Felix Antritter
email: felix.antritter@unibw.de

Franck Cazaurang
email: franck.cazaurang@u-bordeaux1.fr

Jean Lévine
email: jean.levine@mines-paristech.fr

Johannes Middeke
email: jmiddeke@risc.jku.at

On the computation of π-flat outputs for differential-delay systems

Keywords: differential delay systems, differential flatness, π-flatness, Ore polynomials, motion planning, Modules

We introduce a new definition of π-flatness for linear differential delay systems with time-varying coefficients. We characterize πand π-0-flat outputs and provide an algorithm to efficiently compute such outputs. We present an academic example of motion planning to discuss the pertinence of the approach.

Introduction

Differential flatness, roughly speaking, means that all the variables of an under-determined system of ordinary differential equations can be expressed as functions of a particular output, called flat output, and a finite number of its successive time derivatives ( [START_REF] Martin | Contribution à l'étude des systèmes différentiellement plats[END_REF][START_REF] Fliess | Flatness and defect of non-linear systems: introduction theory and examples[END_REF][START_REF] Fliess | A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems[END_REF], see also [START_REF] Sira-Ramirez | Differentially Flat Systems[END_REF][START_REF] Lévine | Analysis and Control of Nonlinear Systems: A Flatness-based Approach[END_REF][START_REF] Lévine | On necessary and sufficient conditions for differential flatness[END_REF] and the references therein).

For time-delay systems and more general classes of infinite-dimensional systems, extensions of this concept have been proposed and thoroughly discussed in [START_REF] Mounier | Propriétés structurelles des systèmes linéaires à retards: aspects théoriques et pratiques[END_REF][START_REF] Fliess | Systèmes linéaires sur les opérateurs de Mikusiński et commande d'une poutre flexible[END_REF][START_REF] Petit | Systèmes à retards. Platitude en génie des procédés et contrôle de certaines équations des ondes[END_REF][START_REF] Rudolph | Flatness Based Control of Distributed Parameter Systems[END_REF]. In a linear context, relations with the notion of system parameterization [START_REF] Pommaret | Localization and parametrization of linear multidimensional control systems[END_REF][START_REF] Pommaret | Partial Differential Control Theory[END_REF] and, in the behavioral approach of [START_REF] Polderman | Introduction to Mathematical Systems Theory: A Behavorial Approach[END_REF], with latent variables of observable image representations [START_REF] Trentelman | On flat systems behaviors and observable image representations[END_REF], have been established. Other theoretic approaches have been proposed e.g. in [START_REF] Rocha | Behavioral controllability of delay-differential systems[END_REF][START_REF] Chyzak | Effective algorithms for parametrizing linear control systems over Ore algebras[END_REF]. Interesting control applications of linear differential-delay systems may be found in [START_REF] Mounier | Propriétés structurelles des systèmes linéaires à retards: aspects théoriques et pratiques[END_REF][START_REF] Petit | Systèmes à retards. Platitude en génie des procédés et contrôle de certaines équations des ondes[END_REF][START_REF] Rudolph | Flatness Based Control of Distributed Parameter Systems[END_REF].

Characterizing differential flatness and flat outputs has been an active topic since the beginning of this theory. The interested reader may find a historical perspective of this question in [START_REF] Lévine | Analysis and Control of Nonlinear Systems: A Flatness-based Approach[END_REF][START_REF] Lévine | On necessary and sufficient conditions for differential flatness[END_REF]. Constructive algorithms, relying on standard computer algebra environments, may be found e.g. in [START_REF] Antritter | Towards a computer algebraic algorithm for flat output determination[END_REF][START_REF] Antritter | On the symbolic computation of flat outputs for differentially flat systems[END_REF] for nonlinear finite-dimensional systems, or [START_REF] Antritter | A toolbox for the analysis of linear systems with delays[END_REF][START_REF] Chyzak | OreModules: a symbolic package for the study of multidimensional linear systems[END_REF] for linear systems over Ore algebras.

For algebraic reasons recalled in Section 4, the notions of π-freeness and π-flatness have been introduced in the context of linear differential-delay systems by [START_REF] Mounier | Propriétés structurelles des systèmes linéaires à retards: aspects théoriques et pratiques[END_REF][START_REF] Petit | δ-freeness of a class of linear systems[END_REF][START_REF] Petit | Systèmes à retards. Platitude en génie des procédés et contrôle de certaines équations des ondes[END_REF][START_REF] Rudolph | Flatness Based Control of Distributed Parameter Systems[END_REF] and then for linear time-varying differential-delay systems, with polynomial time dependence, by [START_REF] Chyzak | Effective algorithms for parametrizing linear control systems over Ore algebras[END_REF][START_REF] Chyzak | OreModules: A symbolic package for the study of multidimensional linear systems[END_REF].

The results and algorithms proposed in this paper concern time-varying differential-delay systems with meromorphic time-dependence. Our approach may be seen as an extension of [START_REF] Petit | δ-freeness of a class of linear systems[END_REF][START_REF] Lévine | Flat output characterization for linear systems using polynomial matrices[END_REF][START_REF] Lévine | Analysis and Control of Nonlinear Systems: A Flatness-based Approach[END_REF] to this context.

Our main contributions are [START_REF] Martin | Contribution à l'étude des systèmes différentiellement plats[END_REF] a new definition of πand π-k-flatness, (2) the characterization of πand π-0-flatness in terms of the hyper-regularity of the system matrices, (3) yielding an elementary algorithm, of polynomial time complexity, to compute πand π-0-flat outputs, based on the row/column reduction of the former matrices.

Let us emphasize that the introduced definitions are directly suitable for motion planning and that our formalism is able to consider state and input variables separately or not. Note that, though the natural algebraic framework used to describe such systems is based on a ring, denoted by O, of polynomials in both the delay and differential operators, which is non principal, the evaluation of our πand π-0-flatness criteria relies upon computations over a principal ideal ring, denoted by O, containing O, thus making the computations much simpler.

The paper is organized as follows. A first section briefly presents recalls on operators and signals (Section 2), followed by recalls on matrices over the non commutative differential ring O (Section 3). In the latter section, the notions of hyper-regularity and row and column reduction are introduced and characterized in this non commutative context. Section 4 then deals with differential algebraic notions of systems, π-flatness and π-k-flatness and contains the main results characterizing π-flat and π-0-flat outputs. Algorithms to compute such π-flat and π-0-flat outputs are then deduced. Finally, the proposed methodology is illustrated by an example of motion planning in Section 5.

Operators & signal space

Recalls on Ore polynomials

In this paper we consider linear mixed differential and time-delay operators. We will model them using the so-called Ore polynomials. These are a class of non-commutative polynomials, named after Øystein Ore who was the first to discuss them in [START_REF] Ore | Theory of non-commutative polynomials[END_REF]. In the following we will give a brief introduction to Ore polynomials, pointing to the appropriate references for details.

Let K be a ring and let σ : K → K be an automorphism. An additive map ϑ : K → K is called a σ-derivation if for all a and b ∈ K the σ-Leibniz rule ϑ(ab) = σ(a)ϑ(b) + ϑ(a)b holds (compare with [START_REF] Cohn | Further Algebra and Applications[END_REF]Sect. 7.3]). Consider the free K-left module generated by the powers of an indeterminate x. We define the right-multiplication of x by an element of K with the commutation rule xa = σ(a)x + ϑ(a) for all a ∈ K.

Assuming associativity and distributivity, this rule allows us to compute arbitrary products. It can be shown (see, e. g., [START_REF] Cohn | Further Algebra and Applications[END_REF]Thm. 7.3.1]) that this makes the free module into a ring which we call the ring of (left) Ore polynomials in x w. r. t. σ and ϑ. In the literature this is usually denoted by K[x; σ, ϑ].

The degree of an Ore polynomial p is defined as the largest exponent n such that x n has a non-zero coefficient in p. We use deg 0 = -∞. If K is a domain, then we have the familiar rule deg(pq) = deg p + deg q for all p and q ∈ K[x; σ, ϑ]. If K is a division ring, then it is possible to divide elements in K[x; σ, ϑ] with remainder in a way which is very similar to the usual polynomial division. See [START_REF] Bronstein | An introduction to pseudo-linear algebra[END_REF] for details. This turns K[x; σ, ϑ] into a (left and right) principal ideal domain and thus into a (left or right) Ore ring (see also [START_REF] Cohn | An introduction to Ring Theory[END_REF]Prop. 5.9]). Thus, we can form the field of (left or right) fractions K(x; σ, ϑ). See for example [START_REF] Ježek | Non-commutative rings of fractions in algebraical approach to control theory[END_REF] for an extensive introduction on how to properly define the various arithmetic operations for such fractions.

If K is commutative, σ = id and ϑ = 0 are the identity and the zero map, respectively, then K[x; id, 0] is just the usual polynomial ring K[x]. Two other important special cases of Ore polynomials are linear differential and delay operators, which we discuss in the following.

Example 1. Let K be the field of meromorphic functions over the real line (see e.g. [29, p. 42])).

1. Assume first that ϑ = 0 and that σ = δ is the time delay operator which is defined by δ f (t) = f (t -τ) for all f ∈ K where τ > 0 is a fixed real number. In an abuse of notation, we will denote the ring K[x; δ, 0] just by K[δ], i. e., we identify δ with the Ore variable x. The commutation rule for K[δ] is δ f (t) = f (t -τ)δ for all f (t) ∈ K. We call it the ring of time delay operators.

2. Assume now, that σ = id. Let ϑ = ∂ be the usual derivation in the sense of calculus. Then, the ring K[x; id, ∂]-which we will just write as K[∂]-has the commutation rule ∂ f (t) = f (t)∂ + ḟ (t) for all f (t) ∈ K. This is the ring of differential operators.

Since the maps ∂ and δ commute, we may extend ∂ to K[δ] by setting ∂(δ) = 0. Thus, the ring O = K[δ, ∂]-or in more complete notation K[x; δ, 0][y; id, ∂]-is well-defined and has the commutation rules

δ f (t) = f (t -τ)δ, ∂ f (t) = f (t)∂ + ḟ (t) and ∂δ = δ∂
where f (t) ∈ K. We call O the ring of time-delay differential operators. The commutation of ∂ and δ means that this ring is an Ore algebra in the sense of [START_REF] Chyzak | Non-commutative elimination in Ore algebras proves multivariate identities[END_REF]Def. 1.2]. Using the formulae in [START_REF] Ježek | Non-commutative rings of fractions in algebraical approach to control theory[END_REF]Thm. 13], we may extend the action of ∂ on the fractions in K(δ). Thus, also the ring O = K(δ)[∂] is a well-defined Ore polynomial ring.

Since δ is an automorphism, by [START_REF] Cohn | Further Algebra and Applications[END_REF]Sect. 7.3] we may form the ring of formal Laurent series K((δ)) in δ with coefficients in K. That is, K((δ)) consists of elements of the form j≥N f j δ j where N ∈ Z and f j ∈ K for all j ≥ N. Moreover, by [START_REF] Cohn | Further Algebra and Applications[END_REF]Prop. 7.3.7] we may embed K(δ) into K((δ)).

Signal space

The signal space has to be contained in the domain of the previously defined operators. Such a space is not unique in general and we may choose it in accordance to our control objectives. A signal space suitable for motion planning tasks should satisfy at least the following requirements. It should:

1. be closed under the action of any polynomial of differentiation (∂) and delay (δ) 2. be closed under advances defined by inverses of δ-polynomials, 3. contain C ∞ functions with compact support in R.

