Sensitivity Enhancement of Remotely Coupled NMR Detectors using Wirelessly Powered Parametric Amplification

Abstract

A completely wireless detection coil with an integrated parametric amplifier has been constructed to provide local amplification and transmission of MR signals. The sample coil is one element of a parametric amplifier using a zero-bias diode that mixes the weak MR signal with a strong pump signal that is obtained from an inductively coupled external loop. The NMR sample coil develops current gain via reduction in the effective coil resistance. Higher gain can be obtained by adjusting the level of the pumping power closer to the oscillation threshold, but the gain is ultimately constrained by the bandwidth requirement of MRI experiments. A feasibility study here shows that on a NaCl/D₂O phantom, ²³Na signals with 20 dB of gain can be readily obtained with a concomitant bandwidth of 144 kHz. This gain is high enough that the integrated coil with parametric amplifier, which is coupled inductively to external loops, can provide sensitivity approaching that of direct wire connection.

Introduction

It has long been a practice in the magnetic resonance community to use small coils for localized spectroscopy and imaging (1,2), such as catheter coils for interventional MRI (3). In the case of implanted or catheter coils, it has been common to run a hard-wired connection from the coil to the preamplifier, sometimes over long distances, leading to sensitivity losses and RF heating along the cables. Mutual inductive coupling between an internal coil and an external pick-up loop can be used to eliminate wires; however, the separation between the external loop and the receive coil must be close if sensitivity is to be maintained (2,4). It would be advantageous to amplify the MRI signal first before it is transmitted wirelessly to an external coil. Most transistor-based low-noise amplifiers require a DC power source that is difficult to provide wirelessly. Parametric amplifiers (5,6), on the other hand, can obtain power from an externally applied RF magnetic field. The pumping signal can be used either to amplify the weak NMR signal at the same frequency or to mix and amplify the signal to a higher frequency. Parametric amplifiers were widely used for low-noise amplification prior to the advent of high-frequency transistors. There has been renewed interest in using parametric amplifiers as MR detectors (7–10) and traveling wave amplifiers in the context of metamaterials (11–13).

Common implementations of a parametric amplifier use a reverse-biased varactor and this generally requires a wire connection to bias the diode. In this work, a parametric amplifier with a zero-biased varactor is demonstrated without bias connections, and amplification can be integrated within a compact, wireless sample circuit. Although the varactor is inaccessible for tuning, the parametric amplifier can provide sufficient gain even though the detection circuit is detuned from the Larmor frequency, as long as the frequency and power of the pumping signal are adjusted to compensate for this detuning. A recently reported wireless design used the upper side-band of a parametric amplifier for gain in phased-array detectors (8). However, our design utilizes the lower mixing sideband that can provide higher gains of up to 27 dB, and pumping power consumption is reduced to only 0 dBm using the zero-biased diode. The integrated detector and amplifier can have greatly enhanced sensitivity when it is remotely coupled to an external loop.

Methods

Fig. 1 illustrates passive and active modes for detecting with inductively coupled resonators. In Fig. 1a, the NMR signal is detected with a passive resonator that is coupled inductively to a larger, external loop. The signal-to-noise (power) ratio (SNR) detected with the external loop is

Fig. 1.

Functional description of the isolated parametric amplifier. In (a), a small volume (V_m) of sample is directly detected by an embedded local resonator of radius b at a distance separation of b. The nuclei spins induce a current flow I_m in the local resonator, which is then inductively coupled to an external pick-up loop at a distance separation of a (b≪a). In (b), the current flow is increased by a factor of n due to parametric frequency mixing, and the embedded local resonator has a power gain G₁ of n².

$${\left(\frac{S}{N}\right)}_0=\frac{S_0{E}_1}{N_1{E}_1+N_{Lp1}}.$$ [1]

