MRI of [2-¹³C]Lactate without J-coupling artifacts

Abstract

Purpose:

Imaging of [2-¹³C]lactate, a metabolic product of [2-¹³C]pyruvate, is over considerable interest in hyperpolarized ¹³C studies. However, artifact-free imaging of a J-coupled nuclear spin species can be challenging due to the peak-splitting induced by the spin-spin interactions. In this work, two new techniques resolving these J-modulated artifacts are presented.

Theory and Methods:

The Product Operator Formalism (POF) of density matrix theory is used to both numerically and analytically derive the coherences arising during RF excitation and readout of a J-coupled spin system. A combination of computer simulations and experiments with [2-¹³C]lactate and ¹³C-formate phantoms are then used to verify the performance of two imaging methods. In the first approach, a quadrature imaging technique is used to eliminate scalar coupling artifacts via the combination of in-phase and quadrature images acquired at echo times differing by 1/2J with an echoplanar readout. The second approach employs a highly narrowband RF excitation pulse to image a single peak from the J-coupled doublet.

Results:

Simulations using a numerical Shepp-Logan phantom, in vitro experiments using thermally polarized [2-¹³C]lactate, thermally and hyperpolarized ¹³C-formate phantoms, and in vivo imaging of [2-¹³C]lactate produced in rat brain following injection of hyperpolarized [2-¹³C]pyruvate show artifact free images and demonstrate potential utility of these methods.

Conclusion:

The quadrature imaging and the narrowband excitation techniques resolve the J-coupling induced ghosting and blurring artifacts present with conventional MRI of J-coupled signals such as [2-¹³C]lactate.

1.1. Introduction

Pyruvate (Pyr) occupies a key position at the intersection of several major pathways of glucose metabolism. In the presence of enzyme lactose dehydrogenase, it can be reduced to lactate (Lac) via glycolysis (GLY) or converted to alanine (Ala) mediated by the enzyme alanine aminotransferase. Alternatively, Pyr is converted to acetyl-coenzyme-A by pyruvate dehydrogenase and enters the tricarboxylic acid (TCA) cycle to generate energy in mitochondria via oxidative phosphorylation (OXPHOS).

[1-¹³C]Pyr is the most commonly used substrate for hyperpolarized ¹³C imaging studies (1). With the ¹³C label on the C₁ carbon, [1-¹³C]Pyr is converted in vivo to [1-¹³C]Lac, [1-¹³C]Ala, and [1-¹³C]bicarbonate. The five orders-of-magnitude signal enhancement provided by dissolution dynamic nuclear polarization (2) can then be used to study in vivo cellular metabolism in real-time by injecting hyperpolarized [1-¹³C]Pyr and imaging its metabolic products (3). To date, this technique has been used to study tumor metabolism (4-6), cardiac energetics (7-10), liver metabolism (11-14) and response to therapy (15-19).

An alternative substrate is [2-¹³C]Pyr. The ¹³C label at the C₂ position is glycolytically converted to [2-¹³C]Lac and, in contrast to [1-¹³C]Pyr, is also observable from the TCA cycle compounds citrate (Cit), glutamate (Glu) and acetylcarnitine, thus providing a real-time window into mitochondrial metabolism and a more direct measurement of OXPHOS (20-22). Recently, the first hyperpolarized [2-¹³C]Pyr study of human brain metabolism reported successful detection of [5-¹³C]Glu, [1-¹³C]Cit and [2-¹³C]Lac (23).

There are two primary approaches for imaging hyperpolarized ¹³C substrates. The first uses long spectroscopic readouts in combination with fast gradient encoding schemes such as EPI (24, 25) or spiral (26-28). The second is to use spectral-spatial RF pulses to sequentially excite each metabolite which is then directly imaged using a single-shot readout, such as echoplanar or spiral (29-32). Given the very wide spectral dispersion of [2-¹³C]Pyr and its metabolic products (134 ppm), sequential metabolite-selective imaging has multiple advantages, particularly in terms of reduced chemical shift misregistration and aliasing artifacts (28).

Unfortunately, the direct imaging of [2-¹³C]Lac, in which J-coupling of the ¹³C₂ nucleus to an adjacent hydrogen proton splits the lactate peak into a doublet, is problematic due to J-modulation induced artifacts. In this paper, we present two imaging techniques for eliminating these artifacts, one based on quadrature imaging and the other using a very narrowband RF excitation. While these techniques are presented in the context of hyperpolarized ¹³C metabolic imaging, they are also applicable to any general spin-1/2 J-coupled spin system.

1.2. Theory

1.2.1. Imaging a J-coupled nuclear spin species

The through-bond interaction between the C₂ labeled carbon and the attached proton in [2-¹³C]Lac (J-coupling, J=140 Hz) causes the C₂ resonance at 74 ppm to split into a doublet. As a result of J-coupling evolution during the signal acquisition in an imaging sequence, a cosine weighting is applied in the spatial frequency domain (k-space) as follows:

$$S^{\prime }[k_x(t),k_y(t)]=\cos (\pi Jt+\phi )S[k_x(t),k_y(t)]$$ [1]

where S[k_x(t), k_y(t)] is the unweighted signal acquisition, J is the J-coupling in Hz, and φ is the phase accumulated between excitation and the start of acquisition. Figure 1A illustrates this weighting in the case of a simple slice-selective excitation followed by an echoplanar imaging (EPI) readout wherein the J-modulation is considerable over the duration of signal acquisition. While the effect is less in the faster readout direction (k_x), the weighting in the phase encode direction (k_y) is significant as seen from the simulation of the applied k-space weighting in Figure 1B. The convolution of the true image with the point spread function of the cosine weighting, a pair of impulses in the image domain, causes ghosting artifact in the resulting image from the EPI sequence (Figure 1D). Two techniques to resolve the J-modulated artifacts arising in [2-¹³C]Lac images are presented in the following sections.

Figure 1.

Schematic illustration of the effect of J-coupling on echoplanar imaging and the quadrature imaging technique to resolve J-coupling modulated artifacts. (A) A typical radiofrequency pulse followed by gradient waveform for echoplanar readout (FOV = 128 mm, 32x32 matrix, TE = 28.4 ms, readout time = 28.8 ms) highlights the evolution of J-modulation as the signal is collected. (B) While, for this example, the applied weighting in k-space due to J-modulation is negligible in the readout direction (kx), it is significant in the slower phase encode direction (ky). (C) J-coupling evolution during readout nulls certain spatial frequencies causing loss of information in the phase encode direction (Acq1). This can be compensated by acquiring a quadrature component by delaying the acquisition time by 1/2J (Acq2). Through a complex combination of these two quadrature components given in (D), the original image can be recovered as illustrated in the reconstruction in (E).

1.2.2. Technique 1: Quadrature imaging for resolving J-modulated artifacts

In typical fast imaging sequences such as EPI, the effect of the temporal cosine modulation due to the J-coupling as given in Eq. [1], is to null signals at specific spatial frequencies as illustrated in Figure 1B and shown in Figure 1C as a 1D weighting function in the k_y direction. This loss of signal can be compensated by acquiring a quadrature component via a second acquisition delayed by 1/2J (Figure 1A, bottom). In the image domain, the two acquisitions can be written as convolutions of the undistorted image, I(x,y), with the inverse Fourier transform (IFT) of the cosine weighting functions in k-space:

$$\begin{gathered} I_1(x,y)=I(x,y)\ast IFT\{\cos (\pi Jt+\phi )\} \\ {I}_2(x,y)=I(x,y)\ast IFT\left\{\cos \left(\pi J\left(t+\frac{1}{2J}\right)+\phi \right)\right\} \end{gathered}$$ [2]

where φ represents the phase accumulation due to the delay between the radiofrequency excitation and the start of acquisition. Therefore, we have:

$$\begin{gathered} I_1(x,y)=I(x,y)\ast IFT\{\cos (\pi Jt+\phi )\} \\ {I}_2(x,y)=I(x,y)\ast IFT\{\sin (\pi Jt+\phi )\} \end{gathered}$$ [3]

Assuming that the weighting in the k_x direction is negligible, the weighting Y along k_y over the readout time is given by the relation $k_{y}Y=\frac{Jt}{2}$, and therefore, πJt = 2πYk_y. The images from the two acquisitions can be written in terms of Y as:

$$\begin{gathered} I_1(x,y)=I(x,y)\ast IFT\{\cos (2\pi Y{k}_y+\phi )\} \\ {I}_2(x,y)=I(x,y)\ast IFT\{\sin (2\pi Y{k}_y+\phi )\} \end{gathered}$$ [4]

Combining the two quadrature components as:

$$\begin{gathered} I_{rec}(x,y)=I_1(x,y)+i{I}_2(x,y) \\ =I(x,y)\ast IFT\left\{e^{i(2\pi Y{k}_y+\phi )}\right\} \\ =I(x,y-Y)e^{i\phi }, \end{gathered}$$ [5]

we can, therefore, recover a shifted version of the original image, with the known shift correctable during post-processing.

1.2.3. Technique 2: Narrowband excitation for imaging [2-¹³C]Lac

While the quadrature imaging technique resolves the J-modulated artifacts, it is a two-shot method requiring additional scan time. Here we describe a single-shot method for imaging [2-¹³C]Lac, based on narrowband excitation, to eliminate the artifacts.

The starting state of a weakly-coupled I-S spin system in a magnetic field, with C₂ carbon as the I-spin and the attached proton as the S-spin, can be written in terms of the spin density operator (see product operator formalism (33-35)) as:

$${\widehat{\sigma }}_0={\widehat{I}}_z+{\widehat{S}}_z$$ [6]

with ${\widehat{I}}_z$ and ${\widehat{S}}_z$ representing the coherences along the z-direction of the I and S spins, respectively.

The Hamiltonian for an RF pulse applied on the I-resonance is (in rotating frame of reference):

$$\widehat{H}=-{\Omega }_I{\widehat{I}}_z-{\Omega }_S{\widehat{S}}_z-{\omega }_1{\widehat{I}}_x+\pi J2{\widehat{I}}_z{\widehat{S}}_z$$ [7]

where Ω_I, Ω_S are the off-resonance angular frequencies of the I, S spins respectively, J is the coupling constant in Hz, and ω₁ = γ_IB₁, with B₁ representing the amplitude of the RF pulse. For this heteronuclear spin system, the $2{\widehat{I}}_x{\widehat{S}}_x$ and $2{\widehat{I}}_y{\widehat{S}}_y$ terms are ignored under the weak-coupling approximation. The ${\widehat{S}}_z$ terms in Eqs. [6] and [7] are ignored in the following analysis since the RF pulse is applied only at the I-spin resonance frequency. The off-resonance and J-coupling terms in Eq. [7], typically ignored in the case of a short duration excitation pulse, are not negligible when a long narrowband frequency-selective pulse is applied and must be included in the analysis. This is further complicated since ${\widehat{I}}_x$ does not commute with ${\widehat{I}}_z$ and $2{\widehat{I}}_z{\widehat{S}}_z$, and therefore this precludes the simple rotation of the initial spin density operator in Eq. [6] about the terms in the Hamiltonian in Eq. [7] as a solution to the Liouville-von Neuman equation to arrive at the state of the system after the application of the RF pulse.

