SIViP (2013) 7:833–842 DOI 10.1007/s11760-011-0272-2
ORIGINAL PAPER
A unified framework for peak detection and alignment: application to HR-MAS 2D NMR spectroscopy Akram Belghith · Christophe Collet · Lucien Rumbach · Jean-Paul Armspach
Received: 18 March 2011 / Revised: 11 October 2011 / Accepted: 12 October 2011 / Published online: 30 October 2011 © Springer-Verlag London Limited 2011
Abstract In this paper, we propose a new scheme to detect and align simultaneously peaks that correspond to different metabolites within a biopsy. The proposed peak detection and alignment scheme is based on the use of evidence theory, which is well suited to model uncertainty and imprecision characterizing the 2D NMR HR-MAS spectra. Consequently, we propose the coupling use of Bayesian and fuzzy set theories to model and quantify the imprecision degree, which is then exploited to define the mass function. We particularly show that our new mass function definition and the use of evidence theory for peak detection and alignment achieve consistently high performance compared to a Bayesian scheme on both synthetic and real spectra. The high quality of peak alignment precision reached by the use of evidence theory allows us to efficiently detect reliable biomarkers, which is an essential step for a better therapeutic and human complement system management in case of multiple sclerosis disease, cancer, etc. Keywords Evidence theory · Fuzzy membership function · Bayesian inference · HSQC HR-MAS 2D NMR · Peak alignment A. Belghith (B) · C. Collet University of Strasbourg, LSIIT-CNRS UMR-7005, Strasbourg, France e-mail:
[email protected] C. Collet e-mail:
[email protected] L. Rumbach · J.-P. Armspach University of Strasbourg, LINC-CNRS UMR-7237, Strasbourg, France e-mail:
[email protected] J.-P. Armspach e-mail:
[email protected]
1 Introduction In the last decade, cancer has become the leading cause of death for people under the age of 85 [1]. According to the American Cancer Society, a total of 1,479,350 cancer cases and 562,340 deaths from cancer were occurred in the United States in 2009 [1]. Generally, tumors are identified according to histological features characteristic of the assumed cell of origin. Nevertheless, diagnosis is frequently controversial since tumors do not follow classic histology enabling pathological diagnosis to be established. Therefore, an objective diagnostic approach that identifies informative cancer biomarkers is needed to improve the tumor identification accuracy. NMR spectroscopy can be used to provide statistically differentiable molecular biomarkers for tumor identification [2]. Indeed, NMR offers the potential to study molecular structures and their associations and interactions. To remove the spectral line broadening resulting from chemical shift anisotropy in the NMR spectroscopy, the High Resolution Magic Angle Spinning (HR-MAS) NMR was developed [3]. 2D 1 H–13 C Heteronuclear Single Quantum Coherence (HSQC) NMR spectra is widely used in metabolic studies. Indeed, almost metabolites contain carbon and hydrogen, and the addition of a second dimension (13 C or 1 H) improves the resolution and enables the identification of a large number of metabolites that are not resolvable in a standard 1D 1 H or 1D 13 C NMR spectrum. Nevertheless, this spectrum analysis requires new image processing tools able to detect the presence of different metabolites in a 2D new framework. Such tools need to be unsupervised to help medical diagnosis. The analysis of metabolite profiling requires comparison of metabolite profiles obtained from multiple replicates of samples exposed to different experimental conditions. What adds difficulty to automating this analysis process is that each
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peak of a given metabolite can be shifted slightly from one sample to the next. The primary causes of chemical shift in peak positions are variations in the pH and the temperature of the sample [4]. Many peak alignment methods have been used to solve problems of 1D NMR spectroscopy [4] such as the point matching [5] and the Beam search [6]. Nevertheless, due to the complexity and the insufficient prior knowledge on the NMR HR-MAS spectra, these methods are no more effective and to this end, a new method of peak alignment was proposed in [4]. It consists in the extraction of all metabolites and then measures the similarity between peaks from different spectra based on an objective function. However, although this method is successfully applied to 2D NMR, its use is constrained by specific prerequisites (e.g., it requires the identification and the extraction of all peaks for the alignment algorithm, which are not always available in the highly complex spectra such as HSQC spectrum). In this paper, we