A Unifying Framework for Spatial and Temporal Diffusion in Diffusion MRI

Abstract

We propose a novel framework to simultaneously represent the diffusion-weighted MRI (dMRI) signal over diffusion times, gradient strengths and gradient directions. Current frameworks such as the 3D Simple Harmonic Oscillator Reconstruction and Estimation basis (3D-SHORE) only represent the signal over the spatial domain, leaving the temporal dependency as a fixed parameter. However, microstructure-focused techniques such as Axcaliber and ActiveAx provide evidence of the importance of sampling the dMRI space over diffusion time. Up to now there exists no generalized framework that simultaneously models the dependence of the dMRI signal in space and time. We use a functional basis to fit the 3D+t spatio-temporal dMRI signal, similarly to the 3D-SHORE basis in three dimensional ’q-space’. The lowest order term in this expansion contains an isotropic diffusion tensor that characterizes the Gaussian displacement distribution, multiplied by a negative exponential. We regularize the signal fitting by minimizing the norm of the analytic Laplacian of the basis, and validate our technique on synthetic data generated using the theoretical model proposed by Callaghan et al. We show that our method is robust to noise and can accurately describe the restricted spatio-temporal signal decay originating from tissue models such as cylindrical pores. From the fitting we can then estimate the axon radius distribution parameters along any direction using approaches similar to AxCaliber. We also apply our method on real data from an ActiveAx acquisition. Overall, our approach allows one to represent the complete 3D+t dMRI signal, which should prove helpful in understanding normal and pathologic nervous tissue.

Rutger Fick and Demian Wassermann contributed equally to this work. This work was partially supported by the MOSIFAH ANR (France) Grant.Marco Pizzolato thanks Olea Medical and the PACA Regional council for support.

A Analytic Laplacian Regularization

Here we compute the analytic form of the Laplacian regularization matrix in Eq. (8). As our basis is separable in $\mathbf{q}$ and $\tau$, we can separate the Laplacian over our basis function $\varXi _i$ in a the spatial and temporal Laplacian as

$$\begin{aligned} \nabla ^2 \varXi _{i}(\mathbf{q},\tau ,u_s,u_t)=\left( \nabla ^2_\mathbf{q}S_i(\mathbf{q},u_s)\right) T_i(\tau ,u_t)+ S_i(\mathbf{q},u_s)\left( \nabla ^2_\tau T_i(\tau ,u_t)\right) \end{aligned}$$

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with $\nabla ^2_\mathbf{q}$ and $\nabla ^2_\tau$ the Laplacian to either $\mathbf{q}$ or $\tau$. We then rewrite Eq. (8) as

$$\begin{aligned} \mathbf{U}_{ik}&= \overbrace{\int _{\mathbb {R}}(\nabla ^2_\mathbf{q} S_i)(\nabla ^2_\mathbf{q} S_k)d\mathbf{q}\int _{\mathbb {R}}T_iT_kd\tau }^{\mathbf{U}^\text {I}_{ik}} +\overbrace{\int _{\mathbb {R}}(\nabla ^2_\mathbf{q} S_i)S_kd\mathbf{q}\int _{\mathbb {R}}T_i(\nabla ^2_\tau T_k)d\tau }^{\mathbf{U}^\text {IIa}_{ik}}\\&+\underbrace{\int _{\mathbb {R}}S_i(\nabla ^2_\mathbf{q} S_k)d\mathbf{q}\int _{\mathbb {R}}(\nabla ^2_\tau T_i)T_kd\tau }_{\mathbf{U}^\text {IIb}_{ik}} +\underbrace{\int _{\mathbb {R}}S_iS_kd\mathbf{q}\int _{\mathbb {R}}(\nabla ^2_\tau T_i)(\nabla ^2_\tau T_k)d\tau }_{\mathbf{U}^\text {III}_{ik}} \end{aligned}$$

where $\mathbf{U}^{\text {IIa}}_{ik}=\mathbf{U}^{\text {IIb}}_{ki}$. In all cases the integrals over $\mathbf{q}$ and $\tau$ can be calculated to a closed form using the orthogonality of the spherical harmonics, and Laguerre polynomials with respect to weighting function $e^{-x}$. The closed form of $\mathbf{U}^\text {I}$ is

