Chemical shift-based water/fat separation in the presence of susceptibility-induced fat resonance shift

Abstract

Chemical shift-based water/fat separation methods have been emerging due to the growing clinical need for fat quantification in different body organs. Accurate quantification of proton-density fat fraction requires the assessment of many confounding factors, including the need of modeling the presence of multiple peaks in the fat spectrum. Most recent quantitative chemical shift-based water/fat separation approaches rely on a multi-peak fat spectrum with pre-calibrated peak locations and pre-calibrated or self-calibrated peak relative amplitudes. However, water/fat susceptibility differences can induce fat spectrum resonance shifts depending on the shape and orientation of the fatty inclusions. The effect is of particular interest in the skeletal muscle due to the anisotropic arrangement of extracellular lipids. In the present work, the effect of susceptibility-induced fat resonance shift on the fat fraction is characterized in a conventional complex-based chemical shift-based water/fat separation approach that does not model the susceptibility-induced fat resonance shift. A novel algorithm is then proposed in order to quantify the resonance shift in a complex-based chemical shift-based water/fat separation approach that considers the fat resonance shift in the signal model, aiming to extract information about the orientation/geometry of lipids. The technique is validated in a phantom and preliminary in vivo results are shown in the calf musculature of healthy and diabetic subjects.

INTRODUCTION

The applications of quantitative chemical shift-based water/fat separation methods have been recently expanding due to the increasing need for robust and accurate fat quantification in different body parts (1–3). In the case of skeletal muscle, the quantification of fatty infiltration is important in assessing risk factors and monitoring therapy for metabolic abnormalities (like obesity and diabetes) (4,5), in monitoring the progression of myopathies (6) and in grading muscle degeneration after injuries (7). Muscular fat includes intramyocellular lipids (IMCLs) that can be approximated as spherical droplets within the myocytes and extramyocellular lipids (EMCLs) that can be approximated as strands of fat in the annular interstitial space between the myocytes. Magnetic resonance spectroscopy measurements have shown that the chemical shift difference between the water peak and the EMCL compartment spectrum changes with the orientation of surrounding myocytes due to bulk magnetic susceptibility effects, whereas the chemical shift difference between the water peak and the IMCL compartment spectrum remains unchanged (8–10).

The work of multiple research groups on the quantification of fat fraction using chemical shift-based water/fat separation (11–13) in different body parts has shown that an accurate proton density fat fraction quantification requires the consideration of multiple confounding factors including: fieldmap variations (14,15), the complexity of fat spectrum (16,17), the effect of T₂* decay (16,18,19), the T₁-induced bias (16,20,21), the noise-induced bias (21), and the effect of eddy currents (22,23).

The consideration of the complexity of fat spectrum has been primarily accomplished by using a multi-peak fat spectrum with pre-calibrated or self-calibrated peak locations and peak relative amplitudes. There are three main approaches proposed in the literature to characterize the complexity of the fat spectrum. First, magnetic resonance spectroscopy measurements have been used to pre-calibrate both the peak locations and relative amplitudes (16,17). Second, multi-echo gradient-echo sequences with a high or low number of echoes have been used to pre-calibrate or self-calibrate the peak relative amplitudes, assuming that the peak locations are known and unchanged based on spectroscopy, and modeling the peak amplitudes as independent variables (17) or variables satisfying certain constraints related to the lipid chemical characterization (24). Third, multi-echo gradient-echo sequences with a high number of echoes have been used to pre-calibrate both the peak locations and relative amplitudes (25).

In most of the recent implementations of chemical shift-based water/fat separation techniques to quantify fat in skeletal muscle (6,20,26–28), a constant pre-calibrated multi-peak fat spectrum model has been adopted (16,17). In general, the contribution of IMCL to the total muscular fat is low and as a result of this, chemical shift-based water/fat separation approaches investigate primarily the EMCL compartment. However, the EMCL compartment fat spectrum shifts due to susceptibility effects and this susceptibility-induced fat resonance shift varies spatially due to the different orientation of muscle fibers with respect to the main magnetic field among different muscles. Therefore, the effect of the susceptibility-induced fat resonance shift on the fat quantification should be considered when a spatially constant pre-calibrated multi-peak fat spectrum model is used in solving the water/fat separation problem. Moreover, the quantification of this fat resonance shift could provide information about the microscopic characteristics of the measured extramyocellular lipids (orientation, shape), which would be complementary to the information about the amount of lipids derived from the fat fraction measures.

Therefore, the first purpose of the present work is to characterize the effect of susceptibility-induced fat resonance shift on quantitative chemical shift-based water/fat separation approaches using a spatially constant pre-calibrated fat spectrum with constant shift of the fat spectrum with respect to water, especially as the number of acquired echoes increases. The second purpose of the present work is to develop an algorithm for the simultaneous quantification of the fat fraction and the fat resonance shift in chemical shift-based water/fat separation approaches using a pre-calibrated fat spectrum with variable shift of the fat spectrum with respect to water, aiming to provide information about the microscopic characteristics (orientation/geometry) of the measured extramyocellular lipids.

THEORY

If ${\mathrm{f}}_{\mathrm{p}}^{\prime }$ are the relative real-valued amplitudes of the P′ peaks of the fat spectrum at frequency locaitons $\Delta {\mathrm{f}}_{\mathrm{p}}^{\prime }$ with respect to water in the absence of susceptibility induced fat resonace shift, and x is the susceptibility-induced resonance shift of all fat peaks, the signal in a voxel containing water (with amplitude M_w and phase ϕ) and fat (with amplitude M_f and phase ϕ) at echo time t becomes:

$$\mathrm{S}\left(\mathrm{t}\right)=[{\mathrm{M}}_{\mathrm{w}}+{\mathrm{M}}_{\mathrm{f}}\sum _{\mathrm{p}=1}^{\mathrm{P}\prime }{\mathrm{a}}_{\mathrm{p}}^{\prime }\exp \left(\mathrm{j}2\pi (\Delta {\mathrm{f}}_{\mathrm{p}}^{\prime }+\mathrm{x})\mathrm{t}\right)]\exp \left(\mathrm{j}\varphi \right)\exp \left(\mathrm{j}2\pi {\mathrm{f}}_{\mathrm{B}}\mathrm{t}\right)\exp (-\mathrm{t}∕{\mathrm{T}}_2^{\ast })$$ [1]

where the T₂* values of water and all fat peaks are assumed to be equal. Eq. [1] takes into account the physical constraint that the water and all the fat peaks should have the same phase ϕ at zero echo time (t=0) (29).

