Departament de Bioquímica i Biología Molecular, Universitat Autònoma de Barcelona (UAB), Cerdanyola del Vallès, Spain

Centro de Investigación Biomédica en Red en Bioingeniería, Biomateriales y Nanomedicina (CIBER-BBN), Barcelona Spain

Institut de Biotecnologia i de Biomedicina, Universitat Autònoma de Barcelona, Cerdanyola del Vallès, Spain

Department of Mathematics and Statistics, Liverpool John Moores University (LJMU), Liverpool, UK

Department of Computer Languages and Systems, Universitat Politècnica de Catalunya (UPC), Barcelona, Spain

Abstract

Background

^{1}H-MRS), coupled with supervised pattern recognition (PR) methods, has been widely used in clinical studies of discrimination of brain tumour types and follow-up of patients bearing abnormal brain masses. SV ^{1}H-MRS provides useful biochemical information about the metabolic state of tumours and can be performed at short (< 45 ms) or long (> 45 ms) echo time (TE), each with particular advantages. Short-TE spectra are more adequate for detecting lipids, while the long-TE provides a much flatter signal baseline in between peaks but also negative signals for metabolites such as lactate. Both, lipids and lactate, are respectively indicative of specific metabolic processes taking place. Ideally, the information provided by both TE should be of use for clinical purposes. In this study, we characterise the performance of a range of Non-negative Matrix Factorisation (NMF) methods in two respects: first, to derive sources correlated with the mean spectra of known tissue types (tumours and normal tissue); second, taking the best performing NMF method for source separation, we compare its accuracy for class assignment when using the mixing matrix directly as a basis for classification, as against using the method for dimensionality reduction (DR). For this, we used SV ^{1}H-MRS data with positive and negative peaks, from a widely tested SV ^{1}H-MRS human brain tumour database.

Results

The results reported in this paper reveal the advantage of using a recently described variant of NMF, namely Convex-NMF, as an unsupervised method of source extraction from SV^{1}H-MRS. Most of the sources extracted in our experiments closely correspond to the mean spectra of some of the analysed tumour types. This similarity allows accurate diagnostic predictions to be made both in fully unsupervised mode and using Convex-NMF as a DR step previous to standard supervised classification. The obtained results are comparable to, or more accurate than those obtained with supervised techniques.

Conclusions

The unsupervised properties of Convex-NMF place this approach one step ahead of classical label-requiring supervised methods for the discrimination of brain tumour types, as it accounts for their increasingly recognised molecular subtype heterogeneity. The application of Convex-NMF in computer assisted decision support systems is expected to facilitate further improvements in the uptake of MRS-derived information by clinicians.

Background

Introduction

The clinical investigation of an abnormal mass in the brain frequently starts with its non-invasive characterisation (localisation, infiltration, etc.), normally with a magnetic resonance imaging (MRI) study. Magnetic resonance spectroscopy (MRS) is another MR technique that, unlike MRI, provides insight into the biochemistry of tissue through a discrete signal in the frequency domain (a spectrum) containing information about the relative abundance of several low molecular weight metabolites, lipids and macromolecules in the millimolar range of concentration.

This MR modality has been used in computer-based systems for diagnostic decision support

The MRS data analysed in the current work are single-voxel. That is, for each patient we have a single spectrum corresponding to a small volume located within the tumour core. The aim of this study is to separate the constituent source signals on the assumption that they are mixed linearly in each single-voxel spectral measurement. This is because, even within a single voxel, an heterogeneous mix of tissue types may be expected. In this way, the main constituents of the voxel could be separately identified and quantified, providing, in turn, a quantification of class (tumour type or healthy tissue) membership for the sources of each single voxel spectrum, as an alternative to the class labelling of the spectrum as a whole.

Linear unsupervised feature extraction PR techniques are commonly used in neuro-oncology for data preprocessing and dimensionality reduction (DR) previous to the diagnostic classification of brain tumours. The usual choices are principal component analysis (PCA)

In this study, we characterise the performance of a range of variants of an unsupervised method of the matrix factorisation family, namely Non-negative Matrix Factorisation (NMF,

The results reported in the current paper reveal the advantage of using one of the recently described NMF variants, namely Convex-NMF ^{1}H-MRS. In contrast with ICA, most of the sources extracted by the proposed technique closely correspond to the mean spectra of some of the analysed tumour types. This similarity allows accurate diagnostic predictions to be made for each patient (that is, for each SV spectrum) both in fully unsupervised mode or using Convex-NMF as a DR step previous to standard supervised classification. These predictions are comparable to or more accurate than those obtained with supervised techniques.

The remaining of the paper is organised as follows. The

Materials

The data analysed in this study were extracted from INTERPRET, an international multi-centre database ^{a }
^{1}H-MRS) data acquired at 1.5T and at two different echo times (short, 20-32 ms (STE) and long, 135-144 ms (LTE)) from brain tumour patients and healthy controls (that is, two spectra, one at STE and another at LTE, are available for each individual).

The importance of using two different signal acquisition conditions (STE and LTE) lies in the different metabolites that are detectable at each of them. STE is more sensitive to those with short T2 (an MR relaxation time parameter) values (it is, for example, more adequate to detect mobile lipids) and, in addition, all signal peaks are positive. On the other hand, in LTE spectra we can find both positive and negative peaks, where the negative peak is due to the inverted Alanine or Lactate doublets. The analysed data set included, at LTE, 20 astrocytomas grade II (A2), 78 glioblastomas (GL), 31 metastases (ME), 55 low-grade meningiomas (MM) and 15 normal brain parenchyma measurements from healthy controls (NO); at STE, it included 22 A2, 86 GL, 38 ME, 58 MM, and 22 NO. Data were processed as in

A further test data set (not used for source extraction, but only for the validation of the obtained results) was gathered from three medical centres: Centre Diagnòstic Pedralbes (CDP), Institut d'Alta Tecnologia (IAT) and Institut de Diagnòstic per la Imatge (IDI)-Badalona in Barcelona, Spain. It was processed in the same conditions as the rest of the data, and consists of STE and LTE spectra from 56 patients and healthy controls: 10 A2, 40 high-grade aggressive tumours (30 GL + 10 ME), 3 MM, and 3 NO subjects.

MRS data were acquired according to the medical ethics regulations of the countries of the medical centres involved, in particular, with the Helsinki Declaration and the Spanish "^{th}, 1999"

Methods

As stated in the introduction, NMF can be seen as a DR technique, functionally similar to source extraction. This section summarily describes some of the existing NMF methods and the different alternatives for their initialisation. The choice of initialisation technique turns out to be a key feature for the success of NMF as a tumour type classification method. The specific way in which these techniques are used and interpreted in the context of MRS data analysis is also described in this section. We later explain how the data can be labelled

Non-negative matrix factorisation methods for source extraction

In the standard NMF description

subject to the non-negativity constraints mentioned above, where ||·||_{
F
}is the Frobenius norm. In this study, the following divergence minimisation methods, which cover a wide palette of algorithmic alternatives, were considered:

• Euclidean distance update equations (herein referred to as

The objective function is optimised with multiplicative update rules for

Monotonic convergence of the algorithm can be proven

• Alternating least squares (

This technique alternately fixes one matrix and improves the other.

where

setting all negative elements in

• Alternating non-negative least squares using projected gradients (

The equations for

where α is the step size, and

The same approach is used to calculate

• Alternating least squares with Optimal Brain Surgeon (OBS)

where, δ_{
W
}and δ_{
H
}act as regularisation terms and are responsible for eliminating the less important elements of

• Convex-NMF (

where (·)^{+ }is the positive part of the matrix, where all negative values become zeros; and (·)^{- }is the negative part of the matrix, where all positive values become zeros. All the algorithms, for all initialisations, were allowed to achieve convergence. Such convergence was qualified as the lack of variation in the reconstruction error, from one iteration to the next, over a common set small threshold of value 10^{-5}.

