Sphere of lorentz and demagnetization factors in white matter

The 1H2O NMR Larmor frequency and R2* relaxation rate constant in white matter tissue show dependencies on the sample orientation in the static magnetic field (B0) (1, 2) that have been attributed in part to sub-voxel magnetic structure associated with so called ‘susceptibility inclusions’ (atoms, molecules or compounds with differing magnetic susceptibility) (3). A number of publications (3–5), the most recent of which is published in this journal (5), have introduced the so-called “Generalized Lorentzian Approach” (GLA) to offer a theoretical basis for this orientation dependence, in particular the frequency shift Δf in major white matter fiber bundles that may have contributions from inclusions with both elongated and isotropic (rotationally symmetric) geometry. Here it is argued that, although sub-voxel magnetic structure can significantly affect frequency and R2*, the GLA per-se is not needed to explain the observations, that its underlying concept is problematic, and its applicability to white matter rather limited. These concerns are elaborated upon in the following. Prior to the observations in white matter, the effect of an anisotropic magnetic structure (i.e., a bulk magnetization that has rotational asymmetry) on the NMR resonance frequency has been studied at various scales, e.g., the shape of the solvent volume constrained in sample tubes (e.g., (6, 7)), and subvoxel structure in trabecular bone (8), lung alveoli (9), red blood cells (10, 11) and muscle fibers (12). Because for biological tissues, the spatial magnetic susceptibility distribution is generally complex and not precisely known, modeling its effects requires some simplification. An established approach for this is to consider a (water hydrogen) proton’s average magnetic environment, which may be spatially non-uniform. However, when averaging over the many other protons and structural details in a voxel, the average environment can be approximated by incorporating an immediate region (or set of regions) with uniform continous magnetization (13). The aggregate near-field effects of the discrete magnetic dipoles in the proton’s immediate environment can be shown to be negligibly small by invoking the ‘sphere of Lorentz (SL)’ concept (see (6, 13) and references therein), whereas the calculation of the effects of the continuum magnetization on the proton frequency is facilitated by the use of demagnetization factors specific for the shapes of the boundaries between regions (e.g., (14, 15), for review see (13, 16)). For example, the average magnetic environment of a water hydrogen proton in white matter, and specifically the excised optic nerve studied in (5), may be approximated by a spherical, mostly water pocket with magnetic susceptibility χw centered on the proton, which is embedded in an elongated (prolate) ellipsoid with uniform magnetic susceptibility χ1, surrounded by nerve tissue with magnetic susceptibility χ2 and suspended in a liquid with magnetic susceptibility χe, with the angle α (degrees) between the ellipsoid’s major axis and the B0 direction [see Figure; Cf. Figure 7 and accompanying text in (13)]. The difference Δω in average 1H2O frequency within and outside the nerve then follows from Δff=13(χ1-χe)-12sin2α(χ1-χe), in analogy with previous results for red blood cells and muscle fibers (7, 11, 12), and consistent with numerical simulations (17). The first term derives from the asserted spherical boundary separating nearby water (and other molecule) electron cloud dipoles from magnetic continuum both inside and outside the nerves, and the spherical boundaries between magnetization areas of water without and with ‘isotropic’ inclusions (volumes with χw and χ1 inside the nerve and χw and χe outside the nerve); the sin2 α term results from the elongated ellipsoidal boundaries between areas without and with oriented, elongated inclusions (volumes with susceptibilities χ1, χ2, and χe). This resembles the results found with the GLA, e.g., Eq. [5] of (5), if one considers that χ1 contains the contributions from isotropic inclusions (with magnetic susceptibility χiso in Eq. [5]). What then, conceptually and computationally, distinguishes these approaches? The GLA applied in (5) appears to assume that the average proton is surrounded by two separately considered distributions of spherical and elongated susceptibility inclusions, whose near-field effects on the proton frequency are further assumed to be approximately zero when inclusions are randomly distributed and a proper averaging volume is used. The authors of (5) then make the point that for this to be the case, the ‘isotropic’ inclusions require a spherical averaging volume (in analogy with the SL), and the elongated inclusions require an elongated ellipsoidal averaging volume. Apart from the fact that the necessity of the latter is not explicitly demonstrated, this approach implicitly assumes infinitely small volume fractions of the two inclusion types and considers their distributions to be independent. Finite inclusion volumes, as may be the case in white matter, may affect not only the randomness of their distribution, but also restrict each type of inclusion to the space that is not occupied by inclusions of the other type. For example, for elongated inclusions with volume fraction VL, the local concentration of ‘isotropic’ inclusions increases by 1/(1−VL). Thus, χ1 above is equivalent to χiso/(1− VL), and this value should replace χiso in Eq. [5] of (5), but not in Eq. [4] of (5). Stated differently, χiso in Eq. [5] of (5) is the contribution of ‘isotropic’ inclusions to the average susceptibility of the whole nerve compartment, whereas the model described above yields a dependence on χ1, [and thus χiso/(1− VL)], the susceptibility of ‘isotropic’ inclusions within the volume that they occupy. Having found that GLA is unnecessary and, in the case of finite inclusion volume fractions, leads to systematic errors, what about the primary rationale for using the GLA, i.e, the presumed presence of elongated susceptibility inclusions in nerve tissue? Luo et al. argue that the different magnitudes of the sin2α dependence outside and within the nerve (Eqs. [4] and [5] of (5), respectively) is evidence for such presence. However, accumulating evidence suggests the magnetic susceptibility of nerve tissue itself is anisotropic (18–21). This obviates the need to assert prolate inclusions. For example, for the simple case of a uniform (but anisotropic) susceptibility (with χ⊥ and χ|| the magnetic susceptibilities along and perpendicular to the fiber, respectively), the standard SL approach leads to coefficients for the sin2α dependences outside and within the nerve being {χ⊥2-χe2} and {χe2-χ⊥+2χ||6} respectively [replacing {χiso+χL2-χe2} and {χe2-χiso2} in Eqs. [4] and [5] of (5)]. Note that our derivation results in different slope magnitudes, without requiring elongated inclusions, the GLA, or even the non-spherical demagnetization factor inside the nerve as used in Fig. 1. Thus, in the presence of anisotropy, the primary contention in (5) that “the frequency observed in the optic nerve ……is inconsistent with Lorentzian sphere approximation” appears unwarranted and non-parsimonious. FIGURE Model of average water environment in optic nerve with both oriented and elongated and isotropic magnetic susceptibility inclusions. Because of correlations between the space occupied by water space and the distribution of elongated inclusions, water .