E Balmes

I Bucher

Sdtools

Accounting for rotation in a multi-stage cyclo-symmetric model -a case study

Rotating structures typically have stages with cyclo-symmetric geometries that are periodic by rotation. The spatial Fourier transform of fields at periodic positions leads to multiple uncoupled low cost problems for each Fourier harmonic. Real rotors are assemblies of stages that have different number of sectors. This motivated the extension of cyclo-symmetry to multistage assemblies. The resulting methodology gives access to low cost predictions of full rotor dynamics. The present paper summarizes the theory and shows that this new ability is useful to analyze gyroscopic and centrifugal softening effects in detail for configurations where beam mass model are traditionally used.

Introduction

Rotating structures typically have stages with cyclo-symmetric geometries. Classically cyclic-symmetry [START_REF] Thomas | Dynamics of rotationally periodic structures[END_REF][START_REF] Wildheim | Excitation of rotationally circumferentially periodic structures[END_REF] considers single stage vibration or assumes the whole rotor to have a single number of sectors. But real rotors are assemblies of stages that have different angular periodicity. This has motivated the extension of cyclo-symmetry to multistage assemblies for mono-harmonic solutions [START_REF] Laxalde | Dynamics of Multistage Bladed Disks Systems[END_REF] and full system models [START_REF] Sternch Üss | Reduction of Multistage disk models : application to an industrial rotor[END_REF][START_REF] Sternch Üss | Multi-level parametric reduced models of rotating bladed disk assemblies[END_REF].

After an introduction of the cyclic symmetry and its extension to multi-stage configurations, the paper summarizes the reduction strategy [START_REF] Sternch Üss | On the reduction of quasi-cyclic disks with variable rotation speeds[END_REF] used to approximate responses at variable rotation speeds using a limited number of full model evaluations. The proposed methodology is then used on the case of a rotor consisting of a heavy impeller with 8 sectors and a turbine with 40 blades. The disks are modeled using detailed volume element meshes, which are typically needed for accurate stress predictions. The full 3D model with 5 million DOF is too large for convenient numerical predictions of global dynamics. The multi-stage cyclic symmetry implemented in the SDT/Rotor toolbox [START_REF] Balmes | Rotor module for SDT, User Manual[END_REF] running in the Matlab environment is used to correlate free-free modes of the rotor and predict Campbell diagrams, where the difference between beam and 3D models are clearly apparent.

Theory

Section 2.1 first introduces classical cyclic symmetry through spatial Fourier transform basics. Following the presentation in [START_REF] Sternch Üss | Multi-level parametric reduced models of rotating bladed disk assemblies[END_REF], case of multi-stage models is treated trough separate transforms of each stage and coupling by inter-stage stiffness. The second aspect of the paper is the use of these models to predict full rotor dynamics. Equations associated with classical effects, gyroscopic coupling, centrifugal softening, ... are given in section 2.3, and the use of fixed basis reductions to predict rotation dependent responses is finally presented.

Cyclic symmetry

Points of a structure with a cyclo-symmetric geometry can be grouped in series of NS (number of sectors) points x s whose location is given by a simple rotation of a nominal sector. Taking any field v, one defines a vector {v} N S that contains all the terms in the spatial series (v(x s )) and a transformed vector { v} N S that contains all the Fourier harmonics

{v} ⊤ = v(x 0 ) • • • v(x s ) • • • v(x Ns-1 ) , { v} ⊤ = v 0 (x 0 ) • • • ℜ v δ (x 0 ) ℑ v δ (x 0 ) • • • v Ns 2 (x 0 ) . ( 1
)
these two vectors are related by the unitary discrete Fourier transform

{ v} = E -1 {v}, (2)
and its inverse

{v} = [E] { v}. (3)
where the Fourier transform is here specified as matrix

[E] [E] = 1 √ N s          1 • • • √ 2 0 • • • 1 . . . . . . . . . . . . 1 • • • √ 2 cos (s δ α) - √ 2 sin (s δ α) • • • (-1) s . . . . . . . . . . . . 1 • • • √ 2 cos ((N s -1) δ α) - √ 2 sin ((N s -1) δ α) • • • (-1) Ns-1          (4)
When considering a full 3D model, matching DOFs in each sector can be arranged as vectors {v} and transformed separately. This transformation generates two harmonic DOF vectors { q} and { f } that are the transforms of full DOF vectors {q} and {f } respectively. The sector number in spatial space is thus replace by harmonic number in wave-number space. The Fourier harmonic index δ happens to correspond to the number of times a given field changes sign over a half circle, so that the classical denomination for δ is diameter number.

Assuming that the properties of each sector are identical (the case of mistuned disks is discussed in [START_REF] Sternch Üss | Multi-level parametric reduced models of rotating bladed disk assemblies[END_REF]), one can show that the transform of the system equations [Z]{q} = {f } takes the block form

           Z 0 • • • 0 0 • • • 0 . . . . . . . . . . . . . . . 0 • • • Z 0 0 • • • 0 0 • • • 0 Z 0 • • • 0 . . . . . . . . . . . . . . . 0 • • • 0 0 • • • Z 0                                   q 0 . . . ℜ ( q δ ) ℑ ( q δ ) . . . q Ns 2                        =                        f 0 . . . ℜ( f δ ) ℑ( f δ ) . . . f Ns 2                        . ( 5
)
The block form is valid for purely disjoint sectors, in reality sectors are connected by edges and the motion on the left edge of one sector is equal to the motion on the right edge of the previous sector. The inter-sector continuity constraint does not couple Fourier harmonics so that the full model ( 5) can be decomposed in a series of independent problems associated with each harmonic (classically called diameter) :

• for δ = 0, { q 0 } is real and satisfies

Z 0 { q 0 } = { f 0 }, [c l -c r ] { q 0 } = {0}. ( 6
)
where c l , c r are observation equations relating the motion on edges to the full set of DOFs in the sector.

For 3D fields such as displacement, the components are projected on a basis local to the edge with a rotation angle corresponding to the number of sectors.

• if δ = 0 and δ = N s /2, { q δ } is complex and satisfies

Z 0 0 0 Z 0 ℜ ( q δ ) ℑ ( q δ ) = ℜ( f δ ) ℑ( f δ ) , c l -c r cos(δ α) c r sin(δ α) -c r sin(δ α) c l -c r cos(δ α) ℜ ( q δ ) ℑ ( q δ ) = 0 0 . (7)
• if δ = N s /2, { q Ns 2
} is real and satisfies

Z 0 q Ns 2 = { f Ns 2 }, [c l + c r ] q Ns 2 = {0}. (8)
When all the individual responses have been computed, the full response is obtained by Fourier synthesis thanks to (3).

Multi-stage cyclic symmetry

For multi-stage cyclic symmetry, the stages each have their own spatial periodicity. Assuming that stages are disjoint, the Fourier transform can be applied to each separately, leading to two problems of form ( 5).

The inter-stage coupling is achieved by a dynamic stiffness Z R δ that is meshed using physical elements shown in red in figure 1. The requirement [START_REF] Sternch Üss | Multi-level parametric reduced models of rotating bladed disk assemblies[END_REF] to use standard elements that are not too distorted, while leading to well posed problems, can be difficult if the disk meshes have very different refinements (see figure 4). It is thus proposed here to achieve coupling by computing the energy of a penalized representation of bilateral contact between surfaces of the two disks. As argued in Refs [START_REF] Laxalde | Dynamics of Multistage Bladed Disks Systems[END_REF][START_REF] Sternch Üss | Multi-level parametric reduced models of rotating bladed disk assemblies[END_REF], one can, for low diameter numbers, assume that shaft modes a composed of a single Fourier harmonic. In cases with complex Fourier harmonics

(δ = 0 [lcm(N 1 s , N 2 s )] and δ = 0 [N 1 s /2] and δ = 0 [N 2 s /2]
), the solutions satisfy system equations for the real and imaginary components of the response of each stage, coupled by the inter-stage stiffness Z R δ and the inter-sector continuity conditions verified independently by each stage

          Z 1,0 0 0 0 0 Z 1,0 0 0 0 0 Z 2,0 0 0 0 0 Z 2,0      + Z R δ               ℜ( q 1 δ ) ℑ( q 1 δ ) ℜ( q 2 δ ) ℑ( q 2 δ )          =          ℜ( f 1 δ ) ℑ( f 1 δ ) ℜ( f 2 δ ) ℑ( f 2 δ )          ,      c 1 l -cos(δ α 1 ) c 1 r sin(δ α 1 ) c 1 r 0 0 -sin(δ α 1 ) c 1 r c 1 l -cos(δ α 1 ) c 1 r 0 0 0 0 c 2 l -cos(δ α 2 ) c 2 r sin(δ α 2 ) c 2 r 0 0 -sin(δ α 2 ) c 2 r c 2 l -cos(δ α 2 ) c 2 r               ℜ( q 1 δ ) ℑ( q 1 δ ) ℜ( q 2 δ ) ℑ( q 2 δ )          = {0}. (9)
Similar equations are derived for cases with real harmonics of one or multiple stages.

The use of mono-harmonic modes to build a shaft model with sector having distinct periodicity will be illustrated here. The key extension in Ref [START_REF] Sternch Üss | Multi-level parametric reduced models of rotating bladed disk assemblies[END_REF] is the use of a few mono-harmonic modes to build reduced sector models that allow predictions of all the modes in the system.

When compared to full 3D simulations, cyclic symmetry uses the assumption of mono-harmonic responses to go from one full 3D problem to multiple one sector solutions. Solving multiple small problems is typically significantly cheaper than addressing the full system predictions. Post processing however requires the use of the Fourier transform (3), which can be something of a difficulty when the full 3D model contains millions of nodes.

In multi-stage analysis, harmonic decoupling is only approximate. In particular, aliasing occurs for diameters above the lowest half spectrum of a given stage. For the application shown here, one however focuses on 1 diameter modes associated with shaft bending and assuming mono-harmonic responses is very accurate.

