Microwave Vertebrae Strength Probe Development: Robust and Fast Phase Unwrapping Technique

Abstract

We have developed a new transmission-based, open-ended coaxial probe for assessing vertebrae strength during spinal fusion surgery. The approach exploits the fact that the probes are within the far field of each other implying that the phase varies linearly with respect to propagation distance. Determining the absolute phase is critical for recovering the associated tissue dielectric properties from which bone strength will be determined. Unfortunately, unwanted multi-path signals corrupt the signals at the lower end of the operating frequency range from which our conventional unwrapping strategy depends. Our new approach requires only three measurements within the prime frequency range and can be determined robustly with a minimum of computations. This will be vital to developing a commercial device since the signal levels will be extremely low power requiring longer than usual data acquisition times, which will be mitigated by measuring the data at only a few frequencies. Fast and efficient operation will be critical for clinical success.

I. Introduction

Phase is a critical feature of different radar and imaging technologies. One of the more important applications is in microwave radars and sensors where the phase varies linearly with distance in the antenna far field, making it particularly sensitive to small distance changes. For instance, spaceborne interferometric SAR techniques can detect surface changes as small as 0.3 – 1 cm [1]. However, the phase embeds the challenging aspect that it also retains an ambiguity of multiples of +/− 360°. Removing this ambiguity usually involves determining the absolute phase through a process called phase unwrapping [2], [3]. Different approaches have been developed for this and are usually unique to the application depending on what external information is conveniently available. Below we summarize just a few techniques to demonstrate the importance and ubiquity of this process.

Synthetic aperture radar (SAR) utilizes advanced synthetic phased array approaches which can produce a narrow beam for analysis of ground targets. It exploits both the amplitudes and absolute phases of the returned signals. Conversely, interferometric SAR exploits the differences in the phases for two or more SAR systems. Spaceborne repeat pass radar interferometry platforms routinely produce topographic change maps as digital displacement models (DDMs) with resolution as small as 0.3-1.0 cm [1]. It depends largely on the fact that in the far field, the phase varies linearly with distance. The technique can be used to measure very small variations associated with natural disasters such as land position changes during volcanoes, earthquakes and others. In these cases it is critical to know the absolute phases with respect to the original transmitting signal – hence the need for unwrapping the phase [4].

In MRI, phase images are used widely for monitoring a range of biological conditions. For instance, these images are used in key motion monitoring such as blood velocity, susceptibility-weighted imaging (SWI) and quantitative susceptibility mapping (QSM) [5] - [7]. They are also used in fat suppression imaging [8]. A key aspect of these techniques is that the raw data from the measurement system naturally restricts the data to a single Riemann sheet – i.e. between −180° and +180°. During the early stages of MR development, this usually meant that the technicians could only use the amplitude portion of the signal [2]. The most conventional unwrapping strategy exploits the Itoh condition that the difference in phase between two neighboring points is less than 180° [9]. Unwrapping the entire space of an MR image using this approach can be tedious and computationally expensive. A range of strategies has been developed for this purpose including spatial [2], temporal [10] and the 3-point Dixon technique [8].

We have previously exploited the phase measurements as part of our microwave breast tomography imaging algorithm. In the simplest terms, the algorithm compares the magnitudes and phases of the measurement data with that of the computed data at each iteration of the algorithm and updates the images until the measured and computed measurements converge with each other [11]. For this approach, the absolute phases of the measurements and computed values need to be calculated. For the measurements, this is performed by acquiring data over a range of frequencies and assuming that the phase difference between adjacent frequencies is less than 180°. Starting at a sufficiently low frequency, we are confident that the initial phases are unwrapped while also residing within the - 180° to +180° interval and can be used as a robust baseline. For the computed phases, it would be possible to exploit the Itoh condition and unwrap the phases for the entire physical domain. However, this would be computationally expensive. Alternatively, our algorithm starts as an initial image estimate of the homogeneous coupling medium. Because we are only interested in the phase changes during the reconstruction process, the phases can be normalized to zero for the first iteration. In addition, the step size of the iteration is set to a low level ensuring that the phase changes at all computation points never varies by more than 180° per iteration [11]. It is especially relevant to note that these phases only need to be unwrapped at the individual receiver locations. Therefore, this imaging technique utilizes unwrapping as functions of frequency and iteration.