Here, S₀ is the total signal power generated by the nuclear spins, N₁ is the noise power originating from the local resonator, N_Lp1 is the noise power originating from the external loop, and E₁ is the transmission efficiency, i.e. the power delivered to the pick-up loop per unit power originating from the local resonator. If the local resonator has a power gain of G₁, the SNR becomes

$${\left(\frac{S}{N}\right)}_1=\frac{S_0{E}_1{G}_1}{N_1{F}_1{E}_1{G}_1+N_{Lp1}},$$ [2]

where F₁ is the noise factor of the amplification process. If the gain is sufficient such that E₁G₁ ≫ 1, the noise of the local resonator can be amplified to a level much larger than that of the external loop, and the SNR is approximately S₀/N₁F₁, which is comparable to that of a local resonator with a direct wire connection.

A local resonator with gain was implemented as a triple frequency parametric resonator shown in Fig. 2a. To promote mixing, strong currents flow through the varactor C₂ at each of its resonance frequencies, ω₁₀, ω₂₀, ω₃₀. The circuit is constructed such that ω₁₀ + ω₂₀ ≈ ω₃₀ with ω₁₀ close to the Larmor frequency ω₁. The weak NMR signal at ω₁ is mixed with a strong pumping input at ω₃ to produce an amplified output at ω₂= ω₃ − ω₁. The output signal at ω₂ is mixed again with the pumping signal at ω₃ to provide a secondary output with gain at ω₁. This secondary signal manifests itself as an increased current flow at ω₁. Previous analysis of parametric amplifiers (5) addressed the case where the circuit is tuned precisely to its driving frequency, i.e. ω₁₀= ω₁, ω₂₀= ω₂ and ω₃₀= ω₃. The analysis can be extended to include the off-resonance behavior (see Appendix), which allows for the situation when the resonator is isolated and cannot be retuned. To optimize the gain, two conditions should be satisfied: first, the frequency matching condition

Fig. 2.

(a) The circuit diagram of the integrated coil with parametric amplifier and two peripheral loops labeled by 1, 3. (b) A snapshot of the resonator showing the sample placement. The actual components used for the triple frequency resonator were: L₁=90 nH (Coilcraft 132-09SM), C₁=3.6 pF (ATC-100A), L₂ is a rectangular copper loop with a dimension of 24 × 8 mm², C₂ is a varactor (Skyworks Inc, SMV-1139) with a zero-bias capacitance of 8 pF, L₃=1.65 nH (Coilcraft 0906-2KL), C₃=33 pF (ATC-100B). The volume dimensions of the entire resonator were 30 × 8 × 6 mm³. L₁ contains a glass cylindrical sample tube with the red cover. (c) For the actual NMR experiments, the triple frequency resonator couples with loop 1 and loop 3 through the inductor L₂. Loop 3 is placed in the same plane as the resonator. Loop 1 is placed directly above the resonator at a certain distance separation, and it is used for both excitation and detection at ω₁. NMR and MRI experiments were performed using an 11.7 T horizontal magnet (Magnex Inc, Oxford, UK) equipped with an Avance 3 spectrometer console (Bruker Inc, Billerica, MA).

$$\frac{2Q_1({\omega }_1-{\omega }_{10})}{{\omega }_{10}}\approx \frac{2Q_2({\omega }_2-{\omega }_{20})}{{\omega }_{20}}\equiv \kappa$$ [3a]

where Q₁ and Q₂ are the quality factors at ω₁₀ and ω₂₀ respectively; second, the resistance matching condition

$$R_N\equiv \frac{M^2}{C_2^2{(1-M^2)}^2{\omega }_1{\omega }_2{R}_2(1+{\kappa }^2)}\approx R_1$$ [3b]

where C₂ is the varactor capacitance at zero bias, M is the modulation index defined by C₂(t)=C₂(1+2M cosω₃t), R₁ and R₂ are the effective resistance of the circuit loop containing the varactor at ω₁₀ and ω₂₀ respectively, and κ is the ratio defined in Eq. [3a]. Eq. [3b] defines the “negative resistance” R_N that can cancel the circuit resistance R₁, and this is done by adjusting the modulation index M. When R₁ = R_N, M = M₀ and the parametric resonator begins to oscillate. To avoid oscillation, R₁ > R_N is required, and the power gain can be estimated from the extent to which the resonator deviates from the oscillation condition

$$G_1={\left(\frac{R_1}{R_1-R_N}\right)}^2\approx \frac{1}{4}{\left(\frac{M_0}{M_0-M}\right)}^2.$$ [4]

