Giovanni Bonny
email: gbonny@sckcen.be

Roberto C Pasianot

Dmitry Terentyev

Lorenzo Malerba

Iron chromium potential to model high-chromium ferritic alloys

Keywords: interatomic potential, atomistic simulation, thermodynamics, dislocations iron chromium alloys interatomic potential, atomistic modelling, iron chromium alloys, thermodynamics, dislocations PACS: 81.30.-t, 81.40.Cd, 61.72.J-, 61.72.Lk 1. Introduction

In this paper we present an Fe-Cr interatomic potential to model high-Cr ferritic steels. The potential is fitted to thermodynamic and point-defect properties obtained from density functional theory (DFT) calculations and experiments. The here developed potential is also benchmarked against other potentials available in literature. It shows particularly good agreement with the DFT obtained mixing enthalpy of the random alloy, the formation energy of intermetallics and experimental excess vibrational entropy and phase diagram. Also DFT calculated point-defect properties, both interstitial and substitutional, are well reproduced as is the screw dislocation core structure. As a first validation of the potential we study the precipitation hardening of Fe-Cr alloys by means of static simulations of the interaction between Cr precipitates and screw dislocations. It is concluded that the description of the dislocation core modification near a precipitate might have a significant influence on the interaction mechanisms observed in dynamic simulations.

Fe-Cr's complex behaviour is related to magnetic interactions [11]. Yet, in the literature two interatomic potentials have been developed in an embedded atom method (EAM) like formalism, which are capable of reproducing Fe-Cr's heat of mixing behaviour as a function of composition, even without explicitly introducing magnetism. Although both formalisms succeed at describing the complexity of the ferro-magnetic phase in an effective way, temperature dependent effects attributed to magnetic transitions cannot be captured within these models. These are the two-band model (2BM) potential developed by Olsson et al. [20] (henceforth OLS) and the concentration dependent model (CDM) potential developed by Caro et al. [21] (henceforth CAR). Both formalisms are adaptations of the EAM introducing local concentration dependence. Both potentials, however, suffer from some important drawbacks regarding thermodynamic properties, point-defect properties and description of screwdislocations. In this work, we develop a 2BM potential to correct those short-comings, while keeping the known useful qualities of both potentials.

The paper is organized as follows. In section 2 we describe the 2BM formalism and the fitting methodology used; in section 3, calculated properties for the pure potentials are presented and compared with existing potentials. Sections 4 and 5 report, respectively, thermodynamic and point-defect properties of our potential, and compare them to experiment and DFT data; also included is a comparison against OLS (strictly, the one fitted to PAW data from [20; 22]) and CAR. In section 6, as a first application of our potential, we study Cr precipitation hardening by analysing the interaction between a screw dislocation and a Crprecipitate. Finally, section 7 summarizes the advantages and drawbacks of the different potentials.

Formalism and Fitting Methodology

It has been shown that both the 2BM and the CDM formalisms are adequate models to describe Fe-Cr's complex mixing enthalpy without accounting for magnetism explicitly [23].

Here we choose the 2BM formalism, where the total energy is given as in the EAM with an extra embedding term, F s , The additional embedding term on the right-hand side is assumed to stem from the s-band electrons, while the standard embedding, F d , is thought of as resulting from the d-band electrons [20]; t i denotes the atom type (in our case Fe or Cr) on site i. The electron densities, ρ d and ρ s , for the d-and s-band, respectively, are calculated as,

∑ ∑ ∑ + + = ≠ i i t i i t j i j i ij t t i i j i F F r V E ) ( ) ( ) ( 2
1 s s d d ) ( , ρ ρ , (1)
∑ ≠ = ) ( ) ( i j j ij t t i r j i λ λ ϕ ρ . ( 2
)
Here ϕ d and ϕ s are the d-and s-density functions, respectively, which comply with the relations,

       = = = = = = = s BA s AB s BB s AA d B d AB d BB d A d BA d AA ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ 0 . ( 3
)
This choice is in line with [20] so that the d-density reduces to the EAM density, and the sdensity samples the local concentration [23]. In this way the s-embedding terms are only relevant for the alloy, and do not contribute to the pure elements' energy, which keeps the standard EAM expression.

Thus, for the Fe-Cr system and within the 2BM, ten functions need to be determined: For pure Fe, we chose the potential developed by Mendelev et al. [24], while the pure Cr potential and the mixed Fe-Cr interactions were fitted as is described further on. As is also better detailed further on, the main reason to refit the potential for pure Cr is to provide a better description of dislocation properties in this metal. Generally, the fitting of an interatomic potential to material properties is a problem of matching data obtained from a trial function to a given data set. This problem can be cast into one of minimizing the overall squared deviation, so-called objective function (OF), between the calculated properties and the associated reference data, possibly also imposing constraints. Within our methodology, those properties become linear in the fitting parameters, reducing the scheme to a quadratic programming problem [START_REF] Nocedal | Numerical Optimization[END_REF] which is exactly solvable; more details are given below.

V FeFe , d

Chromium Potential

Prior to fitting the pair interaction V and the embedding term F, we define a reasonable form for the electron density function ϕ. For this purpose, a simple Thomas-Fermi screening function is selected, which is shifted and truncated for a smooth cut-off between third and fourth nearest neighbour distance (on the bcc equilibrium crystal),

) ( ) exp( ) exp( ) ( cut cut cut 0 x f x x x x x         - - - = β β ϕ ϕ , (4)
       < ≤ < - - - ≤ = x x x x x x x x f cut cut 3 cut 3 cut , 0 1 , ) 1 ( ) 1 ( 1 1 , 1 ) ( . (5)
The pair interaction V is parameterized by the cubic spline expansion,

∑ = - Θ - = N k k k k r r r r a r V 1 3 ) ( ) ( ) ( , (6)
where N denotes the number of knots, a k are the fitting parameters and Θ the Heaviside unit step function. As explained below, the embedding term F is only available in numerical format.

The essential material properties considered in the fit for pure Cr (anti-ferromagnetic phase unless stated otherwise) are the correct stability of different crystallographic structures, the cohesive energy E coh , the equilibrium lattice constant a 0 , the elastic constants (for cubic lattice structures) C 11 , C 12 , C 44 [START_REF] Kittel | Introduction to Solid State Physics[END_REF] (paramagnetic phase, see Section 3), and the vacancy formation energy, V f E . In addition to these, we also included in the fit the formation energy of self-interstitial configurations (dumbbells oriented along the 〈100〉, 〈110〉 and 〈111〉 E , the {110} 〈111〉 gamma surface cut, γ, and the ½ 〈111〉 screw-dislocation core structure. In particular, the gamma surface cut (~10 points)

was fitted through the OF, only using the effective pair potential (i.e. neglecting higher order many-body contributions) [START_REF] Finnis | [END_REF]. The lattice parameter (zero pressure condition), unrelaxed vacancy formation energy, V f E , and the three elastic constants were fitted through equality constraints. The bcc lattice stability against fcc (E coh (bcc)-E coh (fcc)>0) and the Duesbery &

Vitek condition for a compact ½ 〈111〉 screw dislocation core (γ(b/3) -2 γ(b/6) > 0 [28], with b the length of the burgers vector) were fitted through inequality constraints.

Given the effective pair potential, the embedding function is computed by subtraction from Rose's equation [29], which describes the energy per atom E(ã) for the lattice under uniform expansion or contraction. The expression as a function of the nearest neighbour distance, r 1nn , is given as,

) ẽxp( ) 1 ( ) ( coh a a E a E - + - = . (7)
Here

) 1 / ( ~1nn 0 1nn - = r r a α and coh 2 / 9 E B Ω = α
, where Ω is the atomic volume and B is the bulk modulus. This embedding function, however, is not the one entering the potential. In order to ensure a smooth behaviour and particularly, to constrain the function curvature to remain positive throughout and decreasing at high density, the former embedding is re-fitted on a discrete number of points (~100) taken as reference data. The fit is performed in a minimum square sense and by a cubic spline expansion (also fulfilling F'(0)=0). It is the latter expansion that constitutes the embedding function. The price paid for a smooth behaviour is a violation of Rose's equation; however, in the applications of interest the lattice will never be subjected to the large, uniform contractions and dilations that are probed by Rose's equation.

Finally, the last fitting step consisted in accommodating manually (i.e. by trial and error) the short distance region to fit the DFT obtained values for self-interstitial formation energies.

Alloy Potential

Given the EAM potentials for the pure elements and prior to fitting V FeCr ,

r g r r K r ζ ϕ - = . (8)
Here K=20.34075425 is a normalisation constant, ζ=2.5001 and g cut is a cut-off function to provide a smooth cut-off between fifth and sixth nearest neighbour distance, ) (

       < ≤ <             - - ≤ = r r r r r d r r r r r g f c f c i c m i c cut , 0 , ) ( 2 sin 1 2 1 , 1 ) ( π , (9)
ρ ρ ρ A A F s + = , (10)
with A 1 and A 2 fitting parameters. The square root proved to be essential in reproducing the correct trends in Cr-Cr and Cr-vacancy interactions. In fact, former attempts using spline expansions as in [23] were met with failure. In this work we normalized the (maximum) sdensity in such a way that unity corresponds to a single Cr hosted in the Fe matrix.

The mixed pair interaction, as well as both s-embedding functions, were fitted simultaneously to all properties of interest. The mixing enthalpy was fitted in the whole concentration range (~50 points) using a variance expansion as described in [23], while at each concentration a zero pressure condition was imposed to obtain a lattice parameter as close as possible to the linear interpolation between the equilibrium lattice parameters of the pure elements. In addition, the Cr-Cr and Cr-vacancy interaction energies up to second nearest neighbour were fitted, as well as the Cr-vacancy migration energy in Fe, using DFT reference values as guides [12]. All the mentioned properties were fitted using appropriate weights to obtain reasonable values. In addition to the expansion coefficients of the s-embedding functions and the mixed pair potential, there remains one degree of freedom that expresses the relative weight between the d-densities of Fe and Cr, ρ Fe /ρ Cr . By manually changing this fraction and consequently solving the quadratic programming problem, we fitted the excess vibrational entropy, computed in the harmonic approximation [31], to a value as close as possible to the experimental one, without losing the above properties and without stabilizing unphysical intermetallic compounds. Finally, the potential was manually adjusted in the short distance region to reproduce the DFT-obtained binding energy of the mixed 〈110〉 Fe-Cr dumbbell. A description of the full Fe-Cr potential is given in Appendix A.

