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A 3D Vector-Additive Iterative Solver for the Anisotropic Inhomogeneous Poisson Equation
in the Forward EEG problem
V. Volkov1, A. Zherdetsky1, S. Turovets2, Allen D. Malony3
Department of Mathematics and Mechanics1
Belarusian State University, Minsk, Belarus
Neuroinformatics Center2
Department of Computer & Information Science3
University of Oregon
ICCS 2009
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 2009 3D 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 2009 3D 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 physicsComputational head model formulation
Mathematical ConstraintsCriteria (e.g., “smoothness”) to constrain solution
Current solutions limited by Simplistic geometry Assumptions of conductivities, homogeneity, isotropy
ICCS 2009 3D 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-DifferenceFinite-Element
Boundary-ElementFinite-Volume
Spectral
Discrete Nodal Values
TridiagonalADISOR
Gauss-SeidelGaussian elimination
(x,y,z,t)J (x,y,z,t)B (x,y,z,t)
image
mesh
solution
ICCS 2009 3D 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 withwith no-flux Neumann boundary condition
(U)=SJ, in
(U) n = 0 , on
(x,y,z) = head tissues conductivity (known)
SJ = is the current source (known)
U =U( x,y,z,t) is the electrical potential (to find)
ICCS 2009 3D 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 Up given a forward model
F, as nonlinear functional Up =F() Then an appropriate cost function is defined and
minimized against the measurement set V:
Find global minimum using non-linear optimization€
E =1
N(U i
p −Vi)2
i=1
N
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥
1/ 2
ICCS 2009 3D 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:
€
J = σE = σ∇U
€
Jx = σ 11Ex + σ 12Ey + σ 13E z;
Jy = σ 21E x + σ 22Ey + σ 23E z;
Jz = σ 31Ex + σ 32Ey + σ 33E z;
Modeling Anisotropy
ICCS 2009 3D 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 2009 3D 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 t
Diffusion is preferential along white matter tracts
Linear relation between conductivity and diffusion tensor eigenvalues = K (d - d0), λ= 1, 2 , 3
ICCS 2009 3D 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, ij
(U)=SJ in , - scalar function of (x,y.z)
(U) n = 0 on
(ijU)=SJ in , ij - tensor function of (x,y.z)
(U) n = 0 on
ICCS 2009 3D 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:
σjglobal=RT
j σjlocal Rj ,
where rotation matrix Rj is defined by the local Euler angles αβγ, (sine: s and cosine: c) :
€
R =
cα cγ − sα cβ sγ −cα sγ − sα cβ cγ sβ sα
sα cγ + cα cβ sγ −sα sγ + cα cβ cγ −sβ cα
sβ sγ sβ cγ cβ
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
ICCS 2009 3D 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
€
∂
∂xσ xx
∂u
∂x+ σ xy
∂u
∂y+ σ xz
∂u
∂z
⎛
⎝ ⎜
⎞
⎠ ⎟+
∂
∂yσ yy
∂u
∂y+ σ yx
∂u
∂x+ σ yz
∂u
∂z
⎛
⎝ ⎜
⎞
⎠ ⎟+
+∂
∂zσ zz
∂u
∂z+ σ zx
∂u
∂x+ σ zy
∂u
∂y
⎛
⎝ ⎜
⎞
⎠ ⎟= S (x, y,z)
ICCS 2009 3D 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 problemcomputational domain isrepresented by a 3Dcheckerboard lending itselffor domain decomposition(partitioning)
ICCS 2009 3D 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
Am are vectors with components given by coefficients of the finite difference approximation
€
u 9−m
€
um
€
Lu = Amu + Am u
ICCS 2009 3D Vector Additive Iterative Solver for Anisotropic Inhomogeneous Poisson Equation in the Forward EEG Problem 16
Expression for Operators A1 and A8
€
A1u = −σ xx
12
hx2 +
σ yy14
hy2 +
σ zz16
hz2 −
σ xy14
+ σ yx12
2hxhy
−σ xz
16+ σ zx
12
2hxhz
−σ yz
16+ σ zy
14
2hyhz
⎛
⎝ ⎜
⎞
⎠ ⎟u1 +
+σ xx
12
hx2 −
σ xy32
+ σ yx12
2hxhy
−σ xz
25+ σ zx
12
2hxhz
⎛
⎝ ⎜
⎞
⎠ ⎟u2 +
σ xy32
+ σ yx34
2hxhy
u3 +σ yy
14
hy2 −
σ xy14
+ σ yx34
2hxhy
−σ yz
47+ σ zy
14
2hyhz
⎛
⎝ ⎜
⎞
⎠ ⎟u4 +
+σ xz
25+ σ zx
56
2hxhz
u5 −σ xz
16+ σ zx
56
2hxhz
u6 +σ yz
47+ σ zy
67
2hyhz
u7 ;
€
A8u =σ yz
25+ σ zy
23
2hyhz
u2 +σ zz
38
hz2 −
σ xz38
+ σ zx34
2hxhz
⎛
⎝ ⎜
⎞
⎠ ⎟u3
+σ xz
47+ σ zx
34
2hxhz
u4 +σ yy
58
hy2 −
σ xy58
+ σ yx65
2hxhy
⎛
⎝ ⎜
⎞
⎠ ⎟u5 + +
σ xy67
+ σ yx65
2hxhy
u6 +σ xx
78
hx2 −
σ xy67
+ σ yx87
2hxhy
−σ xz
47+ σ zx
78
2hxhz
⎛
⎝ ⎜
⎞
⎠ ⎟u7 −
−σ xx
78
hx2 +
σ yy58
hy2 +
σ zz38
hz2 −
σ xy58
+ σ yx87
2hxhy
−σ xz
38+ σ zx
78
2hxhz
−σ yz
38+ σ zy
58
2hyhz
⎛
⎝ ⎜
⎞
⎠ ⎟u8.
ICCS 2009 3D 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 N3 , 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=NQ0 /8, where Q0 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, Q0 ~ (2/3)83 ≈341 floating operations per–cell at one iteration
ICCS 2009 3D Vector Additive Iterative Solver for Anisotropic Inhomogeneous Poisson Equation in the Forward EEG Problem 18
3D Anisotropic Simulations (88x128x128 voxels)
All tissuesisotropic
Anisotropic skullσr/σt= 1/10
ICCS 2009 3D 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
Heterogeneouscoefficients1e-04 accuracy
ICCS 2009 3D 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 2563 voxels
Currently developing GPGPU implementation