Fig. 2b shows the prototype resonator that was constructed. This configuration was chosen to ensure that significant current passed through the diode at each frequency. It was constructed on a planar Teflon substrate of 1.6 mm thickness. A single, large, rectangular inductor 24 × 8 mm² in size was cut from single-clad Cu PCB, with a single break at the center of one long conductor for the varactor diode. Two parallel circuits were added at each end of the rectangle for triple resonance. The circuit resonance ω₁₀ was chosen to be close to the ²³Na frequency and ω₂₀ was chosen close to the ¹H frequency. The X-channel of the spectrometer was used to observe ²³Na signals, and the proton-channel was used to observe the lower side-band of the up-converted frequency. The sample used for all experiments was a 3 mm tube filled with 1 M NaCl/D₂O solution. The resonance frequencies of the resonator when loaded with the sample were, ω₁₀ =131.7 MHz, ω₂₀ =498.7 MHz and ω₃₀ =628.4 MHz, and the quality factors were Q₁ = 93, Q₂ = 48 and Q₃ = 60. There were two separate single-tuned loops on the periphery that were weakly coupled to this resonator. Loop 1 coupled with L₂ to excite the sodium spins at ω₁, and loop 3 coupled with L₂ for the pumping signal at ω₃.

For NMR experiments, loop 3 was placed on the same plane as the resonator, and loop 1 was placed directly above the rectangular inductor such that it could be used for both excitation and detection at ω₁ (Fig. 2c). ω₁ was 132.1 MHz which was the ²³Na Larmor frequency at 11.7 T, and the correct operating condition was found by empirically adjusting the pumping signal to a proper frequency and power level so that the sharp oscillation peak overlapped the anticipated spectrum peak. The optimal pumping frequency was observed to be ω₃ = 633.9 MHz. According to ω₂= ω₃ − ω₁, ω₂ was 501.8 MHz, which was in good agreement with the frequency predicted by Eq. [3a]. The pumping power level was then reduced and spectra and images were acquired.

Results

The parametric amplifier is predicted to have a bandwidth that is inversely proportional to the square root of the power gain $\sqrt{G_1}$ (5). To test this relation using a zero-biased diode, bench measurements were performed on the parametric resonator with Loop 3 driving the pump signal, and a double pick-up loop placed directly above the resonator was connected to a network analyzer to measure gain and bandwidth. A reference curve was first measured using no pumping power, and the remaining curves were obtained by gradually increasing the pumping power until it reached approximately 95% of the oscillation value. As is shown in Fig. 3a, when the gain increases, the bandwidth decreases. The relative height of each curve reflects the power gain G₁ of the circuit at ω₁, and this is plotted in Fig. 3b to show the reciprocal relationship between the bandwidth and $\sqrt{G_1}$. When the resonator reaches a gain of 20 dB ( $1/\sqrt{G_1}=0.1$), the bandwidth at −3 dB is 144 kHz, which is sufficient for most MRI experiments.

Fig. 3.

(a) The bench measurement of the S₁₂ transmission curve with a double pick-up loop connected to an HP-8715 network analyzer. The bottom-most curve is measured without pumping power. As the pumping power is increased, the peak position rises and the bandwidth decreases. The five curves correspond to the pumping power settings that are 1.16 dB, 1.00 dB, 0.84 dB, 0.66 dB, and 0.46 dB below the oscillation condition. (b) The linear dependence of bandwidth on the reciprocal of the magnitude gain. For each curve shown in Fig. 3a, the bandwidth is measured for the −3 dB points with respect to the peak value, and the power gain (in dB) is obtained by subtracting the peak value with the corresponding S₁₂ value of the bottom-most curve at that particular frequency. (c) The magnitude gain as measured by NMR experiments using the setup detailed in Fig. 2c. M₀ is the modulation index required for oscillation, M is the modulation index used in NMR experiments, and the vertical axis represents the magnitude gain at the output frequency ω₁ excluding the contribution from the unamplified signal.

