Igor V Nikiforov
email: nikiforov@utt.fr

Fouzi Harrou

Rémi Cogranne

Pierre Beauseroy

Edith Grall-Maës

Blaise Kévin Guépié

Lionel Fillatre

Jean-Philippe Jeannot

Edith Grall

Kevin Blaise

Lionel Guépié

Fillatre

Sequential detection of a total instantaneous blockage occurred in a single subassembly of a sodium-cooled fast reactor

Keywords: Nuclear safety, Sodium fast reactor, Subassembly total instantaneous blockage, Core outlet sodium temperature, Model-based statistical detection, Transient changes

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

Introduction and motivation

Online monitoring of critical industrial systems, such as nuclear reactors, plays an important role in improving their safety and productivity. Furthermore, early fault de-5 tection and isolation are vital to maintaining reliable process operation and avoiding expensive maintenance. For instance, the Fukushima accident of 2011 in Japan revealed the highest need for developing accurate and efficient monitoring systems for nuclear plants. Accordingly, this paper focuses on designing an innovative statistical monitoring technique for enhancing nuclear reactor safety.

The central core of Sodium Fast Reactor (SFR) contains the fuel subassemblies bundled in a rectangular or hexagonal array. The fuel elements, called fuel pins (or fuel rods), are formed of a stack of fissile fuel pellets placed in thin cylindrical steel tubes and closed by welding of end plugs at both the ends. The fuel rods are then grouped into subassemblies (see Figure 1). The subassemblies ensure the channelization of the coolant flow that cools the fuel pins. The heat produced by the nuclear reaction is evacuated by the liquid coolant circulating between the fuel pins and in the clearance between the subassemblies. In the case of SFR, the liquid coolant used is sodium in the liquid state. It flows in a closed circuit from the bottom of the subassembly upward through pumps. One or more heat exchangers are provided at the outlet of the assembly for extracting heat from the coolant. Since the fuel pins are closely packed in the central core of SFR, the space between them is kept to a minimum value by using spiral spacer wires placed across the fuel pin in the axial direction [START_REF] Nomoto | Measurement of subassembly outlet coolant temperature in the JOYO experimental fast reactor[END_REF][START_REF] Chellapandi | Thermal hydraulic issues and challenges for current and new generation FBRs[END_REF]. In the safety approach for the core of the 4-th generation French SFR, the TIB is a hypothetic accident scenario belonging to a prevention situation category (probability inferior to 10 -6 per year). This scenario is very unlikely as it assumes a complete and simultaneous blockage of all the subassembly inlet ports when the reactor is at full power. However, the study of a single subassembly inlet port blockage is justified because it is considered as a limiting accident [START_REF] Paumel | R&D on early detection of the Total Instantaneous Blockage for 4 th Generation Reactors -inventory of non-nuclear methods investigated by the CEA[END_REF][START_REF] Sarkar | Investigation of heat transfer from a totally blocked fuel subassembly of fast breeder reactor with 7 and 19 pin bundles[END_REF]. As it follows from [START_REF] Sarkar | Investigation of heat transfer from a totally blocked fuel subassembly of fast breeder reactor with 7 and 19 pin bundles[END_REF], the TIB formation in a subassembly contained a fuel pin could be initiated by various possible ways such as loading of blocked subassemblies, passing of foreign particles such as weld spatter present in primary circuit through core via coolant, clogging by broken spacer wire, etc.

This paper is devoted to the sequential (on-line) statistical detection of a TIB occurred in a single subassembly by detecting an abnormal temperature rise with the TCs at the top of the fuel subassemblies that monitor the neighboring subassemblies. This temperature rise is a consequence of the heating of adjacent subassemblies by conduction through the hexcans. Because the inlet port blockage, the temperature measured above the subassembly with a TIB is not representative for the TIB detection. The early detection of an abnormal temperature rise provoked by a TIB has been previously considered in [START_REF] Straka | Some techniques for computerized LMFBR -subassembly outlet temperature monitoring based on estimation theory[END_REF][START_REF] Hartert | Dynamic detection of nuclear reactor core incident[END_REF][START_REF] Martinez-Martinez | Two neural network based strategies for the detection of a total instantaneous blockage of a sodium-cooled fast reactor[END_REF]. The recursive filtering based on the statistical estimation theory is used to reduce the outlet temperatures uncertainties in [START_REF] Straka | Some techniques for computerized LMFBR -subassembly outlet temperature monitoring based on estimation theory[END_REF]. Next, the filtered TCs signals are used to detect a local temperature rise due to a TIB. The TIB detection algorithms based on the artificial neural network-based strategies are used in [START_REF] Hartert | Dynamic detection of nuclear reactor core incident[END_REF][START_REF] Martinez-Martinez | Two neural network based strategies for the detection of a total instantaneous blockage of a sodium-cooled fast reactor[END_REF]. Nevertheless, some problems hold unsolved, especially, the calculation of the probabilities of missed detection and false alarm. These probabilities are crucially important to estimate the Technological Readiness Level (TRL) for the early TIB detection based on the core outlet temperatures.

The proposed TIB detection algorithm is composed of two parts. The first part is an adaptive estimation of the stochastic-dynamic model of subassembly outlet temperatures. This model is used to generate the residuals sensitive with respect to (w.r.t.) the abnormal temperature rise due to a TIB and, simultaneously, insensitive w.r.t. normal operating temperature variations. The second part is the statistical sequential test, which uses the residuals generated by the stochastic-dynamic model of subassembly outlet temperatures to detect the temperature rise due to a TIB.

The original contributions of this paper are as follows :

• The detection of an abnormal temperature rise due to a TIB is reduced to the problem of reliable detection of transient changes in the temperature signals by using a parametric model of the subassembly outlet temperature.

• The Finite Moving Average (FMA) test for a reliable detection of a TIB is combined with the recursive adaptive Least Squares (LS) estimator. This LS estimator is used to reduce a negative impact of the outlet temperatures uncertainties on the FMA test.

• In contrast to the previous publications, the asymptotic upper bounds for the probabilities of missed detection and false alarm are proposed in the current paper. These upper bounds permit to predict the statistical properties of the proposed solution.

• The performance of the proposed FMA test has been evaluated with records of normal operating outlet temperatures collected on the French SFR "Phénix" by the CEA. The abnormal temperature rise in the neighboring subassemblies has been simulated.

This paper is organized as follows. Section 2 states the problem of a TIB detection.

The temperature records from the TCs above the subassemblies used for this study are presented in Section 3. These records provided to us by the CEA, Cadarache. Next, an explanatory parametric model describing the subassembly outlet temperature is developed in Section 4. This parametric model is non-stationary. An adaptive parameter estimation algorithm is proposed in Section 5. The reliable detection of transient changes, the nuisance parameter rejection and the design of the FMA test combined with an adaptive estimator are discussed in Section 6. The statistical properties of the detector and its evaluation with real data are also discussed in Section 6. Finally, some conclusions are drawn in section 7.

The TIB of a subassembly

The subassembly plays an essentially hydraulic role. It guarantees the channelization of the sodium flow, which cools the fuel pins. Following a TIB, the subassembly is cooled only by the "inter-assembly" sodium flow and the sodium contained in the subassembly very quickly reaches the vaporization temperature. After evaporation of the sodium, the steel tubes of the fuel pins melt and partially solidify in the bottom of the assembly remained cold. The fuel pellets then reach the melting temperature and form a molten bath placed above the steel of the tubes. Lateral cooling by interassembly sodium flow results in the formation of a solid fuel crust along the hexagonal steel casing that temporarily prevents its melting [START_REF] Jeanne | Modélisation des transferts thermiques par conduction et rayonnement, dans une géomtrie quelconque, multicomposant en multiphase[END_REF]. The rest of the incident depends essentially on the existence of a plug in the upper part of the assembly. At the time of melting the fuel pellets, it is highly likely that the fission gases thus released cause molten material to the top of the assembly. As a result of their re-solidification a plug, possibly porous, could be formed.

Several methods of a TIB early detection for the 4-th generation French SFR are considered in [START_REF] Paumel | R&D on early detection of the Total Instantaneous Blockage for 4 th Generation Reactors -inventory of non-nuclear methods investigated by the CEA[END_REF]. As it follows from [START_REF] Paumel | R&D on early detection of the Total Instantaneous Blockage for 4 th Generation Reactors -inventory of non-nuclear methods investigated by the CEA[END_REF], the highest TRL can be attributed to the method of early detection based on the core outlet temperatures collected from the TCs above the individual subassemblies. The outlet temperature is a permanently monitored parameter for obvious safety reasons. The temperature measurements at the top of the fuel subassemblies allow the operator to permanently have a map of the radial distribution of temperature output core. In the French concepts ("Phénix" and "Superphénix"), the core outlet sodium temperature is monitored by TC located inside a thimble (thermowell) above each subassembly [START_REF] Paumel | R&D on early detection of the Total Instantaneous Blockage for 4 th Generation Reactors -inventory of non-nuclear methods investigated by the CEA[END_REF]. The TIB of a subassembly causes a rise in temperature of the heat transfer fluid of neighboring subassemblies while the temperature measured above the subassembly with a TIB is not representative for its detection because the inlet port blockage.

The principle of the early TIB detection based on the core outlet temperatures is to detect an abnormal temperature rise with the TCs that monitor the neighboring subassemblies. This temperature rise is a consequence of the heating of adjacent subassemblies by conduction through the hexcans. It has been estimated that this heating resulted in an increase of the neighboring outlet temperatures of a few tenths of • C/s per second for 10 s (until the hexcans rupture).

Up to our knowledge, the very first recursive filtering method to reduce the outlet temperatures uncertainties based on the statistical estimation theory has been proposed and tested with real data in [START_REF] Straka | Some techniques for computerized LMFBR -subassembly outlet temperature monitoring based on estimation theory[END_REF]. The filtered signals used to detect a local temperature rise by using a rather rudimentary algorithm. The current paper can be seen as a further development of the method proposed in [START_REF] Straka | Some techniques for computerized LMFBR -subassembly outlet temperature monitoring based on estimation theory[END_REF] by using the modern adaptive statistical estimation and detection algorithms based on the temperature measurements from the TCs located above the subassemblies. This involves studying the sensitivity of TCs for the detection of a local temperature rise of at least two neighboring subassemblies while respecting the probability of non-detection, the probability of false alarm and the maximum detection delay. The early detection of the TIB makes it possible to prevent the propagation of this fusion to the entire active core, i.e. adjacent assemblies, which could result in the destruction of the reactor envelope, or even the containment of reactor.

