A 3D Vector-Additive Iterative Solver for the Anisotropic Inhomogeneous Poisson Equation in the Forward EEG problem V. Volkov 1, A. Zherdetsky 1, S. Turovets 2, Allen D. Malony 3 Department of Mathematics and Mechanics 1 Belarusian State University, Minsk, Belarus Neuroinformatics Center 2 Department of Computer & Information Science 3 University of Oregon ICCS 2009

3D Vector Additive Iterative Solver for Anisotropic Inhomogeneous Poisson Equation in the Forward EEG Problem 2 Background: Observing Dynamic Brain Function  Brain activity occurs in cortex  Observing brain activity requires  high temporal and spatial resolution  Cortex activity generates scalp EEG  EEG data (dense-array, 256 channels)  High temporal (1msec) / poor spatial resolution (2D)  MR imaging (fMRI, PET)  Good spatial (3D) / poor temporal resolution (~1.0 sec)  Want both high temporal and spatial resolution  Need to solve source localization problem!!!  Find cortical sources for measured EEG signals

ICCS D Vector Additive Iterative Solver for Anisotropic Inhomogeneous Poisson Equation in the Forward EEG Problem 3 Computational Head Models  Source localization requires modeling  Goal: Full physics modeling of human head electromagnetics  Step 1: Head tissue segmentation  Obtain accurate tissue geometries  Step 2: Numerical forward solution  3D numerical head model  Map current sources to scalp potential  Step 3: Conductivity modeling  Inject currents and measure response  Find accurate tissue conductivities  Step 4: Source localization  Applies to optical transport modeling electrical optical (NIR)

ICCS D Vector Additive Iterative Solver for Anisotropic Inhomogeneous Poisson Equation in the Forward EEG Problem 4 Source Localization  Mapping of scalp potentials to cortical generators  Signal decomposition (addressing superposition)  Anatomical source modeling (localization)  Source modeling  Anatomical Constraints  Accurate head model and physics  Computational head model formulation  Mathematical Constraints  Criteria (e.g., “smoothness”) to constrain solution  Current solutions limited by  Simplistic geometry  Assumptions of conductivities, homogeneity, isotropy

ICCS D Vector Additive Iterative Solver for Anisotropic Inhomogeneous Poisson Equation in the Forward EEG Problem 5 Theoretical and Computational Modeling Governing Equations ICS/BCS Discretization System of Algebraic Equations Equation (Matrix) Solver Approximate Solution Continuous Solutions Finite-Difference Finite-Element Boundary-Element Finite-Volume Spectral Discrete Nodal Values Tridiagonal ADI SOR Gauss-Seidel Gaussian elimination  (x,y,z,t) J (x,y,z,t) B (x,y,z,t) image mesh solution

ICCS D Vector Additive Iterative Solver for Anisotropic Inhomogeneous Poisson Equation in the Forward EEG Problem 6 Governing Equations (Forward Problem)  Given positions and magnitudes of current sources  Given geometry and head volume   Calculate distribution of electrical potential on scalp    Solve linear Poisson equation on   in  with with no-flux Neumann boundary condition  (  U)=S J, in   (  U)  n = 0, on    (x,y,z) = head tissues conductivity (known) S J = is the current source (known) U =U( x,y,z,t) is the electrical potential (to find)

ICCS D Vector Additive Iterative Solver for Anisotropic Inhomogeneous Poisson Equation in the Forward EEG Problem 7 Governing Equations (Inverse Problem)  Inverse problem uses general tomographic structure  Given distribution of the head tissue parameters   Predict measurements values U p given a forward model F, as nonlinear functional U p =F(  )  Then an appropriate cost function is defined and minimized against the measurement set V:  Find global minimum using non-linear optimization

ICCS D Vector Additive Iterative Solver for Anisotropic Inhomogeneous Poisson Equation in the Forward EEG Problem 8  According to Ohm’s law the current density, J, and electrical field, E, are related by  J is in the same direction as E when σ is a scalar  If σ is a tensor (anisotropic), the direction of the current density, J, is different from the direction of the applied electrical field, E: Modeling Anisotropy

ICCS D Vector Additive Iterative Solver for Anisotropic Inhomogeneous Poisson Equation in the Forward EEG Problem 9 Inhomogeneity and Anisotropy in Human Head  Inhomogeneous  Conductivity depends on location  Anisotropy  Conductivity depends on orientation  Human head tissues are inhomogeneous  White matter (WM): includes fiber tracts  Gray matter (GM): cortex mainly  Cerebrospinal fluid (CSF): clear conductive fluid  Skull: highly resistive, different components  Scalp  Image segmentation used to identify head tissues  Conductivities can not be directly measured accurately

ICCS D Vector Additive Iterative Solver for Anisotropic Inhomogeneous Poisson Equation in the Forward EEG Problem 10 Skull and Brain Anisotropy Parameterization  Skull is more conductive tangentially than radially MRI DT brain map (Tuch et al, 2001) rr tt  Diffusion is preferential along white matter tracts  Linear relation between conductivity and diffusion tensor eigenvalues   = K (d - d 0 ), λ= 1, 2, 3

