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An algorithm for the optimum combination of data from arbitrary magnetic resonance phased array probes
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2002 Phys. Med. Biol. 47 N39 (http://iopscience.iop.org/0031-9155/47/2/402) View the table of contents for this issue, or go to the journal homepage for more
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INSTITUTE OF PHYSICS PUBLISHING
PHYSICS IN MEDICINE AND BIOLOGY
Phys. Med. Biol. 47 (2002) N39–N46
PII: S0031-9155(02)29453-1
NOTE
An algorithm for the optimum combination of data from arbitrary magnetic resonance phased array probes T Prock, D J Collins, A S K Dzik-Jurasz and M O Leach CRC Clinical MR Research Group, The Institute of Cancer Research, Royal Marsden NHS Trust, Downs Road, Sutton, Surrey, SM2 5PT, UK E-mail:
[email protected]
Received 4 October 2001 Published 4 January 2002 Online at stacks.iop.org/PMB/47/N39 Abstract When summing the spectra acquired with phased array coils, signals with low signal-to-noise ratio or wrongly corrected phase may degrade the overall signalto-noise ratio (SNR). Here we present a mathematical expression predicting the dependence of combined SNR on the signal-to-noise ratios and errors in phase correction of composite signals. Based on this equation, signals that do not lead to an overall increase in signal-to-noise ratio can be identified and excluded from the weighted sum of signals. This tool is particularly useful for the combination of large numbers of signals. Additionally, a simple and robust algorithm for calculating the complex weighting factors necessary for the signal-to-noise weighted combination of spectroscopic data is presented. Errors in the calculation and correction of relative phase differences between composite spectra are analysed. The errors have a negligible effect on the overall spectral SNR for typical clinical magnetic resonance spectroscopy (MRS). The signal combination routine developed here has been applied to the first in vivo MRS study of human rectal adenocarcinomas at 1.5 T (Dzik-Jurasz A S K, Murphy P S, George M, Prock T, Collins D J, Swift I and Leach M O 2001 Magn. Reson. Med. at press), showing improvements of combined spectral SNR of up to 34% over the maximum SNR from a single element.
1. Introduction To maximize the potential improvements in SNR available from MR phased array probes, the signals received from the array elements have to be combined optimally, taking account of the spatially dependent complex coil sensitivity and correlated background noise (Roemer et al 1990). Although these quantities may be acquired in an MR experiment, their measurement may be experimentally demanding or increase the total examination time. 0031-9155/02/020039+08$30.00
© 2002 IOP Publishing Ltd Printed in the UK
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Signal-to-noise weighted combination (Roemer et al 1990) uses complex weighting factors based on information available from spectroscopic or imaging data while neglecting noise correlation. This technique has found wide use in clinical MRS (i.e. Hayes and Roemer 1990, Wald et al 1995, 1997). We demonstrate that signals with low SNR may degrade the SNR of the signal-tonoise weightedly combined signal. From SNR weighted summation theory, an equation predicting the combined SNR is derived (Prock et al 2000). Based on this equation we demonstrate the ability to identify the signals improving the overall SNR and determine which signals to exclude from the weighted sum to achieve maximum combined SNR. This becomes increasingly important for phased arrays with a large number of elements and receive channels. Additionally, we illustrate the effects of errors in phase correction on combined SNR, using the equation we have derived to evaluate the value of a phase detection and correction algorithm developed in house. The developed algorithm was used to combine the clinical data sets acquired in a MRS study of human rectal adenocarcinomas at 1.5 T (Dzik-Jurasz et al 2001). An example of a typical data set from this study is shown to illustrate the SNR improvements achieved. 2. Methods 2.1. The influence of the SNR of composite signals and errors in phase correction on combined SNR The amplitude of a phase corrected spectral peak can be written as Si = Si0 e jαi
(1)
where Si0 is the actual amplitude of the signal and αi represents the error in phase correction, α i = φ i − θ i, φ i being the constant phase offset of signal i and θi the applied linear phase correction term. The variance σ 2 of the sum of correlated Gaussian noise distributions Ni can be calculated using σ2 =
N coils
σi σj ρi,j
(2)
i,j =1
where σ 2 is the variance of the composite noise distributions, σ i is the standard deviation of Ni and ρ i,j is the noise correlation coefficient between Ni and Nj. Using noise distributions measured with a phased array built in house as well as commercial phased array coils, the Gaussian nature of the received noise was verified. The noise correlation coefficients may be determined from regions of the recorded signal that only contain noise, similar to the determination of the noise standard deviations. This method was used for all calculations in this publication. Alternatively, an additional reference scan with 0◦ flip angle performed after the clinical protocol could provide data for the determination of noise correlation coefficients based on noise distributions with a larger number of points thus providing a stronger statistical basis. Weighting (1) and (2) with the equation for SNR weighted combination of phased array data, reproduced here in a notation similar to that used in (Wright and Wald 1997) Ncoils i=1 (wi Si ) (3) Scom = Ncoils 2 i=1 wi
