Carl-Fredrik Westin^{1}, Filip Szczepankiewicz^{1}, Lipeng Ning^{1}, Yogesh Rathi^{1}, and Markus Nilsson^{2}

Quantitative T2 and diffusion imaging provide important information about tissue microstructure. However, a joint knowledge of quantitative T2 and diffusion-derived measures can provide richer information about the microstructure that is not accessible when using these modalities independently. The standard approach for estimating the joint distributions of the T2-diffusion relies on the inverse Laplace transform. This transform is known to be unstable and difficult to invert. In this work, we introduce an alternative approach based on cumulant expansion, and extend the recently proposed multidimensional diffusion MRI framework ``Q-space Trajectory Imaging" (QTI) to include T2-relaxation modeling. The cumulants of the expansion include estimates of mean diffusion and T2 relaxation, as well as their variance and covariance. We demonstrate the feasibility of this approach in a healthy human brain.

In a system defiend by a distribution of diffusion tensors the signal is given by

$$s({\bf B}) = s_0 \int \rm{e}^{-({\bf D},{\bf B})} \rho({\bf D}) d {\bf D} \quad \text{(1)}$$

where $$$\bf{B}$$$ is a 3x3 matrix determined by the acquisition sequence, $$$\bf{D}$$$ denotes the diffusion tensor, $$$\rho(\bf{D})$$$ represents the diffusion tensor distribution (DTD) function of the underlying tissue, and $$$(\cdot, \cdot)$$$ denotes the standard inner product between matrices or tensors. By performing cumulant expansion we can identify mean and covariance terms of the signal:

\begin{equation}s(\mathbf{B}) \approx s_0\exp\left(-(\bar{\bf{D}}, \mathbf{B})+\frac{1}{2} (\mathbf{C}, \mathbf{B}^{\otimes 2})\right )\quad \text{(2)}\label{eq:cumulantQTI}\end{equation}

where $$$\bar{\bf{D}}= \langle \bf{D} \rangle_\rho,~\rm{and}~\bf{C}=\langle (\mathbf{D}-\bar{\mathbf{D}})^{\otimes 2} \rangle_\rho$$$. Below we will extend this framework to include T2-relaxation, i.e. the signal dependence on the echo time $$$t$$$ (TE), by modeling the MRI signal as

\begin{equation}s(\mathbf{B},t) = s_0 \int e^{-rt}e^{-(\mathbf{D},\mathbf{B})} \rho(r,\mathbf{D})~ dr d \mathbf{D} = s_0 \int e^{-rt -(\mathbf{D},\mathbf{B})} \rho(r,\mathbf{D})~ dr d \mathbf{D}\quad \text{(3)}\label{eq:QTIrelax}\end{equation}

where $$$\bf{B}$$$ is the diffusion measurement tensor $$$(b=\text{Tr}({\bf B}))$$$, $$$r$$$ is the T2-relaxation coefficient, and $$$\rho(r,\mathbf{D})$$$ represents the joint probability distribution of $$$r$$$ and $$$\bf{D}$$$. A typical approach for estimating $$$\rho(r, \mathbf{D})$$$ is to invert the Laplace transform using a large number of measurements acquired with different combinations of $$$\bf{B}$$$ and $$$t$$$. However, this approach requires very long scan time and the inversion is numerically unstable. It turns out that the cumulant expansion method, described in equation 2, can be used as an alternative analysis method to the inverse Laplace transform by cumulant expansion of the joint diffusion-relaxation distribution. In Westin [4], the first cumulant in equation 2, $$${\bf \bar{D}}$$$ is the mean diffusivity (of a diffusion tensor), and the second cumulant $$${\bf C}$$$ is a fourth-order tensor that describes the covariance of the diffusion parameters. When incorporating relaxation time into the signal model (equation 3) and expanding this signal, the same term contains additional covariances related to diffusion-relaxation. These terms can be identified as diffusion-relaxation tensor:\begin{equation}{\bf C}_{{\bf D}r} = \left(\begin{array}{rrr}c_{xx,r} & c_{xy,r} & c_{xz,r} \\c_{xy,r} & c_{yy,r} & c_{yz,r} \\c_{xz,r} & c_{yz,r} & c_{zz,r} \\\end{array}\right)\end{equation}and the variance of the relaxation $$$c_{rr}$$$.

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Examples of diffusion-relaxations parameter maps obtained by the proposed diffusion-relaxation method. Top row shows the mean diffusivity (MD), fractional anisotropy (FA) with and without color from the diffusion tensors obtained by the first moment in the cumulant expansion. The bottom row shows the T2 relaxation (r) obtained from the first moment of the cumulant expansion, and the variance of r $$$c_{rr}$$$, and the covariance between diffusion and relaxation (calculated as the trace of the diffusion-relaxation tensor), from the second cumulant (red-to-green color shows positive correlation between diffusion and relaxation rate).