Synopsis

The widely adopted Standard Model (SM) for diffusion in white matter tissue has been shown to possess intrinsic degeneracies, making parameter estimation from single diffusion encoding (SDE) data ill-conditioned. We extend the SM to a multidimensional diffusion MRI acquisition and analyse the case where the fibre orientation distribution function (fODF) is a Watson distribution. The information contained in the kurtosis tensor, accessed by SDE, is insufficient to recover biophysical model parameters from the MR signal. We prove that Double Diffusion Encoding (DDE), letting us additionally access the full diffusion tensor covariance, makes parameter estimation well-posed.

Introduction

Diffusion MRI (dMRI) probes tissue architecture at scales much smaller than typical MRI resolution. Biophysical modelling can extract microstructural information from the dMRI signal at the expense of making assumptions about main tissue features. For brain white matter (WM), the standard model (SM)¹ has been widely adopted to interpret the dMRI signal. It comprises two impermeable compartments, resembling intra-axonal and extra-axonal environments. Recent work showed that parameter estimation of the SM from single diffusion encoding (SDE) measurements is ill-conditioned^2,3. It has been proposed that complementary information provided by orthogonal measurements, such as echo time dependence⁴, diffusion time dependence⁵, or double diffusion encoding (DDE), might help resolve this degeneracy^6,7,8. We focus on the case where the fibre orientation distribution function (fODF) is a Watson distribution (Watson SM), extend it to a general b-tensor acquisition⁹ and analyse its cumulant expansion. We prove that adding complementary information provided by DDE is sufficient to make parameter estimation well-posed.

Theory

According to the SM, the dMRI signal is the convolution over the unit sphere $$$\mathbb{S}^2$$$

$$S(\boldsymbol{B})=S_0~\int_{\mathbb{S}^2}~\mathcal{P}(\hat{\mathbf{u}})\mathcal{K}(\boldsymbol{B},\hat{\mathbf{u}})\,dS_{\hat{\mathbf{u}}},$$

where $$$S_0=S(\boldsymbol{B}\equiv0)$$$ is the non-weighted diffusion signal, $$$\mathcal{P}(\hat{\mathbf{u}})$$$ the fODF (Watson distribution here), and

$$\mathcal{K}(\boldsymbol{B},\hat{\mathbf{u}})=f\exp\bigl[-D_{\text{a}}B_{ij}u_iu_j\bigr]+(1-f)\exp\bigl[-bD_\text{e}^\perp-\Delta_\text{e}B_{ij} u_i u_j\bigr],$$

is the response signal (kernel) from an individual fibre segment oriented along direction $$$\hat{\mathbf{u}}$$$, acquired with the b-tensor $$$\boldsymbol{B}$$$⁹. $$$f$$$ is the $$$T_2$$$-weighted intra-axonal volume fraction, $$$D_{\text{a}}$$$ the axial intra-axonal, $$$D_\text{e}^\parallel$$$ the axial extra-axonal and $$$D_\text{e}^\perp$$$ the radial extra-axonal diffusivity ($$$\Delta_\text{e}=D_\text{e}^\parallel-D_\text{e}^\perp$$$).

In SDE, $$$\boldsymbol{B}=b\,\hat{\mathbf{n}}\otimes\hat{\mathbf{n}}$$$, thus, $$$\boldsymbol{B}$$$ has one non-zero eigenvalue, viz. linear tensor encoding (LTE)⁹. In DDE, $$$\boldsymbol{B}=b_1\,\hat{\mathbf{n}}_{1}\otimes\hat{\mathbf{n}}_{1}+b_2\,\hat{\mathbf{n}}_{2}\otimes\hat{\mathbf{n}}_{2}$$$, with gradient directions $$$\hat{\mathbf{n}}_1$$$, $$$\hat{\mathbf{n}}_2$$$, and b-values, $$$b_1$$$, $$$b_2$$$ (generally equal). If $$$\hat{\mathbf{n}}_1=\hat{\mathbf{n}}_2$$$ (parallel direction pair) then $$$\boldsymbol{B}_{\text{DDE}}=\boldsymbol{B}_{\text{SDE}}$$$. However, if $$$\hat{\mathbf{n}}_1\perp\hat{\mathbf{n}}_2$$$ (perpendicular direction pair) and $$$b_1=b_2$$$, $$$\boldsymbol{B}_{\text{DDE}}$$$ has two equal non-zero eigenvalues and equals planar tensor encoding (PTE).

