Jérôme Margueron
email: jerome.margueron@ipno.in2p3.fr

Nicolae Sandulescu
email: sandulescu@theory.nipne.ro

Pairing correlations and thermodynamic properties of inner crust matter 1 Chapter 1 PAIRING CORRELATIONS AND THERMODYNAMIC PROPERTIES OF INNER CRUST MATTER

Keywords: 05.45-a, 52.35.Mw, 96.50.Fm Neutron star crust, nuclear superfluidity

In this review paper we discuss the effects of pairing correlations on inner crust matter in the density region where nuclear clusters are supposed to coexist with nonlocalised neutrons. The pairing correlations are treated in the framework of the finite temperature Hartree-Fock-Bogoliubov approach and using zero range nuclear forces. After a short introduction and presentation of the formalism we discuss how the pairing correlations affect the structure of the inner crust matter, i.e., the proton to neutron ratio and the size of Wigner-Seitz cells. Then we show how the pairing correlations influence, though the specific heat of neutrons, the thermalization of the crust in the case of a rapid cooling scenario.

Introduction

The superfluid properties of the inner crust of neutron stars have been considered long ago in connection to the large relaxation times which follow the giant glitches. Thus, according to the present models, the glitches are supposed to be generated by the unpinning of the superfluid vortex lines from the nuclear clusters immersed in the inner crust of neutron stars [1,47]. Later on the superfluidity of the inner crust matter was also considered in relation to the cooling of isolated neutron stars [34,12] and, more recently, in the thermal after-burst relaxation of neutron stars from X-ray transients [START_REF] Shternin | [END_REF]13,[START_REF] Haensel | Neutron Stars I[END_REF].

The superfluid properties of the inner crust are essentially determined by the nonlocalized neutrons. For baryonic densities smaller than about 1.4 x 10 -14 g cm -3 the non-localized neutrons are supposed to coexist with nuclei-type clusters [11,43]. At higher densities, before the nuclear matter becomes uniform, the neutrons and the protons can form other configurations such as rods, plates, tubes and bubbles [46].

A microscopic ab initio calculation of pairing in inner crust matter should take into account the polarization effects induced by the nuclear medium upon the bare nucleonnucleon interaction. This is a very difficult task which is not yet completely solved even for the infinite neutron matter. Thus, compared to BCS calculations with bare nucleon-nucleon forces, most of variational or diagrammatic models predict for infinite matter a substantial reduction of the pairing correlations due to the in-medium polarisation effects [START_REF] Lombardo | Physics of Neutron Star Interiors[END_REF]. On the other hand, calculations based on Monte Carlo techniques predict for dilute neutron matter results closer to the BCS calculations (for a recent study see [23]).

A consistent treatment of polarization effects on pairing is still missing for inner crust matter (for a recent exploratory study see [6]). Therefore at present the most advanced microscopic model applied to inner crust matter remains the Hartree-Fock-Bogoliubov (HFB) approach. Pairing correlations have been also considered in the Quasiparticle Random Phase Approximation (QRPA)(see Section 2.1 below) in relation to the collective modes in inner crust matter [31]. However, a systematic investigation of the effect of collective QRPA excitations on thermodynamic properties of inner crust matter is still missing.

The scope of this chapter is to show how the HFB approach can be used to investigate the effects of pairing correlations on inner crust matter properties. Hence, in the first part of the chapter we will discuss the influence of pairing, treated in HFB approach at zero temperature, on the structure of inner crust matter. Then, using the HFB approach at finite temperature, we will show how the pairing correlations affect the specific heat and the thermalization of the inner crust matter in the case of a rapid cooling scenario.

In the present study we will focus only to the region of the inner crust which is supposed to be formed by a bbc crystal lattice of nuclear clusters embedded in non-localized neutrons. The crystal lattice is divided in elementary cells which are treated in the Wigner-Seitz approximation.

Treatment of pairing in the inner crust of neutron stars 2.1. Finite-temperature Hartree-Fock-Bogoliubov approach

In this section we discuss the finite-temperature HFB approximation for a Wigner-Seitz cell which contains in its center a nuclear cluster surrounded by a neutron gas. The cell contains also relativistic electrons which are considered uniformly distributed.

In principle, the HFB equations should be solved by respecting the bbc symmetry of the inner crust lattice. However, imposing the exact lattice symmetry in microscopic models is a very difficult task (for approximative solutions to this problem see Refs. [18,26] and the references therein). We therefore solve the HFB equations for a spherical WS cell, as commonly done in inner crust studies [43,3]. Since we are interested to describe the thermodynamic properties of the inner crust matter, we present here the HFB approach at finite temperature.

The HFB equations for a spherical WS cells have the same form as for isolated atomic nuclei. Thus, for zero range pairing forces and spherical symmetry, the HFB equations at finite temperature are defined as [24],

h T,q (r) -λ q ∆ T,q (r) ∆ T,q (r) -h T,q (r) + λ q U i,q (r) V i,q (r) = E i,q U i,q (r) V i,q (r) , (1)
where E i,q is the quasiparticle energy, E i,q = (e i,q -λ q ) 2 + ∆ 2 i,q , U i,q (r) and V i,q (r) are the components of the HFB wave function and λ q is the chemical potential ( q = n, p is the index for neutrons and protons) . The quantity h T,q (r) is the thermal averaged mean field hamiltonian and ∆ T,q (r) is the thermal averaged pairing field.

In a self-consistent HFB calculation based on a Skyrme-type force, as used in the present study, h T,q (r) and ∆ T,q (r) are expressed in terms of thermal averaged densities, i.e., particle density ρ T,q (r), kinetic energy density τ T,q (r), spin density J T,q (r) and, respectively, pairing density κ T,q (r). The thermal averaged densities mentioned above are given by [53]:

ρ T,q (r) = 1 4π i g i,q |V i,q (r)| 2 (1 -f i,q ) + |U i,q (r)| 2 f i,q , (2)
τ T,q (r) = 1 4π i g i,q dV i,q (r) dr - V i,q (r) r 2 + l i,q (l i,q + 1) r 2 V i,q (r) 2 (1 -f i,q ) + dU i,q (r) dr - U i,q (r) r 2 + l i,q (l i,q + 1) r 2 U i,q (r) 2 f i,q , (3)
J T,q (r) = 1 4π i g i,q j i,q (j i,q + 1) -l i,q (l i,q + 1) -3 4

× |V i,q (r)| 2 (1 -f i,q ) + |U i,q (r)| 2 f i,q , (4)
κ T,q (r) = 1 4π i g i,q U * i,q (r)V i,q (r)(1 -2f i,q ), (5)
where f i,q = [1 + exp(E i,q /T )] -1 is the Fermi-Dirac distribution of quasiparticles, T is the temperature expressed in energy units, and g i,q = 2j i,q + 1 is the degeneracy of the state i with angular momentum j i . The summations in the equations above are over the spectrum of bound nucleons which form the nuclear cluster and of unbound neutrons which form the neutron gas. A constant density at the edge of the WS cell is obtained imposing Dirichlet-Von Neumann boundary conditions at the edge of the cell [43], i.e., all wave functions of even parity vanish and the derivatives of odd-parity wave functions vanish. The nuclear mean field has the same expression in terms of densities as in finite nuclei [19]. However, for a WS cell the Coulomb mean field of protons has an additional contribution coming from the interaction of the protons with the electrons given by

u pe Coul (r) = -e 2 d 3 r ′ ρ e (r ′ ) 1 |r -r ′ | . (6)
Assuming that the electrons are uniformly distributed inside the cell, with the density ρ e = 3Z/(4πR 3 W S ), one gets

u pe Coul (r) = -2πe 2 ρ e R 2 W S - 1 3 r 2 = Ze 2 2R W S r R W S 2 -3 (7)
It can be seen that inside the WS cell the contribution of the proton-electron interaction to the proton mean field is quadratic in the radial coordinate. The pairing field is calculated with a zero range force of the following form

V Pair,q (r i , r j ) = V 0 g Pair,q [ρ T,n (r), ρ T,p (r)](1 -P σ )δ(r ij ) , (8)
where P σ = (1 + σ1 • σ2 )/2 is the spin exchange operator. For this interaction the pairing field is given by ∆ T,q (r) = V 0 g Pair,q [ρ T,n (r), ρ T,p (r)] κ T,q (r).

