Souleymane Kadri
email: souleymane.kadri-harouna@imag.fr

Valérie Perrier
email: valerie.perrier@imag.fr

Helmholtz-Hodge Decomposition on [0, 1] d by Divergence-free and Curl-free Wavelets

Keywords:

This paper deals with the Helmholtz-Hodge decomposition of a vector field in bounded domain. We present a practical algorithm to compute this decomposition in the context of divergence-free and curlfree wavelets satisfying suitable boundary conditions. The method requires the inversion of divergence-free and curl-free wavelet Gram matrices. We propose an optimal preconditioning which allows to solve the systems with a small number of iterations. Finally, numerical examples prove the accuracy and the efficiency of the method.

Introduction

Vector fields analysis is ubiquitous in engineering, physics or applied mathematics. Most of the solutions of problems arising from these domains are vector fields and they have some compatibility properties related to the nature of the problem. This is the case in the numerical simulation of incompressible fluid flows where the velocity field is divergence-free or in electromagnetism where the electric field contained in the electromagnetic field is curl-free.

The Helmholtz-Hodge decomposition, under certain smoothness assumptions, allows to separate any vector field into the sum of three uniquely defined components : divergence-free, curl-free and gradient of a harmonic function. Thus, the Helmholtz-Hodge decomposition provides a powerful tool for several applications such as the resolution of partial differential equations [START_REF] Girault | Finite element methods for Navier-Stokes equations[END_REF], aerodynamic design [START_REF] Tong | Discrete multiscale vector field decomposition[END_REF], detection of flow features [START_REF] Polthier | Identifying Vector Field Singularities using a Discrete Hodge Decomposition[END_REF] or computer graphics [START_REF] Petronetto | Meshless Helmholtz-Hodge decomposition[END_REF]. Therefore, it is important to have at hand an efficient algorithm to deal with such decomposition numerically.

In case of periodic boundary conditions, Fourier domain offers an ideal setting to compute the Helmholtz-Hodge decomposition, thanks to the Leray projector ⋆. Laboratoire Jean Kuntzmann, University of Grenoble, BP 53 Grenoble Cedex 9, France which writes explicitly [START_REF] Deriaz | Orthogonal Helmholtz decomposition in arbitrary dimension using divergence-free and curl-free wavelets[END_REF]. For more general (physical) boundary conditions, this decomposition is usually achieved by solving a Poisson equation relative to each field component : this is the case when using finite element or finite difference methods [START_REF] Girault | Finite element methods for Navier-Stokes equations[END_REF]. The well known drawback of these methods is their cost, for example in particle-based physical simulations. Then, one resorts to mesh-less method [START_REF] Petronetto | Meshless Helmholtz-Hodge decomposition[END_REF] or methods leading to variational equations [START_REF] Polthier | Identifying Vector Field Singularities using a Discrete Hodge Decomposition[END_REF].

In wavelet setting the Helmholtz-Hodge decomposition will not use the resolution of a Poisson equation, since one knows explicit bases for the divergence-free and curl-free function spaces [START_REF] Lemarié-Rieusset | Analyses multi-résolutions non orthogonales, commutation entre projecteurs et dérivation et ondelettes vecteurs à divergence nulle[END_REF][START_REF] Urban | Wavelet Bases in H(div) and H(curl)[END_REF][START_REF] Dodu | Irrotational or Divergence-Free Interpolation[END_REF][START_REF] Deriaz | Divergence-free and curl-free wavelets in 2D and 3D, application to turbulent flows[END_REF]. In this context, Urban [START_REF] Urban | Wavelets in Numerical Simulation[END_REF], and latter Deriaz and Perrier [START_REF] Deriaz | Orthogonal Helmholtz decomposition in arbitrary dimension using divergence-free and curl-free wavelets[END_REF] introduced methods to compute the orthogonal projections onto divergence-free and curl-free wavelet bases. These wavelet methods have for main interest to provide accuracy for a small number of degrees of freedom, due to the good nonlinear approximation property provided by wavelet bases [START_REF] Cohen | Numerical Analysis of Wavelet Methods[END_REF]. However, the works of [START_REF] Urban | Wavelets in Numerical Simulation[END_REF][START_REF] Deriaz | Orthogonal Helmholtz decomposition in arbitrary dimension using divergence-free and curl-free wavelets[END_REF] are limited to periodic boundary conditions for lack of suitable bases.

The main objective of this paper is to extend the works of [START_REF] Urban | Wavelets in Numerical Simulation[END_REF][START_REF] Deriaz | Orthogonal Helmholtz decomposition in arbitrary dimension using divergence-free and curl-free wavelets[END_REF] to more general physical boundary conditions. We first present the principles of tensorproduct divergence-free and curl-free wavelet construction on the cube. This construction was detailed in [START_REF] Kadri-Harouna | Divergence-free and Curl-free Wavelets on the Square for Numerical Simulation[END_REF] in the 2D case (see also [START_REF] Kadri-Harouna | Ondelettes pour la prise en compte de conditions aux limites en turbulence incompressible[END_REF]). Then we propose an effective method for the Helmholtz-Hodge decomposition based on the computation and inversion of corresponding Gram matrices. The tensor structure of the bases is fully exploited to reduce the computational complexity. Moreover the system is solved with a low complexity, thanks to an optimal preconditioning. The layout of this paper is as follows. In section 2, we recall the theoretical definition and mathematical background of the Helmholtz-Hodge decomposition on a bounded domain of R d . In section 3 we explicit the construction of divergencefree and curl-free wavelets on [0, 1] d with desired boundary conditions. Section 4 is devoted to the description of the numerical method for the Helmholtz-Hodge decomposition, and numerical examples will illustrate its performance.

Helmholtz-Hodge decomposition

We recall in this section some definitions related to the Helmholtz-Hodge decomposition on a bounded domain Ω of R d [START_REF] Girault | Finite element methods for Navier-Stokes equations[END_REF]. We assume that Ω is an open subset of R d , bounded and connected with Lipschitz-continuous boundary Γ (see [START_REF] Girault | Finite element methods for Navier-Stokes equations[END_REF][START_REF] Amrouche | Vector potentials in three dimensional nonsmooth domains[END_REF]).

Definitions

The Helmholtz-Hodge decomposition theorem [START_REF] Girault | Finite element methods for Navier-Stokes equations[END_REF][START_REF] Chorin | A Mathematical Introduction to Fluid Mechanics[END_REF] states that any vector field u ∈ (L 2 (Ω)) d can be uniquely decomposed into the sum of its divergencefree, curl-free and gradient of harmonic function components :

u = u div + u curl + u har (1)
with :

∇ • u div = 0 and ∇ × u curl = 0 (2)
The last component u har is both divergence-free and irrotational :

∇ • u har = 0 and ∇ × u har = 0 (3)
Following [START_REF] Girault | Finite element methods for Navier-Stokes equations[END_REF][START_REF] Amrouche | Vector potentials in three dimensional nonsmooth domains[END_REF], this decomposition may alternatively be written using vector and scalar potentials. There exist a vector potential (scalar if d = 2) Ψ ∈ (H 1 0 (Ω)) d , a scalar potential q ∈ H 1 0 (Ω) and a harmonic potential h ∈ H 1 (Ω) such as :

u div = curl(Ψ) u curl = ∇q u har = ∇h
Moreover, Ψ, q and h are uniquely defined.

In terms of spaces, the decomposition (1) corresponds to an orthogonal splitting of (L 2 (Ω)) d :

(L 2 (Ω)) d = H div (Ω) ⊕ H curl (Ω) ⊕ H har (Ω) (4)
where H div (Ω) is the space of divergence-free vector functions of (L 2 (Ω)) d with vanishing tangential boundary condition :

H div (Ω) = {u ∈ (L 2 (Ω)) d : div(u) = 0, u • n| Γ = 0} (5)
which also coincides (for d = 2, 3) with the curl of potential space :

H div (Ω) = {u = curl(Ψ) : Ψ ∈ (H 1 0 (Ω)) d } (6) (Ψ ∈ H 1 0 (Ω) if d = 2
). On the other side, the space H curl corresponds to the gradient of H 1 (Ω)-potentials which vanish on Γ :

H curl (Ω) = {u = ∇q : q ∈ H 1 0 (Ω)} (7)
Finally, H har corresponds to the gradient of H 1 (Ω)-harmonic potentials :

H har (Ω) = {u = ∇q : q ∈ H 1 (Ω), ∆q = 0} (8)
Other splittings exist, for example in (H 1 0 (Ω)) d to incorporate homogeneous boundary conditions [START_REF] Girault | Finite element methods for Navier-Stokes equations[END_REF].

In the whole space R d or with periodic boundary conditions, the decomposition (1) is explicit in Fourier domain and the third term vanishes : u har = 0 (one obtains the Helmholtz decomposition in this case [START_REF] Deriaz | Orthogonal Helmholtz decomposition in arbitrary dimension using divergence-free and curl-free wavelets[END_REF][START_REF] Urban | Wavelets in Numerical Simulation[END_REF]). In the wavelet context, an iterative procedure was proposed by Deriaz and Perrier [START_REF] Deriaz | Orthogonal Helmholtz decomposition in arbitrary dimension using divergence-free and curl-free wavelets[END_REF]. The purpose here is to extend such method with boundary conditions for the spaces H div (Ω), H curl (Ω) and H har (Ω) introduced in (5, 7, 8).

3 Divergence-free and curl-free wavelets on [0, 1] d This section introduces the principles of the construction and main properties of divergence-free and curl-free wavelets on the hypercube [0, 1] d . The two dimensional case d = 2 has been detailed in [START_REF] Kadri-Harouna | Divergence-free and Curl-free Wavelets on the Square for Numerical Simulation[END_REF], we develop below the extension to more general dimensions. Our construction is based on 1D multiresolution analysis generators (ϕ 1 , φ1 ) and (ϕ 0 , φ0 ) linked by differentiation / integration, and introduced by Lemarié-Rieusset and collaborators in original works [START_REF] Lemarié-Rieusset | Analyses multi-résolutions non orthogonales, commutation entre projecteurs et dérivation et ondelettes vecteurs à divergence nulle[END_REF][START_REF] Jouini | Analyse multirésolution biorthogonale sur l'intervalle et applications[END_REF]. It follows two steps, described in the two forthcoming sections :

(i) Construction of two biorthogonal MRAs of L 2 (0, 1) linked by differentiation / integration.

(ii) Construction of MRAs and wavelet bases of H div (Ω) and H curl (Ω).

3.1 Multiresolution analyses of L 2 (0, 1) linked by differentiation / integration

The construction of regular biorthogonal multiresolution analyses (BMRA) on the interval [0, 1] is now classical (see [START_REF] Cohen | Wavelets on the Interval and Fast Wavelet Transforms[END_REF][START_REF] Chiavassa | On the Effective Construction of Compactly Supported Wavelets Satisfying Homogeneous Boundary Conditions on the Interval[END_REF][START_REF] Monasse | Orthogonal Wavelet Bases Adapted For Partial Differential Equations With Boundary Conditions[END_REF][START_REF] Grivet-Talocia | Wavelet on the interval with optimal localization[END_REF]). It begins with a pair of biorthogonal compactly supported scaling functions (ϕ 1 , φ1 ) [START_REF] Cohen | Biorthogonal bases of compactly supported wavelets[END_REF] of L 2 (R), with some r polynomial reproduction :

x ℓ = k∈Z x ℓ , φ1 (x -k) ϕ 1 (x -k) for 0 ≤ ℓ ≤ r -1 (9)
and similarly for φ1 , with r polynomial reproduction.