We have recently developed a new microwave transmission-based sensor for assessing vertebrae bone strength during spinal fusion surgery. For this invention, the open-ended coaxial probes are inserted into the surgeon-made holes through the pedicles and into the vertebral body. Because, the coaxial apertures are so small, the propagation from one probe to the other can be considered to be in the far field for the full operating frequency range [12]. Acquiring data over a broad range would normally make it possible to unwrap the phases as a function of frequency, starting at the low end where the phases nominally should be within the −180° to +180° range. However, two synergistic phenomena confound this approach. First, at lower frequencies, the open-ended coaxial pairs appear similar to a series capacitor with respect to simple circuit theory. This implies that the amplitude has a substantial roll off at the lower end. Because of this, the technique is vulnerable to disorienting multi-path signals. These signals are most prominent at the lower frequencies and in this situation are sufficient to substantially distort the inherently low level desired signals at the lower frequencies. The constructive and destructive interference of the two signals can actually create nulls where the amplitude goes to zero, and hence the phase is indeterminate. If these phases are utilized in the unwrapping process, it could corrupt the final result.

A solution to this is to start the unwrapping process at an arbitrarily higher frequency where the series capacitor roll off is not so severe and where the multi-path signal contributions are minor. However, it is often the case that one cannot guarantee that the unwrapped phase at the higher starting frequency will be within the standard baseline Riemann sheet. A new technique is needed to robustly utilize only higher frequency data and reliably produce unwrapped phase measurements for the entire frequency range.

The primary application for this new probe is for examining vertebrae strength during spinal fusion surgery [13]. For these procedures, surgeons construct an instrumentation which consists of a pair of rods positioned in parallel with the spine with a set of brackets and corresponding screws inserted into the vertebrae for stability. If the apparatus fails, the fusion between bones can be compromised along with other complications including paralysis and even death [14]-[17]. Conservative estimates suggest that up to 15% of screws in osteoporotic bone loosen, and over a third of patients have at least one loose screw [18]. A high proportion of these pullouts are ultimately due to osteoporosis [19],[20]. The current standard of care is for surgeons to perform dual energy x-ray absorptiometry (DXA) scans prior to surgery to assess the bone strength [21]. However, these scans are fraught with challenges that undermine their success [20].

We have developed a new transmission-based coaxial probe for use during surgery that can quickly and accurately determine the dielectric properties of the bone [12]. The dielectric properties have been shown to correlate well with the bone mineralization and subsequently with bone strength [12]. The design of the probe is simple yet unique. It consists of a pair of open-ended coaxial lines that can be inserted into holes through the pedicles on opposing sides of the vertebra and into the vertebral body. The guide holes are previously formed by the surgeon in preparation for inserting the screws. The standard RG-405 cable is one of few practical structures that can fit within the hole. While the open-ended coaxial lines present a nearly open-circuit impedance, this characteristic is distinctively uniform over a large bandwidth for which we can operate. The low transmission efficiency is readily compensated for by modern, high-dynamic range vector network analyzers [22]. As demonstrated in Meaney et al [12], the phase and magnitude measurements can be readily transformed into the associated dielectric properties.

One technique that could be used to recover the necessary phase and amplitude data while simultaneously solving the phase wrapping challenge is by implementing a bi-static chirp transmission [23]. It would operate well in this configuration because of the broad band characteristics of the coaxial probes. Because it retains the phase information in a continuous form over the entire band, unwrapping is straightforward. The major drawback is its limited dynamic range which is restricted by the noise floor on the low end. The primary way to circumvent this would be to repeat the measurement many times and average the signals. For the dynamic range needed in this approach, the averaging time would be excessive and impractical.

One of the primary challenges with this device is that the measurements need to be performed in roughly 5 seconds or less to minimize the impact on the surgical workflow. Because of the low efficiency of the probes and the severe attenuation of the signals through the bone, the instrument IF bandwidth needs to be set to 1 Hz to provide sufficient dynamic range. This corresponds to roughly 1 second per measurement, substantially limiting the number of frequencies that can be sampled. To accommodate the slow acquisition process and still be able to unwrap the measured phases, we have developed this new approach which only requires three measurements to reliably unwrap the data. Certainly more frequency data points could be used, but when limited in time as we expect, it is vital to understand what the minimum is.

The Methods section describes the new approach, the Results section successfully demonstrates the new technique in comparison with the old, and the Discussion and Conclusions section describes the overall implications.

