle, bone

– Conductivity – FEM discretization

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Getting Arm Geometry

MRI Segmentation

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Inversion Process

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Properties of source q(x)

q(x) made of (1000+) smeared tripoles running along muscle fibers

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Choosing Representation

• q = m, directly reconstruct source

– Simple, resulting eq. easier to solve

• q=m*D with D tripole basis functions
• m expected to be “simpler” than q
• In principle allows detection of depth

– Use apparent size as cue

• Effective outside sparse reconstruction context
• Leads to numerical difficulties
• Results in “numerical lab” favor D by a lot

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Choosing Regularization R(m)

• Decide on R(m)

– L2 norm favors superficial sources – Weighted L2 norm with W s.t. interior sources are equally likely (W is computable with some effort) – (m)2 favors smooth distribution

• Good for bulk activity localizations
• Leads to linear inverse problem
• Can be weighted (less well motivated)

– L1 norm favors superficial sources

• Attempts sparse reconstruction (but is 2000 small?)
• Non linear, hard to deal with time dependent data
• Promising in EEG but many technical hurdles

– L0 norm

• Actual sources all of about equal strength
• Should work best, but is untractable

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Real-time is possible if we stay linear

Extension Flexion

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Numerics (linear case)

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Numerics (L1 case)

• l1_magic (Koh, Kim, Boyd, 2008)

– Recast as convex quadratic problem with linear inequality constraints – Uses interior point method – Newton’s method step involves solving linear system which gets more and more ill- conditioned as problem gets bigger and as we get closer to the solution (in practice) – Works fine for 2D small problems

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Numerical Experiments

• Reconstructions inaccurate
• Formal performance measures for

inversion useless

• Instead attempt a palette of synthetic

problems and use taste and common sense measure

• 3D FEM experiments hampered by

numerical difficulties, so let’s work on a square grid in 2D

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2D “muscle” (and the real thing)

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Smooth, no D (=1)

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L2, with D

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Smooth

Smooth, with D

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L1 (with D)

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Overall

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L2

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L1

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Smooth

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Overall

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L1, no noise, tiny regularization

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Smooth, no noise, tiny regularization

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L1, smooth

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Summary

• Inverse potential problem has no unique solution
• Hard to decide which regularization is best as inversion is

inherently inaccurate

• Numerical experiments seem to favour one particular

formulation (Laplacian weighted 2-norm)

• Tripole basis is necessary
• Depth can be resolved theoretically but in practice only

under unrealistically ideal conditions

• Real-time inversion is possible (it’s linear)
• In the large scale version I don’t know how to solve the

linear equation

• Inaccuracy of inversion remains an obstacle in medical

applications (EEG,ECG,CMG)