Following classical constructions, one define finite dimensional biorthogonal multiresolution spaces :

V 1 j = span{ϕ 1 j,k ; 0 ≤ k ≤ N j -1} and Ṽ 1 j = span{ φ1 j,k ; 0 ≤ k ≤ N j -1} (10)
whose dimension N j ≃ 2 j depends on some free integer parameters (δ 0 , δ 1 ). The scaling functions ϕ 1 j,k satisfy ϕ 1 j,k = 2 j/2 ϕ 1 (2 j x -k) "inside" the interval [0, 1], but this is no more true near the boundaries 0 and 1 (idem for φ1 j,k ). In practice, the scale index j must be great than some index j min , to avoid boundary effects. The biorthogonality between bases writes : < ϕ 1 j,k / φ1 j,k ′ >= δ k,k ′ The approximation order provided by such MRA (V 1 j ) in L 2 (0, 1) is r :

∀ f ∈ H s (0, 1), inf fj ∈V 1 j f -f j L 2 (0,1) ≤ C2 -js , 0 ≤ s ≤ r (11)
whereas ( Ṽ 1 j ) has for approximation order r.

Homogeneous Dirichlet boundary conditions can be simply imposed on (V 1 j ) by removing one scaling function at each boundary 0 and 1 :

V D j = V 1 j ∩ H 1 0 (0, 1) = span{ϕ 1 j,k ; 1 ≤ k ≤ N j -2} (12)
A possibility to adjust the dimension of the biorthogonal space is also to impose Ṽ D j = Ṽ 1 j ∩ H 1 0 (0, 1), and Ṽ D j rewrites (keeping the same notation for the basis functions, which have changed after biorthogonalization) :

Ṽ D j = span{ φ1 j,k ; 1 ≤ k ≤ N j -2} (13)
As usual, biorthogonal wavelet spaces (W 1 j , W 1 j ) are defined by :

W 1 j = V 1 j+1 ∩ ( Ṽ 1 j ) ⊥ W 1 j = Ṽ 1 j+1 ∩ (V 1 j ) ⊥ (14)
and generated by finite dimensional wavelet bases on the interval [START_REF] Monasse | Orthogonal Wavelet Bases Adapted For Partial Differential Equations With Boundary Conditions[END_REF][START_REF] Grivet-Talocia | Wavelet on the interval with optimal localization[END_REF] :

W 1 j = span{ψ 1 j,k ; 0 ≤ k ≤ 2 j -1} and W 1 j = span{ ψ1 j,k ; 0 ≤ k ≤ 2 j -1} (15)
The difficulty now is to derive a new biorthogonal MRA (V 0 j , Ṽ 0 j ) of L 2 (0, 1) such that :

d dx V 1 j = V 0 j
The existence of such biorthogonal MRA was already proved by Jouini and Lemarié-Rieusset [START_REF] Jouini | Analyse multirésolution biorthogonale sur l'intervalle et applications[END_REF] and it should be based on generators (ϕ 0 , φ0 ) introduced in [START_REF] Lemarié-Rieusset | Analyses multi-résolutions non orthogonales, commutation entre projecteurs et dérivation et ondelettes vecteurs à divergence nulle[END_REF] satisfying :

(ϕ 1 (x)) ′ = ϕ 0 (x) -ϕ 0 (x-1) and ( φ0 (x)) ′ = φ1 (x+1) -φ1 (x) (16)
In [START_REF] Kadri-Harouna | Divergence-free and Curl-free Wavelets on the Square for Numerical Simulation[END_REF], we proposed a practical construction of spaces (V 0 j , Ṽ 0 j ) :

V 0 j = span{ϕ 0 j,k ; 0 ≤ k ≤ N j -2} and Ṽ 0 j = span{ φ0 j,k ; 0 ≤ k ≤ N j -2} (17)
satisfying the following proposition.

Proposition 1.

The two BMRAs (V ǫ j , Ṽ ǫ j ) ǫ=0,1 of L 2 (0, 1) constructed in [START_REF] Kadri-Harouna | Divergence-free and Curl-free Wavelets on the Square for Numerical Simulation[END_REF] from biorthogonal generators (ϕ ǫ , φǫ ) ǫ=0,1 satisfying relation [START_REF] Petronetto | Meshless Helmholtz-Hodge decomposition[END_REF], verify :

(i) d dx V 1 j = V 0 j and d dx • P 1 j f = P 0 j • d dx f, ∀ f ∈ H 1 (0, 1) (ii) Ṽ 0 j = H 1 0 (0, 1) ∩ x 0 Ṽ 1 j and d dx • P0 j f = P1 j • d dx f, ∀ f ∈ H 1 0 (0, 1)
where (P ǫ j , Pǫ j ) are the biorthogonal projectors on (V ǫ j , Ṽ ǫ j ).

Wavelet bases of the biorthogonal MRA (V 0 j , Ṽ 0 j ) j≥jmin are simply defined by respectively differentiating and integrating the wavelets of (V 1 j , Ṽ 1 j ) j≥jmin , as stated by the following proposition [START_REF] Jouini | Analyse multirésolution biorthogonale sur l'intervalle et applications[END_REF][START_REF] Kadri-Harouna | Divergence-free and Curl-free Wavelets on the Square for Numerical Simulation[END_REF].

Proposition 2. [START_REF] Jouini | Analyse multirésolution biorthogonale sur l'intervalle et applications[END_REF][START_REF] Kadri-Harouna | Divergence-free and Curl-free Wavelets on the Square for Numerical Simulation[END_REF] Let {ψ 1 j,k } and { ψ1 j,k } be biorthogonal wavelet bases of respectively W 1 j and W 1 j . Then, the wavelets defined by :

ψ 0 j,k = 2 -j (ψ 1 j,k ) ′ and ψ0 j,k = -2 j x 0 ψ1 j,k (18)
are respectively biorthogonal wavelet bases of W 0 j and W 0 j :

W 0 j = V 0 j+1 ∩ ( Ṽ 0 j ) ⊥ and W 0 j = Ṽ 0 j+1 ∩ (V 0 j ) ⊥ (19)
3.2 Divergence-free scaling functions and wavelets on [0, 1] d

Le Ω be the hypercube Ω = [0, 1] d . The objective in this section is to derive wavelet bases of the space H div (Ω), with vanishing outward normal at the boundary Γ = ∂Ω. Following ( 5), H div (Ω) is the curl of the space (H 1 0 (Ω)) d (for d = 3, and H 1 0 (Ω) for d = 2) of vector (scalar for d = 2) stream functions. We begin with the description of basis functions in the 2D case (already detailed in [START_REF] Kadri-Harouna | Divergence-free and Curl-free Wavelets on the Square for Numerical Simulation[END_REF]), then we generalize the construction for d ≥ 3.

Two-dimensional case :

We start with the 2D MRA (V D j ⊗V D j ) j≥jmin of H 1 0 (Ω), where (V D j ) j≥jmin is the 1D MRA of H 1 0 (0, 1) defined in section 3.1 [START_REF] Grivet-Talocia | Wavelet on the interval with optimal localization[END_REF]. For each scale index j ≥ j min , divergence-free scaling functions on Ω = [0, 1] 2 are constructed by taking the curl of scaling functions of V D j ⊗ V D j :

Φ div j,k := curl[ϕ D j,k1 ⊗ ϕ D j,k2 ] = ϕ D j,k1 ⊗ (ϕ D j,k2 ) ′ -(ϕ D j,k1 ) ′ ⊗ ϕ D j,k2 , 1 ≤ k 1 , k 2 ≤ N j -2 (20)
The choice of space V D j ensures that the divergence-free scaling functions satisfy the boundary condition Φ div j,k • n = 0 by construction. Let V div j be the space spanned by these divergence-free scaling functions :

V div j = span{Φ div j,k }, 1 ≤ k 1 , k 2 ≤ N j -2 (21)
By construction, the spaces V div j form a multiresolution analysis of H div (Ω), since it can be proven from proposition 1 that we have [START_REF] Kadri-Harouna | Divergence-free and Curl-free Wavelets on the Square for Numerical Simulation[END_REF] :

V div j = (V D j ⊗ V 0 j ) × (V 0 j ⊗ V D j ) ∩ H div (Ω) (22)
In the same manner, the corresponding anisotropic divergence-free wavelets on Ω are defined by taking the curl of the three types of scalar anisotropic wavelets associated to V D j ⊗ V D j : for j = (j 1 , j 2 ), with j 1 , j 2 > j min ,

Ψ div,1 j,k := curl[ϕ D jmin,k ⊗ ψ D j2,k2 ], 1 ≤ k ≤ N jmin -2, 0 ≤ k 2 ≤ 2 j2 -1 Ψ div,2 j,k := curl[ψ D j1,k1 ⊗ ϕ D jmin,k ], 0 ≤ k 1 ≤ 2 j1 -1, 1 ≤ k ≤ N jmin -2 Ψ div,3 j,k := curl[ψ D j1,k1 ⊗ ψ D j2,k2 ], 0 ≤ k 1 ≤ 2 j1 -1, 0 ≤ k 2 ≤ 2 j2 -1 (ψ D j,k ) being the wavelet basis of W D j = V D j+1 ∩ ( Ṽ D j ) ⊥ .
Three-dimensional case : Divergence-free scaling functions and wavelets on Ω = [0, 1] 3 are constructed by taking the curl of suitable scaling functions and wavelets of (H 1 0 (Ω)) 3 . We will focus on the scaling function construction, since the same technique is used for wavelets. The divergence-free scaling functions are defined by :

Φ div 1,j,k := curl 0 0 ϕ D j,k1 ⊗ ϕ D j,k2 ⊗ (ϕ D j,k3 ) ′ = ϕ D j,k1 ⊗ (ϕ D j,k2 ) ′ ⊗ (ϕ D j,k3 ) ′ -(ϕ D j,k1 ) ′ ⊗ ϕ D j,k2 ⊗ (ϕ D j,k3 ) ′ 0 (23) Φ div 2,j,k := curl (ϕ D j,k1 ) ′ ⊗ ϕ D j,k2 ⊗ ϕ D j,k3 0 0 = 0 (ϕ D j,k1 ) ′ ⊗ ϕ D j,k2 ⊗ (ϕ D j,k3 ) ′ -(ϕ D j,k1 ) ′ ⊗ (ϕ D j,k2 ) ′ ⊗ ϕ D j,k3 (24)
Φ div 3,j,k := curl

0 ϕ D j,k1 ⊗ (ϕ D j,k2 ) ′ ⊗ ϕ D j,k3 0 = -ϕ D j,k1 ⊗ (ϕ D j,k2 ) ′ ⊗ (ϕ D j,k3 ) ′ 0 (ϕ D j,k1 ) ′ ⊗ (ϕ D j,k2 ) ′ ⊗ ϕ D j,k3 (25)
where 1 ≤ k 1 , k 2 , k 3 ≤ N j -2. These functions are contained in H div (Ω) by construction. The space V div j spanned by this family is included into the following standard BMRA of (L 2 (Ω)) 3 :

V j = V 1 j ⊗ V 0 j ⊗ V 0 j × V 0 j ⊗ V 1 j ⊗ V 0 j × V 0 j ⊗ V 0 j ⊗ V 1 j ( 26
)
which makes it easy the coefficient computations on divergence-free bases. To each scaling function, we can associate 7 types of anisotropic divergence-free generating wavelets by taking respectively the curl of wavelets of

{0} × {0} × (V D j ⊗ V D j ⊗ V 0 j ), (V 0 j ⊗ V D j ⊗ V D j ) × {0} × {0} and {0} × (V D j ⊗ V 0 j ⊗ V D j ) × {0}.
As scaling functions Φ div 1,j,k , Φ div 2,j,k and Φ div 3,j,k , their corresponding divergencefree wavelets are also linearly dependent. To construct a multiresolution analysis and wavelet bases of H div (Ω), we need to keep only two scaling functions and their 14 corresponding generating wavelets.

