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Keywords:

  • independent component analysis (ICA);
  • principal component analysis (PCA);
  • automatic decomposition;
  • MRS;
  • model

Abstract

  1. Top of page
  2. Abstract
  3. MATERIALS AND METHODS
  4. RESULTS
  5. DISCUSSION AND CONCLUSIONS
  6. Acknowledgements
  7. REFERENCES

Fully automated methods for analyzing MR spectra would be of great benefit for clinical diagnosis, in particular for the extraction of relevant information from large databases for subsequent pattern recognition analysis. Independent component analysis (ICA) provides a means of decomposing signals into their constituent components. This work investigates the use of ICA for automatically extracting features from in vivo MR spectra. After its limits are assessed on artificial data, the method is applied to a set of brain tumor spectra. ICA automatically, and in an unsupervised fashion, decomposes the signals into interpretable components. Moreover, the spectral decomposition achieved by the ICA leads to the separation of some tissue types, which confirms the biochemical relevance of the components. Magn Reson Med 50:697–703, 2003. © 2003 Wiley-Liss, Inc.

There has been much interest in recent years in the use of statistical methods for analyzing databases of MR spectra, in particular for the diagnosis and grading of tumors, using pattern recognition techniques (1, 2). Statistical methods and algorithms, such as principal component analysis (PCA) and neural networks, have been used but they often operate as “black boxes” that give no easily interpretable outputs other than the assignment to one class or another.

An MR spectrum is the addition of a number of peaks, all of which represent biochemical information. The biochemical interpretation of a spectrum requires its decomposition into these constituent components. This operation can be difficult due to the presence of noise or overlapping peaks, and automated procedures are needed to do it accurately. There exist two types of methods for decomposing signals: model-based and statistics-based.

Modeling techniques can be used to decompose a spectrum into its basis signals. Variable projection (VARPRO) (3), for example, uses simplified lineshape functions (e.g., Lorentzian or Gaussian) that are fitted to manually selected individual peaks. LCModel (4) fits curves to the signal by using a set of ideal spectra from phantoms and mathematical transformations to take physical phenomena (e.g., signal distortion and line broadening) into account. In practice, VARPRO requires a high degree of user input, and both VARPRO and LCModel require assumptions about which biochemicals have signals that contribute to the spectrum. Furthermore, when fitting overlapping peaks by either method in the presence of noise, the accuracy of determining individual metabolite levels is reduced. If the set of selected peaks (e.g., phantom data for LCModel) is incomplete, there is a risk that the system will distort the signal or miscalculate the individual amplitude to force a fit to the spectrum (cf.,5). Thus, there is a need for a method that produces an automatic and objective decomposition, particularly when there is a large collection of spectra to analyze.

When multiple datasets are available, statistical techniques can use the information contained in the whole collection to extract components. A popular method is PCA (6), which decomposes the data according to the largest trends in the whole dataset. This allows the data to be reexpressed in terms of a small number of components. For example, PCA has been used for quantitation by Stoyanova et al. (7). The method was used on selected regions of the spectra. It performed well when the window contained only one peak, but had difficulties when two overlapping peaks were present. The principal components represented various mixtures instead of single metabolites, and a transformation matrix was necessary to disentangle them. Finding this matrix was not trivial and required prior knowledge. Howells (8) used a similar approach and attempted to automatically build the unraveling matrix by using target testing (9) with phantom spectra as the targets. In another approach, Ochs et al. (10) used Bayesian statistics to decompose the spectra. Because that approach was based on a probabilistic model, it was possible to compute the likelihood of the decomposition and thus automatically discard unrealistic or irrelevant basis signals. Unfortunately, the method was sensitive to low SNR and initial conditions.

