Aurélie Fischer
email: aurelie.fischer@upmc.fr

Projection-based curve clustering

Keywords: Clustering, basis expansion, projections, k-means, wavelet packets

This paper focuses on unsupervised curve classification in the context of nuclear industry. At the Commissariat à l'Energie Atomique (CEA), Cadarache (France), the thermal-hydraulic computer code CATHARE is used to study the reliability of reactor vessels. The code inputs are physical parameters and the outputs are time evolution curves of a few other physical quantities. As the CATHARE code is quite complex and CPU-time consuming, it has to be approximated by a regression model. This regression process involves a clustering step. In the present paper, CATHARE output curves are clustered using a k-means scheme, with a projection onto a lower dimensional space. We study the properties of the empirically optimal cluster centers found by the clustering method based on projections, compared to the "true" ones. The choice of the projection basis is discussed, and an algorithm is implemented to select the best projection basis among a library of orthonormal bases. The approach is illustrated on a simulated example and then applied to the industrial problem.

1 Introduction

The CATHARE code

A major concern in nuclear industry is the life span of reactor vessels. To go on using the current nuclear reactors, their reliability has to be proved. For this purpose, complex computer codes are developed to simulate the behavior of the vessel under different sequences of accidents. At the Commissariat à l'Energie Atomique (CEA), Cadarache (France), one of the main types of accident under study is the pressurized thermal shock. This is a problem due to the combined stresses from a rapid temperature and pressure change. More specifically, as a reactor vessel gets older, the potential for failure by cracking when it is cooled rapidly at high pressure increases greatly. The analysis of pressurized thermal shock is made of two main steps. First, a thermal-hydraulic analysis is done to determine the temporal evolutions of temperature, pressure and thermal exchange coefficient in the vessel annular space, since these features have an influence on the mechanical and thermal charge on the vessel inner surface. Some evolution curves x 1 (t), . . . , x n (t) corresponding to the thermal exchange coefficient are depicted in Figure 1 (n = 66). Each curve x i (t) is obtained as the simulation result for a certain vector of input physical parameters. The curves of temperature, pressure and thermal exchange coefficient obtained during this first step are then used as limit conditions in the second part of the analysis, which is a mechanical investigation aiming at checking if some defects on the vessel annular space could propagate and gain importance to such an extent that this would cause a break of the vessel inner surface. For further details on the reliability of reactor vessels, we refer the reader to Auder, De Crecy, Iooss, and Marquès [START_REF] Auder | Screening and metamodeling of computer experiments with functional outputs[END_REF].

The simulation step relies on a computer code called CATHARE (Code Avancé de THermohydraulique pour les Accidents des Réacteurs à Eau, in English Code for Analysis of THermalhylaudrics during an Accident of Reactor and safety Evaluation). The CATHARE code is a system code for pressurized water reactors safety analysis, accident management, definition of plant operating procedures and for research and development. The project is a result of a joint effort of the reactor vendor AREVA, the CEA, EDF (Electricité de France) and the IRSN (Institut de Radioprotection et de Sûreté Nucléaire). The first delivered version V1.3L was available in 1997. The CEA team CATHARE located in Grenoble (France) is in charge of the development, the assessment and the maintenance of the code. (See http://www-cathare.cea.fr.)

The CATHARE code allows to simulate the evolution of temperature, pressure and thermal exchange coefficient, given the physical parameters as inputs. However, this code is so slow (about 6 to 10 hours for one run) that it cannot be used directly for reliability calculations. To bypass this obstacle, the strategy drawn up by the CEA is to build a so-called metamodel which is a fast approximation of the original code, precise enough to carry out statistical computations. The term "metamodel" indicates that a computer code approximating a physical process has already been developed, and now this code is modeled in turn. Here, the purpose is the construction of a regression model based on a few hundreds CATHARE code outputs, obtained during one week of computation on a supercomputer in 2007. The inputs were sampled randomly by latin hypercube methods (see, e.g., McKay, Conover, and Beckman [START_REF] Mckay | A comparison of three methods for selecting values of input variables in the analysis of output from a computer code[END_REF] and Loh [START_REF] Loh | On latin hypercube sampling[END_REF]), so that we have no control over the inputs in the learning sample. As different kinds of behavior for temperature, pressure and thermal exchange coefficient may be observed depending on the physical parameters, a preliminary unsupervised classification of CATHARE code output curves is essential. Once the curves have been clustered in meaningful classes, the regression model can be adjusted for each group of outputs separately. The clustering step is the object of the present paper.

Clustering

Clustering is the problem of partitioning data into a finite number of groups (denoted hereafter by k), or clusters, so that the data items inside each of them are very similar among themselves and as different as possible from the elements of the other clusters (Duda,Hart,and Stork [14,Chapter 10]). In our industrial context, the data is made of evolution curves of temperature, pressure or thermal exchange coefficient. Using a probabilistic point of view, these curves can be seen as independent draws X 1 (t), . . . , X n (t) with the same distribution as a generic random variable X(t) taking values in a functional space (E, • ) -typically, the Hilbert space of square integrable functions.

A widely used clustering method is the so-called k-means clustering, which consists in partitioning the random observations X 1 , . . . , X n ∈ E into k classes by minimizing the empirical distortion

W ∞,n (c) = 1 n n i=1 min =1,...,k X i -c 2 ,
over all possible cluster centers c = (c 1 , . . . , c k ) ∈ E k . Here, µ n denotes the empirical measure associated with the sample X 1 , . . . , X n , i.e.,

µ n (A) = 1 n n i=1 1 {X i ∈A}
for every Borel subset A of E. In other words, we look for a Voronoi partition of E.

The Voronoi partition C 1 , . . . , C k associated with c = (c 1 , . . . , c k ) ∈ E k is defined by letting an element x ∈ E belong to C if it is closer (with respect to the norm • )
to c than to any other c j (ties are broken arbitrarily). The -th cluster is made of the observations X i assigned to c , or equivalently, falling in the Voronoi cell C . In this framework, the accuracy of the clustering scheme is assessed by the distortion or mean squared error

W ∞ (c) = E min =1,...,k X -c 2 ,
where E stands for expectation with respect to the distribution of X. This clustering method is in line with the more general theory of quantization. More specifically, it corresponds to the empirical version of nearest neighbor quantization (Linder [START_REF] Linder | Learning-Theoretic Methods in Vector Quantization[END_REF], Gersho and Gray [START_REF] Gersho | Vector Quantization and Signal Compression[END_REF], Graf and Luschgy [START_REF] Graf | Foundations of Quantization for Probability Distributions[END_REF]). However, the problem of finding a minimizer of the criterion W ∞,n (c) is in general NP-hard, and there is no efficient algorithm to find the optimal solution in reasonable time. That is why several iterative algorithms have been developed to give approximate solutions. One of the first to be described historically is Lloyd's algorithm (Lloyd [25]).

The challenge is to adapt the k-means clustering method to our setting. The main difficulty here is the high dimensionality of the data, which casts the problem into the general class of functional statistics. For a comprehensive introduction to this topic, see Ramsay and Silverman [START_REF] Ramsay | Functional Data Analysis[END_REF] and Ferraty and Vieu [START_REF] Ferraty | Nonparametric Functional Data Analysis: Theory and Practice[END_REF]. A possible approach to reduce the infinite dimension of the observations X 1 , . . . , X n consists in projecting them onto a lower-dimensional subspace. In this context, Abraham, Cornillon, Matzner-Løber, and Molinari [START_REF] Abraham | Unsupervised Curve Clustering using B-Splines[END_REF] project the curves on a B-spline basis, and get clusters with the k-means algorithm applied to the coefficients. These authors argue that projecting onto a smooth spline basis plays the role of a denoising procedure, given that the observed curves could contain measurement errors, and also allows to deal with curves which were not measured at the same time. James and Sugar [START_REF] James | Clustering for sparsely sampled functional data[END_REF] use a B-spline basis to model the centers of the clusters and write each curve in cluster as a main effect defined by spline coefficients plus an error term. This allows for some deviations around a model curve specific to cluster . The authors add a Gaussian error term to model the individual variations among one cluster. This way, the main effect is enriched, and the model can take into account more complex behaviors. The method in Gaffney [START_REF] Gaffney | Probabilistic Curve-Aligned Clustering and Prediction with Mixture Models[END_REF] is similar, also with a B-spline basis. Another option is to use a Self-Organizing Map algorithm on the coefficients (Rossi, Conan-Guez, and El Golli [START_REF] Rossi | Clustering functional data with the som algorithm[END_REF]), again obtained by projecting the functions onto a B-spline basis. These bases are often used because they are easy to implement, and require a relatively minimal number of parametric assumptions. Besides, Biau, Devroye, and Lugosi [START_REF] Biau | On the performance of clustering in Hilbert spaces[END_REF] examine the theoretical performance of clustering with random projections based on the Johnson-Lindenstrauss Lemma, which represent a sound alternative to orthonormal projections thanks to their distance-preserving properties. Chiou and Li [START_REF] Chiou | Functional Clustering of Longitudinal Data[END_REF] propose a method which generalizes the k-means algorithm to some extent, by considering covariance structures via functional principal component analysis. In the approach of these authors, each curve is decomposed on an adaptive local basis (valid for the elements in the cluster), and the clusters are determined according to the full approximation onto each basis. In the wavelet-based method for functional data clustering developed in Antoniadis, Brossat, Cugliari, and Poggi [START_REF] Antoniadis | Clustering functional data with wavelets[END_REF], a smooth curve is reduced to a finite number of representative features, by considering the contribution of each wavelet coefficient to the global energy of the curve.

In the present contribution, we propose to investigate the problem of clustering output curves X 1 , . . . , X n of the CATHARE code, assuming that they arise from a random variable X taking its values in some subset of the space of square integrable functions. As a general strategy, we reduce the infinite dimension of X by considering only the first d coefficients of the expansion on a Hilbertian basis, and then perform clustering in R d . We study the theoretical properties of this clustering method with projection. A bound expressing what is lost when replacing the empirically optimal cluster centers by the centers obtained by projection is offered (Section 2). Since the result may depend on the basis choice, several projection bases are used in practice, and we look for the best one minimizing a given criterion. To this end, an algorithm based on Coifman and Wickerhauser [START_REF] Coifman | Entropy-based algorithms for best basis selection[END_REF] is implemented, searching for an optimal basis among a library of wavelet packet bases available in the R package wmtsa, and this "optimal basis" is compared with the Fourier basis, the Haar basis, and the functional principal component basis (Section 3). Finally, this algorithm is applied to a simulated example and to our industrial problem (Section 4). Proofs are postponed to Section 5.