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where Si are the composite phase corrected signals, Scom the combined signal and wi the applied weighting factor, provides the phasing and weighting dependent amplitude of the combined signal and the standard deviation of the combined noise distribution. Dividing the derived term for the signal amplitude by the expression for the noise standard deviation yields the SNRcom of the combined signal: Ncoils i=1 wi Si0 e jαi (4) SNRcom = . Ncoils i,j =1 wi wj σi σj ρi,j 2.2. Verification of equation (4) To verify the accuracy of equation (4), four spectra were acquired from the Siemens body phased array (Siemens Vision, Siemens AG, Erlangen, Germany) on a phantom with elliptical cross section (axes of 20 cm and 30 cm length), filled with 60 mM NaCl solution. A (2 cm)3 stimulated echo acquisition mode (STEAM) volume of interest (VOI) at 8 cm depth measured from the lower coil was used for data acquisition. The spectra from this measurement were combined using SNR weighted combination. The combined SNR is compared with the combined SNR predicted using equation (4). The SNRs of the four acquired unsuppressed water spectra were 415, 1106, 125 and 4467. The noise correlation coefficients and standard deviations of the background noise required in equation (4) were determined from a manually chosen spectral regions not containing any signal. Two cases were considered, one assuming good phase correction (achieved manually) and the second with phase correction errors of ±30◦ . In the following, equation (4) is used to evaluate the SNR penalty originating from potential inaccuracies of an algorithm for the detection of the complex weighting factors required in equation (3). 2.3. Determination of the complex weighting factors A range of spectral data points around the resonance peak to be used is extracted from the acquired spectra (figure 1(a)) and used to determine these complex weighting factors. As all signals originate from the same VOI they have the same lineshape. However, due to different coil sensitivity at the location of the VOI, they vary in amplitude. Assuming the imaginary part of the spectrum is zero at the maximum of the spectral peak in absorption mode, as is the case for ideal lineshapes, the maximum of the magnitude of the spectrum can be considered a good approximation of the peak height (figure 1(b)). Determining the magnitude weighting factor this way avoids the need for integrating the peak area and thus reduces susceptibility to baseline artefacts as well as eliminating a possible dependence of the magnitude weighting factor on the accuracy of the phase correction algorithm employed. Using the peak height to represent the magnitude of the complex weighting factor required for weighted signal summation will only provide optimum results if the noise levels in all spectra are the same, i.e. all coils have the same geometry and are loaded equivalently and all receive channels have the same noise figure. As this is generally not the case, spectra should be weighted by their signal-to-noise ratio rather than by their maximum peak height to provide the optimum combined SNR. To detect the phase of the spectra, the total area under the real component of the spectrum is initially integrated over the selected chemical shift range (figure 1(c)). This is done repeatedly while changing the phase of the spectrum by φ = 4◦ between integration steps. It is assumed that the phase shift maximizing the integrated area shifts the spectrum into absorption. In a
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Figure 1. Flow chart of the routine used for weighting factor detection. (a) From the four acquired spectra, the range of spectral data points to be used for the detection of the magnitude and phase of the weighting factor is extracted. (b) The maximum of the magnitude of the spectral shift range provides the magnitude of the weighting factor. (c) The peak area over the chemical shift range is computed repeatedly while the phase is shifted by φ (here 4◦ between integration steps. The peak integral with the maximum provides phase term necessary to shift the peak into absorption mode. (d) Subsequent integration of the difference between the magnitude and the real part of the spectrum depending on phase (φ = 2◦ ) provides a more sensitive means of detecting the phase offset term. The smallest integral corresponds to the constant phase offset term of the spectrum.
subsequent step, the root mean square deviation between the absolute part of the spectrum and its real component is calculated (figure 1(d)). This is again done iteratively, this time varying the phase of the spectrum in steps of φ = 2◦ over a smaller range of phase angles centred on the phase offset detected in the first computation step. The phase shift that provides the smallest error between the absolute and the real part of the spectrum is taken to be the phase offset of the spectrum. 2.4. Determining the accuracy of the phase detection routine To determine the error of this routine, a range of spectra with SNRs between 6 and 20 and varying spectral linewidth were simulated and their phase detected using the above routine. Each SNR/linewidth combination (6 different SNR values and 13 different linewidth) was reproduced 256 times with differing phase offsets, overlaid with different Gaussian noise distributions to provide a good basis for statistical evaluation of the phasing accuracy.
