Exploiting gradient-echo frequency evolution: probing white matter microstructure and extracting bulk susceptibility-induced frequency for quantitative susceptibility mapping

Abstract

Purpose:

(1) To investigate the microstructure-induced frequency shift in white matter (WM) with crossing fibers. (2) To separate the microstructure-related frequency shift from the bulk susceptibility-induced frequency shift by model fitting the gradient-echo (GRE) frequency evolution for potentially more accurate QSM.

Methods:

A hollow cylinder fiber model (HCFM) with two fiber populations was developed to investigate GRE frequency evolutions in WM voxels with microstructural orientation dispersion. The simulated and experimentally measured TE-dependent local frequency shift was then fitted to a simplified frequency evolution model to obtain a microstructure-related frequency difference parameter ($\Delta f$) and a TE-independent bulk susceptibility-induced frequency shift ($C_f$). The obtained $C_f$ was then used for QSM reconstruction. Reconstruction performances were evaluated using a numerical head phantom and in vivo data, and compared to other multi-echo combination methods.

Results:

GRE frequency evolutions and $\Delta f$-based tissue parameters in both parallel and crossing fibers determined from our simulations were comparable to those observed in vivo. The TE-dependent frequency fitting method outperformed other multi-echo combination methods in estimating $C_f$ in simulations. The fitted $\Delta f$, $C_f$ and QSM could be improved further by navigator-based B₀ fluctuation correction.

Conclusion:

A HCFM with two fiber populations can be used to characterize microstructure-induced frequency shifts in WM regions with crossing fibers. HCFM-based TE-dependent frequency fitting provides tissue contrast related to microstructure ($\Delta f$) and may in addition help improve the quantification accuracy of the bulk susceptibility-induced frequency shift ($C_f$) and the corresponding QSM.

Introduction

High resolution MR phase images acquired using gradient echo (GRE) sequence at high field contain rich anatomical information related to tissue magnetic susceptibility contrast.^1,2 The susceptibility-induced macroscopic field perturbation follows a well-known relationship of dipole convolution connecting the magnetic susceptibility source and the field.^3,4 QSM is a recently developed technique to solve the field-to-susceptibility dipole inversion that enables the calculation of tissue bulk magnetic susceptibility distributions.^5–7 In the past few years, QSM has served as an important imaging contrast for assessing cerebral iron deposition and myelin content changes in normal aging^8–10 and a series of neurodegenerative diseases such as multiple sclerosis^11–13, Huntington’s disease^14–16, Parkinson’s disease^17–19 and Alzheimer’s disease^20–23.

Besides bulk tissue susceptibility effects, previous studies reported that both GRE magnitude and phase are influenced by white matter (WM) microstructures with multi-compartmental field perturbations and relaxation rates, leading to a non-single-exponential signal decay²⁴ and distinct nonlinear phase evolutions.^25,26 Such microstructure effects have been attributed to the myelin sheath,^27–31 which has anisotropic magnetic susceptibility and water compartments with shorter T₂ and T₂* relaxation compared to axonal and extracellular water.^29,30 The sub-voxel field perturbation depends on the geometry and susceptibility of multiple compartments (i.e., axonal, myelin and extracellular space) and their directions with respect to the B₀ field. To characterize the relationship between the WM microstructure properties and GRE signal evolution, a hollow cylinder fiber model (HCFM)^25,32 was proposed incorporating the following features: (i) isotropic and anisotropic magnetic susceptibility in a hollow cylindrical myelin compartment; (ii) compartment-specific T₂ and T₂* relaxation; (iii) reduced spin density in myelin and (iv) a chemical exchange related frequency shift. HCFMs were used to explain the orientation-dependent R₂* and frequency shift observed in WM.^26,29,30,32 Later developments on HCFM aimed at further improving the characterization of GRE signal evolution with more realistic axon geometry modeling.^33,34 However, existing HCFMs model parallel fibers only, or assume a simple scaling factor (like DTI fractional anisotropy) for potential fiber orientation dispersions in crossing fiber regions.³²

To investigate microstructure-related frequency shifts free from confounding bulk susceptibility-induced non-local frequency shifts, frequency difference mapping (FDM) has been proposed.^32,35 FDM takes the difference between frequency maps acquired at short and long TEs or fits a frequency difference parameter $\Delta f$ from the GRE frequency evolution. As predicted by the HCFM, the fitted $\Delta f$ parameter shows unique WM contrast and depends on the fiber-to-field angle $\theta$ in the form of $\Delta f=a_1{sin}^2\theta +a_2$, where $a_1$ is related to tissue microstructure and myelin susceptibility and $a_2$ represents any $\theta$-independent frequency difference.³²

The microstructure-induced frequency shift cannot be explained by the bulk isotropic or anisotropic magnetic susceptibility models³⁶ used in QSM or susceptibility tensor imaging (STI)^37–39, and would introduce artifacts and a potentially under- or over-estimated tissue magnetic susceptibility if not properly accounted for. This modeling error could lead to the observed TE-dependent QSM values in various brain regions^40–42 and QSM variability across protocols, sites and fields. Separation of the microstructure-induced and bulk susceptibility-induced frequency shifts is therefore important for improving QSM accuracy as well as assessment of microstructure.

The first aim of the present study was to investigate the microstructure-induced frequency shift in WM regions with crossing fibers by using a HCFM with two distinct fiber populations. The second aim was to separate the time-dependent microstructure-induced frequency from the time-independent bulk susceptibility-induced frequency shift for potentially more accurate QSM reconstruction. A TE-dependent frequency fitting method was proposed and compared to other multi-echo combination methods. Reconstruction performances were evaluated using both simulations and in vivo data. Orientation dependence of the frequency difference parameter was also compared in selected parallel and crossing fibers.

Method

Simulations: signal modeling

Full signal modeling of a spoiled GRE sequence can be formulated using the steady state equation,

$$S=S_0{e}^{-TE\bullet R_2^{*}}\bullet \left(1-e^{-TR\bullet R_1}\right)\bullet \frac{sin\left(FA\right)}{1-cos\left(FA\right)\bullet e^{-TR\bullet R_1}}{e}^{i(2\pi \bullet TE\bullet f_{\mathit{\text{total}}}+{\varphi }_0)}$$ (1)

where FA denotes flip angle, ${\varphi }_0$ is an initial phase at TE = 0. The total frequency shift $f_{\mathit{\text{total}}}$ includes contributions from both microstructural and bulk susceptibility sources.

To investigate microstructure-induced frequency shift in crossing fibers, a HCFM incorporating two sub-voxel fiber populations with comparable fiber volume fractions was developed as illustrated in Fig. 1A. Within the coordinate system aligned with the major fiber population (i.e., fiber_1 frame, within which the major fiber is along $\left[\begin{array}{ccc}0& 0& 1\end{array}\right]\prime$), the direction of the 2^nd fiber population (fiber 2) can be set in the y-z plane along $\left[\begin{array}{ccc}0& sin\alpha & cos\alpha \end{array}\right]\prime$. The unit vector of the main magnetic field ${\widehat{H}}_0$ can be described as $\left[\begin{array}{ccc}sin\theta cos\phi & sin\theta sin\phi & cos\theta \end{array}\right]\prime$ in the fiber_1 frame. Denoting the angles between ${\widehat{H}}_0$ and the two fiber populations as ${\beta }_1$ and ${\beta }_2$, we have $\theta ={\beta }_1$ and $\phi ={sin}^{-1}\left(\frac{cos{\beta }_2-cos\alpha cos{\beta }_1}{sin\alpha sin{\beta }_1}\right)$.