The fat spectrum is pre-calibrated using the multi-species iterative decomposition of water and fat with echo asymmetry and least-squares estimation (IDEAL) algorithm (13,15) in a fat-only region by determining P complex-valued relative peak amplitudes a_p at frequency locaitons $\Delta {\mathrm{f}}_{\mathrm{p}}$. A formulation using complex-valued relative peak amplitudes instead of the physically consistent real-valued relative peak amplitudes is adopted. Although a complex amplitude formulation relaxes one of the constraints from the known physically consistent MR signal model, it is implemented since its linearity facilitates the numerical solution of the pre-calibration problem using the multi-species IDEAL algorithm. The resulting non-zero phase of the smaller fat peaks relative to the main fat peak (methylene peak-CH₂) should be attributed to the fact that the employed pre-calibrated P-peak fat spectrum model constitutes an approximation of the real more complicated P′-peak fat spectrum. Specifically, some of the peaks in the employed P-peak pre-calibrated fat spectrum model are the result of a superimposition of multiple peaks overlapping within a narrow frequency range of the real more complicated P′-peak fat spectrum.

Based on basic magnetostatics, the susceptibility-induced resonance shift for cylindrical inclusions of infinite length with susceptibility difference Δχ from the surrounding medium, forming an angle θ with the main magnetic field is given by the equation:

$$\mathrm{x}=\frac{\Delta \chi }{6}(1-3{\cos }^2\left(\theta \right))\frac{\gamma }{2\pi }{\mathrm{B}}_0$$ [2]

For fat inclusions in skeletal muscle Δχ = 0.61 ppm (8) and thus x varies between −0.2 ppm (θ=0°) and +0.1 ppm (θ=90°). Therefore, x is expected to vary between −26 Hz and +13 Hz in skeletal muscle at 3 T.

Two different approaches for fat quantification in the presence of susceptibility-induced fat resonance shift are discussed below.

Method A: approach not modeling the susceptibility-induced fat resonance shift

Let's assume that instead of taking explicitly into account the susceptibility-induced fat resonance shift term in the signal model, the linearly varying with time susceptibility-induced fat resonance shift phase term exp(j2πxt)is approximated by a constant phase term. The water phase ${\widehat{\varphi }}_{\mathrm{w}}$ should then be different than the fat phase ${\widehat{\varphi }}_{\mathrm{f}}$ and the previously proposed multi-peak single T₂* model (17) is derived:

$$\mathrm{S}\left(\mathrm{t}\right)=\left[{\widehat{M}}_{\mathrm{w}}\exp \left(\mathrm{j}{\widehat{\varphi }}_{\mathrm{w}}\right)+{\widehat{M}}_{\mathrm{f}}\exp \left(\mathrm{j}{\widehat{\varphi }}_{\mathrm{f}}\right)\sum _{\mathrm{p}=1}^{\mathrm{P}}{\mathrm{a}}_{\mathrm{p}}\exp \left(\mathrm{j}2\pi \Delta {\mathrm{f}}_{\mathrm{p}}\mathrm{t}\right)\right]\exp \left(\mathrm{j}2\pi {\widehat{f}}_{\mathrm{B}}\mathrm{t}\right)\exp (-\mathrm{t}∕{\widehat{T}}_2^{\ast })$$ [3]

where the ^ notation denotes estimates.

The problem has then three complex unknowns: ${\widehat{\rho }}_{\mathrm{w}}={\widehat{M}}_{\mathrm{w}}\exp \left(\mathrm{j}{\widehat{\varphi }}_{\mathrm{w}}\right)$, ${\widehat{\rho }}_{\mathrm{f}}={\widehat{M}}_{\mathrm{f}}\exp \left(\mathrm{j}{\widehat{\varphi }}_{\mathrm{f}}\right)$ and $\stackrel{⌢}{\psi }={\widehat{f}}_{\mathrm{B}}+\mathrm{j}∕\left(2\pi {\widehat{T}}_2^{\ast }\right)$ and can be solved using the IDEAL algorithm (13,15) with single T₂* correction, as proposed in (17,19).

Let's denote with TE_min and TE_max the minimum and maximum echo times of the multi-echo acquisition respectively. If it is assumed that the phase change due to susceptibility induced shift between the acquired echoes is small, namely 2πx (TE_max −TE_min) / 2 << 1, then the phase angle difference between the water and fat signal can be used to compute the apparent phase difference caused by the susceptibility induced resonance shift x (30):

$$2\pi \mathrm{x}{\mathrm{TE}}_{\mathrm{mid}}\approx {\widehat{\varphi }}_{\mathrm{f}}-{\widehat{\varphi }}_{\mathrm{w}}$$ [4]

where TE_mid =(TE_max + TE_min) / 2. The Appendix describes an analytical derivation for the above approximation at low and high fat fractions.

Therefore, despite the fact that the effect of the fat resonance shift is not modeled, an approximate estimation of the susceptibility-induced fat resonance shift is feasible by computing the phase angle difference between the water and fat signal in an approach that does not explicitly consider the fat resonance shift in the signal model.

Method B: approach modeling the susceptibility-induced fat resonance shift

The water/fat phase term exp(jϕ) in Eq. [1] is a non-linear term. The water and fat peak signals are frequently assumed to have independent phases adopting a complex water/fat signal linear formulation (13,14). Although assuming independent phases for the water and fat peaks relaxes one of the constraints imposed by the physically consistent MR signal model, it facilitates the numerical solution of the problem by reducing the dimension of the non-linear parameter space (13,15).

