``If I ¹ 0 for all nuclei, NMR in its present form would be impossible!''

Derek Shaw (1978)

Fourier Transform NMR Spectroscopy

3.1 History of Magnetic Resonance

The phenomenon of nuclear magnetic resonance (NMR) was independently discovered by Felix Bloch and Edward Purcell in the 1940's and the first commercial NMR spectrometer became available in 1953 [47]. Since this time, magnetic resonance has become a popular mode of non-invasive imaging and magnetic resonance spectroscopy has become a mainstay of analytical chemistry.

NMR spectroscopy is predicated on the ability of certain nuclei to absorb and reradiate radio frequency (RF) energy when placed in a static polarizing magnetic field.

Nuclei excited by an RF pulse at or near their Larmor frequency will radiate energy in the form of electromagnetic waves at frequencies f ± d_i where d _i is the chemical shift of precession frequency caused by the local chemical environment of the nucleus. It is in the chemical shift that the intrinsic structural information about the molecule is contained. It is usually extracted by taking the Fourier transform of the received nuclear free induction decay (FID) and examining the resultant frequency spectrum for patterns characteristic of particular chemical interactions [48, 49]. RF energy is usually delivered to and received from the sample via one or more appropriately designed inductive coils. These coils are essentially RF resonators tuned to the Larmor frequency and inductively coupled to the object under study. The design and analysis of these specialized structures is perhaps one of the most promising areas for fundamental advances in the state of NMR. In this chapter, we explore the design of coils of this sort for use in in vivo ³¹ P NMR spectroscopy.

3.2 The NMR Signal

Nuclei of atoms which have an asymmetric charge distribution may be NMR sensitive. Based on the sum of the motions of their individual particles, these nuclei possess a quantum mechanical property dubbed "spin" [50]. Though not a physical motion but rather a property of a specific wave function solution, spin can be treated quite well with a classical mechanical view as will be done briefly here.

A spinning nucleus with odd half-integral spin number is essentially a spinning electrical charge. As such, it appears as a ring current and therefore possesses a magnetic dipole moment oriented along the axis of spin. When placed in an external magnetic field, this dipole moment will try to align itself in one of two possible configurations: either parallel to the field or anti-parallel to it. The anti-parallel orientation is higher in energy and is thus said to be the excited state of the nucleus. At a particular temperature, the ratio of excited to unexcited or ``ground-state''nuclei in a sample will follow the Boltzman distribution such that

where N_E is the number of excited spins, N_G is the number of ground-state spins, k is Boltzman's constant (1.38·10²³ J/K) and T is absolute temperature [47].

The dipole moment of the spinning charge, in addition to having a net alignment, will also have a precessional frequency prescribed by its angular momentum. That is, the dipole moment will actually wobble in a circle some angle from its central alignment axis. This precessional motion is often compared by analogy to the slow wobbling of a top under the influence of an external gravitational field [48]. The frequency of the nuclear magnetic moment's precession is dependent on both a constant g called the gyromagnetic ratio and on the strength of the external field B₀ . The precession frequency can thus be written as f = gB ₀. For ¹H, g is 42.6 MHz/Tesla [47].

A magnetic dipole in a magnetic field can absorb energy at its resonant frequency. The absorption of energy causes a transition to a higher available energy state. In the case of a spin-½ nucleus, there are only two available energy states (for the parallel and anti-parallel alignments). Once a nucleus has been excited, it decays spontaneously to its lower energy state after some period of time. As it decays, it radiates energy at its characteristic frequency which can be observed with a suitably constructed antenna.

The NMR signal observed is actually due to many spins acting in concert. Once excited, these spins return to equilibrium with an exponential decay characterized by a time-constant T1 known as the spin-lattice relaxation constant. Decay of the NMR signal also occurs as the nuclear spins dephase due to slight differences in their precessional frequencies. These differences are due to very small local differences in the magnetic field [47]. The decay of the signal due to dephasing also follows an exponential decay characterized by a time constant T2, the spin-spin relaxation constant. The action of a spin system in a z-directed magnetic field of strength B ₀ and an x-directed applied RF field at frequency w₀ is described by the Bloch equations:

^{where the initial magnetization is , w0 is the Larmor}

frequency w₀ = 2pgB₀ (rad/sec), h is Plank's constant, and DN = N _G - N_E [47].