propose a new method able to simultaneously detect and align different peaks. In this approach, each peak is parameterized by its position and its shape. These characteristics that are theoretically invariable for the same metabolite are in practice corrupted by a variation of the position and shape. In fact, it corresponds to an imprecision, which is added to the spectra in practice. We will model this imprecision and the uncertainty always present on the observed HR-MAS 2D data so as to obtain optimal peak alignment results. Two notions of uncertainty and imprecision are distinct, and they must be now clearly defined [7]. On the one hand, the uncertainty presents the belief or the doubt we have on the existence or on the validity of the data. This uncertainty comes from the reliability or the unreliability of the observation made by the system: This observation can be uncertain or erroneous [8,9]. On the other hand, when we have not enough knowledge on the data, we describe it with vague terms but its realization is sure: In this case, we speak about imprecision. This phenomenon is due to sensor imperfection where data acquisition process is always corrupted and leads to an error associated with every measurements. In order to take into account both imprecision and uncertainty of the spectra, we propose the use of the evidence theory, which can be well suited to deal with raw data. Moreover, this theory offers combination tools to merge data issued from several sources (spectra exposed to different experimental conditions) while taking into account their complementarity, their redundancy and their possible opposition (conflict information). Evidential peak alignment scheme proposed in this paper is also based on the fuzzy set theory [10,11] to model and quantify the imprecision degree presented in the spectra. In particular, we show that this modeling, used in the mass function definition, increases the performance
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of the alignment scheme in comparison with the Bayesian scheme. This paper is organized as follows: In the next section, the analytical model of a 2D-NMR spectroscopy image formation is exposed. In Sect. 3, we model the imprecision degree present in the spectra by defining three main hypotheses required for imprecision quantification. In Sect. 4, we present the peak alignment scheme based on the imprecision degree previously defined. In Sect. 5, we develop the hyperparameter estimation procedure. Finally, in Sect. 6, some results obtained from synthetic and real spectra are presented and we show, in particular, the robustness and the efficiency of this novel proposed approach in comparison with a Bayesian scheme.
2 Spectrum modeling To model HRMAS-2D image formation, we consider a 2D spectrum image characterized by its intensity distribution Y . It corresponds to the observation of a theoretical 2D spectrum image X through a nuclear magnetic resonance system. In our case, X consists of various peaks corresponding to the metabolites present in the biopsy. If the nuclear magnetic resonance system was perfect (linear and shift invariant), the relation between y(i, j) i = 1. . .M, j = 1. . .N and x on the same location should be expressed as a convolution product [12]: y(i, j) =
K1 K2 k1
x(k1 , k2 ) h(i − k1 , j − k2 ) + b(i, j)
k2
(1) where h is the Point Spread Function (PSF) of the nuclear magnetic resonance system and b is additive noise modeling both acquisition noise and degradation of the biopsy tissues. In the observed 2D spectrum image, every peak is identified by its location and its amplitude, which depends on the metabolite concentration and the parameter of the shape filter h. In case of 2D spectra, h is assumed to be a Lorentzian filter [13] whose continuous expression is parameterized by γ = (γ1 , γ2 ): h(x, y; γ ) =
((1/γ1
1/γ1 2 ) +(x −x
0
)2 )
((1/γ2
1/γ2 2 ) +(y
− y0 )2 ) (2)
3 Imprecision quantification In order to obtain an optimal peak alignment result, both imprecision and uncertainty, always present on the data, have
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to be modeled. The evidence theory offers the potential to model this imprecision through conflict modeling management, whereas the Bayesian inference models perfectly the uncertainty (peak presence or absence). Peak alignment algorithm is based on conflict modeling between two 2D spectrum images: reference image Yref and another spectrum image Y . To facilitate the contradiction expression, we try to fit locally a Lorentzian function (position, amplitude, shape) on each detected peak. Therefore, the estimation of Lorentzian parameters on each detected peak must be made. In the next part, we will define three hypotheses HYP1 , HYP2 and HYP3 in order to model and manage conflicts.