Accounting for rotation effects

Since the geometry is not invariant by rotation, analysis of the rotating system must be performed in the body fixed frame. Matrices associated with motion in a non Galilean frame are well known (see [START_REF] Desceliers | Dynamique non linéaire en déplacements finis des structures tridimensionnelles viscoélastiques en rotation[END_REF][START_REF] Géradin | Mechanical Vibrations. Theory and Application to Structural Dynamics[END_REF] for example) and given by

• [M ] = S 0 ρ 0 [N xyz ] ⊤ [I][N xyz
] dS 0 the mass in the rotating frame

• [D g ] = S 0 2ρ 0 [N xyz ] ⊤ [Ω][N xyz ] dS 0 the gyroscopic coupling • [K c ] = S 0 ρ 0 [N xyz ] ⊤ Ω 2 [N xyz ] dS 0 the spin softening • [K a ] = S 0 ρ 0 [N xyz ] ⊤ ∂[Ω] ∂t [N xyz ] dS 0 the centrifugal acceleration • f c = S 0 ρ 0 [N xyz ] ⊤ Ω 2 {p} dS 0 the centrifugal load • f g = S 0 ρ 0 [N xyz ] ⊤ ∂[Ω]
∂t {p} dS 0 the Coriolis load Coupling with the elastic K e and geometric stiffening K geom matrices, obtains system equations

[M ]{q} + ([C + D g (Ω)]){ q} + K e + K geom + K a ( Ω) + K c (Ω) {q} = f ext -f c (Ω) -f g ( Ω) (10)
In the applications considered in this study, one will use a fixed axis of rotation Ω = ω(t){e z }∧. The matrices and loads are thus proportional to the scalars ω, ω 2 and ω and one can use

[D g (ω)] = ω[D g (1)]
which results in significant computational cost savings since the matrix only needs to be computed for a single velocity. One proceeds similarly for the other matrices and loads.

Computations of interest are the Campbell diagrams where one seeks, as a function of rotation speed, the modes {ψ j } and poles λ j verifying the eigenvalue problem

[M ]λ 2 j + ([C + D g (Ω)])λ j + [K e + K geom + K c (Ω)] {ψ j } = 0 (11)
When considering undamped rotors on isotropic bearings, the eigenvalues are real but modeshapes complex. Specific transformation [START_REF] Wang | New eigensolutions and modal analysis for gyroscopic rotor systems, part 1 undamped systems[END_REF] can be applied to obtain real modes and combining such transformation with cyclic symmetry seems possible by accounting for the phase shift of the mode pair found for non zero diameter modes. The application is however not trivial and the interest limited since the procedure does not apply to damped rotors. The reduction procedure proposed in section 2.4 also strongly diminishes the cost of using complex modes, so that they are used here.

To apply cyclic symmetry, the spatial Fourier transform (2) must thus be applied to the real and imaginary parts of the complex mode shape separately. The numerical implementation must thus choose between using complex numbers to represent the real and imaginary part of the spatial transform or those of the complex field. But other than this software difficulty the process is straight-forward.

Many excitations, imbalance, torque variation, aerodynamic pressure field, are at multiples of the rotation speed. One is thus interested in computing critical speeds where one neglects damping (C = 0) and assumes the modal frequency to be equal to a multiple of the rotation speed (in rev/s), that is λ j = niΩ, thus leading to

-Ω 2 j [M ] + niΩ 2 [D g (1)] + [K e + K geom ] + n 2 Ω 2 K c (1) {ψ j } = 0 (12)

Model reduction for variable speed computations

While cyclic symmetry makes full rotor predictions accessible, computing the response of a range of speeds can however be quite costly. A solution, first proposed in [START_REF] Sternch Üss | On the reduction of quasi-cyclic disks with variable rotation speeds[END_REF], is to build a fixed Ritz basis to approximate solutions at variable speeds.

One thus builds a collection of vectors by assembling multiple modal bases

T = [φ 1:N M (Ω 1 ) ... φ 1:N M (Ω N w )] (13)
with N M modes at N W rotation speeds. The vectors thus generated are not independent, so that a numerical procedure to generate a basis T of the subspace generated by T is used. Classical procedures [START_REF] Sternch Üss | Multi-level parametric reduced models of rotating bladed disk assemblies[END_REF] are Gramm-Schmidt, singular value decomposition and the orthogonal maximum sequence [START_REF] Balmes | Orthogonal Maximum Sequence Sensor Placements Algorithms for modal tests, expansion and visibility[END_REF]. Once a basis generated each of the matrices and loads in ( 10)-( 12) can be reduced using T T M T , T T f , ... While using a fixed basis built using just a few velocities is an approximation, the results are quite sufficient to estimate the response over a large range of rotation speeds. The main development for this paper was to implement the combination of cyclic symmetry and this multi-speed modal basis approach for models including gyroscopic coupling and centrifugal softening.

A simple reference case

As a sample case, one will consider a volume model of the rotor studied in Ref. [START_REF] Lalanne | Rotordynamics prediction in Engineering[END_REF]. As shown in figure 2, the first bending mode has a strong frequency split due to gyroscopic effects and a critical speed around 3159 RPM. At this frequency centrifugal softening makes the stiffness contribution singular, that is K e + Ω 2 (K c (1) + iD g (1)) {ψ} = 0 has a solution and in body fixed coordinates the whirl direction changes. As this simple 3D model only contains 2367 DOF, one could verify that solutions computed using cyclic symmetry gave the same results as the full 3D with relative errors on frequencies below 1e-6. The advantages of cyclic symmetry are faster solutions and better numerical conditioning.

The next verification is a check that gyroscopic and centrifugal softening effects can be well predicted in the basis of nominal elastic modes. As shown in figure 3, the error associated with a computation based on the first bending mode only is below 6.5%. For the first disk bending, which has two diameters and a low split due to gyroscopic effects, the relative error on frequency associated with a single mode computation is below 1e-4.

A realistic application 4.1 Model and free/modes

The rotor considered in this section consists of a slender shaft, a turbine and a large and heavy impeller. The operating speed is 50,000 RPM. The ANSYS study begun by running an automatic meshing process using ANSYS workbench. The result is shown in figure 4. The fine mesh generated by ANSYS contains approximately 5 million DOF. For such a complex body, only non rotating analysis was performed.

A model with two base sectors was imported in to SDT. The impeller (on the left in dark red) is meshed in 8 sectors using 41,736 elements and 69,440 nodes. The turbine is composed of a disk and 45 blade. The turbine and shaft are modeled using 45 sectors and use 18,257 second order volume elements and 41,213 nodes. Accounting for nodes common at interfaces, the full 3D model would have 2 million nodes (6e6 DOF) and 1,155,453 elements. This mesh, with second order elements, represents model that is very expensive to evaluate.

Accounting for cyclic symmetry, one only has 307428 DOF for zero diameter modes, and twice that number for higher diameters that number doubles. Computation of 30 modes takes around 300 s with SDT/Rotor [START_REF] Balmes | Rotor module for SDT, User Manual[END_REF], which is a major speedup compared to the full model computation.

The zoom on the shaft / impeller area shows that significantly different mesh densities were used on the two components. Meshing the connection area, as proposed in [START_REF] Sternch Üss | Multi-level parametric reduced models of rotating bladed disk assemblies[END_REF], is thus impractical and a penalized contact strategy was implemented. very clear and this area has a significant impact on frequencies. A refined modal test would however be needed to confirm the exact modifications needed in particular to predict the global bending at 1866 Hz. The correlation was thus deemed acceptable as is. Inspecting the results, it was observed that the discs show little distortion in the operating range of 0-50,000 RPM. Replacing blades by disc-like elements thus seems appropriate. A simple Matlab-based Timoshenkobeam rotor-dynamics code that incorporates effects of rotation [START_REF] Genta | Dynamics of Rotating Systems[END_REF] was used for comparison. The first two elastic modes are shown in figure 7. Improved frequency match was obtained by slightly modifying elements and material constants around regions with the largest bending stress.

Predictions of rotating response

In order to examine the behavior of the system under realistic conditions, simple self-aligning ball-bearings were added to the beam model to realize a simply supported rotating shaft. Modes of interest are now the first global bending with nodes near the bearings, a turbine bending mode near 980 Hz and an impeller bending at 1924 Hz. Visualizing modeshapes at critical speeds is an important design tool. The first critical speed corresponds to global bending of the shaft. At the second critical speed (circle number 5), both the beam and 3D models (right) show a localization of the mode on the turbine. The shaft area near the turbine was however identified as important for the free-free mode correlation so that further validation of the shaft model is important here.

Conclusion

The use of cyclic symmetry and fixed basis parametric models makes the prediction of responses as a function of rotation speed accessible for full 3D models.

The classical approach based on beam and rigid disk models is a rather elaborate process that requires many adjustments and significant expertise to properly define disks, in particular when representing heavy impellers such as the one considered in the application. This strategy is however well mastered in industry and the proposed methods are well suited to replace beam models by volume elements, which have the clear advantage of allowing studies of disk flexibility. For a volume model with the geometry of the beam model shown earlier, computation of a Campbell diagram runs in a few CPU seconds, so that cyclic symmetry models are directly competitive with beam models.

Multi-stage cyclic symmetry gives access to the actual shaft geometry. In the present application, complete analysis ran in less than 10 mn. While this may be too slow for parametric optimization procedures, the accuracy associated with proper accounting of the local flexibilities is definitely an advantage in validation phases.

Figure 1 :

 1 Figure 1: Sample mesh of a 2 disk application.

Figure 2 :

 2 Figure 2: Left: Campbell diagram in body fixed coordinates. Triangles pointing right indicate forward whirl. Center: backward whirl associated with first bending mode. Right : backward whirl mode associated with 2 diameter bending of the disk.

Figure 3 :

 3 Figure 3: Campbell diagrams. Left: computed with 1 mode or exactly. Right: first disk bending mode.

Figure 4 :

 4 Figure 4: Top: Sample rotor meshed in ANSYS. Top right : mesh section at the impeller shaft interface. Bottom: sector model imported into SDT.

