Coherence pathway analysis of J-coupled lipids and lactate and effective suppression of lipids upon the selective multiple quantum coherence lactate editing sequence

Abstract

Objective:

The selective multiple quantum coherence (Sel-MQC) sequence is a magnetic resonance spectroscopy (MRS) technique used to detect lactate and suppress co-resonant lipid signals in vivo. The coherence pathways of J-coupled lipids upon the sequence, however, have not been studied, hindering a logical design of the sequence to fully attenuate lipid signals. The objective of this study is to elucidate the coherence pathways of J-coupled lipids upon the Sel-MQC sequence and find a strategy to effectively suppress lipid signals from these pathways while keeping the lactate signal.

Approach:

The product operator formalism was used to express the evolutions of the J-coupled spins of lipids and lactate. The transformations of the product operators by the spectrally selective pulses of the sequence were calculated by using the off-resonance rotation matrices. The coherence pathways and the conversion rates of the individual pathways were derived from them. Experiments were performed on phantoms and two human subjects at 3T.

Main results:

The coherence pathways contributing to the various lipid resonance signals by the Sel-MQC sequence depending on the gradient ratios and RF pulse lengths were identified. Theoretical calculations of the signals from the determined coherence pathways and signal attenuations by gradients matched the experimental data very well. Lipid signals from fatty tissues of the subjects were successfully suppressed to the noise level by using the gradient ratio −0.8:−1:2 or 1:0.8:2. The new gradient ratios kept the lactate signal the same as with the previously used gradient ratio 0:−1:2.

Significance:

The study has elucidated the coherence pathways of J-coupled lipids upon the Sel-MQC sequence and demonstrated how lipid signals can be effectively suppressed while keeping lactate signals by using information from the coherence pathway analysis. The findings enable applying the Sel-MQC sequence to lactate detection in an environment of high concentrations of lipids.

1. Introduction

Magnetic resonance spectroscopy (MRS) and magnetic resonance spectroscopic imaging (MRSI) allow for the detection and imaging of key metabolites in the human body non-invasively [1]. Lactate is an end product of glycolysis [2]. A high increase of lactate in cancerous and ischemic tissues suggests the metabolite’s diagnostic utility [3, 4]. However, detection of lactate in vivo by MRS is often hampered by lipid signals resonating at frequencies overlapping with lactate. To solve the problem, a spectral editing technique is required. The J-difference method [5] and the multiple quantum coherence (MQC) method [6] are the two most often used techniques for this purpose. While the J-difference method is more widely utilized [4, 7, 8] than the other, as the technique relies on subtraction of lipid signals from two different scans, the editing is inherently sensitive to motion and, therefore, its application is mostly limited to the brain. Moreover, large lipid signals, e.g., from the skull, are difficult to eliminate by the technique [7]. On the other hand, the MQC method exploits the coherence pathway difference between lactate and lipids during a scan. In this method, lactate is transferred from the single quantum coherence (SQC) state to the double quantum coherence (DQC) state and transferred back to the SQC state. Lipids that do not evolve through the same coherence pathway as lactate are dephased by the coherence selection gradients. Though the MQC method involves a 50% signal loss during the coherence transfer to DQC [6], the method is valuable as it may be used to detect lactate from the body regions or when the J-difference method is not suitable.

Two representative types of MQC lactate editing sequences have been used for in vivo studies. The first one is from Hurd and Freeman [9]. The backbone of the sequence is 90°(hard) – 1/(4J) – 180°(hard) – 1/(4J) – 90°(hard) – t₁ – 90°(CH) –1/(4J) – 180°(hard) – 1/(4J) – acquisition, where 90°(CH) indicates the spectrally selective pulse centered at the lactate methine resonance and J is the spin-spin coupling constant between CH₃ and CH protons of lactate. The SQC→DQC→SQC pathway of lactate is selected by the coherence selection gradients with the ratio of 1:−2. While uncoupled lipid spins do not evolve to DQC, a portion of J-coupled lipid spins [10–12] evolve through the same coherence pathway as lactate and give rise to the observable signal despite the filtering effect of the spectrally selective 90°(CH) pulse. To resolve the issue, various strategies have been combined with the basic structure of the sequence. The use of the two-dimensional (2D) nuclear magnetic resonance (NMR) technique collects data with increments of t₁ evolution time of the MQC sequence to separate lactate and J-coupled lipids in the 2D spectral domain [9]. The t₁ cycling method makes use of the fact that the phases of lactate and J-coupled lipids change differently with the t₁ evolution time of the MQC sequence due to differences in the double quantum coherence frequencies between them. By subtracting data from experiments with two different t₁ times where the lactate phases differ by 180°, the lipid signals are removed [13, 14]. The fat inversion method places a fat inversion pulse before the MQC sequence to suppress lipid signals [15]. Each of these methods has shortcomings, however. The first method is time-inefficient, the second method entails the subtraction artifacts and the third method in part suppresses the lactate signal as well.

The second type of MQC lactate editing sequence, which is called the Selective Multiple Quantum Coherence (Sel-MQC), is from He et al. [16]. It has the structure of 90°(CH₃) – 1/(2J) – 90°(CH) – (1/2)t₁ – 180°(CH₃) – (1/2)t₁ – 90°(CH) – acquisition, where CH₃ and CH indicate methyl and methine resonance frequencies of lactate, respectively. The coherence selection gradient pulses are applied after the second, third and fourth RF pulses of the sequence, respectively. Depending on the area ratio of these pulsed-field gradients at 0:−1:2 or 1:0:2, the SQC→ZQC→DQC→SQC or SQC→DQC→ZQC→SQC pathway of lactate is selected [16]; ZQC, zero quantum coherence. The sequence achieves an improved lipid suppression compared to the basic structure of the first type of MQC sequences [9] by employing multiple spectrally selective RF pulses. The sequence has been utilized to successfully suppress lipid signals and detect lactate from local tumors in numerous in vivo studies [17–26]. However, when tissues with high lipid concentrations, e.g., calf muscles, were studied, significant amounts of lipids passed through the sequence [20, 27]. Such insufficiently suppressed lipid signals may contaminate the lactate signals from ischemic calf muscles or the lactate signals from tumors if the tumor is surrounded by fatty tissues and the chemical shift imaging (CSI) localization is used.

To resolve the limited lipid suppression problem of the Sel-MQC sequence, in a preliminary study we have tested the coherence selection gradient ratio α:−1:2 with nonzero α values [28]. The new gradient ratios improved lipid suppressions up to two orders of magnitude compared to the gradient ratio 0:−1:2 while keeping the lactate signal. However, the suppression highly depended on the resonance peaks of lipids [28]. The unchanged lactate signal is understandable as the additional gradient α is applied during the ZQC state of lactate. However, how the improved lipid suppression occurs with the new gradient ratios and why the suppression depends on the resonance peaks of lipids are difficult to understand without knowing the coherence pathways of J-coupled lipids upon the sequence. Lipids have various J-couplings with corresponding chemical shifts. The spectrally selective RF pulses of the Sel-MQC sequence induce differential rotations on the lipid spins of these resonances, generating various coherence pathways with differential yields. A clear understanding of these coherence pathways will allow for the precise design of the sequence to effectively suppress lipid signals of various chemical shifts while minimally affecting the lactate signal. In this study, we elucidate the coherence pathways of J-coupled lipids and lactate upon the Sel-MQC sequence by theoretical analyses of the spin evolutions and experiments on phantoms and human subjects. We apply the determined coherence pathways for designing an optimal strategy to attenuate lipid signals while keeping the lactate signal.

2. Theory

2.1. Coherence pathway selection by gradients.

In multi-pulse NMR experiments, the desired coherence transfer pathways (CTPs) of the spin system of interest are selected by phase cycling or pulsed-field gradients [29, 30]. For the latter method, the sum of phases accumulated by gradients for the selected CTP should satisfy

$$\sum _{i=1}^n{\psi }_i\left(\overrightarrow{r}\right)=0$$ (1)

where ${\psi }_i\left(\overrightarrow{r}\right)$ is the phase induced by gradients for spins at the position $\overrightarrow{r}$ during the ith period which is defined as the period between the ith pulse and (i + 1)-th pulse or between the last ith pulse and acquisition. The phase ${\psi }_i\left(\overrightarrow{r}\right)$ is calculated as

$${\psi }_i\left(\overrightarrow{r}\right)=\gamma p_i\overrightarrow{r}\cdot \int \overrightarrow{g_i}dt$$ (2)

where γ is the gyromagnetic ratio, p_i is the coherence level (0, ±1, ±2, …) of the spin system in the ith period, and $\overrightarrow{g_i}$ is the gradient vector in the ith period. The product operator formalism [31] allows for finding the coherence levels and expressing evolutions of the weakly coupled spin system during the ith period of a pulse sequence.

2.2. Spin systems of the J-coupled lipids and lactate.

The known spin-spin couplings of the J-coupled lipids are as follows [10–12]: CH₂-CH₂-CH₃ (I₂S₃; I = 1.33 ppm, S = 0.90 ppm; J = 8.0 Hz), OC-CH₂-CH₂-CH₂ (I₂S₂; I = 1.33 ppm, S = 1.65 ppm; J = 7.1 Hz), OC-CH₂-CH₂-CH₂ (I₂S₂; I = 1.65 ppm, S = 2.36 ppm; J = 7.1 Hz), CH-CH₂-CH₂ (I₂S₂; I = 1.33 ppm, S = 2.07 ppm; J = 7.14 Hz), CH-CH₂-CH₂ (I₂S; I = 2.07 ppm, S = 5.38 ppm; J = 6.2 Hz), CH-CH₂-CH (I₂S; I = 2.79 ppm, S = 5.38 ppm; J = 1.0 Hz), O-CH₂-CH-O (I₂S; I = 4.16 ppm, S = 5.29 ppm; I =4.37 ppm, S = 5.29 ppm; J = 7.0 Hz) and O-CH₂-CH-CH₂-O (I₂S₂; I = 4.16 ppm, S = 4.37 ppm; J = 12.4 Hz), where the boldface indicates the J-coupled protons. The kinds of the coupled spin system, chemical shifts of I and S spins and the J-coupling constants are indicated inside the parentheses. Lactate has the spin-spin coupling of CH₃-CH (I₃S; I = 1.33 ppm, S= 4.11 ppm, J = 6.93 Hz). The weak coupling approximation holds when J/(2δ) << 1, where δ is the chemical shift difference between the coupled spins in Hz [32]. For the above couplings, J/(2δ) is 0.01−0.1 at 3T for all couplings except for the coupling of 4.16–4.37 ppm for which J/(2δ) is 0.25. We perform all the analyses in this manuscript with the weak coupling approximation.

2.3. Product operator evolutions of the J-coupled spins.

To analyze the spin evolutions and coherence pathways of the J-coupled lipids and lactate, we express the evolutions of the I₂S, I₂S₂, I₂S₃ and I₃S spin systems using the raising and lowering operator forms of product operators [31]. Equations 4 to 7 were derived by using the product operator evolutions for the IS spin system (Eqs. 3.1-3.3) and the method by Kingsley [33]. The J-coupled evolutions of product operators for the IS spin system are expressed as [1, 31]

$$I^{+}\stackrel{J}{\to }{I}^{+}\cos (\pi Jt)-2i{I}^{+}{S}_0\sin (\pi Jt)$$ (3.1)

$$I^{-}\stackrel{J}{\to }{I}^{-}\cos (\pi Jt)+2i{I}^{-}{S}_0\sin (\pi Jt)$$ (3.2)

$$2I^{-}{S}_0\stackrel{J}{\to }2I^{-}{S}_0\cos (\pi Jt)+i{I}^{-}\sin (\pi Jt)$$ (3.3)

