Isabelle Gallagher
email: isabelle.gallagher@math.jussieu.fr

AND Thierry Paul
email: thierry.paul@math.polytechnique.fr

Laure Saint-Raymond
email: laure.saint-raymond@ens.fr

ON THE PROPAGATION OF OCEANIC WAVES DRIVEN BY A STRONG MACROSCOPIC FLOW

Keywords: 2010 Mathematics Subject Classification. 35Q86, 76M45, 35S30, 81Q20 Semiclassical analysis, microlocal analysis, Mourre estimates, Geophysical flows

, where the profile was assumed to be zonal only and where explicit calculations were made possible due to the 1D setting.

Here the diagonalization of the system, which allows to identify Rossby and Poincaré waves, is proved by an abstract semi-classical approach. The dispersion of Poincaré waves is also obtained by a more abstract and more robust method using Mourre estimates. Only some partial results however are obtained concerning the Rossby propagation, as the two dimensional setting complicates very much the study of the dynamical system.

Introduction

This paper is a continuation of [START_REF] Cheverry | Semiclassical and spectral analysis of oceanic waves[END_REF] so before discussing the matter of this paper (in Section 2) let us review the contents of that work. We shall start by recalling briefly the model, then we shall explain the methods and results obtained in [START_REF] Cheverry | Semiclassical and spectral analysis of oceanic waves[END_REF] and discuss their limitations.

1.1. The model. The goal of [START_REF] Cheverry | Semiclassical and spectral analysis of oceanic waves[END_REF] is to understand, through the study of a toy model, the persistence of oceanic eddies observed long past by physicists among which [START_REF] Gill | Atmosphere-Ocean Dynamics[END_REF][START_REF] Gill | Resonant interactions between planetary waves[END_REF][START_REF] Greenspan | The theory of rotating fluids[END_REF][START_REF] Majda | Introduction to PDEs and waves for the atmosphere and ocean[END_REF][START_REF] Pedlosky | Geophysical fluid dynamics[END_REF][START_REF] Pedlosky | Ocean Circulation Theory[END_REF], who gave heuristic arguments to explain their formation due both to wind forcing and to convection by a macroscopic current.

The ocean is considered in this toy model as an incompressible, inviscid fluid with free surface submitted to gravitation and wind forcing, and we further make the following classical assumptions: we assume that the density of the fluid is homogeneous ρ = ρ 0 = constant, that the pressure law is given by the hydrostatic approximation p = ρ 0 gz, and that the motion is essentially horizontal and does not depend on the vertical coordinate. This leads to the so-called shallow water approximation.

For the sake of simplicity, the effects of the interaction with the boundaries are not discussed and the model is purely horizontal with the longitude x 1 and the latitude x 2 both in R.

The evolution of the water height h and velocity v is then governed by the shallow-water equations with Coriolis force

(1.1) ∂ t (ρ 0 h) + ∇ • (ρ 0 hv) = 0 ∂ t (ρ 0 hv) + ∇ • (ρ 0 hv ⊗ v) + ω(ρ 0 hv) ⊥ + ρ 0 gh∇h = ρ 0 hτ
where ω denotes the vertical component of the Earth rotation vector Ω, v ⊥ := (-v 2 , v 1 ), g is the gravity and τ is the -stationary -forcing responsible for the macroscopic flow. The vertical component of the Earth rotation is therefore Ω sin(x 2 /R), where R is the radius of the Earth; note that it is classical in the physical literature to consider the linearization of ω (known as the betaplane approximation) ω(x 2 ) = Ωx 2 /R. We consider general functions ω in the sequel, with some restrictions that are be made precise later.

We consider small fluctuations around the stationary solution ( h, v) satisfying h = constant, ∇ • (v ⊗ v) + ωv ⊥ = τ, div v = 0.

In [START_REF] Cheverry | Semiclassical and spectral analysis of oceanic waves[END_REF] the study is restricted to the case of a shear flow, in the sense that v(x) = (v 1 (x 2 ), 0), with v1 a smooth, compactly supported function. Moreover some orders of magnitude and scalings allow to transform the previous system into the following one:

(1.2)

∂ t η + 1 ε ∇ • u + ū • ∇η + ε 2 ∇ • (ηu) = 0 , ∂ t u + 1 ε 2 bu ⊥ + 1 ε ∇η + ū • ∇u + u • ∇ū + ε 2 u • ∇u = 0 ,
where b := ω/|Ω| and where ε is a small parameter (of the order of Fr 2 , where Fr is the Froude number, and of Ro 1 2 where Ro is the Rossby number).

1.2. Methods and results in [START_REF] Cheverry | Semiclassical and spectral analysis of oceanic waves[END_REF]. Most of the analysis in [START_REF] Cheverry | Semiclassical and spectral analysis of oceanic waves[END_REF] concerns the linear version of (1.2), namely the following system:

(1.3)

ε 2 i∂ t v + A(x 2 , εD, ε)v = 0 v = (v 0 , v 1 , v 2 ),
where D := 1 i ∂, and the linear propagator is given by A(x 2 , εD, ε)

:= i   εū 1 ε∂ 1 ε∂ 1 ε∂ 2 ε∂ 1 εū 1 ε∂ 1 -b(x 2 ) + ε 2 ū′ 1 ε∂ 2 b(x 2 ) εū 1 ε∂ 1   ,
The first step of the analysis consists in diagonalizing (approximately) the system (1.3). The computation of a kind of characteristic polynomial associated with (1.3), in symbolic form, allows to construct three symbols the quantization of which provides three scalar propagators (this will be explained more explicitly below).

Two of those propagators, called Poincaré propagators, are then proved to satisfy dispersive estimates; that result relies on a spectral analysis (usual semi-classical theory does not operate here due to the very large time scales at play) using Bohr-Sommerfeld quantization, which requires that b has only one, non degenerate critical value and which also uses very much the fact that the motion is translation-invariant in x 1 . A stationary phase argument on the spectral decomposition of any solution to the Poincaré propagation gives the result: Poincaré modes exit any compact set in finite time.

The last propagator is the Rossby one, which is one order of magnitude (in ε) smaller than the Poincaré modes. This allows to analyse the propagation by semi-classical analysis tools.

In particular the precise study of the dynamical system associated with those waves, which is an integrable system due to translation invariance in x 1 , allows to derive a condition on the initial microlocalization of the solution which guarantees that the Rossby waves are trapped for all times in a compact set.

Those results on the linear system (1.3) can finally be transposed to the original system (1.2) due to the high power of ε in front of the nonlinearity, and due to the semi-classical setting, which allows to exhibit vector fields which almost-commute with the linear operator A(x 2 , εD, ε).

1.3. Limitations of the methods of [START_REF] Cheverry | Semiclassical and spectral analysis of oceanic waves[END_REF]. The restriction which is the most used in the analysis described briefly in the previous paragraph is the fact that the stationary flow ū is a shear flow of the type ū = (ū 1 (x 2 ), 0). Indeed

• It allows to Fourier-transform in the direction x 1 , which makes the diagonalization procedure much easier; • It simplifies the spectral analysis of Poincaré waves, again due to the Fourier transform (in particular the dual variable ξ 1 is fixed during the propagation, and there is a wavelike behaviour in x 1 ); • It allows the Rossby dynamical system to be integrable, which is a tremendous help in the analysis.

An additional restriction in the previous arguments is that in order to prove the dispersion of Poincaré waves, the rotation amplitude b should have only one, non degenerate critical value: this allows to use a Bohr-Sommerfeld quantization argument to compute the eigenvalues of the Poincaré operator. This assumption on b is not really restrictive from the physical point of view. On the other hand, it is important for physical reasons to consider 2D convection flows.

1.4. On the nonlinear term. As explained above, most of the analysis in [START_REF] Cheverry | Semiclassical and spectral analysis of oceanic waves[END_REF] is concerned with the linear system (1.3). In order to transpose the linear results to the nonlinear setting, one uses the following arguments (along with the fact that the coupling is vanishes when ε goes to zero):

• Uniform existence which is obtained via an almost-commutation result;

• Bilinear estimates in anisotropic semi-classical spaces;

• A Gronwall lemma, which requires an L ∞ (R 2 ) bound on the linear solution. This is not known in general, due to the bad Sobolev embeddings in semi-classical settings, so the nonlinear result is proved for vanishing couplings only.

It is important to notice that none of those three steps require that ū is a shear flow. In the whole of this paper we shall therefore only focus on the linear equation, and leave to the reader the transposition to the nonlinear equation, using the above steps.

2. Main result of this paper and strategy of the proof 2.1. The model. In this paper we shall be concerned with the linear system (2.1)

ε 2 i∂ t v + A(x, εD, ε)v = 0 v = (v 0 , v 1 , v 2 ),
where the linear propagator is given by

(2.2) A(x, εD, ε) := i   εū • ε∇ ε∂ 1 ε∂ 2 ε∂ 1 εū • ε∇ + ε 2 ∂ 1 ū1 -b(x 2 ) + ε 2 ∂ 2 ū1 ε∂ 2 b(x 2 ) + ε 2 ∂ 1 ū2 εū • ε∇ + ε 2 ∂ 2 ū2   .
We shall assume throughout the paper that b is smooth, with a symbol-like behaviour: for all α ∈ N, there is a constant C α such that for all x 2 ∈ R,

(2.3) |b (α) (x 2 )| ≤ C α 1 + b 2 (x 2 ) 1 2 .
We shall further assume that lim

|x 2 |→∞ b 2 (x 2 ) = ∞,
and that b 2 has only non degenerate critical points.