In a typical NMR experiment, an RF field B₁ (t) at frequency w = w₀ is applied until all of the initial z-directed magnetization has been tipped into the transverse (x-y) plane. The B₁ field is then discontinued and the signal generated as the spins return to equilibrium is recorded. This signal is called the free induction decay (FID) and is generally Fourier transformed in order to obtain information about the sample under study [48].

3.3 The NMR Coil

As has been stated, interaction with the nuclear spin system under study in an NMR experiment is conducted through a suitably designed inductive radio frequency (RF) coil. In general the coil consists not only of a suitable inductive structure but of external circuitry to (1) resonate the structure at the frequency of interest and (2) match the resulting impedance of the circuit to that of the NMR transmitter and receiver system, typically 50 W real. Design of an appropriate NMR coil involves consideration of such factors as (1) signal-to-noise ratio (SNR), (2) sensitivity to the sample of interest, (3) desired field of view (FOV), (4) frequency of operation, (5) homogeneity of the resulting field, and (6) matching the coil impedance to that of the receiver. Many of these issues will be addressed in detail later in the context of the development of particular coils, however, a few are discussed in general terms here.

3.3.1 Signal-to-Noise

For an experiment to be successful, useful information must often be extracted from a measurement result which contains a significant amount of noise. The ratio of the energy in the desired signal divided by the noise energy present in the measurement is often referred to as the signal-to-noise ratio (SNR) of an experiment, and is notoriously low for NMR. The NMR experiment consists of exciting a sample and recording the resulting FID through an RF coil. As such, the signal of interest consists of a voltage, measured at the terminals of the coil, induced by the sample. This voltage will also have a noise component superimposed on it, and the ratio of the signal voltage to the noise voltage is called the signal-to-noise ratio SNR for the coil.

The voltage at the terminals of the coil due to the signal can be written as [51]:

where w is the Larmor frequency and M is the magnetization per unit volume given by [47]

If the volume element of interest is small and the magnetization and coil field are in the same direction, then B₁ (here, the applied field per unit current) and M can be considered constant over the region and the following expression holds [52, 53]:

S µ wMVB₁ (3.5)

In order to find S, we must find B₁. By applying the law of Biot-Savart and integrating along the conductor path we can compute the B₁ field of the coil [54].

In order to compute the SNR, we also need to find the magnitude of the noise voltage seen by the coil. Nyquist developed an expression for the noise power seen at a receiver. The voltage must then be proportional to the square-root of the power, or [55]:

R_eff is the effective resistance seen by the coil and is due to a combination of copper losses in the conducting elements of the coil, R_coil, and losses due to power dissipation in a lossy medium near the coil, R_sample [51].

Large coils with large relatively lossless conductors and large sample FOV's tend to be dominated by sample losses and hence sample noise. Small (micro) coils have smaller, more resistive conductors and less visible sample and are thus dominated by coil losses and conductor noise [56, 57]. Optimization of SNR then depends on the type of loss most important in the coil.

For sample dominated coils, we can write that the power dissipated in the sample is [58]:

where A(r) is again the magnetic potential vector and s is the conductivity of the sample. The magnetic potential can be written [55]:

^{where r' is the distance from the current element to a point in space and}

with a the radius of the coil. For the near-field case (which we are interested in for NMR coils) br' is small and the phase is approximately constant.

The resulting sample resistance R_sample is then found from

The coil resistance is found by noting that for NMR frequencies l = c/f is much greater than the linear dimension of the conductor. In the DC case, we can write the conductor resistance as

where r = 1/s is the resistivity of the conductor in W·m, l is the total length of the conductor and A is the cross-sectional area of the conductor. For a conductor carrying an AC current, currents tend to flow near the surface of the conductor and cause resistance to be somewhat higher, a phenomenon known as the skin-effect. When the conductor radius r is much larger than the skin depth d given by [59]

the resistance of the conductor can be approximated by assuming that all current flows in a shell one skin-depth thick. The cross sectional area is then A = pr² - p(r-d) ² » 2prd, and the resistance is given by:

For copper, r = 17.7 nW·m [54], and at 34.6 MHz, the skin-depth is d = 11.4 mm. For 28 ga. wire, r = 63.5 mm, and from equation (3.12) above, the corresponding resistance per unit length is 3.9 W/m. For large coils, R_sample is typically much larger than R_coil, and hence, thermal noise comes predominantly from the sample. However, as the dimensions of the coil are decreased, the coil may become coil noise dominated when R_coil > R_sample. Such a coil is termed a micro-coil [56, 57] and will be explored in section 3.4.