Finite-dimensional projection for clustering

As mentioned earlier, we are concerned with square integrable functions. Since all results can be adapted to L 2 ([a, b]) by an appropriate rescaling, we consider for the sake of simplicity the space L 2 ([0, 1]). As an infinite-dimensional separable Hilbert space, L 2 ([0, 1]) is isomorphic via the choice of a Hilbertian basis to the space 2 of square-summable sequences. We focus more particularly on functions in L 2 ([0, 1]) whose coefficients in the expansion on a given Hilbertian basis belong to the subset S of 2 given by

S = x = (x j ) j≥1 ∈ 2 : +∞ j=1 ϕ j x 2 j ≤ R 2 , ( 1
)
where R > 0 and (ϕ j ) j≥1 is a nonnegative increasing sequence such that

lim j→+∞ ϕ j = +∞.
It is worth pointing out that S is closely linked with the basis choice, even if the basis does not appear explicitly in the definition. To illustrate this important fact, three examples are discussed below.

Example 2.1 (Sobolev ellipsoids).

For β ∈ N * and L > 0, the periodic Sobolev class

W per (β, L) is the space of all functions f ∈ [0, 1] → R such that f (β-1) is absolutely continuous, 1 0 (f (β) (t)) 2 dt ≤ L 2 and f ( ) (0) = f ( ) (1) for = 0, . . . , β - 1. Let (ψ j ) j≥1 denote the trigonometric basis. Then a function f = +∞ j=1 x j ψ j is in W per (β, L) if and only if the sequence x = (x j ) j≥1 of its Fourier coefficients belongs to S = x ∈ 2 : +∞ j=1 ϕ j x 2 j ≤ R 2 ,
where

ϕ j = j 2β
for even j (j -1) 2β for odd j and R = L π β . For the proof of this result and further details about Sobolev classes, we refer the reader to the book of Tsybakov [START_REF] Tsybakov | Introduction to Nonparametric Estimation[END_REF]. Note that the set S could also be defined by ϕ j = j r e αj with α > 0 and r ≥ -α (Tsybakov [START_REF] Tsybakov | On the best rate of adaptive estimation in some inverse problems[END_REF]).

Example 2.2 (Reproducing Kernel Hilbert Spaces).

Let K : [0, 1] × [0, 1] → R be a Mercer kernel, i.e., K is continuous, symmetric and positive definite. Recall that a kernel K is said to be positive definite if for all finite sets {x 1 , . . . , x m }, the matrix A defined by a ij = K(x i , x j ) for 1 ≤ i, j ≤ m is positive definite. For example, the Gaussian kernel K(x, y) = exp(-(x-y) 2 σ 2 ) and the kernel K(x, y) = (c 2 + (x -y) 2 ) -a with a > 0 are Mercer kernels. For x ∈ [0, 1], let K x : y → K(x, y). Then, Moore-Aronszajn's Theorem (Aronszajn [4]) states that there exists a unique Hilbert space (H K , •, • ) of functions on [0, 1] such that:

1. For all x ∈ [0, 1], K x ∈ H K . 2. The span of the set {K x , x ∈ [0, 1]} is dense in H K . 3. For all f ∈ H K and x ∈ [0, 1], f (x) = K x , f .
The Hilbert space H K is said to be the reproducing kernel Hilbert space (for short, RKHS) associated with the kernel K. Next, the operator K defined by

Kf : y → 1 0 K(x, y)f (x)dx
is self-adjoint, positive and compact. Consequently, there exists a complete orthonormal system (ψ j ) j≥1 of L 2 ([0, 1]) such that Kψ j = λ j ψ j , where the set of eigenvalues {λ j , j ≥ 1} is either finite or a sequence tending to 0 at infinity. Moreover, the λ j are nonnegative. Suppose that K is not of finite rank -so that {λ j , j ≥ 1} is infinite -and that the eigenvalues are sorted in decreasing order, that is λ j ≥ λ j +1 for all j ≥ 1. Clearly, there is no loss of generality in assuming that λ j > 0 for all j ≥ 1. Indeed, if not, L 2 ([0, 1]) is replaced by the linear subspace spanned by the eigenvectors corresponding to non-zero eigenvalues.

According to Mercer's theorem, K has the representation

K(x, y) = +∞ j=1 λ j ψ j (x)ψ j (y),
where the convergence is absolute and uniform (Cucker and Smale [10, Chapter III, Theorem 1]). Moreover, H K may be characterized through the eigenvalues of the operator K by

H K = f ∈ L 2 ([0, 1]) : f = +∞ j=1 x j ψ j , +∞ j=1 x 2 j λ j < ∞ ,
with the inner product

+∞ j=1 x j ψ j , +∞ j=1 y j ψ j = +∞ j=1
x j y j λ j (Cucker and Smale [START_REF] Cucker | On the mathematical foundations of learning[END_REF]Chapter III,Theorem 4]). Then, letting

S = x ∈ 2 , +∞ j=1 x 2 j λ j ≤ R 2 ,
the set S is of the desired form [START_REF] Abraham | Unsupervised Curve Clustering using B-Splines[END_REF], with ϕ j = 1/λ j .

Example 2.3 (Besov ellipsoids and wavelets).

Let α > 0.

For f ∈ L 2 ([0, 1]), the Besov semi-norm |f | B α 2 (L 2 ) is defined by |f | B α 2 (L 2 ) = +∞ j=0 [2 jα ω r (f, 2 -j , [0, 1]) 2 ] 2 1/2
where ω r (f, t, [0, 1]) 2 denotes the modulus of smoothness of f, as defined for instance in DeVore and Lorentz [START_REF] Devore | Constructive Approximation[END_REF], and r = α + 1. Let Λ(j) be an index set at resolution level j and (x j, ) j≥0, ∈Λ(j) the coefficients of the expansion of f in a suitable wavelet basis. Then, for

f such that |f | B α 2 (L 2 ) ≤ ρ, the coefficients x j, satisfy +∞ j=0 ∈Λ(j) 2 2jα x 2 j, ≤ ρ 2 C 2 ,
where C > 0 depends only on the basis. We refer to Donoho and Johnstone [START_REF] Donoho | Minimax estimation via wavelet shrinkage[END_REF] for more details.

Let us now come back to the general setting

S = x = (x j ) j≥1 ∈ 2 : +∞ j=1 ϕ j x 2 j ≤ R 2 ,
and consider the problem of clustering the sample X 1 , . . . , X n with values in S. Some notation and assumptions are in order. First, we will suppose that P { X ≤ R} = 1. Notice that the fact that X takes its values in S is in general not enough to imply P { X ≤ R} = 1. Secondly, let j 0 be the smallest integer j such that ϕ j > 0. To avoid technical difficulties, we require in the sequel d ≥ j 0 . For all d ≥ 1, we will denote by Π d the orthogonal projection on R d and let S d = Π d (S). Lastly, observe that S d identifies with the ellipsoid

x = (x 1 , . . . , x d ) ∈ R d : d j=1 ϕ j x 2 j ≤ R 2 .
As explained in the introduction, the criterion to minimize is

W ∞,n (c) = 1 n n i=1 min =1,...,k X i -c 2 ,
and the performance of the clustering obtained with the centers c = (c 1 , . . . , c k ) ∈ S k is measured by the distortion

W ∞ (c) = E min =1,...,k X -c 2 .
The quantity

W * ∞ = inf c∈S k W ∞ (c)
represents the optimal risk we can achieve. With the intention of performing clustering in the projection space S d , we also introduce the "finite-dimensional" distortion

W d (c) = E min =1,...,k Π d (X) -Π d (c ) 2
and its empirical counterpart

W d,n (c) = 1 n n i=1 min =1,...,k Π d (X i ) -Π d (c ) 2 ,
as well as

W * d = inf c∈S k W d (c).
Let us observe that, as the support of the empirical measure µ n contains at most n points, there exists an element ĉd,n which is a minimizer of W d,n (c) on S k . Moreover, in view of its definition, W d,n (c) only depends on the centers projection

Π d (c) (one has W d,n (c) = W d,n (Π d (c
)) for all c) and we can thus assume that ĉd,n ∈ (S d ) k . Notice also that for all c ∈ S,

Π d (X) -Π d (c) 2 ≤ X -c 2 (the projection Π d is 1-Lipschitz), which implies that W d (c) ≤ W ∞ (c)
for all c.

The following lemma provides an upper bound for the maximal deviation

sup c∈S k [W ∞ (c) -W d (c)].
Lemma 2.1. We have

sup c∈S k [W ∞ (c) -W d (c)] ≤ 4R 2 ϕ d .
We are now in a position to state the main result of this section.

Theorem 2.1. Let ĉd,n ∈ (S d ) k be a minimizer of W d,n (c). Then, E[W ∞ (ĉ d,n )] -W * ∞ ≤ E[W d (ĉ d,n )] -W * d + 8R 2 ϕ d . ( 2
)
Theorem 2.1 expresses the fact that the expected excess clustering risk in the infinite dimensional space is bounded by the corresponding "finite-dimensional risk" plus an additional term representing the price to pay when projecting onto S d . Yet, the first term in the right-hand side of inequality (2) above is known to tend to 0 when n goes to infinity. More precisely, as P { Π d (X) ≤ R} = 1, we have

E[W d (ĉ d,n )] -W * d ≤ Ck √ n ,
where C = 12R 2 (Biau, Devroye, and Lugosi [START_REF] Biau | On the performance of clustering in Hilbert spaces[END_REF]). In our setting, to keep the same rate of convergence O(1/ √ n) in spite of the extra term 8R 2 /ϕ d , ϕ d must be of the order √ n. For Sobolev ellipsoids (Example 2.1), where ϕ j ≥ (j -1) 2β , this means a dimension d of the order n 1/4β . When ϕ j = j r e αj , the rate of convergence is O(1/ √ n) as long as d is chosen of the order ln n/(2α). In the RKHS context (Example 2.2), consider the case of eigenvalues {λ j , j ≥ 1} with polynomial or exponential-polynomial decay, which covers a broad range of kernels (Williamson, Smola, and Schölkopf [START_REF] Williamson | Generalization performance of regularization networks and support vector machines via entropy numbers of compact operators[END_REF]). If

λ j = O(j -(α+1) ), α > 0, then 1/ϕ d = O(d -(α+1)
), and d must be of the order n 1/(2α+2) , whereas λ j = O(e -αj p ), α, p > 0, leads to a projection dimension d of the order (ln n/(2α)) 1/p . Obviously, the upper bound ( 2) is better for large ϕ d , and consequently large d. Nevertheless, from a computational point of view, the projection dimension should not be chosen too large.