Therefore we introduce

S = { f : R → R | ∃E f ⊆ R discrete : f ∈ C ∞ (R \ E f , R) and ∃t 0 ∈ R ∀t ∈ R : t < t 0 =⇒ f (t) = 0}.
which is an adaptation of a space introduced in [START_REF] Zerz | An algebraic analysis approach to linear time-varying systems[END_REF]Sec. 3]. Indeed, S fullfills the first and third property and it is even an O-module.

As far as property 2 is concerned: if π ∈ K[δ] with π 0, let us compute π -1 f for f ∈ S. To this end we can develop π -1 into a formal Laurent series π -1 = j≥N a j δ j where N ∈ Z and a j ∈ K for all j ≥ N (see Appendix A). Then for any f ∈ S and t ∈ R we have

(π -1 f )(t) = j≥N a j δ j f (t) = j≥N a j f (t -jτ).
Since f (tjτ) = 0 for jτ > tt 0 all but finitely many of the summands on the right hand side vanish at any given t. We can check that π -1 f is again in S. Thus, property 2 is satisfied. Moreover, one can show that S is an O-module. Note that changing O to O in conjunction with S will yield significant simplifications in the sequel.

The example discussed in Section 5 also shows that S is relevant for motion planning tasks. Note in addition that S contains splines and step functions.

Matrices & hyper-regularity

We model systems of linear time-delay differential equations using matrices of operators. The set of all n × m matrices with entries in O is denoted by O n×m . Square matrices in O n×n which possess a two-sided inverse that is also in O n×n are called unimodular. The set of all unimodular matrices of O n×n is denoted by Gl n (O). We write 1 n for the n × n identity matrix and 0 n×m for the n × m zero matrix. In both cases we will omit the indices when they are obvious from the context. We use O 1×m for the set of row vectors of length m and O n for the set of column vectors of length n. Given a matrix M ∈ O n×m we denote its O-row space by O 1×n M. For a matrix M = (M i, j ) ∈ O n×m we define deg ∂ M = max{deg ∂ M i, j | i = 1, . . . , n, j = 1, . . . , m}. This will also be applied to row or column vectors regarding them as 1 × m-or n × 1-matrices, respectively. We write M i, * for the i-th row of M and M * , j for the j-th column of M where i = 1, . . . , n and j = 1, . . . , m.

Let M ∈ O n×m . Since K(δ) is a division ring, O is a principal ideal domain. This means we can apply [START_REF] Cohn | Free Rings and Their Relations[END_REF]Thm. 8.1.1] in order to find unimodular matrices S ∈ Gl n (O) and T ∈ Gl m (O) such that S MT is in Smith-Jacobson form. This is a diagonal form with the additional property that each diagonal element is a total divisor of the next one.

Definition 1 (Hyper-regularity [START_REF] Lévine | Analysis and Control of Nonlinear Systems: A Flatness-based Approach[END_REF]). A matrix M ∈ O n×m is called hyper-regular if the diagonal elements of its Smith-Jacobson form are all 1, i. e., if there are unimodular matrices S ∈ Gl n (O) and T ∈ Gl m (O) such that n ≥ m and S MT =

1 m 0 n-m×m or n < m and S MT = (1 n , 0 n×m-n ).
Usually, the given method for obtaining a Smith-Jacobson form involves computing a diagonal form by repeatedly taking greatest common (left or right) divisors as a first step. Since the performance of this approach is difficult to predict and most likely exponential due to degree growth, we will give an alternate, and computationally more efficient, characterisation of hyper-regularity below.

Proposition 1 ([19]).

(i

) A matrix M ∈ O n×m , with n < m, is hyper-regular if, and only if, it possesses a right-inverse, i.e. ∃ T ∈ Gl m (O) such that MT = (1 n , 0 n×(m-n) ). (ii) A matrix M ∈ O n×m , with n ≥ m, is hyper-regular if, and only if, it possesses a left-inverse, i.e. ∃ S ∈ Gl n (O) such that S M = 1 m 0 (n-m)×m .
Proof. We only prove (i). The proof of (ii) follows the same lines and is left to the reader. Let M ∈ O n×m be given with n < m. M is hyper-regular if, and only if, there are matrices S ∈ Gl n (O) and T ∈ Gl m (O) such that S MT = (1 n , 0 n×m-n ). Thus, using the identity

1 n , 0 = S -1 1 n , 0 S 0 0 1 m-n we get 1 n , 0 = S -1 (S MT ) S 0 0 1 m-n = M T S 0 0 1 m-n .
which proves that M is hyper-regular if, and only if, it has T S 0 0 1 m-n as right-inverse.

We will apply the technique of row-reduction for the computation of left-or right-inverses.

A matrix M ∈ O n×m is called row-reduced (or row-proper) if for all row vectors v ∈ O 1×n the so-called predictable degree property

deg ∂ vM = max{deg ∂ v j + deg ∂ M j, * | j = 1, . . . , n}
holds. Note, that this definition differs from the usual one as given in [START_REF] Beckermann | Fraction-free row reduction of matrices of Ore polynomials[END_REF]Sect. 2] or [34, Sect. 2.2], but is shown to be equivalent in [START_REF] Beckermann | Fraction-free row reduction of matrices of Ore polynomials[END_REF]Lem. A.1 (a)] where one can easily check that the proof also works for division rings instead of fields. Similarly, the algorithm outlined in the proof of [33, Thm. 2.2] can be easily transferred to division rings, and we obtain that for every matrix M ∈ O n×m there exists a matrix S ∈ Gl n (O) such that the non-zero rows of S M form a row-reduced submatrix. Using the results in [START_REF] Middeke | A computational view on normal forms of matrices of Ore polynomials[END_REF] row-reduction is of low polynomial complexity in the size of M and its degree. Directly from the definition we obtain that the rows of a row-reduced matrix must be linearly independent.

Row-reducedness is connected to the Popov normal form (see, e. g., [36, Def. 2.2]) which is essentially a row-reduced matrix with additional properties to make it unique. One may prove that for each matrix in O n×m there exists exactly one matrix Popov form having the same row space. Also, row-reduction may be regarded as a special case of Gröbner basis computationsee [START_REF] Middeke | A computational view on normal forms of matrices of Ore polynomials[END_REF].

We consider now the case n ≤ m. Let M ∈ O n×m , and let S ∈ Gl n (O) be such that

S M = M 0 .
where M ∈ O k×m is row-reduced and k is the (left) row-rank of M by [START_REF] Beckermann | Fraction-free row reduction of matrices of Ore polynomials[END_REF]Thm. A.2]. Assume first that M is hyper-regular. As discussed above, this means that all unit vectors of O 1×m are in the row-space of M. Thus, we must have k = m and by [START_REF] Beckermann | Fraction-free row reduction of matrices of Ore polynomials[END_REF]Lem. A.1 (c)] all rows of M have ∂-degree 0 or below. Thus, M ∈ K(δ) m×m . Since all rows of M are linearly independent, we conclude that M has maximal (left) row rank.

Conversely, if row-reduction of M yields a matrix of degree 0 and (left) row-rank m, then clearly M is hyper-regular.

There is also the analogue concept of column-reduction. Each matrix may be brought into column-reduced form (up to zero rows) by right multiplication with a unimodular matrix. All the results cited above hold with the appropriate changes. In total, we have proved the following lemma.

Lemma 1 ([19]

). A matrix M ∈ O n×m is hyper-regular if and only if 1. n ≥ m and row-reduction yields a matrix of ∂-degree 0 and left row-rank m 2. or, n ≤ m and column-reduction yields a matrix of ∂-degree 0 and right column-rank n.

Systems

Consider systems of the kind

Ax = Bu, (1)
with the pseudo state x of dimension n, input u of dimension m and the matrices A ∈ R n×n and B ∈ R n×m , where R is a ring. For the analysis of such systems, two important objects are considered (see e.g. [START_REF] Fliess | Some basic structural properties of generalized linear systems[END_REF][START_REF] Oberst | Multidimensional constant linear systems[END_REF][START_REF] Willems | Paradigms and puzzles in the theory of dynamical systems[END_REF][START_REF] Fliess | A remark on Willems' trajectory characterization of linear controllability[END_REF][START_REF] Mounier | Propriétés structurelles des systèmes linéaires à retards: aspects théoriques et pratiques[END_REF][START_REF] Polderman | Introduction to Mathematical Systems Theory: A Behavorial Approach[END_REF][START_REF] Pommaret | Localization and parametrization of linear multidimensional control systems[END_REF][START_REF] Chyzak | Effective algorithms for parametrizing linear control systems over Ore algebras[END_REF][START_REF] Zerz | State representations of time-varying linear systems[END_REF]):

• its behavior B ker(A, -B), where the kernel is taken with respect to the chosen signal space 2 , where the components of the variables x and u are supposed to live • and its system module M R 1×(n+m) /R 1×n (A, -B), where R 1×p is the set of row vectors of length p, for every p ∈ N, with components in R and where R 1×n (A, -B) is the module generated by the rows of the matrix (A, -B).

Linear time-invariant differential systems, i. e. without delay, the ring R being chosen as R[∂] and being commutative, are shown to be differentially flat if, and only if, their system module M over R is free, and a flat output is, by definition, a basis of the free module M (see e. g. [START_REF] Fliess | Flatness and defect of non-linear systems: introduction theory and examples[END_REF][START_REF] Lévine | Analysis and Control of Nonlinear Systems: A Flatness-based Approach[END_REF]). This property may serve as a flatness definition for linear differential-delay systems ( [START_REF] Mounier | Propriétés structurelles des systèmes linéaires à retards: aspects théoriques et pratiques[END_REF][START_REF] Petit | δ-freeness of a class of linear systems[END_REF][START_REF] Rudolph | Flatness Based Control of Distributed Parameter Systems[END_REF]). Nevertheless, only few systems have a free system module. Moreover, since the considered base ring R = O = K[δ, ∂] is not a principal ideal domain, freeness is in general different e. g. from torsion freeness or projective freeness (see e. g. [START_REF] Mounier | Propriétés structurelles des systèmes linéaires à retards: aspects théoriques et pratiques[END_REF]), with important, but otherwise unclear, pratical consequences on the control possibilities of the system. However, for linear time-invariant differential-delay systems, namely with R = R[δ, ∂], in virtue of the localization property of a commutative Ore algebra, torsion freeness of M is equivalent to the existence of a so-called liberation polynomial π 0, π ∈ R[δ], such that π -1 M is free. This property is called π-freeness (see again [START_REF] Mounier | Propriétés structurelles des systèmes linéaires à retards: aspects théoriques et pratiques[END_REF]) and a basis of the free module π -1 M is called a π-flat output. It can be interpreted as follows: if the system module, finitely generated by m input variables, is torsion free, it admits a basis of the form π -1 {y 1 , . . . , y m } where y i ∈ R, i = 1, . . . , m. Consequently, every system variable can be expressed as a combination of derivatives, delays and advances of (y 1 , . . . , y m ). Note that the action of π -1 on a variable z may be interpreted as a formal power series in δ -k z = z(• + kτ) for k ≥ -N, for some N ∈ N, and thus a combination of advances, or predictions, of z (see Section 2).