Interpretation of the methods

In NMF for the analysis of MRS data, the rows in

Convex-NMF, instead, enforces this non-negative constraint only on ^{1}H-MRS LTE spectra). As in the previous methods,

NMF initialisations

NMF methods unavoidably converge to local minima. As a result, the NMF bases will be different for different initialisations. In this study, six forms of initialisation were considered (with some variations depending on the method). Although a standard procedure to justify the choice of NMF initialisation does not exist, the six alternatives considered here cover a wide array of approaches: from random initialisation, to prototype-based clustering methods (K-means and Fuzzy C-Means, which provide a data density-based sample of initial data locations), and feature extraction techniques (PCA, ICA and NMF itself, which initialise the algorithm according to the basic eigenstructure of the data).

• Random:

[all methods]:

• K-means clustering:

[

[^{(0) }
_{1}, . . ., _{n}
_{ik }
^{(0) }= (^{-1}, where

• Fuzzy C-Means (FCM):

[

[^{(0) }= ^{(0) }= (^{-1 }, where

• PCA:

[

[^{(0) }= ^{T}
^{T}
^{-1}, and then ^{(0) }= (^{+ }+ 0.2^{+}〉 so that the negative elements are removed, where 〈_{
n, k
}|_{n, k}
_{n, k}
_{0}, and where ||_{n, k}
_{0 }is the number of nonzero elements in

• ICA (FastICA

[

[

• Non-negative Matrix Factorisation (NMF,

[

[

In principle, we might expect the different initialisation strategies to behave as follows. Random initialisation might be considered as an uninformed first estimate for NMF methods, which may lead to different outcomes given different initialisation conditions

Tumour type labelling using the mixing matrix and the sources

As explained in the introduction section, NMF is used in this study as an unsupervised method in the sense that labelled MRS cases are not used to create the data model. The obvious advantage of this approach is that the labelling procedure can be made independent of any specific labelled (or mislabelled) MRS dataset that might bias the generalisation capabilities of the model.

In order to determine how well the sources obtained through NMF represent the data, we propose to infer the labels of the data only on the basis of the mixing matrix and the source signals calculated, which will give us an idea of the extent to which the sources contribute to the reconstruction of each MRS observation (or patient case). The calculation of the contribution

where

Source extraction as a dimensionality reduction procedure prior to classification

The description of the MR spectra through a limited number of extracted sources also entails a DR process in the form of feature extraction. As previously mentioned, the use of DR methods in the form of feature selection or extraction is commonplace in the analysis of MRS. The extracted features can then be used for traditional classification, within a standard supervised framework using labelled cases. This was accomplished in the current study using the Gram-Schmidt process _{1}
_{k }
_{1}, ..., _{k }
^{
n
}as

Results

In this section, we compile and present all the experimental results. The objective of the experiments was twofold: first, the assessment of NMF in fully unsupervised mode as a source extraction and tumour type-labelling method and, second, the evaluation of NMF as a DR method prior to standard supervised classification.

NMF as a source extraction method

Here, we provide the comparative results of the application of the five NMF methods for source extraction outlined in the

A2 are low-grade (grade II on a scale I-IV of the WHO

In summary, the choice of these specific problems at both time of echo acquisition conditions ultimately aimed to find answers to the following questions: 1) (A2

Tables

Summary of the results for LTE.

**Experiment: LTE. A2, NO. (2 sources)**

Random

K-means

FCM

PCA

FastICA

NMF (als)

euc

A2: 0.98

A2: 0.97

A2: 0.98

A2: 0.91

A2: 0.98

A2: 0.97

NO: 0.96

NO: 0.99

NO: 0.99

NO: 0.98

NO: 0.87

NO: 0.95

als

A2: 0.97

A2: 0.97

A2: 0.97

A2: 0.97

A2: 0.97

A2: 0.97

NO: 0.98

NO: 1.00

NO: 1.00

NO: 1.00

NO: 0.95

NO: 0.96

alspg

A2: 0.97

A2: 0.97

A2: 0.97

A2: 0.97

A2: 0.97

A2: 0.97

NO: 0.96

NO: 1.00

NO: 1.00

NO: 1.00

NO: 0.95

NO: 0.96

alsobs

A2: 0.97

A2: 0.97

A2: 0.97

A2: 0.97

A2: 0.97

A2: 0.97

NO: 0.96

NO: 1.00

NO: 1.00

NO: 1.00

NO: 0.95

NO: 0.99

convex

A2: 1.00

A2: 1.00

A2: 1.00

A2: 0.99

A2: 1.00

A2: 1.00

NO: 1.00

NO: 1.00

NO: 1.00

NO: 1.00

NO: 1.00

NO: 1.00

**Experiment: LTE. A2, ME, NO. (3 sources)**

Random

K-means

FCM

PCA

FastICA

NMF (als)

euc

A2: 0.94

A2: 0.94

A2: 0.94

A2: 0.88

A2: 0.94

A2: 0.94

ME: 0.85

ME: 0.85

ME: 0.85

ME: 0.78

ME: 0.86

ME: 0.85

NO: 0.95

NO: 1.00

NO: 0.99

NO: 0.90

NO: 0.92

NO: 0.93

als

A2: 0.94

A2: 0.94

A2: 0.94

A2: 0.94

A2: 0.94

A2: 0.94

ME: 0.85

ME: 0.84

ME: 0.84

ME: 0.85

ME: 0.85

ME: 0.85

NO: 0.92

NO: 0.99

NO: 0.99

NO: 0.92

NO: 0.91

NO: 0.91

alspg

A2: 0.94

A2: 0.94

A2: 0.94

A2: 0.94

A2: 0.94

A2: 0.94

ME: 0.85

ME: 0.85

ME: 0.85

ME: 0.85

ME: 0.85

ME: 0.85

NO: 0.99

NO: 0.99

NO: 0.99

NO: 0.95

NO: 0.93

NO: 0.92

alsobs

A2: 0.94

A2: 0.94

A2: 0.94

A2: 0.94

A2: 0.94

A2: 0.94

ME: 0.85

ME: 0.85

ME: 0.85

ME: 0.85

ME: 0.85

ME: 0.85

NO: 0.96

NO: 0.99

NO: 0.99

NO: 0.95

NO: 0.95

NO: 0.95

convex

A2: 0.99

A2: 0.99

A2: 0.98

A2: 0.98

A2: 0.98

A2: 0.98

ME: 0.88

ME: 0.88

ME: 0.90

ME: 0.86

ME: 0.87

ME: 0.87

NO: 1.00

NO: 1.00

NO: 1.00

NO: 1.00

NO: 1.00

NO: 1.00

**Experiment: LTE. A2, GL, NO. (3 sources)**

Random

K-means

FCM

PCA

FastICA

NMF (als)

