Synopsis

With the advent of multi-shell acquisition, there is an increasing need for compact linear orthonormal representations of the DWI signal that extend over the radial as well as angular domain. In this work, we evaluate and compare 3 candidate basis function sets: spherical Bessel functions with and without reparametrization and the SHORE basis. Results show that the reparametrized Bessel functions and SHORE basis can faithfully represent the DWI signal with low numbers of parameters, and can be tuned to the properties of the signal independently of acquisition parameters.

Introduction

Methods

Ideally, basis functions should be orthonormal and separable in spherical coordinates, i.e., $$$\Phi_{nlm}(q,\theta,\phi) = R_n(q) Y_l^m(\theta,\phi)$$$ with $$$Y_l^m$$$ the modified spherical harmonics² of order $$$l$$$. They should also be invariant to allow comparisons across voxels and individuals. In this study, we evaluate and compare 3 candidate function basis sets for the radial domain (Fig.1):

1) SBF: Spherical Bessel functions $$$j_l(k\,q)$$$, with zero-derivative boundary condition. These form a natural complement to the spherical harmonics: combined, they facilitate the Fourier transform in spherical coordinates, hence maintaining a direct relation with the ensemble average propagator³. However, a discrete function basis can only represent the signal in a bounded domain $$$q<q_{max}$$$, with either a zero-value or zero-derivative boundary condition at this edge⁴.

2) RSBF: Reparametrized Bessel functions. By introducing a reparameterisation of $$$q\in[0,+\infty]$$$ to $$$u\in[0,1]$$$ by means of a sigmoid mapping, the discrete Bessel functions can be used with zero-value boundary condition at $$$u=1$$$. This mapping is defined such that free diffusion with diffusivity $$$d$$$ maps exactly onto the 0^th order basis function $$$j_0(k_1\,u)$$$ (Fig.2). The optimal value of this scaling parameter $$$d$$$ is determined by minimizing the fitting residuals over the entire brain.

3) SHORE: In contrast to the original SHORE literature^5,6, we fix and optimize the diffusion scale for the entire image rather than in each voxel separately, to satisfy our requirements for an invariant basis.

All bases were compared in terms of their residual mean square error (RMSE) in simulations (mono-exponential decay with ADC between 0.1-3.0 µm²/ms, sampled at b-values 0, 400, 1000, and 2600 s/mm^{2 7}, 100 instances with SNR=20 Gaussian noise) and in adult brain data (healthy volunteer, 7$$$\times$$$50 diffusion-sensitizing gradients at b=500, 1000, 2000, 3000, 4000, 6000, and 8000 s/mm², 7 b=0 images).

Results

In simulations, the optimal scale parameter is $$$d=$$$ 0.4 for RSBF and $$$d=$$$ 1.1 for SHORE, resulting in RMSE across a dense range of $$$q$$$-values of 46.8 (SBF), 46.0 (RSBF), and 48.3 (SHORE), with a good fit for ADCs between 0.5-3.0 µm²/ms (Figure 3). SBF undershoots at high $$$q$$$ due to the zero-derivative boundary condition.

In the in vivo data, the optimal scale parameters were $$$d=$$$ 0.3 (RSBF) and 0.8 (SHORE) respectively, with RSBF and SHORE exhibiting the lowest RMSE. A good fit was obtained across white and grey matter, with increased residuals in regions of very high or low ADC (Fig.4). RSBF and SHORE performed similarly for varying numbers of parameters (Fig.5).

Discussion

These results underscore the fundamental need to tune the scale of the basis functions to the data at hand. In SHORE, tuning is typically done per voxel, based on the ADC. However, this is undesirable as it results in different basis functions for every voxel, making comparisons difficult. However, our results suggest that a single fixed scale can be used for the entire image with good accuracy in white and grey matter.

The SBF basis is limited by the appropriateness of the boundary conditions necessary to discretise the basis. Neither the zero-value or zero-derivative conditions are physically realistic at reasonable values of $$$q_{max}$$$, and furthermore introduce a dependency of the basis on parameters of the acquisition. In contrast, RSBF and SHORE can be tuned to the properties of the intrinsic signal (via the scale parameter $$$d$$$), allowing for comparisons across different acquisition parameters.

In terms of RMSE, RSBF performed best in simulation, but similarly to SHORE in vivo.

Conclusion

Acknowledgements

References

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