^{*}

The authors have declared that no competing interests exist.

Conceived and designed the experiments: BS SKW. Performed the experiments: BS. Analyzed the data: BS. Contributed reagents/materials/analysis tools: BS SKW. Wrote the paper: BS SKW.

The characterization of the complex diffusion signal arising from the brain remains an open problem. Many representations focus on characterizing the global shape of the diffusion profile at each voxel and are limited to the assessment of connectivity. In contrast, Multiple Fascicle Models (MFM) seek to represent the contribution from each white matter fascicle and may be useful in the investigation of both white matter connectivity and diffusion properties of each individual fascicle. However, the most appropriate representation of multiple fascicles remains unclear. In particular, a multiple tensor representation of multiple fascicles has frequently been reported to be numerically challenging and unstable. We provide here the first analytical demonstration that when using a diffusion MRI acquisition with only one non-zero b-value, such as in conventional single-shell HARDI acquisition, a co-linearity in model parameters makes the precise model estimation impossible. Motivated by this theoretical result, we propose the novel CUSP (CUbe and SPhere) optimal acquisition scheme to achieve multiple non-zero b-values. It combines the gradients of a single-shell HARDI with gradients in its enclosing cube, in which varying b-values can be acquired by modulation of the gradient strength, without modifying the minimum echo time. Compared to a multi-shell HARDI acquisition, our scheme has significantly increased signal-to-noise ratio. We propose a novel estimation algorithm that enables efficient, robust and accurate estimation of the parameters of a multi-tensor model. In conjunction with a CUSP acquisition, it enables

Measuring water diffusion with magnetic resonance diffusion weighted imaging (MR-DWI) has enabled non-invasive investigation and characterization of the white matter architecture and microstructure in the brain. The diffusion in a white matter fascicle has been observed to be highly anisotropic, with primary orientation corresponding to the orientation of the fascicle

Diffusion-weighted acquisition with

DTI and its underlying mono-exponential signal attenuation assumption are generally considered to satisfactorily represent

DTI is however well known to be a poor parametric model for representing the diffusion signal arising at voxels that encompass multiple fascicles with heterogeneous orientation such as fascicle crossing, kissing or fanning. A wide number of approaches have been investigated to overcome this fundamental limitation. They involve both novel

Mainly two q-space sampling strategies have been used for complex fiber structure assessment: Cartesian sampling and spherical sampling. Cartesian sampling is used by diffusion spectrum imaging (DSI)

Other sampling techniques have been proposed for reasons other than assessing complex fiber structures. Sampling using the tetrahedral

A large number of approaches have been investigated to analyze the diffusion signal and represent multiple white-matter fascicles with complex geometry. Both parametric (model-based) and non-parametric (model-free) approaches have been proposed. They generally focus on estimating either (1) the diffusion displacement probability density function (diffusion PDF), (2) the diffusion orientation distribution function (dODF) which is the angular profile of the diffusion PDF or (3) the fiber orientation distribution function (fODF), also known as the fiber orientation density (FOD) and which is of central interest for tractography.

Model-free approaches include diffusion spectrum imaging (DSI)

Q-space approaches such as DSI, QBI, or EQBI are however limited by three major error sources. First they are based on the

In contrast, parametric models describe a predetermined model of diffusion rather than an arbitrary one. They potentially require a smaller number of images to be acquired, leading to a reduced acquisition time. A large number of model-based approaches have been investigated. Among them, generalized diffusion tensor imaging (GDTI)

A major drawback to DSI, QBI, DOT, SD and GDTI is that they focus on describing the

In contrast, multi-fascicle models (MFM) consider at each voxel a mixture of independent fascicles with heterogeneous orientation. Making the assumption of a slow exchange between the fascicles' compartments, the diffusion signal in each voxel is modeled as a mixture of the diffusion signal arising from each individual fascicle. Integration of an isotropic component has also been investigated

The diffusivity of free water is generally considered to be well modeled by an isotropic Gaussian distribution

In a particular case of multi-fascicle model, the ball-and-stick model

In contrast, since an individual fascicle is generally considered to be well represented by a single tensor in DTI, a natural candidate has been to represent each fascicle by a tensor. Considering