In the calculations presented here we use two different functionals for g Pair,q [ρ T,n (r), ρ T,p (r)]. The first one, called below isoscalar (IS) pairing force, depends only on the total baryonic density, ρ T,B (r) = ρ T,n (r) + ρ T,p (r). Its expression is given by

g Pair,q [ρ T,n (r), ρ T,p (r)] = 1 -η ρ T,B (r) ρ 0 α , (10)
where ρ 0 is the saturation density of the nuclear matter. This effective pairing interaction is extensively used in nuclear structure calculations and it was also employed for describing pairing correlations in the inner crust of neutron stars [52,53,54,40]. The parameters are chosen to reproduce in infinite neutron matter two pairing scenarii, i.e., corresponding to a maximum gap of about 3 MeV (strong pairing scenario, hereafter named ISS) and, respectively, to a maximum gap around 1 MeV (weak pairing scenario, called below ISW). These two pairing scenarii are simulated by two values of the pairing strength, i.e., V 0 ={-570,-430} MeV fm -3 . The other parameters are taken the same for the strong and the weak pairing, i.e., α=0.45, η=0.7 and ρ 0 =0.16 fm -3 . The energy cut-off, necessary to cure the divergence associated to the zero range of the pairing force, is introduced through the factor e -E i /100 acting for E i > 20 MeV, where E i are the HFB quasiparticle energies.

The second pairing functional, referred below as isovector strong pairing (IVS), depends explicitly on neutron and proton densities and has the following form in the neutron channel [START_REF] Margueron | [END_REF],

g Pair,n [ρ T,n (r), ρ T,p (r)] = 1 -η s (1 -I(r)) ρ T,B (r) ρ 0 αs -η n I(r) ρ T,B (r) ρ 0 αn , (11)
where I(r) = ρ T,n (r) -ρ T,p (r). This interaction is adjusted to reproduce the neutron 1 S 0 pairing gap in neutron and symmetric nuclear matter provided by the BCS calculations with the bare nucleon-nucleon forces [14]. In addition, the pairing strength V 0 and the cut-off energy are related to each other through the neutron-neutron scattering length according to the procedure described in Ref. [9]. Therefore this interaction is expected to describe properly the pairing for all the nuclear densities of the inner crust matter, including the low density neutron gas. As shown in Refs. [39,10], this pairing functional describes well the two-neutron separation energies and the odd-even mass differences in nuclei with open shells in neutrons. In the present calculations for this pairing functional we have used the parameters V 0 = -703.86 MeV fm -3 , η s =0.7115, α s =0.3865, η n =0.9727, α n =0.3906. The cut-off prescription is the same as for the isoscalar pairing force. The pairing gaps in symmetric matter and neutron matter predicted by the three pairing forces introduced above are represented in Fig. 1 for a wide range of sub-nuclear densities. It can be seen that the isovector IVS interaction gives a maximum gap closer to the strong isoscalar ISS force, and the ISW interaction predict a suppression of the pairing gap up to saturation density.

To illustrate how the pairing correlations are spatially distributed in the Wigner-Seitz cells and how they are affected by the temperature, in Fig. 2 are shown the pairing fields for neutrons in the cells 2 and 5 (see Table 1). It can be noticed that the clusters have a nontrivial influence on the pairing of neutron gas. Thus, depending of the relative intensity of 1) labeled, respectively, as 1800 Sn and 950 Sn. The full and the long-dashed lines correspond to the ISS and ISW pairing interaction. Here the calculations have been done with the strenghts V 0 = {-430, -330} MeV fm 3 and with an energy cut-off of 60 MeV. pairing in the cluster and the gas region, the presence of the cluster can suppress or enhance the pairing in the surface region of the cluster.

Quasiparticle Random Phase Approximation (QRPA)

Pairing correlations affect not only the ground state properties of inner crust matter but also its excitations modes. The non-collective excitations are commonly described by the quasiparticle energies obtained solving the HFB equations. To calculate the collective excitations one needs to take into account the residual interaction between the quasiparticles. In what follows we discuss briefly the collective modes of the inner crust matter in the framework of QRPA, which takes properly into account the pairing correlations [31].

The QRPA can be obtained from the time-dependent HFB approach in the limit of linear response. In the linear response theory the fundamental quantity is the Green function which satisfies the Bethe-Salpeter equation

G = (1 -G 0 V) -1 G 0 = G 0 + G 0 VG. (12)
The unperturbed Green's function G 0 has the form:

G 0 αβ (rσ, r ′ σ ′ ; ω) = ij U α1 ij (rσ) Ū * β1 ij (r ′ σ ′ ) hω -(E i + E j ) + iη - U α2 ij (rσ) Ū * β2 ij (r ′ σ ′ ) hω + (E i + E j ) + iη , (13)
where E i are the HFB quasiparticle energies and U ij are 3 by 2 matrices expressed in term of the two components of the HFB wave functions [32]. The symbol in the equation above indicates that the summation is taken over the bound and unbound quasiparticle states. The latter corresponds here to the non-localised neutrons in the WS cell.

V is the matrix of the residual interaction expressed in terms of the second derivatives of the HFB energy functional, namely: ) Figure 3. Strength distributions for the quadrupole mode in WS cell 2 of Table 1. The full (dashed) line corresponds to the QRPA (HFB) strength. The results are for ISS pairing force with the same parameters as used in Figure 1.

V αβ (rσ, r ′ σ ′ ) = ∂ 2 E ∂ρ β (r ′ σ ′ )∂ρ ᾱ(rσ) , α, β = 1, 2, 3. (14
In the above equation {ρ 1 , ρ 2 , ρ 3 } ≡ {ρ, κ, κ * }, where ρ and κ are, respectively, the particle (2) and pairing (5) densities; the notation ᾱ means that whenever α is 2 or 3 then ᾱ is 3 or 2.

The linear response of the system to external perturbation is commonly described by the strength function. Thus, when the external perturbation is induced by a particle-hole external field F the strength function writes:

S(ω) = - 1 π Im F * (r)G 11 (r, r ′ ; ω)F (r ′ )dr dr ′ (15)
where G 11 is the (ph,ph) component of the QRPA Green's function.

As an example in Fig. 3 it is shown the strength function for the quadrupole response calculated for the WS cell 2 of Table 1 below [31]. The results correspond to the isoscalar pairing force with the strength V 0 =-430 MeV fm -3 and an energy cut-off of 60 MeV. It can be seen that the unperturbed spectrum, distributed over a large energy region, becomes concentrated almost entirely in the peak located at about 3 MeV when the residual interaction between the quasiparticles is introduced. The peak collects more than 99% of the total quadrupole strength and it is extremely collective. An indication of the extreme collectivity of this low-energy mode can be also seen from its reduced transition probability, B(E2), which is equal to 25 × 10 3 Weisskopf units. This value of B(E2) is two orders of magnitude higher than in standard nuclei. This underlines the fact that in this WS cell the collective dynamics of the neutron gas dominates over the cluster contribution. In Ref. [31] it is shown that similar collective modes appears for the monopole and the dipole excitations. A very collective low-energy quadrupole mode it was also found in all the Wigner-Seitz cells with

N cell ρ B k F N Z R W S [g cm -3 ] [fm -1 ] [fm] 1
7.9 10 [43] ρ B is the baryon density, k F = (3π 2 n B ) 1/3 the Fermi momentum calculated as in Ref. [3] where n B is the number of baryons per fm 3 , N and Z are the numbers of neutrons and protons while R W S is the radius of the WS cells. Compared to Ref. [43] here it is not shown the cell with the highest density located at the interface with the pasta phase.

