“Shim pulses” for NMR spectroscopy and imaging

Methods

Theory. The objective was to design a pulse that modulates the phase φ(r) of the transverse magnetization by an amount depending on position r, where the functional form of φ(r) is under software control. To constitute a robust experimental method, φ(r) should be insensitive to frequency offset, including offset due to chemical shift σ, and RF amplitude misset within reasonable ranges. We limit this discussion to noninteracting spins 1/2 with gyromagnetic ratio γ in a strong main magnetic field, implying that the direction of quantization is unaffected by the applied gradients.

If signal damping originating from relaxation and diffusion is neglected, the transformation of the local magnetization vector m(r, t) during a general RF- and gradient-pulse sequence lasting a time t _p can be written as follows: R(r) is a rotation matrix, which depends on the time dependence of the local Larmor frequency ω₀(r, t), local nutation frequency ω₁(r, t), RF phase ϕ(t), and RF carrier frequency ω_RF during the pulse sequence. On modern NMR spectrometers, shaped RF and gradient pulses are implemented as a sequence of square pulses with constant RF amplitude, RF phase, and gradient amplitudes. The total rotation matrix for a shaped pulse can be calculated by sequential multiplication of the rotation matrices for the individual square pulses given in ref. 11, for example.

The local nutation frequency is given by the following: where A _RF(t) is the time-dependent relative RF amplitude and B ₁(r) is the maximum local RF amplitude. It is convenient to express the local Larmor frequency offset Δω₀(r, t) = ω₀(r, t) - ω_RF as follows: where B ₀(r) is the magnetic field produced by the main magnet, and B _{0, n}(r) are the maximum magnetic fields originating from the N auxiliary “gradient” coils. A_n(t) are the relative amplitudes of the field from each of these coils. B ₁(r), B ₀(r), and B _{0, n}(r) are given by the coil geometries, neglecting sample-induced field distortions, whereas ϕ(t), A _RF(t), and A_n(t) are controlled by the spectrometer software. The ω_RF value is constant and close to the average value of γB ₀(r) in the region of space from which the NMR signal is detected.

If the rotation matrix is expressed in Cartesian coordinates, the deviation χ² from ideal pulse performance can be defined as follows: where w ₁, w ₂, and w ₃ are weighting parameters. The w ₁-weighted terms quantify the deviation from the target phase shift, and the terms weighted by w ₂ and w ₃ describe the loss of transverse and longitudinal magnetization. The values of the weighting parameters reflect the relative importance of the different terms, but they are generally of the same order of magnitude. The definition of χ² in Eq. 4 is only one of several different possibilities, but we find this version particularly transparent with respect to the desired result (i.e., a pure rotation of the magnetization around the z axis).

Finding the functions ϕ(t), A _RF(t), and A_n(t) that minimize χ² under the conditions given by B ₁(r), B ₀(r), and B _{0, n}(r) is a straightforward nonlinear least-squares problem. The functions ϕ(t), A _RF(t), and A_n(t) should be well behaved in the sense that implementation on a standard NMR spectrometer is feasible. This condition can be assured by either including a term penalizing unrealistic modulation functions in the definition of χ² or by expressing the modulation as a smooth analytical function with a limited number of adjustable parameters. The calculation of χ² should also be performed over a range of B ₀ offsets and B ₁ amplitude missets to guarantee a certain degree of robustness. Note that favorable B ₁ gradients could be used in the manner of the ex situ methodology (1).

Let us assume that we have designed a pulse that gives rise to the desired φ(r). If we sample the NMR signal at the midpoint of the free precession delays in a train of interleaved shim pulses with length t _p and free-precession delays with length t _dw, it appears as though the spins are precessing under the influence of the B ₀ offset, including chemical shift, for a time t _dw in an apparent field of strength as follows: This fact means that the apparent field experienced by the spins can be controlled by the time modulation of the RF field and the auxiliary coil currents. These modulation functions can then be adjusted to make the field appear to be homogeneous for spectroscopy applications or to have a constant gradient for imaging purposes. The desired field profile can be achieved, although none of the hardware components or any linear combination of them, provide constant main field or linear gradient field. This description disregards J couplings and signal damping due to relaxation and diffusion during t _p.