From them, the J-coupled evolutions of product operators for the I₂S spin system are derived as

$$I^{+}\stackrel{J}{\to }{I}^{+}\cos (\pi Jt)-2i{I}^{+}{S}_0\sin (\pi Jt)$$ (4.1)

$$S^{+}\stackrel{J}{\to }{S}^{+}{\cos }^2(\pi Jt)-2i{S}^{+}{I}_0\cos (\pi Jt)\sin (\pi Jt)-4S^{+}{I}_{10}{I}_{20}{\sin }^2(\pi Jt)$$ (4.2)

$$-2i{I}^{-}{S}_0\stackrel{J}{\to }{I}^{-}\sin (\pi Jt)+(\text{antiphase\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}terms\hspace{0.17em}})$$ (4.3)

$$-2i{S}^{-}{I}_0\stackrel{J}{\to }{S}^{-}\cos (\pi Jt)\sin (\pi Jt)+(\text{antiphase\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}terms\hspace{0.17em}})$$ (4.4)

$$-4S^{-}{I}_{10}{I}_{20}\stackrel{J}{\to }{S}^{-}{\sin }^2(\pi Jt)+(\text{antiphase\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}terms\hspace{0.17em}})$$ (4.5)

where $I^{+}=I_1^{+}+I_2^{+}$, $I_0=I_{10}+I_{20}$, and $I^{-}=I_1^{-}+I_2^{-}$. Similarly, the J-coupled evolutions of the product operators for the I₂S₂ spin system are derived as

$$I^{+}\stackrel{J}{\to }{I}^{+}{\cos }^2(\pi Jt)-2i{I}^{+}{S}_0\cos (\pi Jt)\sin (\pi Jt)-4I^{+}{S}_{10}{S}_{20}{\sin }^2(\pi Jt)$$ (5.1)

$$S^{+}\stackrel{J}{\to }{S}^{+}{\cos }^2(\pi Jt)-2i{S}^{+}{I}_0\cos (\pi Jt)\sin (\pi Jt)-4S^{+}{I}_{10}{I}_{20}{\sin }^2(\pi Jt)$$ (5.2)

$$-2i{I}^{-}{S}_0\stackrel{J}{\to }2I^{-}\cos (\pi Jt)\sin (\pi Jt)+(\text{antiphase\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}terms\hspace{0.17em}})$$ (5.3)

$$-4I^{-}{S}_{10}{S}_{20}\stackrel{J}{\to }{I}^{-}{\sin }^2(\pi Jt)+(\text{antiphase\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}terms\hspace{0.17em}})$$ (5.4)

$$-2i{S}^{-}{I}_0\stackrel{J}{\to }2S^{-}\cos (\pi Jt)\sin (\pi Jt)+(\text{antiphase\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}terms\hspace{0.17em}})$$ (5.5)

$$-4S^{-}{I}_{10}{I}_{20}\stackrel{J}{\to }{S}^{-}{\sin }^2(\pi Jt)+(\text{antiphase\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}terms\hspace{0.17em}})$$ (5.6)

where $I^{+}=I_1^{+}+I_2^{+}$, $I^{-}=I_1^{-}+I_2^{-}$, $I_0=I_{10}+I_{20}$, $S^{+}=S_1^{+}+S_2^{+}$, $S^{-}=S_1^{-}+S_2^{-}$, and $S_0=S_{10}+S_{20}$. The J-coupled evolutions of the product operators for the I₂S₃ spin system are derived as

$$\begin{array}{l}{I}^{+}\stackrel{J}{\to }{I}^{+}{\cos }^3(\pi Jt)-2i{I}^{+}{S}_0{\cos }^2(\pi Jt)\sin (\pi Jt)\\ -4I^{+}(S_{10}{S}_{20}+S_{20}{S}_{30}+S_{30}{S}_{10})\cos (\pi Jt){\sin }^2(\pi Jt)+8i{I}^{+}{S}_{10}{S}_{20}{S}_{30}{\sin }^3(\pi Jt)\end{array}$$ (6.1)

$$S^{+}\stackrel{J}{\to }{S}^{+}{\cos }^2(\pi Jt)-2i{S}^{+}{I}_0\cos (\pi Jt)\sin (\pi Jt)-4S^{+}{I}_{10}{I}_{20}{\sin }^2(\pi Jt)$$ (6.2)

$$-2i{I}^{-}{S}_0\stackrel{J}{\to }3I^{-}{\cos }^2(\pi Jt)\sin (\pi Jt)+(\text{antiphase\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}terms\hspace{0.17em}})$$ (6.3)

$$-4I^{-}(S_{10}{S}_{20}+S_{20}{S}_{30}+S_{30}{S}_{10})\stackrel{J}{\to }3I^{-}{\sin }^2(\pi Jt)+(\text{antiphase\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}terms\hspace{0.17em}})$$ (6.4)

$$8i{I}^{-}{S}_{10}{S}_{20}{S}_{30}\stackrel{J}{\to }{I}^{-}{\sin }^3(\pi Jt)+(\text{antiphase\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}terms\hspace{0.17em}})$$ (6.5)

$$-2i{S}^{-}{I}_0\stackrel{J}{\to }2S^{-}\cos (\pi Jt)\sin (\pi Jt)+(\text{antiphase\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}terms\hspace{0.17em}})$$ (6.6)

$$-4S^{-}{I}_{10}{I}_{20}\stackrel{J}{\to }{S}^{-}{\sin }^2(\pi Jt)+(\text{antiphase\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}terms\hspace{0.17em}})$$ (6.7)

where $I^{+}=I_1^{+}+I_2^{+}$, $I^{-}=I_1^{-}+I_2^{-}$, $I_0=I_{10}+I_{20}$, $S^{+}=S_1^{+}+S_2^{+}+S_3^{+}$, $S^{-}=S_1^{-}+S_2^{-}+S_3^{-}$, and $S_0=S_{10}+S_{20}+S_{30}$. Similarly, the J-coupled evolutions of the product operators for the I₃S spin system are derived as

$$I^{+}\stackrel{J}{\to }{I}^{+}\cos (\pi Jt)-2i{I}^{+}{S}_0\sin (\pi Jt)$$ (7.1)

$$\begin{array}{l}{S}^{+}\stackrel{J}{\to }{S}^{+}{\cos }^3(\pi Jt)-2i{S}^{+}{I}_0{\cos }^2(\pi Jt)\sin (\pi Jt)\\ -4S^{+}(I_{10}{I}_{20}+I_{20}{I}_{30}+I_{30}{I}_{10})\cos (\pi Jt){\sin }^2(\pi Jt)+8i{S}^{+}{I}_{10}{I}_{20}{I}_{30}{\sin }^3(\pi Jt)\end{array}$$ (7.2)

$$-2i{I}^{-}{S}_0\stackrel{J}{\to }{I}^{-}\sin (\pi Jt)-2i{I}^{-}{S}_0\cos (\pi Jt)$$ (7.3)

$$-2i{S}^{-}{I}_0\stackrel{J}{\to }{S}^{-}{\cos }^2(\pi Jt)\sin (\pi Jt)+(\text{antiphase\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}terms\hspace{0.17em}})$$ (7.4)

$$-4S^{-}(I_{10}{I}_{20}+I_{20}{I}_{30}+I_{30}{I}_{10})\stackrel{J}{\to }3S^{-}{\sin }^2(\pi Jt)+(\text{antiphase\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}terms\hspace{0.17em}})$$ (7.5)

$$8i{S}^{-}{I}_{10}{I}_{20}{I}_{30}\stackrel{J}{\to }{S}^{-}{\sin }^3(\pi Jt)+(\text{antiphase\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}terms\hspace{0.17em}})$$ (7.6)

where $I^{+}=I_1^{+}+I_2^{+}+I_3^{+}$, $I^{-}=I_1^{-}+I_2^{-}+I_3^{-}$, and $I_0=I_{10}+I_{20}+I_{30}$.

2.4. Signal attenuation by gradients.

For CTPs that do not satisfy Eq. 1, signals are attenuated by gradients. The signal modulation factor by gradients can be written as

$$f_{\text{g}}=\prod _i\exp \left(-i{\psi }_i(\overrightarrow{r})\right)=\prod _i\exp \left(-i(k_{ix}x+k_{iy}y+k_{iz}z)\right)$$ (8)

where i indicates the ith period of the pulse sequence and $k_{\text{ix, iy, iz}}=\gamma p_i\int g_{ix,iy,iz}dt$. When no gradient is applied, f_g = 1. When the gradients are applied in the z-direction for an object with homogeneous spin distribution from -z_max, to z_max, the signal attenuation factor f_z can be derived from the spatial average of Eq. 8 as

$$f_{\text{z}}=\overline{\exp \left(-i\psi (z)\right)}=\frac{1}{2z_{\max }}\cdot {\int }_{-z\max }^{z\max }\exp \left(-i{k}_{z}z\right)dz=\text{sinc}\left(k_z{z}_{\max }\right)$$ (9)

where $k_{\text{z}}=\sum _i{k}_{iz}$. If the object is cuboidal and the gradients are applied in three orthogonal directions,

$$f_{\text{g}}=f_x\cdot f_y\cdot f_z=\text{sinc}\left(k_x{x}_{\max }\right)\cdot \text{sinc}\left(k_y{y}_{\max }\right)\cdot \text{sinc}\left(k_z{z}_{\max }\right)$$ (10)

where x_max, y_max and z_max are the halves of the object sizes in the x-, y- and z-directions, respectively. If a cylindrical object is placed along the z-axis of the magnet,$f_g=f_{xy}\cdot f_z$ where f_z is as in Eq. 9 and f_xy has the form of the Bessel function [34]. When k_xx_max ≥ π/2, k_yy_max ≥ π/2 and k_zz_max ≥ π/2, the envelopes of the Sinc functions of Eq. 10 have the forms of 1/|k_xx_max|, 1/|k_yy_max| and 1/|k_zz_max|, respectively [34, 35]. In that condition, the signal attenuation factor by gradients can be written as

$$f_{\text{g}}=\frac{\mu }{\left|k_x{x}_{\max }\right|\cdot \left|k_y{y}_{\max }\right|\cdot \left|k_z{z}_{\max }\right|}$$ (11)

Here we introduced the correction factor μ (≥1) to consider any inhomogeneity in the spin distribution within the object. Similarly, when a CSI localization is used, the signal attenuation factor for an individual voxel of the CSI matrix can be written as Eq. 11. In that case, the correction factor μ comes from the combined effect of the spin distributions in both the target voxel and the surrounding voxels, as there is a signal bleed-over effect between neighboring voxels of CSI [1]. For CSI with an axial slice, x_max and y_max are the halves of the voxel sizes in the x and y directions, respectively, and z_max is half the slice thickness.

3. Methods

3.1. MRI system.

A 3T (123.25 MHz) whole-body MRI system (Tim Trio, Siemens, Erlangen, Germany) was used for this study. The body coil of the scanner and a home-built two-channel surface coil (6 cm × 8 cm) were used for the transmitter and the receiver, respectively.

3.2. Pulse sequence.

The pulse sequence was coded in the Siemens IDEA programming environment (VB 17). The sequence (Fig. 1) is a modification of our previously published clinical scanner version of the Sel-MQC sequence at 3T where the coherence selection gradient ratios 0:−1:2 and 1:0:2 had been used [20]. We tested the gradient ratio g₂:g₂:g₃ (area ratio with polarity) of 0:0:0, 0:−1:2, α:−1:2, 1:0:2 and 1:β:2 with various nonzero α and β in this study. The slice gradient was applied directly with the first RF pulse of the Sel-MQC sequence instead of Hadamard slice inversion pulse in the previous study [20]. This removes the need for 2n encoding steps for n slices and the potential artifacts from the addition and subtraction procedure for slice reconstruction. Two different values of τ₁, i.e., 1/(2J_Lac) and 1/(2J_Lac) – t₁, were used depending on the gradient ratio (see the figure caption for Fig. 1) to match the echo time of lactate; explained further in Section 4.3. A Hamming-filtered 7-lobe Sinc pulse with a 3.0 ms length (90° bandwidth = 2,970 Hz, 90° transition width = 603 Hz) was used for the first RF pulse. For the remaining thee RF pulses, Gaussian pulses with 5% truncation cutoff and -π phase (90° R_BW = 2.25, 90° R_TW = 1.42; 180° R_BW = 1.28, 180° R_TW = 1.08) were employed, where R_BW and R_TW are bandwidth (Hz) × pulse length (s) and transition width (Hz) × pulse length (s), respectively. The bandwidths and transition widths were defined as de Graaf [1] and were calculated from the simulation of the off-resonance spin rotation profiles. The center frequencies of the first and third RF pulses were set to 1.33 ppm while the second and fourth pulses were centered at 4.11 ppm. When lactate was included in the experiment, the lactate methyl (CH₃) resonance was referenced at 1.33 ppm. In this condition, the water signal from the lactate phantom appeared at 4.82 ppm in the phantom-only experiment at room temperature (20°C) and at 4.73 ppm in the phantom-plus-subject experiment due to the body temperature of the subject. When only oil was studied, we referenced the biggest peak of lipids at 1.33 ppm. Two sets of pulse lengths for the Gaussian RF pulses were used for this study. Pulse length condition 1: Both 90° and 180° pulses had 7.8 ms lengths. Pulse length condition 2: The 90° pulse had a 12 ms length while the 180° pulse had a 5.6 ms length. An exception was for the experiments of Figs. S5.E and S5.F where the 90° pulse of 13.5 ms length and the 180° pulse of 7.8 ms length were employed. The coherence selection gradient durations and amplitudes used are indicated in the figure captions for the relevant data. The gradient rise and fall times were 0.3 ms, respectively, for all the gradients. The crusher gradients (g_csr) placed before the second pulse and after the fourth pulse had amplitudes of 5 mT/m and lengths of 2 ms. The phase encoding gradients for CSI had 3 ms lengths. The WET water suppression pulses (bandwidth = 35 Hz) [36] were used in front of the Sel-MQC pulse train in the weak water suppression mode of the scanner.

Fig. 1. Slice selective Sel-MQC-CSI sequence.