Properties of the Iron and Chromium Potentials

From the EAM-type potentials available in the literature to describe pure Fe that are capable of reproducing the 〈110〉 self-interstitial configuration as the most stable, we considered those from Refs. [24], [32] and [33], finally selecting "potential 2" developed by Mendelev et al.

[24], which has been widely used and tested. In [34] it has been demonstrated that the potentials from [24] and [32] succeed best in reproducing the properties of iron that are of interest for radiation damage studies. However, of all, our choice is the only one for which a clear proof exists of being able to consistently describe stable glide of a screw dislocation in a {110} plane [35].

Among the potentials for Cr, we considered those from Refs. [36] and [20]. The latter, henceforth O05, was used in combination with the Fe potential by Ackland et al. [32] to produce the OLS and CAR potentials. The basic properties of our potential are summarized and compared to O05 and DFT data in Table 1. There, B=(C 11 +2C 12 )/3 is the bulk modulus,

C'=(C 11 -C 12 )
/2 is the tetragonal shear and C 44 is the shear modulus [START_REF] Kittel | Introduction to Solid State Physics[END_REF]. Clearly, the experimental elastic properties are well reproduced by both potentials. Note, however, that at zero Kelvin Cr is anti-ferromagnetic and has a negative Cauchy pressure [37], which cannot be reproduced within a standard central force framework [38; 39; 40]. Above its Néel temperature (~310 K), however, Cr is paramagnetic with a positive Cauchy pressure [41].

Since we focus on technological applications above room temperatures, our potential was fitted to the elastic constants of paramagnetic Cr, linearly extrapolated down to zero Kelvin (the values given in Table 1, see e.g. [36]). Furthermore, our potential also closely fits a 0 , to DFT data regarding the energy difference between the body-centred cubic (bcc) and facecentred cubic (fcc) phase, thus providing enhanced stability of the bcc phase at high temperature. In summary, from the values presented in Table 1 we can conclude that the basic properties of Cr are equally well reproduced by both potentials.

Instert table 1 around here

The O05 potential, however, does not reproduce the correct structure for a ½ 〈111〉 screw-dislocation core, which is the main reason to refit the Cr potential. This is depicted in Figure 1 via differential displacement maps [42]. Clearly O05 shows a three-fold symmetry core, opposite to the compact one, as predicted by DFT [43] and our potential.

Insert figure 1 around here

Thermodynamic Properties

Prior to presenting the phase diagrams resulting from the different potentials, we compare and discuss statically calculated properties that influence the Fe-Cr phase diagram. In Figure 2 the mixing enthalpy calculated with the three potentials is compared to DFT data [8; 12; 44]. For the mixing enthalpy resulting from the potentials 50 compositions in the complete concentration range were sampled. For each composition the solutes were randomly distributed in cubic bcc crystals containing 2000 atoms and relaxed at constant pressure using a conjugate gradient method. Although different DFT techniques give different values for the mixing enthalpy, the DFT curves show the same behaviour, i.e., a change of sign in the mixing enthalpy, being negative below ~5 at.% Cr and positive above. Both 2BM potentials were fitted to DFT data [12; 44] obtained from special quasi-random structures (SQSs) [45] while CAR was fitted to DFT data [8] obtained from a coherent potential approximation (CPA) [46]. The mixing enthalpy curves from our potential and CAR reproduce the DFT mixing enthalpy very well, both qualitatively and quantitatively. Below ~25 at.% Cr this is also true for OLS; the complete curve, however, takes a symmetric shape, in contrast to the asymmetric DFT shape. As a consequence, the mixing enthalpy takes negative values at the Cr-rich side too, leading to short-range order and high Fe solubility in the Cr-rich phase.

These observations contradict DFT data and have never been reported in experimental studies. The formation energy of intermetallic compounds is another issue to be considered. In Figure 3 the formation energy of 57 intermetallic compounds (as described in [11; 47]) calculated by the three potentials and DFT are compared. In both cases, the configurations were relaxed at constant pressure using a conjugate gradient method. Below 10 at.% Cr DFT predicts stable intermetallic compounds, consistent with the negative mixing enthalpy for the disordered alloy. Above 10 at.% Cr, the formation energy of the intermetallic compounds becomes positive and remains so throughout the composition range.

In the Fe-rich region (< 10 at.% Cr), the most stable compounds predicted by DFT are also the most stable ones predicted by all three potentials (see Figure 3), with most importantly the lowest compound at 6.67 at.% Cr. This compound is described by a unit cell with axes oriented along the [ ] 133 . It should be noted that this compound is consistent with the ordered phase observed in the Monte Carlo simulations by Pareige et al. [48] using OLS. Among the three, both 2BM potentials also give good quantitative agreement in that concentration range. At higher Cr composition (> 10 at.%), OLS generally overstabilizes the compounds thereby even giving negative values at 50 and 93.33 at.% Cr. The latter is consistent with the low (compared to DFT) and negative mixing enthalpy of the disordered alloy at equiatomic and Cr-rich compositions, respectively. Such negative values in turn may lead to unphysical configurations in the course of e.g. Monte Carlo simulations, and thus constitute a drawback of the potential. The other two potentials do not suffer from the latter problems, although ours still somewhat overstabilizes the compounds.

210 , [ ]

Insert figure 3 around here

Experiments have shown the excess vibrational entropy to be non-negligible for the Fe-Cr system [49; 50; 51]; also, it was predicted to have a significant influence on the location of the solubility limit [44]. We calculated the excess vibrational entropy with all three potentials for disordered alloys, based on the harmonic approximation [31] in the Г point applied to cells containing 1024 atoms, prepared as purely random samples.

In Figure 4 gives a close fit to the experimental values. From the three, however, both 2BM potentials agree best, with ours and OLS closest to the data from [49] and [51], respectively. In the former case the experimental data is underestimated by a factor of about 1.5; in the latter it is overestimated (below ~75 at.% Cr) by about the same factor. The CAR potential, on the other hand, underestimates vib xc S by about a factor 3 and gives negative contributions below ~12 at.% Cr, which contradicts experiments (see also [44; 52]).

Insert figure 4 around here

The phase diagrams based on all three potentials are compared to a recently proposed parameterization based on the CALPHAD methodology [53] in Figure 5. The CALPHAD miscibility gap is calculated from an experimentally based [53; 54] Gibbs free energy parameterization. To indicate the ferro to paramagnetic phase transition, the CALPHAD calculated Curie temperature is also added. The phase boundaries for our potential and OLS are obtained from isobaric Monte Carlo simulations in the semi-grand canonical ensemble [55]. In the figure the data points are interpolated by smooth curves to guide the eye. These curves are based on the data points and the fact that full solubility was verified to occur starting from 1200K and 800K for our potential and OLS, respectively. The phase boundaries for CAR were obtained from free energy calculations using a thermodynamic integration technique (see [52] for more details). Thus in all cases vibrational and configurational entropy were accounted for.

When focussed on the Fe-rich side, we observe large Cr solubility at low temperature, due to the negative heat of mixing. As shown in Figure 5, this behaviour is well reproduced by all potentials, with a particularly close agreement for ours. At about 750K the Fe-rich solubility limit increases until full solubility is observed above the critical temperature ~900K within the CALPHAD calculation. The curve resulting from OLS underestimates this critical temperature by ~200K while our potential overestimates it by roughly the same amount. The critical temperature resulting from CAR, on the other hand, lies above the melting temperature. Note that the difference in critical temperature between OLS and ours closely follows the different trends observed in the excess vibrational entropy. Good agreement of the critical temperature obtained with potentials and CALPHAD shoud not be expected since magnetic interactions are not explicitly included in the former. Below ~25 at.%, however, range the Calphad phase boundary is well below the Curie temperature. For higher Cr concentrations, however, care should be taken as the magnetic transition might significantly affect the solubility, thus close agreement between experiment and potential prediction might not be a concern for potentials that effectively describe the ferro-magnetic phase.

On the Cr-rich side, the Fe solubility approaches zero with decreasing temperature, which is well reproduced by both our potential and CAR. The OLS potential, on the other hand, predicts high Fe-solubility as an unphysical artefact of the negative heat of mixing at the Cr-rich side.

Insert figure 5 around here

Point-Defect Properties

The formation, migration and binding energies reported in the current section were calculated in cubic bcc boxes containing 2000 atoms at constant volume (a 0 =2.8553Å) up to a precision of 1meV. For the formation and binding energies the boxes were relaxed using a conjugate gradient method and the migration energies were estimated using the nudged elastic band method [56].

In Figure 6 the Cr-Cr and Cr-vacancy interaction energy (E I ) in bulk Fe calculated using the three potentials and DFT [12] is presented. The DFT data show Cr-Cr repulsion in bulk Fe that decreases fast with increasing distance. This behaviour is mainly responsible for the experimentally observed SRO and is acceptably well reproduced by all potentials. Both 2BM potentials, however, underestimate the difference in repulsion between first and second nearest neighbour Cr pairs; an effect that is well reproduced by CAR. Concerning the Crvacancy interaction, according to both DFT calculations [12] and experiments [57], it is essentially negligible. As shown in Figure 6, this feature is also qualitatively reproduced by all potentials, that predict weak repulsion between Cr and vacancy. It is important, however, that the interaction energy between 1nn and 2nn Cr-vacancy pairs increases, as reproduced by all potentials. This is a prerequisite to obtain the correct order between forward and backward Cr-vacancy exchanges for several chemical environments in the Fe matrix, as discussed further.

Insert figure 6 [58]. The DFT results show that the barriers are largest for both Cr and Fe exchanges when there are three Cr and three Fe atoms occupying the six nearest neighbour positions around the saddle-point. This effect is only reproduced by our potential, although OLS and CAR also give reasonable trends. Note, however, that in the case of the CAR potential, the value for Cr migration in bulk Fe is significantly overestimated. As a consequence, Cr migration barriers calculated with CAR are somewhat higher as compared to the corresponding DFT values. For the Fe jumps, on the other hand, the migration barriers are largely underestimated when locally enriched by three or more Cr atoms. For completeness, a table summarizing the precise configurations and values of both forward and backward jumps on which Figure 8 is based is given in Appendix B.

