Member, IEEE Elhoussin Elbouchikhi

Member, IEEE, Franc ¸ois Auger Vincent Choqueuse

Senior Member, IEEE Mohamed Benbouzid

ON INSTRUMENTATION AND MEASUREMENT Motor Current Signal Analysis based on a Matched Subspace Detector

Keywords: Induction motor, Diagnosis, Fault detection, Eccentricity fault, Broken rotor bars, Bearing faults, Subspace Matched Detector, Oblique projection, MCSA

Motor current signature analysis (MCSA) is a well-proven technique for electrical and mechanical faults detection in induction motors. The appearance of stator current components or increase of magnitude of some components at characteristic frequencies indicates a motor fault condition. In this paper, we propose a new MCSA fault detector based on a matched subspace technique. The proposed detector consists of three steps. First, the influence of the fundamental supply frequency is removed from the current signal using an interference cancellation technique based on oblique projection. Then, the fault-related frequency is estimated from the interference-free signal using the maximum likelihood (ML) principle. Finally, the fault detection is performed using a generalized likelihood ratio test. Simulation and experimental results illustrate the effectiveness of the proposed approach for eccentricity fault, bearing faults and broken rotor bars detection.

I. INTRODUCTION

I NDUCTION motors are one of the most commonly used motors in industrial applications. Despite their efficiency and robustness, induction motors can breakdown due to several faults such as bearings, stator, rotor, or eccentricity faults [START_REF] Tavner | Condition monitoring of rotating electrical machines[END_REF]. These unexpected breakdowns are very expensive in terms of time and money. In this context, reducing operation and maintenance costs is of main concern in such industrial applications. Consequently, preventive maintenance is preferred over the corrective maintenance. In fact, the preventive maintenance is intended to reduce the probability of failure, minimize downtime and maximize productivity. It includes the scheduled and condition-based maintenance [START_REF] Hashemian | State-of-the-art predictive maintenance techniques[END_REF]. Unlike the scheduled maintenance, the condition-based one monitors the current state of the induction machine and schedules the maintenance actions at the optimum time. Several techniques have been proposed for induction machines condition monitoring such as vibration, torque, temperature, current/power monitoring or Elhoussin Elbouchikhi is with the ISEN Brest, FRE CNRS 3744 IRDL -Institut de Recherche Dupuy de L ôme, 20, Rue Cuirass é Bretagne, 29200 Brest, France (email: elbouchikhi@isen-bretagne.fr. Vincent Choqueuse and Mohamed Benbouzid are with the University of Brest, FRE CNRS 3744 IRDL -Institut de Recherche Dupuy de L ôme, Rue de Kergoat, CS 93837, 29238 Brest Cedex 03, France (email: Youness.Trachi@univ-brest.fr, Vincent.Choqueuse@univ-brest.fr, Mohamed.Benbouzid@univ-brest.fr). Mohamed Benbouzid is also with the Shanghai Maritime University, 201306 Shanghai, China.

Franc ¸ois Auger is the University of Nantes/IUT of Saint-Nazaire, EA 4642 IREENA , 58 Rue Michel Ange, 44606 Saint-Nazaire, France.

oil/debris analysis [START_REF] Zhang | A survey of condition monitoring and protection methods for medium-voltage induction motors[END_REF]. Among them, motor current signature analysis (MCSA) is one of the most promising solutions since this technique is non-invasive and avoids the use of extra sensors [START_REF] De | FPGA-based broken bars detection on induction motors under different load using motor current signature analysis and mathematical morphology[END_REF].

In this context, MCSA has been widely investigated for induction machine condition monitoring as the stator currents constitute a reliable information source. In fact, an induction machine is a highly symmetrical electromagnetic system. Consequently, any fault may cause some degree of asymmetry. This asymmetry induces a disturbance of the rotating electromagnetic field in the air gap, which straightforwardly affects the stator currents. Several works have presented numerical machine models under faults based on finite element methods or analytical methods based on modified winding function approach (MWFA). Moreover, the consequences of mechanical and electrical faults have been also theoretically investigated using the magnetomotive force (MMF) and permeance wave approach for airgap magnetic flux density computation [START_REF] Blodt | Distinguishing load torque oscillations and eccentricity faults in induction motors using stator current Wigner distributions[END_REF]. These contributions allow to understand the effect of some phenomena, but do not provide the exact sub-harmonics amplitude introduced by the fault. The faults effects on the stator currents can be grouped into three major contributions, which are: the impact on the stator current power spectral density (PSD) [START_REF] Bellini | Advances in diagnostic techniques for induction machines[END_REF], the amplitude and/or phase modulation of the stator currents [START_REF] Batista | An empirical demodulation for electrical fault detection in induction motors[END_REF], [START_REF] Hu | A novel transform demodulation algorithm for motor incipient fault detection[END_REF], and the impact over the negative-sequence component [START_REF] Bouzid | New expressions of symmetrical components of the induction motor under stator faults[END_REF].

Induction machine faults diagnosis based on MCSA consists of faults characteristics extraction based on signal processing techniques followed by a fault detection procedure. Several approaches have been presented for stator currents processing and reliable fault detection [START_REF] Blodt | Distinguishing load torque oscillations and eccentricity faults in induction motors using stator current Wigner distributions[END_REF], [START_REF] Bellini | Advances in diagnostic techniques for induction machines[END_REF]. Nevertheless, the detection step is often performed manually, based on visual inspection of the fault signature on the stator currents. To automatically detect a fault, several authors have proposed algorithms based on threshold detectors [START_REF] Trachi | Induction machines fault detection based on subspace spectral estimation[END_REF], [START_REF] Bouleux | Oblique projection pre-processing and TLS application for diagnosing rotor bar defects by improving power spectrum estimation[END_REF]. However, the implementation of these techniques requires the user to set a threshold manually, based on the knowledge of the induction machine to be supervised. To avoid the need of manual setting, several pattern recognition techniques have been investigated for decision making. These techniques include Artificial Neural Networks [START_REF] Sadeghian | Online detection of broken rotor bars in induction motors by wavelet packet decomposition and artificial neural networks[END_REF], Support Vector Machines [START_REF] Widodo | Support vector machine in machine condition monitoring and fault diagnosis[END_REF], Fuzzy Logic [START_REF] Rodríguez | Detection of stator winding fault in induction motor using fuzzy logic[END_REF], and combined techniques. Nevertheless, these techniques are black-box methods whose parameters are difficult to tune in practice. Moreover, the detection performance critically depends of the learning database. In fact, the training phase is critical for optimal operation and may be misleading or pro-duce results limited to a particular set of faults. Moreover, the learning database must be sufficiently large depending on the studied faults and the induction machine operating conditions. However, a large learning database may lead to over-fitting problems thus limiting the generalisation capabilities of the detector [START_REF] Bishop | Pattern recognition[END_REF]. Induction motor stator currents contain several spectral components due to the supply voltage, rotor slotting and iron saturation [START_REF] Faiz | Effect of magnetic saturation on static and mixed eccentricity fault diagnosis in induction motor[END_REF]. In case of faults, additional frequency components also appear in the current spectrum. To reveal these additional components, several spectral analysis techniques can be used such as the FFT [START_REF] Yang | Feature knowledge based fault detection of induction motors through the analysis of stator current data[END_REF], MUSIC [START_REF] Naha | A method for detecting half-broken rotor bar in lightly loaded induction motors using current[END_REF], [START_REF] Garcia-Perez | The application of high-resolution spectral analysis for identifying multiple combined faults in induction motors[END_REF], and ESPRIT algorithms [START_REF] Trachi | Induction machines fault detection based on subspace spectral estimation[END_REF], [START_REF] Kim | High-resolution parameter estimation method to identify broken rotor bar faults in induction motors[END_REF]. However, these general spectral estimators do not exploit the particular spectrum structure induced by the fault. Moreover, these approaches only focus on the current spectral analysis and do not provide any algorithm for automatic fault detection. In [START_REF] Trachi | Induction machines fault detection based on subspace spectral estimation[END_REF], [START_REF] Elbouchikhi | Induction machine faults detection using stator current parametric spectral estimation[END_REF], the authors have proposed a Maximum Likelihood (ML) parametric estimator for current spectral analysis, and a simple ad-hoc fault criterion. In [START_REF] Bouleux | Oblique projection pre-processing and TLS application for diagnosing rotor bar defects by improving power spectrum estimation[END_REF], the same fault criterion is used in conjunction with an interference cancellation technique for the removal of the supply component at 50Hz. However, in these two studies, no strategy has been provided to automatically detect a fault from the estimated spectral components.

As compared to the techniques in [START_REF] Bouleux | Oblique projection pre-processing and TLS application for diagnosing rotor bar defects by improving power spectrum estimation[END_REF], [START_REF] Elbouchikhi | Induction machine faults detection using stator current parametric spectral estimation[END_REF], this paper directly focuses on the MCSA fault detection problem. Specifically, we propose a new matched subspace detector that treats the supply frequency components as an interference signal. To remove the supply frequency component, the authors in [START_REF] Naha | A method for detecting half-broken rotor bar in lightly loaded induction motors using current[END_REF] have proposed a Kalman-based technique. However, this technique is suboptimal from a statistical sense when the fundamental frequency is quasi-stationary (i.e. when the frequency and the amplitude vary smoothly with time). In this study, the interference signal is removed from the raw data using an oblique projection. Then, the fault detection problem is formulated as a binary hypothesis test and solved using the Generalized Likelihood Ratio Test (GLRT). In [START_REF] Trachi | A novel induction machine faults detector based on hypothesis testing[END_REF], a general GLRT-based approach has been investigated for induction motors fault detection using the raw data. This approach is time consuming and difficult to implement since it requires many matrix inversions. In this paper, the proposed detector allows reducing the computational cost and can be easily implemented using FFT algorithms, which are almost available on DSP-boards. In our case, the GLRT is referred to as a matched subspace detector [START_REF] Scharf | Matched subspace detectors[END_REF].

The remaining parts of this paper are organized as follows. Section II presents the signal model under faulty conditions. Section III describes the proposed approach for interferencefree signal estimation and the MLE based estimate for fault related frequency. Section IV introduces the proposed fault detection and diagnosis technique, which is performed using a matched subspace detector. Section V deals with the statistical performance evaluation of the proposed approach. Section VI gives experimental results for eccentricity fault, bearing faults, and broken rotor bars. Section VII concludes this paper.