euc

A2: 0.91

A2: 0.91

A2: 0.91

A2: 0.87

A2: 0.92

A2: 0.91

GL: 0.75

GL: 0.76

GL: 0.75

GL: 0.92

GL: 0.78

GL: 0.76

NO: 0.99

NO: 0.99

NO: 0.99

NO: 0.96

NO: 0.94

NO: 0.99

als

A2: 0.92

A2: 0.92

A2: 0.92

A2: 0.92

A2: 0.92

A2: 0.92

GL: 0.76

GL: 0.76

GL: 0.76

GL: 0.76

GL: 0.76

GL: 0.76

NO: 0.95

NO: 0.99

NO: 0.99

NO: 0.99

NO: 0.95

NO: 0.98

alspg

A2: 0.91

A2: 0.91

A2: 0.91

A2: 0.91

A2: 0.91

A2: 0.91

GL: 0.75

GL: 0.75

GL: 0.75

GL: 0.75

GL: 0.75

GL: 0.75

NO: 0.99

NO: 0.99

NO: 0.99

NO: 0.99

NO: 0.97

NO: 0.96

alsobs

A2: 0.91

A2: 0.91

A2: 0.91

A2: 0.91

A2: 0.91

A2: 0.91

GL: 0.75

GL: 0.75

GL: 0.75

GL: 0.75

GL: 0.75

GL: 0.75

NO: 0.99

NO: 0.99

NO: 0.99

NO: 0.99

NO: 0.98

NO: 0.99

convex

A2: 0.94

A2: 0.97

A2: 0.95

A2: 0.94

A2: 0.96

A2: 0.96

GL: 0.80

GL: 0.73

GL: 0.77

GL: 0.81

GL: 0.75

GL: 0.74

NO: 0.98

NO: 1.00

NO: 1.00

NO: 0.99

NO: 1.00

NO: 1.00

**Experiment: LTE. A2, MM, NO. (3 sources)**

Random

K-means

FCM

PCA

FastICA

NMF (als)

euc

A2: 0.96

A2: 0.88

A2: 0.89

A2: 0.97

A2: 0.88

A2: 0.88

MM: 0.92

MM: 0.97

MM: 0.97

MM: 0.91

MM: 0.99

MM: 0.97

NO: 0.95

NO: 0.98

NO: 0.98

NO: 0.72

NO: 0.89

NO: 0.98

als

A2: 0.88

A2: 0.88

A2: 0.88

A2: 0.88

A2: 0.88

A2: 0.88

MM: 0.97

MM: 0.97

MM: 0.97

MM: 0.97

MM: 0.97

MM: 0.97

NO: 0.98

NO: 0.97

NO: 0.97

NO: 0.98

NO: 0.97

NO: 0.98

alspg

A2: 0.88

A2: 0.88

A2: 0.88

A2: 0.88

A2: 0.88

A2: 0.88

MM: 0.97

MM: 0.97

MM: 0.97

MM: 0.97

MM: 0.97

MM: 0.97

NO: 0.97

NO: 0.98

NO: 0.98

NO: 0.98

NO: 0.96

NO: 0.98

alsobs

A2: 0.88

A2: 0.88

A2: 0.88

A2: 0.88

A2: 0.88

A2: 0.88

MM: 0.98

MM: 0.98

MM: 0.97

MM: 0.97

MM: 0.97

MM: 0.98

NO: 0.98

NO: 0.98

NO: 0.98

NO: 0.98

NO: 0.97

NO: 0.98

convex

A2: 0.98

A2: 0.99

A2: 0.98

A2: 0.98

A2: 0.99

A2: 0.88

MM: 1.00

MM: 0.99

MM: 0.99

MM: 0.99

MM: 1.00

MM: 0.98

NO: 1.00

NO: 1.00

NO: 1.00

NO: 1.00

NO: 1.00

NO: 0.99

Summary of the correlation values for the sources most highly correlating with each type of tissue as expressed by its mean spectrum, for different diagnostic problems at LTE, and for all the NMF methods and initialisation conditions in the study. The diagnostic problems are: A2, NO; A2, ME, NO; A2, GL, NO; and A2, MM, NO

Summary of the results for STE.

**Experiment: STE. A2, NO. (2 sources)**

Random

K-means

FCM

PCA

FastICA

NMF (als)

euc

A2: 0.94

A2: 0.94

A2: 0.94

A2: .83

A2: 0.96

A2: 0.92

NO: 0.87

NO: 0.97

NO: 0.96

NO: 0.93

NO: 0.76

NO: 0.75

als

A2: 0.92

A2: 0.95

A2: .95

A2: .92

A2: 0.93

A2: 0.92

NO: 0.75

NO: 0.96

NO: 0.96

NO: 0.96

NO: 0.83

NO: 0.75

alspg

A2: 0.93

A2: 0.95

A2: .95

A2: .92

A2: 0.95

A2: 0.92

NO: 0.75

NO: 0.96

NO: 0.96

NO: 0.96

NO: 0.75

NO: 0.75

alsobs

A2: 0.92

A2: 0.94

A2: .94

A2: .92

A2: 0.95

A2: 0.92

NO: 0.76

NO: 0.96

NO: 0.96

NO: 0.96

NO: 0.75

NO: 0.75

convex

A2: 0.99

A2: 0.99

A2: .99

A2: .98

A2: 0.99

A2: .99

NO: 0.99

NO: 1.00

NO: 1.00

NO: 1.00

NO: 0.99

NO: 1.00

**Experiment: STE. A2, ME, NO. (3 sources)**

Random

K-means

FCM

PCA

FastICA

NMF (als)

euc

A2: 0.93

A2: 0.94

A2: 0.93

A2: 0.91

A2: 0.94

A2: 0.93

ME: 0.98

ME: 0.98

ME: 0.98

ME: 0.86

ME: 0.99

ME: 0.98

NO: 0.83

NO: 0.80

NO: 0.89

NO: 0.87

NO: 0.86

NO: 0.74

als

A2: 0.93

A2: 0.94

A2: 0.94

A2: 0.94

A2: 0.94

A2: 0.93

ME: 0.98

ME: 0.98

ME: 0.98

ME: 0.98

ME: 0.98

ME: 0.98

NO: 0.75

NO: 0.74

NO: 0.74

NO: 0.74

NO: 0.74

NO: 0.74

alspg

A2: 0.94

A2: 0.93

A2: 0.93

A2: 0.94

A2: 0.94

A2: 0.94

ME: 0.98

ME: 0.98

ME: 0.98

ME: 0.98

ME: 0.98

ME: 0.98

NO: 0.69

NO: 0.73

NO: 0.73

NO: 0.69

NO: 0.70

NO: 0.74

alsobs

A2: 0.94

A2: 0.94

A2: 0.94

A2: 0.94

A2: 0.94

A2: 0.94

ME: 0.98

ME: 0.98

ME: 0.98

ME: 0.98

ME: 0.98

ME: 0.98

NO: 0.69

NO: 0.69

NO: 0.72

NO: 0.69

NO: 0.71

NO: 0.69

convex

A2: 0.98

A2: 0.99

A2: 0.99

A2: 0.91

A2: 0.99

A2: 0.99

ME: 1.00

ME: 1.00

ME: 1.00

ME: 0.99

ME: 0.99

ME: 0.99

NO: 0.99

NO: 1.00

NO: 1.00

NO: 0.93

NO: 0.99

NO: 0.99

**Experiment: STE. A2, GL, NO. (3 sources)**

Random

K-means

FCM

PCA

FastICA

NMF (als)

