Horia D Cornean

Thomas G Pedersen

Benjamin Ricaud

Rigorous perturbation theory versus variational methods in the spectral study of carbon nanotubes May 1st, 2007

Recent two-photon photo-luminescence experiments give accurate data for the ground and first excited excitonic energies at different nanotube radii. In this paper we compare the analytic approximations proved in [CDR], with a standard variational approach. We show an excellent agreement at sufficiently small radii.

Introduction

Recent experimental results on carbon nanotubes using two photon photo luminescence [START_REF] Bachilo | Structure-Assigned Optical Spectra of Single-Walled Carbon Nanotubes[END_REF], [W], [MPR] reveal the energy levels of the excitons, especially the ground and first excited states, and point out the dependence of these energies on the radius of the nanotube. As it is, this technique appears to be a promising way to sort out nanotubes. But on the other hand, theoretical results seem to require heavy ab initio calculation like in [MPR] [CBRM], to cite the most recent, in order to find the absorption peaks due to excitons. Nevertheless, a simple formula for the optical response based on excitons levels [HK] can give a good approximation. It has been pointed out [BGEH] [OT] that the exciton binding energy in quantum wires depends on the width of the wires by a relatively simple relation. This property is also valid for nanotubes [P1]. In the first part of the paper, we outline a rigorous justification for this latter fact and write a simple analytic formula for the energy levels of the exciton depending on the radius of the tube, based on the paper [CDR]. In the second part we compare our results with a variational numerical method and show a very good agreement between them.

The exciton model

As first suggested in [P1], we deal with Wannier excitons (a rigorous justification of this procedure is in preparation [R]). We take as configuration space a cylinder of radius r and infinite length, space denoted by C r = R × rS 1 , S 1 being the unit circle. The coordinates on the cylinder are (x, y) ∈ (R × rS 1 ) where x is the variable along the tube axis and y is the transverse coordinate.

The two quasi-particles live in the Hilbert space L 2 (C r × C r ). We formally consider the Hamiltonian Hr = -

2 ∆ x1 2m 1 + ∆ x2 2m 2 + ∆ y1 2m 1 + ∆ y2 2m 2 -V r (x 1 -x 2 , y 1 -y 2 ), (2.1) where V r (x, y) := -e 1 e 2 ε x 2 + 4r 2 sin 2 y 2r (2.2)
1 Dept. of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, 9220 Aalborg, Denmark; e-mail: cornean@math.aau.dk 2 Dept. Phys. and Nanotech., Aalborg University, 9220 Aalborg, Denmark; e-mail: tgp@physics.aau.dk 3 Centre de Physique Théorique UMR 6207 -Unité Mixte de Recherche du CNRS et des Universités Aix-Marseille I, Aix-Marseille II et de l'université du Sud Toulon-Var -Laboratoire affilié à la FRUMAM, Luminy Case 907, F-13288 Marseille Cedex 9 France; e-mail: ricaud@cpt.univ-mrs.fr (x i , y i ) are the coordinates on the cylinder of the two charged particles, m i their masses, and e i their charges. Here ε is the electric permittivity of the material. The potential V r is the three dimensional Coulomb potential simply restricted to the cylinder. We justify the expression of V r by Pythagoras's theorem. The cylinder is embedded in R 3 . The distance ρ from one particle to the other in R 3 is:

ρ 2 = (x 1 -x 2 ) 2 + 4r 2 sin 2 y 1 -y 2 2r
where |2r sin y1-y2

2r

| is the length of the chord joining two points of coordinate y 1 and y 2 on the circle.

"Separation of the center of mass"

Due to the restrictions imposed by the cylindrical geometry, the usual separation of the center of mass does not work here.

The standard separation with Jacobi coordinates only works for the longitudinal variable, and introducing M := m 1 + m 2 and µ := m 1 m 2 /(m 1 + m 2 ), we denote X := (m 1 x 1 + m 2 x 2 )/M and x := x 1 -x 2 . For the transverse variable, we change to atomic coordinates Y = y 2 and y = y 1 -y 2 . Let us also define the effective Rydberg Ry * = µe 4 /2 2 ε 2 and Bohr radius a * B = 2 ε/µe 2 , where we set e = e 1 = e 2 . By a scaling, the new energy and radius will be expressed in multiple of these units. This gives us the Hamiltonian:

H r = - 1 M ∂ 2 X - 1 m 2 ∂ 2 Y - 1 µ ∂ 2 x - 1 µ ∂ 2 y + 2 m 2 ∂ y ∂ Y -2V r (x, y).
First, we can separate the partial center of mass with coordinate X. Second, since on the Y variable there are periodic boundary conditions, let us consider the orthonormal basis of eigenvectors of

-∂ 2 Y , -∂ 2 Y = ∞ n=-∞ E r n |χ r n χ r n | where χ r n (Y ) = 1 √ 2πr e in Y
r and E r n = n 2 r 2 , n ∈ Z. One can see that for small radii, the separation between different transverse levels of energy is high. A recent theoretical study based on ab initio calculations [MPR] shows that the probability density of the exciton is constant along the circumference. So it is reasonable to assume that the radius is so small that the system stays in the ground state of -∂ 2 Y , where n = 0 and the density of probability is constant along the circumference. (Note that this has been rigorously proved in [CDR]). As a consequence, we can approximate the eigenfunctions ψ of H as:

ψ(x, y, Y ) = φ(x, y) • χ r 0 (Y ) = φ(x, y) • 1 √ 2πr .
After this restriction to the lowest transverse mode, we only have to study the following operator:

H r = - 1 µ ∂ 2 x - 1 µ ∂ 2 y -2V r (x, y),
which is two dimensional and acts on L 2 (C r ).

An effective one dimensional operator for the low lying spectrum

It will turn out that the limit r → 0 is too singular and H r does not have a "nice" limit. It is suitable at this point to introduce the quadratic form associated to H r , defined on the Sobolev space H 1 (C r ):

t H (ψ, φ) = 1 µ ∂ x ψ, ∂ x φ + 1 µ ∂ y ψ, ∂ y φ -2 V r (x, y)ψ, V r (x, y)φ .
Reasoning as in the previous subsection, a good approximation of the behavior of eigenfunctions along the transverse variable when the radius is small is given by the ground state of the free Laplacian with periodic boundary conditions. If we restrict the above quadratic form to functions of the type:

φ(x, y) = ϕ(x) • 1 √ 2πr , (2.3)
we have:

t H (φ, φ) = 1 2πrµ ∂ x ϕ, ∂ x ϕ - 1 πr ∞ -∞ πr -πr 1 x 2 + 4r 2 sin 2 y 2r ϕ(x)ϕ(x)dydx = 1 µ ∞ -∞ (∂ x ϕ)(x)(∂ x ϕ)(x)dx -2 ∞ -∞ V r eff (x)ϕ(x)ϕ(x)dx where V r eff (x) = 1 2πr πr -πr 1 x 2 + 4r 2 sin 2 y 2r dy.
For the sake of simplicity, we will put µ = 1. Let us introduce what will be our effective onedimensional comparison operator:

H r eff := - d 2 dx 2 -2V r eff (x).
(2.4)

We now have reduced our problem of two particles on a cylinder to a one dimensional problem describing a particle interacting with an external potential. A complete and detailed mathematical justification of these steps can be found in [CDR].

The one dimensional Coulomb Hamiltonian

Approximation of H r eff

We now try to approximate as accurate as possible the eigenfunctions and eigenvalues of H r eff when the radius is small. Using the notation Y r (x) = 1 √

x 2 +4r 2 , it was shown in [CDR] that if r ≤ 1 and f, g are two smooth functions

f, V r eff g = f, Y r g + ln(4)f (0)g(0) + O(r 4/9 )||f || H 1 (R) ||g|| H 1 (R) .
(2.5)

One can recognize in Y r a form of the regularized one dimensional Coulomb potential. In the following, our approach will be quite similar to the one of Loudon [L] in the sense that we let the regularizing parameter tend to zero. Note though that our parameter has a clear physical interpretation being given by the radius of the nanotube, and it is not just an artifact as in Loudon's case. Hence this will give us the exciton energies as functions of r, and allow us to estimate the errors we make using this approximation. Define the quadratic form C 0 (., .) by

1 C 0 (f, f ) := - ∞ 0 ln(x) • (|f | 2 ) ′ (x)dx + 0 -∞ ln(-x) • (|f | 2 ) ′ (x)dx = 0 -ε ln(-x) • (|f | 2 ) ′ (x)dx - ε 0 ln(x) • (|f | 2 ) ′ (x)dx + ln(ε) |f (ε)| 2 + |f (-ε)| 2 + R\[-ε,ε] 1 |x| • |f (x)| 2 dx.
(2.6)

1 Notice that this definition of C 0 differs from the one of [CDR] in which there is an additional term: 2 ln 2|f (0

)| 2 = 0 -∞ ln 2 • (|f | 2 ) ′ (x)dx + ∞ 0 ln 2 • (|f | 2 ) ′ (x)dx.
This term is here put together with the other term depending on the function at zero.