Real data used in this study

The temperature measurements used in this study come from the French SFR "Phénix "(today at final shutdown). They were collected in 2009 during the end-of-life tests of the plant. These data were provided by the center of the CEA in Cadarache. The core outlet temperatures have been collected from the TCs, placed into the control plug for real-time observation of sodium outlet temperatures from individual subassemblies. The data acquisition system has been equipped with a 8-bit analog-to-digital converter. The spatial distribution of TCs in the core is illustrated in Figure 2(a).

For this study, we have used the dataset containing the temperature measurements from 121 TCs located above the centers of the subassemblies shown in Figure 2(a). The TCs provide the temperature measurement every three seconds, i.e. the sampling period is ∆T = 3 s. This dataset is composed of two samples covering two periods. The instantaneous temperature field measured at the positions (u i , v i ), i = 1, . . . , 121, is represented in Figure 2(b). First of all, it is worth noting that the spatial distribution of the temperature is not symmetrical. We can also notice that the temperature is more important in the center than on the edges. The average temperature in the center is approximately equal to 560 • C and 460 • C in the periphery. The temperatures do not vary in a homogeneous way over the whole core. Since the behavior of the temperature depends on the position of the TC, it seems appropriate to take into account the positions of the TCs to describe the evolution of the temperature.

In the following, we will try to build a model that describes the variations of temperature signals in the absence of local temperature increase associated with a TIB.

Nominal model of the subassembly outlet temperature

This section is devoted to a stochastic-dynamic model of subassembly outlet temperatures measured by a small group of neighboring TCs under normal operating conditions (without a TIB). The goal of this model is to define a projection of the observed data on a specially designed subspace, free from the normal operating variations of the outlet temperatures vector Y ∈ R n measured each ∆T seconds at the positions (u i , v i ), i = 1, . . . , n.

From the preliminary data analysis, it has been established that an adequate description of the local temperature field can be done by the AutoRegressive Moving -Average model with eXogenous inputs, i.e. ARMAX (p, q) model, where p is the order of the AR part and q is the order of the MA part. In the sequel, our attention is restricted to a subclass of ARMAX (p, q) model, i.e. to the AutoRegressive model with eXogenous inputs ARX (p) model. There are two reasons for this :

• Despite the fact that the ARMAX (p, q) model is more flexible than the ARX (p) model, the parameter estimation in the case of ARMAX (p, q) model leads to a nonlinear minimization problem with the upper bound for the number of local minima comparable to the sample size. Adaptive parameter estimation in the 170 case of ARMAX (p, q) model is even a more difficult problem because the effective sample size is constant. Hence, the probability to reach a wrong minimum is positive. In contrast to the ARMAX (p, q) model, the parameter estimation in the case of ARX (p) model is reduced to a quadratic minimization problem (with a single minimum). It can be easily solved by using a conventional recursive LS 175 algorithm.

• The approximation of the MA polynomial 1 -q i=1 B i β i of order q by an AR polynomial 1 -p i=1 B i α i of suitably high order p is frequently used

1 - q i=1 B i β i ≃ 1 1 - p i=1 B i α i , p ≫ q (1)
where B i is the operator of the delay (backshift operator) :

B i Y t = Y t-i .
Hence, the ARX (p), p = p + p 1 , model can be used as an approximation of the AR-MAX (p 1 , q) model. Let us consider that a group of TCs, i.e. a potentially accidental subassembly (TIB) with its neighbors, is composed of n sensors installed at the positions (u i , v i ), i = 1, . . . , n. This situation is represented in Figure 3. We consider that the scalar temperature field Y measured each ∆T seconds at the positions (u i , v i ), i = 1, . . . , n, is locally modeled by the following dynamic-stochastic ARX (p) equation [START_REF] Box | Time Series Analysis : Forecasting and Control[END_REF][START_REF] Anderson | The Statistical Analysis of Time Series[END_REF][START_REF] Brilinger | Time Series, Data Analysis and Theory[END_REF]]

Y t = I n - p i=1 B i A i Y t = H t X + ξ t , t = 1, 2, . . . , T (2)
where Y t = (y t,1 , . . . , y t,n ) T ∈ R n is the vector of observations (temperatures) of n TCs at the discrete time instant t, I n is the identity (n × n) matrix H t X is a exogenous inputs term, which is used to model the behavior of the instantaneous scalar field of temperatures, X ∈ R ℓ is the vector of the unknown inputs (regressors),

p i=1 B i A i ) is the AR polynomial of order p, A i = diag{α i,1 , . . . , α i,n } is a diagonal (n × n) matrix of AR coefficients, α i,j is the AR coefficient, {ξ t } t≥1 ∈ R n is a sequence of i.i.d. random vectors from a zero-mean Gaussian distribution N (0, Σ), Σ = diag{σ 2 1 , . . . , σ 2 n }. (1 -
Let us discuss the main elements of the ARX (p) model given by [START_REF] Chellapandi | Thermal hydraulic issues and challenges for current and new generation FBRs[END_REF]. The role of the linear explanatory term with exogenous inputs is to model the instantaneous behavior of the temperature field

Y t = H t X + ξ t , t = 1, 2, . . . , T (3)
where

Y t = I n - p i=1 B i A i Y t
is the filtered vector of temperatures. To describe the temperature field, a polynomial in u and v has been chosen. Hence, the filtered temperature y t,i measured by the TC i located in the position (u i , v i ) at time t is given by

y t,i = h t,i X + ξ t,i = 2 k=0 2 m=0 u m i v k i a m,k + p t b + ξ t,i , i = 1, . . . , n (4)
where p t is the core thermal power at time t measured in megawatt (MW) and the row vector h t,i of matrix H t is given by

h t,i = (u 2 i v 2 i , u i v 2 i , v 2 i , u 2 i v i , u i v i , v i , u 2 i , u i , 1, p t ), i = 1, . . . , n
The vector X ∈ R ℓ , where ℓ = 10, represents the unknown nuisance parameters composed of the polynomial coefficients and the coefficient of the core thermal power. The first 9 coefficients of the vector

X = (a 2,2 , a 1,2 , a 0,2 , a 2,1 , a 1,1 , a 0,1 , a 2,0 , a 1,0 , a 0,0 , b)
T reflet the geometric factors (spatial correlation between neighboring TCs) and the coefficient b reflet the impact of the core thermal power. The matrix

H t of size (n × 10) is composed of n rows h t,i , i = 1, . . . , n.
Discussion. At first glance, it may seem strange that the last two columns coexist in the matrix H t . But it is worth noting that the roles of the last two columns in the matrix H t are different. The ninth column defining the common factor of the polynomial approximation for the temperature field in the core. It is responsible, together with the first 8 columns, for the best possible instantaneous fitting of this polynomial approximation Therefore, the right-hand side of the ARX (p) model given by ( 2) represents a combination of the regressive part defined by the term H t X and the AR part defined by the term 2) and ( 4) can be re-written in the vector form as follows

p i=1 A i Y t-i . Equations (
Y t = H t X + p i=1 A i Y t-i + ξ t , t = 1, 2, . . . , T (5)
and in the scalar form

y t,i = 2 k=0 2 m=0 u m i v k i a m,k +p t b+ p j=1 α j,i y t-j,i +ξ t,i , i = 1, . . . , n, , t = 1, 2, . . . , T (6)
The traditional method of the ARX model estimation is to re-write the above equations ( 5) and [START_REF] Hartert | Dynamic detection of nuclear reactor core incident[END_REF] in the form of regression model (see [START_REF] Anderson | The Statistical Analysis of Time Series[END_REF])

Yt          yt,1 . . . yt,i . . . yt,n          = Ht        yt-1,1 . . . yt-p,1 0 . . . 0 Ht 0 . . . 0 . . . . . . . . . . . . . . . . . . 0 . . . 0 0 . . . 0 yt-1,n . . . yt-p,n        X                  X α1,1 . . . αp,1 . . . α1,n . . . αp,n                  + ξt     ξt,1 . . . ξt,n     (7)
where t = 1, 2, . . . , T and X is a vector of unknown parameters of size q = ℓ + np.

It consists of two sub-vectors. The sub-vector X of size ℓ represents the regression (exogenous) parameters and the sub-vector (α

H t = H t ( H t , Y t-1 , . . . , Y t-p ), t = 1, . . . , T , stacked together Y 1,T =    Y 1 . . . Y T    =    H 1 . . . H T    X +    ξ 1 . . . ξ T    = H 1,T X + ξ 1,T (8)

Adaptive estimation of the outlet temperature model

This section is devoted to the identification of the subassembly outlet temperature model given by ( 7) -( 8) under normal operating conditions (without a TIB). In the case of time-invariant systems, the statistical properties of the LS estimation for the ARX (p) model have been studied in [START_REF] Box | Time Series Analysis : Forecasting and Control[END_REF][START_REF] Anderson | The Statistical Analysis of Time Series[END_REF][START_REF] Brilinger | Time Series, Data Analysis and Theory[END_REF]. It has been shown that the ARX (p) model is reduced to a conventional regression model. Hence, it possesses all optimal properties of a linear regression (at least asymptotically, when T → ∞). It is assumed that the covariance matrix of the random noise

ξ 1,T ∈ R nT is scalar, i.e. σ 2 1 = • • • = σ 2 n and cov(ξ 1,T ) = σ 2 I nT .
There are two reasons for this : i) all the outlet temperatures are measured by the same K-type TCs and these temperatures belong to the same interval, i.e. all the TCs have approximately the same Standard Deviation (SD) σ i of instrumental errors; ii) the heteroscedasticity has no negative impact on the detection algorithm (see details in Section 6.2). Hence, the LS estimation and its residual vector are given by using the batch of T data

X = H T 1,T H 1,T -1 H T 1,T Y 1,T , Ψ 1,T = P H1,T Y 1,T = Y 1,T -H 1,T X (9)
where Ψ 1,T = (ε T 1 . . . ε T T ) T is the vector formed from residuals ε 1 , . . . , ε T stacked together and P H1,

T = I nT -H 1,T (H T 1,T H 1,T ) -1 H T 1,
T is a projection matrix. Putting together equations ( 7) -( 9), we get the covariance matrix of the residual ε t :

Σ ε = cov (ε t ) = (I n -B) Σ, B = E H T (H T 1,T H 1,T ) -1 H T T (10)
The outlet temperature model estimation should be causal because it will be used for the sequential detection of a TIB. A preliminary analysis of the real data revealed that the behavior of the temperature vector {Y t } t≥1 of n TCs at geometric positions (u i , v i ), i = 1, . . . , n is non-stationary. Therefore, for this reason, the regression model ( 7) -( 8) must be not only sequentially estimated but also permanently adapted in realtime. To avoid the computational burden, it is preferable to use the Recursive LS (RLS) algorithm to estimate/adapt the ARX (p) model given by ( 7) - [START_REF] Jeanne | Modélisation des transferts thermiques par conduction et rayonnement, dans une géomtrie quelconque, multicomposant en multiphase[END_REF]. The RLS algorithm minimizes the following quadratic function of the observations (for details see [START_REF] Ljung | Theory and practice of recursive identification[END_REF])

