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