Methods

The fourth order cumulant expansion of the SM dMRI signal for a general b-tensor has the form

$$\log(S)=\log(S_0)-B_{ij}D_{ij}+\frac12B_{ij}B_{k\ell}Z_{ijk\ell}+{\cal~O}(\boldsymbol{B}^3),$$

where $$$\boldsymbol{D}$$$ is the diffusion tensor and $$$\boldsymbol{Z}$$$, for multicomponent Gaussian compartments (MGC) as SM, is the diffusion tensor covariance¹⁰. In our case,

$$D_{ij}=\bigl\langle~D_{ij}\bigr\rangle=\sum_\alpha~f_\alpha~D_{ij}^{(\alpha)},\quad~Z_{ijk\ell}=\Bigl\langle\bigl(D_{ij}-\langle~D_{ij}\rangle\bigr)\bigl(D_{k\ell}-\langle~D_{k\ell}\rangle\bigr)\Bigr\rangle=\sum_\alpha~f_\alpha~D_{ij}^{(\alpha)}D_{k\ell}^{(\alpha)}-D_{ij}D_{k\ell},$$

where $$$f_\alpha$$$ and $$$D_{ij}^{(\alpha)}$$$ denote the fraction and diffusion tensor of compartment $$$\alpha$$$. LTE accesses $$$\boldsymbol{D}$$$ and the totally symmetric part of $$$\boldsymbol{Z}$$$, identical to $$$\boldsymbol{W}$$$, the kurtosis tensor in MGCs: $$$\bar{D}^2 W_{ijk\ell}=3Z_{(ijk\ell)}$$$. Jespersen et al. showed $$$\boldsymbol{D}$$$ and $$$\boldsymbol{W}$$$ are insufficient to compute the kernel parameters due to an intrinsic degeneracy of the model⁵. The purpose of this work is to assess whether adding the remaining information in $$$\boldsymbol{Z}$$$ resolves this degeneracy.

Owing to the axial symmetry of the Watson distribution, $$$\boldsymbol{D}$$$ has 2 independent components, $$$\boldsymbol{W}$$$ has 3 and $$$\boldsymbol{Z}$$$ has 5. We can write them as

$$\boldsymbol{D}=D^\perp\delta_{ij}+\Delta\mu_i\mu_j,\qquad\boldsymbol{W}=\omega_1\boldsymbol{P}+\omega_2\boldsymbol{Q}+\omega_3\boldsymbol{I}\qquad\text{and}\qquad\boldsymbol{Z}=\frac13\bar{D}^2\boldsymbol{W}+\zeta_1\boldsymbol{R}+\zeta_2\boldsymbol{J},$$

with

$$P_{ijk\ell}=\mu_i\mu_j\mu_k\mu_\ell,\\Q_{ijk\ell}=\mu_{(ij}\delta_{k\ell)},\\I_{ijk\ell}=\delta_{(ij}\delta_{k\ell)},\\R_{ijk\ell}=\tfrac12\bigl(\mu_i\mu_j\delta_{k\ell}+\mu_k\mu_\ell\delta_{ij}\bigr)-\tfrac14\bigl(\mu_i\mu_k\delta_{j\ell}+\mu_j\mu_k\delta_{i\ell}+\mu_i\mu_\ell\delta_{jk}+\mu_j\mu_\ell\delta_{ik}\bigr),\\J_{ijk\ell}=\delta_{ij}\delta_{k\ell}-\tfrac12\bigl(\delta_{ik}\delta_{j\ell}+\delta_{i\ell}\delta_{jk}\bigr),$$