Gain measurements were also obtained from the parametric circuit using ²³Na NMR. The modulation index M₀ was determined from the point of oscillation, and the ratio M₀/M₀(− M) was monitored as the gain of the NMR signal was being measured (Note: M is not the magnetization of the nuclear spins). In Fig. 3c the magnitude of the signal gain, $\sqrt{G_1}-1$, was plotted against M₀/M₀(− M) to show the linear relationship predicted by Eq. [4]. $\sqrt{G_1}$ was obtained by taking the ratio of spectra intensities at ω₁ with and without parametric amplification. In the plot, we use $\sqrt{G_1}-1$ instead of $\sqrt{G_1}$ to remove the source contribution to the output signal. This is an important modification to Eq. [4], especially when the gain is low. The line in the figure has a slope of 0.473, which is in good agreement with the theoretical value 0.5 in Eq. [4], and the intercept at the vertical axis is close to the slope, which reconfirms $\sqrt{G_1}=1$ when M = 0.

In Fig. 4a the sensitivity of the parametric resonator is compared with that of a single tuned coil matched to 50 ohms and connected by cable to the spectrometer preamp; both used the same sample and identical sample coils. The parametric resonator was positioned a short distance (1.1 cm) away from the external pickup loop, and its gain was set to 20 dB. The spectrum and images obtained by the parametric resonator are displayed in the left column of Fig. 4a, and those obtained by the direct connected coil are displayed in the right column. The comparison of spectral intensity in the first row demonstrates that at short distance separation, the parametric resonator is only 10% less sensitive than a coil with a direct connection. The comparison of MRI images in the bottom rows demonstrates that the parametric resonator can be used to obtain high quality images. Fig. 4b demonstrates the sensitivity advantage of parametric amplification under weak coupling conditions where the distance separation between the external pickup loop and the parametric resonator was 2.8 cm. Images collected in the absence of pumping power are shown in the left column, and images acquired in the presence of pumping power are shown in the right column. The pumping power was approximately 0.4 dB below oscillation. The images in the bottom row are 16 dB more sensitive than the images in the top row. This greatly enhanced sensitivity demonstrates the advantage of a parametric resonator over a passive resonator when signals are remotely detected.

Fig. 4.

(a) A performance comparison between the parametric resonator and a local resonator with direct wire connection. The images in the left column were acquired by the parametric resonator according as the experimental setup detailed in Fig. 2c, with the distance separation between the resonator and the external pick-up loop to be 1.1 cm. The images in the right column were acquired by the same inductor coil matched to 50 ohms and directly connected to the spectrometer preamp. The first row shows the intensity comparison for single pulse experiments when the peak height is normalized to the same noise level. The second row and third row show the comparison of longitudinal and axial slices of 3D FLASH experiments. The sample used in all experiments is 1 M NaCl/D2O solution contained in 3 mm tube, and the parameters for FLASH images are: TR = 100 ms, TE = 3.1 ms, NEX = 1, FOV = 1.9 × 1.9 × 1.9 cm³, matrix size 64 × 64 × 64. (b) A 3D FLASH images acquired by the weakly coupled parametric resonator without amplification (left column) and with 20 dB gain (right column). The distance separation between the parametric resonator and the external pick-up loop is 2.8 cm. Other experimental parameters are the same as those in Fig. 4a.