Remark 2.1. Throughout, we assumed that P { X ≤ R} = 1. This requirement, called the peak power constraint, is standard in the clustering and signal processing literature. We do not consider in this paper the case where this assumption is not satisfied, which is feasible but leads to technical complications (see Merhav and Ziv [START_REF] Merhav | On the amount of statistical side information required for lossy data compression[END_REF], Biau, Devroye, and Lugosi [START_REF] Biau | On the performance of clustering in Hilbert spaces[END_REF] for results in this direction). Besides, the number of clusters is assumed to be fixed throughout the paper. Several methods for estimating k have been proposed in the literature (see, e.g., Milligan and Cooper [START_REF] Milligan | An examination of procedures for determining the number of clusters in a data set[END_REF] and Gordon [START_REF] Gordon | Classification. Monographs on Statistics and Applied Probability[END_REF]).

As already mentioned, the subset of coefficients S is intimately connected to the underlining Hilbertian basis. As a consequence, all the results presented strongly depend on the orthonormal system considered. Therefore, the choice of a proper basis is crucial and is discussed in the next section.

Basis selection

Wavelet packet best basis algorithm In this section, we describe an algorithm searching for the best projection basis among a "library".

If {ψ α , α ∈ I} ⊂ L 2 ([0, 1]) is a collection of elements in L 2 ([0, 1]) which span L 2 ([0, 1]
) and allow to build several different bases by choosing various subsets {ψ α , α ∈ I β } ⊂ L 2 ([0, 1]), the collection of bases built this way is called a library of bases. Here, I is some index set, and β runs over some other index set.

More specifically, we focus on the best basis algorithm of Coifman and Wickerhauser [START_REF] Coifman | Entropy-based algorithms for best basis selection[END_REF] (see also Wickerhauser [START_REF] Wickerhauser | Adapted Wavelet Analysis from Theory to Software[END_REF]), which yields an optimal basis among a library of wavelet packets. Wavelets are functions which cut up a signal into different frequency components to study each component with a resolution matched to its scale. Unlike the Fourier basis, wavelets are localized both in time and frequency. Hence, they have advantages over traditional Fourier methods when the signal contains discontinuities as well as noise. For detailed expositions of the mathematical aspects of wavelets, see the books of Daubechies [START_REF] Daubechies | Ten Lectures on Wavelets[END_REF], Mallat [START_REF] Mallat | A wavelet tour of signal processing, The Sparse Way[END_REF] and Meyer [START_REF] Meyer | Wavelet and Operators[END_REF].

Let the sequence of functions (ψ ν ) ν≥0 be defined by

ψ 0 (t) = Hψ 0 (t), R ψ 0 (t)dt = 1, ψ 2ν (t) = Hψ ν (t) = √ 2 p∈Z h(p)ψ ν (2t -p), ψ 2ν+1 (t) = Gψ ν (t) = √ 2 p∈Z g(p)ψ ν (2t -p),
where H and G are orthogonal quadrature filters, i.e., convolution-decimation operators satisfying some algebraic properties (see, e.g., Wickerhauser [START_REF] Wickerhauser | Adapted Wavelet Analysis from Theory to Software[END_REF]). Let Λ ν denote the closed linear span of the translates ψ ν (• -p), p ∈ Z, of ψ ν , and

σ s Λ ν = {2 -s/2 x(2 -s t), x ∈ Λ ν }.
To every such subspace of L 2 (R) corresponds a dyadic interval

I sν = ν 2 s , ν + 1 2 s .
For all (s, ν), these subspaces give an orthogonal decomposition

σ s Λ ν = σ s+1 Λ 2ν σ s+1 Λ 2ν+1 .
Observe that for ν = 0, . . . , 2 s -1, the I sν are dyadic subintervals of [0, 1[ .

The next proposition provides a library of orthonormal bases built with functions of the form ψ sνp = 2 -s/2 ψ ν (2 -s t -p), called wavelet packets of scale index s, frequency index ν and position index p.

Proposition 3.1 (Wickerhauser [37]). If s ≤ L for some finite maximum L, H and G are orthogonal quadrature filters and I is a collection of disjoint dyadic intervals whose union is R + , then B I = {ψ sνp , p ∈ Z, I sν ∈ I} is an orthonormal basis for

L 2 (R). Moreover, if I is a disjoint dyadic cover of [0, 1[, then B I is an orthonormal basis of Λ 0 .
This construction yields orthonormal bases of L 2 (R). Some changes must be made to obtain bases of L 2 ([0, 1]). Roughly, they consist in considering not all scales and shifts, and adapting the wavelets which overlap the boundary of [0, 1] (see for instance Cohen, Daubechies, and Vial [START_REF] Cohen | Wavelets on the interval and fast wavelet transforms[END_REF]).

The library can be seen as a binary tree whose nodes are the spaces σ s Λ ν (Figure 3 and4). An orthonormal basis is given by the leaves of some subtree. Figure 5 and6 show two examples of bases which can be obtained in this way.

To define an optimal basis, a notion of information cost is needed. Coifman and Wickerhauser [START_REF] Coifman | Entropy-based algorithms for best basis selection[END_REF] propose to use Shannon entropy. In our context, the basis choice will be done with respect to some reference curve x 0 which has to be representative of the data. We compute, for each basis in the library, the Shannon entropy of the coefficients of x 0 in this basis, and select the basis minimizing this entropy. The Λ 0 [0, 1[ construction of this best basis, relying on the binary tree structure of the library, is achieved by comparing, at each node, starting from the bottom of the tree, parents with their two children. We decided to take for x 0 the median curve in the sample in the L 2 norm sense. Indeed, it is more likely to present characteristic behaviors of the other curves than the mean, because the mean curve is a smooth average representative, which is probably too easy to approximate with a few basis functions. However, we observed that these two functions surprisingly give rise to almost the same basis. Both choices are thus possible. The mean curve is useful if we know that some noise has to be removed, whereas the median curve seems a better choice to reflect the small-scale irregularities.

σ 1 Λ 0 σ 1 Λ 1 σ 2 Λ 0 σ 2 Λ 1 σ 2 Λ 2 σ 2 Λ 3 σ 3 Λ 0 σ 3 Λ 1 σ 3 Λ 2 σ 3 Λ 3 σ 3 Λ 4 σ 3 Λ 5 σ 3 Λ 6 σ 3 Λ 7
[0, 1 2 [ [ 1 2 , 1[ [0, 1 4 [ [ 1 4 , 1 2 [ [ 1 2 , 3 4 [ [ 3 4 , 1[ [0, 1 8 [ [ 1 8 , 1 4 [ [ 1 4 , 3 8 [ [ 3 8 , 1 2 [ [ 1 2 , 5 8 [ [ 5 8 , 3 4 [ [ 3 4 , 7 8 [ [ 7 8 , 1[
Λ 0 σ 1 Λ 0 σ 1 Λ 1 σ 2 Λ 0 σ 2 Λ 1 σ 2 Λ 2 σ 2 Λ 3 σ 3 Λ 0 σ 3 Λ 1 σ 3 Λ 2 σ 3 Λ 3 σ 3 Λ 4 σ 3 Λ 5 σ 3 Λ 6 σ 3 Λ 7
Λ 0 σ 1 Λ 0 σ 1 Λ 1 σ 2 Λ 0 σ 2 Λ 1 σ 2 Λ 2 σ 2 Λ 3 σ 3 Λ 0 σ 3 Λ 1 σ 3 Λ 2 σ 3 Λ 3 σ 3 Λ 4 σ 3 Λ 5 σ 3 Λ 6 σ 3 Λ 7
We have implemented the algorithm in R, and it has been run through all filters available in the R package wmtsa. These filters belong to four families, extremal phase family (Daubechies wavelets), including Haar basis, least asymmetric family (Symmlets), "best localized" wavelets and Coiflets. For example, the least asymmetric family contains ten different filters "s2", "s4", "s6", "s8", "s10", "s12", "s14", "s16", "s18", "s20". Finally, we keep the clustering result obtained with the basis minimizing the distortion among the various filters. In the sequel, this basis will be called Best-Entropy basis.

In the applications, the performance of the Best-Entropy basis will be compared with the Haar wavelet basis, the Fourier basis and the functional principal component analysis basis. For the sake of completeness, we recall here the definition of these bases.

The Haar wavelet basis

Let φ(t) = 1 [0,1] (t) and ψ(t) = 1 [0,1/2[ (t) -1 [1/2,1] (t).
Then, the family {φ, ψ j, }, where ψ j, (t) = 2 j/2 ψ(2 j t -), j ≥ 0, 0 ≤ ≤ 2 j -1, constitutes a Hilbertian basis of L 2 ([0, 1]), called Haar basis. The Fourier basis The Fourier basis on [0, 1] is the complete orthonormal system of L 2 ([0, 1]) built with the trigonometric functions

ψ 1 (t) = 1, ψ 2j (t) = √ 2 cos(2πjt), ψ 2j+1 (t) = √ 2 sin(2πjt), j ≥ 1.
Functional principal component analysis Principal component analysis for functional data (for short, functional PCA) is the generalization of the usual principal component analysis for vector data. The Euclidean inner product is replaced by the inner product in L 2 ([0, 1]). More precisely, functional PCA consists in writing X i (t) under the form

X i (t) = EX(t) + +∞ j=1 x ij ψ j (t),
where the (x ij ) j≥1 and the functions (ψ j ) j≥1 are defined as follows. At the first step, the function ψ 1 is chosen to maximize

1 n n i=1 x 2 i1 = 1 n n i=1 ψ 1 (t)X i (t)dt 2 subject to ψ 1 (t) 2 dt = 1.
Then, each ψ j is computed by maximizing

1 n n i=1 x 2 ij subject to ψ j (t) 2 dt = 1
and to the orthogonality constraints

ψ (t)ψ j (t)dt = 0, 1 ≤ ≤ j -1.
Functional PCA can be characterized in terms of the eigenanalysis of covariance operators. If (λ j ) j≥1 denotes the eigenvalues and (ψ j ) j≥1 the eigenfunctions of the operator C defined by C(f

)(s) = 1 0 C(s, t)f (t)dt, where C(s, t) = cov(X(s), X(t)), then X i (t) = EX(t) + +∞ j=1 x ij ψ j (t),
where the x j are uncorrelated centered random variables with variance E[x 2 ij ] = λ j . There are similarities with the context of Example 2.2, but here the kernel depends on X. In practice, the decomposition can easily be computed with discrete matrix operations, replacing C(s, t) by the covariance matrix of the X i . This basis has some nice properties. In particular, considering a fixed number of coefficients, it minimizes among all orthogonal bases the average squared L 2 distance between the original curve and its linear representation (see, e.g., the book of Ghanem and Spanos [START_REF] Ghanem | Stochastic Finite Elements: A Spectral Approach[END_REF]). The two main advantages using this synthetic data set is that we can simulate as many curves as we wish and we know the right clusters.