J T (X, A 1 , . . . , A p ) = T t=1 λ T -t Y t -H t X - p i=1 A i Y t-i 2 2 (11)
where 0 < λ ≤ 1 is the forgetting factor. For the time-variant (non-stationary) system, it is necessary to have a prior model of system dynamics to design an optimal estimation algorithm. In the case of the subassembly outlet temperature, such a priori information is absent. For this reason, the tracking properties of the RLS is defined by a single tuning parameter λ, (for details see [START_REF] Ljung | Theory and practice of recursive identification[END_REF]). Usually, the forgetting factor is λ ∈ [0.99, 0.9999]. The RLS algorithm is defined as follows [START_REF] Ljung | Theory and practice of recursive identification[END_REF][START_REF] Gustafsson | Adaptive Filtering and Change Detection[END_REF]

                 K t = 1 λ P t H T t I n + 1 λ H t P t-1 H T t -1 P t = 1 λ (P t -K t H t P t-1 ) X t = X t-1 + K t Y t -H t X t-1 ε t = Y t -H t X t , t = 1, 2, . . . , T (12)
where the initial estimation X 0 is assumed to be Gaussian X 0 ∼ N (X 0 , P 0 ). If there is no a priori information available, it can be assumed that X 0 = 0 and P 0 = γI q with 205 γ large. Let us consider a group of 9 subassemblies with the following TC numbers (see Figure 2 The role of the ARX (20) model in the TIB detection algorithm is the generation of the residuals {ε t } t≥p+1 . First of all, let us simplify the covariance matrix Σ ε of residuals given by [START_REF] Anderson | The Statistical Analysis of Time Series[END_REF]. Let δ = max 1≤i,j≤n |b i,j |, where b i,j is the entry in the i-th row and j-th column of the matrix B. As it follows from [START_REF] Ljung | Theory and practice of recursive identification[END_REF][START_REF] Gustafsson | Adaptive Filtering and Change Detection[END_REF], the efficient sample size T of the LS estimator using a batch of T data is given as a function of the forgetting factor λ : T = (1 + λ)/(1λ). For the forgetting factor λ taken from the interval [0.999, 0.9999], the efficient sample size T belongs to the interval [2

• 10 3 , 2 • 10 4 ].
For such interval of T , typical values of δ obtained by using two 7-day periods of the sodium outlet temperatures belong to the interval [10 -2 , 10 -3 ]. Hence, it is assumed that Σ ε ≃ Σ = diag{σ 2 1 , . . . , σ 2 n } in the sequel. In order to check the main statistical assumptions about the residuals {ε t } t≥p+1 of the ARX (20) model [START_REF] Box | Time Series Analysis : Forecasting and Control[END_REF][START_REF] Anderson | The Statistical Analysis of Time Series[END_REF][START_REF] Ljung | Theory and practice of recursive identification[END_REF][START_REF] Gustafsson | Adaptive Filtering and Change Detection[END_REF], i.e. their Gaussianity and the absence of serial and cross correlations, the following statistics have been calculated for the group of 9 TCs (see [START_REF] Gustafsson | Adaptive Filtering and Change Detection[END_REF]) :

• The histogram of the ARX (20) model residuals {ε t } t≥p+1 .

• The empirical AutoCorrelation Function (ACF) of the residuals {ε t } t≥p+1 .

• The empirical correlation matrix of the residuals {ε t } t≥p+1 .

The typical histograms of residuals (corresponding to TCs 24 -21 and 25 -19) are shown in Figure 5(a) and (b). The red line shows the superimposed fitted Gaussian Probability Density Function (PDF) calculated by using the empirical mean and SD. The residuals of TC 24 -21 are better fitted to a Gaussian PDF than the residuals of TC Fig. 6. A one hour registration of the temperatures of 9 TCs (see ( 13)).

25 -19. This fact is explained by a relatively low resolution of 8-bit analog-to-digital converters. This situation is illustrated by Figure 6. As it follows from Figure 6, the quantization step size of some TCs is too rough w.r.t the SD of signals. The empirical ACFs of residuals are shown in Figure 7(a) and (b). As it follows from Figure 7, the empirical ACFs of residuals {ε t } t≥p+1 look like delta-functions. Hence, the hypothesis about the absence of serial correlations seems to be likely. Finally, the empirical correlation matrix corr(ε t ) of the residuals {ε t } t≥p+1 and its determinant are

corr(ε t ) =                1
               (14)
230

The non-diagonal coefficients of the matrix corr(ε t ) are close to zero and det [corr(ε t )] = 0.88 is close to one. From these two facts, the hypothesis about the absence of cross-correlations between the components {ξ t,i } t≥p+1 and {ξ t,j } t≥p+1 , where 1 ≤ i = j ≤ 9, of the random noise {ξ t } t≥1 also seems to be likely.

Let us switch to the results of the ARX (20) model estimation. The RLS estimations

235

(with λ = 0.9995) of the X (exogenous) and AR part coefficients as functions of elapsed time in hours are shown in Figures 89. For the AR part of the model, only the first five coefficients α 1,i , . . . , α 5,i , i = 1, . . . , n, calculated for TC 23 -20, are shown in Figures 89. Other AR coefficients, α 6,i , . . . , α 20,i , i = 1, . . . , n, are less significant. It is clear that the variations of the RLS estimations are more important for the second sample, due to some changes in the core thermal power of the SFR, as compared to those from the first sample. It can be concluded from Figures 89that the outlet temperatures represent strongly non-stationary signals even for a stationary operating mode of the SFR.

The crucially important parameter for the design of a TIB detection algorithm is the SD of the random noise {ξ t } t≥1 . This parameter defines the Signal-to-Noise Ratio (SNR) in the sequel. Obviously, the true SD is unknown. For this reason, the unknown SD will be replaced with its empirical estimation. The empirical SD of the residuals {ε t } t≥p+1 for the group of 9 TCs (see [START_REF] Gustafsson | Adaptive Filtering and Change Detection[END_REF]) are collected in Table 1

Statistical detection of a single subassembly TIB

The aim of this section is to study the feasibility of early statistical detection of a single subassembly TIB. We will propose a sequential test detecting a local increase in the sodium outlet temperatures from individual subassemblies neighboring a potential TIB and calculate the probability of missed detection and the probability of false alarm 260 provided that the delay for detection is upper bounded by a given constant for this test. The final goal is therefore to provide a complementary tool to improve the early detection of a single subassembly incident (TIB) in the context of nuclear safety.

Sequential reliable detection of abrupt changes

Let us briefly introduce the sequential detection of abrupt changes in the properties of stochastic processes [START_REF] Basseville | Detection of Abrupt Changes: Theory and Application[END_REF][START_REF] Lai | Sequential analysis : some classical problems and new challenges (with discussion)[END_REF][START_REF] Poor | Quickest detection[END_REF][START_REF] Tartakovsky | Sequential Analysis : Hypothesis Testing and Changepoint Detection[END_REF]. We begin with Lorden's minimax criterion of optimality [START_REF] Lorden | Procedures for reacting to a change in distribution[END_REF], which involves the minimization of the (worst-case) mean detection delay provided that a prescribed level of false alarms is respected 3 . Let us consider the non-Bayesian framework, where the "change-point" t 0 is unknown but non-random. This non-Bayesian framework is realistic for many safety-critical systems. In opposition the Bayesian framework, it is assumed that there is no a priori information on the distribution of t 0 . Let {y t } t≥1 be a sequence of independent random variables and let t 0 be the index of the first post-change observation where F 0 is the pre-change Cumulative Distribution Function (CDF) and F 1 is the post-change CDF. Let P t0 be the joint distribution of the observations y 1 , y 2 , • • • , y t0 , y t0+1 , • • • when t 0 < ∞. Let P 0 denote the same when t 0 = ∞, i.e. there is no change and all the observations y 1 , y 2 , • • • are i.i.d. with CDF F 0 . Let E t0 (resp. E 0 ) and P t0 (resp. P 0 ) be the expectation and probability w.r.t. the distribution P t0 (resp. P 0 ). Lorden [START_REF] Lorden | Procedures for reacting to a change in distribution[END_REF] proposed an optimality criterion, which involves the minimization of the worst-case mean detection delay

y t ∼ F 0 if n < t 0 F 1 if n ≥ t 0 (15)
inf T ∈Cη E(T ) def = sup t0≥1 esssup E t0 (T -t 0 + 1) + |y 1 , . . . , y t0-1 (16)
where (x) + = max(0, x), T ∈ {1, 2, . . .} is a stopping time w.r.t. the sequence of random variables {y t } t≥1 , i.e. an integer random variable such that, for every t ≥ 1, the event {T = t} depends only on the variables y 1 , y 2 , ..., y t , among all stopping times T belonging to the class

C η = {T : E 0 (T ) ≥ η} ( 17
)
where η > 0 is a prescribed value of the Average Run Length (ARL) to a false alarm.

The traditional criterion, like ( 16) -( 17), involves the minimization of the (worstcase) mean detection delay provided that the ARL to a false alarm is lower bounded by a given constant. It is based on the idea initially proposed by Wald [START_REF] Wald | Sequential Analysis, ser[END_REF] and motivated by the economic criteria in quality control when the price of each new observation is constant. Using criterion ( 16) -( 17), we accept that some run lengths will be very long, some other -very short, but, in the mean, the detection delay will be acceptable. In safety-critical applications (like a TIB detection), if the delay for detection is greater than the required time-to-alert N , the price of each new observation is infinitely more important than the price of observation if the delay for detection is less than or equal to N . For this reason, we propose to use another criterion of reliable detection, which involves the minimization of the worst-case probability of missed detection provided that the worst-case probability of false alarm per a given reference period m α is upper bounded.