Two possible approaches to solve for the state of the system after the application of the radiofrequency pulse, one utilizing the analytical solution, and the other based on numerical simulation using the full-density matrix, are presented in this work. In the analytical approach, the initial spin density operator in Eq. [6] must be transformed to a tilted frame of reference, wherein the terms in the transformed Hamiltonian once again commute, rotated about the Hamiltonian in this tilted reference, and then transformed back to the original reference. The resulting state of the system at the end of the RF pulse is given by (see Appendix A for the complete derivation):

$$\begin{gathered} {\widehat{\sigma }}_{RF}(t)=\frac{1}{2}(2+{\cos }^2{\theta }_1(\cos D_{1}t-1)+{\cos }^2{\theta }_2(\cos D_{2}t-1)){\widehat{I}}_z \\ +\frac{1}{2}({\cos }^2{\theta }_1(\cos D_{1}t-1)-{\cos }^2{\theta }_2(\cos D_{2}t-1))2{\widehat{I}}_z{\widehat{S}}_z \\ +\frac{1}{2}(\cos {\theta }_1\sin D_{1}t+\cos {\theta }_2\sin D_{2}t){\widehat{I}}_y \\ +\frac{1}{2}(\cos {\theta }_1\sin D_{1}t-\cos {\theta }_2\sin D_{2}t)2{\widehat{I}}_y{\widehat{S}}_z \\ +\frac{1}{2}(\cos {\theta }_1\sin {\theta }_1(1-\cos D_{1}t)+\cos {\theta }_2\sin {\theta }_2(1-\cos D_{2}t)){\widehat{I}}_x \\ +\frac{1}{2}(\cos {\theta }_1\sin {\theta }_1(1-\cos D_{1}t)-\cos {\theta }_2\sin {\theta }_2(1-\cos D_{2}t))2{\widehat{I}}_x{\widehat{S}}_z \end{gathered}$$ [8]

where,

$$\begin{gathered} \Delta =({\Omega }_I-\pi J),D_1=\sqrt{{\omega }_1^2+{\Delta }^2},\cos {\theta }_1=\frac{{\omega }_1}{D_1},\sin {\theta }_1=\frac{\Delta }{D_1} \\ S=({\Omega }_I+\pi J),D_2\sqrt{{\omega }_1^2+{\mathrm{S}}^2},\cos {\theta }_2=\frac{{\omega }_1}{D_2},\sin {\theta }_2=\frac{\mathrm{S}}{D_2} \end{gathered}$$ [9]

With this analytical expression for the spin density operator describing the evolution of off-resonance and J-coupling during the application of the RF pulse, we now analyze the following cases:

Case 1: A short 90° RF pulse (broadband) is applied on resonance (i.e. Ω_I = 0 and ω₁ ≫ πJ). Under these conditions:

$$\begin{gathered} \cos {\theta }_1=\cos {\theta }_2\approx 1,\text{which implies}{\theta }_1={\theta }_2\approx 0,\text{i.e.}\sin {\theta }_1=\sin {\theta }_2\approx 0, \\ {D}_1=D_2\approx {\omega }_1\text{and for}\mathrm{a}{90}^0\text{RF pulse, we have},D_{1}t=D_{2}t\approx \frac{\pi }{2} \end{gathered}$$

The spin density operator at the end of the RF pulse, Eq. [8], reduces to:

$${\widehat{\sigma }}_{RF}={\widehat{I}}_y$$ [10]

giving rise to the familiar doublet during free-induction decay (FID) signal collection.

Case 2: Next consider a long-duration, narrowband RF pulse applied on resonance. For this long pulse, the amplitude B₁ required for a 90°, and hence ω₁, is very small. In this case, Ω_I = 0 and ω₁ ≪ πJ, cos θ₁ = cos θ₂ ≈ 0, therefore ${\theta }_1={\theta }_2\approx \frac{\pi }{2}$, i.e. sin θ₁ = sin θ₂ ≈ 1

Substituting these into Eq. [8], we have:

$${\widehat{\sigma }}_{RF}={\widehat{I}}_z$$ [11]

and in this case, there is no excitation.

Case 3: A long-duration, narrowband RF pulse is applied at off-resonance such that Ω_I = πJ. Since the pulse is long, the amplitude of the RF pulse and hence ω₁ ≪ πJ. With this we have the following conditions:

$$\begin{gathered} \Delta ={\Omega }_{\mathrm{I}}-\pi J=0,D_1={\omega }_1,\text{hence cos}{\theta }_1=1,\sin {\theta }_1=0, \\ \text{since}{\omega }_1\ll D_2,D_2\approx 2\pi J,\cos {\theta }_2\approx 0,\sin {\theta }_2\approx 1, \end{gathered}$$

and for a 90° RF pulse, ${\omega }_{1}t=\frac{\pi }{2}$ which gives sin D₁t = 1 and cos D₁t = 0.

Substituting these values into Eq. [8], we have:

$${\widehat{\sigma }}_{RF}=\left(\frac{1}{2}\right){\widehat{I}}_z-\left(\frac{1}{2}\right)2{\widehat{I}}_z{\widehat{S}}_z+\left(\frac{1}{2}\right){\widehat{I}}_y+\left(\frac{1}{2}\right)2{\widehat{I}}_y{\widehat{S}}_z$$ [12]

In this case of narrowband excitation at off-resonance frequency Ω_I = πJ, it can be seen that during the signal acquisition, the sum of the in-phase doublet arising from the ${\widehat{I}}_y$ term combines with anti-phase doublet from $2{\widehat{I}}_y{\widehat{S}}_z$ to produce a singlet, eliminating all J-coupling associated artifacts. Therefore, a narrowband spectrally selective pulse centered at 70 Hz (J/2) away from the [2-¹³C]Lac resonance, followed by a fast EPI readout, can be used to image [2-¹³C]Lac artifact free. This idea is schematically presented in Figure 2.

Figure 2.

Narrowband excitation for resolving J-coupling induced doublet in [2-¹³C]Lac. Evolution of J-coupling during a short RF pulse (500 μs, 2 kHz bandwidth) is negligible and hence results in a doublet during signal collection (top row). A long narrow bandwidth pulse (13.33 ms, 75 Hz), J/2 Hz off-resonance (70 Hz), on the other hand, generates in-phase and anti-phase coherences that combine to result in a singlet during signal acquisition.

1.2.4. SNR Considerations

For a spectroscopic acquisition (well-resolved spectra along the k_t-axis), the signal-to-noise ratio (SNR) is defined as the ratio of the total area under the peak(s) to the standard deviation of noise over the spectral bandwidth of the peak(s). Therefore, for a J-coupling induced split of a peak into a doublet, while the total signal under the two peaks remains the same, the collection of signal over two bands doubles the noise variance resulting in a loss of SNR by $\sqrt{2}$. The relative SNRs of the proposed quadrature and narrowband techniques, normalized to the SNR of a singlet as acquired in a heteronuclear decoupled free-induction decay (FID) experiment, are presented in Table 1. Since the quadrature imaging technique is a two-shot method, for a fair comparison, relative SNRs are provided for both single and double shots for all methods.

Table 1. Theoretical Relative SNRs of [2-¹³C] Lactate Detection Techniques.

Theoretical relative SNRs of the different methods for [2-¹³C]lactate detection. In the thermal equilibrium case, the two-shot acquisitions employ 90° pulses successively, whereas, in the case of hyperpolarization, the flip angles of the RF pulses for the two shots are 45°-90°, to preserve magnetization. In the case of narrowband excitation of hyperpolarized compounds, a 90° narrowband pulse followed by a 90° broadband pulse results in maximum SNR. The effects of T₁, T₂, and T₂* are ignored, and it is assumed that the time between the two acquisitions are minimized such that metabolite conversion dynamics are negligible.

	Single shot			Two-shot (Thermal)			Two-shot (Hyperpolarized)
Method	Signal	Noise (std. dev)	SNR	Signal	Noise (std. dev)	SNR	Signal	Noise (std. dev)	SNR
FID	1	$\sqrt{2}$	$1∕\sqrt{2}$	2	2	1	$\sqrt{2}$	2	$1\sqrt{2}$
FID with decoupling	1	1	1	2	$\sqrt{2}$	$\sqrt{2}$	$\sqrt{2}$	$\sqrt{2}$	1
Quadrature technique	-	-	-	2	2	1	$\sqrt{2}$	2	$1∕\sqrt{2}$
Narrowband excitation	1/2	1	1/2	1	$\sqrt{2}$	$1∕\sqrt{2}$	1	$\sqrt{2}$	$1∕\sqrt{2}$

Due to the non-replenishable decay of magnetization in a hyperpolarized experiment, unlike the case of thermal polarization wherein 90° flip angle pulses can be applied for both the shots, the optimal strategy to maximize SNR in a two-shot hyperpolarized quadrature imaging experiment is to apply 45° flip angle for the first excitation, immediately followed by a 90° for the second in order to use all available magnetization and minimize T₁ losses. For the narrowband excitation case, the residual z-coherences after the first 90° excitation ($\frac{1}{2}{\widehat{I}}_z$ and $\frac{1}{2}2{\widehat{I}}_z{\widehat{S}}_z$, see Eq. [12]) can be fully utilized via a subsequent broadband 90° RF pulse. Again, the time between acquisitions should be as short as possible to minimize T₁ losses. We further note that the $2{\widehat{I}}_z{\widehat{S}}_z$ coherence relaxes as 1/T₁ = 1/T_1,S + 1/T_1,I (36).

1.3. Methods

All simulations were done in MATLAB (Mathworks Inc., MA, USA) using custom developed scripts. To test the lactate imaging methods, a syringe (length 65 mm, dia. 15 mm) filled with 10 ml of 0.5 M [2-¹³C]Lac in 60% D₂O, doped with 3 μl/ml of Gadolinium agent (Magnevist, Bayer Schering Pharma, Berlin-Wedding, Germany) was placed on a single loop transmit-receive coil (dia. 40 mm) tuned to the carbon resonance and imaged using a clinical 3T PET-MR scanner (GE Healthcare). After acquiring proton images using a ¹H T/R birdcage coil (dia. 100 mm) and a 3-plane fast GRE sequence (FOV=128 mm, slice thickness=5 mm, freq. x phase 256x128, receiver BW=31.2 kHz) to localize the phantom, the carbon transmit and receive gains were manually adjusted to obtain maximum signal from the phantom for a 90°-flip angle. To further demonstrate the effectiveness of these techniques for chemical species with larger J-coupling, a ¹³C-labeled formate phantom (2.5 M in D₂O + μl of Gadolinium, 3 ml in a 50 mm long, 10 mm dia. syringe, J_CH=192 Hz), was also used.

1.3.1. Technique 1: Quadrature imaging

A 32x32 Shepp-Logan phantom (37) with intensity variations normalized to a 256 gray scale image was used in simulations. In this context, the phantom represents the spatial distribution of spin ½ J-coupled spin system. To simulate the effect of J-coupling on an EPI sequence, a cosine weighting as given in Eq. [1] was applied to the simulated k-space data. Parameters used in the simulations were matched to those used in typical hyperpolarized in vivo experiments (FOV=128 mm, matrix size 32x32, TE=28.4 ms, readout time, T_RO=28.8 ms, J=140 Hz). The phantom image was then generated via an inverse 2D FFT of the weighted k-space data. Similarly, a second image representing the quadrature component was simulated by delaying the echo time by 1/2J=3.6 ms. The weighting matrices in k-space due to the evolution of J-coupling during the two acquisitions, and the resulting artifact in the reconstructed image are shown in Figure 1B and D respectively.

Quadrature imaging was then tested using both [2-¹³C]Lac and ¹³C-formate phantoms. Two images whose echo times differed by 1/2J=3.6 ms (J_CH=140 Hz for [2-¹³C]Lac) were acquired using a slice-selective fast echo-planar imaging sequence (gradient echo (GRE) EPI, FOV=128 mm, slice thickness = 20 mm, TR=130 ms, TE=22.8 ms, flip angle=80°, 32x32 matrix, receiver BW=41.7 kHz, NEX=150). For the analogous hyperpolarized ¹³C formate phantom imaging experiments, a 10° flip angle was used and NEX set to 3 to acquire the in-phase and quadrature images with an echo-time difference of 2.6 ms.

1.3.2. Technique 2: Narrowband excitation

A full density matrix simulator for a J-coupled I-S system was used to simulate the evolution of coherences during RF excitation. Starting with the coherences given in Eq. [6], the time evolution of the coherences under the Hamiltonian in Eq. [7] was computed using a numerical solution to the Liouville-Von Neumann equation (see (38)) at each time step (RF duration, T_RF = 28.5 ms, Δt = 10 μs, off-resonance frequency of I-spins, Ω_I = 2π * 70 rad/s, J = 140 Hz, ${\omega }_1{T}_{RF}=\frac{\pi }{2}$). The coherences at the end of the excitation pulse, Eq. [8], are then propagated under free-precession Hamiltonian (ω₁ = 0 in Eq. [7]) to simulate the signal acquisition, and a 1D FFT performed on the data to obtain the spectrum.