G_csr indicates the crusher gradient. g₂, g₃ and g₄ are the coherence selection gradients. WS, water suppression pulses; CSI, the chemical shift imaging gradients; ACQ, the acquisition window. The center frequencies of the RF pulses were set to either 1.33 ppm (lactate methyl resonance) or 4.11 ppm (lactate methine resonance) as indicated in the figure. τ₁ was set to 1/(2J_Lac) when g₂:g₃:g₄ = 0:0:0, 0:−1:2 or α:−1:2. τ₁ was set to 1/(2J_Lac) – t₁ when g₂:g₃:g₄ = 1:0:2 or 1:β:2. 1/(2J_Lac) = 72 ms was used. The lactate echo is formed at τ₂ = 1/(2J_Lac) – t₁ for the former τ₁ condition and at τ₂ = 1/(2J_Lac) for the latter τ₁ condition. For the gradient ratio 0:0:0, 0:−1:2 or α:−1:2, the acquisition window started immediately after the CSI gradients. For other gradient ratios, the acquisition window was adjusted so that the number of data points from the beginning of the acquisition to the echo is the same as the comparison experiments. The sequence is displayed for g₂:g₃:g₄ = 0.25:−1:2.

3.3. Phantom-only experiments.

Lactate was prepared at a concentration of 10 mM by using lithium lactate and phosphate-buffered saline. The safflower seed oil (S8281, Sigma-Aldrich, St. Louis, MO) was used for the lipid sample. 50 ml centrifuge tubes (inner diameter = 28 mm, outer diameter = 30 mm) were filled with either lactate or the safflower seed oil. The tubes were separately immersed in a water-filled container (experiments for Fig. 3) or placed directly on the scanner bed (experiments for Figs. 6, 7 and 10 and Fig. S7). The phantoms used for the data of Figs. S5 and S6 are described in their figure captions. The long axis of the tube was aligned along the z-direction of the magnet. After a 2D fast low angle shot (FLASH) imaging (TR/TE = 350/2.46 ms), localized shimming was performed for the target voxel of interest. The 90° RF power calibration was performed on the selected voxel using a STEAM sequence (TR = 8,000 ms, TE = 20 ms, TM = 20 ms). The Sel-MQC experiments were run with the axial slices. The acquisition parameters were: CSI FOV = 16×16 cm or 32×16 cm, 8×8 matrix, slice thickness = 2 cm or 4 cm, SW = 1,500 Hz, vector size = 512 or 1,024, TR = 1.5 s, NA = 2, TA = 3:12 min.

Fig. 3. Dependence of lipid and lactate signals on the coherence selection gradient ratio and the Gaussian RF pulse lengths.

(A-E) Lipid signals in the following conditions: (A) g₂:g₃:g₄ = 0:0:0; 90°, 7.8 ms; 180°, 7.8 ms; t₁ = 18.6 ms. (B) g₂:g₃:g₄ = 0:−1:2; 90°, 7.8 ms; 180°, 7.8 ms; t₁ = 18.6 ms. (C) g₂:g₃:g₄ = 0.25:−1:2; 90°, 7.8 ms; 180°, 7.8 ms; t₁ = 18.6 ms. (D) g₂:g₃:g₄ = 0:−1:2; 90°, 12 ms; 180°, 5.6 ms; t₁ = 20.6 ms. (E) g₂:g₃:g₄ = 0.25:−1:2; 90°, 12 ms; 180°, 5.6 ms; t₁ = 20.6 ms. (F) A lactate phantom MR image, the CSI matrix and the studied voxel. (G-J) Lactate signals with the experimental parameters corresponding to Figs. 3B–3E. A 2×2×2 cm voxel inside either the oil or lactate phantom tube was used for the study. The SNRs in the lipid spectra are from the 1.33 ppm peak. The SNRs in the lactate spectra were averaged from the lactate doublet. The durations of g₂, g₃ and g₄ gradients were 0.9 ms, 0.9 ms and 1.8 ms, respectively. When g₂:g₃:g₄ = 0, the amplitudes of g₂, g₃ and g₄ gradients were all set to zero. When g₂:g₃:g₄ = 0:−1:2, the following gradient amplitudes were used: g₂ = 0, g₃ = −26 mT/m and g₄ = 26 mT/m. When g₂:g₃:g₄ = 0.25:−1:2, the following gradient amplitudes were used: g₂ = 6.5 mT/m, g₃ = −26 mT/m and g₄ = 26 mT/m. Acquisition and processed vector sizes = 512 for the oil experiments. Acquisition vector size = 512 and processed vector size = 2,048 for the lactate experiments. The lipid spectra are presented in the magnitude mode and the lactate spectra are presented in the phase-sensitive mode.

Fig. 6. Dependence of lipid signals on the gradient ratio.

(A-F) Lipid spectra at the gradient ratio α:−1:2 with α = −1, −0.5, 0, 0.25, 0.5 and 1, respectively. The spectra are presented in the magnitude mode. (G) Normalized lipid signal intensity (max of the 1.3–1.5 ppm region) with varied α of the gradient ratio α:−1:2. Each of the plotted experimental data is a normalization to the lipid signal at α = 1. Theoretical signal attenuation by gradients (Eq. 11) for the CTPs with the coherence levels of p(1, 1, −1, −1), p(1, 1, −2, −1) and p(1, 2, −1, −1) are also plotted using μ = 6. The durations and amplitudes of the g₂, g₃ and g₄ gradients were the same as in the experiments for Fig. 3 except that the g₂ gradient amplitudes were varied according to α × 26 mT/m. Pulse length condition 2 was used. Slice thickness = 4 cm. Signals from a 2×2×4 cm voxel inside the oil phantom tube were used for analysis. Acquisition and processed vector sizes = 512.

Fig. 7. Dependence of lactate signals on the gradient ratio.

(A-F) Spectra from a 2×2×2 cm voxel of the 10 mM lactate phantom at the gradient ratio α:−1:2 with α = −1, −0.8, −0.4, 0, 0.5 and 0.9, respectively. (G) Lactate signal integrals (0–3 ppm) at varied α normalized to the lactate signal integral (0–3 ppm) at α = 0. Experimental parameters were the same as in Fig. 6 except for the following: slice thickness = 2 cm, acquisition vector size = 1,024 and processed vector size = 2,048. The spectra are presented in the phase-sensitive mode.

Fig. 10. Lactate spectra from various editing conditions and the editing efficiencies.

(A) Unedited; TE = 1/(J_Lac). (B) ZQC→DQC; g = −0.8:−1:2, τ₁ = 1/(2J_Lac) (C) DQC→ZQC; g = 1:0.8:2, τ₁ = 1/(2J_Lac) - t₁. (D) DQC→ZQC; g = 1:0:2, τ₁ = 1/(2J_Lac) - t₁. (E) DQC→ZQC; g = 1:0:2, τ₁ = 1/(2J_Lac) - t₁. The gradient durations and amplitudes were defined in the same way as in Fig. 8 except (E) where the gradient amplitudes were 10% of (D). Pulse length condition 2 was used. t₁ = 21.8 ms. Slice thickness = 2 cm. Acquisition vector size = 1,024 and processed vector size = 2,048. The echo is formed at TE = 1/(J_Lac) for all the data. (F) Lactate editing efficiencies (%). Editing efficiency was calculated by comparing the integration of an edited spectrum (0–3 ppm) with the integration of the unedited spectrum (0–3 ppm) in (A). The data are presented as the averages of two separate experiments. The standard deviation was less than 1% of the average for each data point. All spectra are presented in the phase-sensitive mode.

3.4. Calculation of the off-resonance spin rotation and spin operator transformations.

The off-resonance frequency dependences of spin rotation upon the employed RF pulses were simulated by using the equations for the magnetization rotation about an effective magnetic field on the Riemann sphere as derived by Yao et al. [37]. The effective magnetic field B_eff is given by

$$\overrightarrow{B_{eff}}=(\Omega /\gamma )\stackrel{\wedge }{z}+B_1\stackrel{\wedge }{x}$$ (12)

where Ω is the off-resonance frequency, γ is the gyromagnetic ratio and B₁ is the transverse magnetic field applied along the x-axis in the rotating frame. The simulation was performed in Matlab (Mathworks Inc., Natick, MA) with the initial magnetization of unit M_x, unit M_y and unit M_z, respectively, for the 90° and 180° Gaussian RF pulses of the pulse length conditions 1 and 2. Figure 2 shows the resultant magnetizations in the pulse length condition 1. The off-resonance rotation matrices for the I and S spins of J-coupled lipids and lactate were derived from these simulations (Tables S1, S3 and S5-S10). The Cartesian spin operator transformations by these matrices were converted to the transformations of the spin operators having (±1) and 0 quantum coherences (Tables S2, S4 and S5-S10) using the following relations: I⁺ = I_x + iI_y, I⁻ = I_x – iI_y, I_x = (1/2)(I⁺ + I⁻), I_y = (1/2i)(I⁺ - I⁻), I₀ = I_z, where i = √(−1). The same relations apply to the S spin operators.

Fig. 2. Off-resonance frequency dependence of spin rotation upon the Gaussian RF pulses.

(A) Response of the unit M_x (M_x = 1, M_y = 0, M_z = 0) to a 90° pulse. (B) Response of the unit M_y (M_x = 0, M_y = 1, M_z = 0) to a 90° pulse. (C) Response of the unit M_z (M_x = 0, M_y = 0, M_z = 1) to a 90° pulse. (D) Response of the unit M_x to a 180° pulse. (E) Response of the M_y to a 180° pulse. (F) Response of the unit M_z to a 180° pulse. Gaussian pulses of 7.8 ms length with 5% truncation were used for both 90° and 180° pulses. The black dashed line, the solid gray line and the solid black line represent the resultant M_x, M_y and M_z, i.e., M_x′, M_y′ and M_z′, respectively.

3.5. Determination of the coherence transfer pathways.

Each CTP consisted of four periods that are separated by the timings of the second, third and fourth RF pulses. The property that the coherence level (order) is unchanged between RF pulses and is only changed by application of an RF pulse [33] was used. The coherence level of the fourth period was set to (−1) from the quadrature detection condition [32]. In the first period, among (+1) and (−1) quantum coherences generated by the first RF pulse, (+1) quantum coherence was chosen due to the crusher gradients in the first and fourth periods. The J-coupled spins of lipids and lactate were treated to be uniformly rotated by the first RF pulse regardless of their chemical shifts due to the large bandwidth of the pulse. The antiphase terms that evolve after the first pulse (see Eqs. 4 to 7) were followed. Transformations of spin operators by the second pulse (Tables S2, S4 and S5-S10) were applied to the I and S spins of the antiphase terms. Similarly, transformations of spin operators by the third and fourth pulses were applied to the product operators resulting after the second and third pulses, respectively. The coherence level at each period for a CTP was determined by summing the coherence levels of spin operators constituting the traced product operator at that period. The CTPs that start with (+1) coherence level, finish with (−1) coherence level and satisfy Eqs. 1 and 2 for the given coherence selection gradient ratios were identified and labeled in the CTP maps (Figs. 4 to 5 and Figs. S1 to S4).

Fig. 4. Coherence transfer pathway maps for the J-coupled lipid spins from the couplings of 1.33–0.90, 1.33–1.65 and 1.33–2.07 ppm.

p = coherence level; g = g₂:g₃:g₄. The coherence transfer pathways corresponding to the indicated gradient ratios are color-labeled. At g = 0:0:0, all the indicated CTPs are allowed regardless of color.

Fig. 5. Coherence transfer pathway maps for the lactate spins from the coupling of 1.33–4.11 ppm.

p = coherence level; g = g₂:g₃:g₄. The coherence transfer pathways corresponding to the indicated gradient ratios are color-labeled. At g = 0:0:0, all the indicated CTPs are allowed regardless of color.

3.6. Calculation of conversion rates.

The conversion rates were calculated for τ₁ = 1/(2J_Lac). The following steps were used for the calculation of the conversion rate for a specific CTP. Step 1: An antiphase term that evolved from I⁺ or S⁺ during the first period was written from Eqs. 4 to 7 with the trigonometric coefficient at t = 1/(2J_Lac) = 72 ms. Step 2: The transformations of I and S spin operators corresponding to the product operator transformations by the second, third and fourth RF pulses for the defined CTP were found from Tables S2, S4 and S5-S10. These are a total of six spin operator transformations. Step 3: The in-phase term that evolves from the (−1) quantum coherence antiphase term in the fourth period was identified from Eqs. 4 to 7. The maximum of the time-dependent in-phase term was found using Matlab. The value was multiplied by the trigonometric coefficient found in Step 1 and the magnitudes of the six spin operator transformations found in Step 2. The result was called the conversion rate for the CTP. We ignored the minor signal modulation by the evolution of J-coupled lipids during the t₁ time of the sequence. For lactate, there is no net signal change during the t₁ time because the I-spin selective 180° pulse in the middle of the t₁ time refocuses the J-coupled evolution between S spin and the passive I spins [16, 31, 38].