3.3.2 Sensitivity

In order for an NMR coil to be useful, it should be sensitive to signals from the nuclei of the sample under investigation. One useful definition of the sensitivity of an NMR coil is the SNR of a particular coil per unit volume [56], that is, the SNR attained in a particular coil divided by the volume of sample which yields that measurement. Thus, the sensitivity of a coil may be increased by creating a coil with a higher SNR, or by maintaining SNR as the effective sample volume of the coil is decreased. Examination of the latter case, leads to a number of interesting results which allow one to maximize sensitivity when the sample size is small [56, 60, 61, 62, 63, 64].

3.3.3 Field of View

The field of view for an NMR coil in a uniform magnetic field is equivalent to the volume over which the coil receives signals from the sample. By reciprocity, this implies that the useful FOV of a particular coil is equivalent to the region inside of which the quantity B₁ /I, the magnetic field of the coil normalized with respect to current, is greater than some threshold value relative to the maximum. For a simple loop coil placed on a semi-infinite sample, this corresponds approximately to a hemispherical region with radius equal to that of the loop [1]. For a solenoidal NMR coil, the FOV is generally considered to be the cylinder enclosed by the solenoid [65].

3.3.4 Impedance Matching

Maximum power is transmitted between the coil and the receiver or transmitter when the impedance of the coil and its associated matching network is equal to that of the NMR system and a matched transmission line, generally 50 W real. In order to transform the (almost) purely reactive impedance of the inductive coil to that of the NMR system, a number of creative options have been employed. One standard method consists of matching the coil with two capacitors as shown in Figure 3.1.

Figure 3.1: NMR Coil Match Network. Capacitor C ₁ is used to tune the coil to near resonance at w ₀ while C₂ is used to cancel the resulting reactance, presenting Z_in =50W.

The resulting RLC circuit formed by L_coil , C₁ and the combined coil and sample resistance has an associated resonance with center frequency given by w = (C ₁L_coil )^-1/2 and quality factor given by Q = (w L_coil)/R. When C₁ is adjusted appropriately, w can be set slightly higher than w ₀ so that the parallel impedance at the coil terminals is 50 + jX W. The series impedance of C₂ given by -j/wC₂ is then adjusted to cancel the reactance jX and present an impedance Z_in = 50 W. Typically, a slight variation on the above method is used in which a third capacitance is added in series with the bottom terminal of the coil. This so-called ``balanced'' case is similar in analysis to the slightly simpler case presented in Figure 3.1.

3.4 Design of a Coil Array for NMR Spectroscopy

From the preceding discussion, it is apparent that as the size of an NMR coil is decreased, the FOV and hence the sample resistance R_sample become smaller. At some point, the sample resistance will become smaller than the loss resistance of the coil such that R_sample < R_coil . When this is the case, the coil itself and not the sample is the dominant source of noise. If the region of interest (ROI) for a particular experiment is small, then the SNR over that region can be maximized by choosing a small coil (which is likely to be coil noise dominated because of conductor resistance) whose FOV is matched to the ROI. Similarly, if the ROI can be effectively covered by an array of smaller coils and if data from multiple coils can be obtained, then SNR can be maximized over this ROI by creating an array of smaller coil-noise dominated sample coils whose combined FOV adds up to cover the ROI. This is the case which will be explored for developing a coil array for ³¹P NMR spectroscopy of the in vivo turtle heart. The use of such an array has been suggested previously [66, 64], but a rigorous design treatment has not been attempted.

3.4.1 Depth Resolved Spectroscopy

In cross section the reptilian heart ventricle is as shown in Figure 3.2. A large chamber of blood is surrounded by a muscular wall and partially divided by a septum whose prominence varies across species. In Chrysemys , the ventricular wall is composed of a spongy myocardium substantially thicker in proportion than that of mammalian myocardium. The ventricular cardiomyocytes, or cells of the myocardium are the ones from which an NMR signal is desired in order to study their function. However, the blood also contains some phosphorus, and the signal from this phosphorus is essentially a confounder imposed on the desired signal from ventricular phosphorus species. Therefore, the problem is to design an experiment in which ³¹ P signal from the ventricular cardiomyocytes is maximized and that from the blood is minimized. Several techniques for this type of spatially localized or depth resolved spectroscopy exist including the use of specially designed pulses through surface coils [67] or gradient imaging spectroscopy techniques [68, 69]. However, the surface coil localization technique does not solve the problem that the coil is still sample noise dominated, and spatial localization methods involving gradients are not available on NMR spectroscopy systems which do not have gradient coils.