II. STATOR CURRENT SIGNAL MODEL

Three induction machine faults are considered in this study: eccentricity fault, bearing faults, and broken rotor bars.

A. Fault Related Frequencies 1) Eccentricity fault effect over stator currents: When eccentricity occurs in a rotating machine, the effective airgap function varies in a sinusoidal manner with respect to angular position θ s and time in the stator reference frame [START_REF] Blodt | Distinguishing load torque oscillations and eccentricity faults in induction motors using stator current Wigner distributions[END_REF], [START_REF] Joksimovic | Dynamic simulation of dynamic eccentricity in induction machines-winding function approach[END_REF]. In the literature, two types of eccentricity are commonly considered, which are static and dynamic eccentricities. They can jointly occur leading to the so-called a mixed eccentricity.

Airgap variations have an effect over the airgap permanence function and consequently on the motor inductances. Eccentricity fault effect has been investigated to model the fault impact on stator current [START_REF] Joksimovic | Dynamic simulation of dynamic eccentricity in induction machines-winding function approach[END_REF], [START_REF] Faiz | Finite-element transient analysis of induction motors under mixed eccentricity fault[END_REF]. It has been proved, that under eccentricity faults, the stator currents contain the frequencies given by [START_REF] Tavner | Condition monitoring of rotating electrical machines[END_REF].

ω k = ω s 1 ± k 1 -s p ( 1
)
where p is the pole pairs number. Moreover, eccentricity fault leads to an increase of oscillating torque components at ω r =

1-s p ω s [5]
2) Bearing faults effects on stator currents: Bearing support the rotor of an induction machine. Consequently, any bearing fault can induce two different effects, which are the introduction of a particular radial rotor movement and some load torque variations [START_REF] Immovilli | Bearing fault model for induction motor with externally induced vibration[END_REF], [START_REF] Blodt | Models for bearing damage detection in induction motors using stator current monitoring[END_REF]. Bearing single-point defects can be monitored by supervising some frequency components around the supply frequency, which are given by

ω k = (ω s + kω f ), (2)
where ω f is one of the characteristic vibration angular frequencies given in [START_REF] Elbouchikhi | Induction machine faults detection using stator current parametric spectral estimation[END_REF]. Amplitude and frequency of such frequency components depend on shaft rotational speed, fault location, and bearing dimensions.

3) Broken rotor bars effects on stator currents: Broken rotor bar induces a bar resistance increase, which leads to asymmetry of the resistance in rotor equivalent phases. Consequently, broken rotor bars fault induces asymmetry of the rotating electromagnetic field in the air gap. Since stator currents are linked to the airgap electromagnetic field, any broken rotor bar may have an effect over the stator current waveform [START_REF] Shi | A new diagnosis of broken rotor bar fault extent in three phase squirrel cage induction motor[END_REF]. This effect is modeled by adding some frequency components on the stator currents PSD [START_REF] Gu | A new method of accurate broken rotor bar diagnosis based on modulation signal bispectrum of motor current signals[END_REF], [START_REF] Saidi | Diagnosis of broken-bars fault in induction machines using higher order spectral analysis[END_REF], which are located at

ω k = (1 + 2ks)ω s , (3)
where is s is the per unit slip, ω s is the normalized electrical supply fundamental angular frequency and k ∈ Z * .

B. Stator Current Model Under Faulty Condition

For the previously discussed three faults , the fault signature can be modeled as additional (normalized) angular frequencies that can be expressed by

ω k = ω s + kω d , (4)
where ω d is the normalized fault-related angular frequency, and

-k 1 ≤ k ≤ k 2 (k ∈ Z * ).
For eccentricity fault, bearing faults and broken rotor bars, the normalized faultrelated angular frequency are ω d = 1-s p ω s , ω d = ω f and ω d = 2sω s , respectively. In this study, we assume that the supply frequency is equal to its nominal value i.e. 50Hz (or 60Hz). In this context, the normalized electrical supply angular frequency is equal to ω d = 100π/F s rad/sample, where F s corresponds to the sampling rate.

By separating the supply and fault related components, the stator current can be decomposed as

x[n] = s 1 [n] + s 2 [n] + b[n], (5)
where:

• s 1 [n] = k2 k=-k1 k =0 a k cos ((ω s + kω d ) n + φ k ) corre- sponds to fault spectral components, • s 2 [n] = a 0 cos (ω s n + φ 0 ) corresponds to the supply fundamental component, • a k ∈ R + and
, i.e. b[n] ∼ N (0, σ 2
). This choice is motivated by the central limit theorem [START_REF] Rosenblatt | A central limit theorem and a strong mixing condition[END_REF], and by the fact that the Gaussian noise assumption leads to largest Cramér-Rao bound [START_REF] Stoica | The gaussian data assumption leads to the largest cramér-rao bound [lecture notes[END_REF]. In practice, the signal is observed during a time slice of N ≥ L samples, where L = 2(k 1 + k 2 + 1). By noting x [x[0], . . . , x[N -1]] T the observed current signal, the signal model in ( 5) can be written under a matrix form as follows

x = G 1 (ω d )θ 1 + G 2 θ 2 + b (6)
where:

• G 1 (ω d )θ 1 is the additive term that models the influence of the fault-related components. More precisely,

G 1 (ω d ) ∈ R N ×(L-2) is a matrix given by G 1 (ω d ) H(ω s -k 1 ω d ) • • • H(ω s + k 2 ω d ) (7)
where

H(ω)        1 0 cos(ω) -sin(ω) . . . . . . cos(ω(N -1)) -sin(ω(N -1))        (8)
and θ 1 ∈ R (L-2)×1 is a column vector defined as

θ 1          a -k1 cos(φ -k1 ) a -k1 sin(φ -k1 ) . . . a k2 cos(φ k2 ) a k2 sin(φ k2 )          . ( 9
)
It should be emphasized that the matrix H(ω s ) and the vector θ 2 = a 0 cos(φ 0 ) a 0 sin(φ 0 ) T are excluded from ( 7) and ( 9), respectively. • G 2 θ 2 is an interference term that models the influence of the supply component. Specifically, G 2 ∈ R N ×2 is a matrix given by G 2 H(ω s ) and θ 2 ∈ R 2×1 is a column vector, which is defined by

θ 2 a 0 cos(φ 0 ) a 0 sin(φ 0 ) T . ( 10
) • b [b[0], . . . , b[N -1]]
T contains the additive white noise. The goal of this paper is to detect a fault, i.e. θ 1 = 0, from the stator current. It should be mentioned that this paper assumes that the values of k 1 and k 2 are known. If this condition is not satisfied, these values can be estimated using a model-order selection technique [START_REF] Stoica | Model-order selection: a review of information criterion rules[END_REF].

III. ESTIMATION OF ω d Let us denote G(ω d ) [G 1 (ω d ) G 2 ] the N × L matrix obtained by concatenating the matrices G 1 (ω d ) and G 2 and θ [θ T 1 , θ T 2 ]
T the L × 1 vector obtained by concatenating the vectors θ 1 and θ 2 . Under the assumption that b ∼ N (0, σ 2 I), the ML estimator of ω d is given by the minimizing argument of the Least Squares function x -G(ω d )θ 2 with respect to ω d and θ [START_REF] Kay | Fundamentals of Statistical Signal Processing: Estimation Theory[END_REF]. In this section, we derive the exact and approximate estimators of ω d . Furthermore, we show that these two estimators require the estimation of the interference-free signal.

A. Exact Frequency Estimator

The minimizing argument of the Least Squares function x -G(ω d )θ 2 with respect to ω d is given by [START_REF] Golub | The differentiation of pseudo-inverses and nonlinear least squares problems whose variables separate[END_REF]

ω d = arg min ω d x T P ⊥ G (ω d )x (11)
where

P ⊥ G (ω d ) I N -G(ω d ) G T (ω d )G(ω d ) -1 G T (ω d )
is the orthogonal projector into the kernel of G T (ω d ). Due to the particular structure of G(ω d ), the estimator of ω d can also be simplified as [START_REF] Scharf | Matched subspace detectors[END_REF]Eq 3.6]

ω d = arg min ω d y T (I N -E G1G2 (ω d ))y (12)
where y P ⊥ G2 x is the interference-free signal obtained by removing the fundamental angular frequency component, P ⊥ G2 is the orthogonal projector into the kernel of G T 2 , and E G1G2 (ω d ) corresponds to the oblique projector onto the range of G 1 (ω d ) along G 2 and is defined as:

E G1G2 (ω d ) G 1 (ω d ) G T 1 (ω d )P ⊥ G2 G 1 (ω d ) -1 G T 1 (ω d )P ⊥ G2 (13) As G 2 is a N × 2 matrix,
it is interesting to note that the expression of the interference-free signal y = P ⊥ G2 x can be derived analytically. Indeed, the interference-free signal expression can be written as

y = P ⊥ G2 x = x -G 2 θ 2 ( 14
)
where

θ 2 = (G 2 T G 2 ) -1 G 2 T
x corresponds to the estimator of θ 2 under the assumption of a healthy electrical motor. Performing some computations leads to the following result

θ 2 = e( c) m( c) (15)
where c is defined as

c = 2 N 2 -|q(ω s )| 2 (N X(ω s ) -q * (ω s )X * (ω s )) (16)
where (.) * corresponds to the complex conjugate, and

q(ω s ) = sin(N ω s ) sin(ω s ) e jωs(N -1) (17a) X(ω s ) N -1 n=0 x[n]e -jωsn (17b)
After some computations, it can be checked that the ele-

ments of y = [y[0], • • • , y[N -1]] T are given by y[n] = x[n] -| c| cos (ω s n + arg[ c]) (18)
where | c| and arg[ c] correspond to the modulus and argument of c, which is the estimate of c = a 0 e jφ0 . Note that the complex scalar X(ω s ) simply corresponds to the Discrete-Time Fourier Transform (DTFT) of x[n] evaluated at the normalized angular frequency ω s . Concerning the signal length, it should be mentioned that the particular values N = kπ/ω s (k ∈ N) are attractive from a computational point of view. Indeed, for these particular values, the complex scalar c reduces to c = 2X(ω s )/N

B. Approximate Frequency Estimator

The evaluation of the cost function in [START_REF] Sadeghian | Online detection of broken rotor bars in induction motors by wavelet packet decomposition and artificial neural networks[END_REF] requires the inversion of a (L -2) × (L -2) matrix. For N 1, it can be shown that this cost function has a simple closed-form expression. Indeed, for N 1, the matrix products G T 1 (ω d )G 1 (ω d ) and G T 2 G 2 can be approximated by (N/2)I L-2 and (N/2)I 2 , respectively. It follows that P G2 ≈ I N , and so the oblique projector can be simplified as

E G1G2 (ω d ) ≈ (2/N )G 1 (ω d )G T 1 (ω d )
, which does not depend on G 2 . Consequently, the minimizing argument of ( 12) is asymptotically equal to

ω d = arg max ω d k2 k=-k1 k =0 P y (ω s + kω d ) (19)
where P y (ω) is the periodogram of y[n], which is defined as

P y (ω) 1 N N -1 n=0 y[n]e -jωn 2 . ( 20
)
As compared to the exact cost function, the approximate one does not require any matrix inversion and explicitly depends on the periodogram of y These sidelobes critically affect the values of P x (ω s + kω d ) for k = 0. Using the interference-free signal instead of the original one allows to overcome this issue.