3.7. Study on the human subjects.

The study was approved by the Institutional Review Board (IRB) of the University of Pennsylvania and was performed following informed, written consent from two normal volunteers who participated in the study. Each subject lay on the scanner bed and a glass tube (diameter = 2 cm, length = 8 cm) filled with 10 mM lactate was placed on the right upper leg of the subject. The surface coil was placed on top of it. MR experiments were performed as in the phantom experiment. Localized shimming was performed for the 2×2×2 cm voxel centered in the glass tube. The CSI acquisition parameters were: slice thickness = 2 cm, FOV = 16×16 cm or 32×32 cm, 8×8 matrix, vector size = 1,024. The used RF pulses and coherence selection gradients are indicated in the figure captions.

3.8. Analysis of spectral data.

The analysis was performed using the data from one of the two channels of the RF coil except for the CSI presentation of the human subject studies where the sum-of-squares combined mode data from the scanner was used. The .rda files retrieved from the scanner were read by the 3DiCSI software [39]. The free induction decay (FID) data from the interested voxel was extracted from the CSI matrix as an ASCII file and analyzed by the NUTS software (Acorn NMR Inc., Livermore, CA). The spectra were presented and analyzed in the magnitude mode or the phase-sensitive mode; indicated in the figure captions. For the magnitude mode spectra, the full FID data of the target voxel was Fourier transformed without zero-filling. For the phase-sensitive mode spectra, data points before the echo of the FID were discarded and the remaining half echo data was zero-filled to 1,024 or 2,048, Fourier transformed and phase-corrected. The numbers of data points discarded were 62 (pulse length condition 1) and 56 (pulse length condition 2), respectively, for the lactate data of Fig. 3. The number was 56 for the data of Fig. 7, Figs. S5.B-S5.D and Fig. S6. The number was 52 for the data of Figs. 8–10 and Fig. S7. The number was 49 for the data of Figs. S5.E-S5.F. The acquisition and processed vector sizes are indicated in the figure captions.

Fig. 8. Sel-MQC CSI from a human subject with a lactate phantom.

Coherence selection gradient ratio = 0:−1:2 (A), 0.25:−1:2 (B) −0.8:−1:2 (C) and 1:0.8:2 (D). Tissue voxels that exhibited high lipid intensities in (A) are labeled yellow while the lactate phantom voxel is labeled red. Lactate spectra from the phantom voxels of (C) and (D) are displayed in (E) and (F). The durations of g₂, g₃ and g₄ gradients were 1.5 ms, 1.5 ms and 3 ms, respectively. The g₂, g₃ and g₄ gradient amplitudes are calculated similarly to the experiments in Fig. 3 from the gradient ratios and the maximum gradient amplitude of 26 mT/m. Pulse length condition 2 was used. t₁ = 21.8 ms. τ₁ = 1/(2J_Lac) for (A) to (C). τ₁ = 1/(2J_Lac) – t₁ for (D). Slice thickness = 2 cm, CSI FOV = 16×16 cm, 8×8 matrix, TR = 1.5 s, NA = 2. Acquisition vector size = 1,024 and processed vector size = 2,048. All spectra are presented in the phase-sensitive mode.

4. Results and Discussion

4.1. Experimental data for the dependence of lipid and lactate signals on the coherence selection gradients and RF pulses

Figure 3 shows lipid and lactate signals under varied conditions of the pulse sequence. τ₁ = 1/(2J_Lac) and the gradient ratios 0:0:0, 0:−1:2 and α:−1:2 with α = 0.25 were used. The α value was selected from the α value range that experimentally exhibited similar lipid suppressions from the oil phantom, as is presented in a later subsection. Figures 3A–3E are from the oil phantom and Figs. 3G–3J are from the lactate phantom. Figure 3F shows the lactate phantom image, a part of the CSI matrix and the selected voxel. Figures 3A–3C and Figs. 3G–3H were obtained with the pulse length condition 1 (90° and 180° Gaussian RF pulse lengths = 7.8 ms), while Figs. 3D–3E and Figs. 3I–3J were obtained with the pulse length condition 2 (90° Gaussian pulse length = 12 ms, 180° Gaussian pulse length = 5.6 ms). Under the pulse length condition 1, the lipid 1.33 ppm signal decreased 290-fold and the lipid 2.07 ppm signal decreased 26-fold at the gradient ratio 0:−1:2 compared to the respective signals at the gradient ratio 0:0:0 (Fig. 3A vs. Fig. 3B). When the gradient ratio 0.25:−1:2 was used, the 1.33 ppm signal dropped 70-fold further while the 2.07 ppm signal increased by 20% (Fig. 3B vs. Fig. 3C). The total decrease of the 1.33 ppm signal was 20,300-fold while the total decrease of the 2.07 ppm signal was 21-fold. Therefore, while the 1.33 ppm peak disappeared in Fig. 3C, the 2.07 ppm region exhibited a strong peak in the same figure. When the pulse length condition 2 and the gradient ratio 0:−1:2 were used, the 1.33 ppm signal decreased 1,400-fold and the 2.07 ppm signal decreased 135-fold (Fig. 3D) compared to the respective signals in Fig. 3A. When the gradient ratio 0.25:−1:2 was used, the 1.33 ppm signal decreased by a total of 15,260-fold and the 2.07 ppm signal decreased by a total of 1,320-fold compared to signals in Fig. 3A, resulting in suppression of both signals to the noise level (Fig. 3E). Lipid signals at 0.90 ppm and around 4.2 ppm decreased to the noise level at the gradient ratio 0.25:−1:2 under both RF pulse length conditions (Figs. 3C and 3E). The lactate signal was unaffected by the change of gradient ratio from 0:−1:2 to 0.25:−1:2 while it decreased slightly (2%) by the change in RF pulse lengths (see Figs. 3G–3J). The varied dependence of lipid and lactate signals on the gradient ratios and RF pulse lengths are explained by the coherence pathway analysis in the following subsections.

4.2. Coherence pathway analyses of the J-coupled lipids

4.2.1. Coupling of 1.33–0.90 ppm (I₂S₃; I = 1.33 ppm, S = 0.90 ppm, J = 8.0 Hz).

Equations 6.1- 6.7 are used. At t = τ₁ = 1/(2J_Lac), the trigonometric coefficients for the antiphase terms of Eqs. 6.1-6.2 are: cos²(πJτ₁)sin(πJτ₁) = 0.054, cos(πJτ₁)sin²(πJτ₁) = −0.22, sin³(πJτ₁) = 0.92, cos(πJτ₁)sin(πJτ₁) = −0.23, sin²(πJτ₁) = 0.94. Using these coefficients, the antiphase terms of Eq. 6.1 are (−2iI⁺S₀)×0.054 + (−4I⁺(S₁₀S₂₀ + S₂₀S₃₀ + S₃₀S₁₀))×(−0.22) + (8iI⁺S₁₀S₂₀S₃₀)×0.92. The antiphase terms of Eq. 6.2 are (−2iS⁺I₀)×(−0.23) + (−4S⁺I₁₀I₂₀)×0.94.

From Tables S2.A and S4.A, the following properties are found for the transformations of the I and S spin operators by 90° and 180° pulses. P₂ (90°) and P₄ (90°): I⁺ is converted to I⁺ and I₀. I⁻ is converted to I⁻ and I₀. I₀ is converted to I⁺, I₀ and I⁻. S⁺ is converted to S⁺ and S₀. S⁻ is converted to S⁻ and S₀. S₀ is converted to S⁺, S₀ and S⁻. P₃ (180°): I⁺ is converted to I⁻. I⁻ is converted to I⁺. I₀ is converted to -I₀. Each of S⁺, S₀ and S⁻ is converted to a combination of S⁺, S₀ and S⁻. Using these properties, the evolutions through the antiphase terms are written as in Eqs. 13.1-13.5 and the CTP maps are drawn as in Fig. S1. A portion of these maps is presented in Fig. 4. Depending on the gradient ratios, different CTPs that satisfy Eqs. 1 and 2 are allowed. We labeled them with different colors. When the gradient ratio is 0:0:0, all the pathways in these maps are allowed regardless of color. Among them, the blue pathways corresponding to the gradient ratio 1:−1:2 have conversion rates several orders bigger than other pathways. Thus, the signal and conversion rate for the gradient ratio 0:0:0 are explained by using the blue pathways. The red, gray and pink pathways are selected when the gradient ratios are 0:−1:2, 0.5:−1:2 and −1:−1:2, respectively. The green pathway is allowed when the gradient ratio is α:−1:2. Here, α can be any value because the gradient α is applied when the coherence level is zero. That is, if α is not one of −1, 0, 0.5 and 1, only the green pathway is allowed. In Eqs. 13.1-13.5 and Fig. 4, each spin operator indicated that of an individual spin not the sum of operators as used in Eqs. 4 to 7. This convention was applied to all the evolution equations and CTP maps of this manuscript. For the evolutions involving the double and triple antiphase terms, only one pathway was written for the respective evolution, as deviations to other pathways were very small.

$$\begin{array}{l}\hfill I^{+}\stackrel{J}{\to }{I}^{+}{S}_0\stackrel{P_2}{\to }\left(\begin{array}{l}{I}^{+}\\ I_0\end{array}\right)\left(\begin{array}{l}\hfill S^{+}\\ S_0\\ S^{-}\end{array}\right)\stackrel{P_3}{\to }\left(\begin{array}{l}\hfill I^{-}\\ I_0\end{array}\right)\left(\begin{array}{l}\hfill S^{+}\\ S_0\\ S^{-}\end{array}\right)\stackrel{P_4}{\to }{I}^{-}{S}_0\stackrel{J}{\to }{I}^{-}\\ \hfill \stackrel{P_4}{\to }{I}_0{S}^{-}\stackrel{J}{\to }{S}^{-}\end{array}$$ (13.1)

$$I^{+}\stackrel{J}{\to }{I}^{+}{S}_0{S}_0\stackrel{P_2}{\to }{I}^{+}{S}_0{S}_0\stackrel{P_3}{\to }{I}^{-}{S}_0{S}_0\stackrel{P_4}{\to }{I}^{-}{S}_0{S}_0\stackrel{J}{\to }{I}^{-}$$ (13.2)

$$I^{+}\stackrel{J}{\to }{I}^{+}{S}_0{S}_0{S}_0\stackrel{P_2}{\to }{I}^{+}{S}_0{S}_0{S}_0\stackrel{P_3}{\to }{I}^{-}{S}_0{S}_0{S}_0\stackrel{P_4}{\to }{I}^{-}{S}_0{S}_0{S}_0\stackrel{J}{\to }{I}^{-}$$ (13.3)

$$\begin{array}{l}\hfill S^{+}\stackrel{J}{\to }{S}^{+}{I}_0\stackrel{P_2}{\to }\left(\begin{array}{l}\hfill S^{+}\\ S_0\end{array}\right)\left(\begin{array}{l}\hfill I^{+}\\ I_0\\ I^{-}\end{array}\right)\stackrel{P_3}{\to }\left(\begin{array}{l}\hfill S^{+}\\ S_0\\ S^{-}\end{array}\right)\left(\begin{array}{l}\hfill I^{-}\\ I_0\\ I^{+}\end{array}\right)\stackrel{P_4}{\to }{S}^{-}{I}_0\stackrel{J}{\to }{S}^{-}\\ \hfill \stackrel{P_4}{\to }{S}_0{I}^{-}\stackrel{J}{\to }{I}^{-}\end{array}$$ (13.4)

$$S^{+}\stackrel{J}{\to }{S}^{+}{I}_0{I}_0\stackrel{P_2}{\to }{S}^{+}{I}_0{I}_0\stackrel{P_3}{\to }{S}^{-}{I}_0{I}_0\stackrel{P_4}{\to }{S}^{-}{I}_0{I}_0\stackrel{J}{\to }{S}^{-}$$ (13.5)

We calculate the conversion rates at three different gradient ratios used for the experiments in Fig. 3. The coherence pathways for the I spin signal (1.33 ppm) are analyzed first in the pulse length condition 1. Spin operator transformations in Table S2.A were applied to Eqs. 13.1-13.3. When the gradient ratio was 0:0:0, the evolution by the triple antiphase term (Eq. 13.3; the blue pathway in Fig. 4F) contributed the most, the double antiphase term (Eq. 13.2; the blue pathway in Fig. 4E) contributed the next, and the single antiphase term contributed the least (Eq. 13.1; the blue pathway of Fig. 4B, I⁺→I⁺S₀→I⁺S₀→I⁻S₀→I⁻S₀→I⁻). The total conversion rate from these three CTPs was 0.13. For this calculation, the time-dependent in-phase terms evolved from three different antiphase terms after the fourth pulse (Eqs. 6.3 to 6.5) were added in Matlab and the maximum was chosen. When the gradient ratio was 0:−1:2, the red pathway of Fig. 4B, I⁺→I⁺S₀→I⁺S₀→I⁻S⁻→I⁻S₀→I⁻, contributed the most. The conversion rate was 9.6×10^-‍5 which is 1,350-fold smaller than that at the gradient ratio 0:0:0. When the gradient ratio is 0.25:−1:2, only the green pathway of Fig. 4C, I⁺→I⁺S₀→I⁺S⁻→I⁻S⁻→I⁻S₀→I⁻, is allowed. The conversion rate was 1.2×10⁻⁸ which is 8,000 times smaller than that at the gradient ratio 0:−1:2. Under the pulse length condition 2, the spin operator transformations in Table S4.A were used. At the gradient ratio 0:−1:2 (the red pathway of Fig. 4B), the conversion rate was 1.7×10⁻⁴ which is 770-fold smaller than that at the gradient ratio 0:0:0 in the pulse length condition 1. At the gradient ratio 0.25:−1:2 (the green pathway of Fig. 4C), the conversion rate was 2.5×10⁻⁷ which is 680 times smaller than at the gradient rato 0:−1:2. These conversion rate changes explain the observed 1.33 ppm signal changes in Figs. 3A–3E but not fully. To better understand the signal changes at 1.33 ppm, the contributions from other couplings, i.e., from 1.33–1.65 ppm and 1.33–2.07 ppm, should also be considered. These are provided in the next subsections.