Note the last equality is obtained by an integration by parts, and holds for all ε > 0; also note the appearance of a Coulomb potential in one dimension. Again from [CDR] we have

f, Y r g = -2 ln(r)f (0)g(0) + C 0 (f, g) + O(r 4/9 )||f || H 1 (R) ||g|| H 1 (R) .
(2.7)

Now let us define a new "potential" via a quadratic form

f, V C g := -2 ln( r 2 )f (0)g(0) + C 0 (f, g),
which is close to V r eff when r is small and is exactly the Coulomb potential when we look away from the origin. We will see in the following that this particular potential gives a solvable eigenvalue problem for the associated Hamiltonian.

Boundary conditions of the Coulomb Hamiltonian

The operator H C we will now consider is given by its associated quadratic form defined on H 1 (R):

t C (ψ, φ) := R φ ′ (x)ψ ′ (x)dx + 2 2 ln( r 2 )φ(0)ψ(0) -C 0 (φ, ψ) .
One can recognize the kinetic energy in the first term and the potential V C in the second term.

The general theory of closed symmetric quadratic forms gives us the existence of an associated operator

H C defined by t C (φ, ψ) = φ, H C ψ , (2.8)
whenever ψ is in the domain of H C . The difficulty here is that V C is not a usual Schrödinger-type multiplication potential, but due to various Sobolev embeddings it turns out that if ψ is in the domain of H C then ψ ′′ is square integrable outside the origin, and we still have:

(H C ψ)(x) = -ψ ′′ (x) - 2ψ(x) |x| , x = 0.
(2.9)

Now if ψ is an eigenfunction of H C corresponding to an energy E, then it obeys the differential equation:

-ψ ′′ (x) - 2ψ(x) |x| = Eψ(x), x = 0.
(2.10)

In order to get the behavior at the origin of the eigenfunctions of H C , we integrate by parts and use (2.10). For ε > 0 we have:

R φ ′ (x)ψ ′ (x)dx (2.11) = φ(-ε)ψ ′ (-ε) - -ε -∞ φ(x)ψ ′′ (x)dx -φ(ε)ψ ′ (ε) - ∞ ε φ(x)ψ ′′ (x)dx + ε -ε φ ′ (x)ψ ′ (x)dx
where the last integral will converge to zero as ε goes to zero. On the other hand,

C 0 (φ, ψ) = 0 -ε ln(-x) • (d x (φψ))(x)dx - ε 0 ln(x) • (d x (φψ))(x)dx + ln(ε) φ(ε)ψ(ε) + φ(-ε)ψ(-ε) + R\[-ε,ε] 1 |x| • φ(x)ψ(x)dx.
(2.12)

Then adding (2.11) with (2.12), using (2.10) and letting ε tend to zero (see [CDR] for technical details) we have:

lim ε→0 t C (φ, ψ) = φ, Eψ + 2 lim ε→0 φ(0) ψ ′ (-ε) -ψ ′ (ε) 2 + 2 ln( r 2 )ψ(0) -ln(ε) (2ψ(0)) .
Now using (2.8) and the eigenvalue equation H C ψ = Eψ we get the boundary condition:

lim ε→0 ψ ′ (-ε) -ψ ′ (ε) 2 + 2 ln( r 2 )ψ(0) -2 ln(ε)ψ(0) = 0.
(2.13)

Eigenvalues and eigenfunctions

We now have to solve the equation

-∂ 2 x ψ - 2 |x| ψ = Eψ, x = 0, (2.14)
with the boundary condition (2.13). Similarly to Loudon in [L], we introduce a dimensionless quantity α and the change of variables

E = - 1 α 2 and x = α 2 z, (2.15)
then we obtain

d 2 dz 2 ψ - 1 4 ψ + α |z| ψ = 0, z = 0.
(2.16)

The solutions are known for z > 0 and z < 0, see [AS,chap. 13] for example. The second thing is to see what condition at z = 0 should the eigenfunctions ψ obey. If we scale (2.13),

lim ε→0 ψ ′ (-ε) -ψ ′ (ε) 2 + α(ln r -ln(αε)) ψ(0) = 0.
(2.17)

We only give here the results. Details of calculations can be found in [CDR]. If we take the only square integrable solution, we have two cases: If α = N is a positive integer, then the eigenstates are the odd functions ψ nαp with associated eigenvalues E nαp , where n α = N + 1,

ψ nαp (z) = e -1 2 |z| z 1 √ 2N L 1 N -1 (|z|), E nαp = - 1 N 2 ,
where L 1 N -1 is an associated Laguerre polynomial. Notice that these energies are independent of the radius of the tube. If α is not an integer, the eigenstates are the even functions ψ nαs with associated eigenvalues E nαs , where n α is the smallest integer larger than α and:

ψ nαs = C α W α, 1 2 (|z|) = C α |z|e -1 2 |z| U (1 -α, 2, |z|) E nαs = - 1 α 2 ,
where C α is a normalizing constant, W is the Whittaker function and U is the confluent hypergeometric function or Kummer function of the second kind. We denote with Γ(z) and Ψ(z) = Γ ′ (z)/Γ(z) the usual gamma and digamma functions, and we get from (2.17) that for even solutions α must satisfy the relation:

Ψ(1 -α) + 2γ + 1 2α -ln α + ln r = 0. (2.18)
From this relation, which contains an implicit expression for α(r), one can deduce several important facts. For all integers N and for α in between N and N + 1, there is only one solution of (2.16) satisfying the boundary condition. Furthermore, the energies associated with the non integer α tend to their closest lower integer when r tends to zero. A special case is the one of the ground state E 1s of the exciton which tends to minus infinity as the radius tends to zero. The behavior for small r is E 1s r→0 = -4(ln r) Notice that equation (2. 18) is exactly what one gets by explicitly calculating the condition requiring the derivative of even states to vanish at the origin, relation (3.22) in Loudon's paper, and replacing his parameter a by r • a 0 /2, for small r. We can also compare with the result from [BGEH] where Banyai et al. found numerically a = 0.3a 0 • r for the exciton problem in a quantum wire of radius r. The energies associated to the exciton are drawn on figure 1 and we calculated numerically the lowest four eigenvalues with respect to the radius on figure 3, along with a comparison of the ground state given by the variational method of [P1]. On figure 4 a zoom was made around the first and second excited states.

E E -0 E E E E (r) (r) 2 =-1/9 =-1/4 =-1 (r) ~ -4(ln(r))
One can notice that the odd states possess a constant energy, independent of the radius. This is due to the fact that the odd states vanish at zero where there is the singularity of the potential. Indeed, the potential becomes, for an odd function f :

f, V r eff f ≃ C 0 (f, f ),
which is independent of r.

Variational approach and comparison

The variational method operates by minimizing the energy of a trial function and is therefore usually applied to approximate the ground state. However, by restricting the trial function to forms that are orthogonal to the ground state, low lying excited states can be obtained variationally as well. In Ref. [P1], the variational method was applied to the ground state and in Ref. [P3] a similar approach was applied to calculate the 2p oscillator strength of interest for two-photon absorption. The 2p state is especially important because this state is used for excitation in twophoton fluorescence measurements [W]. By recording the energy of photons emitted from the lowest (1s) exciton, a direct measure of the 2p -1s energy difference is obtained. In turn, the 1s exciton binding energy (i.e the 1s excitation energy measured relative to the band gap) can be derived if a reliable model for the exciton energy spectrum is invoked.

In the present work, we wish to compare results of the relatively complicated variational approach to the straight-forward and analytical Coulomb model. Hence, in the following we briefly explain the reasoning behind our variational estimate of the lowest excited (2p) state. In practical applications, the trial function must be sufficiently simple that calculation of the energy is manageable. This implies that relatively few adjustable parameters should be considered. At the same time, a certain flexibility is required to provide reasonable accuracy. A useful strategy consists in constructing trial functions so that they correctly accommodate the known solutions in limiting cases of the general problem. Thus, we are guided by the analytical solution for the plane, i.e. for nanotube radii much larger than the effective Bohr radius. In our units, this state is simply ϕ 2p ∝ x exp{-2/3(x 2 + y 2 ) 1/2 }. On the other hand, in the extreme 1D limit we expect ϕ 2p ∝ x exp{-|x|}. To accommodate both limits, we consequently suggest the (un-normalized) trial form ϕ 2p (x, y) = x exp{-(x 2 /k 2 + y 2 /q 2 ) 1/2 }, (3.1)

where q and k are variational parameters to be determined by minimizing the expectation value of the energy . This expectation value is found as E 2p = (K -V )/N , where K, V and N are the kinetic energy, potential energy and normalization constant, respectively. The integrations are quite cumbersome and only K and N can be obtained in closed form in terms of Struve and modified Bessel functions. The potential energy V is evaluated numerically using Gaussian quadrature.