Therefore, if the signal model takes explicitly into account the susceptibility induced fat resonance shift and assumes that the water and fat signal have different phase, the real and imaginary parts of the signal Eq. [1] can be written as:

$$\begin{array}{cc}\hfill & {\mathrm{S}}_{\mathrm{r}}\left(\mathrm{t}\right)=\left[{\rho }_{\mathrm{w},\mathrm{r}}\cos \left(2\pi {\mathrm{f}}_{\mathrm{B}}\mathrm{t}\right)-{\rho }_{\mathrm{w},\mathrm{i}}\sin \left(2\pi {\mathrm{f}}_{\mathrm{B}}\mathrm{t}\right)+{\rho }_{\mathrm{f},\mathrm{r}}\sum _{\mathrm{p}=1}^{\mathrm{P}}\mid {\mathrm{a}}_{\mathrm{p}}\mid \cos [{\epsilon }_{\mathrm{p}}+2\pi ({\mathrm{f}}_{\mathrm{B}}+\Delta {\mathrm{f}}_{\mathrm{p}}+\mathrm{x})\mathrm{t}]-{\rho }_{\mathrm{f},\mathrm{i}}\sum _{\mathrm{p}=1}^{\mathrm{P}}\mid {\mathrm{a}}_{\mathrm{p}}\mid \sin [{\epsilon }_{\mathrm{p}}+2\pi ({\mathrm{f}}_{\mathrm{B}}+\Delta {\mathrm{f}}_{\mathrm{p}}+\mathrm{x})\mathrm{t}]\right]\exp (-\mathrm{t}∕{\mathrm{T}}_2^{\ast })\hfill \\ \hfill & {\mathrm{S}}_{\mathrm{i}}\left(\mathrm{t}\right)=\left[{\rho }_{\mathrm{w},\mathrm{r}}\sin \left(2\pi {\mathrm{f}}_{\mathrm{B}}\mathrm{t}\right)-{\rho }_{\mathrm{w},\mathrm{i}}\cos \left(2\pi {\mathrm{f}}_{\mathrm{B}}\mathrm{t}\right)+{\rho }_{\mathrm{f},\mathrm{r}}\sum _{\mathrm{p}=1}^{\mathrm{P}}\mid {\mathrm{a}}_{\mathrm{p}}\mid \sin [{\epsilon }_{\mathrm{p}}+2\pi ({\mathrm{f}}_{\mathrm{B}}+\Delta {\mathrm{f}}_{\mathrm{p}}+\mathrm{x})\mathrm{t}]-{\rho }_{\mathrm{f},\mathrm{i}}\sum _{\mathrm{p}=1}^{\mathrm{P}}\mid {\mathrm{a}}_{\mathrm{p}}\mid \cos [{\epsilon }_{\mathrm{p}}+2\pi ({\mathrm{f}}_{\mathrm{B}}+\Delta {\mathrm{f}}_{\mathrm{p}}+\mathrm{x})\mathrm{t}]\right]\exp (-\mathrm{t}∕{\mathrm{T}}_2^{\ast })\hfill \end{array}$$ [5]

where the complex-valued relative peak amplitudes are decomposed as ${\mathrm{a}}_{\mathrm{p}}=\mid {\mathrm{a}}_{\mathrm{p}}\mid \exp \left(\mathrm{j}{\epsilon }_{\mathrm{p}}\right)$.

Because of the low values of x (betwen −0.2 and 0.1 ppm) relative to the water-methylene fat peak chemical shift difference, the signal is affected by the presence of the resonance shift primarily at longer echo times (TEs). Therefore, the problem of Eq. [1] can be solved in two steps. The first step assumes x=0 and solves for complex water signal (ρ_w), comlex fat signal (ρ_f), and complex fieldmap (ψ) using the multi-peak T₂* IDEAL algorithm (17,19), considering the signal only at short TEs (in the present implementation the first six TEs, as reconstructions of six-echo data have been previously frequently used in T₂*-corrected water/fat separation). The second step solves the full problem of Eq. [1] using the real and imaginary decomposition of the signal as it is shown in Eq. [5]. The problem of Eq. [5] has seven real unknowns: the real and imaginary part of the complex water signal (ρ_w,r and ρ_w,i), the real and imaginary part of the complex fat signal (ρ_f,r and ρ_f,i), the fieldmap f_B, the T₂* relaxation time value and the fat resonance shift x. The second step uses as initial estimates the results of the first step and x=0, and solves the underlying non-linear problem based on a Gauss-Newton method, similarly to the way it has been previously proposed in solving the water/fat separation problem assuming different T₂* values for water and fat (18). The second step uses the signal values at all the acquired TEs.

MATERIALS AND METHODS

Numerical simulations

Numerical simulations were performed to examine the performance of the two above approaches in the estimation of fat fraction and susceptibility-induced fat resonance shift. The simulation parameters included TE_min=1.8 ms, ΔTE=0.9 ms (representative values for 3 T experiments), T₂*=25 ms, ψ=0 Hz and ϕ_w = ϕ_f = 0°. The multi-peak fat spectrum measured in the soybean oil region of the phantom (see below) was used and the chemical shift difference between the water and the main fat peak in the absence of susceptibility-induced shifts was considered equal to the chemical shift difference between the water and the main fat peak in the fat emulsion region of the phantom (see below). The effect of a susceptibility induced fat resonance shift x=–26 Hz was then considered in simulations with different number of TEs: N=6 and N=16. Gaussian noise was added in quadrature to the simulated data assuming SNR=100 for the first echo at zero fat fraction. 10000 repetitions were generated for fat fractions varying between 0.1% and 99.9%. The data was then analyzed using both the approach not modeling the fat resonance shift and the approach modeling the fat resonance shift. The mean of all estimated parameters (including the fat fraction and the susceptibility-induced fat resonance shift) were computed after extreme outlier removal. A value is considered to be an extreme outlier if its distance to the interquartile interval (i.e., interval between the 25th and 75th percentiles) exceeds three times the length of the interquartile interval.

Phantom experiment

Phantom construction

A water-fat phantom was constructed containing 3 samples. Sample 1 (small cylindrical vial in the left of Fig. 1) contained soybean oil. Sample 2 (large cylindrical vial in the center of Fig. 1) included a small cylindrical vial of soybean oil immersed in a large cylindrical water bath to simulate a water/fat region where the effect of susceptibility-induced fat resonance shift is important. Sample 3 (small cylindrical vial in the right of Fig. 1) contained 20% Intralipid fat emulsion (Baxter Healthcare Corporation, Deerfield, IL) to simulate a water/fat region where the effect of susceptibility-induced fat resonance shift is zero. Intralipid is an intravenous fat emulsion of fine droplets (no larger than 0.5 μm in diameter) of soybean oil and has been also previously assumed to be composed of spherical lipid droplets (10).

