Grant Yang^{1,2} and Jennifer McNab^{2}

We demonstrate through simulations and empirical data that it is possible to simultaneously estimate the variance of the voxel-wise diffusion tensor shape and size distributions using efficient isotropic and linear diffusion encodings on a whole-body clinical MRI scanner with whole-brain coverage at 3mm isotropic resolution in under 2 minutes.

_{}The DTD model approximates the diffusion signal using
a fourth-order diffusion covariance tensor:

$$S(\mathbf{B}){\approx}S_0{\exp}(-{\langle}\mathbf{B},\langle{\mathbf{D}\rangle\rangle+\frac{1}{2}\langle}\mathbb{B},\mathbb{C}\rangle)\hspace{3cm}(1)$$

where $$$\mathbf{B}=\int_0^\tau\mathbf{q}(t)\mathbf{q}^\mathrm{T}\,d\tau$$$ is the b-tensor, $$${\langle}\mathbf{D}{\rangle}$$$ is the average DT, $$$\mathbb{B}=\mathbf{B}^{\otimes2}$$$, and $$$\mathbb{C}\in\mathbb{R}^{3\times3\times3\times3}$$$ is the diffusion covariance tensor. Equation 1 can be written in matrix vector form:

$$\left(\begin{array}{c}{\log}S_1\\\vdots \\{\log}S_m\end{array}\right)=\left( \begin{array}{ccc}1&-\mathbf{b}_1^\mathrm{T}&\frac{1}{2}\mathbb{b}_1^\mathrm{T} \\\vdots&\vdots&\vdots\\1& -\mathbf{b}_m^\mathrm{T}&\frac{1}{2}\mathbb{b}_m^\mathrm{T} \end{array}\right)\left(\begin{array}{ccc}\log S_0&\langle\mathbf{d}\rangle&\mathbb{c}\end{array}\right)^\mathrm{T}\hspace{3cm}(2)$$

where $$$\mathbf{b}=\left(\begin{array}{cccccc}b_{xx}&b_{yy} & b_{zz}&\sqrt{2}b_{yz}&\sqrt{2}b_{xz} &\sqrt{2} b_{xy}\end{array}\right)^\mathrm{T}$$$, $$$\mathbb{b}\in\mathbb{R}^{21\times1}$$$ contains the independent elements of $$$\mathbf{b}\mathbf{b}^{\mathrm{T}}$$$, $$$ \mathbf{d}=\left(\begin{array}{cccccc}d_{xx}&d_{yy}&d_{zz}&\sqrt{2}d_{yz}& \sqrt{2}d_{xz}&\sqrt{2}d_{xy}\end{array}\right)^\mathrm{T}$$$ are
independent elements of $$$\langle \mathbf{D}\rangle$$$, and $$$\mathbb{c}\in\mathbb{R}^{21\times1}$$$ contains independent elements of $$$\mathbb{C}$$$.
However, since C_{MD} and μFA are rotationally
invariant, they can be estimated from $$$\mathbb{C}$$$ averaged over all orientations ( $$$\mathbb{C}_{iso}$$$).
Since $$$\mathbb{C}_{iso}$$$ describes an isotropic medium, the same signal
($$$S_{iso}$$$) is
expected from all diffusion encoding orientations.
Therefore $$$\mathbb{b}_{linear}^\mathrm{T}\mathbb{c}_{iso}=\alpha $$$ and $$$\mathbb{b}_{planar}^\mathrm{T}\mathbb{c}_{iso}=\alpha$$$.
This can be rewritten as the homogeneous equation: $$$\mathbf{A}\left(\begin{array}{ccc}\mathbb{c}_{iso}&\alpha&\beta\end{array}\right)=0$$$. The null space of $$$\mathbf{A}$$$ can be spanned by two vectors and
fully parameterizes $$$\mathbb{c}_{iso}$$$ such that $$$\mathbb{c}_{iso}=\mathbf{V}\mathbf{v}$$$,
where $$$\mathbf{V}\in\mathbb{R}^{21\times2}$$$, $$$ \mathbf{v}\in\mathbb{R}^{2\times1}$$$.
Therefore, Equation 2 simplifies to:

$$\left(\begin{array}{c}{\log}S_{iso,1}\\\vdots\\{\log}S_{iso,m}\end{array}\right)=\left( \begin{array}{ccc}1&-b_1&\frac{1}{2}b_1^2\mathbf{r}_1\\\vdots&\vdots&\vdots\\1&-b_m&\frac{1}{2}b_m^2\mathbf{r}_m\end{array}\right)\left(\begin{array}{ccc}\log S_0 &\mathrm{MD}&\mathbf{v}\end{array}\right)^\mathrm{T} \hspace{3cm}(3)$$