An alternate solution for matching FOV to a particular ROI while still maintaining optimal SNR is to design an array of small coils, each of which is coil noise dominated and whose individual FOV's add up to cover the desired ROI. Such an array may be constructed of small surface coils and acquisition of the resulting signals may be accomplished via an inexpensive multiplexed receiver [70, 71]. Figure 3.3 depicts one arrangement in which three coils are used to cover an ROI larger than the FOV of a single coil.

Figure 3.3: Use of an Array to Match FOV to ROI . An array of three small surface coils exhibits a combined coverage of the ventricle approximately equal to that of a single larger coil, however, the smaller coils do not receive signals from the blood space.

The first step in the design of an appropriate array consists of choosing an appropriately sized coil array element. That is, we must choose the geometry and dimensions of the individual coils which will make up the array such that SNR from the desired ROI is maximized while signal from the unwanted region is rejected. This problem is illustrated in Figure 3.3. Here the lightly shaded region indicates the ventricular wall which is assumed to contain some homogeneous solution of phosphorus from which we wish to extract an NMR signal. The darkly shaded region corresponds to the blood space which also has a uniform concentration of phosphorus but from which we do not want to acquire signal. In order to analyze this situation we define the a relative signal-to-noise figure and attempt to maximize this quantity. In order to choose a good metric for relative signal to noise, further exploration of the problem is required.

3.4.2 Analysis of an Array for Depth Resolved Spectroscopy

If, for simplicity, we restrict our coil to be a single loop of radius a, oriented transverse to the z-axis, then we can write the axial B ₁ field for the coil as

where m is the magnetic succeptibility, a is the coil radius, z is the distance from the center of the coil, and I is the current in the coil [54]. The signal to noise of such a coil at a particular point in space is proportional to the strength of the B₁ field of the coil at that point divided by the square root of the effective resistance R_eff for the coil and sample. We can evaluate equation (3.7) for the dissipated power by first evaluating equation (3.8) with r' the distance from the current element to a point in space and dl = a df. For the near-field case (which we are interested in for NMR coils) br' is small and the phase is approximately constant. Then equation (3.8) can be written as:

where R is the distance |r - r'|. This quantity varies as a, so the integral of its square will then vary as a ³ times a dimensional integral which will be the same for all similarly shaped flat circular coils [72]. Hence, we will have

If we plug this result into equation (3.6) and substitute equation (3.13) into equation (3.5) we can write an expression for SNR:

This is the case for the sample dominated coil and is in agreement with [57, 64]. If we substitute equation (3.12) for the resistance, as is the case in the coil noise dominated situation, and use l = 2pa for the loop coil, then we get

which is an expression for the SNR along the axis of a single loop coil of radius a when the coil is small and is not sample noise dominated [64]. In either case, it is apparent that the signal to noise of the coil depends on both the distance from the coil and the diameter of the coil [64, 71]. It is this property which will be exploited in order to design an array element of optimal size.

At first glance, it is tempting to simply try to maximize the ratio of the signal from the heart wall S_wall to the signal from the blood S_blood . Since the coil is coil noise dominated, dividing each of these quantities by the coil noise yields SNR_wall /SNR_blood. However, from analysis of equations (3.16) and (3.17) above, it is apparent that such a definition is inadequate since, as a approaches zero, SNR_blood will tend toward zero faster than SNR_wall and no local maximum will occur. If we instead consider the ratio of both the signal plus the noise from the heart wall to the signal plus the noise from the blood, then we will obtain a quantity which is well behaved even when a goes to zero. This quantity can be written as

Since the dominant source of noise is from the coil itself, N _blood and N_wall are equivalent and SNR_rel can be related to the relative SNR obtained from each compartment by dividing both the numerator and denominator by the noise N to yield

For a particular coil, SNR_wall is the average signal-to-noise ratio obtained from the ventricular wall only, and SNR_blood is the average signal-to-noise ratio obtained from the blood space only.