IV. MATCHED SUBSPACE FAULT DETECTOR

Let us consider the following hypothesis testing problem: the induction motor is healthy (H 0 ) or a fault is present (H 1 ). Mathematically, this hypothesis testing problem can be formulated as

H 0 : θ 1 = 0, H 1 : θ 1 = 0. ( 21
)
This problem can be solved by a Generalized Likelihood Ratio Test (GLRT). For the particular model in [START_REF] Bellini | Advances in diagnostic techniques for induction machines[END_REF] with ω d known, the GLRT reduces to a matched subspace detector [START_REF] Scharf | Matched subspace detectors[END_REF], [START_REF] Behrens | Signal processing applications of oblique projection operators[END_REF].

A. Exact Fault Detector

Let us consider the interference-free signal y computed by [START_REF] Naha | A method for detecting half-broken rotor bar in lightly loaded induction motors using current[END_REF]. When the noise variance is unknown, the matched subspace detector decides

H 1 if [23] T (y) = N -L L -2 y T E G1G2 (ω d )y y T (I N -E G1G2 (ω d ))y > γ ( 22
)
where γ is the test threshold. Note that when the fault-related (normalized) angular frequency ω d is unknown, it can be replaced in [START_REF] Trachi | A novel induction machine faults detector based on hypothesis testing[END_REF] by its Maximum Likelihood estimator ω d given in [START_REF] Sadeghian | Online detection of broken rotor bars in induction motors by wavelet packet decomposition and artificial neural networks[END_REF].

Regarding the detector performance, the probability of false alarm is given by

P F A = Q F L-2,N -L (γ), (23)
where Q F L-2,N -L (.) is the complementary cumulative distribution function of a F distribution with L -2 numerator degrees of freedom and N -L denominator degrees of freedom. Furthermore, the probability of detection is given by

P D = Q F L-2,N -2 (λ) (γ), (24)
where Q F L-2,N -L (λ) (.) corresponds to the complementary cumulative distribution function of a non-central F distribution with L -2 numerator degrees of freedom, N -L denominator degrees of freedom, and a noncentrality parameter λ equal to

λ = θ T 1 G T 1 (ω d )P ⊥ G2 G 1 (ω d )θ 1 σ 2 . ( 25
)
Note that the probability of detection P D depends on several parameters such as the noise variance σ 2 , the value of θ 1 , the sample length N , and the detection threshold, γ.

In practice, the detection threshold γ can be set according to a desired probability of false alarm, P F A , using [START_REF] Scharf | Matched subspace detectors[END_REF]. Since P F A does not depend on σ 2 , this test leads to a constant false alarm rate (CFAR) detector.

Parameter k = -2 k = -1 k = 0 k = 1 k = 2 a k 0.

B. Approximate Fault Detector

The implementation of the exact fault detector requires a matrix inversion. For N 1, it can be shown that the statistical test T (y) has a simple closed-form that only depends on the periodogram of y[n].

As provided in subsection III-B, the oblique projector

E G1G2 (ω d ) can be approximated by (2/N )G 1 G T 1 when N 1.
It follows that the matched subspace detector reduces to

T a (y) = N -L L -2 k2 k=-k1 k =0 P y (ω s + kω d ) 1 2 y 2 - k2 k=-k1 k =0 P y (ω s + kω d ) > γ ( 26
)
where P y (ω) is defined in [START_REF] Kim | High-resolution parameter estimation method to identify broken rotor bar faults in induction motors[END_REF] and corresponds to the periodogram of y[n] evaluated at ω.

C. Fault Detection Methodology

The proposed induction machine fault detection methodology is based on four steps, which are summarized by Fig. 1. The proposed approach performance is tested on synthesized signals. Afterwards, its effectiveness is confirmed on experimental data for eccentricity fault, bearing faults, and broken rotor bars.

V. SIMULATION RESULTS

In this section, we evaluate the performance of the proposed techniques with synthetic signals. These signals are generated according to the signal model in [START_REF] Blodt | Distinguishing load torque oscillations and eccentricity faults in induction motors using stator current Wigner distributions[END_REF] with the parameters provided in Table I. The (normalized) angular frequencies ω s and ω d are respectively set to 2π × 50/F s rad/sample and 2π × 10.2/F s rad/sample, with F s = 2400 Hz.

A. Estimation of ω d

In this subsection, we evaluate the performance of three estimators of ω d : the exact estimator provided in [START_REF] Sadeghian | Online detection of broken rotor bars in induction motors by wavelet packet decomposition and artificial neural networks[END_REF], the approximate estimator provided in [START_REF] Garcia-Perez | The application of high-resolution spectral analysis for identifying multiple combined faults in induction motors[END_REF] and a simple technique without interference rejection, obtained by replacing P y (ω) by P x (ω) in [START_REF] Garcia-Perez | The application of high-resolution spectral analysis for identifying multiple combined faults in induction motors[END_REF]. For these estimators, the optimization of the cost function is performed using the Nelder-Mead method with an initial value equal to ω d = 20π/F s rad/samples. The estimation performance is evaluated in terms of Mean Square Error (MSE), which is estimated using M = 1000 Monte Carlo trials as

M SE ω d = 1 M M -1 m=0 (ω d -ω d [m]) 2 . ( 27
)
where ω d and ω d [n] correspond to the true and estimated (normalized) fault-related angular frequencies, respectively. The

Stator current samples x[n]

Compute the interference-free signal y[n] based on ( 18)

y[n]
Estimate ω d and θ 1 based on [START_REF] Sadeghian | Online detection of broken rotor bars in induction motors by wavelet packet decomposition and artificial neural networks[END_REF] and(??)

ω d and θ 1
Compute fault detection criterion based on [START_REF] Trachi | A novel induction machine faults detector based on hypothesis testing[END_REF] or ( 26) MSE is also compared with exact and approximate Cramér-Rao Bounds (CRBs). While the exact CRB must be evaluated numerically, it can be approximated for a large number of samples by

T (y) or T a (y) T (y) > γ
CRB a (ω d ) = 24σ 2 N 3 k2 k=-k1 k 2 a 2 k . (28)
Note that the CRB for angular frequency decreases as the noise variance σ 2 increases and that the bound decreases as 1 /N 3 , making it quite sensitive to data record length N . This fact makes the frequency estimates more sensitive to data acquisition length N than noise variance σ 2 .

Figure 2 displays the MSE versus signal length, N , for a signal to noise ratio equal to SNR = 30dB. We observe that the MSE of the exact estimator approaches the CRB when N > 300 samples. Furthermore, this figure clearly shows the benefit of using the interference-free signal, y[n]. Indeed, the approximate estimator based on y[n] clearly outperforms the approximate one without interference rejection. Furthermore, we note that the approximate estimator based on the interference-free signal, y[n], periodically approaches the CRB.

B. Fault Detection

In this subsection, the performance of the matched subspace detector is assessed on synthesized signals. Two detector types are considered in the following, which are the clairvoyant matched subspace detector that requires a perfect knowledge of θ 1 and ω d and the blind matched subspace detector that replaces θ 1 and ω d with their estimates θ 1 and ω d . The detection performance is usually assessed through two parameters, which are the probability of detection P D (i.e. the probability to detect H 1 when H 1 is true), and the probability of false alarm P F a (i.e. the probability to detect H 1 when H 0 is true).

Figure 3 presents the performance of three fault detectors for the discrimination between H 0 and H 1 with respect to N (SNR = 30dB). For H 0 , the signal parameters are obtained from Table I by setting a k = 0 for k = 0. The three considered fault detectors are the clairvoyant exact detector provided by [START_REF] Trachi | A novel induction machine faults detector based on hypothesis testing[END_REF], the clairvoyant approximate detector provided by [START_REF] Immovilli | Bearing fault model for induction motor with externally induced vibration[END_REF], and the blind approximate detector given by ( 26) (obtained by replacing ω d by its approximate estimator in ( 19)). For these three detectors, the threshold γ is computed from the probability of false alarm, which is set to 0.1. The detector performances are assessed through the Probability of Detection, P d , and Probability of False Alarm, P F A , which are estimated using M = 1000 Monte Carlo trials. These probabilities are compared with the theoretical values of P F A and P D provided in section IV.

For the three detectors, we observe that the probability of detection, P D , becomes equal to 1 for N ≥ 100 samples. Regarding P F A , we note that the probability of false alarm obtained with the approximate clairvoyant detector is close to the desired one for N ≥ 200 samples. When the (normalized) fault-related angular frequency is unknown, Figure 3 also shows that the blind approximate detector seems to increase the false alarm rate since P F A ≥ 0.1 when N ≥ 250 samples.

VI. EXPERIMENTAL RESULTS

This section illustrates the behavior of the exact and approximate blind detectors for eccentricity fault, bearing faults and broken rotor bars detection using experimental signals issued from different induction motors and for different load conditions.

A. Experimental Setup Description

Two mechanical faults and one electrical fault are considered in this section. These faults include eccentricity fault, rolling-element bearing faults, and broken rotor bars. The stator currents are measured using a data acquisition card and processed off-line on a standard desktop PC using Matlab ® .

