inverse MEG-problems Authors: Galchenkova Marina, Demidov - - PowerPoint PPT Presentation

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inverse MEG-problems Authors: Galchenkova Marina, Demidov - - PowerPoint PPT Presentation

Oct 30, 2022 521 likes β€’720 views

A flat approximation of inverse MEG-problems Authors: Galchenkova Marina, Demidov Alexander, Kochurov Alexander Plan What is magnetoencephalography (MEG)? Inverse problem The reason of our interest in this problem Steps of

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A flat approximation of inverse MEG-problems

Authors: Galchenkova Marina, Demidov Alexander, Kochurov Alexander

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Plan

  • What is magnetoencephalography (MEG)?
  • Inverse problem
  • The reason of our interest in this problem
  • Steps of solution
  • Subsequent goals

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What is MEG?

Magnetoencephalography is a noninvasive technique for investigating neuronal activity in the living human brain.

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Inverse problem

  • a problem of finding

electrical impulses’ distribution in some area Y (associated with cortex), that based on data of its induced magnetic field in another place X that we obtain by MEG system.

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Forward computation Inverse computation

The reason of our interest in this problem

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According to Biot–Savart law

,

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At first we observe a following model:

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π‘Œ = ℝ2 βˆ‹ 𝑦 = 𝑦1, 𝑦2 , 𝑦𝑙 < ∞ (ℝ3 βŠƒ 𝑍) βˆ‹ 𝑧 = 𝑧1, 𝑧2, βˆ’πœ : 𝑧𝑙 < ∞ 𝜁 = 1

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The equation assumes the following form:

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𝑛=1 3

πΏπ‘šπ‘› 𝑦 βˆ’ 𝑧 𝑅𝑛 𝑧 𝑒𝑧 = πΆπ‘š 𝑦 , π‘š = 1,2,3 Y

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Lemma 1

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𝐿12 𝜊 = 𝐹 𝜊 , where 𝐹 𝜊 = 2πœŒπ‘“βˆ’2𝜌|𝜊| ∢ 𝜊 = 𝜊1, 𝜊2 , 𝜊 = 𝜊1

2 + 𝜊2 2, and

𝐿23 𝜊 = βˆ’π‘— 𝜊1 𝜊 𝐹 𝜊 , 𝐿31 𝜊 = βˆ’π‘— 𝜊2 𝜊 𝐹 𝜊 , where 𝐿 𝜊 =β„±π‘‘β†’πœŠ 𝐿(𝑑)

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A consequence of Lemma 1

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(1) π‘ƒπ‘ž 𝐿 𝜊 𝑅 𝑨 = 𝐢(𝑦), where π‘ƒπ‘ž 𝐿 𝜊 = β„±πœŠβ†’π‘¦

βˆ’1

𝐿 𝜊 β„±π‘¨β†’πœŠ , 𝐿 𝜊 =β„±π‘‘β†’πœŠ 𝐿(𝑑) 𝐿 𝜊 - is a symbol of pseudodifferential operator 2 𝐿 𝜊 𝑅 𝜊 = 𝐢 𝜊 , 𝜊 = (𝜊1, 𝜊2)

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Lemma 2

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Equation form Basis Details 𝑳 𝝄 𝑹 𝝄 = π‘ͺ 𝝄 𝑓1 = 1,0,0 , 𝑓2 = 0,1,0 𝑓3 = 0,0,1 𝑅 𝜊 =( 𝑅1, 𝑅2, 𝑅3) 𝐢 𝜊 =( 𝐢1, 𝐢2, 𝐢3) 𝝉 𝝄 𝒗 𝝄 = 𝒉 𝝄 𝝉 𝝄 = 1 1 𝑓1

β€²=(βˆ’ β…ˆπœŠ1 𝜊 , βˆ’ β…ˆπœŠ2 𝜊 , 1)

𝑓2

β€²=(β…ˆπœŠ2 𝜊 , βˆ’ β…ˆπœŠ1 𝜊 , 0)

𝑓3

β€²=( β…ˆπœŠ1 𝜊 , β…ˆπœŠ2 𝜊 , 0)

𝑣 = ( 𝑣1, 𝑣2, 𝑣3) 𝑕 = ( 𝑕1, 𝑕2, 0) Οƒ = (𝑇𝑒)βˆ’1 𝐿(𝜊)𝑇𝑒 , where 𝑇 -amplication matrix

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Lemma 3

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𝜏(𝜊) 𝑣 𝜊 = 𝑕 𝜊 𝑣2 𝜊 = 𝑕1(𝜊)𝑓2𝜌 𝜊 2𝜌 𝑣3 𝜊 = 𝑕2(𝜊)𝑓2𝜌 𝜊 2𝜌

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The algorithm of obtaining π’—πŸ

2nd step Making the inverse Fourier transform 𝑅 𝑨, 𝑣1 = β„±πœŠβ†’π‘¨

βˆ’1

𝑅 𝜊 1st step Transition from the basis 𝒇′ to 𝒇 𝑓′ = 𝑇𝑓; 𝛾𝑓= 𝑇𝑒𝛽𝑓′; 𝑣 𝜊 β†’ 𝑅 𝜊

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The algorithm of obtaining π’—πŸ

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3rd step According to Biot–Savart law, calculate all components of the magnetic field

𝑛=1 3

πΏπ‘šπ‘› 𝑦 βˆ’ 𝑧 𝑅𝑛 𝑧 𝑒𝑧 = πΆπ‘š 𝑦 , π‘š = 1,2,3

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The algorithm of obtaining π’—πŸ

4th step

Finding the magnitude of the vector B 𝐢 2 = 𝐢1

2 + 𝐢2 2 + 𝐢3 2

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The algorithm of obtaining π’—πŸ

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5th step

Getting the final integral equation for π’—πŸ 𝐢 2 =

π‘š=1 3 𝑛=1 3

πΏπ‘šπ‘› 𝑦 βˆ’ 𝑧 (𝐻𝑛 𝑧 + π‘ƒπ‘ž βˆ’π‘— πœŠπ‘› 𝜊 𝑣1(𝑧))𝑒𝑧

2

Y

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Lemma 4

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Suppose that 𝑧1 = 𝑠 cos 2𝜌Θ , 𝑧2 = 𝑠 sin 2𝜌Θ . 𝐻 𝑠, Θ = 𝑕 𝑧1, 𝑧2 = π‘›βˆˆβ„€ 𝐻𝑛(𝑠)π‘“β…ˆ2πœŒπ‘›Ξ˜, where 𝐻𝑛 𝑠 ∈ β„‚. π”π’πŸπ¨ 𝐻 𝜊 , πœ• =

π‘œβˆˆβ„€

π‘“β…ˆ2𝜌 πœ•βˆ’1

4 π‘œ ∞

π‘ π»π‘œ 𝑠 πΎπ‘œ 2𝜌 𝜊 𝑠 𝑒𝑠 , where 𝐻 𝜊 , πœ• = β„± π‘§β†’πœŠπ‘• 𝑧 , 𝜊1 = 𝜊 cos 2πœŒπœ• , 𝜊2 = 𝜊 sin 2πœŒπœ•

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Subsequent goals

  • 1. To get the computational solution of
  • ur equation for π’—πŸ
  • 2. To make the model suitable for the

human brain topology

  • 3. To true up our calculations

corresponding to MEG data

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Thank You for Your attention!

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