To test this imaging technique in vitro, a narrowband spectrally selective radiofrequency excitation pulse (pulse width=32 ms, BW=100 Hz, slice thickness=20 mm) followed by a 2D GRE EPI image acquisition (FOV=128 mm, TR=130 ms, TE=22.8 ms, flip angle=80°, freq. x phase = 32x32, receiver BW = 41.7 kHz, NEX=150 ) was used. Similar to the quadrature imaging, a small tip angle of 10° and NEX=3 was used for the narrowband imaging of hyperpolarized formate phantom, and the bandwidth of the pulse set at 133 Hz (pulse width=24 ms) to accommodate variations in spin frequencies from off-resonance effects.

1.3.3. Hyperpolarized sample preparation

345 mg of ¹³C labeled sodium formate (Sigma Aldrich) dissolved in 1 ml of water was mixed with 25 mg of trityl radical AH11141(GE Healthcare) in 1 ml glycerol, to prepare a 2.5 M formate solution. 150 μL of this solution was polarized in SPINLab system (GE Healthcare) for 2-3 hours and dissolved using 16 g solution of 40 mM tris (hydroxymethyl aminomethane), 100 mg/L ethylenediaminetetraacetic acid (EDTA), and 50 mM NaCl. 4 ml of the resulting ~80 mM solution of sodium formate drawn into a syringe (length 40 mm, dia. 12.5 mm).

For the in vivo imaging experiment, 54 μL of 15.5 M [2-¹³C]Pyruvic acid (Sigma Aldrich) mixed with the 15 mM trityl radical was polarized between 3-4 hours in SPINLab system and dissolved using 16 g solution of 40 mM tris ( 100 mg/L EDTA) and 50 mM NaCl. The resulting 100 mM pyruvate solution was neutralized with 640 μL of 125 mM NaOH buffer. 2.5-3.0 ml of this solution (dose=1 mmol/kg body weight), was injected at a rate of 0.25 mL/s via a tail-vein catheter and imaged on 3T PET/MR scanner starting 20 s post injection.

1.3.4. In vivo imaging

A custom-built birdcage T/R coil (dia. 100 mm) tuned to the ¹H resonance was used to excite and acquire proton MR images from Wistar rats (male and female, 250-300g). After a fast GRE sequence for localization (similar to the one used for phantoms), a dual-echo T₂-weighted fast spin echo (FSE) sequence (TR/TE1/TE2=5000/11.3/56.7 ms; slice thickness=2 mm; 0.25 mm in-plane resolution; 256x256 matrix size; 8 echo train length; 25 slices) in the axial plane was used to determine the slice placement in the rat brain. The homogeneity of the B₀ field over the imaging slice was optimized using linear shim currents. For the [2-¹³C]Lac imaging, a custom-built ¹³C surface T/R coil (dia. 40 mm) was used to acquire data from an axial slice of the brain using the same set of sequences as described in the imaging of the phantom. All the experiments were conducted under protocol reviewed and approved by the Stanford Institutional Animal Care and Use Committee (IACUC Protocol number 11396).

1.3.5. Reconstruction and display

For the quadrature imaging reconstruction, the complex images from acquisitions 1 and 2 were combined as given in Eq. [5] to obtain the reconstructed image. In the case of hyperpolarized formate, an empirical scale factor, A_corr, was used for the second acquisition to compensate for the T₁ and flip angle related losses. Additionally, a relative phase factor ϕ_corr was introduced to counter the phase accumulation arising from off-resonance effects so that the reconstruction in Eq. [5] is modified as I_rec(x,y) = I₁(x,y) + iA_corrI₂(x,y)e^iϕ_corr. The resulting image was shifted by $\left(\frac{J}{2}+f_{offres}\right)\ast T\_RO$ pixels in the slower k-space acquisition direction to offset the shift arising from the quadrature reconstruction, where f_offres is the off-resonance frequency in Hz. In the case of narrowband acquisition, the native EPI reconstruction from the scanner was used without any further processing.

Magnitude [2-¹³C]Lac (or ¹³C-formate) images, mapped to a “hot” colormap, were overlaid at 50% transparency on a grayscale reference proton image of the phantom or animal, with pixel values less than 10% of the maximum value in the image set to zero.

1.4. Results

1.4.1. Quadrature Imaging

Representative simulations using the quadrature imaging method are shown in Figure 1B, D and E respectively. Figure 3 shows reference proton coronal and axial images, thermally-polarized ¹³C EPI results from both lactate and formate phantoms, and hyperpolarized imaging results using ¹³C labeled formate phantom. Prominent imaging ghosts are largely eliminated in the quadrature reconstruction combining them as per modified Eq. [5] given in the methods section, wherein a scale factor of A_corr = 2.3 was used for the second acquisition to compensate for relaxation and flip angle related losses. The residual artifact was further reduced by applying a relative phase, ϕ_corr = −0.9 radians, as seen from the phase compensated image on the right in Figure 3C.

Figure 3.

Quadrature imaging reconstruction of thermally polarized and hyperpolarized phantoms. (A) The coronal (top row) and axial (2^nd row) images from Acq1 and Acq2 spaced 1/2J = 3.5 ms apart from a thermally polarized 0.5 M [2-¹³C]Lac phantom overlaid on the proton image, are shown. The quadrature reconstruction (3^rd column) resolves the J-modulated ghosting artifacts seen in Acq1 and Acq2. In the last column, proton MRI of the phantom depicts its location. (B) Similarly, in the next two rows, coronal and axial acquisitions Acq1 and Acq2 from a thermal 2.5 M ¹³C-formate phantom (J_CH = 192 Hz) spaced 2.6 ms apart, when combined, produce artifact free images of the formate phantom (3^rd column), highlighting the utility of the method even for high values of J. (C) EPI reconstructions from two acquisitions spaced 2.6 ms apart overlaid on the proton image, of a 20 mm axial slice from a hyperpolarized ¹³C-formate phantom (~80 mM, 4 ml) demonstrate the degree of ghosting artifact due to J-coupling. The quadrature reconstruction eliminates this artifact, and the residual ghost is completely removed with a relative phase correction between the two acquisitions.

To investigate the sensitivity to ΔTE, simulations of the Shepp-Logan phantom were performed with ΔTE values ranging from 0.9 ms to 7.2 ms, and J-coupling values from 92 Hz to 278 Hz. As seen from the residual error map (Figure 4A), errors are minimized for ΔTE = 1/2J. In vitro experiments confirmed this trend as well, wherein, images acquired with ΔTE = 1/2J for both lactate and formate, J=140 and 192 Hz respectively, showed complete elimination of the ghosting artifact (Figure 4B, center images). Appreciable ghosts start appearing for deviations ≥ ±0.9 ms from the optimal values of ΔTE.

Figure 4.

Effect of changes in echo time, TE, on quadrature imaging. (A) The residual error from simulations, defined as the norm of the difference between the quadrature reconstruction and original Shepp-Logan phantom, is shown for varying amounts of ΔTE and for different J-coupling values. This error is minimized for ΔTE = 1/2J, as highlighted by the images along the diagonal enclosed in white rectangles. (B) Quadrature reconstructions from [2-¹³C]Lac (top row) and ¹³C-formate (bottom row) demonstrates the same effect in phantoms as well, wherein, the error in the middle images is least compared to the other images acquired with suboptimal ΔTEs.

Off-resonance results in the incomplete cancellation of the ghost as seen from the [2-¹³C]Lac phantom images (Figure 5, top row). The center frequency of acquisition was varied from −80 to +80 Hz relative to the resonance frequency of [2-¹³C]Lac in steps of 20 Hz, and the resulting artifacts are presented in Figure 5 for the two extreme deviations. In this case of known bulk frequency shift, a phase correction factor applied during the reconstruction process (${\varphi }_{corr}=\mp 1.79$ radians, for Δf = ±80 Hz respectively) completely eliminated the artifacts (Figure 5, bottom row).

Figure 5.

Effect of off-resonance on quadrature imaging. In the absence of any off-resonance effect, Δf = 0 (top row, center image), the quadrature reconstruction fully compensates for the effect of J-coupling modulated artifacts as demonstrated in this coronal image from a 0.5 M [2-¹³C]Lac thermal phantom. However, when the Larmor frequency of the spins deviates from the receive frequency, the quadrature reconstruction is unable to eliminate the artifacts (top left and right images). A simple relative phase term between the two acquisitions, applied during post-processing, compensates for this effect and results in artifact free images even in the case of substantial off-resonance of Δf = ± 80 Hz (bottom row, left and right images).

1.4.2. Narrowband Imaging

Analytical time evolutions of the relevant coherences (see Eq. [8]: ${\widehat{I}}_x$, ${\widehat{I}}_y$, ${\widehat{I}}_z$, $2{\widehat{I}}_x{\widehat{S}}_z$, $2{\widehat{I}}_y{\widehat{S}}_z$, $2{\widehat{I}}_z{\widehat{S}}_z$) agree very well with the density matrix simulation (Figure 6A). The combination of the in-phase doublet from the ${\widehat{I}}_y$ term and the anti-phase doublet arising from the $2{\widehat{I}}_y{\widehat{S}}_z$ term in Eq. [12] during signal collection, results in a singlet as seen from the simulation result in Figure 6B.

Figure 6.

Broadband vs Narrowband excitation. (A) Analytical (solid) and simulated (dashed) evolution of coherences during a 1 ms on-resonance broadband pulse (left) and a 28.5 ms long, 70 Hz off-resonance narrowband RF pulse (right), J = 140 Hz. The resulting spectra from simulations, (B), agree very well with (C), spectra acquired from a 0.5 M [2-¹³C]Lac phantom (J = 140 Hz). The doublet resulting from a short duration RF pulse (left) is resolved into a singlet when a long-duration narrowband RF pulse is used (right). This scheme eliminates any J-coupling associated artifact during imaging, thus enabling fast single-shot acquisition as seen from the EPI images in the bottom panel (D) acquired from the phantom. The sequence parameters used for narrowband imaging are pulse width = 32 ms, BW = 100 Hz, slice thickness = 20 mm, FOV = 128 mm, TR = 130 ms, TE = 22.8 ms, flip angle = 80°, freq. x phase = 32x32, receiver BW = 41.7 kHz. The placement of the phantom is illustrated in the proton grayscale image on the right.

Spectrum from the numerical simulation and in vitro [2-¹³C]Lac using nominal bandwidth RF pulse (2.2 kHz, 1.8 ms) demonstrates the splitting caused due to the J-coupling between the ¹³C labeled carbon and its attached proton (left column, Figure 6B,C, respectively). Use of a narrow bandwidth RF pulse (100 Hz, 32 ms) results in a singlet as seen in the simulation (Figure 6B, right column) and the in vitro spectrum (Figure 6C, right column). This singlet can then be imaged using a single-shot fast imaging EPI readout as seen in Figure 6D.

The results of using narrowband excitation followed by an EPI readout to image hyperpolarized nuclei are shown in Figure 7. The carbon image on the left from a 20 mm axial slice of a hyperpolarized ¹³C formate phantom shows no residual artifacts. The in vivo [2-¹³C]Lac map acquired from an axial slice in the rat brain after an injection of hyperpolarized [2-¹³C]Pyr, overlaid on an anatomical proton MRI reference image, illustrates similar performance.

Figure 7.

Artifact-free imaging of J-coupled hyperpolarized nuclei using narrowband imaging demonstrated in phantom and in vivo. A frequency selective RF pulse (pulse width 24 ms, bandwidth 133 Hz, slice thickness 20 mm) followed by an EPI readout (FOV = 128 mm, TR = 130 ms, TE = 22.8 ms, flip angle = 80°, freq. x phase = 32x32, receiver BW = 41.7 kHz) produces artifact-free images of an axial slice of (A) hyperpolarized ¹³C-formate phantom (dia. 12.5 mm), and (B) in vivo [2-¹³C]Lac generated in the brain of a male Wistar rat after an injection of 2.5 ml of hyperpolarized [2-¹³C]Pyr. The artifact-free image of the reference [2-¹³C]Lac phantom placed on top of the rat head is also visible (indicated by the arrow).