To extend this approach to time-varying differential-delay systems without restricting the time dependence of the system coefficients to be polynomial, as in [START_REF] Chyzak | Effective algorithms for parametrizing linear control systems over Ore algebras[END_REF][START_REF] Chyzak | OreModules: A symbolic package for the study of multidimensional linear systems[END_REF] where effective Gröbner bases techniques are used, many new difficulties appear, and in particular the fact that the set made of the powers of a given polynomial is not in general an Ore set. Therefore, we are lead to propose a different definition of π-flatness, based on an extension of the "practical" flatness definition, in the spirit of [START_REF] Petit | δ-freeness of a class of linear systems[END_REF][START_REF] Petit | Systèmes à retards. Platitude en génie des procédés et contrôle de certaines équations des ondes[END_REF], saying that all the system variables are expressible in terms of a flat output, its derivatives, delays and advances in finite number. Moreover we introduce the definition of π-k-flatness, which is an analogue, in our differential-delay context, to the notion of k-flatness (see [START_REF] Martin | Flat systems[END_REF][START_REF] Pomet | On dynamic feedback linearization of four-dimensional affine control systems with two inputs[END_REF][START_REF] Rathinam | Configuration flatness of Lagrangian systems underactuated by one control[END_REF][START_REF] Pereira Da Silva | Flatness of nonlinear control systems : a Cartan-Kähler approach[END_REF]). This viewpoint is developed in the next two subsections.

Recalls on the framework

From now on, we take System (1) with R = O = K[∂, δ] as in Section 2.1. Recall that δ, the delay operator, is defined by δ f (t) = f (t -τ) for all t ∈ R, where τ is a given positive real number and f ∈ K, and that ∂ d dt is the ordinary time derivative operator. The ground field K is the field of meromorphic functions over the real line and the notation O = K(δ)[∂] is defined in Section 2.1. Furthermore, we will consider only the signal space S from Section 2.2.

We assume that the matrix (A, -B) has full (left) row rank. In this case, System (1) is a differential-delay system, possibly with time-varying coefficients, the coefficients being meromorphic functions of time. An example of such system is provided by the following: Example 2. Consider the system

ẋ1 (t) = a(t) x 2 (t -1) -x 2 (t -2) ẋ2 (t) = u(t -1),
where the pseudo-state x (x 1 , x 2 ) belongs to S 2 , the control u belongs to S, and the function a is meromorphic with respect to time. Here, the matrices A and B are given by

A ∂ -a(δ -δ 2 ) 0 ∂ , B 0

π-flatness

One usual way to consider System (1) is to bring all the variables in one side, i.e. to consider the system (A, -B)

x u = 0 or, in other words,

Fξ = 0 (2)
with F (A, -B) and ξ x u .

Definition 2 (π-flatness). The system (2) is called π-flat if there exist π ∈ K[δ] and matrices P ∈ O m×(n+m) and Q ∈ O (n+m)×m such that

π -1 QS m = B = ker F and π -1 P π -1 Q = 1 m .
Equivalently, there exist a polynomial π ∈ K[δ] and matrices P and Q of suitable dimensions over the ring O, such that ξ = π -1 Qy, with y = π -1 Pξ for all ξ ∈ S n+m , where y ∈ S m is a π-flat output. In other words, π-flatness means that all the system variables can be expressed as linear combinations of y and a finite number of its delays, advances and derivatives, in an invertible way, namely the matrix π -1 P admits the matrix π -1 Q as right-pseudo-inverse.

If π ∈ K, then the system is simply called flat and y a flat output. This definition is thus an extension of the one proposed by [START_REF] Fliess | Flatness and defect of non-linear systems: introduction theory and examples[END_REF][START_REF] Fliess | A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems[END_REF] for finite-dimensional nonlinear ordinary differential systems or by [START_REF] Petit | δ-freeness of a class of linear systems[END_REF] for time-invariant delay-differential systems. Other definitions for time-varying linear differential-delay systems with polynomial time dependence are proposed by Chyzak, Quadrat and Robertz [START_REF] Chyzak | Effective algorithms for parametrizing linear control systems over Ore algebras[END_REF]. Remark 2. To every π-flat system there obviously corresponds a polynomial π 0 ∈ K[δ] of minimal degree such that the system is π 0 -flat.

To characterize π-flat systems, we introduce the following definition:

Definition 3. We call M O n+m /O
n F the extended system module.

Theorem 1. We have the following equivalences:

(i) System (2) is π-flat;
(ii) The extended system module M is free;

(iii) The matrix F is hyper-regular over O.

Remark 3. Note that in the linear time-invariant case Definition 2 is equivalent to the definitions of [START_REF] Mounier | Propriétés structurelles des systèmes linéaires à retards: aspects théoriques et pratiques[END_REF][START_REF] Chyzak | Effective algorithms for parametrizing linear control systems over Ore algebras[END_REF].

For the proof we need the following lemma:

Lemma 2. Let Λ be an arbitrary ring and M ∈ Λ p×q , and let S ∈ Gl p (Λ) and T ∈ Gl q (Λ) be unimodular. Then Λ 1×q /Λ 1×p M Λ 1×q /Λ 1×p (S MT ) as left Λ-modules.

Proof. The map ϕ = v → vT is an automorphism of Λ 1×q because T is unimodular. The kernel of the composition π • ϕ of the projection π :

Λ 1×q → Λ 1×q /Λ 1×p S MT with ϕ is ϕ -1 (Λ 1×p S MT ) = Λ 1×p S M = Λ 1×p
M where the second identity follows from the unimodularity of S . Since π • ϕ is also surjective, by the first isomorphism theorem for modules (see, e. g., [27, Theorem 1.17]) there is an isomorphism

Λ 1×q /Λ 1×p M → Λ 1×q /Λ 1×p (S MT ).
Proof (of Theorem 1). We prove first that (i) is equivalent to (iii). If the system Fξ = 0 is π-flat,

there are matrices P ∈ O m×(n+m) and Q ∈ O (n+m)×m
with common denominator π ∈ K[δ] \ {0} such that QS m = B = ker F and P Q = 1 m . That means that Q is surjective and that the following diagramme is exact.

0 S m S n+m im F 0 Q F P Ē
Since P Q = 1, by the splitting lemma (see, e. g., [START_REF] Lang | Algebra[END_REF]Prop. 3.2]) one can show (see [START_REF] Middeke | Description of all π-flat outputs of linear differential-delay systems with time-varying coefficients, Linear Algebra and its ApplicationsSoon to be submitted[END_REF]) that there is a matrix Ē ∈ O k×n such that F Ē = 1 n where n is the (left) rank of im F because the rows of F are linearly independent by assumption. Therefore F is hyper-regular.

Conversely, let F be hyper-regular. Using Lemma 1 we may thus compute W ∈ Gl n+m (O) such that F W = (1 n , 0). If we let Q be the last m columns of W and P the last m rows of W-1 , then we have F Q = 0 and P Q = 1 m . Extracting a common denominator π of P and Q, we have proved that Fξ = 0 is π-flat.

Next we show that (iii) is equivalent to (ii). Assume first that F is hyper-regular. Since the unit vectors are in the column space of F, by column-reduction we obtain an invertible matrix T ∈ Gl n+m (O) such that F T = (1 n , 0). By Lemma 2, this means that

M = O 1×(n+m) O 1×n F O 1×(n+m) O 1×n F T O 1×(n+m) O 1×n (1 n , 0) O 1×m
which is free. Conversely, assume that M is free. Using, e. g., the method in [32, Thm. 8.1.1] we may obtain unimodular matrices S ∈ Gl n (O) and T ∈ Gl n+m (O) such that S F T = ( ∆, 0) where

∆ = diag(a 1 , . . . , a n ) ∈ O n×n is a diagonal matrix. By Lemma 2, M = O 1×(n+m) O 1×n F O 1×(n+m) O 1×n S F T O 1×(n+m) O 1×n ( ∆, 0) n j=1 O Oa j ⊕ O 1×m .
Since this module is free by assumption we conclude that all a j must be units, i. e., we may assume w. l. o. g. that ∆ = 1 n . By Definition 1, F is hyper-regular.

Algorithm 1 (Computation of a π-flat output).

Input: A matrix F ∈ O n×(n+m) representing the system (2).

Output: An Ore polynomial π ∈ K[δ] together with matrices P ∈ O m×(n+m) and Q ∈ O (n+m)×m as in Definition 2 or fail if such matrices do not exist.

Procedure:

1. Use column-reduction to check whether F is hyper-regular. If not, then return fail.

2. Else, let W ∈ Gl n+m (O) be such that

F W = 1 n 0 n×m . 3. Let Q W 0 n×m 1 m and P 0 m×n 1 m W-1 .
4. Let π ∈ K[δ] be a common denominator of P and Q.

Set P

π P ∈ O m×(n+m) and Q π Q ∈ O (n+m)×m .
6. Return π, P and Q.

Remark 4. In the above algorithm we exploit the fact that W-1 can be computed at the same time as W by inverting the elementary actions that compose W.

π-0-flatness

In contrast to the considerations of the previous subsection, it is sometimes necessary to keep the state and input variables separate: in the theory of linear time-invariant systems, controllability is equivalent to the existence of Brunovský's canonical form (see e.g. [START_REF] Brunovský | A classification of linear controllable systems[END_REF][START_REF] Kailath | Linear Systems[END_REF]); the interpretation of some of their states as flat output (see e.g. [START_REF] Fliess | Flatness and defect of non-linear systems: introduction theory and examples[END_REF]) shows that flat outputs do not need to depend on u, i. e. there exist P ∈ R m×n and Q ∈ R n×m such that y = Px, x = Qy and PQ = 1 m . This property is called 0-flatness. The fact that u can be expressed as a function of y is, in this case, an immediate consequence of the system equation with x = Qy. More generally: Definition 4. We say that a system is π-k-flat, with k ≥ 1, if and only if there exists a π-flat output y such that the maximal degree with respect to ∂ of the matrix P

0 n-m 1 m is equal to k -1.
We set k = 0, by convention, if P 0 n-m 1 m = 0, i. e. y does not depend on u.

Note that π-0-flatness is equivalent to the existence of (π, P, Q) as in Definition 2 such that

P = P 1 0 m,m with P 1 ∈ O m×n and π -1 P 1 π -1 Q 1 = 1 m where Q 1
1 n 0 n,m Q. A linear flat system is not necessarily 0-flat, as shown by the following elementary example:

Example 3. Consider the system 1 -∂ x u = 0 or x = u.
It can be easily seen that y = u is a flat output. Hence the system is 1-flat.

Remark 5. Note that elementary considerations show that 1-flatness is not preserved under the action of the group of unimodular matrices. Indeed, setting x u = 0 1 1 0

x u in the previous example, we get the system -∂ 1 x u = 0, orẋ = u , which admits y = x as flat output and is thus obviously 0-flat. However, if we restrict the transformation group to be Gl n (O) ⊗ Gl m (O), a group preserving the control variables, then 1-flatness is preserved.

Example 3 and Remark 5 thus show that obtaining a characterization of π-0-flat systems is of interest.

Lemma 3 (Elimination). If B in (1) is hyper-regular, the system can be decomposed according

to R ϕ -1 F x = MAx = MBu = 1 m 0 (n-m)×m u. ( 3
)
where F ∈ O (n-m)×n and ϕ ∈ K[δ] with ϕ 0.