The construction extends to larger dimensions d > 0 readily. As in the isotropic construction of Lemarié-Rieusset [START_REF] Lemarié-Rieusset | Analyses multi-résolutions non orthogonales, commutation entre projecteurs et dérivation et ondelettes vecteurs à divergence nulle[END_REF], we obtain in this case (d -1) types of linear independent divergence-free scaling functions and (d -1)(2 d -1) types of corresponding divergence-free wavelets. For 1 ≤ i ≤ d -1, the general formula of these scaling function is given by :

Φ div i,j,k := 0 . . . 0 arrow i → (ϕ D j,k1 ) ′ ⊗ • • • ⊗ (ϕ D j,ki-1 ) ′ ⊗ ϕ D j,ki ⊗ (ϕ D j,ki+1 ) ′ ⊗ • • • ⊗ (ϕ D j,k d ) ′ arrow i + 1 → -(ϕ D j,k1 ) ′ ⊗ • • • ⊗ (ϕ D j,ki ) ′ ⊗ ϕ D j,ki+1 ⊗ (ϕ D j,ki+2 ) ′ ⊗ • • • ⊗ (ϕ D j,k d ) ′ 0 . . . 0
(27) The scaling functions Φ div i,j,k satisfy the boundary condition : Φ div i,j,k • n = 0, by construction. The space V div j spanned by this family is included into the following standard BMRA of (L 2 (Ω)) d :

V j = V 1 j × • • • × V d j with V i j = V δ1,i j ⊗ • • • ⊗ V δ d,i j , 1 ≤ i ≤ d (28)
δ i,j denotes the Kronecker symbol.

Curl-free scaling functions and wavelets on [0, 1] d

The construction of irrotational scaling functions and wavelets is easier than in the case of divergence-free functions, since it does not depend on the dimension d. According to the definition ( 7) of H curl (Ω), basis functions will be constructed by taking the gradient of scaling functions and wavelets of a multiresolution analysis of H 1 0 (Ω).

The starting point is again a regular multiresolution analysis of H 1 0 (Ω) given by d tensor-product of V D j :

V j = V D j ⊗ • • • ⊗ V D j (29)
Then, the curl-free scaling functions of H curl (Ω) are defined by :

Φ ∇ j,k = ∇[ϕ D j,k1 ⊗ • • • ⊗ ϕ D j,k d ] and V ∇ j = span{Φ ∇ j,k } (30)
where 1 ≤ k i ≤ N j -2 for 1 ≤ i ≤ d. By construction we have :

V ∇ j = ∇[V D j ⊗ • • • ⊗ V D j ]
The multiresolution decomposition of the space V ∇ j leads to :

V ∇ j = ∇   V D jmin ⊗ • • • ⊗ V D jmin jmin≤ji≤j-1   ω∈Ω * d W ω j     , 1 ≤ i ≤ d (31)
with :

Ω * d = {0, 1} d \ (0, • • • , 0) and W ω j = W ω1 j1 ⊗ • • • ⊗ W ω d j d
where the spaces W ωi ji correspond to :

W ωi ji = W D ji if ω i = 1, W ωi ji = V D jmin if ω i = 0
Denoting by Ψ ω j,k the wavelets of W ω j , we define the curl-free wavelets and spaces by :

Ψ ω,∇ j,k = ∇ [Ψ ω j,k ] and W ω,∇ j = span{Ψ ω,∇ j,k } (32)
where j i ≥ j min and 0

≤ k i ≤ 2 j -1, for 1 ≤ i ≤ d.
From proposition 1, the spaces spanned by these curl-free functions are contained in the following BMRA of (L 2 (Ω)) d :

V + j = V 1 j × • • • × V d j with V i j = V 1-δ1,i j ⊗ • • • ⊗ V 1-δ d,i j , 1 ≤ i ≤ d δ i,j
denotes the Kronecker symbol. This property allows fast coefficient computations on irrotational bases.

Since the spaces V j defined in (29) constitute a multiresolution analysis of H 1 0 (Ω), we get :

H 1 0 (Ω) = V jmin ji≥jmin   ω∈Ω * d W ω j   , 1 ≤ i ≤ d (33)
Taking the gradient of relation (33) and using again proposition 1, we obtain :

H curl (Ω) = V ∇ jmin ji≥jmin   ω∈Ω * d W ω,∇ j   , 1 ≤ i ≤ d (34)
This relation (34) was proved in [START_REF] Kadri-Harouna | Divergence-free and Curl-free Wavelets on the Square for Numerical Simulation[END_REF] in the case of 2D construction.

4 Wavelet Helmholtz-Hodge decomposition

Description of the method

The Helmholtz-Hodge decomposition, introduced in section 2, provides the orthogonal splitting of any vector field u ∈ (L 2 (Ω)) d into a divergence-free part, a curl-free part, and a gradient of a harmonic function :

u = u div + u curl + u har (35)
where :

∇ • u div = 0 and u div • n = 0 ∇ × u curl = 0 and u curl • τ = 0 ∇ × u har = 0 and ∇ • u har = 0
n and τ are respectively the unit outward normal and tangent to the boundary ∂Ω.

Our aim in this section is to describe a practical way to compute the components u div and u curl . Our method uses the divergence-free and curl-free scaling functions and wavelet bases constructed in the previous section.

Since H div (Ω) = span{Ψ div j,k } and H curl (Ω) = span{Ψ ∇ j,k } (we adopt a unified notation for the wavelet bases), the components u div and u curl are searched under the form of their wavelet series :

u div = j,k d div j,k Ψ div j,k and u curl = j,k d ∇ j,k Ψ ∇ j,k (36)
By orthogonality of the decomposition (35) in (L 2 (Ω)) d , we obtain :

u, Ψ div j,k = u div , Ψ div j,k and u, Ψ ∇ j,k = u curl , Ψ ∇ j,k (37)
Accordingly the computation of coefficients (d div j,k ) and (d ∇ j,k ) is reduced to the resolution of two linear systems :

M div (d div j,k ) = ( u, Ψ div j,k ) and M curl (d ∇ j,k ) = ( u, Ψ ∇ j,k ) ( 38
)
where M div and M curl are respectively the Gram matrices of the bases {Ψ div j,k } and {Ψ ∇ j,k }.

The above method is nothing but orthogonal projections from (L 2 (Ω)) d to H div (Ω) and H curl (Ω) respectively. In practice, u div is searched as u j div ∈ V div j for some j, and u curl as u j curl ∈ V curl j . Then we recover from the usual Jacksontype estimations :

∀ u ∈ H div (Ω) ∩ (H s (Ω)) d , u -u j div (L 2 (Ω)) d ≤ C2 -js , 0 ≤ s ≤ r -1 and ∀ u ∈ H curl (Ω) ∩ (H s (Ω)) d , u -u j curl (L 2 (Ω)) d ≤ C2 -js , 0 ≤ s ≤ r -1
where r denotes the approximation order provided by the generator ϕ 1 .

The last component u har of the decomposition ( 35) is computed by subtracting u div and u curl from u :

u har = u -u div -u curl (39)
4.2 Divergence-free and curl-free Gram matrices computation

In this section we present a practical computation of matrices M div and M curl . For easy reading, we focus on the matrix M curl in the 2D case. The extension to larger dimensions d > 2 follows readily from this two-dimensional case.

The key idea is to use the tensor structure of M curl to reduce the computation. Let M j and R j denote respectively the Gram and stiffness matrices of the 1D basis {ψ D j,k } :

[M j ] k,k ′ = ψ D j,k , ψ D j,k ′ and [R j ] k,k ′ = (ψ D j,k ) ′ , (ψ D j,k ′ ) ′ (40)
The tensor structure of the basis {Ψ ∇ j,k } allows to express the inner product Ψ ∇ j,k , Ψ ∇ j ′ ,k ′ in terms of matrix elements (40). By definition of the basis functions we get :

Ψ ∇ j,k , Ψ ∇ j ′ ,k ′ = (ψ D j 1 ,k 1 ) ′ ⊗ψ D j 2 ,k 2 , (ψ D j ′ 1 ,k ′ 1 ) ′ ⊗ψ D j ′ 2 ,k ′ 2 + ψ D j 1 ,k 1 ⊗(ψ D j 2 ,k 2 ) ′ , ψ D j ′ 1 ,k ′ 1 ⊗(ψ D j ′ 2 ,k ′ 2 ) ′
which rewrites :

Ψ ∇ j,k , Ψ ∇ j ′ ,k ′ = [M j ] k1,k ′ 1 • [R j ] k2,k ′ 2 + [R j ] k1,k ′ 1 • [M j ] k2,k ′ 2 (41)
Then M curl can be decomposed as :

M curl = M j ⊗ R j + R j ⊗ M j ( 42
)
The tensorial decomposition (42) has for main interest to reduce a 2D matrixvector product with M curl to matrix-matrix products with M j and R j . More precisely, if (d ∇ j,k ) denotes the vector of curl-free wavelet coefficients of u curl , defined in (36), equation (42) leads to :

[M curl (d ∇ j,k )] = M j [d ∇ j,k ]R j + R j [d ∇ j,k ]M j (43)
where [d ∇ j,k ] denotes the matrix of elements d ∇ j,k . In practice the matrices only needed to compute and to store are the 1D matrices M j and R j .

Finally, the matrix M curl has a sparse structure, due to the compact support of basis functions. Figure 1 shows the shape of M curl , for j = 6, in the case of Daubechies generators with r = 3 vanishing moments. Remark that in 2D M div = M curl (which is also the stiffness matrix of the Laplacian onto the wavelet basis {ψ D j1,k1 ⊗ ψ D j2,k2 }), since we have :

∀ u, v ∈ H 1 0 (Ω); Ω curl(u) • curl(v) dx = Ω ∇u • ∇v dx (44)

Right-hand side computations

To solve system (38), we need to compute efficiently inner products u, Ψ div j,k and u, Ψ ∇ j,k . This is achieved by using the decomposition of u in the wavelet bases in the suitable multiresolution analyses of (L 2 (Ω)) d that contain alternatively the divergence-free or curl-free functions. To illustrate, we will explain the computation of u, Ψ ∇ j,k in the two dimensional case.

Let (d 1 j,k ) and (d 2 j,k ) denote respectively the coefficients of the decomposition of u = (u 1 , u 2 ) on the wavelet basis of (V 0 j ⊗ V D j ) × (V D j ⊗ V 0 j ) :

u 1 = j,k d 1 j,k ψ 0 j1,k1 ⊗ ψ D j2,k2 u 2 = j,k d 2 j,k ψ D j1,k1 ⊗ ψ 0 j2,k2
The computation of inner product u, Ψ ∇ j ′ ,k ′ writes :

u, Ψ ∇ j ′ ,k ′ = j,k d 1 j,k ψ 0 j1,k1 ⊗ ψ D j2,k2 , (ψ D j ′ 1 ,k ′ 1 ) ′ ⊗ ψ D j ′ 2 ,k ′ 2 + j,k d 2 j,k ψ D j1,k1 ⊗ ψ 0 j2,k2 , ψ D j ′ 1 ,k ′ 1 ⊗ (ψ D j ′ 2 ,k ′ 2 ) ′ In terms of coefficient matrices [d 1 j,k ] and [d 1 j,k ], it becomes : [ u, Ψ ∇ j ′ ,k ′ ] = C 0 j [d 1 j,k ] M j + M j [d 2 j,k ] (C 0 j ) t
where C 0 j is the stiffness matrix of elements :

ψ 0 j,k , (ψ D j ′ ,k ′ ) ′ .
The computation of the Gram and stiffness matrices M j , C 0 j is classical [START_REF] Beylkin | On the representation of operator in bases of compactly supported wavelets[END_REF] (see also [START_REF] Monasse | Orthogonal Wavelet Bases Adapted For Partial Differential Equations With Boundary Conditions[END_REF][START_REF] Kadri-Harouna | Ondelettes pour la prise en compte de conditions aux limites en turbulence incompressible[END_REF]).