ICCS D Vector Additive Iterative Solver for Anisotropic Inhomogeneous Poisson Equation in the Forward EEG Problem 11 Modeling Head Electromagnetics  Current forward problem (isotropic Poisson equation)  Used in Salman et al., ICCS 2005 / 2007  Anisotropic Forward Problem  If we model anisotropy with existing principal axes then the tensor is symmetrical - 6 independent terms:  ij =  ji  Numerical implementation so far deals with the orthotropic case:  ii are different, but all other components of  ij = 0, i  j  (  U)=S J in ,  - scalar function of (x,y.z)  (  U)  n = 0 on    (  ij  U)=S J in ,  ij - tensor function of (x,y.z)  (  U)  n = 0on  

ICCS D Vector Additive Iterative Solver for Anisotropic Inhomogeneous Poisson Equation in the Forward EEG Problem 12 Transformation to the Global XYZ  We assume that the conductivity tensor is diagonal in the local coordinate system (for every voxel)  The transformation from the local to the global Cartesian system for any voxel j: σ j global =R T j σ j local R j, where rotation matrix R j is defined by the local Euler angles αβγ, (sine: s and cosine: c) :

ICCS D Vector Additive Iterative Solver for Anisotropic Inhomogeneous Poisson Equation in the Forward EEG Problem 13 Anisotropic Poisson Equation  In the global Cartesian coordinate system, the anisotropic Poisson equation is expressed as  If we model anisotropy with the existing principal axes the tensor is symmetrical - 6 independent terms:  ij =  ji

ICCS D Vector Additive Iterative Solver for Anisotropic Inhomogeneous Poisson Equation in the Forward EEG Problem 14 Finite Difference Modeling of Poisson Equation  Finite difference approximation of the second order accuracy with mixed derivatives can be made with a minimal stencil of 7 points in  2D (7 point stencil) [Volkov, Diff. Equations, 1997]  Generalization to 3D leads to a 13-point stencil [Volkov, ICCS 2009]  The whole problem computational domain is represented by a 3D checkerboard lending itself for domain decomposition (partitioning)

ICCS D Vector Additive Iterative Solver for Anisotropic Inhomogeneous Poisson Equation in the Forward EEG Problem 15 Numerical Equations  Consider even (or only odd) mesh cells, each of them having eight neighboring computational cells  Every internal node of this checkerboard grid belongs simultaneously to two neighboring cells  Natural to introduce two components of an approximate numerical solution, (, ), where m=1, …, 8  In these notations, the finite difference approximation, L, of the differential operator in the Poisson equation in an arbitrary node of the grid can be represented as  A m are vectors with components given by coefficients of the finite difference approximation

ICCS D Vector Additive Iterative Solver for Anisotropic Inhomogeneous Poisson Equation in the Forward EEG Problem 16 Expression for Operators A1 and A8

ICCS D Vector Additive Iterative Solver for Anisotropic Inhomogeneous Poisson Equation in the Forward EEG Problem 17 Iterative Solution  Numerical scheme is equivalent to a system of finite difference equations with a 13 diagonal system matrix and dimension N 3, where N is a total number of voxels  Elementary per-voxel step of the iterative process solves a system of linear algebraic equations  Computational complexity per iteration is Q=NQ 0 /8, where Q 0 is the computational cost for solving the linear system with a matrix 8 X 8  N/8 is a number of computational cells in the checkerboard discretization  Assuming Gaussian elimination algorithm, Q 0 ~ (2/3)8 3 ≈341 floating operations per–cell at one iteration

ICCS D Vector Additive Iterative Solver for Anisotropic Inhomogeneous Poisson Equation in the Forward EEG Problem 18 3D Anisotropic Simulations (88x128x128 voxels) All tissues isotropic Anisotropic skull σ r /σ t = 1/10

ICCS D Vector Additive Iterative Solver for Anisotropic Inhomogeneous Poisson Equation in the Forward EEG Problem 19 (1) Quasi-Minimal Residual (QMR) (2) BiConjugate Gradient (BiCG) (3) Vector-additive method (a,b,c) preconditioners (none, Jacobi, incomplete Cholesky) Vector-additive method not optimized Matlab Prototype Performance Heterogeneous coefficients 1e-04 accuracy

ICCS D Vector Additive Iterative Solver for Anisotropic Inhomogeneous Poisson Equation in the Forward EEG Problem 20 Summary  3D finite volume algorithm for solving the anisotropic heterogenious Poisson equation based on the vector- additive implicit methods with a 13-points stencil  Variable iterative parameters to improve the convergence rate in the heterogeneous case  First attempt to implement the vector-additive numerical scheme for a 3D anisotropic problem  Believe the 3D vector additive method has better parallelism potential due to its cell-level data decomposition, especially as head volumes scale to voxels  Currently developing GPGPU implementation