Figure 8 shows the effects of the bandwidth and frequency variations of the applied RF pulse on the performance. The simulated frequency profile of the RF pulse included in the top and third rows (plots in red) show the bandwidth and center frequency relative the center of the [2-¹³C]Lac doublet (blue peaks). A relatively short pulse with bandwidth spanning both the doublet peaks, generates only the ${\widehat{I}}_y$ coherence after the pulse (Eq. [10]), resulting in the ghosting artifact seen in Figure 8. As the duration of the pulse is increased, bandwidth gets narrower, and the transverse coherence starts disappearing, until eventually causing no excitation as seen in the image on the right in Figure 8A, consistent with Eq. [11]. Off-resonance effects are illustrated in Figure 8B via images acquired by varying center-frequency of the RF pulse. When the off-resonance frequency Δf = J/2, the combination of the in-phase and anti-phase terms in Eq. [12] eliminates the doublet and the associated artifacts, as confirmed in the center image of Figure 8B. As the excitation frequency shifts away from this optimal value, the amplitude of RF excitation is reduced as seen from the frequency profile plots and lactate images in Figure 8B, ultimately leading to no excitation when the peak and the RF pulse profile no longer overlap.

Figure 8.

Effect of bandwidth and frequency of the applied RF pulse on narrowband imaging demonstrated in an axial slice of a 0.5 M [2-¹³C]Lac phantom. (A) The simulated frequency profile of the applied RF pulse (red plot) is shown in the top row to highlight the bandwidth and center frequency of the applied pulse relative to the [2-¹³C]Lac doublet (blue peaks separated by J = 140 Hz). The images from the EPI acquisition, post excitation by an on-resonance RF pulse, overlaid on a proton reference image of the phantom, demonstrate that a broad bandwidth pulse spanning both the peaks results in a ghosting artifact (left image), whereas, a narrowbandwidth pulse produces no excitation of lactate (image on the right), consistent with Eq. [11]. (B) A sweep of the RF frequencies in increments of Δf = 35 Hz for a fixed bandwidth of the pulse (50 Hz), bottom row, highlights that the maximum lactate signal is detected for Δf = J/2 (center image), when the frequency coincides with one of the peaks.

1.5. Discussion

The use of hyperpolarized [2-¹³C]Pyr, versus [1-¹³C]Pyr, has the advantage of simultaneously probing both glycolysis and OXPHOS. However direct imaging of the resulting [2-¹³C]Lac is complicated by the J-coupling induced artifacts discussed in this paper. The effect of J-modulation is to apply a cosine weighting during the signal acquisition resulting in blurring and/or ghosting, depending on the k-space trajectory. Values of J on the order of the readout duration result in noticeably degraded image quality. Artifacts can be somewhat mitigated by increasing the size of the imaging matrix (higher resolution) so that the number of pixels shifted relative to the overall image is small. However, the resolution of images in hyperpolarized studies is severely limited by the available SNR, with typical in-plane resolution of 5 mmx5 mm, and in this case the effect of J-coupling can no longer be ignored.

The two strategies described in this work to eliminate the J-modulated artifacts, in fact, use the same underlying principle of combining the in-phase and quadrature components to recover the signal free of J-coupling artifacts. While the quadrature imaging technique acquires images at two different echo times, the analogous result is achieved in the second method by combining the in-phase and anti-phase magnetizations simultaneously generated during excitation by the long-duration narrowband RF pulse.

The proposed schemes, demonstrated here with EPI readouts, are equally suitable for use with other fast imaging readout trajectories; the quadrature imaging technique being a pointwise combination of the in-phase and quadrature components and the narrowband excitation approach independent of the readout. Depending on the application, one approach may be preferable over the other.

The single-shot narrowband excitation is faster but requires a long-duration RF pulse sensitive to B₀ inhomogeneities and T₂* losses. However, in hyperpolarized ¹³C experiments, T₁ decay (and motion) may adversely affect two-shot acquisitions. The choice of flip-angles for the two acquisitions should be optimized based on T₁, pyruvate inflow, and lactate production rate. In the narrowband case, the residual $\left(\frac{1}{2}\right){\widehat{I}}_z$ and $\left(\frac{1}{2}\right)2{\widehat{I}}_z{\widehat{S}}_z$ magnetization remaining after the RF pulse can further be utilized by immediately adding a second readout using a broadband RF pulse to image the remaining magnetization. Experimental verification of this approach was not explored in this work and subject to further investigation.

The quadrature detection technique is sensitive to unwanted relative phase accumulation, e.g. due to B₀ inhomogeneity, between the two acquisitions. Here, we addressed this issue by applying a relative phase factor to correct for a bulk frequency shift. This approach can be easily extended to spatially-varying B₀ inhomogeneities by acquiring a B₀ field map and applying a pointwise correction. The effect of B₁ inhomogeneities is not specific to either of the imaging methods proposed here, and the resulting shading in the image due to spatially varying flip angles is same as seen in conventional imaging.

1.6. Conclusions

In this work, two new techniques for eliminating artifacts in the imaging of a spin-1/2 J-coupled spin system, one based on the idea of quadrature imaging and the other using narrowband RF excitation, are presented. Computer simulations, in vitro, and in vivo results demonstrate the potential utility of these methods particularly in the context of hyperpolarized imaging of [2-¹³C]Lac.

1.8. Appendix A

Rewriting the Hamiltonian, Eq.[7], as sum and difference of the coherences ${\widehat{I}}_x$, ${\widehat{I}}_z$, $2{\widehat{I}}_x{\widehat{S}}_z$ and $2{\widehat{I}}_z{\widehat{S}}_z$ (ignoring ${\widehat{S}}_z$ term):

$$\begin{gathered} \widehat{H}=-\frac{{\Omega }_I}{2}({\widehat{I}}_z+2{\widehat{I}}_z{\widehat{S}}_z)-\frac{{\Omega }_I}{2}({\widehat{I}}_z-2{\widehat{I}}_z{\widehat{S}}_z) \\ +\frac{\pi J}{2}({\widehat{I}}_z+2{\widehat{I}}_z{\widehat{S}}_z)-\frac{\pi J}{2}({\widehat{I}}_z-2{\widehat{I}}_z{\widehat{S}}_z) \\ -\frac{{\omega }_1}{2}({\widehat{I}}_x+2{\widehat{I}}_x{\widehat{S}}_z)-\frac{{\omega }_1}{2}({\widehat{I}}_x-2{\widehat{I}}_x{\widehat{S}}_z) \end{gathered}$$ [13]

resulting in:

$$\widehat{H}=-\frac{\Delta }{2}({\widehat{I}}_z+2{\widehat{I}}_z{\widehat{S}}_z)-\frac{S}{2}({\widehat{I}}_z-2{\widehat{I}}_z{\widehat{S}}_z)-\frac{{\omega }_1}{2}({\widehat{I}}_x+2{\widehat{I}}_x{\widehat{S}}_z)-\frac{{\omega }_1}{2}({\widehat{I}}_x-2{\widehat{I}}_x{\widehat{S}}_z)$$ [14]

where Δ = (Ω_I − πJ) and S = (Ω_I + πJ) Noting that [${\widehat{I}}_p+2{\widehat{I}}_p{\widehat{S}}_z$, ${\widehat{I}}_q-2{\widehat{I}}_q{\widehat{S}}_z$] = 0 for p, q = x, y, z, the Hamiltonian in Eq.[14] can be written as a sum of two commuting parts, i.e.:

$$\widehat{H}={\widehat{H}}_1+{\widehat{H}}_2$$ [15]

with $[{\widehat{H}}_1,{\widehat{H}}_2]=0$ and,

$$\begin{gathered} {\widehat{H}}_1=-\frac{\Delta }{2}({\widehat{I}}_z+2{\widehat{I}}_z{\widehat{S}}_z)-\frac{{\omega }_1}{2}({\widehat{I}}_x+2{\widehat{I}}_x{\widehat{S}}_z) \\ {\widehat{H}}_2=-\frac{S}{2}({\widehat{I}}_z-2{\widehat{I}}_z{\widehat{S}}_z)-\frac{{\omega }_1}{2}({\widehat{I}}_x-2{\widehat{I}}_x{\widehat{S}}_z) \end{gathered}$$ [16]

The spin density operator after the application of an RF pulse of duration t can then be written as a solution to the Liouville-von Neuman equation as follows:

$$\begin{gathered} {\widehat{\sigma }}_{RF}(t)=e^{-i\widehat{\widehat{H}}t}{\widehat{\sigma }}_0 \\ =e^{-i\left({\widehat{\widehat{H}}}_1+{\widehat{\widehat{H}}}_2\right)t}{\widehat{\sigma }}_0 \\ =e^{-i{\widehat{\widehat{H}}}_{2}t}{e}^{-i{\widehat{\widehat{H}}}_{1}t}{\widehat{\sigma }}_0,\text{since}[{\widehat{H}}_1,{\widehat{H}}_2]=0 \end{gathered}$$ [17]

However, the normalized terms $\frac{1}{2}({\widehat{I}}_z+2{\widehat{I}}_z{\widehat{S}}_z)$ and $\frac{1}{2}({\widehat{I}}_x+2{\widehat{I}}_x{\widehat{S}}_z)$ in ${\widehat{H}}_1$, and similarly $\frac{1}{2}({\widehat{I}}_z-2{\widehat{I}}_z{\widehat{S}}_z)$ and $\frac{1}{2}({\widehat{I}}_x-2{\widehat{I}}_x{\widehat{S}}_z)$ in ${\widehat{H}}_2$, do not commute, and therefore we must transform the coherences to a tilted plane of reference. By rotating the coherences about an axis given by $i\left[\frac{1}{2}({\widehat{I}}_x+2{\widehat{I}}_x{\widehat{S}}_z),\frac{1}{2}({\widehat{I}}_z+2{\widehat{I}}_z{\widehat{S}}_z)\right]=\frac{1}{2}({\widehat{I}}_y+2{\widehat{I}}_y{\widehat{S}}_z)$ by an angle θ₁ shown in Figure A1, we get:

$$\begin{gathered} {\widehat{H}}_{1t}=-\Delta \left(\frac{1}{2}({\widehat{I}}_z+2{\widehat{I}}_z{\widehat{S}}_z)\cos {\theta }_1+\frac{1}{2}({\widehat{I}}_x+2{\widehat{I}}_x{\widehat{S}}_z)\sin {\theta }_1\right) \\ -{\omega }_1\left(\frac{1}{2}({\widehat{I}}_x+2{\widehat{I}}_x{\widehat{S}}_z)\cos {\theta }_1-\frac{1}{2}({\widehat{I}}_z+2{\widehat{I}}_z{\widehat{S}}_z)\sin {\theta }_1\right) \end{gathered}$$ [18]

Substituting $\cos {\theta }_1=\frac{{\omega }_1}{D_1}$ and $\sin {\theta }_1=\frac{\Delta }{D_1}$, where $D_1=\sqrt{{\omega }_1^2+{\Delta }^2}$, we have:

$$\begin{gathered} {\widehat{H}}_{1t}=-\Delta \left(\frac{1}{2}({\widehat{I}}_z+2{\widehat{I}}_z{\widehat{S}}_z)\left(\frac{{\omega }_1}{D_1}\right)+\frac{1}{2}({\widehat{I}}_x+2{\widehat{I}}_x{\widehat{S}}_z)\left(\frac{\Delta }{D_1}\right)\right) \\ -{\omega }_1\left(\frac{1}{2}({\widehat{I}}_x+2{\widehat{I}}_x{\widehat{S}}_z)\left(\frac{{\omega }_1}{D_1}\right)-\frac{1}{2}({\widehat{I}}_z+2{\widehat{I}}_z{\widehat{S}}_z)\left(\frac{\Delta }{D_1}\right)\right) \end{gathered}$$ [19]

which simplifies to:

$${\widehat{H}}_{1t}=-D_1\left(\frac{1}{2}({\widehat{I}}_x+2{\widehat{I}}_x{\widehat{S}}_z)\right)$$ [20]

