Benoit Scherrer, Simon K. Warfield

Benoit Scherrer, ISBI 2010, RotterdamBenoit 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

Benoit Scherrer, ISBI 2010, RotterdamBenoit Scherrer, ISBI 2010, RotterdamBenoit Scherrer, ISBI 2010, RotterdamBenoit Scherrer, ISBI 2010, Rotterdam

Diffusion imaging

Diffusion tensor imaging (DTI) Gaussian assumption for the diffusion PDF of water molecules

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

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.


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 ImagingDrawbacks: 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


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


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


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


Diffusion signal modeling Models fitting

yk : 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


By choosing and we verify that

Why several b-values are requiredDemonstration

For anyWe consider a single b-value acquisition and


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 , andis a solution as well

Non-degenerate tensor for

If is a solution,


Why several b-values are required

Single b-value acquisitionsLeads to a colinearity in the parameters conflates the tensor size and the fractions of each tensor

Two-tensor models:

Multiple b-value acquisitionsThe system is better determined, leading to a unique solution


A novel constrained log-euclidean two-tensor approach


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



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]


A novel two-tensor approachConstrained 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



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)



Two-Tensor fitting

Differentiation in the log-euclidean framework:

Solving

Conjugate gradient descent algorithm (Fletcher-Reeves-Polak-Ribiere algorithm)


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


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


Evaluation


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/mm2

2 shells 45 images.30 dir. b1=1000s/mm2

+ 15 dir. b2=7000s/mm2

2 shells 90 images.30 dir. b1=1000s/mm2

+ 30 dir. b2=7000s/mm2

Qualitative evaluation

45 images with 2 b-values provides better results than 90 images with one b-value


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)


Evaluation Quantitative evaluation

Even an acquisition with 282 directions provides lower results than (45,45)


DiscussionConclusion 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 ?


DiscussionNovel 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.


Thank you for your attention,

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Benoit Scherrer, Simon K. Warfield