The k-means algorithm on the projected coefficients has been run for all four bases (Best-Entropy, Haar, functional PCA and Fourier basis), with varying sample size n and projection dimension d. Since the result of a k-means algorithm may depend on the choice of the initial centers, the algorithm is restarted 100 times. The maximum number of iterations per run is set to 500. This program tries to globally minimize the projected empirical distortion W d,n (c). To evaluate its performance, we compute an approximation W (d, n) of the distortion W ∞ (ĉ d,n ) using a set of 18000 sample curves. This is possible in this simulated example, since we can generate as many curves as we want. The distortion W (d, n) is computed for d varying from 2 to 50 and n ranging from 100 to 3100. We restricted ourselves to the case d ≤ 50, since there are only 128 discretization points. Moreover, as pointed out earlier, d must not be too large for computational complexity reasons. Indeed, a projection dimension d = 50 is already high for a practical use. provides good results and is reasonably low for many applications. The evolution of the distortion for the Fourier basis looks quite odd: it shows a first decreasing step before increasing again. However, this increasing can be explained in the following way. The centers chosen first are wrong, but seem to give a better distortion than the "real" clustering. When the dimension grows large enough, these first wrong centers no longer represent a local minimum, and the k-means algorithm moves slowly toward the "right" clustering, although losing a bit in distortion. This interpretation is confirmed if we look at the clusters corresponding to each distortion. Furthermore, when the dimension is high relatively to the number of discretization points, some basis functions which oscillate a lot may not be sampled correctly. As a result, the coefficients estimated by approximating the inner products can become very inaccurate as d increases. For very high d, these computed coefficients confound with some noise. Consequently, data becoming more noisy without adding any information, the distortion will increase. The other bases tend to oscillate too, so that they would probably show the same behavior if d were increased above 50. Besides, the small fluctuations observed for the Best-Entropy basis indicate that this basis is not suitable for clustering of control chart time series. The functional PCA basis always gives the lowest distortion. However, the distortions obtained for the three other bases are quite similar, with a preference for the Haar basis over the Best-Entropy wavelet basis, the Fourier basis being the worst choice. As an example, Table 1

Industrial code examples

Let us now turn to the industrial issue which motivated our study. As explained in the introduction, the computer codes used in nuclear engineering have become very complex and costly in CPU-time consumption. That is why we try to approximate them with a cheap function substituted to the code. In order to build a regression model, a preliminary analysis of the different types of outputs is essential. This leads to data clustering, applied here to a computer code with functional outputs. Two different kinds of outputs are presented, the temperature evolution with a data set containing 100 curves, and the thermal exchange coefficient evolution with a data set of 200 curves.

Temperature curves

The data is made of 100 CATHARE code outputs representing the evolution of the temperature in the vessel annular space (Figure 15). Here, the sample size is fixed to n 0 = 100. However, the discretization can be controlled to some extent with spline interpolation. In this case, 256 discretization points are used.

Figure 15: The 100 temperature curves.

Observe that all curves converge in the long-time limit to the same value, corresponding to the temperature of the cold water injected. These curves have been clustered, for physical reasons pertaining to nuclear engineering, in two groups. More precisely, there is a critical set of physical parameters beyond which the thermal shock is more violent and the temperature changes more rapidly (see Auder, De Crecy, Iooss, and Marquès [START_REF] Auder | Screening and metamodeling of computer experiments with functional outputs[END_REF] for more details). The algorithm on the projected coefficients has been run for the Best-Entropy, Haar, functional PCA and Fourier bases, with varying dimension, with the same settings as before. Since we consider real-life data, it is not possible to compute an approximation of W ∞ (ĉ d,n ) as in the simulated example. Hence, the distortion W (d, n 0 ) is simply the output W d,n 0 (ĉ d,n 0 ) of the clustering algorithm, with fixed n 0 = 100. This distortion is computed for d varying from 2 to 50. Figure 16 shows the curve corresponding to the evolution of W (d, n) as a function of d. As expected, it is decreasing in d. We note that until d = 16, the Haar basis provides lower distortion, but for larger values of d, the Best-Entropy basis is better. As before, the Fourier basis is the worst and the functional PCA basis is the best. This can also be checked from Table 2, which presents the distortion obtained for each basis. Although the functional PCA basis gives the best result in terms of distortion, we see that using any of the other three bases is not that bad. Indeed, the same partitioning is found every time (Figure 17). selected as centers. Figure 22 shows the three centers obtained with the Fourier basis, functional PCA basis, Haar basis and Best-Entropy basis, for d = 30.

Conclusion

These clusters allow to build accurate models for the industrial application. The partitioning method presented in this article has been integrated in our metamodel written in R. More specifically, given an array of n input vectors corresponding to n output curves, the purpose is to learn a function φ : z → x mapping an input vector to a continuous curve. In order to improve the accuracy of this task, we begin with a clustering step and then look for a regression model in each cluster separately.

The metamodel lets the user choose between several clustering techniques, either assuming some clusters shapes (like this projected k-means) or trying to discover them in data (like the ascendant hierarchical clustering). The latter are attractive as they do not make assumptions about the results, but they generally need a relatively good sampling of the data. Consequently, the k-means-like techniques are useful in many of our industrial applications, where only a few samples are available. Moreover, these methods provide easily interpretable clusters. In each cluster, after a dimension reduction step, which can either be achieved through the decomposition on an orthonormal basis (linear), or any manifold learning algorithm (nonlinear, with the assumption that the outputs lie on a functional manifold), a statistical learning method is applied to predict representation within this cluster. The mainly used method at this stage is the Projection Pursuit Regression algorithm (see Friedman, Jacobson, and Stuetzle [START_REF] Friedman | Projection pursuit regression[END_REF]). Finally, a simple k-nearest neighbors classifier gives the most probable cluster for a new input, the corresponding regression function is applied, and the curve can be reconstruct from its predicted representation.

For the moment, our metamodel with the clustering method presented have successfully been used on two different scenarios involving the CATHARE code (minor or major break, for each we get temperature, pressure and thermal exchange coefficient curves).

Another research track could consist in considering other types of distances between curves. Distances involving derivatives might be hard to estimate on the thermal exchange coefficient dataset, because several curves are varying rapidly over short period of time, contrasting for instance with the Tecator dataset (http: //lib.stat.cmu.edu/datasets/tecator), on which such distances proved successful (Ferraty and Vieu [15,Chapter 8], Rossi and Villa [START_REF] Rossi | Support vector machine for functional data classification[END_REF]). However, further investigations are needed to know if a smoothing step before clustering based on m-order derivatives would lead to improved results. As the "true" classes are unknown, such a procedure can only be validated within a cross validation framework involving the full metamodel. Experiments with the L 1 distance or some mixed distances related to functions shapes (Heckman and Zamar [START_REF] Heckman | Comparing the shapes of regression functions[END_REF]) could also be studied in future research. (since Π d is the orthogonal projection on R d )

≤ E max =1,...,k R d (X) -R d (c ) 2 ≤ 4R 2 ϕ d .
Hence,

sup c∈S k [W ∞ (c) -W d (c)] ≤ 4R 2 ϕ d ,
as desired.

Proof of Theorem 2.1

We have

W ∞ (ĉ d,n ) -W * ∞ = W ∞ (ĉ d,n ) -W d (ĉ d,n ) + W d (ĉ d,n ) -W * d + W * d -W * ∞ .
According to Lemma 2.1, on the one hand,

W ∞ (ĉ d,n ) -W d (ĉ d,n ) ≤ sup c∈S k [W ∞ (c) -W d (c)] ≤ 4R 2 ϕ d ,
and on the other hand,

W * d -W * ∞ = inf c∈S k W d (c) -inf c∈S k W ∞ (c) ≤ sup c∈S k [W ∞ (c) -W d (c)] ≤ 4R 2 ϕ d ,
and the theorem is proved.

Figure 1 :Figure 2 :

 12 Figure 1: 66 evolution curves of thermal exchange coefficient in a nuclear vessel.

Figure 3 :

 3 Figure 3: Tree structure of wavelet packet bases.

Figure 4 :

 4 Figure 4: Correspondence with dyadic covers of [0,1[ .

Figure 5 :

 5 Figure 5: The wavelet basis.

Figure 6 :

 6 Figure 6: An example of fixed level wavelet packet basis.

Figure 7 :

 7 Figure 7: Haar scaling function φ and mother wavelet function ψ.

Figure 8 :

 8 Figure 8: 10 example curves for each of the 6 types of control chart.

Figures 9

 9 Figures 9 and 10 show the contours plots corresponding to the evolution of W (d, n) as a function of d and n, for the functional PCA and the Haar basis. We remark that the norm of the gradient of W (d, n) vanishes when d and n are close to their maximal values. Hence, as expected according to Theorem 2.1, W (d, n) is decreasing in d and n. When d or n is too small (for instance d = 2 or n = 100), the clustering results are inaccurate. Besides, they are not stable with respect to the n observations chosen. However, for larger values of these parameters, the partitions obtained quickly become satisfactory. The choice n ≥ 300 together with d ≥ 6 generally

Figure 9 :

 9 Figure 9: Contour plot of W (d, n) for the functional PCA basis.

Figure 10 :

 10 Figure 10: Contour plot of W (d, n) for the Haar basis.

Figure 11

 11 Figure 11 shows the curve corresponding to the evolution of W (d, n) as a function of d for n = 500, for all four bases, whereas Figure 12 represents the evolution of W (d, n) versus n for d = 10. According to Section 2, ϕ d must be of the order √ n.For n = 500, √ n is about 22. Considering that ϕ d and d are approximately of the same order, a projection dimension close to 22 should thus be suitable. Indeed, we see via Figure11that W (d, n) does not decrease much more after this value.

Figure 11 :

 11 Figure 11: Evolution of W (d, n) for n = 500 and d ranging from 2 to 50, for (a) functional PCA basis, (b) Haar basis, (c) Fourier basis and (d) Best-Entropy basis.

Figure 12 :

 12 Figure 12: Evolution of W (d, n) for d = 10 and n ranging from 100 to 3100, for (a) functional PCA basis, (b) Haar basis, (c) Fourier basis and (d) Best-Entropy basis.

Figure 13 Figure 13 :Figure 14 :

 131314 Figure 13 represents the 6 clusters for the Fourier basis, for n = 300 and d = 10, whereas Figure 14 shows them for d = 30. The classes obtained with the algorithm are shown in colors, and the real clusters are indicated in the caption.For relatively small values of d, the normal and cyclic patterns are merged into one big cluster, and one cluster corresponding to increasing (or decreasing) shift pattern is split in two. For large enough d, the normal and cyclic designs are well detected, and the overall clustering is correct despite some mixing increasing-upward shift or decreasing-downward shift. We also tested the algorithm for smaller values of the number k of classes. As expected, for the particular choice k = 3, clusters A and B are merged into one single group, and the same occurs for the cluster pairs {C, E} and {D, F }.