We shall also suppose that the initial data is microlocalized in some compact set

C of T * R 2 satisfying (2.4) C ∩ {ξ 2 1 + ξ 2 2 + b 2 (x 2 ) = 0} = ∅ or actually rather (2.5) C ∩ {ξ 1 = 0} = ∅ .
We shall prove that assumption (2.4) is propagated by the flow, while (2.5) is propagated only by the Poincaré component. We recall (see for instance [START_REF] Cheverry | Semiclassical and spectral analysis of oceanic waves[END_REF], Appendix B) that a function f is microlocalized in a compact set C of T * R 2 if for any (x 0 , ξ 0 ) in the complement of C in R 4 (we shall identify T * R 2 to R 4 in the following), there is a smooth function χ 0 , bounded as well as all its derivatives and equal to one at (x 0 , ξ 0 ), satisfying

(2.6) Op W ε (χ 0 )u 0 L 2 (R 2 ) = O(ε ∞ ),
where Op W ε denotes the Weyl quantization:

(2.7) Op W ε (χ 0 )u 0 (x) := 1 (2πε) 4 e i(x-y)•ξ/ε χ 0 ( x + y 2 , ξ)u 0 (y) dydξ.

We also recall that (2.6) means that for any N ∈ N, there are ε 0 and C such that

∀ε ∈]0, ε 0 ], Op W ε (χ 0 )u 0 L 2 (R 2 ) ≤ Cε N .
In the following, to simplify some formulations, we shall denote by (µ)Supp ⋆ f the projection of the (micro)support of f onto the ⋆ = 0 axis.

2.2.

Statement of the main result and organization of the paper. Let us state the main theorem proved in this paper.

Theorem 1. Let v ε,0 be a family of initial data, microlocalized in a compact set C satisfying Assumptions (2.4)- (2.5). For any parameter ε > 0, denote by v ε the associate solution to (2.1). Then for all t ≥ 0 one can write v ε (t) as the sum of a "Rossby" vector field and a "Poincaré" vector field:

v ε (t) = v R ε (t) + v P ε (t)
, satisfying the following properties:

(1) µSupp v R ε (t) and µSupp v P ε (t) satisfy (2.4) for all times. (2) For any compact set Ω in R 2 , one has

∀t > 0, v P ε (t) L 2 (Ω) = O(ε ∞ ).
(3) µSupp x 2 v R ε (t) lies in a bounded subset of R uniformly in time.

Compared to [START_REF] Cheverry | Semiclassical and spectral analysis of oceanic waves[END_REF], the main difficulties are due to the presence of a x 1 -dependent underlying flow ū. The diagonalization of the system (exhibiting Rossby and Poincaré-type waves, with very different qualitative features) must be revised, and obtained in a less explicit way. Moreover the proof of (2) in Theorem 1, namely the dispersion of Poincaré waves can also not be proved in the same way (note that it is not assumed here that b 2 has only one non degenerate critical value). Finally the trapping of Rossby waves seems much harder to obtain since the underlying dynamical system no more decouples; the behaviour of the Rossby waves is therefore much less precise than in [START_REF] Cheverry | Semiclassical and spectral analysis of oceanic waves[END_REF].

Let us explain our strategy here, compared with that in [START_REF] Cheverry | Semiclassical and spectral analysis of oceanic waves[END_REF] described above.

2.2.1. The diagonalization. The construction of the Rossby and Poincaré modes is not as direct as in [START_REF] Cheverry | Semiclassical and spectral analysis of oceanic waves[END_REF] due to the lack of translation invariance in x 1 . We choose therefore to follow a more abstract way to recover those modes in Section 3, which relies on semi-classical analysis, and normal forms (instead of explicit computations as in [START_REF] Cheverry | Semiclassical and spectral analysis of oceanic waves[END_REF]). Finding the propagators associated with those modes requires a microlocalisation assumption of the type (2.4), in order for the eigenvalues of the matrix of principal symbols to be well separated. The diagonalization result is therefore in this paragraph conditional to the fact that the solution to the propagation equation is correctly microlocalized (that corresponds to Point 1 of Theorem 1).

Dispersion of Poincaré waves and propagation of the nondegeneracy assumption (2.5).

In order to prove (2) in Theorem 1 we again rely on a more abstract, and more efficient method than that followed in [START_REF] Cheverry | Semiclassical and spectral analysis of oceanic waves[END_REF]. It is based on Mourre estimates and the assumption (2.5) on the initial data: we start by proving, by a semi-classical argument, that after a very short time (of the order of ε) the support in x 1 of the solution escapes the support of ū. Then we use Mourre estimates to prove that the solution remains outside the support of ū for all times, and actually escapes any compact set in x 1 in finite time (to prove this last point we use the fact that the equation reduces to a translation-invariant equation in x 1 since the support of the solution is outside the support in x 1 of ū). This allows finally to check that the nondegeneracy assumption (2.5) does hold for all times. This analysis is achieved in Section 4.

Study of Rossby waves and propagation of the nondegeneracy assumption (2.4).

In Section 5 we first prove that the nondegeneracy assumption (2.4) does hold during the propagation of Rossby waves. That is due to semi-classical analysis, by the study of the dynamical system associated with those waves. The study of that system is also the key to the proof of Point (3), which is also proved in Section 5.

Reduction to scalar propagators

In this section we shall construct three operators T + , T -and T R diagonalizing A(x, εD, ε).

We shall start by proving a general diagonalization result, and at the end we shall apply the general result to our context.

Before stating the general result, let us give some notation. A semi-classical symbol is a function a = a(x, ξ; ε) defined on R 2d ×]0, ε 0 ] for some ε 0 > 0, which depends smoothly on (x, ξ) and such that for any α ∈ N 2d and any compact K ⊂ R 2d , there is a constant C such that for any ((x, ξ), ε) ∈ K×]0, ε 0 ],

|∂ α a((x, ξ), ε)| ≤ C.
We shall consider the Weyl quantization of such symbols, as recalled in (2.7): for all u is in D(R d ),

Op W ε (a)u(x) := 1 (2πε) d e i(x-y)•ξ/ε a( x + y 2 , ξ)u(y) dydξ.
We shall say that a pseudodifferential operator Op W ε (a) is supported in a set K if for any smooth function χ equal to one in a neighborhood of K one has aχ = a.

Finally we shall say that a matrix is pseudodifferential if each of its entries is a pseudodifferential operator.

Let us first prove the following general result. Theorem 2. Let K be a compact subset of R 2d , and consider a N × N hermitian pseudodifferential matrix A ε = A(x, εD, ε), supported in K. Assume that • the (matrix) principal symbol of A(x, εD, 0), denoted by A 0 , is diagonalizable, in the sense that there are some unitary and diagonal matrices of symbols, U and D, such that

U -1 A 0 U = D,
• the eigenvalues (δ 1 (x, ξ), . . . , δ N (x, ξ)) satisfy

(3.1) ∀i = j, inf (x,ξ)∈K |δ i (x, ξ) -δ j (x, ξ)| ≥ C > 0.
Then there exists a family of unitary and diagonal pseudodifferential operators V ε and D ε supported in K, such that:

(3.2) V * ε A ε V ε = D ε + O(ε ∞ ), V * ε V ε = I + O(ε ∞ ). Moreover one has (3.3) D ε = Op W ε (D) + εD 1 + O(ε 2 )
, where the principal symbol of D 1 is given by

D 1 = diag ∆ 1 - D 0 I 1 + I 1 D 0 2
with the notations

(3.4) ∆ 1 = 1 ε Op W ε (U * )A ε Op W ε (U ) -D 0 , I 1 = 1 ε Op W ε (U * )Op W ε (U ) -I
More explicitly, let us denote by a ij (x, ξ) the matrix elements of A 1 , subsymbol of A(x, εD, ε) defined by:

A 1 (x, ξ) := ∂A ∂ε (x, ξ, 0)
and by u nj (x, ξ), i = 1 . . . N, the coordinates of any unit eigenvector of A 0 (x, ξ) of eigenvalue δ n (x, ξ). We have

(3.5) (D 1 ) nn = jk=1...N ℑ (u jn {a jk , u kn }) + a jk {u jn , u kn } 2i + (U * A 1 U ) nn ,
where {f, g}

:= ∇ ξ f ∇ x g -∇ x f ∇ ξ g is the Poisson bracket on T * R n .
Here and in all the sequel, we say that a pseudo-differential operator V is unitary if it satisfies

V * V = I + O(ε ∞ ).
The proof is divided into two parts: in Section 3.1 we present the formal construction and in Section 3.2 we show that the symbols of the various operators formally constructed are indeed symbols. Finally Section 3.3 is devoted to the case of the matrix given by (2.2).

3.1. The formal construction. The proof of Theorem 2 is a combination of semiclassical and perturbation methods. Let us start by defining

U 0 = Op W ε (U ). Elementary properties of the Weyl quantization imply then that U * 0 A ε U 0 = D 0 + O(ε).
The following proposition shows that one can construct a unitary pseudodifferential operator

U ∞ such that U * ∞ A ε U ∞ = D 0 + O(ε). Lemma 3.1. Let U be a pseudodifferential matrix such that U * U = I + εI 1 , where I is the identity. Then one can find V ∼ ∞ k=0 ε k V k such that (3.6) (U + εV ) * (U + εV ) = I + O(ε ∞ ). Proof. Let us denote V 0 := - 1 2 I 1 U . On easily check that (U + εV 0 ) * (U + εV 0 ) = I + O(ε 2 ). Indeed (U + εV 0 ) * (U + εV 0 ) = U * U + ε 2 (I 1 U * U + U * U I 1 ) + O(ε 2 ) = I + εI 1 -εI 1 + O(ε 2 ).
Then one concludes by iteration.