The coherence pathways leading to the S spin signal (0.90 ppm) were analyzed similarly. We describe the analysis first in the pulse length condition 1. When the gradient ratio was 0:0:0, the double antiphase term (Fig. 4H) contributed the most, followed by the single antiphase term (the blue pathway of Fig. 4G; S⁺→S⁺I₀→S⁺I₀→S⁻I₀→S⁻I₀→S⁻). The total conversion rate from these two pathways was 0.74. When the gradient ratio was 0:−1:2, the red pathway of Fig. 4B (I⁺→I⁺S₀→I⁺S₀→I⁻S⁻→I₀S⁻→S⁻) contributed the most. Polarization transfer from I spin to S spin occurs for this pathway. The conversion rate was 1.1×10^-3. As there are two I spins per three S spins for this coupling, a factor of 2/3 should be multiplied to compare with the initial S spin magnetization. The conversion rate relative to the initial S spin magnetization was 7.3×10^-4. This is 1,010-fold smaller than when the gradient ratio is 0:0:0. This number closely matched the experimentally measured 1,050-fold change of the 0.90 ppm signal from Fig. 3A to Fig. 3B, demonstrating the ability of our analysis. The S spin signal at the gradient ratio 0.25:−1:2 comes from the green pathway of Fig. 4C, I⁺→I⁺S₀→I⁺S⁻→I⁻S⁻→I₀S⁻→S⁻. Polarization transfer occurs again. The conversion rate was 4.4×10^-6. The conversion rate relative to the initial S spin magnetization was 2.9×10^-6. The disappearance of the 0.90 ppm signal in Fig. 3C matches the calculation. In the pulse length condition 2, the conversion rates relative to the initial S spin magnetization at the gradient ratio 0:−1:2 and 0.25:−1:2 were 3.8×10⁻⁵ and 5.8×10⁻⁸, respectively. These rates are consistent with the very small or unobserved signals at 0.90 ppm in Figs. 3D and 3E.

4.2.2. Coupling of 1.33–1.65 ppm (I₂S₂; I = 1.33 ppm, S = 1.65 ppm, J = 7.1 Hz).

Equations 5.1-5.6 are used. At t = τ₁ = 1/(2J_Lac), the trigonometric coefficients for the antiphase terms of Eqs. 5.1-5.2 are: cos(πJτ₁)sin(πJτ₁) = −0.035, sin²(πJτ₁) = 1. Using these coefficients, the antiphase terms of Eq. 5.1 are (−2iI⁺S₀)×(−0.035) + (−4I⁺S₁₀S₂₀). The antiphase terms of Eq. 5.2 are (−2iS⁺I₀)×(−0.035) + (−4S⁺I₁₀I₂₀). From Tables S2.B and S4.B, similar transformation properties as the spin operators of the previous coupling are found. Therefore, we utilize the CTP maps of Fig. 4.

We first analyze the I spin signal (1.33 ppm) in the pulse length condition 1 (Table S2.B). When the gradient ratio was 0:0:0, the evolution of the double antiphase term (Eq. 13.2 and the blue pathway of Fig. 4E) contributed the most, followed by the evolution of the single antiphase term (the blue pathway of Fig. 4B). The combined conversion rate was 0.72. When the gradient ratio was 0:−1:2, the red pathway of Fig. 4B contributed the most, with the conversion rate of 2.8×10⁻⁴ which is 2,600-fold smaller than at the gradient ratio 0:0:0. When the gradient ratio is 0.25:−1:2, only the green pathway of Fig. 4C is allowed, with the conversion rate of 1.2×10⁻⁶ which is 230-fold smaller than at the gradient ratio 0:−1:2. In the pulse length condition 2 (Table S4.B), the conversion rates at the gradient ratio 0:−1:2 and 0.25:−1:2 were 1.4×10⁻⁴ and 2.6×10⁻⁷, respectively, the latter being 540-fold smaller than the former.

We next analyze the S spin signal (1.65 ppm) in the pulse length condition 1. When the gradient ratio was 0:0:0, the evolution of the double antiphase term (Fig. 4H) contributed the most, followed by the evolution of the single antiphase term (the blue pathway of Fig 4G; S⁺→S⁺I₀→S⁺I₀→S⁻I₀→S⁻I₀→S⁻). The combined conversion rate was 0.86. When the gradient ratio was 0:−1:2, the red pathway of Fig. 4B, I⁺→I⁺S₀→I⁺S₀→I⁻S⁻→I₀S⁻→S⁻, contributed the most. The conversion rate was 1.9×10⁻⁴ which is 4,600-fold smaller than at the gradient ratio 0:0:0. When the gradient ratio is 0.25:−1:2, the signal comes from the green pathway of Fig. 4C, I⁺→I⁺S₀→I⁺S⁻→I⁻S⁻→I₀S⁻→S⁻. The conversion rate was 8.7×10⁻⁷ which is 220-fold smaller than at the gradient ratio 0:−1:2. In the pulse length condition 2, the conversion rates at the gradient ratio 0:−1:2 and 0.25:−1:2 were 9.8×10⁻⁶ and 1.9×10⁻⁸, respectively, the latter being 520-fold smaller than the former. While the 1.65 ppm signals that are not detected in Figs. 3C–3E match the low conversion rates calculated above, the 1.65 ppm signal in Fig. 3B is much higher than expected from the conversion rate when the spectra in Fig. 3A and Fig. 3B are compared. This is explainable by the 1.65 ppm signal coming from the coupling of 1.65–2.36 ppm (I₂S₂ spin system). For the relevant CTP, the red pathway of Fig. 4B applies. The spins at 2.36 ppm will be much more rotated than the spins at 1.33 ppm by the last RF pulse centered at 4.11 ppm. This results in more signals at 1.65 ppm from the I spins of the coupling of 1.65–2.36 ppm (I = 1.65 ppm, S = 2.36 ppm) than from the S spins of the coupling of 1.33–1.65 ppm (I = 1.33 ppm, S = 1.65 ppm). The signals from the red pathway (Fig. 4B) are again dephased at the gradient ratio 0.25:−1:2, consistent with the disappearance of the 1.65 ppm signal in Fig. 3C.

4.2.3. Coupling of 1.33–2.07 ppm (I₂S₂; I = 1.33 ppm, S = 2.07 ppm, J = 7.14 Hz).

Equations 5.1-5.6 are used. At t = τ₁ = 1/(2J_Lac), the trigonometric coefficients for the antiphase terms of Eqs. 5.1-5.2 are: cos(πJτ₁)sin(πJτ₁) = −0.044, sin²(πJτ₁) = 1. Using these coefficients, the antiphase terms of Eq. 5.1 are (−2iI⁺S₀)×(−0.044) + (−4I⁺S₁₀S₂₀) while the antiphase terms of Eq. 5.2 are (−2iS⁺I₀)×(−0.044) + (−4S⁺I₁₀I₂₀). From transformations in Tables S2.C and S4.C, the same types of evolutions and CTP maps as the previous coupling are applied.

In the pulse length condition 1, I spin signals (1.33 ppm) from this coupling had the conversion rates of 0.16, 8.8×10⁻⁴ and 2.1×10⁻⁵ at the gradient ratio 0:0:0, 0:−1:2 and 0.25:−1:2, respectively. The conversion rate at the gradient ratio 0:−1:2 was 180-fold smaller than at the gradient ratio 0:0:0, and the conversion rate at the gradient ratio 0.25:−1:2 was 42 times smaller than at the gradient ratio 0:−1:2. In the pulse length condition 2, the I spin signal conversion rates were 2.1×10⁻⁴ and 7.8×10⁻⁷ at the gradient ratio 0:−1:2 and 0.25:−1:2, respectively, the latter being 270-fold smaller than the former. The conversion rates in the pulse length condition 2 were, therefore, 4.2-fold and 27-fold smaller than the pulse length condition 1 at the gradient ratios 0:−1:2 and 0.25:−1:2, respectively. If we compare conversion rates from three different couplings (1.33–0.90 ppm, 1.33–1.65 ppm and 1.33–2.07 ppm), we can see that the coupling of 1.33–2.07 ppm contributes the most to the 1.33 ppm signal at the gradient ratio 0:−1:2 or 0.25:−1:2. The coupling of 1.33–1.65 ppm contributed the next most. The observed 290-fold decrease of the 1.33 ppm signal from the gradient ratio 0:0:0 to 0:−1:2 in the pulse length condition 1 (Fig. 3A vs. Fig. 3B) is explained by the combined effect from these three couplings. The observed 70-fold decrease of the 1.33 ppm signal by change of the gradient ratio from 0:−1:2 to 0.25:−1:2 (Fig. 3B vs. Fig. 3C) is explained in the same way. The 4.8-fold change of the 1.33 ppm signal in Fig. 3D compared to Fig. 3B and disappearance of the signal in Fig. 3E are also explained.

For the S spin signal (2.07 ppm), the conversion rates in the pulse length condition 1 were 4.2×10⁻¹, 3.5×10⁻⁴ and 7.5×10⁻⁶ at the gradient ratio 0:0:0, 0:−1:2 and 0.25:−1:2, respectively. The same pathways used for the analysis of S spin signals in the previous coupling of 1.33–1.65 ppm were used. The conversion rates under the pulse length condition 2 were 2.1×10⁻⁵, and 7.8×10⁻⁸ at the gradient ratio 0:−1:2 and 0.25:−1:2, respectively. While the absence of the 2.07 ppm signal in Fig. 3E matches the very low conversion rate, the large signals at 2.07 ppm in Figs. 3B–3D do not match the small conversion rates. To understand the phenomenon, we examine the coupling of 2.07–5.38 ppm in the next subsection.

4.2.4. Coupling of 2.07–5.38 ppm (I₂S; I = 2.07 ppm, S = 5.38 ppm, J = 6.2 Hz).

Equations 4.1- 4.5 are used. At t = τ₁ = 1/(2J_Lac), the trigonometric coefficients for the antiphase terms of Eqs. 4.1-4.2 are: sin(πJτ₁) = 0.99, cos(πJτ₁)sin(πJτ₁) = 0.17, sin²(πJτ₁) = 0.97. Using these coefficients, the antiphase term of Eq. 4.1 is (−2iI⁺S₀)×0.99 and the antiphase terms of Eq. 4.2 are (−2iS⁺I₀)×0.17 + (−4S⁺I₁₀I₂₀)×0.97.