Figure 1.

In phase image of susceptibility effect phantom, showing soybean oil-water region (ROI A), Intralipid region (ROI B), and soybean oil-only region (ROI C).

MRI/MRS measurements

A quadrature transmit-receive knee coil was used to scan the phantom on a 3 T Signa HDx system (GE Healthcare, Waukesha, WI). The phantom was rotated inside the coil to achieve seven different orientations with respect to B_o (quantified by the angle θ between the phantom axis and B_o) and low resolution gradient echo images were acquired to quantify the angle θ. The phantom was scanned at every orientation with a multi-echo gradient-echo sequence and a single-voxel spectroscopy sequence.

An investigational version of 16-point IDEAL in a multi-shot 3D SPGR sequence with monopolar readout gradients was used with parameters: 4 interleaved TRs each with ETL=4, TR/TE/ΔTE=18.4/1.96/0.88 ms, flip angle= 2°, matrix 150×150, bandwidth= 31.25 kHz, N_ex=2, FOV=13 cm, phase FOV=0.7, 4 mm slice thickness, 20 slice locations. The phantom was scanned with the multi-echo gradient-echo sequence in an axial slice orientation at the seven different orientations with respect to B_o, in order to study the orientation dependence of the susceptibility-induced fat resonance shift. The phantom was also scanned in an oblique-sagittal slice orientation at θ=0, in order to form an ROI in the water/oil interface that could be used for comparing the results of the approach not modeling the shift and the approach modeling the shift at different fat fractions.

Single voxel magnetic resonance spectroscopy (MRS) measurements using a position resolved spectroscopy (PRESS) sequence with TE=28ms, TR=2000ms, no regional saturation bands, and 40 averages without water suppression were performed to measure peak locations. The PRESS sequence was first scanned with a 12×8×8 mm³ voxel inside the vial with 20% fat Intralipid (ROI B in Fig. 1) and then with a 12×8×8 mm³ voxel inside the soybean oil-only vial in the left sample (ROI C in Fig. 1) at θ=0 for pre-calibrating the fat spectrum. The PRESS sequence was then scanned with a 12×20×20 mm³ voxel surrounding the vial of soybean oil in the center of the phantom (ROI A in Fig. 1) and with a 12×8×8 mm³ voxel inside the vial with 20% fat Intralipid (ROI B in Fig. 1) at all the seven different orientations to measure the orientation-dependent susceptibility-induced fat resonance shift.

Pre-calibration of phantom fat spectra

Based on the acquired spectrum for the soybean oil-only region (ROI C) at θ=0, the frequency locations of the fat peaks relative to the main fat peak (Δf_p − Δf_CH2) were determined. Based on the acquired spectrum for the Intralipid region (ROI B) at θ=0, the frequency location Δf_CH2 of the main fat peak relative to the water peak was determined. Using the frequency location Δf_CH2 of the main fat peak relative to the water peak, the frequency locations Δf_p of all soybean oil peaks in the absence of susceptibility-induced shifts were determined. Given the knowledge of the fat peaks Δf_p, the soybean oil relative peak amplitudes a_p values were derived using the multi-species IDEAL algorithm (13) (with complex amplitudes for numerical convenience) on the 16-point data in the soybean oil-only region (ROI C) at θ=0. The frequency locations of peaks in the Intralipid were taken the same as the frequency locations of peaks in the soybean oil with the additional consideration of the presence of glycerin at 148 Hz from the water. The Intralipid relative peak amplitudes were computed based on the soybean oil relative peak amplitudes and the known content of glycerin in the Intralipid (2.25% by weight).

In vivo experiment

Subjects

The middle calf musculature of three subjects was scanned in accordance with the local Institutional Review Board, in order to investigate the feasibility of measuring the susceptibility-induced fat resonance shift in vivo using the proposed approaches. Subject A (age 32 years, BMI BMI 26.5 kg/m²) was a healthy male volunteer. Subjects B (age 61 years, BMI 32.9 kg/m²) and C (age 62 years, BMI 25.7 kg/m²) were female patients with a history of type 2 diabetes melitus.

MRI/MRS measurements

An eight-channel transmit-receive knee coil was used to scan the middle calf muscle of all subjects using an investigational version of 16-point IDEAL in a multi-shot 3D SPGR sequence with monopolar readout gradients with parameters: 4 interleaved TRs each with ETL=4, TR/TE/ΔTE=17.8/1.49/0.64 ms, flip angle= 3°, matrix 180×180, bandwidth= 62.5 kHz, N_ex=2, FOV=18 cm, phase FOV=0.75, 4 mm slice thickness, 12 slice locations (scan time=5 min 10 s). The same muscle volume was also scanned using an axial fast spin-echo T₁-weighted sequence with parameters: TR/TE=600/9.4 ms, ETL=7, matrix 384×192.

Single voxel magnetic resonance spectroscopy measurements using a position resolved spectroscopy (PRESS) sequence with TE=28ms, TR=2000ms, 12×8×8 mm³ voxel, no regional saturation bands, and 40 averages were also performed in the healthy volunteer for pre-calibrating the fat spectrum and measuring the susceptibility-induced fat resonance shift in specific muscle regions. The PRESS sequence (with water suppression) was first scanned with a voxel in the tibialis anterior muscle not-containing obvious EMCL signal on the T₁-weighted image to measure the frequency Δf_CH2 of the main IMCL peak relative to the water peak. The PRESS sequence (without water suppression) was then scanned with a voxel in the bone marrow region to measure the fat peak locations. The PRESS sequence (with water suppression) was also scanned with a voxel in the medial gastrocnemius (MG) region containing obvious EMCL signal on a T₁-weighted image in order to measure the chemical shift difference between water and the main fat peak in this specific MG region.