Q Q1 R Q1 satisfy P Q = P1 Q1 = 1 m and M A -B Q = M A Q1 -B R Q1 = R ϕ -1 F Q1 - 1 m 0 R Q1 = 0. (4)
Equation ( 4) implies that QS m ⊆ B. To prove the other inclusion, let γ = γ 1 γ 2 ∈ B. Then analogously

0 = M A -B γ = R ϕ -1 F γ 1 - 1 m 0 γ 2 .
Since ker ϕ -1 F = ker F, we obtain γ 1 ∈ ker F and Rγ 1 = γ 2 . Consequently, there exists ζ ∈ S m such that γ 1 = Q1 ζ and thus also γ 2 = R Q1 ζ. Hence, γ ∈ QS m and the other inclusion also holds. Thus, extracting a common denominator π such that P = π -1 P and Q = π -1 Q where P ∈ O m×n and Q ∈ O n×m , we see that π, P and Q fulfill Definition 4. Thus, (iii) implies (i).

Conversely, if (i) holds with π, P and Q being as in Definition 4, then with Q 1 = (1 n , 0)Q and P 1 = P 1 n 0 , by the same calculation (4) we obtain ϕ -1 Fπ -1 Q 1 = 0 and π -1 P 1 π -1 Q 1 = 1 m . Thus, F x = 0 is π-flat. Thus, by Theorem 1 we have (iii).

If the matrix B of system (1) is hyper-regular, we can directly check for the existence of a π-0-flat output, as shown in Theorem 2. A suitable algorithm is as follows: Else, if the system defined by A and B is not π-0-flat, then FAIL.

Procedure:

1. Compute M ∈ Gl n (O), s.t. MB = 1 m 0 (n-m)×m . 2. Write MA = R ϕ -1 F where R ∈ O m×n and F ∈ O (n-m)×n . 3. If Algorithm 1 applied to F returns π 1 , P 1 and Q 1 , then (a) Set P = (π -1 1 P 1 , 0) and Q = π -1 1 Q 1 Rπ -1 1 Q 1 . (

Else, return fail

If the matrix B of system (1) is not hyper-regular, we can check for π-flatness by applying Algorithm 1.

Remark 7. Note that Algorithms 1 and 2 do not necessarily yield a polynomial π of minimal degree in δ. Especially, it is not guaranteed, that we obtain π ∈ K if the system is flat.

Example

To illustrate the results of the preceding sections and the usefulness of the concept for the feedforward controller design for linear time-delay differential systems, we demonstrate all steps for the system of Example 2. Note that all the necessary computations can be done with a package for the computer algebra system Maple, which has been presented in [START_REF] Antritter | A toolbox for the analysis of linear systems with delays[END_REF]. The package can be obtained from the first author upon request.

The matrix B is hyper-regular and following Algorithm 2 we compute M such that MB = 1 0 .

We get

M = 0 δ -1 1 0 . Thus: R F = MA = 0 δ -1 ∂ ∂ a(δ 2 -δ)
We now follow Algorithm 1 and compute W such that F W = 1 0 to

W = 0 1 (δ 2 -δ) -1 1 a -(δ 2 -δ) -1 1 a ∂ . The inverse is W-1 = ∂ a(δ 2 -δ) 1 0 .
Subsuming the remaining computations of Algorithm 1 we directly obtain the following matrices in step 3 a) of Algorithm 2, namely

P = 0 1 W-1 0 = 1 0 0 , (5) and Q
= 1 2 R W 0 1 =            1 -(δ 2 -δ) -1 1 a ∂ -δ 3 -δ 2 -1 1 a ∂ 2 -ȧ a 2 ∂            . (6)
We readily get π = δ 3 -δ 2 . From ( 5), a π-0-flat output is given by y = x 1 . This can also be deduced from the first entry in Q. From [START_REF] Lévine | On necessary and sufficient conditions for differential flatness[END_REF] we deduce that x 2 = -(δ 2 -δ) -1 1 a ẏ and u = -δ 3 -δ 2 -1 1 a ÿ -ȧ a 2 ẏ . We now use these formulas to generate feedforward trajectories corresponding to the following motion planning problem: we are looking for a trajectory of (x 1 , x 2 , u) starting from x(0) = (x 1 (0), x 2 (0)) = (0 0) and arriving at the final state x(2) = (1 0) at time t = 2, while x 1 has to be equal to zero before t = 0 and to 1 after t = 2.

In order to compute explicit feedforward state and input trajectories without integrating the system differential-delay equations, we compute a reference trajectory for the flat output y, which is not constrained by any differential-delay equation thanks to the freeness of the extended system module, and then deduce the state and input trajectories by x u = Qy with Q defined by [START_REF] Lévine | On necessary and sufficient conditions for differential flatness[END_REF]. We take a(t) = t + 3 and thus ȧ(t) = 1. Furthermore, we take τ = 1, i.e. δa(t) = a(t -1).

In order to obtain an explicit expression for x 2 = -(δ 2 -δ) -1 1 a ẏ , we compute the series expansion

δ 2 -δ -1 = j≥-1 -δ j .
Taking into account that x 1 is required to be constant for t < 0 and t > 2, the desired trajectory y d for y = x 1 will be constant at these points of time and thus ẏd = ÿd = 0 for t < 0 and t > 2. This yields

x 2,d =         j≥-1 δ j         1 a ẏd = X 2,
where X 2,d (t) = 1 a(t-t ) ẏd (tt ). 13

We make the ansatz y d (t) = (a 0 + a 1 t + a 2 t 2 + a 3 t 3 + a 4 t 4 + a 5 t 5 )χ [0,2[ (t) + χ [2,∞[ (t) the coefficients a i , i = 0, . . . , 5 being such that y(0) = x 1 (0) = 0, y(2) = x 1 (2) = 1, ẏ(0) = 0, ẏ(2) = 0, ÿ(0) = 0 and ÿ(2) = 0. We readily get:

y d (t) = - 45
We now insert this expression in [START_REF] Mounier | Propriétés structurelles des systèmes linéaires à retards: aspects théoriques et pratiques[END_REF] to obtain the corresponding feedforward control. Since

π -1 = δ 3 -δ 2 -1 = j≥-2 δ j (
where U d (t) = -1 a(t-t )) ÿd (tt ) + 1 a 2 (t-t ) ẏd (tt ), which completes the solution of this motion planning problem. The resulting trajectories are shown in Figure 1. Though the trajectory for y is relatively smooth, it is generated by an input u which is quite irregular and complicated. We also notice that u has to start 2τ before the beginning of the y-trajectory in order to satisfy the above requirements, which corresponds to the order of the Laurent series associated to π -1 .

Conclusion

In this paper we have introduced a new definition of π-flatness and the important particular case of π-0-flatness. These notions have been characterized by means of the properties of matrices over Ore polynomial rings, such as hyper-regularity or row-and column-reducedness. This characterization has yielded an efficient and simple algorithm to test the existence of πand π-0-flat outputs and compute them, via an algorithm that is implemented in the computer algebra system Maple [START_REF] Antritter | A toolbox for the analysis of linear systems with delays[END_REF]. We have illustrated the pertinence of this approach by an academic example.

Note that the description of all possible π-0-flat outputs can be achieved; it will be the subject of a forthcoming paper. This will lead to a method to determine the minimal order of the polynomial π.

Another potential direction to extend this approach may be to investigate other possible signal spaces.

Finally, the question of increasing the efficiency of the proposed algorithm by using quotient free computations may also be addressed.

  δ and, as announced, they are matrices over the ring O = K[δ, ∂].

Remark 1 .Proposition 2 .

 12 Note that Definition 2 is equivalent to the existence of matrices P ∈ O m×(n+m) and Q ∈ O (n+m)×m such that QS m = B and P Q = 1 m . We recover π by computation of the left common denominator of P and Q [27, Prop. 5.3]. Therefore all computations can be done over O.We have the following proposition: Assume that y is a π-flat output of system[START_REF] Fliess | Flatness and defect of non-linear systems: introduction theory and examples[END_REF]. Let us set y = T z with T ∈ Gl m (O). There exists a polynomial κ ∈ K[δ] such that z is a κ-flat output of system (2).Proof. By Remark 1, it is sufficient to prove that T -1 P and Q T are related to the transformed flat output z. Obviously, we have T -1 P Q T = 1 m and moreover Q T S m = QS m = B. With κ being a common denominator of T -1 P and Q T , the claim follows.

Remark 6 .

 6 Let F ∈ O p×q . Let ϕ ∈ K[δ] be its common denominator. We have F = ϕ -1 F with F ∈ O p×q . Thus, ker F = ker F. Theorem 2. If B is hyper-regular, we have the following equivalences: (i) The control system (1) is π-0-flat; (ii) The extended system module O 1×n /O 1×(n-m) F is free, with F defined in (3); (iii) F is hyper-regular over O. Proof. Let F be hyper-regular over O. From representation (3) combined with Theorem 1, there exists P1 ∈ O m×n and Q1 ∈ O n×m such that F Q1 = 0 and P1 Q1 = 1 m . Thus, P P1 , 0 and

Algorithm 2 (

 2 Computation of a π-0-flat output). Input: Matrices A ∈ O n×n and B ∈ O n×m with B hyper-regular, representing (1). Output: If system (1) is π-0-flat, the polynomial π ∈ K[δ] and the triple (P, Q, R) of the matrices P ∈ O m×n , Q ∈ O n×m and R ∈ O m×m from Definition 2.

  b) Let π be a common denominator of P and Q and set P = π P ∈ O m×n and Q = π Q ∈ O n×m . (c) Return π, P and Q.

Figure 1 :

 1 Figure 1: Graphs for x 1 = y, x 2 and u

  d (t)χ [-1,0[ (t) + (X 2,d (t) + X 2,d (t + 1))χ [0,∞[ (t + 1),

this space is not uniquely defined and may be chosen according to the specific application one is interested in, such as motion planning, tracking, etc. See section 2

(Johannes Middeke)

1 J. Middeke was supported by the Austrian Science Foundation (FWF) under the project DIFFOP (P 20 336-N18) and by the strategic programme "Innovatives OÖ 2010plus" by the Upper Austrian government.

Appendix A. Inversion of a δ-Polynomial

Let π = a n δ n + . . .

where k < n and a k , . . . , a n ∈ K. Assume that a k 0, i.e. that the order of π is k. The inverse π -1 in K((δ)) may be computed in the following way: First, 16

using the fact that

we may assume w. l. o. g. that k = 0. We make an ansatz j≥0 c j δ j for π -1 and compute

Comparing coefficients and using a 0 0, we can compute

for all ≥ 1. Note, that the left hand side depends only on those c i which are already computed.


Manuscript

A Niwińska

K Pogoda

M Murawska

P Niwiński

in patients with breast cancer and single or solitary brain metastasis

Keywords: breast cancer, RPA RTOG prognostic class, single brain metastasis, solitary brain metastasis, systemic therapy

Factors influencing survival

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

Introduction:

Approximately 15-30% of patients with cancer will develop cerebral metastases over the course of their disease [START_REF] Patchell | Neurologic complications of systemic cancer[END_REF][START_REF] Mintz | Management of single brain metastasis: a practice guideline[END_REF], but in 30-40% of those patients, only one metastatic lesion in the brain will be detected [START_REF] Wen | Management of brain metastases[END_REF]. "Solitary brain metastasis" is a relatively rare occurrence of a single parenchymal brain metastasis and it is the only known site of metastatic cancer in the body, with no systemic metastatic disease. The term "single brain metastasis" refers to a single cerebral lesion, without any implications connected to the extent of extracranial disease [START_REF] Wen | Management of brain metastases[END_REF]. In the present study, it means the presence of cancer lesions outside the brain.

