1. Benoit Scherrer, ISBI 2010, Rotterdam Why multiple b-values are required for multi-tensor models. Evaluation with a constrained log- Euclidean model. Benoit Scherrer, Simon K. Warfield

2. Benoit Scherrer, ISBI 2010, Rotterdam Diffusion imaging  Diffusion tensor imaging (DTI) Gaussian assumption for the diffusion PDF of water molecules  Diffusion imaging Provides 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  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  But inappropriate for assessing multiple fibers orientations.

3. Benoit Scherrer, ISBI 2010, Rotterdam 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)  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 General aim: estimate an approximate of the underlying fiber orientation distribution

4. Benoit Scherrer, ISBI 2010, Rotterdam Diffusion imaging – parametric approaches  Parametric approaches Describe a predetermined model of diffusion  Spherical decomposition, Generalized Tensor Imaging, CHARMED…  Two-tensor approaches An 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. Benoit Scherrer, ISBI 2010, Rotterdam 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. Benoit Scherrer, ISBI 2010, Rotterdam Diffusion signal modeling  Homogeneous Gaussian model (DTI) Diffusion weighted signal S k along a gradient g k (||g k ||=1) : D: 3x3 diffusion tensor, S 0 : signal with no diffusion gradients, b k : 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 f 1, f 2 : Apparent volume fraction of each subregion, f 1 + f 2 =1

7. Benoit Scherrer, ISBI 2010, Rotterdam Diffusion signal modeling  Models fitting y k : measured diffusion signal for direction k.  Manipulating the exponential because Least square approach by considering the K gradient directions: For one gradient direction: α>0

8. Benoit Scherrer, ISBI 2010, Rotterdam By choosing and we verify that Why several b-values are required Demonstration For any We consider a single b-value acquisition and Then for any, and is a solution as well Non-degenerate tensor for

9. Benoit Scherrer, ISBI 2010, Rotterdam Why several b-values are required  Infinite number of solutions  The fractions and the tensor size (eigen-values) are colinear  With several b-values Then for any, and is a solution as well Non-degenerate tensor for If is a solution,

10. Benoit Scherrer, ISBI 2010, Rotterdam Why several b-values are required  Single b-value acquisitions Leads to a colinearity in the parameters  conflates the tensor size and the fractions of each tensor Two-tensor models:  Multiple b-value acquisitions The system is better determined, leading to a unique solution

11. Benoit Scherrer, ISBI 2010, Rotterdam A novel constrained log-euclidean two-tensor approach

12. Benoit Scherrer, ISBI 2010, Rotterdam A novel two-tensor approach  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] 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

13. Benoit Scherrer, ISBI 2010, Rotterdam 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. Benoit Scherrer, ISBI 2010, Rotterdam 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  One tensor solution:   2D minimization problem. Estimate 2D tensors subsequently rotated by V.  Only 4 parameters per tensor  Formulation

15. Benoit Scherrer, ISBI 2010, Rotterdam A novel two-tensor approach  Two-Tensor fitting  Differentiation in the log-euclidean framework for the constrained model:  Solving  Conjugate gradient descent algorithm (Fletcher-Reeves-Polak-Ribiere algorithm) (Iterative algorithm)

16. Benoit Scherrer, ISBI 2010, Rotterdam A novel two-tensor approach  Two-Tensor fitting  Differentiation in the log-euclidean framework:  Solving  Conjugate gradient descent algorithm (Fletcher-Reeves-Polak-Ribiere algorithm)

17. Benoit Scherrer, ISBI 2010, Rotterdam A novel two-tensor approach  Initial position  We consider the one-tensor solution  Initial tensors: rotation of angle in the plane formed by  Initial tensors almost parallel  Initial tensors perpendicular  The final two-tensor are obtained by:  Formulation

18. Benoit Scherrer, ISBI 2010, Rotterdam A novel two-tensor approach  Initial position  We consider the one-tensor solution  Initial tensors: rotation of angle in the plane formed by  Initial tensors almost parallel  Initial tensors perpendicular  The final two-tensor are obtained by:  Formulation

19. Benoit Scherrer, ISBI 2010, Rotterdam Evaluation

20. Benoit Scherrer, ISBI 2010, Rotterdam Evaluation  Simulations  Phantom representing two fibers crossing at 70°  Simulation of the DW signal, corrupted by a Rician noise  Evaluation of different acquisition schemes 1 shell 90 images. 90 dir. b=1000s/mm 2 2 shells 45 images. 30 dir. b 1 =1000s/mm 2 + 15 dir. b 2 =7000s/mm 2 2 shells 90 images. 30 dir. b 1 =1000s/mm 2 + 30 dir. b 2 =7000s/mm 2  Qualitative evaluation  45 images with 2 b-values provides better results than 90 images with one b-value

21. Benoit Scherrer, ISBI 2010, Rotterdam Evaluation  Quantitative evaluation  Two-shells acquisitions: b 1 =1000s/mm 2, D 1 =30 directions and different values for b 2, D 2 (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. Benoit Scherrer, ISBI 2010, Rotterdam Evaluation  Quantitative evaluation  Even an acquisition with 282 directions provides lower results than (45,45)

23. Benoit Scherrer, ISBI 2010, Rotterdam 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. Benoit Scherrer, ISBI 2010, Rotterdam 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 framework Not 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. Benoit Scherrer, ISBI 2010, Rotterdam Thank you for your attention,