Benoit Scherrer, Simon K. Warfield

1. Why multiple b-values are required for multi-tensor models.Evaluation with a constrained log-Euclidean model. Benoit Scherrer, Simon K. Warfield

2. Diffusion imaging • Diffusion imagingProvides insight into the 3-D diffusion of water molecules in the human brain. Depends on cell membranes, myelination, …  Central imaging modality to study the neural architecture • Diffusion tensor imaging (DTI) Gaussian assumption for the diffusion PDF of water molecules But inappropriate for assessing multiple fibers orientations • Models local diffusion by a 3D tensor • Widely used (short acquisitions) • Reveals major fiber bundles = “highways” in the brain Good approximation for voxels containing a single fiber bundle direction  But inappropriate for assessing multiple fibers orientations.

3. Diffusion imaging - HARDI • High Angular Resolution Diffusion Imaging (HARDI) Cartesian q-space imaging (DSI), Spherical q-space imaging • Introduce many gradient directions. • One gradient strength (single-shell) or several (multiple-shell) General aim: estimate an approximate of the underlying fiber orientation distribution • Non-parametric approaches • Diffusion Spectrum Imaging, Q-Ball Imaging • Drawbacks: • Narrow pulse approximation. • Need to truncate the Fourier representation  quantization artifacts [Canales-Rodriguez, 2009] • Broad distributions of individual fibers at moderate b-values • Lots of data need to be acquired  limited use for clinical applications

4. Diffusion imaging – parametric approaches • Parametric approaches Describe a predetermined model of diffusion Spherical decomposition, Generalized Tensor Imaging, CHARMED… • Two-tensor approachesAn individual fiber is well represented by a single tensor  multiple fiber orientation expected to be well represented by a set of tensors. Limited number of parameters: a good candidate for clinical applications • BUT: known to be numerically instable

5. Contributions In this work • Show that the multi-tensor models parameters are colinear when using single-shell acquisitions. Demonstrate the need of multiple-shells acquisitions. • Verify these findings with a novel constrained log-euclidean two-tensor model

6. Diffusion signal modeling • Homogeneous Gaussian model (DTI)Diffusion weighted signal Sk along a gradient gk(||gk||=1) : D: 3x3 diffusion tensor, S0: signal with no diffusion gradients, bk: b-value for the gradient direction k. • Multi-fiber models (multi-tensor models) • Each voxel can be divided into a discrete number of homogeneous subregions • Subregions assumed to be in slow exchange • Molecule displacement within each subregion assumed to be Gaussian f1, f2: Apparent volume fraction of each subregion, f1+ f2=1

7. Diffusion signal modeling • Models fitting For one gradient direction: Least square approach by considering the K gradient directions: yk: measured diffusion signal for direction k. • Manipulating the exponential α>0 because

8. Why several b-values are required Demonstration We consider a single b-value acquisition and For any Then for any , and is a solution as well Non-degenerate tensor for By choosing and we verify that

9. Why several b-values are required If is a solution, Then for any , and is a solution as well  Infinite number of solutions  The fractions and the tensor size (eigen-values) are colinear Non-degenerate tensor for • With several b-values

10. Why several b-values are required Two-tensor models:  Single b-value acquisitions Leads to a colinearity in the parameters  conflates the tensor size and the fractions of each tensor  Multiple b-value acquisitions The system is better determined, leading to a unique solution

11. A novel constrained log-euclidean two-tensor approach

12. A novel two-tensor approach • Tensor estimation • Care must be taken to ensure non-degenerate tensors(Cholesky parameterization, Bayesian prior on the eigen values, …) • Elegant approach: consider an adapted mathematical space •  Symmetric definite positive (SPD) matrices: elements of a Riemannian manifold… •  … with a particular metric: null and negative eigen values at an infinite distance • Elegant but at a extremely high computational cost. • Log-euclidean framework • Efficient and close approximation [Arsigny et al, 2006] • Has been applied to the one-tensor estimation [Fillard et al., 2007]

13. A novel two-tensor approach • Two-tensor log-euclidean model • We consider • And the predicted signal for a gradient direction k: • Fractions: parameterized through a softmax transformation [Tuch et al, 2002]

14. A novel two-tensor approach Constrained two-tensor log-euclidean model • To reduce the number of parameters:Introduction of a geometrical constraint [Peled et al, 2006] •  each tensor is constraint to lie in the same plane •  defined by the two largest eigenvalues of the one-tensor solution • Formulation • One tensor solution:  • 2D minimization problem. Estimate 2D tensors subsequently rotated by V. • Only 4 parameters per tensor

15. A novel two-tensor approach • Two-Tensor fitting • Solving • Conjugate gradient descent algorithm (Fletcher-Reeves-Polak-Ribiere algorithm)(Iterative algorithm) • Differentiation in the log-euclidean framework for the constrained model:

16. A novel two-tensor approach • Two-Tensor fitting • Solving • Conjugate gradient descent algorithm (Fletcher-Reeves-Polak-Ribiere algorithm) • Differentiation in the log-euclidean framework:

17. A novel two-tensor approach • Initial position • We consider the one-tensor solution  Initial tensors almost parallel  Initial tensors perpendicular • Initial tensors: rotation of angle in the plane formed by • Formulation • The final two-tensor are obtained by:

18. A novel two-tensor approach • Initial position • We consider the one-tensor solution  Initial tensors almost parallel  Initial tensors perpendicular • Initial tensors: rotation of angle in the plane formed by • Formulation • The final two-tensor are obtained by:

19. Evaluation

20. Evaluation • Simulations • Phantom representing two fibers crossing at 70° • Simulation of the DW signal, corrupted by a Rician noise • Evaluation of different acquisition schemes • Qualitative evaluation 2 shells 45 images. 30 dir. b1=1000s/mm2 + 15 dir. b2=7000s/mm2 1 shell 90 images. 90 dir. b=1000s/mm2 2 shells 90 images. 30 dir. b1=1000s/mm2 + 30 dir. b2=7000s/mm2 • 45 images with 2 b-values provides better results than 90 images with one b-value

21. Evaluation • Quantitative evaluation • Two-shells acquisitions: b1=1000s/mm2 , D1=30 directionsand different values for b2 , D2 (1034 experiments) • The introduction of high b-values helps in stabilizing the estimation • tAMD: Average Minimum LE distance • Fractions compared in term of Average Absolute Difference (AAD)

22. Evaluation • Quantitative evaluation • Even an acquisition with 282 directions provides lower results than (45,45)

23. Discussion Conclusion • Analytical demonstration that multi-tensor require at least two b-value acquisitions for their estimations • Verified these findings on simulations with a novel log-euclidean constrained two-tensor model Need of several b-values • Already observed experimentally. But here theoretically demonstrated • A number of two-tensors approaches are evaluated with one b-value acquisition conflates the tensor size and the fractions A uniform fiber bundle may appear to grow & shrink due to PVE(But generally, tractography algorithms take into account only the principal direction) • High b-values: provides better results. Possibly numerical reasons (reduce the number of local minima?). • Three tensors : requires three b-value ?

24. Discussion Novel log-euclidean two-tensor model • Log-euclidean: elegant and efficient framework to avoid degenerate tensors • Constrained: reduce the number of free parameters (only 8) • Preliminary evaluations: a limited number of acquisitions appears as sufficient Two-tensor estimation from 5-10min acquisitions? (clinically compatible scan time) In the future • Fully take advantage of the log-euclidean frameworkNot only to avoid degenerate tensors, also to provide a distance between tensors. Tensor regularization • Full characterization of such as model • Noise and angle robustness • Evaluation on real data with different b-value strategies.

25. Thank you for your attention,