3.1 Hypotheses description Hypothesis HYP1 Let us denote H1 the hypothesis corresponding to the absence of a peak at location (i, j) and H2 the hypothesis of the presence of a peak (detection) at the same position. We are interested with the a posteriori probability of the hypotheses Hk , k ∈ {1, 2} of the observation Y . The estimation of these probabilities pi, j (Hk /Y ) at every position (i, j) will be presented in Sect. 5. For a given hypothesis Hk estimated in (i, j) in both images Yref and Y , we will assume that the more the a posteriori probability are close the more the imprecision on the data is small. Let us take the extreme case where pi, j (H2 /Yref ) = 1 and pi, j (H2 /Y ) = 1. The contradiction in this case is absent because the peak appears at the same position in both images. This is based on the assumption that the higher is the conflict, the higher is the imprecision.
Hypothesis HYP2 Let us denote γ 1 (i, j) the shape parameter of a peak at position (i, j) belonging to Yref and γ 2 (i 2 , j2 ) the shape parameter of a peak at position (i 2 , j2 ) belonging to Y . We will assume that the more the parameters of both peaks are close, the more the imprecision on the data is small.
Hypothesis HYP3 We will assume that the more the peaks are far, the more the contradiction is big. Indeed, the peak position variations are limited by a fuzzy neighborhood denoted V around the expected position. Outside of this neighborhood, two peaks cannot be assigned as corresponding.
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3.2 Membership function These hypotheses are defined to quantify the degree of the imprecision in the data, which may be modeled using the fuzzy approach. This is based on the assumption that the concept of the imprecision is an ambiguous concept, i.e, all data are considered as imprecise with a certain degree of membership in this fuzzy set denoted E imprecise (e.g., the imprecise dataset). Fuzzy set is defined as a collection of ordered pairs of elements and its membership degree to the set (from interval [0, 1]). In our case, the degree of membership μi, j denotes how much the pixel with specific a posteriori probability is imprecise, given different hypotheses. The link between certain domain and fuzzy domain is given with an S-membership function f whose expression is given in Eq. 3. Note that the range [a, c] defines the fuzzy region. ⎧ 0 x ≤a ⎪ ⎪ ⎪ ⎨ (x−a)2 a ≤x ≤b (3) f (x) = (b−a)(c−a) 2 (x−c) ⎪ 1 − (c−b)(c−a) b ≤ x ≤ c ⎪ ⎪ ⎩ 1 otherwise where a ≤ b ≤ c. To calculate μi, j for each observed coefficient y(i, j), we will define in the next subsection, a S-membership function associated with each hypothesis. 3.3 Imprecision modeling through member function Let us describe in this part the three S-membership functions associated with three hypotheses: Modeling of HYP1 HYP1 models the contradiction between two a posteriori probabilities for the same hypothesis of peak presence/ absence. The modeling of HYP1 requires the definition of 1 ∈ E imprecise for every pixel a S-membership degree, μi,HYP j of Yref and Y . To measure the distance between two variables, we generally use their ratio; however, this approach leads sometimes to obtain undefined fraction (dividing by zero). To avoid such difficulty, it is better to manipulate the exponential of these two variables before computing their ratio. Therefore, membership degree computing in E imprecise means here calculating the ratio of the exponential of the a posteriori probabilities, then finding its projection by the S-membership function (defined in Eq. 3). This function allows us to quantify, from this ratio of exponential, the membership to the fuzzy set E imprecise . 1 The proposed μi,HYP defined by the exponential ratio of j the smallest and the biggest probabilities of the couple
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Pi, j (H2 /Yref ), Pi, j (H2 /Y ) in order to keep a ratio smaller than one (< 1) is given by:
expmin Pi, j (H2 /Yref ), Pi, j (H2 /Y ) HYP1 μi, j = f hyp1 expmax Pi, j (H2 /Yref ), Pi, j (H2 /Y ) (4)
be presented in Sect. 5 with a Bayesian approach (MCMC optimization). The quantification of μi, j ∈ E imprecise of the imprecision allows us to define the mass functions, a crucial step in the evidence theory. This mass function will be used then to detect and align peaks as we will see in the next section.
In practice, the values of the coefficients a, b and c for f hyp1 are estimated by a supervised learning step using a database of images with known alignment results.