That lemma allows to define the pseudo-differential operator of (semiclassical) order 0

∆ 1 = 1 ε (U * ∞ A ε U ∞ -D 0 ) ,
where U ∞ is a unitary operator. Now our aim is to find a unitary operator

V ∞ (up to O(ε ∞ )) such that (U ∞ V ∞ ) * A ε (U ∞ V ∞ ) = D ∞ + O(ε ∞ ),
where D ∞ = D 0 + εD 1 + . . . is a diagonal matrix satisfying the conclusions of the theorem.

We shall write V ∞ = e iεW , with W selfadjoint (so V ∞ thus constructed is automatically unitary). We recall that if W is a pseudodifferential operator, then so is e iεW (simply by writing

e iεW ∼ ∞ 0 (iε) k k! W k ).
We look for W under the form W ∼ ∞ 0 ε k W k , and compute the W k recursively. Since

V * ∞ (D 0 + ε∆ 1 )V ∞ = (D 0 + ε∆ 1 ) + iε[(D 0 + ε∆ 1 ), W ] + (iε) 2 2 [[(D 0 + ε∆ 1 ), W ], W ] + . . . we see that, if W 1 satisfies (3.7) [D 0 , W 1 ] + ∆ 1 = D 1 + O(ε 2 ), D 1 diagonal,
then we have that

(3.8) e -iεW 1 (D 0 + ε∆ 1 )e iεW 1 = D 0 + εD 1 + ε 2 ∆ 2 ,
where ∆ 2 is a zero order pseudodifferential operator. The following lemma is a typical normal form type result, and is crucial for the following.

Lemma 3.2. Let D 0 be a diagonal pseudodifferential matrix whose principal symbol D 0 has a spectrum satisfying (3.1) and let ∆ 1 be a pseudodifferential matrix.

Then there exist two pseudodifferential matrices W and D 1 , with D 1 diagonal, such that:

(3.9) [D 0 , W ] + ∆ 1 = D 1 + ε∆ 2 ,
where ∆ 2 is a pseudodifferential matrix of order 0.

Moreover the principal symbol of D 1 is the diagonal part of the principal symbol of ∆ 1 .

Proof. By the non degeneracy condition of the spectrum of D 0 we know, by standard arguments (see [START_REF] Reed | Methods of Modern Mathematical Physics[END_REF] for instance), that there exists a matrix W 0 and a diagonal one

D 1 such that [D 0 , W 0 ] + D 1,0 = D 1 ,
where D 1,0 is the principal symbol of ∆ 1 .

Indeed it is enough to take D 1 as the diagonal part of D 1,0 and

(3.10) (W 0 (x, ξ)) i,j = (D 1,0 (x, ξ)) i,j δ i (x, ξ) -δ j (x, ξ)
and notice that the Weyl quantization of W 0 satisfies (3.9). By Lemma 3.2 we know that there exists W 1 satisfying (3.7). Developing (3.8) as

e -iεW 1 (D 0 + ε∆ 1 )e iεW 1 = D 0 + ε(∆ 1 + [D 0 , W 1 ]) + ε 2 ∆ 2 ,
we get immediately (3.8).

It is easy to get convinced that all the W k will satisfy recursively an equation of the form

[D 0 , W k ] + ∆ k = D k + O(ε),
which can be solved by Lemma 3.2.

The expression for the principal symbol of D 1 follows by construction and the following well known lemma (see [START_REF] Martinez | An introduction to semiclassical and microlocal analysis[END_REF] for instance): In order to derive (3.5) we have to compute the subprincipal symbol of the diagonal part of the right-hand side of (3.4), that is, for each n = 1 . . . N ,

jk Op W ε (U jn )Op W ε ((A 0 + εA 1 ) jk )Op W ε (U kn ),
since U is unitary.

The term εA 1 is obviously responsible for the second term in the right-hand side of (3.5). Using Lemma 3.3 and the distributivity of the Poisson bracket, we get the following expression for the first one:

jk 1 2i {U jn , (A 0 ) jk U kn } + U jn {(A 0 ) jk , U kn } = jk 1 2i U jn {(A 0 ) jk , U kn } + (A 0 ) jk {U jn , U kn } + U kn {U jn , (A 0 ) jk }
Interverting j and k in half of the terms and noticing that, since A 0 is Hermitian, (A 0 ) jk = (A 0 ) kj , we get easily (3.5).

Symbolic properties.

With the hypothesis that both D and U are pseudodifferential matrices it is quite obvious that V ε and D ε are pseudodifferential matrices as well. Indeed the formal construction in the preceding section shows that the iterative process uses only three things: multiplications of pseudodifferential operators, computation of subprincipal symbols and solving equation (3.9).

For (3.9), the formula (3.10) used in the proof of Lemma 3.2, together with the non-degeneracy condition (3.1) which shows clearly that (δ i (x, ξ)δ j (x, ξ)) -1 is a symbol, implies that W 0 is a pseudodifferential operator.

Note that the microlocalization assumption is crucial in order that the expansions obtained by this iterative construction do define symbols. We have indeed no uniform control on the growth at infinity.

The Rossby-Poincaré case.

In the case of oceanic waves A(x, εD, ε) is given by (2.2):

A(x, εD) := i   εū • ε∇ ε∂ 1 ε∂ 2 ε∂ 1 εū • ε∇ + ε 2 ∂ 1 ū1 -b(x 2 ) + ε 2 ∂ 2 ū1 ε∂ 2 b(x 2 ) + ε 2 ∂ 1 ū2 εū • ε∇ + ε 2 ∂ 2 ū2   . Therefore A 0 (x, ξ) :=   0 ξ 1 ξ 2 ξ 1 0 -ib(x 2 ) ξ 2 ib(x 2 ) 0   ,
and

A 1 (x, ξ) :=   ū • ξ 0 0 0 ū • ξ 0 0 0 ū • ξ   = ū • ξ   1 0 0 0 1 0 0 0 1   .
A straightforward computation shows that the spectrum of A 0 is

0, ξ 2 1 + ξ 2 2 + b 2 (x 2 ), -ξ 2 1 + ξ 2 2 + b 2 (x 2 ) .
3.3.1. Microlocalization. The three eigenvalues of A 0 are separated if and only if

ξ 2 1 + ξ 2 2 + b 2 (x 2 ) = 0. Therefore, considering a compact subset K of R 2d such that K ∩ {(x 1 , x 2 , ξ 1 , ξ 2 ) / ξ 2 1 + ξ 2 2 + b 2 (x 2 ) = 0} = ∅ ensures that
• the eigenvalues do not cross, so that it is possible to get a unitary diagonalizing matrix with regular entries; • the non degeneracy condition (3.1) is satisfied.

In other words, A(x, εD, ε) satisfies the assumptions of Theorem 2 provided that one considers only its action on vector fields which are suitably microlocalized.

We assume of course that this microlocalization condition is satisfied by the initial datum, which is the condition (2.4).

Furthermore, we will prove in the next two sections that the propagation by the scalar operators T ± and T R (to be defined now) preserves this suitable microlocalization, thus justifying a posteriori the diagonalization procedure for all times.

Computation of the Poincaré and Rossby Hamiltonians. The above computations show that one can define the two Poincaré Hamiltonians as follows:

τ ± := ± ξ 2 1 + ξ 2 2 + b 2 (x 2
) and we shall denote the associate operator constructed via Theorem 2 by T ± . Now let us consider the Rossby Hamiltonian. In all this paragraph, for the sake of readability, we will denote

ξ b = ξ 2 1 + ξ 2 2 + b 2 (x 2
) . An easy computation shows that a (normalized) eigenvector of A 0 (x, ξ) of zero eigenvalue is

u 0 = 1 ξ b   b iξ 2 -iξ 1  
By Theorem 2, the Rossby Hamiltonian is then given by the formula

(3.11) τ R = j,k=1...3 ℑ (u j0 {a jk , u k0 }) + a jk {u j0 , u k0 } 2i + j,k=1...3 (A 1 ) jk u j0 u k0 .
In order to compute the different Lie brackets, we start with a couple of simple remarks :

{ξ j , f } = ∂ x j f and {b(x 2 ), f } = -b ′ (x 2 )∂ ξ 2 f .
In particular, if f does not depend on x 1 , then {ξ 1 , f } = 0.

The contribution of the first term in the parenthesis in (3.11) is j,k=1...3

(u j0 {a jk , u k0 }) = b ξ b ξ 2 , -iξ 1 ξ b + iξ 2 ξ b ib, -iξ 1 ξ b + iξ 1 ξ b ξ 2 , b ξ b + ib, iξ 2 ξ b = -ibξ 1 ξ b ∂ x 2 1 ξ b - iξ 2 ξ 1 b ′ ξ b ∂ ξ 2 1 ξ b + iξ 1 ξ b ∂ x 2 b ξ b + iξ 1 b ′ ξ b ∂ ξ 2 ξ 2 ξ b = 2iξ 1 b ′ ξ 2 b •
Using the distributivity of the Poisson brackets, we get the contribution of the second term in a very similar way j,k=1...3

a jk {u j0 , u k0 } 2 = ξ 1 b ξ b , iξ 2 ξ b -ξ 2 b ξ b , iξ 1 ξ b + ib iξ 1 ξ b , iξ 2 ξ b = ξ 1 ib ξ b 1 ξ b , ξ 2 + iξ 2 ξ b b, 1 ξ b + i ξ 2 b {b, ξ 2 } - iξ 2 ξ 1 ξ b b, 1 ξ b - ibξ 1 ξ b 1 ξ b , ξ 2 = -ib ′ ξ 1 ξ 2 b •
The computation of the second term of the right hand side of (3.11) is trivial since A 1 is a multiple of the identity. Adding with the two previous expressions we get finally

τ R = ξ 1 b ′ ξ 2 1 + ξ 2 2 + b(x 2 ) 2 + ū • ξ
and the associate operator will be denoted by T R .