Z=50 [27]. However, a systematic investigation of the influence of these collective modes on the thermodynamic properties of inner crust matter is still missing.

The effect of pairing on inner crust structure

The first microscopic calculation of the inner-crust structure was performed by Negele and Vautherin in 1973 [43]. In this work the crystal lattice is divided in spherical cells which are treated in the Wigner-Seitz (WS) approximation. The nuclear matter from each cell is described in the framework of Hartree-Fock (HF) and the pairing is neglected. The properties of the WS cells found in Ref. [43], determined for a limited set of densities, are shown in Table I. The most remarkable result of this calculation is that the majority of the cells have semi-magic and magic proton numbers, i.e., Z=40,50. This indicates that in these calculations there are strong proton shell effects, as in isolated atomic nuclei.

The effect of pairing correlations on the structure of Wigner-Seitz cells was first investigated in Refs. [3,4,5] within the Hartree-Fock BCS (HFBCS) approach. In this section we shall discuss the results of a recent calculations based on Hartree-Fock-Bogoliubov approach [28]. This approach offers better grounds than HF+BCS approximation for treating pairing correlations in non-uniform nuclear matter with both bound and unbound neutrons.

As in Ref. [43], the lattice structure of the inner crust is described as a set of independent cells of spherical symmetry treated in the WS approximation. For baryonic densities below ρ ≈ 1.4 × 10 14 g/cm 3 , each cell has in its center a nuclear cluster (bound protons and neutrons) surrounded by low-density and delocalized neutrons and immersed in a uniform gas of ultra-relativistic electrons which assure the charge neutrality. At a given baryonic density the structure of the cell, i.e., the N/Z ratio and the cell radius is determined from the minimization over N and Z of the total energy under the condition of beta equilibrium. The energy of the cell, relevant for determining the cell structure, has contributions from the nuclear and the Coulomb interactions. Its expression is written in the following form

E = E M + E N + T e + E L . (16)
The first term is the mass difference E M = Z(m p + m e ) + (N -A)m n where N and Z are the number of neutrons and protons in the cell and A=N+Z. E N is the binding energy of the nucleons, which includes the contribution of proton-proton Coulomb interaction inside the nuclear cluster. T e is the kinetic energy of the electrons while E L is the lattice energy which takes into account the electron-electron and electron-proton interactions. The contribution to the total energy coming from the interaction between the WS cells [44] it is not considered since it is very small compared to the other terms of Eq.( 1). We shall now discuss the effect of pairing correlations on the structure of the WS cells. To study the influence of pairing correlations we have performed HFB calculations with the three pairing interactions introduced in Section 2.1. The structure of the WS cells obtained in the HFB approach is given in Table II while in Fig. 4 it is shown the dependence of the binding energies, at beta equilibrium, on protons number. The contribution of pairing energy, nuclear energy and electron kinetic energy to the total energy are given in Fig. 5 for the cells 2 and 6. From this figure we can notice that the nuclear binding energy is almost compensated by the kinetic energy of the electrons which explains the weak dependence of the total energy on Z seen in Fig. 4. In Fig. 4 we observe also that pairing is smoothing significantly the variation of the HF energy with Z. For this reason the HFB absolute minima are very little pronounced compared to the other neighboring energy values. The results corresponds to the isoscalar weak (ISW), isoscalar strong (ISS) and isovectorisoscalar (IVS) pairing forces. In the last 4 columns are shown the proton numbers obtained with the finite size corrections. In the table are shown only the structures of the cells which could be well-defined by the present calculations.

N cell N Z
From Fig. 4 we observe that in the cells 1-2 the binding energy does not converge to a minimum at low values of Z. For the cells 3-4, although absolute minima can be found for HF or/and HFB calculations, these minima are very close to the value of binding energy at the lowest values of Z we could explore. Therefore the structure of these cells is ambiguous. The situation is different in the cells 5-10 where the binding energies converge to absolute minima which are well below the energies of the configurations with the lowest Z values.

From Table II it can be observed that the structure of some cells becomes very different when the pairing is included. Moreover, these differences depend significantly on the intensity of pairing (see the results for ISW and ISS). However, as seen in Fig. 4, for the majority of cells the absolute minima are very little pronounced relative to the other energy values, especially for the HFB calculations. Therefore one cannot draw clear quantitative conclusions on how much pairing is changing the proton fraction in the cells.

When the radius of the cell becomes too small the boundary conditions imposed at the cell border through the WS approximation generate an artificial large distance between the energy levels of the nonlocalized neutrons. Consequently, the binding energy of the neutron gas is significantly underestimated. An estimation of how large could be the errors in the binding energy induced by the WS approximation can be obtained from the quantity

f (ρ n , R W S ) ≡ B inf. (ρ n ) -B W S-inf. (ρ n , R W S ) , ( 17
)
where the first term is the binding energy per neutron for infinite neutron matter of density ρ n and the second term is the binding energy of neutron matter with the same density for the HF (left pannel) and HFB (right pannel) calculations. Are shown: the total energy (solid line), the nuclear energy (dashed line), the kinetic energy of the electrons (dasheddotted line), and the pairing energy for the ISS pairing interaction (dotted line). The pairing energies are shifted up as indicated in the figure .   calculated inside the cell of radius R W S and employing the same boundary conditions as in HF or HFB calculations. In Ref. [START_REF] Margueron | Proceeding of the International Symposium EXOCT07[END_REF] it was proposed for the finte size energy correction, Eq. 26, the following parametrisation

f (ρ ng , R W S ) = 89.05(ρ ng /ρ 0 ) 0.1425 R -2 W S , (18)
where ρ ng is the average density of neutrons in the gas region extracted from a calculation in which the cell contains both the nuclear cluster and the nonlocalized neutrons while ρ 0 is the nuclear matter saturation density. How the energy corrections described by Eq. ( 18) influences the HF (HFB) results can be seen in Fig. 4 (right pannel) and in Table II (last four columns). As expected, the influence of the corrections is more important for the cells 1-5, in which the neutron gas has a higher density, and for those configurations corresponding to small cell radii

Effect of pairing on the thermodynamic properties in the inner crust matter 4.1. Specific heat of the baryonic matter

The specific heat of the inner crust has contributions from the electrons, the lattice (or Ion) and the baryonic matter (essentially the de-localized neutrons). The contribution of these 3 components is shown in Fig. 6 where different scenarios for the neutrons are considered.

If neutrons are non-superfluid they dominate the contribution of the other component in the inner crust. The effect of the neutron superfluidity is the reduce strongly the contribution of

log10 (C V [erg cm -3 K -1 ]) log10 (ρ [g cm - 3
]) Ions Electrons Protons Non-superfluid neutrons Weakly paired neutrons Strongly paired neutrons Figure 6. Specific heat in the crust of the neutron star for the different components of the star matter and for a temperature of 10 9 K. the neutrons. It is therefore expected a very large effect induced by the neutron superfluidity that we now discuss. The specific heat of the neutrons is calculated from the quasiparticle energies obtained solving the HFB equations at finite-temperature in the Wigner-Seitz approximation, as discussed in Section 2.1. For the WS cells we use the structure determined in Ref. [43] (see Table 1).