Figure 16 :

 16 Figure 16: Evolution of W (d, n 0 ) for the 100 temperature curves, d ranging from 2 to 50, for (a) functional PCA basis, (b) Haar basis, (c) Fourier basis and (d) Best-Entropy basis.

Figure 17 :Figure 18 :

 1718 Figure 17: The temperature curves divided in two groups.

Figure 21 :Figure 22 :

 2122 Figure 21: Three clusters obtained with d = 14, functional PCA basis.

6. 1 1

 11 Proof of Lemma 2.If we define the remainderR d by R d (x) = x -Π d (x) for all x ∈ S, then for (x, y) ∈ S 2 , R d (x -y) 2 ≤ 2 R d (x) 2 + 2 R d (y) > 0 for all j ≥ d, since d > j 0 ) c ∈ S k , W ∞ (c) -W d (c) = E min =1,...,k X -c 2 -min =1,...,k Π d (X) -Π d (c ) 2 = E min =1,...,k Π d (X) + R d (X) -Π d (c ) -R d (c ) 2 -min =1,...,k Π d (X) -Π d (c ) 2 = E min =1,...,k Π d (X) -Π d (c ) 2 + R d (X) -R d (c ) 2 -min =1,...,k Π d (X) -Π d (c ) 2

Table 1 :

 1 gives the values of W (d, n) for n = 1100 and d = 30. Distortion W (d, n) for n = 1100 and d = 30.

	Basis	Fourier Functional PCA Haar Best-Entropy
	Distortion	35.3	32.3	32.8	33.4

Table 2 :

 2 Distortion values for d = 30.

	Basis	Fourier Functional PCA Haar Best-Entropy
	Distortion 480.8	125.3	254.9	151.5

For more details on functional PCA, we refer the reader to Ramsay and Silverman [START_REF] Ramsay | Functional Data Analysis[END_REF].

Observe that since the functional PCA basis is a stochastic basis and the Best-Entropy basis algorithm also uses the data, rigorously the sample should be divided into two subsamples, one to build the basis, and the other for clustering.

Experimental results and analysis

We evaluated the performance of the clustering method with projection, using the various bases described in the previous section, for two different kinds of curves. First, a simulated example where the right clusters are known is discussed, to illustrate the efficiency of the method. Then, we focus on our industrial problem and cluster output curves of a "black box" computer code.

Observe that, although the curves considered in Section 2 and Section 3 were true functions, in practice, we have to deal with curves sampled on a finite number of discretization points. Therefore, a preprocessing step based on spline interpolation is necessary.

Synthetic control chart time series

Control chart time series are used for monitoring process environments, to achieve appropriate control and to produce high quality products. Different types of series can be encountered, but only one, a kind of white noise, indicates a normal working. All the other types of series must be detected, because they correspond to abnormal behavior of the process.

The data set contains a few hundreds to a few thousands curves generated by the process described in Alcock and Manolopoulos [START_REF] Alcock | Time-series similarity queries employing a feature-based approach[END_REF], discretized on 128 time points. There are six types of curves: normal, cyclic, increasing trend, decreasing trend, upward shift and downward shift, which are represented in Figure 8. The equations which generated the data are indicated below. We can see via Figure 20 that W (d, n 0 ) is decreasing in d again. The functional PCA basis is still the best choice, with a fast convergence (stabilization from d = 10). Interestingly, the Fourier basis shows smaller distortion values than the two wavelets basis in this case, as Table 3 indicates. All the partitions obtained are very similar. A typical example is given by Figure 21. However, as for the temperature curves, it is interesting to look at the curves


Marco Heide

Prof. Dr Martina Karola Hasseler

Dipl.-Pflegelehrer Marco Von Der Heide

Staatsexamen Pflegewissenscha

Sonja Indefrey

Haw Hamburg

Resources and barriers of an effective diabetes care management -experiences and perspectives of people with type 2 diabetes

Keywords: diabetes management, nurse-led diabetes clinic, primary care, prevention, health promotion Blinded

Resources for and barriers to effective diabetes care management-experiences and perspectives of people with type 2 diabetes

Martina K. Hasseler,

7. Methodology: I corrected Methodology in Methods. Furthermore, I added a paragraph on following information: when was study carried out, socio-demographic characteristics of the sample and sampling "As part of the inclusion criteria for this project, individuals had to be over the age of 50, diagnosed with type II diabetes for at least 2 years, able to give informed consent, able to understand and speak German fluently and living inside or around a city in the north of Germany. Participants were recruited over a period of three months from March 2008 until May 2008. The process of recruiting was carried out at GP (General Practitioner) surgeries and diabetes support groups. Overall, a convenience sample was drawn and seven patients with type 2 diabetes willing to participate in the study were interviewed. Overall, five female and two male diabetics between 55 and 76 years were interviewed. Three of the participants were married, one was single and three were widowed. All of them learned a profession but six were retired with only one still working in his profession. The income level ranged between 500 € and 1.500 € monthly." 8. Conclusion: I added information on limitations of the study and future proceedings: See page 3: The findings of the project form the basis for developing a nurse-led diabetes care management program in primary care settings in Germany. The aim of the further project is an individual-centered concept of care which considers the resources as well as the daily difficulties of patients and attempts to develop need-orientated solutions in co-operation with the patients in order to enhance permanent changes in life-style and disease-related behaviour. The design and structure of the nurse-led diabetes clinics follows a preventative home visit approach with regular home visits. Therefore, the perspective of patients with type 2 diabetes is of high importance. For that reason, a qualitative and explorative design based on symbolic interactionism was chosen. This approach enables to explain the subjective behaviour as well as people's consciousness of social processes [14]. See page 10: "Since only seven diabetics were interviewed the validity and reliability of the results are limited. However, they indicate that diabetes management programs need revisions and more personcentered and health promotional approaches. For that reason, more research is required to gain further insight into the perspectives of people concerned in order to integrate these findings into more effective diabetes management programs. It is planned to develop on the basis of the findings a more standardised questionnaire for people with type 2 diabetes and to examine their perspectives of useful and meaningful diabetes counselling and management. Furthermore, the findings of the project form the basis for developing a nurse-led diabetes care management program in primary care settings in Germany as described above."

Yours sincerely

Background

At present, approximately 6 million people in Germany have a diagnosis of type 2 diabetes. The prevalence rate is increasing with the current rate being 5% a year. The health care costs for patients with diabetes mellitus type 2 place a financial pressure on the German health care system costing approximately 16 billons Euros a year. The majority of the money is allocated to treat the long-term effects of diabetes and related complications [START_REF] Liebl | Kosten des Typ-2-Diabetes in Deutschland[END_REF][START_REF] Lange | Ältere Menschen mit Diabetes: Stärken erkennen und nutzen[END_REF]. However, the prevalence rate increases in lower socioeconomic levels, the reasons for which are manifold. In lower socioeconomic levels more risk factors can be identified such as obesity or level of education, which are linked to health-related behavior as well as inadequate utilization of health care measurements [START_REF] Brown | Socioeconomic position and health among persons with diabetes mellitus: a conceptual framework and review of the literature[END_REF]. Furthermore, research indicates that people with a lower education level participate less in preventative measures, health-related education and training programs [START_REF] Mielck | Soziale Ungleichheit und Gesundheit[END_REF]. In light of these developments, health care concepts and programs for patients with type 2 diabetes face new challenges.

Effective diabetes care management programs aim at changing the life-style of those concerned. Beneficial factors for a permanent change of life-style are support networks as well as on-going diabetes-relating counseling, education and training programs [START_REF] Nafdm | Nationales Aktionsforum Diabetes mellitus: Leitfaden Prävention Diabetes mellitus Typ 2[END_REF][START_REF] Hasseler | Pflegesprechstunden in der Primärversorgung -Prävention in Diabetes mellitus Typ 2 durch integrierte, wohnortnahe und berufsgruppenübergreifende Schulung, Anleitung und Beratung[END_REF]. Research findings in public health indicate that measurements for behaviour modification are more effective in terms of scope and consistency if they are well integrated in the circumstances, social environment and mode of life of those concerned. Especially, measurements following a setting-oriented approach are considered much more effective (Rosenbrock and Gerlinger 2004). The health care situation of patients with type 2 diabetes is viewed as problematic in the primary care setting in Germany. The workload as well as time pressure in primary care has increased considerably within the last years. Studies reveal that diabetes care management is in many cases uncoordinated, integrated health care concepts are absent and patients rarely receive a systematic consultation and education [START_REF] Gabbay | Nurse case management improves blood pressure, emotional distress and diabetes complication screening[END_REF][START_REF] Möller | Patientenschulung in der Hausarztpraxis als Beitrag zum Disease Management. Qualitative Studie über Motive und Hindernisse bei der Durchführung von Schulungen für Typ-2-Diabetiker[END_REF][START_REF] Taylor | Promoting Health in Type 2 Diabetes: Nurse-Physician Collaboration in Primary Care[END_REF]. Social, psychological and biographical aspects of the lives of diabetics are infrequently taken into consideration. Current education and training modules for patients with type 2 diabetes are focused on a traditional concept of teaching and education with an aim of mere knowledge transfer without considering individual circumstances. An individualand setting-orientated concept of diabetes management in primary care is missing, even though the majority of diabetics are cared for in the primary care setting (DDU/NAFDM 2007; [START_REF] Hasseler | Pflegesprechstunden in der Primärversorgung -Prävention in Diabetes mellitus Typ 2 durch integrierte, wohnortnahe und berufsgruppenübergreifende Schulung, Anleitung und Beratung[END_REF][START_REF] Nafdm | Nationales Aktionsforum Diabetes mellitus: Leitfaden Prävention Diabetes mellitus Typ 2[END_REF].

The objective of the project was to identify diabetes-related coping strategies and problems of adjustment to the disease from the perspective of patients suffering from diabetes in order to reveal needs for support with their daily routine and life in general. In particular, barriers and facilitators or resources which enhance or hamper an adjustment to health-related behavior were of interest in this project.

Against this background following research question were formulated:

1. How do patients with diabetes cope with the disease with regard to their health/disease-related behaviour? 2. Which resources do patients with type 2 diabetes use? What meanings have the resources in order to implement a healthy behavior and lifestyle?

3. What are the barriers hindering the implementation of a healthy behavior and lifestyle in patients with type 2 diabetes?