Fig. 5 shows the dependence of sensitivity upon the separation between the resonator and external pick-up loop. For each distance, the pumping power was adjusted to three levels: 1.70 dB, 0.46 dB and 0.20 dB below oscillation, which corresponded to gain levels of 10 dB, 20 dB and 27 dB respectively. The sensitivity measured under each condition was then normalized against a reference spectrum obtained with the single-tuned coil having a direct cable connection. The normalized sensitivities of Na spectra acquired without parametric amplification are plotted as the blue curve at the bottom, which is described by Eq. [1] and falls rapidly at the rate of −8.4 dB/cm. As pumping power is applied, the gain overcomes the attenuation at short distances, and the sensitivity curves flatten. Most notable is the sensitivity profile at 27 dB gain (red curve), which remains above −3.4 dB for distance separations up to 3.7 cm. This distance separation is much larger than the dimension of the resonator and the pick-up loop. No experiments were attempted beyond 27 dB gain, because the −3 dB bandwidth falls to 67 kHz according to Fig. 3b, which is a lower limit for many MRI experiments. For larger separations, the attenuation eventually dominates, and the sensitivity curves fall off at the same rate as the passive circuit.

Fig. 5.

Dependence of the normalized detection sensitivity on the separation between the parametric resonator and the output loop. The resonator has overall dimensions of 3 × 0.8 × 0.6 cm³, while the rectangular output receiving loop has a dimension of 6.5 × 6.5 mm². The blue curve represents the normalized sensitivity when signal is detected without parametric amplification, the green curve represents the normalized sensitivity when pumping power is 1.70 dB below oscillation and achieves a gain of 10 dB, the cyan curve represents the normalized sensitivity when the pumping power is 0.46 dB below oscillation and achieves a gain of 20 dB, and the red curve represents the normalized sensitivity when the pumping power is 0.20 dB below oscillation and achieves a gain of 27 dB. The bandwidth at each gain level is estimated from the linear relation obtained in Fig. 3b.

Discussion

A detector with an integrated parametric amplifier has been constructed to demonstrate enhanced detection sensitivity of remotely coupled detectors. The detection coil had a larger dimension than the pick-up loop, but according to the principle of reciprocity, this is equivalent to the case when the parametric resonator has a small dimension, as would be used for implantation, and the external loop could have a much larger dimension. The radius of the external pick-up loop approximates the required penetration depth, while the dimensions of the local resonator should be large enough to have sufficient field-of-view and small enough to fit into the internal space within the tissue. A parametric resonator on the mm scale should be straightforward to make.

The pumping signal used to power the parametric circuit creates an additional heat source besides the NMR excitation pulse (14). For our prototype circuit, the output power level from the pumping loop was 13 dBm, which yielded an effective power received by the resonator of 0 dBm after attenuation was taken into account. A zero-biased varactor requires less pumping power owing to its greater nonlinearity, and this power is far less than that required by a low-noise transistor amplifier and too small to induce local heating. If the resonator is implanted such that a passively coupled circuit experiences a 30 dB reduction in SNR, the pumping power required to recover this sensitivity can also be estimated as 30 dBm, which is a relatively low power. However, further studies will be needed to evaluate the RF safety of the wirelessly powered parametric amplifier for in-vivo applications (15). In this context, according to Eq. [3b], the modulation index M₀ for oscillation is proportional to $1/\sqrt{Q_1{Q}_2}$, thus the required pumping power is proportional to 1/(Q₁Q₂). Therefore, when a parametric resonator with smaller size is constructed, it is important to design for good quality factors to limit the required power.

In summary, a parametric resonator with a zero-biased varactor circuit operating efficiently in a de-tuned condition has been demonstrated. A simple triple frequency resonator can wirelessly harvest external power and amplify signals for detection via inductive coupling. Excellent gain with acceptable noise figure was achieved. The parametric resonator should find applications in catheter and implanted coils where a wire connection is unfavorable or impossible. It can serve as a detector to couple to travelling wave MRI devices (16,17) when the imaging object is remotely excited by an antenna. It may also increase the sensitivity of specialized probes where the detection coil needs to reorient (18) or rotate (19) with the sample, making wire connections difficult to implement.