In previous studies, the fact that an MR spectrum is a linear combination of statistically independent signals (i.e., metabolites) was ignored. PCA is a very general statistical technique that can be applied to any type of data, and the Bayesian approach only assumed positivity of the components. We propose using a multivariate method whose model is closer to that of an MR spectrum. To our knowledge, the first attempt to use independent component analysis (ICA) for NMR spectra was in 1998, when Nuzillard et al. (11) analyzed 1D and 2D in vitro data to separate mixtures of three chemical components. The results were not completely conclusive; although the data were simple three components, with no noise), the technique seemed to be difficult to automate, and sophisticated preprocessing was needed in order to obtained the required results. Lee et al. (12) used ICA to extract features for classifying brain tumor 1H spectra. The algorithm they used was based on well-founded principles, and although the main objective was classification, the extracted components bore some resemblance to those found in average tissue types. In another study (13), ICA was used to decompose the chemical shift imaging (CSI) data of one patient into three biochemically plausible components.

The present study investigates a method that combines PCA and ICA to automatically decompose a large (N = 333) collection of spectra into biochemically significant components. The next section provides a description of PCA and ICA. We first test the method on artificial data to assess its potentialities and pitfalls, and then apply it to real data. The experiments are detailed in the Materials and Methods section, and results are presented in the subsequent section. A discussion concludes the study.

MATERIALS AND METHODS

  1. Top of page
  2. Abstract
  3. MATERIALS AND METHODS
  4. RESULTS
  5. DISCUSSION AND CONCLUSIONS
  6. Acknowledgements
  7. REFERENCES

Theory of PCA and ICA

In this study, we used PCA to denoise and compress the data, and ICA to extract significant independent features. These are two statistical methods that decompose the data into a linear combination of basis signals si, as seen in Eq. [1]. Such a decomposition is not unique, and the difference between the two methods lies in the definition of those constituting signals. In this section, both techniques are briefly introduced.

  • equation image(1)

PCA (14) is probably the most common statistical method to analyze and compress or simplify high-dimensional data. It consists of finding a linear combination of the attributes so that variance is maximized. Examples are then expressed as a weighted sum of a small number of vectors, or “principal components” (PCs) that capture the largest trends of the data. The PCs are easily extracted by classical linear algebra techniques; they are the eigenvectors of the data covariance matrix.

PCs are ranked by the amount of variance they explain, and the number of PCs is determined by the cumulative variance explained by the first components: as more components are used, the reconstructed data get closer to the original data. A scree plot (plot showing the relative and cumulative contribution of each component, see Fig. 1) can be used to help the user make a decision. By construction, PCs are uncorrelated.

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Figure 1. Scree plot corresponding to the PCA of the real data. The bars and the line respectively represent the percent variability explained by each component and the cumulative variability explained by the first components.

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ICA (15) is a multivariate technique that decomposes data into linear combinations of statistically independent signals. After the extraction of the independent components (ICs) si, each data sample x is written as in Eq. [1]. Equation [1] could represent an MR spectrum that consists of a linear combination of metabolite spectra that are independent from each other. Provided that the dataset contains enough variation of metabolite amplitudes (e.g., a large dataset of different tissue types), ICA should be able to decompose the sample into the original metabolites on the basis of their statistical properties alone.

Statistical independence is a much stronger constraint than noncorrelation. Whereas a correlation coefficient is a measure of the linear relationship between two random variables, statistical dependence infers any kind of relationship between two random variables — it can be considered as a nonlinear correlation. If two random variables are independent, they are uncorrelated but the converse is false: if they are uncorrelated, there is no linear relationship between them but they could be linked by a more complicated connection. For example, if x is uniformly distributed on [−1; +1] and y = x2, then the correlation between x and y is zero (i.e., they are not correlated), although y is a function of x (so they are dependent on each other). However, the formal definition of independence is not practical because it requires testing all the possible relationships, and an approximation is needed.

It can be shown that mathematical independence and non-Gaussianity are strongly related, and statistical measures (e.g., kurtosis) can be used to estimate independence. A better means of testing for non-Gaussianity is negentropy, a concept derived from information theory. However, calculating negentropy involves acquiring knowledge of the variables' distribution, which is generally not possible. We thus have to rely on an estimator. Hyvärinen (16) developed an algorithm (FastICA) that takes advantage of such approximation. This is the package we used for the experiments in this study (matlab™ toolbox FastICA (17) on a Solaris station).