Figure 5 :

 5 Figure 5: Free/free test of the rotor modes of the rotor.

Figure 6 :

 6 Figure 6: Free/free modes of the rotor. Multistage cyclic symmetry in SDT. Color is x motion for torsion and z otherwise.

Figure 7 :

 7 Figure 7: Free/free modes of the non rotating beam model. 461 and 1270 Hz

Figure 8 :

 8 Figure 8: Campbell diagram predicted with beam (left) and 3D model (right).

Figure 9 Figure 9 :

 99 Figure9compares the Campbell diagrams obtained with the beam and the 3D cyclic symmetry models. For 3D prediction, frequencies are shifted to account for the fact that predictions are performed in the body fixed frame.The effect of rotation is clearly visible. A number of critical frequencies are indicated as circles in the diagram of the beam model. The critical frequencies associated with forward whirl modes (indicated as solid circles) are typically excited by imbalance[START_REF] Lalanne | Rotordynamics prediction in Engineering[END_REF] and thus important to analyze. The same two critical speeds

Figure 10 :

 10 Figure 10: Forward critical modes. Left : beam mode 2 (left), volume modes 1 (center) 2 (right)

Table 1 :

 1 Free/free modes of the rotor.

	Name	ANSYS Full SDT cyclic Beam model Measured
		Hz	Hz	Hz	Hz,%
	First bending	528.53	514.3	462	460 1.6%
	Torsion	588.2	567.3
	Second bending	1724	1586	1270	1405 2.6%
	Global bending	3452	2952	2876	1866 1.6%


J.-P Bianchi
email: bianchi@sdtools.combalmes@sdtools.comvermot@sdtools.com

E Balmes

G Vermot Des Roches

A Bobillot
email: adrien.bobillot@sncf.fr

Using modal damping for full model transient analysis. Application to pantograph/catenary vibration

Experimentally, modal damping is known to allow a relatively accurate representation of damping for wide frequency ranges. For transient simulation of full finite element models, viscous damping is very often the only formulation that is associated with an acceptable computation time. In the absence of a proper material damping model, it is common to assume Rayleigh damping, where the viscous matrix is a linear combination of the mass and stiffness, or piece-wise Rayleigh damping. This representation very often results in modal damping ratios that do not correspond to the physical reality that can be tested. One thus introduces an implicit representation of the viscous matrix that allows transient time simulation with minor time penalty, while using a much more appropriate damping representation. Modal amplitudes are also considered to postprocess time simulations and analyze damping levels. When studying vibrations induced by the passage of pantographs under a catenary, the high modal density of catenaries and the load moving over a large part of the model is a strong motivation to use full model transient analysis. Modal damping is shown to be a practical tool to analyze the properties of this complex system.

Introduction

Damping is the general term used to talk about physical dissipation mechanisms: material damping, friction, fluid/structure interaction, . . . Dissipation in materials is linked to viscoelasticity or other non linear constitutive relations. Linear viscoelasticity [START_REF] Bert | Material damping: An introductory review of mathematical models, measures, and experimental techniques[END_REF][START_REF] Nashif | Vibration Damping[END_REF][START_REF] Balmes | Viscoelastic vibration toolbox, User Manual[END_REF] is classically represented through complex moduli, which do not have time domain equivalents, or internal states, which cannot easily be incorporated in standard FEM simulations. Friction typically occurs in joints that are either not modeled or not with sufficient detail to allow prediction.

As a result of these constraints, the physical mechanisms of damping are only modeled when such simulation is a primary objective. In other cases, one only seeks to reproduce system level behavior. In the frequency domain, one classically specifies modal damping ratio. While in the time domain Rayleigh damping is used. The objective of this paper is to demonstrate that modal damping can efficiently and usefully be used in time domain transient analysis.

OSCAR (Outil de Simulation du CAptage pour la Reconnaissance de défauts) is a pantograph-catenary dynamics software developed by SNCF and SDTools and certified with respect to the EN50318 standard. Its first objective is to provide signatures of defects and singularities (road bridges, turnouts, overlaps) to help analyzing test data by extracting the relevant information. The second is to help designers to optimize existing or future OCL (Overhead Contact Lines). OSCAR uses a fully 3D FEM description of OCL geometries, including stagger, tensioned beam elements for the contact wire(s) (CW) and messenger wire(s) (MW), bar elements for steady arms and unilateral droppers. The pantograph can be modeled using both lumped mass or multi-body models with unilateral contact law.

Computing the static shape under tension and gravity loads is a first major difficulty. Starting from design data, that is the lengths of each component and tensions in CWs and MWs, tensions are applied first and non linear iterations are then used to converge on a static state. At this stage one can access to catenary elasticity, tension in each pretensioned element, static sag of the CW, modes, etc.

Pantograph description includes unilateral contact with the catenary, on one or multiple points when there are several CWs, and possibly non linearities such as end stops, friction, active control ... Time integration is performed using a classical implicit non linear Newmark scheme [START_REF] Géradin | Mechanical Vibrations. Theory and Application to Structural Dynamics[END_REF].

OSCAR provides the possibility of simulating the pantograph passage through a section overlap, which is a very attractive feature for assessing the different design possibilities. In this particular zone, the pantograph contacts one CW and then another, with a short period where it contacts both. One of the main issues for the modeling is then to have contact on each wire, taking into account the wire staggers.

Section 2 details the equations used for classical Rayleigh, modal and the proposed full model modal damping. Modal amplitudes are then defined to be used as post-processing tools in transient analysis.

In section 3, the proposed models are used to analyze the influence of damping on catenary vibrations using a simple pretensioned wire, an impact test on a full model and a standard pantograph passage.

Theory

Linear damping models

Viscous damping is a force that is proportional to velocity and leads to dynamic equations of the form

M { q (t)} + C{ q(t)} + K{q(t)} = {f (t)} (1)
where the viscous damping matrix C is generally a symmetric semi-positive definite matrix thus ensuring a passive system.

Viscous damping is not physically representative of material damping, which is better represented by hysteretic models where the dissipation force is proportional to displacement, or friction, which tends to only depend on the direction of velocity. The use of viscous damping in time domain simulations should thus be considered as an equivalent approach. Viscous damping is however the only dissipation mechanism that is linear and thus low cost, so that the motivation to use it in models is very strong.

The most common viscous damping description is uniform Rayleigh model, where a linear combination of mass and stiffness matrices

C = α[M ] + β[K] is used.
It is typically used when one has no real knowledge of damping: there are only two coefficients (or even one as α = 0 is often considered) to adjust to match some response. More elaborate approaches use a piece-wise Rayleigh damping where distinct coefficients are used for various parts of the model

[C] = e α e [M e ] + β e [K e ]. (2)
The weighting α e and β e account for the fact that certain components are thought to be more dissipative than others and have dissipation proportional to the time derivative of displacement or deformation respectively.

Illustrations will be given in 3.2. A major limitation of this approach is that damping often occurs in junctions that are modeled as perfect and are thus not associated with matrices.

Real modeshapes {φ j } and pulsations ω j are solutions of the eigenvalue problem

(-ω 2 j [M ] + [K]){φ j } = {0}. (3)
When taking all modes, one has the mass orthogonality conditions and can use the fact that a matrix and its inverse can be permuted to obtain the following relations

([Φ] T [M ])[Φ] = [Φ] T ([M ][Φ]) = I = [Φ]([Φ] T [M ]) = ([M ][Φ])[Φ] T (4)
One also has the stiffness orthogonality condition, which for mass normalized modes is given by

[Φ] T [K][Φ] = \ ω 2 j \ . (5)
Modal damping [START_REF] Balmes | New results on the identification of normal modes from experimental complex modes[END_REF][START_REF] Caughey | Classical normal modes in damped linear dynamic systems[END_REF][START_REF] Hasselman | Modal coupling in lightly damped structures[END_REF] corresponds to the assumption that the viscous damping matrix is diagonal in the real mode basis, that is to say that

[Φ] T [C][Φ] = \ 2ω j ζ j \ . ( 6
)
where ζ j is the modal damping ratio of mode j. Using orthonormalization conditions (4)-( 5), one can easily show that for uniform Rayleigh damping, one has

ζ j = α 2 1 ω j + β 2 ω j . (7)
Damping proportional to mass cannot be applied to cases with rigid body modes, since they would have infinite damping, and is limited to damp the first few modes since the damping decreases with frequency. Damping proportional to stiffness increases linearly with frequency which rarely corresponds to the physical behavior, where damping ratios are mostly constant over large frequency range.

If modal damping is a valid assumption, the Modal Strain Energy method [START_REF] Rogers | The modal strain energy finite element method and its application to damped laminated beams[END_REF] can be used and leads for modeshape {φ j } to

2ζ j ω j = e α e φ T j [M e ]φ j + β e φ T j [K e ]φ j (8)
In reality modes are really complex, so that the damping ratio is not associated with the real modeshape as assumed by the MSE method. As shown in [START_REF] Balmes | New results on the identification of normal modes from experimental complex modes[END_REF][START_REF] Hasselman | Modal coupling in lightly damped structures[END_REF] and illustrated in figure 10, modal damping is however always a good approximation for well separated modes and often a close approximation otherwise.

An important consequence of this equation is that modes where most of the energy is concentrated in element group e will have a damping ratio given by αe 2 1 ω j + βe 2 ω j . It also follows, as will be illustrated in figure 7, that piece-wise Rayleigh damping generates damping ratios that are limited by the various weighting coefficients.

In the modal basis, viscous damping is easily expressed as a diagonal matrix [START_REF] Caughey | Classical normal modes in damped linear dynamic systems[END_REF]. Going back to the physical basis the modal diagonal matrix becomes full. Using (4), it clearly appears that

[Φ] -1 = [Φ] T [M ] and [Φ T ] -1 = [M ][Φ].
One can thus invert (6) and obtain

[C] = [M Φ] N ×Nm \ 2ζ j ω j \ Nm×Nm [M Φ] T Nm×N ( 9
)
The equation is exact when all modes are kept but is typically applied using a truncated modal series of N m modes. A key aspect of this simplification is that rows of [Φ] -1 are given by {φ j } T [M ]. The C matrix in ( 9) is full and thus too large for practical applications. For time simulations one however needs to compute [C]{v} products, which can be organized as

[C]{v} = [M Φ] \ 2ζ j ω j \ [M Φ] T {v} . ( 10
)
which involves a full transposed matrix/vector product [M Φ] T {v}, a multiplication by a diagonal matrix and a second matrix vector products. One thus only stores [M Φ] for selected modes and uses BLAS (Basic Linear Algebra Subroutines) to perform these operations. These routines are now very well parallelized and very good performance can be achieved.

