Internal gradient distributions: A susceptibility-derived tensor delivering morphologies by magnetic resonance

Nuclear magnetic resonance is a powerful tool for probing the structures of chemical and biological systems. Combined with field gradients it leads to NMR imaging (MRI), a widespread tool in non-invasive examinations. Sensitivity usually limits MRI’s spatial resolution to tens of micrometers, but other sources of information like those delivered by constrained diffusion processes, enable one extract morphological information down to micron and sub-micron scales. We report here on a new method that also exploits diffusion – isotropic or anisotropic– to sense morphological parameters in the nm-mm range, based on distributions of susceptibility-induced magnetic field gradients. A theoretical framework is developed to define this source of information, leading to the proposition of internal gradient-distribution tensors. Gradient-based spin-echo sequences are designed to measure these new observables. These methods can be used to map orientations even when dealing with unconstrained diffusion, as is here demonstrated with studies of structured systems, including tissues.

M (t) = exp − 1 2 φ 2 (t) , the signal will evidence a decay depending on the attenuation factor β(t) = 1 2 φ 2 (t) . With most sources of decoherence normalized out by the constanttime, constant-pulses-number, fixed-number-of-gradients nature of the NOGSE sequences assayed [20,21,44], we ascribe to diffusion effects as the sole source of this attenuation.
It is then convenient to describe the β-factor in terms of the gradient modulating function G tot (t ) [45][46][47]: where in the second equation we redefined the gradient modulation function such that G tot (t , T E) = 0 if t < 0 or t > T E (i.e., outside the total evolution time range). The evolution is given in terms of a tensorial correlation function reflecting the displacements' where G tot (ω, T E) = G(ω, T E) + G 0 (ω, T E) , is the filter function introduced in Eq. (1) of the main text.
Considering the applied gradient modulation G(t , T E), the internal background gradient modulation G 0 (t , T E), and their respective filter functions G(ω, T E) and G 0 (ω, T E), the argument of the integral defining this attenuation factor can then be expanded as

cross-term (S.4)
This leads to Eq. (2) of the main text, where β(T E) = β G 2 (T E) + β G 2 0 (T E) + β G· G 0 (T E) and the normalized spin magnetization becomes and D G (ω) = G † · D(ω) · G /G 2 . Likewise, the pure background gradient decay is independent of the applied gradient . Finally, the cross-term attenuation will be . Our derivations also assumed that G 0 can be described by a Gaussian distribution. The cross-term contribution to the attenuation factor turns out to be . This second term is always positive, since it is a quadratic term, while the first term depends on the relative sign of the parallel component of G † ·D to the background gradient G 0 .
For an isotropic diffusion D(ω) = D(ω)I, the attenuation factor get the simplified form As an example on the use of this formalism, we consider the sequence of Fig. 1 of the main text and assume free diffusion to derive Eq. (3-5) of the main text. For free diffusion only the tail of the displacement power spectrum D(ω) ∝ 1/ω 2 is important [20,44]. The purely applied-gradient diffusion term M G 2 (T E) is as derived for a CPMG sequence [44] where the delay x = T E/N . The pure background gradient decay term is in turn the one that corresponds to a spin-echo modulation [44] M G 2 0 (T E, N ) = exp − 1 12 which is independent of x. The cross-term signal-decay contribution is calculated from Eq.
(S.10) leading to Supporting Information 2: sNOGSE/aNOGSE's: Analytical attenuation expressions for the general case of anisotropic diffusion We calculate next the normalized spin signal arising from Eq. (S.5), for the symmetric and asymmetric non-uniform gradient spin echo modulations ( s a NOGSE) introduced in Fig. 2. As described in the main text, as a result of in Eq. (S.7). The pure background gradient signal contribution is therefore independent of the applied gradient modulation and direction, providing the same weight for both NOGSE sequences. The cross-term in the attenuation factor for sNOGSE is zero but that for aNOGSE is not, as the products F sN OGSE † (ω, T E) F 0 (ω, T E) and This cross-term between the aNOGSE-modulated applied gradient and the background gradient G 0 will be As explained in the main text, the measured spin signal decays for the sNOGSE and aNOGSE sequences as described in Fig. 2e, factor out all non-diffusing sources of decoherence after normalizing them by the single-echo signal [20,21,44]. The amplitude of the NOGSE modulation is then where the amplitude contrast of the attenuation factors ∆β = β CP M G − β Single−echo . As the contribution to the attenuation factor that purely depends of the background gradient is independent of the applied gradient modulation its contribution ∆β G 2 0 is null, and the amplitude of the attenuation factors is then For the sNOGSE sequence ∆β s = ∆β G 2 as the cross-term is null, and ∆β a = ∆β G 2 + ∆β G· G 0 for the aNOGSE modulation curve. Notice that the contribution of the term that only depends of the applied gradient ∆β G 2 is the same for both sequences according to Eqs.
(S.17) and (S.18). Then by subtracting ∆β a and ∆β s , the ∆β G· G 0 cross-term contribution to the amplitude modulation is obtained, where ω 2 F 0 (ω, T E) . Assuming as before a Gaussian distribution for G 0 , where the first term depends on the relative sign of the parallel component of G · D(ω) to the background gradient G 0 , which depends of the anisotropic restricted-diffusion weighting.
Notice that the second term is always positive and it contains the IGDT ∆ G 0 ∆ G 0 . This was the expression used to evaluate the results presented in Fig. 4 after being normalized by ∆β s = ∆β G 2 to remove the anisotropic weighting due to restricted diffusion effects. For an isotropic diffusion D(ω) = D(ω)I, this gets simplified to For free diffusion only the tail of the displacement power spectrum D(ω) ∝ 1/ω 2 will be important, as it is this short-times regime active before restriction effects are seen, that matters [20,44]. The purely applied-gradient diffusion term M ( s a )NOGSE 26) where (N − 2) x + 2y = T E N OGSE = T E/2 (see Fig. 2 of the main text for definitions). The pure background gradient decay term is in turn which is independent of x and y, and therefore of the applied gradient modulation as was mentioned in the manuscript.
The cross-term signal-decay contribution for sNOGSE is zero as described before, and the one for aNOGSE will be (S.28) Notice that the sign of the attenuation factor for this cross-term contribution depends of the relative sign of G G 0 , where G is the applied component of the G-gradient that is parallel to the background gradient vector. The extremes of this attenuation arise for x = y = T E N OGSE /N (CPMG-like modulation) and for y = T E N OGSE /2 and x = 0 (single-echo modulation) i.e., it depends on the relative sign of G and G 0 . If N/2 is odd β CP M G G· G 0 (T E) = 0; but for N/2 odd the attenuation decays with 1/N 2 , and this makes the contrast lower. Assuming a Gaussian distribution for G 0 , The attenuation factor contrast amplitude is then where ∆ G 0 ∆ G 0 is the IGDT.