Figure 1.

(A) Diagram of a hollow cylinder fiber model (HCFM) with two fiber populations in a coordinate system aligned with fiber 1 (fiber_1 frame). An axon packing algorithm was implemented in each fiber population. (B) Plots of simulated voxel averaged frequency shifts as a function of TE at 7T using the HCFM in three different fiber configurations for ${\widehat{H}}_0={\left[100\right]}^{\prime }$ and ${\widehat{H}}_0={\left[010\right]}^{\prime }$. (C,D) The simulated frequency perturbations in the x-y plane for parallel fibers (left) and two-fiber populations crossing at 45° (middle) and at 90° (right), corresponding to the curves shown in (B) when ${\widehat{H}}_0={\left[100\right]}^{\prime }$ (C) and ${\widehat{H}}_0={\left[010\right]}^{\prime }$ (D). The blue squares in (C) and (D) represent the ROI from which the signal S was sampled and averaged.

Using the 3D Fourier-based method and ignoring the bulk susceptibility-induced non-local frequency shift from neighboring voxels, the complex signal of this HCFM can be calculated (Supporting Information S1 (SI S1)). Following Wharton and Bowtell³², using a small phase approximation (SI S2), the WM microstructure-induced frequency shift evolution can be simplified to a TE-dependent function in Eq. (2), with $\Delta f$ being the amplitude of frequency change over TE.

$$f=\Delta f\left(\frac{\text{exp}\left(-p_{1}TE\right)}{p_2\left(\text{exp}\left(-p_{1}TE\right)-1\right)+1}\right)+C_f$$ (2)

where $p_1=\frac{1}{T_{2,m}^{*}}-\frac{1}{T_{2,nm}^{*}}$, $p_2=\frac{v_m{\rho }_m}{v_m{\rho }_m+v_{ax}+v_{ex}}$ and $C_f=\frac{v_{ax}\Delta {\omega }_{ax}+v_{ex}\mathrm{\Delta }{\omega }_{ex}}{v_{ax}+v_{ex}}$. $v_{ax}$, $v_m$ and $v_{ex}$ are the volume fractions of the axonal, myelin and extracellular water compartments in a voxel. ${\rho }_{ax}$, ${\rho }_m$ and ${\rho }_{ex}$ are the water proton density of the corresponding compartments. $T_{2,m}^{*}$ and $T_{2,nm}^{*}{=T}_{2,ax}^{*}=T_{2,ex}^{*}$ represent the $T_2^{*}$ relaxation times in myelin and non-myelin (axonal and extracellular) water. $\Delta {\omega }_{ax}$ and $\Delta {\omega }_{ex}$ are the frequency shifts in the axonal and extracellular compartment, respectively. Note that $\Delta f$ is physically defined as the difference between the frequency measured in the short- and long-TE regime³², and $p_2$ is equivalent to the myelin water fraction (MWF) commonly used in the literature^30,43,44. $C_f$ is a time-independent frequency shift, which can further include the non-local frequency shift contribution from bulk tissue magnetic susceptibility sources (isotropic and anisotropic).

Using multi-echo GRE signals, $\Delta f$ and $C_f$ maps can be reconstructed using nonlinear least squares fitting of Eq. (2) to the effective tissue frequency estimated at each TE. $C_f$ can be used for QSM reconstruction, in which microstructure-related effects can be expected to be minimal.

Simulations: frequency shift from both microstructural and bulk susceptibility sources

A numerical head phantom incorporating both microstructure and bulk susceptibility-induced frequency shifts with a volume size of 224×224×126 mm³ and 1×1×1 mm³ resolution was generated using brain atlas data acquired on a single participant from a previous study⁴⁵ (Fig. 3A).

Figure 3.

(A) Schematic diagram of the numerical phantom data generation process. Microstructure-induced echo-time dependent frequency shift and bulk susceptibility-induced time- independent frequency shift were added together to simulate the full GRE signal. (B) Schematic of the pipeline used to generate the microstructure-related frequency difference parameter $\Delta f$ and the time-independent bulk susceptibility-induced frequency shift $C_f$, which can be used to obtain QSM.

The HCFM with two fiber populations was used to simulate the microstructure-induced frequency shift inside each WM voxel. Each model had a 200×200×200 sub-voxel grid, with 156 (fiber 1) and 161 (fiber 2) cylinders simulated for the two fiber populations, respectively (Fig. 1A). An axon packing algorithm⁴⁶ was used with the outer radii $r_{\mathit{\text{outer}}}$ of the myelinated axon following a Gamma distribution with mean of 5 and SD of 1 sub-voxel unit length. The ratio of inner to outer radii $g_{\mathit{\text{ratio}}}$ depends on the diameter of the myelinated axon⁴⁷ as $g_{\mathit{\text{ratio}}}=0.22\times \text{log}\left(2*r_{\mathit{\text{outer}}}\right)+0.508$. Other parameters were chosen as, isotropic and anisotropic susceptibility of myelin lipid: ${\chi }_m^i=-0.1$ ppm, ${\chi }_m^a=-0.1$ ppm; chemical exchange $E=0.02ppm$, ${\rho }_m=0.7,$ $T_{2,m}^{*}=8$ ms, $T_{2,nm}^{*}=36$ ms (SI S1). For the head phantom, fiber orientation distribution function (ODF) peaks from multi-shell diffusion MRI (dMRI) acquired on the same participant were used to determine the angles between the two fiber populations $\alpha$, the angles between B₀ field and each fiber population ${\beta }_1$ and ${\beta }_2$, respectively. ${\widehat{H}}_0$ in the fiber_1 frame can then be determined. To estimate the ODF peaks, the dMRI images were preprocessed using MRtrix3⁴⁸ and affinely co-registered to the GRE first-echo magnitude image using Advanced Normalization Tools (ANTs)⁴⁹. Fiber ODFs were then reconstructed using multi-shell multi-tissue CSD⁵⁰, and the ODF peaks of each voxel were extracted.⁵¹ Crossing fiber voxels were assigned when the amplitude of ODF peak2 was greater than 30% of peak1⁵². Otherwise, the WM voxels were considered to have parallel fibers (i.e., set crossing angle $\alpha =0°$). The crossing angle $\alpha$ was further divided into 19 groups with 5° span for each group to simplify the computations, i.e., each $\alpha$ group shares the same HCFM but may have different ${\widehat{H}}_0$ in the fiber_1 frame for different voxels.

Intra-voxel field perturbations in the HCFM and the complex signal due to microstructure were simulated (see SI S1) for TEs of 1–30 ms with 1 ms step using an isotropic cubic signal sampling mask (Fig. 1A). The microstructure-induced frequency shift $f_{\mathit{\text{micro}}}$ at each TE can then be estimated (Eq. S9). $\Delta f$, $p_1$ and $p_2$ were then calculated by voxel-wise nonlinear least squares fitting of Eq. (2) to the calculated frequency evolution (see SI S2) and treated as the ground truth ${\Delta f}_{\_gt}$, $p_{1\_gt}$ and $p_{2\_gt}$ parameter maps. Orientation dependence of ${\Delta f}_{\_gt}$ was also investigated. This was done by least squares fitting of ${\Delta f}_{\_gt}$ using ${sin}^2\theta$ of the mean $\theta$ values in each angular group (37 groups, spans $\left[0°:5°:180°\right]$) in selected WM regions including corpus callosum (CC), cingulum (CG), superior longitudinal fasciculus (SLF) and corona radiata (CR).