Pulse Design. As the point of departure for the shim-pulse design, we used the well known fact that an adiabatic full passage applied to transverse magnetization induces a phase shift depending on the frequency offset (12). A second passage reverses this phase shift and restores any longitudinal magnetization to its initial state. However, a residual phase shift will be the result of an adiabatic double passage in the presence of a B ₀ inhomogeneity, which is changing between the first and the second passage. Intuitively, it seems possible to tailor the spatial shape of the phase shift by a smooth modulation of the current applied to the coil producing the B ₀ inhomogeneity. In the examples given below, we verify this assumption.

For the adiabatic pulses, we chose to use the sech and tanh amplitude and frequency modulation functions (13). The amplitude modulation of A_n(t) was expressed as a sum of M sine functions as follows: where a_mn is the amplitude of each sine component. Typically, M was chosen to be between 10 and 20. This approach assures that the modulation is zero at the beginning and end of the pulse and reasonably smooth in between. The pulses were discretized in 200 or 500 steps, depending on the overall pulse length. The set of a_mn and, occasionally, t _p and the width of the adiabatic frequency sweep were used as adjustable parameters. The pulses were designed to be offset insensitive over a range of ±5 ppm and independent of B ₁ inhomogeneity or missetting in the range of ±25% the nominal value obtained with standard 90° pulse-length calibration. Numerical optimizations were implemented in matlab 5.2 (MathWorks, Natick, MA) by using the “constr” routine in the Optimization toolbox. As soon as reasonable initial values of the adjustable parameters were estimated, each pulse optimization took on the order of minutes to hours on a 900-MHz iBook (Macintosh).

Experiments and Results. Experiments were carried out on DMX-300 and Avance-700 spectrometers (Bruker, Billerica, MA) by using Bruker probes with coils producing three orthogonal gradients with a maximum gradient strength of 0.5 T/m at a current of 10 A. The probe for the Avance-700 was a standard high-resolution triple-resonance HCN probe, whereas the probe for the DMX-300 was modified to locate the active volume of the RF coil in a region with nonconstant B ₀ gradients. The maximum gradient coil current during the shim pulses was ≈5% of the maximum possible value. The ¹H 90° pulse length was ≈10 μs for both spectrometers, corresponding to an RF field strength of 25 kHz. Unless otherwise stated, the relaxation delay was 5 s, and four scans were accumulated throughout. Experimental details for the individual examples are given in the corresponding figure legends.

Saddle-Shaped Phase Shift with Two Orthogonal Linear Gradients. As a first demonstration of the approach, we use two gradient coils producing linear gradients in the x and y directions to induce a saddle-shaped phase shift in the x, y plane. According to the notation given above, these conditions can be expressed as follows: Note that such a phase shift is impossible when using traditional NMR methods in which the shape of the phase shift is necessarily a linear combination of the shapes of the fields from the gradient coils. A 2D image of the phase shift in the x, y plane was recorded with the pulse sequence in Fig. 1a by using a cylindrical water phantom (i.d., 4 mm) aligned along the z direction in a homogeneous B ₀ field. This pulse sequence is a modification of a standard chemical shift or B ₀ inhomogeneity imaging sequence, yielding an NMR spectrum for each pixel in the image. In the direct dimension, the signal is recorded stroboscopically during a train of shim pulses and free-precession delays. In the indirect dimensions, the strength of the phase encoding gradients are varied. Note that the same gradient coils are used for both the linear phase encoding to obtain the image and the nonlinear phase shifts during detection.

The shim pulse was optimized according to the principles outlined above. As shown in Fig. 1b, the resulting phase shift is saddle shaped. The overall constant phase shift is a result of evolution under the offset for a time t _dw during the delay in which the signal is recorded. The spins behave as though they evolve under the influence of frequency offset, or chemical shift, in a magnetic field made inhomogeneous by using an x ²–y ² shim coil, although the main field is homogeneous and we only use x and y gradient coils.