Current treatment approaches to local therapy of brain metastasis include surgery, whole brain radiotherapy (WBRT), stereotactic radiosurgery (SRS) or a combination of these modalities [START_REF] Suh | ACR appropriateness criteria: single brain metastasis[END_REF]. Patients with single or solitary brain metastasis generally undergo more aggressive local treatment than those with multiple metastases. Despite this, majority of patients with breast cancer and single brain metastases die because of progressive extracranial disease [START_REF] Patchell | Postoperative radiotherapy in the treatment of single metastases to the brain: a randomized trial[END_REF], therefore, a continuation of systemic therapy after local treatment in patients with active systemic disease is justified. However, the role of systemic therapy after local treatment in patients with breast cancer and solitary brain metastasis has not been studied.

Due to the fact that survival of patients with brain metastases depends on many factors that are related to patient's characteristics, breast disease and method of treatment, the aim of the study was to assess prognostic factors influencing survival from brain metastasis in 100 patients with breast cancer who had single or solitary brain lesion.

Patients and Methods:

Between 01.01.2003 and 31.12.2009, 441 patients with breast cancer and brain metastases, were treated in Breast Cancer Department at the Cancer Center and Institute of Oncology in Warsaw, Poland. Out of 441, one hundred patients had solitary or single brain metastasis. The prospective observation of patients started at the time of the detection of brain metastases and all data was collected in data base. In each case, treatment options were approved by team of medical oncologists, radiation oncologist (A.N.), neurologist and neurosurgeons and were performed after patients had signed written consent.

Without having the possibility to perform gene expression profiling, biological subtypes of brain metastases were defined based on the expression of estrogen (ER), progesterone (PgR) and HER2 receptors, which were proposed by Hugh et al [START_REF] Hugh | Breast cancer subtypes and response to docetaxel in node-positive breast cancer: use of an immunohistochemical definition in the BCIRG 001 trial[END_REF]. Immunohistochemistry (IHC) was performed in order to evaluate the level of ER, PgR and

Statistical analysis:

Descriptive statistics were determined for the patient demographics and clinical characteristics. All hypotheses tests were conducted at the α = 0.05 level with a 95% confidence interval. In order to compare categorical tumor features in 4 biological subgroups of patients, the chi-square test was used. For those categorical variables in which the chisquare test was inappropriate because of small numbers, the Fisher exact test was used. In order to determine factors influencing survival from brain metastases multivariate Cox proportional hazards model was developed. Proportional hazards assumption for the final Cox model was tested using Schoenfeld's residuals. The following factors were analyzed: RPA RTOG prognostic class (III vs I and III vs II), biological subtype of breast cancer (triplenegative vs luminal A; triple-negative vs luminal B and triple-negative vs HER2), localization of brain lesion (infratentorial vs supratentorial), leptomeningeal carcinomatosis appearing as a relapse after definitive treatment (yes vs no), neurosurgery (yes vs no), WBRT (40Gy vs 30Gy vs 20Gy) and systemic treatment after WBRT (yes vs no). Survivals were estimated using the Kaplan-Meier method and compared using the log-rank test.

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

Results:

Patients' demographics and treatment In all patients, brain metastases were detected in magnetic resonance (MR) after clinical symptoms appeared. In 57% of patients neurosurgery with curative intention was performed. In 95% of patients WBRT was performed, using 4-6 MV photon beam by two lateral opposed standard fields covering all intracranial contents. In 5 patients neither surgery nor WBRT was performed because of poor performance status. After local treatment of brain metastases, systemic treatment was ordered in 67% of patients. Stereotactic radiosurgery was performed in 3 patients, after local recurrence.

A comparison of clinicopathological features between patients with single and solitary brain metastasis is presented in Table 1. Solitary brain metastasis (one lesion within the brain without any extracranial metastatic site) was detected in 29% of patients and single brain metastasis (one lesion in the brain with any extracranial metastases) was detected in the remaining 71% of patients. Patients with solitary brain metastasis were more likely to be HER2-negative and belonged to RPA RTOG prognostic class I. They were often treated with neurosurgery and WBRT on schedule 40 Gy in 20 fractions. Systemic treatment was ordered less frequently and in most cases, hormonal therapy was given.

A comparison of clinical features of four biological subsets is presented in Table 2.

Almost half of patients were HER2-positive (HER2 and luminal B subtypes). Biological subsets of patients did not differ in relation to age at initial diagnosis, initial TNM stage, KPS, rates of neurosurgery, WBRT and systemic treatment after WBRT. The only difference was the pattern of metastatic spread. In HER2-positive breast cancer patients (HER2 and luminal B subtypes), metastases in many organs were detected and strong correlation between brain metastases and metastases to the viscera was observed. In 42% of patients with triple-negative breast cancer, the first site of dissemination of breast cancer could be found in the brain. In luminal A subtype, bones were the most frequent site of metastasis. Locoregional failure was observed in all biological subtypes, but it was the most frequent in luminal B subset of patients.

Survivals

In the entire group, median disease free survival (DFS) was 2 years and overall survival (OS) was 4.8 years. Median survival from the detection of brain metastasis was 13 months and 28% of patients survived for 2 years. Median survival from brain metastasis in

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 6
patients with RPA RTOG prognostic class I, II and III was 22 months, 13 months and 6 months, respectively, p <0.0001. Median survival from brain metastasis in 29 patients with solitary brain metastasis was 20 months and in 71 patients with single brain metastasis was 11 months, p = 0.01 (Figure1). Median survival in patients with solitary brain metastasis, treated and not treated systemically after WBRT, was 34 months and 8 months, respectively and in patients with single brain metastasis it was was 13 months and 4 months, respectively, p <0.001. Median survival from brain metastases in triple-negative, HER2, luminal B and luminal A subtypes was 11months, 13 months, 16 months and 15 months, respectively (p = 0.60) and 2-year-survival was 12%, 20%, 17% and 32%, respectively. Median survivals of patients are presented in Table 3.

Factors influencing survival after WBRT -multivariate analys is

The Cox multivariate analysis revealed that patients with RPA RTOG prognostic class I as well as those who underwent neurosurgery and those who were treated systemically after WBRT, were at a lower risk of death. On the contrary, RPA RTOG prognostic class III, lack of surgical resection and systemic treatment as well as central nervous system relapse in the form of carcinomatous meningitis correlated with poor survival. Biological subtype of breast cancer (triple-negative vs luminal A, p = 0.51; triple-negative vs luminal B, p = 0.51, and triple-negative vs HER2, p = 0.78), localization of the lesion in the brain (p = 0.34) and the schedule of WBRT (p = 0.41) did not correlate with survival. The results of final model are presented in Table 4.

Discussion

Survival from brain metastases depends on factors related to patient's characteristics (age, performance status), the disease (biological subtype of breast cancer, number of brain metastases, presence and advancement of extracranial disease) and type of treatment given (surgery, WBRT, SRS and systemic treatment after WBRT).

Patient-related factors

In the present study, affiliation to the RPA RTOG prognostic class was an important prognostic factor influencing survival from the detection of brain metastasis. Our results are

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

7 comparable to the results that had been already published [START_REF] Tendulkar | RPA classification has prognostic significance for surgically resected single brain metastasis[END_REF][START_REF] Paek | Reevaluation of surgery for the treatment of brain metastases: review of 208 patients with single or multiple brain metastases treated at one institution with modern neurosurgical techniques[END_REF] and to our previous study concerning survival of patients with operated brain metastases [10].

Tendulkar et al [START_REF] Tendulkar | RPA classification has prognostic significance for surgically resected single brain metastasis[END_REF] analyzed 271 patients with a single brain metastasis who underwent surgical resection and adjuvant therapy. Survival of patients in RPA class I was 21.4 months, in RPA class II -9.0 months and in RPA class III -8.9 months. This analysis supported the prognostic significance of the RPA classification in patients with a single/solitary brain metastasis who underwent surgical resection and adjuvant therapy.

Cancer-related factors

Biological subtypes

Biological subtypes of breast cancer are prognostic factors that give information about natural history of the disease. The survival of HER2-positive (HER2 and luminal B) and triple-negative breast cancer patients was the worst while survival for luminal A subtype was the best [START_REF] Niwińska | Breast cancer brain metastases: differences in survival depending on biological subtype, RPA RTOG prognostic class and systemic treatment after whole-brain radiotherapy (WBRT)[END_REF], however, new therapeutic approaches can improve the outcome of patients with poor prognosis. In the post-trastuzumab era, the survival of HER2-positive breast cancer patients is comparable or even better, than that observed in patients with luminal A subtype [START_REF] Niwińska | Breast cancer subtypes and response to systemic treatment after whole-brain radiotherapy in patients with brain metastases[END_REF][START_REF] Matsunaga | Gamma Knife surgery for metastatic brain tumors from primary breast cancer: treatment indication based on number of tumors and breast cancer phenotype[END_REF]. In the present study, the rate of HER2-positive (HER2 and luminal B), triplenegative and luminal A subsets was comparable to our previous study concerning the whole group of 420 patients with brain metastases treated in the years 2003-2009 [START_REF] Niwińska | Breast cancer subtypes and response to systemic treatment after whole-brain radiotherapy in patients with brain metastases[END_REF].

We failed to demonstrate the influence of biological subtype of breast cancer on survival in single or solitary brain metastasis. The possible explanation of this result is that the extent of metastatic disease of those patients was relatively small and intensive local and systemic treatment could be effective enough to overcome poor prognosis attributed to particular biological subtype. However, two-year survival from brain metastases in triplenegative, HER2, luminal B and luminal A subtypes was 12%, 20%, 17% and 32%, respectively. Although the difference is not statistically significant (p = 0.60), this suggests, that triple-negative breast cancer remains the worst and that luminal A is the best biological subtype. Unfortunately, the presented subgroups are too small to definitively solve this problem.

Treatment-related factors

Surgery

In the present study, neurosurgery positively correlated with survival.

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The publication of several prospective studies established the value of combining surgical resection and radiation therapy for brain metastases. In the trial by Patchell et al. [START_REF] Patchell | A randomized trial of surgery in the treatment of single metastases to the brain[END_REF], median survival for patients who received surgery and WBRT was 9.2 months as compared to 3.5 months for patients who received WBRT alone. In the trial by Vecht et al. [START_REF] Vecht | Treatment of single brain metastasis: radiotherapy alone or combined with neurosurgery?[END_REF] median survival was 10 months as compared to 6 months. The difference in survival was most robust in a subgroup of patients with stable or absent extracranial disease (median survival 12 versus 7 months). No significant survival difference was observed in patients with active extracranial disease (median survival 5 months in both treatment groups). In the trial by Mintz et al. [START_REF] Mintz | A randomized trial to assess the efficacy of surgery in addition to radiotherapy in patients with a single cerebral metastasis[END_REF] median survival between the surgery plus WBRT arm and the WBRT only arm was not statistically different (5.6 months and 6.3 months, respectively). In this study, the systemic extent of primary disease was identified as a major contributing factor and predictor of mortality. The results of the above mentioned studies suggests that surgical excision should be considered for patients with good performance status, with minimal or lack of extracranial disease, and in cases where single brain metastasis can be amenable to complete excision.

