FEM modeling for tDCS

• 对于 tDCS 电流在大脑中的建模，主要是靠解决下面这个等式来的：
  
  M⋅U=I
  
  
  • M 是tDCS的forward matrix
  • I 是injected current vector
  • U 是在已知以上输入电流情况下，要求解的potential vector
  • M combines volume conduction properties (FEM stiffness matrix) and electrical boundary conditions， including the complete electrode model to simulate current injection
    
    ∇⋅(σ∇u)=0,(x∈Ω)
    
    
    u+zlσ∂u∂n=Ul,(on∂Ωel,x∈el)
    
    
    ∫elσ∂u∂n=Il(l=1,2,⋯,L)
    
    
    σ∂u∂n=0,(x∈∂Ω∖∪Ll=1el)
  • In more detail, M is composed of the regular FEM stiffness (A∈N×N, with N representing the number of FEM nodes) and additional electrical boundary conditions specified as submatrices A2,B,C (B∈N×L,C∈L×L, with L representing the number of electrodes)
  • M is composed such as:
    
    M=(A−B−BTC)
    
    
    (A−B−BTC)(UnUe)=I=(InIe)
    
    
    A(i,j)=A1(i,j)+A2(i,j)=∫Ωσ∇ϕi⋅∇ϕjdΩ+∑l=1L∫el1zlϕiϕjdΩel
    
    
    B(i,l)=1zl∫elϕidΩel
    
    
    C(i,l)=1zl∫eldΩel
  • M∈RN+L×N+L,A1,A2∈RN×N,B∈RN×L,C∈RL×L
  • Un represents potentials at nodes, Ue represents potentials at electrodes, Ie is currents at electrodes, ϕ denotes linear basis functions, zl resistive electrode impedance.
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Reference:
https://www.sci.utah.edu/images/software/SCIRun/ElectricalBrainStimulationTutorial.pdf