Shimming a Quadratic Magnetic Field Inhomogeneity Using a Linear Gradient. As a second demonstration, we use the shim-pulse methodology to correct the line broadening from an inhomogeneous magnetic field with linear and quadratic components by using a coil producing a linear magnetic field. Effective (i.e., nonrefocused) chemical shift evolution takes place during a delay of length t _dw between the individual shim pulses. The shim pulse should induce a phase shift that exactly cancels the evolution in the inhomogeneity during t _dw. The conditions for which the shim pulse was optimized can be stated as follows: where a is a constant. The current modulation was optimized for the quadratic part of the inhomogeneity making use of symmetry arguments to make the pulse scalable and improve the offset independence. When this modulation is added to a constant baseline current, simultaneous linear and quadratic corrections are possible.

The sample was 5 mg/ml thiamine (vitamin B₁) hydrochloride (purchased from Sigma–Aldrich and used without further purification) in ²H₂O in a standard 5-mm o.d. sample tube. Accumulating 64 transients by using a recycle delay that was short in comparison with the relaxation time of water suppressed the signal from the residual ¹H²HO. A limited amount of sample was used to assure RF homogeneity. NMR spectra were recorded with the pulse sequence shown in Fig. 2a. The experimental spectra are shown in Fig. 2b. The spectrum obtained with inhomogeneous B ₀ without applied gradients (Fig. 2bi) has a poor signal-to-noise ratio and does not reveal the fine structure. A constant current corrects for the z ¹ term of the inhomogeneity, yielding a spectrum with highly asymmetric peaks (Fig. 2bii), which are typical for a z ² inhomogeneity. With full current modulation, as shown in the pulse sequence in Fig. 2a, the homogeneous spectrum is recovered (Fig. 2biii). The spectra recorded in a homogeneous field with the adiabatic pulse train (Fig. 2biv), and with a one-pulse experiment (Fig. 2b), are shown for comparison. Note that all spectra in Fig. 2b, except v, are recorded with the adiabatic RF pulses an, thus, suffer from the same relaxation-induced line broadening (≈50 Hz). The different effective time scales for the sampling of chemical shifts and J couplings (chemical shift evolution is refocused at the end of the pulse, whereas the evolution of J couplings is not) leads to the expanded multiplet patterns for shim pulses in comparison with standard detection, as can be seen by comparing the peaks at 3.0 and 3.7 ppm in the spectra shown in iv and v.

Linearizing a Gradient with Linear and Cubic Components. The final example consists of correcting the image obtained with a setup in which the gradient coil produces a nonlinear gradient. A schematic representation of the experimental setup is shown in Fig. 3a. The RF coil is located at the edge of the gradient coil and, thus, will pick up signal from regions with nonlinear gradients. The sample was water in 1.25-mm-deep grooves cut in a Delrin rod (o.d., 4 mm). The correlation between B ₁(z) and B _0,z(z) was estimated with the pulse sequence shown in Fig. 3b. Because the sample gives rise to signal from well defined positions along z, B ₁(z) and B _0,z(z) can be estimated from the peak positions shown in Fig. 3c. The five peaks with γB ₁ ≈ 25 kHz correspond to the grooves within the active volume of the RF coil. The compressed distance between the upper peaks results from the smaller gradient strength at the edge of the gradient coil. Above the RF coil, the gradient even reverses direction and B _0,z(z) starts to decrease. The experimental B _0,z(z) is shown in Fig. 3d, together with the fit of a third-order polynomial.

The shim pulse was optimized for the range of z containing the five grooves with an almost-constant B ₁. To a good approximation, the pulse-design problem can be stated as follows: where a is a constant. Images using shim pulses for phase encoding were recorded with the pulse sequence shown in Fig. 4a. Results for a shim pulse with φ(z) ∝ B _0,z(z), which was used as a starting point for the iterative pulse design, and for an optimized shim pulse with φ(z) ∝ z are shown in Fig. 4 b and c, respectively. The compressed distance between the peaks in the top part of Fig. 4b is similar to the results in Fig. 3c. This compression is corrected in Fig. 4c, demonstrating the ability to “linearize” imperfect gradient fields through the use of the shim-pulse methodology.