WBRT

In the present study the analysis of survival from brain metastasis revealed the same equivalence for all radiotherapy schedules used. The dose of 40 Gy in 20 fractions was not better than a dose of 30 Gy in 10 fractions and 20 Gy in 5 fractions.

In the study by Patchell et al [START_REF] Patchell | Postoperative radiotherapy in the treatment of single metastases to the brain: a randomized trial[END_REF] the role of WBRT in 95 patients with single brain metastasis after surgical resection was assessed. Patients, who received treatment with surgical resection and WBRT, had fewer recurrences in the brain (18% vs 70%) and were less likely to die due to neurological causes (14% vs 44%). There was no difference in overall survival between two groups (48 vs 43 weeks). The study confirms the opinion, that postoperative WBRT should be considered in patients who underwent resection of a single brain metastasis in order to reduce the risk of tumor recurrence.

Currently, there is no consensus on the optimal radiation scheme for patients with brain metastases. The optimal dose-fractionation schedules for patients with brain metastases have been evaluated in randomized trials conducted by the RTOG [START_REF] Borgelt | The palliation of brain metastases: final results of the first two studies by the Radiation Therapy Oncology Group[END_REF][START_REF] Kurtz | The palliation of brain metastases in a favorable patient population: a randomized clinical trial by the Radiation Therapy Oncology Group[END_REF]. The overall response and median survival rates were equivalent in arm of 40 Gy in 4 weeks, 30 Gy in 2 weeks and 20 Gy in 1 week. There were no survival benefits regardless of which fractionation scheme was used. The results of the present study confirm the above mentioned data. Long term side effects of WBRT are a significant issue in the treatment of patients with single or solitary brain metastases because of the relatively long survival time of these patients. In some

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 9
previous reports, increased risk of neurotoxicity after daily fractions 3-6 Gy was shown [START_REF] Deangelis | Radiation-induced dementia in patients cured of brain metastases[END_REF],

but recent class I evidence suggests that, altered dose/fractionation schedules of WBRT do not result in significant differences in neurocognitive function when compared to 30 Gy in 10 fractions [START_REF] Gaspar | The role of whole brain radiation therapy in the management of newly diagnosed brain metastases: a systematic review and evidence-based clinical practice guideline[END_REF].

Systemic treatment

In the present study, the Cox multivariate analysis revealed, that systemic treatment ordered after WBRT statistically significantly correlated with survival.

Based on the recent publication, systemic therapy is not a standard treatment in patients with brain metastases; however, this recommendation was established mostly by the analysis of patients with non small cell lung cancer [21]. Breast cancer cells are relatively sensitive to chemotherapy, hormonal therapy and targeted therapy and tailored treatment can be effective in particular biological subsets of patients after WBRT [START_REF] Niwińska | Breast cancer subtypes and response to systemic treatment after whole-brain radiotherapy in patients with brain metastases[END_REF]. Moreover, about 46% -71% of patients with brain metastases and active extracranial disease die because of progression in the viscera and not due to neurological causes [START_REF] Patchell | Postoperative radiotherapy in the treatment of single metastases to the brain: a randomized trial[END_REF][START_REF] Vecht | Treatment of single brain metastasis: radiotherapy alone or combined with neurosurgery?[END_REF][START_REF] Mintz | A randomized trial to assess the efficacy of surgery in addition to radiotherapy in patients with a single cerebral metastasis[END_REF][START_REF] Borgelt | The palliation of brain metastases: final results of the first two studies by the Radiation Therapy Oncology Group[END_REF], therefore, continuation of systemic treatment after WBRT in patients with active systemic disease is justified [START_REF] Paek | Reevaluation of surgery for the treatment of brain metastases: review of 208 patients with single or multiple brain metastases treated at one institution with modern neurosurgical techniques[END_REF][START_REF] Niwińska | Breast cancer brain metastases: differences in survival depending on biological subtype, RPA RTOG prognostic class and systemic treatment after whole-brain radiotherapy (WBRT)[END_REF]22].

Unexpectedly, in the present study, a strong correlation between systemic therapy and survival of patients with solitary brain metastasis has been shown. One possible explanation of this phenomenon is that systemic therapy can control occult metastatic lesions outside the brain and, in this way, it can prevent distant recurrence. The second explanation can be the fact, that WBRT disrupts the blood-brain barrier. The level of systemic drugs in central nervous system after WBRT can be high enough, to control brain lesions. To our knowledge, this is the first data demonstrating the activity of systemic therapy in patients with solitary brain metastasis from breast cancer, however, this study has some limitations and the results require confirmation by other authors.

Conclusions

In our study, clear difference in survival between single and solitary brain metastasis was demonstrated. RPA RTOG prognostic class was the most reliable indicator of survival in patients with one metastatic lesion in the brain. Neurosurgery as a part of multidisciplinary approach and systemic therapy used after local treatment prolonged survival in both subgroups. Although the proportion of particular biological subsets is comparable to the one,

  Fluorescence in situ hybridization (FISH) analysis for HER2 amplification was performed in the case of IHC 2+. Four biological subtypes were selected: triple-negative (ER/PgR negative, HER2 negative), HER2 (HER2 positive, ER/PgR negative), luminal B (HER2 positive, ER/PgR positive) and luminal A (ER/PgR positive, HER2 negative). Triplenegative and luminal A subsets were HER2-negative. HER2 and luminal B subsets were HER2-positive.Overall survival (OS) was calculated from the diagnosis of breast cancer to a death from any cause. Disease free survival (DFS) was calculated from the diagnosis of breast cancer to distant or local-regional cancer recurrence or to a death from any cause, whichever occurred first. DFS and OS were calculated in the entire group and within 4 biological subgroups. Survival calculated from the detection of brain metastasis was analyzed in the entire group, including 4 biological subgroups, in three Recursive Partitioning Analysis Radiation Therapy Oncology Group (RPA RTOG) prognostic classes, and in both solitary and single brain metastasis subsets. RPA RTOG prognostic class I included patients below 65 years of age, with Karnofsky Performance Status (KPS) score of 70 or more and with controlled disease at the primary site without metastases outside the brain. Prognostic class III included patients with a KPS score below 70, regardless of other factors, while prognostic class II included all the remaining patients [7].

  with multiple brain metastases, we failed to show significant correlation between biological subtype of breast cancer and the outcome. MP, Paleologos NA, Mikkelsen T et al. The role of chemotherapy of newly diagnosed brain metastases: a systemic review and evidence-based clinical practice guideline. J Neurooncol 2010;96: 71-83. 22. Lee SS, Ahn JH, Kim MK et al. Brain metastases in breast cancer: prognostic factors and management. Breast Cancer Res Treat 2008; 111:523-30.

Figure legends:Figure 1 .

 1 Figure legends:Figure 1. Survival from brain metastasis in patients with solitary and other single brain metastasis.

Table legends :

 legends Table1. A comparison of clinicopathological features between patients with single and solitary brain metastasis.

Table 2 .

 2 Differences in clinical features depending on biological subtype.

Table 3 .

 3 Survival from the detection of brain metastasis.

Table 4 .

 4 Factors influencing survival from brain metastasis: multivariate analysis, final model.

  Table1. A comparison of clinicopathological features between patients with single and solitary brain metastasis. In some patients more than one line of systemic therapy was ordered Abbreviations: HER2, human epidermal growth factor receptor 2; RPA RTOG, recursive partitioning analysis of Radiation Therapy Oncology Group; WBRT, whole-brain radiotherapy.

		ACCEPTED MANUSCRIPT
	Schedule of WBRT:
	40 Gy/20 fractions	16 (55%)	16 (22%)
	30 Gy/10 fractions	11 (39%)	35 (49%)
	20 Gy/5 fractions	1 (3%)	16 (23%)
	No radiotherapy	1 (3%)	4 (6%)	0.002
	Characteristic Neurosurgery of brain metastasis: Number of patients Age at initial diagnosis, years Median Range Initial TNM stage: I II III IV Histological type: Ductal carcinoma Lobular carcinoma Papilar or neuroendocrine carcinomas Cancer cells or invasive cancer after chemotherapy Yes No Systemic therapy after brain metastasis: Yes No Type of systemic therapy ª Hormonal therapy Chemotherapy Targeted therapy Type of chemotherapy; schedules withª Taxanes Anthracycline Capecitabine Vinorelbine Platinum Fluorouracil Cyclophosphamide Estrogen/Progesterone-receptor status: Positive Negative HER2 status: Positive Negative Biological subtype: Triple-negative HER2 Luminal B Luminal A RPA RTOG class: I Type of endocrine therapy Aromatase inhibitors Tamoxifen Megestrol acetate Type of targeted therapy Trastuzumab Lapatinib A C C E P T E D Solitary brain metastasis 29 (29%) 51 40-78 1 (3%) 12 (43%) 15 (52%) 1 (3%) 19 (66%) 2 (7%) 1 (3%) 7 (24%) 25 (86%) 4 (14%) 14 (48%) 15 (52%) 10/14 (71%) 5/14 (36%) 2/14 (14%) 2/5 (40%) 2/5 (40%) 2/5 (40%) 1/5 (20%) A N U S C R I P T Single brain metastasis p-value 71 (71%) 52 26-73 6 (9%) 31 (44%) 22 (31%) 11 (16%) 54 (77%) 5 (7%) 2 (3%) 9 (13%) 0.380 0.151 0.569 32 (45%) 39 (55%) 53 (75%) 18 (25%) 14/53 (26%) 41/53 (77%) 23/53 (43%) 10/41 (24%) 5/41 (12%) 4/41 (12%) 5/41 (12%) <0.001 0.011 0.002 0.003 0.045 0.453 0.102 0.058 0.625 1/5 (20%) 1/5 (20%) 7/41 (17%) 5/41 (12%) 0.869 0.625 M 1/5 (20%) 2/41 (5%) 0.196 12 (43%) 17 (57%) 10 (34%) 19 (66%) 11 (42%) 6 (24%) 4 (16%) 8 (32%) 20 (69%) 37 (52%) 34 (48%) 39 (56%) 32 (44%) 15 (21%) 19 (27%) 20 (28%) 17 (24%) 0 0.329 0.063 0.219 7/10 (70%) 2/10 (20%) 1/10 (10%) 1/2 (50%) 1/2 (50%) 10/12 (83%) 2/12 (17%) 0 21/22 (95%) 1/22 (5%) 0.507 0.026 a-
	II	4 (14%)	57 (80%)
	III	5 (17%)	14 (20%)	<0.001
	Karnofsky Performance Status:
	>=70	24 (83%)	57 (80%)
	<70	5 (17%)	14 (20%)	1

Table 2 .