From Tables S2.D and S4.D, we find the following properties on the transformation of I and S spin operators of this coupling by 90° and 180° pulses. P₂ (90°) and P₄ (90°): I⁺ is converted to I⁺ and I₀. I⁻ is converted to I⁻ and I₀. I₀ is converted to I⁺, I₀ and I⁻. Each of S⁺, S₀ and S⁻ is converted to a combination of S⁺, S₀ and S⁻. P₃ (180°): Each of I⁺, I₀ and I⁻ is converted to a combination of I⁺, I₀ and I⁻. Coherence orders of S⁺, S₀ and S⁻ are unchanged. Using these properties, Eqs. 14.1-14.3 are derived and the CTP maps are drawn as in Fig. S2. As before, minor deviation to other pathways in the evolution of the double antiphase term was neglected.

$$\begin{array}{l}\hfill I^{+}\stackrel{J}{\to }{I}^{+}{S}_0\stackrel{P_2}{\to }\left(\begin{array}{l}\hfill I^{+}\\ I_0\end{array}\right)\left(\begin{array}{l}\hfill S^{+}\\ S_0\\ S^{-}\end{array}\right)\stackrel{P_3}{\to }\left(\begin{array}{l}\hfill I^{+}\\ I_0\\ I^{-}\end{array}\right)\left(\begin{array}{l}\hfill S^{+}\\ S_0\\ S^{-}\end{array}\right)\stackrel{P_4}{\to }{I}^{-}{S}_0\stackrel{J}{\to }{I}^{-}\\ \hfill \stackrel{P_4}{\to }{I}_0{S}^{-}\stackrel{J}{\to }{S}^{-}\end{array}$$ (14.1)

$$\begin{array}{l}\hfill S^{+}\stackrel{J}{\to }{S}^{+}{I}_0\stackrel{P_2}{\to }\left(\begin{array}{l}\hfill S^{+}\\ S_0\\ S^{-}\end{array}\right)\left(\begin{array}{l}\hfill I^{+}\\ I_0\\ I^{-}\end{array}\right)\stackrel{P_3}{\to }\left(\begin{array}{l}\hfill S^{+}\\ S_0\\ S^{-}\end{array}\right)\left(\begin{array}{l}\hfill I^{+}\\ I_0\\ I^{-}\end{array}\right)\stackrel{P_4}{\to }{S}^{-}{I}_0\stackrel{J}{\to }{S}^{-}\\ \hfill \stackrel{P_4}{\to }{S}_0{I}^{-}\stackrel{J}{\to }{I}^{-}\end{array}$$ (14.2)

$$S^{+}\stackrel{J}{\to }{S}^{+}{I}_0{I}_0\stackrel{P_2}{\to }{S}^{+}{I}_0{I}_0\stackrel{P_3}{\to }{S}^{-}{I}_0{I}_0\stackrel{P_4}{\to }{S}^{-}{I}_0{I}_0\stackrel{J}{\to }{S}^{-}$$ (14.3)

We first analyze the I spin signal in the pulse length condition 1. When the gradient ratio was 0:0:0, most I spin signals came from the blue pathway of Fig. S2.B, I⁺→I⁺S₀→I⁺S₀→I⁻S₀→I⁻S₀→I⁻. The conversion rate was 0.34. When the gradient ratio was 0:−1:2, the biggest contribution to the I spin signal came from the green pathway of Fig. S2.C, I⁺→I⁺S₀→I⁺S⁻→I⁻S⁻→I⁻S₀→I⁻. The conversion rate was 3.5×10^-2. This is only a 10-fold drop from the conversion rate at the gradient ratio of 0:0:0 and partially explains why we see a large signal at 2.07 ppm in Fig. 3B. When the gradient ratio is 0.25:−1:2, the same green pathway is selected. Thus, the 2.07 ppm signal is not further reduced. Interestingly, the 2.07 ppm signal increased at the gradient ratio 0.25:−1:2 compared to the signal at the gradient ratio 0:−1:2 (Fig. 3B vs. Fig. 3C). This phenomenon can be understood by considering contributions from two different couplings. One is from the I spin of the coupling of 2.07–5.38 ppm (I = 2.07 ppm, S = 5.38 ppm). The other is from the S spin of the coupling of 1.33–2.07 ppm (I = 1.33 ppm, S = 2.07 ppm). At the gradient ratio 0:−1:2, the I spin signal from the green pathway of the former coupling and the S spin signal from the red pathway of the latter coupling have opposite signs with the same orders of magnitudes, resulting in a partial cancellation of the signal. At the gradient ratio of 0.25:−1:2, the S spin signal from the green pathway of the latter coupling is much smaller than the I spin signal from the green pathway of the former coupling. Therefore, the signal cancellation is negligible. This results in a bigger 2.07 ppm signal at the gradient ratio 0.25:−1:2 than the signal at the gradient ratio of 0:−1:2. The 2.07 ppm signals were able to be reduced by using the pulse length condition 2 (Figs. 3D–3E). In this condition (Table S4.D), the conversion rate for the green pathway was 1.1×10^-3. This is 32-fold smaller than that in the pulse length condition 1, which matches the decrease of the 2.07 ppm signal to the noise level observed in Fig. 3E. A puzzle is that it does not explain a significant amount of signal remaining at the gradient ratio 0:−1:2 in the pulse length condition 2 (Fig. 3D), which was ~15% of the signal of Fig. 3C. Besides, the 10-fold drop in the conversion rate from the gradient ratio 0:0:0 to 0.25:−1:2 in the pulse length condition 1 does not match the experimentally observed 21-fold decrease for the 2.07 ppm signal (Fig. 3A vs. Fig. 3C). The discrepancy is explainable by introducing the contribution from another coupling. A correlated 2D spectroscopy (COSY) NMR data of corn oil by Claxson et al. [40] indicates that there is spin-spin coupling between protons at 2.07 ppm and 2.79 ppm, though they did not determine the J-coupling constant. By calculating the off-resonance rotation matrices for the S spins at 2.79 ppm, the following transformations are derived in the pulse length condition 2. 90° pulse: S⁺ S⁺(0.99) + S_z(−0.0001 + 0.04i), S⁻ S⁻(0.99) + S_z(−0.0001 – 0.04i), S_z (S⁺/2)(0.005 + 0.04i) + (S⁻/2)(0.005 – 0.04i) + S_z. 180° pulse: S⁺ (S⁺/2)(0.35 – 1.64i) + (S⁻/2)(0.33 – 0.02i) + S_z(−0.49 – 0.56i), S⁻ (S⁺/2)(0.33 + 0.03i) + (S⁻/2)(0.35 + 1.64i) + S_z(−0.49 + 0.56i), S_z (S⁺/2)(−0.44 – 0.59i) + (S⁻/2)(−0.44 + 0.59i) + 0.67S_z. Using these and the I spin transformations in Table S4.D, we calculated the conversion rates. The red pathway of Fig. 4B and the green pathway of Fig. 4C were used. The contribution from the red pathway of Fig. 4A is much smaller than that from the red pathway of Fig. 4B, thus it was ignored. When the unknown J-coupling constant for this I₂S₂ spin system was set to 10 Hz, the conversion rate for the red pathway of Fig. 4B was 4.7×10⁻³ and the conversion rate for the green pathway of Fig. 4C was 9.3×10^-5. Similarly, the conversion rates in the pulse length condition 1 for the indicated red and green pathways were calculated to be 1.4×10⁻² and 2.8×10⁻³, respectively. The experimentally observed 6-fold change in the 2.07 ppm signal between Fig. 3C and Fig. 3D is explained by these conversion rates. The 2.07 ppm signal in the former condition (Fig. 3C) was dominated by the conversion rate from the coupling of 2.07–5.38 ppm while the 2.07 ppm signal in the latter condition (Fig. 3D) was dominated by the conversion rate from the coupling of 2.07–2.79 ppm. Similarly, the 2.07 ppm signals in Fig. 3B and Fig. 3E were dominated by the conversion rates from the coupling of 2.07–5.38 ppm. The coupling of 2.07–2.79 ppm also contributes to the 2.07 ppm signal at the gradient ratio 0:0:0 by the blue pathways in Figs. 4D and 4E. This explains how the 21-fold change in the 2.07 ppm signal between Fig. 3A and Fig. 3C is possible.

For the S spin signal (5.38 ppm), the biggest contribution at the gradient ratio 0:−1:2 came from the green pathway of Fig. S2.C, i.e., I⁺→I⁺S₀→I⁺S⁻→I⁻S⁻→I₀S⁻→S⁻. The conversion rates relative to the initial S spin magnetization were 4.9×10⁻⁴ in the pulse length condition 1 and 8.8×10⁻⁵ in the pulse length condition 2, respectively. The use of the gradient ratio 0.25:−1:2 did not change the conversion rates because the same green pathways were selected. The calculated low conversion rates match the negligible or undetectable signals in the 5.38 ppm region in Figs. 3C–3E. They, however, do not match a large signal in the same region in Fig. 3B. These signals and their changes with the gradient ratio and RF pulse length can be explained by the coupling of 4.16–5.29 ppm (see SM-1 and Fig. S3 of the Supplementary Materials). Also, the signals around 4.2 ppm and their changes with the gradient ratio and RF pulse length can be explained by the coupling of 4.16–4.37 ppm (see SM-1 and Fig. S4 of the Supplementary Materials).

4.3. Coherence pathway analysis of lactate (I₃S; I = 1.33 ppm, S = 4.11 ppm, J = 6.93 Hz)

Equations 7.1-7.6 are used. At t = τ₁ = 1/(2J_Lac), the trigonometric coefficients for the antiphase terms of Eq. 7.1-7.2 are: sin(πJτ₁) = 1.00, cos²(πJτ₁)sin(πJτ₁) = 0.00, cos(πJτ₁)sin²(πJτ₁) = 0.00, sin³(πJτ₁) = 1.00. Using these coefficients, the antiphase term of Eq. 7.1 is −2iI⁺S₀ and the antiphase term of Eq. 7.2 is 8iS⁺I₁₀I₂₀I₃₀. From Tables S5 and S6, we find the following properties of the transformation of I and S spin operators by 90° and 180° pulses. P₂ and P₄ (90°): The coherence orders of I⁺, I₀ and I⁻ are not changed. Each of S⁺ and S⁻ is transformed into a combination of S⁺, S₀ and S⁻. S₀ is transformed into S⁺ and S⁻. P₃ (180°): I⁺ is transformed to I⁻. I⁻ is transformed to I⁺. I₀ is unchanged. The coherence orders of S⁺, S₀ and S⁻ are not changed. We ignored minute deviations. From these properties, the evolutions of I and S spin operators are written as Eqs. 15.1-15.2 and the CTP maps are drawn as in Fig. 5.

$$I^{+}\stackrel{J}{\to }{I}^{+}{S}_0\stackrel{P_2}{\to }{I}^{+}\left(\begin{array}{l}{S}^{+}\\ S^{-}\end{array}\right)\stackrel{P_3}{\to }{I}^{-}\left(\begin{array}{l}{S}^{+}\\ S^{-}\end{array}\right)\stackrel{P_4}{\to }{I}^{-}{S}_0\stackrel{J}{\to }{I}^{-}$$ (15.1)

$$S^{+}\stackrel{J}{\to }{S}^{+}{I}_0{I}_0{I}_0\stackrel{P_2}{\to }\left(\begin{array}{l}{S}^{+}\\ S_0\\ S^{-}\end{array}\right)I_0{I}_0{I}_0\stackrel{P_3}{\to }\left(\begin{array}{l}{S}^{+}\\ S_0\\ S^{-}\end{array}\right)I_0{I}_0{I}_0\stackrel{P_4}{\to }{S}^{-}{I}_0{I}_0{I}_0\stackrel{J}{\to }{S}^{-}$$ (15.2)

When the gradient ratio is 0:−1:2, just one CTP is allowed for I spins. The pathway is I⁺→I⁺S₀→I⁺S⁻→I⁻S⁻→I⁻S₀→I⁻, labeled green in Fig. 5A. The same green pathway is selected when the gradient ratio is 0.25:−1:2. The conversion rates for the I spin signal by this pathway were 0.50 for both pulse length conditions using the transformations in Tables S5 and S6. Let us write the CTP as SQC→ZQC→DQC→SQC. The second RF pulse transfers 50% of SQC to ZQC, the third pulse fully converts ZQC to DQC and the fourth pulse transfers 100% of DQC to SQC. As the gradient α is applied during ZQC (see Fig. 5A), the allowed coherence pathways and the conversion rates are not affected by the value of α. This explains why there is no difference in the lactate signal between the gradient ratio condition 0:−1:2 and 0.25:−1:2 (Fig. 3G vs. Fig. 3H or Fig. 3I vs. Fig. 3J). In contrast, lipid signals had big changes between the two gradient ratio conditions, as we saw from the analysis of the conversion rates and the experimental data (Fig. 3B vs. Fig. 3C, and Fig. 3D vs. Fig. 3E).

In Fig. 3, we see a slight decrease (2%) in the lactate signal intensities when the pulse length condition 2 is used compared to the pulse length condition 1. This is explainable from the theoretical echo signal intensity. The lactate echo is formed when the following equation is met: g₀×(1/2J_lac) + 0×g₀×(t₁/2)+ (−2)×g₀×(t₁/2)+ (−1)×g₀×τ₂ = 0, where g₀ is the background gradient. To derive this equation, Eqs. 1 and 2 and the coherence levels of lactate, p(1, 0, −2, −1), were used. The equation is met at τ₂ = (1/2J_Lac) – t₁, which leads to TE = τ₁ + t₁ + τ₂ = 1/(J_Lac). From Eq. 7.3, the in-phase lactate signal at the echo has the modulation factor of sin(πJ_Lacτ₂) = sin(πJ_lac×[(1/2J_Lac) – t₁]) = cos(πJ_Lact₁). In our study, t₁ is 18.6 ms in the pulse length condition 1 and 20.6 ms in the pulse length condition 2. With these t₁ values, cos(πJ_lact₁) = 0.92 and 0.90, respectively. This matches the 2% difference in the lactate signal (see Figs. 3G–3J). The S spin signal (4.11 ppm) is allowed when the gradient ratio is 0:0:0 or −1:−1:2 as indicated in Figs. 5B-D. When the gradient ratio is 0:−1:2 or 0.25:−1:2, no pathway for the S spin signal is allowed (see Fig. 5).

