Department of Computer Science

California Institute of Technology

9/10/11 Computer Graphics Group 1

Oriented Tensor Reconstruction: Tracing Neural Pathways from DT-MRI

Leonid Zhukov Alan H. Barr

9/10/11 Computer Graphics Group 2

Talk outline •  Introduction

–  Tensor visualization: previous work –  Motivation: brain anatomy –  Diffusion tensor DT-MRI overview

•  Algorithm for directional tensor reconstruction: –  Data interpolation and filtering –  Moving Least Squares method –  Fiber tracing algorithm

•  Results: –  Extracted anatomical structures: corona radiata, corpus callosum, cingulum

bundle, U-shape fibers etc

•  Conclusions

9/10/11 Computer Graphics Group 3

Previous work •  Tensor visualization

–  Tensor fields (stress–strain tensors) (Delmarcelle & Hesselink 92)

•  Diffusion tensor based segmentation –  Anisotropy measures ( Basser 96 ) –  Ellipsoid classification (Westin 97)

•  Diffusion tensor visualization –  DT-MRI 2D – ellipsoids (Laidlaw 98) –  DT-MRI 3D volume rendering (Kindlmann 99)

•  Diffusion tensor based fiber tracing – streamline integration –  Tensorlines, streamtubes (Weinstein 98, Laildlaw 01) –  In vivo fiber tractography (Basser 2000) –  Anatomical brain connectivity (Parker 01)

9/10/11 Computer Graphics Group 4

Brain structure

Photo:University of Iowa Virtual Hospital

9/10/11 Computer Graphics Group 5

Diffusion tensor

•  Diffusion – random thermal motion (Brownian motion) of water molecules:

•  Diffusion equation:

x

y

9/10/11 Computer Graphics Group 6

Dxx Dxy Dxz Dyx Dyy Dyz Dzx Dzy Dzz

DT- MRI

•  Diffusion tensor data

Data: SCI Institute, University of Utah

9/10/11 Computer Graphics Group 7

Eigenvalues/vectors

•  Eigenvalues/eigenvectors basis

•  In e1,e2,e3 local Cartesian frame - tensor diagonal

•  Interpretation: ellipsoid = D * sphere •  Bilinear form –invariant

e1 e2

e3

every voxel

D

9/10/11 Computer Graphics Group 8

Diffusion ellipsoids

9/10/11 Computer Graphics Group 9

DT-MRI & fibers

9/10/11 Computer Graphics Group 10

DT-MRI & fibers

9/10/11 Computer Graphics Group 11

Fiber tracing

1) continues representation 2) local averaging filter “with memory” and look ahead (oriented anisotropic)

9/10/11 Computer Graphics Group 12

Method

•  Build continues representation (super-sampling) for tensor data –  Static preprocessing –  Component-wise filtering –  Tri-linear interpolation

•  Dynamic adaptive local filtering + fibertracing –  Anisotropic local filter, orientation determined by the fiber –  Local least squares approximation to the data (MLS) –  Forward Euler type integration

9/10/11 Computer Graphics Group 13

Super-sampling

Continues tensor field – component-wise tri-linear interpolation

Kindlmann, Weinstein, 2000

9/10/11 Computer Graphics Group 14

Moving filter

Local filter – moving oriented least squares (MLS) filter for tensors

9/10/11 Computer Graphics Group 15

Moving Least Squares

•  Polynomial approximation: tensor tensor

•  Find best approximation in LS sense - minimizing functional:

tensor tensor scalar scalar

•  Minimization:

tensor tensor scalar

(every tensor component separately!)

9/10/11 Computer Graphics Group 16

Moving Least Squares

•  Polynomial approximation :

•  Approximated tensor : tensor tensor

•  Approximated tensor- zero-order polynomial :

tensor tensor scalar

tensor tensor

9/10/11 Computer Graphics Group 17

Integration

Forward Euler (RG) integration (diverging) :

vector vector vector

Inverse Euler –implicit scheme integration (converging):

vector vector vector

Streamline integration:

vector vector

9/10/11 Computer Graphics Group 18

Diffusion ellipsoids

9/10/11 Computer Graphics Group 19

Anisotropy measures

C Westin, 97

9/10/11 Computer Graphics Group 20

Anisotropy

Anisotropy Cl

DT MRI

9/10/11 Computer Graphics Group 21

Tracing algorithm

Tracing Procedure: for (every starting point P) { Tp = filter(T,P,sphere); cl = anisotropy(Tp); if (cl > eps) { e1 = direction(Tp); trace1 = fiber_trace(P, e1); trace2 = fiber_trace(P,-e1); trace = trace1 + trace2; } }

trace = fiber_trace(P,e) { trace->add(P); do { Pn = integrate_forward(P,e1,dt); Tp = filter(T,Pn,ellipsoid,e1); cl = anisotropy(Tp) if ( c1 > eps ) { trace->add(Pn); P = Pn; e1 = direction(Tp); } } while (cl >eps) return(trace); }

9/10/11 Computer Graphics Group 22

Tracing algorithm

9/10/11 Computer Graphics Group 24

Results

9/10/11 Computer Graphics Group 25

MLS effect

9/10/11 Computer Graphics Group 26

Results

9/10/11 Computer Graphics Group 28

Results

9/10/11 Computer Graphics Group 30

Results

9/10/11 Computer Graphics Group 31

Results

9/10/11 Computer Graphics Group 33

Invariant volumes

Diffusivity I Anisotropy Cl

DT MRI

9/10/11 Computer Graphics Group 36

Conclusions

•  Contributions: –  New method for non-linear tensor filtering –  Smooth reconstruction of anatomically recognizable brain

structures

•  Future work: –  additional analytic developments –  needs a good validation

9/10/11 Computer Graphics Group 37

Acknowledgements

•  Gordon Kindlmann and SCI institute for brain dataset •  Yarden Livnat and David Breen •  Supported by NSF grants •  Human Brain Project