The number N m of modes and associated damping coefficients, directly impact computation time. Part of the procedure is thus to select target modes that are of interest and combine with a Rayleigh model for the rest of frequencies. In that case, Rayleigh damping should be compensated in order to obtain desired modal damping ratios. For a desired ζj , one thus uses (10) with

2ζ j ω j = 2 ζj ω j -φ T j C 0 φ j (11)
where C 0 is the Rayleigh damping. Compensation is exact if modal damping hypothesis ( 6) is verified and approximate otherwise.

Modal amplitudes and energies

An idea closely linked with modal damping is that modal amplitudes are quantities of direct interest in transient analysis. By definition of modal coordinates, the response can be described as a linear combination of modes

{q(t)} = [Φ]{α(t)} (12)
where {α(t)} is the vector of the modal amplitudes. Using the inverse of [Φ] implied by ( 4), one finds

{α(t)} = [M Φ] T {q(t)} (13) { α(t)} = [M Φ] T { q(t)} (14)
which can easily be computed either as a post-processing step if all components of q are saved or during the time simulation as a generalized sensor. This is a direct application of the modal sensor concept known in the active vibration control literature.

Modal energy is the mechanical energy associated with a given mode and is given by

Em j (t) = 1 2 (α j (t)φ T j Kα j (t)φ j + αj (t)φ T j M αj (t)φ j ) = 1 2 (ω 2 j α 2 j + α2 j ) (15)
In modal coordinates, mass and stiffness are diagonal matrices. It thus clearly appears that the total mechanical energy is the sum of modal energies.

Validation

This section illustrates the use of modal coordinates and damping for transient analysis of catenaries. Section 3.1 uses a simple pretensioned wire to illustrate the use of modal energies and the modal damping properties of Rayleigh damping. In section 3.2 an impact test is used to show how modal damping ratio can be used to get the transient model to fit global behavior better. Finally section 3.3 illustrates how modal damping can be used to analyze full transient analyzes.

Pretensioned wire

The example of a simple wire is first considered to illustrate the relation between transient analysis and modal quantities. The model is a tensioned wire (l=60m, T=10kN) hung every 6m to springs (K=50kN). An initial global Raleigh with α = 0 and β = 3.25e -3 is defined. The passage at a speed of 83m/s of a pantograph with mean contact force of 150 N is simulated.

The pantograph-catenary interaction is a moving load problem. Modal amplitudes and modal energies of the wire depend on modeshapes and modal energies can be used to analyze this dependence. Modes, see figure 1, can be separated into horizontal (5,7 and 9) and vertical modes (1,2,3,4,6,8 and 10). Only vertical modes are excited by the pantograph when sliding under the wire. This is directly confirmed by the maximum modal energies shown in the figure 1. In more complex cases, the sorting of modal energies can be used to identify excited modes. This is important for modal damping implementation, since the damping of modes that do not respond has no influence on the response. To get a better understanding of how modal energies evolve, figure 2 compares modal energies through the passage and modeshapes using the same horizontal scale. A decrease of modal energies, when the pantograph is around the nodes of modes 2 and 3 is clearly visible. The fact that energies do not drop to zero indicates that decrease due to damping is not fast compared to the time that the pantograph takes to pass a modeshape node. Classification of the vertical modes can also be interesting. Figure 3 illustrates the maximum of Em response for each exited mode. The highest energies are found at low frequencies, with maximum for mode 13 at 7.2 Hz. The energy then decreases regularly. The max(Em) curve illustrates that the energy input into a mode is associated to a match between wavelength and pantograph speed. Once the pantograph leaves, the wire becomes free and modal energies decrease, and one will seek to verify the ability to use modal coordinates to estimate damping ratio. The Logarithmic Decrement method is classically used for single mode systems. Measuring the decrease of two successive peaks in the modal amplitude one deduces damping ratio for sub-critical systems.

To eliminate peak detection difficulties, it is proposed here to track the modal energies. Considering a given modal amplitude α j (t) the free response to an initial condition is that of a single DOF mass/stiffness/damper system with

m = 1, k = ω 2 j , c = φ T j Cφ j = 2ζ j ω j (16)
and dynamic equation αj (t) + 2ζ j ω j αj (t) + ω 2 j α j (t) = 0 (17)

whose characteristic equation has discriminant ∆ = 4ω 2 j (ζ 2 j -1). Sub-critical damping is found for ζ j < 1 (negative discriminant). There are two conjugate poles λ = ω j (-ζ j ± ı 1 -ζ 2 j ) and a time response of the form α j (t) = A j cos(ω j 1 -ζ 2 j t + θ j ) exp(-ζ j ω j t) where A j and θ j are two constants depending on initial conditions. Modal energy can be computed using (15) and is of the form

Em j (t) = (A + f (t)) exp(-2ζ j ω j t)
where f (t) << A (f is composed of sin 2 , cos 2 and sin cos products that explains minor oscillations).

The energy decrease in a log scale plot is thus a straight line showing minor oscillations and can be estimated with a simple curve fit which gives an estimate of modal damping. Critical damping is found for ζ j = 1. The system then has a double pole λ = -ω j and a time response of the form (A j + B j t) exp(-ω j t) where A j and B j are two constants depending on initial conditions. Modal energy is of the form Em j (t) = P (t) exp(-2γ j t) where P is a second order polynomial. Supercritical damping is found for ζ j > 1. Then there are two real poles λ = ω j (-ζ j ± ζ 2 j -1) and a time response α j (t) = (A j exp(ω j ζ 2 j -1t) + B j exp(-ω j ζ 2 j -1t)) exp(-ζ j ω j t) where A j and B j are two constants depending on initial conditions. The exponential associated with the lowest frequency pole decreases much more slowly so that the modal energy takes the form Em j (t) ≈ (A + f (t)) exp(-2ω j (ζ jζ 2 j -1)t) where f is small compared to A. From the decay of modal energies log(Em j ), shown in figure 4, a simple least square methods was used to estimate the exponential constant γ j (ζ j for sub-critical damping and ζ j -ζ 2 j -1 for supercritical damping). One clearly finds the theoretical behavior for Rayleigh damping of a linear increase up to critical damping and a decrease of γ j above that threshold.

The low frequency at which critical damping occurs is a property of Rayleigh damping that is rarely discussed. In the present application, it implies that wave propagation can only be studied at frequencies much below that limit. Explaining the variations in estimated damping around critical damping would be interesting. In order to validate the implementation of modal damping in SDT, two passage simulations with and without modal damping are compared. In the second, a modal damping of 50 % is added to mode 3 only in addition to the initial Rayleigh damping. The total damping ratio expected on mode 3 is thus β 2 ω 3 +0.5 = 55.5%. For other modes, the β 2 ω j value remains unchanged. From the decay a value of 55.37% (relative error of 0.24 %) is found for mode 3. For other modes the maximum relative error is 0.2%. During the passage, figure 5 shows that modal energy remains almost unchanged for mode 2 and 4, and only changes for mode 3. The figure 5 also shows that damping ratio of mode 3 does not impact the response a lot since the modal energy only decreases by 56% whereas modal damping has increased by a 9 factor.

Impact test on a real catenary

A significant objective of using modal damping in transient analysis is to introduce a simple mechanism to control damping related system behavior. In catenaries, impact or drop tests can be used to verify global behavior and will be discussed here.

The SNCF LN2 catenary, shown in figure 6, is considered with an Initial piecewise Rayleigh damping proportional to stiffness (β CW = 3.25e -3, β M W = 1.75e -2, β dropper = 1.5e -8 and every α e = 0). These values are not optimal in many respects. One first seeks to illustrate implications of (8) (the MSE method). Figure 7 (right) shows the fraction of energy in the contact wire (CW) or in the messenger wire (MW), distinguishing between compression (A), bending (I1), torsion (J), and pretension (ten). For most of the modes the major part of elastic energy is pretension energy (ten). Some modes are mostly localized in the CW or in the MW (when energy fraction of the component is near 1). For these modes, the damping ratio, shown left in the figure, match the damping ratio of the component where the energy is localized. For impact tests, frequency and time domain analyses can easily be compared. From the time domain analysis, a transfer is obtained as ratio of response and force input spectra. The results shown in figure 8 correlate perfectly at low frequencies. Differences at higher frequencies are attributed to the periodicity error induced by the mean acceleration Newmark scheme.

The response shows peaks for the first four groups of modes at 1 Hz, 2 Hz, 3 Hz and 4 Hz (first and second group vertical modes correspond respectively mainly to first and second span modes). Within each group, individual modal resonances cannot be separated. The Initial damping model is thus too damped. Frequencies higher than 4 Hz are not seen, and modes are not clearly separated. Tests can be used to obtain a physical reference for such response. In this study, one will take this reference to be a modal damping of 1% on modes below 3.5 Hz. For higher modes, Rayleigh damping is still used but coefficients are decreased to have 1% at 3.5 Hz on the contact wire. Values are shown in figure 9 (left). This damping model will be called Modal in the rest of the study.

The first mode groups now show individual peaks (as seen in figure 9 -right) and CW modes only reach 5% damping at 17.5 Hz.

OSCAR time simulation

The same LN2 catenary is now considered in a full OSCAR time simulation. A simple pantograph, composed of 3 mass/stiffness/damper stages, is added. The time simulation corresponding to the passage of the pantograph at a speed of 83 m/s with a mean contact force of 170N is analyzed.