1) Experimental setup for mechanical faults: Healthy machine and faulty ones with eccentricity fault and bearing faults have been tested. Each machine is a 230/400V, 0.75kW, threephase induction machine with one pole pairs and 2780 rpm rated speed. The machines under study are fed by a PWM inverter with a fundamental frequency that can be tuned from 0 to 60 Hz. A DC generator is used to load the induction machines. The Experimental test bed is illustrated by Fig. 4. The induction machines have two 6204-2 ZR type bearings (single row and deep groove ball bearings) with the following parameters: outside diameter is 47 mm, inside diameter is 20 mm, and pitch diameter D is 31.85 mm. Bearings have 8 balls with an approximate diameter d of 12 mm and a contact angle of 0 • . In the eccentricity-based faulty machine, eccentricity is introduced by acting on jack bolts on the circumference of each end-bell. Indeed, this will introduce a non-uniform air gap. Bearing faults are obtained by drilling holes of several diameters in the inner raceway (faults ranging from 0.178 mm in diameter to 1.016 mm).

All the experiments were done in steady-state conditions. Induction machines stator currents have been measured using closed-loop (compensated) current transducers using Hall effect. These transducers are advantageous in terms of accuracy, linearity, low-temperature drift, wide frequency bandwidth, and high immunity to external interference. The stator currents acquisition is performed by a 24 bits LabJack UE9 acquisition card with 20 kHz sampling frequency as illustrated by Fig. 5. The bipolar analog-to-digital converter (ADC) of the LabJack

B. Eccentricity Fault Detection

For illustration purposes, the stator current spectral signature of a motor with eccentricity fault is shown in Fig 6 . The current spectra shows an increase of the fault-related frequencies around the fundamental frequency. This behavior is identical for several load conditions. Consequently, it can be clearly recognized that a monitoring strategy based on these components can be efficiently used for detection purposes. Moreover, we observe that the noise component level is about -110dB, which may eventually hide the low-amplitude fault frequency components for small data lengths. This noise component is due to the measurement chain, quantization errors and the induction machine operating conditions. Finally, we note that that the amplitude of the fundamental frequency is much more larger than those at the fault-related frequencies. When using a classical periodogram, the large amplitude of the fundamental component can mask the amplitude of the fault related components. To overcome this issue, the proposed technique explicitly removes the fundamental component from the raw signal using an oblique projection, and estimates the amplitude of the fault related components from the interference-free signal.

The original stator current signal has been low-pass filtered and down-sampled to 400 Hz. Then, it has been processed using the proposed approach. The evolution of the fault detection criteria T (y) with respect to load conditions for several rotational speeds is given in Fig. 7. The detector threshold γ obtained by setting P F A = 0.1 is also shown for comparison. For the healthy motor, we observe that the fault criteria T (y) is below the threshold γ whatever the load and speed conditions are. For a motor with eccentricity fault, the detector allows to correctly detect the fault regardless of the rotational speed conditions. However, the fault may not be detected for an unloaded motor since it has a small influence on the stator current under small load. This is mainly due to the loading impact on the fault signature.

C. Bearing Fault Detection

The proposed fault detection criteria sensitivity has been evaluated according to bearing faults level. In fact, several fault severity degrees are considered in this study as summarized by table II. The stator current PSD for healthy and faulty induction machines is given by Fig. 8. This figure shows clearly that bearing fault introduce some frequency components around the supply fundamental frequency. These components already exist on the stator current spectrum for healthy machine, which may be due to inherent eccentricity at the manufacturing stage, or harsh operating conditions. However, these components are extremely low for healthy machines and their amplitude increases when a fault is present. The results after the PSD computation using the periodogram for various fault conditions are given in table III. We observe that the noise level is about -160dB. Furthermore, this table clearly shows that the fundamental frequency has higher amplitude than the fault related frequencies. Consequently, removing this component from the raw data can significantly enhance the detection of the fault related components estimates.

The raw data has been low-pass filtered and down-sampled to 400 Hz. Then, it has been processed using the proposed approach. The evolution of the exact and approximate fault detection criteria T (y) and T a (y) with respect to N for several fault degrees is given by Fig. 9. The detector threshold γ obtained by setting P F A = 0.1, is also shown for comparison. For the healthy motor, we observe that the fault criteria T (y) and T a (y) are below the threshold γ whatever the number of samples, N . For bearing fault, the detector requires at least N = 600 samples to correctly detect the fault. For a severe bearing fault, we observe that a smaller number of samples is required. Moreover, the fault detection criteria increase as the fault degree increases.

D. Broken Rotor Bars Fault Detection

The healthy and the two faulty machines have been operated under steady state conditions with a fundamental frequency equal to 50 Hz at 50% load. The stator current has been acquired using a data acquisition board, with F s = 20 kHz. The signal is further low-pass filtered, down-sampled to 400Hz.

The stator current power spectral density (PSD) for healthy and faulty induction machines is given by Fig. 10. This figure clearly shows that broken rotor bars introduce several frequency components around the supply fundamental frequency. The fundamental frequency and the fault related frequency components for various fault degrees are summarized in table IV. In this table, only the upper f u and the lower f l sidebands are considered. It shows that the amplitude of the frequency components, signature of the fault, evolve according to the fault severity. Moreover, these frequencies are too close to the fundamental frequency which make it difficult to detect using the classical DTFT for short data measurement. Moreover, this table gives the noise level, which appears to be more important in the case of faulty machines. Visually, these differences can make the broken rotor bars more difficult to detect. The evolution of the exact and approximate fault detection criteria T (y) and T a (y) with respect to N for several fault degrees is given by Fig. 11. The detector threshold γ obtained by setting P f a = 0.1 is also shown for comparison. For the healthy motor, we observe that the criteria T (y) and T a (y) are below the threshold whatever the number of samples N . For one broken bar, the detector requires at least N = 400 samples to correctly detect the fault. For two broken rotor bars, we observe that a smaller number of samples is required.

VII. CONCLUSION

This paper has proposed a new faults detector for induction machines. The proposed detector is based on three steps, which are the estimation of the interference-free signal, the estimation of the fault-related angular frequency, and the application of a matched subspace detector. For the two last steps, we have derived exact and approximate algorithms. The approximate algorithm is based on the periodogram of the interference-free signal and does not require any matrix inversion, making it well suited for practical implementations. The effectiveness of the proposed approach has been verified for eccentricity fault, bearing faults, and broken rotor bars detection.

  [n]. Note that it is really important to use the periodogram of the interference-free signal y[n] instead of the periodogram of the original signal x[n]. Indeed, the supply frequency, which has usually the largest amplitude, can introduce strong sidelobes in the periodogram of x[n].

Fig. 1 .

 1 Fig. 1. Flowchart of the matched subspace detector.

Fig. 2 .

 2 Fig. 2. Estimation of ω d : Cramer-Rao Bounds (CRB) and MSE for frequency versus signal length N (SNR = 30dB).

Fig. 3 .

 3 Fig. 3. Fault detection: Probability of detection, P d , and false alarm, P F A , versus signal length N (P F A = 0.1, SNR = 30dB).

Fig. 4 .Fig. 5 .

 45 Fig. 4. Machinery fault simulator.

  PSD for eccentricity failure.

Fig. 6 .

 6 Fig. 6. Eccentricity fault: Healthy and faulty phase motor current data collected from the experimental setup (N = 10000, Hanning window).

  Exact detector for 30 Hz.

  Exact detector for 40 Hz.

  Exact detector for 50 Hz.

Fig. 7 .

 7 Fig. 7. Eccentricity fault: T (y) versus induction motor load for several rotational speeds.

  PSD for bearing failure.

Fig. 8 .

 8 Fig. 8. Bearing fault: Healthy and faulty phase motor current data collected from the experimental setup (N= 10000, Hanning window).

Fig. 9 .

 9 Fig. 9. Bearing faults: T (y) and Ta(y) versus N (L = 6).

  PSD for healthy machine.

  PSD for 2 broken rotor bars.

Fig. 10 .

 10 Fig. 10. Broken rotor bars: Healthy and faulty phase motor current data collected from the experimental setup (N= 20000, Hanning window).

Fig. 11 .

 11 Fig. 11. Broken rotor bars: T (y) and Ta(y) versus N (L = 18).

  φ k are the amplitude and initial phase of the k th component, respectively,

• b[n] corresponds to the noise component. This random component is introduced to model the uncertainty errors that may occur during the measurement, conversion, storage or processing stages. This component is assumed to be a white, Gaussian noise with zero mean and variance σ 2

TABLE I SIGNAL

 I PARAMETERS UNDER ASSUMPTION H 1 (L = 10).

TABLE II BEARING

 II FAULT DEGREE VERSUS INNER RACEWAY HOLE DIAMETER.

TABLE III AMPLITUDE

 III OF THE BEARING FAULT RELATED COMPONENTS.

	Fault	Fundamental Frequency PSD	Fault related components Frequency PSD	Noise level (dB)
		(Hz)	(dB)	(Hz)		(dB)
	Healthy	fs = 50	6.58	f l = 25.33 fu = 74.65	-115.4 -116.4	-166.9
	Sev.1	fs = 50	6.17	f l = 25.3 fu = 74.68	-99.1 -106.7	-162.3
	Sev. 2	fs = 50	6.35	f l = 25.33 fu = 74.65	-92.72 -94.57	-167.6
	Sev. 3	fs = 50	6.33	f l = 25.33 fu = 74.65	-86.23 -87.69	-160.4
	Sev. 4	fs = 50	6.42	f l = 25.33 fu = 74.65	-83.16 -84.89	-161.3
			Healthy
		20	Severity 1
			Severity 2
			Severity 3
	T (y)	10	Severity 4 γ
		0
		200	400	600		800	1,000
				N (samples)
			(a) detector.
			Healthy
		20	Severity 1 Severity 2
	T a (y)	10	Severity 3 Severity 4 γ
		0
		200	400	600		800	1,000
				N (samples)
			(b) Approximate detector.

TABLE IV AMPLITUDE

 IV OF THE BROKEN ROTOR BARS RELATED COMPONENTS.