Pre-calibration of in vivo fat spectrum

Based on the acquired spectrum for the bone marrow ROI, the frequency locations of the fat peaks relative to the main fat peak (Δf_p - Δf_CH2) were determined. Based on the acquired spectrum in the IMCL region, the frequency location Δf_CH2 of the main fat peak relative to the water peak in the absence of susceptibility-induced shifts was determined. Using the frequency location Δf_CH2 of the main fat peak relative to the water peak, the frequency locations Δf_p of all fat peaks in the absence of susceptibility-induced shifts were determined. Given the knowledge of the fat peaks Δf_p, the relative peak amplitudes a_p values were derived using the 16-point multi-species IDEAL algorithm (13) (with complex amplitudes for numerical convenience) in the bone marrow ROI. The fat spectrum pre-calibration procedure was performed only on the healthy volunteer. The spectrum used in the water/fat separation in the two diabetes patients was the same as the pre-calibrated fat spectrum in the healthy volunteer.

RESULTS

Numerical simulations

Figure 2 shows the mean results for the water/fat amplitude and phase, the fieldmap and T₂* values using N=6 echoes with method A (the approach that does not model the fat resonance shift) in the presence of a fat resonance shift x=−26 Hz, as a function of the fat fraction (varying between 0.1% and 99.9%). The model mismatch induces differences between the estimated parameters and the corresponding nominal values. The deviation of the water/fat amplitudes from the corresponding nominal values is low at fat fractions close to 0 and 100% and becomes significant at moderate fat fractions. The deviation of the water phase from its nominal value increases as the fat fraction increases, whereas the deviation of the fat phase from its nominal value decreases as the fat fraction increases. However, the phase difference between water and fat shows a small variation with fat fraction around the imposed susceptibility-induced fat resonance shift value of −26 Hz. The deviation of the fieldmap from its nominal value increases as the fat fraction increases from 0 to the imposed susceptibility-induced fat resonance shift value of −26 Hz at 100% fat fraction. Finally, the deviation of the estimated T₂* from its corresponding nominal value is low at fat fractions close to 0 and 100% and becomes significant at moderate fat fractions.

Figure 2.

Results of method A (approach not modeling the fat resonance shift) in the presence of a shift x=−26 Hz using N=6 echoes: water/fat amplitude, water/fat phase, fieldmap and T₂* values. Solid lines plot the estimated parameter results and the dashed lines plot the nominal values.

Figure 3 shows a comparison of the fat fraction and fat resonance shift bias results from the two methods under study based on numerical simulations using N=6 and N=16 respectively, as a function of the fat fraction (varying between 5% and 95%). Method B (the approach modeling the fat resonance shift) shows no bias in fat fraction and fat resonance shift irrespective of the number of echoes (Fig. 3), as it is expected. The maximum bias in fat fraction using method A (the approach not modeling the shift) is below 2% using N=6 (Fig. 3), but increases up to 5% as the number of echoes increases to N=16 (Fig. 3). The calculation of the fat resonance shift, based on the water/fat phase difference in method A, shows small bias in the shift for both N=6 (maximum bias of the order of 2 Hz) and N=16 (maximum bias of the order of 1 Hz).

Figure 3.

Comparison of the fat fraction and fat resonance shift bias results from the two approaches under study based on numerical simulations using N=6 and N=16 echoes: The first column shows the fat fraction results and the second column the fat resonance shift results. The first row shows results using N=6 and the second row shows results using N=16. The circles represent the results with method A (approach not modeling the fat resonance shift) and the squares represent the results with method B (approach modeling the fat resonance shift).

Phantom measurements

The fat spectrum pre-calibration in the soybean oil in the absence of susceptibility-induced shifts derived peaks at locations [505, 452, 356, 268, 88, −57] Hz relative to the water peak with relative complex amplitudes [7.8e^j0.11π, 60.7, 12.9e^−j0.11π, 3.7e^−j0.22π, 3.9e^−j0.16π, 11.0 e^−j0.07π]/100 (keeping the phase of the main fat peak at zero). The fat spectrum pre-calibration in the Intralipid derived peaks at locations [505, 452, 356, 268, 148, 88, −57] Hz relative to the water peak with relative complex amplitudes [7.5e^j0.11π, 57.8, 12.3e^−j0.11π, 3.5 e^−j0.23π, 5.1, 3.7 e^−j0.16π, 10.5 e^−j0.07π]/100.

Figure 4 shows the phantom results of the susceptibility-induced fat resonance shift in the region surrounding the soybean oil vial (Fig. 4a) and inside the Intralipid vial (Fig. 4b) at different orientations with respect to the main magnetic field (quantified by the angle θ). Susceptibility-induced fat resonance shift values were derived based on the MRS measurements (by measuring the chemical shift difference between water and the main fat peak) and the multi-echo gradient-echo measurements (by using method B with N=16). The MRS results for the susceptibility-induced fat resonance shift in the region surrounding the soybean oil vial were also fitted to the theoretical expression in Eq. [2] using non-linear least squares, resulting in an estimation of Δχ equal to 1 ppm. The variation of the susceptibility-induced fat resonance shift x with the orientation angle θ using Eq. [2] and Δχ=1 ppm is also plotted in Fig. 4a. There was a good agreement for the susceptibility-induced fat resonance shift using the proposed chemical shift-based water/fat separation approach (method B) with the MRS-derived fat resonance shift in the water/fat region surrounding the soybean oil vial (Fig. 4a). The fat resonance shift values using both MRS and the proposed chemical shift-based water/fat separation approach were close to zero for the ROI inside the Intralipid region (Fig. 4b), where no susceptibility-induced shift is expected (due to the spherical shape of lipids in the Intralipid).

Figure 4.

Phantom rotation results for the susceptibility-induced fat resonance shift using MRS and the proposed chemical shift-based water/fat separation method B (approach modeling the fat resonance shift) with N=16: (a) in the water/fat region surrounding the soybean oil vial (ROI A in Fig. 1), and (b) inside the Intralipid vial (ROI B in Fig. 1).

