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

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 solution • Subsequent goals 2

What is MEG? Magnetoencephalography is a noninvasive technique for investigating neuronal activity in the living human brain. 3

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. 4

The reason of our interest in this problem Forward computation Inverse computation

According to Biot – Savart law , 6

At first we observe a following model: 𝑌 = ℝ 2 ∋ 𝑦 = 𝑦 1 , 𝑦 2 , 𝑦 𝑙 < ∞ ( ℝ 3 ⊃ 𝑍) ∋ 𝑧 = 𝑧 1 , 𝑧 2 , −𝜁 : 𝑧 𝑙 < ∞ 𝜁 = 1 7

The equation assumes the following form: 3 𝐿 𝑚𝑛 𝑦 − 𝑧 𝑅 𝑛 𝑧 𝑒𝑧 = 𝐶 𝑚 𝑦 , 𝑚 = 1,2,3 𝑛=1 Y 8

Lemma 1 𝐿 12 𝜊 = 𝐹 𝜊 , where 𝐹 𝜊 = 2𝜌𝑓 −2𝜌|𝜊| ∶ 2 + 𝜊 2 2 , and 𝜊 = 𝜊 1 , 𝜊 2 , 𝜊 = 𝜊 1 𝐿 23 𝜊 = −𝑗 𝜊 1 𝐿 31 𝜊 = −𝑗 𝜊 2 𝜊 𝐹 𝜊 , 𝜊 𝐹 𝜊 , where 𝐿 𝜊 = ℱ 𝑡→𝜊 𝐿(𝑡) 9

A consequence of Lemma 1 (1) 𝑃𝑞 𝐿 𝜊 𝑅 𝑨 = 𝐶(𝑦) , where −1 𝑃𝑞 𝐿 𝜊 = ℱ 𝜊→𝑦 𝐿 𝜊 ℱ 𝑨→𝜊 , 𝐿 𝜊 = ℱ 𝑡→𝜊 𝐿(𝑡) 𝐿 𝜊 - is a symbol of pseudodifferential operator 2 𝐿 𝜊 𝑅 𝜊 = 𝐶 𝜊 , 𝜊 = (𝜊 1 , 𝜊 2 ) 10

Lemma 2 Equation form Basis Details 𝑓 1 = 1,0,0 , 𝑅 𝜊 =( 𝑅 1 , 𝑅 2 , 𝑅 3 ) 𝑳 𝝄 𝑹 𝝄 = 𝑓 2 = 0,1,0 𝑪 𝝄 𝐶 𝜊 =( 𝐶 1 , 𝐶 2 , 𝐶 3 ) 𝑓 3 = 0,0,1 𝑣 = ( 𝑣 1 , 𝑣 2 , 𝑣 3 ) ⅈ𝜊 1 ⅈ𝜊 2 𝝉 𝝄 𝒗 𝝄 = 𝒉 𝝄 ′ =( − 𝑓 1 𝜊 , − 𝜊 , 1 ) 𝑕 = ( 𝑕 1 , 𝑕 2 , 0) ′ = ( ⅈ𝜊 2 𝜊 , − ⅈ𝜊 1 σ = (𝑇 𝑢 ) −1 𝐿(𝜊)𝑇 𝑢 , 𝑓 2 𝜊 , 0) 0 1 0 𝝉 𝝄 = where 𝑇 - amplication 0 0 1 ⅈ𝜊 1 ⅈ𝜊 2 ′ =( 𝑓 3 𝜊 , 𝜊 , 0 ) 0 0 0 matrix 11

Lemma 3 𝑕 1 (𝜊)𝑓 2𝜌 𝜊 𝑣 2 𝜊 = 2𝜌 𝜏(𝜊) 𝑣 𝜊 = 𝑕 𝜊 𝑕 2 (𝜊)𝑓 2𝜌 𝜊 𝑣 3 𝜊 = 2𝜌 12

The algorithm of obtaining 𝒗 𝟐 1 st step 𝑓 ′ = 𝑇𝑓 ; 𝛾 𝑓 = 𝑇 𝑢 𝛽 𝑓 ′ ; Transition from the basis 𝒇 ′ to 𝒇 𝑣 𝜊 → 𝑅 𝜊 2 nd step −1 Making the inverse Fourier 𝑅 𝑨, 𝑣 1 = ℱ 𝜊→𝑨 𝑅 𝜊 transform 13

The algorithm of obtaining 𝒗 𝟐 3 rd step 3 According to Biot – Savart 𝐿 𝑚𝑛 𝑦 − 𝑧 𝑅 𝑛 𝑧 𝑒𝑧 = 𝐶 𝑚 𝑦 , 𝑚 = 1,2,3 law, calculate all components of the magnetic 𝑛=1 field 14

The algorithm of obtaining 𝒗 𝟐 4 th step 𝐶 2 = 𝐶 1 2 + 𝐶 2 2 + 𝐶 3 2 Finding the magnitude of the vector B 15

The algorithm of obtaining 𝒗 𝟐 5 th step 2 3 3 𝐿 𝑚𝑛 𝑦 − 𝑧 (𝐻 𝑛 𝑧 + 𝑃𝑞 −𝑗 𝜊 𝑛 𝐶 2 = 𝑣 1 (𝑧))𝑒𝑧 Getting the final integral 𝜊 equation for 𝒗 𝟐 𝑚=1 𝑛=1 Y 16

Lemma 4 Suppose that 𝑧 1 = 𝑠 cos 2𝜌Θ , 𝑧 2 = 𝑠 sin 2𝜌Θ . 𝐻 𝑠, Θ = 𝑕 𝑧 1 , 𝑧 2 = 𝑛∈ℤ 𝐻 𝑛 (𝑠)𝑓 ⅈ2𝜌𝑛Θ , where 𝐻 𝑛 𝑠 ∈ ℂ. 𝐔𝐢𝐟𝐨 ∞ 𝑓 ⅈ2𝜌 𝜕− 1 4 𝑜 𝐻 𝜊 , 𝜕 = 𝑠𝐻 𝑜 𝑠 𝐾 𝑜 2𝜌 𝜊 𝑠 𝑒𝑠 , 𝑜∈ℤ 0 where 𝐻 𝜊 , 𝜕 = ℱ 𝑧→𝜊 𝑕 𝑧 , 𝜊 1 = 𝜊 cos 2𝜌𝜕 , 𝜊 2 = 𝜊 sin 2𝜌𝜕 17

Subsequent goals 1. To get the computational solution of our equation for 𝒗 𝟐 2. To make the model suitable for the human brain topology 3. To true up our calculations corresponding to MEG data 18

Thank You for Your attention! 19