euc

A2: 0.94

A2: 0.91

A2: 0.91

A2: 0.70

A2: 0.95

A2: 0.91

GL: 0.95

GL: 0.91

GL: 0.94

GL: 0.56

GL: 0.96

GL: 0.95

NO: 0.81

NO: 0.92

NO: 0.92

NO: 0.65

NO: 0.85

NO: 0.80

als

A2: 0.91

A2: 0.90

A2: 0.90

A2: 0.90

A2: 0.91

A2: 0.90

GL: 0.95

GL: 0.95

GL: 0.95

GL: 0.95

GL: 0.95

GL: 0.95

NO: 0.76

NO: 0.90

NO: 0.89

NO: 0.83

NO: 0.79

NO: 0.82

alspg

A2: 0.90

A2: 0.91

A2: 0.91

A2: 0.90

A2: 0.93

A2: 0.90

GL: 0.95

GL: 0.93

GL: 0.93

GL: 0.95

GL: 0.95

GL: 0.95

NO: 0.84

NO: 0.93

NO: 0.93

NO: 0.85

NO: 0.72

NO: 0.83

alsobs

A2: 0.93

A2: 0.91

A2: 0.91

A2: 0.91

A2: 0.92

A2: 0.92

GL: 0.95

GL: 0.95

GL: 0.93

GL: 0.95

GL: 0.95

GL: 0.95

NO: 0.72

NO: 0.80

NO: 0.93

NO: 0.80

NO: 0.72

NO: 0.74

convex

A2: 0.95

A2: 0.98

A2: 0.94

A2: 0.94

A2: 0.99

A2: 0.99

GL: 0.98

GL: 0.98

GL: 0.98

GL: 0.98

GL: 0.98

GL: 0.98

NO: 0.94

NO: 1.00

NO: 0.94

NO: 0.95

NO: 0.99

NO: 1.00

**Experiment: STE. A2, MM, NO. (3 sources)**

Random

K-means

FCM

PCA

FastICA

NMF (als)

euc

A2: 0.93

A2: 0.91

A2: 0.95

A2: 0.83

A2: 0.94

A2: 0.92

MM: 0.57

MM: 0.56

MM: 0.54

MM: 0.74

MM: 0.64

MM: 0.57

NO: 0.83

NO: 0.95

NO: 0.73

NO: 0.89

NO: 0.77

NO: 0.79

als

A2: 0.93

A2: 0.93

A2: 0.93

A2: 0.92

A2: 0.93

A2: 0.92

MM: 0.57

MM: 0.57

MM: 0.57

MM: 0.57

MM: 0.57

MM: 0.57

NO: 0.76

NO: 0.79

NO: 0.77

NO: 0.75

NO: 0.76

NO: 0.75

alspg

A2: 0.92

A2: 0.93

A2: 0.91

A2: 0.91

A2: 0.92

A2: 0.92

MM: 0.57

MM: 0.57

MM: 0.57

MM: 0.57

MM: 0.57

MM: 0.57

NO: 0.77

NO: 0.80

NO: 0.81

NO: 0.81

NO: 0.80

NO: 0.75

alsobs

A2: 0.92

A2: 0.92

A2: 0.92

A2: 0.92

A2: 0.92

A2: 0.92

MM: 0.57

MM: 0.57

MM: 0.57

MM: 0.57

MM: 0.57

MM: 0.57

NO: 0.77

NO: 0.80

NO: 0.80

NO: 0.80

NO: 0.76

NO: 0.77

convex

A2: 0.95

A2: 0.98

A2: 0.93

A2: 0.96

A2: 0.98

A2: 0.98

MM: 0.98

MM: 0.90

MM: 0.91

MM: 0.79

MM: 0.86

MM: 0.85

NO: 0.91

NO: 1.00

NO: 0.95

NO: 0.98

NO: 0.99

NO: 0.98

Summary of the correlation values for the sources most highly correlating with each type of tissue as expressed by its mean spectrum, for different diagnostic problems at STE, and for all the NMF methods and the initialisation conditions in the study. The diagnostic problems are: A2, NO; A2, ME, NO; A2, GL, NO; and A2, MM, NO

Figure

Sources extracted in the experiment A2, MM, NO at LTE

**Sources extracted in the experiment A2, MM, NO at LTE**. The first five rows show the source signals obtained in the experiments with A2, MM and NO at LTE, for all the methods under study and K-means clustering initialisation. The last row shows, from left to right, the mean spectra of A2, MM and NO, at LTE. Horizontal axis, for all plots: frequency in ppm scale. Vertical axis, for all plots: UL2 normalised intensity. The range of the vertical scales is fixed for each experiment and they are the same as those of the mean spectra of the last row, for comparative purposes.

The computation times for the different methods used in this study, in a personal computer (memory (RAM): 4 GB, processor: Pentium Dual-Core T4400, 64-bit operating system), were less than one second in almost all cases, with the exception of

Labelling using convex-NMF

The results summarised in Tables

We next report the results of the unsupervised labelling process: That is, the assignment of class labels (tumour types and healthy tissue) to each of the cases using the extracted sources and without modelling explicitly the relationship between the sources and the class labels. Table

Labelling accuracy results obtained using Convex-NMF.

**LTE**

**A2, NO(2SS)**

**A2, ME, NO(3SS)**

**A2, GL, NO(3SS)**

**A2, MM, NO(3SS)**

**A2, AG, NO(4SS)**

**A2, AG, MM(4SS)**

Total:100%(35/35)

Total:84.8%(56/66)

Total:71.1%(81/113)

Total:96.7%(87/90)

Total:77.8%(112/144)

Total:73.9%(136/184)

A2:100%(20/20)

A2:100%(20/20)

A2:100%(20/20)

A2:100%(20/20)

A2:100%(20/20)

A2:95.0%(19/20)

NO:100%(15/15)

ME:67.7%(21/31)

GL:59.0%(46/78)

MM:94.5%(52/55)

AG:70.6%(77/109)

AG:64.2%(70/109)

NO:100%(15/15)

NO:100%(15/15)

NO:100%(15/15)

NO:100%(15/15)

MM:85.5%(47/55)

**STE**

**A2, NO(2SS)**

**A2, ME, NO(3SS)**

**A2, GL, NO(3SS)**

**A2, MM, NO(3SS)**

**A2, AG, NO(4SS)**

**A2, AG, MM(4SS)**

Total:93.2%(41/44)

Total:91.5%(75/82)

Total:88.5%(115/130)

Total:89.2%(91/102)

Total:92.9%(156/168)

Total:86.3%(176/204)

A2:86.4%(19/22)

A2:77.3%(17/22)

A2:81.8%(18/22)

A2:77.3%(17/22)

A2:81.8%(18/22)

A2:90.9%(20/22)

NO:100%(22/22)

NO:100%(22/22)

GL:87.2%(75/86)

MM:89.7%(52/58)

AG:93.5%(116/124)

AG:87.9%(109/124)

ME:94.7%(36/38)