The reliable detection of (transient) changes is motivated by two possible scenarios. The first scenario corresponds to the situation when the observed phenomena is of short and maybe unknown (and random) duration Γ. It is a so-called transient change detection problem. Sometimes even the "latent" detection (i.e. the detection after the end of transient change) is acceptable [START_REF] Han | A detection optimal minmax test for transient signals[END_REF][START_REF] Streit | Detection of random transient signals via hyperparameter estimation[END_REF][START_REF] Han | Some methods to evaluate the performance of Page's test as used to detect transient signals[END_REF][START_REF] Wang | A performance study of some transient detectors[END_REF][START_REF] Wang | A variable threshold Page procedure for detection of transient signals[END_REF]. The second scenario arises when the observed anomaly (a TIB, for example) leads to serious degradation of the system performance/safety when the anomaly is detected with the delay greater than the required time-to-alert N . It is a so-called reliable detection of (transient) changes. In the framework of this second scenario, the duration Γ is assumed to be sufficient, i.e. N ≤ Γ. A change detected with the delay greater than N , i.e. Tt 0 + 1 > N , is assumed to be missed [START_REF] Bakhache | Reliable detection of faults in measurement systems[END_REF][START_REF] Guepie | Sequential Monitoring of Water Distribution Network[END_REF][START_REF] Guépié | Sequential detection of transient changes[END_REF][START_REF] Guépié | Detecting a suddenly arriving dynamic profile of finite duration[END_REF]. On the other hand, if the true duration Γ of the transient change is smaller, than the required time-to-alert N , such a transient change is considered as less dangerous because its impact on the system is limited or negligible.

Let us formalize the reliable change detection problem. We consider the sequence of random variables {y t } t≥1 . Let t 0 be the index of the first post-change observation (unknown and non-random) and the post-change period is of sufficient duration Γ such that N ≤ Γ. Let us define the generative model of the transient change for the prescribed duration N

y t ∼ F 0 if 1 ≤ t < t 0 F θt-t 0 +1 if t 0 ≤ t ≤ t 0 + N -1 (18)
where F 0 is the pre-change CDF and F θ1 , . . . , F θN are the known post-change distributions during the period N . It is worth noting that the transient change profile is defined only for N observations after change because all what happens after the time t 0 + N -1 is considered as a missed detection.

As previously, it is assumed that P t0 is the joint distribution of the observations

y 1 , • • • , y t0 , y t0+1 , • • • when t 0 < ∞.
Because the considered subclass of stopping times is based on the variable-threshold truncated Sequential Probability Ratio Tests (SPRT), the existence of a short "pre-heating" period N is assumed. This short period is necessary to accumulate the first N observations y 1 , y 2 , ..., y N in order to avoid the situation when the truncated SPRT performs by using an insufficient number of observations. The quality of a statistical decision cannot be guaranteed if the number of observations is less than the required time-to-alert N , which is also equal to the minimum duration of the post-change period Γ, i.e. N ≤ Γ. Finally, the optimality criterion utilized in this report is given by [START_REF] Guepie | Sequential Monitoring of Water Distribution Network[END_REF][START_REF] Guépié | Sequential detection of transient changes[END_REF][START_REF] Guépié | Detecting a suddenly arriving dynamic profile of finite duration[END_REF] inf

T ∈Cα P md (T ) def = sup t0≥N P t0 (T -t 0 + 1 > N | T ≥ t 0 ) (19)
where P md (T ) is the worst-case probability of missed detection, among all stopping times T ∈ C α satisfying

C α = T : P fa (T ; m α ) def = sup ℓ≥N P 0 (ℓ ≤ T < ℓ+m α ) ≤ α (20)
where P fa (T ; m α ) is the worst-case probability of false alarm during the reference period m α measured in discrete time.

The design of the transient change detector is discussed in [START_REF] Guepie | Sequential Monitoring of Water Distribution Network[END_REF][START_REF] Guépié | Sequential detection of transient changes[END_REF][START_REF] Guépié | Detecting a suddenly arriving dynamic profile of finite duration[END_REF]. Let us briefly define this FMA test, which has been obtained as a result of optimisation in a subclass of truncated SPRT. The stopping time T FMA of the FMA test is given as follows

T FMA = inf t ≥ N : Λ t t-N +1 ≥ h , Λ t t-N +1 = t i=t-N +1 log f θN-t+i (y i ) f 0 (y i ) (21)
where Λ t t-N +1 denotes the Log-Likelihood Ratio (LLR) calculated for the moving window y t-N +1 , . . . , y t , f 0 denotes the pre-change probability density function (PDF), f θN-t+i denotes the post-change PDF and h is the threshold. In the context of a TIB detection, f 0 is the zero-mean Gaussian PDF of the ARX(p) pre-change residuals and f θN-t+i is the Gaussian PDF of the ARX(p) post-change residuals.

To get upper bounds for the probabilities P md (T FMA ) and P fa (T FMA ; m α ) defined in criterion ( 19) - [START_REF] Tartakovsky | Sequential Analysis : Hypothesis Testing and Changepoint Detection[END_REF], it is necessary to respect some technical conditions and define several probabilities and their bounds. Further details and results can be found in [START_REF] Guépié | Sequential detection of transient changes[END_REF][START_REF] Guépié | Detecting a suddenly arriving dynamic profile of finite duration[END_REF]. The worst-case probability of missed detection P md (T FMA ) is upper bounded as follows [START_REF] Guépié | Detecting a suddenly arriving dynamic profile of finite duration[END_REF]

P md (T FMA ) ≤ G(h) def = P t0 Λ N +t0-1 t0 < h , t 0 ≥ N (22)
where P t0 Λ N +t0-1 t0 < h corresponds to the probability of missed detection of the Neyman-Pearson LLR test calculated for the time window [t 0 , t 0 + N -1] and the threshold h. It is assumed that the CDF of the LLR Λ t t-N +1

x → F (x)

def = P 0 Λ t t-N +1 ≤ x (23)
is a continuous function on ] -∞; ∞[ under the measure P 0 . The smallest value α 1 of the upper bound G(h) provided that the upper bound for the worst-case probability of false alarm is equal to a pre-assigned value α 0 , is given by [START_REF] Guépié | Detecting a suddenly arriving dynamic profile of finite duration[END_REF] α

1 = G F -1 (1 -α 0 ) 1 mα (24)
Finally, the optimal threshold is given by [START_REF] Guépié | Detecting a suddenly arriving dynamic profile of finite duration[END_REF]

h = F -1 (1 -α 0 ) 1 mα (25)

Rejection of nuisance parameters and adaptive detection

A key issue in fault detection is to state the significance of the observed deviations (e.g., a local temperature rise due to a TIB) w.r.t. random noises and nuisance parameters. Handling the presence of nuisance parameters is indeed an important issue in this framework. Distinguishing two subsets of components of the parameter vector, the parameters of interest and the nuisance parameters, is necessary because some parameters of no interest for monitoring. The nuisance parameters, if not of no physical meaning, may appear in the model for its flexibility or specification, or for data interpretation reasons [START_REF] Basseville | Handling nuisance parameters in systems monitoring[END_REF].

Let us consider a group of n TCs with a potentially accidental subassembly (TIB). We will propose a sequential test detecting a local increase in the sodium outlet temperatures of two subassemblies neighboring a potential TIB. First of all, let us define the generative model of a local increase in the sodium outlet temperatures by using equation ( 5) of the ARX (p) model

Z t = Y t if 1 ≤ t < t 0 Y t + θ t-t0+1 if t 0 ≤ t ≤ t 0 + N -1 , Y t = H t X + p i=1 A i Y t-i + ξ t (26)
where Z t is the vector of sodium outlet temperatures measured by a group of TCs, t 0 is the TIB arrival time, θ 1 , . . . , θ N are the vectors of additive outlet temperature profiles due to a TIB, θ i ∈ R n . Obviously, the period of a local temperature rise due to a TIB is not limited by N but, as it follows from the criterion ( 19) - [START_REF] Tartakovsky | Sequential Analysis : Hypothesis Testing and Changepoint Detection[END_REF], all what happens after t 0 + N -1 has no impact on the criterion of reliable change detection. For this reason, we limit the discussion to the vectors θ 1 , . . . , θ N in the sequel. The vectors θ 1 , . . . , θ N are parameters of interest (informative parameters) of the transient change model. The vector X = X T , α 1,1 . . . α p,1 . . . α 1,n . . . α p,n T is a nuisance parameter of this model.

An efficient method to manage the nuisance parameters and the parameter of interest is an invariant hypothesis testing approach (if the original problem is invariant) or an adaptive testing method (such as the generalized likelihood ratio test) [START_REF] Scharf | Matched Subspace Detectors[END_REF][START_REF] Fouladirad | Optimal statistical fault detection with nuisance parameters[END_REF][START_REF] Fouladirad | Optimal fault detection with nuisance parameters and a general covariance matrix[END_REF]. Key features of these statistical methods are their ability to handle noises and uncertainties, i.e. to reject nuisance parameters, in order to decide between two hypotheses H 0 (no local temperature rise due to a TIB) and H 1 (there exists a local temperature rise due to a TIB). Let us go back to equation [START_REF] Jeanne | Modélisation des transferts thermiques par conduction et rayonnement, dans une géomtrie quelconque, multicomposant en multiphase[END_REF] and re-write it taking into account the generative model [START_REF] Streit | Detection of random transient signals via hyperparameter estimation[END_REF] with the stacked vectors Z 1,t , Y 1,t , ξ 1,t ∈ R nt and the stacked matrix H 1,t of size

(nt × ℓ + np) Z 1,t = Y 1,t + θ 1,t = H 1,t X + θ 1,t + ξ 1,t (27)
where the stacked vector θ t 1 is defined as follows

θ 1,t = (0 . . . 0) T if 1 ≤ t < t 0 0 . . . 0 θ T 1 . . . θ T t-t0+1 T if t 0 ≤ t ≤ t 0 + N -1 ( 28
)
where 0 is the zero row vector of appropriate size, and apply the method of invariant tests developed in [START_REF] Fouladirad | Optimal statistical fault detection with nuisance parameters[END_REF][START_REF] Fouladirad | Optimal fault detection with nuisance parameters and a general covariance matrix[END_REF] to this regression model [START_REF] Han | Some methods to evaluate the performance of Page's test as used to detect transient signals[END_REF].

The idea of the invariant hypotheses testing approach is based on the existence of the natural invariance of the detection problem w.r.t. a certain group of transformation [START_REF] Lehmann | Testing Statistical Hypotheses, ser. Springer Texts in Statistics[END_REF]Ch. 6]. In contrast to the Bayesian approach, the invariant hypotheses testing theory is based on the nuisance X rejection and, therefore, it does not use any a priori information on the distribution of X. It is worth noting that the drawback of the Bayesian approach in the case of a TIB detection is the following : this approach exploits some a priori information on the distribution of X but this information may be unreliable. Hence, the Bayesian approach is irrelevant to the case of nuisance parameters governed by an unknown environment.