ICA of Artificial Data

We began work with ICA on a dataset of artificial spectra. This was necessary to assess the behavior of the ICA in the presence of real spectral characteristics, such as noise or line broadening, with all of those factors under control. Once possible problems were identified, we applied the method to real data.

Each spectrum was modeled as a linear combination of actual metabolite spectra acquired from pure solutions (50 mM concentration acquired using point-resolved spectroscopy (PRESS) with TE 30-ms to obtain very high SNR) and was designed with three sets of parameters: the contributions of each metabolite, the line broadening of each metabolite, and the desired SNR of the spectrum, defined as the ratio of the highest peak in the spectrum and the standard deviation (SD) of the noise. The contribution of each metabolite was random (uniformly distributed between 0 and 1) and no attempt was made to model actual diseases. The line-broadening factor of each metabolite was also randomly distributed, with a value of 4 ± 0.5 Hz that might be expected in vivo. Gaussian noise was then added to the combination, and its SD was set in order to obtain the predetermined SNR. Thus it was possible to generate a whole dataset of spectra that would share the same SNR. This was necessary to rule out the possibility that ICA elicits peaks or patterns by using only the less noisy spectra of the dataset. Since all of the metabolite parameters were exactly known, we were able to study the capacity of ICA to retrieve the original constituents of the data in situations comparable to that found from in vivo clinical brain 1H MRS.

We conducted three types of experiments. The first objective was to investigate how practical the method was, and, in particular, if some preprocessing was needed. After that, we wanted to know if the method was robust against low SNR. And finally we wanted to assess the sensitivity of the algorithm to detect peaks that only rarely occur.

Optimising Peak Extraction Accuracy

In the first experiment, we generated a dataset of 100 spectra, with an SNR of 10, from the four metabolites shown in Fig. 2a. To extract the ICs, we used the method of Hyvärinen (18), in which after a PCA, the data are reconstructed using the most significant PCs to reduce the noise. In effect, due to the particular form of the ICA objective function in this method, signals composed of a single sharp peak are much more likely to be selected if the noise accounts for a significant proportion. An efficient way to avoid these artifacts is to reduce the noise beforehand.

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Figure 2. a: The four metabolites used for creating artificial in vivo spectra. PCh is phosphocholine, Cr is creatine, NAA is N-acetyl aspartic acid, and Ala stands for alanine. Bottom: ICs extracted by the ICA from a set of 100 generated in vivo spectra. b: six of the 100 extracted components from a direct ICA. c: The entire set of ICs found by the ICA when the data was denoised beforehand.

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Assessment of Peak Quantitation

In a second experiment, we used the previous set (100 spectra, SNR = 10) and the extracted ICs. Since the main ICs could be matched with the original spectra, we were able to compare the amounts of the different ICs required to make up each spectrum (its scores) with the correct amplitude of the corresponding metabolite. In Fig. 3a, one can see a strong correlation between the actual and the calculated values. Note that the ICs are found with an unknown scalar multiplier that can be either positive or negative, hence individual ICs may appear upside down. So if one wants to compare the calculated values and the expected ones, one has to normalize the ICs to the highest peak and invert them if necessary; the absolute value of the correlation will remain unchanged, since it is invariant under linear transformation, but the newly scaled scores will be comparable to the actual quantities (in arbitrary units).

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Figure 3. Robustness of ICA. a: Plots of the true amplitudes of the metabolite vs. the scores of the corresponding IC. The SNR of the dataset is 10. b: Goodness of fit (mean of the correlation coefficients between the amplitudes and the IC scores for each spectrum) vs. SNR of the dataset.

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Detection of Signals With Low Occurence

The method worked well on a dataset with good SNR. In order to assess its robustness to low SNR, the same experiment was carried out on 30 similar sets of 100 spectra, all with different SNRs, ranging from 3 to 20. For each set, the four (for four metabolites) correlation coefficients between the amplitudes and the IC scores were averaged to produce a “goodness-of-fit” value. We ensured that each extracted component corresponded to one of the basis spectra by monitoring their similarity, measured by the correlation between the two signals.