Remark 3.4. Since the elementary steps of the diagonalization process use only multiplications, computations of subprincipal symbols and solving normal forms equations, all the subsymbols of T R and T ± depend on x 1 only through ū and its derivatives.

Study of the Poincaré waves

In Section 3 we constructed two linear operators, called T ± , whose principal symbols are

τ ± = ± ξ 2 1 + ξ 2 2 + b 2 (x 2
). We now want to study the propagation equation associated to those operators, namely the linear equation in R × R 2 (4.1)

iε 2 ∂ t ϕ ± = T ± ϕ ± , ϕ ±|t=0 = ϕ 0 ±
where ϕ 0 ± are microlocalized in a compact set C satisfying Assumption (2.4). Before studying that equation we need to check that it makes sense, since a priori T ± is only defined on vector fields microlocalized on such a compact set. This is achieved in the coming section, where we check that the separation of eigenvalues (3.1) required in the statement of Theorem 2 holds because ξ 2 1 + ξ 2 2 + b 2 (x 2 ) remains bounded away from zero during the propagation.

Then we shall show that the solutions to these equations exit any compact set in finite time (Point (2) of Theorem 1).

4.1. Microlocalization. Let us prove the following result, which provides the first part of Point (1) in Theorem 1 and allows to make sense of Equation (4.1) for all times.

Proposition 4.1. Under the assumptions of Theorem 1, the operators T ± are self-adjoint, and the function ϕ(t) = e i t ε 2 T ± ϕ 0 ± are such that µSuppϕ ± (t) satisfies (2.4) for all times.

Proof. The proof of that result relies on a spectral argument. Due to the form of the principal symbols of T ± recalled above, the operators T ± are self-adjoint. We can therefore define two families (ψ ± n ) n∈N of (pseudo)-eigenvectors of T ± in L 2 (R 2 ) and two sequences of eigenvalues λ ± n such that if the initial data writes

ϕ 0 ± (x) = n c ±,0 n ψ ± n (x), then ϕ ± (t, x) = n e i λ ± n t ε 2 c ±,0 n ψ ± n (x).
Since the eigenfunctions ψ ± n are microlocalized on the energy surfaces of the Poincaré Hamiltonians, the result follows.

Dispersion.

In this paragraph we shall prove Point (2) of Theorem 1. The strategy is the following.

In Section 4.2.1 we prove using semi-classical analysis that for a very short time, the solutions to (4.1) remain microlocalized in a compact set satisfying assumption (2.5), and such that µSupp x 1 ϕ ± become disjoint from Supp x 1 ū. Section 4.2.2 is then devoted to the long-time behaviour of the solution, and Mourre estimates allow to prove that the solution exits any compact set after some time, and that it remains microlocalized far from ξ 1 = 0. 4.2.1. Short time behaviour. The aim of this paragraph is to prove the following result. It shows that the solutions of (4.1) exit the support of ū after a time t exit ε, for |t exit | large enough (independent of ε). We only state the forward in time result: the backwards result is identical, up to changing the sign of time. We shall further restrict the analysis to T + since the argument for T -is identical, up to some sign changes. Proposition 4.2. Let ϕ 0 be a function, microlocalized in a compact set C satisfying Assumption (2.4), and let ϕ be the associate solution of (4.1). Let [u -, u + ] be a closed interval of R containing Supp x 1 ū. There exists a constant t exit > 0 such that for any

ε ∈]0, 1[, the function ϕ(εt exit , •) is microlocalized in a compact set K such that the projection of K onto the x 1 -axis does not interesect [u -, u + ]. Moreover µSupp ξ 1 ϕ is unchanged. More precisely, if µSupp ξ 1 ϕ 0 ⊂ R + \{0}, then µSupp x 1 ϕ(εt exit , •) ⊂]u + , +∞[, and if µSupp ξ 1 ϕ 0 ⊂ R -\ {0}, then µSupp x 1 ϕ(εt exit , •) ⊂] -∞, u -[.
Proof. Define the function ψ(s) := ϕ(εs). Then (4.1) reads (4.2)

iε∂ s ψ = T + ψ, ψ |s=0 = ϕ 0 ,
and any result proved on ψ on [0, T ] will yield the same result for ϕ on [0, T ε]. Notice that (4.2) is written in a semi-classical setting, so by the propagation of the microsupport theorem (see for instance [START_REF] Martinez | An introduction to semiclassical and microlocal analysis[END_REF] and the references therein), the microsupport of ψ is propagated by the bicharacteristics, which are the integral curves of the principal symbol. Recall that the principal symbol of T + is

τ + (ξ 1 , x 2 , ξ 2 ) = ξ 2 1 + ξ 2 2 + b 2 (x 2 )
and the bicharacteristics are given by the following set of ODEs:

ẋt = ∇ ξ τ + (ξ t 1 , x t 2 , ξ t 2 ), x 0 = (x 0 1 , x 0 2 ) ξt = -∇ x τ + (ξ t 1 , x t 2 , ξ t 2 ), ξ 0 = (ξ 0 1 , ξ 0 2 ).
Notice that τ + is independent of x 1 , so ξt 1 is identically zero and therefore ξ t 1 ≡ ξ 0 1 . So for all s ≥ 0, the microlocal support in ξ 1 of ψ(s) remains unchanged, and in particular is far from ξ 1 = 0. Moreover one has

ẋt 1 = ξ 0 1 (ξ 0 1 ) 2 + (ξ t 2 ) 2 + b 2 (x t 2 )
• Now we recall that the bicharacteristic curves lie on energy surfaces, meaning that on each bicharacteristic, τ

+ (ξ 0 1 , x t 2 , ξ t 2 ) is a constant. That implies that (ξ t 2 ) 2 + b 2 (x t
2 ) is a constant on each bicharacteristic, so that for all times,

ẋt 1 ≡ ξ 0 1 (ξ 0 1 ) 2 + (ξ 0 2 ) 2 + b 2 (x 0 2 )
• If ξ 0 1 > 0, then x 1 is propagated to the right and eventually escapes to the right of the support in x 1 of ū, whereas if ξ 0 1 < 0, the converse (to the left) occurs. Proposition 4.2 is proved. Proof. Before going into the proof, we shall simplify the analysis by only studying the case of T + (the case T -is obtained by identical arguments), and we shall only deal with the case when the support in ξ 1 of the data lies in the positive half space. The other case is obtained similarly.

The proof is based on Mourre's theory which we shall now briefly recall, and we refer to [START_REF] Jensen | Multiple commutator estimates and resolvent smoothness in quantum scattering theory[END_REF] and [START_REF] Hunziker | Minimal escape velocities[END_REF] for all details. Let us consider two self-adjoint operators H and A on a Hilbert space H. We make the following assumptions:

(1) the intersection of the domains of A and H is dense in the domain D(H) of H .

(2) t → e itA maps D(H) to itself, and for all ϕ 0 ∈ D(H),

sup t∈[0,1]
He itA ϕ 0 < ∞. 3)

E ∆ i[H, A]E ∆ ≥ θE ∆ .
Note that Assumptions (1 -3) can be replaced by the fact that [f (H), A] is bounded for any smooth, compactly supported function f (see [START_REF] Hunziker | Minimal escape velocities[END_REF]).

Under those assumptions, for any integer m ∈ N and for any θ ′ ∈]0, θ[, there is a constant C such that χ -(Aaθ ′ t)e -iHt g(H)χ + (Aa) ≤ Ct -m where χ ± is the characteristic function of R ± , g is any smooth compactly supported function in ∆, and the above bound is uniform in a ∈ R. As pointed out in [START_REF] Hunziker | Minimal escape velocities[END_REF], this implies in particular that for any ϕ 0 in the image of E ∆ , the function e -iHt ϕ 0 has spectral support in [a + tθ ′ , ∞[ with respect to A, up to t -∞ .

Let us apply this theory to our situation. We consider equation (4.1) with data e i t exit ε T + ϕ 0 + , and let us define the operator T 0 + as the operator T + where ū has been chosen identically zero. We shall start by studying the equation (4.4) iε 2 ∂ t ϕ = T 0 + ϕ, ϕ |t=εt exit = e i t exit ε T + ϕ 0 + , for which we shall prove Proposition 4.3. Then we shall prove that the solution ϕ actually solves the original equation (4.1) with the same data e i t exit ε T + ϕ 0 + at t = εt exit up to O(ε ∞ ), because its support in x 1 lies outside the support of ū and because the symbolic expansion of T + depends on x 1 only through ū and its derivatives (see Remark 3.4).