Divergence-free and curl-free Gram matrices preconditioning

The tensorial decomposition (42) of M curl is used to deduce a preconditioner from those of matrices M j ⊗ R j and R j ⊗ M j . Let I j be the identity matrix of dimension (N j -2) and I R be the diagonal matrix of R j :

[I j ] k,k ′ = δ k,k ′ and [I R ] k,k ′ = [R j ] k,k ′ δ k,k ′ , 1 ≤ k, k ′ ≤ N j -2
On one hand, as an optimal (and diagonal) preconditioner of R j is given by the inverse of I R (see [START_REF] Cohen | Numerical Analysis of Wavelet Methods[END_REF]), readily we deduce optimal diagonal preconditioners of matrices M j ⊗ R j and R j ⊗ M j by respectively the inverse of matrices I j ⊗ I R and I R ⊗ I j .

On the other hand, since M curl is the 2D stiffness matrix of a scalar Laplacian on the basis {ψ D j1,k1 ⊗ ψ D j2,k2 }, an optimal preconditioner is given by the inverse of its diagonal matrix [START_REF] Cohen | Numerical Analysis of Wavelet Methods[END_REF]. Then, the inverse of the diagonal matrix D j defined by :

D j = I j ⊗ I R + I R ⊗ I j
is an optimal diagonal preconditioner for M curl .

However, the matrix D j has the same size as M curl . To reduce the complexity, we replace the 2D matrix-vector product D j (d ∇ j,k ) by the following matrix-matrix products :

[D j (d ∇ j,k )] = [d ∇ j,k ]I R + I R [d ∇ j,k ] (45)
Because of the diagonal structure of I R , equation ( 45) is then reduced to term by term matrix product :

[D j (d ∇ j,k )] k,k ′ = [d ∇ j,k ] k,k ′ • [I * R ] k,k ′ where [I * R ] k,k ′ = [I R ] k,k + [I R ] k ′ ,k ′ (46)
From equation (46), multiplying (d ∇ j,k ) by the matrix D -1 j is therefore equivalent to divide term by term the matrix [d ∇ j,k ] by I * R .

The preconditioner I * R is also valid for the matrix M div in dimension two (d = 2) since M div = M curl .

The performance of the above preconditioner for M curl was tested in two and three dimensions, using a preconditioned conjugate gradient method to solve system (43). Then we study the number of iterations needed to reach a given residual, first with respect to the dimension index j, second with respect to the regularity (approximation order r) of the basis functions. Figure 2 shows that the number of iterations does not increase significantly with the dimension index j, in the periodic and non periodic cases, which indicates that our preconditioner is quasi-optimal.

Examples of Helmholtz-Hodge and Helmholtz decomposition

In this section, we carry out some experiments to illustrate and study the convergence rate of the Helmholtz-Hodge decomposition. First we show in dimension two, the Helmholtz decomposition of a vector field u (Figure 5), and its Helmholtz-Hodge decomposition (Figure 7). The vector field u was constructed analytically :

u 2D = u div + u curl + u har where :

u div = sin(2πx) 2 sin(4πy) -sin(4πx) sin(2πy) 2 , u curl = sin(4πx) sin(2πy) 2 sin(2πx) 2 sin(4πy) , u har = (1/2, -1/4)
The terms of the decompositions are computed using the method described previoulsy.

Then we investigate the convergence rate of the projection error onto the divergence-free vector space V div j , in two en three dimensions. The tests have been performed on analytic fields, which we know the exact solutions. We used u 2D in two dimensions and in three dimensions we used : u 3D = sin(2πx) 2 sin(4πy) sin(4πz) + sin(4πx) sin(2πy) 2 sin(2πz) 2 sin(4πx) sin(2πy) 2 sin(4πz) + sin(2πx) 2 sin(4πy) sin(2πz) 2 -2 sin(4πx) sin(4πy) sin(2πz) 2 + sin(2πx) 2 sin(2πy) 2 sin(4πz) (47) The solutions verify homogeneous Dirichlet boundary conditions by construction. Figure 7 plot the ℓ 2 -projection errors in terms of the dimension index j with generators of approximation order r = 3. For both experiments (2D and 3D), the convergence rate follows the theoretical law of -2 predicted in [START_REF] Girault | Finite element methods for Navier-Stokes equations[END_REF].

Conclusion

In this paper, we have presented a practical algorithm to compute the Helmholtz-Hodge decomposition of a vector field in the hypercube. Our method is based on the existence of divergence-free and irrotational wavelet bases satisfying boundary conditions. After presenting the principles of their construction in any dimension, we have detailed the computation of each term of the decomposition, which requires the inversion of divergence-free and curl-free wavelet Gram matrices. We have used the tensorial structure of the bases to propose an optimal and diagonal preconditioning, to invert the system using a preconditioned conjugate gradient. Numerical tests on 2D and 3D analytical vector fields illustrate the potential of the approach, in terms of complexity and storage. Since the Helmholtz-Hodge decomposition is a key ingredient for the analysis and simulation of incompressible flows, future works will present its application in numerical schemes for the Stokes and Navier-Stokes equations [START_REF] Kadri-Harouna | Ondelettes pour la prise en compte de conditions aux limites en turbulence incompressible[END_REF].

Références

Figure 1 .

 1 Figure 1. Gram matrices of divergence-free scaling functions (left) and wavelets (right) built from Daubechies generators with r = 3, jmin = 4, j = 6.

Figure 2 .

 2 Figure 2. Preconditioned conjugate gradient residual versus iteration number, for different values of the dimension index j : periodic case (left) and non periodic case (right). Daubechies generators ψ 1 with r = 3 vanishing moments, jmin = 3 in non periodic (2D case).

Figure 3 (

 3 Figure3(two-dimensional case) and figure4(three-dimensional case) highlight that the approximation order r speed up the convergence of the resolution. The behaviour of the slopes are less regular in the non periodic case, because of the influence of the smallest scale j min > 0.

Figure 3 .

 3 Figure 3. Preconditioned conjugate gradient residuals versus iteration number, for different values of the approximation order r : periodic case (left) and non periodic case (right). Daubechies generators ψ 1 with r = 3 and r = 4 vanishing moments. The resolution is j = 10 in two dimension (d = 2).

Figure 4 .

 4 Figure 4. Preconditioned conjugate gradient residuals versus iteration number, for different values of the approximation order r : periodic case (left) and non periodic case (right). Daubechies generators ψ 1 with r = 3 and r = 4 vanishing moments. The resolution is j = 7 in three dimension (d = 3).

Figure 5 .Figure 6 .

 56 Figure 5. Example of Helmholtz decomposition.

Figure 7 .

 7 Figure 7. ℓ2-projection error onto V div j versus j. Two-dimensional case (left) and threedimensional case (right). The generators (ϕ 1 , φ1 ) correspond to biorthogonal splines with : r = r = 3.


François Bordes

Charles Dortch

Jean-Paul Raynal

Claude Thibault

Claude Thibault The

The 1978 season of the Franco-Australian Quaternary Project in the Murchison Basin, Western Australia

come   L'archive ouverte pluridisciplinaire


Lotte E Elshof

Emiel J Th

Eline E Rutgers

Claudette E Deurloo

Jelle Loo

Kenneth E Wesseling

Kenneth G A Pengel

Gilhuijs

Rutgers

Eline E Deurloo

Claudette E Loo

Jelle Wesseling

Kenneth E Pengel

Kenneth G A Gilhuijs
email: k.gilhuijs@nki.nl

A practical approach to manage additional lesions at preoperative breast MRI in patients eligible for breast conserving therapy: results

Keywords:

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

Introduction

Contrast-enhanced magnetic resonance imaging (MRI) of the breast has shown superior sensitivity to detect invasive breast cancer compared to conventional imaging (mammography, ultrasonography) and clinical examination [START_REF] Berg | Diagnostic accuracy of mammography, clinical examination, US, and MR imaging in preoperative assessment of breast cancer[END_REF][START_REF] Boetes | Breast tumors: comparative accuracy of MR imaging relative to mammography and US for demonstrating extent[END_REF][START_REF] Deurloo | Contrast-enhanced MRI in breast cancer patients eligible for breast-conserving therapy: complementary value for subgroups of patients[END_REF][START_REF] Hata | Magnetic resonance imaging for preoperative evaluation of breast cancer: a comparative study with mammography and ultrasonography[END_REF]. MRI detects additional foci of disease in the ipsilateral breast of patients with known breast cancer in 10-30% of cases [START_REF] Mann | Breast MRI: guidelines from the European Society of Breast Imaging[END_REF]. Clinically and mammographically occult cancer in the contralateral breast is detected in 3-5% of patients who undergo preoperative breast MRI [START_REF] Mann | Breast MRI: guidelines from the European Society of Breast Imaging[END_REF]. As a result, MRI frequently changes the therapy of patients initially eligible for breast-conserving therapy (BCT) [5, 8, 11-13, 15, 18, 22, 27]. Because the majority of the additional cancer may, however, be controlled with radiotherapy [START_REF] Bleicher | MRI and breast cancer: role in detection, diagnosis, and staging[END_REF][START_REF] Solin | Relationship of breast magnetic resonance imaging to outcome after breast-conservation treatment with radiation for women with[END_REF], the clinical relevance of these findings is currently debated [START_REF] Houssami | Accuracy and surgical impact of magnetic resonance imaging in breast cancer staging: systematic review and metaanalysis in detection of multifocal and multicentric cancer[END_REF][START_REF] Morrow | Magnetic resonance imaging in breast cancer: one step forward, two steps back?[END_REF][START_REF] Morrow | Magnetic resonance imaging in the preoperative evaluation of breast cancer: primum non nocere[END_REF][START_REF] Morrow | Magnetic resonance imaging in the breast cancer patient: curb your enthusiasm[END_REF][START_REF] Orel | Who should have breast magnetic resonance imaging evaluation?[END_REF]. Consequently, the clinical indications for breast MRI are still subject of research and discussion. The use of MRI, specifically for breast cancer staging prior to BCT, is recommended by a number of studies [START_REF] Bedrosian | Changes in the surgical management of patients with breast carcinoma based on preoperative magnetic resonance imaging[END_REF][START_REF] Braun | Influence of preoperative MRI on the surgical management of patients with operable breast cancer[END_REF][START_REF] Fischer | The influence of preoperative MRI of the breasts on recurrence rate in patients with breast cancer[END_REF][START_REF] Grobmyer | Is there a role for routine use of MRI in selection of patients for breast-conserving cancer therapy?[END_REF][START_REF] Sh | Contrast-enhanced MRI of the breast: accuracy, value, controversies, solutions[END_REF][START_REF] Hollingsworth | Preoperative breast MRI for locoregional staging[END_REF], but remains controversial in others [START_REF] Bleicher | MRI and breast cancer: role in detection, diagnosis, and staging[END_REF][START_REF] Houssami | Accuracy and surgical impact of magnetic resonance imaging in breast cancer staging: systematic review and metaanalysis in detection of multifocal and multicentric cancer[END_REF][START_REF] Morrow | Magnetic resonance imaging in breast cancer: one step forward, two steps back?[END_REF][START_REF] Morrow | Magnetic resonance imaging in the preoperative evaluation of breast cancer: primum non nocere[END_REF][START_REF] Morrow | Magnetic resonance imaging in the breast cancer patient: curb your enthusiasm[END_REF][START_REF] Orel | Who should have breast magnetic resonance imaging evaluation?[END_REF].