The specific heat of the neutrons is calculated by

C V = T V ∂S ∂T , ( 19
)
where V is the volume of the Wigner-Seitz cell, T is the temperature and S is the entropy. The latter is given by

S = -k B i,q g i,q (f i,q ln f i,q + (1 -f i,q ) ln(1 -f i,q )). (20)
In Fig. 7 are shown the results obtained for the specific heat in the case of a strong isoscalar pairing interaction. To illustrate the particular behavior of the specific heat in non-uniform matter and the validity of various approximations, in what follows we shall discuss in more detail the results for the cell number 6, which contains N=460 neutrons and Z=40 protons (see Table 1). In this cell the HFB calculations predict 378 unbound neutrons. The specific heat given by the HFB spectrum, in which the contribution of the cluster is included, is shown in Fig. 8 by full line. In the same figure are shown also the specific heats corresponding to two approximations employed in some studies [34,48,45]. In these approximations the non-uniform distribution of neutrons is replaced with a uniform gas formed either by the total number of neutrons in the cell (N=460, dashed How these approximations work is seen in Fig. 8. To make the comparison meaningful, the calculations for the uniform neutron gas are done solving the HFB equations with the same boundary conditions as for the non-uniform system, i.e., neutrons+cluster. As seen in Fig. 8, the transition from the superfluid phase to the normal phase is taking place at a lower temperature in the case of uniform neutron gas, especially when are considered only the unbound neutrons. The critical temperature is therefore lower in uniform matter (for the two prescriptions often used) than in non-uniform matter.

Pairing and thermalization time of the inner crust

Several studies have shown that the thermalization time depends significantly on the superfluid properties of the inner crust baryonic matter [34,25,48,40,54]. This dependence is induced through the specific heat of unbound neutrons, strongly affected by the pairing energy gap. As an example we refer to Fig. 6 where it is seen that neutron superfluidity strongly suppress the neutron specific heat which, in the absence of the pairing would dominate over the specific heat of electrons and the lattice.

To estimate the effect of neutron superfluidity to the thermalization time of inner crust matter we use a rapid cooling scenario in which the core arrives quickly to a much smaller temperature than the crust due to the direct URCA colling process. The thermalization time is defined as the time for the core nd the crust temperatures to equilibrate. The heat diffusion is described by the relativistic heat equation [57]:

∂ ∂r Kr 2 Γ(r) e φ ∂ ∂r (e φ T ) = r 2 Γ(r)e φ C V ∂T ∂t + e φ Q ν , ( 21
)
where t is the time, K is the thermal conductivity, C V is the specific heat and Q ν is the neutrino emissivity. The effect of the gravity is given through the gravitational potential φ, which enters in the definition of the redshifted temperature T = T e φ , and the quantity

Γ(r) = 1 -2Gm(r)/rc 2 -1/2
, where G is the gravitational constant and m(r) is the gravitational mass included in a sphere of radius r. The latter is obtained from the Tolman-Oppenheimer-Volkoff (TOV) equations [START_REF] Haensel | Neutron Stars I[END_REF] based on an equation of state obtained from SLy4 Skyrme interaction [20]. More details on the cooling model are given in Ref. [22]. The time evolution of the apparent surface temperature T eff ∞ is displayed in Fig. 9 for the initial temperatures of the crust T i = 500 keV [22]. The effective surface temperature shown in Fig. 9 is obtained from the temperature at the bottom of the crust, T b = T (ρ b ), where ρ b = 10 10 g.cm -3 , using the relationship given in Ref. [50] for a non-accreted envelope. The results shown in Fig. 9 corresponds to a neutron star of mass 1.6 M ⊙ in which the inner crust extends from R c =10.72 km, which is the radius at the core-crust interface, to 11.19 km. As can be noticed, the pairing enhances significantly the cooling at the surface of the star. In Fig. 9 are also shown the apparent surface temperatures obtained neglecting the effect of the clusters, i.e., supposing that the neutron specific heat is given solely by that of the neutron gas. In this case the neutron specific heat is calculated from the quasiparticle spectrum of BCS equations solved for infinite neutron matter at a density corresponding to that of the external neutrons in the WS cell [35]. In the case of weakly pairing scenario (ISW), the apparent surface temperature is dropping faster for superfluid non uniform matter than for superfluid uniform matter. For the strong pairing scenario, since the pairing correlations suppress the role of the neutron in almost the entire inner crust, the effect of the clusters is less important [22].

A simple random walk model for the cooling is sometimes used to estimate the thermalization time [12,48,40]. In this model the diffusion of the cooling wave towards the core is calculated without taking into account the dynamical change of the temperature through the whole crust and neglecting the neutrino emissivity in Eq. ( 21). The crust is divided into shells of thickness R i for which the thermal diffusively D i = K i /C V,i is considered as constant. From a dimensional analysis of the heat equation ( 21), the relaxation time of each of these shells is defined as,

τ i = Γ(R) 3 R 2 i /(γD i ), (22)
where γ is a geometrical factor and R is the radius of the neutron star. The total relaxation through the crust is given by,

τ th = i √ τ i 2 . (23)
The thermalization time defined as (23) satisfy the condition that in a uniform density case, τ th is independent of the shell discretization. Using this model, the effect of the neutron superfluidity has also been estimated to be large [48,40], and is qualitatively similar to the result deduced from solving of the heat diffusion equation (21).

In some previous studies [34,25] it was found that the thermalization time, defined as the time needed for the crust to arrive to the same temperature with the core, satisfies the scaling relation t th = t 1 α β where

α = ∆R crust 1 km 2 1 -2GM/Rc 2 -3/2 (24)
depends solely on the global properties of the neutron star, i.e., the crust thickness ∆R crust , the star radius R and the mass of the star M . The scaling relation can be inferred from the random walk model presented above. In Fig. 10 are shown as a function of the scaling parameter α the cooling times we have obtained from the solution of the heat diffusion equation ( 21) for various neutron stars masses lying between 1.4 and 2.0 M ⊙ with a step of 0.1M ⊙ . In this figure are displayed two sets of fits for t w versus α, i.e., a linear fit with β = 1 (dashed line) and a fit with a fractional value for β (solid line). It can be noticed that the best fit is obtained with a fractional value for β, which is equal to 0.86 for the normal neutrons, to 0.85 for weakly paired neutrons and to 0.89 for strongly paired neutrons. Considering the simpler linear fit, i.e., t w ≈ αt 1 , as done in [34,25], we get for the normalized time t 1 the values t 1 = {68.9, 39.3, 22.3} corresponding, respectively, to the normal neutrons, neutrons with weak pairing and neutrons with strong pairing. For a 1.5 M ⊙ neutron star with α = 1.15, we obtain t 1 = {66.4, 37.4, 26.6}. These values for t 1 are larger than that of Ref. [25] (Table 2) by a factor 2.3 in the non-superfluid case, and 3.4 (3.0) for the weak (strong) pairing scenario. These differences could be explained by the effects of the nuclear clusters on the neutron specific heat, disregarded in Ref. [25], and by different neutrinos processes and thermal conductivities in the core matter used in the two calculations.

Summary and Conclusions

In the first part of this chapter we have discussed the influence of pairing correlations on the structure of inner crust of neutron stars. The study was done for the region of the inner crust which is supposed to be formed by a lattice of spherical clusters embedded in a gas of neutrons. The lattice was treated as a set of independent cells described in the Wigner-Seitz approximation. To determine the structure of a cell we have used the nuclear binding energy given by the Skyrme-HFB approach. For the cells with high density and small radii the binding energies do not converge to a minimum when the proton number has small values. We believe that it is related to the discretization of the continuum. For a small radius of the cell the average distance between the energy levels of the non-localised neutrons becomes artificially large which cause an underestimation of the binding energy.

To correct this drawback we have used an empirical expression based on the comparison between the binding energy of neutrons calculated in infinite matter and in a finite-size spherical cell [START_REF] Margueron | Proceeding of the International Symposium EXOCT07[END_REF]. We found that the finite size corrections to the binding energies are significant for the high density cells with small proton numbers. This conclusion indicates the need of a more accurate evaluation of the errors induced by the finite size of the WS cells on nuclear binding energy, which requires to go beyond the empirical expression used here.

