Quantication of Resonances in Magnetic Resonance Spectra. via Principal Component Analysis and Hankel Total Least. Squares.

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1 Quantication of Resonances in Magnetic Resonance Spectra via Principal Component Analysis and Hankel Total Least Squares Yu Wang, Sabine Van Huel and Nicola Mastronardi October 9, 999 Abstract A careful comparison between the Principal Component Analysis (PCA) and the Hankel Total Least Squares (HTLS) based methods for estimating the resonances in sets of MR spectra is presented. After a description of the methods and their relationships, we compare their performance on simulated data sets of Magnetic Resonance Spectroscopy (MRS) signals and discuss their advantages and limitations. Key words: Quantication, Magnetic Resonance Spectroscopy, Principal Component Analysis, Total Least Squares, Singular Value Decomposition Introduction Quantication of Magnetic Resonance Spectroscopy (MRS) data, yielding information about metabolite concentrations, is hampered by low signal-to-noise ratios. S. Van Huel et al. [] applied the total least squares method to MRS signal quantication, assuming that the data are modeled as a sum of exponentially damped sinusoids. Their algorithm, called Hankel Total Least Squares (HTLS), was shown to improve the accuracy of all MRS parameter estimates as compared to HSVD, a non-iterative black box time domain method. Recently, HTLS was generalized [] to the quantication of sets of MR spectra, resulting into the HTLSstack and HTLSsum algorithms. Department of Electrical Engineering / Division SISTA/COSIC / Katholieke Universiteit Leuven / K. Mercierlaan 94 / Leuven-Heverlee / Belgium / Tel: FAX: / {yu.wang, sabine.vanhuel,nicola.mastronardi}@esat.kuleuven.ac.be As a widely used statistical technique, PCA was introduced in MRS data quantication by R. Stoyanova et al. []. Only the real part of the MRS data is considered and transformed to the frequency domain. An interval surrounding the peak of interest is selected and the points within this interval are stored row by row in a matrix, to which PCA is applied. In order to improve the accuracy and eliminate the inuence of the phase dierence, PCA is performed on complex data using a complex SVD [4]. Mathematically, PCA decomposes the data set in order to extract the basic features, called principal components (PCs). It has been shown that under some conditions PCA can successfully extract quantitative metabolite information from data sets without any prior knowledge about the line shape [][4][7][8]. Methods. MRS Data model function It is assumed that, basically, MRS data can be modeled as a sum of exponentially damped complex-valued sinusoids : y n = y n + e n = P K k= a ke jk(?dk+jfk)tn + e n() n = ; ; ; N? where y n is the nth measured data point of a MRS signal, y n is the exact value by applying the model function, j = p?, a k is the amplitude, k the Also called Lorentzian. Note that other model functions, such as Gauss or Voigt line shapes, are sometimes used to model MRS data.

2 phase, d k the damping factor and f k the frequency of the kth sinusoid (k = ; ; ; K), K is the number of the sinusoids; t n = nt +t with t the sampling interval, t the time between the eective time origin and the rst data point to be included in the analysis and e n is complex white Gaussian noise. The amplitude a k is directly related to the concentration of a certain metabolite, and should be estimated as precisely as possible.. Hankel Total Least Squares Arranging the data points y n, n = ; ; ; N? of one MRS signal y in a Hankel matrix H of dimensions LM; L K; M K; N = L+M?, yields: H = y y y M? y y y M ().... y L? y L? y N? Computing the SVD of the Hankel matrix H; we obtain H LM = U LL LMV H MM () where = diag( ; ; ; q ), q, q = min(m; L), the superscript H denotes the hermitian conjugate. According to [], H should be chosen as square as possible in order to get the best parameter accuracy. Truncate H to a matrix H K of rank K ( K is the estimated number of signal poles ): H K = U K K V H K (4) U K and V K are the rst K columns of U and V, respectively. K is the KK upper-left sub-matrix of. Now compute the Total Least Squares (TLS) [5] solution ^E of the following (incompatible) set: V (t) K EH V (b) K (5) V (b) (t) K and V K are derived from V K by omitting its rst and last row respectively. The K eigenvalues of E yield the signal pole estimates: Filling in the estimated frequencies ^f k and damping factors ^d k into the model equation, yields the set: y n P K k= c ke (? ^d k+j ^f k)t n n = ; ; ; N? From its least squares solution ^c k = ^a k e j ^ k, we nd ^a k and ^ k, the estimated amplitudes and phases. A more detailed description of the HTLS algorithm and underlying principles is given in [5].. HTLSsum and HTLSstack Two extensions of the HTLS algorithms quantifying simultaneously sets of MRS signals are presented now. A thorough discussion of these algorithms can be found in []. Suppose we have P MRS signals y ; y ; ; yp of N data points each. Let y = y + y + + yp Now we can apply the HTLS algorithm to the sum y of these signals to rstly obtain the signal poles and afterwards calculate the corresponding amplitudes and phases for each signal separately. This algorithm is called HTLSsum. The HTLSstack algorithm is as follows: Arrange N data points of each MRS signal y p into a Hankel matrix as in Eq. : H p = y p y p y (M?)p y p y p y Mp.... (7) y (L?)p y (L?)p y (N?)p where p = ; ; ; P. Stack vertically these P Hankel matrices to obtain: H stack = H H. H P The equations,4,5 can be solved using H stack in order to obtain frequencies and damping factors for all MRS signals together. Note that also here signal poles need to be assigned to corresponding signals. The same procedure is used as in HTLSsum []. ^z k = e (? ^d k+j ^f k)t k = ; ; K (6) How to assign the signal poles to corresponding signals is not trival. A procedure is outlined in [].

3 .4 Principal Component Analysis Principal Component Analysis (PCA) is usually performed by means of an eigenvalue decomposition. Here we use the complex SVD. Arrange the P signals y ; y ; ; yp as rows in a matrix D: y D =. T (8) yp T Compute the complex SVD of the matrix D: y T D PN = U PP PN V H NN where = diag( ; ; ; q ); q = min(p; N ). The columns of V contain the basic shapes of the signals, called principal components. The elements in S PN = U PP PN are called the scores of each principal component (PC). In fact, Eq. 8 can be rewritten as follows: y T y D =. T = S PN VNN H (9) yp T From Eq. 9, we can clearly see that PCA orthogonally transforms the original coordinate system into a new one, which is spanned by the PCs - the columns of V. Now, suppose our set of P MRS signals is composed of a single resonance (peak), which means that the matrix D has rank one. Therefore, only 6=, = = = q =. Now the score matrix S contains only one column S = s s s P T. The rst PC, i.e. the rst column v of V, represents the basic line shape, which, in our case, contains only one resonance. Each signal y p in the set can be expressed in the following way: y p = s p v p = ; ; P () This PCA can be performed on the time domain data as well as on the corresponding frequency domain data. Both methods are equivalent except the normalization. Using time domain data, the rst PC v needs to be normalized in such a way that it represents a unit amplitude signal. To normalize v, we need to divide v by its rst element v so that Eq. becomes (in matrix notation) D = [y y P ] T = ~ S ~v H = (Sv )(v H =v ) The modied scores ~ S = Sv then represent the amplitudes of the original signals, from which the corresponding metabolite concentration can be obtained. Using frequency domain data, obtained by applying the DFT to the rows of MRS signals, we need to normalize v so that it has unit area, i.e., P N i= v i = In this case, Eq. becomes (in matrix notation) D = ~ S ~v H = (SN i= v i)(v H =N i= v i) The modied scores ~ S = S N i=v i then represent the areas under the spectra, from which the corresponding metabolite concentration can be obtained. Using the properties of the DFT, it is easy to prove that PCA performed in the frequency domain is equivalent to its counterpart in the time domain, which agrees with our experimental results. Quantication of MRS signals containing more than one resonance or basic line shape using PCA is much more complex. Since PCA does not enable automated quantication of each resonance or line shape separately (each PC always contains a mixture of both resonances), we need to lter out each basic line shape before applying PCA. The best results are obtained using a maximum phase time domain FIR lter [9]..5 Theoretical Relationships between HTLSstack and PCA Suppose the signals are noise free and modeled by Eq. with t =. Comparing the data matrices H stack and D, as used by HTLSstack and PCA respectively, we observe the similarities and dierences between both methods when time domain data are used (PCA also applies to frequency domain data). While PCA arranges each signal as one row in its matrix, HTLSstack takes L truncated (to M samples) and shifted (the time shift between consecutive rows is t) versions y ip ; ; y (i+m?)p;