4.4. Lipid and lactate signals at varied α values

Experiments on lipids were performed using an oil-filled tube at varied α of the gradient ratio α:−1:2 in the pulse length condition 2. Figures 6A–6F show the spectra at α = −1, −0.5, 0, 0.25, 0.5 and 1, respectively. In Fig. 6A, big signals are observed in the 4.0–4.6 ppm region. These signals are consistent with the allowed CTPs (labeled pink) for the 4.16 ppm and 4.37 ppm signals at α = −1 (Figs. S3 and S4). The spectra in Fig. 6B (α = −0.5) and Fig. 6D (α = 0.25) correspond to the green pathway of Fig. 4C. The negligible signals agree with the very small conversion rates. The spectral pattern of the lipid signals at α = 0 (Fig. 6C) was explained in the previous subsections. The lipid spectrum in Fig. 6E (α = 0.5) can be explained by the gray pathway of Fig. 4A; I⁺→I⁺S₀→I⁺S⁺→I⁻S₀→I⁻S₀→I⁻. The conversion rate of this pathway for the coupling of 1.33–2.07 ppm is calculated to be 2.1×10^-4. This is identical to the conversion rate of the red pathway (α = 0; Fig. 4B) for the same coupling. This matches the similar amplitudes of the 1.33 ppm peak observed in Fig. 6C (α = 0) and Fig. 6E (α = 0.5). In Fig. 6E, we see a significant amount of signal at 2.07 ppm. As the conversion rate for the 2.07 ppm signal from the green pathway of Fig. S2.C in the pulse length condition 2 is very small, the observed signal is not explainable from the coupling of 2.07–5.58 ppm. The signal, however, can be explained by the coupling of 2.07–2.79 ppm. The gray pathway in Fig. 4A that we used for the analysis of the 1.33 ppm signal is allowed at α = 0.5 for this coupling. The conversion rate of the 2.07 ppm signal from this pathway is calculated to be 4.8×10⁻³, assuming the J-coupling constant of 10 Hz. This number is close to 4.7×10⁻³, the conversion rate of the 2.07 ppm signal from this coupling at the gradient ratio 0:−1:2, calculated in a previous subsection. This explains why the 2.07 ppm signal in Fig. 6E (α = 0.5) has a similar amplitude as the 2.07 ppm signal in Fig. 6C (α = 0). The spectrum in Fig. 6F (α = 1) shows large signals at 1.33 ppm and 0.90 ppm as well as at 2.07 ppm. This matches the large conversion rates for the blue pathways in Figs. 4, S1 and S2 that are selected with the gradient ratio 1:−1:2 as presented in the previous subsections.

In Fig. 6G we plotted the lipid signal intensities of the 1.33 ppm region with α varied from −1 to 1, normalized to the lipid signal intensity at α = 1. The lipid signals had maxima at α = 1, 0 and 0.5, as was explained by the allowed CTPs in the above paragraph. For α values in the range −0.9 ≤ α ≤ −0.05 or 0.05 ≤ α ≤ 0.4, the lipid signals were in the noise level. At α = −1, the lipid signal slightly increased due to the influence of the big signals at 4.16 and 4.37 ppm (Fig. 6A). For α values around 1, 0 and 0.5, signal attenuations by gradients occurred. We also plotted the theoretical signal attenuation by Eq. 11 using the solid and dashed lines for the CTPs with coherence levels of p(1, 1, −1, −1), p(1, 1, −2, −1) and p(1, 2, −1, −1) that are selected with α = 1, 0 and 0.5, respectively. Using μ = 6 in Eq. 11, the experimental data and theoretical signal attenuation matched very well, confirming the validity of Eq. 11. The value μ bigger than 1 is explainable by the geometry of the CSI voxels on the phantom (Fig. 3F). While the target voxel is homogeneously filled with oil, the nearby voxels are only partially filled with oil, reducing the effective sizes of those voxels. Signals in these voxels are less attenuated by the gradients than the target voxel. The combined effect has resulted in μ = 6 for this geometry.

Similar experiments were performed with a 10 mM lactate phantom. Figures 7A–7F show lactate spectra with the gradient ratio α:−1:2 at α = −1, −0.8, −0.4, 0, 0.5 and 0.9, respectively. Figure 7G is a plot of the lactate signals at −1 ≤ α ≤ 0.9 normalized to the lactate signal at α = 0. The integrations of the signals for 0–3 ppm were used for the analysis. Lactate signals were almost constant; the coefficient of variation = 1%. This is consistent with the theoretical prediction that lactate spins satisfy Eq. 1 regardless of α of the gradient ratio α:−1:2 by the green pathway of Fig. 5A that has the coherence levels of p(1, 0, −2, −1). We excluded data at α = 1 because α = 1 corresponds to the large lipid signals (see Fig. 6F) and two lactate echoes with different echo times [16] will be formed by two allowed CTPs (Fig. 5A). The Infiltrated water signal was the biggest when α = −1 and decreased as α increased. This is consistent with the signal attenuation expected from Eq. 11 for the water CTP with the coherence levels of p(−1, −1, −1, −1).

4.5. Study on the human subjects

The pulse sequence with the pulse length condition 2 was applied to two normal volunteers. A small 10 mM lactate phantom (see the Methods section) was placed on the upper leg of each subject. To improve the lipid signal attenuation, the base gradient duration for the coherence selection gradients was increased from 0.9 ms to 1.5 ms. This increases k_x, k_y and k_z of Eq. 11 for the coherence pathways that need to be dephased by gradients. Figures 8A–8D show Sel-MQC-CSI multi-voxel spectra in the chemical shift range of 0–4 ppm at four different coherence selection gradient ratio conditions overlaid on a T1-weighted image of a subject. Voxels for the lipid-rich tissues are labeled yellow while the lactate voxel is labeled red. When the gradient ratio 0:−1:2 was used (Fig. 8A), the intensities of lipid signals from the yellow voxels were similar to the lactate signal from the red voxel. When the gradient ratio 0.25:−1:2 was applied, lipid signals were suppressed substantially but the suppression was incomplete and voxel-dependent (Fig. 8B). The limited lipid suppression is attributable to inhomogeneous spin distribution in vivo which increases the correction factor μ of Eq. 11. The lipid signals were suppressed to the noise level by applying the gradient ratio −0.8:−1:2 (Fig. 8C) which increases the denominator of Eq. 11 by (0.8/0.25)³ = 33 times compared to the gradient ratio 0.25:−1:2 for the coherence pathway p(1, 1, −2, −1), the dominant contributor to lipid signals at the gradient ratio 0:−1:2 that we identified in Section 4.2.

In Fig. 8A, the resonances of the lipid peaks appeared downfield shifted compared to the assignments in Fig. 3 when the lactate peak in the phantom voxel was referenced to be 1.33 ppm. In voxel a, the lipid peaks that had been at 0.90 ppm, 1.33 ppm, 2.07 ppm and 2.79 ppm in Fig. 3 appeared 0.32 ppm downfield from them. Lipid peaks in voxels b, c, d and e also exhibited resonance shifts by 0.36, 0.43, 0.66 and 0.76 ppm, respectively. These shifts are attributed to field inhomogeneity left after localized shimming on the phantom, as lactate and lipids have almost the same chemical shifts in vivo; 1.33 ppm vs. about 1.30 ppm [41]. Although such resonance shifts do not change the coherence pathways selected by the gradient ratio 0:−1:2 or α:−1:2, the shifts alter the conversion rates of these coherence pathways. Let us consider the lipid signals in voxel a. We have shown in Section 4.2 that the biggest contribution to the 1.33 ppm lipid signal was the coupling of 1.33–2.07 ppm. With the 0.32 ppm downfield shift of resonances, both the S₀→S⁻ and S⁻→S₀ transformations of the S spin (the shifted 2.07 ppm resonance) by the second and fourth RF pulses centered at 4.11 ppm are increased. This results in increased conversion rates for the coherence pathways, p(1, 1, −2, −1) and p(1, 0, −2, −1), on the shifted 1.33 ppm resonance signal allowed at α = 0. Likewise, due to an increased effect on the shifted 2.79 ppm resonance of the coupling of 2.07–2.79 ppm by the second and fourth RF pulses, the conversion rates for both coherence pathways, p(1, 1, −2, −1) and p(1, 0, −2, −1), on the shifted 2.07 ppm resonance signal are increased. The conversion rates for the coherence pathways on the shifted 0.90 ppm and 2.79 ppm resonance signals for the couplings of 1.33–0.90 ppm and 2.07–2.79 ppm, respectively, are also increased. This explains the reason for the large lipid signals in Fig. 8A. The lipid signals in voxels b, c, d and e did not increase further compared to the signals in voxel a, despite bigger resonance shifts. This is because the I⁺ to I⁻ transformations by the third RF pulse centered at 1.33 ppm that are needed for the above-mentioned coherence pathways of lipids are decreased with the downfield shift of the I spin resonances. In voxels d and e, the shifted 2.07 ppm and 2.79 ppm peaks have disappeared. This is explained by the inability of the third RF pulse to induce sufficient I⁺ to I⁻ transformations to these lipids due to increased chemical shift differences between the resonance peaks of these lipids and the center frequency of the third RF pulse.

In Fig. 8C (α = −0.8), most lipid signals in Fig. 8A (α = 0) were suppressed to the noise level. This indicates that the coherence pathway p(1, 1, −2, −1) was still the dominant contributor to the signals at the gradient ratio 0:−1:2 after the resonance shifts and that the conversion rates for the coherence pathway p(1, 0, −2, −1) that is not affected by α were very low. However, there were small remnant signals observable at the shifted 2.07 ppm and 2.79 ppm resonances whereas the signals at the shifted 0.90 ppm and 1.33 ppm resonances were below the noise level. This indicates that the contributions from the coherence pathway p(1, 0, −2, −1) to the shifted 2.07 ppm and 2.79 ppm resonance signals were non-negligible. Such signals can be suppressed by increasing the RF pulse lengths, thereby reducing the conversion rates for this coherence pathway. A phantom study demonstrated that by using the 90° RF pulse of 13.5 ms and the 180° RF pulse of 7.8 ms, the signals from the shifted 2.07 ppm and 2.79 ppm resonances of lipids were fully suppressed to below the noise level when the resonance shift of lipids was 0.30 ppm (Fig. S5). In another phantom study where the resonance shift of lipids was 0.14 ppm, the pulse length condition 2 was sufficient to suppress lipid signals from these resonances (Fig. S6).

We also tested the gradient ratio 1:β:2 with β = 0.8 (Fig. 8D). This gradient ratio selects the SQC→DQC→ZQC→SQC (I⁺S₀→I⁺S⁺→I⁻S⁺→I⁻S₀; p(1, 2, 0, −1)) pathway of lactate (Fig. 5A; orange pathway) while effectively dephasing the coherence pathways of the J-coupled lipids allowed at the gradient ratio 1:β:2 with β = 0, −0.5 and −1 (see Fig. S7); the corresponding CTPs are I⁺S₀→I⁺S⁺→I⁻S₀→I⁻S₀ (p(1, 2, −1, −1)), I⁺S₀→I⁺S₀→I⁻S⁻→I⁻S₀ (p(1, 1, −2, −1)) and I⁺S₀→I⁺S₀→I⁻S₀→I⁻S₀ (p(1, 1, −1, −1)), respectively. We set τ₁ at 1/(2J_Lac) – t₁ instead of 1/(2J_Lac) to match the echo time (TE) of lactate the same as the experiment of Fig. 8C. The echo position is calculated by the equation, g₀×[(1/2J_lac) – t₁] + 2×g₀×(t₁/2) + 0×g₀×(t₁/2) + (-1)×g₀×τ₂ = 0, which is met at τ₂ = 1/(2J_Lac) and leads to TE = 1/J_Lac. Lipid signals in the yellow voxels were suppressed to the noise level (Fig. 8D), similarly to Fig. 8C. Figure 8E and Fig. 8F show the spectra from the phantom voxels of Fig. 8C and Fig. 8D, respectively. The clean lactate spectra were observed in both conditions owing to the suppression of lipid signals from the nearby voxels. The amplitudes and lineshapes of the lactate signals were different between the two spectra, however. This comes from the different contributions of the antiphase component of lactate from the two experimental conditions, which are studied in more detail in the next subsection.