The findings of the project form the basis for developing a nurse-led diabetes care management program in primary care settings in Germany. The aim of the further project is an individualcentered concept of care which considers the resources as well as the daily difficulties of patients and attempts to develop need-orientated solutions in co-operation with the patients in order to enhance permanent changes in life-style and disease-related behavior. The design and structure of the nurse-led diabetes clinics follows a preventative home visit approach with regular home visits. Therefore, the perspective of patients with type 2 diabetes is of high importance. For that reason, a qualitative and explorative design based on symbolic interactionism was chosen. This approach enables to explain the subjective behavior as well as people's consciousness of social processes [START_REF] Lamnek | Qualitative Sozialforschung[END_REF].

Methods

Problem-centered interviews according to [START_REF] Witzel | Das problemzentrierte Interview [Absätze[END_REF] were used for data collection. This tool includes a short questionnaire on socio-demographic data such as age, gender, profession/occupation, qualification and year of diabetes diagnosis. Furthermore an interview guide with open-ended questions was used to ensure that certain information was collected. The questions were focused on lifestyle, experiences in daily life, support measures, experiences with health professionals, social support, important resources. The inductive-deductive approach of this data collection tool ensures to understand individual and subjective actions and meanings of actions as well as social reality in the interpretation of the individuals [START_REF] Mayring | Einführung in die qualitative Sozialforschung: Eine Anleitung zum qualitativen Denken[END_REF][START_REF] Witzel | Das problemzentrierte Interview [Absätze[END_REF].

The interviews were tape-recorded and transcribed verbatim. A qualitative content analysis was applied for data analysis in order to identify major themes from the data [START_REF] Mayring | Einführung in die qualitative Sozialforschung: Eine Anleitung zum qualitativen Denken[END_REF]). According to rules and theoretical principles a coding system was developed. The interviews were structured, paraphrased, categorized as well as analyzed on the basis of the coding system. Overall, 12 main categories could be developed: 1. well-being/satisfaction, 2. individual concepts of disease and experience of disease, 3. life-style, 4. education/knowledge, 5. physical perception , 6. individual coping strategies, 7. monitoring, 8. social support, 9. family life, 10. religion/spirituality, 11. social life, 12. health care.

As part of the inclusion criteria for this project, individuals had to be over the age of 50, diagnosed with type II diabetes for at least 2 years, able to give informed consent, able to understand and speak German fluently and living inside or around a city in the north of Germany. Participants were recruited over a period of three months from March 2008 until May 2008. The process of recruiting was carried out at GP (General Practitioner) surgeries and diabetes support groups. Overall, a convenience sample was drawn and seven patients with type 2 diabetes willing to participate in the study were interviewed. Overall, five female and two male diabetics between 55 and 76 years were interviewed. Three of the participants were married, one was single and three were widowed. All of them learned a profession but six were retired with only one still working in his profession. The income level ranged between 500 € and 1.500 € monthly.

Results and Discussion

The starting point of this project was to question how patients with diabetes view and perceive resources, barriers and support in order to change and to adjust life-style and health-related behavior. In summary, patients with type 2 diabetes apply a variety of coping styles. The necessary implementation of changes of life-style and health-related behavior causes challenges in the daily lives of these patients. The most important categories and discussions are presented below.

Well-being and satisfaction

The study participants have been diagnosed with type 2 diabetes for several years. In retrospect they conclude that subjective satisfaction and well-being have not changed since being diagnosed with diabetes. They perceive variation in blood sugar levels as unpleasant but it does not influence their subjective satisfaction and well-being. All of the interviewed participants state that they do not follow a strict diet in order to enhance their subjective satisfaction. An enhancement of subjective satisfaction and well-being is associated with improved physical activity and mobility. A decrease in subjective satisfaction and well/being is attributed to diabetes-related complications or general uncertainty about how to handle the disease. The participants perceive physical restraints as an important reason for less satisfaction and well-being. They refer particularly to physical restrictions and pain as important factors influencing their performance of daily activities.

It seems that with regard to subjective well-being and satisfaction the disease is only of secondary concern. It becomes obvious, as Schneider et al pointed out [START_REF] Schneider | Krankheitsbewältigung und Krankheitsbelastung älterer Menschen mit Diabetes mellitus[END_REF], that the disease influences the subjective well-being and satisfaction only to a minor degree.

On one hand, the results suggest that diabetics have integrated the disease in their life. It seems that they have obtained an insight into the disease as well as treatment which results in an acceptance of the chronic disease, an increased diabetes-specific knowledge and an active participation in therapy and treatment. On the other hand, the results could implicate that they have a low insight into the disease and treatment as they perceive diabetes only of secondary influence to their live, satisfaction and well-being. It is possible that diabetes-related knowledge is not applied appropriately to their lives and that consistent diabetes treatment and life-style are not adhered to. This could result in diabetes-related complications which could negatively influence satisfaction and well-being [START_REF] Woods-Büggeln | Psychologische Standards in der Diabetesschulung: die Patienten verstehen[END_REF]. A lack of acceptance of the disease could constitute a barrier to follow a healthy life-style and behavior. A specific need for support for patients with type 2 diabetes emerges from these findings. Diabetes clinics that are focused on a preventative and health promotional approach, aiming to empower patients and supporting them in their self-management should assist diabetics in developing an improved and encompassing awareness of the disease as well as insight into the disease.

Individual concepts of disease and experience of disease

This category is focused on the question how people explain and how they experience their disease. Health-and disease-related behavior is subjected to different determinants, e.g.for instance subjective theories of diseases, which are of importance for disease-related behavior and physical perception. People suffering from a disease are generally prepared to change harmful behavior when they are of the opinion that they can positively influence the course of disease and the outcomes. Harmful health-related behavior will be maintained or cultivated if it appears that the onset and progression of the disease can not be influenced, e.g. due to genetic pre-disposition or 'fate' [START_REF] Fürstler | Psychologie und Sozialwissenschaften für Pflegeberufe 2, Klinische Psychologie, Behinderung, Soziologie[END_REF].

The participants in this project offer different subjective concepts of diseases. Genetic heredity was the most mentioned concept.

In addition to that some participants are of the opinion that other diseases have triggered the disease or that deep emotional experiences have caused it.

With the exception of the genetic pre-disposition as a causal explanation factor for type 2 diabetes [START_REF] Kerner | Definition, Klassifikation und Diagnostik des Diabetes mellitus[END_REF] no other references in the literature relating to the expressed individual concepts of disease could be found.

In summary, these results show that the participants have limited knowledge about diabetes with regards to pathophysiology and aetiology. For that reason, they tend to apply a laypersons' concept. This could indicate a lack of insight into the disease and a need for increased support in order to improve an awareness of the disease as well as related side-effects, complications and necessary changes of life-style.

Life-Style

The category "life-style" includes the sub-categories diet, physical activity and daily routine. The most significant changes have occurred within the sub-category "diet". Most of the participants consider the adjustment to a new diet as very difficult. Only after many years of adaptation to it, the diet is not longer regarded as problematic and is integrated into the life style of the participant. [START_REF] Kulzer | Verhaltensmedizinische Ansätze zur Gewichtsreduktion bei Typ II Diabetikern[END_REF] have also referred to the process of internalization of new diets for people with diabetes which obviously takes a longer period of time. Furthermore, the data suggest a gender difference with regards to knowledge about a diabetes adjusted diet. For instance, the female diabetics demonstrated a differentiated knowledge and revealed recipes in the interviews. On the contrary, the male diabetics have made no specific comments on the new diets.

This finding suggests that female and male diabetics have different needs in teaching, education and information. Gender could influence the adjustment and adaptation to different aspects of diabetes-related life-style and the outcomes of diabetes clinics. The role of gender needs further consideration and research in how to develop effective diabetes clinics and consultations.

Another very interesting finding is related to insulin. Some of the participants use insulin purposefully in order to compensate for maladaptive eating patterns that are not in keeping with the diet, as it allows quick adjustment of blood sugar levels.

Literature suggests that the use of insulin liberalizes diabetes management and therapy which facilitates individual autonomy [START_REF] Chantelau | Diät bei Diabetes mellitus[END_REF]. The reclamation of autonomy can be regarded as positive in diabetes management and therapy however; it carries the inherent danger that necessary adjustments in diet are not taken seriously, which make a relapse into previous eating patterns and behavior more likely. On the basis of this finding, it can be concluded that patients with type 2 diabetes receiving a medication-based therapy need special support and counseling which stresses the importance of behavior-and life-style modification (Seidl et al. 2007).

With regards to the sub-category physical activity it can be summarized that all participants engage in physical activity and consider it as very important for their diabetes management and disease-related quality of life. Physical restraints are regarded as negative and they can result in restrictions on daily activities as well as social contacts. On the other hand, increased physical activity causes wellbeing and higher quality of life. The importance and effects of physical activity coincides with guidelines on evidence-based diabetes management [START_REF] Hader | Diagnostik, Therapie und Verlaufskontrolle des Diabetes mellitus im Alter[END_REF]).

The findings in sub-category changes in daily routine indicate that activities in the daily routine are planned well ahead according to the diabetes therapy.

According to the "health-belief-model" [START_REF] Schwarzer | Diabetes -Der Umgang mit einer chronischen Krankheit[END_REF]) people with a lack in awareness of problems are less prone to change their health behavior compared with people with a higher awareness of problems who actively try to influence their situation. Thereby, self-efficacy plays an important role in adjusting and adapting the daily routine. It mirrors the expectation that human-beings possess the necessary competencies and are able to use them adequately in order to deal and cope with certain situations [START_REF] Vogt | Psychologische Grundlagen der Gesundheitswissenschaften[END_REF]. The collected data display the important significance of self-efficacy in self-management. A lacking disease-related awareness of problems and a low self-efficacy hampers modification of behavior.

Education and knowledge

Special training courses for diabetics represent an important part in coping with the disease. Overall, the participants felt well informed about the disease after participating in a training course.

However, in retrospect and compared with previous results it is obvious that the study participants show deficits in diabetes specific knowledge which lead to the conclusion that special training courses do not result in the desired outcomes.

All participants reported positive as well as negative aspects of the special training courses they attended. They rated new knowledge about diet and open discussion of the disease as positive. Furthermore, they appreciated group sessions and the opportunity to talk to other diabetics about the disease. From the perspective of the participants, individual training courses were perceived as positive when individual problems needed solving. In summary, special training courses can be regarded as social resources in order to integrate diabetes in the daily routine.

The negative aspects of special training courses refered to difficulties in transferring contents of the training courses into daily life. Modifications of behavior are hampered by these difficulties.

Despite the fact that most participants attended the courses several times it becomes obvious that they have deficits in diabetes self-management.

Overall, the data suggest that traditional diabetes training courses only offer solutions to the problems of coping with a disease to a certain degree. New concepts of education and counseling have to be developed, which follow the strategy and objective to support patients with diabetes in their daily routines. Some interviews give the impression that cognitive impairment leads to difficulties in applying new knowledge to the daily routine. In addition, individual barriers such as lack of motivation can hamper the application of new knowledge in the daily routine.