Figure 5 shows a comparison of the results between the chemical shift-based water/fat separation approach not modeling the fat resonance and the approach modeling the fat resonance in the multi-echo acquisition with N=16 in the water/soybean oil interface of the phantom. Fig. 5a shows an ROI in the water/oil interface in the sample in the center of the phantom when the phantom was scanned with an oblique-sagittal orientation at θ=0. A variable fat fraction was achieved by averaging the complex signal across x at every y location in the water/soybean oil interface (Fig. 5a). Reference fat fraction values were derived using method A (the approach not modeling the fat resonance shift) with the first N=6 echoes, since for N=6 the bias in the fat fraction induced by the fat resonance shift is expected to be small. Figs. 5b and 5c show the fat fraction and fat resonance shift results derived by methods A and B using all the acquired N=16 echoes. Method A induced significant bias in the fat fraction, whereas method B (the approach modeling the fat resonance shift) induced small bias in the fat fraction (compared to the reference fat fraction values using N=6 echoes, as shown in Fig. 5b). Method B resulted in a fat resonance shift estimate of the order of −40 Hz almost independent of the fat fraction (Fig. 5b). This value is close to the susceptibility-induced fat resonance estimate of −42 Hz at θ=0, derived by the MRS measurement (Fig. 4a). Method A by taking the phase difference between water and fat signals resulted in a fat resonance shift dependent on the fat fraction, but close to the shift derived using method B especially at moderate fat fractions (Fig. 5c).

Figure 5.

Phantom results comparing the chemical shift-based water/fat separation method A (approach not modeling the fat resonance shift) with method B (approach modeling the fat resonance shift) in an acquisition with N=16. A variable fat fraction is achieved by averaging the complex signal across x at every y location in an ROI in the water/soybean oil interface, as shown in (a). The results of method A and method B with N=16 are shown for (b) the fat fraction, and (c) the susceptibility-induced fat resonance shift, as a function of the fat fraction derived using method A with N=6.

In vivo measurements

The in vivo fat spectrum pre-calibration in the absence of susceptibility-induced shifts derived peaks at locations [487, 434, 324, 245, 57, −82] Hz relative to the water peak with relative complex amplitudes [5.6e^−j0.14π, 65.9, 12.8e^j0.09π, 3.3e^j0.23π, 6.3e^−j0.15π, 6.2e^j0.10π]/100 (keeping the phase of the main fat peak at zero).

Figure 6 shows the results in one slice of the calf musculature of the healthy volunteer (subject A). Fig. 6a shows the fat fraction map and Fig. 6b shows the fat resonance shift map using method B (the approach modeling the fat resonance shift) with N=16 for voxels with fat fraction between 7% and 95%. Figs. 6c and 6d show the time evolution of the in vivo gradient echo magnitude signal for an ROI in the medial gastrocnemius (MG) muscle and an ROI in the soleus (SOL) muscle respectively (white boxes in Fig. 6b). The fitted magnitude signal curves using both methods A and B are also plotted in Figs. 6c and 6d. A better agreement was observed between the experimental data and method B than between the experimental data and the method A, as it was expected due to the model mismatch using the approach not modeling the shift for both ROIs. However, the disagreement between the two methods is larger in the case of the MG ROI than in the case of the SOL ROI, since the absolute susceptibility-induced fat resonance shift is higher in MG than in SOL due to the lower pennation angle of MG compared to SOL. Specifically, the results for the susceptibility-induced fat resonance shift using method A and method B were −28 Hz and −26 Hz respectively for the MG ROI, close to the value of −25 Hz measured by MR spectroscopy. The results for the susceptibility-induced fat resonance shift using method A and method B were −1 Hz and −6 Hz respectively for the SOL ROI.

Figure 6.

In vivo results in healthy volunteer (subject A): (a) fat fraction map (values in color bar are in %), (b) fat resonance shift map using method B (approach modeling the fat resonance shift) with N=16 (values are shown for voxels with fat fraction between 7% and 95%, values in color bar are in Hz), (c) signal evolution curve for ROI in medial gastrocnemius (MG) muscle including experimental points and fitted curves using both method A and method B, and (d) signal evolution curve for ROI in soleus (SOL) muscle including experimental points and fitted curves using both method A and method B.

Figure 7 shows the results in one slice of the calf musculature in the two patients with type 2 diabetes mellitus (subjects B and C). Figs. 7a and 7c show fat fraction results and Figs. 7b and 7d show fat resonance shift maps using method B (the approach modeling the fat resonance shift) with N=16. The fat fraction maps show more regions with increased fat content in the diabetic patients compared to the healthy volunteer, which is consistent with the expected changes in the distribution of intermuscular adipose tissue in diabetic patients compared to healthy volunteers (4,31). Lower absolute values of the fat resonance shift are also observed in the soleus muscle compared to the medial gastrocnemius muscle in all subjects, a finding consistent with the expected lower pennation angle of the medial gastrocnemius muscle compared to the soleus muscle.

Figure 7.

In vivo results in two patients with type 2 diabetes mellitus (subjects B and C): (a) and (c) fat fraction maps for subjects B and C respectively (values in color bar are in %), (b) and (d) fat resonance shift maps using method B (approach modeling the fat resonance shift) with N=16 for subjects B and C respectively (values are shown for voxels with fat fraction between 7% and 95%, values in color bar are in Hz).

DISCUSSION

The effect of different confounding factors on fat fraction quantification using chemical shift-based water/fat separation techniques has been addressed in previous studies (14–23). The present work focuses on the study of the effect of the susceptibility-induced fat resonance shift on quantitative fat fraction measurements using chemical shift-based water/fat separation, which is of particular interest in skeletal muscle applications. It is shown that susceptibility induced fat resonance shift leads to low fat fraction bias in standard quantitative chemical shift-based water/fat separation acquisitions where a low number of echo time points and short TEs are used. However, the effect is amplified in acquisitions with high number of echo time points and longer TEs. The correction of the bias induced in the fat fraction by not explicitly modeling the effect of the susceptibility-induced fat resonance shift is the first purpose of the present work. An algorithm is developed for solving the problem where the susceptibility induced fat resonance shift is taken into consideration in the signal model. The second purpose of the present work is to investigate the possibility of measuring the shift during the process of water-fat separation, aiming to provide additional information about the microscopic characteristics of the measured lipids (orientation, shape). It is also shown that even using the approach that does not model the fat resonance shift, the shift can be estimated with relatively small bias by taking the phase difference between the water and fat signals.