 2 Differences in clinical features depending on biological subtype.

	Characteristic	Luminal A	HER2-positive		Triple-negative	p-value
		(HER2-negative,	N = 49		(HER2-negative,
		ER/PgR-positive) Luminal B	HER2/neu (HER2-	ER-negative,
			(HER2-positive,	positive, ER/PgR-	PgR-negative)
			ER/PgR-positive)	negative)
	Number of patients	25	24	25	26	-
	Age at initial
	diagnosis (years):
	Median	51	51.5	54	50
	Range	(36-73)	(33-67)	(26-73)	(28-78)	0.850
	Initial TNM stage:
	I	2 (8.3%)	2 (8.3%)	0	3 (11.5%)
	II	12 (50%)	10 (41.7%)	11 (44%)	10 (38.5%)
	III	8 (33.3%)	8 (33.3%)	11 (44%)	10 (38.5%)
	IV	2 (8.3%)	4 (16.7%)	3 (12%)	3 (11.5%)	0.909
	Histological type:
	Ductal carcinoma
	grade 3	5 (20.8%)	6 (26.1%)	12 (48%)	16 (64%)
	Lobular carcinoma	7 (29.2%)	1 (4.3%)	0	0
	Other types	12 (50%)	16 (69.6%)	13 (52%)	9 (36%)	0.015
	Localization of other
	metastases:
	Brain as only site	9 (36%)	4 (16%)	7 (28%)	11 (42%)	0.235
	Liver	6 (24%)	5 (20.8%)	9 (36%)	4 (15.4%)	0.367
	Lung	5 (20%)	14 (58.3%)	9 (36%)	7 (26.9%)	0.030
	Bone	11 (44%)	9 (37.5%)	10 (40%)	2 (7.7%)	0.021
	Local recurrence	4 (16%)	13 (54.2%)	5 (20%)	7 (26.9%)	0.015
	Systemic treatment
	after recurrence:
	Yes	17 (68%)	19 (79.2%)	18 (72%)	13 (50%)
	No	8 (32%)	5 (20.8%)	7 (28%)	13 (50%)	0.151
	Karnofsky
	performance status
	≥70	20 (80%)	19 (80%)	20 (80%)	22 (85%)
	<70	5 (20%)	5 (20.8%)	5 (20%)	4 (15%)	0.975
	Leptomeningeal
	carcinomatosis
	Yes	1 (4%)	1 (4.2%)	0	2 (7.7%)
	No	25 (96%)	23 (95.8%)	25 (100%)	24 (92.3%)	0.579

Abbreviations: HER2, human epidermal growth factor receptor 2; ER, estrogen receptor; PgR, progesterone receptor; Rth, radiotherapy; WBRT, whole-brain radiotherapy.

Table 3 .

 3 Survival from the detection of brain metastasis.

	Abbreviations: HER2, human epidermal growth factor receptor 2; RPA RTOG, recursive partitioning
	analysis of Radiation Therapy Oncology Group

Table 4 .

 4 Factors influencing survival from brain metastasis: multivariate analysis, final model.

	Factor	Hazard Ratio	95% Confidence	p-value
			Interval
	RPA RTOG prognostic class	6.32	2.58-15.50	<0.0001
	III vs I
	RPA RTOG prognostic class	2.85	1.33-6.15	0.007
	III vs II
	Surgical excision (yes vs no)	0.56	0.32-0.97	0.039
	Systemic treatment after
	WBRT (yes vs no)	0.28	0.16-0.49	<0.0001
	Leptomeningeal
	carcinomatosis (yes vs no)	5.17	1.17-22.81	0.030

Abbreviations: RPA RTOG, recursive partitioning analysis of Radiation Therapy Oncology Group; WBRT, whole-brain radiotherapy.

Conflict of interest statement

There have been no involvements that might raise the question of bias in the work reported.


L Traldi Macedo

& J Ribeiro

MD Giuseppe Curigliano
email: giuseppe.curigliano@ieo.it

L Fumagalli

M Locatelli

J Barreto

Campello Carvalheira

A Quintela

S Bertelli

O De Cobelli

Multidisciplinary Approach in the Treatment of Patients with Small Cell Bladder Carcinoma

Keywords: Small cell carcinoma, Urinary bladder neoplasm, Chemotherapy, Radiotherapy, Cystectomy

Small cell carcinoma of the urinary bladder (SCCUB) is considered to be a tumor with a neuroendocrine phenotype characterised by aggressive behaviour and poor prognosis. Small cell carcinoma of the urinary bladder comprises 0,35 to 1% of all bladder cancers and is frequently observed in combination with other histological subtypes of carcinoma. Clinical presentation is characterized by advanced stage at diagnosis and rapidly progressive disease. In daily clinical practice there is no gold standard for the management of patients affected by this disease. Treatment of patients with limited disease combines neoadjuvant platinum-based chemotherapy followed by specific local treatment of the primary tumour. Cystectomy or radiotherapy should be proposed on an individual basis. In the metastatic setting, prognosis remains poor with a potential benefit from chemotherapy containing platinum compounds. Treatment of small cell carcinoma of the urinary bladder is based on evidence obtained from case reports and retrospective analyses. Due to low disease frequency there is a lack of randomized trials to provide guidance as to optimal therapy. Thus, systemic and local approaches are extrapolated from the literature available for the treatment of small cell carcinomas at other (nonurological) sites. We provide an overview of the currently available literature with it's main focus on the treatment of either locally advanced or metastatic small cell carcinoma of the urinary bladder.

M A N U S C R I P T A C C E P T E D

Introduction

Small Cell Carcinoma of the Urinary Bladder (SCCUB) is a rare pathological presentation within the group of extrapulmonary small cell carcinomas. It is usually characterized by an advanced stage at diagnosis and rapid clinical progression. The mean survival time for all stages is 19.6 months, and the percentage of patients alive at 5-years is 8.1% [START_REF] Cheng | Small cell carcinoma of the urinary bladder: A clinicopathologic analysis of 64 patients[END_REF] . Patients with localized disease are prone to develop distant metastases after treatment with a short time to progression.

There is no standard approach in the management of SCCUB diagnosed at an early stage and the currently available evidence supports a multidisciplinary approach incorporating neoadjuvant chemotherapy followed by local therapy (radiotherapy or surgery) in order to optimize clinical outcomes. Chemotherapy is usually based on platinum regimens in both the localized and metastatic settings. The prognosis of this disease remains poor and multi-institutional trials are needed to better define the best treatment for this rare disease.

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

Methods

Data were obtained by a search within PubMed using the following search term "bladder cancer" and "small cell" and "bladder small cell carcinoma". There were no limits applied for language, methodological characteristics, or year of publication. Case reports and case series were also selected.

Epidemiology

The incidence of SCCUB has been reported to be between 0,35 to 1% [1-4] of all bladder cancers. In several pathology based retrospective studies small cell histology usually coexists with tumour types such as transitional cell carcinoma [START_REF] Cheng | Small cell carcinoma of the urinary bladder: A clinicopathologic analysis of 64 patients[END_REF][START_REF] Abrahams | Small cell carcinoma of the bladder: a contemporary clinicopathological study of 51 cases[END_REF] . The coexistence of distinct tumours in the same pathological sample can be considered as supporting a pluripotent stem-cell origin [START_REF] Abrahams | Small cell carcinoma of the bladder: a contemporary clinicopathological study of 51 cases[END_REF][START_REF] Cheng | Molecular genetic evidence for a common clonal origin of urinary bladder small cell carcinoma and coexisting urothelial carcinoma[END_REF] . The risks factors most frequently associated with SCCUB are smoking (65-79% reported) [1, 7] , age (median: 66 years) and sex (male: female ratio up to 4:1) [START_REF] Cheng | Small cell carcinoma of the urinary bladder: A clinicopathologic analysis of 64 patients[END_REF][START_REF] Abrahams | Small cell carcinoma of the bladder: a contemporary clinicopathological study of 51 cases[END_REF] . Some retrospective studies also describe other risk factors associated with chronic inflammation such as stone disease, and recurrent urinary infection present before diagnosis [START_REF] Barrington | Tumors in bladder remnant after augmentation enterocystoplasty[END_REF][START_REF] Swanson | Small cell carcinoma of urinary bladder[END_REF] .

Clinical Presentation

The clinical presentation of SCCB is similar to other bladder cancers. The most common presenting symptom reported in various series is hematuria (68% -88%) [START_REF] Cheng | Small cell carcinoma of the urinary bladder: A clinicopathologic analysis of 64 patients[END_REF][START_REF] Choong | Small Cell Carcinoma of the Urinary Bladder The Mayo Clinic Experience[END_REF] .

Other reported symptoms include dysuria, urinary frequency, flank pain or acute renal failure due to uretheral obstruction, hematospermia or recurrent urinary tract infection hypercalcemia and hyperphosphatemia, although they are less frequently described than in small cell carcinoma of the lung 11,[START_REF] Trias | Small cell carcinoma of the urinary bladder. Presentation of 23 cases and review of 134 published cases[END_REF] .

Staging

Complete staging workup includes history, physical examination and imaging studies with chest, abdomen and pelvis computed tomography (CT) to rule out metastatic disease. Brain magnetic resonance imaging (MRI) or brain CT scan should also be performed. The use of 18F-Fluoro-deoxyglucose positron emission tomography (18F-FDG PET) has also been described as an aid to staging although it is not adequate for detecting brain metastases [START_REF] Podoloff | NCCN Task Force: Clinical Utility of PET in a variety of tumour types[END_REF] .

The staging system for SCCUB used in clinical practice is that advocated by the 2010 American Joint Committee on Cancer (AJCC) -TNM staging system for urothelial carcinoma [START_REF]Urinary bladder[END_REF] .

Table 1 summarizes histology and stage at clinical presentation of 2 major retrospective analyses.

Treatment

Limited disease

There is no standard treatment for SCCUB due to a lack of prospective randomized clinical trials to provide definitive recommendations. The suggested treatment approach is mainly based on retrospectives studies from single centers. The standard of care may include surgery, surgery plus chemotherapy either in the adjuvant or neoadjuvant setting and radiation therapy plus chemotherapy.

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

Surgery

Despite the absence of clinical evidence surgical resection has been considered for many years to be the main standard for the treatment of limited-stage SCCUB. Cystectomy alone is, however, characterized by a relatively poor prognosis. In a retrospective study from the MD Anderson Cancer Center (MDACC) 25 patients were treated with cystectomy only [START_REF] Siefker-Ratdke | Evidence supporting preoperative chemotherapy for small cell carcinoma of the bladder: a retrospective review of the M. D. Anderson cancer experience[END_REF] . The 5-year cancer-specific survival was 36%. Cheng et al. reported a case series of 64 patients; they observed no difference in survival between the cystectomy vs no cystectomy groups [START_REF] Cheng | Small cell carcinoma of the urinary bladder: A clinicopathologic analysis of 64 patients[END_REF] .

Another retrospective analysis by Choong et al. found that patients with localized disease may have long term survival after cystectomy alone with 41% (7 of 17) of patients being disease-free [START_REF] Choong | Small Cell Carcinoma of the Urinary Bladder The Mayo Clinic Experience[END_REF] .

Neoadjuvant chemotherapy

Neoadjuvant chemotherapy followed by local treatment, when disease response is documented, can be considered an optimal strategy for patients with limited SCCUB. The chemotherapy regimens typically used are platinum based as these have known activity against neuroendocrine or small cell tumours. The rationale behind this approach relies on the frequent observation of upstaging and rapid growth rates after initial surgery . When considering local treatment after chemotherapy there is insufficient data to recommend whether this should be cystectomy or radiation therapy.