Insert figure 7 around here

In Figure 8 we show the formation energy of interstitial defects in bulk Fe, taking the 〈110〉 mixed dumbbell as reference; values predicted by DFT [12] and the three potentials are compared. From the DFT data [12] it follows that the 〈110〉 mixed dumbbell configuration in bulk Fe has the lowest formation energy. However, the inclusion of Cr in the defect diminishes the difference in formation energy between the 〈111〉 and 〈110〉 configurations.

In the case of pure Fe, all potentials follow the DFT trend reasonably well, with a proper reproduction of the relative differences between the formation energies of the 〈100〉, 〈110〉 and 〈111〉 configurations being the most important feature. With the introduction of Cr, the data from our potential still captures the shape of the DFT curves reasonably well, while the data from OLS give almost degenerate values for all configurations. The CAR potential, on the other hand, fails to reproduce the 〈110〉 mixed dumbbell as the lowest energy configuration.

Insert figure 8 around here

Next, we compare the binding energy of Cr interstitials in bulk Fe obtained from the three potentials with the corresponding DFT values [12]. Such a comparison is presented in DFT data. Given that both 2BM potentials were only fitted to the binding energy between a single Cr and the 〈110〉 dumbbell, they show reasonable agreement with DFT. In particular, our potential is the only one that correctly reproduces the sign for all reported binding energies.

Insert figure 9 around here

Dislocation-Precipitate Interaction

Here we address the problem of dislocations in Fe and Cr and of the interaction of a ½〈111〉 screw dislocation with Cr precipitates in the Fe matrix, as obtained from static simulations. The dislocation core energy and radius, shear modulus, Poisson ratio and Peierls stress for a ½〈111〉 screw and a ½〈111〉{110} edge dislocation in the pure metals are summarized in Table 2. Details regarding the calculations can be found in Ref. [59], where the interaction of an edge dislocation with Cr precipitates was characterized. From Table 2 it appears that the two Fe potentials provide very similar results, whereas the Cr potentials exhibit at least two essential differences. The shear modulus calculated with our potential is ~30% smaller, while the Poisson ratio is two times larger than according to OLS. The characteristics of the core of both edge and screw dislocations are similar with both potentials.

Finally, we note that the two Cr potentials predict approximately the same Peierls stress (τ P ) for the screw dislocation, which is higher than τ P in Fe by 35% and 55% according to OLS and our potentials, respectively.

Insert table 2 around here

The interaction of a screw dislocation with Cr precipitates was studied in a bcc simulation box with dimensions 20×12×28 nm 3 . Here E D+P , E D and E P are the total energy of the crystal containing the dislocation interacting with the precipitate, dislocation only and precipitate only, respectively. E Fe is the total energy of the Fe crystal with the same geometry and boundary conditions as the crystals used to calculate the above mentioned energies. All crystals were relaxed to reach a convergence of at least 5meV in the total energy.

As mentioned in section 3 the two Cr potentials predict different core structures for the screw dislocation in pure Cr. Therefore it is of interest to investigate whether or not the nondegenerate-to-degenerate core structure modification occurs at the precipitate-matrix interface and inside the precipitate. It should be noted that such a modification was found to occur with OLS in our previous study [61]. To address this issue, the core structure of different dislocation segments, selected according to the scheme presented in Table 3, was constructed using the differential displacement (DD) method [42] from the atomic positions obtained after relaxation. The corresponding DD maps are also presented in Table 3. Firstly, we observe that the core structure of the segment located near the centre of the precipitate was found to be the same as in pure Cr for the corresponding potentials, i.e. the three-fold split core for OLS and isotropic for our potential (Figure 1). The core of the segment penetrating the precipitate normal to its surface (#1 of Table 3) has a two-fold structure with pronounced split over the precipitate surface, similar for both potentials. However, a difference in the core structure was found for the segments penetrating the precipitate at a tangent to its surface (#2 and #3 in Table 3 as the limiting case). With our potential no isotropic or three fold structure modification occurs, contrary to OLS. Additional calculations, not reported here, proved that, with OLS, the isotropic-to-degenerate core transformation occurs not only inside the precipitate but also everywhere near the precipitate matrix interface. The latter implies that in MD simulations such core transformation may manifest itself as emission of kink pairs in the {112} plane, thus causing cross-slip movement instead of stable glide (see [43] for details).

Insert table 3 around here

The interaction energy of the dislocation with precipitates of different sizes as a function of distance between precipitate centre and dislocation line is presented in Figure 10.

Qualitatively, the curves look similar for both potentials, showing a maximum in the centre of the precipitate and a fast decrease beyond the precipitate radius. Quantitatively, however, the maximum interaction energy estimated with the OLS is about twice as high considering precipitates of the same size. The derivative of the interaction energy with respect to the distance gives an estimate for the interaction force, from which the maximum force (and hence stress) necessary for the dislocation to shear a precipitate can be calculated. Irrespective of the precipitate size and applied potential, the maximum interaction force occurs at ~65% from the precipitate centre.

Insert figure 10 around here

The difference in magnitude of the interaction energy between the two potentials can be attributed to the difference in the shear modulus of pure Cr (see Table 2). For a more correct treatment, however, contributions from the chemical energy (due to additional Fe-Cr interface) and dislocation core energy (which differs in Fe and Cr) should not be neglected.

Therefore, we have performed additional calculations allowing the total interaction energy to be decomposed into the contributions coming from the shear modulus misfit, chemical and relaxation (related to the strain created around the precipitate) excess energy. The chemical component, E S , was calculated statically, by shearing a precipitate in the ) 0 1 1 ( plane at its centre by the distance b, in a number of steps, each equal to 2R P /b, followed by relaxation. In this way we mimic the shearing of the precipitate due to the advance of the screw dislocation.

The shear modulus misfit component, E µ , was calculated by taking the volume integral of the dislocation strain energy between the particle centred at (h x ,h y ,0) and a dislocation line parallel to the z axis, following the expression [62]:

∫ ⋅ ⋅ ⋅ ⋅ - = P M Z M M P d d e D E θ ρ θ ρ θ ρ ρ µ µ µ µ ) , ( ) , ( . (11)
Here the subscripts M and P indicate respectively matrix and precipitate,

y x P Z h h R D - - - - = θ ρ θ ρ θ ρ . ( 12
)
In turn, the excess relaxation energy was calculated as As a representative example, the three above determined components and their derivatives, calculated for a screw dislocation interacting with a 4 nm precipitate, are shown in Figure 11. The smallest contribution comes from the chemical component, whose curve is magnified ten times for better observation while the contribution from the shear modulus misfit clearly dominates. As expected, its absolute contribution is the highest for OLS.

E xc =E I -(E S +E µ ).
Interestingly, the third term, i.e. the relaxation excess energy, appears to be approximately the same and negative for both potentials. In absolute terms, however, E xc is higher for OLS.

Insert figure 11 around here

Summary and Concluding Remarks

Based on the phase diagram, all potentials are capable of reproducing short-range order and α' precipitation. Our potential, however, seems to be the best compromise for thermodynamic modelling. It gives an asymmetric mixing enthalpy, does not stabilize unobserved intermetallic compounds (unlike OLS), provides (though underestimated) a significant amount of excess vibrational entropy (unlike CAR) and a reasonable agreement with the experimental miscibility gap, given that magnetic interactions are not explicitly included in the formalism.

Based on the description of substitutional defect interactions and the phase diagram, both OLS and our potential a priori seem to be the most appropriate choice to simulate thermal annealing in Fe-Cr alloys. Besides a proper reproduction of the thermodynamic limit, a reasonable reproduction of vacancy migration barriers, Cr-Cr and Cr-vacancy interactions assure that the correct kinetic path is followed using the correct mechanisms. All potentials provide a reasonable description of the Cr-Cr and Cr-vacancy interactions, but only the 2BM potentials provide reasonable values for vacancy migration barriers, with ours following the DFT trends the closest among the two.

Based on the description of interstitial and substitutional defect interactions and the phase diagram, both 2BM potentials a priori seem to be the most appropriate choice to simulate thermal annealing of Fe-Cr alloys under irradiation. Besides a proper description of thermal annealing, a correct stabilization of interstitial complexes is needed to assure that the correct micro-structure is obtained under irradiation. Both 2BM potentials provide reasonable interstitial stabilities (ours in particular), while CAR poorly reproduces interstitial properties. Based on the description of the ½ 〈111〉 screw dislocation core structure (in bulk Fe and Cr) and its stable glide in a {110} plane in bulk Fe, our potential a priori seems to be the most suitable choice for the simulation of precipitate-screw dislocation interaction. From the three potentials, ours is the only one reproducing the compact ½ 〈111〉 screw dislocation core structure in bulk Cr. In addition, our static simulations have revealed an isotropic-todegenerate core transition in every position near the precipitate with OLS that is not seen with our potential. Such a core modification may lead to different interaction mechanisms in dynamic simulations, such as cross slip.

In summary, we have shown that the here developed Fe-Cr potential reached the goal of removing the main shortcomings of previously existing ones, while conserving their "good"

properties. It should be noted, however, that it is not stiffened to the screened Coulomb interaction and is therefore not suitable to simulate collision cascades in its present form. CALPHAD based calculation. The curves for the OLS and our potential were taken from [55], while for CAR and CALPHAD they were taken from [52] and [53], respectively. Cr Fraction (at.%)

F

Tables

R P § b ] 12 1 [ _ ] 111 [ ) 0 1 1 ( _ § § #3 R P ) 0 1 1 ( _ § b ] 12 1 [ _ ] 111 [ § §

Fe

  first three functions determine the Fe potential, the second three the Cr potential and the last four the mixed Fe-Cr interactions.

j

  =0.0676504617 a normalization factor and β=5. The cut-off function f cut is defined as,

  the vacancy migration energy, V m

  dimensions 5 a 0 × 5 a 0 ×3a 0 . It contains 30 atoms, of which 2 Cr are separated by 7nn distance ( 2 / 19 a 0 ) along [ ]

  excess vibrational entropies for all three potentials are compared with experimental values for the ferro-magnetic phase[49; 51]. The experimental curves suggest a contribution to the total excess entropy, amounting to 30-40% of the configurational entropy for the disordered alloy. As shown in the figure none of the potentials

  the Calphad phase diagram is justified since in this concentration

13 In Figure 7

 137 the average (of forward and backward) vacancy migration barriers for different local Cr configurations calculated with the three potentials are compared to the corresponding DFT values

Figure 9 .