4 i = ; ; L? of each signal y p, p = ; ; P, which are arranged as rows into a Hankel matrix H p. Both methods use the complex SVD, denoted by U V H ; as kernel computation. The columns of V, corresponding to nonzero singular values, are called PCs and represent the basic line shapes. The main advantage of using HTLSstack lies in the fact that this method is able to quantify signals with multiple resonances automatically while PCA can only quantify single resonance spectra. As soon as two dierent line shapes (resonances) are present in the data set (this is the case when the rank of D >, or equivalently K > in Eq. ), each PC consists of a mixture of both line shapes and there is no automatic way to separate both contributions. This is needed in order to quantify each resonance, corresponding to the concentration of one metabolite, separately. HTLSstack however exploits the assumption that each signal is modeled by Eq., i.e., each y p is a linear combination of Lorentzians satisfying: y np = K k=c kp z n k () where c kp = a kp e jkp and z k = e (?dk+jfk)t ; n = ; ; ; N? : Here K is the number of dierent signal poles in the entire data set, z k is the kth signal pole while a kp ; kp are the amplitude and phase corresponding to the kth signal pole in the pth signal y p. This assumption enables to decompose the corresponding Hankel matrix H p of y p in Eq.7 as follows: H p = z z zk... z L? z L? zk L? c p c p... ckp z z zk... z M? z M? zk M? = S K C Kp T T K T where T denotes matrix transposition, S K and T K are Vandermonde matrices composed of the K signal poles, and C Kp is a diagonal matrix composed of K complex-valued amplitudes of the pth signal y p. Note that S K and T K have the shift-invariant structure, i.e., S (b) K = S(t) K Z K; T (b) K = T (t) K Z K, where Z K is a diagonal matrix composed of the K signal poles, z k ; k = ; ; K. On the other hand, the rank-k property of H stack enables a truncated SVD Eq.4 of H stack. Equating the two decompositions, we have the shift-invariant property of V K, given by Eq.5, where E is a similarity transform of Z K. This enables us to compute the individual signal poles by solving Eqs.5-6. In this way, each signal y p, written as a linear combination of PCs (the rst few columns of V ), can be rewritten as a linear combination of K separate resonances. The corresponding coecient c kp contains the amplitude a kp which enables to quantify automatically the kth resonance and corresponding metabolite concentration in the pth signal of the set. Assume now that the signals have an arbitrary non-lorentzian line shape. In this case, the same PCA method can still be used to quantify the area under this line shape (provided the set is characterized by only one line shape). However, HTLSstack can be applied too since each arbitrary line shape can always be approximated by a number of Lorentzians. Assume J Lorentzian are needed, with corresponding signal poles z ; z ; ; z J and amplitudes c p ; ; c Jp, then the area under the corresponding line shape a p of the pth signal is obtained by linearly combining the amplitudes as follows [6]: a p = J i=a ip cos ip () where c ip = a ip e jip. This computational procedure is illustrated for a Gaussian line shape in the next section. Results. MRS data set of Lorentzian line shape We generate a complex signal of 8 data points composed by one exponentially damped sinusoid (frequency = 6 Hz, damping factor = Hz, 4

5 Time Domain: Real part of signal.8 Magnitude spectrum of the first PC Magnitude spectrum of the second PC 5 Frequency domain: Real part of FFT Magnitude spectrum of the third PC Figure : Noise free MRS signal (one peak) amplitude = arbitrary units (a.u.), phase = o ). This signal (Figure ) is multiplied by to 9 to produce a data set which contains 9 signals having the same Lorentzian line shape. Random noise with variance 5 and mean is added to the real and imaginary part of the signal (Figure ) Time Domain: Real part of signal+noise Frequency domain: Real part of FFT Figure : Noisy MRS signal We test the amplitude estimation of PCA, HTLS, HTLSsum and HTLSstack. PCA processes whole data set at once, while HTLS quanties every signal separately. HTLSstack and HTLSsum use whole Magnitude spectrum of the fourth PC Figure : Magnitude spectrum of the rst four principal components data set to nd the damping factor and frequency of the signal, then treat every signal individually