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT  Neoadjuvant chemotherapy plus cystectomy

The potential efficacy of neoadjuvant chemotherapy was demonstrated in the MDACC phase II trial. Investigators included 18 patients with SCCUB staged ad cT2 or cT3b N0 disease. Patients underwent four cycles of chemotherapy, with a regimen including ifosfamide plus doxorubicin alternating with the combination of etoposide plus cisplatin [START_REF] Siefker-Radtke | Phase II clinical trial of neoadjuvant alternating doublet chemotherapy with ifosfamide/doxorubicine and etoposide/cisplatin in small-cell urothelial cancer[END_REF] .

The pathological downstaging to stage pT1N0M0 occurred in 14 (78%) out of 18 patients with a mean overall survival (OS) of 58 months. A retrospective analysis from MDACC , observed a 5-year disease specific survival rate of 78% in 21 patients treated with preoperative chemotherapy, compared to 36% in 25 patients treated with initial cystectomy (regardless of adjuvant chemotherapy) [START_REF] Siefker-Ratdke | Evidence supporting preoperative chemotherapy for small cell carcinoma of the bladder: a retrospective review of the M. D. Anderson cancer experience[END_REF] . In another retrospective analysis Walther noted that five of seven patients were alive and cancer free at 36 months mostly attributed to the use of preoperative chemotherapy [START_REF] Walther | Adjunctive adjuvant or neoadjuvant chemotherapy for locally advanced bladder cancer: a critical appraisal of the present status[END_REF] . All these data suggest that a bladder sparing approach with neoadjuvant systemic chemotherapy followed by radiotherapy should be considered as primary option in limited disease and should be further investigated in this disease.

 Neoadjuvant chemotherapy plus radiotherapy

Metastatic or recurrent disease

Patients with pure small cell metastatic carcinoma of the bladder are supposed to have a poor prognosis and worse survival compared to those with mixed subtypes or transitional cell carcinomas [START_REF] Siefker-Radtke | Phase II clinical trial of neoadjuvant alternating doublet chemotherapy with ifosfamide/doxorubicine and etoposide/cisplatin in small-cell urothelial cancer[END_REF] .

In the metastatic setting optimal chemotherapy should be platinum-based mostly combining cisplatin with etoposide [START_REF] Cheng | Small cell carcinoma of the urinary bladder: A clinicopathologic analysis of 64 patients[END_REF][START_REF] Trias | Small cell carcinoma of the urinary bladder. Presentation of 23 cases and review of 134 published cases[END_REF][START_REF] Ismaili | Small cell cancer of the bladder: pathology, diagnosis, treatment and prognosis[END_REF] .

A recently published phase II trial from Siefker-Radtke et al. treated 30 patients, of whom 12 had metastatic disease, with alternating doublet chemotherapy using ifosfamide/doxorubicin and etoposide/cisplatin [START_REF] Siefker-Radtke | Phase II clinical trial of neoadjuvant alternating doublet chemotherapy with ifosfamide/doxorubicine and etoposide/cisplatin in small-cell urothelial cancer[END_REF] . In the metastatic arm the response rate was 100%, with 72% complete (eight patients) and 28% partial responses; one patient was excluded from the analysis due to refusal for continuing treatment after the first cycle.

Despite the response rate, the median OS was only 13.3 months [START_REF] Siefker-Radtke | Phase II clinical trial of neoadjuvant alternating doublet chemotherapy with ifosfamide/doxorubicine and etoposide/cisplatin in small-cell urothelial cancer[END_REF] .

Another prospective study, from Bex et al., included 25 patients of whom 8 were considered to have extensive disease 19 . They received cisplatin and etoposide. The median OS was 5 months.

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

Although SCCUB, like pulmonary small cell carcinoma, is considered highly chemosensitive the progression free survival obtained is still poor and relapse is observed in the vast majority of cases.

Treatment for relapsed disease after first line chemotherapy is rarely described. A report of three cases using weekly vinorelbine as second line therapy showed one complete response and two partial responses with no safety concerns [START_REF] June | Significant activity of single agent vinorelbine against small-cell cancer of the bladder as second line chemotherapy: A case series and review of the literature[END_REF] .

Patients with Brain Metastasis or Relapse

In metastatic SCCUB the presence of brain metastasis is less frequently reported, even though described in16% of patients [START_REF] Choong | Small Cell Carcinoma of the Urinary Bladder The Mayo Clinic Experience[END_REF][START_REF] Ismaili | Outcome of recurrent and metastatic small cell carcinoma of the bladder[END_REF][START_REF] Mukesh | Small cell carcinoma of the urinary bladder: a 15-year retrospective review of treatment and survival in the Anglian Cancer Network[END_REF] .

Two studies performed prophylactic cranial irradiation in few selected patients [START_REF] Lohrisch | Small cell carcinoma of the bladder: long term outcome with integrated chemoradiation[END_REF][START_REF] Mackey | Genitourinary small cell carcinoma: determination of clinical and therapeutic factors associated with survival[END_REF] , but prospective and randomized studies confirming the benefit from such an approach are still awaited.

Prognostic Factors

Patients with SCCUB are most likely to have advanced or metastatic disease at first presentation. The proportion of patients with advanced disease at diagnosis ranges from 32% to 56% in most retrospective analyses [START_REF] Trias | Small cell carcinoma of the urinary bladder. Presentation of 23 cases and review of 134 published cases[END_REF]19,[START_REF] Mukesh | Small cell carcinoma of the urinary bladder: a 15-year retrospective review of treatment and survival in the Anglian Cancer Network[END_REF] . Early metastatic spread is usually observed in small cell urinary carcinoma explaining the high relapse rate and the short disease free survival [START_REF] Lohrisch | Small cell carcinoma of the bladder: long term outcome with integrated chemoradiation[END_REF][START_REF] Holman | Primary small cell carcinoma of the bladder: a report of 25 cases[END_REF]19,[START_REF] Mackey | Genitourinary small cell carcinoma: determination of clinical and therapeutic factors associated with survival[END_REF][START_REF] Ismaili | Small cell cancer of the bladder: The Leon-Berard cancer centre experience[END_REF] .

The main prognostic factorobserved is stage with a median OS from 7,3 to 59,2 months for limited disease and from 5 to 8,1 months for advanced disease [START_REF] Holman | Primary small cell carcinoma of the bladder: a report of 25 cases[END_REF][START_REF] Siefker-Radtke | Phase II clinical trial of neoadjuvant alternating doublet chemotherapy with ifosfamide/doxorubicine and etoposide/cisplatin in small-cell urothelial cancer[END_REF]19,27 . Other

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

possible prognostic factors associated with SCCUB include performance status prior to therapy 19 and tumour histology as pure SCCUB histology has been described as having a lower survival than mixed tumour subtypes [START_REF] Siefker-Radtke | Phase II clinical trial of neoadjuvant alternating doublet chemotherapy with ifosfamide/doxorubicine and etoposide/cisplatin in small-cell urothelial cancer[END_REF] .

Conclusions

Small cell carcinoma of the urinary bladder is a rare condition. Tumours are chemoresponsive although short times to progression are usually described. The management of limited disease remains a challenge and the integration of a neoadjuvant chemotherapy(with a cisplatin based regimen) followed by a local approach (surgery or radiotherapy) should be considered optimal. The choice between cystectomy or radiotherapy as local treatment should be decided on an individual basis. Cystectomy alone or cystectomy followed by adjuvant chemotherapy resulted in inferior long-term disease free-survival and OS. Prospective, multi-institutional, randomized studies are required to assess the role of neoadjuvant chemotherapy followed either by surgery or radiotherapy.

The rarity of this disease is a major limitation to patient accrual in such hypothetical trials.

In metastatic disease prognosis remains poor but with a trend of benefit from platinumbased chemotherapy. Another dilemma that remains to be answered is whether mixed small cell carcinoma should be managed with the same therapeutic approach as pure small cell carcinoma since these histological subtypes have been reported to have different prognoses.

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

In conclusion, a multidisciplinary approach involving urologist, radiation therapist and medical oncologist, should drive clinical decision making in patients with small cell carcinoma of the urinary bladder. Asmis TR, Reaume NM, Dahrouge S, Malone S. Genitourinary small cell carcinoma: a retrospective review of treatment and survival patterns at The Ottawa Hospital Regional Cancer Center. BJU Int 2006; 97(4):711-5.

Table legends

Table1: Histopathological characteristics of two major retrospective studies.

[ 1 ]

 1 . Partanen et al. 11 reported paraneoplastic syndromes with ectopic adrenocorticotropic hormone (ACTH) production leading to Cushing´s syndrome, M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

  Radiation therapy (RT) has been used as an alternative to cystectomy in many centers. The largest retrospective study to report on chemo-radiotherapy in SCCUB reviewed 17 patients with limited disease treated at The Netherlands Cancer Institute between 1993-2007 19 . All patients received platinum based chemotherapy prior to local radiotherapy with 56-70 Gy. A complete response was achieved in 15 patients (88%) and the median OS was 32,5 months and (?). Four patients had local recurrence with 3 requiring a cystectomy. In another retrospective analysis 10 patients were treated with multidrug chemotherapy combined with local irradiation. The overall survival rate was 70% at 2-year and 44% at 5-year 3 . Bastus et al. treated 5 consecutive patients with M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT systemic chemotherapy followed by external radiotherapy in those achieving a complete response 20 . Patients were alive and free of disease 60, 48 and 27 months after diagnosis.

Table 2 :

 2 Multidisciplinary approach and clinical outcome according to stage in prospective and retrospective studies.

	M A N U S C R I P T

Table 2 :

 2 Multidisciplinary approach and clinical outcome according to stage in prospective and retrospective studies.

	Author/Year	Type of Study	Number	Treatment	Results
			of
			Patients
		Localized
	Siefker-Ratdke	Prospective	18	Ifosfamide+Doxorubicin and	OS: 58 months
	2009			Etoposide+Cisplatin	RR*:78%
				alternated plus cystectomy
	Siefker-Ratdke	retrospective	46	Upfront cystectomy VS	DFS 5 years: 36% vs
	2004			Preoperative chemotherapy	78%
	Bex 2009	retrospective	17	TUR + Platinum -based	OS: 32.5 months
				chemotherapy + local RTx(56-	RR: 100%
				70 Gy)
	Lorisch 1999	retrospective	10	Cisplatin+Etoposide followed	OS 2-y: 70%
				by local RTx	OS 5-y: 44%
					DFS 2y/ 5y: 70%
		Metastatic
	Siefker-Ratdke	prospective	12	Ifosfamide+Doxorubicin and	Response: 72% CR,
	2009			Etoposide+Cisplatin	28% PR;
				alternated	OS: 13.3 months
	Bex 2005	prospective	8	Cisplatin+Etoposide (3	OS: 6, 16 and 17
				patients)	months
	Mukesh 2009	retrospective	9	-Cyclophosphamide+	Response:
				doxorubicin+vincristine,	50% CR,25% PR ;
				-carboplatin+etoposide (3),	OS> 33 months vs 3
				-platinum-based	months with no
				chemotherapy.	chemo
	Asmis 2006	retrospective	4	Platinum+etoposide;	Response not
					stated;
					OS: from 3.7 to 20
					months
	Choong 2005	retrospective	5	Platinum+etoposide	100% response for
					carbo+etoposide

* Response rate -considered pathologic downstaging to < pT1N0M0.