The impact of nuisance parameters, expressed by the term H 1,t X in equation ( 27), defines a subspace in the observation space Z = R nt , i.e. the column space R(H 1,t ) of the matrix H 1,t [START_REF] Fouladirad | Optimal statistical fault detection with nuisance parameters[END_REF][START_REF] Fouladirad | Optimal fault detection with nuisance parameters and a general covariance matrix[END_REF]. Because the unknown nuisance parameter X is nonrandom, the only solution is to eliminate any impact of X on the decision function. Consequently, this solution leads to a projection of Z 1,t on the orthogonal complement R(H 1,t ) ⊥ of the column space R(H 1,t ). The space R(H 1,t ) ⊥ is also well-known under the name "parity space" in the analytical redundancy literature [START_REF] Frank | Fault diagnosis in dynamic systems using analytical and knowledge-based redundancy: A survey and some new results[END_REF]. It is shown in [START_REF] Lehmann | Testing Statistical Hypotheses, ser. Springer Texts in Statistics[END_REF]Ch. 6], that the optimal invariant tests are based on the maximal invariants (principle of invariance). In the case of ( 27) the maximal invariant is given by the projection of Z 1,t onto the left null space of the matrix H 1,t (for details see [START_REF] Fouladirad | Optimal statistical fault detection with nuisance parameters[END_REF][START_REF] Fouladirad | Optimal fault detection with nuisance parameters and a general covariance matrix[END_REF]) :

Ψ 1,t = P H1,t Z 1,t = Z 1,t -H 1,t X, X = (H 1,t T H 1,t ) -1 H T 1,t Z 1,t (29)
where P H1,t is a projection matrix defined in [START_REF] Box | Time Series Analysis : Forecasting and Control[END_REF]. Hence, starting from the changepoint t 0 , the observations Z t and the matrices H t , and H 1,t are "contaminated" by the vectors θ 1 , . . . , θ N of additive outlet temperature profiles due to a TIB.

As we have mentioned above, to design an adaptive detector of transient changes, we use the residual {ε t } t≥p calculated recursively by the RLS [START_REF] Ljung | Theory and practice of recursive identification[END_REF]. Let us re-write the generative model of transient change [START_REF] Streit | Detection of random transient signals via hyperparameter estimation[END_REF] for this residual

ε t ∼ N (0, Σ ε ) if 1 ≤ t < t 0 N (M t-t0+1 , Σ ε ) if t 0 ≤ t ≤ t 0 + N -1 (30)
where Σ ε ≃ Σ = diag{σ 2 1 , . . . , σ 2 n } is the covariance matrix of residuals and M t-t0+1 = E (ε t≥t0 ) is the transient profile (expectation) of residuals due to a TIB.

Finally, let us add the following comment concerning the impact of the heteroscedasticity on the detection algorithm. As it follows from [37, Lemma 1], in the case of an arbitrary covariance matrix Σ of the random noise ξ t in the regression model ( 5) -( 8), the LLR of the maximal invariant statistics can be calculated in two steps. The first step is a projection of Z 1,t onto the "parity space" by using the orthogonal complement R(H 1,t ) ⊥ of the column space R(H 1,t ) assuming that the covariance matrix Σ is scalar. The second step consists in calculating the LLR of an invariant test by using the covariance matrix of the residuals obtained after the first step. The empirical estimations of the diagonal elements σ 2 i of this covariance matrix is given in Table 1. This means that the assumption about a scalar covariance matrix of the random noise ξ in the definitions of the LS and RLS algorithms ( 9) -( 12) has no negative impact on the invariant statistics.

As it follows from Sections 4 and 5, the ARX (p) model defined by equations ( 5) -( 8) requires a permanent adjustment to the real data due to the fact that the SFR is a time-variant (non-stationary) system. Hence, the significance of observed deviations should be established by using an adaptive estimation algorithm described in Section 5.

The idea to associate a change detection algorithm to an adaptive estimation algorithm (e. g., an adaptive RLS estimator) has been considered in [START_REF] Benveniste | The asymptotic local approach to change detection and model validation[END_REF][START_REF] Benveniste | Adaptive Algorithms and Stochastic Approximations[END_REF][START_REF] Gustafsson | Adaptive Filtering and Change Detection[END_REF]. The goal of an adaptive detector of transient changes is to detect the additive outlet temperature profiles θ 1 , . . . , θ N due to a TIB while considering the nuisance parameters as unknown.

To solve this problem, it is proposed to use the residuals {ε t } t≥p calculated by an adaptive RLS estimator given by [START_REF] Ljung | Theory and practice of recursive identification[END_REF].

T FMA = inf t ≥ N : Λ t t-N +1 ≥ h Λ t t-N +1 = t i=t-N +1 j∈J 1 σ 2 j ε i,j m i-t+N,j - m 2 i-t+N,j 2σ 2 j ( 31
)
where J is a subset of TCs neighboring a potential TIB, m t-t0+1,j > 0 is a residual profile for t ≥ t 0 and j ∈ J . As it follows from [START_REF] Guépié | Detecting a suddenly arriving dynamic profile of finite duration[END_REF] :

P md (T FMA ) ≤ Φ h -1 2 d J d J , P fa (T FMA ; m α ) ≤ 1 -Φ h + 1 2 d J d J mα (32)
where

d J = j∈J M j 2 2 σ 2 j
is the total SNR and M j 2 2 = N t=1 m 2 t,j .

Statistical properties of the sequential adaptive detector

The statistical properties of the adaptive detector defined in [START_REF] Guepie | Sequential Monitoring of Water Distribution Network[END_REF] have been examined by using :

• Asymptotic upper bounds for the probabilities of missed detection P md (T FMA ) and false alarm P fa (T FMA ; m α ) given by [START_REF] Guépié | Sequential detection of transient changes[END_REF]. The non-zero components θ t,j ∈ J of the vector θ t of additive outlet temperature rising profiles are defined as follows : θ t-t0+1,j = κ∆T (tt 0 + 1) for t ≥ t 0 and j ∈ J Other parameters of the TIB detection scenario are the following ∆T = 3 s, N = 3, 4, m α = 3600/∆T, σ i ∈ [0.16, 0.35], λ = 0.9995

The asymptotic upper bounds for the probability of missed detection P md (T FMA ) are calculated as functions of the upper bounds for the probability of false alarm The second test realized to evaluate the proposed adaptive detector defined in (31) has been done by using the sodium outlet temperature samples of 9 TCs (see [START_REF] Gustafsson | Adaptive Filtering and Change Detection[END_REF]) covering two 7-day periods together with the corresponding samples of the core thermal power. The decision function Λ t t-N +1 defined in [START_REF] Guepie | Sequential Monitoring of Water Distribution Network[END_REF] for these two samples is presented in Figure 13 detection delay is 6 s for both samples; no false alarms have been reported.

Conclusion

This paper addresses the adaptive detection of an abnormal temperature rise due to a TIB in a single subassembly in the core of a SFR. This problem has been reduced to the reliable detection of transient changes in the residuals of a subassembly outlet temperature parametric model. The proposed adaptive recursive algorithm is composed of two parts : i) an adaptive RLS algorithm, which estimates the ARX (p) model of outlet temperatures measured by a local group of TCs installed at the top of the fuel subassemblies and computes the residuals of this model; ii) the FMA test for the reliable transient change detection, which uses these residuals to calculate the LLR in a moving window and to compare the LLR to a predefined threshold. The statistical properties of this algorithm have been studied by using the asymptotic upper bounds for the probabilities of missed detection and false alarm and by using the real data provided by the center of the CEA in Cadarache.

The following conclusions and perspectives can be drawn :

• The proposed detection algorithm is recursive, numerically stable and efficient.

• If the required time-to-alert is equal to 9 s, the required probability of missed detection is upper bounded by 10 -6 and the required probability of false alarm is upper bounded by 10 -6 per hour, then the minimal detectable temperature rising rate in two subassemblies neighboring a potential TIB is equal to κ ≃ 0.27 • C/s. If the required time-to-alert is 12 s, then the minimal detectable temperature rising rate is equal to κ ≃ 0.2 • C/s.

• The sampling period of temperature records used in this study is ∆T = 3 s. A reduction of the sampling period ∆T to 1 s or 0.5 s can improve the statistical properties of the detection algorithm.

• Several temperature anomalies have been reported during the second 7-day period from 01/03/2009 to 07/03/2009. These anomalies are not very disturbing for the proposed TIB detection algorithm but it can be interesting to examine this phenomena carefully in the case of a future study.

• Several hundred (thousand) local groups of TCs have to be monitored simultaneously to cover a fuel assembly. It is interesting to compute the probability of false alarm for such a stream of parallel data in the case of a future study.

Fig. 1 .

 1 Fig. 1. Fuel assembly of the French SFR "Phénix".

Fig. 2 .

 2 Fig. 2. a) The assembly of 121 fuel rods. b) The instantaneous temperature field measured in • C. Blue crosses indicate the positions (u i , v i ) of TC. Black numbers indicate the temperatures in • C corresponding to the level curves.

Fig. 3 .

 3 Fig.3. Representation of two groups with 9 TCs : a potentially accidental subassembly (TIB) with its neighboring subassemblies.

195

  to the temperature field. The tenth column is responsible for the linear relationship between the outlet temperatures and the core thermal power, which can rapidly variate between the minimum and maximum values as it happens in the case of the second 7day period from 01/03/2009 to 07/03/2009. The coefficients a 0,0 and b of the vector X have quite different dynamics (see Figures8(a) and 9(a)). This coexistence improves 200 the quality of the ARX (p) model given by (2), especially for the second 7-day period from 01/03/2009 to 07/03/2009.

Fig. 4 .

 4 Fig. 4. a) The temperatures of 9 TCs (see (13)) and the reactor thermal power for the first sample covering a 7-day period from 15/02/2009 to 21/02/2009. b) The same parameters for the second sample covering a 7-day period from 01/03/2009 to 07/03/2009.

  (a))23-20, 23-21, 23-22, 24-19, 24-20, 24-21, 25-18, 25-19, 25-20 (13) The outlet temperatures of 9 TCs (see[START_REF] Gustafsson | Adaptive Filtering and Change Detection[END_REF]) and the reactor thermal power for two samples covering 7-day periods from 15/02/2009 to 21/02/2009 and from 01/03/2009 to 07/03/2009 are shown in Figure4(a) and (b). From preliminary analysis, it has been concluded that an adequate description of the outlet temperatures of a group of 9 TCs 210 can be done by using the order of AR-part p = 20. The forgetting factor λ has been chosen from the interval [0.999, 0.9999].

Fig. 5 .

 5 Fig. 5. a) The histogram of the ARX (20) model residuals corresponding to the TC 24 -21 and its Gaussian approximation. b)The same parameters corresponding to the TC 25 -19.

  67 67.1 67.2 67.3 67.4 67.5 67.6 67.7 67.8 67.

Fig. 7 .