Figure 11 compares the contact force for the Initial (piece-wise Rayleigh damping) and Modal (adjusted Rayleigh for f >3.5Hz + modal damping 1% for f <3.5Hz) models of section 3.2. The contact force mostly shows the excitation seen by the pantograph: dropper periodicity, span periodicity etc. at current speed. In the figure, the main peak is associated with dropper repetition (distance between droppers at a train velocity of 83 m/s correspond to frequencies around 1.6 Hz as shown in gray in the FFT). The uplift measured at the middle of the catenary is shown in figure 12 and is more sensitive to damping. With modal damping, residual vibration after the pantograph passage is significantly more accurate. The spectrum is comparable to the impact spectrum shown in section 3.2, so that the influence of modal damping is not a surprise. Finally, uplift is not necessarily the response most sensitive to damping. For fatigue analysis, the amplitude between the maximum and the minimum stress at a given point is more appropriate. Stress is linked to curvature by σ = Ezκ (18

)
where E is the Young modulus, z the distance to the neutral fiber and κ the curvature. Extracting the observation matrices for curvature at integration points, one can compute stresses at the wire surface where it is maximal. Figure 15 compares stresses and uplifts for the two damping models. Stress is clearly more sensitive to damping with a 50 % increase of stress levels at lower damping.

Conclusion

Modal damping was shown to be a practical tool that can be used to analyze systems showing complex transient behavior that requires time domain integration. The concept is directly linked to the use of modal amplitudes that can also be used to post-process classical transients. The proposed procedure has been used here to diagnose the impact of damping in simulations, which is rarely done in detail. In Ref. [START_REF] Vermot Des Roches | Designed oriented time/frequency analysis of contact friction instabilities in application to automotive brake squeal[END_REF] the same concept was used as a design tool.

Figure 1 :

 1 Figure 1: First 10 modes and corresponding maximum modal energy during pantograph passage.

Figure 2 :

 2 Figure 2: First 3 modal energies for a time simulation compared to the first 3 modes z deflection.

Figure 3 :

 3 Figure 3: When pantograph is sliding under the wire. Left: modal energies. Right: Max of Em.

Figure 4 :

 4 Figure 4: When wire is free. Left: modal energies in ylog. Right: Identified γ j /ω j .

Figure 5 :

 5 Figure 5: When pantograph is sliding, time evolution of the energies for modes 2, 3 and 4. Blue: nominal Rayleigh damping. Green: nominal+modal damping ratio of 50% added to mode 3.

Figure 6 :

 6 Figure 6: LN2 catenary (transverse dimensions with a 100 scale)

Figure 7 :

 7 Figure 7: LN2 catenary, initial piece-wise Rayleigh damping (Initial). Left: Modal damping ratios. Right: Strain energy fraction for each mode.

Figure 8 :

 8 Figure 8: LN2 catenary, Initial piece-wise Rayleigh damping. FFT (blue) and FRF (green).

Figure 9 :

 9 Figure 9: LN2 catenary. Left: modal damping ratios (Modal). Right: FRF for Initial (blue) and Modal (green) damping models.

  /(c ii c jj )0.5

Figure 10 :

 10 Figure 10: c ij √ c ii c jj for the first 50 modes. Left : color map. Right : histogram (scale log).

Figure 11 :

 11 Figure 11: Contact force. LN2 catenary, Initial (blue) and Modal (green) damping models.

Figure 13 :

 13 Figure 13: Top: some unkept modes. Down left: contact force. Down right: uplift at the middle of the catenary.

Figure 14 :

 14 Figure 14: Residual computation time.

Figure 15 :

 15 Figure 15: LN2 catenary, Initial (blue) and Modal (green) damping models. Top: stress criteria and stress at PK=206m when pantograph is passing. Down: Uplift at PK=206m.

  Uplift at the middle of the catenary. LN2 catenary, Initial (blue) and Modal (green) damping models compared to measurement (red).The Modal model with 1% damping shows better correlation for the peaks at 2Hz, 3 Hz and 4 Hz. The first resonance is however not predicted properly. It was seen in section 3.1, that the moving load and low frequencies are important driver factors in the level of a given resonance. In this case, reducing the damping ratio from 1% to 0.02% around 1 Hz has almost no impact. The ability to control modal damping can thus be used to analyze the influence of various factors.

		0.08	Initial							Initial
	Uplift [m]	0 0.02 0.04 0.06	Modal Measured	Uplift	10 -3 10 -2				Modal Measured
		-0.02
		5	10	15		0	1	2	3	4	5
			Time [s]					Frequency [Hz]
	Figure 12: The numerical cost of transient modal damping is directly associated with the number of retained modes.
	Modal energies provide a simple mechanism to target modes that have no influence. Figure 13 illustrates
	modes that don't excite contact wire or have horizontal contact wire motion. A second simulation retaining
	133 modes out of 185 shows nearly identical results in figure 13 (the contact force relative error is 8.9e-3 %
	and uplift relative error is 0.17 %). For the same precision, a 22.6% decrease of CPU time (49.5s to 38.3s
	for 2000 time steps) is obtained.

Table 1

 1 shows that time added for residual computation by the modal damping (comparing to Initial residual) is proportional to the number of retained modes. Selection of target modes to be damped is thus an important aspect of the proposed methodology.

	Damping model	Total time [s] Residual computation time [s]
	Initial	39.4	9.73
	Selective Modal (133 modes)	68.7	38.28
	Modal (185 modes)	80.2	49.49

Table 1 :

 1 Computation time for 2,000 steps of time

		60
	Residual time [s]	20 40
		0 0	50	100	150	200
			Number of modes


Gunter Minckwitz

Michael Untch

Eveline Nüesch

Sibylle Loibl

Manfred Kaufmann

Sherko Kümmel

Peter A Fasching

Wolfgang Eiermann

Jens-Uwe Blohmer

Serban Dan Costa

Gunter Von Minckwitz
email: gunter.vonminckwitz@germanbreastgroup.de

Keyur Mehta

Jörn Hilfrich

Christian Jackisch

Bernd Gerber

Andreas Du Bois

Jens Huober

Claus Hanusch

Gottfried Konecny

Werner Fett

Elmar Stickeler

Nadia Harbeck

Volkmar Müller

Peter Jüni

Impact of treatment characteristics on response of different breast cancer phenotypes: pooled analysis of the German neo-adjuvant chemotherapy trials

Keywords: neoadjuvant, breast cancer, pooled analysis, taxanes, anthracyclines trastuzumab

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

Introduction

Several anthracycline and taxane-based chemotherapy regimens are established for treatment of early breast cancer [START_REF] Peto | The worldwide overview updated (2005-6) meta-analysis of trial results[END_REF][START_REF]Effects of chemotherapy and hormonal therapy for early breast cancer on recurrence and 15-year survival: an overview of the randomised trials[END_REF][START_REF] Ebctcg) | Adjuvant chemotherapy in oestrogen-receptor-poor breast cancer: patientlevel meta-analysis of randomised trials[END_REF]. Specific dosing characteristics of regimens, including the cumulative dose of agents and the number of cycles, is largely dependent on local guidelines or preference of individual physicians. A recent guideline on the diagnosis and treatment of early and locally advanced breast cancer published by the UK National Collaborating Centre for Cancer on behalf of NICE [START_REF]Early and locally advanced breast cancer: diagnosis and treatment[END_REF], for example, did not provide any recommendation on the dosages and numbers of cycles, and may contribute to the already existing, considerable variation in treatment standards across regions and countries. Similarly, collaborative analyses of randomized trials [START_REF] Peto | The worldwide overview updated (2005-6) meta-analysis of trial results[END_REF][START_REF]Effects of chemotherapy and hormonal therapy for early breast cancer on recurrence and 15-year survival: an overview of the randomised trials[END_REF][START_REF] Ebctcg) | Adjuvant chemotherapy in oestrogen-receptor-poor breast cancer: patientlevel meta-analysis of randomised trials[END_REF] were forced to ignore the issue of dosing and mainly reported average effects across many different regimens with considerable heterogeneity in dosing characteristics.

Neoadjuvant regimens are considered equivalent to adjuvant regimens with identical dosing schemes [START_REF] Kaufmann | Recommendations from an international expert panel on the use of neoadjuvant (primary) systemic treatment of operable breast cancer: an update[END_REF][START_REF] Mieog | Neoadjuvant chemotherapy for operable breast cancer[END_REF][START_REF] Mauri | Neoadjuvant versus adjuvant systemic treatment in breast cancer: a meta-analysis[END_REF].

However, in contrast to adjuvant therapy the neoadjuvant approach allows to immediately ascertain the effects of chemotherapy based on the histopathological assessment of surgically removed tissue. Pathologically confirmed complete response (pCR) after chemotherapy highly correlates with favorable long-term outcome and was therefore suggested as a surrogate marker for survival [START_REF] Kaufmann | Recommendations from an international expert panel on the use of neoadjuvant (primary) systemic treatment of operable breast cancer: an update[END_REF][START_REF] Tao | Clinical and biomarker endpoint analysis in neoadjuvant endocrine therapy trials[END_REF]. Between 1998 and[START_REF] Kaufmann | Recommendations from an international expert panel on the use of neoadjuvant (primary) systemic treatment of operable breast cancer: an update[END_REF], the German Breast Group (GBG) and the Arbeitsgemeinschaft Gynäkologische Onkologie (AGO) "Breast", conducted a series of neoadjuvant trials. To address the current uncertainty surrounding dosing characteristics, we performed a pooled analysis of these trials and determined whether variations in cumulative dose of taxanes, anthracyclines and trastuzumab, number of administered cycles and administration of additional drugs, including tamoxifen and capecitabine, are associated with treatment response overall and across different breast cancer phenotypes.