II. Methods

A. Summary of transmission probe and unwrapping challenges

For testing the vertebrae for spinal fusion surgery, we have developed a technique for recovering the bone dielectric properties from just the transmission data (both magnitude and phase). The approach exploits the fact that in the far field, the phases vary linearly with respect to separation distance between the probes, while the magnitude also varies linearly except for a 1/R² term, where R is the separation distance. We utilize simple open-ended coaxial probes which are inserted into holes that the surgeon makes through the pedicles from the posterior on both sides, leading into the vertebral body (Figure 1a). These are electrically small over a wide frequency range such that the furthest the far field begins is roughly 1.5 mm from the tip of the probe – for our purposes this is essentially the full range of possible separation distances. Signals are measured over a broad frequency range – typically between 0.5 and 8 GHz – while the probes are in the trabecular bone and also as a calibration reference when outside the bone and nearly touching. The slopes of the amplitude and phase as a function of distance are then used to compute the dielectric properties of the vertebral trabecular bone – for which there is a good correlation between bone mineral density. The slopes can be computed when the magnitude and phases are measured at a minimum of two separate, but known separation distances. The details of the algorithm are described in Meaney et al [12].

Fig. 1.

Photograph of the two probe handles. (a) The open-ended RG405 semi-rigid cables are inserted into the holes in the pedicles of the 3D printed vertebrae. The handles are equipped with both a machine vision approach (camera on the right handle and printed tag on the left handle) and surgical navigation-based method for computing the probe tip separation distance. (b) The probe tips are inserted into a dielectric mixture comprised of 20% glycerin and 80% water for validation testing.

A key to the approach is that the phases need to be unwrapped. Figures 2a and b show the magnitude and phases (wrapped) as a function of frequency for several different probe tip separation distances in a dielectric liquid mixture of 20% glycerin and 80% water (Figure 1b). Probe positions 1 – 8 are 8.0, 10.0, 14.0, 16.0, 18.0, 20.0, 22.0, and 24.0 mm apart, respectively. For readability purposes, only three of the phase curves are shown (for positions 4, 6, and 8, respectively). As can be seen, the magnitudes all have the characteristics of a roll off at the lower frequencies. By this, we mean that the magnitudes peak just below 2 GHz and then gradually decline down below - 100 dB at frequencies below 0.1 GHz (the multi-path signals do cause large spikes where they add constructively and destructively with the desired signal at these lower frequencies). These are primarily due to unwanted surface waves propagating along the coaxial lines and across the probe mounting fixtures where the air/material interfaces present a nearly lossless propagation path. These signals are generally low and are essentially only visible at the lower end because the desired signal is so low. It should also be noted that there is a slight ripple in the measured data. This is caused by a standing wave set up between the apertures of the two probes. In this case, the probes are nearly open circuits across the band, and the standing waves are excited across the distances between the discontinuities. The phases exhibit multiple wrappings and are bound between −180° and +180°. While the phases mainly exhibit predictable, repeating wrapping patterns, at the lower frequencies there are examples of sharp discontinuities that closely correlate with the magnitude nulls associated with destructive interference between the desired signals and the multi-path signals. In addition, there are also sharp spikes in the phases at frequencies above about 7 GHz for position 8. In these cases, the magnitude is essentially in the noise and the associated phases are undetermined which accounts for the randomness.

Fig. 2.

(a) magnitude and (b) phase measurements of a 20% glycerin: water mixture at eight different probe spacings as a function of frequency. The phases have not been unwrapped and are only shown for three of the spacings for visibility purposes.

B. New umwrappimg strategy

It is evident that the phase behavior at roughly 2 GHz and above is reasonably predictable. However, it is also clear from Figure 1b that the phases at 2 GHz have also been wrapped at least once for the representative cases. The new approach only requires the phases at three frequencies. For simplicity, we utilize, 2, 3, and 4 GHz. For those three data points, we add either 0 or −360 and generate seven permutations: (0, 0, 0), (0, −360, 0), (0, 0, −360), (0, −360, −360), (−360, 0, 0), (−360, −360, 0), and (−360, 0, −360) (the (−360, −360, −360) is essentially equivalent to the (0, 0, 0) for the purposes of unwrapping). These permutations are subsequently plotted as a function of frequency for the different separation distances between the probe tips. A trendline is computed along with the associated coefficient of determination (R²). The curve with a negative slope and the highest R² is the optimal permutation.

One challenge with this new approach is that it is possible for the new, optimal curve to be completely offset up or down by 360° from the ideal. However, we have empirically found that when the phase is plotted as a function of frequency for each new curve and extrapolated down to zero frequency, the resulting phase should be within the −180° to +180° range. If not, it is a simple matter of adding or subtracting multiples of 360° to each term.