So let us start by applying Mourre's theory to (4.4). Let us write the projection of K onto the ξ 1 -axis as included in [d 0 , d 1 ] with 0 < d 0 < d 1 < ∞. We recall that on the support of e i t exit ε T + ϕ 0 + , x 1 remains to the right of the support of ū. Then we apply the theory to H = T 0 + and to A = x 1 (the pointwise multiplication). Assumptions (1) to (3) are easy to check, in particular because this is a semiclassical setting, so only the principal symbols need to be computed. Similarly finding a lower bound for

E ∆ i[T 0 + , x 1 ]E ∆ boils down to computing the Poisson bracket {τ + , x 1 } where {f, g} = ∇ ξ f • ∇ x g -∇ x f • ∇ ξ g,
and one finds

(4.5) {τ + , x 1 } = ξ 1 ξ 2 1 + ξ 2 2 + b 2 (x 2 )
• Since T 0 + has constant coefficients in x 1 , one can take the Fourier transform of (4.4) and ξ 1 is preserved, so in particular for all times one has µSupp ξ 1 ϕ(t) ⊂ [d 0 , d 1 ]. One can furthermore choose for ∆ an interval of R of the type ]D 0 , D 1 [ where the constants D 0 and D 1 are chosen so that for any (x, ξ) ∈ K, one has

(4.6) D 0 < ξ 2 1 + ξ 2 2 + b 2 (x 2 ) < D 1 .
As the microlocal supports of the eigenfunctions of T 0 + lie on energy surfaces, we know that the solution to (4.4) will remain in E ∆ for all times. Now let us apply the results of [START_REF] Jensen | Multiple commutator estimates and resolvent smoothness in quantum scattering theory[END_REF] and [START_REF] Hunziker | Minimal escape velocities[END_REF]. By Lemma 3.3, (4.5), (4.6) and the assumption on ξ 1 written above, we have that

E ∆ i[H, A]E ∆ ≥ ε d 0 D 0 E ∆ , so (4
ε 2 T 0 + e i t exit ε T + ϕ 0 + = e i (t-εt exit ) ε 2 T + e i t exit ε T + ϕ 0 + in L 2
locally uniformly in t (see Appendix).

The proposition follows. Using the fact that ū has compact support, and simple properties of the Rossby dynamics in the absence of zonal flow, we can prove the following Proposition 5.2. Let C be some compact subset of R 4 such that

C ∩ {(x 1 , x 2 , ξ 1 , ξ 2 ) / ξ 2 1 + ξ 2 2 + b 2 (x 2 ) = 0} = ∅ .
Then the bicharacteristics of the Rossby Hamiltonian starting from any point of C are bounded in x 2 : sup

(x 0 1 ,x 0 2 ,ξ 0 1 ,ξ 0 2 )∈C |x 2 (t)| < ∞.
Proof. • Let us start by describing the dynamics in the absence of zonal flow: ξ 1 is then an invariant of the motion, so that the dynamics in (x 2 , ξ 2 ) can be decoupled. Furthermore, as the energy surfaces are compact

τ = ξ 1 b ′ (x 2 ) ξ 2 1 + ξ 2 2 + b 2 (x 2
) the motion along x 2 is periodic (with infinite period for homoclinic and heteroclinic orbits).

The motion along x 1 is then determined by the equation

dx 1 dt = b ′ (x 2 ) ξ 2 b -2ξ 2 1 ξ 4 b •
It is trapped if and only if the average of the right-hand side over one period is zero. Outside from saddle points, this quantity depends continuously on ξ 1 , so that we expect the initial data leading to trapped trajectories to belong to a manifold of codimension 1. This can be proved rigorously if b 2 has only one non degenerate critical points (see [START_REF] Cheverry | Semiclassical and spectral analysis of oceanic waves[END_REF]).

• Let us now turn to the influence of the zonal flow. We will first check that the only possible escape direction is again x 1 . Indeed the energy surfaces corresponding to τ = 0 are bounded in the x 2 direction : as

x 2 → ±∞, b ′ (x 2 )ξ 1 ξ 2 b + ū(x) • ξ → 0 .
Consider now a trajectory on the energy level τ R = 0, and some point of this trajectory (y 1 , y 2 , ξ 1 , ξ 2 ) such that y 2 / ∈ Supp x 2 ū. One has b ′ (y 2 )ξ 1 = 0 .

-If b ′ (y 2 ) = 0, then

dx 1 dt = dx 2 dt = dξ 1 dt = dξ 2 dt = 0 .
The uniqueness in Cauchy-Lipschitz theorem implies then that the trajectory is nothing else than a fixed point, and therefore in particular is bounded.

-If ξ 1 = 0, then , meaning that the trajectory is a uniform straight motion along x 1 . In particular, it is bounded in the x 2 -direction.

Finally, we conclude that trajectories on the energy level τ R = 0 are either trapped in the support Supp x 2 ū, or trivial in the x 2 -direction.

5.3. Perspectives. As recalled in the introduction, it is generally believed that in the situation depicted in this paper (a flow around a large macroscopic current) Rossby waves are trapped. However due to the 2-dimensional setting (compared to the work in [START_REF] Cheverry | Semiclassical and spectral analysis of oceanic waves[END_REF]) the trapping in the x 1 direction seems difficult to prove, outside some specific cases studied in the previous paragraph. One way to be convinced of the trapping phenomenon should be by implementing the dynamical system numerically. It should be pointed out however that actually in order to get physically relevant predictions for the oceanic eddies, one should consider 3D models, or at least 2D models involving the influence of stratification. The methods presented here seem to be robust and should be extended to such complex models, up to again the study of the hamiltonian system describing the Rossby dynamics.

Appendix : a comparison result

For the sake of completeness, we state here the result which shows the stability of the propagation under a O(ε ∞ ) error on the propagator. This result has been implicitly used in the proof of the diagonalization when comparing A and T ± , T R , and in the proof of dispersion when comparing T ± and T 0 ± .

Proposition 5.3. Let A ε and Ãε be two pseudo-differential operators such that

• iA ε is hermitian in L 2 (R d ), • A ε -Ãε = O(ε ∞ ) microlocally on Ω ⊂ R 2d .
Let φ be a solution to i∂ t φ + Ãε φ = 0 microlocalized in Ω, and ϕ be the solution to

i∂ t ϕ + A ε ϕ = 0
with the same initial data. Then, for all N ∈ N,

sup t≤ε -N ϕ(t) -φ(t) L 2 (R d ) = O(ε ∞ ) .
Proof. The proof is based on a simple energy inequality and is completely straightforward. We have

d dt ϕ -φ 2 L 2 (R d ) = 2 iA ε ϕ -i Ãε φ|ϕ -φ = 2 (iA ε -i Ãε ) φ|ϕ -φ ≤ 2 (A ε -Ãε ) φ L 2 (R d ) ϕ -φ L 2 (R d )
. This leads to ϕ(t)φ(t) 2 L 2 (R d ) = O(ε ∞ )t , which concludes the proof.

Lemma 3 . 3 .

 33 Let a and b two symbols. Then the principal symbol of Op W ε (a)Op W ε (b) is ab and its subprincipal symbol is 1 2i {a, b}.

( 3 )

 3 The operator i[H, A] is bounded from below and closable, and the domain D(B 1 ) where iB 1 is its closure, contains D(H). More generally for all n ∈ N the operator i[iB n , A] is bounded from below and closable and the domain D(B n+1 ) of its closure iB n+1 contains D(H), and finally B n+1 extends to a bounded operator from D(H) to its dual. (4) There exists θ > 0 and an open interval ∆ of R such that if E ∆ is the corresponding spectral projection of H, then (4.

5 . 5 . 2 .

 552 Propagation of the Rossby waves 5.1. Semiclassical transport equations and microlocalization. In any case, we have the lower bound ξ b ≥ η min 1, 1 |τ | + ū ∞ η , which can be made uniform for initial data in C replacing |τ | by max C |τ |. Dynamics outside from the support of ū.

b 2 (y 2 ) -ξ 2 1 ξ 4 b

 2214

  4.2.2.Long time behaviour. The aim of this paragraph is to prove the following result, which again is only proved for positive times for simplicity. Proposition 4.3. Under the assumptions of Proposition 4.2, let ϕ + be the solution of (4.1) associated with the data ϕ(εt exit , •). Then µSupp x 1 ϕ + (t) does not intersect Supp x 1 ū for t ≥ εt exit , and µSupp ξ 1 ϕ + (t) remains unchanged for t ≥ εt exit . Finally µSupp x 1 ϕ + (t) exits any compact set in x 1 in finite time.

  .3) holds with θ = εd 0 /D 0 . It follows that the solution e i (t-εt exit )

				ε 2	T 0 + e i t exit ε T + ϕ 0 + to (4.4)
	has a support in x 1 such that
		x 1 > u + +	d 0 D 0	t ε
	which proves the result for (4.4).
	Since µSupp x 1 e i (t-εt exit ) ε 2	T 0 + e i t exit ε T + ϕ 0 + does not cross Supp x 1 ū, one has actually
	e i (t-εt exit )

Acknowledgements. The authors are grateful to J.-F. Bony and N. Burq for introducing them to Mourre estimates, and for interesting discussions. I. Gallagher and L. Saint-Raymond are partially supported by the French Ministry of Research grant ANR-08-BLAN-0301-01.

Because of the scaling of the Rossby hamiltonian (which is smaller than the Poincaré hamiltonians by one order of magnitude), on the times scales considered here, the propagation of energy by Rossby waves is described by the hamiltonian dynamics

which can be written explicitly (5.1)

where we recall that

. In order for the dynamics to be well defined and also in order to justify the diagonalization process, we need the quantity ξ b to remain bounded from below for all times.

Proposition 5.1. Let C be some compact subset of R 4 such that

Then the bicharacteristics t → (x(t), ξ(t)) of the Rossby Hamiltonian starting from any point (x 0 1 , x 0 2 , ξ 0 1 , ξ 0 2 ) of C are defined globally in time, and ∀t ∈ R, inf

Proof. As b ′ , b ′′ , u and Du are Lipschitz, by the Cauchy-Lipschitz theorem the system of ODEs (5.1) has a unique maximal solution. In order to prove that this solution is defined globally, it is enough to prove that the time derivative of this solution is uniformly bounded. This comes from assumption (2.3) giving an upper bound on b

, and from the lower bound on ξ b to be established now.

The crucial assumption here is the fact that b ′ and b do not vanish simultaneously, and more precisely the existence of η, β > 0 such that for all x 2 ∈ R.