4 The alignment scheme
Modeling of HYP2
4.1 An overview of the evidence theory
HYP2 models the contradiction between shape parameters of two peaks belonging to two spectrum images. The modeling of HYP2 requires the definition of a membership degree 2 ∈ E imprecise using a S-membership function f hyp2 μi,HYP j expressed as:
min γ 1 (i, j), γ 2 (i 2 , j2 ) HYP2 μi, j = f hyp2 (5) max γ 1 (i, j), γ 2 (i 2 , j2 ) where γ 1 (i, j) = γ1 .γ2 , (γ1 , γ2 ) stands for the shape parameters of the peak at (i, j) belonging to Yref , γ 2 (i 2 , j2 ) = γ1,1 .γ1,2 and (γ1,1 , γ1,2 ) stands for the shape parameters of the peak at the position (i 2 , j2 ) belonging to Y .
Dempster Shafer (DS) Theory is a mathematical theory of evidence. In a finite discrete space, DS theory may be read as a generalization of probability theory. Indeed, the probabilities are assigned to sets as against to mutually exclusive singletons. In probability theory, evidence is only related to one hypothesis [14,15]. In DS theory, evidence is related to sets of events. The principal feature of DS theory lies in the fact that the model is designed to cope with varying levels of precision regarding the information. Therefore, DS theory provides tools to represent the uncertainty of data where an imprecise may be characterized by a set or an interval and the resulting output is a set or an interval.
Modeling of HYP3
The mass function
HYP3 models a neighborhood where the possibility to assign two peaks is highly encouraged. The modeling of HYP3 3 requires the definition of a membership degree μi,HYP ∈ j E imprecise using an S-membership function f hyp3 : 3 μi,HYP = f hyp3 ((i − i 2 )2 + ( j − j2 )2 ) j
Let us denote the frame of discernment, which is defined as: = {H1 , H2 , . . ., H N }
(6)
where (i, j) stands for the position of the peak belonging to Yref and (i 2 , j2 ) stands for the position of the peak belonging to Y .
It is composed of N exhaustive and exclusive hypotheses H j , j = 1, . . . , N . From the frame of discernment, let 2 be the power set composed of the 2 N propositions A of :
Hypotheses fusion
2 = {∅, {H1 }, {H2 }, . . ., {H N }, {H1 , H2 }, . . ., }
Our aim now is at calculating the overall membership degree μi, j ∈ E imprecise . To this end, we opted for the average operator as fusion operator: 1 HYP2 HYP3 1 (7) + μ + μ μi, j = . μi,HYP j i, j i, j 3 Note that μi, j ∈ [0 1]. In conclusion, we proposed in this section an imprecision modeling scheme of spectrum images through three intuitive hypotheses, mathematically translated and fused in order to obtain the overall membership degree μi, j . Nevertheless, we need a posteriori probability and peak shape parameters to estimate μi, j : A proposed estimation scheme will
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The DS evidence theory provides a representation of both imprecision and uncertainty through the definition of two functions: plausibility (Pls) and belief (Bel), which are both derived from a mass function (m) where m is defined for every element A of 2 and observation Y , such that the mass value m(A; Y ) belongs to the [0, 1] interval with respect to: ⎧ ⎨m(∅; Y ) = 0
m: m(A; Y ) = 1 ⎩ A⊂2
where ∅ is the empty set.
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The belief and plausibility functions, derived from m, are, respectively, defined from 2 to [0, 1]: Bel(A/Y ) = m(B; Y ) (8) A⊂2 ,B⊆A
Pls(A/Y ) =
m(B; Y )
(9)
A⊂2 ,B∩A=∅
hypotheses are non-zero. (2) The second situation is characterized by the total ignorance (μi, j = 1): All mass functions of the simple hypothesis are null. The expression of the proposed non-normalized mass function m i, j (Y ) for a given observation Y is: m i, j ({H1 }; Y ) = (1 − μi, j ).Pi, j (H1 /Y )
(13)
m i, j ({H2 }; Y ) = (1 − μi, j ).Pi, j (H2 /Y )
(14)
m i, j ({H1 , H2 }; Y ) = μi, j .max(Pi, j (H1 /Y ), Pi, j (H2 /Y ))
DS combination In case of problems taking into account both uncertain and imprecise data, it is often useful to combined the information emanated from several sources in order to get more relevant information. DS theory offers tools to combine the knowledge given by different sources. The orthogonal rule also called Dempster’s rule of combination is the first combination defined within the framework of evidence theory. Let us denote m(Y1 ), . . ., m(Y L ), L masses of belief coming from L distinct sources Yl , l = 1. . .L. The belief function m resulting from the combination of L sources by means of Dempster’s combination rule is defined by: m(A) = m(A; Y1 ) ⊕ m(A; Y2 ) ⊕ · · · ⊕ m(A; Y L )
(10)
(15) When we have two or several sources on the same frame of discernment built by various hypotheses ( = {H1 , H2 , {H1 , H2 }} in our case), we can associate for every image Y a mass function m i, j (Y ), which quantify knowledge brought by the observation. The combination rule of Dempster–Shafer consists in supplying a single mass function from all mass functions m i, j (Y ) associated with each observation Y . The combined mass function m(i, j) is then calculated using the DS combination (Eq. 10). This mass function will be used in the following paragraph to estimate the chemical shift of a detected peak. We hence be able to propose a method realizing simultaneous detection and alinement of peaks.
