Oriented Tensor Reconstruction. Tracing Neural Pathways from DT-MRI
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Department of Computer Science
California Institute of Technology
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Oriented Tensor Reconstruction: Tracing Neural Pathways from DT-MRI
Leonid Zhukov Alan H. Barr
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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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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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Brain structure
Photo:University of Iowa Virtual Hospital
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Diffusion tensor
• Diffusion – random thermal motion (Brownian motion) of water molecules:
• Diffusion equation:
x
y
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Dxx Dxy Dxz Dyx Dyy Dyz Dzx Dzy Dzz
DT- MRI
• Diffusion tensor data
Data: SCI Institute, University of Utah
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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
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Diffusion ellipsoids
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DT-MRI & fibers
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DT-MRI & fibers
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Fiber tracing
1) continues representation 2) local averaging filter “with memory” and look ahead (oriented anisotropic)
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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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Super-sampling
Continues tensor field – component-wise tri-linear interpolation
Kindlmann, Weinstein, 2000
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Moving filter
Local filter – moving oriented least squares (MLS) filter for tensors
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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!)
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Moving Least Squares
• Polynomial approximation :
• Approximated tensor : tensor tensor
• Approximated tensor- zero-order polynomial :
tensor tensor scalar
tensor tensor
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Integration
Forward Euler (RG) integration (diverging) :
vector vector vector
Inverse Euler –implicit scheme integration (converging):
vector vector vector
Streamline integration:
vector vector
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Diffusion ellipsoids
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Anisotropy measures
C Westin, 97
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Anisotropy
Anisotropy Cl
DT MRI
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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); }
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Tracing algorithm
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Results
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MLS effect
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Results
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Results
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Results
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Results
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Invariant volumes
Diffusivity I Anisotropy Cl
DT MRI
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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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Acknowledgements
• Gordon Kindlmann and SCI institute for brain dataset • Yarden Livnat and David Breen • Supported by NSF grants • Human Brain Project