Methods

Patients

Between 1998 and 2006, a total of eight major prospective neoadjuvant trials in 6625 breast cancer patients were conducted in Germany [START_REF] Von Minckwitz | Dose-dense Doxorubicin and Docetaxel with or without Tamoxifen as Preoperative Therapy in Patients with Operable Carcinoma of the Breast: a Randomized, Controlled, Open Phase IIb Study[END_REF][START_REF] Von Minckwitz | Doxorubicin with cyclophosphamide followed by docetaxel every 21 days compared with doxorubicin and docetaxel every 14 days as preoperative treatment in operable breast cancer: the GEPARDUO study of the German Breast Group[END_REF][START_REF] Von Minckwitz | In vivo chemosensitivity-adapted preoperative chemotherapy in patients with early-stage breast cancer: the GEPARTRIO pilot study[END_REF][START_REF] Von Minckwitz | Intensified neoadjuvant chemotherapy in early-responding breast cancer: phase III randomized GeparTrio study[END_REF][START_REF] Von Minckwitz | Neoadjuvant vinorelbine-capecitabine versus docetaxel-doxorubicin-cyclophosphamide in early nonresponsive breast cancer: phase III randomized GeparTrio trial[END_REF][START_REF] Von Minckwitz | Capecitabine in Addition to Anthracycline / Taxane-based Neoadjuvant Treatment in Patients with Primary Breast Cancer: The Phase III GeparQuattro Study[END_REF]15,16,[START_REF] Untch | Dose dense sequential epirubicin-paclitaxel as preoperative treatment of breast cancer: Results of a randomised AGO study[END_REF][START_REF] Untch | A multicenter phase II study of preoperative epirubicin, cyclophosphamide (EC) followed by paclitaxel (P) plus trastuzumab (T) in Her2 positive primary breast cancer[END_REF]. Details of design and conduct of individual trial were previously reported (Table 1). Seven trials were phase III randomized parallel-group trials (6408 patients) [START_REF] Von Minckwitz | Dose-dense Doxorubicin and Docetaxel with or without Tamoxifen as Preoperative Therapy in Patients with Operable Carcinoma of the Breast: a Randomized, Controlled, Open Phase IIb Study[END_REF][START_REF] Von Minckwitz | Doxorubicin with cyclophosphamide followed by docetaxel every 21 days compared with doxorubicin and docetaxel every 14 days as preoperative treatment in operable breast cancer: the GEPARDUO study of the German Breast Group[END_REF][START_REF] Von Minckwitz | In vivo chemosensitivity-adapted preoperative chemotherapy in patients with early-stage breast cancer: the GEPARTRIO pilot study[END_REF][START_REF] Von Minckwitz | Intensified neoadjuvant chemotherapy in early-responding breast cancer: phase III randomized GeparTrio study[END_REF][START_REF] Von Minckwitz | Neoadjuvant vinorelbine-capecitabine versus docetaxel-doxorubicin-cyclophosphamide in early nonresponsive breast cancer: phase III randomized GeparTrio trial[END_REF][START_REF] Von Minckwitz | Capecitabine in Addition to Anthracycline / Taxane-based Neoadjuvant Treatment in Patients with Primary Breast Cancer: The Phase III GeparQuattro Study[END_REF][15][16][START_REF] Untch | Dose dense sequential epirubicin-paclitaxel as preoperative treatment of breast cancer: Results of a randomised AGO study[END_REF] and one was a phase II case series (217 patients) [START_REF] Untch | A multicenter phase II study of preoperative epirubicin, cyclophosphamide (EC) followed by paclitaxel (P) plus trastuzumab (T) in Her2 positive primary breast cancer[END_REF] In two trials (2357 patients), allocation to intervention groups depended on an intermediate clinical assessment of tumor response after two cycles of TAC (docetaxel [Taxotere®], doxorubicin [adriamycin], cyclophosphamide): in the GeparTrio pilot trial [START_REF] Von Minckwitz | In vivo chemosensitivity-adapted preoperative chemotherapy in patients with early-stage breast cancer: the GEPARTRIO pilot study[END_REF] the 74 nonresponders were randomly allocated to 4 further cycles of TAC (41 patients) or to 4 cycles of NX (vinorelbine [Navelbine®] and capecitabine [Xeloda®], 33 patients). The 211 responders all received a further four cycles of TAC. In the GeparTrio main trial [START_REF] Von Minckwitz | Intensified neoadjuvant chemotherapy in early-responding breast cancer: phase III randomized GeparTrio study[END_REF][START_REF] Von Minckwitz | Neoadjuvant vinorelbine-capecitabine versus docetaxel-doxorubicin-cyclophosphamide in early nonresponsive breast cancer: phase III randomized GeparTrio trial[END_REF], the 622 non-responders were randomized to 4 further cycles of TAC (321 patients) or 4 cycles of NX (301 patients), whereas the 1390 responders were allocated to 4 or 6 further cycles of TAC (704 and 686 patients, respectively). To avoid confounding by treatment selection according to intermediate clinical assessment of tumor response, we excluded the 334 patients allocated to NX, a proportional random sample of 95 responders, who received a total of 6 cycles of TAC in the GeparTrio pilot trial [START_REF] Von Minckwitz | In vivo chemosensitivity-adapted preoperative chemotherapy in patients with early-stage breast cancer: the GEPARTRIO pilot study[END_REF], and the 686 responders, who received a total of 8 cycles of TAC in the GeparTrio main trial [START_REF] Von Minckwitz | Intensified neoadjuvant chemotherapy in early-responding breast cancer: phase III randomized GeparTrio study[END_REF][START_REF] Von Minckwitz | Neoadjuvant vinorelbine-capecitabine versus docetaxel-doxorubicin-cyclophosphamide in early nonresponsive breast cancer: phase III randomized GeparTrio trial[END_REF]. 134 patients included in the run-in period of GeparTrio [START_REF] Von Minckwitz | Intensified neoadjuvant chemotherapy in early-responding breast cancer: phase III randomized GeparTrio study[END_REF][START_REF] Von Minckwitz | Neoadjuvant vinorelbine-capecitabine versus docetaxel-doxorubicin-cyclophosphamide in early nonresponsive breast cancer: phase III randomized GeparTrio trial[END_REF] and GeparQuattro [START_REF] Von Minckwitz | Capecitabine in Addition to Anthracycline / Taxane-based Neoadjuvant Treatment in Patients with Primary Breast Cancer: The Phase III GeparQuattro Study[END_REF]15] who dropped out before randomization because of progression, malcompliance, or side-effects, were analyzed in actual treatment arms as determined by computer-generated random allocation. Characterization of tumor subtypes by human epidermal growth factor receptor 2 (HER2) was not protocol mandated in the early trials. We therefore excluded one trial, [ 16 ], in which none of the 668 randomized patients had undergone HER2 testing. Of the remaining 4813 patients we excluded another 1114 patients with unknown HER2 status and 367 patients with other unknown baseline parameters. Our analysis is therefore based on patient-level data from the remaining 3332 patients from 7 trials with 12 treatment arms. Baseline characteristics of the 3332 analyzed patients and the 3293 excluded patients were similar (Supplemental Table ).

All trials used uniform definitions for patient selection: diagnosis of invasive breast cancer had to be histologically confirmed by core biopsy. Patients needed to have measurable tumor dimensions either by palpation, sonography or mammography. Locally advanced (cT4a-d) and inflammatory breast cancers were eligible for all trials except GeparDo [START_REF] Von Minckwitz | Dose-dense Doxorubicin and Docetaxel with or without Tamoxifen as Preoperative Therapy in Patients with Operable Carcinoma of the Breast: a Randomized, Controlled, Open Phase IIb Study[END_REF] and GeparDuo [START_REF] Von Minckwitz | Doxorubicin with cyclophosphamide followed by docetaxel every 21 days compared with doxorubicin and docetaxel every 14 days as preoperative treatment in operable breast cancer: the GEPARDUO study of the German Breast Group[END_REF]. Patients were eligible irrespective of hormone receptor and HER2 receptor status, except in Techno [START_REF] Untch | A multicenter phase II study of preoperative epirubicin, cyclophosphamide (EC) followed by paclitaxel (P) plus trastuzumab (T) in Her2 positive primary breast cancer[END_REF], in which only HER2-positive patients were allowed.

All trials were approved by the respective ethics committees.

Tumor size was determined before start of systemic treatment by palpation or sonography. Histological type, tumor grade, and estrogen-, progesterone, and HER2 receptor status were determined locally on pre-treatment core-biopsies.

To calculate cumulative anthracycline doses we converted epirubicin doses into doxorubicin equivalence doses based on a pre-specified ratio of 3:2, assuming, e.g., 90mg/m 2 epirubicin to be equivalent to 60mg/m 2 doxorubicin. For taxanes, we converted paclitaxel doses to docetaxel equivalence doses based on a pre-specified ratio of 7:4, 175mg/m 2 paclitaxel, e.g., were considered equivalent to 100 mg/m 2 docetaxel. We calculated cumulative equivalence doses administered across all cycles as pre-specified in individual trial protocols and considered ≥300mg/m² cumulative doxorubicin equivalents as a higher cumulative anthracycline dose, and ≥400mg/m² cumulative docetaxel equivalents as a higher cumulative taxane dose. A standardized definition of pCR was used for all trials: undetectable invasive residual tumor in the excised tissue of breast and axillary lymph nodes, corresponding to ypT0/is ypN0 [START_REF] Gralow | Preoperative therapy in invasive breast cancer: pathologic assessment and systemic therapy issues in operable disease[END_REF].

Statistical methods

We used multi-level logistic regression models with random effects at the level of trials and trial arms. All analyses were performed according to the intention-to-treat as specified in the protocol. In univariable analyses, we compared baseline characteristics of patients and tumors and treatment parameters between patients with and without pCR. The primary analysis in all 3332 patients was a multivariable analysis including all baseline characteristics and treatment parameters to determine the association of pCR with number of cycles of chemotherapy (per 2 cycles), cumulative dose of taxanes and anthracyclines (high versus low dose), administration of capecitabine, tamoxifen and trastuzumab (yes versus no). The effect of trastuzumab was estimated for HER2-positive patients only. Secondary analyses were stratified according to HR status, HER2 status, and biological subtype (HER2-negative/HR-positive, HER2-positive/HR-positive, HER2-positive/HRnegative, HER2-negative/HR-negative), with P-values for interaction between effect of the treatment variable and stratum derived from appropriate interaction terms in the multi-level model. Finally, we determined the association of the cumulative number of trastuzumab cycles (4 versus 8 to 12), overall and stratified according to age (<40 versus ≥40 years), tumor stage (stages 1 to 2 versus 3 to 4), grading (grade 1 to 2 versus 3), and HR status. All p-values are two-sided. Statistical analyses were performed using Stata Version 10.1 (Stata Corporation, College Station, Texas).