In the last experiment, we investigated whether the ICA was sensitive enough to detect those peaks with low occurrence in the dataset. In effect, in a large collection of spectra, it might be that only a very few samples contain some particular peak—for example, the ones corresponding to a rare pathology in a brain tumor database. The accuracy of extracting such a peak is determined by two parameters: 1) the percentage of spectra in the whole dataset containing that peak, and 2) the level of noise present in the spectra. We therefore performed the following experiment: 200 spectra were generated with a fixed SNR using only the three first metabolites of Fig. 2. A smaller set of spectra was also created, using the same SNR and the four metabolites. A series of ICAs were then performed on a dataset composed of a combination of “three metabolites” spectra and with various percentages of the “four metabolites” spectra. ICA was said to have correctly extracted the fourth metabolite signal when the correlation between the fourth metabolite and its corresponding IC was above 0.8 and the SNR of the IC was above 3.5. The minimum percentage to correctly extract an IC corresponding to the extra metabolite was thus obtained.

ICA of In Vivo Human Data

During the last three years, INTERPRET (19), a European project, has been building a large collection of formally validated brain tumor MR data. Formal validation means that spectra were only included when the diagnosis and grade of the tumor had been agreed upon by a panel of radiologists and histopathologists, after reviewing the clinical (patient's history), MRI, and histological data. In this ICA study, we used single-voxel spectra from the INTERPRET database that had been acquired from 333 patients and normal volunteers. The study was performed with the patients' informed consent and with approval from the local ethics committees. The dataset is fairly heterogeneous: spectra were acquired in four centers across Europe, on three types of machines (GE (N = 196), Philips (N = 116), Siemens (N = 21)) and with two types of data acquisition (PRESS (N = 197) and stimulated echo acquisition mode (STEAM) (N = 136)). All spectra are short-echo (TE = 20–30 ms) phased spectra. The data had 220 dimensions, representing the ppm range of [4.1827, −0.0212] ppm. Details of the data acquisition and preprocessing steps can be found in Ref. 20. Twenty-two tissue types are present: glioblastoma (N = 88), meningioma (N = 64), metastases (N = 49), normal brain (N = 26), astrocytoma grade II (N = 26), oligodendroglioma (N = 10), astrocytoma grade III (N = 10), lymphoma (N = 9), pnet (N = 8), abcess (N = 8), schwanoma (N = 5), and a few samples of chordoma (N = 1), anaplastic oligoastrocytoma (N = 2), anaplastic oligodendroglioma (N = 2), atypical meningoma (N = 4), oligoastrocytoma (N = 5), piloccytic astrocytoma (N = 4), ganglioma (N = 1), germinoma (N = 1), hemangioblastoma (N = 8), anaplastic meningioma (N = 1), and melanoma (N = 1).

We used the same analysis protocol as with the artificial data: first, data were denoised with PCA, and then the ICA was performed. The choice of how many PCs were used to describe the data was made by observing the variance plot and by the number of ICs extracted. When 10 were used, 10 ICs were found, and when 15 PCs were found only 13 independent components were found. Using more PCs to describe the data carried the risk of adding too much noise (cf. next section), hence we decided to use those 13 ICs.

RESULTS

  1. Top of page
  2. Abstract
  3. MATERIALS AND METHODS
  4. RESULTS
  5. DISCUSSION AND CONCLUSIONS
  6. Acknowledgements
  7. REFERENCES

Experiments on Artificial Data

Figure 2 shows some ICs extracted from a direct ICA, and from an ICA preceded by a PCA. In the former case, 100 components were found—many more than the four original signals we were looking for. Each IC consisted of a single peak at a different frequency, even in the region of the spectrum where no metabolites are present. In the latter case, only five ICs could be found: the first four matched the original signals, and the fifth one clearly represented noise.