Similarly defining $\cos {\theta }_2=\frac{{\omega }_1}{D_2}$ and $\sin {\theta }_2=\frac{S}{D_2}$, where $D_2=\sqrt{{\omega }_1^2+{\mathrm{S}}^2}$, and rotating the coherences by an angle θ₂ (Figure A1) about an axis $i\left[\frac{1}{2}({\widehat{I}}_x-2{\widehat{I}}_x{\widehat{S}}_z),\frac{1}{2}({\widehat{I}}_z-2{\widehat{I}}_z{\widehat{S}}_z)\right]=\frac{1}{2}({\widehat{I}}_y-2{\widehat{I}}_y{\widehat{S}}_z)$, ${\widehat{H}}_2$ transforms to:

$${\widehat{H}}_{2t}=-D_2\left(\frac{1}{2}({\widehat{I}}_x-2{\widehat{I}}_x{\widehat{S}}_z)\right)$$ [21]

Therefore, to find the state after the application of the RF, the initial spin density operator in Eq.[6] must be transformed to the first tilted frame, rotated about the Hamiltonian ${\widehat{H}}_{1t}$ in this tilted reference, transformed back to the original rotating frame, followed by another transformation to the second frame of reference, rotation about ${\widehat{H}}_{2t}$, and a subsequent rotation back to the original reference frame by a series of operations as follows:

$${\widehat{\sigma }}_{RF}(t)=\left({\widehat{\widehat{R}}}_{2t}^{{}_{†}}{e}^{-i{\widehat{\widehat{H}}}_{2t}t}{\widehat{\widehat{R}}}_{2t}\right)\left({\widehat{\widehat{R}}}_{1t}^{{}_{†}}{e}^{-i{\widehat{\widehat{H}}}_{1t}t}{\widehat{\widehat{R}}}_{1t}\right){\widehat{\sigma }}_0$$ [22]

where the rotation superoperators ${\widehat{\widehat{R}}}_{1t}$, ${\widehat{\widehat{R}}}_{2t}$ and their adjoints ${\widehat{\widehat{R}}}_{1t}^{{}_{†}}$, ${\widehat{\widehat{R}}}_{2t}^{{}_{†}}$, to transform to the tilted reference frames 1,2, and back, respectively, are given by:

$$\begin{gathered} {\widehat{\widehat{R}}}_{1t}=e^{-i\frac{{\theta }_1}{2}\left({\widehat{\widehat{I}}}_y+2{\widehat{\widehat{I}}}_y{\widehat{\widehat{S}}}_z\right)} \\ {\widehat{\widehat{R}}}_{1t}^{{}_{†}}=e^{+i\frac{{\theta }_1}{2}\left({\widehat{\widehat{I}}}_y+2{\widehat{\widehat{I}}}_y{\widehat{\widehat{S}}}_z\right)} \\ {\widehat{\widehat{R}}}_{2t}=e^{-i\frac{{\theta }_2}{2}\left({\widehat{\widehat{I}}}_y-2{\widehat{\widehat{I}}}_y{\widehat{\widehat{S}}}_z\right)} \\ {\widehat{\widehat{R}}}_{2t}^{{}_{†}}=e^{+i\frac{{\theta }_2}{2}\left({\widehat{\widehat{I}}}_y-2{\widehat{\widehat{I}}}_y{\widehat{\widehat{S}}}_z\right)} \end{gathered}$$ [23]

Substituting Eqs.[20],[21] and [23] into [22]], we have:

$$\begin{gathered} {\widehat{\sigma }}_{RF}(t) \\ =\left(e^{+i\frac{{\theta }_2}{2}\left({\widehat{\widehat{I}}}_y-2{\widehat{\widehat{I}}}_y{\widehat{\widehat{S}}}_z\right)}{e}^{i\frac{D_{2}t}{2}\left({\widehat{\widehat{I}}}_x-2{\widehat{\widehat{I}}}_x{\widehat{\widehat{S}}}_z\right)}{e}^{-i\frac{{\theta }_2}{2}\left({\widehat{\widehat{I}}}_y-2{\widehat{\widehat{I}}}_y{\widehat{\widehat{S}}}_z\right)}\right)\left(e^{+i\frac{{\theta }_1}{2}\left({\widehat{\widehat{I}}}_y+2{\widehat{\widehat{I}}}_y{\widehat{\widehat{S}}}_z\right)}{e}^{i\frac{D_{1}t}{2}\left({\widehat{\widehat{I}}}_x+2{\widehat{\widehat{I}}}_x{\widehat{\widehat{S}}}_z\right)}{e}^{-i\frac{{\theta }_1}{2}\left({\widehat{\widehat{I}}}_y+2{\widehat{\widehat{I}}}_y{\widehat{\widehat{S}}}_z\right)}\right){\widehat{\sigma }}_0 \end{gathered}$$ [24]

Let ${\alpha }_1=\frac{{\theta }_1}{2}$, ${\alpha }_2=\frac{{\theta }_2}{2}$, ${\beta }_1=\frac{D_{1}t}{2}$ and ${\beta }_2=\frac{D_{2}t}{2}$. Eq.[24] reduces to:

$$\begin{gathered} {\widehat{\sigma }}_{RF}(t)=\left(e^{+i{\alpha }_2\left({\widehat{\widehat{I}}}_y-2{\widehat{\widehat{I}}}_y{\widehat{\widehat{S}}}_z\right)}{e}^{i{\beta }_2\left({\widehat{\widehat{I}}}_x-2{\widehat{\widehat{I}}}_x{\widehat{\widehat{S}}}_z\right)}{e}^{-i{\alpha }_2\left({\widehat{\widehat{I}}}_y-2{\widehat{\widehat{I}}}_y{\widehat{\widehat{S}}}_z\right)}\right)\left(e^{+i{\alpha }_1\left({\widehat{\widehat{I}}}_y+2{\widehat{\widehat{I}}}_y{\widehat{\widehat{S}}}_z\right)}{e}^{i{\beta }_1\left({\widehat{\widehat{I}}}_x+2{\widehat{\widehat{I}}}_x{\widehat{\widehat{S}}}_z\right)}{e}^{-i{\alpha }_1\left({\widehat{\widehat{I}}}_y+2{\widehat{\widehat{I}}}_y{\widehat{\widehat{S}}}_z\right)}\right){\widehat{\sigma }}_0 \\ =\left(e^{+i{\alpha }_2{\widehat{\widehat{I}}}_y}{e}^{-i{\alpha }_{2}2{\widehat{\widehat{I}}}_y{\widehat{\widehat{S}}}_z}{e}^{i{\beta }_2{\widehat{\widehat{I}}}_x}{e}^{-i{\beta }_{2}2{\widehat{\widehat{I}}}_x{\widehat{\widehat{S}}}_z}{e}^{-i{\alpha }_2{\widehat{\widehat{I}}}_y}{e}^{+i{\alpha }_{2}2{\widehat{\widehat{I}}}_y{\widehat{\widehat{S}}}_z}\right)\left(e^{+i{\alpha }_1{\widehat{\widehat{I}}}_y}{e}^{+i{\alpha }_{1}2{\widehat{\widehat{I}}}_y{\widehat{\widehat{S}}}_z}{e}^{i{\beta }_1{\widehat{\widehat{I}}}_x}{e}^{i{\beta }_{1}2{\widehat{\widehat{I}}}_x{\widehat{\widehat{S}}}_z}{e}^{-i{\alpha }_1{\widehat{\widehat{I}}}_y}{e}^{-i{\alpha }_{1}2{\widehat{\widehat{I}}}_y{\widehat{\widehat{S}}}_z}\right){\widehat{\sigma }}_0 \end{gathered}$$ [25]

Starting with initial spin density operator ${\widehat{\sigma }}_0={\widehat{I}}_z$ (ignoring the ${\widehat{S}}_z$ term since RF pulse is applied only at I-spin resonance), we have:

$$\begin{gathered} {\widehat{\sigma }}_{RF}(t) \\ =\left(e^{+i{\alpha }_2{\widehat{\widehat{I}}}_y}{e}^{-i{\alpha }_{2}2{\widehat{\widehat{I}}}_y{\widehat{\widehat{S}}}_z}{e}^{i{\beta }_2{\widehat{\widehat{I}}}_x}{e}^{-i{\beta }_{2}2{\widehat{\widehat{I}}}_x{\widehat{\widehat{S}}}_z}{e}^{-i{\alpha }_2{\widehat{\widehat{I}}}_y}{e}^{+i{\alpha }_{2}2{\widehat{\widehat{I}}}_y{\widehat{\widehat{S}}}_z}\right)\left(e^{+i{\alpha }_1{\widehat{\widehat{I}}}_y}{e}^{+i{\alpha }_{1}2{\widehat{\widehat{I}}}_y{\widehat{\widehat{S}}}_z}{e}^{i{\beta }_1{\widehat{\widehat{I}}}_x}{e}^{i{\beta }_{1}2{\widehat{\widehat{I}}}_x{\widehat{\widehat{S}}}_z}{e}^{-i{\alpha }_1{\widehat{\widehat{I}}}_y}{e}^{-i{\alpha }_{1}2{\widehat{\widehat{I}}}_y{\widehat{\widehat{S}}}_z}\right){\widehat{I}}_z \end{gathered}$$ [26]

Branch diagram for the first four rotations:

$${\widehat{I}}_z\stackrel{2{\widehat{I}}_y{\widehat{S}}_z(-{\alpha }_1)}{\to }\{\begin{array}{c}{\widehat{I}}_z\stackrel{{\widehat{I}}_y(-{\alpha }_1)}{\to }\{\begin{array}{c}{\widehat{I}}_z\stackrel{2{\widehat{I}}_x{\widehat{S}}_z({\beta }_1)}{\to }\{\begin{array}{c}{\widehat{I}}_z\stackrel{{\widehat{I}}_x({\beta }_1)}{\to }\{\begin{array}{c}{\widehat{I}}_z\hfill \\ {\widehat{I}}_y\hfill \end{array}\phantom{\}}\hfill \\ 2{\widehat{I}}_y{\widehat{S}}_z\stackrel{{\widehat{I}}_x({\beta }_1)}{\to }\{\begin{array}{c}2{\widehat{I}}_y{\widehat{S}}_z\hfill \\ -2{\widehat{I}}_z{\widehat{S}}_z\hfill \end{array}\phantom{\}}\hfill \end{array}\phantom{\}}\hfill \\ {\widehat{I}}_x\stackrel{2{\widehat{I}}_x{\widehat{S}}_z({\beta }_1)}{\to }{\widehat{I}}_x\stackrel{{\widehat{I}}_x({\beta }_1)}{\to }{\widehat{I}}_x\hfill \end{array}\phantom{\}}\hfill \\ 2{\widehat{I}}_x{\widehat{S}}_z\stackrel{{\widehat{I}}_y(-{\alpha }_1)}{\to }\{\begin{array}{c}2{\widehat{I}}_x{\widehat{S}}_z\stackrel{2{\widehat{I}}_x{\widehat{S}}_z({\beta }_1)}{\to }2{\widehat{I}}_x{\widehat{S}}_z\stackrel{{\widehat{I}}_x({\beta }_1)}{\to }2{\widehat{I}}_x{\widehat{S}}_z\hfill \\ -2{\widehat{I}}_z{\widehat{S}}_z\stackrel{2{\widehat{I}}_x{\widehat{S}}_z({\beta }_1)}{\to }\{\begin{array}{c}-2{\widehat{I}}_z{\widehat{S}}_z\stackrel{{\widehat{I}}_x({\beta }_1)}{\to }\{\begin{array}{c}-2{\widehat{I}}_z{\widehat{S}}_z\hfill \\ -2{\widehat{I}}_y{\widehat{S}}_z\hfill \end{array}\phantom{\}}\hfill \\ -{\widehat{I}}_y\stackrel{{\widehat{I}}_x({\beta }_1)}{\to }\{\begin{array}{c}-{\widehat{I}}_y\hfill \\ {\widehat{I}}_z\hfill \end{array}\phantom{\}}\hfill \end{array}\phantom{\}}\hfill \end{array}\phantom{\}}\hfill \end{array}\phantom{\}}$$ [27]