 9 Figure 9. As expected from the above results, the CAR potential shows poor agreement with

A

  with axes oriented along the[1 12], [1 10] and[111] directions. Periodic boundary conditions were applied in the [111] direction and a straight screw dislocation with left-handed thread and Burgers vector b= ] Cr precipitate was inserted at a defined distance from the dislocation line and placed so that its centre coincides with the ) the screw dislocation. Then the relaxation was performed to calculate the total energy of the crystal as a function of the distance between the precipitate centre and dislocation line. The precipitate-dislocation interaction energy E I E I =(E D+P +E Fe )-(E D -E P ), following[60] (positive values indicate repulsive interaction)

Figure 1 -

 1 Figure 1 -Comparison of the screw component of differential displacement maps obtained with our potential (a) and the O05 potential (b).

Figure 2 -

 2 Figure 2 -Comparison of the mixing enthalpies obtained from the potentials with DFT data [8; 12; 44].

Figure 3 -

 3 Figure 3 -Comparison of the formation energy obtained by the different potentials with the corresponding DFT values [11; 47; 48] for various intermetallic compounds.

Figure 4 -

 4 Figure 4 -Comparison of the excess vibrational entropy obtained from the different potentials with the experimental ones from [49; 51].

Figure 5 -

 5 Figure 5 -Comparison of the phase diagrams resulting from the different potentials against a

Figure 6 - 23 Figure 7 -

 6237 Figure 6 -Comparison between DFT [12] and all three potentials of the interaction energy between Cr-Cr and Cr-vacancy pairs in bulk Fe.

Figure 8 -

 8 Figure 8 -Formation energies of different interstitial configurations in bulk Fe and Cr, calculated using DFT[12] and the three different potentials.

Figure 9 -

 9 Figure 9 -Comparison between DFT [12] and the three potentials of the binding energy of Cr interstitial configurations.

Figure 10 -

 10 Figure 10 -Precipitate dislocation interaction energy as function of the distance for our (left) and the OLS potential (right). The position of maximal force between precipitate and dislocation is indicated by F MAX .

Figure 11 -

 11 Figure 11 -Different contributions to the interaction energy (top row) and interaction force (bottom row) for our (left) and the OLS potential (right), for a 4 nm Cr precipitate. The position of the precipitate edge and maximal force between precipitate and dislocation are indicated by R P and F MAX , respectively.

  Figure 1

  Figure 6

Table 1 -

 1 In this appendix the migration barriers for Fe-Vacancy and Cr-Vacancy exchange for different local chemical environments are provided. The latter are built by adding Cr atoms at the six nearest neighbour positions around the migration saddle point. The details are as reported in Table B-1 next. Summary of the basic properties of both Cr potentials.

	Insert table B-1 around here
	o r
	P
	e
	e r
	R
	e
	v i e
	w
	O n l
	y
	19

Table 2 -

 2 Properties of Fe and Cr as obtained from the potentials.

Table 3 -

 3 Differential displacement maps drawn in the (111) zone, for a screw dislocation interacting with a 2nm precipitate. The thick dashed line shows the edge of the precipitate and a double arrow shows the directions of the preferential split of the dislocation core. The centre of the dislocation core is shown by symbol §. Table B-1 -The vacancy migration barrier for different local Cr configurations, with the first and second value denoting the forward and backward jump, respectively. In the figure black circles are Fe, white ones are Cr, in grey is the migrating atom, and the square represents the vacancy.

Table 1

 1

									Experiment or DFT	This Work	O05
	a 0 (Å)						2.878 a / 2.834 b	2.866	2.878
	B (GPa)					208 c	215	208
	C' (GPa)					152 c	155	152
	C 44 (GPa)				105 c	108	105
	E coh (eV)					4.10	4.10	3.84
	V f E (eV)					2.0 ± 0.2 d / 2.59 b	2.52	2.56
	V m E (eV)					0.95 e	0.94	0.99
	fcc coh E	-	bcc coh E	(eV)	-0.4 f	-0.16	-0.03
	E	f <	110 >	-	E	f <	111 >	(eV)	-0.02 b	-0.07	-0.02
	E	f <	110 >	-	E	f <	100 >	(eV)	-1.12 b	-0.58	-1.23
		e Experiment, Ref. [65]

a Experiment, Ref.

[START_REF] Pearson | A Handbook of Lattice Spacings and Structures of Metals and Alloys[END_REF]

b DFT, Ref

[20]

c Experiment, Ref.

[41]

d Experiment, Ref.

[START_REF] Loper | Positron Annihilation[END_REF]

* The shear modulus was estimated by straining a crystal on a {110} plane in a 〈111〉 direction, as was done in Ref.[67]. ** The Poisson ratio was determined following the method used in Ref.[68] for pure Fe.

Table 3

 3

	Configuration	OLS	This Work
				#1
				] 111 [	b
	_
	12 1 [	]
					§	§	§
	_
	1 1 (	0	)	R P

Table A - 1

 A1

		Cr m E	0.74 / 0.56	0.66 / 0.62		0.54 / 0.52	0.76 / 0.72
		Fe m E	0.70 / 0.61	0.67 / 0.65		0.65 / 0.64	0.50 / 0.50
		Cr m E	0.56 / 0.56	0.64 / 0.64		0.52 / 0.52	0.67 / 0.67
		Fe m E	0.64 / 0.64	0.66 / 0.66		0.64 / 0.64	0.47 / 0.47
	Figures
		F o r
			P
	Configuration (a)	Cr m Fe m E E	DFT 0.57 / 0.57 0.64 / 0.64 e This Work 0.57 / 0.57 0.63 / 0.63 r e		OLS 0.56 / 0.56 0.64 / 0.64	CAR 0.90 / 0.90 0.63 / 0.63
		Cr m E	0.72 / 0.54	0.63 / 0.57 R	0.61 / 0.53	0.97 / 0.87
		Fe m E	0.66 / 0.60	0.65 / 0.63 e 0.67 / 0.63	0.64 / 0.62
		Cr m E Fe m E	0.79 / 0.52 0.73 / 0.57	0.66 / 0.58 0.67 / 0.63	0.62 / 0.51 0.68 / 0.61 v i e	1.00 / 0.85 0.65 / 0.61
		Cr m E Fe E m Cr m E Fe m E	1.09 / 0.69 0.84 / 0.55 0.88 / 0.55 0.78 / 0.58	0.68 / 0.58 0.69 / 0.63 0.67 / 0.61 0.68 / 0.64		0.59 / 0.49 w 0.67 / 0.60 0.56 / 0.51 0.66 / 0.62 O 0.99 / 0.82 0.65 / 0.59 0.86 / 0.77 0.57 / 0.55 n l
						y
						26

http://mc.manuscriptcentral.com/pm-pml Philosophical Magazine & Philosophical Magazine Letters

Acknowledgements

The authors thank Drs. D. Nguyen-Manh, M.Yu. Lavrentiev and S. Cottenier for fruitful discussions and their help in preparation of this paper. This research has received partial funding from the European Atomic Energy Community's 7 th Framework Programme (FP7/2007-2011), under grant agreement number 212175 (GetMat project). It also contributes to the European Fusion Technology programme (EFDA). The work was partially sponsored by the belgo-argentine MINCYT-FWO bilateral cooperation agreement, Project FW/07/EXII/002. RCP acknowledges support from CONICET-PICT 5062.

Appendix A: Tabulation of the Potential Functions

Tabulations containing up to 5000 points are available on-line as supplementary material related to this paper or at: http://www.ctcms.nist.gov/potentials/ or upon request to the authors. Note that a tabulation of the s-embedding functions is not provided due to the singular slope in the origin, so that care should be exercised if used in tabular form. The parameters for the s-embedding functions are A 1 = -0.217009784, A 2 = 0.388002579 and A 1 = -0.00977557632, A 2 = 0.374570104 for Fe and Cr, respectively.

Appendix B: Vacancy Migration Barriers


Jesús Rodríguez-Díaz

Rebeca Montava

Rosa Viana

Javier Buesa

Gaspar Pérez-Martínez

Vicente Monedero

Section: Animal Cell Technology Oral immunization of mice with Lactococcus lactis expressing the rotavirus VP8* protein

Keywords: lactic acid bacteria, cell-wall, vaccine, rotavirus

The efficacy of recombinant Lactococcus lactis as a delivery vehicle for a rotavirus antigen was evaluated in a mouse model. The rotavirus VP8* protein was expressed intracellularly and extracellularly in L. lactis wild type and in an alr mutant deficient in alanine racemase activity, necessary for the synthesis of the cell-wall component D-alanine. When the mucosal immune response was evaluated by measuring VP8*-specific IgA antibody in faeces, wild-type L. lactis triggered a low IgA synthesis only when the secreting strain was used. In contrast, VP8*-specific IgA was detected in faeces of both groups of mice orally given the alr mutant expressing extracellular VP8* and intracellular VP8*, which reached levels similar to that obtained with the wild type secreting strain. However, oral administration of the recombinant strains did not induce serum IgG or IgA responses. L. lactis cell-wall mutants may therefore provide certain advantages when low-antigenic proteins are expressed intracellularly. However, the low immune response obtained by using this antigen-bacterial host combination prompts to the use of new strains and vaccination protocols in order to develop acceptable rotavirus immunization levels.

INTRODUCTION

Rotaviruses are enteric viruses and are a main cause of diarrhea in children worldwide. They are responsible of more than 500,000 deaths of children under 5 years each year [START_REF] Parashar | Global mortality associated with rotavirus disease among children in 2004[END_REF]). The VP4 protein from the outside viral layer is the viral hemagglutinin and is digested by trypsin into two polypeptides, the N-terminal VP8* (28 kDa) and the C-terminal VP5* (60 kDa) proteins. Mucosal immunity against rotavirus infections is based on the production of IgA antibodies at the intestinal mucosa [START_REF] Kapikian | A rotavirus vaccine for prevention of severe diarrhoea of infants and young children: development, utilization and withdrawal[END_REF][START_REF] Ward | Possible mechanisms of protection elicited by candidate rotavirus vaccines as determined with the adult mouse model[END_REF]. In fact, antibodies to VP8* produced in vivo can protect against rotavirus produced diarrhoea [START_REF] Ruggeri | Antibodies to the trypsin cleavage peptide VP8 neutralize rotavirus by inhibiting binding of virions to target cells in culture[END_REF][START_REF] Gil | Homotypic protection against rotavirusinduced diarrhea in infant mice breast-fed by dams immunized with the recombinant VP8* subunit of the VP4 capsid protein[END_REF], making VP8* a good target for the design of new rotavirus vaccines.