	Broken	Fundamental	Fault related components	Noise level
	bars	Frequency	PSD	Frequency	PSD	(dB)
		(Hz)	(dB)	(Hz)	(dB)
	0	fs = 50	23.87	f l = 46.84 fu = 53.25	-89.15 -100.4	-132.7
	1	fs = 50	24.85	f l = 47.3 fu = 53.1	-71.26 -67.72	-97.99
	2	fs = 50	24.93	f l = 46.54 fu = 53.71	-36.94 -34.53	-92.73


Julien Fernandes

Valérie Michel

Margarita Camorlinga-Ponce

Alejandro Gómez

Carmen Maldonado

Hilde De Reuse

Javier Torres

Dr Eliette Touati
email: etouati@pasteur.fr.

Alejandro Gomez

Circulating Mitochondrial DNA Level, a Noninvasive Biomarker for the Early Detection of Gastric Cancer

Keywords: Mitochondrial DNA, Biomarkers, Gastric cancer, Prevention/detection, Diagnostic test

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Gastric cancer (GC) represents a major health burden worldwide, affecting about onemillion people per year (1), [START_REF] Ferlay | GLOBOCAN 2012 v1.0, Cancer incidence and mortality worldwide: IARC CancerBase N°11[END_REF]. GC is often diagnosed at an advanced stage and consequently carries a poor prognosis [START_REF] Kamangar | Opposing risks of gastric cardia and noncardia gastric adenocarcinomas associated with Helicobacter pylori seropositivity[END_REF]. Importantly, if it is detected at an early asymptomatic stage, it can be curable (4); e.g. in Japan, a country with the highest incidence of GC, nation-wide strategies based on improved tests for detection of early GC or precancerous lesions have decreased the incidence of GC and increased survival rate [START_REF] Asaka | A new approach for elimination of gastric cancer deaths in Japan[END_REF].

Although GC arises from the complex interplay of environmental and host genetic factors [START_REF] Resende | Gastric cancer: Basic aspects[END_REF][START_REF] Bornschein | Gastric cancer: Clinical aspects, Epidemiology and Molecular Background[END_REF], the major risk factor is Helicobacter pylori infection, which is associated with more than 80% of all distal GC cases [START_REF] De Martel | Global burden of cancers attributable to infections in 2008: a review and synthetic analysis[END_REF]. The prevalence of H. pylori infection is high, with 80-95% of the population infected in developing countries and up to 30%-40% of adults in industrialized countries [START_REF] Resende | Gastric cancer: Basic aspects[END_REF]. All infected individuals develop a gastritis which evolve to peptic ulcer diseases in about 10% of the cases, while gastric adenocarcinoma and mucosaassociated lymphoid tissue (MALT) lymphoma develop in <3% and 0.3% of infected subjects, respectively [START_REF] Blaser | Helicobacter pylori persistence: biology and disease[END_REF][START_REF] Atherton | The pathogenesis of Helicobacter pylori-induced gastro-duodenal diseases[END_REF]. Two types of GC can be distinguished, the intestinal and diffuse types. The intestinal type develops through progressive changes in the gastric mucosa from non-atrophic gastritis (NAG), atrophic gastritis, intestinal metaplasia (IM), dysplasia and gastric cancer (GC) [START_REF] Correa | Human gastric carcinogenesis : a multistep and multifactorial process. First American Cancer Society Award Lecture on Cancer Epidemiology and Prevention[END_REF]. It has been shown that eradication of the infection at an early stage can reverse gastric lesions and more importantly, prevent the development of preneoplasia ((7), [START_REF] Zhou | A five-year follow-up study on the pathological changes of gastric mucosa after H. pylori eradication[END_REF], [START_REF] Ley | Helicobacter pylori eradication and gastric preneoplastic conditions: a randomized, double-blind, placebo-controlled trial[END_REF]). A clinical study conducted in Japan demonstrated the efficacy of H. pylori eradication to reduce the incidence of GC [START_REF] Fukase | Effect of eradication of Helicobacter pylori on incidence of metachronous gastric carcinoma after endoscopic resection of early gastric cancer: an open-label, randomised controlled trial[END_REF], and confirmed that it was not enough to prevent all GC cases. These data raise the need for the development of biomarkers to detect precancerous lesions or early GC; the test should be a simple and non-invasive method, applicable in large scale screening programs. In the case of intestinal type GC, the measure of pepsinogens levels has been shown to be useful to detect gastric atrophy, although its utility in non-Asian countries is controversial [START_REF] Di Mario | Non-invasive tests in gastric diseases[END_REF]. Thus, in spite of several efforts to develop biomarkers to identify patients at risk for distal GC [START_REF] Cooke | Biomarkers of Helicobacter pylori-associated gastric cancer[END_REF], no efficacious screening test is yet available.

Mitochondria are essential organelles of eukaryotic cells that possess their own genome.

Mitochondrial DNA (mtDNA) is a circular molecule present at 2-10 copies per organelle [START_REF] Wallace | Mitochondrial DNA sequence variation in human evolution and disease[END_REF].

Both, mutations and alterations of mtDNA content have been described in many different cancer types [START_REF] Meng | Mitochondrial DNA copy number alteration in human cancers[END_REF][START_REF] Yu | Mitochondrial DNA copy number alterations in human cancers[END_REF][START_REF] Carew | Mitochondrial defects in cancer[END_REF]. MtDNA mutations have been detected at early stages of gastric carcinogenesis [START_REF] Kamalidehghan | Tumoral cell mtDNA approximately 8.9kb deletion is more common than other deletions in gastric cancer[END_REF][START_REF] Maximo | Microsatellite instability, mitochondrial DNA large deletions and mitochondrial DNA mutations in gastric carcinoma[END_REF]. In H. pylori-infected patients, mtDNA mutations are significantly more frequent in GC patients than in cancer free patients [START_REF] Hiyama | Somatic mutation of mitochondrial DNA in Helicobacter pylori-associated chronic gastritis in patients with and without gastric cancer[END_REF]. According to our previous studies, mtDNA mutations are induced in vitro in H. pylori-infected gastric epithelial cells and in the gastric mucosa of chronically infected mice [START_REF] Machado | Helicobacter pylori infection downregulates nuclear and mitochondrial DNA repair in gastric cells[END_REF], [START_REF] Touati | When bacteria become mutagenic and carcinogenic: lessons from H. pylori[END_REF]. A decrease in mtDNA content has also been described in most tumor tissues of advanced GC, compared with nearby non-tumor control tissue [START_REF] Lee | Mitochondrial genome instability and mtDNA depletion in human cancers[END_REF], [START_REF] Zhang | Variable copy number of mitochondrial DNA (mtDNA) predicts worse prognosis in advanced gastric cancer patients[END_REF], [START_REF] Wen | Decreased copy number of mitochondrial DNA: A potential diagnostic criterion for gastric cancer[END_REF]. Changes in peripheral blood mtDNA levels have recently emerged as a potential preventive/diagnosis biomarker associated with cancer risk [START_REF] Meng | Mitochondrial DNA copy number alteration in human cancers[END_REF]. Circulating mtDNA levels were significantly higher in patients with urologic malignancies [START_REF] Ellinger | Circulating mitochondrial DNA in serum: a universal diagnostic biomarker for patients with urological malignancies[END_REF], breast [START_REF] Shen | Mitochondrial copy number and risk of breast cancer: a pilot study[END_REF] colorectal [START_REF] Qu | Association between mitochondrial DNA content in leukocytes and colorectal cancer risk: a case-control analysis[END_REF] and lung cancer [START_REF] Bonner | Mitochondrial DNA content and lung cancer risk in Xuan Wei, China[END_REF][START_REF] Hosgood Hd 3rd | Mitochondrial DNA copy number and lung cancer risk in a prospective cohort study[END_REF]. These studies indicated that increased mtDNA content in peripheral blood is associated with elevated cancer risk. However, contrasting reports have found mtDNA depletion in the blood of patients with stage I breast tumors [START_REF] Xia | Decreased mitochondrial DNA content in blood samples of patients with stage I breast cancer[END_REF]. A recent large prospective cohort study in women from Shanghai, found no association between the leukocyte mtDNA copy number and the presence of gastric tumor [START_REF] Liao | Mitochondrial DNA copy number and risk of gastric cancer: a report from the Shangai women's health study[END_REF]. However, they observed a positive association between a risk of developing GC and low mtDNA copy numbers in blood collected within the two years prior to cancer diagnosis. To our knowledge, there are no other reports on the variation of circulating mtDNA during the gastric carcinogenesis cascade, in particular comparing pre-neoplastic and neoplastic stages. To address this issue, in the present study we measured levels of mtDNA in the peripheral leucocytes of NAG, IM and GC patients. Our data showed significant variations in mtDNA levels during the progression from NAG to IM and to GC, supporting the notion that circulating mtDNA levels can be useful as potential biomarkers for the identification of early steps of gastric carcinogenesis.

Material and methods

Study population

Two cohorts of Mexican adult patients were studied. Cohort 1 included 48 healthy asymptomatic H. pylori negative blood-donors, 28 NAG and 74 GC patients for a total of 150 adults recruited during the period 2009-2011. The blood donors were recruited at the blood bank of the Instituto Mexicano del Seguro Social (IMSS), Medical Center SXXI in Mexico City. Cohort 2 was meant to validate results in cohort 1, and included 46 NAG patients, 31 patients with IM and 49 with GC for a total of 126 patients recruited during the period 1999-2002. Patients from both cohorts were adults who were attended for gastroduodenal diseases at IMSS. We selected patients who were not under treatment for cancer and who had not been treated with antibiotics, bismuth compounds, proton pump inhibitors and non-steroidal antiinflammatory drugs for at least two preceding weeks. Diagnosis was based on endoscopic examination and histopathology analysis [START_REF] Camorlinga-Ponce | Age and severity of mucosal lesions influence the performance of serologic markers in Helicobacter pylori-associated gastroduodenal pathologies[END_REF]. All patients and asymptomatic controls were informed and asked to sign a consent letter. The study was approved by the ethical committee from the National Council for Research on Health, IMSS.

Collection of clinical samples and histological analysis

For each patient, 10 ml of blood were collected and gastric tissue specimens isolated. For patients with NAG or IM, gastric biopsies collected from both the antrum and the corpus were taken. For patients with GC, one fraction of tumoral and adjacent tissues were collected during surgery. Biopsies were immersed in formalin and processed for haematoxilin-eosin (H&E) staining for histology analysis and diagnosis of gastric lesions. The presence of H. pylori was confirmed by Giemsa staining and serology.