Mathieu Hatt
email: hatt@univ-brest.fr

Catherine Cheze-Le Rest

Eric O Aboagye

Laura M Kenny

Lula Rosso

Federico E Turkheimer

Nidal M Albarghach

Jean-Philippe Metges

Olivier Pradier

Dimitris Visvikis

Hatt Mathieu

Latim

Reproducibility of 18F-FDG and 3'-deoxy-3'-18F-fluorothymidine PET tumor volume measurements

come

Introduction

Most of current PET clinical practice for diagnosis, staging, prognosis, therapy response assessment and patient follow-up rely on manual and visual analysis [START_REF] Kelloff | Progress and promise of FDG PET imaging for cancer patient management and oncologic drug development[END_REF].

The index most commonly employed in PET clinical studies is the standardized uptake value (SUV). In order to obtain this index of activity accumulation a region of interest (ROI) should be determined, usually drawn manually or using some fixed threshold. Despite not being the only factor that can affect the accuracy of SUVs, the type and size of ROI is a large contributor to the variability of such measures as has been previously demonstrated [START_REF] Visvikis | Influence of OSEM and Segmented Attenuation Correction in the Calculation of Standardised Uptake Values for 18FDG-PET[END_REF][START_REF] Krak | Effects of ROI definition and reconstruction method on quantitative outcome and applicability in a response monitoring trial[END_REF]. A popular alternative is the use of the pixel with the maximum activity value, usually referred to as SUV max . A large number of studies have demonstrated its prognostic and predictive value, despite the fact that it is sensitive to image noise [START_REF] Lucignani | Doctor, what does my future hold? The prognostic values of FDG PET in solid tumours[END_REF][START_REF] Wahl | From RECIST to PERCIST: Evolving Considerations for PET Response Criteria in Solid Tumors[END_REF]. On the other hand, there are a limited number of mostly recent studies that have explored the use of overall tumor volume (TV) as an index for prognosis and response assessment [START_REF] Seol | Measurement of tumor volume by PET to evaluate prognosis in patients with head and neck cancer treated by chemo-radiation therapy[END_REF][START_REF] Chung | Metabolic tumor volume of (18F)fluorodeoxyglucose positron emission tomography/computed tomography predicts short-term outcome to radiotherapy with or without chemotherapy in pharyngeal cancer[END_REF][START_REF] Hyun | Prognostic value of metabolic tumor volume measured by 18F-fluorodeoxyglucose positron emission tomography in patients with esophageal carcinoma[END_REF] considering either the TV alone or in combination with the mean SUV (SUV mean ) to form the total glycolytic or proliferative (for FDG and FLT respectively) volume (TGV/TPV), defined as the product of TV x SUV mean [START_REF] Larson | Tumor treatment response based on visual and quantitative changes in global tumor glycolysis using PET FDG imaging: the visual response score and the change in total lesion glycolysis[END_REF][START_REF] Francis | Early Prediction of Response to Chemotherapy and Survival in Malignant Pleural Mesothelioma Using a Novel Semiautomated 3-Dimensional Volume-Based Analysis of Serial 18F-FDG PET Scans[END_REF][START_REF] Cazaentre | Pre-therapy 18F-FDG PET quantitative parameters help in predicting the response to radioimmunotherapy in non-Hodgkin lymphoma[END_REF].

Clearly one of the issues associated with the use of functional volumes derived from PET images, which is directly responsible for the reduced use of such indices, is the accuracy, robustness, repeatability and reproducibility considering the delineation. On the one hand, manual delineation of functional volumes using PET images leads to high inter-and intra-observer variability (3), principally arising from the poor quality of PET images. On the other hand, current state-of-the-art algorithms for functional volume segmentation consist of fixed [START_REF] Erdi | Segmentation of Lung Lesion Volume by Adaptative Positron Emission Tomography Image Thresholding[END_REF] or adaptive thresholding approaches [START_REF] Daisne | Tri-dimensional automatic segmentation of PET volumes based on measured source-to-background ratios: influence of reconstruction algorithms[END_REF][START_REF] Nestle | Comparison of Different Methods for Delineation of 18F-FDG PET-Positive Tissue for Target Volume Definition in Radiotherapy of Patients with Non-Small Cell Lung Cancer[END_REF]. Although attractive as a result of the easiness of use, the drawbacks of fixed threshold approaches are numerous, since the value of the threshold to be used for each lesion clearly depends on multiple factors, such as lesion contrast and size as well as image noise [START_REF] Biehl | 18F-FDG PET definition of gross tumor volume for radiotherapy of non-small cell lung cancer: is a single standardized uptake value threshold approach appropriate?[END_REF]. The solutions based on the use of adaptive thresholding consider the contrast between the object to delineate and its surrounding background. However, they require imaging system specific optimisation carried out considering uniformly filled spherical lesions reducing the robustness of the approach, particularly in the case of multi-centre trials. In addition, their performance depends on the background ROI choice which can in turn lead to reduced inter-observer reproducibility for functional volume determination. A few automatic algorithms have been recently proposed [START_REF] El Naqa | Concurrent multimodality image segmentation by active contours for radiotherapy treatment planning[END_REF][START_REF] Montgomery | Fully automated segmentation of oncological PET volumes using a combined multiscale and statistical model[END_REF][START_REF] Geets | A gradient-based method for segmenting FDG-PET images: methodology and validation[END_REF][START_REF] Hatt | A fuzzy Bayesian locally adaptive segmentation approach for volume determination in PET[END_REF]. The main difference between these algorithms and the threshold-based approaches is that they automatically estimate the parameters of interest and find the optimal regions' characteristics in a given image without system-dependent parameters. This may allow reducing issues associated with deterministic approaches based on thresholding, potentially increasing the robustness and reproducibility of PET functional volumes determination [START_REF] Hatt | PET functional volume segmentation: a robustness study[END_REF].

Establishing the level of reproducibility and repeatability is essential in the use of any image derived index and its use in prognostic or therapy response studies, allowing the evaluation of which change between two studies can be considered significant. To date there have only been a limited number of reproducibility studies [START_REF] Minn | In vitro comparison of cell proliferation kinetics and uptake of tritiated fluorodeoxyglucose and L-methionine in squamouscell carcinoma of the head and neck[END_REF][START_REF] Weber | Reproducibility of metabolic measurements in malignant tumors using FDG PET[END_REF][START_REF] Nahmias | Reproducibility of standardized uptake value measurements determined by 18F-FDG PET in malignant tumors[END_REF][START_REF] Paquet | Within patient variability of FDG standardised uptake values in normal tissues[END_REF][START_REF] Velasquez | Repeatibility of 18F-FDG PET in a multicenter phase I study of patients with advanced gastrointestinal malignancies[END_REF], almost exclusively concentrating on SUV max and SUV mean variability in double baseline FDG PET scans, showing a relative absolute percentage difference of up to 13% with a standard deviation of 10%. The reproducibility of quantitative indices (Patlak influx constant, Ki), associated with the acquisition of dynamic datasets, have been also assessed [START_REF] Minn | In vitro comparison of cell proliferation kinetics and uptake of tritiated fluorodeoxyglucose and L-methionine in squamouscell carcinoma of the head and neck[END_REF][START_REF] Weber | Reproducibility of metabolic measurements in malignant tumors using FDG PET[END_REF] showing similar levels of reproducibility (mean percentage difference of 8%-10%). Studies on the reproducibility of such indices in the case of FLT PET imaging has shown that changes larger than 15-20% and 25-30% may be considered significant in SUV mean (obtained using a 41% fixed threshold) and SUV max or Ki respectively [START_REF] De Langen | Reproducibility of quantitative 18FLT measurements using positron emission tomography[END_REF][START_REF] Kenny | Imaging early changes in proliferation at 1 week post chemotherapy: a pilot study in breast cancer patients with FLT positron emission tomography[END_REF].

In the majority of these studies, SUV mean values have been calculated using manually drawn ROIs or a single fixed threshold (varying from 40% to 75% of the maximum activity). Among these studies only one has considered the reproducibility of metabolic functional volumes, using fixed threshold. Krak et al (3) have shown a mean percentage difference in the ROI volumes of 23±20% and 55±35% for a fixed threshold of 50% and 75% respectively. Finally, according to our knowledge there has been no published study evaluating the reproducibility of the TGV/TPV. To date, despite numerous studies assessing the accuracy of different segmentation algorithms there is a lack of evaluation of the repeatability and reproducibility of these algorithms relative to different threshold and automatic based delineation approaches. Therefore the main objective of this study was to assess the repeatability and reproducibility in determining 3D functional volumes and associated indices (SUV mean , TGV/TPV) in PET imaging using different algorithms. The reproducibility on SUV max was also included since it represents the index mostly used today in clinical practice as well as in order to facilitate a direct comparison with previous studies. This evaluation was carried out on double baseline FDG and FLT clinical PET datasets.

Materials and methods

Segmentation algorithms considered

Four approaches were used in this work. Two different fixed thresholds [START_REF] Erdi | Segmentation of Lung Lesion Volume by Adaptative Positron Emission Tomography Image Thresholding[END_REF] were considered, at 42% (T42) and 50% (T50) of the maximum voxel value, using a region growing algorithm with the maximum intensity voxel as seed.

An adaptive thresholding (TSBR) [START_REF] Daisne | Tri-dimensional automatic segmentation of PET volumes based on measured source-to-background ratios: influence of reconstruction algorithms[END_REF] was also included:

1 threshold I a b S B R   (1)
SBR is the source-to-background ratio, defined as the contrast between a manually defined background region of interest (ROI) and the mean of the maximum intensity voxel and its eight surrounding neighbours in the same slice. The parameters (a,b) are optimised through linear regression analysis for a given scanner using phantom acquisitions of various sphere sizes and contrast.