Multiple works have pointed out that a non-monoexponential decay may be observed

(a): Illustration of the monoexponential decay arising from a single tensor (FA =

Importantly, compartmentalization is not a prerequisite for the presence of a non-monoexponential decay. Schwarcz

Multiple approaches have been investigated to account for the non-Gaussianity of the diffusion signal in a voxel

To the best of our knowledge, all approaches accounting for the non-monoexponential signal decay have considered the case of a single fascicle in each voxel. For example, Cheung

Therefore, in this work, we focused on a representation of each individual fascicle by a single tensor model. The

The contributions of this work are three-fold. First we provide the

Second, we propose a novel multi-tensor optimization technique based on the maximum

Third, we propose to employ a novel acquisition scheme that enables estimation of a

The paper is organized as follows. We provide in Section 0.1 the theoretical demonstration that multi-tensor models require multiple non-zero b-values to be fully estimated. We describe our novel algorithm for estimating the parameters of the multi-fascicle model (MFM) in Section 2. We detail our Cube and Sphere (CUSP) imaging technique in Section 0.3. The CUSP-MFM evaluation includes several qualitative and quantitative experiments with both synthetic and in vivo data: angular resolution performance, comparison with the state-of-the-art ball-and-stick model and bootstrap experiments. We show that CUSP-MFM enables the characterization of multiple white-matter fascicles from short duration acquisitions compatible with routine clinical practice.

We demonstrate in this section that the tensors and fractions of occupancy of a multi-tensor model cannot be uniquely determined when using a single shell HARDI acquisition

Consequently, when using a single non-zero

It is not the case with multiple non-zero b-values

We consider the image domain

We denote by

In this work we did not considered any prior knowledge on the estimated fractions and considered

We parameterize each tensor's orientations with the Euler angle. We experimentally found this representation to enable a more efficient optimization of the parameters. In addition, it enables the choice of introducing various constraints to further reduce the number of parameters: symmetry of the eigenvalues (

The model parameters are estimated by performing an iterative minimization which requires a starting point. As in

The solution of

We have demonstrated in Section 0.1 that multiple non-zero b-values are required to fully estimate multi-tensor models. In this section we provide an optimal gradient encoding scheme which satisfies this requirement.

In diffusion weighted imaging, a key parameter in controlling the signal-to-noise ratio (SNR) is the echo time (TE). An increase in TE leads to a signal dropout due to T2 relaxation and therefore to a decrease in SNR (see

Single-shell HARDI as used in

We propose instead the novel CUbe and SPhere (CUSP) acquisition technique. We incorporate multiple non-zero b-values by combining a single-shell HARDI acquisition at a specified

We envisage three ways to construct a CUSP acquisition which are based on a generalization of a multi-shell HARDI (see

(a): In CUSP-T (Truncated), we consider a multi-shell HARDI with uniformly spaced radius (blue, green, red) and truncate those parts of the shells that project outside of the cube of constant TE of the inner shell. (b): CUSP-xT (eXponential Truncated) employs portion of multiple shells with exponentially spaced radius to achieve an improved uniformity of SNR. (c): In CUSP-P (Projected), we consider an inner shell at

Our work is the first report of utilizing such a CUbe and SPhere acquisition to enable the full estimation of a multiple fascicle model. The strength of our technique is to provide multiple non-zero b-values and higher b-values than the nominal b-value

We provided the theoretical demonstration that only the tensor orientation can be uniquely estimated when using a single non-zero b-value. Multiple non-zero b-values are required to fully estimate the tensors' direction in addition to the tensors' size and their respective fraction of occupancy. We proposed a novel algorithm for the estimation of the parameters of a multi-fascicle model (MFM). It is formulated as a log-Euclidean

The multi-tensor estimation algorithm was implemented in C++ and parallelized over the image space. The model parameters were set as follows. The diffusivity of free water at

We generated various synthetic phantoms to evaluate CUSP-MFM. The tensor profile ^{2} images.