For the bulk/macroscopic magnetic susceptibility-induced frequency shift, the isotropic component ${\chi }_{\mathit{\text{macro}}}^I$ in different regions of interest (ROIs) were set according to the literature and listed in S4 Table S1. The bulk anisotropic susceptibility of any WM voxel was set as ${\mathit{\chi }}_{\mathit{a}\mathit{p}\mathit{p}}^{\mathit{A}}=\left[\begin{array}{ccc}0.014& 0& 0\\ 0& -0.009& 0\\ 0& 0& -0.009\end{array}\right]$ to satisfy mean magnetic susceptibility (MMS) = −0.031 ppm and magnetic susceptibility anisotropy (MSA) = 0.023 ppm (SI S5). As the bulk susceptibility anisotropy is believed to be affected by fiber orientation dispersion (OD), we further modulated the magnitude of ${\mathit{\chi }}_{\mathit{a}\mathit{p}\mathit{p}}^{\mathit{A}}$ based on OD measured from neurite orientation dispersion and density imaging (NODDI)⁵³ as ${\mathit{\chi }}_{\mathit{m}\mathit{a}\mathit{c}\mathit{r}\mathit{o}}^{\mathit{A}}{=\mathit{\chi }}_{\mathit{a}\mathit{p}\mathit{p}}^{\mathit{A}}\bullet \left(1-OD\right)$. The full bulk susceptibility tensor can then be written as ${\mathit{\chi }}_{\mathit{m}\mathit{a}\mathit{c}\mathit{r}\mathit{o}}={\chi }_{\mathit{\text{macro}}}^I+{\mathit{\chi }}_{\mathit{m}\mathit{a}\mathit{c}\mathit{r}\mathit{o}}^{\mathit{A}}$.Using a tensor representation in a common subject frame of reference³⁸, the frequency shift due to these bulk susceptibility sources can then be calculated using the STI forward model³⁷ $f_{\mathit{\text{macro}}}\left(\mathit{r}\right)=\gamma \bullet B_0\bullet {FT}^{-1}\left\{\frac{1}{3}{\widehat{\mathit{H}}}^{T}FT\left\{{\mathit{\chi }}_{\mathit{m}\mathit{a}\mathit{c}\mathit{r}\mathit{o}}\left(\mathit{r}\right)\right\}\widehat{\mathit{H}}-{\mathit{k}}^T\widehat{\mathit{H}}\frac{{\mathit{k}}^{T}FT\left\{{\mathit{\chi }}_{\mathit{m}\mathit{a}\mathit{c}\mathit{r}\mathit{o}}\left(\mathit{r}\right)\right\}\widehat{\mathit{H}}}{k^2}\right\}$.

Full complex GRE signal S was then calculated using Eq. (1). ${\varphi }_0$ was modeled as a 3D second order polynomial ensuring $\pi$ phase variation inside the brain region. $S_0$ and $R_1$ were set in each ROI as in Table S1, TR was set as 75 ms, and FA = $15°$. R₂* was set according to data previously acquired on the same participant.⁴⁵ The total frequency shift $f_{\mathit{\text{total}}}=f_{\mathit{\text{micro}}}+f_{\mathit{\text{macro}}}$. Note that since the time-independent term $C_f$ in Eq. (2) could be replaced by $f_{\mathit{\text{macro}}}$ induced by the bulk susceptibility sources, we dropped the original $C_f$ term when calculating the $f_{\mathit{\text{micro}}}$, i.e., $f_{\mathit{\text{micro}}}=\Delta f\left(\frac{\text{exp}\left(-p_{1}TE\right)}{p_2\left(\text{exp}\left(-p_{1}TE\right)-1\right)+1}\right).$ The complex signal S was simulated for TEs of 2–30 ms with a 2 ms step, which is close to our in vivo experimental setup. Finally, zero mean Gaussian white noise was added to both real and imaginary parts of the complex data with SNR set to 100.

Simulations: estimation of frequency parameters and QSM reconstruction

Simulated GRE phase maps were first unwrapped by a rapid opensource minimum spanning tree algorithm (ROMEO) with template unwrapping⁵⁴. The initial phase offset ${\varphi }_0$ was estimated and removed using the MCPC-3D method⁵⁵. Frequency maps were then calculated by scaling the resulting phase maps by $2\pi {TE}_n$. Background fields were removed by Laplacian boundary value (LBV) method for each echo⁵⁶. $\mathrm{\Delta }f$ and $C_f$ were then fitted (using ‘lsqnonlin” function in Matlab, default options) with phase SNR weights⁵⁷ as $w_n={TE}_n{e}^{-\frac{{TE}_n}{T_2^{*}}}/{\sum }_{n=1}^N{TE}_n{e}^{-\frac{{TE}_n}{T_2^{*}}}$. The initial values for $\Delta f$, $p_1$, $p_2$ and $C_f$ were set as 1 Hz, 100 s⁻¹, 0 and 0 Hz, respectively. The lower and upper bounds of $\Delta f$ were set as (−10, 10) Hz based on the HCFM simulation. Regarding $p_1$ and $p_2$, the lower and upper bounds were set as (80, 1000) s⁻¹ and (0, 1), based on the reported T₂* values in myelin and axon/extracellular compartments at 7T.^29,58 No bounds were set for $C_f$. We termed this approach as TE-dependent frequency fitting.

Orientation dependence of the fitted $\Delta f$ (${\Delta f}_{\_fit}$) was also investigated in selected WM ROIs. The fitted $C_f$ map was compared to its corresponding ground truth ${C_{f\_gt}=f}_{\mathit{\text{macro}}}$. For comparison, tissue frequency maps were also estimated using other common echo combination methods, including nonlinear complex data fitting ($f_{NL}$)⁵⁹ and weighted echo averaging⁵⁷ with different combinations of echoes, i.e., using the first to the tenth echoes $\left(f_{{\mathit{\text{wavg}}}_{1-10}}\right)$, all echoes $\left(f_{{\mathit{\text{wavg}}}_{1-15}}\right)$, sixth to fifteenth echoes $\left(f_{{\mathit{\text{wavg}}}_{6-15}}\right)$.

We further compared the QSM calculated from the $C_f$ maps and other tissue frequency maps in comparison to the ground truth MMS. QSM images were reconstructed using iLSQR⁶⁰ without strong a-priori regularization and referenced to the whole brain.

Root-mean-square error (RMSE) percentage, structural similarity index (SSIM) and high frequency error norm (HFEN) were used as evaluation metrics.⁶¹

MRI data acquisition

For in vivo experiments, five healthy participants (34 ± 6 years old, 3 females, participant #1–5) were scanned at 7T (Philips Healthcare, Best, Netherlands) using a 32 channel NovaMedical head coil. A 3D multi-echo GRE sequence was used with parameters: FOV = 220×192×100 mm³, voxel size = 1×1×1 mm³, TR/TE/ΔTE = 77/2.8/2.2 ms, number of echoes = 15, SENSE factor of 1.5×1×2, flip angle of 10°, water selective ProSet 121 pulse for fat suppression and a scan time of 9 min.

To test the effects of physiological noise (e.g., respiration) compensation on $\Delta f$ and $C_f$ maps, one participant (31 years old, male, participant #6) was scanned at a 7T MRI scanner (Magnetom, Siemens Healthcare, Erlangen, Germany) equipped with a 32 channel Nova Medical head coil. A 2D multi-echo GRE sequence⁵⁸ was used with parameters: FOV = 256×256×64 mm³, voxel size = 1×1×1 mm³, TR/TE/ΔTE = 2400/2.5/1.6 ms, echo number = 17, flip angle of 83° and scan time of 20 min 30 sec.