For automatic segmentation approaches, the Fuzzy C-Means (FCM) [START_REF] Dunn | A fuzzy relative of the isodata process and its use in detecting compact well-separeted clusters[END_REF] clustering algorithm was considered with two clusters (background and lesion). This algorithm has been previously used for functional volume segmentation tasks in both brain and oncology applications [START_REF] Zhu | Automation segmentation of PET image for brain tumours[END_REF][START_REF] Belhassen | Segmentation of heterogeneous tumors in PET using a novel fuzzy C-means algorithm[END_REF] and iteratively minimizes a cost function of the voxels intensity values in order to estimate the centre of each cluster and membership of each voxel to these clusters. The second automatic algorithm considered was the fuzzy locally adaptive Bayesian (FLAB) [START_REF] Hatt | A fuzzy Bayesian locally adaptive segmentation approach for volume determination in PET[END_REF] methodology, based on a combination of statistical models with a fuzzy measure in order to simultaneously address both issues of noise and blur resulting from partial volume effects (PVE) in PET images. FLAB is also able to deal with strongly heterogeneous uptake in complex-shaped tumours and generate non binary segmented volumes by considering three classes and the associated fuzzy transitions [START_REF] Hatt | Accurate automatic delineation of heterogeneous functional volumes in positron emission tomography for oncology applications[END_REF]. Estimation of the parameters required for the segmentation (Gaussian mean and variance of each class and spatial priors for each voxels) are estimated using the iterative Stochastic Expectation Maximization (SEM) procedure. For all approaches, the tumours were delineated after having been isolated in a 3D box of interest previously defined and fixed for all segmentation methodologies (manual and automatic).

Repeatability, reproducibility: definitions

Within the context of this study repeatability is defined as the ability of a given segmentation algorithm to reach the same result regarding the definition of a functional volume when applied multiple times on a single image. In such a task entirely deterministic fixed threshold approaches (T42,T50) will always give the same result. On the other hand, more advanced methods, like adaptive thresholding or automatic algorithms, such as FCM and FLAB considered here, are susceptible to give different results when applied multiple times on the same image. The adaptive thresholding segmentation, for instance, depends on a manually drawn background ROI and may thus result in variable delineation depending on the choice of this ROI.

On the other hand, FCM and FLAB are iterative procedures that may not converge to the same result at each execution. Finally, manual delineation may be considered as the least repeatable, even when considering a single operator (intra-operator variability). A second aspect considered in this study was in terms of the impact of a segmentation algorithm on the reproducibility of determining functional volumes from two baseline PET scans.

Two different clinical datasets were used comprising of esophageal and breast cancer patients scanned with 18F-FDG and 18F-FLT respectively. In both cases, two consecutive PET scans were acquired at few days interval (see section 2.3). We therefore studied the differences in derived functional tumour volumes, lesion SUV mean , and total glycolytic/proliferative volumes extracted from both images.

Repeatability of measuring tumour volumes using the various delineation approaches considered in this study was investigated on the same clinical datasets.

Validation studies

Fourteen Patients were scanned on a CTI/Siemens ECAT962/HR+ PET scanner and data was reconstructed using OSEM (360 iterations, 6 subsets, no post-filtering).

In both cases two baseline scans were acquired within an average of 3-4 days from each other. As no treatment was administered between the two baseline scans, and considering the short time between the two acquisitions the assumption is that no significant physiological changes occurred in between. A similar assumption has been previously used in all other studies evaluating the reproducibility and repeatability of different SUV measures in PET imaging with double baseline scans carried out within 5-10 days from each other [START_REF] Minn | In vitro comparison of cell proliferation kinetics and uptake of tritiated fluorodeoxyglucose and L-methionine in squamouscell carcinoma of the head and neck[END_REF][START_REF] Weber | Reproducibility of metabolic measurements in malignant tumors using FDG PET[END_REF][START_REF] Nahmias | Reproducibility of standardized uptake value measurements determined by 18F-FDG PET in malignant tumors[END_REF][START_REF] Paquet | Within patient variability of FDG standardised uptake values in normal tissues[END_REF][START_REF] Velasquez | Repeatibility of 18F-FDG PET in a multicenter phase I study of patients with advanced gastrointestinal malignancies[END_REF]. Figure 1 shows the two baseline scans for one (a) esophageal and (b) breast cancer patient.

Analysis

For the repeatability evaluation, the tumours in the first image for each patient were segmented ten times each with FCM, FLAB, and TSBR. In addition, manual delineation was carried out by two nuclear medicine experts. More specifically the two experts performed ten different slice-by-slice manual delineations for the different lesions considered in a randomised fashion, ensuring a minimum of a week between two consecutive delineations of the same lesion. All these manual segmentations were carried out under the same conditions of full range contrast display. The mean percentage variability and associated standard deviation with respect to the mean segmented volume was computed for each of the lesions and segmentation approaches across the ten executions and across the ten manual delineations, in order to assess the repeatability of the approaches. The repeatability of the manual delineations from the two experts were compared separately (intra-observer variability) and to each other (inter-observer variability) using intra-class coefficients (ICCs).

To study the relative impact of the different segmentation algorithms on the reproducibility of deriving different PET image indices, tumour volumes were segmented independently on both baseline scan images for each lesion, using all the different automatic segmentation approaches considered (see section 2.1).

Subsequently, TV (in cm 3 ), SUV mean , TGV and SUV max quantitative values M were computed for each delineated lesion and compared between the two scans using the mean percentage difference relative to the mean of both baseline scans:

1 2 2 1 ( ) ( ) / 1 0 0 2 scan scan scan scan M M M M    (2)
The distribution of the differences between each pair of measurements was assessed for each of the considered index using the Kolmogorov-Smirnov test,

showing no significant differences from a normal distribution (see figure 2). Bland-Altman analysis [START_REF] Bland | Statistical methods for assessing agreement between two methods of clinical measurement[END_REF] was subsequently used to highlight differences between segmentation methodologies. Mean and standard deviation (SD) of differences as well as the respective 95% confidence intervals (CI) were obtained. In order to define the reproducibility limits (normal range of spontaneous changes) the 95% CI for the difference between two measurements were computed as the mean difference ±1.96 times the SD of the difference. In order to investigate any potential correlations in the measured reproducibility the magnitude of the percentage difference for the TV, SUV max and SUV mean measurements were compared to the average of the tumour volumes using Pearson correlation coefficient r. This analysis was repeated to investigate the correlation of the reproducibility of the different parameters with the SUV mean . Tables II and III contain a summary of the reproducibility results for the different parameters computed from Bland-Altman plots on the two consecutive baseline scans for FDG esophageal and FLT breast lesions respectively. The observed reproducibility of SUV max and SUV mean measurements for the volumes obtained using TSBR and FLAB is illustrated in figure 3. The corresponding plots for TV are shown in figures 4(A) and 4(B) using TSBR and FLAB respectively.

Results

Table

Concerning the reproducibility of SUV max similar percentage differences were measured for the FDG and FLT datasets with a SD of the mean percentage difference of 16.7% and 14.9% respectively. The upper and lower percentage reproducibility limits for the SUV max was -31% to 35% and -30% to 28% for the FDG and FLT datasets respectively. On the other hand the automatic approaches led to FDG TV measurement reproducibility limits of -21% to 31% and -51% to 52% for the FLAB and the FCM algorithms respectively. A poorer reproducibility of the FDG TV measurements was observed for the threshold based approaches with upper and lower reproducibility limits of -90% to 51% and -69% to 73% for the adaptive and T42 respectively. In the case of FLT TV measurements, the reproducibility was similar to FDG for the threshold based approaches, while a deterioration in the reproducibility obtained with the automatic approaches was observed particularly for the FCM algorithm with reproducibility limits of -66% to 74%.

SUV mean measurements using FLAB exhibited reproducibility levels of similar magnitude to that for the TV definition, with a SD of the mean percentage difference of 15.6% and 14.1% for the FDG and FLT datasets respectively. This was however not the case for the other tumour delineation algorithms considered, with the larger SUV mean reproducibility limits using the FCM tumour definition (-77% to 62% and -59% to 59% for the FDG and FLT datasets respectively). Finally, the smaller SUV mean reproducibility for the threshold based approaches was obtained using T50, for both the FDG and FLT datasets with a mean percentage difference of -10.5±23% and -13.3±16.8% respectively.

The reproducibility of TGV/TPV, being the product of TV and SUV mean , was dependent on the direction of changes for both TV and SUV mean . As an increase (respectively decrease) of TV was correlated with a decrease (respectively increase) of SUV mean (p<0.002, r=0.54, 0.67, 0.72 for FLAB, TSBR and T42 respectively), TGV/TPV reproducibility levels were generally similar in magnitude to the TV and SUV mean considered separately. However, in certain cases there were more increases or decreases of both TV and SUV mean for a given patient, resulting in larger variability of the TGV/TPV measurements (for example the TSBR measurements of the FLT breast lesions, with 22.1±48.9% for the TPV whereas TV and SUV mean were 11.3±31.4% and -3.2±26.5% respectively).