Lactococcus lactis is a well-characterized lactic acid bacterium which has been the chosen candidate vehicle for the delivery of active molecules by mucosal routes, including numerous bacterial and viral antigens for the development of live vaccines [START_REF] Wells | Mucosal delivery of therapeutic and prophylactic molecules using lactic acid bacteria[END_REF][START_REF] Bermudez-Humaran | Lactococcus lactis as a live vector for mucosal delivery of therapeutic proteins[END_REF][START_REF] Rottiers | Modulation of gut-associated lymphoid tissue functions with genetically modified Lactococcus lactis[END_REF].

Earlier work has addressed the effects of expressing antigens at different levels and locations (cytoplasmic, secreted or cell-wall anchored), showing a clear dependence of the amount of expressed antigen in the elicited immune response in animal models when the oral route was employed [START_REF] Grangette | Protection against tetanus toxin after intragastric administration of two recombinant lactic acid bacteria: impact of strain viability and in vivo persistence[END_REF]. This led to the evaluation of mutants altered in cell-wall synthesis with the aim of increasing the delivery capacity to the mucosal immune system by promoting the in vivo release of antigens. Strains of L. lactis and Lactobacillus plantarum deficient in the synthesis of D-alanine (alanine racemase mutants; alr), essential for the synthesis of peptidoglycan, showed enhanced capacity to deliver the tetanus toxin fragment C (TTFC) model antigen to the mucosal immune system in mice [START_REF] Grangette | Enhanced mucosal delivery of antigen with cell wall mutants of lactic acid bacteria[END_REF].

The purpose of the present study was to assess the efficacy of L. lactis and its alr mutant in the mucosal delivery of an antigen different to TTFC and with a potential use as rotavirus vaccine. To this end, wild-type L. lactis and its alr mutant producing rotavirus VP8* intracellularly and extracellularly were assayed for immunization in a murine model.

MATERIAL AND METHODS

Bacterial strains and growth conditions

Lactococcus lactis MG1363 and L. lactis PH3960 (alr), an MG1363 derivative deficient in the alanine racemase gene [START_REF] Hols | Conversion of Lactococcus lactis from homolactic to homoalanine fermentation through metabolic engineering[END_REF], were used in this study. Both strains were routinely grown in M17 medium (Difco) supplemented with 0.5 % (w/v) glucose at 30ºC in static conditions. For growth of the PH3960 strain the medium was supplemented with 200 g D-alanine ml -1 . E. coli DH5 was used as cloning host and E. coli M15 was used for protein expression. They were grown in LB medium at 37ºC under shaking. For plasmid selection, erythromycin was used at 5 g ml -1 for L. lactis and ampicillin was used at 100 g ml -1 for E. coli.

Expression and purification of VP8* in E. coli and anti-VP8* serum preparation

The VP8* cDNA from the rotavirus simian strain SA11 (P5B[2] G3 genotype) was obtained from plasmid pGALVP8 [START_REF] Gil | Secretion of the rotavirus VP8* protein in Lactococcus lactis[END_REF]) by digestion with BamHI and KpnI and cloned into pQE30 plasmid (Qiagen) digested with the same enzymes. The resulting pQE-VP8 plasmid was used to transform E. coli M15 and protein expression was induced with 1 mM IPTG. VP8* was totally present as inclusion bodies and it was dissolved from the pellet by treatment with 8 M urea in 100 mM Tris/HCl pH 7.4 for 2 h at room temperature. The denatured protein was then purified in Ni-NTA columns according to the supplier (Qiagen) including 8 M urea in all the buffers. Urea was removed by sequential dialysis with decreasing urea concentrations followed by a final dialysis in 50 mM Tris/HCl pH 7.4, 100 mM NaCl, 1mM EDTA and the protein was stored at -80ºC. Anti-VP8* serum was obtained by intramuscular injection of three doses of 100 g of the purified protein to rabbits at the animal facilities of the Instituto de Biomedicina of Valencia.

Construction of plasmid vectors expressing VP8*

The VP8* cDNA was obtained by treating plasmid pGALVP8 [START_REF] Gil | Secretion of the rotavirus VP8* protein in Lactococcus lactis[END_REF]) with XbaI/Klenow followed by digestion with BamHI and cloned into the pT1NX vector [START_REF] Schotte | Secretion of biologically active murine interleukin-10 by Lactococcus lactis[END_REF] digested with SpeI/Klenow and BamHI, giving pRo157. In this construct the VP8* cDNA was fused to the sequence coding the Usp45 lactococcal secretion signal and expressed from the constitutive P1 lactococcal promoter [START_REF] Waterfield | The isolation of lactococcal promoters and their use in investigating bacterial luciferase synthesis in Lactococcus lactis[END_REF]. A vector for intracellular expression was constructed based in pT1NX. The P1 promoter region and the ribosome-binding site of usp45 were obtained from pT1NX by PCR with oligonucleotides 5'-AATAGAATTCGATTAAGTCATCT-3' and 5'-TGAGGATCCGTCGCCGGCCATGCGTATATCTCCTTT-3' (restriction sites introduced for cloning are underlined). The PCR fragment was digested with EcoRI and BamHI and used to replace the EcoRI-BamHI fragment from pT1NX which contained the promoter and usp45 secretion signal. The resulting plasmid was used for cloning the VP8* cDNA as a XbaI/Klenow and BamHI fragment, giving pRo298. All plasmid constructions were performed in L. lactis, which was transformed by electroporation in a GenePulser apparatus (BioRad).

VP8* expression in Lactococcus and inocula preparation

The day before immunizations, 5 ml GM17 were inoculated with 10 l frozen bacterial stock and were incubated for 16 h at 30 ºC. The cultures were washed in GM9 medium [1X minimal M9 salts, 0.5 % (w/v) glucose, 0.5 % (w/v) Casitone, 20 mM MgSO 4 , 2 mM CaCl 2 , 50 mM carbonate buffer pH 7.4] and diluted 1/10 in the same medium and further incubated for 4 h at 37ºC. For the L. lactis alr strain a suboptimal concentration of D-alanine (25 g ml -1 ) was added to allow bacterial growth. For inocula preparation, bacteria were recovered by centrifugation washed three times with GM9 and resuspended to 10 9 c.f.u. in 100 l of GM9 containing 10 g the B subunit of cholera toxin (CTB) as adjuvant. CTB was purified using plasmid pAE-ctxB as described (Areas et al. 2002). For western blot analysis, cells corresponding to 10 9 cfu were washed and resuspended in 50 l of 50 mM Tris/HCl pH 7.5 containing 0.5 M sucrose and 50 mg lysozyme ml -1 . After 1 h at 37ºC, cells were washed with 1 ml 50 mM Tris/HCl pH 7.5 and resuspended in 50 l Laemmli loading buffer and boiled for 5 min. One ml growth medium supernatants were treated with trichloroacetic acid at 10 % (w/v), incubated on ice for 1 h and proteins were precipitated at 12,000 x g 10 min. Cell lysates and supernatant precipitated proteins corresponding to 5 x 10 8 cfu were separated in 10 % SDS-PAGE gels and expression of VP8* was analyzed by western blot with rabbit anti-VP8* serum and the ECL system (GE Healthcare).

Mice inoculation and sample collection

Each immunization group consisted of 5 female BALB/c mice, 6 to 10 weeks old. Each individual was analyzed by ELISA to confirm the absence of anti-rotavirus antibodies prior to the immunization schedule. The inoculation was performed intragastrically using a mice intragastric probe (Harvard Apparatus) to ensure equal administration (10 9 c.f.u. in 100 l) to each mouse. The immunization consisted of three inoculation cycles of three doses in three consecutive days spaced by three weeks intervals. Serum samples and faeces were collected 10 days after each immunization cycle. Blood samples were obtained from the tail vein and sera were obtained by centrifugation at 3000 x g and stored at -32ºC. Stool samples were collected and dissolved to at100 mg ml -1 in PBS containing the Mini EDTA-free Protease inhibitor cocktail (Roche) and frozen at -32ºC until use. Prior to be used the faecal suspensions were homogenized and centrifuged at 5000 x g for 10 min.

Viral challenge

After the last blood and faecal sampling, mice were inoculated by the intragastric route with 50 l suspension of rotavirus SA11 strain (6 x 10 7 pfu ml -1 ). Faeces were recovered from individual mice for seven days and blood samples were taken at 15 days post-infection. Samples were tested by ELISA for rotavirus antigens and rotavirus-specific antibodies as described [START_REF] Garcia-Diaz | Nasal immunization of mice with a rotavirus DNA vaccine that induces protective intestinal IgA antibodies[END_REF])

ELISA assays

Polystyrene plates were coated with 1 g purified (6XHis)-VP8* protein ml -1 in 0.1 M carbonate/bicarbonate buffer pH 9.6 incubated at 37 ºC for 2 h and stored at 4ºC until use. The serum samples were serially diluted in PBS with 0.5 % Tween 20 (v/v) (PBS-T) plus 1 % BSA (w/v) as blocking agent. 100 l of undiluted faecal suspensions were used. After washing with PBS-T, a 1:2000 dilution of anti-mouse IgG-or anti-mouse IgA-HRP-conjugate (Sigma) in PBS-T containing 1 % BSA (w/v) was added as secondary antibody. The colorimetric reaction was developed with o-phenylenediamine and H 2 O 2 for 10 min and the reactions stopped with 50 l 3 M H 2 SO 4 . The final absorbance was read at 492 nm. The statistical significance was calculated using the Mann-Whitney U-test with SPSS 17.0 software (SPSS, Inc., Chicago, IL)

RESULTS

VP8* expression in L. lactis

Two vectors were constructed for intracellular and extracellular expression of VP8* from the rotavirus strain SA11 in L. lactis (Fig. 1a). The extracellular construct (pRo157) was based on the widely used lactococcal Usp45 secretion signal. As can be seen in Figure 1B, this signal efficiently promoted the secretion of VP8*, with low amounts of VP8* detected in the cellular fractions. Secreted VP8* was partially degraded into smaller peptide products. When the intracellular vector (pRo298) was used, most of VP8* was detected in the cellular fraction, with a small portion present in the supernatant probably as a result of cell lysis (Fig. 1b). Expression was assayed in L. lactis wild type and an alr deficient mutant which was totally dependent on the addition of Dalanine for growth. However, to ensure differences in cell-wall characteristics between both strains, a suboptimal D-alanine concentration was used in the expression and inoculation experiments [START_REF] Grangette | Enhanced mucosal delivery of antigen with cell wall mutants of lactic acid bacteria[END_REF]). Compared to the wild type, the presence of the alr mutation did not affect VP8* expression in either the extracellular or intracellular constructs.