The minimized energy as a function of radius r is illustrated in Fig. 2. The limiting values are E 2p = -1 and E 2p = -4/9 for small and large r, respectively, in agreement with the analytic solutions in these limits. In between these limits, the curve interpolates smoothly between the limiting values and the dominant correction at small r is -8(1 + γ + ln(r))r 2 .

In order to judge the usefulness of the different approaches used in the present work it is essential to determine the appropriate nanotube radius r in excitonic units, i.e. in units of a *

B . An important point in this respect is that, in fact, a * B varies between different nanotubes. The relation needed for conversion is a * B = 0.529 Åε/µ, where ε is the dielectric constant screening the Coulomb interaction and µ is the reduced effective mass. Whereas ε may be assumed the same for all nanotubes, µ must be derived from the curvature of the band structure and, hence, depends on both radius and chirality of the nanotube. However, detailed studies [P2] show that a * B is roughly proportional to the nanotube diameter and as a consequence r (in units of a * B ) is nearly constant and given by r ≈ 0.1a * B if ε ≈ 3.5. It should be noted, though, that in media with little screening a larger r/a * B is expected. The smallness of r means that approximations based on expansion around r = 0 are expected to be accurate.

In figure 3 and4, a comparison of variational energies and the results of the Coulomb model is given. In both the 1s and 2p cases, reasonable agreement between the two approaches is found around r ≈ 0.1a * B . Also, in both cases, the variational result lies slightly higher than the Coulomb model. The ground state has deviated by more than 100% from the plane value for radii around 0.1a *

B . An error of less than 5% is seen between the curves at this point. Note that if the curve from the Coulomb model is lower in energy than the variational one, this does not means that the approximation is better. Although in the variational approach the lower is the better since the exact solution is always below the variational result, in the Coulomb model the exact value is somewhere around the approximation, bounded by an error bound. In this work the error bound was not calculated because, unfortunately, the compromises made to get a simple formula implied a difficult calculus to optimize on the bound, even by numerical computations. For the first excited state, the Coulomb model approximation for the energy is independent of the radius of the tube. This must be a good approximation for very small radii. However, this approximation becomes increasingly inaccurate as the radius increases, as the exact value should tend to the energy of the problem on the plane with energy equal to -4/9a * B as do the variational curve.

Conclusion

The Coulomb model applied to excitons in carbon nanotubes demonstrates that the energy associated with each even state decreases to the energy of its closest odd state as the radius tends to zero. Since each odd state energy is independent of the size of the tube in this approach, all energies associated to excited states stay in the range (-1, 0) (in effective Rydberg energy units). So only the ground state diverges when the radius gets smaller. This is confirmed by the behavior of the first variational excited state which converges to -1. The simple exciton model proposed in this paper and studied both by means of rigorous perturbation theory and by a variational approach is a good starting point for attacking the difficult problem of electron-electron interactions in low dimensional structures such as carbon nanotubes.

5 Acknowledgments.

Figure 1 :

 1 Figure 1: Energy of the bound states of the exciton. Energies are expressed in multiples of the effective Rydberg of the exciton.

Figure 2 :

 2 Figure 2: Variational 2p state energy as a function of nanotube radius. Inset: the r-dependence of decay lengths k and q of the trial wave function.

  H.C. acknowledges support from the Danish F.N.U. grant Mathematical Physics and Partial Differential Equations.

Figure 4 :

 4 Figure 4: Zoom around the energy of the 2p state showing both variational and Coulomb model results. In addition, the 2s state obtained with Coulomb model is included for comparison.

  Energy of the four lowest bound states of the exciton with respect to the radius. Even states were calculated numerically using (2.18). Notice that the energy of the second s state is equal to the one of the first p state at zero. In red: graph of the ground state computed with the variational method on the cylinder from[P1].

	-16 -12 -8 -4 0 0 Exciton binding energy E [Ry*]	0,1	0,2	0,3 Second p state E 3p 0,4 0,5 Second s state E 2s First p state E 2p Ground state E 1s Variational E 1s	0,6
			Nanotube radius r [ a* B ]
	Figure 3:


D De Ruysscher

B Lueza

C Le Péchoux

D H Johnson

M O'brien

N Murray

S Spiro

X Wang

M Takada

B Lebeau

W Blackstock

D Skarlos

P Baas

H Choy

A Price

L Seymour

R Arriagada

J-P Pignon

Dr Jean-Pierre Pignon
email: jean-pierre.pignon@gustaveroussy.fr

Impact of thoracic radiotherapy timing in limited-stage small-cell lung cancer: usefulness of the individual patient data meta-analysis

Keywords: individual participant data meta-analysis, randomised clinical trials, thoracic radiotherapy, radiotherapy timing, small-cell lung cancer, chemotherapy compliance

Impact of thoracic radiotherapy timing in limited-stage

small-cell lung cancer: usefulness of the individual patient data meta-analysis

INTRODUCTION

Small-cell lung cancer (SCLC) is a rapidly disseminating cancer so that its primary treatment is chemotherapy, whatever the stage [START_REF] Van Meerbeeck | Small-cell lung cancer[END_REF]. Approximately 25% of patients present with localised disease, formerly known as "limited-stage" disease, now called stage I-IIIB [START_REF] Shepherd | The International Association for the Study of Lung Cancer lung cancer staging project: proposals regarding the clinical staging of small cell lung cancer in the forthcoming (seventh) edition of the tumor, node, metastasis classification for lung cancer[END_REF]. It is well known that optimal survival is achieved when chemotherapy can be administered at the total intended dose and at the required intervals [START_REF] Van Meerbeeck | Small-cell lung cancer[END_REF][START_REF] Arriagada | Initial chemotherapy doses and survival in limited small cell lung cancer[END_REF]. Nevertheless, due to loco-regional failures after chemotherapy alone, the adjunction of thoracic radiotherapy was investigated. A worldwide meta-analysis showed that adding thoracic radiotherapy to chemotherapy improved long-term survival [START_REF] Pignon | A meta-analysis of thoracic radiotherapy for smallcell lung cancer[END_REF]. Concurrent chemotherapy comprising cisplatin and etoposide and thoracic radiotherapy has become the standard of care [START_REF] Van Meerbeeck | Small-cell lung cancer[END_REF][START_REF] Stahel | 1st ESMO Consensus Conference in lung cancer; Lugano 2010: small-cell lung cancer[END_REF][START_REF]Practice Guidelines in Oncology: Small Cell Lung Cancer[END_REF]. In nonprogressing patients, this can be followed by prophylactic cranial irradiation, at the optimal dose of 25 Gy, as this treatment further prolongs survival [START_REF] Aupérin | Prophylactic cranial irradiation for patients with small-cell lung cancer in complete remission[END_REF][START_REF] Péchoux | Standard-dose versus higher-dose prophylactic cranial irradiation (PCI) in patients with limited-stage small-cell lung cancer in complete remission after chemotherapy and thoracic radiotherapy (PCI 99-01, EORTC 22003-08004, RTOG 0212, and IFCT 99-01): a randomised clinical trial[END_REF].

However, the optimal timing and sequencing of thoracic radiotherapy with chemotherapy has fuelled debate for many years. When all trials were pooled together, no survival gain was detected whether thoracic radiotherapy was delivered early with chemotherapy or later [START_REF] Mc | Early versus late chest radiotherapy for limited stage small cell lung cancer[END_REF][START_REF] De Ruysscher | Systematic review and metaanalysis of randomised, controlled trials of the timing of chest radiotherapy in patients with limited-stage, small-cell lung cancer[END_REF][START_REF] Pijls-Johannesma | Timing of chest radiotherapy in patients with limited stage small cell lung cancer: a systematic review and meta-analysis of randomised controlled trials[END_REF][START_REF] Spiro | Early compared with late radiotherapy in combined modality treatment for limited disease small-cell lung cancer: a London Lung Cancer Group multicenter randomized clinical trial and meta-analysis[END_REF].