NO:100%(22/22)

NO:100%(22/22)

NO:100%(22/22)

MM:81.0%(47/58)

Summary of the labelling accuracy using Convex-NMF with K-means initialisation for A2, NO; A2, ME, NO; A2, GL, NO; A2, MM, NO; A2, AG, NO; and A2, AG, MM, at LTE and STE. They include the accuracy (total and by tumour type), and the number of correctly labelled samples from the total, in parentheses. The number of source signals (SS) used in the experiments is indicated in parentheses

Convex-NMF was also initialised with K-means clustering, and a total of 4 source signals were calculated for these two problems, given that 4 classes were involved. The predicted labels were then used to determine to what extent each observation was correctly labelled, according to the INTERPRET database information. The results of the six diagnostic problems are compiled in Table

Sources extracted in the experiment A2, AG, NO at LTE and STE

**Sources extracted in the experiment A2, AG, NO at LTE and STE**. Source signals obtained in the experiments with A2, AG (GL+ME) and NO at LTE (first row) and STE (second row), calculated with Convex-NMF, and initialised with K-means clustering. The sources in the first column (S1) represent A2, the ones in the second column (S2) represent NO, and the ones in the last two columns (S3 and S4) mainly represent AG. Axes labels and representation as in Figure 1.

Sources extracted in the experiment A2, AG, MM at LTE and STE

**Sources extracted in the experiment A2, AG, MM at LTE and STE**. Source signals obtained in the experiments with A2, AG (GL+ME) and MM at LTE (first row) and STE (second row), calculated with Convex-NMF, and initialised with K-means clustering. The sources in the first column (S1) again represent A2, the ones in the second column (S2) represent MM, and the ones in the last two columns (S3 and S4) again mainly represent AG. Axes labels and representation as in previous figures.

In the next section we use the sources in the context of supervised classification, and compare the results with equivalent classifiers, using the same settings.

NMF for classification

Using convex-NMF extracted source signals for dimensionality reduction prior to classification

We now switch to experiments that analyse the use of Convex-NMF as a dimensionality reduction technique to preprocess the MRS data prior to standard classification. For this, we used the orthogonal set corresponding to the source signals obtained, and projected the data onto this basis. The SpectraClassifier^{b }software

Classification results using Convex-NMF for DR prior to classification with Fisher LDA.

**LTE**

**A2, NO(2SS)**

**A2, ME, NO(3SS)**

**A2, GL, NO(3SS)**

**A2, MM, NO(3SS)**

**A2, AG, NO(4SS)**

**A2, AG, MM(4SS)**

Total:100% ± 0.0

Total:92.6% ± 3.3

Total:85.1% ± 3.4

Total:97.7% ± 1.6

Total:90.9% ± 2.4

Total:79.4% ± 3.0

A2:100% ± 0.0

A2:100% ± 0.0

A2:84.9% ± 8.5

A2:94.8% ± 5.0

A2:100% ± 0.0

A2:94.9% ± 5.2

NO:100% ± 0.0

ME:84.1% ± 6.8

GL:82.3% ± 4.3

GL:98.2% ± 1.9

AG:88.0% ± 3.1

AG:72.5% ± 4.3

NO:100% ± 0.0

NO:100% ± 0.0

NO:100% ± 0.0

NO:100% ± 0.0

MM:87.5% ± 4.3

**STE**

**A2, NO(2SS)**

**A2, ME, NO(3SS)**

**A2, GL, NO(3SS)**

**A2, MM, NO(3SS)**

**A2, AG, NO(4SS)**

**A2, AG, MM(4SS)**

Total:95.5% ± 3.1

Total:94.0% ± 2.6

Total:91.0% ± 2.5

Total:92.2% ± 2.7

Total:92.3% ± 2.0

TOTAL:87.7% ± 2.3

A2:91.2% ± 6.0

A2:86.3% ± 7.4

A2:82.5% ± 8.1

A2:86.3% ± 7.7

A2:81.9% ± 8.2

A2:95.5% ± 4.6

NO:100% ± 0.0

ME:95.0% ± 3.5

GL:90.9% ± 3.1

GL:91.5% ± 3.7

AG:92.8% ± 2.3

AG:86.3% ± 3.2

NO:100% ± 0.0

NO:100% ± 0.0

NO:100% ± 0.0

NO:100% ± 0.0

MM:87.7% ± 4.3

Classification results (accuracy ± standard deviation) obtained with Fisher LDA (implemented in SpectraClassifier) for six diagnostic problems, from the source signals obtained by Convex-NMF, for data acquired at LTE and STE. Classifier results were validated through bootstrap. The number of source signals (SS) used in the experiments is indicated in parentheses

In order to compare these results with those of a traditional feature extraction method, we replicated all experiments using the SpectraClassifier software with PCA as data preprocessing feature extraction method (extracting a number of principal components equal to the number of source signals calculated for the corresponding NMF experiment). This was again followed by Fisher LDA classification and evaluated through bootstrap with 1,000 repetitions, as in the experiments in Table

Classification results using PCA for DR prior to classification with Fisher LDA.

**LTE**

**A2, NO(2PC)**

**A2, ME, NO(3PC)**

**A2, GL, NO(3PC)**

**A2, MM, NO(3PC)**

**A2, AG, NO(4PC)**

**A2, AG, MM(4PC)**

Total:100% ± 0.0

Total:93.8% ± 3.1

Total:82.1% ± 3.6

Total:95.4% ± 2.2

Total:84.6% ± 3.0

Total:80.2% ± 2.9

A2:100% ± 0.0

A2:100% ± 0.0

A2:100.0% ± 0.0

A2:95.0% ± 5.1

A2:100% ± 0.0

A2:100% ± 0.0

NO:100% ± 0.0

ME:86.9% ± 6.1

GL:75.4% ± 4.9

MM:94.4% ± 3.1

AG:79.7% ± 3.9

AG:75.9% ± 4.2

NO:100% ± 0.0

NO:93.2% ± 6.8

NO:100% ± 0.0

NO:100% ± 0.0

MM:81.6% ± 5.2

**STE**

**A2, NO(2PC)**

**A2, ME, NO(3PC)**

**A2, GL, NO(3PC)**

**A2, MM, NO(3PC)**

**A2, AG, NO(4PC)**

**A2, AG, MM(4PC)**

Total:93.2% ± 3.9

Total:90.2% ± 3.3

Total:84.4% ± 3.2

Total:88.1% ± 3.2

Total:86.2% ± 2.7

Total:81.3% ± 2.7

A2:86.4% ± 7.4

A2:86.2% ± 7.5

A2:81.4% ± 8.4

A2:81.6% ± 8.4

A2:72.5% ± 9.6

A2:90.7% ± 6.4

NO:100% ± 0.0

ME:91.9% ± 4.4

GL:84.6% ± 3.8

MM:87.8% ± 4.2

AG:89.5% ± 2.8

AG:80.6% ± 3.5

NO:91.1% ± 6.0

NO:86.2% ± 7.5

NO:95.5% ± 4.5

NO:81.5% ± 8.6

MM:79.2% ± 5.3

Classification results (accuracy ± standard deviation) obtained with Fisher LDA (implemented in SpectraClassifier) for six diagnostic problems, from the source signals obtained by PCA, for data acquired at LTE and STE. Classifier results were validated through bootstrap. The number of principal components (PC) in the experiments is indicated in parentheses

Classification accuracies for the independent test set.