$$\begin{aligned} {}{\mathbf{U}^\text {I}_{ik}=\frac{u_s}{u_t}\delta _{o(k)}^{o(i)}\delta _{l(k)}^{l(i)}\delta _{m(k)}^{m(i)} {\left\{ \begin{array}{ll} \delta _{(j(i),j(k)+2)}\frac{2^{2-l}\pi ^2 \varGamma (\frac{5}{2}+j(k)+l)}{\varGamma (j(k))} &{} \\ \delta _{(j(i),j(k)+1)}\frac{2^{2-l}\pi ^2 (-3+4j(i)+2l)\varGamma (\frac{3}{2}+j(k)+l)}{\varGamma (j(k))} &{} \\ \delta _{(j(i),j(k))}\frac{2^{-l}\pi ^2\left( 3+24j(i)^2+4(-2+l)l+12j(i)(-1+2l)\right) \varGamma (\frac{1}{2}+j(i)+l)}{\varGamma (j(i))} &{} \\ \delta _{(j(i),j(k)-1)}\frac{2^{2-l}\pi ^2 (-3+4j(k)+2l)\varGamma (\frac{3}{2}+j(i)+l)}{\varGamma (j(i))} &{} \\ \delta _{(j(i),j(k)-2)}\frac{2^{2-l}\pi ^2 \varGamma (\frac{5}{2}+j(i)+l)}{\varGamma (j(i))} &{} \end{array}\right. }} \end{aligned}$$

where $\delta$ is the Kronecker delta. Similarly computing $\mathbf{U}^\text {II}_{ik}=\mathbf{U}^\text {IIa}_{ik}+\mathbf{U}^\text {IIb}_{ki}$ gives

$$\begin{aligned} \mathbf{U}^\text {II}_{ik}&=\frac{u_t}{u_s}\delta _{l(k)}^{l(i)}\delta _{m(k)}^{m(i)} {\left\{ \begin{array}{ll} \delta _{(j(i),j(k)+1)}\frac{2^{-l}\varGamma (\frac{3}{2}+j(k)+l)}{\varGamma (j(k))} &{} \\ \delta _{(j(i),j(k))}\frac{2^{-(l+1)}\left( 1-4j(i)-2l\right) \varGamma (\frac{1}{2}+j(i)+l)}{\varGamma (j(i))} &{} \\ \delta _{(j(i),j(k)-1)}\frac{2^{-l}\varGamma (\frac{3}{2}+j(i)+l)}{\varGamma (j(i))} &{} \end{array}\right. }\\&\times \left( \frac{1}{2}\delta _{o(i)}^{o(k)}+(1-\delta _{o(i)}^{o(k)})\cdot |o(i)-o(k)|\right) \nonumber \end{aligned}$$

where $|\cdot |$ is the absolute sign. We now denote the operator $M_{x_1}^{x_2}=\text {min}(x_1,x_2)$ for the minimal value of $x_1,x_2$ and $H_x$ the Heaviside step function with $H_x=1\,\text {iff}\,x\ge 0$. The last term $\mathbf{U}^\text {III}_{ik}$ evaluates to

$$\begin{aligned} \mathbf{U}^\text {III}_{ik}&=\frac{u_t^3}{u_s^3}\delta _{j(k)}^{j(i)}\delta _{l(k)}^{l(i)}\delta _{m(k)}^{m(i)}\frac{2^{-(l+2)}\varGamma (j(i)+l+1/2)}{\pi ^2\varGamma (j)} \times \Bigg (\frac{1}{4}|o(i)-o(k)| + \frac{1}{16}\delta _{o(i)}^{o(k)}+M_{o(i)}^{o(k)}\\&+\textstyle {\sum _{p=1}^{M_{o(i)}^{o(k)}+1}}(o(i)-p)(o(k)-p)H_{M_{o(i)}^{o(k)}-p}+H_{o(i)-1}H_{o(k)-1}\bigg (o(i)+o(k)+2\\&+ \textstyle {\sum _{p=0}^{M_{o(i)-1}^{o(k)-2}}}p +\textstyle {\sum _{p=0}^{M_{o(i)-2}^{o(k)-1}}}p+M_{o(i)-1}^{o(k)-1}\left( |o(i)-o(k)|-1\right) H_{\left( |o(i)-o(k)|-1\right) }\bigg )\Bigg ) \end{aligned}$$

We finally compute the complete 3D+t Laplacian regularization matrix as

$$\begin{aligned} \mathbf{U}=\mathbf{U}^\text {I}+\mathbf{U}^\text {II}+\mathbf{U}^\text {III} \end{aligned}$$

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