where ⊕ is defined by: m(A; Y1 )⊕m(A; Y2 ) =
1 1− K
B
4.3 Peak alignment scheme m(B; Y1 ).m(C; Y2 )
C=A
(11) and
K = B
m(B).m(C)
(12)
C=∅
K is often interpreted as a measure of conflict between different sources and is introduced as a normalization factor. The larger is K , the more the sources are conflicting and the less sense has their combination. The K factor indicates the amount of evidential conflict. If K = 0, this shows a complete compatibility, and if 0 < K < 1, it shows partial compatibility. Finally, the orthogonal sum does not exist when K = 1. In this case, the sources are totally contradictory, and it is no longer possible to combine them.
In order to model the peak chemical shifts, we areinterested di for each in recovering a displacement vector di, j = dj peak. As a matter of fact, the peak alignment task returns to estimate the peak chemical shift between peaks belonging to a reference spectrum and their corresponding peaks belonging to other spectra. Adopting an evidence strategy previously defined, we formulate dˆ estimation using belief function (Eq. 8) as: Beli, j (H2 /Yref , (Y + d) argmax (16) dˆi, j = d/(i+di , j+d j )∈V
To maximize this function, we need the a posteriori probabilities as well as the parameters of the shape filters. An analytical solution of this problem is unfortunately impossible, and we decide to use a MCMC procedure to realize such estimation.
4.2 Proposed mass function Determination of the proposed mass function requires the a posteriori probability and the imprecision degree μi, j already defined and with respect to specific properties, in particular the coherence between the Bayesian model and the evidential model. Indeed, two extreme situations appear: (1) The first one is characterized by the total imprecision absence (μi, j = 0); in this case, only the mass functions of the simple
5 MCMC procedure for hyperparameter estimation In this section, we develop a Monte Carlo Markov Chain (MCMC) procedure to estimate the peak locations, amplitudes and shapes required for the proposed peak annotation scheme. The principle of MCMC method is to generate samples drawn from the posterior densities and then to be able to
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achieve hyperparameter estimation using the marginal mean (MPM) estimator: Xˆ = E [X/Y ]
(17)
where X is the variable to estimate from the observation Y . We use a Gibbs sampler based on a stationary ergodic Markov chain allowing to draw samples whose distribution asymptotically follows the a posteriori density p(X, θ/Y ), θ = {θf , θb , θx }) where: – θf stands for the pdf hyperparameters of the Lorentzian shape filter (Eq. 2); – θb stands for the hyperparameters of additive noise; – θx represents the hyperparameters of the theoretical 2D spectrum image X (location, shape, amplitude (Eq. 1)). 5.1 Noise model MCMC method requires the definition of a noise model that can be based in a Bayesian framework as additive, white and Gaussian. However, the hypothesis of a white gaussian noise is not always entirely justified [16]. Thus, we propose here to keep the Gaussian behavior but to take into account the correlation of the additive noise. Indeed, vertical lines appear sometimes in the observed spectrum (raw available HSQC HRMAS-2D observation), and they are due to experimental condition [13]: It led us to introduce a correlated noise modeling this kind of artifact. Then, we adopt a multivariate Gaussian distribution with covariance matrix b . The expression of the noise density is given by Eq. 18. f (b; μb , b ) = ⎛