In the second part we have discussed how thermalization of neutron stars crust depends on pairing properties and on cluster structure of the inner crust matter. The thermal evolution was obtained by solving the relativistic heat equation with initial conditions specific to a rapid cooling process. The specific heat of neutrons was calculated from the HFB spectrum. The results show that the crust thermalization is strongly influenced by the intensity of pairing correlation. It is also shown that the cluster structure of the inner crust affects significantly the time evolution of the surface temperature.

The thermodynamic properties of the inner crust discussed in this chapter are based on the excitation spectrum of HFB equations. As it was shown in Section 2.2, the QRPA approach predicts in some cells collective excitations located at low energies, comparable with the pairing energy gap. It is thus expected a significant contribution of the collective modes to the specific heat and to the thermalization process of inner crust matter. This is an open issue which deserves further investigations.

Figure 1 .

 1 Figure 1. (color online) Neutron pairing gap for the interactions ISW (isoscalar weak), ISS (isoscalar strong) and IVS (isovector strong) in symmetric nuclear matter and in neutron matter.

Figure 2 .

 2 Figure 2. Neutron pairing fields for the Wigner-Seitz cells 2 and 5 (see Table1) labeled, respectively, as 1800 Sn and 950 Sn. The full and the long-dashed lines correspond to the ISS and ISW pairing interaction. Here the calculations have been done with the strenghts V 0 = {-430, -330} MeV fm3 and with an energy cut-off of 60 MeV.

Figure 4 .

 4 Figure 4. (color online) The HFB energies per particle as function of proton number for the pairing forces ISW (dotted line), ISS (dashed line) and IVS (dashed-dotted line). The solid lines represent the HF results. In the left pannel are shown the results obtained including the finite size corrections.

Figure 5 .

 5 Figure 5. (color online) The different contributions to the total energy in the cells 2 and 6 for the HF (left pannel) and HFB (right pannel) calculations. Are shown: the total energy (solid line), the nuclear energy (dashed line), the kinetic energy of the electrons (dasheddotted line), and the pairing energy for the ISS pairing interaction (dotted line). The pairing energies are shifted up as indicated in the figure.

Figure 7 .

 7 Figure 7. Specific heat of neutrons in various WS cells for isoscalar strong ISS pairing. The specific heat is given in units of Boltzmann constant k B .

Figure 8 .

 8 Figure 8. Neutron-specific heat in the the WS cell 6 for isoscalar strong pairing. The results corresponds to various approximations discussed in the paper. The specific heat is given in units of Boltzman constant k B .

Figure 9 .

 9 Figure 9. Time evolution of the apparent surface temperature for the initial temperatures T i = 500 keV. NC indicates the results of the calculations obtained by neglecting the effect of the clusters.

Figure 10 .

 10 Figure 10. Cooling times t w versus scaling parameter α for three pairing scenarii as discussed in the text. The results correspond to neutron stars with masses between 1.4 and 2.0 M ⊙ . The fitting curves are given for the case of a linear scale (dashed lines, right side) and for a fractional power of the scaling parameter α (solid lines, left side).

Table 1 .

 1 The structure of the Wigner-Seitz cells obtained from a density matrix expansion (DME)

		13	1.12	1460	40	19.7
	2	3.4 10 13	0.84	1750	50	27.7
	3	1.5 10 13	0.64	1300	50	33.2
	4	9.6 10 12	0.55	1050	50	35.8
	5	6.2 10 12	0.48	900	50	39.4
	6	2.6 10 12	0.36	460	40	42.3
	7	1.5 10 12	0.30	280	40	44.4
	8	1.0 10 12	0.26	210	40	46.5
	9	6.6 10 11	0.23	160	40	49.4
	10	4.6 10 11	0.20	140	40	53.8

Table 2 .

 2 The structure of Wigner-Seitz cells obtained in the HF and HFB approximation.

	Z corr

Acknowledgements:

We thank to our young collaborators, Fabrizzo Grill and Morgane Fortin for having done most of the calculations presented in Sections 3 and 4. The work presented in this review paper was partially financed by the European Science Foundation through the project "New Physics of Compact Stars", by the Romanian Ministry of Research and Education through the grant Idei nr. 270, by the French-Romanian collaboration IN2P3-IFIN and by the ANR NExEN contract. The calculations have mostly been performed on the GRIF cluster (http://www.grif.fr).


D Dabli
email: dabli@clermont.in2p3.fr

G Montarou

M Beuve

C Rodriguez-Lafrasse

RBE modelization: Present Status and Future Prospects

The main objective of this presentation is to review the RBE simulation models considered for application in hadrontherapy planning treatment. We focus on the two most advanced models for clinical application which are the Microdosimetric Kinetic Model and the Local effect Model. We present the formalism of the both models and a comparison of their basic concepts. Then we apply the MKM using the published data for HSG cell lines in order to compare our results with other published MKM results for the same cell line. Finally we apply the MKM for head and neck squamous cell carcinomas (SCC61 and SQ20B Cells) exposed to high LET ions. The results obtained with the Microdosimetric Kinetic model are analyzed and the model parameters of the SCC61 and SQ20B cell lines are compared in term of radiosensitivity to high-LET ions. We have seen that the parameters obtained in this study for both cell lines reflect the difference in their radiosensitivity. All these results are discussed in terms of parameter analysis and comparison between cell lines, benefits and limits of the MK model and leads to some prospects.

Conceptual comparison of the MKM with LEM

(Y. Kase et al 2008)

Introduction

Cell survival to ionizing radiations is a relevant biological endpoint to plan radiotherapy and hadrontherapy treatments since it can be linked to the probability of tumor control. Generally, cell survival is estimated by invitro measurements of cell-survival curves. To be integrated into a treatment planning system, experimental data have to be accurately reproduced by a model, which can predict survival values at any dose. In the field of radiotherapy with light ions (hadrontherapy), cell survival depends not only on dose, but also on ion species and ion energy. Consequently, a complex model is needed in order to take into account for the biological dependence of the radiation. Then, several models have been proposed to describe the biological efficiency of radiation on cells. The assumptions and formalisms of these models evolve with the development of knowledge in radiobiology. However the complexity of the radiochemical and radiobiological mechanisms involved in the formation of biological damages complicates the elaboration of such models. Consequently, some models, based on assumptions and approximations extrapolated from experimental results allow useful predictions. Most of the recent models are based on micrometric and nanometric scale calculation. Among others, two models have been the subject of study and development in the context of applications to hadrontherapy treatment planning. These models are the Local Effect Model (LEM) [START_REF] Scholz | Track structure and the calculation of biological effects of heavy charged particles[END_REF][2] and the Microdosimetric Kinetic Model (MKM) [START_REF] Hawkins | A Statistical Theory of Cell Killing by Radiation of Varying Linear Energy Transfer[END_REF] [START_REF] Hawkins | A Microdosimetric Kinetic Model of cell death from exposure to ionizing radiation of any LET, with experimental and clinical applications[END_REF]. The LEM is already used in clinical routine in Germany [START_REF] Krämer | Treatment planning for heavy-ion radiotherapy: calculation and optimization of biologically effective dose[END_REF] and the MKM is seriously considered by NIRS group, which has published several studies testing this model in order to adapt it to their treatment planning system [START_REF] Kase | Biological dose calculation with Monte Carlo physics simulation for heavy-ion radiotherapy[END_REF] [START_REF] Sato | Biological dose estimation for charged-particle therapy using an improved PHITS code coupled with a microdosimetric kinetic model[END_REF]. Therefore, we focus in this paper on both these models and applied the MKM model to three cell lines.

