Multi-Spherical Diffusion MRI: Exploring Diffusion Time Using Signal Sparsity

Abstract

Effective representation of the diffusion signal’s dependence on diffusion time is a sought-after, yet still unsolved, challenge in diffusion MRI (dMRI). We propose a functional basis approach that is specifically designed to represent the dMRI signal in this four-dimensional space—varying over gradient strength, direction and diffusion time. In particular, we provide regularization tools imposing signal sparsity and signal smoothness to drastically reduce the number of measurements we need to probe the properties of this multi-spherical space. We illustrate a novel application of our approach, which is the estimation of time-dependent q -space indices , on both synthetic data generated using Monte-Carlo simulations and in vivo data acquired from a C57Bl6 wild-type mouse. In both cases, we find that our regularization approach stabilizes the signal fit and index estimation as we remove samples, which may bring multi-spherical diffusion MRI within the reach of clinical application.

Appendix: Analytic Laplacian Regularization

We provide the analytic form of the Laplacian regularization matrix in Eq. (9). As our basis is separable in q and τ, the Laplacian of our basis function Ξ _i is

$$\displaystyle{ \nabla ^{2}\varXi _{ i}(\mathbf{q},\tau,u_{s},u_{t}) = \left (\nabla _{\mathbf{q}}^{2}\varPhi _{ i}(\mathbf{q},u_{s})\right )T_{i}(\tau,u_{t}) +\varPhi _{i}(\mathbf{q},u_{s})\left (\nabla _{\tau }^{2}T_{ i}(\tau,u_{t})\right ) }$$

(12)

with ∇ _q ² and ∇ _τ ² the Laplacian to either q or τ. We then rewrite Eq. (9) as

$$\displaystyle{ \begin{array}{rl} \mathbf{U}_{ik}& =\int _{\mathbb{R}}(\nabla _{\mathbf{q}}^{2}\varPhi _{i})(\nabla _{\mathbf{q}}^{2}\varPhi _{k})d\mathbf{q}\int _{\mathbb{R}}T_{i}T_{k}d\tau +\int _{\mathbb{R}}\varPhi _{i}\varPhi _{k}d\mathbf{q}\int _{\mathbb{R}}(\nabla _{\tau }^{2}T_{i})(\nabla _{\tau }^{2}T_{k})d\tau \\ & +\int _{\mathbb{R}}(\nabla _{\mathbf{q}}^{2}\varPhi _{i})\varPhi _{k}d\mathbf{q}\left (\int _{\mathbb{R}}T_{i}(\nabla _{\tau }^{2}T_{k})d\tau +\int _{\mathbb{R}}(\nabla _{\tau }^{2}T_{i})T_{k}d\tau \right ) \end{array} }$$

(13)

Equation (13) can be calculated to a closed form using the orthogonality of physicists’ Hermite polynomials with respect to weighting function $e^{-x^{2} }$ on [−∞, ∞]. Let us first consider the integrals with respect to q, which all parts of the Laplacian regularization functional of the MAP basis [14]. Writing the second order derivative as a double apostrophe^{′ ′}, the Laplacian of the spatial basis is given in terms of the 1D-SHORE functions as $\nabla _{\mathbf{q}}^{2}\varPhi _{i} =\phi _{ n_{x}}^{{\prime\prime}}\phi _{n_{y}}\phi _{n_{z}} +\phi _{n_{x}}\phi _{n_{y}}^{{\prime\prime}}\phi _{n_{z}} +\phi _{n_{x}}\phi _{n_{y}}\phi _{n_{z}}^{{\prime\prime}}$. The integral of the product of two Laplacians therefore becomes a sum of nine terms, but can be described using the following three equations:

$$\displaystyle\begin{array}{rcl} & & \begin{array}{rl} \mathop{\mathrm{U}}\nolimits _{n}^{m}(u)& =\int _{\mathbb{R}}\phi _{n}^{{\prime\prime}}\phi _{m}^{{\prime\prime}}d\mathbf{q} = u^{3}2(-1)^{n}\pi ^{7/2}\Bigg(\delta _{n}^{m}3(2n^{2} + 2n + 1) +\delta _{ n}^{m+4}\sqrt{n!/m!} \\ & +\delta _{ n+2}^{m}(6 + 4n)\sqrt{m!/n!} +\delta _{ n+4}^{m}\sqrt{m!/n!} +\delta _{ n}^{m+2}(6 + 4m)\sqrt{n!/m!}\Bigg) \end{array} \\ & & \begin{array}{rll} \mathop{\mathrm{V}}\nolimits _{n}^{m}(u)& =\int _{\mathbb{R}}\phi _{n}^{{\prime\prime}}\phi _{m}d\mathbf{q} = u(-1)^{n+1}\pi ^{3/2}\bigg(\delta _{n}^{m}(1 + 2n)& +\delta _{ n}^{m+2}\sqrt{n(n - 1)} +\delta _{ n+2}^{m}\sqrt{m(m - 1)}\bigg) \\ \end{array}{} \\ & & \begin{array}{rl} \mathop{\mathrm{W}}\nolimits _{n}^{m}(u)& =\int _{\mathbb{R}}\phi _{n}\phi _{m}d\mathbf{q} = u^{-1}\delta _{n}^{m}(-1)^{n}/(2\pi ^{1/2}) \end{array} \\ \end{array}$$

(14)

Using the functions in Eq. (14) we define the q-dependent parts of Eq. (13):

$$\displaystyle\begin{array}{rcl} & & \begin{array}{rl} \int _{\mathbb{R}}(\nabla _{\mathbf{q}}^{2}\varPhi _{i})(\nabla _{\mathbf{q}}^{2}\varPhi _{k})d\mathbf{q}& = \frac{u_{x}^{3}} {u_{y}u_{z}}\mathop{\mathrm{U}}\nolimits _{ x_{i}}^{x_{k}}\mathop{\mathrm{W}}\nolimits _{ y_{ i}}^{y_{k}}\mathop{\mathrm{W}}\nolimits _{ z_{ i}}^{z_{k}} + 2\frac{u_{x}u_{y}} {u_{z}} \mathop{\mathrm{V}}\nolimits _{ x_{i}}^{x_{k}}\mathop{\mathrm{V}}\nolimits _{ y_{ i}}^{y_{k}}\mathop{\mathrm{W}}\nolimits _{ z_{ i}}^{z_{k}} + \frac{u_{y}^{3}} {u_{z}u_{x}}\mathop{\mathrm{U}}\nolimits _{ y_{i}}^{y_{k}}\mathop{\mathrm{W}}\nolimits _{ z_{ i}}^{z_{k}}\mathop{\mathrm{W}}\nolimits _{ x_{ i}}^{x_{k}} \\ & + 2\frac{u_{y}u_{z}} {u_{x}} \mathop{\mathrm{V}}\nolimits _{ y_{i}}^{y_{k}}\mathop{\mathrm{V}}\nolimits _{ z_{ i}}^{z_{k}}\mathop{\mathrm{W}}\nolimits _{ x_{ i}}^{x_{k}} + \frac{u_{z}^{3}} {u_{x}u_{y}}\mathop{\mathrm{U}}\nolimits _{ z_{i}}^{z_{k}}\mathop{\mathrm{W}}\nolimits _{ x_{ i}}^{x_{k}}\mathop{\mathrm{W}}\nolimits _{ y_{ i}}^{y_{k}} + 2\frac{u_{x}u_{z}} {u_{y}} \mathop{\mathrm{V}}\nolimits _{ x_{i}}^{x_{k}}\mathop{\mathrm{V}}\nolimits _{ z_{ i}}^{z_{k}}\mathop{\mathrm{W}}\nolimits _{ y_{ i}}^{y_{k}} \end{array} {}\\ & & \begin{array}{rl} \int _{\mathbb{R}}(\nabla _{\mathbf{q}}^{2}\varPhi _{i})(\varPhi _{k})d\mathbf{q}& = \frac{u_{x}} {u_{y}u_{z}}\mathop{\mathrm{V}}\nolimits _{ x_{i}}^{x_{k}}\mathop{\mathrm{W}}\nolimits _{ y_{ i}}^{y_{k}}\mathop{\mathrm{W}}\nolimits _{ z_{ i}}^{z_{k}} + \frac{u_{y}} {u_{x}u_{z}}\mathop{\mathrm{V}}\nolimits _{ x_{i}}^{x_{k}}\mathop{\mathrm{W}}\nolimits _{ y_{ i}}^{y_{k}}\mathop{\mathrm{W}}\nolimits _{ z_{ i}}^{z_{k}} + \frac{u_{z}} {u_{x}u_{y}}\mathop{\mathrm{V}}\nolimits _{ x_{i}}^{x_{k}}\mathop{\mathrm{W}}\nolimits _{ y_{ i}}^{y_{k}}\mathop{\mathrm{W}}\nolimits _{ z_{ i}}^{z_{k}} \end{array} {}\\ & & \begin{array}{rl} \int _{\mathbb{R}}\varPhi _{i}\varPhi _{k}d\mathbf{q}& = \frac{1} {u_{x}u_{y}u_{z}}\mathop{\mathrm{W}}\nolimits _{ x_{i}}^{x_{k}}\mathop{\mathrm{W}}\nolimits _{ y_{ i}}^{y_{k}}\mathop{\mathrm{W}}\nolimits _{ z_{ i}}^{z_{k}} \end{array} {}\\ \end{array}$$