⎡
N j=1
prior p X/Y, θxvs , θx
⎤T
⎤⎞ b(1, j) ⎢ b(2, j) ⎥⎟ ⎢ ⎥⎟ ⎢ ⎥⎟ . −1 ⎢ ⎥⎟ b ⎢ ⎥⎟ . ⎢ ⎥⎟ ⎣ ⎦⎟ . ⎠ b(M, j)
N
(20)
with " ! θxvs = X [l] , b[l] , γ [l] $ # prior = α [l] , β [l] θx
(21)
$ # where α [l] , β [l] are the hyperparameters of the Gamma distribution modeling the a priori on X . The gamma distribution is an exponential family distribution, which is used for fitting non-negative data [18]. Indeed, the advantage of the gamma distribution lies in the fact that its shape $ # parameters ( α [l] , β [l] ) allow to fit spectral data that may present some sparsity and possibly a background [19]. The gamma density G is expressed as: p(X ; α, β) =
M N
G(x(i, j), α, β) x(i, j) > 0
i=1 j=1 M N
x(i, j)(α−1)
i=1 j=1
βα exp(−βx(i, j)) (α) (22)
⎡
(18)
N (y j − xγ , j , θb )
(19)
j=1
where y j = [y(1, j), y(2, j), . . . , y(M, j)]T , T x j = xγ , j (1), . . . , xγ , j (i), . . . , xγ , j (M) , K1 K2
x(k1 , k2 ) h(i − k1 , j − k2 ) and θb = {b }. xγ , j (i) =
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1. sampling the theoretical 2D spectrum image X [l+1] from
=
The likelihood corresponding to the data-driven term is expressed as:
k1 k2
We present in this paragraph the implementation of the iterative Gibbs algorithm [17] we used for sampling. We introduce prior the variables θx and θxvs that represent the parameter of the prior a priori probability p(X/θx ) and the likelihood probabilvs ity P(Y/ X, θx ), respectively. To sample p(X, γ , θ ), at every iteration l, the main steps consist in:
1 (2π ) M/2 |b |1/2
b(1, j) ⎜ ⎢ b(2, j) ⎥ ⎜ ⎥ ⎜ 1⎢ ⎢ ⎥ . ⎜ ⎢ ⎥ ×exp ⎜− ⎢ ⎥ . ⎜ 2⎢ ⎥ ⎜ ⎣ ⎦ . ⎝ b(M, j)
p(Y/ X, γ , θb ) =
5.2 Simulation scheme
2. sampling the hyperparameters of the Lorentzian shape filter γ [l+1] from prior p γ /Y, θγvs , θγ
(23)
where " ! θγvs = X [l+1] , b[l] , γ [l] # $ prior = σ [l] θγ
(24)
# $ where σ [l] is the variance of the gaussian distribution modeling our a priori on γ p(γ ; σ ) =
c=1,2
√
1 2π σ [l]
exp
% % && )2 − (γc[l] 2σ
(25)
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Fig. 1 Overview diagram of the treatment chain
3. sampling the covariance matrix of the noise b[l+1] from p b /Y, X [l+1] , γ [l+1]
(26)
4. sampling the hyperparameter α [l+1] from p α/Y, X [l+1] , γ [l+1] , β [l]
(27)
5. sampling the hyperparameter β [l+1] from p β/Y, X [l+1] , γ [l+1] , α [l]
(28)
6.1 Peak detection and alignment results on synthetic data The main advantage of using simulated data is that we perfectly know the characteristics of the data such as the number of peaks presented in every spectrum and the peak chemical shift values. In order to validate and emphasize the benefit of the proposed approach, we have defined the accuracy level of estimated parameters. It can be defined as the difference between computed and theoretical values known from a ground truth. In our case, we use the mean chemical shift error function. For each 2D spectrum image, we calculate the carbon mean error chemical shift c and the hydrogen mean error chemical shift h : c = E[ Xˆ c − X rc ] h = E[ Xˆ h − X rh ]
6. sampling the hyperparameter % p
1 /Y, X [l+1] , γ [l+1] σ
σ [l+1] from
& (29)
After lmax iterations, Xˆ and γˆ are given by ⎧ l max ⎪ 1 ⎪ ⎪ X [l] ⎨ Xˆ = lmax −lmin ⎪ ⎪ ⎪ ⎩γˆ =
1 lmax −lmin
l=lmin +1 l max
l=lmin +1
(30) γ [l]
where lmin stands for the number of iterations corresponding to the burn-in time of the Markov chain [20].