 7 Fig. 7. a) The ACFs of residuals {εt} t≥p+1 for the group of 9 TCs (see (13)) for the first sample (15/02/2009 to 21/02/2009). b) The ACFs of residuals for the second sample (01/03/2009 to 07/03/2009).

Fig. 8 .

 8 Fig. 8. a) The X-part coefficients (components of the vector X numbered from 1 to 10), calculated for the first sample (15/02/2009 to 21/02/2009). b) The first 5 AR-coefficients α 1,1 , . . . , α 5,1 , calculated for TC 23 -20 and for the first sample (15/02/2009 to 21/02/2009).

Fig. 9 .

 9 Fig. 9. a) The X-part coefficients (components of the vector X numbered from 1 to 10), calculated for the second sample (01/03/2009 to 07/03/2009). b) The first 5 AR-coefficients α 1,1 , . . . , α 5,1 , calculated for TC 23 -20 and for the second sample (01/03/2009 to 07/03/2009).

6 1 ) 19 Fig. 10 .

 11910 Fig. 10. The profiles of the residuals corresponding to a local increase in the sodium outlet temperatures of two subassemblies 24 -21 and 25 -19 neighboring a potential TIB at the subassembly 24 -20 or 25 -20.The rate of temperature rising is 0.5 • C/s. The temperature sampling period is 3 s.

Fig. 11 .

 11 Fig. 11. The probability of missed detection P md (T FMA ) as a function of the probability of false alarm per hour P fa (T FMA ; mα) for different values of the temperature rising rate κ ∈ {0.200, ..., 0.335} • C/s for two subassemblies 24 -21 and 25 -19 neighboring a potential TIB at the subassembly 24 -20 or 25 -20. The required time-to-alert is 9 s (N = 3).

  (a) and (b). It is assumed that the simulated TIB onset time is 120 h for both samples. The TIB location is the subassembly 24 -20 or 25 -20. The outlet temperature rise in the measurements of the TC 24 -21 and TC 25 -19 with a rising rate of κ ∈ 0.5 • C/s. The transient change profiles are shown in Figure 10. The

Fig. 12 .

 12 Fig. 12. The probability of missed detection P md (T FMA ) as a function of the probability of false alarm per hour P fa (T FMA ; mα) for different values of the temperature rising rate κ ∈ {0.130, ..., 0.275} • C/s for two subassemblies 24 -21 and 25 -19 neighboring a potential TIB at the subassembly 24 -20 or 25 -20. The required time-to-alert is 12 s (N = 4).

Fig. 13 .

 13 Fig. 13. a) The decision function Λ t t-N+1 calculated for the first 7-day period from 15/02/2009 to 21/02/2009. b) The decision function Λ t t-N+1 calculated for the second 7-day period from 01/03/2009 to 07/03/2009. The simulated TIB onset time is 120 h for both periods. The detection delay is 6 s for both periods.

  1,1 . . . , α p,1 . . . α 1,n . . . , α p,n ) T of size np represents the AR parameters. Therefore, after elapsed time T , the total regression model is defined by the following equation with the vectors Y 1,T ∈ R nT (resp. ξ 1,T ∈ R nT ) formed from the vectors Y 1 , . . . , Y T (resp. ξ 1 , . . . ξ T ) stacked together and the matrix H 1,T of size (nT × ℓ + np) formed from the matrices

Table 1 .

 1 . It follows from Sample TC 23 -20 23 -21 23 -22 24 -19 24 -20 24 -21 25 -18 25 -19 25 -20 The estimated SDs of the ARX (20) model residuals.Table1that the residuals' SDs vary between 0.16 and 0.35. The amplitude of their variations is moderate. Hence it can be concluded that the initial hypothesis about a scalar covariance matrix cov(ξ 1,t ) = σ 2 I nt of the random noise ξ 1,t is rather realistic. It is worth noting that the residuals' SDs from Table1correspond approximately to the random error SD obtained for the K-type TCs after a special calibration procedure for

	First	σi 0.17	0.20	0.25	0.16	0.21	0.35	0.22	0.25	0.16
	Second σi 0.18	0.21	0.24	0.17	0.21	0.35	0.22	0.25	0.16

the same temperature range [14, Ch. 12], [15, Ch. 5],

[START_REF] Nakos | Uncertainty Analysis of Thermocouple Measurements Used in Normal and Abnormal Thermal Environment Experiments at Sandia's Radiant Heat Facility and Lurance Canyon Burn Site[END_REF]

.

•

  The sodium outlet temperatures samples covering two periods from 15/02/2009 to 21/02/2009 and from 01/03/2009 to 07/03/2009 together with two samples of the core thermal power synchronized with these temperature samples.

The Bayesian approach to this problem and its optimal solution have been proposed by Shiryaev[START_REF] Shiryaev | The detection of spontaneous effects[END_REF][START_REF] Shiryaev | On optimum methods in quickest detection problems[END_REF].

Acknowledgements

The authors gratefully acknowledge the research and financial support of this work from the Commissariat à l'Energie Atomique et aux Energies Alternatives (CEA), from the group of scientific interest GIS 3SGS, and from the project AAP NEEDS of CNRS, France.


Vanessa Da Cruz

MD Violaine Yvorel

MD François Casteillo

MD Claire Tissot

MD Antoine Luchez

MD Sophie Bayle-Bleuez

MD Pierre Fournel

MD Olivier Tiffet

PhD Michel Péoc'h

PhD Fabien Forest
email: f.forest@univ-st-etienne.fr

Histopathological subtyping is a prognostic factor in stage IV lung adenocarcinoma

Keywords: Lung adenocarcinoma, metastases, KRAS, solid subtype, EGFR

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

Introduction

Lung adenocarcinoma, the most prevalent lung tumor, is a heterogeneous tumor. In primary tumors, lung adenocarcinoma is often made of several histopathological subtypes. In the 2004 World Health Organization (WHO) classification, most lung adenocarcinomas were classified as 'lung adenocarcinoma, mixed subtype' [START_REF] Who | Pathology and Genetics of Tumours of the Lung, Pleura[END_REF]. This classification was replaced by the 2015 WHO classification of thoracic tumors [START_REF] Travis | Weltgesundheitsorganisation, International Agency for Research on Cancer[END_REF].The 2015 classification recommends a subtyping of lung adenocarcinoma based on the predominant histopathological subtype. The recognized subtypes are lepidic, acinar, papillary, micropapillary, and solid, based on the corresponding predominant pattern. These subtypes are more relevant than the former 'mixed subtype' because they are related to the prognosis. For primary tumors, solid and micropapillary subtypes have the worst prognosis, the acinar subtype has an intermediate prognosis, while the lepidic subtype has the most favorable prognosis of all the subtypes.

The following question was raised by the IASLC/ATS/ERS (International Association for the Study of Lung Cancer/American Thoracic Society/European Respiratory Society) in pathology research recommendations: "In specimens from metastatic sites, is there any clinical significance to recognizing histologic patterns, including the predominant pattern?" [START_REF] Travis | International Association for the Study of Lung Cancer / American Thoracic Society / European Respiratory Society International Multidisciplinary Classification of Lung Adenocarcinoma[END_REF].

Between 2000 and 2010, a retrospective study of 100 patients with different sites of metastases and types of samples (70% with visceral metastasis) showed a worse survival for the major solid subtype compared with acinar and micropapillary subtypes [START_REF] Clay | The clinical relevance of pathologic subtypes in metastatic lung adenocarcinoma[END_REF]. In a retrospective study of 97 patients with resected brain metastases of lung adenocarcinoma, the predominant pattern evaluated on the metastatic sample was a major predictor of survival [START_REF] Casteillo | Pathologic Subtypes of Lung Adenocarcinoma Brain Metastasis Is a Strong Predictor of Survival After Resection[END_REF]. The prognostic relevance of histopathological subtypes on visceral metastases of lung adenocarcinoma is not well known and requires validation [START_REF] Clay | The clinical relevance of pathologic subtypes in metastatic lung adenocarcinoma[END_REF][START_REF] Campos-Parra | Relevance of the novel IASLC/ATS/ERS classification of lung adenocarcinoma in advanced disease[END_REF]. This information is relevant because lung adenocarcinoma is often diagnosed at stage IV, and metastatic tissue is often the only material available for the diagnosis, biomarker testing, and the evaluation of prognosis.

The goals of this work on surgically sampled metastases of lung adenocarcinoma were (a) to evaluate the prognostic significance of lung adenocarcinoma subtypes evaluated on visceral metastases, and (b) to compare histopathological subtypes to clinical and molecular factors.

Materials and methods

Patients

The Ethics Committee of the University Hospital of Saint Etienne (IRBN142018/CHUSTE) approved this study. All patients with a metastatic lung adenocarcinoma diagnosed on surgically resected samples in our institution from 2010 to 2019 were included in this study.

For lymph-node metastases, N1, N2, and N3 lymph nodes were not included; only distant lymph nodes were considered. The clinical data of 253 patients were collected from medical and pathological records. The clinical variables were date of birth, sex, date of diagnosis, date of death or last follow-up, smoking status, tumor location, treatment received, and performance status (PS).

Histopathological evaluation

The diagnosis of lung adenocarcinoma was confirmed according to the current WHO classification. Whole sets of slides stained with hematoxylin and eosin (H&E) for each patient were reviewed separately by two pathologists (VDC, FF) and evaluated jointly until consensus was reached. The histopathological patterns were recorded as acinar, papillary, micropapillary, and solid, and the proportion of each subtype was recorded by 5% increments as previously described [START_REF] Travis | International Association for the Study of Lung Cancer / American Thoracic Society / European Respiratory Society International Multidisciplinary Classification of Lung Adenocarcinoma[END_REF]. We also recorded the presence or absence of each pattern. For some cases, the histopathological subtypes could not be recorded and remained unclassifiable.

Immunohistochemistry with anti-thyroid transcription factor 1 (TTF-1) antibody (dilution1/50; clone8G7G3/1; Dako Corporation, Trappes, France) was recorded as negative or positive according to nuclear staining of tumor cells.

Molecular pathology

Molecular testing was performed on the metastatic tumor as described previously [START_REF] Forest | BRAF and HER2 testing in metastatic lung adenocarcinoma: Value of testing on samples with poor specimen adequacy and analysis of discrepancies[END_REF].

Epidermal growth factor receptor (EGFR) mutations were detected with a peptic nucleic acid (PNA) clamp detection kit (Entrogen detection kit, Entrogen Inc., Woddland Hill, CA, USA) or next-generation sequencing (NGS) with the Personal Genome Machine (Thermofisher, Dardilly, France). KRAS (V-Ki-ras2 Kirsten rat sarcoma viral oncogene homolog) mutations were screened with Snapshot, PNA clamp, or NGS, always including exon 2. Exon 15 mutations of BRAF (v-Raf murine sarcoma viral oncogene homolog B) mutations were analyzed with Snapshot or NGS. ERBB2 (HER2) mutations of exon 20 were searched with direct sequencing or NGS.