Circulating mtDNA

Peripheral blood (10ml) was taken from each patient, and mononuclear cells were purified by centrifugation through a Ficoll-Hypaque density gradient. DNA was isolated from these cells using the salting-out micro-technique and frozen at -70°C until tested for mtDNA quantification. The serum fraction was separated from cells and frozen at -20ºC until tested for serology to H. pylori antigens and IL-8 levels.

Determination of IgG against H. pylori antigens and of IL-8 in the serum fractions

H. pylori serology was determined with IgG antibodies against H. pylori whole-cell antigens and against CagA H. pylori protein using an enzyme-linked immune-absorbent assays (ELISA) previously validated by us [START_REF] Camorlinga-Ponce | Validation of a serologic test for the diagnosis of Helicobacter pylori infection and the immune response to urease and CagA in children[END_REF]. Serological levels of IL-8 were measured by ELISA (BD Biosciences).

Quantification of mtDNA

MtDNA levels were measured on DNA isolated from circulating leukocytes by quantitative Polymerase Chain Reaction (q-PCR) using the StepOne TM Plus Real-Time PCR system and FastStart Universal SYBR Green Master (Applied Biosystems) as previously described [START_REF] Chatre | Large heterogeneity of mitochondrial DNA transcription and initiation of replication exposed by single-cell imaging[END_REF]. Ct mtDNA ) as 2 ΔCt as previously described [START_REF] Fan | Mitochondrial DNA content in paired normal and cancerous breast tissue samples from patients with breast cancer[END_REF].

Data analysis.

The Student's t-test and Pearson's chi 2 test were used to compare mtDNA level in peripheral blood between healthy subjects and patients at various stages of the gastric pathologies.

Differences were considered significant for p-values<0.05. For Odd Ratio (OR) determination the 95% confidence intervals were determined according to the Woolf's method [START_REF] Woolf | On estimating the relation between blood group and disease[END_REF]. The sensitivity and specificity of mtDNA and IL-8 tests were calculated using receiver operator characteristic (ROC) analyses. Statistical tests performed using GraphPad Prism4 for Macintosh (GraphPad Inc., La Jolla, CA).

Results

Characteristics of the studied cohorts

The general characteristics of the patients included in this study are described in Table 1

Levels of mtDNA in peripheral leukocytes from patients with NAG and GC

In Cohort 1, mtDNA was quantified in NAG and GC patients as described in material and methods, and compared with asymptomatic H. pylori negative subjects. In asymptomatic controls, the relative mtDNA values ranged from 2 to 17.09, whereas in NAG group minimum and maximum mtDNA levels were 0.05 and 29.3 respectively. Even though in GC patients minimum mtDNA level was 1.74, the maximum value measured was 60 (Figure 1A).

Mean mtDNA levels were significantly higher in GC patients compared to asymptomatic controls (2-fold) (p-value=0.0013) and to NAG patients (2.2-fold) (p-value=0.0095). As reported in Figure 1A, three intervals of mtDNA values can be distinguished: whereas for all healthy individuals, mtDNA values grouped between 2 and 20 (interval II); in the mtDNA <2.0 group (interval I) only NAG (46% of the cases) and GC (14% of the cases) patients were observed. Importantly, samples with mtDNA<0.5 corresponded exclusively to NAG cases. In contrast, mtDNA values >20 (interval III) were not observed in healthy individuals, and only in 7% of NAG, but in 28% of GC cases. We performed a ROC analyses [START_REF] Delacour | ROC (receiver operating characteristics) curve: principles and application in biology[END_REF] to determine a cut-off value of mtDNA that will differentiate GC cases from the asymptomatic H. pylorinegative controls and found a value of 8.46, with a specificity of 80%, but a sensitivity of 47%. However, using this value we found that mtDNA>8.46 was significantly associated with GC (OR=3.93; 95% CI=1.75-8.81) but not with NAG (OR=1.24; 95% CI=0.42-3.67) (Table 2), in agreement with an association between the presence of gastric malignancy and increased peripheral leukocytes mtDNA content.

An analysis of GC subgroups according to GC types, showed that the mean mtDNA values were similar among diffuse (n=28) and intestinal types (n=20) of GC patients, and significantly higher compared to healthy and NAG samples (Figure 1B). The severity of GC did not affect mtDNA as no significant differences were identified between cases with hyperplasia, metastasis or late-stage (unresectable) (Figure 1B). The presence of H. pylori infection had no effect on mtDNA content neither in NAG or in GC patients (Supplementary Figure S1). In addition, age had no influence on mtDNA levels among asymptomatic controls, NAG and GC groups, although a weak correlation was observed when considering the whole cohort (Supplementary Figure S2A). However, no effect of age was detected on mtDNA on both healthy and NAG samples, despite the presence of old individuals (Supplementary Figure S2A, insert graph Cohort 1). Furthermore, a slight mtDNA decrease in smokers compared to non-smokers was noticed (Supplementary Figure S3A).

In order to validate results in Cohort 1, and to further test whether mtDNA levels differentiate patients with pre-neoplasia from NAG patients, mtDNA was quantified in Cohort 2 which included patients with NAG (n=46), intestinal metaplasia (IM) (n=31) and GC (n=49) (Table 1). Samples from Cohort 2 were collected 10 years earlier (1999)(2000)(2001)(2002) than Cohort 1 (2009Cohort 1 ( -2011)). This time lapse might be the reason why mtDNA levels in all patients of Cohort 2 were lower than in Cohort 1. Still, behavior between groups was similar and mean mtDNA levels were significantly higher in GC than in both NAG (p-value=0.0012) and IM patients (p-value=0.0022) (Supplementary Figure S4). Whereas no significant differences were observed between NAG and IM patients. In this cohort, higher mtDNA values (>2.0) were significantly associated with GC, as compared with NAG (OR=8.48; 95% CI 2.22-32.46) and with IM patients (OR=5.87; 95% CI 1.51-22.84). Thus, in agreement with data from Cohort 1, peripheral leucocytes mtDNA levels shifted to higher values in GC patients as compared to pre-neoplastic NAG and IM patients. Age had no influence on mtDNA levels, in NAG, IM and GC groups, although a weak association was found for the whole cohort (p-value=0.04) (Supplementary Figure S2C). As for Cohort 1, the same analysis including only non cancer groups did not show any effect of age on mtDNA levels, despite the advanced age of some patients in these groups, specially among IM patients for which mean age was similar to GC patients (Supplementary Figure S2C, insert graph Cohort 2).

Comparison of serological levels of IL-8 and mtDNA in NAG and GC patients

Chronic gastric inflammation is associated with the promotion of carcinogenesis and inflammation can be correlated with plasma mtDNA levels [START_REF] Cossarizza | Increased plasma levels of extracellular mitochondrial DNA during HIV infection: a new role for mitochondrial damage-associated molecular patterns during inflammation[END_REF]. We wondered whether there was any correlation between plasma IL-8 and mtDNA in patients with NAG and GC using samples from Cohort 1. IL-8 levels are not affected by parameters as age (Supplementary Figure S2B). Compared to NAG patients and controls, IL-8 levels were significantly higher in GC patients (4-and 2.3-fold respectively) (p-value=0.016) (Figure 2A) either with diffuse (p-value=0.019) or intestinal type (p-value=0.036) of cancer and with metastasis (p-value=0.003) or unresectable (p-value=0.0015) (Figure 2B). However, no correlation was observed between IL-8 and mtDNA levels in GC patients. It should be noticed that among GC patients, IL-8 levels were higher in smokers than in non-smokers (Supplementary Figure S3B). Eighty percent of samples with IL-8 >50 pg/ml corresponded to GC cases, among which 67% were in the mtDNA interval II. Moreover, 89% of GC samples with the highest mtDNA level (interval III), showed IL-8<50 pg/ml. For NAG patients, a weak but significant correlation was observed between IL-8 and mtDNA (r=0.586; P=0.0041), although the analysis included only 25 samples. Thus, in most of the cases higher concentration of IL-8 differentiated GC patients with "normal" mtDNA levels (interval II), in spite of the lack of correlation between mtDNA and IL-8 levels (Figure 3).

Discussion

Gastric cancer remains the second cause of death by cancer, often detected at an advanced stage. Thus, there is a real need to develop appropriate screening strategies for its early detection. In Japan, one of the countries with the highest incidence of GC, strategies that include the eradication of H. pylori among young people, and periodic endoscopic examination have been proposed to reduce GC deaths [START_REF] Asaka | A new approach for elimination of gastric cancer deaths in Japan[END_REF]. However, endoscopy is an invasive and costly method that requires special equipment and well-trained personnel to offer reliable diagnosis. Despite all recent developments in molecular diagnostic ( 16), identification of biomarkers with reliable predictive value are still unavailable. Peripheral blood mtDNA copy number has recently emerged as a potential preventive/diagnosis biomarker associated with the risk of various cancers [START_REF] Meng | Mitochondrial DNA copy number alteration in human cancers[END_REF], [START_REF] Yu | Mitochondrial DNA copy number alterations in human cancers[END_REF]. Levels of circulating mtDNA were found significantly higher in patients with various type of tumors ( 29),( 30),( 31),( 32), [START_REF] Xu | Association of mitochondrial DNA copy number in peripheral blood leukocytes with risk of esophageal adenocarcinoma[END_REF]. Peripheral mtDNA content has also been proposed as a novel molecular marker for tracing tumor progression [START_REF] Xia | Decreased mitochondrial DNA content in blood samples of patients with stage I breast cancer[END_REF]. In the present study, we found more variation of circulating leukocytes mtDNA level in patients with NAG or GC as compared with healthy subjects. We identify a distribution of samples according to three intervals of mtDNA levels measured as previously described In a recent large prospective study, no association between mtDNA copy number and GC was found [START_REF] Liao | Mitochondrial DNA copy number and risk of gastric cancer: a report from the Shangai women's health study[END_REF]. However a positive association between low mtDNA copy number and risk of GC was observed in blood collected within the two years prior to cancer diagnosis, suggesting that a decrease in mtDNA level is an early indicator of the malignant transformation of gastritis lesions. In agreement, in our study the lowest mtDNA levels correspond to NAG patients. Gastritis is characterized by a chronic inflammation of the gastric mucosa, associated with high production of oxidative species known to damage DNA, rendering mtDNA highly susceptible to damage and depletion. This is probably the reason for the lower mtDNA levels observed in some NAG cases. Although the role of mtDNA depletion in the cancer process remains unknown, it might decrease mitochondrial genes expression and impair the oxidative phosphorylation chain activity, leading to an increased production of ATP by glycolysis, a mechanism occurring during tumorigenesis, known as the Warburg effect [START_REF] Warburg | On respiratory impairment in cancer cells[END_REF]. Low mtDNA copy number has also been reported associated to an increased risk in various cancers [START_REF] Purdue | A casecontrol study of peripheral blood mitochondrial DNA copy number and risk of renal cell carcinoma[END_REF], [START_REF] Xing | Mitochondrial DNA content: Its genetic heritability and association with renal cell carcinoma[END_REF]. A lower blood mtDNA level has been described in stage I breast cancer as compared to advanced stages II to IV, leading to suggest that cancer cells deplete a larger amount of mtDNA in circulating cells at an early stage of cancer development. The initial decline in mtDNA content should be compensated at later stage of the cancer process to restore respiratory functions [START_REF] Xia | Decreased mitochondrial DNA content in blood samples of patients with stage I breast cancer[END_REF], which is in agreement with the increase we observed in many of the CG cases studied. This kinetic of mtDNA levels during the carcinogenic process would also be in agreement with results from Cohort 2 patients showing significant lower mtDNA levels in NAG and in precancerous IM patients, compared to most GC patients. This tendency was observed in Cohort 2 in spite of lower levels of mtDNA in all patients studied, which was probably due to the long time lapse of storage samples before testing. This observation would argue to the importance of using only fresh specimens to have reliable and clinically useful values. It also suggest that mtDNA levels cannot distinguish NAG patients and patients with early pre-neoplastic lesions.