Oral immunization of mice with L. lactis expressing VP8* and viral challenge

An inoculation schedule was designed consisting in three consecutive daily oral doses of 10 9 bacterial cells repeated three times at three weeks intervals. Serum samples were analyzed by ELISA for VP8*-specific IgG and IgA. After the three immunization cycles no antibody response was obtained in sera with either of the strains. However, as seen in Figure 2, administration of L. lactis expressing VP8* led to a significant increase in the VP8*-specific IgA response in faeces, indicating that the recombinant strains only triggered a local immune response. However, strains producing secreted VP8* resulted in IgA levels that were already significant after the second inoculation cycle for the wild type (P=0.01) and after the third inoculation cycle for the alr mutant (P=0.004). When the intracellular constructs were used, the wild type induced specific IgA production only after the third inoculation cycle which did not reach statistical significance (P=0.134). On the contrary, a significant IgA production was already measured with the alr mutant after the second and third cycle (P=0.04 and 0.01, respectively). This indicated that the low immunogenicity of the wild type was increased by the alr mutation, which resulted in IgA levels similar to that obtained with the strains producing secreted VP8*.

Mice treated with recombinant L. lactis were orally challenged with rotavirus strain SA11. Although this simian strain is able to infect and induce diarrhea in mice pups, its limited infection capacity in adult mice resulted in background levels of rotavirus antigens in faeces with no differences between groups (data not shown). ELISA analysis of sera from infected mice showed that some groups of mice were primed. This was reflected by significantly (P=0.013) higher anti-rotavirus IgG levels in mice treated with wild-type L. lactis expressing extracellular VP8*. The same effect was observed for the alr mutant expressing intracellular VP8*, although differences were not significant (P=0.052) (Fig. 3).

DISCUSSION

Vaccination against rotavirus is currently regarded as the most efficient preventive measure against rotavirus severe diarrhoea [START_REF] Parashar | Global illness and deaths caused by rotavirus disease in children[END_REF]. The present work aimed to investigate if L. lactis was a suitable vector to confer mucosal immunity against a rotavirus antigen that has previously shown to elicit a rotavirus neutralizing response and to prove if cell-wall mutants of this bacterium behaved as better antigen delivery vehicles. We showed that orally administered recombinant L. lactis to mice was able to deliver the VP8* antigen to the intestinal mucosa. VP8* displayed a limited immunogenicity, as reflected by the need of a repeated inoculation scheme and the use of CTB as adjuvant to trigger specific IgA production. When CTB was omitted in the inoculations, no immune responses were observed (data not shown). Additionally, the low immune response against VP8* can be the result of other factors. The P1 promoter has been successfully employed for the delivery of recombinant proteins to mice intestinal mucosa by L. lactis [START_REF] Steidler | Treatment of murine colitis by Lactococcus lactis secreting interleukin-10[END_REF][START_REF] Robinson | Mucosal and cellular immune responses elicited by recombinant Lactococcus lactis strains expressing tetanus toxin fragment C[END_REF][START_REF] Vandenbroucke | Active delivery of trefoil factors by genetically modified Lactococcus lactis prevents and heals acute colitis in mice[END_REF]). Nevertheless, the actual strength of this promoter under the conditions of the different segments of mice gastrointestinal tract is not known and the possibility exists that VP8* in situ expression was a limiting factor.

When immunized mice were challenged with rotavirus SA11, a priming effect was only observed in the groups that showed the higher local immune response. However, the use of the simian SA11 strain makes difficult to assess the efficacy of protection and further experiments using murine VP8* antigens and virus strains are required to address this point.

L. lactis producing VP8* elicited only a local response, with no production of serum specific antibodies. These results are in contrast with very recent data reported by Quiao et al., who showed that Lactobacillus casei expressing porcine VP8* produced an IgG humoral response when given orally to mice [START_REF] Qiao | Recombinant porcine rotavirus VP4 and VP4-LTB expressed in Lactobacillus casei induced mucosal and systemic antibody responses in mice[END_REF]. This suggests that immune responses to antigens expressed by lactic acid bacteria depend not only on the nature of the antigen but also in the intrinsic adjuvant characteristics of the host strain and questions the suitability of L. lactis as a host for rotavirus VP8* delivery. However, in agreement to the results reported here, a VP8*-LTB (heat-labile toxin B subunit from E. coli) fusion was required to increase adjuvancy and obtain a strong immune response in mice [START_REF] Qiao | Recombinant porcine rotavirus VP4 and VP4-LTB expressed in Lactobacillus casei induced mucosal and systemic antibody responses in mice[END_REF].

When wild-type L. lactis producing VP8* was used, an IgA response was only observed with the extracellular expression vector. The amount of secreted VP8* in culture medium was higher than that observed in the intracellular constructs. This, together with the fact that soluble antigens are generally less immunogenic, suggested that the lack of immune response elicited by the wild type with the intracellular construct was possibly due to the limited amounts of VP8* produced. We tried to circumvent this by following a similar strategy to that reported for the delivery of the TTFC model antigen by L. lactis and L. plantarum [START_REF] Grangette | Enhanced mucosal delivery of antigen with cell wall mutants of lactic acid bacteria[END_REF]. The use of an alr mutant expressing intracellular VP8* only resulted in a slight increase of specific IgA in faeces compared to the wild type expressing VP8* at the same location. Nevertheless, in comparison to the wild-type strain producing secreted VP8*, the use of the alr strain did not result in a stronger immune response. The introduction of this mutation causes L. lactis cells to lyse upon D-alanine depletion [START_REF] Steen | Autolysis of Lactococcus lactis is increased upon D-alanine depletion of peptidoglycan and lipoteichoic acids[END_REF]. It is thus conceivable that the release of intracellular VP8* during the gastrointestinal passage of recombinant L. lactis alr could be the reason for the improved delivery capacity of intracellularly-expressed VP8* obtained with this mutant. Following this reasoning, no clear differences could be observed between the wild type and the alr mutant when the VP8* antigen was produced extracellularly. However, as previously suggested [START_REF] Grangette | Enhanced mucosal delivery of antigen with cell wall mutants of lactic acid bacteria[END_REF], other reasons, such as a different interaction with the host immune cells and the activation of antigen-presenting cells, cannot be ruled out.

In summary, L. lactis expressing VP8* induces a low local response in mice when given orally. This response could not be enhanced by the use of L. lactis cell-wall mutants. This implies that new lactic acid bacteria strains with better adjuvant/delivery capacity, in combination with appropriated immunization protocols and viral strains, must be tested in order to develop an efficient rotavirus vaccine based on VP8*.

Fig 1 VP8*

 1 Fig 1 VP8* expression in L. lactis. a Schematic representation of the plasmid constructs for intracellular and extracellular production of VP8*. P1, lactococcal constitutive promoter; SD, Shine-Dalgarno sequence from usp45; Usp45; secretion signal from the L. lactis Usp45 protein. b Western blot analysis of VP8* production.Cell extracts and supernatants corresponding to 5x10 8 cfu were loaded onto each lane and VP8* was detected with a rabbit antiserum. wt is wild-type L. lactis and alr is the alr mutant strain. C is an intracellular extract of wild-type L. lactis transformed with the pT1NX vector. MW indicates the position of molecular weight markers in kDa. The asterisks point to degradation products of extracellular VP8*. Usp45::VP8* is non-secreted unprocessed VP8*

Fig 3

 3 Fig 3Serum anti-SA11 IgG response in mice treated with recombinant L. lactis and challenged with rotavirus SA11 strain. Serum samples were tested at 1:100 dilution for rotavirus-specific IgG by ELISA in rotavirus SA11 antigen-coated wells.

ACKNOWLEDGEMENTS

This work was supported by European project DEPROHEALTH QLRT-2000-00146 and Consolider Fun-c-Food CSD2007-00063 from the Spanish Ministry of Science and Innovation. Jesús Rodríguez-Díaz was recipient of a FPU grant and a JAE-post contract from the Consejo Superior de Investigaciones Científicas. We are indebted to Dr. Annick Mercenier and Dr. Catherine Daniel for making L. lactis alr mutant available to us.

Authors also wish to thank Dr. L. Steidler for the kind gift of plasmid pT1NX.


Paola Baiardi
email: pbaiardi@cvbf.net

Carlo Giaquinto

Silvia Girotto

Cristina Manfredi

Adriana Ceci

INNOVATIVE STUDY DESIGN FOR PAEDIATRIC CLINICAL TRIALS 1

Keywords: clinical trials, paediatrics, clinical trial design, TEDDY

Despite representing a fundamental step to an efficacious and safe utilisation of drugs in the paediatric population, the conduct of clinical trials in children poses several problems. Methodological issues and ethical concerns represent the major obstacles that have traditionally limited paediatric research.

The randomised clinical trial, mainstay of clinical studies to assess the effects of any therapeutic intervention, shows some weaknesses which make it scarcely applicable to the paediatric population.

Alternative and innovative approaches to clinical trial design in small populations have been developed in the last decades with the aim of overcoming the limits related to small samples and to the acceptability of the trial. These features make them particularly appealing for the paediatric population.

This systematic review describes a variety of alternative designs to assess efficacy and safety in the paediatric population, including their applicability, advantages, disadvantages and real case examples.

Approaches include sequential and adaptive designs, Bayesian methods and more innovative approaches. Thanks to their features, these methods may rationally limit the amount of experimentation in children to what is achievable, necessary and ethical, and represent a reliable way of ultimately improving paediatric care.

INTRODUCTION

Despite representing a fundamental step to an efficacious and safe utilisation of drugs in the paediatric population, the conduct of clinical trials in children still poses several problems: methodological issues, mainly related to the small sample sizes, and ethical concerns, deriving from children exposure to the potential risks of a trial, represent the major obstacles that have traditionally limited paediatric research. In addition, economical concerns in consideration of the limited paediatric medicines market, unable to guarantee adequate return on investments, have had also a crucial role in limiting drug development in children.