The presented numerical simulation results study the accuracy for the fat fraction and the susceptibility-induced fat resonance shift estimation using the approach not modeling the fat resonance shift (method A) and the approach modeling the fat resonance shift (method B) in acquisitions with different number of echoes. In acquisitions with low number of echoes (N=6 in the present study), the bias in the fat fraction and the fat resonance shift (by taking the phase difference between water and fat signals) are relatively small using method A. As expected, the bias in both fat fraction and fat resonance shift becomes zero using method B. In acquisitions with high number of points (N=16 in the present study), the bias in the fat fraction is relatively large using method A. Therefore, in acquisitions with low number of points method A would induce small, acceptable, bias in the estimation of the fat fraction. However, in acquisitions with high number of points method B would be necessary for unbiased estimation of the fat fraction. It should be emphasized that the high number of echoes used in the present study (N=16) was selected heuristically. A characterization of the echo time step (ΔTE) and number of echoes (N) required in order to optimize noise performance for the estimation of the susceptibility-induced fat resonance shift would depend on the value of the measured shift and would require further investigation using numerical simulations.

A comparison of the noise performance of the two proposed methods was not presented, as a consistent comparison of the noise performance of the two proposed methods would require constraining the phases of water and fat to be equal for method B (approach modeling the susceptibility-induced fat resonance shift) (29). However, imposing the phase constraint on method B would transform Eq. [5] such that the water/fat phase becomes a non-linear parameter, increasing further the dimension of the nonlinear parameter space (29). The investigation of an approach incorporating this phase constraint into method B was outside the scope of the present work.

The estimation of the susceptibility-induced fat resonance using the proposed two approaches (modeling the shift and not modeling the shift) has some limitations. First, a reliable estimation of the shift is feasible only in areas where both water and fat are present. The noise performance of the estimation of the shift using both approaches should be expected to degrade at high and low fat fractions. Second, the extraction of the orientation based on the susceptibility-induced shift can be affected by the positioning of the subject's leg relative to the main magnetic field, as well as temperature-induced differences in potentially poorly circulated extremities. A standardization of the leg positioning would help minimizing the variability in the leg orientation with respect to the main magnetic field. The acquisition of a reference spectrum in a region with only IMCLs would help overcoming the issues related to the temperature-induced water frequency shifts. Third, a single T₂* model was employed throughout the present analysis. Assuming independent T₂* values for the water and fat compartments might be important for improving the accuracy of fat quantification in acquisitions with high number of echoes in fibrotic muscles (32), but it would add one more unknown to the signal model and would degrade the noise performance of the two proposed methods (25,33).

Chemical shift-based water/fat separation approaches traditionally focus on the quantification of the amount of lipids. The determination of the susceptibility-induced fat resonance shift at different tissue (leg) orientations could be used to obtain a fat magnetic susceptibility tensor map, as it has been proposed in (34). Therefore, the measurement of the susceptibility-induced fat resonance shift using the proposed approaches could have several potential applications towards extracting additional information about the microscopic characteristics of the lipids, including the orientation and the shape/organization of the fatty inclusions. First, measuring muscle fiber orientation in skeletal muscles with moderate fat fractions using diffusion tensor imaging can be challenging due to the incomplete suppression of the aliphatic and olefinic fat peaks (35). In areas where the EMCLs are forming islands of fats confined in the interstitial space between muscle fibers and show an anisotropic shape following the orientation of fibers, the measurement of susceptibility-induced fat resonance shift at different leg orientations could be used as a way to measure the orientation of the surrounding muscle fibers. Second, the quantification of the susceptibility-induced fat resonance shift at different tissue (leg) orientations could provide information about the shape/organization of the lipid strands. Markers of the organization of lipid strands would be very important in monitoring the progression of fatty infiltration in skeletal muscle from early to advanced stages and would help towards understanding the pathogenesis of fatty infiltration (36). Third, the quantification of the susceptibility-induced fat resonance shift could be helpful in the discrimination of increased intracellular lipid storage from extracellular fatty infiltration. A non-invasive assessment of the discrimination between extracellular and intracellular fat accumulation could be helpful in the study of disorders associated with intracellular lipid storage, like lipid storage myopathies (37).

Finally, the algorithm for solving the complex signal model using the approach modeling the fat resonance shift is quite general and could be also used to detect temperature-induced frequency shifts in water/fat regions. Previous studies have used the multi-echo gradient-echo magnitude signal (38) or complex signal (39) to measure the temperature-induced water frequency shifts in thermometry experiments. The proposed two-step algorithm for solving the problem using the approach modeling the fat resonance shift constitutes a technique solving the full problem of estimating the water and fat signals, the fieldmap, the T₂* relaxation time and the frequency shift, which has not been addressed in previous thermometry studies.

CONCLUSION

The effect of susceptibility-induced fat resonance shift on chemical shift-based water/fat separation was studied in order to characterize the significance of the effect on the fat fraction quantification and to propose two novel approaches to quantify the resonance shift, aiming to provide information about the orientation/geometry of the measured lipids. It was shown that the fat fraction bias induced by the fat resonance shift is small in multi-echo acquisitions with low number of echoes, but it becomes large in acquisitions with high number of echoes. The two proposed methods to quantify the fat resonance shift were validated with MRS results in a water/fat phantom scanned at different orientations with respect to the main magnetic field. Preliminary in vivo results in the calf musculature of one healthy volunteer and two subjects with type 2 diabetes mellitus were also reported, showing the feasibility of measuring the susceptibility-induced fat resonance shift in vivo.

APPENDIX

The signal model of Eq. [1] can be reformulated, as it follows:

$$\mathrm{S}\left(\mathrm{t}\right)=\left[{\mathrm{M}}_{\mathrm{w}}\exp (-\mathrm{jA}\left(\mathrm{t}\right))+{\mathrm{M}}_{\mathrm{f}}\exp \left(\mathrm{jA}\left(\mathrm{t}\right)\right)\exp \left(\mathrm{j}2\pi {\mathrm{xt}}_{\mathrm{mid}}\right)\sum _{\mathrm{p}=1}^{\mathrm{P}}{\mathrm{a}}_{\mathrm{p}}\exp \left(\mathrm{j}2\pi \Delta {\mathrm{f}}_{\mathrm{p}}\mathrm{t}\right)\right]\exp \left(\mathrm{j}\varphi \right)\exp \left(\mathrm{jA}\left(\mathrm{t}\right)\right)\exp \left(\mathrm{j}2\pi {\mathrm{f}}_{\mathrm{B}}\mathrm{t}\right)\exp (-\mathrm{t}∕{\mathrm{T}}_2^{\ast })$$ [6]

where $\mathrm{A}\left(\mathrm{t}\right)=2\pi \mathrm{x}\left(\frac{\mathrm{t}-{\mathrm{t}}_{\mathrm{mid}}}{2}\right)$.