to nd the corresponding amplitude and phase. To get an average estimation error, the test is repeated one hundred times. Each time random noise with the same mean and variance is added to the noise free signals. Figure shows the magnitude spectrum of the rst four principal components of one data set. Only the rst PC represents the basic shape of the signal. The remaining PCs are due to noise. Figure 4 shows the average relative error of the amplitude as estimated by HTLS, HTLSstack, HTLSsum and PCA versus the signal-to-noise ratio (SNR) which varies from. to.. It clearly shows that at every SNR, HTLSstack and HTLSsum perform the best. Only at low SNR, PCA is better than HTLS. This is due to the fact that HTLSstack, HTLSsum and PCA operate on the whole data set, taking advantage of the correlation between the signals to lter out noise. Since HTLSstack and HTLSsum also exploit prior knowledge of the Lorentzian signal shape, they provide more accurate information than PCA. HTLSstack yields, for most SNRs, slightly better results than HTLSsum. The dierence in relative 5

6 Relative Error PCA HTLSstack HTLSsum HTLS shape can be approximated by a linear combination of several Lorentzians. If no noise is present, the approximation error can be negligibly small, provided J in Eq. is large enough. We generate data sets of Gaussian line shape (gure 5) in the same way as described in the previous section, except that the model function of the noise free set is now Gaussian. This model is mathematically described by Eq. in which the damping factor d k is replaced by g k t n (g k = Hz in our data set) SNR Figure 4: Relative error of the amplitude, averaged over all signals of the noisy data sets, as computed by 4 dierent methods versus the SNR. The SNR is given by SNR = log (A = ), where A is the amplitude and is the noise variance. The curves of HTLSstack and HTLSsum nearly coincide. Relative Error PCA HTLSstack HTLSsum HTLS error between both methods is below?. The reason might be that HTLSstack better exploits the correlation between the signals by arranging every individual signal in a Hankel matrix, which is benecial for noise suppression.. MRS data set of Gaussian line shape Time Domain: Real part of signal+noise Frequency domain:real part of FFT Figure 5: MRS signal of Gaussian line shape As mentioned in Section.5, a Gaussian line SNR Figure 6: Average relative error of the amplitude versus the SNR. A Gaussian line shape is used. In this experiment, we approximate the Gaussian line shape by up to 5 Lorentzians lying in its frequency range, i.e., K is set to 5 in Eq. implying that J 5 in Eq.. At high SNR the 5 signal poles computed by the HTLS based methods fall within this range, i.e., K = J = 5. However, with decreasing SNR noise peaks are tted too, resulting in a decrease of J (J < K): J is typically equal to at low SNR. Figure 6 shows the advantage of PCA at very low SNR. Without any prior knowledge about the model function, HTLS based algorithms are not as good as PCA at very low SNR, although HTLSstack and HTLSsum yield better results than PCA at high SNR. HTLS shows the worst performance except for high SNR.. MRS data set of two Lorentzians In this section, we demonstrate that PCA is unsuitable for direct automated analysis of signals composed of more than two dierent line shapes. The 6

7 data set is produced in the following way: rst, we generate a signal composed of two Lorentzian peaks. To evaluate better the estimation error of the method, we do not add any noise in this experiment. We generate 5 dierent data sets, in which the two peaks are moved towards each other (see Table ). Each data set contains signals. All the signals in one data set contain the same parameters except for the amplitude. For each signal, the damping factor is Hz and the phase o. The amplitude of the rst peak is?p a.u. while the amplitude of the second peak equals p a.u. for the pth signal in the data set (p = ; ; ; ) Magnitude spectrum of the first PC Magnitude spectrum of the second PC Magnitude spectrum of the third PC. experiment peak peak 5 Hz 95 Hz 5 Hz 85 Hz 5 Hz Hz 4 5 Hz 65 Hz 5 45 Hz 55 Hz Table : Frequency of the two Lorentzian peaks Figure 7 plots the magnitude spectrum of the rst four PCs of the third experiment. Since we can not separate the two peaks directly by applying PCA, as mentioned in Section.4, we use a maximum phase time domain FIR lter [9] to lter out each peak, which are then separately quantied by PCA. The relative error is plotted in Figure 8. The amplitude estimation with