Going Deep Into q-Space
Ileana Jelescu1
1Ecole Polytechnique Fédérale de Lausanne, Switzerland

Synopsis

The overall diffusion weighting, also referred to as the b-value, is the resulting diffusion attenuation from two different contributions: the spatial dephasing q imparted by the diffusion gradient pulse and the time t given to molecules to diffuse before they are rephased. Here, we will focus on increasing the q-vector for a fixed diffusion time t. This is what is commonly implied by “increasing the b-value”. Going “deep into q-space” opens entirely new doors for tissue microstructure mapping and brain tractography. The former are covered in this lecture.

Target audience

Scientists and clinicians who use, or would like to use, quantitative metrics derived from advanced diffusion acquisitions.

Objectives

This lecture aims at:
  1. Defining q-space and clarifying the distinction between varying the diffusion vector q and the diffusion time t
  2. Clarifying the different diffusion regimes with increasing q
  3. Spelling out the advantages and limitations of going deep into q-space
  4. Reviewing the most recent applications of high q acquisitions and analysis

The b-value – a “big bag” for two distinct concepts: q-space and diffusion time

The overall diffusion weighting, also referred to as the b-value, is the resulting diffusion attenuation from two different contributions:
$$b=(\gamma G\delta)^{2}t=q^{2}t$$
where q is the spatial dephasing imparted by the diffusion gradient pulse and t is the time given to the molecules with their spatially-encoded phase to diffuse before they are rephased. These two dimensions, q-space and time domain, can be used to explore and probe distinct features of the diffusion process in biological tissue.
Here, we will focus on increasing the q-vector for a fixed diffusion time t. This is what is commonly implied by “increasing the b-value”. Since diffusion in biological tissues is not Gaussian, quantities such as diffusivity and kurtosis are time-dependent (D(t) and K(t)). Thus it is paramount, when exploring q-space, to keep the diffusion time t constant, otherwise the data becomes difficult to interpret. Varying the diffusion time will be covered in the following lecture.

Within the convergence radius of the cumulant expansion

The MRI signal as a function of diffusion weighting can be described using mathematical formulae that are independent of the underlying medium in which diffusion is taking place, i.e. the tissue. The most widespread choice of such signal representations is the cumulant expansion, i.e. an expansion of the logarithm of the signal in polynomials in b (or q).
The first order approximation is known as diffusion tensor imaging (DTI)1, and holds until around b = 1 ms/μm2 for the brain, in vivo. In this regime, for a given diffusion time, the tissue cannot be distinguished from a Gaussian medium and a diffusion tensor is sufficient to fully capture the diffusion behavior. In this approximation, more information about the tissue microstructure and how it differs from a Gaussian medium can only be gleaned by varying the diffusion time instead2–5. However, some information about the contribution of capillary perfusion to pseudo-diffusion (also referred to as Intra-Voxel Incoherent Motion) can also be extracted in this low q regime6,7.
Going beyond the Gaussian approximation provides valuable information about tissue complexity. The cumulant expansion can be used to estimate higher order terms, such as the kurtosis of the diffusion probability distribution function8. One crucial element is that the series approximates the diffusion signal well only up to a certain convergence radius, beyond which it diverges regardless of the number of terms included9,10. This cut-off value is typically around b = 2.5 ms/μm2 for the brain, in vivo.

Going further in q-space: gain in sensitivity & specificity

Going beyond the convergence radius of the cumulant expansion opens entirely new doors for tissue microstructure mapping and tractography. In this lecture we will focus on microstructure applications. High q-values typically provide sensitivity to very small displacements, and thereby to smaller-scale structures. In the brain, high q-values are beneficial for preferentially attenuating relatively fast-diffusing extra-cellular water and obtaining a signal that stems predominantly from intra-cellular water: axons, cell processes and cell bodies, which largely determine the microarchitecture.
Very high q-values can be achieved using dedicated field gradient systems. For very small samples, custom gradients covering a small volume can be built to produce over 10 T/m11. On small animal pre-clinical systems, recent designs allow for 1 – 3 T/m. And in terms of clinical systems, the Connectom scanners can achieve 300 mT/m, which is an amplitude 4 to 10 times higher than regular clinical MRI systems.
Insights from model-free approaches
Q-space Imaging is a model-free technique that provides an estimation of the diffusion propagator based on the Fourier transform of the signal attenuation measured across q-space12,13. The constraint of having a short diffusion gradient pulse (δ << t) during which displacement is negligible was later relaxed in Diffusion Spectrum Imaging (DSI)14, while the constraint of Cartesian q-space coverage was very recently relaxed in Generalized DSI, making the technique compatible with most widespread multi-shell diffusion data15. While DSI has been largely used to estimate the diffusion orientation density function (dODF) for tractography purposes, it can also provide information about the microstructure in the form of direction-specific displacement probabilities. This method can be useful in complex regions for which appropriate biophysical modeling is not yet accessible (e.g. thalamus, striatum).
Insights from model-based approaches
Biophysical models assume a given simplified geometry – a “sketch” of the underlying tissue and rely on the analytical expression of the diffusion signal in the chosen environment. As opposed to the cumulant expansion, the analytical expression of the model is valid over a broad range of q-values. While within the radius of convergence of the cumulant expansion, microstructure parameters from biophysical models can be directly derived from the signal cumulants or moments16–20, one significant advantage of fitting the model signal equations directly to the data is precisely to go “deep into q-space” and exploit accrued sensitivity at higher diffusion weightings21.
However, multi-compartment models of diffusion in the long diffusion time limit typically assume each compartment to be Gaussian16,18,20,22–24, an approximation which may break down for very high q-values, where intra-compartment non-Gaussian effects should be accounted for.
Very strong diffusion weightings have recently been instrumental in:
  • Selectively suppressing extra-axonal signal in white matter, and retaining diffusion signal from water inside axons exclusively25,26
  • Gaining sensitivity to potential “immobile water”, trapped in small spherical compartments27,28
  • Gaining sensitivity to very small length scales and probing axon diameters29–31 or cell bodies32
  • Show-casing the impact of inter-compartment water exchange (i.e. exchange across the cellular membrane) in gray matter11,33,34.

Acknowledgements

No acknowledgement found.

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Proc. Intl. Soc. Mag. Reson. Med. 28 (2020)