3.4.3 Computer Simulations

In order to evaluate the quantity SNR_rel for a particular coil, we also need information about the geometry and composition of the sample. Specifically we must know the conductivity s for each compartment (wall and blood), the concentration of ³¹ P nuclei in each compartment, and the dimensions of each compartment.

The ventricular wall in the hearts of animals used in this research is about 2-2.5 mm thick. For the purposes of analysis, the blood space is assumed to extend essentially infinitely beyond the ventricle. The approximate composition of blood plasma in turtles is summarized in Table 3.1.

Table 3.1: Turtle Plasma Composition (source: Gans [23]).

Phosphorus concentration in the blood is essentially due to inorganic phosphates in the plasma and is thus estimated to be 1 mM [23]. The concentration of ³¹P in the ventricular wall can be estimated by summing the concentrations of major constituent species. Wasser estimates that cytosolic phosphodiester (PDE) concentrations in turtle hearts from Chrysemys are conservatively 15-20 mM [73]. Extrapolation of other concentrations from published spectra [33] then yields maximum concentration levels of [P_i] ~ 3.5 mM, total phosphomonoesters ~ 4 mM, [a-ATP] ~ 8 mM, [b-ATP] ~ 8 mM, [PCr] ~ 7 mM, and [g-ATP] ~ 5 mM. Finally, the conductivity of the blood is taken to be approximately 0.7 S/m [54]. This value reflects a slight decrease from accepted value of 0.72 S/m in human tissues [71], since from Table 3.1 it is evident that electrolyte concentration in reptilian plasma is slightly lower than 300 mOsm for humans [23]. Krauss lists the conductivity of animal muscle to be significantly lower than that of blood with s = 0.4 siemens per meter [54]. Adapting this value for a first approximation to turtle cardiac muscle is probably reasonable, though the spongy myocardium of the turtle may impart to it a slightly higher conductance due to the increased concentration of blood.

With this information in hand, a model of the sample can be formulated and used to analyze the quantity SNR_rel for coils of varying size. A quasistatic computational electromagnetics program (QSS) was used to evaluate SNR_wall and _SNRblood for each coil [74]. Input to the program included dimensions of a slab for each separate tissue, conductivity, coil radius and position relative to the slab, and frequency of operation (34.635 MHz for ³¹P at 2T). In all cases, the coil conductor was copper wire of radius 63.5 mm (28 Ga.) The actual program parameters are included in APPENDIX A.

The QSS program returns a value representing the average SNR over the entire volume analyzed for each coil. These results are included in APPENDIX A. In order to evaluate SNR_rel, the SNR returned for each constituent slab is scaled by the relative concentration of phosphorus and substituted into equation (3.19). SNR calculations are based on coil and sample losses only and assume perfect lossless match networks for each coil. The resulting values of SNR_rel for coils of varying size are summarized in Table 3.2 and are plotted versus coil diameter in Figure 3.4 which shows that the relative SNR is maximized when the coil radius is 1.5 mm, or equivalently when the diameter is 3 mm. SNR calculations are based on coil and sample losses only and assume perfect identical match networks for each coil.

Table 3.2: Calculated SNR Values.

Thus, the SNR in an experiment designed to extract ³¹ P spectra from the ventricular wall of the turtle heart while rejecting unwanted signals from the blood will be optimum when an array of small 3 mm diameter coils is used.

Figure 3.4: Plot of SNR_rel versus Coil Diameter.

3.4.4 Phantom Experiments

In the preceding section, it was demonstrated theoretically that the local FOV of an NMR experiment could be tailored by the construction of an appropriately designed array of small surface coils, alleviating the need for gradients or calibrated RF pulses. It was found that for the particular geometry and physiological parameters of the turtle heart under investigation, such an array would maximize relative SNR of the ROI over the unwanted bloodspace signal when the coil diameter was 3 mm. In order to investigate the efficacy of this technique, a series of experiments were designed to verify the theoretical results presented.

A two-chambered phantom depicted in Figure 3.5 was built which allowed the investigation of a particular surface coil's SNR in each chamber by alternately filling the chambers with a phosphorus containing phantom and a standard (no phosphorus) conductive phantom.

