Synopsis

Diffusion magnetic resonance imaging of biological systems most often results in non-monoexponential signal, due to their complexity and heterogeneity. One approach to interpreting the data without imposing microstructural models is to fit it to a multiexponential function, and to display the coefficients as a distribution of the diffusivities. Here we suggest parameterizing the measured water mobility spectra using a bimodal lognormal function. This approach allows for a compact representation of the spectrum, while also resolving overlapping spectral peaks. We apply the method on a spinal cord sample and use it to generate robust intensity images of slow and fast-diffusion components.

Introduction

An approach similar to the one used to investigate porous media with nuclear magnetic resonance (NMR) relaxation-based methods,¹ multiexponential modeling of the diffusion signal can be used to characterize heterogenous biological systems.^2,3 Finding the coefficients of a multiexponential function is, in theory, equivalent to performing an inverse Laplace transformation (ILT), and, therefore, these methods are often termed Laplace NMR or MRI. Treating the diffusion signal decay as a multiexponent provides a spectrum of diffusivities, which leads to a model-free description of the water mobility. Because of the heavy data requirements, diffusion-weighted Laplace approaches have most often been limited to NMR. As methods that make Laplace MRI more feasible and accessible emerge⁴ and are applied,^5-7 there is a concomitant growing need for dimensionality reduction. Whereas traditional Laplace NMR results in one- to three-dimensional spectral data, three additional spatial dimensions are added when this approach is combined with imaging. The increased dimensionality, along with the requirement to visualize the data in a summarized manner, creates a need for a more compact representation of the spectral information. In many instances the spectral peaks are overlapping, making it hard to robustly determine their limits. In this work, we present a method to reduce the dimensionality of the nonparametric Laplace-based diffusivity spectrum by fitting it to a bimodal parametric function.

Methods

The diffusion-weighted signal attenuation, $$$A$$$, can be described by the following discrete sum:

$$A(b_{i})=\sum_{n=1}^{N}{P(D_{n})\,\exp(-b_{i}D_{n})}+\epsilon(b), (1)$$

where $$$\epsilon(b)$$$ is the experimental noise, which is assumed to be Gaussian. The measured diffusion parameter, $$$D$$$, is distributed according to $$$P(D)$$$ with $$$N$$$ discrete components.

Reports of two dominant diffusivity regimes in nerve tissue are very common in the diffusion MR literature.^8,9 These studies have led Ronen et al. to use a bimodal lognormal function to describe the estimated diffusivity distributions in the brain.³ Here we are using the same parametric function to reduce the dimensionality of the nonparametric Laplace-based diffusivity spectrum and to achieve a robust separation of overlapping spectral peaks. A bimodal lognormal distribution as a function of the diffusivity is given by

$$P_{bilog}(D) =\frac{f_{slow}}{S_{slow}\sqrt{2\pi}D}e^{-(\ln D - M_{slow})^{2}/(2S_{slow}^2)} \\ + \frac{f_{fast}}{S_{fast}\sqrt{2\pi}D}e^{-(\ln D - M_{fast})^{2}/(2S_{fast}^2)}, (2)$$

where the volume fractions, $$$f$$$, and the lognormal distribution variables, $$$S$$$ and $$$M$$$, of the slow and fast diffusion components are used. Once the nonparametric diffusivity spectrum, $$$P(D)$$$, was estimated by inverting a regularized version¹⁰ of Eq. 1, it was fed into a subsequent optimization procedure that fit it to Eq. 2.

Results and Discussion

The arrows in Fig. 2 point to potential $$$D_{lim}$$$ values, which change and move as a function of tissue type and orientation. Thus, a global $$$D_{lim}$$$ value does not exist, and the choice of such value would significantly bias the resulting signal fraction. Although the nonparametric distribution (i.e., the one obtained directly from solving Eq. 1 contains the richest information compared with a parametric model, it may be challenging to interpret. Specifically, the overlapping spectral components are often hard to resolve, and the fact that their location on the spectrum is not constant, make it very difficult to automatically determine where one peak ends and the other begins. Fitting the diffusion spectrum in each voxel to the bimodal lognormal function in Eq. 2 allowed for a more robust interpretation of the data. Examples from white and gray matter, and from parallel and perpendicular orientations are shown in Fig. 3. Regardless of the tissue type or orientation, the parametrization captured most of the information contained within the nonparametric distributions.

When a voxelwise analysis is desired, the large number of voxels requires a completely automated and robust method to identify the peaks and their point of separation and to resolve them. Knowing the parametric form of the slow and the fast diffusion spectral components allows for a simple integration over their entire range, which results in their respective signal fractions images, shown in Fig. 4. The parametric $$$X$$$ diffusion images demonstrated a typical white-gray matter contrast, which is attributed to the higher density and content of barriers perpendicular to the spine's axis of symmetry in white matter, compared with gray matter.⁶

Conclusions

In this paper we suggested parameterizing the empirically measured diffusivity spectra obtained from Laplace dMRI spinal cord data. The advantage is twofold: (1) the parametric form has a compact representation of the spectra, which can be used to infer useful statistical quantities, and (2) it provides a robust method to resolve overlapping spectral peaks. We note that this parametrization approach is not limited to diffusion-based Laplace applications but can also be used to process relaxation spectra (e.g., $$$T_1$$$ and $$$T_2$$$ distributions), and to characterize and reduce the dimensionality of multidimensional Laplace MRI data, such as $$$D$$$-$$$T_1$$$.⁶

Acknowledgements

References

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