Results

In included trials, number of cycles ranged from 4 to 12, cumulative anthracycline dose from 200 to 400mg/m 2 , and cumulative taxane dose from 300 to 450mg/m 2 . Capecitabine was administered in two treatment arms (966 patients), tamoxifen in 3 arms (215 patients), and trastuzumab in 4 arms (614 HER2-positive patients, see Table 1). Table 2 shows the association of baseline characteristics of patients and tumors and treatment parameters with pCR in univariable analyses. Higher pCR rates were observed in younger patients, lower stages, ductalinvasive carcinomas, less differentiated tumors, HR-negative and HER2-positive tumors (p for trend or interaction all ≤0.001). There was an association between pCR and number of cycles (p=0.005), but little evidence for differences in treatment response according to cumulative anthracycline or taxane dose, use of capecitabine or tamoxifen in the univariable analysis. The test for trend was formally positive for the association between number of trastuzumab cycles and pCR, but this was driven by the difference in pCR between patients with and without trastuzumab (p for difference<0.001), but not by an association with number of trastuzumab cycles: the odds ratios comparing 4 cycles with no trastuzumab (2.84, 95%-CI 1.93-4.18), and 8 to 12 cycles with no trastuzumab (3.14, 95%-CI 2.28-4.32) were similar and confidence intervals overlapped widely.

Figure 1 presents results from multivariable analysis of treatment characteristics after adjustment for all potential confounders. PCR became more likely with an increase in the number of cycles: the odds of a pCR increased approximately 1.2fold with every two additional treatment cycles (p=0.009). The odds of pCR were approximately 1.6fold higher in regimens with higher cumulative anthracycline doses (p=0.002), regimens with higher cumulative taxane doses (p=0.009), and capecitabine containing regimens (p=0.022). There was little evidence that concurrent tamoxifen use impacted on pCR (p=0.69). Simultaneous trastuzumab treatment in patients with HER2 positive tumors, however, increased the odds of pCR 3.2fold (p<0.001).

Figure 2 presents stratified analyses of the association of pCR with treatment characteristics by receptor status.

Stratification by HR status (Panel A) indicated that the association of pCR with number of cycles was particularly pronounced in patients with HR-positive tumors, as compared with HR-negative tumors, with a positive test for interaction between treatment effect and HR status (p=0.039). The effect of trastuzumab appeared more pronounced in patients with HR-negative tumors, but confidence intervals were wide and a test for interaction negative (p for interaction=0.20). There was little evidence that the association of pCR with remaining treatment characteristics varied according to HR status (p for interaction≥0.44). Figure 2 (Panel B) presents results from analyses stratified by HER2 status. The effects of increasing numbers of cycles, cumulative taxane dose and use of capecitabine appeared not to vary according to HER2 status (p for interaction≥0.45). The estimated odds ratios of pCR for tamoxifen were below one in patients with HER2-positive tumors and above one in patients with HER2-negative tumors, but confidence intervals were wide, overlapped the line of no difference at 1.0 and a test for interaction was negative (p for interaction=0.28). The association of pCR with cumulative anthracycline dose was pronounced in patients with HER2-negative tumors (p=0.004), whereas no association was apparent in patients with HER2-positive tumors (p=0.66), confidence intervals of estimates were wide and overlapped again between strata and the test for interaction did not reach significance (p for interaction=0.15).

Figure 3 shows the association of pCR with treatment characteristics in biological breast cancer subtypes. In general, the patterns observed for the 4 subtypes were in concordance with analyses stratified according to HR and HER2 status even though tests for interaction were negative. The effect of number of cycles was pronounced in HR-positive tumors, irrespective of HER2 status. The effect of the cumulative anthracycline dose was pronounced HER2-negative tumors irrespective of HR status. No clear pattern evolved for cumulative taxane dose, administration of capecitabine, tamoxifen and trastuzumab, with little evidence to suggest variation across subgroups.

Figure 4 presents analyses of the association of pCR with number of trastuzumab cycles (4 versus 8 to 12 cycles). Overall, we found no evidence for an association of pCR with number of cycles (OR 1.20, 95%-CI 0.79 to 1.81, P=0.39). In stratified analyses, there was no evidence that the effects of the number of cycles varied according to age, stage and grade (p for interaction ≥0.24). However, the effect of number of cycles appeared to be more pronounced in HR-negative compared to HR-positive tumors (p for interaction=0.023).

Discussion

In this pooled analysis of German neoadjuvant breast cancer trials, we found that higher numbers of cycles and higher cumulative doses of both, anthracyclines and taxanes, and use of capecitabine and trastuzumab were associated with a clinically relevant increase in pCR. When stratifying according to hormone receptor and HER2 status, we found number of cycles to be particularly relevant in patients with HR-positive tumors, and cumulative anthracycline dose in patients with HER2-negative tumors. Further stratification by biological breast cancer subtype revealed imprecise results, but mainly confirmed effects found in overall analyses and those stratified according to HR and HER2 status. There was no evidence that the benefit of higher cumulative taxane doses and capecitabine varied according to biological subtypes. In the triple-negative subgroup, higher cumulative doses of anthracycline and taxane and administration of capecitabine, but not more treatment cycles appeared beneficial, but it remained unclear whether this subgroup also benefits from an increase in the number of cycles. Finally, in an analysis restricted to patients with HER2-positive tumors receiving trastuzumab, we found little evidence to suggest that an increasing number of trastuzumab cycles (from 4 to 8-12) is associated with an advantage regarding pCR in HER2-positive patients.

Our statistical model allowed a coherent analysis of the association of pCR with treatment characteristics, implicitly assigned more weight to randomized within-trial comparisons than to comparisons of treatment characteristics between trials, and fully accounted for the heterogeneity in pCR between trials. However, our analysis is observational in nature, and we cannot exclude confounding as a potential explanation of observed results. Confounding by indication may be important. To minimize this source of systematic error, we implemented two measures which led to the exclusion of almost 50% of trial participants. First, our ascertainment of treatment characteristics was based on the pre-specified, protocol-defined "intention to treat", not on the treatment actually received. Second, we omitted treatment arms, in which confounding by indication was induced by treatment selection depending on an intermediate clinical assessment of tumor response at the end of a run-in period after administration of an initial series of chemotherapy cycles. Confounding by indication introduced by an association between treatment selection and prognosis is therefore unlikely. HER2 status is of a major prognostic importance in breast cancer. In some of the earlier trials, HER2 status was not determined. To avoid confounding by this unobserved tumor characteristic, we restricted the analysis to trials with available HER2 status [16]. Since availability of HER2 status is a mere function of time and not associated with prognostically important patient or tumor characteristics (Table 2), this restriction is unlikely to have introduced bias. Other unmeasured confounders might be tumor characteristics associated with time, however median tumor size and nodal status in the early Gepardo trial [START_REF] Von Minckwitz | Dose-dense Doxorubicin and Docetaxel with or without Tamoxifen as Preoperative Therapy in Patients with Operable Carcinoma of the Breast: a Randomized, Controlled, Open Phase IIb Study[END_REF] was similar to the most recent GeparQuattro [START_REF] Von Minckwitz | Capecitabine in Addition to Anthracycline / Taxane-based Neoadjuvant Treatment in Patients with Primary Breast Cancer: The Phase III GeparQuattro Study[END_REF] trial.

It cannot be fully excluded that the observed higher pCR rate with the longer therapies in the HR-positive subset, which are in general more slowly proliferating, may be in part due to a longer time interval from chemotherapy initiation to breast resection with the longer treatments. Dividing regimens into lower and higher according to their cumulative dose may be considered arbitrary.

However, according to the results achieved, regimens with a lower cumulative dose have to be considered as underdosed, whereas those with a higher cumulative dose are to be considered as adequately dosed. It has to be recognized, that many regimen recommended today as standard options do not qualify for such an adequate cumulative dose of anthracyclines, taxanes, or both. Sufficiently powered non-inferiority trials are warranted befor such lower doses should be used in routine care of patients.

Equivalence doses for paclitaxel and docetaxel were chosen according to labels. However, a post-hoc sensitivity analysis assuming 240mg/m² paclitaxel to be equivalent to 100mg/m² docetaxel yielded nearly identical overall results (data available on request). The final, but most important limitation of this analysis is the use of pCR (the primary endpoint of all 7 trials) as a surrogate outcome. Even though pCR was found to be associated with prognosis [START_REF] Kaufmann | Recommendations from an international expert panel on the use of neoadjuvant (primary) systemic treatment of operable breast cancer: an update[END_REF][START_REF] Mieog | Neoadjuvant chemotherapy for operable breast cancer[END_REF][START_REF] Mauri | Neoadjuvant versus adjuvant systemic treatment in breast cancer: a meta-analysis[END_REF], the associations with pCR observed in this analysis may not necessarily translate into associations of treatment characteristics with recurrence-free or overall survival.