Surface coils of diameter 20 mm, 3 mm, and 1 mm were fabricated and tuned and matched to 50 W at 34.63 MHz. With each coil, two experiments were performed. In the first, chamber A was filled with a phantom solution consisting of 1 M NaH₂PO₄ and 0.075 M Na₄HP₂O ₇ phosphorus phantom, and chamber B was filled with a saline solution with conductivity 0.72 Siemens/m and doped with CuSO ₄, a T1 relaxing agent. In the second experiment, the contents of the two chambers were reversed. Though the conductivity of the phosphorus solution was not measured, it was assumed that the effect of solution conductivity is negligible in the coil loss dominated case.

The surface coil was affixed to the bottom edge of the phantom below chamber B and a large ¹H volume coil was used for shimming the sample. Phosphorus spectra were then taken using the surface coil in both the transmit and receive mode with 256 averages, 4096 points, and a 90° tip angle, and a ±50000 Hz spectral width. The resulting spectra were then used to calculate SNR_A (when chamber A contained phosphorus) and SNR_B (when chamber B contained phosphorus). SNR was calculated by taking the ratio of the maximum spectral peak height to the RMS value of the spectrum noise. In order to compare these results meaningfully to those of section 3.4.3, these raw SNR figures were converted to ``average'' SNR figures by dividing SNR <sub>A</sub> by the relative volume ratio between the two chambers (which is about 6.8). These ``average'' SNR values are represented as the ``Adjusted'' values in Table 3.3. SNR_Relative was then computed as (SNR_B(Adjusted) + 1)/(SNR_A(Adjusted) + 1). The results are summarized in Table 3.3.

Table 3.3: Measured SNR Values.

Unfortunately, data from the 1 mm coil were unusable, since as the coil radius decreases, the SNR of the experiment becomes too small to extract useful data. Part of this may have been due to the fact that as coil radius decreases, inductance decreases as well. The task of matching and tuning a coil with extremely small inductance is difficult at low frequencies and essentially becomes one of matching a short circuit (which is impossible). For these reasons, even with 1024 acquisitions, no signal was ever evident from the 1 mm coil.

3.4.5 Discussion of the Coil Array

It is evident from Table-3.3 that the 3 mm coil did indeed exhibit the expected improvement in relative SNR. However, it can also bee seen from both Table 3.2 and Table 3.3 that while decreased coil radius increases the relative SNR, the overall SNR of the experiment suffers. In the case of in vivo phosphorus levels at 2T, the SNR of such a small coil is so small as to render the coil useless since it is apparent from Table 3.3 that an array of (274.20/15.244)² » 320 decoupled coils would be required to obtain the same SNR per unit time over the heart wall region [58, 75]. In fact, even in experiments with larger (20 mm) surface coils placed on the turtle heart, the SNR at 2T was so low that an alternate coil had to be designed. It should be noted, however, that the theoretical analysis of relative SNR is applicable to hydrogen spectroscopy as well as phosphorus spectroscopy at higher field strengths. In each of these cases, the SNR of a small coil is improved, and the matching and tuning of such a coil is also facilitated by the increase in operation frequency.

3.5 Design of a Doubly-Tuned, Inductively Coupled Resonator

An alternate means for tailoring the FOV of an NMR experiment is to design a coil which is sensitive only to a particular ROI. Solenoid coils and birdcage resonators both have FOV's which are essentially cylindrical and enclosed by the coil structure [63, 65, 76]. Creation of such a coil which encloses the heart will guarantee an FOV which is matched to the ROI. Although such a coil does not eliminate the unwanted P_i signal from the blood, the plasma P_i signal is sufficiently down field from PCr and b-ATP peaks that it should not interfere with quantification of these analytes and is only a nuisance for tracking intracellular pH via chemical shift of the intracellular P _i peak [2, 3] and for quantitating intracellular inorganic phosphate concentration [P_i ]_i. Fabrication of a solenoid coil tuned for ³¹P spectroscopy and which encloses the turtle heart is a fairly straight-forward matter and provides good sensitivity and SNR for phosphorus spectroscopy of the heart. However, the concentration of phosphorus is sufficiently low such that shimming (the process by which the magnetic B₀ field is iteratively adjusted to yield maximum homogeneity and hence signal amplitude when a sample is present) directly on the ³¹P signal is impossible. In such cases, the much more abundant ¹ H nucleus is used for shimming prior to collection of ³¹P data.