We focused here on short duration acquisitions with a low number of directions which are of practical interest for clinical applications. We considered a CUSP-T acquisition consisting of a total of thirty-five images, referred to as CUSP35 (

We compared the CUSP35 acquisition scheme to a single-shell HARDI acquisition, referred to as HARDI35, which includes five

For each experiment we reported both qualitative and quantitative results. For the quantitative analysis, we compared the estimated multi-fascicle model to the synthetic ground truth by means of different metrics. The tensors were compared in term of average log-Euclidean distance (tALED), taking into account a possible permutation between the estimated and the reference tensors:

The performance of CUSP-MFM was assessed on in vivo data acquired on a Siemens 3T Trio scanner with a ^{3}. Eddy current distortion was minimized by utilizing a twice-refocused spin echo sequence

We acquired a multi-shell HARDI composed of

We acquired a CUSP-P acquisition referred to as CUSP65 and constructed from a

We employed a generalization of the optimization algorithm of

Finally, a T1-weighted MPRAGE image was acquired with the following parameters: ^{3}, TE =

The diffusion weighted images were corrected for head motion during the scan by rigid registration of the DW-images to the

We performed an experiment to examine the effect of CUSP-MFM on the assessment of tensor diffusion parameters. We applied

Finally, a

We generated a set of phantoms containing one hundred two-tensor models separated by a given angle

(a) One hundred synthetic tensors crossing at

Quantitative evaluation of the estimation accuracy for the fractions (first line, fAAD metric) and the tensors (second line, tALED metric). Each plot shows the quality metric (fAAD, tALED) in function of the crossing angle for various gradient encoding scheme and various signal-to-noise ratios. It shows that employing a large number of directions (HARDI256) does not dramatically improve the results whereas introducing multiple non-zero b-values does (CUSP35). CUSP35-MFM consistently provides the best estimation accuracy.

Evaluation of the angle detection accuracy in term of average minimum angle error (tAMA) and comparison with the ball-and-stick model of FSL. CUSP35-MFM provides on average the best angular resolution, particularly for angles lower than

Finally, we investigated whether or not CUSP introduces an angular preference for certain spatial directions when characterizing fascicles (

The DW-images for a single tensor with constant FA (FA =

We then generated a phantom representing two uniform fascicles (

Estimation of two synthetic crossing fascicles (angle =

These findings were quantitatively verified by simulating one hundred times the diffusion signal for different signal-to-noise ratios (

Quantitative evaluation of diffusion parameters along the horizontal synthetic tract of

We report in this section the results of experiments on in vivo data. In

Comparison of the HARDI35 (first column) and CUSP35 (second column) acquisitions. Fig.a and Fig.b: in contrast to CUSP35-MFM (b), HARDI35-MFM (a) leads to degenerate tensors (area 1) and confounds CSF contamination and fascicles (area 2). Fig.c and Fig.d: when ignoring the estimation of the isotropic compartment, the performance of CUSP35-MFM (d) are strongly affected. The diffusion of unrestricted water cannot be ignored when using a multiple b-values acquisitions. Fig.e and Fig.f: FSL estimates sticks with noisy orientation (area 4), and leads to non-aligned sticks in a single fascicle region of the corpus callosum (area 3). Fig.g and Fig.h: FSL estimation after denoising the DW images (dHARDI35 and dCUSP35).

Estimation of

In

(a) Estimated MFM superimposed on the T1-weighted anatomical image. Particularly, three tensor were correctly estimated in the centrum semiovale, which is a known brain region in which three fascicles are crossing. (b) Illustration of the tractography streamlines passing through the voxel encircled in yellow in (a), showing the three crossing fascicles.

We computed the FA along a same tract (Fig.a) for various artificial rotations of the diffusion-weighted images. For each streamline point, the most aligned anisotropic tensor with the streamline orientation was selected and its FA assessed. Fig.b shows the variance of the FA along the tract across the rotations, when using the CUSP or the HARDI acquisition and the MFM estimator without regularization and with the same parameters. HARDI has dramatically increased variance, as it conflates tensor size with partial voluming. CUSP does not. Fig.c shows the FA variance when adding the regularization to the estimation with both CUSP and HARDI. Fig.d shows the corresponding value of the FA along the tract. It shows that CUSP-MFM enables estimation of diffusion tensor parameters which do not vary with the partial volume fractions nor the regularization.

Finally, we report in

(a) T1-weighted image showing the anatomy. (b) Standard deviation of the maximum FA when using MSHARDI-65-MFM. (c) Standard deviation of the maximum FA when using CUSP-65-MFM. The standard deviation of the maximum FA is significantly lower when using CUSP, showing a lower uncertainty in the MFM estimates.