**LTE, FE method:PCA**

**A2, NO(2PC)**

**A2, ME, NO(3PC)**

**A2, GL, NO(3PC)**

**A2, MM, NO(3PC)**

**A2, AG, NO(4PC)**

**A2, AG, MM(4PC)**

Total:92.3%(12/13)

Total:82.6%(19/23)

Total:65.1%(28/43)

Total:81.3%(13/16)

Total:64.2%(34/53)

Total:67.9%(36/53)

A2:100%(10/10)

A2:100%(10/10)

A2:90%(9/10)

A2:80.0%(8/10)

A2:90%(9/10)

A2:80.0%(8/10)

NO:66.7%(2/3)

ME:70.0%(7/10)

GL:53.3%(16/30)

MM:66.7%(2/3)

AG:57.5%(23/40)

AG:62.5%(25/40)

NO:66.7%(2/3)

NO:100%(3/3)

NO:100%(3/3)

NO:66.7%(2/3)

MM:100%(3/3)

BER:0.17

BER:0.21

BER:0.19

BER:0.18

BER:0.29

BER:0.19

**LTE, FE method:Convex-NMF**

**A2, NO(2SS)**

**A2, ME, NO(3SS)**

**A2, GL, NO(3SS)**

**A2, MM, NO(3SS)**

**A2, AG, NO(4SS)**

**A2, AG, MM(4SS)**

Total:92.3%(12/13)

Total:82.6%(19/23)

Total:67.4%(29/43)

Total:68.8(11/16)

Total:71.7%(38/53)

Total:64.2%(34/53)

A2:90.0%(9/10)

A2:90.0%(9/10)

A2:70.0%(7/10)

A2:50.0%(5/10)

A2:70.0%(7/10)

A2:60%(6/10)

NO:100%(3/3)

ME:70.0%(7/10)

GL:63.3%(19/30)

MM:100%(3/3)

AG:70.0%(28/40)

AG:62.5%(25/40)

NO:100%(3/3)

NO:100%(3/3)

NO:100%(3/3)

NO:100%(3/3)

MM:100%(3/3)

BER:0.05

BER:0.13

BER:0.22

BER:0.17

BER:0.20

BER:0.26

**STE, FE method:PCA**

**A2, NO(2PC)**

**A2, ME, NO(3PC)**

**A2, GL, NO(3PC)**

**A2, MM, NO(3PC)**

**A2, AG, NO(4PC)**

**A2, AG, MM(4PC)**

Total:92.3%(12/13)

Total:73.9%(17/23)

Total:76.7%(33/43)

Total:75.0(12/16)

Total:83.0%(44/53)

Total:73.6%(39/53)

A2:90.0%(9/10)

A2:80.0%(8/10)

A2:60.0%(6/10)

A2:60.0%(6/10)

A2:80.0%(8/10)

A2:70.0%(7/10)

NO:100%(3/3)

ME:70.0%(7/10)

GL:80.0%(24/30)

MM:100%(3/3)

AG:87.5%(35/40)

AG:72.5%(29/40)

NO:66.7%(2/3)

NO:100%(3/3)

NO:100%(3/3)

NO:33.3%(1/3)

MM:100%(3/3)

BER:0.05

BER:0.28

BER:0.20

BER:0.13

BER:0.33

BER:0.19

**STE, FE method:Convex-NMF**

**A2, NO(2SS)**

**A2, ME, NO(3SS)**

**A2, GL, NO(3SS)**

**A2, MM,NO(3SS)**

**A2, AG, NO(4SS)**

**A2, AG, MM(4SS)**

Total:92.3%(12/13)

Total:91.3%(21/23)

Total:90.7%(39/43)

Total:87.5(14/16)

Total:90.6%(48/53)

Total:83.0%(44/53)

A2:90.0%(9/10)

A2:90.0%(9/10)

A2:90.0%(9/10)

A2:80.0%(8/10)

A2:90.0%(9/10)

A2:90.0%(9/10)

NO:100%(3/3)

ME:90.0%(9/10)

GL:90.0%(27/30)

MM:100%(3/3)

AG:90.0%(36/40)

AG:80.0%(32/40)

NO:100%(3/3)

NO:100%(3/3)

NO:100%(3/3)

NO:100%(3/3)

MM:100%(3/3)

BER:0.05

BER:0.07

BER:0.07

BER:0.07

BER:0.07

BER:0.10

Classification accuracies (total and by tumour type) and balanced error rates (BER) for the independent test set, using all the classification settings from Tables 4 and 5, for data at LTE and STE

Determining the most adequate number of sources

One of the issues to which attention should be paid is the determination of the most appropriate number of sources for each problem. For this, we investigate the effect of varying the number of extracted sources on the classification results. For illustration, results for only one of the six previously investigated problems, namely A2, AG, MM, are presented. This problem is the most complex of those studied since it encompasses tumour type and grade, as well as extra or intra-axial origin discrimination: low grade neuroepithelial

Figures

Problem A2, AG, MM at LTE, varying the number of sources calculated

**Problem A2, AG, MM at LTE, varying the number of sources calculated**. Sources of the problem A2, AG, MM at LTE in different experiments with varying number of extracted sources. The first 4 rows show the sources corresponding to experiments in which 3, 4, 5 and 6 sources were calculated. Horizontal axis in the first four rows: frequency in ppm scale. The last row shows the percentage of contribution of each source to each tumour type, for each experiment. Horizontal axis in the last row: source signals. Vertical axes labels and representation of the sources as in previous figures.

Problem A2, AG, MM at STE, varying the number of sources calculated

**Problem A2, AG, MM at STE, varying the number of sources calculated**. Sources of the problem A2, AG, MM at STE in different experiments with varying number of extracted sources. The first 4 rows show the sources corresponding to experiments in which 3, 4, 5 and 6 sources were calculated. Horizontal axis in the first four rows: frequency in ppm scale. The last row shows the percentage of contribution of each source to each tumour type, for each experiment. Horizontal axis in the last row: source signals. Axes labels and representation as in previous figures.

Classification results of A2, AG, MM for the training set, varying the number of extracted features.