In a study on another subject (Fig. 9), we prescribed the in-plane voxel sizes of the Sel-MQC-CSI twice as big as those in Fig. 8 so that one of the voxels contains both the lactate phantom and the fatty tissue of the subject. We varied α of the gradient ratio α:−1:2. Figures 9A–9G show the CSI spectra from six voxels at α = −0.8, −0.5, −0.25, 0, 0.25, 0.5 and 0.8 in the chemical shift range of 0–4 ppm. The lactate signals along with lipid signals of varying intensities were observed depending on α. Consistent with theoretical expectation in Fig. 6G, the best lipid suppression was achieved at α = −0.8 and the least suppression was observed at α = 0.8. The spectrum at the orange voxel in Fig. 9D (α = 0) shows the superposition of lactate signal from the coherence pathway p(1, 0, −2, −1) and the lipid signals from the coherence pathway p(1, 1, −2, −1). When the lactate peak was referenced at 1.33 ppm, lipid peaks in this voxel appeared 0.18 ppm downfield shifted from the assignments in Fig. 3. The lipid signals in this voxel were much smaller than those from voxel a of Fig. 8A (α = 0) when compared to the lactate signal. This is explained by less increase in the conversion rates for the coherence pathway p(1, 1, −2, −1) of lipids due to a smaller resonance shift. The spectrum at the orange voxel in Fig. 9F (α = 0.5) represents the superposition of lactate signal from the coherence pathway p(1, 0, −2, −1) and the lipid signals from the coherence pathway p(1, 2, −1, −1). At α = −0.8, the above two coherence pathways of lipids were fully dephased by gradients while the coherence pathway of lactate was not affected, resulting in a lactate-only spectrum as displayed in Fig. 9H. While we used the pulse length condition 2, employing longer RF pulse lengths as in Figs. S5.E-S5.F could be an even better option, considering the resonance shift of lipids. Lipid signals from the coherence pathway p(1, 0, −2, −1) are not dephased by gradients and may have slightly affected the baseline of the spectrum. This coherence pathway is better suppressed with the longer RF pulses. There are two issues with this approach, however. One is that there will be a signal loss of lactate due to cos(πJ_Lact₁) dependence of signal. The theoretically predicted lactate signal difference between the mentioned two pulse length conditions is 2%. Thus, the signal decrease is not much of a problem. A bigger issue is that the doublet shape of lactate can be significantly damaged due to an increased antiphase component coming from the increase in t₁ time. The problem can be avoided by using the scheme used for Fig. 8D, as discussed in the next subsection.

Fig. 9. Sel-MQC CSI from a human subject with a lactate phantom at varied α of the gradient ratio α:−1:2.

(A-G) Spectra of the 1–4 ppm region in six voxels at α = −0.8, −0.5, −0.25, 0, 0.25, 0.5 and 0.8, respectively. A 10 mM lactate phantom was placed on the upper leg of the subject. The CSI FOV = 32×32 cm; 8×8 matrix; voxel size = 4×4×2 cm. Other parameters were the same as those for Figs. 8A–8C. (H) Spectrum from the red voxel in (A). All spectra are presented in the phase-sensitive mode.

In the in vivo lactate detection condition, the chemical shifts of lactate and the biggest peak of lipids almost coincide [41]. In that case, for a locally shimmed voxel of the Sel-MQC-CSI, there will be no resonance shift of lipids and associated effects on the coherence pathways as long as the center frequencies of the RF pulses of the sequence are correctly set at the lactate resonances. If multiple voxels are of interest and there is field inhomogeneity in them, chemical shifts of lactate and lipid will be shifted from the on-resonance condition in some voxels. Lipid signals can increase upon the resonance shifts. However, our study in this subsection demonstrated that the increased lipid signals can be suppressed by using the effective gradient combinations and enlargement of RF pulse lengths. Hindrance to signal attenuation by in vivo spin distribution was shown to be able to be overcome by improving signal attenuation with optimal gradient combinations. The base gradient duration of the coherence selection gradients that we used can be further increased if needed; the signal attenuation factor is proportional to 1/(t_G)³ by Eq.11 where t_G is the base gradient duration. While lactate signals will decrease in voxels having resonance shifts, the decreases can be estimated theoretically. By using the revised off-resonance rotation matrices (Tables S7 to S10), the estimated lactate signals have a 2% reduction for the +0.15 ppm resonance shift and an 11% reduction for the +0.30 ppm resonance shift compared to the on-resonance condition in the pulse length condition 2. The estimated lactate signals have a 4% reduction for the +0.15 ppm resonance shift and a 21% reduction for the +0.30 ppm resonance shift when the 90° pulse of 13.5 m and the 180° pulse of 7.8 ms are used.

4.6. Comparison of the lactate signals from the two different editing conditions and evaluation of the editing efficiencies

To compare the lactate editing efficiencies and the lineshapes from the two different editing conditions studied in the previous subsection, additional experiments were performed on a 10 mM lactate phantom without a subject. The unedited lactate experiment was performed by modifying the sequence of Fig. 1. The center frequencies of the second and fourth pulses were moved to 15 ppm to remove the effects of these pulses. The gradients g₂, g₃ and g₄ were all set to zero. The interval between the first and third pulses was set at 1/(2J_Lac) so that the echo is formed at TE = 1/J_Lac and the J-coupled evolution effect is canceled out. Figure 10A shows the obtained lactate spectrum from the unedited lactate experiment. Figure 10B and Fig. 10C are spectra obtained using the sequences used for Fig. 8C and Fig. 8D, respectively. The spectrum of Fig. 10B reflects a combination of the in-phase and antiphase terms of Eq. 7.3 at t = τ₂ = 1/(2J_Lac) – t₁. On the other hand, the spectrum of Fig. 10C reflects the pure in-phase term as the antiphase term in Eq. 7.3 is removed at t = τ₂ = 1/(2J_Lac), resulting in the same lineshape as the unedited spectrum (Fig. 10A). The removal of the antiphase term has resulted in a much-improved lineshape.

The theoretically predicted in-phase lactate signal at the echo can be calculated by multiplying the trigonometric coefficients of the antiphase term of Eq. 7.1 and the in-phase term of Eq. 7.3. A factor 0.5, which arises when the antiphase SQC term of the first period of the sequence is transformed to ZQC and DQC by the second RF pulse, is further multiplied to it. There is no signal loss by J-evolution during the t₁ time, as explained in Section 3.6. The signal modulation factor for the in-phase term of lactate is, then, 0.5×sin(πJ_Lacτ₁)×sin(πJ_Lacτ₂) = 0.5×cos(πJ_Lact₁) for the conditions of both Figs. 10B and 10C, excluding the T2 relaxation and diffusion effects.

We evaluated the lactate editing efficiencies by comparing the areas of the edited spectra (Figs. 10B and 10C) with that of the unedited spectrum (Fig. 10A). The editing efficiencies were 42% for both spectra of Fig. 10B and Fig. 10C. The identical editing efficiency in the two conditions is consistent with the theoretical prediction of the in-phase terms in these conditions. The antiphase term at the echo for the experiment of Fig. 10B altered the lactate lineshape from the one without the antiphase term (Fig. 10C), but it did not affect the total spectral area as the antiphase term has a zero net area [32]. Using the sequence parameter t₁ = 21.8 ms, the theoretical in-phase modulation factor 0.5×cos(πJ_Lact₁) is 44.5%. Thus, there is a 5% difference between the theoretical and experimental editing efficiencies. Using the gradient ratio 1:0:2 (Fig. 10D) or reducing the strengths of the editing gradients to 10% of the original condition (Fig. 10E) did not change the measured lactate editing efficiency (Fig. 10F). This indicates that diffusion did not affect the measured lactate signal. The 5% difference is, then, reasonably inferred to have come from the difference in T2 relaxation times of lactate between the SQC and the ZQC/DQC states.

4.7. Summary and further discussions

In this study, we inspected the coherence pathways of J-coupled lipids and lactate upon the Sel-MQC lactate editing sequence by a novel product operator analysis. We also investigated the strategy to effectively attenuate signals from J-coupled lipids while keeping the lactate signal. The coherence pathways responsible for the lipid signals of various chemical shifts were revealed and the conversion rates of the individual coherence pathways were calculated by the product operator analysis. Lipid signals were suppressed by dephasing the coherence pathways with optimal gradient combinations and by reducing the conversion rates of the coherence pathways with changes in the RF pulse lengths. Lactate signals were only slightly decreased by the RF pulse length changes and remained unchanged by the gradient ratio changes. Lipid signals from the fatty tissues of the volunteers were successfully suppressed to the noise level.

While product operator analysis is commonly used to explain MQC and polarization transfer experiments [32], most previous studies were performed on the cases where hard RF pulses were applied or when the spectrally selective pulses were applied on-resonance to one partner of J-coupled spins at a time not affecting the other [6, 16, 32, 42–44]. In the Sel-MQC sequence, the coupling partners of J-coupled lipid spins receive off-resonance partial rotations by the spectrally selective pulses centered on lactate resonance frequencies. Product operator analysis in such off-resonance rotation conditions has not been reported. We accomplished it by calculating the off-resonance rotation matrices and applying them to the product operators. This enabled us to draw the CTP maps of J-coupled lipids with various spin couplings and calculate the conversion rates of the individual CTPs.

There are many numerical analysis packages for simulating NMR signals upon a pulse sequence, e.g., GAVA, FID-A and jMRUI [45–47]. They simulate NMR signals by the propagation of the density matrix to a time-varying Hamiltonian [48]. However, as the methods do not use the product operator formalism, they do not provide information on the coherence orders or coherence pathways, the genuine properties of the product operator formalism [31]. Moreover, most of the available packages do not have an option to enter gradient pulses needed for simulating MQC experiments. In contrast, our analysis based on product operators provided detailed information on the coherence pathways, which was essential to understand various lipid resonance signals upon the Sel-MQC sequence and to intelligently design the sequence for suppressing lipid signals.

In this study, we have derived the equation for the coherence pathway-dependent signal attenuation by gradients for the CSI voxels and have experimentally verified it. The coherence pathway information of J-coupled lipids and the mentioned equation allowed us to exactly understand the effect of the gradient ratios, amplitudes and durations on suppression of lipid signals from the individual coherence pathways. The correction factor μ of the equation allowed us to understand the effect of the spin distribution on the signal attenuation by gradients. Such understandings led us to design and apply optimal gradient combinations to suppress lipid signals in vivo.

We have shown in this study that the antiphase term of the lactate signal at the echo can be removed by using τ₁ = 1/(2J_Lac) – t₁ and the gradient ratio 1:β:2. In contrast, He et al.’s original scheme [16] and all other studies using the Sel-MQC sequence [17–26] have employed τ₁ = 1/(2J_Lac) regardless of the gradient ratio. In such conditions, the existence of the antiphase term at the echo is inevitable, which results in distortion of the doublet shape of lactate. The distortion increases with the t₁ time. Removal of the antiphase term allows for the lactate signal with a clean doublet. Showing such a doublet is a persuasive demonstration that the detected signal is lactate, not lipids. Also, keeping the doublet shape is useful for spectral fitting and quantification. We suggest that all future Sel-MQC studies exploit the condition for removing the antiphase component of lactate; let us call it the revised Sel-MQC sequence.

The revised Sel-MQC sequence with optimal gradient combinations and RF pulse lengths could be applied for detecting lactate from ischemic or tumor tissues of human subjects when contamination by lipids is concerned. Instead of CSI localization employed in this study, a single voxel localization using the LASER technique could also be used similarly to Payne et al. [25, 26]. When an optimized Sel-MQC-CSI sequence is applied to the brain, we expect that lactate detection from the tumor or ischemic tissues very close to the skull will be possible because the lipid signals from the subcutaneous fat of the skull will be able to be suppressed to the noise level. When an optimized Sel-MQC-CSI sequence is applied to bodily tumors, we can expect that the signals from the fatty tissues surrounding the tumor will all be suppressed, leaving the lactate signal from the tumor. When an optimized Sel-MQC-LASER sequence is applied to the ischemic or tumor tissues in the brain or body regions, we can expect that even when the chosen voxel contains the fatty tissue, the lipid signal will all be suppressed, leaving lactate. A motion correction technique [49, 50] could also be combined with the sequence to obtain high-quality data in clinical patient studies.

An issue to be further considered when applying the optimized Sel-MQC sequence to tumors is that some tumors, e.g., meningioma and diffuse large B-cell lymphoma, often contain alanine with concentrations comparable to lactate [18, 51]. Alanine is an important indicator of glutaminolysis [52]. As alanine has similar J-couplings and chemical shifts as lactate, both lactate and alanine can be detected by the revised Sel-MQC sequence. However, their amplitudes and relative phases will vary depending on the sequence parameters such as the center frequencies of the spectrally selective pulses, pulse lengths and pulse intervals. Our product operator analysis method can be used to design the sequence so that the phase difference of the lactate and alanine signals is removed and the maximal signals of lactate and alanine are acquired while suppressing lipid signals. These will be dealt with in our next study.

5. Conclusion

We thoroughly investigated the coherence pathways of J-coupled lipids and lactate upon the Sel-MQC lactate editing sequence by using the product operator formalism and off-resonance rotation matrices. The coherence pathways of various J-coupled lipids and lactate upon the Sel-MQC sequence were revealed and the roles of the varied coherence selection gradient combinations and RF pulse lengths on these pathways and the resultant signals were understood. Lipid signals from the human subjects were successfully suppressed to the noise level while retaining the lactate signal from the phantom by a revised design of the Sel-MQC sequence. The study provided a solid theoretical basis supported by experimental data for an optimal design of the Sel-MQC sequence to effectively suppress J-coupled lipid signals and keep the lactate signal. This enables lactate detection in an environment of high concentrations of lipids.