Diffusion tensor DT-MRI tractography. Talk at Vis 2002

1. Oriented Tensor Reconstruction:
Tracing Neural Pathways from DT-MRI
Leonid Zhukov
Alan H. Barr
Department of Computer Science
California Institute of Technology
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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
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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)
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4. Brain structure
Photo:University of Iowa Virtual Hospital
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5. Diffusion tensor
y
• Diffusion – random thermal motion
(Brownian motion) of water molecules:
x
• Diffusion equation:
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6. DT- MRI
• Diffusion tensor data
Dxx Dxy Dxz
Dyx Dyy Dyz
Dzx Dzy Dzz
Data: SCI Institute, University of Utah
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7. Eigenvalues/vectors
• Eigenvalues/eigenvectors basis
e3
• In e1,e2,e3 local Cartesian frame - tensor diagonal
e2
e1
D
every voxel
• Interpretation: ellipsoid = D * sphere
• Bilinear form –invariant
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8. Diffusion ellipsoids
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9. DT-MRI & fibers
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10. DT-MRI & fibers
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11. Fiber tracing
1) continues representation
2) local averaging filter “with memory”
and look ahead (oriented anisotropic)
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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
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13. Super-sampling
Continues tensor field – component-wise tri-linear interpolation
Kindlmann, Weinstein, 2000
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14. Moving filter
Local filter – moving oriented least squares (MLS) filter for tensors
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15. Moving Least Squares
• Find best approximation in LS sense - minimizing functional:
scalar scalar tensor tensor
• Polynomial approximation:
tensor tensor
• Minimization:
scalar tensor tensor
(every tensor component separately!)
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16. Moving Least Squares
• Polynomial approximation :
tensor tensor
• Approximated tensor :
tensor tensor
• Approximated tensor-
zero-order polynomial :
tensor scalar tensor
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17. Integration
Streamline integration:
vector vector
Forward Euler (RG) integration (diverging) :
vector vector vector
Inverse Euler –implicit scheme integration (converging):
vector vector vector
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18. Diffusion ellipsoids
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19. Anisotropy measures
C Westin, 97
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20. Anisotropy
DT MRI
Anisotropy Cl
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21. Tracing algorithm
Tracing Procedure: trace = fiber_trace(P,e) {
trace->add(P);
for (every starting point P) {
do {
Tp = filter(T,P,sphere); Pn = integrate_forward(P,e1,dt);
cl = anisotropy(Tp); Tp = filter(T,Pn,ellipsoid,e1);
if (cl > eps) { cl = anisotropy(Tp)
e1 = direction(Tp); if ( c1 > eps ) {
trace->add(Pn);
trace1 = fiber_trace(P, e1);
P = Pn;
trace2 = fiber_trace(P,-e1); e1 = direction(Tp);
trace = trace1 + trace2; }
} } while (cl >eps)
} return(trace);
}
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22. Tracing algorithm
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24. Results
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25. MLS effect
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26. Results
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28. Results
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30. Results
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31. Results
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33. Invariant volumes
DT MRI
Diffusivity I Anisotropy Cl
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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
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37. Acknowledgements
• Gordon Kindlmann and SCI institute for brain dataset
• Yarden Livnat and David Breen
• Supported by NSF grants
• Human Brain Project
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