In order to find the optimal number of PCs required for the preprocessing phase, we used the common method of looking at the drop in the scree plot (see Fig. 1). Moreover, we realized that ICA was good at producing the true number of components: when the data were noiseless, ICA was able to track the original signals back and nothing else, and in the presence of noise, the extra ICs clearly represented noise, as long as a not too large number of PCs was used. Data reconstructed with n PCs lie in a subspace of n dimensions, so no more than n ICs can be found. Still, ICA often found fewer ICs, suggesting that there were no more to extract from the data.

In the second experiment, we assessed the robustness by measuring a goodness of fit when the SNR worsens. For the whole range of SNRs, the correlation between extracted ICs and the basis signals was always higher than 0.8, indicating that the two signals had similar shape. Figure 3b shows how the goodness of fit drops when the SNR is reduced. Even in the worst case (SNR = 3), the average correlation is good (0.7), and an SNR of 4.5 is sufficient to reach a very strong correlation (0.92). The method is thus suitable for noisy in vivo spectra.

The last experiment with artificial data was aimed at investigating the sensitivity of the method. Figure 4 shows the minimum number of spectra containing the extra metabolite vs. the SNR. As expected, the higher the noise, the more samples were needed. However, in practical situations, such as SNR = 5, a proportion of 14% is enough to elicit the right signal. Moreover, a real collection of spectra would contain signals with a wide range of SNRs, and even fewer samples could be needed if some of the spectra containing the extra peak were not too noisy and thus allowed an easier extraction.

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Figure 4. Minimum frequency of spectra with an additional peak needed for the additional peak to be correctly determined when the SNR of the dataset worsens.

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From these experiments concluded that: 1) we needed to denoise the data before performing the ICA, 2) the method was particularly robust to low SNR, and 3) the method was able to extract independent signals with low occurrence.

Experiment on In Vivo Human Data

Figure 5 shows some of the ICs extracted from the human data. They are easy to read (SNR > 20) and most of them can be interpreted in terms of metabolites. For example, IC#9 consists of mostly NAA with a small creatine contribution; IC#4 covers the spectral region containing myoinositol and glycine; IC#8 has a broad peak representing glutamine, glutamate, (Glx) and macromolecules; IC#11 has a sharp lipid peak; IC#3 covers the region centered around the Glx multiplets at 3.78 ppm; and IC#7 represents the choline resonances. Some ICs appear to come from spectra of poor quality: IC#1 is most probably from spectra with poor linewidths resulting in residual water signal in the processed spectra. It was not difficult to identify the problem: only a few samples contained this particular component (i.e., its contribution was zero for most of the spectra), and visual inspection of the faulty spectra was sufficient.

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Figure 5. a, b, and d: ICs extracted from the whole dataset. Labels indicate the metabolites that are most probably represented by the ICs. Cr, creatine; Cho, choline; mIG, Myo-inositol and glycine; Glx, glutamate/glutamine; NAA, N-acetyl aspartic acid; Ala, alanine; MM, macromolecule. A typical spectrum (astrocytoma II) and the mean spectrum of the whole dataset (± SD) are displayed in c.

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The Glx resonances at 3.78 ppm and 2–2.5 ppm appear as contributions to IC#3 and IC#8, respectively, rather than being found in a single IC as might be expected. However, macromolecules also resonate in the 2–2.5 ppm region, which may explain why the Glx resonances appear to be independent.

IC#6 and IC#7 both describe the choline and creatine peaks, with IC#6 representing a shifted component due to slight variations in the exact referencing (currently to the unsuppressed water signal) of the spectra. The separation of metabolites is not always perfect, though: IC#10 to IC#13 have peaks representing lipids, lactate, and alanine, with some significant overlapping. In particular, the doublet for alanine splits on two ICs: 10 and 12.

Each spectrum can be accurately described by using its scores and the ICs. It is also possible to compare the scores of different tumor types on one component at a time. Figure 6 shows their distribution for pathologies that were most extensively represented in the dataset. The tissue types considered are: aggressive tumors (metastases and glioblastomas), astrocytoma low grade, meningioma, and normal white matter. Some separations stand out: e.g., discrimination of normal brain spectra from the tumors can be obtained by looking at their scores on IC#9, which represent NAA. Similarly, most aggressive tumors separate out from the rest on IC#11, which represents lipids. Astrocytomas are characterized by IC#8, and it is possible to discriminate between meningioma and astrocytoma on IC#4, which mostly represents myo-inositol and glycine. For each IC that provided separation, a t-test (21) between the IC score of the separated group and that of the other groups had P < 10−4 in all cases.