Multi-shell dMRI for estimating fiber ODF maps and T1-weighted MPRAGE for anatomical referencing were also acquired (details in SI S3).

IRB approval from the local committees and written informed consent were obtained before all participants were scanned.

In vivo data processing

Processing pipeline for generating $C_f$ and $\Delta f$ maps in vivo is similar as that used in simulation. To suppress imaging artifacts caused by unreliable fitting in some voxels, we added a Tikhonov regularization term as

$${\Delta f,p_1,p_2,C_f=\mathit{\text{argmin}}}_{\Delta f,p_1,p_2,C_f}{\sum }_{n=1}^N{‖w_n\left(\Delta f\left(\frac{\text{exp}\left(-p_1{TE}_n\right)}{p_2\left(\text{exp}\left(-p_1{TE}_n\right)-1\right)+1}\right)+C_f-f_n\right)‖}_2^2+\lambda {‖C_f-f_{{\mathit{\text{wavg}}}_{6-15}}‖}_2^2$$ (3)

where $f_n$ is the effective frequency at n-th TE, regularization parameter $\lambda$ was empirically determined as 0.01 to obtain a trade-off between over-fitting and bias. Susceptibility maps were reconstructed from $C_f$, $f_{{\mathit{\text{wavg}}}_{1-15}}$ and $f_{NL}$ using iLSQR-QSM. Due to unresolved phase aliasing artifacts possibly caused by low SNR at the first TE, we used second to last echoes to calculate $f_{NL}$ for all participants except participant #6 with navigator correction. To compare with Wharton and Bowtell³⁶, susceptibility map was also calculated using local frequency map estimated at a selected short TE (third echo with TE = 7.2 ms for participant #1–5 and 5.7 ms for #6 as $f_{\mathit{\text{echo}}3}$). In addition, the orientation dependence of $\Delta f$ was investigated in CC, CG, SLF and CR, which were automatically segmented based on affinely co-registered MPRAGE images using a multi-atlas matching approach (mricloud.org)⁶².

Results

Simulations

The assigned myelin volume fractions (MVFs) were 0.301, 0.297, 0.301 and fiber volume fractions (FVFs) were 0.652, 0.647, 0.652 for the two fiber populations in the HCFM crossing at 0°, 45° and 90°, respectively. Fig. 1(B) shows the simulated signal frequency as a function of TE for ${\widehat{H}}_0$ of $\left[\begin{array}{ccc}1& 0& 0\end{array}\right]\prime$ and $\left[\begin{array}{ccc}0& 1& 0\end{array}\right]\prime$ in parallel fibers, and for two fiber populations crossing at 45° and 90°. When ${\widehat{H}}_0=\left[\begin{array}{ccc}1& 0& 0\end{array}\right]\prime$, the parallel fibers and 90°-crossed fibers yielded the same frequency evolution curve, and 45°-crossed fibers showed only a slight difference (<1 Hz). In comparison, when ${\widehat{H}}_0$ is along $\left[\begin{array}{ccc}0& 1& 0\end{array}\right]\prime$, temporal frequency evolutions among different fibers showed much larger differences, with 5.4 Hz change over TE for parallel fibers, but only 1.4 Hz change over TE for 90°-crossed fibers. Figs. 1(C, D) show the corresponding simulated field perturbation in the central x-y plane of the HCFM.

Figure 2(A) shows selected WM ROIs overlaid on a T1w image. Fig. 2(B) shows color maps of the first and second ODF peaks. A representative slice of ${\Delta f}_{\_gt}$ is shown in Fig. 2(C). Two red arrows in Fig. 2(C) point to corticospinal tract (label 1) and SLF (label 2) with striking contrast, which mostly run parallel and perpendicular to B₀, respectively. Figs. 2(D, E) further demonstrate the orientation dependence of ${\Delta f}_{\_gt}$ in CC, CG, SLF and CR. In the parallel fiber dominated CC and CG regions, an excellent fit of $\Delta f=a_1{sin}^2\theta +a_2$ was achieved (R² > 0.95). In addition, larger a₁ and smaller a₂ were fitted when excluding crossing fibers in these two bundles (blue lines in Fig. 2D), e.g., for CC, a₁ of 4.20 Hz and a₂ of 0.71 Hz were fitted in all voxels versus a₁ of 5.08 Hz and a₂ of 0.10 Hz in voxels with only parallel fibers. In comparison, a larger portion of voxels in SLF and CR are composed of crossing fibers. Smaller a₁ and larger a₂ were fitted when excluding parallel fibers in these structures (blue lines in Fig. 2E). For SLF, a₁ of 3.85 Hz and a₂ of 0.77 Hz were fitted in all voxels versus a₁ of 3.34 Hz and a₂ of 1.05 Hz in voxels with only crossing fibers. In addition, CC and CG exhibit larger $\Delta f$ (~5 Hz) variations over the relevant $\theta$-range as compared with SLF and CR (~3 Hz).

Figure 2.

Simulation results of HCFM with two fiber populations. (A) The selected regions of interest (ROIs) including corpus callosum (CC), cingulum (CG), superior longitudinal fasciculus (SLF) and corona radiata (CR) are marked on a T1w image. (B) Representative slices showing color maps of the first and second peaks of the fiber orientation distribution function (ODF) measured using multi-shell multi-orientation diffusion MRI in vivo and used to determine the sub-voxel fiber configurations in the numerical head phantom. (C) Representative slice showing the fitted frequency difference parameter ${\Delta f}_{\_gt}$ from simulated noise-free frequency evolution considering only the microstructure related frequency shift (HCFM). Two red arrows point to the corticospinal tract (label 1) and SLF (label 2), which run mostly parallel and perpendicular to the B_0, respectively, as also seen in (B). (D, E) Orientation dependence of ${\Delta f}_{\_gt}$ in parallel fibers dominated ROIs such as CC and CG (D), and crossing fibers dominated ROIs such as SLF and CR (E). Note that the fiber-to-field angle $\theta$ of a crossing fiber voxel refers to the orientation of the major fiber population in that voxel, i.e., peak 1 of the ODF. The dots with error bars correspond to mean and SD of ${\Delta f}_{\_gt}$ measures at mean $\theta$ values in each angular group (5° bin). The solid lines represent the least squares fit using ${sin}^2\theta$. Red dots/lines represent data using all voxels in the selected ROIs, while blue dots/lines represent data using only voxels containing the dominant fiber type, e.g., parallel fibers in the parallel fiber dominated ROIs (by excluding crossing fibers).

Figure 3 (A) illustrates the schematic diagram of the numerical phantom data generation process. The pipeline for the estimation of $\Delta f$ and $C_f$ parameters is shown in Fig. 3(B).

Figure 4 (A) shows the fitted $\Delta f$, $C_f$, $f_{{\mathit{\text{wavg}}}_{1-15}}$ and $f_{NL}$ as well as their differences with respect to the ground truth. Though with some underestimation (1–2 Hz) and noise effects, the fitted ${\Delta f}_{\_fit}$ showed anatomical features and WM orientation dependent contrast like ${\Delta f}_{\_gt}$ (the first row in Fig. 4(A)). The orientation dependence of the fitted ${\Delta f}_{\_fit}$ is shown in Fig. S2. The fitted $C_{f\_fit}$ showed good resemblance to its ground truth, giving a relatively uniform error map. In comparison, without considering the microstructure-induced frequency shift, larger error was observed in $f_{{\mathit{\text{wavg}}}_{1-15}}$ and $f_{NL}$ with higher RMSE and lower SSIM. Fibers with different orientations exhibited different error patterns (red and yellow arrows in Fig. 4(A)) and such patterns differed in $f_{NL}$ as compared to $f_{{\mathit{\text{wavg}}}_{1-15}}$. Fig. 4(B) shows the QSM maps reconstructed from different tissue frequency estimations using iLSQR. The difference map between QSM($C_{f\_gt}$) and MMS shows different error patterns for different fibers (red and yellow arrows in the first row of Fig.4 (B)) due to the fact that the susceptibility anisotropy effect was not accounted for in the QSM deconvolution. QSM calculated from $C_{f\_fit}$, i.e., QSM($C_{f\_fit}$) outperformed QSM($f_{{\mathit{\text{wavg}}}_{1-15}}$) and QSM($f_{NL}$) with lower RMSE and higher SSIM. Fig. 4 (C) displays the fitted curve shape parameters $p_1$ and $p_2$ maps. The $p_{1\_fit}$ and $p_{2\_fit}$ maps are generally noisy and do not contain obvious anatomical information.