6 Experiments This part describes some peak alignment results obtained with the proposed evidential alignment scheme coupled with MCMC method for hyperparameter estimation, and a summary of the method is proposed in Fig. 1
(31)
where ( Xˆ c , Xˆ h ) stands for the estimated coordinates of the the peak X , whereas (X rc , X rh ) stands for the theoretical location of the same peak X . We generate firstly a synthetic theoretical 2D spectrum image X ref with M = 500 and N = 500, which contains Np = 200 peaks. The distance between two closest peaks obtained by the simulation is 4 pixels in x axes (hydrogen chemical shift axes) and 3 pixels in y axes (carbon chemical shift axes). Note that the pixel resolution is 0.16 ppm (resp. 0.0068 ppm) for the 13 C chemical shift axes, (resp. the 1 H chemical shift axes). The positions of different peaks and the hyperparameters of the shape filter γ for each peak are generated randomly. This synthetic spectrum will be used as a reference to register other synthetic spectra. Three other synthetic theoretical 2D spectrum images X 1 , X 2 and X 3 are generated from X ref by modifying the location and the shape hyperparameters of each peak belonged to X ref . The values of chemical shift vectors are assumed random and follow a Gaussian distribution with zero mean and variance matrix 0.02 0 d = . The new shape peak hyperparameter for each 0 0.25 peak is calculated by adding randomly a white gaussian
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Table 1 The mean chemical shift errors c H expressed in ppm obtained by the proposed and the Bayesian methods
Table 2 Estimated Kappa coefficient associated with the coherence of experienced investigator opinions
PSNR (dB)
Proposed scheme c H
Technique
M1
M2
M3
0.833
0.872
0.923
0.0051
9.1 10−5
Dataset1
30
0.097
0.0061
0.791
0.818
0.897
0.0121
5.1 10−4
Dataset2
28
0.139
0.0086
25
0.1098
0.0025
0.2584
0.0191
23
0.1874
0.00935
0.3278
0.02031
Bayesian scheme c H
error of variance 0.005. A correlated noise b with covariance matrix b was added to each spectrum to obtain the synthetic spectra used in the simulation Yref , Y1 , Y2 and Y3 . The Peak Signal to Noise Ration PSNR of Yref is 30 dB (PSNR = 10 log10 (max(X ref )2 /E[(b)2 ])). This value of PSNR was chosen to fit at best the real spectra. The PSNR of the real spectrum is ∼ 30 dB. In order to emphasize the benefit of the proposed approach, two different methods were applied to the synthetic spectrum Y3 with different values of PSNR: a Bayesian method [21] and our alignment method. The peak alignment results are presented in Table 1. We can easily observe that the proposed method performed better than the Bayesian method. Indeed, even with a PSNR = 25 dB, we obtained a subpixel precision for the mean chemical shift errors, which is 2 times smaller compared to that obtained by the Bayesian method. This can be explained by the fact that we took into account in our alignment scheme both the uncertainty (the a posteriori probability) and the imprecision in the spectra (conflict information). It is important to note that the Bayesian scheme provides only tools to handle the uncertainty, and for this reason, the use of evidence theory was proposed. 6.2 Peak detection and alignment results on real data We now deal with the problem of peak detection and alignment validation on real HR-MAS 2D spectra. Our database contains two datasets. The first one (dataset1) is composed of twenty-five 2D spectra from healthy colorectal tissue biopsies and twenty-five other spectra from cancerous colorectal tissue biopsies. The second one (dataset2) is composed of thirty 2D spectra of multiple sclerosis pathology rat tissue and thirty other spectra from healthy rat tissue biopsies. Note that we have used the experimental autoimmune encephalomyelitis (EAE) as a model of multiple sclerosis pathology. Before applying the detection and alignment algorithm, we calibrate our whole data with a peak reference (lactate in our case) which location is known (22.7 ppm for carbon 13 C axis and 1.33 ppm for proton 1 H axis).The reference spectrum was chosen randomly from the healthy tissue spectra. We emphasize that no a priori on peak positions and number was considered.