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Figure 6. Box plots of the scores associated with IC#4, 8, 9, and 11 (ad, respectively). Note that despite the large heterogeneity of the dataset, each tissue type has a consistent behavior.

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Some of the signals extracted by the ICA show surprisingly broad peaks (IC#5 and IC#8). A closer inspection shows that these humps are actually composed of more than one peak, overlapping with each other. The reason these peaks are not separated is that they correlate over the whole dataset. For example, the cross-correlations of the three peaks comprising IC#5 are all beyond 0.8, which means they always appear in roughly the same proportion and constitute an independent signal. Also, other peaks, such as the lipid peak identified on IC#13, are effectively quite broad in the original spectra, as seen in the mean spectrum of Fig. 5.

DISCUSSION AND CONCLUSIONS

  1. Top of page
  2. Abstract
  3. MATERIALS AND METHODS
  4. RESULTS
  5. DISCUSSION AND CONCLUSIONS
  6. Acknowledgements
  7. REFERENCES

Using artificial data, we showed that after reducing the noise with PCA, ICA was able to identify signals with low occurrence and low SNR. We also demonstrated on real MRS data of brain tumors that the method was able to automatically extract biochemical features out of a large dataset, with no prior knowledge or attempt to separate tissue types. Indeed, ICA performed similarly to combined PCA and factor analysis, but did not require an expert's input to rotate the PCs and make them meaningful. Rather, ICA selected projection axes on the grounds of their statistical properties alone.

Some of these biochemical differences may be expected and have been previously reported (22). Previous studies involved prior knowledge or were simply concerned with classification and gave no information about the physical basis of the distinctions. It is important to note that in the current work, no a priori information was given on which particular peaks or spectral regions were most likely to represent actual biochemicals or be relevant for classifying different tumor types. Thus ICA provided information on biochemical differences automatically, supporting its use for exploratory data analysis of large complex 1H MRS datasets. However, it should be noted that the relationship between individual ICs and the different metabolites is more complex than simply one-to-one. This could be due to numerical approximations, and to the fact that the number of mixtures (MR signals) greatly exceeds the number of possible components (be it metabolites or pathologies). We are currently addressing this problem with specific methods (23).

We envisage at least two possible applications of ICA. In the first one, we use ICA as a good starting point, selected on a sound statistical basis, to analyze large MR datasets. Once the ICA is performed, it is possible to interpret the ICs and plot the spectra with respect to their decomposition. As shown in this study, differences can appear and lead to a more quantitative analysis. The second application is classification using the IC scores as input variables. Since their number is small, and the corresponding ICs can easily be interpreted, it should be possible to explain the decision made by the computer on the basis of biochemical differences and thus make it more intelligible for the user. We are currently developing such tree-based classifiers to discriminate tumor types with lower occurrence than the ones considered in the present study.

Acknowledgements

  1. Top of page
  2. Abstract
  3. MATERIALS AND METHODS
  4. RESULTS
  5. DISCUSSION AND CONCLUSIONS
  6. Acknowledgements
  7. REFERENCES

This work was funded by the European Commission (IST-1999-10310). We thank our INTERPRET partners for providing data, in particular C. Majós and C. Arús from the Universitat Autonòma de Barcelona, and A. Capdevila, B.A. Bell, P. Wilkins, A. Loosemore, and D.J.O. McIntyre from St. George's Hospital Medical School, London. The spectra used in the first part of this study were generated by a program by Kirstie Opstad from St. George's Hospital. We also thank Jianfeng Feng from the University of Sussex for fruitful discussions.

REFERENCES

  1. Top of page
  2. Abstract
  3. MATERIALS AND METHODS
  4. RESULTS
  5. DISCUSSION AND CONCLUSIONS
  6. Acknowledgements
  7. REFERENCES
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