Figure 4.

Simulation results. (A) Representative slices of $\Delta f$ (1^st row), $C_f$(2^nd row), echo-combined local frequency using weighted echo averaging with 1^st-15^th echoes $f_{{\mathit{\text{wavg}}}_{1-15}}$(3^rd row), and nonlinear complex data fitting $f_{NL}$ (bottom row), as well as the corresponding difference maps. (B) Representative slices of QSM reconstructed using ground truth $C_{f\_gt}$(1^st row), fitted $C_{f\_fit}$ (2^nd row), $f_{{\mathit{\text{wavg}}}_{1-15}}$(3^rd row), $f_{NL}$ (bottom row) and the corresponding difference maps as compared to the ground truth mean magnetic susceptibility (MMS). Fibers with different orientations exhibited different error patterns (red and yellow arrows). (C) Representative slices of ground truth curve shape parameters $p_1$ (left half) and $p_2$ (right half), the fitted maps and their differences.

The reconstruction performances in terms of evaluation metrics for the fitted $\Delta f$ and tissue frequency maps as well as the corresponding iLSQR QSMs are summarized in Table 1. $C_{f\_fit}$ outperformed $f_{NL}$ and all $f_{\mathit{\text{wavg}}}$ with the lowest RMSE, HFEN and highest SSIM in whole brain as well as lower RMSE than $f_{NL}$ and $f_{{\mathit{\text{wavg}}}_{1-10}}$ in WM. Weighted averaging using early echoes gave a more inferior performance (i.e., $f_{{\mathit{\text{wavg}}}_{1-10}}$ gave highest RMSE, HFEN and lowest SSIM among all $f_{\mathit{\text{wavg}}}$). QSM($C_f$) outperformed QSM($f_{NL}$) and all QSM($f_{\mathit{\text{wavg}}}$) with lower RMSE and HFEN in whole brain/WM as well as higher SSIM in WM. Consistently, QSM maps calculated from weighted averaging using early echoes showed larger errors.

Table 1.

Quality metrics of fitted frequency difference parameter (${\mathrm{\Delta }f}_{\_fit}$) and tissue frequency maps estimated using TE-dependent frequency fitting ($C_{f\_fit}$), nonlinear complex data fitting ($f_{NL}$), weighted echo averaging with 1^st-10^th echoes ($f_{{\mathit{\text{wavg}}}_{1-10}}$), 1^st-15^th echoes ($f_{{\mathit{\text{wavg}}}_{1-15}}$), 6^th-15^th echoes ($f_{{\mathit{\text{wavg}}}_{6-15}}$) from the simulation, as compared to their corresponding ground truth ${\mathrm{\Delta }f}_{\_gt}$ and $C_{f\_gt}$, respectively. The corresponding QSM reconstruction using the iLSQR algorithm is compared with the underlying mean magnetic susceptibility (MMS). Quality metrics with subscript WM, e.g., RMSE_WM, indicate comparisons in white matter (WM) region, otherwise in whole brain.

Frequency	RMSE	SSIM	HFEN	RMSEWM	SSIMWM	HFENWM
${\Delta f}_{\_fit}$	72.09	0.220	50.41	62.05	0.799	56.36
$C_{f\_fit}$	39.82	0.988	24.19	32.87	0.995	28.16
$f_{NL}$	44.64	0.882	31.35	39.05	0.972	35.41
$f_{{\mathit{\text{wavg}}}_{1-10}}$	44.50	0.866	33.67	34.32	0.962	30.97
$f_{{\mathit{\text{wavg}}}_{1-15}}$	43.91	0.874	32.14	32.14	0.967	31.31
$f_{{\mathit{\text{wavg}}}_{6-15}}$	43.73	0.877	31.48	31.48	0.970	32.36
iLSQR	RMSE	SSIM	HFEN	RMSEWM	SSIMWM	HFENWM
$QSM\left(C_{f\_gt}\right)$	34.42	0.972	31.89	30.92	0.982	43.45
$QSM\left(C_{f\_fit}\right)$	42.27	0.937	39.12	38.31	0.974	52.57
$QSM\left(f_{NL}\right)$	44.50	0.937	39.18	42.08	0.960	54.42
$QSM\left(f_{{\mathit{\text{wavg}}}_{1-10}}\right)$	50.62	0.889	45.41	51.84	0.935	66.38
$QSM\left(f_{{\mathit{\text{wavg}}}_{1-15}}\right)$	47.06	0.907	42.64	45.79	0.942	59.82
$QSM\left(f_{{\mathit{\text{wavg}}}_{6-15}}\right)$	44.71	0.941	40.71	41.51	0.947	55.36

In vivo results

Figure 5(A) shows three representative slices with fitted $\Delta f$ in vivo and the corresponding fiber ODF peak1 for comparison. $\Delta f$ shows large contrast differences in different oriented fibers, e.g., higher $\Delta f$ in the red and green encoded fibers than that in the blue encoded fibers. Fig. 5(B) compares the fitted $C_f$ and $f_{{\mathit{\text{wavg}}}_{1-15}}$ as well as their QSM reconstructions. Fig. 5(C) compares the $C_f$ and $f_{\mathit{\text{echo}}3}$ estimated at the third echo (TE = 7.2 ms). Differences to the fitted $C_f$ show that $f_{\mathit{\text{echo}}3}$ likely contains larger microstructure-induced frequency residuals (>2 Hz) than $f_{{\mathit{\text{wavg}}}_{1-15}}$ (~1 Hz), which also leads to larger differences in QSM. $f_{NL}$ provides some biased estimation of tissue frequency and QSM in some brain regions (red arrows in Fig. 5D).

Figure 5.

In vivo results of participant #1. (A) Three representative slices of fitted $\Delta f$ map and the corresponding colormap of the first peak of the fiber orientation distribution function (ODF). (B) Representative slice of fitted $C_f$ and tissue frequency estimated using weighted echo averaging with 1^st-15^th echoes $f_{{\mathit{\text{wavg}}}_{1-15}}$ and their corresponding difference map. The corresponding iLSQR reconstructed QSM from $C_f$ and $f_{{\mathit{\text{wavg}}}_{1-15}}$ and their difference map are shown on the right. (C) Representative slice of fitted $C_f$, tissue frequency acquired at third echo (TE = 7.2 ms) $f_{\mathit{\text{echo}}3}$, their difference map as well as the corresponding iLSQR QSMs and difference. (D) Representative slice of fitted $C_f$, tissue frequency estimated using nonlinear complex data fitting $f_{NL}$, their difference map as well as the corresponding iLSQR QSMs and difference.