Parallel to these efforts to define the role of preoperative MRI in the ipsilateral breast, both the European Society of Breast Imaging and the American College of Radiology have recommended the use of MRI to screen the contralateral breast in patients with proven cancer [START_REF] Gilhuijs | ACR practice guideline for the performance of contrast-enhanced magnetic resonance imaging (MRI) of the breast[END_REF][START_REF] Mann | Breast MRI: guidelines from the European Society of Breast Imaging[END_REF]. These reports recommend that simultaneous bilateral MRI should be performed, because the breasts are symmetric organs, and there is negligible time penalty for imaging both breasts [START_REF] Gilhuijs | ACR practice guideline for the performance of contrast-enhanced magnetic resonance imaging (MRI) of the breast[END_REF]. Consequently, preoperative breast MRI is thus increasingly used, and the need for clinical guidelines rises.

In our hospital, prospective directives were established to handle additional findings in the breast detected at MRI [START_REF] Deurloo | Additional breast lesions in patients eligible for breast-conserving therapy by MRI: impact on preoperative management and potential benefit of computerised analysis[END_REF][START_REF] Pengel | The impact of preoperative MRI on breastconserving surgery of invasive cancer: a comparative cohort study[END_REF]. The aim of this study was twofold. First, to prospectively evaluate the efficacy of these directives in a large patient population. Secondly, to establish recommendations to further improve the efficacy of this approach.

Magnetic resonance imaging

Initially MRI was performed with a 1.5-tesla scanner (Magnetom, Siemens, Erlangen, Germany) using a dedicated double-breast array coil. From April 2007, MRI was performed with a 3.0-tesla scanner (Achieva, Philips, Best, The Netherlands) using a dedicated 7-elements Sense breast coil. Both breasts were simultaneously imaged in prone position. An unenhanced coronal 3D thrive sense T1-weighed sequence was acquired before the administration of contrast agent. Subsequently, four consecutive series were acquired after the intravenous administration of contrast. A bolus (14mL) of gadolinium containing contrast agent (gadoteridol, gadopentetate dimeglumine or gadoteric acid) was administered intravenously at 3mL/s using a power injector followed by a bolus of 30ml of saline solution. These series were acquired with a voxel size of 1. A viewing station that permitted simultaneous viewing of two series reformatted and linked in three orthogonal directions was used for the interpretation of the breast MRI [START_REF] Gilhuijs | Breast MR imaging in women at increased lifetime risk of breast cancer: clinical system for computerized assessment of breast lesions initial results[END_REF]. The viewing station displayed all image series (unenhanced and contrast enhanced), subtraction images at initial, and at late enhancement, and maximum intensity projection of both breasts. The subtraction images were also colour coded, representing different degrees and curve types of enhancement. The largest tumour diameter was assessed in the three reformatted planes (sagittal, axial and coronal) at initial enhancement.

Additional lesions

Additional lesions at MRI were defined as lesions separate from the index tumour, categorized BIRADS (Breast Imaging Reporting And Data System [START_REF]Breast Imaging Reporting and Data System Atlas (BI-RADS® Atlas)[END_REF]) 3, 4 or 5 and occult at clinical examination and initial conventional imaging.

Additional lesions were classified in three groups, based on the localization with respect to the index lesion: (1) multifocal (maximum diameter of volume including index tumour and additional lesions < 3 cm), (2) multicentric (ipsilateral, maximum diameter of volume including index tumour and additional lesion > 3 cm), and (3) contralateral. The first group included additional lesions close to the index tumour (total lesion bearing region < 3 cm), whereas the last two groups represented additional lesions far from the index tumour.

Unidentified Breast Objects (UBOs) were defined as additionally enhancing findings, from which no pathology proof was available before surgery.

They were pathology-proven after surgery or considered benign by follow-up. For all other additional lesions, pathology proof was available preoperatively using second-look ultrasound and FNA or core biopsy.

Directives for management of additional lesions at MRI Directives were established to handle additional lesions detected at MRI, aiming to minimize additional procedures and treatment changes due to benign findings [START_REF] Deurloo | Additional breast lesions in patients eligible for breast-conserving therapy by MRI: impact on preoperative management and potential benefit of computerised analysis[END_REF][START_REF] Pengel | The impact of preoperative MRI on breastconserving surgery of invasive cancer: a comparative cohort study[END_REF]. In short, attempts were made to obtain proof of malignancy for multicentric and contralateral additional lesions by second-look targeted ultrasound and FNA or core biopsy. If pathology confirmed malignant disease over a region too large to allow cosmetically acceptable BCT, a conversion to mastectomy was advised. If pathology proof could not be obtained, the therapy plan was not changed and follow-up by MRI was advised. For multifocal additional lesions additional second-look ultrasound was not advised and these findings typically led to BCT with larger wide-local excision margins to include the additional finding. Depending on the size of the breast, the total diameter of disease in patients eligible for BCT typically did not exceed 3cm. The final treatment plan was implemented after consultation with the patient including the advice after multidisciplinary discussion of the MRI findings.

Surgery

The aim of the surgical procedures was to achieve tumour-free margins with the best possible cosmetic outcome [START_REF] Rutgers | Quality control in the locoregional treatment of breast cancer[END_REF][START_REF] Rutgers | Sentinel node procedure in breast carcinoma: a valid tool to omit unnecessary axillary treatment or even more?[END_REF]. The procedures were performed according to accepted surgical standards and were carried out or directly supervised by fully trained surgeons specialized in breast surgery. Wide local excisions in our hospital are generally performed according to the technique described by Aspegren et al. [START_REF] Aspegren | Standardization of the surgical technique in breast-conserving treatment of mammary cancer[END_REF]. Non-palpable lesions were typically marked preoperatively by using intra-lesional injection of 99m Tc-nanocolloid (Nanocoll; GE-Healthcare, Eindhoven, The Netherlands) administered under ultrasound guidance, and detected and removed intraoperatively, guided by a gamma-ray probe [40]. Results of MRI and other preoperative diagnostic imaging were available for review in the operating theatre. The number of patients in which surgical treatment was changed due to additional lesions was assessed.

Pathology

Excision specimens were handled according to a protocol adopted from the approach described by Egan [START_REF] Egan | Multicentric breast carcinomas: clinical-radiographicpathologic whole organ studies and 10-year survival[END_REF]. Briefly, each specimen was cut into 3-4 mm slices and fixed in 4% formalin overnight. Subsequently, a radiograph from the slices was obtained. Based on macroscopic, radiographic, and MRI findings, samples were taken to enable adequate microscopic investigation of the lesions and their surrounding.

Additional procedures

The number of additional procedures caused by additional lesions was assessed.

These procedures included preoperative second-look targeted ultrasound, ultrasound-guided FNA, ultrasound-guided or stereotactic core biopsy, second MRI and follow-up.

Statistics

The ability of preoperative breast MRI to detect malignant additional lesions was defined by the positive-predictive value (PPV), i.e., the number of malignant lesions divided by the total number of additional lesions. UBOs that were lost to follow-up were not included in the calculation of PPV. Chi-squared tests were performed to explore differences in diagnosis between multifocal and distant lesions. To compare age in different groups Mann-Whitney U tests were used. SPSS Version 15.0; SPSS Chicago, Ill, was used for the analyses. A P-value of less than 0.05 was considered to be significant.

Results

Patient cohort

Between November 2000 and January 2008, 715 patients participated in the MARGINS study. In the current study, 690 patients with 698 cancers (8 bilateral) passed the inclusion criteria. The pathological types of the 698 index cancers were as follow: IDC: 505, ILC: 93, DCIS: 36, tubular carcinoma: 17, colloid carcinoma: 8, medullary carcinoma: 6, other: 33.

Additional lesions

At preoperative MRI, 141 additional lesions were detected in 121 patients (121/690 = 17.5%; Table 1, Figure 1). UBOs were found in 81 patients (81/690 = 11.7 %).

Of a total of 141 additional lesions, 44 multicentric and contralateral findings were followed up and considered benign (table 1), i.e., in none of the patients these UBOs were found to be malignant during follow-up (44/141 = 31.2%). Forty of these UBOs were categorized BIRADS-3 at MRI. Follow-up was performed using conventional x-ray mammography in all cases. In addition, 16

UBOs were followed up by MRI. The patients who were followed up by MRI were significantly younger than the patients who were followed up by conventional imaging alone (52.3 years vs 57.7 years, p = 0.01). The mean follow-up time was 57.1 months (55.0 months median, range: 22-103 months). Approximately half the number of these UBOs had follow-up time ≥60 months (21/44 = 47.7%). Four patients, each with one additional finding were lost to follow-up after BCT. A pathological substrate could not be found for seven multifocal additional findings in the wider-local excision specimens. These findings at MRI were assumed to be caused by benign fibrocystic changes (Table 1). Twenty-four lesions were proven to be benign (Table 1), i.e. fibroadenoma, cyst, benign lymph node or intraductal papilloma: 3 multifocal, 13 multicentric, and 8 contralateral.

Surgery

Preoperative MRI resulted in a change in surgical management in 66 out of 690 patients (9.6%). Due to multicentric malignant additional lesions in 18 patients, treatment was changed to mastectomy.

In one patient, the MRI showed that the index tumour was closer to the capsula of the protheses than could be appreciated at conventional imaging. As a result the therapy was converted to mastectomy. Moreover, this patient also showed an additional lesion that was reported malignant by FNA cytology prior to surgery. At final pathology the additional tumour was not found in the mastectomy specimen.

This lesion was proven benign (fibroadenoma) after revision of the cytology.

In another patient, with a multifocal malignant additional lesion, treatment was changed to mastectomy. Because this patient also had a breast augmentation and the tumour was located near the capsula of the protheses, BCT was expected to result in a bad cosmetic outcome. Moreover a complete wide local excision of both cancers was expected to result in incomplete margins.

Forty-one patients underwent more extensive wide local excision for multifocal disease, of whom 33 were confirmed to have one or more malignant additional lesions. In one of these patients a mastectomy was done after an incomplete surgical excision.

In four patients a contralateral additional lesion was proven malignant. One mastectomy was performed according to the wish of the patient. The other three patients underwent wide local excision.

One patient underwent bilateral mastectomy, because of a malignant additional multicentric lesion and an additional lesion in the contralateral breast which turned out to be invasive lobular carcinoma with extensive lobular carcinoma in situ.

Additional procedures

As a result of the detection of additional lesions at preoperative MRI, 114 additional procedures were performed preoperatively and 34 procedures were done during follow-up (Figure 2, Figure 3).

Positive predictive value

A total of 62 malignant lesions were detected at MRI in 56 patients (56/690 = 8.1%). To remind, these series do not include unexpected larger extent of the index tumour. Four UBOs were lost to follow-up and excluded from the PPV calculations. More than two-fifth of MRI detected additional lesions was proven malignant (PPV: 62/137 = 45.3%; Table 1). Multifocal lesions were more often malignant than multicentric and contralateral lesions (PPV: 35/45 = 77.8% vs.

22/66 = 33.3% and 5/26 = 19.2%, p < 0.0001). Notably, none of the multicentric and contralateral UBOs resulted in malignant disease during follow-up after radiotherapy.

Discussion

We evaluated directives for the management of additional lesions detected at MRI. The guidelines were sensitive to include malignant lesions in surgical planning without causing conversions to mastectomy for benign lesions. Despite this high sensitivity, the fraction of additional findings for which follow-up procedures were applied and that did not yield malignant disease is still relatively high. In this section we discuss additional recommendations to enhance the directives.