Several methods have been investigated to overcome the limitations of DTI and to represent multiple white matter fascicles from diffusion-weighted imaging. Approaches such as DSI, QBI, EQBI, DOT, SD or GDTI focus on estimating the global shape of the diffusion profile resulting from multiple fascicles present in each voxel. The major drawback is they do not consider each fascicle independently. Consequently, they do not enable characterization of each fascicle, and do not enable comparison of the fascicle characteristics between individuals. The assessment of parameters such as the generalized fractional anisotropy (GFA) or the generalized mean diffusivity (GMD) has also been proposed

In contrast, our model enables the determination of the orientation of the white matter fascicles, measures of their local diffusion properties and the characterization of an unrestricted water component that is important in assessing edema and inflammation. Multi-fascicle approaches generally require the determination of the number of white matter fascicles at each voxel. This and only this enables characterization of each fascicle in addition to the orientation information, which is of central interest to study the white matter development or degeneration in research and clinical practice. Recently

Multi-tensor models have however frequently been reported to be numerically challenging and unstable

Our solution lies in a novel multi-fascicle estimation framework, CUSP-MFM, which is the combination of a novel multi-tensor estimation algorithm and an optimal acquisition scheme which satisfies the need of multiple non-zero b-values. The characteristics of our multi-tensor estimation procedure were driven by the objective to make it possible to represent multiple fascicles in clinical DWI with a relatively short acquisition time, compatible with pediatric and adult imaging. The

We show that when using an optimized acquisition scheme and when estimating the diffusion of unrestricted water, we can accurately estimate the fascicle orientation from the hindered diffusion. In this work, we have relied on the assumption that,

We have formulated our multi-tensor estimation approach in the log-Euclidean framework

We proposed a log-Euclidean regularization scheme which is not the direct extension of the one-tensor regularization. We suggested a particular approximation of the spatial gradient for multi-tensor fields (

In this work we propose an acquisition scheme designed to enable accurate assessment of multiple white matter fascicles. Our CUSP (CUbe and SPhere) imaging technique combines a single-shell HARDI with gradients in its enclosing cube. The single-shell HARDI provides a full spherical sampling with the optimal SNR and the optimal TE for the chosen nominal b-value. Any gradient in the enclosing cube of the single-shell HARDI can be acquired

Our evaluation shows clear evidence that the estimation of both the tensors and the fractions of occupancy are improved when using CUSP instead of a single shell acquisition (

The CUSP-MFM's performance was assessed via various experiments on both synthetic and in vivo data. We focused on short acquisitions suitable for routine clinical use, especially for pediatric MRI. The angular resolution is substantially superior to the state-of-the-art ball-and-stick model implemented in FSL (

The qualitative evaluation on real data (

Finally, we demonstrated that the estimation uncertainty is higher when using a multi-shell HARDI instead of CUSP (

Future work will focus on assessing different gradient schemes for the CUSP acquisition. Particularly, we will investigate the optimal ordering of the gradient directions. CUSP-MFM performance will be compared to Q-Ball Imaging and Spherical Deconvolution approaches for the assessment of connectivity.

Robust estimation will be also explored. It enables to reduce the influence of large residuals, making the estimation less sensitive to outliers than when using the least square criteria. It may provide a better robustness to patient motion and will be of particular interest for pediatric imaging.

We demonstrated and experimentally verified that multiple non-zero b-values are required to fully estimate multi-tensor models. As a solution we proposed CUSP-MFM which combines an optimal CUbe and SPhere (CUSP) acquisition technique with a novel algorithm for the estimation of the parameters of a Multi-Fascicle Model (MFM). Our proposed CUSP acquisition technique provides multiple high b-values with the optimal achievable TE. It does not increase the imaging time nor the eddy current distortion compared to a single-shell HARDI. Additionally, it does provide the optimal signal-to-noise ratio, leading to estimates with higher certainty. Our novel multi-fascicle fitting algorithm MFM is formulated as a

We provide the CUSP65 gradient encoding schemes in the Siemens format. On a Siemens scanner, this requires to set the imaged b-value to

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