Statistical analysis

Statistical analysis was performed using R software (version 3.4.1) with R studio software for Windows (version 1.0.143) [START_REF] Core | R: A language and environment for statistical computing[END_REF]. For survival analysis, the 'survival' package was used (version 2.44-1.1) [START_REF] Terry | A Package for Survival Analysis in S[END_REF].

Means and standard deviations for continuous variable were used. The Fisher and chi-square tests were used to compare categorical variables. Overall survival was calculated by the Kaplan-Meier method, and log-rank p values were used to assess significance. Survival curves were constructed with the Kaplan-Meier method. The date of diagnosis to the date of death (or censoring if the patient was alive at the time of last follow-up) was used to calculate overall survival. Median overall survival and upper and lower confidence intervals (CIs) are given. All reported p values are two-tailed; p <0.05 was considered to be significant.

Prognostic significance of clinical and pathological characteristics was analyzed with univariate Cox regression. A multivariate Cox model including parameters significantly

associated with overall survival in univariate analysis was calculated and used to determine hazard ratios (HRs).

Results

Patient characteristics

Two hundred fifty-three patients with a mean age at diagnosis of 69 ± 0.7 years were included; 66% were males. Among all the patients, 161 (63%) were active or passive smokers, 61 (24%) were not smokers, and for 31 patients (13%) the data were not available.

The metastatic sites sampled were pleura for 169 patients (67%), distant lymph node for 26 patients (10%), bone for 23 patients (9%), pericardium for 12 patients (5%), skin for nine patients (3%), adrenal glands for four patients (2%), and other sites (ovary, soft tissue, mouth, liver, testicle, peritoneum) for ten patients (4%). The first-line treatment was chemotherapy for 146 patients (58%), tyrosine kinase inhibitor or immunotherapy for 31 patients (12%), palliative care for 54 patients (21%), and unknown for 22 patients (9%). The Eastern Cooperative Oncology Group (ECOG) performance status at the time of metastasis resection was <2 for 136 patients (54%), ≥2 for 53 patients (21%), and not specified for 64 patients (25%).

Histopathological characteristics

The acinar subtype was predominant in 95 patients (37%), the micropapillary subtype was predominant in 23 patients (9%), the papillary subtype was predominant in 19 patients (7%), and the solid subtype was predominant in 106 patients (42%) (Figure 1). For ten patients (4%) the main histopathological subtype remained unclassifiable due to the paucity of tumor cells or cauterized tissue. The Cohen kappa for major histopathological subtype was 0.90. Seven patients had an ALK rearranged tumor in 202 patients tested, one patient had a ROS1rearranged tumor in 65 tested, three patients had an ERBB2 mutation in exon 20 in 110 tumors tested, and five patients had a p.V600 mutation of BRAF in 153 samples tested.

Correlation between overall survival and clinical data

The correlation between clinical data and overall survival is showed in Table 1 andFigure Sex, metastatic sites, and intrathoracic versus extrathoracic metastatic sites were not related to overall survival.

Correlation between overall survival and histopathological and molecular data

The correlation between histopathological and molecular data and overall survival is shown in Table 1 and Figure 2. The presence of solid subtype is correlated with overall survival (p = 0.05). Patients with the solid subtype have a shorter overall survival (6.8 months) than patients without the solid pattern (11.1 months). The presence of papillary subtype is correlated with a longer overall survival: 17.5 months when present and 6.9 months when absent (p = 0.01). The presence of acinar or micropapillary subtypes was not related to overall survival (p = 1 and p = 0.8, respectively). Major architectural pattern was not related to overall survival (p = 0.5) (Table 1). TTF-1 positivity was related to overall survival. Patients who were TTF-1-positive had a median overall survival of 11.2 months, whereas patients whose tumor was negative for TTF-1 had a shorter overall survival of 4 months (p <0.001).

The presence of KRAS mutation was significantly correlated with overall survival. Patients with a mutation in exon 2 had a shorter survival of 4.6 months, whereas patients without a mutation had a median overall survival of 10.4 months (p <0.001).

The presence of rearranged ALK was significantly correlated with overall survival. Patients with a rearranged ALK had a longer survival of 57.3 months, whereas patients without mutation had a median overall survival of 7.6 months (p = 0.007).

EGFR, BRAF and ERBB2 mutations and ROS1 rearrangement were not related to overall survival (Table 1).

Association between clinicopathological factors and survival

We used univariate and multivariate Cox proportional hazards regression models to identify independent prognostic factors for overall survival. In univariate analysis, age, ECOG performance status, the type of first-line treatment, the presence of solid subtype, KRAS mutation, ALK translocation, and TTF-1 status were all related to overall survival (Table 2).

The multivariate analysis included all factors significantly related to overall survival in univariate analysis. On multivariate analysis, age >60 years (HR 1.89; 95%CI 1.12-3.21; p = 0.017), chemotherapy treatment (HR 0.54; 95%CI 0.31-0.95; p = 0.033), tyrosine kinase inhibitor treatment (HR 0.37; 95%CI 0.17-0.81; p = 0.013), the presence of the solid subtype (HR 1.55; 95%CI: 1.03-2.34; p = 0.036), and TTF-1 positivity (HR 0.64; 95%CI 0.42-0.98; p = 0.044) were correlated with overall survival (Table 3).

Discussion

Lung adenocarcinoma is a heterogeneous tumor and was classified as 'lung adenocarcinoma, mixed subtype' in the 2004 WHO classification. The 2015 WHO classification and the 2011 IASLC/ATS/ERS consensus recommend a subtyping of primary lung adenocarcinoma based on histopathological patterns and the recording of each minor component based on the relative surface occupied by 5% increments [START_REF] Travis | International Association for the Study of Lung Cancer / American Thoracic Society / European Respiratory Society International Multidisciplinary Classification of Lung Adenocarcinoma[END_REF]. Several studies on primary tumors have shown that histopathological subtypes and patterns are related to prognosis [START_REF] Casteillo | Pathologic Subtypes of Lung Adenocarcinoma Brain Metastasis Is a Strong Predictor of Survival After Resection[END_REF][START_REF] Warth | The novel histologic International Association for the Study of Lung Cancer/American Thoracic Society/European Respiratory Society classification system of lung adenocarcinoma is a stageindependent predictor of survival[END_REF][START_REF] Mansuet-Lupo | The new histologic classification of lung primary adenocarcinoma subtypes is a reliable prognostic marker and identifies tumors with different mutation status: The experience of a French cohort[END_REF][START_REF] Russell | Does lung adenocarcinoma subtype predict patient survival?: A clinicopathologic study based on the new international association for the study of lung cancer/American thoracic society/European respiratory society international multidisciplinary lung adeno[END_REF][START_REF] Hung | Predictive value of the International Association for the Study of Lung Cancer/American Thoracic Society/European Respiratory Society classification of lung adenocarcinoma in tumor recurrence and patient survival[END_REF][START_REF] Boland | Growth pattern-based grading of pulmonary adenocarcinoma-Analysis of 534 cases with comparison between observers and survival analysis[END_REF][START_REF] Yoshizawa | Validation of the IASLC/ATS/ERS lung adenocarcinoma classification for prognosis and association with EGFR and KRAS gene mutations: Analysis of 440 Japanese patients[END_REF]. The solid and micropapillary subtypes have the worst prognosis while the lepidic subtype has the best prognosis. The acinar subtype has an intermediate prognosis. The prognosis of the papillary subtype is more often favorable, but other studies have found this subtype to have an intermediate or even an adverse prognosis [START_REF] Warth | The novel histologic International Association for the Study of Lung Cancer/American Thoracic Society/European Respiratory Society classification system of lung adenocarcinoma is a stageindependent predictor of survival[END_REF][START_REF] Hung | Predictive value of the International Association for the Study of Lung Cancer/American Thoracic Society/European Respiratory Society classification of lung adenocarcinoma in tumor recurrence and patient survival[END_REF][START_REF] Kadota | A grading system combining architectural features and mitotic count predicts recurrence in stage I lung adenocarcinoma[END_REF]. A recent meta-analysis showed that the solid subtype in surgically resected primary tumors is associated with a worse overall survival and shorter disease-free survival [START_REF] Miyahara | Solid predominant subtype in lung adenocarcinoma is related to poor prognosis after surgical resection: A systematic review and meta-analysis[END_REF].

Few studies have evaluated the impact of histopathological subtypes in metastatic lung adenocarcinoma. A study of 100 patients, including 70 visceral metastases diagnosed between 2000 and 2010, showed that a predominantly solid histopathological subtype correlates with a worse overall survival compared with the acinar and papillary subtypes [START_REF] Clay | The clinical relevance of pathologic subtypes in metastatic lung adenocarcinoma[END_REF].

This study was carried out before the era of tyrosine kinase inhibitors, and the authors suggested that their results require validation with larger cohorts. A previous work on a homogeneous cohort of 97 resected brain metastases showed that the solid subtype is correlated with shorter overall survival [START_REF] Casteillo | Pathologic Subtypes of Lung Adenocarcinoma Brain Metastasis Is a Strong Predictor of Survival After Resection[END_REF]. Two works on lung adenocarcinoma at different stages included 15/487 and 7/407 stage IV patients; however, the numbers of patients with metastasis in these studies are too low for the specific study of histopathological patterns on metastases [START_REF] Warth | The novel histologic International Association for the Study of Lung Cancer/American Thoracic Society/European Respiratory Society classification system of lung adenocarcinoma is a stageindependent predictor of survival[END_REF][START_REF] Mansuet-Lupo | The new histologic classification of lung primary adenocarcinoma subtypes is a reliable prognostic marker and identifies tumors with different mutation status: The experience of a French cohort[END_REF]. Another work showed that prognostic differences in metastases may not apply to patients with advanced disease, but this work was made on computed tomographyguided biopsies [START_REF] Campos-Parra | Relevance of the novel IASLC/ATS/ERS classification of lung adenocarcinoma in advanced disease[END_REF].

Our work on visceral metastases-regardless of the classical and well-known clinical factors such as age, performance status, and type of therapy received-showed that the presence of the solid subtype is independent of the classical and well-known clinical factors related to survival. Furthermore, we also highlight the fact that TTF-1 negativity is a predictor of shorter survival. We can hypothesize that TTF-1 negativity might reflect a dedifferentiated tumor state. We also showed that TTF-1 immunohistochemistry is a major prognostic factor alongside the classic clinical factors, and this finding confirms the findings of other groups [START_REF] Schilsky | Prognostic impact of TTF-1 expression in patients with stage IV lung adenocarcinomas[END_REF].