The application of contents and knowledge of training courses decreases with increased therapy requirements. Furthermore, intervention and therapy measurements are less often transferred into daily lives when they require a higher level of self-control. Since the sample in this project includes mostly elderly participants it can be concluded that impairment of cognitive skills and competencies represent a barrier in the adequate application and implementation of new knowledge to the lives of those concerned [START_REF] Lange | Ältere Menschen mit Diabetes: Stärken erkennen und nutzen[END_REF].

Most of the participants were of the opinion that the inclusion of families in the training courses would be very helpful. As one participant pointed out: It seems to be important that families receive the same information as the diabetics in order to enable them to understand the disease and the necessary interventions and measurements. On the basis of this finding it can be concluded that families represents a very important social resource which needs to be included in education, training and counseling.

Physical perception

The study participants notice physical changes, especially obesity, hypoglycaemia or comorbidities as a result of the diabetes.

Nevertheless, it seems to be more difficult for them to sense complications such as neuropathy. Only one participant educated herself explicitly about possible complications and she elaborated the information in the interview. The other interviewees did not offer much information on this specific topic. Compared with data analyzed from the short questionnaire it can be assumed that the participants either play the effects, risks and complications of diabetes down or the perception of risks and complications in diabetes is not very well developed. It becomes evident that future consultations and clinics in diabetes care should integrate physical changes and complications of diabetes.

Social support

According to Kuzla the understanding of social support occurs on emotional and spiritual as well as instrumental and physical/practical levels [START_REF] Fürstler | Psychologie und Sozialwissenschaften für Pflegeberufe 2, Klinische Psychologie, Behinderung, Soziologie[END_REF].

These data suggest that the participants in this study received social support from different sources. Family represents the most important source. The interviewees viewed support from their families as positive and individualised. Family support seems to be much more directed at emotional and physical/practical levels. Especially support in coping with the diseases, uncertainties in detecting symptoms and changes in diet were raised.

Besides the individuals' families, support groups were rated as an important source of support. They offer spiritual and instrumental support in particular and enable diabetics to talk about their diseases and to get reassurance.

The data gives the impression that support groups assist the participants to shed their role as a chronically ill patient and thus to regain a normal everyday routine. Participation in self-helf groups could contribute to support self-management and represents that therefore a very important resource of social support [START_REF] Petermann | Verhaltensmedizinische Ansätze bei Diabetes mellituseine Übersicht[END_REF].

Health care professionals offer a predominantly instrumental assistance in the process of social support.

It is of high importance for diabetics to have access to diabetes-related knowledge. The first port of contact is usually their GP (General Practitioner) or clinical based diabetes adviser. These finding mirrors the current health care situation for people with diabetes in Germany. Generally, GPs (General Practitioner) treat and counsel diabetics in their practices in primary care settings. In addition, diabetes-specialized surgeries offer diabetes management and therapy. Each specialized practice treats up to 400 people with diabetes (Deutsche Diabetes Union 2006; Gesundheitsziele.de. Forum für Gesundheitsziele Deutschland 2003; [START_REF] Möller | Patientenschulung in der Hausarztpraxis als Beitrag zum Disease Management. Qualitative Studie über Motive und Hindernisse bei der Durchführung von Schulungen für Typ-2-Diabetiker[END_REF]). However, due to the high amount of people suffering from diabetes in Germany and the high workloads of GPs (General Practitioner) and diabetes specialists, individual-centered counseling is hardly feasible [START_REF] Gabbay | Nurse case management improves blood pressure, emotional distress and diabetes complication screening[END_REF][START_REF] Taylor | Promoting Health in Type 2 Diabetes: Nurse-Physician Collaboration in Primary Care[END_REF][START_REF] Möller | Patientenschulung in der Hausarztpraxis als Beitrag zum Disease Management. Qualitative Studie über Motive und Hindernisse bei der Durchführung von Schulungen für Typ-2-Diabetiker[END_REF]). Based on these findings, nurse-led diabetes clinics within the primary care setting following a regular preventative home visit concept could improve the access to health care and patient-centred consultations. It could furthermore provide the integration of the family in the consultation process which is not possible in GP surgeries (General Practitioner) or specialist practices. A recent report on diabetes in Germany stresses the significance of health professionals working with diabetics to strengthen their efforts to support people suffering from diabetes (DDU/NAFDM 2007). Obviously, the traditional structures and infrastructures of diabetes therapy and management are not sufficient and effective in Germany`s primary care setting.

Health Care

For the category health care three subcategories were formed: helpful interventions and measurements, unnecessary interventions and measurements, wishes/demands.



Generally, satisfaction with medical care seems to be high. The participants value particularly the care they received in diabetes-specialized hospitals as very helpful.



However, they very often perceive compulsory interventions and measurements as unnecessary. In particular if these are prescribed and the participants can not decide independently if they want to obtain them or not. Another disadvantage seems to be a lack of information with regard to the health system. On one hand, the interviewees point to services offered by health care insurances which are not transparent to them. On the other hand, they are only aware of training courses as diabetes-relevant health care services.



Concerning the sub-category requests and demands for the health system they require more preventative interventions and measurements which have the potential to support them in coping with the disease and to offer those regularly.

It has to be taken into consideration that it is a difficult task to measure satisfaction in health care. Especially, if those concerned do not receive enough information about measurements and interventions. However, the data give the impression that there is a lack in individual-focused and preventative orientated measurements and initiatives for patients with type 2 diabetes in Germany. Furthermore, the finding that some of the participants have felt some measurements and interventions as obligatory needs to be to be critically discussed. Taking into consideration that some participants perceive difficulties in self-management and a need for more information and preventative measures, it can be concluded that so far the measurements and interventions in diabetes care management are not effective and do not fulfill the needs of patients with diabetes. Subjective ideas about health, disease, depression etc., influence the self-management of these patients. Empowering people and supporting them in their needs is time-consuming. Due to high workloads and economic pressures GP's (General Practitioners) and specialists in private practices very often lack time to offer individual-centered, comprehensive discussions, consultations and instructions for the clients and their families [START_REF] Gabbay | Nurse case management improves blood pressure, emotional distress and diabetes complication screening[END_REF][START_REF] Taylor | Promoting Health in Type 2 Diabetes: Nurse-Physician Collaboration in Primary Care[END_REF][START_REF] Möller | Patientenschulung in der Hausarztpraxis als Beitrag zum Disease Management. Qualitative Studie über Motive und Hindernisse bei der Durchführung von Schulungen für Typ-2-Diabetiker[END_REF]).

In the light of theories in health promotion and the necessity of empowerment it is questionable whether the current approach in diabetes care management in primary care settings in Germany is effective. In order to achieve sustainable effects in education, training, instruction and counseling in diabetes care, it is imperative to introduce integrative models of care which take the knowledge, expectations, feeling of well-being of those concerned into consideration [START_REF] Hirsch | Empowerment bei Diabetes: den eigenen Weg finden[END_REF]. These concepts of care should support empowerment and self-management of diabetics.

Conclusion

The results indicate that people with type 2 diabetes need support in the primary care and home setting. The current services in the German health system do not fulfill the needs of patients with diabetes to receive individual-centered care, family integration as well as comprehensive consultations, counseling and instructions.

The support needs to be focused on knowledge application, modification of disease-and healthrelated behavior as well as assistance in dealing with daily difficulties. Furthermore it needs to address challenges and perception of physical changes and complications due to diabetes. Besides a special diabetes-related knowledge and resources are of high importance for these patients in order to cope with the disease in their day-to-day life. The integration of families in consultations or training courses could be valuable in the diabetes care management process. Families represent very important social resources and could support diabetics in their efforts to change life-style and health-related behavior. The extracted data suggest that diabetes clinics need a gender oriented approach. Women and men suffering from diabetes deal differently with the disease and take different approaches. Health professionals could support people with diabetes coping with the disease by applying the resources and reducing the barriers perceived by patients. It is of high importance to strengthen the responsibility and the autonomy of patients with diabetes in order to enhance diabetes self-management and diabetes-related quality of life. In addition, research has shown that nurse-led clinics can be an effective concept of care in primary and in partnership with GPs (General Practitioner) for people suffering from type 2 diabetes [START_REF] Gabbay | Nurse case management improves blood pressure, emotional distress and diabetes complication screening[END_REF][START_REF] Taylor | Promoting Health in Type 2 Diabetes: Nurse-Physician Collaboration in Primary Care[END_REF][START_REF] Wong | Establishing a definition for a nurse-led clinic: structure, process, and outcome[END_REF]. Since only seven diabetics were interviewed the validity and reliability of the results are limited. However, they indicate that diabetes management programs need revisions and more person-centered and health promotional approaches. For that reason, more research is required to gain further insight into the perspectives of people concerned in order to integrate these findings into more effective diabetes management programs. It is planned to develop on the basis of the findings a more standardized questionnaire for people with type 2 diabetes and to examine their perspectives of useful and meaningful diabetes counseling and management. Furthermore, the findings of the project form the basis for developing a nurse-led diabetes care management program in primary care settings in Germany as described above.

Acknowledgment:

The study was performed at the University of Osnabrueck, Faculty of Human Sciences, Department of Public Health and Nursing Science, Germany. No external grants supported this study. The authors would like to thank all interviewees participating in this study.

Conflict of interest: None


MD Belén Merck
email: belenmerck@yahoo.es

Contralateral risk-reducing mastectomy in young women

Introduction

Contralateral risk-reducing mastectomy consists in the surgical removal of the contralateral breast to reduce the risk of breast cancer there [START_REF] Anderson | Prophylactic surgery to reduce breast cancer risk: a brief literature review[END_REF].

Patients with unilateral primary breast cancer have an increased risk of developing a contralateral breast cancer [START_REF] Herrinton | Efficacy of prophylactic mastectomy in women with unilateral breast cancer: a cancer research network project[END_REF]. In a large-scale study, Quan et al [START_REF] Quan | Incidence and outcomes of contralateral breast cancers[END_REF] evaluated the incidence, stage and outcomes of women with contralateral synchronous (defined as tumors diagnosed within 12 months of the initial diagnosis) and metachronous (tumors diagnosed beyond 12 months after the initial diagnosis) breast cancer over a 9-year study period. In the 28787 cases of breast cancer identified, the mean annual incidence of contralateral breast cancer was 0.1%, while the cumulative incidence of contralateral synchronous cancer was 2.1%, and the cumulative incidence of metachronous cancer was 1.2%.