Figure 6 shows the effects of navigator-based physiological noise correction on $\Delta f$, $C_f$ and QSM reconstruction. As indicated by the red arrows in Fig. 6(A, B), respiration-induced B₀ fluctuation can lead to severe artifacts in $\Delta f$ maps and large fitting errors in some voxels in $C_f$ maps. When the navigator-based correction was applied (Fig. 6 (C, D)), the fitted $\Delta f$ map demonstrated reduced artifacts and largely improved GM/WM contrast that better revealed small fiber bundles (cyan arrows in Fig. 6(C)); the differences between the fitted $C_f$ map and other tissue frequency maps also became clearer, for example, $f_{{\mathit{\text{wavg}}}_{1-17}}$ showed slightly positive differences (Fig. 6D). Comparisons between $C_f$ and $f_{\mathit{\text{echo}}3}$, $C_f$ and $f_{NL}$, along with their corresponding QSM reconstruction are illustrated in Fig. S3. Example frequency evolution in vivo in different parallel and crossing fibers can be found in SI S8 and Fig. S4.

Figure 6.

In vivo results of participant #6 before and after navigator correction for B₀ fluctuations. Representative slices of fitted $\Delta f$ before and after corrections are shown in (A, C). The fitted $C_f$ and tissue frequency estimated using weighted echo averaging with 1^st-17^th echoes $f_{{\mathit{\text{wavg}}}_{1-17}}$ and their corresponding difference maps are shown in (B, D, top row). Comparison of the corresponding iLSQR QSMs from $C_f$ and $f_{{\mathit{\text{wavg}}}_{1-17}}$ and their difference maps are shown in the bottom row (B, D, bottom row).

Figure 7(A) displays the orientation dependence of in vivo $\Delta f$ in CC and CG, as well as in SLF and CR (Fig. 7(B)). In all ROIs, $\Delta f$ exhibits a consistent relationship over $\theta$ as $\Delta f=a_1{sin}^2\theta +a_2$. Larger a₁ and smaller a₂ were fitted when excluding crossing fibers in CC and CG (blue lines in Fig. 7A). For example, a₁ of 1.96 Hz and a₂ of 0.34 Hz were fitted in all voxels in CC versus a₁ of 2.40 Hz and a₂ of 0.06 Hz in voxels with only parallel fibers. In comparison, smaller a₁ and larger a₂ were fitted when excluding parallel fibers in SLF and CR (blue lines in Fig. 7B). For example, for SLF, the fitted a₁ and a₂ are 1.76 Hz and 0.57 Hz in all voxels versus 1.60 Hz and 0.68 Hz in voxels with only crossing fibers. The fitted coefficients for each participant can be found in Fig. S5.

Figure 7.

(A) Orientation dependence of in vivo $\mathrm{\Delta }f$ in corpus callosum (CC) and cingulum (CG) with all voxels (red) and voxels with only parallel fibers (blue); (B) $\mathrm{\Delta }f$ in superior longitudinal fasciculus (SLF) and corona radiata (CR) with all voxels (red) and voxels with only crossing fibers (blue). The histograms displayed above the graphs represent the group-averaged distribution of the fiber-to-field angle $\theta$ within the corresponding ROIs. Note that the $\theta$ of a crossing fiber voxel refers to the orientation of the major fiber population in that voxel, i.e., angle between peak1 of the ODF and B₀ field. Thin light lines represent the binned data of each individual participant, while the dots with error bars represent the mean and the SD across all participants at mean $\theta$ values in each angular group (5° bin). The solid lines represent the least squares fit using ${sin}^2\theta$. The voxel number threshold for each binned group was set to 50.

Discussion

For simplicity, the HCFM with two fiber populations used in our simulations assumed almost equal volume fraction for the two sub-voxel fiber populations. However, in real brain WM, the ratio of ODF peak2 to peak1 can be 30%–100% in crossing fibers voxels. We compared the simulated frequency evolution in WM voxels with two 90°-crossed fiber populations but different ODF peak2 to peak1 ratio (SI S10 and Fig. S6) and the results suggested a larger $\Delta f$ at smaller peak2 to peak1 ratio when ${\widehat{H}}_0$ is $\left[\begin{array}{ccc}0& 1& 0\end{array}\right]\prime$. In comparison, when ${\widehat{H}}_0$ is $\left[\begin{array}{ccc}1& 0& 0\end{array}\right]\prime$, $\Delta f$ is relatively insensitive to such ratio of fiber volume fractions.

The simulated $\Delta f$ using our HCFM follows the relationship of $\Delta f=a_1{sin}^2\theta +a_2$. Larger $a_1$ values were observed in parallel fiber bundles (CC and CG) as compared to bundles with more crossing fibers (SLF and CR), agreeing in general with the principle that $a_1$ accounts for the underlying distribution of fiber orientations as in Eq. (10) of Wharton and Bowtell.³² This is also supported by our in vivo data (Fig. 7 and Fig. S5). The study of Wharton and Bowtell³² also suggested that the fiber orientation dispersion effect in the microstructure-related frequency difference can be described with a simple scaling factor. However, as shown in our simulation study, this effect may be more complicated than expected as it also depends on the relative direction of the field with respect to each individual fiber population within the crossing fibers rather than the angle between the fibers alone (see Fig. 1 and Fig. S6). The selection of the two field directions in our simulations, i.e., $\left[\begin{array}{ccc}0& 1& 0\end{array}\right]\prime$ and $\left[\begin{array}{ccc}1& 0& 0\end{array}\right]\prime$ for ${\widehat{H}}_0$, was an attempt to better demonstrate this effect.

The fitted $\Delta f$ values in simulations with full signal modeling and their orientation dependence (Fig. 4 and Fig. S2) were found to be 1–2 Hz smaller than the values predicted by the HCFM (Fig. 2). In vivo $\Delta f$ showed even lower values (1–3 Hz difference, Figs. 5–7). This underestimation could be explained in part by the bias and errors generated during the phase preprocessing steps including ROMEO, MCPC-3D and LBV. Fig. S7 suggests that $\Delta f$ underestimation may originate mostly from the biased estimation of ${\varphi }_0$ during MCPC-3D. MCPC-3D assumes a linear phase evolution (constant frequency shift) and usually estimates ${\varphi }_0$ using phase images at two short TEs. However, since the microstructure-induced nonlinear frequency evolution is largest at short TEs, the estimated ${\varphi }_0$ by MCPC-3D may contain significant errors, which in turn affects the estimated effective frequency evolution and the fitted $\Delta f$ values. To better estimate ${\varphi }_0$, the two TEs chosen should be as short as possible but sufficiently far apart to allow phase evolution between echoes⁶³. Including the initial phase component in the nonlinear signal modeling and fitting may be worth exploring in future studies.

Partial volume effects could be considered as another potential cause for underestimated $\Delta f$ in vivo as compared to our simulation. In our HCFM, parallel fibers regions were determined by voxels where the ODF peak2 was smaller than 30% of peak1 for simplicity, but a 0–30% ODF peak2 may still lead to smaller $\Delta f$ estimation. For example, as shown in our simulations, the frequency may change 5 Hz over TE in voxel with parallel fibers (Fig. 1B) versus only 3 Hz in voxels with 90°-crossed fiber and ODF Peak2/Peak1 = 20% when ${\widehat{H}}_0$ is $\left[\begin{array}{ccc}0& 1& 0\end{array}\right]\prime$ as in Fig. S6(B).