Material and method

The LEM formalism

To predict cell survival, the Local Effect Model (LEM), considers that cell killing arises from the induction of lethal events by the ionizing radiation. Assuming that the distribution of lethal events obeys a Poisson distribution, the probability for the cell to survive reads:

Where lethal N (D) is the mean number of lethal events induced in the cell after a dose D. The first key assumption of the LEM is to consider lethal events as point-like events generated by the local dose deposited by the radiation. Thus, the number of lethal events in the cell is the summation of the local lethal events over the cell sensitive volume:

lethal lethal Sensitive Volume N (D) ( )d ρ = ∫∫∫ r r (2)
Where the local density of lethal events is assumed to be a simple function of the local dose d(r):

lethal lethal ( ) (d( )) ρ ρ = r r (3)
In the LEM, the local dose is calculated by cumulative effects, superimposing the local dose deposited by each ion, which is represented by the radial dose R d :

R i i d( ) d (r ) = ∑ r (4)
Where i r is the radial distance of the point r to the trajectory of the i th ion in the transversal plane to the beam axis.

The second key assumption of the LEM consists in extracting the relation between the density of lethal events and the local dose from survival measurements performed with X-ray radiation. Indeed, the local dose deposited by X-ray radiation is considered as uniform within the cell. Neglecting stochastic effects, it is therefore equal to the macroscopic dose D , which is delivered to the sample by the X-ray source: d( ) D = r [START_REF] Krämer | Treatment planning for heavy-ion radiotherapy: calculation and optimization of biologically effective dose[END_REF] Therefore for X-ray irradiation, Eq. ( 2) becomes simply:

lethal lethal sensitive N (D) (D).V ρ ≈ (6)
According to equation (2), N lethal (D), and therefore lethal (D) ρ can be deduced from the measurement of cell survival to X-ray irradiation (described by the α and β parameters) and from an estimation of the cell sensitive volume V sensitive . This latter is assumed to be uniformly distributed over the cell nucleus. The diameter of the sensitive volume depends on the cell and ranges from 5-20 µm. An explicit expression for the average number of lethal events can thus be obtained as:

X lethal sensitive Sensitive Volume ln S (d(r)) N d V - = ∫∫∫ r (7)
The photon dose-effect relation is parameterized using an additional parameter, the threshold dose , allowing extrapolating the radiation effect to very high dose. This special parameterization is presented in equation ( 8

) : 2 X X 2 X X seuil X seuil D (r)+ D (r) si D (r) ln ( ( )) ( + 2 D )D (r)-D si D (r)> seuil seu il D S D r D α β α β β  ≤  - =    (8)
Practically, the threshold dose cannot be measures and is fitted using experimental data measured for high-LET ions [START_REF] Beuve | Parameters and Local Effect Model predictions for head and neck squamous cell carcinomas exposed to High Linear Energy Transfer ions[END_REF].

The LEM authors have changed their model by incorporating the effect of ionization clusters [START_REF] Elsasser | Cluster effects within the local effect model[END_REF] and by modifying the parameterization of the radial dose in integrating an expression of the minimum radial distance r min dependent on ion energy. However, the basic formalism of the LEM is still the same. The basic concepts of this model was extensively analyzed by Beuve et al [10][11].

The MKM formalism

The MK model is based on the statistical theory and the microdosimetric formalisms and quantities. In the MK model, the lesions are produced in sub-volumes of the cell nucleus called "domains". The dose deposited in these domains is quantified by the so-called "specific energy" stochastic variable z . The lesions are classified into two different types [START_REF] Hawkins | A Microdosimetric Kinetic Model for Effect of Non Poisson Distribution of Lethal Lesions on the Variation of RBE with LET[END_REF], Type I are lethal and non-repairable, type II are initially not lethal but may become lethal if they undergo some specific transformations. The probability of forming a type I lesion in a domain is proportional to the dose absorbed by the domain z . The non-lethal type II lesions may be repaired or transformed to lethal lesions. The processes that the type II lesions may undergo are 1°) a monomolecular-like process with first-order rate constant a. 2°) a pairwise combination with another type II lesion located in the same domain to form a lethal unrepairable lesion with second-order rate constant b. 3°) a repair process by a monomolecular-like process with first-order rate constant c.4°) a persistence for a period of time tr, after which it becomes lethal and not repairable. These possibilities are resumed in figure 1. Therefore the expression of 1P RBE in the limit of the zero dose is given by:

1 0 1 D p R R R B E z α β α α - = + (10)
0 1 0.2 ² D p R R R B E y d α β α α - = + (11)
This expression allows deducing the two parameters 0 α and the domain diameter d by fitting experimental data for LET below the saturation effect. The value of β is deduced from the X rays radiation and assumed to be unchanged with changing LET or particle type.

However, for the very high-LET ions, the distribution of lethal lesions in the nucleus cannot be described by a Poisson distribution and the RBE value decreases with increasing LET. Then, Equation [START_REF] Beuve | Statistical effects of dose deposition in track-structure modelling of radiobiology efficiency[END_REF] requires a correction in order to take into account for this saturation effect in RBE. This correction was proposed by R.B Hawkins [START_REF] Hawkins | A Microdosimetric Kinetic Model for Effect of Non Poisson Distribution of Lethal Lesions on the Variation of RBE with LET[END_REF]:

1 1 1 1 (1-exp( )) Dn Dn p P R p z RBE RBE z α α α α - - - = = (12)
Where 1Dn z is the dose mean specific energy in the nucleus. It is expressed as a function of D y by:

1 0 . 1 6 D D n y z σ - - = Gy ( 13
)
Where σ is the cross sectional area of the sensitive nuclear volume expressed in µm²

If one replaces 1Dn z -, p α and 1P RBE given respectively by equations ( 13), ( 9) and ( 11), in the equation ( 12), one can express the corrected linear coefficient of dose by equation [START_REF] Kase | Biophysical calculation of cell survival probabilities using amorphous track structure models for heavy-ion irradiation[END_REF].

0 * 0.2 0.16( ) ² 1-exp 0.16 D D D y y d y α β σ α σ - - -        - +                                   = (14)
Finally the equation ( 14) is fitted with experimental data for large LET range in order to deduce the value of σ .

The LEM vs MKM comparison

An interesting conceptual comparison was published by Y. Kase et al 2008 [START_REF] Kase | Biophysical calculation of cell survival probabilities using amorphous track structure models for heavy-ion irradiation[END_REF]. In their work, Kase et al have adapted the MKM to amorphous track calculation in order to compare it to the LEM. We present here some conclusions of this comparison.

In the LEM, the biological effect is related to an energy deposition in infinitesimal small nuclear sub-volume assimilated to point-like target (Scholz et al 1996 [START_REF] Scholz | Track structure and the calculation of biological effects of heavy charged particles[END_REF]). Regarding the MKM, the biological effect is related to an energy deposition in nuclear sub-volumes with micrometric dimensions named "Domains". The domain is a small homogeneous reaction vessel with boundary that is impermeable to lesions. (Hawkins 2003 [12]). Both models need the photon dose-effect relation as an input parameter. In addition, the LEM needs the radial dose distribution of the incident ions. The LEM parameterization of this dose effect relation is presented in the equation ( 8). In the case of LEM the parameter is considered as dependent of the LET and the particle type. For the MKM the photon dose-effect relation is parameterized by the linear quadratic model whatever the dose range and the parameter is considered independent of the particle LET and type. In the case of the MKM the β β additional input parameters are the lineal energy distribution in the domain and the experimental RBE-LET relation for at least two different LET values.