Three situations can be distinguished in the management of additional lesions:

(1) the additional lesion is preoperatively proven to be malignant, (2) the additional lesion is preoperatively proven to be benign, and (3) no pathology proof of the additional finding can be obtained prior to surgery (UBO).

Preoperatively proven malignant additional lesions led to changes in treatment.

In patients with multifocal malignant disease, BCT could still be performed, but larger wide-local margins were excised to include additional lesions and avoid potentially incomplete excisions. Multicentric and contralateral malignant additional lesions led to more radical changes in patient treatment (e.g., change to mastectomy or bilateral surgery). It was considered that the detection of these lesions at preoperative MRI could result in treatment at an earlier stage and prevent multiple operations.

The assumption that surgery of additional tumour corresponding to the additional MRI abnormality is beneficial for the patient is debated. Small additional tumour foci in the ipsilateral breast may not be of clinical relevance, because patients treated with BCT generally receive postoperative externalbeam radiation of the whole breast [START_REF] Bleicher | MRI and breast cancer: role in detection, diagnosis, and staging[END_REF][START_REF] Morrow | Magnetic resonance imaging in breast cancer: one step forward, two steps back?[END_REF][START_REF] Morrow | Magnetic resonance imaging in the breast cancer patient: curb your enthusiasm[END_REF]. Nonetheless, it is also known that radiotherapy is ineffective to eradicate large residual tumour burden [START_REF] Bartelink | Recurrence rates after treatment of breast cancer with standard radiotherapy with or without additional radiation[END_REF][START_REF]Effects of radiotherapy and of differences in the extent of surgery for early breast cancer on local recurrence and 15-year survival: an overview of the randomised trials[END_REF]41].

Moreover, the contralateral breast does not receive radiotherapy.

Local failure after BCT is an uncommon event: 96% of women without a preoperative MRI undergo BCT without local recurrence at 8 years [START_REF] Solin | Relationship of breast magnetic resonance imaging to outcome after breast-conservation treatment with radiation for women with[END_REF].

Furthermore, the prevalence of additional malignant lesions exceeds the current local recurrence rates after BCT [START_REF] Solin | Relationship of breast magnetic resonance imaging to outcome after breast-conservation treatment with radiation for women with[END_REF]. On the other hand, long-term impact of additional lesions on local recurrence is currently unknown.

The issue whether preoperative MRI reduces the rate of incomplete tumour excision has also recently been addressed [START_REF] Mann | The impact of preoperative breast MRI on the reexcision rate in invasive lobular carcinoma of the breast[END_REF][START_REF] Pengel | The impact of preoperative MRI on breastconserving surgery of invasive cancer: a comparative cohort study[END_REF]39] Preoperatively proven benign additional lesions contributed to a different aspect of preoperative MRI. To obtain pathology-proof for these lesions, additional procedures were performed, which may have resulted in redundant inconvenience and expense. On the other hand, some patients may have benefited from the assurance that no other malignant disease was detected at MRI. The combination of radiological reading and a computer system is able to reliably identify benign lesions and may be helpful to reduce the number of lesions that require further workup [START_REF] Deurloo | Additional breast lesions in patients eligible for breast-conserving therapy by MRI: impact on preoperative management and potential benefit of computerised analysis[END_REF].

The last group of UBOs causes the greatest clinical dilemma. Our study suggests that multifocal UBOs should be considered separately from multicentric and contralateral UBOs, because the large majority of the multifocal lesions were malignant. These findings are in agreement with the pathology studies performed by Holland [START_REF] Holland | Extent, distribution, and mammographic/ histological correlations of breast ductal carcinoma in situ[END_REF]. It does not appear to be necessary to evaluate these findings prior to surgery, because BCT can be pursued with larger wide-local excisions.

Preoperative evaluation of multifocal findings may lead to redundant additional procedures and unnecessary expense.

Our results suggest that it may be safe to follow up findings considered to be benign on targeted ultrasound after BCT. Moreover, the use of MRI to followup multicentric and contralateral UBOs may be limited, because no malignant disease associated with UBOs could be detected at follow-up (mean of 57.1 months). In the current study, follow-up was performed using conventional x-ray mammography in all cases, but not all findings were followed up using MRI.

Patient preference, breast density, and low suspicion of presence of malignant disease on targeted ultrasound following MRI were the most important underlying reasons for this discrepancy. Patients who were followed up by MRI were significantly younger than patients who were followed up by conventional imaging alone (52.3 years vs 57.7 years, p = 0.01). The decision was always based on consensus by the multidisciplinary team of breast-cancer specialists. Our observations suggest that patients with additional breast findings may be followed up after BCT in the same fashion as patients without additional findings.

Previous studies do not recommend MRI as routine breast cancer surveillance in patients without a high clinical suspicion of local recurrence following breast conservation surgery and radiotherapy as it will incur significant cost and will be unlikely to improve overall survival [START_REF] Gorechlad | Screening for Recurrences in Patients Treated with Breast-Conserving Surgery: Is there a Role for MRI?[END_REF][START_REF] Khatcheressian | American Society of Clinical Oncology 2006 Update of the Breast Cancer Follow-Up and Management Guidelines in the Adjuvant Setting[END_REF]. The use of preoperative MRI may further reduce the need for screening MRI at the time of annual follow-up evaluations [START_REF] Gorechlad | Screening for Recurrences in Patients Treated with Breast-Conserving Surgery: Is there a Role for MRI?[END_REF].

Several studies described the use of MRI-guided biopsies to sample regions containing MRI-detected abnormalities [START_REF] Bedrosian | Magnetic resonance imaging-guided biopsy of mammographically and clinically occult breast lesions[END_REF][START_REF] Grobmyer | Is there a role for routine use of MRI in selection of patients for breast-conserving cancer therapy?[END_REF]. The European Society of Breast Imaging emphasizes the importance of this diagnostic device and states that MRI should only be offered by institutions that can also offer MRI-guided biopsy [START_REF] Mann | Breast MRI: guidelines from the European Society of Breast Imaging[END_REF].

In the current study approximately half the number of additional lesions further away from the index tumour were detected by targeted ultrasound and FNA or core biopsy. If not visible at targeted ultrasound, these findings did not manifest as malignant disease at follow-up after BCT. Again these results suggest that MRI-guided biopsy of BIRADS-3 multicentric and contralateral additional findings following targeted ultrasound may have limited impact on patient outcome.

A prior study discussed the differences in incidence of additional findings and malignant additional findings in patients with breast cancer, ranging from 8% to 51% and 4% to 28% respectively [START_REF] Deurloo | Additional breast lesions in patients eligible for breast-conserving therapy by MRI: impact on preoperative management and potential benefit of computerised analysis[END_REF]. Differences in study design (retrospective vs. prospective), inclusion criteria and focus are mentioned as a cause. In the present study, only lesions separate from the index tumour and previously occult were evaluated. More extensive enhancement of the index lesion, was evaluated in a prior study [START_REF] Pengel | The impact of preoperative MRI on breastconserving surgery of invasive cancer: a comparative cohort study[END_REF]. Also, BIRADS-2 findings (benign) were not defined as additional lesions, because they do not require further workup and typically do not cause problems in patient management due to their low suspicion of harbouring malignant disease.

Our study also has some limitations. First, not all additional findings could be identified as malignant or benign lesions. Without the use of MRI-guided biopsy, findings not visible at conventional imaging could not be biopsied and therefore remained unverified before surgery. Hence, we are unable to discriminate between preoperative benign lesions and potentially malignant disease that was eradicated by radiotherapy. Our results should therefore always be interpreted with postoperative radiotherapy in mind. With regard to multifocal findings, two were excised and proved benign after surgery. These two lesions might have 20 been identified using MRI-guided biopsy, thus potentially avoiding two wider local excisions. On the other hand, 35 of the 45 wider local excisions showed additional disease at pathology, leaving relatively little room for MRI-guided biopsy to improve these results while incurring additional expense and an invasive procedure.

Another limitation of our study is that four patients with unverified findings were lost to follow-up. Furthermore, patients were not randomized to compare workup with and without MRI, but were consecutively recruited for an MRI prior to surgery. Moreover, this is a single-institution study. Although some of our findings are in agreement with those from other studies, details may vary depending on clinical implementation. In this study, the MRI findings for each individual patient were discussed and translated into clinical actions by a multi-disciplinary team of breast cancer specialists based on prior established directives.

Conclusions

Prospective directives to manage additional lesions at preoperative breast MRI proved to be sensitive to include malignant disease in the surgical planning without causing conversions to mastectomy for benign lesions. The strategy to pursue BCT with larger wide-local excisions for multifocal UBOs and to follow up multicentric and contralateral UBOs with conventional imaging is effective to exclude malignancy at follow-up. After second-look targeted ultrasound has been

  in the MARGINS (Multi-modality Analysis and Radiological Guidance IN breast conServing therapy, 2000-2008) singleinstitution trial were included. In this study, patients with pathology-proven breast cancer planned for BCT on the basis of clinical assessment and conventional imaging by mammography and breast-ultrasound, were recruited for an additional preoperative breast MRI. The MARGINS study was performed after approval of the institutional review board and written informed consent of all patients. Patients were excluded if therapy was changed to neoadjuvant chemotherapy. Proof of breast cancer was obtained using image directed fineneedle aspiration (FNA) or core biopsy. Treatment plans were established in consensus by a multi-disciplinary team of breast cancer specialists in all patients.

35 x 1 .

 1 35 x 1.35 mm3 (1.5T) or 1.1 x 1.1 x 1.1 mm3 (3.0T). The following scanning parameters were used: acquisition time 90 s (1.5T and 3.0 T); TR/TE: 8.1/4.0 1.5T (4.4/2.3 3.0T); flip angle 20° 1.5T (10° 3.0T); FOV 310 1.5T (360 3.0T).

  . The results from the COMICE trial, which randomized between women with and without preoperative MRI, show no significant difference in rate of incomplete tumour excision with or without preoperative MRI. Nonetheless, MRI was reported to be helpful to improve the localization of breast cancer. Two retrospective cohort studies indicate that preoperative MRI may be helpful to reduce incomplete excision of breast cancer in subgroups of tumours. Pengel et al. noted a significant reduction in incomplete excision of IDC in the MRI group. In a cohort study including 267 patients with ILC, Mann et al. showed significantly lower reexcision rates in the preoperative MRI group.

Fig. 1 2 Fig. 2 2 Fig. 3

 12223 Fig. 1 Work-up and outcome of additional lesions

  performed, MRI-guided biopsy of BIRADS-3 multicentric and contralateral additional findingsmay have limited complementary clinical value. early-stage invasive breast carcinoma or ductal carcinoma in situ. J Clin Oncol 26:386-391 39. Turnbull L et al (2010) Comparative effectiveness of MRI in breast cancer (COMICE) trial: a randomised controlled trial. The Lancet 375:563-571 40. van Rijk MC et al (2007) Sentinel node biopsy and concomitant probeguided tumor excision of nonpalpable breast cancer. Ann Surg Oncol 14:627-632 41. Vrieling C et al (2003) Can patient-, treatment-and pathology-related characteristics explain the high local recurrence rate following breastconserving therapy in young patients? European Journal of Cancer 39:932-944

					690 patients
					121 patients
				141 additional lesions
		42 patients			57 patients				27 patients
	45 multifocal lesions		67 multicentric lesions			29 contralateral lesions
	Preoperative diagnosis		Preoperative diagnosis			Preoperative diagnosis
	Malignant: 4	Benign: 1	UBO: 40	Malignant: 22*	Benign: 12**	UBO: 33	Malignant: 5	Benign: 6	UBO: 18
	Postoperative diagnosis			Follow-up
	38 patients, 40 lesions				40 patients, 47 lesions***
	Malignant: 31	Benign: 2	UBO: 7		Malignant: 0	Benign: 3	Unknown: 44

Acknowledgements

We thank Anita Paape and Angelique Schlief for the data management. Jelle Teertstra, Robert Kröger, Charlotte Lange, Warner Prevoo, Peter Besnard, Wim Koops, Frank Pameijer, Marie-Jeanne Vrancken Peeters, Hester Oldenburg, Saar Muller, Marc van de Vijver and Hans Peterse † are thanked for their dedicated input.