Realistic geometry of myelin sheath and angular dispersion within fiber populations can modulate signal magnitude decay and frequency evolution, thus potentially account for the $\Delta f$ discrepancy observed between our simulation and in vivo data (SI S12 and S13). For example, the WM model with realistic geometry generated a slowly varying signal phase and slightly smaller frequency change over TE compared with HCFM (i.e., 4.7 Hz versus 5.4 Hz for parallel fibers and 90°-crossed fibers; 4.4 Hz versus 4.8 Hz for 45°-crossed fibers when ${\widehat{H}}_0$ = $\left[\begin{array}{ccc}1& 0& 0\end{array}\right]\prime$, Fig. S8). This is consistent with previous findings showing circular models give greater phase shift than models using realistic myelin geometry³³. In addition, the model with more angular dispersion results in a reduction in both signal magnitude decay and frequency change over TE in almost all the cases (Fig. S10).

The sensitivity of frequency evolution to the four parameters used in the TE-dependent frequency fitting was investigated in Fig. S11. The dependence of frequency evolution on $\Delta f$ is strong only at short TE (e.g., 0.78 Hz shift per 1 Hz change of $\Delta f$ at TE = 3 ms, at 7T). Among all 4 parameters, the strongest dependence of frequency evolution curve is on $C_f$ (1 Hz shift at all TEs per 1 Hz change of $C_f$). Regarding $p_1$ and $p_2$, the frequency evolutions do not show large changes when $p_1$ is larger than 300 s⁻¹, or when $p_2$ is between 0.1 and 0.8. These results may partly explain the unreliable and noisy $p_1$ and $p_2$ fittings in WM. In addition, when considering GM and CSF regions with small or negligible $\Delta f$ values, the fitting of $p_1$ and $p_2$ would be unreliable as expected. This also applies to fibers parallel to the magnetic field, e.g., CST, as shown in the $p_{1\_gt}$ and $p_{2\_gt}$ maps (Fig. 4). Potentially improved fitting accuracy of $\Delta f$ could be achieved with more data sampled at shorter TE, e.g., with TE < 10 ms at 7T. It should be noted that $p_1$ and $p_2$ contain the signal intensity and signal decay information of the sub-voxel compartments. Since the TE-dependent frequency fitting is solely based on signal phase, it is not surprising that the fitted $p_1$ and $p_2$ parameters are noisy. A better estimation of $p_1$ and $p_2$ parameters may be achieved by utilizing the complex multi-echo GRE signal with compartmental volume fractions, T₂*, and frequency shifts (i.e., similar to the myelin water imaging optimization problem). Furthermore, the field perturbation and signal evolution calculated using the 3D Fourier method and the analytic method^25,32 were compared in SI S15. Both methods give similar field perturbation patterns with some small deviations. The analytic method gives a slower decay of signal magnitude and smaller phase accumulation (Fig. S12). However, both methods show comparable frequency change over TE, resulting in similar $\Delta f$ but different $C_f$.

Since the microstructure-induced frequency shift is time-dependent and its contribution is larger at shorter TE, conventional echo combination methods used for QSM, such as the weighted echo averaging would include less microstructure-related contribution when using later echoes only, e.g., ${QSM(f}_{\mathit{\text{wavg}}6-15})$ versus ${QSM(f}_{wag1-10})$.

For in vivo data, original TE-dependent frequency fitting yielded large $C_f$ fitting errors (see SI S16 and Fig. S13) in regions with potential phase artifacts, e.g., due to physiological noise or phase preprocessing errors. Tikhonov regularization was thus used to suppress such errors leading to Eq. (3). As shown in our study, navigator-based B₀ fluctuation correction is a promising approach⁶⁴ to suppress such errors and help improve the fitted $\Delta f,C_f$ and the QSM images. Future developments toward more sophisticated processing and fitting strategies for more robust extraction of the WM microstructure- and bulk susceptibility-induced frequencies are warranted.

In the derivation of Eq. (2), we ignored the effects of diffusion (assuming static dephasing), chemical exchange, and the possible larger axonal T₂* compared extracellular T₂*,^29,30,65 which is a limitation of our current model. Fig. S14 and Table S2 demonstrate that a difference between axonal T₂* and extracellular T₂* results in slightly larger $\Delta f$ and smaller $C_f$ values (more negative). In addition, the relationships between susceptibility sources and Larmor frequency shift may contain different mesoscopic contributions depending on the magnetic microstructure.^66,67 This differs from the classical Lorentz sphere correction used in the dipole kernel as in QSM and STI, and may need further investigations.^2,68 Improved models of the susceptibility to frequency shift relationship incorporating these contributions along with the consideration of orientation dispersion will help obtain more accurate susceptibility estimation and better characterize the WM microstructure.

Conclusion

In this work, a HCFM with two fiber populations was proposed to better characterize the microstructure-induced frequency shift in regions with parallel and crossing fibers. While tissue frequency and QSM maps calculated using different TEs or different echo combination methods may be affected differently by the microstructure-induced frequency shift, TE-dependent frequency fitting enables the separation of the microstructure-induced frequency (time-dependent component with amplitude of $\Delta f$) from bulk susceptibility-induced frequency (time-independent component $C_f$) and may help generate more accurate QSM images.

Supplementary Material

Supinfo

Figure S1. Rotations around z and x axis are applied to transform any radial tensors into the common frame of reference aligned with fiber population 1 (fiber_1 frame), where $\varphi$ is the azimuthal angle of the radial tensor along the fiber and $\alpha$ is the zenith angle of the fiber bundle in the y-z plane.

Figure S2. Orientation dependence of fitted $\mathrm{\Delta }f$ (${\Delta f}_{\_fit})$ from the simulated effective frequency evolution when considering both HCFM and bulk susceptibility sources as in Fig. 3(A) (main text). Results are shown in parallel fibers dominated ROIs in the CC and CG (A), and crossing fibers dominated ROIs in the SLF and CR (B). Note that the fiber-to-field angle $\theta$ of a crossing fiber voxel refers to the orientation of the major fiber population in that voxel, i.e., peak1 of the ODF. The dots with error bars correspond to mean and standard deviation of $\Delta f$ measures at mean $\theta$ values in each angular group (5° bin). The solid lines represent the least squares fit using ${sin}^2\theta$. Red dots/lines represent data using all voxels in the selected ROIs, while blue dots/lines represent data using only voxels containing the dominant fiber type, e.g., parallel fibers in the parallel fiber dominated ROIs (by excluding crossing fibers).

Figure S3. The fitted C_f and tissue frequency estimated at the third echo (TE = 5.7 ms) f_echo33 before and after navigator correction and the corresponding iLSQR QSMs are illustrated in (A, B). The fitted C_f and tissue frequency estimated using nonlinear complex data fitting f_NL as well as their corresponding iLSQR QSMs before and after navigator correction are shown in (C, D).

Figure S4. Example in vivo frequency evolution. Plots are local tissue frequency as a function of TE at 7T averaged in selected voxels for ${\widehat{H}}_0={\left[100\right]}^{\prime }$ and ${\widehat{H}}_0={\left[010\right]}^{\prime }$, respectively.

Figure S5. The fitted coefficients of the $\Delta f$ orientation dependence, using least squares fit of $\Delta f=a_1{sin}^2\theta +a_2$, for each participant in corpus callosum (CC) and cingulum (CG) using all voxels or using voxels with only parallel fibers, as well as in superior longitudinal fasciculus (SLF) and corona radiata (CR) using all voxels or using voxels with only crossing fibers.

Figure S6. Plots of simulated frequency perturbations in HCFMs with two-fiber populations crossing at 90° with different cubic signal sampling masks (squares of different colors) and the corresponding voxel averaged frequency shifts as a function of TE (lines of different colors) for ${\widehat{H}}_0={\left[100\right]}^{\prime }$ (A) and ${\widehat{H}}_0={\left[010\right]}^{\prime }$ (B), respectively.