However, in trials where patients were treated with cisplatin-based chemotherapy at full dose, early administration of thoracic radiotherapy seemed to confer a long-term survival advantage. There is considerable variation in the definition of early or late radiotherapy : early radiotherapy was defined as starting before 9 weeks following the beginning of chemotherapy and before the third cycle of chemotherapy in two previous literature-based meta-analyses [START_REF] Spiro | Early compared with late radiotherapy in combined modality treatment for limited disease small-cell lung cancer: a London Lung Cancer Group multicenter randomized clinical trial and meta-analysis[END_REF][START_REF] Fried | Systematic review evaluating the timing of thoracic radiation therapy in combined modality therapy for limited-stage small-cell lung cancer[END_REF], whilst a 30-day cut-off was used in other literature-based meta-analyses [START_REF] Mc | Early versus late chest radiotherapy for limited stage small cell lung cancer[END_REF][START_REF] De Ruysscher | Systematic review and metaanalysis of randomised, controlled trials of the timing of chest radiotherapy in patients with limited-stage, small-cell lung cancer[END_REF][START_REF] Pijls-Johannesma | Timing of chest radiotherapy in patients with limited stage small cell lung cancer: a systematic review and meta-analysis of randomised controlled trials[END_REF][START_REF] Huncharek | A meta-analysis of the timing of chest irradiation in the combined modality treatment of limited-stage small cell lung cancer[END_REF] (Table S1 for description of previous meta-analyses). One of these meta-analyses suggested that early delivery of thoracic radiotherapy yielded higher survival rates if all the intended cycles of chemotherapy could be administered [START_REF] Spiro | Early compared with late radiotherapy in combined modality treatment for limited disease small-cell lung cancer: a London Lung Cancer Group multicenter randomized clinical trial and meta-analysis[END_REF], implying that the question of optimal radiotherapy timing and fractionation [START_REF] Mauguen | Hyperfractionated or accelerated radiotherapy in lung cancer: an individual patient data meta-analysis[END_REF][START_REF] Turrisi | Twice-daily compared with once-daily thoracic radiotherapy in limited small-cell lung cancer treated concurrently with cisplatin and etoposide[END_REF] could only be addressed with precise information on individual patient compliance with chemotherapy administration. Such information can only be provided by an individual patient data (IPD) meta-analysis. We therefore undertook such a study, aiming to define the best approach for combining thoracic radiotherapy with chemotherapy in stage I-IIIB SCLC.

METHODS-MATERIAL

The meta-analysis was performed according to a pre-specified protocol that is available on the Gustave Roussy website (http://www.gustaveroussy.fr/sites/default/files/meta-analysesprotocol-rtt-sclc.pdf).

Selection criteria and search strategy

To be eligible, trials had to compare two timing schedules of curative thoracic radiotherapy, i.e. earlier versus later within an individual trial in patients with limited-stage SCLC treated with chemo-radiotherapy. Our post-hoc criterion to define early radiotherapy was similar to the one used by Fried et al [START_REF] Fried | Systematic review evaluating the timing of thoracic radiation therapy in combined modality therapy for limited-stage small-cell lung cancer[END_REF] and Spiro et al [START_REF] Spiro | Early compared with late radiotherapy in combined modality treatment for limited disease small-cell lung cancer: a London Lung Cancer Group multicenter randomized clinical trial and meta-analysis[END_REF]: radiotherapy should have been initiated before 9 weeks after randomisation and before the third cycle of chemotherapy. Trials comparing two radiotherapy durations, i.e. a shorter vs. a longer course within an individual trial with at least a two-week treatment difference observed between the two arms, were also eligible. In this paper, we will use the term "earlier or shorter" for arms where earlier and/or shorter radiotherapy was used and the term "later or longer" for later and/or longer radiotherapy arms. Trials had to start after 1969 and to end before 2006, and be properly randomised. The planned chemotherapy schedule (drugs, doses, number of cycles) had to be the same in both arms, but radiotherapy modalities could be different. The total dose of radiotherapy had to be at least 30 Gy. Orthovoltage radiotherapy was an exclusion criterion.

Eligible patients should have had a WHO (or equivalent) performance status of 0-2 and should not have received previous treatment for this cancer. To limit publication bias, we searched for both published and unpublished trials without language restriction (see Web-Appendix 1 for search strategy).

Statistical Analysis

We describe IPD collection and quality control in Web-Appendix 2. The main endpoint was overall survival and the secondary endpoints were progression-free survival and severe acute toxicities. Overall survival was defined as the time from randomisation until death from any cause or the last follow-up for surviving patients. Progression-free survival was defined as the time from randomisation until first progression or death from any cause, or the last follow-up for surviving patients without progression. We did not perform analyses on loco-regional control, cancer deaths and late toxicities due to lack of data. The median follow-up was estimated using the reverse Kaplan-Meier method [START_REF] Schemper | A note on quantifying follow-up in studies of failure time[END_REF].

We carried out all analyses on an intention-to-treat basis. Survival analyses were stratified by trial, and the log-rank expected number of deaths and variance were used to calculate individual and overall pooled hazard ratios (HR) by the fixed effect model [START_REF] Mauguen | Hyperfractionated or accelerated radiotherapy in lung cancer: an individual patient data meta-analysis[END_REF]. A similar model was used to estimate odds ratios (OR) for the comparison of toxicity between arms. χ² tests and the I² statistic were used to study heterogeneity between trials [START_REF] Higgins | Quantifying heterogeneity in a meta-analysis[END_REF]. Hazard ratios were calculated using a DerSimonian-Laird random effects model if heterogeneity had a pvalue <0.10 [START_REF] Dersimonian | Meta-analysis in clinical trials[END_REF]. Stratified survival curves were estimated for control and experimental groups, using annual death rates and the pooled hazard ratio, and were used to estimate the absolute benefit at 3 and 5 years with their 95% confidence intervals [START_REF]Systemic treatment of early breast cancer by hormonal, cytotoxic or immune therapy: 133 randomised trials involving 31,000 recurrences and 24,000 deaths among 75,000 women[END_REF]. Five-year mean survival times, parameters commonly used in economic evaluation, were also estimated (Web-Appendix 3) [START_REF] Wei | Meta-analysis of time-to-event outcomes from randomized trials using restricted mean survival time: application to individual participant data[END_REF][START_REF] Lueza | Difference in Restricted Mean Survival Time for Cost-Effectiveness Analysis Using Individual Patient Data Meta-Analysis: Evidence from a Case Study[END_REF][START_REF] Lueza | Bias and precision of methods for estimating the difference in restricted mean survival time from an individual patient data meta-analysis[END_REF].

Subsets analyses according to trial characteristics were pre-planned. We investigated whether the treatment effect was dependent on any difference in the proportion of patients who were compliant with chemotherapy between the treatment arms within each trial. A patient was defined as compliant if he/she received 100% or more of the planned number of CT cycles, except for the CALGB8083 trial in which patients receiving 6 CT cycles or more were considered as compliant. A trial was considered as having different "between-arm" compliance if the difference was ≥10% and as having similar "between-arm" compliance if it was <10% [START_REF] Spiro | Early compared with late radiotherapy in combined modality treatment for limited disease small-cell lung cancer: a London Lung Cancer Group multicenter randomized clinical trial and meta-analysis[END_REF]. No other information on chemotherapy administration, such as the actual drug dose received or delays in chemotherapy administration, was available. χ² tests for interaction or trend were used to assess treatment effects across trial subsets. Overall heterogeneity was decomposed into the sum of between-subset and residual (within-subset) heterogeneity: the lower the residual heterogeneity, the greater the overall heterogeneity of the treatment effect between trials was explained by the trial characteristic [START_REF] Cell | Chemotherapy in non-small cell lung cancer: a meta-analysis using updated data on individual patients from 52 randomised clinical trials[END_REF]. χ² tests for interaction or trend were also used to test whether there was any evidence that a particular type of patient benefited more or less from "earlier or shorter" radiotherapy according to predefined subgroups. If there was substantial overall heterogeneity, then subgroup analyses were planned within treatment categories. All p-values were two-sided. Analyses were performed using SAS version 9.3.

Role of the funding source

The funding sources had no role in the study design, data collection, data analysis, data interpretation, or manuscript writing. BL and J-PP had full access to all the raw data. The corresponding author had the final responsibility for the decision to submit for publication.