Previous analyses of neoadjuvant trials compared neoadjuvant to adjuvant chemotherapy [START_REF] Mieog | Neoadjuvant chemotherapy for operable breast cancer[END_REF][START_REF] Mauri | Neoadjuvant versus adjuvant systemic treatment in breast cancer: a meta-analysis[END_REF] or taxanecontaining versus non-taxane containing neoadjuvant treatments [START_REF] Cuppone | Taxanes as primary chemotherapy for early breast cancer: meta-analysis of randomized trials[END_REF] and were based on aggregate level data only. Two recent pooled analyses explored the interaction between treatment and HER2 status. Gennari et al [START_REF] Gennari | HER2 status and efficacy of adjuvant anthracyclines in early breast cancer: a pooled analysis of randomized trials[END_REF] analyzed 8 randomized adjuvant trials comparing anthracycline-based with non-anthracycline-based regimens in early breast cancer, which reported clinical effectiveness according to HER2 status. Anthracyclines appeared beneficial in patients with HER2-positive, but not in patients with HER2-negative tumors, with positive tests for interaction for both, disease-free and overall survival. Dhesy-Thind et al [22] included trials of tamoxifen, anthracyclines, and taxanes, and found no evidence for a treatment by HER2 status interaction for tamoxifen, but positive interactions for anthracyclines and taxanes regarding disease-free survival, with both types of drugs being more effective in HER2-positive tumors. In an additional analysis of 3 trials comparing higher with lower cumulative anthracycline doses, the authors found some evidence that higher anthracycline doses were more beneficial regarding disease-free survival only for HER2-positive tumors, but a test for interaction did not reach statistical significance. These analyses of anthracycline effects [START_REF] Cuppone | Taxanes as primary chemotherapy for early breast cancer: meta-analysis of randomized trials[END_REF][START_REF] Gennari | HER2 status and efficacy of adjuvant anthracyclines in early breast cancer: a pooled analysis of randomized trials[END_REF] were performed in trials without concomitant taxane administration and cannot necessarily be generalized to current anthracycline-taxane combinations, whereas the analysis of taxane effects 21 did not address the impact of different cumulative doses.

The observed effect of capecitabine is in opposite to the results of the individual study (GeparQuattro) [START_REF] Von Minckwitz | Capecitabine in Addition to Anthracycline / Taxane-based Neoadjuvant Treatment in Patients with Primary Breast Cancer: The Phase III GeparQuattro Study[END_REF], which showed no differences regarding pCR rates for the arms containing capecitabine and the control arm.

However, the cumulative taxane dose was 300mg/m² in the capecitabine arms and 400mg/m² in the control arm, so capecitabine may have only compensated for the loss of taxane effectiveness. Other trials, which were not confounded by a lower taxane dose, showed a benefit for the addition of capecitabine [23,[START_REF] Joensuu | Adjuvant capecitabine in combination with docetaxel and cyclophosphamide plus epirubicin for breast cancer: an open-label, randomised controlled trial[END_REF]. The simultaneous use of trastuzumab during neoadjuvant chemotherapy has consistently shown higher pCR rates in HER2-positive breast cancer [START_REF] Buzdar | Significantly higher pathologic complete remission rate after neoadjuvant therapy with trastuzumab, paclitaxel, and epirubicin chemotherapy: results of a randomized trial in human epidermal growth factor receptor 2positive operable breast cancer[END_REF][START_REF] Gianni | Neoadjuvant chemotherapy with trastuzumab followed by adjuvant trastuzumab versus neoadjuvant chemotherapy alone, in patients with HER2-positive locally advanced breast cancer (the NOAH trial): a randomised controlled superiority trial with a parallel HER2-negative cohort[END_REF]. In how far our observation regarding the lack of additional benefit for more than 4 trastuzumab cycles, with respect to tumor response, translates into similar results for clinical long term outcome will be answered by several adjuvant trials comparing different durations of trastuzumab in the coming years [START_REF] Piccart-Gebhart | Trastuzumab after adjuvant chemotherapy in HER2-positive breast cancer[END_REF]28,29].

In conclusion, dosing characteristics appear important for the successful treatment of breast cancer. Higher (or adequate) cumulative doses of anthracyclines and taxanes and longer treatment are associated with better response. Our results lend support to the concept of tailoring chemotherapy regimens according to biological tumor type. In contrast to current practice, we submit that tumors with assumed partial resistance to chemotherapy (e.g. HR-positive tumors) require more cycles of adequately dosed chemotherapy to achieve an optimal effect. Less aggressive treatment may be sufficient in highly sensitive tumors: HER2-positive cancers might achieve optimal response already at a low cumulative dose of anthracyclines and at a low number of trastuzumab cycles, and HR-negative tumors may not need long treatment durations. Multivariable analysis of the association of pathological complete response (pCR) with treatment characteristics stratified by subgroups according to hormone receptor (HR) status and HER2 status, after adjustment for age, tumor stage, grade, type, HR and HER2 status as applicable.

Figure 1 :

 1 Figure 1:

Figure 2 :

 2 Figure 2:

Figure 3 :

 3 Figure 3:

Figure 4 :

 4 Figure 4:

2

 2 Figure 2

of cycles (per 2 additional cycles ) Antracycline (higher vs lower dose) Taxan (higher vs lower dose) Capecitabine ( yes vs no) Tamoxifen ( yes vs no) Trastuzumab ( yes vs no)

	Figure 1
	Number of cycles (per 2 additional cycles )		P=0.039
	Number of cycles (per 2 additional cycles ) Number of cycles (per 2 additional cycles ) HR positive 1.35 (1.12 to 1.63) 1.18 (1.04 to 1.34) 1.18 (1.04 to 1.34) HR negative 1.04 (0.87 to 1.24)	0.009 0.009
	Antracycline (high vs Antracycline (higher Antracycline (higher vs lower dose) lower dose)	1.55 (1.18 to 2.03) 1.55 (1.18 to 2.03)	P=0.94	0.002 0.002
	Taxane (higher HR positive vs HR negative lower dose)	1.58 (1.12 to 2.22) 1.55 (1.02 to 2.35) 1.58 (1.12 to 2.22) 1.52 (1.05 to 2.21)	0.009 0.009
	Capecitabine ( yes vs no) Capecitabine ( yes vs no) Taxane (higher vs lower dose	1.62 (1.07 to 2.45) 1.62 (1.07 to 2.45)	P=0.99	0.022 0.022
	Tamoxifen ( yes vs no) Tamoxifen ( yes vs no) HR positive HR negative		1.12 (0.65 to 1.94) 1.60 (0.96 to 2.66) 1.12 (0.65 to 1.94) 1.55 (0.97 to 2.49)	0.69 0.69
	Trastuzumab § ( yes vs no) Trastuzumab § ( yes vs no) Capecitabine ( yes vs no)			3.20 (2.19 to 4.67) 3.20 (2.19 to 4.67)	P=0.44	<0.001 <0.001
	HR positive		1.41 (0.77 to 2.58)
	HR negative		1.79 (1.00 to 3.19)
	HR positive Tamoxifen ( yes vs no)		1.45 (0.65 to 3.25)	Odds of pCR higher with less treatment 0.25 0.5 1.0 with less treatment Odds of pCR higher 0.25 0.5 1.0	Odds of pCR higher with more treatment 2.0 4.0 Odds of pCR higher with more treatment 2.0 4.0 P=0.50	Odds ratio
	HR negative		0.88 (0.42 to 1.85)
	Trastuzumab § ( yes vs no)					P=0.20
	HR positive		2.49 (1.46 to 4.24)
	HR negative	4.08 (2.38 to 6.98)
	Number HER2 positive		1.24 (0.99 to 1.54)		P=0.62
	HER2 negative		1.18 (1.01 to 1.38)
						P=0.15
	HER2 positive		0.83 (0.37 to 1.87)
	HER2 negative		1.61 (1.17 to 2.22)
						P=0.48
	HER2 positive		1.99 (0.87 to 4.53)
	HER2 negative		1.60 (1.08 to 2.36)
						P=0.45
	HER2 positive		2.00 (0.79 to 5.02)
	HER2 negative		1.63 (0.98 to 2.71)
						P=0.28
	HER2 positive		0.60 (0.17 to 2.05)
	HER2 negative	1.28 (0.68 to 2.41)
						not estimable
	HER2 positive		1.62 (0.71 to 3.68)
	HER2 negative		not estimable

Table 1 :

 1 Overview of the German neo-adjuvant breast cancer studies including reasons for exclusion of patients from analysis.

	Figure 3
	Number of cycles (per 2 additional cycles )					P=0.28
	HER2 -/ HR +	1.30 (1.02 to 1.65)
	HER2 + / HR +	1.42 (1.04 to 1.94)
	HER2 + / HR -	1.00 (0.71 to 1.41)
	HER2 -/ HR -	1.09 (0.88 to 1.35)
	Antracycline (higher vs lower dose)					P=0.55
	HER2 -/ HR +	1.92 (1.14 to 3.21)
	HER2 + / HR +	0.94 (0.31 to 2.85)
	HER2 + / HR -	0.72 (0.20 to 2.58)
	HER2 -/ HR -	1.49 (0.98 to 2.27)
	Taxane (higher vs lower dose)					P=0.92
	HER2 -/ HR +	1.52 (0.84 to 2.76)
	HER2 + / HR +	2.23 (0.75 to 6.61)
	HER2 + / HR -	1.87 (0.51 to 6.92)
	HER2 -/ HR -	1.73 (1.02 to 2.94)
	Capecitabine ( yes vs no)					P=0.70
	HER2 -/ HR +	1.88 (0.87 to 4.02)
	HER2 + / HR +	1.44 (0.42 to 4.93)
	HER2 + / HR -	3.06 (0.71 to 13.18)
	HER2 -/ HR -	1.50 (0.75 to 2.99)
	Tamoxifen ( yes vs no)					P=0.55
	HER2 -/ HR +	1.78 (0.66 to 4.82)
	HER2 + / HR +	1.19 (0.23 to 6.27)
	HER2 + / HR -	0.29 (0.04 to 1.84)
	HER2 -/ HR -	1.02 (0.45 to 2.34)
	Trastuzumab ( yes vs no)					P=0.99
	HER2 -/ HR + HER2 -/ HR +	not estimable not estimable
	HER2 + / HR + HER2 + / HR +	1.73 (0.57 to 5.20) 1.73 (0.57 to 5.20)
	HER2 + / HR -HER2 + / HR -	1.74 (0.47 to 6.47) 1.74 (0.47 to 6.47)
	HER2 -/ HR -HER2 -/ HR -	not estimable not estimable
		0.25	0.5	1.0	2.0	4.0	Odds ratio
		Odds of pCR higher	Odds of pCR higher
		with less treatment	with more treatment

Acknowledgment:

The pooled analysis was supported by Clinical Trials Unit (CTU) and University of Bern.

The conduct of trials was supported by Amgen, Bristol Meyers Squibb, Roche, and Sanofi-Aventis, Germany by providing unrestricted grants and trial medication.