**PC/SS**

**LTE. PCA**

**LTE. Convex-NMF**

**STE. PCA**

**STE. Convex-NMF**

2

Total:68.4% ± 3.4

Total:62.2% ± 3.6

Total:83.7% ± 2.6

Total:80.6% ± 2.7

A2:75.0% ± 9.8

A2:55.4% ± 11.7

A2:86.8% ± 7.6

A2:78.0% ± 8.9

AG:65.9% ± 4.1

AG:71.6% ± 4.2

AG:82.9% ± 3.3

AG:84.0% ± 3.3

MM:71.0% ± 6.1

MM:45.7% ± 7.0

MM:84.3% ± 4.9

MM:74.3% ± 5.7

3

Total:77.6% ± 3.0

Total:73.0% ± 3.2

Total:81.7% ± 2.8

Total:83.4% ± 2.6

A2: 95.0% ± 4.8

A2:90.2% ± 6.8

A2:85.7% ± 7.6

A2:95.4% ± 4.4

AG: 71.5% ± 4.3

AG:63.6% ± 4.6

AG:81.3% ± 3.5

AG:81.7% ± 3.6

MM: 83.5% ± 4.9

MM:85.4% ± 5.0

MM:80.8% ± 5.3

MM:82.7% ± 4.9

4

Total:80.2% ± 2.9

Total:79.4% ± 3.0

Total:81.3% ± 2.7

Total:87.7% ± 2.3

A2:100% ± 0.0

A2:94.9% ± 5.2

A2:90.7% ± 6.4

A2:95.5% ± 4.6

AG:75.9% ± 4.2

AG:72.5% ± 4.3

AG:80.6% ± 3.5

AG:86.3% ± 3.2

MM:81.6% ± 5.2

MM:87.5% ± 4.3

MM:79.2% ± 5.3

MM:87.7% ± 4.3

5

Total:83.6% ± 2.7

Total:82.2% ± 2.9

Total:81.8% ± 2.7

Total:86.3% ± 2.4

A2:100% ± 0.0

A2:100% ± 0.0

A2:90.8% ± 6.3

A2:90.6% ± 6.4

AG:79.7% ± 3.9

AG:80.0% ± 3.9

AG:80.5% ± 3.6

AG:86.3% ± 3.1

MM:85.5% ± 4.6

MM:80.3% ± 5.4

MM:81.2% ± 5.3

MM:84.6% ± 4.6

6

Total:84.8% ± 2.5

Total:84.9% ± 2.6

Total:92.1% ± 1.9

Total:91.8% ± 1.9

A2:100% ± 0.0

A2:100% ± 0.0

A2:95.3% ± 4.6

A2:95.7% ± 4.1

AG:81.7% ± 3.5

AG:82.8% ± 3.6

AG:92.7% ± 2.3

AG:91.2% ± 2.6

MM:85.4% ± 4.8

MM:83.7% ± 4.9

MM:89.7% ± 4.1

MM:91.4 ± 3.8

7

Total:84.0% ± 2.7

Total:83.2% ± 2.7

Total:92.5% ± 1.9

Total:92.3% ± 1.9

A2:100% ± 0.0

A2:100% ± 0.0

A2:95.6% ± 4.4

A2:91.2% ± 6.1

AG:80.6% ± 3.8

AG:79.0% ± 4.0

AG:92.6% ± 2.3

AG:92.1% ± 2.4

MM:85.2% ± 4.8

MM:85.7% ± 4.6

MM:91.2% ± 3.9

MM:93.0% ± 3.4

8

Total:83.0% ± 2.7

Total:85.3% ± 2.7

Total:93.5% ± 1.7

Total:92.2% ± 1.9

A2:100% ± 0.0

A2:100% ± 0.0

A2:95.3% ± 4.6

A2:95.6% ± 4.5

AG:78.7% ± 3.8

AG:80.7% ± 3.8

AG:93.0% ± 2.2

AG:91.2% ± 2.5

MM:85.4% ± 4.9

MM:89.0% ± 4.3

MM:92.9% ± 3.4

MM:93.2% ± 3.3

9

Total:84.3% ± 2.6

Total:85.3% ± 2.6

Total:93.5% ± 1.7

Total:94.2% ± 1.7

A2:100% ± 0.0

A2:100% ± 0.0

A2:95.3% ± 4.6

A2:95.5% ± 4.6

AG:80.7% ± 3.6

AG:82.6% ± 3.5

AG:93.5% ± 2.2

AG:95.2% ± 1.9

MM:85.5% ± 4.7

MM:85.5% ± 4.8

MM:92.9% ± 3.4

MM:91.4% ± 3.7

10

Total:82.7% ± 2.8

Total:88.4% ± 2.3

Total:92.6% ± 1.9

Total:93.7% ± 1.7

A2:100% ± 0.0

A2:100% ± 0.0

A2:95.5% ± 4.5

A2:95.7% ± 4.5

AG:79.1% ± 3.9

AG:87.0% ± 3.2

AG:91.9% ± 2.5

AG:93.5% ± 2.2

MM:83.6% ± 4.9

MM:87.1% ± 4.5

MM:93.1% ± 3.4

MM:93.1% ± 3.3

Classification results (accuracy ± standard deviation) for the training set, at LTE and STE, obtained when varying the number of extracted features (principal components -PC- and source signals -SS-) from 4 to 10, for the problem A2, AG, MM. Fisher LDA was the classification method, and results were validated through bootstrap. The second and fourth columns show the results for PCA, and the third and fifth columns the results for Convex-NMF

Classification results of A2, AG, MM for the independent test set, varying the number of extracted features.

**PC/SS**

**LTE. PCA**

**LTE. Convex-NMF**

**STE. PCA**

**STE. Convex-NMF**

2

Total:54.7%(29/53)

Total:54.7%(29/53)

Total:73.6%(39/53)

Total:71.7%(38/53)

A2:60.0%(6/10)

A2:60.0%(6/10)

A2:90.0%(9/10)

A2:60.0%(6/10)

AG:52.5% (21/40)

AG:50.0%(20/40)

AG:67.5%(27/40)

AG:72.5% (29/40)

MM:66.7% (2/3)

MM:100% (3/3)

MM:100% (3/3)

MM:100% (3/3)

BER:0.40

BER:0.30

BER:0.14

BER:0.23

3

Total: 60.4%(32/53)

Total:52.8%(28/53)

Total:69.8%(37/53)

Total:75.5%(40/53)

A2:70.0%(7/10)

A2:60.0%(6/10)

A2:80.0%(8/10)

A2:80.0%(8/10)

AG:55.0% (22/40)

AG:50.0%(20/40)

AG:65.0%(26/40)

AG:75.0%(30/40)

MM:100% (3/3)

MM:66.7% (2/3)

MM:100% (3/3)

MM:66.7% (2/3)

BER:0.25

BER:0.41

BER:0.18

BER:0.26

4

Total:67.9%(36/53)

Total:64.2%(34/53)

Total:73.6%(39/53)

Total:83.0%(44/53)

A2:80%(8/10)

A2:60.0%(6/10)

A2:70.0%(7/10)

A2:90.0%(9/10)

AG:62.5%(25/40)

AG:62.5%(25/40)

AG:72.5%(29/40)

AG:80%(32/40)

MM:100%(3/3)

MM:100%(3/3)

MM:100%(3/3)

MM:100%(3/3)

BER:0.19

BER:0.26

BER:0.19

BER:0.10

5

Total:67.9%(36/53)

Total:75.5%(40/53)

Total:73.6%(39/53)

Total:79.2%(42/53)

A2:80%(8/10)

A2:70.0%(7/10)

A2:70.0%(7/10)

A2:80.0%(8/10)

AG:62.5%(25/40)

AG:75.0%(30/40)

AG:72.5%(29/40)

AG:77.5%(31/40)

MM:100%(3/3)

MM:100%(3/3)

MM:100%(3/3)

MM:100%(3/3)

BER:0.19

BER:0.18

BER:0.19

BER:0.14

6

Total:67.9%(36/53)

Total:73.6%(39/53)

Total:79.2%(42/53)

Total:83.0%(44/53)

A2:80%(8/10)

A2:70.0%(7/10)

A2:70.0%(7/10)

A2:90.0%(9/10)

AG:62.5%(25/40)

AG:72.5%(29/40)

AG:82.5%(33/40)

AG:82.5%(33/40)