RESULTS

Twelve randomised trials [START_REF] Spiro | Early compared with late radiotherapy in combined modality treatment for limited disease small-cell lung cancer: a London Lung Cancer Group multicenter randomized clinical trial and meta-analysis[END_REF][START_REF] Turrisi | Twice-daily compared with once-daily thoracic radiotherapy in limited small-cell lung cancer treated concurrently with cisplatin and etoposide[END_REF][START_REF] Perry | Thoracic radiation therapy added to chemotherapy for small-cell lung cancer: an update of Cancer and Leukemia Group B Study 8083[END_REF][START_REF] Murray | Importance of timing for thoracic irradiation in the combined modality treatment of limited-stage small-cell lung cancer. The National Cancer Institute of Canada Clinical Trials Group[END_REF][START_REF] Gregor | Randomized trial of alternating versus sequential radiotherapy/chemotherapy in limited-disease patients with small-cell lung cancer: a European Organization for Research and Treatment of Cancer Lung Cancer Cooperative Group Study[END_REF][START_REF] Takada | Phase III study of concurrent versus sequential thoracic radiotherapy in combination with cisplatin and etoposide for limited-stage small-cell lung cancer: results of the Japan Clinical Oncology Group Study 9104[END_REF][START_REF] Skarlos | Randomized comparison of early versus late hyperfractionated thoracic irradiation concurrently with chemotherapy in limited disease small-cell lung cancer: a randomized phase II study of the Hellenic Cooperative Oncology Group (HeCOG)[END_REF][START_REF] Blackstock | Split-course versus continuous thoracic radiation therapy for limited-stage small-cell lung cancer: final report of a randomized phase III trial[END_REF][START_REF] Lebeau | A randomized clinical trial comparing concurrent and alternating thoracic irradiation for patients with limited small cell lung carcinoma[END_REF][START_REF] Work | Randomized study of initial versus late chest irradiation combined with chemotherapy in limited-stage small-cell lung cancer. Aarhus Lung Cancer Group[END_REF][START_REF] Park | The effects according to the timing of thoracic radiotherapy in limited stage small cell lung cancer[END_REF][START_REF] Jeremic | Initial versus delayed accelerated hyperfractionated radiation therapy and concurrent chemotherapy in limited small-cell lung cancer: a randomized study[END_REF] including 2,668 patients were eligible. Data on nine trials and 2,305 patients (86% of potentially eligible patients) were available for this IPD meta-analysis (Figure S1). Data from one trial were lost [START_REF] Work | Randomized study of initial versus late chest irradiation combined with chemotherapy in limited-stage small-cell lung cancer. Aarhus Lung Cancer Group[END_REF] and we did not succeed in contacting the investigator of two other trials [START_REF] Park | The effects according to the timing of thoracic radiotherapy in limited stage small cell lung cancer[END_REF][START_REF] Jeremic | Initial versus delayed accelerated hyperfractionated radiation therapy and concurrent chemotherapy in limited small-cell lung cancer: a randomized study[END_REF]. Table 1 depicts the nine trials included [START_REF] Spiro | Early compared with late radiotherapy in combined modality treatment for limited disease small-cell lung cancer: a London Lung Cancer Group multicenter randomized clinical trial and meta-analysis[END_REF][START_REF] Turrisi | Twice-daily compared with once-daily thoracic radiotherapy in limited small-cell lung cancer treated concurrently with cisplatin and etoposide[END_REF][START_REF] Perry | Thoracic radiation therapy added to chemotherapy for small-cell lung cancer: an update of Cancer and Leukemia Group B Study 8083[END_REF][START_REF] Murray | Importance of timing for thoracic irradiation in the combined modality treatment of limited-stage small-cell lung cancer. The National Cancer Institute of Canada Clinical Trials Group[END_REF][START_REF] Gregor | Randomized trial of alternating versus sequential radiotherapy/chemotherapy in limited-disease patients with small-cell lung cancer: a European Organization for Research and Treatment of Cancer Lung Cancer Cooperative Group Study[END_REF][START_REF] Takada | Phase III study of concurrent versus sequential thoracic radiotherapy in combination with cisplatin and etoposide for limited-stage small-cell lung cancer: results of the Japan Clinical Oncology Group Study 9104[END_REF][START_REF] Skarlos | Randomized comparison of early versus late hyperfractionated thoracic irradiation concurrently with chemotherapy in limited disease small-cell lung cancer: a randomized phase II study of the Hellenic Cooperative Oncology Group (HeCOG)[END_REF][START_REF] Blackstock | Split-course versus continuous thoracic radiation therapy for limited-stage small-cell lung cancer: final report of a randomized phase III trial[END_REF][START_REF] Lebeau | A randomized clinical trial comparing concurrent and alternating thoracic irradiation for patients with limited small cell lung carcinoma[END_REF] and Table S2 summarises the trials with no available data. Four trials [START_REF] Turrisi | Twice-daily compared with once-daily thoracic radiotherapy in limited small-cell lung cancer treated concurrently with cisplatin and etoposide[END_REF][START_REF] Gregor | Randomized trial of alternating versus sequential radiotherapy/chemotherapy in limited-disease patients with small-cell lung cancer: a European Organization for Research and Treatment of Cancer Lung Cancer Cooperative Group Study[END_REF][START_REF] Blackstock | Split-course versus continuous thoracic radiation therapy for limited-stage small-cell lung cancer: final report of a randomized phase III trial[END_REF][START_REF] Lebeau | A randomized clinical trial comparing concurrent and alternating thoracic irradiation for patients with limited small cell lung carcinoma[END_REF] had different radiotherapy modalities between the two arms, including three trials [START_REF] Turrisi | Twice-daily compared with once-daily thoracic radiotherapy in limited small-cell lung cancer treated concurrently with cisplatin and etoposide[END_REF][START_REF] Blackstock | Split-course versus continuous thoracic radiation therapy for limited-stage small-cell lung cancer: final report of a randomized phase III trial[END_REF][START_REF] Lebeau | A randomized clinical trial comparing concurrent and alternating thoracic irradiation for patients with limited small cell lung carcinoma[END_REF] comparing shorter vs. longer radiotherapy duration. Central randomisation was used in all trials, except one that used sealed envelopes [START_REF] Perry | Thoracic radiation therapy added to chemotherapy for small-cell lung cancer: an update of Cancer and Leukemia Group B Study 8083[END_REF]. In total, out of the 80 patients initially excluded from the individual trial analyses, data concerning 75 patients were recovered. The median follow-up was 10 years without any difference between the treatment arms. Patient characteristics were well balanced between the two arms of the analysis (Table S3). Three trials [START_REF] Turrisi | Twice-daily compared with once-daily thoracic radiotherapy in limited small-cell lung cancer treated concurrently with cisplatin and etoposide[END_REF][START_REF] Murray | Importance of timing for thoracic irradiation in the combined modality treatment of limited-stage small-cell lung cancer. The National Cancer Institute of Canada Clinical Trials Group[END_REF][START_REF] Takada | Phase III study of concurrent versus sequential thoracic radiotherapy in combination with cisplatin and etoposide for limited-stage small-cell lung cancer: results of the Japan Clinical Oncology Group Study 9104[END_REF] were categorised as having similar chemotherapy compliance in both arms, and they had a proportion of at least 79% of patients who were compliant with chemotherapy (i.e. receiving all their cycles) (Table S4). Five trials [START_REF] Spiro | Early compared with late radiotherapy in combined modality treatment for limited disease small-cell lung cancer: a London Lung Cancer Group multicenter randomized clinical trial and meta-analysis[END_REF][START_REF] Perry | Thoracic radiation therapy added to chemotherapy for small-cell lung cancer: an update of Cancer and Leukemia Group B Study 8083[END_REF][START_REF] Gregor | Randomized trial of alternating versus sequential radiotherapy/chemotherapy in limited-disease patients with small-cell lung cancer: a European Organization for Research and Treatment of Cancer Lung Cancer Cooperative Group Study[END_REF][START_REF] Skarlos | Randomized comparison of early versus late hyperfractionated thoracic irradiation concurrently with chemotherapy in limited disease small-cell lung cancer: a randomized phase II study of the Hellenic Cooperative Oncology Group (HeCOG)[END_REF][START_REF] Lebeau | A randomized clinical trial comparing concurrent and alternating thoracic irradiation for patients with limited small cell lung carcinoma[END_REF] had different chemotherapy (CT) compliance, with all of them exhibiting a lower compliance rate in the "earlier or shorter" arm. For the CCWFU62286 trial, we had no data available on individual CT compliance neither in the patient-level data provided by the investigator nor in the publication [START_REF] Blackstock | Split-course versus continuous thoracic radiation therapy for limited-stage small-cell lung cancer: final report of a randomized phase III trial[END_REF]: the CCWFU62286 trial was thus excluded from the trial subset analysis based on CT compliance. In the "later or longer" arm, 88% of patients started radiotherapy as compared to 93% in the "earlier or shorter" arm (Table S5). Among the five trials [START_REF] Spiro | Early compared with late radiotherapy in combined modality treatment for limited disease small-cell lung cancer: a London Lung Cancer Group multicenter randomized clinical trial and meta-analysis[END_REF][START_REF] Perry | Thoracic radiation therapy added to chemotherapy for small-cell lung cancer: an update of Cancer and Leukemia Group B Study 8083[END_REF][START_REF] Murray | Importance of timing for thoracic irradiation in the combined modality treatment of limited-stage small-cell lung cancer. The National Cancer Institute of Canada Clinical Trials Group[END_REF][START_REF] Gregor | Randomized trial of alternating versus sequential radiotherapy/chemotherapy in limited-disease patients with small-cell lung cancer: a European Organization for Research and Treatment of Cancer Lung Cancer Cooperative Group Study[END_REF][START_REF] Skarlos | Randomized comparison of early versus late hyperfractionated thoracic irradiation concurrently with chemotherapy in limited disease small-cell lung cancer: a randomized phase II study of the Hellenic Cooperative Oncology Group (HeCOG)[END_REF] comparing earlier and later radiotherapy with individual data on radiotherapy compliance, the observed difference in median times between the two arms from randomisation to the start of radiotherapy ranged from 63 to 93 days compared with 56 to 84 days for the planned difference (Table S6). There was also a significant association between individual RT compliance and CT compliance (Cochran-Mantel-Haenszel test stratified by trial: p < 0.0001). The more a patient was compliant with CT (i.e. receiving all their cycles), the more he/she was compliant with RT (i.e. receiving 90% of the total RT dose).