Using the Surveillance, Epidemiology and End Results database (SEER), Gao et al [START_REF] Gao | Risk of second primary cancer in the contralateral breast in women treated for early-stage breast cancer: a population-based study[END_REF] found that the 5-, 10-, 15-, and 20-year actuarial incidence of developing contralateral breast cancer was 3%, 6%, 9% and 12%, respectively. The average incidence of contralateral breast cancer per year of follow-up per patient was estimated at 0.6%.

Based on these, and other retrospective studies, the incidence rate of contralateral breast cancer in breast cancer patients has been estimated to range from approximately 0.5% to 1.8% per year.

Benefits of contralateral risk-reducing mastectomy

Existing data suggest that contralateral risk-reducing mastectomy may diminish by as much as 95% the risk of developing breast cancer in moderate-and high-risk women, particularly in women with ipsilateral breast cancer or known BRCA1/BRCA2 mutation [START_REF] Herrinton | Efficacy of prophylactic mastectomy in women with unilateral breast cancer: a cancer research network project[END_REF][START_REF] Goldflam | Contralateral prophylactic mastectomy. Predictors of significant histologic findings[END_REF][START_REF] Meijers-Heijboer | Breast cancer after prophylactic bilateral mastectomy in women with a BRCA1 or BRCA2 mutation[END_REF]. Using the Anderson model to predict the risk of developing contralateral breast cancer, McDonnell et al [START_REF] Mcdonnell | Efficacy of contralateral prophylactic mastectomy in women with a personal and family history of breast cancer[END_REF] estimated that contralateral risk-reducing mastectomy resulted in a risk reduction of 64%.

However, contralateral risk-reducing mastectomy does not completely eliminate the risk of contralateral cancer, and to date, due to lack of prospective and randomized trials, there is no evidence to support that it improves overall survival [START_REF] Van Sprundel | Risk reduction of contralateral breast cancer and survival after contralateral prophylactic mastectomy in BRCA1 or BRCA2 mutation carriers[END_REF][START_REF] Lostumbo | Prophylactic mastectomy for the prevention of breast cancer[END_REF]. Despite these inconclusive data, with increasing frequency women are undergoing riskreducing mastectomy. Using the SEER database, Tuttle et al [10] reviewed the treatment trends of patients with unilateral breast cancer diagnosed from 1998 through 2003, and found that the overall rate of contralateral risk-reducing mastectomy more than doubled during the 6-year period, increasing from 2% in 1998 to 5% in 2003 in all cancer stages. In their study, being non-Hispanic white, having a tumor with lobular histology, a previous diagnosis of cancer and/or younger age were associated with significantly higher rates of contralateral risk-reducing mastectomy. Logistic regression analysis demonstrated that young age was associated with higher rates of contralateral risk-reducing mastectomy (7% of all surgically treated patients < 39 years old underwent contralateral risk-reducing mastectomy compared with only 1.3% of women ≥ 70 years old) [10].

Additional factors influencing the rate of contralateral risk-reducing mastectomy identified in other population-based studies are a large tumor size (> 5 cm vs. 2 cm), the presence of multicentric disease, a positive lymph node status, being treated by a female surgeon [START_REF] Arrington | Patient and surgeon characteristics associated with increased use of contralateral prophylactic mastectomy in patients with breast cancer[END_REF], physician advice regarding the risk of contralateral breast cancer, patient's fear of developing another tumor, desire for cosmetic symmetry, family history of breast cancer, difficulty in screening for breast cancer, or a combination of all these reasons [START_REF] Yi | Predictors of contralateral breast cancer in patients with unilateral breast cancer undergoing contralateral prophylactic mastectomy[END_REF][START_REF] Montgomery | Issues of regret in women with contralateral prophylactic mastectomies[END_REF].

Risk factors for contralateral breast cancer

Recently, Yi et al [START_REF] Yi | Predictors of contralateral breast cancer in patients with unilateral breast cancer undergoing contralateral prophylactic mastectomy[END_REF] performed a matched case-control study to identify the clinical and pathologic factors of primary breast cancer that would predict a breast cancer, as well as moderate to high-risk histological features in the contralateral breast. They found that patient's age at diagnosis, a 5-year Gail risk ≥ 1.67%, an invasive lobular histology, an additional ipsilateral moderate to high-risk pathological findings or an ipsilateral multicentric tumor were significant predictors of contralateral breast cancer.

Of these, an invasive lobular histology or multicentric tumor and a 5-year Gail risk ≥ 1.67% were identified as independent predictors of malignancy in the opposite breast.

Conversely, estrogen and progesterone receptor status, previous hormone replacement therapy and first-degree family history of breast cancer were not associated with an increased risk of contralateral breast cancer. Multivariate analysis also revealed that an age ≥ 50 years at the time of the initial cancer diagnosis and an additional ipsilateral moderate to high-risk lesions were independent predictors of moderate-to high-risk histologic findings in the contralateral breast [START_REF] Yi | Predictors of contralateral breast cancer in patients with unilateral breast cancer undergoing contralateral prophylactic mastectomy[END_REF].

The BRCA gene mutation status should always be considered when evaluating a patient's risk of contralateral breast cancer, particularly in young women. BRCA1/2 mutation carriers have a significantly higher risk of developing contralateral breast cancer, with an estimated 10-year risk ranging from 20-42% compared with a 5-6% risk in no-carriers. These patients usually tend to be much younger at diagnosis than sporadic cases of breast cancer [START_REF] Bordeleau | Prognosis of BRCA-associated breast cancer: a summary of evidence[END_REF].

In a prospective trial involving 491 women with stage I or II breast cancer and a BRCA1 or BRCA2 mutation, factors predictive of a reduced risk were the presence of a BRCA2 mutation, age over 50 years at first diagnosis, use of tamoxifen and bilateral oophorectomy. This latter factor was particularly strong in women diagnosed before age 49. In women without bilateral oophorectomy or not receiving adjuvant tamoxifen, the risk of developing contralateral breast cancer was 43% for BRCA1 and 35% for BRCA2 mutation carriers at 10 years [START_REF] Metcalfe | Contralateral breast cancer in BRCA1 and BRCA2 mutation carriers[END_REF].

Recommendations of the Society of Surgical Oncology

According to the Society of Surgical Oncology [START_REF] Giuliano | Society of Surgical Oncology: position statement on prophylactic mastectomy. Approved by the Society of Surgical Oncology Executive Council[END_REF], contralateral risk-reducing mastectomy is indicated in patients with a current or previous diagnosis of breast cancer, for: (i) risk reduction in patients at high risk of contralateral breast neoplasm (i.e. patients with known mutations of BRCA1/2 or other strongly predisposing breast cancer susceptibility genes; patients with a family history of breast cancer in multiple first-degree relatives and/or multiple successive generations of family members with breast and/or ovarian cancer; and a high-risk histology); (ii) patients in whom subsequent surveillance of the contralateral breast would be difficult because of mammographically dense breasts or diffuse indeterminate calcifications in the contralateral breast; and (iii) improved symmetry in patients undergoing mastectomy with reconstruction for the index cancer.

Surgical techniques for contralateral risk-reducing mastectomy

The surgical techniques most frequently used for contralateral risk-reducing mastectomy include:

(i) Total mastectomy, which involves the resection of the entire breast gland including the overlying skin and the nipple-areola complex. The axillary lymph nodes and pectoral muscle are left intact (Figure 1). This technique is generally perceived as a more robust procedure for breast cancer surgical prophylaxis However, due to cosmetic issues, it is being performed less often [START_REF] Newman | Prophylactic mastectomy[END_REF]. Breast reconstruction is optional and can be performed at the time of the mastectomy (immediate) or weeks to years afterwards (delayed breast reconstruction).

(ii) Skin-sparing mastectomy is a technical modification of total mastectomy (Figure 2).

In this procedure, all the skin overlying the breast is preserved, and only the gland and the nipple-areola complex are excised. The main difference from total mastectomy is that it avoids skin excision, thereby achieving a more natural breast reconstruction, and a more favorable aesthetic outcome. The most widely used incisions for this technique include periareolar with or without lateral and/or medial extensions, periareolar elliptical, and mastopexy-type incisions [START_REF] Boyero | Skin-sparing mastectomy: an alternative to conventional mastectomy in breast cancer[END_REF].

(iii) Subcutaneous mastectomy, which is always associated with immediate breast volume replacement. Complete resection of the breast tissue is done through an inframammary or periareolar incision that spares the overlying skin and nipple-areola complex (Figure 3) [START_REF] Newman | Prophylactic mastectomy[END_REF]. In this procedure, the skin and nipple-areola complex are preserved. Although this technique can achieve excellent aesthetic results, major concerns exist regarding its efficacy in controlling local disease [START_REF] Spear | Prophylactic mastectomy and reconstruction: clinical outcomes and patient satisfaction[END_REF].

The complication rate in all techniques is mostly related to the associated reconstructive procedure, and include wound infections, seroma formation, hematoma, delayed wound healing, partial or complete necrosis of the nipple-areola complex, and implant-or miocutaneous flap-related complications [START_REF] Vitug | Complications in breast surgery[END_REF]. Unfortunately, there are no randomized, prospective studies comparing the different techniques of risk-reducing mastectomy.

Conclusions

Contralateral risk-reducing mastectomy significantly reduces the risk of developing contralateral breast cancer. However, the procedure is aggressive, irreversible, and probably unnecessary in patients who are not at high risk to develop contralateral disease. When facing a patient who wishes to undergo contralateral risk-reducing mastectomy, we have to be sure that the benefits outweigh complications. It is essential to identify the subgroup of patients whose contralateral breast cancer risk is high enough to benefit from surgery. Patients should be aware of the risk-reducing benefits of systemic adjuvant therapy, which reduces not only the risk of ipsilateral local failure, but also the incidence of contralateral breast neoplasm. Ac cording to the Society of Surgical Oncology, current indications for contralateral risk-reducing mastectomy include patients at high risk of contralateral breast cancer patients in whom subsequent surveillance of the contralateral breast would be difficult, and the requirement for improved symmetry. Recent publications indicate that young patients are especially prone to undergoing contralateral risk-reducing mastectomy. Therefore, selecting a surgical technique that yields favorable cosmetic results, yet at the same time fulfills its risk-reducing role, is imperative in these patients.

Figure 1 .

 1 Figure 1. Right breast: Status after radical modified mastectomy, adjuvant radiotherapy

Figure 2 .

 2 Figure 2. Skin-sparing mastectomy through a periareolar incision with lateral extension.

Figure 3 .

 3 Figure 3. Left breast: Total mastectomy and sentinel node biopsy and immediate

Acknowledgements

The authors acknowledge the support of Pfizer Spain, which has facilitated the necessary meetings to evaluate and discuss all the data presented in this review, and Dr. Ximena Alvira from HealthCo SL (Madrid, Spain) for assistance in the preparation of this manuscript.