The intrinsic contrast in MR imaging of biological tissues mostly comes from spatial variations in water proton magnetic relaxation rates [1], which are governed by various molecular motions on different timescales reflecting the local unique environments [2,3]. In principle, the relaxation measurements can provide the basic microstructural information particularly for tissues with highly organized microarchitectures. For instance, myelinated axons in the human brain white matter (WM) are anisotropic and inhomogeneous in nature; and longitudinal (i.e., R1=1/T1) and transverse (i.e., R2=1/T2) relaxation rates have been revealed varying with the orientations of axon fibers relative to an external static field B0 [[4], [5], [6], [7]].

Diffusion tensor imaging (DTI) can provide voxel-size axon orientation information in WM in vivo [8,9]. According to the standard DTI model, three orthogonal translational diffusivities (i.e., eigenvalues) and the corresponding directions (i.e., eigenvectors) relative to B0 can be determined. While the magnitude of anisotropic diffusion is well defined, it remains ambiguous whether the direction of the principal diffusivity can accurately represent an axon fiber direction in practice [10,11]. When the direction of the principal diffusivity was used as an internal orientation gold standard, some discrepancies are identified when compared with an axon orientation derived from either susceptibility tensor imaging [[12], [13], [14], [15]] or nanostructure-specific X-ray tomography [16]. Similarly, when this internal reference was used to guide the orientation dependence of proton transverse (R2 and R2∗) relaxation in vivo, the observed orientation dependence profile manifested an angle offset ε0 that has not yet been accounted for [4,[17], [18], [19]]. In this work, R2 and R2∗ were treated equally, for convenience, regarding an axon fiber orientation dependence on B0.

In the past years [15,[20], [21], [22]], susceptibility-based relaxation mechanisms have been proposed for characterizing anisotropic R2 and R2∗ in WM, expressed either by A1 + A2 cos 2θ + A3 cos 4θ or by B1 + B2 sin2θ + B3 sin4θ. Here, θ is the angle between an axon fiber and B0, Ai and Bi (i = 1, 2, 3) the model parameters. As recently revealed [7,19], both expressions are mathematically equivalent albeit with different trigonometric coefficients. More importantly, these coefficients are not mutually independent as cos4θ (or sin4θ) can be expressed by cos2θ (or sin2θ) and vice versa. Thus, any proposed biophysical interpretations of the fits from these prior models will be ambiguous [23].

When considered as a potential relaxation mechanism, the magic angle effect (MAE) had not been appropriately evaluated in the literature [21,24]. The standard MAE function is often written as (3cos2θ − 1)2, indicating that MAE will disappear with θ=54.7° (i.e., “magic angle”) and become four times less when θ=90° than that when θ=0°. Unexpectedly, the experimental observations contradicted the theoretical predication. More specifically, the reported R2 and R2∗ appeared larger when θ changed from 0° to 90°. Consequently, MAE had been dropped out as a potential relaxation pathway. It should be emphasized that the standard MAE function implicitly assumes that all restricted water molecules are uniformly orientated along the same direction, which might not be the case in brain WM [25]. A general form of MAE function in an axially symmetric system has long been available [26], and lately reformatted to gain further insight into proton transverse relaxation orientation dependences [27].

Molecular translational and rotational diffusion (or reorientation) should be intimately linked as both originated from thermally driven Brownian motions [7,28,29], where the former can be probed by DTI and the latter by MR relaxation measurements. To our knowledge, no direct connection exists to date between the two drastically different measurements for studying the same anisotropic microstructures in WM [30]. This split might impede better characterization of rotationally restricted molecules both on the surface (water, H2O) and in the interior (lipid methylene, CH2) of phospholipid bilayers in WM [31,32]. In the literature [33], an ultrashort transverse relaxation time (T2b∼10–15 μs) of semisolid lipid CH2 protons was reported in a quantitative magnetization transfer (qMT) imaging study, revealing comparable orientation dependence with respect to that of surface water from prior orientation-dependent transverse relaxation studies [4,7,18,19,34].

The goal of this work was to introduce an angle offset ε0, determined by DTI directional diffusivities, into a cylindrical helix model based on a generalized MAE function for characterizing anisotropic transverse relaxation of ordered water and semisolid CH2 protons in WM. The proposed theoretical framework was validated by a high-resolution Connectome DTI dataset and then applied to previously published anisotropic R2 and R2∗ profiles at 3 T in vivo from the human brain WM of neonates, healthy and diseased adults. The results demonstrate that the proposed model can better characterize the documented orientation-dependent proton transverse relaxation profiles.