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The phasing accuracy for in vivo data was evaluated by comparing the computed phase offset of data from the clinical MRS study outlined below (Dzik-Jurasz et al 2001) to the datas phase offset determined manually using the spectroscopic post processing package (LUISE) supplied with the Siemens Vision operating platform. 2.5. Evaluating the influence of the determined phase correction errors on combined SNR Based on the determined phase correction accuracy, the phantom data of section 2.2 and equation (4), the penalty in combined SNR due to imperfect phasing is evaluated. 2.6. SNR weighted combination of clinical MRS phased array data The SNR in a clinical MRS study of human adenocarcinomas in vivo at 1.5 T (Dzik-Jurasz et al 2001) was improved by combining the acquired phased array spectra SNR weightedly using the above method for the detection of complex weighting factors. Twenty one patients with rectal adenocarcinomas were studied in a Siemens Vision MR Scanner (Siemens AG, Erlangen, Germany) using the Siemens body phased array coil. STEAM single voxel measurements (128 acquisitions, voxel size (1 cm)3 to (2 cm)3, shim better than 20 Hz for all cases/14 Hz mean) were performed with short (TE = 20 ms) and long (TE = 135 ms) echo times with a fixed repetition time TR of 1500 ms. 3. Results 3.1. Verifying the accuracy of equation (4) The combined SNR of the weighted sum of the unsuppressed water spectra acquired in section 2.2 was calculated using equation (4). Assuming good phase correction provided an expected SNR of 4244 for the combined spectrum. The combined SNR measured from the SNR weighted spectral sum was 4276, showing the calculated result to be accurate to 0.6%. For the second case where phasing errors of ±30◦ were assumed, the SNR measured in the calculated weighted sum was 4136 while equation (4) provided a value of 4126 (accurate to 0.2%). These examples not only demonstrate the accuracy of equation (4) but have also been selected for presentation in this context to show that exclusion of one or more signals from the weighted sum may improve the SNR of the weighted sum of signals. From the SNR values shown in section 2.2, it is clear that the combined SNR of the spectral sum of perfectly phase corrected signals (4276) is 4.2% lower than the SNR of the best composite signal (4467). This is due to the dominance of low SNR signals in the spectral sum. Determining the spectral peak height, noise standard deviations and the noise correlation coefficients from phased array MRS data and using these data in equation (4), it permits identification of which potential spectra may be excluded to improve the overall SNR. In the example provided, exclusion of the second spectrum will improve the overall SNR to 4416. 3.2. Accuracy of the phase detection routine determined using simulated spectra and in vivo data On evaluating the accuracy of the phasing algorithm an expected dependence on SNR is found. For SNR larger than eight, the phasing accuracy is consistently better than ±6◦ independent of spectral linewidth. For lower SNR the phasing accuracy shows a strong dependence on linewidth. While being accurate to ±15◦ for linewidth smaller than 1.5% of the total spectral bandwidth, the phase offset of wider spectral lines was detected with an error of up
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to ±40◦. Most spectra, however, will yield peaks with a signal-to-noise ratio larger than eight to permit phase determination with good accuracy using the above algorithm. To aid spectral quantification, an unsuppressed water reference scan is usually acquired. The use of this water peak will provide accurate phase offset information. Distortions of the spectral lineshape can lead to errors in the detection of absolute phase. This was evident from the evaluation of phasing accuracy of the in vivo spectra. Relative phase differences between spectra, however, were determined to within ±13◦ , averaging at 7◦ . Accordingly, phase correction with good accuracy may be achieved using the above algorithm. As distortions in lineshape are equivalent for all spectra the errors introduced are similar and do not affect the detection of phase differences. 3.3. SNR penalty associated with the typical error of the phase detection routine Based on equation (4), the typical phase correction error of the phase detection algorithm of ±6◦ would decrease the SNR of the weighted sum of the four spectra presented in section 2.2 by less than 1.0%. The accuracy of the phase correction algorithm was accordingly found to be sufficient for signals with an SNR greater than eight, found in prominent spectral peaks or easily achieved with an unsuppressed water scan that may be integral part of the MRS study protocol regardless of the requirements of phased array signal combination. 3.4. SNR improvements in a MRS study of human adenocarcinomas Combining the four acquired phased array spectra provided an SNR improvement of up to 34% over the composite spectrum with the best SNR. Figures 2(a)–(d) show an example of the four phase corrected pre-contrast (TE = 20 ms) spectra from one of the patients in this study, acquired with the Siemens body phased array. In figure 2(e), the signal-to-noise weighted sum of the spectra is illustrated, yielding an improvement in SNR of the trimethylamine proton resonance at 3.21 ppm (‘total choline’) of 17% over the signal-to-noise ratios of the same peak in the spectrum from the element with highest SNR. Summing the non-phase corrected spectra with magnitude weighting factors of one results in the spectrum shown in figure 2(f). In this case the SNR of the trimethylamine resonance is reduced by 41% when compared to the best SNR in the composite spectra, rendering spectral quantification less accurate. Please note that the resonance seen at ∼6 ppm in figures 2(c) and (d) are due to errors in the extraction of the residual water resonance. 4. Discussion Equation (4) permits accurate prediction of the combined signal-to-noise ratios of phased array spectra. It has been shown that exclusion of low SNR signals may improve the combined SNR. While the best way of combining the composite signals may be determined by simply performing all possible signal combinations for four channel phased arrays, this is impractical for arrays with larger numbers of elements (up to 16 for new scanner generations (Wright et al 1995, 1997)). Equation (4) provides an easy means of determining the combined SNR values in cases where a large number of signals needs to be considered. The accuracy of detection of phase offset using the above method is shown to be good for spectral peaks with SNR larger than eight. This is the case for prominent spectral peaks as well as the water resonance in unsuppressed water scans that may be part of the MRS protocol. The derived relationship between SNR and errors in phase correction is valid only for the signal-to-noise weighted combination of phased array MRS data (Roemer et al 1990).