Collecting the terms for each of the coherences:

$$\begin{array}{c}\hfill ({\cos }^2{\alpha }_1{\cos }^2{\beta }_1+{\sin }^2{\alpha }_1{\sin }^2{\beta }_1){\widehat{I}}_z\hfill \\ \hfill -({\cos }^2{\alpha }_1{\sin }^2{\beta }_1+{\sin }^2{\alpha }_1{\cos }^2{\beta }_1)2{\widehat{I}}_z{\widehat{S}}_z\hfill \\ \hfill ({\cos }^2{\alpha }_1-{\sin }^2{\alpha }_1)\cos {\beta }_1\sin {\beta }_1{\widehat{I}}_y\hfill \\ \hfill ({\cos }^2{\alpha }_1-{\sin }^2{\alpha }_1)\cos {\beta }_1\sin {\beta }_{1}2{\widehat{I}}_y{\widehat{S}}_z\hfill \\ \hfill \cos {\alpha }_1\sin {\alpha }_1{\widehat{I}}_x\hfill \\ \hfill \cos {\alpha }_1\sin {\alpha }_{1}2{\widehat{I}}_x{\widehat{S}}_z\hfill \end{array}$$ [28]

For the next four rotations given by $e^{-i{\alpha }_2{\widehat{\widehat{I}}}_y}{e}^{+i{\alpha }_{2}2{\widehat{\widehat{I}}}_y{\widehat{\widehat{S}}}_z}{e}^{+i{\alpha }_1{\widehat{\widehat{I}}}_y}{e}^{+i{\alpha }_{1}2{\widehat{\widehat{I}}}_y{\widehat{\widehat{S}}}_z}$, since ${\widehat{I}}_y$ and $2{\widehat{I}}_y{\widehat{S}}_z$ commute, we rewrite them as a sequence of two rotations, $e^{+i({\alpha }_1-{\alpha }_2){\widehat{\widehat{I}}}_y}{e}^{+i({\alpha }_1+{\alpha }_2)2{\widehat{\widehat{I}}}_y{\widehat{\widehat{S}}}_z}$. Therefore, each of the coherences in Eq.[28] evolve under these rotations as:

$$\begin{gathered} ({\cos }^2{\alpha }_1{\cos }^2{\beta }_1+{\sin }^2{\alpha }_1{\sin }^2{\beta }_1){\widehat{I}}_z\stackrel{2{\widehat{I}}_y{\widehat{S}}_z({\alpha }_1+{\alpha }_2)}{\to }\{\begin{array}{c} \\ {\widehat{I}}_z\stackrel{{\widehat{I}}_y({\alpha }_1-{\alpha }_2)}{\to }\{\begin{array}{c}{\widehat{I}}_z\hfill \\ -{\widehat{I}}_x\hfill \end{array}\phantom{\}}\hfill \\ -2{\widehat{I}}_x{\widehat{S}}_z\stackrel{{\widehat{I}}_y({\alpha }_1-{\alpha }_2)}{\to }\{\begin{array}{c}-2{\widehat{I}}_x{\widehat{S}}_z\hfill \\ -2{\widehat{I}}_z{\widehat{S}}_z\hfill \end{array}\phantom{\}}\hfill \end{array}\phantom{\}} \\ -({\cos }^2{\alpha }_1{\sin }^2{\beta }_1+{\sin }^2{\alpha }_1{\cos }^2{\beta }_1)2{\widehat{I}}_z{\widehat{S}}_z\stackrel{2{\widehat{I}}_y{\widehat{S}}_z({\alpha }_1+{\alpha }_2)}{\to }\{\begin{array}{c}2{\widehat{I}}_z{\widehat{S}}_z\stackrel{{\widehat{I}}_y({\alpha }_1-{\alpha }_2)}{\to }\{\begin{array}{c}2{\widehat{I}}_z{\widehat{S}}_z\hfill \\ -2{\widehat{I}}_x{\widehat{S}}_z\hfill \end{array}\phantom{\}}\hfill \\ -{\widehat{I}}_x\stackrel{{\widehat{I}}_y({\alpha }_1-{\alpha }_2)}{\to }\{\begin{array}{c}-{\widehat{I}}_x\hfill \\ -{\widehat{I}}_z\hfill \end{array}\phantom{\}}\hfill \end{array}\phantom{\}} \\ ({\cos }^2{\alpha }_1-{\sin }^2{\alpha }_1)\cos {\beta }_1\sin {\beta }_1{\widehat{I}}_y\stackrel{2{\widehat{I}}_y{\widehat{S}}_z({\alpha }_1+{\alpha }_2)}{\to }{\widehat{I}}_y\stackrel{{\widehat{I}}_y({\alpha }_1-{\alpha }_2)}{\to }{\widehat{I}}_y \\ ({\cos }^2{\alpha }_1-{\sin }^2{\alpha }_1)\cos {\beta }_1\sin {\beta }_{1}2{\widehat{I}}_y{\widehat{S}}_z\stackrel{2{\widehat{I}}_y{\widehat{S}}_z({\alpha }_1+{\alpha }_2)}{\to }2{\widehat{I}}_y{\widehat{S}}_z\stackrel{{\widehat{I}}_y({\alpha }_1-{\alpha }_2)}{\to }2{\widehat{I}}_y{\widehat{S}}_z \\ \cos {\alpha }_1\sin {\alpha }_1{\widehat{I}}_x\stackrel{2{\widehat{I}}_y{\widehat{S}}_z({\alpha }_1+{\alpha }_2)}{\to }\left\{\begin{array}{c} \\ {\widehat{I}}_x\stackrel{{\widehat{I}}_y({\alpha }_1-{\alpha }_2)}{\to }\{\begin{array}{c}{\widehat{I}}_x\hfill \\ {\widehat{I}}_z\hfill \end{array}\phantom{\}}\hfill \\ 2{\widehat{I}}_z{\widehat{S}}_z\stackrel{{\widehat{I}}_y({\alpha }_1-{\alpha }_2)}{\to }\{\begin{array}{c}2{\widehat{I}}_z{\widehat{S}}_z\hfill \\ -2{\widehat{I}}_x{\widehat{S}}_z\hfill \end{array}\phantom{\}}\hfill \end{array}\phantom{\}} \\ \cos {\alpha }_1\sin {\alpha }_{1}2{\widehat{I}}_x{\widehat{S}}_z\stackrel{2{\widehat{I}}_y{\widehat{S}}_z({\alpha }_1+{\alpha }_2)}{\to }\right\{\begin{array}{c}2{\widehat{I}}_x{\widehat{S}}_z\stackrel{{\widehat{I}}_y({\alpha }_1-{\alpha }_2)}{\to }\{\begin{array}{c}2{\widehat{I}}_x{\widehat{S}}_z\hfill \\ 2{\widehat{I}}_z{\widehat{S}}_z\hfill \end{array}\phantom{\}}\hfill \\ {\widehat{I}}_z\stackrel{{\widehat{I}}_y({\alpha }_1-{\alpha }_2)}{\to }\{\begin{array}{c}{\widehat{I}}_z\hfill \\ -{\widehat{I}}_x\hfill \end{array}\phantom{\}}\hfill \end{array}\phantom{\}} \end{gathered}$$ [29]

Combining the terms from Eq.[29], we get:

$$\begin{array}{c}\hfill (({\cos }^2{\alpha }_1{\cos }^2{\beta }_1+{\sin }^2{\alpha }_1{\sin }^2{\beta }_1)\cos ({\alpha }_1+{\alpha }_2)\cos ({\alpha }_1-{\alpha }_2)\hfill \\ \hfill +({\cos }^2{\alpha }_1{\sin }^2{\beta }_1+{\sin }^2{\alpha }_1{\cos }^2{\beta }_1)\sin ({\alpha }_1+{\alpha }_2)\sin ({\alpha }_1-{\alpha }_2)\hfill \\ \hfill +\cos {\alpha }_1\sin {\alpha }_1\cos ({\alpha }_1+{\alpha }_2)\sin ({\alpha }_1-{\alpha }_2)\hfill \\ \hfill +\cos {\alpha }_1\sin {\alpha }_1\sin ({\alpha }_1+{\alpha }_2)\cos ({\alpha }_1-{\alpha }_2)){\widehat{I}}_z\hfill \\ \hfill \hfill \\ \hfill (-({\cos }^2{\alpha }_1{\cos }^2{\beta }_1+{\sin }^2{\alpha }_1{\sin }^2{\beta }_1)\sin ({\alpha }_1+{\alpha }_2)\sin ({\alpha }_1-{\alpha }_2)\hfill \\ \hfill -({\cos }^2{\alpha }_1{\sin }^2{\beta }_1+{\sin }^2{\alpha }_1{\cos }^2{\beta }_1)\cos ({\alpha }_1+{\alpha }_2)\cos ({\alpha }_1-{\alpha }_2)\hfill \\ \hfill +\cos {\alpha }_1\sin {\alpha }_1\sin ({\alpha }_1+{\alpha }_2)\cos ({\alpha }_1-{\alpha }_2)\hfill \\ \hfill +\cos {\alpha }_1\sin {\alpha }_1\cos ({\alpha }_1+{\alpha }_2)\sin ({\alpha }_1-{\alpha }_2))2{\widehat{I}}_z{\widehat{S}}_z\hfill \\ \hfill \hfill \\ \hfill ({\cos }^2{\alpha }_1-{\sin }^2{\alpha }_1)\cos {\beta }_1\sin {\beta }_1{\widehat{I}}_y\hfill \\ \hfill ({\cos }^2{\alpha }_1-{\sin }^2{\alpha }_1)\cos {\beta }_1\sin {\beta }_{1}2{\widehat{I}}_y{\widehat{S}}_z\hfill \\ \hfill \hfill \\ \hfill (-({\cos }^2{\alpha }_1{\cos }^2{\beta }_1+{\sin }^2{\alpha }_1{\sin }^2{\beta }_1)\cos ({\alpha }_1+{\alpha }_2)\sin ({\alpha }_1-{\alpha }_2)\hfill \\ \hfill +({\cos }^2{\alpha }_1{\sin }^2{\beta }_1+{\sin }^2{\alpha }_1{\cos }^2{\beta }_1)\sin ({\alpha }_1+{\alpha }_2)\cos ({\alpha }_1-{\alpha }_2)\hfill \\ \hfill +\cos {\alpha }_1\sin {\alpha }_1\cos ({\alpha }_1+{\alpha }_2)\cos ({\alpha }_1-{\alpha }_2)\hfill \\ \hfill -\cos {\alpha }_1\sin {\alpha }_1\sin ({\alpha }_1+{\alpha }_2)\sin ({\alpha }_1-{\alpha }_2)){\widehat{I}}_x\hfill \\ \hfill \hfill \\ \hfill (-({\cos }^2{\alpha }_1{\cos }^2{\beta }_1+{\sin }^2{\alpha }_1{\sin }^2{\beta }_1)\sin ({\alpha }_1+{\alpha }_2)\cos ({\alpha }_1-{\alpha }_2)\hfill \\ \hfill +({\cos }^2{\alpha }_1{\sin }^2{\beta }_1+{\sin }^2{\alpha }_1{\cos }^2{\beta }_1)\cos ({\alpha }_1+{\alpha }_2)\sin ({\alpha }_1-{\alpha }_2)\hfill \\ \hfill -\cos {\alpha }_1\sin {\alpha }_1\sin ({\alpha }_1+{\alpha }_2)\sin ({\alpha }_1-{\alpha }_2)\hfill \\ \hfill +\cos {\alpha }_1\sin {\alpha }_1\cos ({\alpha }_1+{\alpha }_2)\cos ({\alpha }_1-{\alpha }_2))2{\widehat{I}}_x{\widehat{S}}_z\hfill \end{array}$$ [30]