Overall survival and progression-free survival

In our main analysis, when all trials were pooled together, "earlier or shorter" radiotherapy did not have a significant impact on overall survival compared to "later or longer" radiotherapy (HR 0.99, 95% CI 0.91-1.08, p=0.78) (Figure S2). Treatment effect heterogeneity was observed (p=0.006, I²=63%). With a random effects model, the HR was not significant (0.99, 0.85-1.15, p=0.90).

Data on tumour progression were not available for two trials [START_REF] Gregor | Randomized trial of alternating versus sequential radiotherapy/chemotherapy in limited-disease patients with small-cell lung cancer: a European Organization for Research and Treatment of Cancer Lung Cancer Cooperative Group Study[END_REF][START_REF] Lebeau | A randomized clinical trial comparing concurrent and alternating thoracic irradiation for patients with limited small cell lung carcinoma[END_REF], thus the progression-free survival analysis concerned only seven trials comprising 1,764 patients and 1,596 events.

There was no significant impact of radiotherapy timing on progression-free survival (HR 0.93, 95% CI 0.84-1.02, p=0.13) (Figure S3).

Trial subsets

Table 2 shows the HRs for overall survival according to the different pre-planned subsets analyses, described in Table S7, with overall between-trial heterogeneity decomposed into the sum of between-subset and residual (within-subset) heterogeneity. Trial subsets were in decreasing order of residual heterogeneity: the lower the residual heterogeneity for one trial subset, the greater studied characteristic (CT compliance, RT dose per fraction, etc.) explained overall heterogeneity. In Table 2, between-subset heterogeneity was associated with an interaction test between the treatment received ("earlier or shorter" RT vs "later or longer" RT) and the studied characteristic of the subset, and also with a trend test when the studied subset categories were ordinal (RT dose per fraction and RT overall treatment time). Five trial characteristics were found to be associated with an improvement in overall survival with "earlier or shorter" radiotherapy (Table 2): similar CT compliance in both arms, a dose per fraction lower than 1.8 Gy, hyperfractionated radiotherapy, overall treatment time of less than 30 days, and platin-based chemotherapy. It should be emphasised that trials using hyperfractionated radiotherapy delivered fractions of less than 1.8 Gy, and overall treatment time was less than 30 days.

The "between-arm" CT compliance (number of cycles actually given) is the factor that best explained between-trial heterogeneity, i.e. with the lowest residual heterogeneity (Table 2).

Chemotherapy compliance and overall survival

The HR for overall survival was significantly in favour of "earlier or shorter" radiotherapy among trials in which the defined chemotherapy compliance was similar in both arms (Figure 1; HR 0.79, 95% CI 0.69-0.91) and in favour of "later or longer" radiotherapy among trials with different CT compliance: (1.19, 1.05-1.34). There was a significant interaction between chemotherapy compliance and the treatment effect (interaction test, p<0.0001). In trials with similar CT compliance in both arms, "earlier or shorter" radiotherapy compared to "later or longer" radiotherapy increased the absolute 3-year and 5-year overall survival rate by 5.7% (from 24.4% to 30.1%) and by 7.7% (from 16.5% to 24.2%), respectively (Figure 2). In trials with different CT compliance, "earlier or shorter" radiotherapy decreased the absolute 3-year and 5-year overall survival rate respectively by 3.8% (from 16.1% to 12.3%) and 2.2% (from 10.5% to 8.3%) (Figure 2). In other words, "earlier or shorter" radiotherapy extended the 5year mean survival time by 4.2 months (95% CI 1.8-6.7) from 24.7 to 28.9 months in trials with similar CT compliance. In trials with different CT compliance, "earlier or shorter" radiotherapy shortened the 5-year mean survival time by 3.1 months (95% CI 1.3-4.9) from 20.6 to 17.5 months.

Compliance with chemotherapy and progression-free survival

The HR for progression-free survival favours trials in which "earlier or shorter" radiotherapy was delivered with similar CT compliance in both arms (HR for similar CT compliance: 0.81, 95% CI 0.71-0.92; for different CT compliance: 1.12, 0.95-1.31) (Figure 3). In trials in which CT compliance was similar, "earlier or shorter" radiotherapy increased the 3-year progression-free survival rate by 6.3% (95% CI 1.0-11.6%) and the 5-year progression-free survival rate by 5.6% (0.7-10.5%) (Figure S4).

Compliance with chemotherapy and landmark analysis

As the observed effect of CT compliance may be due to early treatment interruption because of progression or death, a post-hoc landmark analysis on the impact of individual CT compliance on overall survival and progression-free survival was performed among patients who survived (or had no disease progression) for at least 120 days. This landmark was chosen because most of the patients finished their chemo-radiation treatment at 120 days. Patients with good CT compliance, i.e. those receiving the planned total number of chemotherapy cycles had higher overall survival and progression-free survival than those with poor CT compliance (HR: 0.56, 95% CI 0.49-0.64 and 0.70, 0.59-0.83 respectively; Table S8).

Sub-group analyses

When the two subsets of trials with similar and different CT compliance were considered separately, no variation in the treatment effect was seen according to age, sex or the performance status (Figure S5).

Sensitivity analyses

Table S9 shows the results of pre-planned sensitivity analyses after excluding some trials. The results were similar to those of the main analysis, in particular to those related to chemotherapy compliance.

Toxicity

Three types of severe acute toxicities were significantly more frequent in patients receiving "earlier or shorter" thoracic radiotherapy: neutrophil, oesophageal and cardiac toxicity (Table 3) [START_REF] Stewart | Meta-analysis of the literature or of individual patient data: is there a difference?[END_REF]. The toxicity odds ratios according to trial subsets based on CT compliance are shown in Table S10. We did not perform analyses on late toxicities as IPD were available only for two trials [START_REF] Murray | Importance of timing for thoracic irradiation in the combined modality treatment of limited-stage small-cell lung cancer. The National Cancer Institute of Canada Clinical Trials Group[END_REF][START_REF] Gregor | Randomized trial of alternating versus sequential radiotherapy/chemotherapy in limited-disease patients with small-cell lung cancer: a European Organization for Research and Treatment of Cancer Lung Cancer Cooperative Group Study[END_REF].

DISCUSSION

Based on this IPD meta-analysis of nine trials evaluating the optimal timing of thoracic radiotherapy in SCLC, overall there was no survival difference between "earlier or shorter" and "later or longer" thoracic radiotherapy (HR=0.99; p=0.78). As individual trials favoured either "earlier or shorter" or "later or longer" thoracic radiotherapy, it seemed relevant to further analyse these data and perform a subset analysis focusing on CT compliance. For trials with different CT compliance, in which lower compliance was always observed in the "earlier or shorter" arm, "earlier or shorter" delivery had a deleterious effect on survival compared to "later or longer" radiotherapy (HR 1.19, 95% CI 1.05-1.34). For trials that had similar (and good, i.e. at least 79% of compliant patients per arm) CT compliance, "earlier or shorter" delivery of thoracic radiotherapy improved overall survival (HR 0.79, 0.69-0.91). "Earlier or shorter" thoracic radiotherapy, when delivered with similar and good CT compliance, yielded an absolute survival gain of 5.7% at 3 years and 7.7% at 5 years compared with "later or longer" thoracic radiotherapy. Similar results were found for progression-free survival. We performed sensitivity analyses by only taking into account trials in which patients received concomitant chemoradiation and trials that exclusively addressed the timing of thoracic radiotherapy in their design. In these sensitivity analyses, the survival gain of delivering "earlier or shorter" thoracic radiotherapy with similar CT compliance remained significant (Table S9). Using a landmark analysis it was possible to confirm with IPD that good CT compliance was associated with longer survival. Of note, there was a significant association at patient-level between RT compliance and CT compliance which could explain our results. Hyperfractionated accelerated radiotherapy also improved survival when delivered "earlier or shorter", but this finding was driven by two large trials, JCOG9104 [START_REF] Takada | Phase III study of concurrent versus sequential thoracic radiotherapy in combination with cisplatin and etoposide for limited-stage small-cell lung cancer: results of the Japan Clinical Oncology Group Study 9104[END_REF] and ECOG3588 [START_REF] Turrisi | Twice-daily compared with once-daily thoracic radiotherapy in limited small-cell lung cancer treated concurrently with cisplatin and etoposide[END_REF],