Figure S7. Visualization of frequency evolution when using different signal modeling and phase preprocessing methods and their effects on the fitted frequency difference parameter $\Delta f$. Two example voxels with different levels of estimation errors in $\Delta f$ are shown in A (within parallel fibers) and B (within fibers crossing at $50°$), respectively. Red circles represent sampled effective frequency and blue curves show TE-dependent frequency fitting curves. From left to right: gray panels are using simulated signal considering only microstructure-induced frequency shift and infinite SNR (HCFM with two fiber populations), with perfect fit of $\Delta f$ parameter; yellow panels are using signal modeling as in Eq. (1) but with only microstructure-induced frequency component and with added noise giving SNR = 100; blue panels are with added bulk magnetic susceptibility-induced frequency shift, ROMEO and LBV for phase unwrapping and background field removal; green panels are with added phase offset ${\varphi }_0$ and MCPC-3D for phase offset removal.

Figure S8. Simulations using realistic axon geometry model with two fiber populations. (A, D) The simulated frequency perturbations in the x-y plan for parallel fibers (left) and two fiber populations crossing at 45° and two fiber populations crossing at 90° when ${\widehat{H}}_0={\left[100\right]}^{\prime }$ (A) and ${\widehat{H}}_0={\left[010\right]}^{\prime }$ (D). (B, E) Corresponding histograms of frequency perturbations computed within the blue square. Frequency distributions from axonal, myelin and extracellular water compartments are shown in red, blue and black curves, respectively when ${\widehat{H}}_0={\left[100\right]}^{\prime }$ (B) and ${\widehat{H}}_0={\left[010\right]}^{\prime }$ (E). (C, F) Plots of simulated voxel averaged signal magnitude (left), phase (middle) and frequency shift (right) as a function of TE at 7T in three different fiber configurations for ${\widehat{H}}_0={\left[100\right]}^{\prime }$ (C) and ${\widehat{H}}_0={\left[010\right]}^{\prime }$ (F).

Figure S9. Simulations using hollow cylinder fiber model with two fiber populations and identical configurations as described in Figure 1. (A, D) The simulated frequency perturbations in the x-y plan for parallel fibers (left) and two fiber populations crossing at 45° and two fiber populations crossing at 90° when ${\widehat{H}}_0={\left[100\right]}^{\prime }$ (A) and ${\widehat{H}}_0={\left[010\right]}^{\prime }$ (D). (B, E) Corresponding histograms of frequency perturbations computed within the blue square. Frequency distributions from axonal, myelin and extracellular water compartments are shown in red, blue and black curves, respectively when ${\widehat{H}}_0={\left[100\right]}^{\prime }$ (B) and ${\widehat{H}}_0={\left[010\right]}^{\prime }$ (E). (C, F) Plots of simulated voxel averaged signal magnitude (left), phase (middle) and frequency shift (right) as a function of TE at 7T in three different fiber configurations for ${\widehat{H}}_0={\left[100\right]}^{\prime }$ (C) and ${\widehat{H}}_0={\left[010\right]}^{\prime }$ (F).

Figure S10. Comparisons between models of fiber populations without and with angular dispersion in three different fiber configurations. The simulated frequency perturbations in the x-y plane (top row in each sub-panel) and plots of simulated voxel averaged signal magnitude and frequency as a function of TE (bottom row of each sub-panel) at 7T are shown when ${\widehat{H}}_0={\left[100\right]}^{\prime }$ (left half) and ${\widehat{H}}_0={\left[010\right]}^{\prime }$ (right half) for parallel fibers (A), two-fiber populations crossing at 45° (B) and at 90° (C).

Figure S11. Illustration of the dependence of the frequency evolution on different curve shape parameters as in Eq. (1). Black circles represent the original frequency evolution with $\Delta f$ = 5.76 Hz, $p_1$= 174.65 s⁻¹, $p_2$ = 0.59, $C_f$ = −6.84 Hz. The solid curves in each panel are obtained by: (A) varying $\Delta f$ from 1 Hz to 9 Hz; (B) varying $p_1$ from 100 s⁻¹ to 900 s⁻¹; (C) varying $p_2$ from 0.1 to 0.9; (D) varying $C_f$ from −8 Hz to 0 Hz while keeping the other three parameters unchanged. Black arrow indicates curve change direction corresponding to the defined parameter changes (blue line to green line).

Figure S12. (A) Field perturbation of four axons calculated using the 3D Fourier method, analytic method and their difference. The magnetic field is set as perpendicular to the axons. (B) The simulated frequency perturbation and the corresponding histograms computed using similar HCFM model with parallel fibers as in Fig. 1D but using the analytic method. Frequency distributions from axonal, myelin and extracellular water compartments are shown in red, blue and black curves, respectively with ${\widehat{H}}_0={\left[010\right]}^{\prime }$. (C) Plots of simulated voxel averaged signal magnitude (left), phase (middle) and frequency shift (right) as a function of TE.

Figure S13. Illustration of the regularization effects on TE-dependent frequency fitting using in vivo data. (A, B) Representative slices of fitted $C_f$ map using regularization parameter $\lambda =0$ and $\lambda =0.005$, respectively. Small regularization parameters yielded poor fitting in some voxels as highlighted by the yellow arrows. (C, D) TE-dependent frequency fitting in point 1 and point 2 (marked in A) with regularization parameter $\lambda =0$, $\lambda =0.005$, $\lambda =0.01$ and $\lambda =0.02$. Red circles represent effective frequency at different TEs, and blue lines represent fitted nonlinear curves.

Figure S14. Plots of simulated voxel averaged signal magnitude (left), phase (middle) and frequency shift (right) as a function of TE at 7T using the HCFM with two fiber populations and different T₂* values for the axon and extracellular compartments, when ${\widehat{H}}_0={\left[100\right]}^{\prime }$ (A) and ${\widehat{H}}_0={\left[010\right]}^{\prime }$ (B).

Figure S15. Representative slices of fitted $\Delta f$ (left half) and $C_f$ (right half) using different echo numbers with the same TE₁=2 ms and different echo spacing, and the corresponding difference maps and quality matrices as compared to the ground truth ${\Delta f}_{\_gt}$ and $C_{f\_gt}$ using the numerical head phantom.

Figure S16. Box plots of susceptibility values in selected ROIs for the iLSQR QSM using fitted $C_f$, tissue frequency estimated using weighted echo averaging with all echoes $f_{\mathit{\text{wavg}}}$, at the third echo $f_{\mathit{\text{echo}}3}$ and using nonlinear complex data fitting $f_{NL}$. The fitted $f_{NL}$ maps in two participants giving unresolved aliasing artifacts were excluded. CN: caudate nucleus; GP: globus pallidus; CC: corpus callosum; CG: cingulum; SLF: superior longitudinal fasciculus; CR: corona radiata, GM: grey matter, WM: white matter. GM/WM includes all GM/WM regions defined in the brain atlas. * denotes p < 0.05.

Table S1. Isotropic susceptibility ${\chi }_{\mathit{\text{macro}}}^I$, intensity-normalized magnitude (S₀) and R₁ relaxation in regions of interest (ROI)

Table S2 Comparisons of $\Delta f$ and $C_f$ values from simulated frequency evolution using HCFM with two fiber populations when the axonal T₂* and extracellular T₂* were set to be different versus when using the same T₂* value for the two compartments. Simulations were done for ${\widehat{H}}_0={\left[100\right]}^{\prime }$ and ${\widehat{H}}_0={\left[010\right]}^{\prime }$