This work was financially supported by The Dutch Cancer Foundation, grant number NKB 2004-3082.


Paridaens

Hans Wildiers
email: hans.wildiers@uzleuven.be

C Fontaine

P Vuylsteke

M Martens

J L Canon

W Wynendaele

C Focan

J De Greve

P Squifflet

R Paridaens

Keywords: Breast cancer, angiogenesis, sunitinib, consolidation, taxanes 25-6-2010 SUCON trial: SUnitinib CONsolidation therapy in metastatic breast cancer

Multicenter phase II randomized trial evaluating antiangiogenic therapy with sunitinib as consolidation after objective response to taxane chemotherapy in women with HER2-negative metastatic breast cancer

Introduction

Although early breast can often be cured, there remains a significant risk of developing metastases, a clinical situation that is generally incurable. Several antitumor therapies, such as hormone therapy for hormone-sensitive disease, or chemotherapy, are capable of providing significant benefit, including increased survival and quality of life. A n interesting category of drugs has emerged in cancer treatment, targeting the blood supply. This strategy, called antiangiogenic therapy, is rapidly evolving, and the anti-vascular endothelial growth factor (VEGF) antibody bevacizumab has recently been approved by the FDA and EMEA for use in combination with chemotherapy in patients with metastatic breast cancer.

New drugs are generally tested in patients with metastatic disease, at a stage associated with a bulky tumor mass. However, antiangiogenic compounds may be more beneficial in the prevention of malignant progression when the tumor load has been reduced. Metastatic breast tumors are relatively chemosensitive and a significant proportion of patients achieve an objective clinical response. Taxanes are considered the most active chemotherapeutic drugs in breast cancer. Tumor shrinkage following induction chemotherapy provides an excellent model to test the hypothesis that antiangiogenic drugs can prevent revascularization and malignant regrowth.

Sunitinib is an orally administered small molecule inhibitor of multiple tyrosine kinase receptors including VEGF, platelet-derived growth factor (PDGF), c-KIT, and RET.

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Sunitinib is established as a first-line therapy in metastatic renal cancer, in which the angiogenic pathway is constitutively activated, and as second-line therapy after imatinib for inoperable gastrointestinal stromal tumors (GIST), by targeting the mutant c-KIT receptor. In other cancers including breast cancer, modest single-agent antitumor activity has been observed [START_REF] Burstein | Phase II study of sunitinib malate, an oral multitargeted tyrosine kinase inhibitor, in patients with metastatic breast cancer previously treated with an anthracycline and a taxane[END_REF]. This study was designed to test the hypothesis that sunitinib, a compound with a clinically proven antiangiogenic effect, can delay tumor progression after a taxane-induced objective partial or complete response (PR or CR).

Materials and methods

Patients with metastatic breast cancer achieving an objective response with taxane chemotherapy were included in this two-arm, open-label, randomized (randomization 2:1), multicenter, phase II clinical trial. The purpose of the study was to evaluate the ability of sunitinib consolidation therapy (arm A) to delay tumor progression in the setting of reduced tumor burden following induction chemotherapy. Arm B (no therapy) was used as control.

Patient population

Eligible patients had histologically proven, HER2-negative, measurable metastatic breast cancer at the start of taxane therapy. Patients achieving a PR or CR according to RECIST criteria after completing taxane-based therapy were included. No more than two lines of chemotherapy in the metastatic setting (with taxanes included as last line) were allowed. Patients received four to six cycles of taxane therapy on a 3weekly schedule or 8 to 18 doses on a weekly schedule. The recommended taxane 25-6-2010 schedules were six cycles every 3 weeks (six doses over 18 weeks) or 12 to 16 weekly doses. Hormonal therapy (tamoxifen, aromatase inhibitors, or other hormone suppressing therapies) was not allowed during the stud y period.

All patients gave written informed consent and the study was approved by the ethics committees of the participating institutions.

Randomization

Eligible patients were randomly allocated to arm A (sunitinib) or arm B (control) between 3 and 5 weeks after the last 3-weekly taxane administration or between 2 and 4 weeks after completing weekly taxane therapy. Patients were stratified according to the institution, disease-free interval (0 = primary metastatic disease; 0-2 years; 2-5 years; > 5 years), dominant site of disease (non-visceral, single or multiple visceral sites), and response to taxanes (CR or PR). Chemotherapy and hormone therapy were not allowed in arm B during the study period. At the time of disease progression, patients in arm B were allowed to cross over to sunitinib.

Dosing

In arm A, sunitinib was initially administered orally at a daily dose of 50 mg for 4 weeks followed by a drug-free interval of 2 weeks (1 cycle = 6 weeks). However, the protocol was amended on 25 January 2008 in light of new data showing tumor regrowth during the 2-week interval in sunitinib dosing [START_REF] Burstein | Phase II study of sunitinib malate, an oral multitargeted tyrosine kinase inhibitor, in patients with metastatic breast cancer previously treated with an anthracycline and a taxane[END_REF] and the occurrence of toxicity in the first cohort of 32 patients treated with 50 mg sunitinib in arm A or after progression in arm B. Following the protocol amendment, sunitinib was given orally at a dose of 37.5 mg/d continuously in 6-week cycles. Continuous once-daily 25-6-2010 administration of sunitinib 37.5 mg has demonstrated very similar activity to sunitinib 50 mg/d on the 4/2 week intermittent schedule [START_REF] George | Clinical evaluation of continuous daily dosing of sunitinib malate in patients with advanced gastrointestinal stromal tumour after imatinib failure[END_REF]. In addition, the manageable safety profile of continuous dosing provides additional flexibility in dose scheduling. The reduced, continuous-dose regimen has been adopted as the regimen of choice in subsequent phase III studies.

Sunitinib was continued until disease progression or for as long as either the patient or the treating oncologist considered continuation of therapy to be in the patient's best interest. The dosing schedule in arm A was also used for patients from arm B starting sunitinib after progression.

Sunitinib was interrupted temporarily in patients experiencing severe treatmentrelated toxicity and restarted at a reduced dose (no lower than 25 mg) if appropriate.

Study procedures

Radiological assessments of tumors were performed at baseline and in the week before starting new cycles of sunitinib. After four cycles, imaging studies were repeated every two cycles until disease progression or withdrawal from the study.

Imaging studies were also performed to confirm suspected disease progression.

Endpoints

The primary endpoint was the proportion of patients alive and progression-free 5 months after starting sunitinib. The trial was not powered to allow formal comparison between the investigational arm and the control arm (which was intended for descriptive purposes only). Secondary endpoints included other measures of antitumor efficacy (overall survival) in both treatment arms and the safety and tolerability of sunitinib.

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PFS was defined as the time from randomization to first documentation of objective tumor progression, or to death from any cause. PFS data were censored on the day following the date of the last tumor assessment documenting absence of progressive disease. Toxicity and tolerability were graded according to the National Cancer Institute (NCI) Common Terminology Criteria for Adverse Events (CTCAE, Version 3.0). All patients who received at least one dose of study treatment were evaluated for efficacy, toxicity and safety.

The total duration of taxane chemotherapy allowed was approximately 10 to 20 weeks (four to six cycles of 3-weekly therapy or 8 to 18 doses of weekly therapy). A longer duration of taxane therapy was not allowed because of inadequate data to support an improvement in outcome with prolonged chemotherapy in a palliative setting in patients with metastatic breast cancer [START_REF] Gennari | Lack of benefit of maintenance paclitaxel in first-line chemotherapy in metastatic breast cancer[END_REF]. Moreover prolongation of chemotherapy is associated with increased toxicity. Based on data from historic studies, the duration of taxane response was estimated to be about 7 months, depending on the type of taxane (paclitaxel or docetaxel) and the clinical setting (firstor second-line treatment of metastatic disease) [START_REF] Sledge | Phase III trial of doxorubicin, paclitaxel, and the combination of doxorubicin and paclitaxel as front-line chemotherapy for metastatic breast cancer: an intergroup trial (E1193)[END_REF][START_REF] Davidson | Single-agent paclitaxel as first-line treatment of metastatic breast cancer: the British experience[END_REF][START_REF] Gianni | Paclitaxel in metastatic breast cancer: a trial of two doses by a 3-hour infusion in patients with disease recurrence after prior therapy with anthracyclines[END_REF][START_REF] Fountzilas | Paclitaxel by 3-h infusion and carboplatin in anthracycline-resistant advanced breast cancer. A phase II study conducted by the Hellenic Cooperative Oncology Group[END_REF][START_REF] Vici | Paclitaxel activity in anthracycline refractory breast cancer patients[END_REF][START_REF] Seidman | Phase II trial of paclitaxel by 3-hour infusion as initial and salvage chemotherapy for metastatic breast cancer[END_REF][START_REF] Ravdin | Phase II trial of docetaxel in advanced anthracycline-resistant or anthracenedione-resistant breast cancer[END_REF][START_REF] Valero | A phase II study of docetaxel in patients with paclitaxel-resistant metastatic breast cancer[END_REF][START_REF] Paridaens | Paclitaxel versus doxorubicin as first-line single-agent chemotherapy for metastatic breast cancer: A European organization for research and treatment of cancer randomized study with cross-over[END_REF]. In practice, taxanes are generally continued for a few months after an objective response is achieved, and therefore, we estimated the median PFS after discontinuation of taxane therapy would be 5 months in these patients.

Statistics

Evaluable patients were classified as having met the primary endpoint if they were alive and free of tumor progression 5 months after starting study therapy. The study was designed as a one-stage, three-outcome phase II trial [START_REF] Sargent | A three-outcome design for phase II clinical trials[END_REF]. The cut-off values for ineffectiveness and effectiveness were 45% and 65%, respectively. The type I ( ) 25-6-2010 and type II ( ) error rates were set at 0.05. Therefore 36 patients treated with sunitinib were required for the study. If 18 patients or fewer were progression-free and alive at 5 months, sunitinib would be declared insufficiently active, whereas, if at least 22 patients were progression-free and alive at 5 months, sunitinib would be declared active and continuation of the trial using a phase III design would be recommended. If, however, between 19 and 21 patients were progression-free and alive at 5 months, no formal statistical conclusion would be possible. The control group of 18 patients receiving no treatment was intended for descriptive purposes (no formal comparison was planned).

Results

Patient population

Fifty-six patients were enrolled between February 7, 2006 andMarch 17, 2009. Patient and treatment characteristics are shown in table 1. One patient randomized to receive sunitinib progressed before starting treatment and died a few weeks later.

The remaining 55 patients were evaluated for efficacy and toxicity. In four patients in arm A (11%) and two in arm B (11%) the taxane regimen leading to objective response was administered for longer than the protocol-specified 20 weeks (range 22-27 weeks); however, the analysis was intent to treat, the differences were balanced, and median number of taxane administrations was identical in both arms.

Two patients in arm A who had received two lines of chemotherapy before taxaneinduced objective response were also included in the analysis.

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Efficacy

The primary endpoint (progression-free at 5 months) was achieved by 10/36 patients (28%) in arm A and 4/19 (21%) in arm B. Median PFS was 2.8 months in arm A and 3.1 months in arm B. Kaplan-Meier curves for PFS and OS are shown in Figure 1.

In arm B, 12/19 patients received crossover treatment with sunitinib after progression. The response rate was 25% in this group.