In the LEM the critical parameter is the threshold dose , used in the parameterization of the experimental photon dose-effect relation for high dose. It is practically difficult to precisely determine the photon dose effect at very high dose. Then, is kept as an adjustable parameter in order to allow the best representation of the experimental data [START_REF] Beuve | Parameters and Local Effect Model predictions for head and neck squamous cell carcinomas exposed to High Linear Energy Transfer ions[END_REF]. In the case of the MKM, Kase et al [START_REF] Kase | Biophysical calculation of cell survival probabilities using amorphous track structure models for heavy-ion irradiation[END_REF] have reported that the model predictions are sensitive to relative variations of the domain size. Finally, both models provide a good description for different sets of experimental data. However, for some LET-energy combinations, the agreement with the experimental results is better with the MKM. Finally, Kase et al reported in their paper that there is a tendency for overestimation of RBE by the LEM model for high energetic ions at comparably low LET (Kase et al [START_REF] Kase | Biophysical calculation of cell survival probabilities using amorphous track structure models for heavy-ion irradiation[END_REF]) .

Results

Application of the MKM with experimental data of Furusawa et al [START_REF] Furusawa | Inactivation of aerobic and hypoxic cells from three different cell lines by accelerated 3He-,12C-and 20Ne-ion beams[END_REF] The user needs to establish two experimental relations to calculate the necessary parameters of the MK model. These parameters are used later to characterize the cell radiosenstivity and predict the response of this cell to ion irradiation. These two experimental relations are 1°) the photon dose-effect relation of the cell. This relation is used to extract the domain diameter d. 2°) The RBE-LET relation which allow to calculate 0 α and σ . Then, we followed these steps to apply the MKM on experimental data published by Furusawa et al [START_REF] Furusawa | Inactivation of aerobic and hypoxic cells from three different cell lines by accelerated 3He-,12C-and 20Ne-ion beams[END_REF] for HSG cell line irradiated with carbon ion in aerobic conditions. The fitting of the photon dose-effect relation by the linear quadratic model resulted in the determination of the parameters: X α =0,313Gy -1 and β =0,062 Gy -2 . The index x indicates that the alpha parameter is calculated for the X rays irradiation. The MKM was also applied for HSG cell by [START_REF] Kase | Microdosimetric measurements and estimation of human cell survival for heavy-ion beams[END_REF] [START_REF] Kase | Microdosimetric measurements and estimation of human cell survival for heavy-ion beams[END_REF]. In their work Kase et al. have used the dose-effect of 200 KV X rays and of 290 MeV/n carbon ions to extract the domain diameter and 0 α parameter. However, they used another formula to correct the saturation effect and extract the nucleus effective area using the lineal energy saturation (details are presented in reference [START_REF] Kase | Microdosimetric measurements and estimation of human cell survival for heavy-ion beams[END_REF]). The table 1 resumes the MKM parameters obtained by our application of the original MKM and those obtained by Kase et al [START_REF] Kase | Microdosimetric measurements and estimation of human cell survival for heavy-ion beams[END_REF].

Table 1.

Comparison of our application of the MKM with those of Kase et al [START_REF] Kase | Microdosimetric measurements and estimation of human cell survival for heavy-ion beams[END_REF] The modified formulation used by Kase and al [START_REF] Kase | Microdosimetric measurements and estimation of human cell survival for heavy-ion beams[END_REF] can explain the difference between our calculation of n R and that of Kase et al . Concerning d r and 0 α , these parameters are extracted using the original formulation of the MKM in both cases. We note that Kase et al [START_REF] Kase | Microdosimetric measurements and estimation of human cell survival for heavy-ion beams[END_REF] used as input, the dose-effect relation based on two experimental relations determined by an irradiation with 200 kV photons and with 290 MeV /n carbon ions, in order to fit the linear equation expressed in equation [START_REF] Beuve | Statistical effects of dose deposition in track-structure modelling of radiobiology efficiency[END_REF]. Instead we used a larger set of experimental data. Despite this, the differences between the both calculation results are acceptable taking into account the uncertainties associated to each calculation.

Application of the MKM to characterize two cell lines with different radio-sensitivities

The MK model is applied here to reproduce the experimental data of the irradiation of two cell lines. The cell lines used in this study are extracted from the same histological type of head and neck carcinoma. The first one is the radiosensitive SCC61 cells, extracted from a pharynx carcinoma. The second line is the radioresistant SQ20B cells, extracted from a neck carcinoma. These cells are prepared and cultured as described in reference [START_REF]Ceramide induces activation of the mitochondrial/caspases pathway in Jurkat and SCC61 cells sensitive to gamma-radiation but activation of this sequence is defective in radioresistant SQ20B cells[END_REF]. The irradiation procedures and the results of theses irradiations was reported by Beuve et al ( 2008) [START_REF] Beuve | Parameters and Local Effect Model predictions for head and neck squamous cell carcinomas exposed to High Linear Energy Transfer ions[END_REF]. The cells were irradiated with carbon ions of 72 MeV/n and with 85 MeV/n argon ions at GANIL facility in France. The energies mentioned above are the ion energies at the entrance of the cells. X-ray irradiations of both cell lines were performed using the same experimental protocols as used for ion irradiations. This is very important in order to compare the high-LET radiation results with the X rays ones. The approach used previously to calculate the MKM parameters for the HSG cell line is followed here in order to calculate the MKM parameters for SQ20B and SSC61 cell lines. The figure 3 below presents the data adjustment of the alpha parameter versus the dose mean lineal energy in the domain according to the equation ( 14). The calculated parameters, with this fit, are summarized in the Table 2. We note that in this case, we do an MKM application similar to that made by Y Kase et al [START_REF] Kase | Microdosimetric measurements and estimation of human cell survival for heavy-ion beams[END_REF]. This application is based on the use of two experimental dose-response relations obtained with photon irradiation and with low LET ion (72 MeV/n carbon ions), in order to estimate the domain size. However, for the estimation of the effective nucleus area, we used the original formulation of the MKM with an additional experimental relation obtained using a very high LET (85 MeV/n argon ions).

Table 2 : MKM parameters calculated for SCC61 and SQ20B cell lines

Discussion

The radiobiological significance of the MK model parameters allows an analysis of the biological response of the cells and comparison between different cell lines. In the case of SCC61 and SQ20B cell lines, the parameters obtained in this study reflect the difference in their radiosensitivity. The first difference consists in α 0 . A mathematical description of this parameter is given by Hawkins [START_REF] Hawkins | A Microdosimetric Kinetic Model for Effect of Non Poisson Distribution of Lethal Lesions on the Variation of RBE with LET[END_REF] . It depend on the formation rate of nonreparable type I lesions, the formation rate of the lethal lesions by the monomolecular transformation of type II lesions and the transformation of the type II lesions to lethal lesion after a period r t (see section 2 in method and result). Therefore, the cell lines which have low or zero value of 0 α are dominated by the formation of type I lesions or the monomolecular transformation of the type II lesions. In our case, the 0 α value for SCC61 cells is higher than for SQ20B. This implies that the lethal lesions produced in SCC61 cells are the result of the type I lesions or the monomolecular transformation of the type II lesions. The β parameter of the SQ20B cells is higher than those of SCC61, this means that the transformation rate of the type II lesions into lethal lesion by the bimolecular processes (interaction of two sublesions) is higher in the SQ20B cells. However, the MK model assumes that the production rate of type II lesions is constant regardless of the LET. While, the author of the MKM relay the type II lesions to the DSB lesions [START_REF] Hawkins | A Microdosimetric Kinetic Model for Effect of Non Poisson Distribution of Lethal Lesions on the Variation of RBE with LET[END_REF], consequently any change in the LET may affect the production rate of type II lesions and probably the β . Concerning the parameter σ, which reflects the nuclear sensitive volume, is more important in the SCC61 than in SQ20B cell.

Finally, this behavior of the MKM parameters can be exploited clinically to adapt the radiation depending on the degree of radiosensitivity tumor cells based on the model parameters. In this optic, it will be interesting to determine a common experimental protocol in order to calculate the MKM parameters and construct a parameter tables for each cell line considered in hadrontherapy treatement.