HTLS, HTLSstack and HTLSsum is very accurate since no noise is added. For both peaks, the amplitudes are computed exactly, up to machine precision. Finally, we want to mention that similar conclusion hold for unphased data sets, i.e. signals with unequal phases (e.g. randomly chosen). In other words, the performance of the PCA and HTLS based methods using complex data is independent from the phase of the MRS signals contained in the set. 4 Conclusions This paper compares the PCA and HTLS based methods for estimating the resonances in sets con Magnitude spectrum of the fourth PC Figure 7: Magnitude spectrum of the rst four Principal Components the relative error on peak one the relative error on peak two Figure 8: Relative error of each peak's amplitude estimation (averaged over the signals of each data set) versus the distance between the two peaks (in Hz) taining multiple MRS signals. For data sets consisting of a single line shape, PCA demonstrates good performance both in terms of accuracy and 7

8 computational eciency. However, HTLSstack and HTLSsum outperform PCA, in particular when the line shape is Lorentzian, since they not only exploit the correlation between the signals to lter out noise but also use the prior knowledge about the line shape. Only at low SNR, PCA can beat the HTLS based methods if the line shape of the MRS data is non-lorentzian. PCA is unsuitable for direct automated quantication of sets of MRS signals composed of more than one line shape. In this case, the scores of the PCs do not reect the areas under the separate peaks. Only after appropriate ltering, we can apply PCA to such data sets. HTLSstack, HTLSsum and HTLS are computationally less ecient, but can quantify MRS signals with more than two peaks. This makes them suitable for automated quantication of sets of real world MRS data (e.g. in MRS imaging). Acknowledgments S. Van Huel is a Senior Research Associate with the F.W.O. (Fund for Scientic Research Flanders). This paper presents research results of the EC Training and Mobility for Researchers project entitled Advanced Signal Processing for Medical Magnetic Resonance Imaging and Spectroscopy (contract ERBFMRXCT976), the Belgian Programme on Interuniversity Poles of Attraction (IUAP Phase IV/ &4), initiated by the Belgian State, Prime Minister's Oce for Science, Technology and Culture, and of a Concerted Research Action (GOA) project of the Flemish Community, entitled Model-based Information Processing Systems. References [] S. Van Huel, H. Chen, C. Decanniere, and P. Van Hecke, Algorithm for Time-Domain NMR Data Fitting Based on Total Least Squares, Journal of Magnetic Resonance, Series A, 8-7 (994). [] L. Vanhamme and S. Van Huel, Multichannel Quantication of Biomedical Magnetic Resonance Spectroscopy Signals, Advanced Signal Processing Algorithms, Architectures, and Implementations VIII, Editor F. T. Luk, -4, San Diego, California, Proceedings of SPIE, Volume 46, 7-48 (998). [] R. Stoyanova, A. C. Kuesel, and T. R. Brown, Application of Principal-Component Analysis for NMR Spectral Quantitation, Journal of Magnetic Resonance, Series A 5, (995). [4] M. A. Elliott, G. A. Walter, A. Swift, K. Vandenborne, J. D. Schotland, and J. S. Leigh, Spectral Quantitation by Principal Component Analysis Using Complex Singular Value Decomposition, Magnetic Resonance in Medicine 4, (999). [5] S. Van Huel and J. Vandewalle, The Total Least Squares Problem: Computational Aspects and Analysis, Frontiers in Application Mathematics Series, Vol 9. SIAM, Philadelphia, PA (99). [6] C. Decanniere, P. Van Hecke, F. Vanstapel, H. Chen, S. Van Huel, C. Van Der Voort, B. Van Tongeren, and D. Van Ormondt, Evaluation of Signal Processing Methods for the Quantication of Strongly Overlapping Peaks in P NMR Spectra, Journal of Magnetic Resonance, Series B 5, -7 (994). [7] T. R. Brown and R. Stoyanova, NMR Spectral Quantitation by Principal-Component Analysis II. Determination of Frequency and Phase Shifts, Journal of Magnetic Resonance, Series B, -4 (996). [8] A. C. Kuesel, R. Stoyanova, N. R. Aiken, C. Li, B. S. Szwergold, C. Shaller and T. Brown, Quantitation of Resonances in Biological P NMR Spectra via Principal Component Analysis: Potential and Limitations, NMR in Biomedicine, Vol. 9, 9-4 (996). [9] L. Vanhamme, T. Sundin, P. Van Hecke, S. Van Huel, R. Pintelon, Frequency selective quantication of biomedical magnetic resonance spectroscopy data. ESAT-SISTA Report TR 98-, ESAT Laboratory, K.U. Leuven, Belgium, December 998 (submitted to the Journal of Magnetic Resonance). 8