The requirement that shimming be performed on the ¹ H signal necessitates that either a separate coil tuned to hydrogen resonances be used, or that a coil resonated at both hydrogen and phosphorus frequencies be used. The former is the simpler arrangement. However, for effective shimming the FOV of the ³¹ P coil and the ¹H shim coil should be identical. When separate coils are used this constraint can lead to a situation where the coils exhibit significant mutual inductance and hence tuning of one coil may drastically effect the tuning of the other [77]. In the case of the doubly-tuned coil coincident FOV's are assured, and interaction between coils is eliminated. Such coils can be fabricated [78, 79, 80, 81], but construction of a coil appropriate for both ³¹ P and ¹H NMR spectroscopy can be somewhat difficult.

However, if ¹H shimming and not spectroscopy is the goal, and if simultaneous multinuclear spectral acquisition is not required, then a simpler coil which is functional at both frequencies (though not simultaneously) can be constructed fairly easily. LeRoy Willig et al. present one design for such a coil based on combination of inductive matching and transmission line filters [77]. Here we present an alternate design in which a solenoid coil is connected to external circuitry which causes it to resonate at both ¹H and ³¹P frequencies. RF energy is transmitted to and from the coil via two separate inductively coupled feed coils. Each is tuned and matched through a combination of geometric positioning and adjustment of a series capacitor.

A schematic representation of the coil is shown in Figure 3.6. The sensing coil consists of a 5-turn 11 mm diameter solenoid which is parallel resonated at both 34.63 MHz and 85.56 MHz for phosphorus and hydrogen at 2 Tesla, respectively. Radio frequency (RF) energy is communicated to and from the resonator via separate feed coils for both hydrogen and phosphorus. In this case, the hydrogen feed is a 3-turn 8 mm diameter loop coil and the phosphorus feed is a 4-turn 20 mm diameter loop both located concentric and distil to the axis of the resonator solenoid. The trap inductor is a 9-turn toroid wound on a plastic former approximately 4 mm in diameter.

Figure 3.6: Schematic of Inductively Fed Doubly-Tuned Resonator. The sensing coil, L_coil is parallel resonated at both 85.562 MHz and 34.635 MHz by the attached circuitry. Separate hydrogen and phosphorus channels are established by inductively coupled loop coils which are matched to 50 W transmission lines with tunable series capacitances C_tune.

The design of the resonator was carried out by measuring the inductance of the solenoidal sensing coil and then using Matlab to experiment with component values which yielded appropriate resonances by examining the equivalent impedance between the terminals of the sensing coil. The equation for this impedance is

Figure 3.7 shows a plot of |Z_equiv | vs frequency for component values listed in Table 3.4. In designing the resonator, the trap inductance L_trap was first estimated in the Matlab model. Then, a physical inductor having roughly the desired inductance was fabricated using the equation of inductance of an N-turn toroid: L = ½mN² r²/R where r is the turn radius and R is the radius of the toroid. The actual inductance of the resulting structure was measured and placed back into the model. Capacitive values were then changed to yield the desired impedance resonances. Note that in the design method, we ^{choose where w1H is the hydrogen resonance frequency. When this condition is met, the impedance of the trap becomes infinite at and at significantly} lower frequencies, it is essentially a short. In the latter case, capacitances C ₂, C_p, and C ₁ are used to tune the resonance at the lower frequency. By tuning the resonance frequency of the trap slightly higher, the impedance of the series combination of the trap and C₂ becomes inductive and controls the frequency of the higher resonance. Adjustment of C_trap tunes the higher resonance essentially independently of the low-frequency resonance making this coil particularly easy to tune. Once the probe was constructed, an HP4195 analyzer operating with crossed coils in the S₁₂ mode was used to verify the double resonance. The resulting plot is shown in Figure 3.8.

Table 3.4: Resulting Doubly-Tuned Resonator Design.

In order to match the resonator to the 50 W input impedance of the pre-amplifier, we use an inductively coupled loop coil in series with a tunable capacitor. The impedance of this circuit then depends on the coupling constant k between the sample coil and and the feed coil (and thus the relative geometric placement of the coils) and on the value of the series capacitance [82, 83]. Note that although the two feed coils are concentric loops, they do not significantly couple due to the series capacitances. This gives rise to a situation in which the two coils may be tuned independently without complications due to mutual coupling between the feed circuits.