This situation has been tackled in the recent years at both scientific and regulatory levels: the first formal step was established in 2001 with the European Directive on clinical trials [START_REF]regulations and administrative provisions of the Member States relating to the implementation of good clinical practice in the conduct of clinical trials on medicinal products for human use[END_REF] and in particular with its Article 4 on the conduct of clinical trials in minors. A revolutionary change was started recently, in 2007, with the coming into force of the Paediatric Regulation [START_REF]European Parliament and the[END_REF][START_REF]Regulation (EC) No 1902/2006 of the European Parliament and of the Council of 20 December 2006 amending[END_REF] whose aim is to increase knowledge on the use of paediatric drugs, by boostingthrough a system of obligations and incentivesthe development of new drugs for children and ensuring at the same time a safer use of already marketed products.

The obligation of performing paediatric studies and consequently, the increased number of studies expected, and partially already observed as impact of the 2007 Regulation, has stimulated the attention on the adequate methodology for clinical trials in paediatrics. No methods exist that are relevant to paediatric trials and not also applicable to adult studies, stated that children are entitled to receive treatments tested with the same accuracy as those for adults. Clinical trial methodology to study paediatric treatments should not be different from the general methodology whenever paediatric conditions and prevalence of the diseases can ensure the use of standard methods. However, it may be that in conditions affecting small populations, innovative approaches may be pursued to ensure the feasibility of the trial and the validity of the results.

The need for alternative approaches is reported in the ICH Topic E11 guideline [START_REF]ICH/Topic E11. Clinical Investigation of Medicinal Products in the Paediatric Population. Step 5 -Note for guidance on clinical investigation of medicinal products in the paediatric population[END_REF], reference document for planning and conducting clinical trials in paediatrics, and discussed in details in the guideline for clinical trials in small populations [START_REF]Guideline on clinical trials in small populations[END_REF], that generally faces the issue of limited sample sizes. The still open methodological debate on paediatric research is furthermore corroborated by the great number of disease-specific guidelines that EMA has been delivered in the last years [START_REF]Guideline on the clinical development of medicinal products for the treatment of cystic fibrosis[END_REF][START_REF]Guideline on the requirements for clinical documentation for orally inhaled products (OIP) including the requirements for demonstration of therapeutic equivalence between two inhaled products for use in the treatment of asthma and chronic obstructive pulmonary disease (COPD) in adults and for use in the treatment of asthma in children and adolescents[END_REF][START_REF]Paediatric addendum to CHMP guideline on the clinical investigations of medicinal products for the treatment of pulmonary arterial hypertension[END_REF][START_REF]clinical investigation of medicinal products in the treatment of lipid disorders[END_REF][START_REF]Note for guidance on evaluation of anticancer medicinal products in man. Addendum on paediatric oncology[END_REF][START_REF]Guideline on the clinical development of medicinal products for the treatment of HIV infection[END_REF].

CONVENTIONAL VERSUS ALTERNATIVE DESIGN IN PAEDIATRICS

The randomised clinical trial (RCT), mainstay of clinical studies, represents the gold standard for the assessment of the effect of any therapeutic intervention. Its strengths lie in the randomisation, aimed at avoiding bias in the allocation of subjects to treatments, the blindness, aimed at avoiding bias in the evaluation of compared treatments, and in the a priori choice of acceptable margins of error, specifically type I and type II errors.

However, it shows some weaknesses which make it scarcely applicable to the paediatric population.

The RCT, in fact, implies a precise estimate of the sample size to guarantee that the study is powered enough to demonstrate the therapeutic superiority of one treatment compared to another. Usually such estimate leads to the determination of a bigger sample size than the samples usually available in paediatric studies. As a consequence, a paediatric RCT is often underpowered. Multicentric studies can overcome this issue, but while guaranteeing a higher generalisability of results, they introduce bigger difficulties in the management of the study and expose it to likely deviations from the research protocol.

Moreover, in the conventional approach, the noise deriving from non-systematic errors can be easily minimized in case of large samples; on the other hand, it could become serious in case of small samples and sometimes it could lead to a bias towards failing to show a difference between treatments [START_REF]Guideline on clinical trials in small populations[END_REF].

Another limitation of RCT, when applied in paediatrics, is represented by ethics, the other crucial point in designing paediatric studies: in the past ethical concerns represented a crucial obstacle to paediatric research and this is a valid reason to explore alternatives to the typical RCT design [START_REF] Goodman | Ethics and evidence in clinical trials[END_REF].

From a statistical point of view, moral matters can be dichotomised in two categories: individual and collective ethics. When a subject is assigned to a treatment found more unfavourable at the trial end, individual ethics is sacrificed for the benefit of collective ethics [START_REF] Palmer | Ethics and practice: alternative designs for phase III randomized clinical trials[END_REF].

If there is reason to believe that one treatment is superior to the other, the acceptability of a trial could be compromised and if the expectations for the investigational drug are so high that equipoise is lacking, a control group with placebo could be totally unaccepted. Problems with the acceptability of a study make it harder to recruit subjects, especially in case of limited populations, as in paediatrics. A trial extension, where the experimental drug is given to all patients, or a trial with unbalanced randomisation represent the traditional way of facing these issues.

Alternative approaches to clinical trial design share an underlying philosophy that combines ethical and methodological issues, and make them particularly suitable for the paediatric population. Due to their flexibility, innovative designs fit very well the requirements in the paediatric population [START_REF] Pasqua | Study and Protocol Design for Paediatric Patients of Different Ages[END_REF], for a drug development characterised by less strict separation between phases as stated by ICH Topic E11 guideline.

Below, we describe a variety of alternative designs to determine efficacy and safety in the paediatric population, including their applicability, advantages, disadvantages and real case examples. These approaches include data-dependent designs, old methods scarcely applied both in adult and paediatric research, and innovative approaches, some of which were never applied in real contexts, to our knowledge. This paper builds on the principles of the main reference guidelines [START_REF]ICH/Topic E11. Clinical Investigation of Medicinal Products in the Paediatric Population. Step 5 -Note for guidance on clinical investigation of medicinal products in the paediatric population[END_REF][START_REF]Guideline on clinical trials in small populations[END_REF] in paediatrics and places itself as systematic review for clinical trial design in children.

SEQUENTIAL DESIGN

This is a study design with an a priori non fixed sample size. The trial can stop at any time during its course as soon as the scientific evidence of a superiority of one treatment against the other is proven.

The number of repeated analyses during the trial has to be planned initially so that the control of the type I and II errors is ensured together with the necessary power of the trial results. According to the scheduled data monitoring, after groups of subjects respond or after each response, we have different designs, group sequential or fully sequential designs.

Due to its nature, sequential design requires treatment outcomes to be available quickly in relation to the patient recruitment rate and this characteristic partially limits its applicability. It is not suitable for studying survival, but it could be useful to evaluate short treatments through surrogate endpoints.

The expected number of subjects is calculable and it is below the number of a comparable fixed sample size design. Some open-ended designs can continue to recruit patients until a conclusion about the treatments can be achieved, while closed designs have a fixed upper limit in the number of patients to be recruited. In any case, fewer patients are generally necessary to reach a conclusion compared to a fixed sample size design, thus guaranteeing some ethical advantages over a classical approach.

Although the approach was developed as early as the 1960s, it is not widely used in CTs. A systematic review [START_REF] Goodman | Stopping trials for efficacy: an almost unbiased view[END_REF] reports the application of the design in the neonatal intensive care setting: 24 trials have been performed from 1963 to 2005 that have lead to save, on average, 35% of the enrolled subjects when compared to a fixed sample size approach.

ADAPTIVE DESIGN

Adaptive design is a trial design that allows modifications to some aspects of the trial after its initiation, without invalidating the validity and integrity of the trial. Modifications of the trial can refer among others, to sample size re-estimation, early stopping and adaptive randomization.

Sample size re-estimation becomes necessary when limited evidence is available on the expected treatment effect size. In such cases, it is desirable to adjust the sample size according to the effect size for the ongoing trial thus reducing the risk of trial failure. Early stopping is pursued when the efficacy or futility of the experimental drug becomes obvious during the trial [START_REF] Van Der Lee | Sequential design with boundaries approach in pediatric intervention research reduces sample size[END_REF]. Finally, through responseadaptive randomization, allocation probability may change during the trial based on the responses of the previous patients and more patients are expected to be randomised to the group that is proving superiority.

The possibility to make adjustments as one learns during the trial makes this approach particularly appealing in paediatric trials, due to the usual limited knowledge in this field. The flexibility of the design, however, entails more complex data analysis methods and, according to the modification of the inappropriate assumptions, the unblinding of data or the knowledge of treatment outcome before the randomization of the next patient.

The potential uses of adaptive design methods have attracted much attention, especially in the recent years and across all the development phases of drugs. Examples of applications have been reported for dose-finding studies but also for phase III trials, mainly in case of interim analyses followed by sample size re-estimation. The approach works well also in case of phase IV trials in which the method allows the saving of up to a half of the required subjects according to a traditional design [START_REF] Chow | Adaptive design methods in clinical trials -a review[END_REF].

BAYESIAN APPROACH

Bayesian design is the data dependent design par excellence. In essence, data from prior studies may be gainfully quantified to form an a priori probability distribution for treatment effect, and combined with the current trial data to provide an a posteriori distribution on which conclusions may be drawn.

The approach is particularly suitable for the paediatric population. In fact, since a pediatrician's observation that a drug is effective in adults increases his/her belief in its effectiveness in the paediatric population, a quite natural approach to inference is to make these considerations explicit using Bayesian methods. Adult data can be used in designing a paediatric trial by exploiting prior information for the sample size calculation or by directly including them into the study to generate the prior distribution.

Bayesian approach can improve the power of statistical inference or can reduce the width of confidence intervals and increase the precision of statistical estimates: this is what was shown in a recently published trial to assess the impact of a diet regimen in the management of acute lung injury and acute respiratory distress syndrome in children [START_REF] Schoenfeld | Bayesian design using adult data to augment pediatric trials[END_REF]. In the same way, a trial comparing the efficacy of immune globulin therapy with plasmapheresis in children affected by Guillan-Barré syndrome [START_REF] Goodman | A Bayesian approach to randomized controlled trials in children utilizing information from adults: the case of Guillain-Barré syndrome[END_REF] proved that the construction of an evidence-based a priori distribution starting from adult data allowed information from adults to formally increase data from children, thus minimizing unnecessary paediatric experimentations. The consequence could be potentially smaller, more informative trials and for patients, the possibility of receiving better treatments.