Toxicity

The main side effects in both arms of the study are described in table 2.

During the study, new data emerged about a dosing schedule of sunitinib 37.5 mg continuously instead of sunitinib 50 mg 4 weeks on, 2 weeks off. Since we observed significant grade III toxicity and dose reductions in the first 32 patients treated with the intermittent 50 mg dosing schedule, a protocol amendment was made in January 2008 to change the starting dose of sunitinib to 37.5 mg continuously. Sunitinib dose reductions were required in 53% (17/32) of patients treated at a starting dose of 50 mg and in 44% (7/16) treated with the continuous dosing schedule.

Dose reductions and delays

Before the dosing amendment, 26 patients in arm A and six from arm B after crossover received sunitinib at 50 mg. Of the patients treated with the 50 mg dosing schedule, 47% (15/32) received one cycle of sunitinib, 25% two cycles, 13% three cycles, and 15% four or more. In addition, 53% (17/32) required at least one dose reduction and 63% required at least one dose interruption.

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After the protocol amendment, 10 patients in arm A and 6 from arm B after crossover received sunitinib 37.5 mg continuously. Of patients treated with the continuous schedule, 44% (7/16) received 1 cycle of sunitinib, 25% two cycles, 0% three cycles, and 31% four or more. At least one dose reduction was required in 47% (7/16) and at least one dose interruption in 56%.

Discussion

In this study we tested the hypothesis that an antiangiogenic agent such as sunitinib can significantly prolong PFS following a meaningful reduction in tumor mass (PR or CR) induced by chemotherapy (taxanes). It was hypothesized that antiangiogenic agents are most effective in the setting where the tumor has been reduced and regrowth occurs after stopping chemotherapy. It is thought that neoangiogenesis is required for tumor regrowth after cytoreduction and, therefore, blocking angiogenesis could lead to prolonged tumor control. Unfortunately, the results of this proof-ofprinciple study do not support this hypothesis. This was not a phase III trial comparing sunitinib consolidation therapy with placebo and no formal comparison can be made between study arms. However, the median PFS with sunitinib in a consolidation setting was only 2.8 months, and only 28% of patients achieved the 5month PFS endpoint, which was far below the initially defined and presumed clinically relevant cutoff of at least 45% and preferably more than 65%. It should be acknowledged that median PFS in the control arm (3.1 months) was also lower than hypothesized based on previous studies (5 months). The Kaplan-Meier curve for PFS seems to indicate that the curves slightly diverge after 4 months, but there were too few patients with prolonged disease control in the sunitinib arm to allow any definitive 25-6-2010 conclusions to be made. One of the major challenges with angiogenesis inhibitors is the lack of reliable predictive biomarkers to select patients most likely to benefit. Another important consideration is the significant toxicity of sunitinib, which resulted in dose reduction in more than 50% of patients. Following the protocol-specified dose amendment, 47% of patients still required dose reduction. This degree of dose reduction appears to be relatively higher than rates reported for single-agent sunitinib in other settings, such as renal cancer [START_REF] Motzer | Sunitinib versus interferon alfa in metastatic renal-cell carcinoma[END_REF]. It can be hypothesized that cumulative toxicity associated with 3 to 4 months of taxane therapy only partially resolved and that side effects were exacerbated when sunitinib was started. Fatigue is a known side effect of taxanes and the main toxicity associated with sunitinib. It is possible, therefore, that sequential administration of taxanes and sunitinib leads to exacerbation or accumulation of fatigue.

Although no patient discontinued sunitinib permanently before progression, a large proportion of patients required dose interruptions, sometimes prolonged, before a lower dose could be restarted. In non-clinical models, dose interruption of antiangiogenic agents can lead to reactivation [START_REF] Ebos | Accelerated Metastasis after Short-Term Treatment with a Potent Inhibitor of Tumor Angiogenesis[END_REF] and even acceleration of angiogenesis. It is conceivable that such a scenario may have happened in this study population. On the other hand, a recent preliminary analysis of the AVADO trial did not show any accelerated growth after bevacizumab was discontinued, suggesting that metastatic spread does not accelerate in breast cancer after stopping antiangiogenic agents [START_REF] Harbeck | No Clinical Evidence for Increase in Tumour Malignant Potential in Patients (Pts) with Metastatic Breast Cancer (mBC) Treated with Bevacizumab (BV) and Docetaxel (D) in the Phase III AVADO Study[END_REF]. The hypothesis that continuous and sustained inhibition of angiogenesis inhibition can prolong PFS without causing excessive toxicity could be tested with sunitinib administered at a lower starting dose (e.g. 25 mg daily continuously) or with other angiogenesis inhibitors. It is important to answer this 25-6-2010 question definitively because continuous inhibition of angiogenesis after induction chemotherapy combined with antiangiogenic therapy is considered to be standard practice in all previous and ongoing studies. While it has not been shown that this approach is effective, it is evident that toxicity is clinically relevant with some classes of agent, and the cost of therapy is high. Furthermore, the present study does not support the concept of consolidation antiangiogenic therapy in patients with metastatic breast cancer responding to taxanes.

In addition to insufficient dose intensity and possible regrowth of tumor during interruption of sunitinib treatment, there are several other reasons for the lack of efficacy observed. For instance, alternative angiogenic pathways can be activated, or tumors can obtain oxygen and nutrients through co-option of blood vessels. Liver metastases were present in the large majority of patients in this study and it has been clearly described that such tumors can grow in a liver replacement pattern by exploiting the existing liver blood supply [START_REF] Stessels | Breast adenocarcinoma liver metastases, in contrast to colorectal cancer liver metastases, display a non-angiogenic growth pattern that preserves the stroma and lacks hypoxia[END_REF]. Another plausible explanation for the lack of important single-agent activity of sunitinib in breast cancer, in contrast to renal cell cancer, is that there is no documented constitutive activation of the VEGF pathway. Moreover, there is no proof that other sunitinib targets (PDGF, c-KIT) are activated in breast cancer. It seems that in breast cancer (in contrast to renal cancer and GIST), antiangiogenic drugs work essentially in combination with chemotherapy, as shown by the E2100 [START_REF] Miller | Paclitaxel plus bevacizumab versus paclitaxel alone for metastatic breast cancer[END_REF], AVADO and RIBBON trials. One of the possible mechanisms for this phenomenon is normalization of the tumor vasculature , thereby increasing chemotherapy delivery to the tumor site [START_REF] Wildiers | Effect of antivascular endothelial growth factor treatment on the intratumoral uptake of CPT-11[END_REF]. Response rates with bevacizumab [START_REF] Cobleigh | A phase I/II dose-escalation trial of bevacizumab in previously treated metastatic breast cancer[END_REF] and sunitinib [START_REF] Burstein | Phase II study of sunitinib malate, an oral multitargeted tyrosine kinase inhibitor, in patients with metastatic breast cancer previously treated with an anthracycline and a taxane[END_REF] monotherapy have been around 10% in pretreated patients. A recent randomized phase III study [START_REF] Cobleigh | A phase I/II dose-escalation trial of bevacizumab in previously treated metastatic breast cancer[END_REF] showed that the PFS for sunitinib (2.8 months) was lower than for 25-6-2010 capecitabine (4.2 months) in patients with metastatic breast cancer failing anthracyclines and taxanes. The latter study suggests that antiangiogenic drugs should be used and further developed in combination with chemotherapy. Furthermore, such findings bring into question the use of single-agent antiangiogenic drugs in the adjuvant setting, which is being assessed in ongoing phase III studies.

Two further phase III trials of sunitinib in patients with metastatic breast cancer were presented at ASCO 2010. In a first-line study, the addition of sunitinib to docetaxel significantly improved response rate (51% compared with 39%), but did not improve PFS, the primary endpoint of the study (hazard ratio 0.92; median 8.6 versus 8.3 months) [START_REF] Bergh | Sunitinib (SU) in combination with docetaxel (D) versus D alone for the first-line treatment of advanced breast cancer (ABC)[END_REF]. In the setting of second-line therapy or beyond, the addition of sunitinib to capecitabine did not improve PFS or response rate [START_REF] Crown | Phase III trial of sunitinib (SU) in combination with capecitabine (C) versus C in previously treated advanced breast cancer (ABC)[END_REF]. It is remarkable to observe in our study that 3/12 patients had an objective response to sunitinib after progression in the control arm (crossover). Although numbers are very small, this finding is consistent with the study of Burstein and colleagues [START_REF] Burstein | Phase II study of sunitinib malate, an oral multitargeted tyrosine kinase inhibitor, in patients with metastatic breast cancer previously treated with an anthracycline and a taxane[END_REF] that showed an objective response rate of 11% for single-agent sunitinib. It appears that sunitinib has antitumor activity in a minority of patients, but this does not lead to improved PFS for the majority of patients. There is a great need to find biomarkers to predict the efficacy of antiangiogenic therapy. It is possible that antiangiogenic drugs like sunitinib are active within a small, specific subgroup of patients. Nevertheless, accumulated data from the present study and other recent clinical trials do not support the use of sunitinib in the treatment of breast cancer.

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Table 1 :

 1 Patient, tumor, and previous treatment characteristics

	Arm A (Sunitinib)	Arm B (No therapy)
	N=36	N=19

Table 2 :

 2 Side effects Anemia, dizziness and somnolence were the only grade IV toxicities, all in 1 patient in arm A. 1 death occurred during the study period within 2 weeks after stopping sunitinib (autopsy revealed massive progression in the liver which likely caused hepatic failure, encephalopathy, and death).

			All grades	Grade III*
		Arm A	Arm B**	Arm A	Arm B**
		(sunitinib)	(no therapy)	(sunitinib)	(no therapy)
		N=36	N=19	N=36	N=19
	All	100%	68%	69%	11%
	Fatigue	78%	16%	31%	5%
	Neutropenia	14%	0%	8%	0%
	Thrombopenia	25%	0%	11%	0%
	Musculoskeletal pain 69%	32%	11%	5%
	Anorexia	53%	11%	3%	0%
	Diarrhea	44%	0%	3%	0%
	Dyspnea	25%	5%	3%	0%
	Facial edema	14%	0%	0%	0%
	Limb edema	25%	0%	0%	0%
	Hemorrhage	19%	0%	0%	0%
	Hypertension	17%	0%	0%	0%
	Conjunctivitis	14%	0%	0%	0%
	Mucositis	47%	0%	0%	0%
	Nausea	22%	5%	3%	0%
	Vomiting	17%	0%	0%	0%
	Rash	19%	0%	0%	0%

*

Acknowledgements:

We would like to thank all the participating patients, all the centres who contributed to the study, and Liza Van Eenoo who helped in the coordination of the study and data collection.

List of recruiting centres and principal investigators: University Hospitals Leuven

(Robert Paridaens and Hans Wildiers), Imelda Bonheiden (Wim Wynendaele), AZ St Jan Brugge (Nele Claes), Ziekenhuis Oost Limburg (Jeroen Mebis, Daisy Luyten, Guy Debrock), St-Elisabeth Turnhout (Marc Martens), St-Elisabeth Namen (Peter Vuylsteke and Jean-Charles Goemine), UZ Gent (Hannelore Denys and Veronique Cocquyt), St-Niklaas (Willem Lybaert), Charleroi (Jean-Luc Canon), UZ Brussel (Christel Fontaine and Jacques De Grève), CHC-Liège (Drs C.Focan; MP Graas and F. Kreutz), CHU Sart Tilman Liège (Guy Jerusalem), Clinique St-pierre Ottignies (Lionel Duck)

Performed under the auspices of the Belgian Society of Medical Oncology (BSMO).

Funding:

This work was supported by an unrestricted grant by Pfizer N.V. and used for the performance of this study. All authors have declared no conflict of interest.