The TV reproducibility results were dependent on the measured TV with a larger variability seen for smaller tumours. This dependence was statistically significant for the adaptive thresholding (r=0.37, p=0.046, see figure 5(A)) with differences higher than 30% on average (up to 75%) in several of the tumours below 50cm 3 . On the other hand this dependence was not significant for FLAB (r=0.27, p=0.16, figure 5(B)) with most differences <30% irrespective of TV, further demonstrating improved robustness as previously shown [START_REF] Hatt | A fuzzy Bayesian locally adaptive segmentation approach for volume determination in PET[END_REF][START_REF] Hatt | PET functional volume segmentation: a robustness study[END_REF]. In terms of the SUV max reproducibility results there was no statistically significant trend with either the lesion size (r=0.016, p=0.93, figure 5(C)) or the mean of the two measured SUV mean values (r=0.14, p=0.49). Finally no statistically significant trends were found for the SUV mean reproducibility depending on the lesion size irrespective of the segmentation algorithm used (r=0.2, p=0.3 and r=0.23, p=0.23 for TSBR and FLAB respectively).

Discussion

Functional volume delineation represents today an area of interest for multiple clinical (routine and research) applications of PET imaging (prognosis, response prediction, therapy assessment, radiotherapy treatment planning). In all of these applications, the repeatability and reproducibility with which functional volumes can be determined under different imaging conditions plays a predominant role, allowing a level of confidence to be established in the use of such tumour volume measurements. Volume definition methodologies currently used in clinical practice are based on the use of manual delineation or fixed and adaptive thresholding [START_REF] Erdi | Segmentation of Lung Lesion Volume by Adaptative Positron Emission Tomography Image Thresholding[END_REF][START_REF] Daisne | Tri-dimensional automatic segmentation of PET volumes based on measured source-to-background ratios: influence of reconstruction algorithms[END_REF][START_REF] Nestle | Comparison of Different Methods for Delineation of 18F-FDG PET-Positive Tissue for Target Volume Definition in Radiotherapy of Patients with Non-Small Cell Lung Cancer[END_REF], while several promising automatic algorithms have been recently proposed [START_REF] El Naqa | Concurrent multimodality image segmentation by active contours for radiotherapy treatment planning[END_REF][START_REF] Montgomery | Fully automated segmentation of oncological PET volumes using a combined multiscale and statistical model[END_REF][START_REF] Geets | A gradient-based method for segmenting FDG-PET images: methodology and validation[END_REF][START_REF] Hatt | A fuzzy Bayesian locally adaptive segmentation approach for volume determination in PET[END_REF]. The major drawback of manual delineation is high inter-and intra-observer variability in addition to being time consuming. On the other hand, currently considered state of the art adaptive threshold based algorithms have been shown to accurately define functional volumes under certain imaging conditions of spherical and homogeneous activity distribution lesions. However, adaptive thresholding approaches usually involve some user interaction to select background regions of interest, which can potentially lead to user introduced variability. Although signal intensity reproducibility, predominantly considering the use of SUV max , has been previously assessed, the potential of new indices such as tumour volume and/or TGV/TPV can be only considered following the assessment of their reproducibility which has not been previously widely assessed. Therefore in this study the reproducibility limits of these indices, in comparison to other indices considered as the current gold standard, have been assessed using different tumour delineation methodologies on double baseline FDG and FLT datasets.

In terms of repeatability, all algorithms considered exhibited mean differences <5%, with automatic approaches coming closer to the perfect repeatability that can be achieved by deterministic approaches such as a fixed threshold. The repeatability of both threshold and automatic segmentation approaches was superior to that of manual delineation. This of course should be considered within the context of the limited absolute accuracy of thresholding, particularly for non-homogeneous in form and activity distribution lesions [START_REF] Hatt | Accurate automatic delineation of heterogeneous functional volumes in positron emission tomography for oncology applications[END_REF].

The variability in the SUV max observed in this work is similar to that measured in previous reproducibility studies, with similar percentage differences for FDG and FLT datasets, suggesting that differences larger than -30% can be considered as significant in treatment response, while changes above 35% (30% for FLT) may be indicative of no response. Depending on the delineation algorithm used, the mean percentage difference and corresponding SD for TV measured on the two baseline scans varied from 5%±13% (4%±16%) to -19%±36% (10%±35%) for the FDG (FLT) datasets. The smallest TV reproducibility limits obtained were similar to those for SUV max , ranging from -21% to 31% and -27% to 35% for FDG and FLT respectively, suggesting in turn that, depending on the segmentation algorithm used, similar to SUV max confidence intervals may be considered for monitoring therapy response based on functional TV. Similarly in the case of TGV/TPV the smallest reproducibility limits measured were between -16% to 26% and -30% to 37% for FDG and FLT respectively. On the other hand, the largest reproducibility limits for the FDG TV and TGV ranged from -90% to 73% and -68% to 52% respectively.

Reproducibility ranges obtained on the FDG esophageal lesions were almost systematically smaller than the ones obtained on the FLT breast lesions dataset, which can be attributed to the higher level of noise and overall lower contrast observed in the FLT cases, resulting in less robust delineations. In addition, FDG esophageal lesions tended to appear more homogeneous than breast lesions. For instance, FCM which incorporates neither noise nor spatial modelling is associated with a larger mean TV variability on the FLT dataset relative to FDG, whereas FLAB exhibits similar reproducibility levels for both. This highlights the need for a robust delineation tool ensuring high reproducibility in an environment of substantial image quality variability, likely for example to be encountered in multi-center trials where the use of functional TV as a measure of response to therapy may be considered.

T50 uses a more restrictive threshold than 42% and is therefore less prone to large over-evaluation of low contrast (<4:1) and/or small size (<2cm in diameter)

tumor volumes. It led to systematically lower variability than T42. Finally, the adaptive thresholding methodology did not demonstrate better reproducibility than fixed thresholding, which can be attributed to the use of the background ROI placed manually on both scans, combined with the fact that background activity may also vary between the two scans.

Although a potential criticism for the current study can be the lack of groundtruth for the functional volumes, the aim of this work was not to assess the absolute accuracy of algorithms, which has been previously assessed for the approaches used in this work [START_REF] Hatt | A fuzzy Bayesian locally adaptive segmentation approach for volume determination in PET[END_REF][START_REF] Hatt | Accurate automatic delineation of heterogeneous functional volumes in positron emission tomography for oncology applications[END_REF]. The objective was to assess the reproducibility limits of functional volume related indices that can be attained depending on the algorithm.

Within this context, the repeated studies of the double baseline acquisitions have been performed within an average of 3-4 days from each other without any treatment between them, matching that used by all other reproducibility studies to date [START_REF] Minn | In vitro comparison of cell proliferation kinetics and uptake of tritiated fluorodeoxyglucose and L-methionine in squamouscell carcinoma of the head and neck[END_REF][START_REF] Weber | Reproducibility of metabolic measurements in malignant tumors using FDG PET[END_REF][START_REF] Nahmias | Reproducibility of standardized uptake value measurements determined by 18F-FDG PET in malignant tumors[END_REF][START_REF] Paquet | Within patient variability of FDG standardised uptake values in normal tissues[END_REF][START_REF] Velasquez | Repeatibility of 18F-FDG PET in a multicenter phase I study of patients with advanced gastrointestinal malignancies[END_REF].

Finally the reproducibility of the SUV max was included in this work as the current gold standard facilitating at the same time the comparison of our reproducibility study to those performed previously. The SUV max reproducibility limits obtained in this work for both FDG and FLT agree closely with those of previous studies.

Conclusion

The

Figure 1 :

 1 Figure 1

Figure 2 :

 2 Figure 2: Normal plots showing that the distributions of differences for (A). SUV mean

Figure 3 :

 3 Figure 3: Bland-Altman plots of (A). SUV max , (B). SUV mean (adaptive thresholding),

Figure 4 :

 4 Figure 4: Bland-Altman plots of (A). TV (adaptive thresholding), and (B). TV (FLAB)

Figure 5 :

 5 Figure 5: Differences between (A-B). tumor volumes and (C). SUV max measured in

  I contains the mean variability and standard deviation around the mean segmented volume across the ten manual delineations performed from each of the two nuclear medicine experts, and 10 repeated executions of the FLAB, FCM and TSBR algorithms. Results on both clinical datasets are presented separately. FLAB demonstrated highly repeatable results in all of the studied cases, with negligible variability (1%) around the mean segmented 3D volumes across the different repeated executions. FCM also lead to satisfactory repeatability results (1.4±1.6% for the FDG cases and 2.3±1.9% for the FLT cases). In comparison, the use of the TSBR led to more than twice as high variability (2.9±2.7% and 4.7±3.6% for the FDG and FLT cases respectively). By contrast manual segmentation by the two experts

	showed high intra-observer variability for FDG esophageal lesions (14.1±12.1% and
	16.4±11.3% for expert 1 and 2 respectively). Inter-observer variability was
	17.1±14.3% with an ICC of 0.67 (CI: 0.39-0.89). In the case of FLT, this variability
	was even higher, with intra-observer variability of 22.1±18.7% and 23.8±17.8% for
	expert 1 and 2 respectively and an inter-observer variability of 27.4±21.9% with ICC
	0.59 (CI: 0.31-0.84).

Table II

 II

Table captions Table I :

 captionsI Repeatability evaluation: mean variability and standard deviation around the mean segmented volume for repeated delineations of 17 esophageal and 12 breast lesions on the first baseline FDG and FLT scans respectively.

Table II :

 II Reproducibility results concerning the FDG esophageal lesions with differences of scan #2 measurements with respect to scan #1.

Table III :

 III Reproducibility results concerning the FLT breast lesions with differences of scan #2 measurements with respect to scan #1.

Acknowledgments

We gratefully acknowledge funding by the Ligue Contre le Cancer (Finistére Committee), the French National Research Agency (ANR-08-ETEC-005-01), Cancéropôle Grand Ouest (R05014NG), and the CR-UK&EPSRC Cancer Imaging Centre (Imperial College, London), the UK Medical Research Council and the Department of Health (C2536/A10337, U.1200.02.005.00001.01).

Method