However, it has to be considered that potential difficulties may arise at the end of the trial, when physicians are presented with updates on a priori distributions in the form of a posteriori distributions.

In fact, since results are strongly based on the level of confidence on a priori distributions and on the extent to which these hypotheses apply to the current trial, some uncertainty could remain about how to translate the findings in ready to use results for changing their clinical practice.

RANDOMIZED WITHDRAWAL DESIGN

This design belongs to the category of the so-called "enrichment designs" in which the experimental drug is tested against the comparator on an "enriched population" of responders to the test compound [START_REF] Honkanen | A three-stage clinical trial design for rare disorders[END_REF]. With this approach, enrolled subjects receive initially and for an a priori defined period of time the trial drug in an open manner. Then, non-responders stop the trial, while responders are randomised, according to a standard design, to placebo or the trial drug after an adequate wash-out period (Figure 1).

This approach shows at least two advantages particularly appealing for the paediatric population: on the one hand it offers the patient the opportunity to experience the potential benefits of the active treatment [START_REF] Pasqua | Study and Protocol Design for Paediatric Patients of Different Ages[END_REF], on the other hand, the design minimizes the amount of time that the individual receives a placebo . The result may be a better patient accrual, particularly in indications for which the use of placebo poses ethical concerns. In addition, the test of the experimental drug against placebo on the selected population of responders increases the power of the comparison or alternatively, requires a smaller sample size to achieve the same power of results.

Disadvantages include carryover effects and difficulties in assessing whether the underlying disease process is still active: in fact, the design is suitable for testing drugs with short half-life and not inducing a permanent modification in the condition of the subject. Moreover, it may raise ethical concerns since the efficacy of the experimental drug can only be demonstrated by temporarily depriving patients of the benefit they had already obtained from the active drug [START_REF] Chiron | Antiepileptic drug development in children: considerations for a revisited strategy[END_REF].

This design has been successfully employed in studies for chronic disorders of both adults and children; it has been applied for clinical trials in childhood epilepsy [START_REF] Chiron | Vigabatrin withdrawal randomized study in children[END_REF][START_REF] Chiron | Stiripentol in childhood partial epilepsy: randomized placebo-controlled trial with enrichment and withdrawal design[END_REF] and in juvenile idiopathic arthritis [START_REF] Ruperto | Abatacept in children with juvenile idiopathic arthritis: a randomised, double-blind, placebo-controlled withdrawal trial[END_REF][START_REF] Lovell | Adalimumab with or without methotrexate in juvenile rheumatoid arthritis[END_REF]. In particular, the approach has been recognised as the most appropriate for developing new treatments for juvenile idiopathic arthritis at the Paediatric Rheumatology Expert Meeting held in London in December 2009. Its disadvantages are outweighed by its advantages even if the design does not represent the ideal method for confirmatory studies.

RANDOMIZED PLACEBO-PHASE DESIGN (RPPD)

As already mentioned, a clinical trial in paediatrics is generally more acceptable when the number of subjects assigned to the placebo arm is the least numerous possible, but also when the duration of the placebo trial is the shortest possible. The Randomized Placebo-Phase Design (RPPD) sets its innovation on this second strategy [START_REF] Feldman | The randomized placebo-phase design for clinical trials[END_REF].

The RPPD approach is based on the consideration that if the trial drug is active, the earlier it begins, the higher the probability to observe a response in short times. Therefore, in this type of study subjects receive placebo at different times according to the treatment scheme illustrated in Figure 2.

Subjects are randomised to placebo for periods of different duration. At the end of those periods, all subjects receive the active treatment until it is possible to observe a response. This study design belongs to the RCTs class since the treatment scheme guarantees: the presence of a control group according to an intra-patient scheme (each subject is the control of him/herself); the blindness, because subjects may begin the study directly assuming the active drug (this is the case when the placebo administration time is 0); the randomisation, made at the beginning of the active treatment.

Simulation studies have shown good potential of the RPPD to evaluate treatments for chronic conditions or highly potent therapies for rare diseases [START_REF] Feldman | The randomized placebo-phase design for clinical trials[END_REF]. Highly potent therapies simulations showed that the power of results from RPPD is comparable to that of a classical RCT, but power decreases in case of drugs with low potency. In such cases, the decision of implementing an RPPD trial would depend on the perceived acceptability of a standard RCT: with RPPD all subjects know they will receive the experimental treatment and no subjects are required to take a placebo for more than a relatively short time. Therefore, it might represent a good method for early assessment of new therapies when accrual to an RCT would be difficult or as an alternative approach to open or controlled study designs. Finally, given the characteristics of the design, it could be more accepted than a parallel group study for remission-inducing therapies using survival endpoints.

THREE-STAGE CLINICAL TRIAL DESIGN

The three-stage trial design is a methodological approach that combines the classical RCT with the randomised withdrawal trial on the same sample in order to get the maximum level of information available from each subject. The treatment scheme is depicted in Figure 3.

In the first phase, subjects are randomised according to a traditional RCT to the experimental drug or placebo. Subjects responding to placebo and those not responding to the trial treatment withdraw from the study, while responders to the active treatment enter the second phase of the trial and nonresponders to placebo are assigned to the third. Subject undergoing the second phase are randomised again to placebo or the trial drug. Subjects initially not responding to placebo, and therefore undergoing the third step, enter a randomised withdrawal scheme: they are initially assigned to an open treatment period with the experimental drug and subsequently, only those who respond to the drug in this phase continue the study and are randomised to take either placebo or the active treatment [START_REF] Honkanen | A three-stage clinical trial design for rare disorders[END_REF].

The approach has some strengths: first of all the possibility to obtain in the same study three different evaluations of efficacy, one for each phase, and to derive a global efficacy evaluation by pooling the results. Second, the trial has improved statistical power compared to a traditional RCT: simulation studies have highlighted that this approach reduces, ceteris paribus, the sample size by 20-30% compared to a RCT. Third, this type of design offers some features that may address the concerns about the unnecessary exposure to placebo. In fact, all patients entered in the study have a chance to derive a therapeutic benefit from their participation in the trial (patients not responding to placebo are given the opportunity to take the active treatment in the following phases of the study) and at the same time, it is possible to avoid treating patients responding to the placebo. All these advantages make the approach particularly suitable for situations when difficulties in recruiting an adequate number of subjects have to be faced or where patient numbers are limited, both contexts being typical of the paediatric population.

The design appears useful in chronic conditions for which it is expected to return to initial conditions when the active treatment is suspended; in this case the response rates from the three phases can be pooled without bias. Other situations where the study may be potentially useful is in the determination of the therapeutic efficacy in sub-populations, when efficacy in the general population has already been proved, or in the initial stages of the drug development when it is necessary to find dosages in small patients cohorts [START_REF] Honkanen | A three-stage clinical trial design for rare disorders[END_REF].

DISCUSSION AND CONCLUSIONS

The scientific methodology and logistics of clinical research for paediatrics have greatly changed in the recent years, also following the obligation of performing paediatric clinical trials introduced by the European Paediatric Regulation [START_REF]European Parliament and the[END_REF][START_REF]Regulation (EC) No 1902/2006 of the European Parliament and of the Council of 20 December 2006 amending[END_REF] that lays down rules concerning the development of medicinal products in order to meet the needs of the paediatric population, with the fundamental point being the avoidance of unnecessary studies in children. The binomial of obtaining evidence for drug use in paediatrics associated with the ethical issues of not exposing the paediatric population to unnecessary trials highlights more than in the past, the need for addressing new methodologies when designing paediatric trials.

When planning and analysing a clinical trial in children, crucial issues are the definition of the research question, the choice of the optimal design and outcome measure to achieve and demonstrate the objective of the study, the feasibility of the trial and the statistical validity of the findings: all characteristics that do not differ from those of adult studies, but that can pose obstacles in case of paediatric trials.

In this context, alternative approaches to study design that combine the enrolment of a limited number of subjects with measures for increasing the trial acceptability, can facilitate the carrying out of the study, particularly in those situations where a traditional RCT may be difficult to perform.

The trial designs discussed in the present paper offer opportunities for research in paediatrics: even if there is no unique rule of thumb for choosing a specific approach, each design has intrinsic features that meet the requirements of the paediatric population. Adaptive designs are very attractive due to their flexibility and can be useful especially in early clinical development. Bayesian approach generally allows to achieve greater certainty with fewer children: in this case the use of adult evidence can be also of help both in the design and in the analysis of the results. The randomised withdrawal design has been widely studied and applied in real contexts; its features make it particularly suitable for chronic conditions, specifically for juvenile arthritis. More innovative approaches are promising thanks to the optimal results found in simulation studies. However, despite the clear advantages of these innovative approaches to clinical trials design, their application in practice remains still an exception. This is probably due, among others, to obstacles in having the protocol approved [START_REF] Palmer | Ethics and practice: alternative designs for phase III randomized clinical trials[END_REF]. Despite the increasing attention given to paediatric research at scientific, regulatory and legal levels, competencies in the field still need to be improved and deepen at every level. Protocol design requires not only solid knowledge of the methodology, but also in-depth understanding of child physiology, psychology, the social embedding of children and many other factors [START_REF] Pasqua | Study and Protocol Design for Paediatric Patients of Different Ages[END_REF]. A role in limiting the use of such approaches may also be attributable to reluctance by Regulatory Authorities in accepting the new study designs for registrative paediatric trials. The Paediatric Committee established with the European Regulation is making a big work in assessing paediatric research at the European level, including provisions to ensure the performance of welldesigned clinical trials. These provisions cannot avoid the consideration of alternative and innovative approaches to trial design.

In summary, the availability and implementation of new methodologies for the design of paediatric clinical trial represent an opportunity for paediatric research. Thanks to their features, these methods may rationally limit the amount of experimentation in children to what is achievable, necessary and ethical and represent a reliable way of ultimately improving paediatric care.

Figure 1 : 20 Figure 2 : 21 Figure 3 :

 1202213 Figure 1: The randomized withdrawal design trial

ACKNOWLEDGEMENTS

This report is part of the Task-force in Europe for Drug Development for the Young (TEDDY) Network of Excellence supported by the European Commission's Sixth Framework Program (Contract n. 0005216 LSHBCT-2005-005126).