DT-MRI Visualization
Fiber tractography Diffusion tensor filtering and interpolation Leonid Zhukov
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Fiber tractography
n Fiber tractography – computing and following directions of fiber bundles within the tissue based on DT-MRI data • functional connectivity studies • function to structure
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Fiber tractography
n Difficulties: • voxelization / resolution • noise • ill-posedness of the problem
n Algorithms: • Deterministic algorithms • Probabilistic methods • PDE based methods
n Data: • Discrete • Continious
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Deterministic algorithms
n Mori et al. 1999, Jones et al. 1999, Conturo et al. 1999 • Follow local main diffusion direction from voxel to voxel, heuristics
n Westin et al. 1999, 2002 • Diffusion tensors are projection operators rotating and scaling tracing “velocity”
n Weinstein et al. 1999, Lasar et al, 2000,2003 • Tensor deflection
n Basser et al. 2000 • Continues spline approximation to tensor field and integral curves
n Gossl et al. 2001
• State space model , Kalman filtering
n Zhukov et al. 2002 • Moving Least Squares filter , integral curves
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Probabilistic & PDE based methods
Probabilistic methods: n Poupon et al. 2000, 2001
• regularization of tensor field, Markovian fields
n Hagmann et al. 2003 • random walk , random direction distributed according to local diffusion properties,
regularization terms, coliniarity with previous step
PDE based methods: n Parker et al., 2002
• Level set methods, diffusion front propagation
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Fiber tractography
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Data: anisotropy
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Data: anisotropy
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Fiber tracing
2) continues representation 3) local averaging filter “with memory” and look ahead (oriented anisotropic)
1) noise filtering
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Streamline integration fibertracking
n Main steps: • Interpolate (approximate) the data, make it continuous • Smooth and filter the data • Tensor filed –> vector field • Streamline integration (integral curve)
n Typical algorithm: • Select starting points (region) • Integrate forward from every point • Stop if outside of domain • Controlled by anisotropy • Prevent sharp turns
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MLS method
n Continues tensor field by interpolation n Evaluation of local vector field direction is delayed until tracking
(eigen-computations) n Local tensor filtering by polynomial approximation n Look ahead / memory, local weighted average n Filtering is simultaneous with tracing n Tuned up level of smoothing n EU1, RK2,4 integration n Anisotropy controlled
Zhukov and Barr, 2002
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Interpolation
Continues tensor field representation – component-wise interpolation
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MLS filter
• smooth varying variable, corrupted by noise • low–pass filter • window: replace data point by local average • preserves area under the curve
• higher order polynomial • least squares fit
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MLS filter
Local filter: moving oriented least squares (MLS) tensor filter
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Integration
Forward Euler (RG-2,4) type integration (diverging field) :
vector vector vector
Inverse Euler –implicit scheme integration (converging field):
vector vector vector
Streamline integration (vector field):
vector vector
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Tracing algorithm
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); }
Tracing Procedure:
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Tracing algorithm
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Example: Gordon’s brain data
Data: SCI Institute, University of Utah
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Brain structure: corona radiata
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MLS effect
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Brain structure: singulum bundle
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Data: Dr Edward Hsu, Dept. of Bioengineering, Duke University
Example: canine heart data
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Canine heart myofibers
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New developments
n Fiber grouping n Initial value problem, boundary value problem n Fiber merging and splitting n Additional constraints – model surface etc n Fiber distribution analysis
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