where $$$\delta_{ij}$$$ is the Kronecker delta and $$$\hat{\pmb{\mu}}$$$ the Watson distribution main direction. $$$D^\parallel$$$ and $$$D^\perp$$$ are the axial and radial diffusivities ($$$\bar{D} = \frac13 D^\parallel+\frac23 D^\perp$$$, $$$\Delta=D^\parallel - D^\perp$$$), and $$$\omega_1$$$, $$$\omega_2$$$ and $$$\omega_3$$$ are functions of axial, radial, and mean kurtosis ($$$W_\parallel$$$, $$$W_\perp$$$, $$$\bar{W}$$$)¹¹. Information from the non-symmetric part of $$$\boldsymbol{Z}$$$ is parameterised by $$$\zeta_1$$$ and $$$\zeta_2$$$.

Three datasets considering all combinations of $$$f=[0.1,0.3,0.5,0.7,0.9]$$$, $$$D_\text{a}=[0.3,0.8,1.3,1.8,2.3]\mu\!\,m^2/ms$$$, $$$D_{\text{e}}^{\parallel}=[0.8,1.3,1.8]\mu\!\,m^2/ms$$$, $$$D_{\text{e}}^{\perp}=[0.5,1,1.5]\mu\!\,m^2/ms$$$, and $$$\kappa=[0.84,2.58,4.75,9.27,15.53,33.70]$$$ were simulated. Acquisition protocols (LTE, LTE+PTE, PTE) had equal number of measurements (60) and b-values ($$$1-2 ms/\mu\!\,m^2$$$). Rician noise was added ($$$\text{SNR}=\text{50}$$$) and nonlinear least squares fitting of Eq. 1 was performed to the data.

Results

Expressing $$$\zeta_1$$$ and $$$\zeta_2$$$ as a function of the model parameters gives two new equations independent to those in $$$\boldsymbol{D}$$$ and $$$\boldsymbol{W}$$$:

$$\tfrac34\zeta_1=(1-f)D_\text{e}^{\perp}\Delta_{\text{e}}\,p_2-D^\perp\Delta,\qquad\text{and}\qquad\tfrac32\zeta_2=(1-f)\Bigl[\tfrac23D_\text{e}^{\perp}\Delta_{\text{e}}(1-p_2)+{D_\text{e}^{\perp}}^2\Bigr]-{D^\perp}^2,$$

where $$$p_2=\frac{1}{4}\bigl[\frac{3}{\sqrt{\kappa}F(\sqrt{\kappa})}-2-\frac{3}{\kappa}\bigr]$$$ and $$$F$$$ denotes Dawson's function¹². Combining these equations with those of $$$\boldsymbol{D}$$$ and $$$\boldsymbol{W}$$$⁵, and solving for $$$p_2$$$ (if $$$f\,D_\text{a}^2+(1-f)\,\Delta_\text{e}^2\neq0$$$) gives a unique solution

$$p_2=\frac{\frac13(W_\parallel-W_\perp)\bar{D}^2-\frac32\zeta_1+\Delta^2}{\frac13(W_\parallel+2W_\perp)\bar{D}^2-\frac32(\zeta_1+3\zeta_2)+\Delta^2}.$$

From this, we uniquely recover the kernel parameters³ (if $$$p_2\neq0$$$). See ¹³ for a similar result. Figure 1 shows LTE+PTE has the lowest root mean squared error in the nonlinear estimation of all model parameters.

Combining LTE and PTE provides full access to $$$\boldsymbol{D}$$$ and $$$\boldsymbol{Z}$$$. This enables unique mapping between the dMRI signal sensitive up to $$$\mathcal{O}(b^2)$$$ and the model parameters. In contrast, LTE or PTE data alone are insufficient to get full access to $$$\boldsymbol{Z}$$$.

Discussion and Conclusion

Acknowledgements

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