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Figure 2. In the four separate phased array spectra (a)–(d), trimethylamine resonances (3.21 ppm) as well as lipid methylene (1.30 ppm) and lipid methyl resonances (0.90 ppm) and an unknown resonance (3.00 ppm) are detectable. The ‘resonance’ at 6 ppm, only present in (c) and (d) are artefacts that occurred when extracting the residual water peak from these particular data sets. In (e), the phase corrected and SNR weighted spectral sum is depicted indicating the gain in SNR over the maximum SNR found in the composite spectra (d). (f) The un-weighted sum of un-phased composite spectra. It is evident that the good SNR found in (e) is reduced by destructive interference between spectral peaks with large relative phase differences upon summation. All spectra are normalized to have equal noise standard deviations for better appreciation of the differences in the signal-to-noise ratios.
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It may, however, be easily adapted to other combination methods by substituting equation (3) by the equation used for signal weighting. The routine presented here reliably detects relative phase offsets between spectra. While other routines (Brown and Stoyanova 1996) and commercial programs (van den Boogaart et al 1994) for the detection of absolute phase offset are available, the above method also allows accurate, robust, fast and simple detection of the complex weighting factors necessary for SNR weighted combination of phased array MRS data, allowing fully automated, fast and efficient data processing. 5. Conclusions The influence of low SNR signals on the overall SNR was analysed and it was demonstrated that exclusion of low SNR signals may improve overall SNR. An equation to determine these signals is provided. Additionally, an algorithm for the detection of the complex weighting factors necessary for the signal-to-noise weighted combination of MRS data is given. The SNR penalty incurred from errors in phase correction were shown to be below 1%, demonstrating that the phase correction errors are negligible for most cases encountered in a clinical study. Finally, the feasibility of SNR weighted signal combination in conjunction with the above phase detection routine was verified by its use in a study of human rectal adenocarcinomas in vivo (Dzik-Jurasz et al 2001), yielding significant improvement in SNR. Acknowledgments We are thankful to Elie Bassouls for helpful discussions and comments and gratefully acknowledge the support of the Cancer Research Campaign [CRC], grant no SP1780/0103. References Brown T R and Stoyanova R 1996 J. Magn. Reson. B 112 32–43 Dzik-Jurasz A S K, Murphy P S, George M, Prock T, Collins D J, Swift I and Leach M O 2001 Magn. Reson. Med. at press Hayes C E and Roemer P B 1990 Magn. Reson. Med. 16 181–91 Prock T, Bassouls E, Collins D J and Leach M O 2000 Proc. ISMRM, 8th Annual Meeting p 562 Roemer P B, Edelstein W A, Hayes C E, Souza S P and Mueller O M 1990 Magn. Reson. Med. 16 192–225 van den Boogaart A, Ormondt van D, Pijnapel W W F, Beer de R and Ala-Korpela M 1994 Mathematics in Signal Processing (Oxford: Clarendon) Wald L L, Moyher S E, Day M R, Nelson S J and Vigneron D B 1995 Magn. Reson. Med. 34 440–5 Wright S M, Porter J R, Reykowski A, Finkenstaedt M and Naul L G 1995 Proc. IEEE-EMBC and CMBEC, 17th Annual Meeting pp 473–4 Wright S M and Wald L L 1997 NMR Biomed. 10 394–410