III. Results

Figure 3 shows the same phase plots as a function of frequency but having been unwrapped as a function of frequency and utilizing the lowest frequency as the baseline reference – i.e. that its phase value within the −180° to +180° range is already unwrapped. Some of the curves (especially for Positions 1 and 2 which have the shortest probe separation distances) are well behaved and exhibit similar patterns. This is primarily due to the fact that these have the highest overall magnitudes such that the distorting impact of the multi-path signals is diminished. However, for the ones with larger separation distances, these demonstrate sharp discontinuities near 1-2 GHz, after which they also exhibit a similar sloping pattern except for offsets by multiples of 360°. Because these operate in the far field, it is expected that the phases should increase monotonically as a function of distance – which they don’t in the current representation. These offsets would substantially distort the desired phase slopes as a function of distance.

Fig. 3.

Unwrapped phase data for the same data set used in Figure 2b. The technique assumed that the phase never varied more than 180° from one frequency to the next. The technique assumed that the data at the lowest frequency was unwrapped while within the band from −180° to +180°.

The Rohde and Schwartz ZNBT8 VNA (Munich, Germany) was calibrated using their standard calibration process such that the phase reference planes were set at the interfaces of the open ends of two RG405 flexible cables connected to Ports 1 and 2 of the VNA. We then applied the port extension technique to extend the reference planes to the tips of the probes. For each probe, the phase was nominally 0° when the probes were exposed to air.

While the phase curves are nominally well behaved above 2 GHz, the curves for the larger separation distances show abrupt shifts of 360° below about 1 GHz. These correspond with the magnitude nulls and are representative of the situations where the multipath signals have corrupted the desired signal.

Figures 4a, b, and c show the different graphs for the seven different multiple 360° permutations for Positions 4, 6, and 8, respectively. For most of the curves, it is quite evident that most are not straight lines and only a few have a negative slope. Table 1 lists the R² values for the different curves, with the most optimal choices in bold font.

Fig. 4.

Plots of the phases for three frequencies (2, 3, and 4 GHz) for the different permutations of either subtracting 360° or not to each. The graphs are shown for positions 4, 6, and 8, respectively. The (0, 0, 0) case is highlighted using a thick black line to underscore the problems if no unwrapping is performed.

Table 1.

R² errors for the fitted curves for the different 360° permutations for three measurement positions – 4, 6, and 8, respectively.

Position/permutation	0, 0, 0	0, −360, 0	0, −360, −360	0, 0, −360	−360, 0, 0	−360, −360, 0	−360, 0, −360
4	0.9930	0.2549	0.8734	0.9202	0.2084	0.1722	0.3070
6	0.0010	0.0003	0.3868	0.9944	0.9973	0.4202	0.0005
8	0.1620	0.0815	0.9908	0.5725	0.3487	0.9885	0.0176

Figure 5 shows the optimal curves from the graphs in Figure 4 extrapolated to zero frequency for the baseline case and the associated configuration where they have been shifted by a multiple of 360°. The phases at zero frequency for the latter are well within the range of −180° to +180°. The extrapolated portion of the curves were based on least squares fits to the initial three points at 2, 3, and 4 GHz.

Fig. 5.

Optimal permutation data from figures 4a, b, and c extrapolated down to zero frequency. The corrected curves incorporating the shift of - 360° are also shown.

Figure 6 shows the same phase data as in Figure 3b but with the phases unwrapped utilizing the new technique. Once the phases at 2, 3, and 4 GHz are unwrapped, the values for the remaining lower, intermediate, and higher frequencies can all be unwrapped utilizing the Itoh condition. It should be noted that some of the phase values extend well above 0° for frequencies below 2 GHz. These values are unreliable and not used. For frequencies above 2 GHz, the phases increase monotonically as a function of distance as is expected for far field propagation. It should be noted that the phase values for the last three positions have been truncated because the magnitudes extend into the noise floor where the phases are undetermined.

Fig. 6.

Unwrapped phase data for the same data set used in Figure 2b utilizing the new unwrapping strategy.

IV. Discussion and Conclusions

We have demonstrated a simple, but robust approach for unwrapping the phases for transmission-based probes. Absolute phases are critical for exploiting the far field approximation in calculating the dielectric properties. These probes are inherently vulnerable to phase ambiguities because the low frequency signals can often be corrupted by multi-path signals since the desired signal magnitudes are often quite low themselves.

In terms of implementing these probes in a clinical situation measuring the vertebrae dielectric properties during surgery, it will be critical to acquire the data quickly. However, because the magnitudes of the signals can often extend as low as −140 dBm, the VNA IF bandwidth needs to be set extremely low to sufficiently suppress the noise floor. The process implies that the data acquisition will be considerably longer than normal. To accommodate this, it would be highly desired to acquire data for only a few frequencies to keep the probing time minimal. Because this new technique only requires data for three frequencies, the probing time can be kept short even when the signals are very low. The ability to keep the measurement time short will be critical to ensure its clinical usefulness.