For terms with τ, we denote the operator $M_{x_{1}}^{x_{2}} =\mathrm{ min}(x_{1},x_{2})$ for the minimal value of x ₁, x ₂ and H _x the Heaviside step function with H _x  = 1 iff x ≥ 0.

$$\displaystyle\begin{array}{rcl} & & \begin{array}{rl} \int _{\mathbb{R}}(\nabla _{\tau }^{2}T_{i})(\nabla _{\tau }^{2}T_{k})d\tau & =\Bigg (\frac{1} {4}\vert o(i) - o(k)\vert + \frac{1} {16}\delta _{o(i)}^{o(k)} + M_{o(i)}^{o(k)} \\ & + \sum _{p=1}^{M_{o(i)}^{o(k)}+1}(o(i) - p)(o(k) - p)H_{ M_{o(i)}^{o(k)}-p} + H_{o(i)-1}H_{o(k)-1}\bigg(o(i) + o(k) - 2 \\ & + \sum _{p=0}^{M_{o(i)-1}^{o(k)-2} }p + \sum _{p=0}^{M_{o(i)-2}^{o(k)-1} }p + M_{o(i)-1}^{o(k)-1}\left (\vert o(i) - o(k)\vert - 1\right )H_{\left (\vert o(i)-o(k)\vert -1\right )}\bigg)\Bigg) \end{array} {}\\ & & \begin{array}{rl} \left (\int _{\mathbb{R}}T_{i}(\nabla _{\tau }^{2}T_{k})d\tau +\int _{\mathbb{R}}(\nabla _{\tau }^{2}T_{i})T_{k}d\tau \right ) = u_{t}\left (\frac{1} {2}\delta _{o(i)}^{o(k)} + (1 -\delta _{o(i)}^{o(k)}) \cdot \vert o(i) - o(k)\vert \right ) \end{array} {}\\ & & \begin{array}{rl} \int _{\mathbb{R}}T_{i}T_{k}d\tau = 1/u_{t}\delta _{o(k)}^{o(i)} \end{array} {}\\ \end{array}$$