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In order to emphasize the benefit of the proposed approach, three different methods were applied to our spectrum database : the Bayesian method [21] (M1), the method presented in [4] (M2) and our evidential method (M3). The peak detection and alignment results we obtained were manually examined by three experienced investigators, and the coherence of their opinions have undergone to Cohen’s K-test. The results of validation are summarized in Table 2. As one can see, our method performed the best. This means that our method is a robust unsupervised alternative to manual peak detection and alignment task requiring high expert time. Figure 2a displays two false peak assignments obtained with the method presented in [4]: The real location is presented by a continuous arrow, and the estimated position is presented by dotted arrow. Note that these peaks were correctly aligned by the proposed scheme. As one can see, when the imprecision is important, the method proposed in [4] performs false peak assignments. Figure 2b shows an example of missed peak obtained with our method. As one would suspect, the missed detections correspond to weak events, which are strongly noised. Indeed, the average amplitude of missed 1 maximum noise amplitude. peaks is ∼ 15 Figure 3 displays the peak detection and alignment results on the same zone of a healthy spectrum and cancerous spectrum. As we can see, some peaks like peak number 10 are visually very difficult to be detected because of the high noise level; yet, our method is actually able to detect and align it. Note that the spectra are presented as contour plots with the same number of levels, which explain the display absence of the peak number 10 in Fig. 3b.
7 Discussion We described an automated method for peak detection and alignment to 2D HSQC NMR spectrum application. The high complexity of this type of spectra brought by the noise and the artifacts make this task more delicate and difficult [13]. In addition, the precision in the alignment procedure is necessary since the interpretation of the results by the NMR experts or physicians depends on it. In order to increase the precision of the alignment result, we proposed a new vision of the registration problem. This vision is based on the assumption that any modification of the peak characteristics (peak chemical shift and peak shape) is an imprecision, which is added to
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Fig. 2 a False peak assignment obtained with the method proposed in [4]: real location in continuous arrow and estimated position in dotted arrow. These peaks were correctly aligned by the proposed scheme. b Missed peak obtained with the proposed method which correspond to weak events that are strongly noised. Note that the method proposed in [4] only allows the alignment of peaks, which are assumed to be detected in advance
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Fig. 3 Detection and alignment results obtained with the proposed scheme on (a) a healthy spectrum and on (b) a cancerous spectrum. Both spectra are presented as contour plots with the same number of levels. All peaks were correctly detected and aligned by the proposed scheme
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the spectra. In order to model and handle both imprecision and uncertainty, the use of the evidence theory was proposed, which present a generalization of the classic Bayesian theory [7]. The bottleneck for any method based on evidence theory is the definition of the mass function. For this, we have proposed a new approach to model the mass function based on the fuzzy quantification of the imprecision degree. Although the techniques used are classical in the field of fuzzy modeling and Bayesian estimation (MCMC), the main novel contributions of this paper are as follows:
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8 Conclusion
1. the fuzzy imprecision quantification through the definition of several trivial assumptions; 2. the mass function construction method; 3. the basic idea (which consists in the use of evidence theory to handle synergistically both the imprecision and the uncertainty).
In this paper, we proposed a new evidential peak detection and alignment scheme. This method combines the modeling of the knowledge through the evidence theory and the quantification of the imprecision degree through the fuzzy theory. The handling of both imprecision and uncertainty by the evidence theory increased the robustness of the proposed alignment scheme compared to the Bayesian scheme. In addition, we have used the deconvolution model to achieve a better fit of the HSQC spectrum and the multivariate Gaussian distribution to model the noise correlation. The synthetic validation of the proposed approach has shown its features such as its robustness to the high level of noise, one of the delicate issues in HSQC spectra and its ability to align peaks even if they are inseparable manually. This method was validated on real spectra with the collaboration of NMR experts.
The validation experiments on both synthetic and real spectra showed that the evidence theory provides more accurate peak detection and alignment than the Bayesian scheme.
Acknowledgments The authors would like to thank the Region Alsace and ARSEP for support of this research project and Mr. Karim Elbayed and Mr. François-Marie MOUSSALLIEH from Institut de Chimie, University of strasbourg and Dr Izzie Jacque Namer from
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842 Department of Biophysics and Nuclear Medicine, University Hospitals of Strasbourg, LINC- UMR CNRS 7237, for HSQ,C biopsy spectra and for result validation.
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