a 3d vector-additive iterative solver for the anisotropic inhomogeneous poisson equation in the...

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


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)


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


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


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)


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


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


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


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


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


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

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥


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)


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)


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


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.


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


3D Anisotropic Simulations (88x128x128 voxels)

All tissuesisotropic

Anisotropic skullσr/σt= 1/10


(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


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

a 3d vector-additive iterative solver for the anisotropic inhomogeneous poisson equation in the...

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