Summarizing results from both cohorts, in healthy H. pylori negative individuals we observed a limited range of mtDNA levels (interval II) and values below this range were found mostly in patients with precancerous stages NAG and IM lesions (interval I); whereas values above this range were observed mostly in patients with GC (interval III). Based on these results and in agreement with a previous study [START_REF] Liao | Mitochondrial DNA copy number and risk of gastric cancer: a report from the Shangai women's health study[END_REF], patients with decreased mtDNA levels potentially represent individuals with increased risk for GC, which should be monitored periodically to effectively prevent GC development. Whereas in patients with increased mtDNA, the presence of GC lesions should be suspected and further tested by endoscopy.

However, in both, precancerous and GC groups we observed patients with mtDNA levels in the same range as healthy individuals. These patients need to be further studied to identify if they represent a group with a different carcinogenesis process or cancer stage. Among GC cases, we noticed a tendency, although not statistically significant, for mtDNA to increase according to the severity of the disease, from patients with hyperplasia or metastasis to laterstage (unresectable). In any case, it should be studied in a larger number of patients, to confirm if mtDNA may help differentiate GC stages and subtypes.

Chronic inflammation contributes to the promotion of cancer. Recently, a positive association has been reported between plasma level of pro-inflammatory cytokines, IL-6 and IL-8, and high GC incidence, suggesting them as potential biomarkers for GC [START_REF] Epplein | Circulating cytokines and gastric cancer risk[END_REF]. We did not find any correlation between IL-8 and mtDNA levels, indicating that the inflammatory process does not always drive the observed increased mtDNA levels in GC. In fact, we found mostly low IL-8 levels in GC cases with high mtDNA level (interval III). In addition, higher values of IL-8 were detected mostly in GC cases with "normal" mtDNA level (interval II), which supports that determination of both, mtDNA and IL-8, improves the identification of patients with GC in interval II. It is important to notice that most of these GC patients (mtDNA-interval II and IL-8>50 pg/ml) were smokers. MtDNA levels can be also affected by other parameters as infections, that should be taken into account [START_REF] Zhao | Association of mitochondrial DNA content in peripheral blood leukocyte with hepatitis B virus-related hepatocellular carcinoma in a Chinese Han population[END_REF].

In conclusion, we identified three different ranges of peripheral leucocytes mtDNA levels, one present in all healthy H. pylori uninfected subjects and considered as "normal", one with lower mtDNA observed mainly in patients with gastric precancerous lesions and a third increased range corresponding almost exclusively to GC patients. According to our data, quantification of mtDNA by a simple blood sampling would permit an early detection of the presence of lesions and to design a personalized clinical follow-up of patients. Patients with low mtDNA level are likely to present precancerous or GC lesions and should be periodically monitored by endoscopy. Patients with high mtDNA level are more likely to already present GC and should be thoroughly studied by endoscopy. We should notice that there were patients with precancerous and with GC lesions with mtDNA levels in the same range as the asymptomatic group, which deserve more studies. Detection can be improved by testing in addition IL-8 levels, particularly in GC patients that present "normal" mtDNA (interval II).

These data should be further validated in larger patient groups, also including patients from different geographical origin. Our findings indicate that testing for circulating mtDNA and IL-8 might offer reliable minimaly invasive biomarkers, to screen populations at risk for GC.

They pave the way to the development of circulating mtDNA measure as a predictive/earlydiagnostic biomarkers that are particularly needed in developing countries in Asia and Latin America, regions with the highest GC mortality rates.

  MtDNA was quantified using a region in the 12S ribosomal RNA gene, and the nuclear encoded 18S ribosomal RNA gene as an endogenous reference. Primers used were: 12SrRNA (forward): 5'-GCTCGCCAGAACACTACGAG; (reverse): 5'-CAGGGTTTGCTGAAGATGGCG; 18SrRNA (forward): 5'-GAGAAACGGCTACCACATCC; (reverse): 5'-GCCTCGAAAGAGTCCTGTAT (39). The qPCR reaction was carried out in 20μl of total volume containing 5μl of DNA (200 pg), 10μl of FastStart Universal SYBR Green Master, 0.2µl of primers (10µM) using an initial denaturation step at 95°C for 10 minutes and 40 cycles of 15 secondes at 95°C and 1 minute at 60°C. Samples were analyzed in triplicate. The average threshold cycle number values for nDNA (18SrRNA) and mtDNA (12SrRNA) were obtained. The relative mtDNA level was calculated using the delta Ct (ΔCt) of average Ct of nDNA and mtDNA (ΔCt = Ct nDNA -

( 38 )

 38 ,[START_REF] Fan | Mitochondrial DNA content in paired normal and cancerous breast tissue samples from patients with breast cancer[END_REF], intervals I: mtDNA<2.0; II with mtDNA comprised between 2.0 and 20.0 and III corresponding to mtDNA values >20.0. All asymptomatic individuals felt in the interval II, together with some NAG and GC patients. Importantly, in the interval I near half of NAG and very few GC patients but no healthy individuals were observed, whereas in the interval III almost only GC patients were found. Interestingly, low mtDNA values (<0.5) corresponded only to NAG patients whereas high mtDNA levels (>30) were exclusively observed in GC patients. According to these data, by a simple blood sampling, the classification of an individual in intervals I or III would be an indication of the presence of gastric inflammation and probable GC, respectively. This detection would constitute a first indication for the patients, upstream endoscopic investigation and specific follow-up.

Figure 1 .

 1 Figure 1. Circulating mtDNA content in leukocytes of NAG and GC patients compared to healthy blood donors in Cohort 1. A) MtDNA was quantified by quantitative real-time PCR (40), from DNA isolated from peripheral leukocytes isolated from healthy blood donors H. pylori-negative (open triangles), NAG (open circles) and GC (closed circles) patients. (40)Intervals I, II and III correspond to range of mtDNA level: <2, 2-20 and >20, respectively. B) Peripheral leukocytes mtDNA level in NAG and GC patients according to the type of cancer either diffuse (n=31) or intestinal (n=20) (left part) or the presence of hyperplasia (n=19), metastasis (n=19) or late-stage cancer judged unresectable by surgeons (n=17) (right part). As in control group, mean mtDNA levels are lower in NAG patients compared to GC of diffuse (*) intestinal type (**) with hyperplasia (*), metastasis (**) and late-stage cancer (*). Each symbol corresponds to a single patient. Bars indicate mean ± SEM. Samples were tested in triplicate in three independent experiments. *p-value<0.05; ** <0.01: ***<0.001.

Figure 2 .Figure 3 .

 23 Figure 2. IL-8 serological levels in NAG and GC patients. A) IL-8 was measured by ELISA in sera from controls, NAG and GC patients from cohort 1. B) IL-8 levels were compared between diffuse and intestinal type of GC, as well as among GC with hyperplasia, metastasis or at late-stage. Bars correspond to mean ± SEM. *p-value<0.05; **<0.01

  as diffuse-type, of which 6 had metastasis and 9 considered unresectable by surgeons because of a very advanced stage (late-stage). Twenty GC cases were of intestinal type, 7 presented hyperplasia, 7 metastasis and 1 case unresectable. In Cohort 2, 78% of the NAG patients were H. pylori positive, compared to 84% in IM and 59% in GC. Overall, IM and GC patients were older compared to NAG patients.

. In Cohort 1, 50% of the NAG patients were H. pylori-positive compared to 71% in the GC group. In the control group, we only selected H. pylori negative blood donors (n=48). Patients in the NAG and GC groups were older (mean age 56 and 62 respectively), compared to individuals from the asymptomatic group (mean age 32) (p-value<0.0001). Twenty-eight GC patients were diagnosed

Table 1 . Characteristics of the study population

 1 Two different groups of patients are analysed in the study. In each groups, samples were collected around the same period of time, from 2009-2011 for Cohort 1 and 1999-2002 for Cohort 2. a Healthy subjects are H. pylori negative blood donors. P-values in the « Mean age » column represent statistical analyses for age distribution in, b NAG and GC patients compared to healthy individuals; c GC patients compared to the NAG group (Cohort 1); d IM and GC patients compared to the NAG group (Cohort 2).

	Number	Mean age	Sex ratio	H. pylori
	of	(range)	(M/F)	positive
	patients			%

Table 2 . Increased mtDNA is associated with significant risk for gastric cancer

 2

			mtDNA*
	Group	<8.462 >8.462	OR (CI 95%)	X 2	p-value
	Healthy	40	8	1.00
	NAG	21	7	1.24 (0.42-3.67)	0.15	0.69
	GC	36	38	3.93 (1.75-8.81)	11.7	0.0006
	*cutoff value for mtDNA was >8.462 based on a ROC analyses

Odyssey-Re Fundation and Fonds d'Aide à la Recherche, FAR project from the French National Society of Gastroenterology (SNFGE) to E. Touati, CONACYT, Mexico to J. Torres (Grant 2007-C01-69450). J. Torres is a recipient of a exclusivity scholarship from Fundacion IMSS, Mexico.


