Synopsis

b-tensor encoding enables the separation of isotropic and anisotropic tensors. However, little consideration has been given as to how to design a b-tensor encoding sampling scheme. In this work, we propose the first 4D basis for representing the diffusion signal acquired with b-tensor encoding. We study the properties of the diffusion signal in this basis to give recommendations for optimally sampling the space of axisymmetric b-tensors. We show, using simulations, that the proposed sampling scheme enables accurate reconstruction of the diffusion signal by expansion in this basis using a clinically feasible number of samples.

Introduction

MD-dMRI Signal Basis and b-tensor Sampling Scheme

$$\frac{S}{S_0}(\mathbf{B})=\int{P(\mathbf{D})\exp(-\mathbf{B}:\mathbf{D})d\mathbf{D}}=\langle\exp(-\mathbf{B}:\mathbf{D})\rangle,$$ is the diffusion signal acquired with b-tensor encoding, where $P(\mathbf{D})$ is the DTD, $\mathbf{D},\mathbf{B}$ are second order symmetric positive-definite diffusion and b-tensors respectively, and $\mathbf{B}:\mathbf{D}=\sum_i\sum_jb_{ij}D_{ij}$. For a discrete set of diffusion tensor populations, $$\frac{S}{S_0}(\mathbf{B})=\sum_{d=1}^{N_{D}}w_{d}\exp(-\mathbf{B}:\mathbf{D}_{d}),$$ where $w_{d}$ is the proportion of tensor population $\mathbf{D}_{d}$ and $N_{D}$ is the number of populations. The axisymmetric diffusion tensor can be parameterised as^5,6,

$$\mathbf{D}=\mathbf{R}(\theta,\phi){D_{iso}}\Bigg(\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}+D_\Delta\begin{pmatrix}-1&0&0\\0&-1&0\\0&0&2\end{pmatrix}\Bigg)\mathbf{R}(\theta,\phi)^T,$$ where $\mathbf{R}$ is a rotation operator. The b-tensor can be parametrised in terms of $(b_s,b_l,\Theta,\Phi)$^2,6, $$\mathbf{B}=\mathbf{R}(\Theta,\Phi)\Bigg(\frac{b_{s}}{3}\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}+b_{l}\begin{pmatrix}0&0&0\\0&0&0\\0&0&1\end{pmatrix}\Bigg)\mathbf{R}(\Theta,\Phi)^T.$$ Expanding $\frac{S}{S_0}(\mathbf{B})$, using the above parameterisation, gives a separable expression in $b_{s}$, and the other three parameters, $b_l,\Theta,\Phi$²:

$$\frac{S}{S_0}(\mathbf{B})=\sum_{d=1}^{N_{D}}w_{d}\exp(-b_{s}(D_{iso})_d)\exp(-b_{l}(D_{iso})_d)\exp(-2b_{l}(D_{iso})_{d}(D_\Delta)_{d}P_{2}(\cos\beta_d)),$$ $\cos\beta_d=\cos{\Theta}\cos{\theta_d}+\sin{\Theta}\sin{\theta_d}\cos({\Phi-\phi_d})$. This enables $\frac{S}{S_0}(\mathbf{B})$ to be expanded in a separable and orthogonal 4D basis which is the product of a 1D basis for $b_{s}$ dimension and a 3D basis for $b_{l},\Theta$ and $\Phi$ dimensions. As $\frac{S}{S_0}(\mathbf{B})$ is a function of negative exponential of $b_{s}$, we use an exponential modulated with a Laguerre polynomial, which together form an orthogonal basis over $b_{s}$ dimension⁷. Similarly, we use the spherical Laguerre basis⁸, a 3D orthogonal basis with an exponential weighting function in the radial direction, for $b_{l},\Theta,\Phi$ dimensions, leading to expansion: $$\frac{S}{S_0}(b_{s},b_{l},\Theta,\Phi)=\sum_{p=0}^{P}\sum_{n=0}^{N}\sum_{\ell=0}^{L} \sum_{m=-\ell}^{\ell}c_{pn\ell m}Z_p(b_{s})B_{n\ell m}(b_l,\Theta,\Phi)$$ where $B_{n\ell m}(b_l,\Theta,\Phi)=X_n(b_l)Y^m_\ell(\Theta,\Phi)$ is the spherical Laguerre basis, with $X_n(b_{l})=\sqrt{\frac{n!}{{\zeta_l}^3(n+2)!}}\exp(\frac{-b_{l}}{2\zeta_{l}})L^2_n(\frac{b_l}{\zeta_{l}}),$ and $Z_p(b_{s}) = \frac{1}{\sqrt{\zeta_s}}\exp(\frac{-b_s}{2\zeta_{s}})L_{p}(\frac{b_{s}}{\zeta_{s}})$. $Y^m_\ell(\Theta,\Phi)$ are spherical harmonics of maximum degree $L$, $\zeta_{l},\zeta_{s}$ are scale factors and $N,P$ are maximum orders for $b_{l}$ and $b_s$ bases respectively. Coefficients $c_{pn\ell{m}}$ are defined by inner product $c_{pn\ell{m}}=\langle \frac{S}{S_0}(b_{s},b_{l},\Theta,\Phi),Z_p(b_s)B_{n\ell{m}}(b_{l},\Theta,\Phi)\rangle$. Due to the separability of this basis, we can design a separable transform,

$$c_{pn\ell{m}}=\sum_{j=0}^{J}w_jZ_p(b_{s}(j))\sum_{i=0}^{I}w_iX_n(b_{l}(i))\int_{\Theta=0}^{\pi}\int_{\Phi=0}^{2\pi}\frac{S}{S_0}(b_{s}(j),b_{l}(i),\Theta,\Phi)Y^m_\ell(\Theta,\Phi)\sin{\Theta}d\Theta{d}\Phi.$$ with Gauss-Laguerre quadrature⁹ used to choose the weights $w_j,w_i$, and sample locations $b_{s}(j),b_{l}(i)$ for the $b_{s}$ and $b_{l}$ dimensions. We use the scheme^10,11 to place the samples in the angular dimension. For a band-limited signal, this quadrature enables exact reconstruction and efficient sampling with the number of samples equal to number of coefficients.

Methods

Results and Discussion

Conclusion

Acknowledgements

Alice Bates was supported by the Australian Research Council’s Discovery Projects funding scheme (Project no. DP170101897).

Alessandro Daducci is supported by the Rita Levi Montalcini, MIUR for the recruitment of young researchers.

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