, where $$$\mathbf{r}_n = \mathbb{b}_n^\mathrm{T}\mathbf{V}/\mathrm{b}_n^2$$$ and $$$S_{iso, n}$$$ can be computed for anisotropic mediums by
averaging over multiple orientations.
The advantages of Equation 3 over Equation 2 include:
1) it is computationally and numerically advantageous to fit 4 variables
compared to 28, 2) estimating $$$S_{iso}$$$ requires fewer encoding
directions compared to estimating $$$\mathbb{C}$$$ and 3) both C_{MD} and μFA can be estimated
using only linear and isotropic b-tensors, thereby avoiding ellipsoidal or planar
b-tensors, which require longer TEs.

The diffusion environments shown in Figure 3a-c were
simulated with the diffusion encoding scheme described in Table 1. Rician noise was introduced to the
signal to produce 10,000 noisy measurements with SNRs
of 15, 50, and 100 for the non-diffusion weighted signal. Equation 3 was fitted
to the noisy signal and the mean and
standard deviation of the C_{MD} and μFA were computed^{6}.

Orientation averaging to
approximate $$$S_{iso}$$$ was tested by simulating the worst-case
scenario (D_{axial}=3μm^{2}/ms, D_{radial}=0μm^{2}/ms). The coefficient of variation(COV) of
the diffusion signal across 150 tensor orientations equally spaced on a sphere
was computed for diffusion schemes containing 6 to 60 diffusion
directions.

Two healthy subjects were scanned with IRB approval using
a 3T whole-body MR system(Premier, GE Healthcare) equipped with a 32-channel head
coil(Nova Medical). Each subject was examined with linear and isotropic
diffusion encoding sequences(Fig. 1) using diffusion acquisition schemes shown
in Table 1. The eddy current and motion corrected^{10,11} data was fit to Equation 3 to compute C_{MD} and μFA^{6}.

Figure
2 quantifies the rotational invariance achievable using a given number of
linear diffusion encodings. Figure 3d-i display μFA and C_{MD}
estimates from simulated diffusion signals from linear and isotropic diffusion
encodings.
Figure 4 shows full brain estimates of μFA and CMD at $$$3\times3\times3$$$mm^{3} resolution with acquisition times under
2min.

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Figure 1: Q-space trajectory imaging employs time varying diffusion waveforms to produce isotropic (a) planar (b) and linear (c) diffusion encoding as described by the appearance of the resulting b-tensor. Red, green, and blue gradient waveforms represent orthogonal gradient axes. By varying the b-tensor shape, information can be obtained about the diffusion tensor distribution (d), which is inaccessible using only conventional linear diffusion encoding. The diffusion tensor distribution can be used to disentangle microstructural shape and size distributions (d). Pulse sequence parameters for the linear and isotropic diffusion encoding pulse sequences used in this study (e).

Table 1: Diffusion acquisition
schemes for simulated and acquired data. The table specifies the number of
measurements for each b-tensor shape and magnitude. Measurements for linear
diffusion encodings are uniformly spaced over a sphere.

Figure 2: The
coefficient of variation over 150 orientations of a stick diffusion tensor with
axial diffusivity 3μm^{2}/ms after orientation averaging of a
linear diffusion encoding scheme consisting of 6-60 orientations equally spaced
over a sphere with b=2ms/μm^{2}.

Figure 3: Simulation
results for voxels with net isotropic diffusion and mean diffusivity of 1 μm^{2}/ms. Estimated
μFA (d,e,f) and C_{MD} (g,h,i) for (a) a change in tensor shape with no size
variation, (b) a change in size variation with isotropic tensors, and (c) a
change in size variation for elongated tensors.

Figure 4: Maps of microscopic
fractional anisotropy (μFA) and the normalized size variance (C_{MD}) measured
using linear and isotropic diffusion encoding pulse sequences. The reduced
sampling requirements of modeling the orientation averaged diffusion signal enables
whole-brain mapping in 1 minute 26 seconds.