Simplifying these expressions:

$$\begin{array}{c}\hfill (({\cos }^2{\alpha }_1{\cos }^2{\beta }_1+{\sin }^2{\alpha }_1{\sin }^2{\beta }_1)\cos ({\alpha }_1+{\alpha }_2)\cos ({\alpha }_1-{\alpha }_2)\hfill \\ \hfill +({\cos }^2{\alpha }_1{\sin }^2{\beta }_1+{\sin }^2{\alpha }_1{\cos }^2{\beta }_1)\sin ({\alpha }_1+{\alpha }_2)\sin ({\alpha }_1-{\alpha }_2)\hfill \\ \hfill +\cos {\alpha }_1\sin {\alpha }_1\sin (2{\alpha }_1)){\widehat{I}}_z\hfill \\ \hfill \hfill \\ \hfill (-({\cos }^2{\alpha }_1{\cos }^2{\beta }_1+{\sin }^2{\alpha }_1{\sin }^2{\beta }_1)\sin ({\alpha }_1+{\alpha }_2)\sin ({\alpha }_1-{\alpha }_2)\hfill \\ \hfill -({\cos }^2{\alpha }_1{\sin }^2{\beta }_1+{\sin }^2{\alpha }_1{\cos }^2{\beta }_1)\cos ({\alpha }_1+{\alpha }_2)\cos ({\alpha }_1-{\alpha }_2)\hfill \\ \hfill +\cos {\alpha }_1\sin {\alpha }_1\sin (2{\alpha }_1))2{\widehat{I}}_z{\widehat{S}}_z\hfill \\ \hfill \hfill \\ \hfill ({\cos }^2{\alpha }_1-{\sin }^2{\alpha }_1)\cos {\beta }_1\sin {\beta }_1{\widehat{I}}_y\hfill \\ \hfill ({\cos }^2{\alpha }_1-{\sin }^2{\alpha }_1)\cos {\beta }_1\sin {\beta }_{1}2{\widehat{I}}_y{\widehat{S}}_z\hfill \\ \hfill \hfill \\ \hfill (-({\cos }^2{\alpha }_1{\cos }^2{\beta }_1+{\sin }^2{\alpha }_1{\sin }^2{\beta }_1)\cos ({\alpha }_1+{\alpha }_2)\sin ({\alpha }_1-{\alpha }_2)\hfill \\ \hfill +({\cos }^2{\alpha }_1{\sin }^2{\beta }_1+{\sin }^2{\alpha }_1{\cos }^2{\beta }_1)\sin ({\alpha }_1+{\alpha }_2)\cos ({\alpha }_1-{\alpha }_2)\hfill \\ \hfill +\cos {\alpha }_1\sin {\alpha }_1\cos (2{\alpha }_1)){\widehat{I}}_x\hfill \\ \hfill \hfill \\ \hfill (-({\cos }^2{\alpha }_1{\cos }^2{\beta }_1+{\sin }^2{\alpha }_1{\sin }^2{\beta }_1)\sin ({\alpha }_1+{\alpha }_2)\cos ({\alpha }_1-{\alpha }_2)\hfill \\ \hfill +({\cos }^2{\alpha }_1{\sin }^2{\beta }_1+{\sin }^2{\alpha }_1{\cos }^2{\beta }_1)\cos ({\alpha }_1+{\alpha }_2)\sin ({\alpha }_1-{\alpha }_2)\hfill \\ \hfill +\cos {\alpha }_1\sin {\alpha }_1\cos (2{\alpha }_1))2{\widehat{I}}_x{\widehat{S}}_z\hfill \end{array}$$ [31]

Labeling the coefficients of the operators in Eq.[31] as follows:

$$\begin{array}{c}\hfill A=({\cos }^2{\alpha }_1{\cos }^2{\beta }_1+{\sin }^2{\alpha }_1{\sin }^2{\beta }_1)\cos ({\alpha }_1+{\alpha }_2)\cos ({\alpha }_1-{\alpha }_2)\hfill \\ \hfill +({\cos }^2{\alpha }_1{\sin }^2{\beta }_1+{\sin }^2{\alpha }_1{\cos }^2{\beta }_1)\sin ({\alpha }_1+{\alpha }_2)\sin ({\alpha }_1-{\alpha }_2)\hfill \\ \hfill +\cos {\alpha }_1\sin {\alpha }_1\sin (2{\alpha }_1)\hfill \\ \hfill \hfill \\ \hfill B=-({\cos }^2{\alpha }_1{\cos }^2{\beta }_1+{\sin }^2{\alpha }_1{\sin }^2{\beta }_1)\sin ({\alpha }_1+{\alpha }_2)\sin ({\alpha }_1-{\alpha }_2)\hfill \\ \hfill -({\cos }^2{\alpha }_1{\sin }^2{\beta }_1+{\sin }^2{\alpha }_1{\cos }^2{\beta }_1)\cos ({\alpha }_1+{\alpha }_2)\cos ({\alpha }_1-{\alpha }_2)\hfill \\ \hfill +\cos {\alpha }_1\sin {\alpha }_1\sin (2{\alpha }_1)\hfill \\ \hfill \hfill \\ \hfill C=({\cos }^2{\alpha }_1-{\sin }^2{\alpha }_1)\cos {\beta }_1\sin {\beta }_1\hfill \\ \hfill D=({\cos }^2{\alpha }_1-{\sin }^2{\alpha }_1)\cos {\beta }_1\sin {\beta }_1\hfill \\ \hfill \hfill \\ \hfill E=-({\cos }^2{\alpha }_1{\cos }^2{\beta }_1+{\sin }^2{\alpha }_1{\sin }^2{\beta }_1)\cos ({\alpha }_1+{\alpha }_2)\sin ({\alpha }_1-{\alpha }_2)\hfill \\ \hfill +({\cos }^2{\alpha }_1{\sin }^2{\beta }_1+{\sin }^2{\alpha }_1{\cos }^2{\beta }_1)\sin ({\alpha }_1+{\alpha }_2)\cos ({\alpha }_1-{\alpha }_2)\hfill \\ \hfill +\cos {\alpha }_1\sin {\alpha }_1\cos (2{\alpha }_1)\hfill \\ \hfill \hfill \\ \hfill F=-({\cos }^2{\alpha }_1{\cos }^2{\beta }_1+{\sin }^2{\alpha }_1{\sin }^2{\beta }_1)\sin ({\alpha }_1+{\alpha }_2)\cos ({\alpha }_1-{\alpha }_2)\hfill \\ \hfill +({\cos }^2{\alpha }_1{\sin }^2{\beta }_1+{\sin }^2{\alpha }_1{\cos }^2{\beta }_1)\cos ({\alpha }_1+{\alpha }_2)\sin ({\alpha }_1-{\alpha }_2)\hfill \\ \hfill +\cos {\alpha }_1\sin {\alpha }_1\cos (2{\alpha }_1)\hfill \end{array}$$ [32]

Further simplification using trigonometric identities yields:

$$\begin{gathered} A=\frac{1}{2}({\cos }^{2}2{\alpha }_1\cos 2{\beta }_1+{\sin }^{2}2{\alpha }_1+\cos 2{\alpha }_2) \\ B=\frac{1}{2}({\cos }^{2}2{\alpha }_1\cos 2{\beta }_1+{\sin }^{2}2{\alpha }_1-\cos 2{\alpha }_2) \\ C=\frac{1}{2}(\cos 2{\alpha }_1\sin 2{\beta }_1) \\ D=\frac{1}{2}(\cos 2{\alpha }_1\sin 2{\beta }_1) \\ E=\frac{1}{2}(\cos 2{\alpha }_1\sin 2{\alpha }_1(1-\cos 2{\beta }_1)+\sin 2{\alpha }_2) \\ F=\frac{1}{2}(\cos 2{\alpha }_1\sin 2{\alpha }_1(1-\cos 2{\beta }_1)-\sin 2{\alpha }_2) \end{gathered}$$ [33]

Rewriting Eq.[31] and proceeding with the rotations represented by $e^{i{\beta }_2{\widehat{\widehat{I}}}_x}{e}^{-i{\beta }_{2}2{\widehat{\widehat{I}}}_x{\widehat{\widehat{S}}}_z}$ in Eq.[26]:

$$\begin{gathered} A{\widehat{I}}_z\stackrel{2{\widehat{I}}_x{\widehat{S}}_z(-{\beta }_2)}{\to }\left\{\begin{array}{c}{\widehat{I}}_z\stackrel{{\widehat{I}}_x(-{\beta }_2)}{\to }\{\begin{array}{c}{\widehat{I}}_z\hfill \\ {\widehat{I}}_y\hfill \end{array}\phantom{\}}\hfill \\ -2{\widehat{I}}_y{\widehat{S}}_z\stackrel{{\widehat{I}}_x({\beta }_2)}{\to }\{\begin{array}{c}-2{\widehat{I}}_y{\widehat{S}}_z\hfill \\ 2{\widehat{I}}_z{\widehat{S}}_z\hfill \end{array}\phantom{\}}\hfill \end{array}\phantom{\}} \\ B2{\widehat{I}}_z{\widehat{S}}_z\stackrel{2{\widehat{I}}_x{\widehat{S}}_z(-{\beta }_2)}{\to }\right\{\begin{array}{c}2{\widehat{I}}_z{\widehat{S}}_z\stackrel{{\widehat{I}}_x({\beta }_2)}{\to }\{\begin{array}{c}2{\widehat{I}}_z{\widehat{S}}_z\hfill \\ 2{\widehat{I}}_y{\widehat{S}}_z\hfill \end{array}\phantom{\}}\hfill \\ -{\widehat{I}}_y\stackrel{{\widehat{I}}_x({\beta }_2)}{\to }\{\begin{array}{c}-{\widehat{I}}_y\hfill \\ {\widehat{I}}_z\hfill \end{array}\phantom{\}}\hfill \end{array}\phantom{\}} \\ C{\widehat{I}}_y\stackrel{2{\widehat{I}}_x{\widehat{S}}_z(-{\beta }_2)}{\to }\left\{\begin{array}{c} \\ {\widehat{I}}_y\stackrel{{\widehat{I}}_x({\beta }_2)}{\to }\{\begin{array}{c}{\widehat{I}}_y\hfill \\ -{\widehat{I}}_z\hfill \end{array}\phantom{\}}\hfill \\ -2{\widehat{I}}_z{\widehat{S}}_z\stackrel{{\widehat{I}}_x({\beta }_2)}{\to }\{\begin{array}{c}2{\widehat{I}}_z{\widehat{S}}_z\hfill \\ 2{\widehat{I}}_y{\widehat{S}}_z\hfill \end{array}\phantom{\}}\hfill \end{array}\phantom{\}} \\ D2{\widehat{I}}_y{\widehat{S}}_z\stackrel{2{\widehat{I}}_x{\widehat{S}}_z(-{\beta }_2)}{\to }\right\{\begin{array}{c}2{\widehat{I}}_y{\widehat{S}}_z\stackrel{{\widehat{I}}_x({\beta }_2)}{\to }\{\begin{array}{c}2{\widehat{I}}_y{\widehat{S}}_z\hfill \\ -2{\widehat{I}}_z{\widehat{S}}_z\hfill \end{array}\phantom{\}}\hfill \\ {\widehat{I}}_z\stackrel{{\widehat{I}}_x({\beta }_2)}{\to }\{\begin{array}{c}{\widehat{I}}_z\hfill \\ {\widehat{I}}_y\hfill \end{array}\phantom{\}}\hfill \end{array}\phantom{\}} \\ E{\widehat{I}}_x\stackrel{2{\widehat{I}}_x{\widehat{S}}_z(-{\beta }_2)}{\to }{\widehat{I}}_x\stackrel{{\widehat{I}}_x({\beta }_2)}{\to }{\widehat{I}}_x \\ F2{\widehat{I}}_x{\widehat{S}}_z\stackrel{2{\widehat{I}}_x{\widehat{S}}_z(-{\beta }_2)}{\to }2{\widehat{I}}_x{\widehat{S}}_z\stackrel{{\widehat{I}}_x({\beta }_2)}{\to }2{\widehat{I}}_x{\widehat{S}}_z \end{gathered}$$ [34]