with good CT compliance. In the ECOG3588 trial [START_REF] Turrisi | Twice-daily compared with once-daily thoracic radiotherapy in limited small-cell lung cancer treated concurrently with cisplatin and etoposide[END_REF], no dose adjustment was allowed for the first two cycles. Cisplatin-based chemotherapy seems to be more beneficial when combined with "earlier or shorter" thoracic radiotherapy. Issues such as the total radiotherapy dose and the dose per fraction are more difficult to interpret, because they are tightly correlated (Tables 1 and2). "Earlier or shorter" thoracic radiotherapy was associated with a higher incidence of acute severe oesophagitis than "later or longer" radiotherapy (OR 1.93 [1.45-2.56]), but had no consequence on compliance with either chemotherapy or radiotherapy. Mauguen et al [START_REF] Mauguen | Hyperfractionated or accelerated radiotherapy in lung cancer: an individual patient data meta-analysis[END_REF] also showed that hyperfractionated accelerated radiotherapy increased oesophageal toxicity.

In this IPD meta-analysis, neutropenia was more frequent with "earlier or shorter" radiotherapy (OR 1.54, 95% CI: 1.19-2.00) and this effect was observed exclusively in trials with similar CT compliance (Table S10). Acute severe pulmonary toxicity was similar in "earlier or shorter" or "later or longer" thoracic radiotherapy groups, while acute severe cardiac toxicity was higher when "earlier or shorter" radiotherapy was delivered (OR 3.12, 1.46-6.68). The latter finding should be interpreted with caution, for it is based on only 26 cardiac events occurring in 1,648 patients among whom this toxicity was documented.

The results of this IPD meta-analysis primarily reinforce the evidence that chemotherapy should be delivered as intended whenever possible [START_REF] Van Meerbeeck | Small-cell lung cancer[END_REF][START_REF] Pelayo | Chemotherapy versus best supportive care for extensive small cell lung cancer[END_REF]. Cisplatin-based chemotherapy administered with good CT compliance appeared to be the best treatment when combined with "earlier or shorter" thoracic radiotherapy as all the three trials [START_REF] Turrisi | Twice-daily compared with once-daily thoracic radiotherapy in limited small-cell lung cancer treated concurrently with cisplatin and etoposide[END_REF][START_REF] Murray | Importance of timing for thoracic irradiation in the combined modality treatment of limited-stage small-cell lung cancer. The National Cancer Institute of Canada Clinical Trials Group[END_REF][START_REF] Takada | Phase III study of concurrent versus sequential thoracic radiotherapy in combination with cisplatin and etoposide for limited-stage small-cell lung cancer: results of the Japan Clinical Oncology Group Study 9104[END_REF] with similar CT compliance used this regimen. This is in line with previous literature-based meta-analyses [START_REF] Mc | Early versus late chest radiotherapy for limited stage small cell lung cancer[END_REF][START_REF] De Ruysscher | Systematic review and metaanalysis of randomised, controlled trials of the timing of chest radiotherapy in patients with limited-stage, small-cell lung cancer[END_REF][START_REF] Pijls-Johannesma | Timing of chest radiotherapy in patients with limited stage small cell lung cancer: a systematic review and meta-analysis of randomised controlled trials[END_REF][START_REF] Spiro | Early compared with late radiotherapy in combined modality treatment for limited disease small-cell lung cancer: a London Lung Cancer Group multicenter randomized clinical trial and meta-analysis[END_REF][START_REF] Fried | Systematic review evaluating the timing of thoracic radiation therapy in combined modality therapy for limited-stage small-cell lung cancer[END_REF][START_REF] Huncharek | A meta-analysis of the timing of chest irradiation in the combined modality treatment of limited-stage small cell lung cancer[END_REF] in particular that reported by Spiro et al [START_REF] Spiro | Early compared with late radiotherapy in combined modality treatment for limited disease small-cell lung cancer: a London Lung Cancer Group multicenter randomized clinical trial and meta-analysis[END_REF] which focused on CT compliance (Table S1).

Interestingly, a recently published randomised trial [START_REF] Kubota | Etoposide and cisplatin versus irinotecan and cisplatin in patients with limited-stage small-cell lung cancer treated with etoposide and cisplatin plus concurrent accelerated hyperfractionated thoracic radiotherapy (JCOG0202): a randomised phase 3 study[END_REF], where all patients had early hyperfractionated radiotherapy given concomitantly with the first cycle of etoposide, showed a 5-year survival rate of 34.3% that the authors attributed to better patient selection and radiotherapy quality control. It will be interesting to observe the results of the on-going CALGB 30610 (NCT00632853) and the completed CONVERT (NCT00433563) randomised trials comparing early hyperfractionated radiotherapy to early standard radiotherapy with a higher total dose and concomitant cisplatin plus etoposide in both arms.

The present IPD meta-analysis has some shortcomings. First, the trials were conducted at a time when imaging was not as advanced as it is today. However, the observed 5-year survival rate of about 25%, when "earlier or shorter" thoracic radiotherapy was combined with good chemotherapy compliance, remains among the best published results. These results continue to support their applicability today, as there has been no major change in the standard of care of SCLC (NCCN and ESMO guidelines) [START_REF]Practice Guidelines in Oncology: Small Cell Lung Cancer[END_REF]. A recently published Korean phase III trial [START_REF] Sun | Phase III trial of concurrent thoracic radiotherapy with either first-or third-cycle chemotherapy for limited-disease small-cell lung cancer[END_REF], which was not included in this meta-analysis as it was closed to accrual in 2010 (Table S2), showed a similar 5-year survival rate of approximately 24%. This trial did not show a significant difference in terms of overall survival between the two arms (HR 0.93, 0.67-1.29), but the study included only 222 patients. Secondly, data were not available for two other trials [START_REF] Work | Randomized study of initial versus late chest irradiation combined with chemotherapy in limited-stage small-cell lung cancer. Aarhus Lung Cancer Group[END_REF][START_REF] Jeremic | Initial versus delayed accelerated hyperfractionated radiation therapy and concurrent chemotherapy in limited small-cell lung cancer: a randomized study[END_REF] (Table S2). However, when we included these three trials for which we have only published data (two in the similar CT compliance group [START_REF] Jeremic | Initial versus delayed accelerated hyperfractionated radiation therapy and concurrent chemotherapy in limited small-cell lung cancer: a randomized study[END_REF][START_REF] Sun | Phase III trial of concurrent thoracic radiotherapy with either first-or third-cycle chemotherapy for limited-disease small-cell lung cancer[END_REF] and one in a different CT compliance group [START_REF] Work | Randomized study of initial versus late chest irradiation combined with chemotherapy in limited-stage small-cell lung cancer. Aarhus Lung Cancer Group[END_REF]) in a post-hoc analysis, we found similar effects on overall survival (HR 0.81, 95% CI 0.72-0.90 vs. 1.18, 1.06-1.32 for similar and different CT compliance subsets respectively). Third, only the number of chemotherapy cycles administered were available, but not doses or delays in treatment. However, consistency across endpoints and between the main analysis and sensitivity analyses underscore the robustness of our results.

Another limitation is that data on long-term toxicity were not available, but less toxicity would be expected with the newer radiotherapy techniques. Lastly, the quality of radiotherapy could not be addressed in this meta-analysis as it was not explored in the studies included.

To improve the still dismal prognosis of patients with stage I-IIIB SCLC, we postulate that the optimal treatment should be full-dose but acceptable chemotherapy combined with "earlier or shorter" thoracic radiotherapy (i.e. before 9 weeks) preferably within a short overall treatment time. Our IPD meta-analysis provides the best evidence of the beneficial effect of "earlier or shorter" radiotherapy when chemotherapy is administered with good compliance.

Figure 1 .Figure 2 .Figure 3 .