Using Euler's formula exp $(\pm \mathrm{jA}\left(\mathrm{t}\right))=\cos \left(\mathrm{A}\left(\mathrm{t}\right)\right)\pm \mathrm{jsin}\left(\mathrm{A}\left(\mathrm{t}\right)\right)$ and Eq. [6] after some rearrangement becomes:

$$\mathrm{S}\left(\mathrm{t}\right)=\left[{\mathrm{M}}_{\mathrm{w}}+{\mathrm{M}}_{\mathrm{f}}\exp \left(\mathrm{j}2\pi {\mathrm{xt}}_{\mathrm{mid}}\right)\sum _{\mathrm{p}=1}^{\mathrm{P}}{\mathrm{a}}_{\mathrm{p}}\exp \left(\mathrm{j}2\pi \Delta {\mathrm{f}}_{\mathrm{p}}\mathrm{t}\right)\right]\mathrm{B}\left(\mathrm{t}\right)\exp \left(\mathrm{j}\varphi \right)\exp \left(\mathrm{jA}\left(t\right)\right)\exp \left(\mathrm{j}2\pi {\mathrm{f}}_{\mathrm{B}}\mathrm{t}\right)\exp (-\mathrm{t}∕{\mathrm{T}}_2^{\ast })$$ [7]

where

$$\mathrm{B}\left(\mathrm{t}\right)=\cos \left(\mathrm{A}\left(\mathrm{t}\right)\right)+\mathrm{jsin}\left(\mathrm{A}\left(\mathrm{t}\right)\right)\frac{\mathrm{f}\sum _{\mathrm{p}=1}^{\mathrm{P}}{\mathrm{a}}_{\mathrm{p}}\exp \left(\mathrm{j}2\pi \Delta {\mathrm{f}}_{\mathrm{p}}\mathrm{t}\right)-(1-\mathrm{f})}{\mathrm{f}\sum _{\mathrm{p}=1}^{\mathrm{P}}{\mathrm{a}}_{\mathrm{p}}\exp \left(\mathrm{j}2\pi \Delta {\mathrm{f}}_{\mathrm{p}}\mathrm{t}\right)+(1-\mathrm{f})}$$ [8]

with f representing the fat fraction f = M_f/(M_w + M_f).

For low fat fractions: B(t) ≅ cos(A(t))−jsin(A(t)) ≅ exp(−jA(t)) and then the signal Eq. [7] becomes:

$$\mathrm{S}\left(\mathrm{f}\right)\cong \left[{\mathrm{M}}_{\mathrm{w}}\exp \left(\mathrm{j}\varphi \right)+{\mathrm{M}}_{\mathrm{f}}\exp \left(\mathrm{j}(\varphi +2\pi {\mathrm{xt}}_{\mathrm{mid}})\right)\sum _{\mathrm{p}=1}^{\mathrm{P}}{\mathrm{a}}_{\mathrm{p}}\exp \left(\mathrm{j}2\pi \Delta {\mathrm{f}}_{\mathrm{p}}\mathrm{t}\right)\right]\exp \left(\mathrm{j}2\pi {\mathrm{f}}_{\mathrm{B}}\mathrm{t}\right)\exp (-\mathrm{t}∕{\mathrm{T}}_2^{\ast })$$ [9]

By comparing Eq. [9] with Eq. [3], it can be concluded that ${\widehat{M}}_{\mathrm{w}}={\mathrm{M}}_{\mathrm{w}},{\widehat{M}}_{\mathrm{f}}={\mathrm{M}}_{\mathrm{f}}$, ${\widehat{\varphi }}_{\mathrm{w}}\cong \varphi ,{\widehat{\varphi }}_{\mathrm{f}}\cong \varphi +2\pi {\mathrm{xt}}_{\mathrm{mid}}$,${\widehat{f}}_{\mathrm{B}}\cong {\mathrm{f}}_{\mathrm{B}}$ and ${\widehat{T}}_2^{\ast }\cong {\mathrm{T}}_2^{\ast }$.

For high fat fractions: B(t) ≅ cos(A(t))+jsin(A(t)) ≅ exp(jA(t)) and then the signal Eq. [7] becomes:

$$\mathrm{S}\left(\mathrm{t}\right)\cong \left[{\mathrm{M}}_{\mathrm{w}}\exp \left(\mathrm{j}(\varphi +2\pi {\mathrm{xt}}_{\mathrm{mid}})\right)+{\mathrm{M}}_{\mathrm{f}}\exp \left(\mathrm{j}\varphi \right)\sum _{\mathrm{p}=1}^{\mathrm{P}}{\mathrm{a}}_{\mathrm{p}}\exp \left(\mathrm{j}2\pi \Delta {\mathrm{f}}_{\mathrm{p}}\mathrm{t}\right)\right]\exp \left(\mathrm{j}2\pi ({\mathrm{f}}_{\mathrm{B}}+\mathrm{x})\mathrm{t}\right)\exp (-\mathrm{t}∕{\mathrm{T}}_2^{\ast })$$ [10]

By comparing Eq. [10] with Eq. [3], we conclude that ${\widehat{M}}_{\mathrm{w}}={\mathrm{M}}_{\mathrm{w}},{\widehat{M}}_{\mathrm{f}}={\mathrm{M}}_{\mathrm{f}}$, ${\widehat{\varphi }}_{\mathrm{w}}\cong \varphi -2\pi {\mathrm{xt}}_{\mathrm{mid}},\widehat{\varphi }\cong \varphi$,${\widehat{f}}_{\mathrm{B}}\cong {\mathrm{f}}_{\mathrm{B}}+\mathrm{x}$ and ${\widehat{T}}_2^{\ast }\cong {\mathrm{T}}_2^{\ast }$.

Therefore for both low and high fat fractions: ${\widehat{\varphi }}_{\mathrm{f}}-{\widehat{\varphi }}_{\mathrm{w}}\cong 2\pi {\mathrm{xt}}_{\mathrm{mid}}$.