Recent developments in energy-discriminating Photon-Counting Detector (PCD) enable new horizons for spectral CT. With PCDs, new reconstruction methods take advantage of the spectral information measured through energy measurement bins. However PCDs have serious spectral distortion issues due to charge-sharing, fluorescence escape, pileup effect… Spectral CT with PCDs can be decomposed into two problems: a noisy geometric inversion problem (as in standard CT) and an additional PCD spectral degradation problem. The aim of the present study is to introduce a reconstruction method which solves both problems simultaneously: a "one-step" approach. An explicit linear detector model is used and characterized by a Detector Response Matrix (DRM). The algorithm reconstructs two basis material maps from energy-window transmission data. The results prove that the simultaneous inversion of both problems is well performed for simulation data. For comparison, we also perform a standard "two-step" approach: an advanced polynomial decomposition of measured sinograms combined with a filtered-back projection reconstruction. The results demonstrate the potential uses of this method for medical imaging or for non-destructive control in industry.

Spectral CT reconstruction with an explicit photon-counting detector model: a "one-step" approach

Pierre-Antoine Rodesch* a , V. Rebuffel a , C. Fournier a , F. Forbes b and L. Verger a a CEA, LETI, Univ. Grenoble Alpes, MINATEC Campus, F-38054 Grenoble, France b Univ. Grenoble Alpes, INRIA, Laboratoire Jean Kuntzman, Equipe Mistis, France.

INTRODUCTION

Spectral CT using energy-discriminating PCDs has already shown good perspectives for effective use [START_REF] Schlomka | Experimental feasibility of multi-energy photon-counting K-edge imaging in pre-clinical computed tomography[END_REF][START_REF] Brambilla | CdTe Linear Pixel X-Ray Detector With Enhanced Spectrometric Performance for High Flux X-Ray Imaging[END_REF] . New methods are emerging in order to perform spectral CT for medical diagnosis or detect defaults in high-performance devices. The complete forward model of spectral CT can be divided into two parts: a geometrical projection through the object and the distortion of the attenuated spectrum by the detector response. To inverse this forward model two types of strategy can be adopted. The first one, inverses in a first step the detector response and then performs a reconstruction from converted data [START_REF] Schlomka | Experimental feasibility of multi-energy photon-counting K-edge imaging in pre-clinical computed tomography[END_REF] . This approach is referred to as "two-step" in this paper. The second strategy consists of a simultaneous inversion of both parts of the complete forward model and is referred to as "one-step" approach [START_REF] Barber | An algorithm for constrained one-step inversion of spectral CT data[END_REF][START_REF] Liu | Material reconstruction for spectral computed tomography with detector response function[END_REF] .

A key point of the problem is the spectral response of the detector which has to be taken into account. In the past few years, methods dealing with this issue have been investigated. We can distinguish two categories of solutions. The first one uses a calibration basis to decompose measured sinograms to basis material length sinograms. The second needs an explicit detector response model incorporated in the spectral distortion forward model [START_REF] Schlomka | Experimental feasibility of multi-energy photon-counting K-edge imaging in pre-clinical computed tomography[END_REF] .

In this work a "one-step" method has been developed with explicit detector response function. This response is characterized by a Detector Response Matrix (DRM). This method allows to invert the spectral distortion caused by the detector and at the same time to apply a spatial constraint in the reconstruction process. The purpose is to obtain quantitative basis material maps while reducing artifacts (cupping artifact, striking artifact…).

METHOD

The attenuation coefficient of the volume is decomposed as the linear combination of two known basis material attenuation coefficients [START_REF] Alvarez | Energy-selective reconstructions in x-ray computerised tomography[END_REF] . The method aims at reconstructing those two basis material maps. The detector response is modeled with a linear behavior: the attenuated spectrum is multiplied by a DRM to obtain the photon numbers.

The number of photons in measurement bin c for ray i is:

= ( ; ) ( ) ∑ ∑ ( ) (1)
where ( ; ) is interpreted as the probability that a photon coming in the detector at energy is detected in measurement bin ; is indexing the voxels and m the basis materials; ( ) is the raw flux (number of photons coming out of the x-ray generator at energy ); is the linear attenuation coefficient of basis material , is the projection matrix, and is the basis material map, the unknowns of the spectral CT problem.

We assume that , the measured photon number in bin c for ray i, follows a Poisson distribution with mean :

~ ( ) (2)
A spatial penalization is applied to the material basis maps:

{ } = - - ∈ (3)
where si the weight of the penalization, the voxel index, the neighbor voxels of voxel , the weight of each neighbor and the Tukey function:

( ) = 1 -1 - | | < 1 ℎ . ( 4
)
Using the Bayesian framework, the reconstructed basis material maps are the solution of this problem:

{ } = ; ( { } ) + { } (5)
with the Poisson log-likelihood depending on the actual measurements = { } , , on the estimated measurements = { } , from basis material maps and the weight given to the regularization. To solve this problem, the algorithm ML-TR [START_REF] De Man | An iterative maximum-likelihood polychromatic algorithm for CT[END_REF] (Maximum Likelihood-Transmission) has been adapted to our problem. The reconstruction is initialized with a standard polynomial "two-step" method in order to reach faster convergence to an acceptable solution.

We tested our method on slice-CT simulation data. Measurements were generated with the software Sindbad-SFFD [START_REF] Rebuffel | New functionalities of SINDBAD simulation software for spectral X-ray imaging using counting detectors with energy discrimination[END_REF] . The study phantom is a 30 cm diameter cylinder of water with six 5 cm diameter inserts. Each insert is made of a different material such as cortical bone, a less attenuating bone material named compact bone, aluminum, adipose tissue, Polyethylene and Plexiglas. The source was set to 140kV with 0.1 cm aluminum filtering for an x-ray exposure of 5x10 7 per pixel. The spectral convolution for this simulation was performed using a realistic DRM (cf Fig. 1). Two measurements bin have been used [15keV-65keV] and [66keV -115keV]. Poisson noise has been added to simulated data. The proposed method performed the reconstruction of two basis material maps: water and cortical bone.

RESULTS

Results are compared with a "two-step" method. We employed an advanced polynomial approach to decompose measured sinograms into water and cortical length sinograms. The attenuation range is divided in several zones and then in each zone a different second order polynomial is calculated. This improves decomposition results compared to a standard polynomial approach for a large attenuation range. Then a filtered-back projection was applied to the basis material length sinograms.

Figure 2 displays the reconstructed basis material maps for both "two-step" method for comparison and our "one-step" method. The truth image is also shown for reference. The material decomposition is well performed with both methods. However streaking artifacts are visible in the map 2.b) in the middle of the object and in the more attenuating inserts. These imperfections are corrected in our maps 1.c) and 2.c). The complete description of the forward model in the "one -step" method helps to correct those artifacts. Moreover, due to the robust penalization of the Tukey function, our results show very smooth reconstruction while preserving edges. The parameter of the spatial regularization (σ_t in equation ( 4)) was set to 0.1 for this phantom. In the map 2 c), some reconstructed inserts present minors shape imperfections.

The seven regions of interest (ROI) we studied are the inside of the inserts and the water cylinder. Border voxels between two materials are not part of the analysis in Table 1 and Figure 3. Table 1 lists the mean density value for each ROI. The "two-step" approach presents relevant values in each material (around 5-10% of error). But our method has better results (close to 1% of error in ROIs).

Figure 3 display two maps where voxels of the seven ROIs are represented using the coordinate system (Cortical Bone density; Water density). The orientations of the scatter plots of the inserts are depending on the choice of the basis materials. We can observe the different inserts are distinguishable in the representation of figure 3 b). That is not the case for "two-step" method for comparison for softer inserts.

CONCLUSION & PERSPECTIVES

We developed a "one-step" method reconstruction for spectral CT with robust spatial prior. The decomposition into two basis material maps provides good results for simulated data with a realistic linear detector response function. The advantage of the "one-step" method class is that material decomposition can be performed from raw data with a spatial prior defined on the reconstructed volume. However one of the key point is the parameterization of the regularization used. Further experiments show that the efficiency of the prior term depends on the case studied and the field of application. Future work will investigate how to adapt the prior on other phantoms closer to medical applications which present more complex geometry and materials (especially soft tissues). With our algorithmic framework we will try to implement non spatial priors based on tissue-type identification or correlation between angles of view for the physics of detector. Some aspects of the detector will also be studied such as the confidence in the DRM, the non-linearity due to pile-up effect, the optimal choice of bin number.

Figure 1 :

 1 Figure 1: Left: Detector Response Matrix for a CdTe detector. Right: Response of the detector for an incoming photon at 80keV.

Figure 2 :

 2 Figure 2: Reconstructed basis material maps (Top: Cortical Bone maps; Bottom: water maps) (Inserts: 1: Compact bone, 2: Aluminium, 3: Polyethylene, 4: Plexiglas, 5: Cortical cone, 6: Adipose tissue)

Figure 3 :

 3 Figure 3: Representation of each ROIs reconstructed in a Cortical Bone/Water densities map. (Border voxels between two materials are not displayed for visibility). Left a): reference method; Right b): our method. Black markers '+' are the truth mean values.

Table 1 :

 1 Table with the mean values in each material basis map for each type of material (C. B. = Cortical Bone).

			Water map	C. B. map	Water map	C. B. map	Water map
	Water	0.	1.	0.01	0.97	0.00	1.00
	Cortical Bone	1.	0.	0.91	0.18	0.99	0.02
	Compact Bone	0.66	0.60	0.65	0.63	0.65	0.64
	Aluminum	1.25	0.15	1.18	0.31	1.20	0.29
	Polyethylene	-0.06	1.07	-0.03	1.00	-0.05	1.04
	Plexiglas	-0.04	1.22	0.02	1.10	-0.03	1.20
	Adipose Tissue	-0.04	0.99	-0.01	0.94	-0.04	0.98

True Image ''Two-step'' Method Rodesch et al. ''One-Step'' Method Type of material C. B. map