The solid predominant subtype is the most frequently encountered subtype in our work, representing 42% of patients, which is higher than the percentage encountered for primary tumors [START_REF] Forest | Papillary Components and EGFR Mutations are Frequent in Patients With Lung Adenocarcinoma Who are Over 75 Years Old[END_REF]. Although two distinct unmatched cohorts cannot be directly compared, this might reflect the aggressiveness of tumors with solid subtypes. Furthermore, it has been shown that the solid subtype is preponderant in autologous metastasis of lung adenocarcinoma [START_REF] Takahashi | Preponderance of High-Grade Histologic Subtype in Autologous Metastases in Lung Adenocarcinoma[END_REF].

Rearranged ALK and KRAS mutations were found to be related to overall survival in univariate analysis but not in multivariate analysis. For ALK, the very low number of patients with a translocation limits the power of statistical analysis for patients with this translocation. Mutations of KRAS have been shown to be associated with solid subtype and TTF-1 negativity, which are strong prognostic factors in our work.

The question of whether the prognosis is due to the histopathological subtypes themselves or whether the subtype determines a better response to treatment is unknown. Two studies have suggested that solid and micropapillary subtypes might benefit from chemotherapy [START_REF] Campos-Parra | Relevance of the novel IASLC/ATS/ERS classification of lung adenocarcinoma in advanced disease[END_REF][START_REF] Tsao | Subtype Classification of Lung Adenocarcinoma Predicts Benefit From Adjuvant Chemotherapy in Patients Undergoing Complete Resection[END_REF].

Our work has some inherent bias due to its retrospective design. Furthermore, the study covers a period of nearly 10 years, over which the treatment of lung cancer has changed.

There is also a predominance of pleural metastasis, partly because the pleurae are a frequent site of lung cancer metastasis and partly because pleural surgical biopsy allows a pleurodesis. To limit this bias, we have included consecutively all diagnosed patients without selection. Another possible bias is that we have focused our work on resected metastases.

Most metastases are diagnosed on biopsies or cytological samples, but architectural patterns are not evaluable on all biopsies and are not evaluable on cytological samples. Three recent studies have showed that architectural pattern can predict outcome on a core biopsy specimen [START_REF] Leeman | Histologic Subtype in Core Lung Biopsies of Early-Stage Lung Adenocarcinoma is a Prognostic Factor for Treatment Response and Failure Patterns After Stereotactic Body Radiation Therapy[END_REF][START_REF] Gao | Micropapillary and/or Solid Histologic Subtype Based on Pre-Treatment Biopsy Predicts Local Recurrence After Thermal Ablation of Lung Adenocarcinoma[END_REF][START_REF] Kim | Utility of Core Biopsy Specimen to Identify Histologic Subtype and Predict Outcome for Lung Adenocarcinoma[END_REF]. The cribriform pattern and filigree pattern are new patterns recently described and are predictors of recurrence; we have chosen to limit our work to the patterns recognized in the WHO classification [START_REF] Emoto | Expansion of the Concept of Micropapillary Adenocarcinoma to Include a Newly Recognized Filigree Pattern as Well as the Classical Pattern Based on 1468 Stage I Lung Adenocarcinomas[END_REF][START_REF] Kadota | Cribriform Subtype is an Independent Predictor of Recurrence and Survival After Adjustment for the Eighth Edition of TNM Staging System in Patients With Resected Lung Adenocarcinoma[END_REF].

To conclude, as with primary tumors, we show that the presence of the solid subtype is a major adverse independent prognostic factor for overall survival when evaluated on metastatic samples. This finding brings further clues for the evaluation of histopathological patterns on all resected samples of lung adenocarcinoma. This information is not used for the selection of treatment but can provide further information on patient prognosis and

TTF- 1

 1 immunohistochemistry was available in 253 cases. The TTF-1 was positive in 56/106 of the predominant solid subtype, 68/95 of the acinar subtype, 20/23 of the micropapillary subtype and 15/19 of the papillary subtype. TTF-1 could be assessed on all unclassifiable tumors and was positive in 6/10 of them. TTF-1 expression was associated with predominant histopathological subtype (p = 0.003, Fisher exact test). Mutations in exons 18-21 of EGFR were found in 36/248 patients with known EGFR results. EGFR was mutated in 16/95 of the acinar predominant subtype, 1/10 of the unclassifiable tumors, 6/23 of the micropapillary predominant subtype, 9/19 of the papillary subtype, and 4/106 of the solid predominant subtype. EGFR mutations were associated with predominant histopathological subtype (p <0.001, Fisher exact test). KRAS was mutated in exon 2 in 57/217 patients with known KRAS results. KRAS was mutated in 23/95 of the acinar predominant subtype, 3/10 of the unclassifiable tumors, 3/23 of the micropapillary predominant subtype, 2/19 of the papillary subtype, and 26/106 of the solid predominant subtype. KRAS mutations were not associated with predominant histopathological subtype (p = 0.62, Fisher exact test).

2 .

 2 Two hundred seven patients (82%) were dead at the time of analysis. Patients ≤60 years of age at diagnosis had a significantly longer overall survival compared with patients >60 years; the median times were 13.3 and 7.4 months, respectively (p = 0.03). Patients with a performance status (PS) <2 had a longer survival compared with patients with a PS ≥2; the median times were 12 and 3.2 months, respectively (p <0.001). Patients treated by tyrosine kinase inhibitor had a longer survival (22.5 months) than patients treated by chemotherapy (11.1 months); patients with palliative care had the shortest survival (1.3 months) (p <0.001).

Table 1 .

 1 Clinical and histopathological data related to overall survival

	Variables	Patients (%)	Median overall	p
			survival (mo)
			(95%CI)
	All patients	253 (100)	8.0 (6.6-10.9)
	Age (y)			0.03
	≤60	56	13.3 (6.6-26.7)
	>60	197	7.4 (5.9-10.4)
	Sex			0.5
	Female	85	10.9 (6.6-17.6)
	Male	168	8 (5.2-10.5)
	Performance status			<0.001
	0 or 1	136	12 (10.4-17.7)
	≥2	53	3.2 (2.0-7.2)
	Unknown	64	3.3 (1.8-11)
	First-line systemic			<0.001
	anticancer therapy
	Chemotherapy	146	11.1 (8.4-15.4)
	Tyrosine kinase	31	22.5 (16.2-27.2)
	inhibitor or
	immunotherapy
	Palliative care	54	1.3 (0.9-3.3)
	Unknown	22	4.0 (2.2-)
	Major architectural			0.5
	pattern
	Solid	106	6.8 (4.1-11.2)
	Acinar	95	8.4 (6.0-16.3)
	Papillary	19	17.5 (7.6-)
	Micropapillary	23	9.7 (5.9-22.2)
	Unclassifiable	10	9.6 (0.7-)
	Solid subtype			0.045
	Present	162	6.8 (4.4-9.1)
	Absent	91	11.1 (8.6-21.3)
	Acinar subtype			1
	Present	159	8.3 (6.6-11.2)
	Absent	94	7.2 (4.6-12)
	Papillary subtype			0.01
	Present	50	17.5 (10.4-25.4)
	Absent	203	6.9 (4.8-9.6)
	Micropapillary			0.8
	subtype
	Present	119	9.1 (6.6-13.3)
	Absent	134	7.5 (4.8-11.0)
	EGFR			0.5
	Mutated	36	16.3 (7.4-24.8)
	Not mutated	212	7.4 (5.2-10.4)
	Unknown	5	9.8 (6.6-)
	KRAS			0.01
	Mutated	57	4.6 (2.2-9.1)
	Not mutated	160	10.4 (7.4-15.4)
	Unknown	36	8.1 (5.2-17.7)

Table 2 .

 2 Cox univariate analysis of clinical and histopathological factors in relation to overall

	survival
		HR (95% CI)	p
	Age >60 versus ≤60 (y)	1.46 (1.03-2.07)	0.032
	Sex M versus F	1.10 (0.82-1.47)	0.52
	Performance status 0 or 1	0.43 (0.30-0.61)	<0.001
	versus ≥2
	First-line systemic anticancer
	therapy
	Palliative care	-
	Chemotherapy	0.41 (0.29-0.57)	<0.001
	Tyrosine kinase inhibitor or	0.29 (0.17-0.48)	<0.001
	immunotherapy
	Major architectural pattern
	Acinar	-
	Micropapillary	1.00 (0.61-1.64)	0.98
	Papillary	0.62 (0.35-1.01)	0.099
	Solid	1.02 (0.75-1.39)	0.89
	Solid subtype present versus	1.34 (1.01-1.79)	0.046
	absent
	Acinar subtype present versus	0.99 (0.75-1.32)	0.97
	absent
	Papillary subtype present	0.62 (0.43-0.90)	0.011
	versus absent
	Micropapillary subtype present	1.03 (0.78-1.35)	0.84
	versus absent
	EGFR not mutated versus	1.25 (0.84-1.83)	0.27
	mutated
	KRAS not mutated versus	0.64 (0.46-0.88)	0.007
	mutated
	BRAF not mutated versus	1.72 (0.63-4.67)	0.29
	mutated
	ERBB2 not mutated versus	0.81 (0.11-5.96)	0.84
	mutated
	ALK rearranged versus not	0.20 (0.06-0.64)	0.007
	rearranged
	TTF-1-positive versus -negative	0.57 (0.43-0.76)	<0.001

Table 3 .

 3 Multivariate analysis of clinical and histopathological factors in relation to overall

	survival
		HR (95% CI)	p
	Age >60 versus ≤60 (y)	1.89 (1.12-3.21)	0.017
	Performance status 0 or1	0.57 (0.36-0.91)	0.017
	versus ≥2
	First-line systemic anticancer
	therapy
	Palliative care	-
	Chemotherapy	0.54 (0.31-0.95)	0.033
	Tyrosine kinase inhibitor or	0.37 (0.17-0.81)	0.013
	immunotherapy
	Solid subtype present versus	1.55 (1.03-2.34)	0.036
	absent
	Papillary subtype present	0.66 (0.40-1.11)	0.119
	versus absent
	KRAS not mutated versus	0.69 (0.43-1.09)	0.116
	mutated
	ALK rearranged versus not	0.31 (0.09-1.05)	0.059
	rearranged
	TTF-1 positive versus negative	0.64 (0.42-0.98)	0.044

Acknowledgments

The authors thank Philippe Cosmo from Centre de Ressources Biologiques de CHU Saint-Etienne (BRIF no. BB-0033-00041) for his assistance in data retrieval.

does not require an ancillary technique and additional cost. This information is important for the evaluation of patients' overall survival in clinical trials.