3.3.2.1

Model-Free Techniques

The simplest model-free approach is q -space imaging (QSI), often referred to in 3D as diffusion spectrum imaging (DSI). For an extensive treatment of q -space theory and applications, see also Chapter 3.2 . This kind of analysis relies on the very basic signal model in which the diffusion-weighted signal is the Fourier transform of the diffusion propagator. By varying the length, strength, and orientation of the gradient pulses in the diffusion imaging acquisition sequence, we can sample the signal at various wavevectors. QSI and DSI work by acquiring a set of measurements typically with wavevectors in a grid configuration and inverting the Fourier transform to obtain a discrete representation of the diffusion propagator.

Several parameters can be extracted from such analysis. If measured along a particular direction, the displacement distribution function can be characterized by its width (mean displacement < D >) and height (probability for zero displacement P ₀ ). These parameters were found to be extremely useful for characterization of the human brain and excised spinal cords (see section 3.3.3 following). If the QSI experiment is done in 3D, as in DSI, the analysis provides the diffusion orientation density function (ODF) by projecting the propagator onto a sphere via a radial integration. This ODF has peaks in each of potentially many different fiber directions, so provides a better basis for tractography than DTI.

For tractography alone, DSI is uneconomical in terms of acquisition time, and various other methods try to estimate the diffusion ODF and related functions from more economical high angular resolution diffusion imaging (HARDI) acquisition; see for a review. Q-ball imaging uses the Funk–Radon transform as an approximate mapping from HARDI data to the diffusion ODF. PAS-MRI estimates a diffusion ODF with greater sensitivity to fiber orientations. Spherical deconvolution techniques combine the model-based techniques described in the next section with a model-free description of the ODF to obtain a fiber ODF. See Ref. for an explanation and comparison of these various ideas.

Hybrid diffusion imaging (HYDI) provides a framework to estimate all of the common DSI parameters, including < D >, P ₀ , and the diffusion ODF, from multishell HARDI data, which is more economical than full DSI. Diffusion kurtosis imaging (see also Chapters 3.1 , Chapters 3.2 ) uses multishell HARDI and estimates an additional parameter, the diffusion kurtosis tensor, which characterizes the departure from the Gaussian model used in DTI.

All these parameters are useful markers of tissue integrity that often provide complementary information to the basic DTI indices. However, they remain nonspecific and difficult to associate with specific features of the tissue microstructure.

3.3.2.2

Model-Based Techniques

The model-based approaches include several frameworks with increasing complexity of the model. The most basic model includes a multiexponential decay function, anomalous diffusion, or a stretched exponential function. Such models can adequately characterize the deviation from Gaussian diffusion, such as the non-Gaussian diffusion that stems from restricted diffusion, in a similar way to model-free techniques like diffusion kurtosis. A biophysical interpretation can be attached to some parameters; for example, associating a separate tissue compartment to each exponential component provides an estimate of the orientation and volume fraction of each compartment.

More directly, biophysical models employ a geometric description of the tissue compartments and exploit restricted diffusion as a window on the geometry. These models are built from assumptions on the diffusion properties of water in different compartments of the tissue. For example, Stanisz et al. proposed a model with three compartments: axonal, glial, and extracellular space. The axons are elongated ellipsoids, and the glia are spherical. Each has a semipermeable membrane and unique volume fractions and internal diffusivity. The composite hindered and restricted model of diffusion (CHARMED) assumes that the diffusion within axons is restricted perpendicular to the axons, while elsewhere (cells, extra cellular matrix, and parallel to the axons) it is hindered or free. The minimal model of white matter diffusion (MMWMD) combines elements of both CHARMED and Stanisz’s model in an attempt to identify the simplest model that explains diffusion data from white matter. Zhang et al., following the earlier work of Jespersen et al. and Kaden et al., extend the MMWMD to include fiber orientation dispersion, which is another significant influence on the signal. Panagiotaki et al. define a taxonomy of models for diffusion in white matter, which relates all those mentioned herein within a broader framework of compartment models and performs a statistical comparison of their ability to explain measured signals.

From such models parameters such as the axonal density, diameter distribution, and intra/extra cellular diffusivity can be estimated and mapped. The models also extend to include other parameters such as cell permeability and shape. However, the modeling process rapidly becomes complex and the number of parameters increases rapidly rendering estimation ill posed if signal-to-noise ratio is not sufficient (as is likely in high b -value diffusion imaging). Despite that, successful demonstrations and systems are now in the literature. For example, the restricted diffusion volume fraction (axonal density) maps from CHARMED analysis were found to be extremely useful and sensitive in the brain and provide an alternative to FA that relates more specifically to axon density. The Neurite Orientation Dispersion and Density Imaging (NODDI) technique extends the set of specific parameters to include indices of fiber orientation dispersion and CSF volume fraction, thus separating the three key parameters that influence FA in brain imaging.

AxCaliber, an application of CHARMED, models, aside from axonal density, can also estimate the axon diameter distribution. The model assumes that this distribution has a gamma function shape. Based on a multi diffusion time experiment where in each diffusion time a different population of axons will experience restricted diffusion, the estimation of the gamma function becomes feasible. This method was demonstrated on the rat corpus callosum and on excised spinal cord (see following).

The ActiveAx technique uses the MMWMD together with an optimized multishell HARDI acquisition protocol to map an axon diameter index over the brain. The simple model and optimized experiment design enable the technique to be orientationally invariant. Thus, unlike earlier techniques, which require the fiber to have known and fixed orientation, ActiveAx provides indices for fibers in all orientations from a single acquisition. However, the index provided is less quantitative than the AxCaliber axon diameter distribution because it provides only a single summary statistic of the full distribution. In theory, the axon diameter index reflects the average axon diameter weighted by axon volume, although there are some departures from this complicated interpretation. Nevertheless, in fixed monkey brains using data from a 4.7 T high field scanner with 140 mT/m and 300 mT/m, the technique provides maps that reflect what we expect from histology. In vivo human results are noisier, but an ROI analysis also reflects the expected trends. As an alternative to ActiveAx, Barazany et al. suggested an extension of AxCaliber into 3D to allow estimation of the full axon diameter distribution function for any orientated fiber system.

While current axon diameter estimation techniques show good results in experimental animal systems with high gradients available, on human systems they largely remain scientific demonstrations rather than tools for large-scale studies; the human data sets in Alexander et al. took over 1 h to acquire. Further work is required to realize the potential of these techniques for human studies. Modeling advances, such as those described in Panagiotaki et al. and Zhang et al. make incremental but significant improvements. Greater advances are likely to come from improvements in data acquisition to increase sensitivity to the specific parameters of interest. For example, oscillating gradients and generalized waveforms can provide orders of magnitude more sensitivity to small axon diameters. Exploitation of such pulse sequences potentially offers much greater sensitivity and practical axon diameter mapping techniques for human studies in the near future.

In the spinal cord, the orientational invariance of ActiveAx is less of an advantage than in the brain because of the simpler orientational structure of the cord. However, the model and experiment design optimization that underpin the technique also provide advantages when the orientation is known; Schneider et al. demonstrate this in the corpus callosum, and that extension is directly applicable to the spinal cord. Measures of orientation dispersion from NODDI are likely to find use in the spinal cord as well as the brain both for studying the intersections of peripheral nerves with the cord, where orientation dispersion can be significant, and for pinpointing different kinds of pathology more precisely. The benefits of alternative pulse sequences are equally applicable in the spinal cord and will help provide more precise information in more manageable acquisition times as these techniques develop.