Collecting the terms of the coherences:

$$\begin{array}{c}\hfill (A{\cos }^2{\beta }_2+B{\sin }^2{\beta }_2-C\cos {\beta }_2\sin {\beta }_2+D\cos {\beta }_2\sin {\beta }_2){\widehat{I}}_z\hfill \\ \hfill (A{\sin }^2{\beta }_2+B{\cos }^2{\beta }_2+C\cos {\beta }_2\sin {\beta }_2-D\cos {\beta }_2\sin {\beta }_2)2{\widehat{I}}_z{\widehat{S}}_z\hfill \\ \hfill (A\cos {\beta }_2\sin {\beta }_2-B\cos {\beta }_2\sin {\beta }_2+C{\cos }^2{\beta }_2+D{\sin }^2{\beta }_2){\widehat{I}}_y\hfill \\ \hfill (-A\cos {\beta }_2\sin {\beta }_2+B\cos {\beta }_2\sin {\beta }_2+C{\sin }^2{\beta }_2+D{\cos }^2{\beta }_2)2{\widehat{I}}_y{\widehat{S}}_z\hfill \\ \hfill E{\widehat{I}}_x\hfill \\ \hfill F2{\widehat{I}}_x{\widehat{S}}_z\hfill \end{array}$$ [35]

Defining new constants as the coefficients of the operators:

$$\begin{array}{c}\hfill A^{\prime }=(A{\cos }^2{\beta }_2+B{\sin }^2{\beta }_2-C\cos {\beta }_2\sin {\beta }_2+D\cos {\beta }_2\sin {\beta }_2)\hfill \\ \hfill B^{\prime }=(A{\sin }^2{\beta }_2+B{\cos }^2{\beta }_2+C\cos {\beta }_2\sin {\beta }_2-D\cos {\beta }_2\sin {\beta }_2)\hfill \\ \hfill C^{\prime }=(A\cos {\beta }_2\sin {\beta }_2-B\cos {\beta }_2\sin {\beta }_2+C{\cos }^2{\beta }_2+D{\sin }^2{\beta }_2)\hfill \\ \hfill D^{\prime }=(-A\cos {\beta }_2\sin {\beta }_2+B\cos {\beta }_2\sin {\beta }_2+C{\sin }^2{\beta }_2+D{\cos }^2{\beta }_2)\hfill \\ \hfill E^{\prime }=E\hfill \\ \hfill F^{\prime }=F\hfill \end{array}$$ [36]

Substituting for A, B, C, D, E, F from Eq.[33] and using trigonometric identities, these reduce to:

$$\begin{gathered} A^{\prime }=\frac{1}{2}({\cos }^{2}2{\alpha }_1\cos 2{\beta }_1+{\sin }^{2}2{\alpha }_1+\cos 2{\alpha }_2\cos 2{\beta }_2) \\ {B}^{\prime }=\frac{1}{2}({\cos }^{2}2{\alpha }_1\cos 2{\beta }_1+{\sin }^{2}2{\alpha }_1-\cos 2{\alpha }_2\cos 2{\beta }_2) \\ {C}^{\prime }=\frac{1}{2}(\cos 2{\alpha }_1\sin 2{\beta }_1+\cos 2{\alpha }_2\sin 2{\beta }_2) \\ {D}^{\prime }=\frac{1}{2}(\cos 2{\alpha }_1\sin 2{\beta }_1-\cos 2{\alpha }_2\sin 2{\beta }_2) \\ {E}^{\prime }=\frac{1}{2}(\cos 2{\alpha }_1\sin 2{\alpha }_1(1-\cos 2{\beta }_1)+\sin 2{\alpha }_2) \\ {F}^{\prime }=\frac{1}{2}(\cos 2{\alpha }_1\sin 2{\alpha }_1(1-\cos 2{\beta }_1)-\sin 2{\alpha }_2) \end{gathered}$$ [37]

We now perform the operations to transform back to the original frame of reference using the rotations $e^{+i{\alpha }_2{\widehat{\widehat{I}}}_y}{e}^{-i{\alpha }_{2}2{\widehat{\widehat{I}}}_y{\widehat{\widehat{S}}}_z}$:

$$\begin{gathered} A^{\prime }{\widehat{I}}_z\stackrel{2{\widehat{I}}_y{\widehat{S}}_z(-{\alpha }_2)}{\to }\left\{\begin{array}{c}{\widehat{I}}_z\stackrel{{\widehat{I}}_y({\alpha }_2)}{\to }\{\begin{array}{c}{\widehat{I}}_z\hfill \\ -{\widehat{I}}_x\hfill \end{array}\phantom{\}}\hfill \\ 2{\widehat{I}}_x{\widehat{S}}_z\stackrel{{\widehat{I}}_y({\alpha }_2)}{\to }\{\begin{array}{c}2{\widehat{I}}_x{\widehat{S}}_z\hfill \\ 2{\widehat{I}}_z{\widehat{S}}_z\hfill \end{array}\phantom{\}}\hfill \end{array}\phantom{\}} \\ {B}^{\prime }2{\widehat{I}}_z{\widehat{S}}_z\stackrel{2{\widehat{I}}_y{\widehat{S}}_z(-{\alpha }_2)}{\to }\right\{\begin{array}{c}2{\widehat{I}}_z{\widehat{S}}_z\stackrel{{\widehat{I}}_y({\alpha }_2)}{\to }\{\begin{array}{c}2{\widehat{I}}_z{\widehat{S}}_z\hfill \\ -2{\widehat{I}}_x{\widehat{S}}_z\hfill \end{array}\phantom{\}}\hfill \\ {\widehat{I}}_x\stackrel{{\widehat{I}}_y({\alpha }_2)}{\to }\{\begin{array}{c}{\widehat{I}}_x\hfill \\ {\widehat{I}}_z\hfill \end{array}\phantom{\}}\hfill \end{array}\phantom{\}} \\ {C}^{\prime }{\widehat{I}}_y\stackrel{2{\widehat{I}}_y{\widehat{S}}_z(-{\alpha }_2)}{\to }{\widehat{I}}_y\stackrel{{\widehat{I}}_y({\alpha }_2)}{\to }{\widehat{I}}_y \\ {D}^{\prime }2{\widehat{I}}_y{\widehat{S}}_z\stackrel{2{\widehat{I}}_y{\widehat{S}}_z(-{\alpha }_2)}{\to }2{\widehat{I}}_y{\widehat{S}}_z\stackrel{{\widehat{I}}_y({\alpha }_2)}{\to }2{\widehat{I}}_y{\widehat{S}}_z \\ {E}^{\prime }{\widehat{I}}_x\stackrel{2{\widehat{I}}_y{\widehat{S}}_z(-{\alpha }_2)}{\to }\left\{\begin{array}{c} \\ {\widehat{I}}_x\stackrel{{\widehat{I}}_y({\alpha }_2)}{\to }\{\begin{array}{c}{\widehat{I}}_x\hfill \\ {\widehat{I}}_z\hfill \end{array}\phantom{\}}\hfill \\ -2{\widehat{I}}_z{\widehat{S}}_z\stackrel{{\widehat{I}}_y({\alpha }_2)}{\to }\{\begin{array}{c}-2{\widehat{I}}_z{\widehat{S}}_z\hfill \\ 2{\widehat{I}}_x{\widehat{S}}_z\hfill \end{array}\phantom{\}}\hfill \end{array}\phantom{\}} \\ {F}^{\prime }2{\widehat{I}}_x{\widehat{S}}_z\stackrel{2{\widehat{I}}_y{\widehat{S}}_z(-{\alpha }_2)}{\to }\right\{\begin{array}{c}2{\widehat{I}}_x{\widehat{S}}_z\stackrel{{\widehat{I}}_y({\alpha }_2)}{\to }\{\begin{array}{c}2{\widehat{I}}_x{\widehat{S}}_z\hfill \\ 2{\widehat{I}}_z{\widehat{S}}_z\hfill \end{array}\phantom{\}}\hfill \\ -{\widehat{I}}_z\stackrel{{\widehat{I}}_y({\alpha }_2)}{\to }\{\begin{array}{c}-{\widehat{I}}_z\hfill \\ {\widehat{I}}_x\hfill \end{array}\phantom{\}}\hfill \end{array}\phantom{\}} \end{gathered}$$ [38]

Collecting all the terms, the spin density operator at the end of the RF pulse sums up to:

$$\begin{gathered} {\widehat{\sigma }}_{RF}(t)=(A^{\prime }{\cos }^2{\alpha }_2+B^{\prime }{\sin }^2{\alpha }_2+E^{\prime }\cos {\alpha }_2\sin {\alpha }_2-F^{\prime }\cos {\alpha }_2\sin {\alpha }_2){\widehat{I}}_z \\ +(A^{\prime }{\sin }^2{\alpha }_2+B^{\prime }{\cos }^2{\alpha }_2-E^{\prime }\cos {\alpha }_2\sin {\alpha }_2+F^{\prime }\cos {\alpha }_2\sin {\alpha }_2)2{\widehat{I}}_z{\widehat{S}}_z \\ +C^{\prime }{\widehat{I}}_y \\ +D^{\prime }2{\widehat{I}}_y{\widehat{S}}_z \\ +(-A^{\prime }\cos {\alpha }_2\sin {\alpha }_2+B^{\prime }\cos {\alpha }_2\sin {\alpha }_2+E^{\prime }{\cos }^2{\alpha }_2+F^{\prime }{\sin }^2{\alpha }_2){\widehat{I}}_x \\ +(A^{\prime }\cos {\alpha }_2\sin {\alpha }_2-B^{\prime }\cos {\alpha }_2\sin {\alpha }_2+E^{\prime }{\sin }^2{\alpha }_2+F^{\prime }{\cos }^2{\alpha }_2)2{\widehat{I}}_x{\widehat{S}}_z \end{gathered}$$ [39]

Substituting for A′, B′, C′, D′, E′, F′ from Eq.[37], setting 2α₁ = θ₁, 2α₂ = θ₂, 2β₁ = D₁t and 2β₂ = D₂t, and reducing the expressions using trigonometric identities, we get:

$$\begin{gathered} {\widehat{\sigma }}_{RF}(t)=\frac{1}{2}(2+{\cos }^2{\theta }_1(\cos D_{1}t-1)+{\cos }^2{\theta }_2(\cos D_{2}t-1)){\widehat{I}}_z \\ +\frac{1}{2}({\cos }^2{\theta }_1(\cos D_{1}t-1)-{\cos }^2{\theta }_2(\cos D_{2}t-1))2{\widehat{I}}_z{\widehat{S}}_z \\ +\frac{1}{2}(\cos {\theta }_1\sin D_{1}t+\cos {\theta }_2\sin D_{2}t){\widehat{I}}_y \\ +\frac{1}{2}(\cos {\theta }_1\sin D_{1}t-\cos {\theta }_2\sin D_{2}t)2{\widehat{I}}_y{\widehat{S}}_z \\ +\frac{1}{2}(\cos {\theta }_1\sin {\theta }_1(1-\cos D_{1}t)+\cos {\theta }_2\sin {\theta }_2(1-\cos D_{2}t)){\widehat{I}}_x \\ +\frac{1}{2}(\cos {\theta }_1\sin {\theta }_1(1-\cos D_{1}t)-\cos {\theta }_2\sin {\theta }_2(1-\cos D_{2}t))2{\widehat{I}}_x{\widehat{S}}_z \end{gathered}$$ [40]

which gives the analytical expression for the spin density operator considering the evolution of off-resonance and J-coupling during the application of the RF pulse.

Figure A1.

Definition of the two tilted frames of reference, defined by θ₁ and θ₂ respectively.