MM:100%(3/3)

MM:100%(3/3)

MM:66.7%(2/3)

MM:66.7%(2/3)

BER:0.19

BER:0.19

BER:0.27

BER:0.20

7

Total:67.9%(36/53)

Total:73.6%(39/53)

Total:79.2%(42/53)

Total:83.0%(44/53)

A2:80%(8/10)

A2:70.0%(7/10)

A2:70.0%(7/10)

A2:90.0%(9/10)

AG:62.5%(25/40)

AG:72.5%(29/40)

AG:82.5%(33/40)

AG:82.5%(33/40)

MM:100%(3/3)

MM:100%(3/3)

MM:66.7%(2/3)

MM:66.7%(2/3)

BER:0.19

BER:0.19

BER:0.27

BER:0.20

8

Total:75.5%(40/53)

Total:69.8%(37/53)

Total:81.1%(43/53)

Total:84.9%(45/53)

A2:80%(8/10)

A2:70.0%(7/10)

A2:80%(8/10)

A2:90.0%(9/10)

AG:72.5%(29/40)

AG:67.5%(27/40)

AG:80%(32/40)

AG:85%(34/40)

MM:100%(3/3)

MM:100%(3/3)

MM:100%(3/3)

MM:66.7%(2/3)

BER:0.16

BER:0.21

BER:0.13

BER:0.19

9

Total:75.5%(40/53)

Total:71.7%(38/53)

Total:84.9%(45/53)

Total:86.8%(46/53)

A2:80%(8/10)

A2:70.0%(7/10)

A2:90%(9/10)

A2:90%(9/10)

AG:72.5%(29/40)

AG:70.0%(28/40)

AG:82.5%(33/40)

AG:87.5%(35/40)

MM:100%(3/3)

MM:100%(3/3)

MM:100%(3/3)

MM:66.7%(2/3)

BER:0.16

BER:0.20

BER:0.09

BER:0.19

10

Total:73.6%(39/53)

Total:69.8%(37/53)

Total:86.8%(46/53)

Total:84.9%(45/53)

A2:80%(8/10)

A2:70.0%(7/10)

A2:90%(9/10)

A2:90.0%(9/10)

AG:70%(28/40)

AG:67.5%(27/40)

AG:85%(34/40)

AG:85%(34/40)

MM:100%(3/3)

MM:100%(3/3)

MM:100%(3/3)

MM:66.7%(2/3)

BER:0.17

BER:0.21

BER:0.08

BER:0.19

Classification accuracies (total and by tumour type) and BER for the independent test set, at LTE and STE, using the corresponding classification settings from Table 7. The second and fourth columns show the results for PCA, and the third and fifth columns the results for Convex-NMF

Classification results for the problem A2, AG, MM at LTE and STE

**Classification results for the problem A2, AG, MM at LTE and STE**. Plot for the comparison of the classification results for the problem A2, AG, MM at both LTE and STE, when using either PCA or Convex-NMF for DR, previous to classification with Fisher LDA. The left-hand side column corresponds to LTE results, and the right-hand side column to STE results. The first row displays the accuracy of the classification for all the methods, for training and test data sets. The second row displays the balanced error rate (BER) estimates for the test data sets. Horizontal axis: number of principal components or source signals. Vertical axis: accuracy and BER, respectively.

Discussion

NMF as a source extraction method

The results reported in Tables

The illustrative example of Figure

In stark contrast, we can also conclude from Figure

Labelling using convex-NMF

The results reported in Table

• Problem A2

• Problem A2

• Problem A2

• Problem A2

The results for the AG superclass illustrate that Convex-NMF is not always successful in extracting tumour type-specific sources. Two inherent characteristics of AG may explain this: first, AG has been artificially built using two tumour types (ME andGL) and, second, GL by itself is a rather heterogeneous type in which plenty of substructure can be found

This does not preclude the interpretation of the sources. According to the signal profile and its metabolic interpretation, one of the sources representing AG (Figure

Convex-NMF as DR method prior to classification

The comparison of the results of Tables

An independent test set was then used to further validate the robustness of the developed classifiers for data preprocessed with both FE methods: PCA and the orthogonal Convex-NMF sources. Table

Other studies have addressed similar problems in the existing literature, for similar data. We report next some of these results for comparative purposes, although the techniques and the evaluation criteria involved are not always the same.

• In

• In

Determining the most adequate number of sources

Figures

The bar plots for 4 sources, at both times of echo, show the extent to which sources 3 (necrotic tissue) and 4 (proliferative tumour) are representing the AG superclass. At LTE, when calculating 5 sources, the first four look very similar to those calculated in the experiment with 4 sources, while the new one seems to express part of the AG superclass, which is now in fact split into the last three sources. The non-necrotic 4th and 5th sources would show an inverted trend for total choline (ca. 3.21 ppm) versus ML/Lactate (ca. 1.3 ppm). Then, decreasing choline would be matched by increased ML/Lactate, suggesting sampling of aggressive tumour subtypes with variable proliferation rate (total Choline), with concomitant effects on the lactate and ML accumulation. At STE, when calculating 5 sources, the first four also look very similar to those obtained in the experiment with only 4 sources, but the new one is not only part of AG, but also partly of MM.

Six sources at LTE already seem to be too many, given that the contribution of the last one is comparatively very small and completely unspecific. Six sources at STE also seem to be too many. In this case, the MM class is less represented by the second source, while the 5th does contribute both to AG and MM. This could have contributions from class outlier cases (atypical meningiomas), for which mobile lipids could be starting to increase. The last one could contain some artefactual bad water suppression above 3.7 ppm. Up to this point, and based solely on the patterns of the sources, and the percentages of contribution of these to each class, choosing 4 or 5 sources seems to be best option, at both times of echo, to maintain the correspondence between source, or set of sources, and individual tumour types.

Tables

Conclusions

The unsupervised analysis of SV ^{1}H-MRS data from human brain tumours using Convex-NMF has been shown to produce a reduced number of sources that can be confidently recognised as representing brain tumour types or healthy tissue in a way that other source extraction methods, including other NMF variants, cannot. Importantly, this result allows us to produce class assignments for unlabelled spectra in fully unsupervised mode, using the mixing matrix directly as a basis for classification, with results that are comparable to those obtained in fully supervised mode. The use of the sources extracted by Convex-NMF for dimensionality reduction leads to simple LDA-based classifiers with independent test performances that are comparable with, and are often better than previously described strategies. In summary, the unsupervised properties of Convex-NMF place this approach one step ahead of classical label-requiring supervised methods for detection of the increasingly recognised molecular subtype heterogeneity within human brain tumours. The application of Convex-NMF in computer assisted decision support systems is expected to facilitate further improvements in the uptake of MRS-derived information by clinicians.

Authors' contributions

SOM, PJGL and CA conceived the overall scope of the study. MJS participated in the data processing. SOM implemented the methods and carried out the experiments. SOM, PJGL, and AV designed the set of experiments, and analysed the results from the machine learning viewpoint. MJS and CA contributed the biochemical and spectroscopic analysis of the results. PJGL, AV and CA coordinated the work. All authors helped to draft the manuscript and approved its final version.

Endnote

^{a }

^{b }

Acknowledgements

Research funded by Spanish MICINN TIN2009-13895-C02-01 and SAF2008-03323 projects.