 123 Figure 1. Effect of "earlier or shorter" radiotherapy versus "later or longer" radiotherapy on overall survival according to chemotherapy complianceEach trial is represented by a square, the centre of which denotes the hazard ratio of death for that trial comparison with the horizontal lines showing the 95% confidence intervals (CIs). The size of the square is directly proportional to the amount of information contributed by the trial. The clear diamonds represent pooled hazard ratios for the trial groups and the black diamond the overall hazard ratio, with the centre denoting the hazard ratio and the extremities the 95% CI. The fixed effect model was used. Trials were chronologically ordered within each category of trials. Of note, data on CT compliance were not available for the CCCWFU62286 trial which is thus not included in this analysis. Abbreviations: CI = Confidence Interval; CT = Chemotherapy; HR = Hazard ratio; O-E = Observed-Expected; RT = Radiotherapy

Table 1 . Description of trials

 1 Number of patients analysed equals the number of patients randomised, except for the HeCOG trial for which data on 81 patients were available out of the 86 randomised patients. £ Publication[START_REF] Gregor | Randomized trial of alternating versus sequential radiotherapy/chemotherapy in limited-disease patients with small-cell lung cancer: a European Organization for Research and Treatment of Cancer Lung Cancer Cooperative Group Study[END_REF] stated that: "RT started […] on the 14 th day of the second and subsequent courses of chemotherapy in arm Earlier RT"

	Trials	Inclusion period	Initiation of thoracic radiation (day)	RT dose (Gy)/ fraction/ duration (wks)	CT (mg/m²)	Nb CT cycles (before RT, during RT, after RT)	Nb of patients randomised $	Median follow-up (years)
	Earlier vs. Later Radiotherapy
	CALGB8083 [25]	1981-84 EoS: Day 1	50 Gy / 24 fr / 5 wks	C: 1000 mg/m², every 3 wks	about 26 cycles	292	17.2
			LoL: Day 64		V: 1.4 mg/m², every 3 wks	EoS: 2 cycles during RT, up to 24 cycles after RT
					E: 80 x 3 mg/m², every 3 wks	LoL: 3 cycles before RT, 2 cycles during RT, up to 21
					Starting at cycle 7 for odd-numbered cycles:	cycles after RT
					C: 1000 mg/m², every 3 wks
					V: 1.4 mg/m², every 3 wks
					A: 50 mg/m², every 3 wks
	BR.6 [26]	1985-88 EoS: Day 22	40 Gy / 15 fr / 3 wks	EoS: P: 25 mg/m² x 3days, wks4,11,17	EoS: 6 cycles (1 before RT, 1 during RT, 4 after RT)	332	11.2
			LoL: Day 106		E: 100 mg/m² x 3days, wks4,11,17	LoL: 6 cycles (5 before RT, 1 during RT)
					alternating with
					C: 1000 mg/m², wks1,8,14
					A: 50mg/m², wks1,8,14
					V: 2 mg, wks1,8,14
					LoL: P: 25 mg/m² x3days, wks4,10,16
					E: 100 mg/m² x3days, wks4,10,16
					alternating with
					C: 1000 mg/m², wks1,7,13
					A: 50mg/m², wks1,7,13
					V: 2 mg, wks1,7,13
	EORTC08877 [27]	1989-95 EoS: Day 43	EoS: 12.5 Gy / 5 fr / 1 wk + break 3wks
			LoL: Day 99	+ 12.5 Gy / 5 fr / 1wk + break 3wks
				+ 12.5 Gy / 5 fr / 1 wk + break 3wks
				+ 12.5 Gy / 5 fr / 1 wk
				LoL: 50 Gy / 20 fr / 4 wks

$

Table 2 . Effect of "earlier or shorter" radiotherapy versus "later or longer" radiotherapy on overall survival according to different trial subsets

 2 Total heterogeneity is the sum of between-subset and residual (within-subset) heterogeneity, and is equal to 21.6 (analysis based on 9 trials) except for CT compliance 21.4 (8 trials). The test associated with between-subset heterogeneity corresponds to the interaction test. The lower residual heterogeneity, the greater overall heterogeneity of the treatment effect between trials was explained by the trial characteristic.

				Heterogeneity $
	Trials Characteristics	HR [95% CI] £	Between-subset	Residual (or within-subset)
	CT compliance between arms	Similar Different	0.79 [0.69-0.91] 1.19 [1.05-1.34]	19.5***	1.9
		< 1.8 Gy	0.82 [0.71-0.96]
	RT dose per fraction	1.8-2.4 Gy	1.11 [0.90-1.35]	7.5* (p trend = 0.02) $$	14.1
		> 2.4 Gy	1.06 [0.94-1.20]
		Hyperfractionated	0.82 [0.71-0.96]
	Type of RT			7.4**	14.2
		Standard	1.07 [0.96-1.19]
		≤ 30 days in both arms	0.89 [0.78-1.02]
	RT Overall Treatment Time	One arm ≤ 30 days, one > 30 days	0.99 [0.85-1.15]	5.6 (p trend = 0.02) $$	16.0
		> 30 days in both arms	1.16 [0.98-1.38]
	Platin-based CT during RT in both arms	Yes No	0.89 [0.79-1.01] 1.09 [0.97-1.24]	5.5**	16.1
		Yes	0.95 [0.85-1.06]
	Concurrent CT in both arms			1.5	20.1
		No	1.06 [0.92-1.22]
		Yes	0.96 [0.85-1.08]
	Same RT in the 2 arms			0.5	21.1
		No	1.02 [0.90-1.16]
	* : p<0.05 ; ** : p<0.01 ; *** : p<0.001 ;
	£ : Hazard ratio of death following "earlier or shorter" versus "later or longer" radiotherapy.

$ : $$ : test for trend; Abbreviations: CI = Confidence Interval; CT = Chemotherapy; HR = Hazard Ratio; RT = Radiotherapy

Table 3 . Acute toxicity according to radiotherapy arm

 3

		Availability	Toxicity rate	Results
	Severe toxicity (grade 3-5)	No. of trials (patients)	"Later or longer" RT	"Earlier or shorter" RT*	OR [95% CI]	p-value efficacy	I²	p-value heterogeneity
	Neutrophil	6 (1,453)	59	69	1.54 [1.19-2.00]	0.001	79%	<0.001
	Haemoglobin	6 (1,476)	21	24	1.17 [0.91-1.52]	0.22	31%	0.21
	Platelets	7 (1,817)	18	21	1.22 [0.96-1.55]	0.11	45%	0.09
	Oesophageal	8 (1,950)	8	14	1.93 [1.45-2.56]	<0.001	45%	0.08
	Pulmonary	5 (1,207)	4	6	1.50 [0.86-2.62]	0.16	0%	0.68
	Cardiac	6 (1,648)	1	3	3.12 [1.46-6.68]	0.003	0%	0.95
	Toxicity was graded according to the National Cancer Institute Common Toxicity Criteria, World Health Organization criteria, or Eastern
	Cooperative Oncology Group Common Toxicity Criteria depending on the trials. Severe toxicity was defined as grade 3 to 5 toxicity. Grade 5 was
	present only for pulmonary toxicity (n=4) and cardiac toxicity (n=1).
	Radiotherapy

*: The difference in the rate of toxicity between the two treatment arms was computed based on the rate in the "later or longer" radiotherapy arm and the odds ratio

[START_REF] Stewart | Meta-analysis of the literature or of individual patient data: is there a difference?[END_REF]

.

Abbreviations: CI = Confidence Interval; OR = Odds Ratio of the "earlier or shorter" RT arm compared with "later or longer" RT arm; RT =

The optimal timing and sequencing of thoracic radiotherapy and chemotherapy, which is the standard treatment of "limited-stage" small-cell lung cancer, has fuelled debate for many years. This individual patient data meta-analysis provides the best evidence of the beneficial effect of "earlier or shorter" radiotherapy when chemotherapy is administered with good compliance.

ACKNOWLEDGEMENTS

We thank the patients and the clinical investigators who took part in the trials and contributed to this research. The meta-analysis would not have been possible without their participation or without the collaborating institutions or groups that provided their trial data: The Alliance for Clinical Trials in Oncology (formerly Cancer and Leukemia Group B), Comprehensive

FUNDING

The meta-analysis was funded by the French National Cancer Institute (Programme Hospitalier de Recherche Clinique), the Ligue Nationale Contre le Cancer, and partly by Sanofi-Aventis (unrestricted grants). The investigators meeting was also funded by Gustave Roussy, Lilly and Astra-Zeneca (unrestricted grants). No grant number is applicable.

Cancer Centre of Wake Forest University, ECOG-ACRIN Cancer Research Group (formerly Eastern Cooperative Oncology Group), European Organization for Research and Treatment of Cancer, Hellenic Cooperative Oncology Group, Japan Clinical Oncology Group, London Lung Cancer Group, National Cancer Institute of Canada Clinical Trial Group, "Petites Cellules" Group. We are grateful to Lorna Saint Ange for editorial assistance.

RTT-SCLC Collaborative Group