Conclusion

We have seen that the application of the MKM in its original version gives comparable results with the MKM version used by Kase et al [START_REF] Kase | Microdosimetric measurements and estimation of human cell survival for heavy-ion beams[END_REF] except for the determination of the effective nucleus area where Kase et al [START_REF] Kase | Microdosimetric measurements and estimation of human cell survival for heavy-ion beams[END_REF] included a new formula to correct the saturation effect using the lineal energy saturation. We have also seen that the MKM parameters allow characterizing the response mechanisms and the radiosensitivity of a cell lines. The MKM parameters calculated for the SCC61 and SQ20B cell lines reflect well their different radiosensitivity. Then, this characteristic of the MKM parameters can be beneficial in clinical application of the MKM in order to improve treatments by adapting the radiation depending on radiosensitivity of tumor cells. The MK model still has some weaknesses especially regarding the change of beta parameter with LET since the MKM assumes that the β is independent of particle type and energy used. The α and β parameters are expressed in the MKM as a function of several other parameters related to production rates of type II and type I lesions and to their kinetic evolution in the cell. It assumes also that the production rate of type II lesions is constant regardless of the LET and is equal to 30 lesions / Gy. However, the MKM relay the type II lesions to the DSB lesions [START_REF] Hawkins | A Microdosimetric Kinetic Model for Effect of Non Poisson Distribution of Lethal Lesions on the Variation of RBE with LET[END_REF], consequently any change in the LET may affect the production rate of type II lesions and probably the β .

For the prospects of RBE modeling for hadrontherapy treatments, we adopted two approaches. The first approach is to continue the investigations of MKM in order to better understand the behavior of its parameters in different biological conditions. We are working on changing the expression given the production rate of type II lesions in order to include the influence of the LET. For this, we rely on the experimental data we are producing by irradiation with x-rays and carbon ions (at GANIL) of several types of cell lines following a well-defined protocol.

The second approach is to develop a new model, which in particular will include concept or element inspired from the models presently available in the literature and we considered as key points.

I. Introduction

•Radiobiological models are necessary to predict the biological effect of radiation in order to perform a rigorous treatment planning in hadrontherapy.

• There are several models, based on assumptions and approximations extrapolated from experimental results.

3

•There is two most advanced models for clinical application which are :

-The Microdosimetric Kinetic Model (MKM) Hawkins 1994 and 1996.

-The Local effect Model (LEM) Scholz et al 1996

Basic formulation of MKM and LEM

The total effect is obtained by summation of al local effects through total sensitive volume :

ln( ( ( )) ( ) 1. Radiosensitive sites

• Both models suppose that the radiosensitive structures are contained in the nucleus. These structures considered as the sub-volumes of the nucleus.

LEM MKM

The biological effect is related to an energy deposition in infinitesimal small nuclear sub-volume

The biological effect is related to an energy deposition in nuclear sub-volume with micrometric dimension

LEM MKM

The radial distribution of the dose:

Distribution of the dose at distance r from the ion trajectory (In the LEM I).

Microdosimetric spectra :

Distribution of energy deposition in the domain calculation of or

LEM MKM

In the LEM, the biological response of the cell is taken into account by using the photon dose-effect relation.

In the MKM, several parameters are used to describe the cell response and the kinetic evolution of the initial 5. Cell response 12 the photon dose-effect relation.

There are no parameters describing the formation and the kinetic evolution of lesions.

and the kinetic evolution of the initial lesions.

, al of these parameters are considered as independent from LET.

d σ 0 α β

Application of models in treatment planning

• Both models provide a good description for different sets of experimental data.

• However, For some LET-energy combinations, the agreement with the experimental results is better with the MKM. •Domain = distance a type II lesion can travel through the nucleus before it is removed by repair.

•The low difference of the domain size between the two cell lines can not explain the difference in radiosensitivity of the two cell lines. The difference in σ value of both cell lines reflect probably Difference in the distribution of the nuclear sensitive structures between the two cell lines.

• A strength of both models is that due to the simplicity and the computational speed of the calculations, they are both applicable for treatment planning.

• the MKM and the LEM rely on the same three basic constituents of target geometry, photon survival curve and track structure.

Conclusions and prospects:

21 however, their implementation of these constituents is significantly different.

• In both models, the photon dose-effect relation is not sufficient to calculate the ion response.

Additional information about the biological response to ion irradiation is necessary to determine the domain size in MKM or the threshold dose in LEM.

Conclusions and prospects :

• The MKM parameters calculated reflect the difference in cell radiosensitivity.

this characteristic can be beneficial in clinical application of the MKM in order to improve treatments.

• The MK model still has some weaknesses especially regarding the change of beta parameter with LET.

22 parameter with LET.

• We are working on changing the expression given the production rate of type II lesions in order to include the influence of the LET on the parameter.

• To use Geant4 DNA to apply MKM in microdosimetrical context . • Develop a new model, which in particular will include concept or element inspired from the models presently available in the literature and we considered as key points.

β

Figure 1 :α

 1 Figure 1: Schematic representation of the MKM formalism For a short time irradiation and assuming a Poisson distribution of lethal lesions, the mean number of lesions in nucleus is expressed by:2

  The second step of the MKM-parameter calculation is the determination of the domain diameter and the effective area of the sensitive volume σ . These two parameters are extracted from the RBE-LET or RBE-lineal energy relation. Then we determine the domain diameter according to the equation (11) and the effective area of the HSG cells according to the equation (14). The result of these adjustments are shown in figure2.

Figure 2 :.

 2 Figure 2: results of our MKM parameter calculation for HSG cell line irradiated with carbon ions of different energies. The experimental data are extracted from Furusawa et al. [14].

  on the HSG cell line

Figure 3 :

 3 Figure 3: Results of MKM parameter calculation for SQ20B and SCC61 cell lines irradiated with carbon ions of 72 MeV/n and argon ions of 85MeV/n. The experimental data are extracted from Beuve et al. [8].

8 2 .D

 2 Assimilated to point-like target[START_REF] Scholz | Track structure and the calculation of biological effects of heavy charged particles[END_REF] DomainSmall homogeneous reaction vessel with boundary that is impermeable to lesions.(Hawkins 2003)Local effect Photon dose-effect relation• Both models use the photon dose-effect relation in the determination of the ion effect.LEM MKM: Threshold dose, obtained by adjusting the experimental data of The photon dose-effect relation.tThe linear quadratic function of the dose is used for any value of dose. 9is not dependent in LET and particle type β changes with changing LET and particle type.

β 3 .

 3 The energy deposition• Both models need physical input parameter to calculate the cell response.

•D

  Both models use parameters that could not be directly measured :Threshold dose :The domain size :It is practically difficult to precisely determine the photon dose at very high dose. t Domain can not be identified with any known structure in the cell. an adjustable parameter in order to allow the best representation of the experimental data. Obtained by fitting the RBE1-LET relation at low LET range. Linear approximation of the RBE1-LET relation in this range of LET It's value depends on the LEM version considered.

13 • 1 .d• 2 . 16 •

 131216 Solid lines : MKM calculations dotted lines : initial version LEM calculations. There is a tendency for overestimation of RBE by the LEM model for high energetic ions. Calculation of the biological parameters for HSG cell line : • experimental data extracted from Furusawa et al 2000, for HSG cell line irradiated with carbon ions in aerobic conditions 15 ² d To deduce : : initial slop of the survival curve. : domain's diameter. 0 α To deduce the effective area of the nuclear sensitive volume: σ Calculation of the MKM parameters for HSG cell line by Kase et al (2006) using as input data de dose-effect relations obtained with: • X rays of 200 KV and carbon ions of 290 MeV/n Comparison of the parameters obtained by Kase et al for HSG cells:The correction of the saturation effect is done using the lineal energy saturation in Kase et al 2006, calculation. The differences between the both calculation results are acceptable taking into account the uncertainties associated to each calculation.• Except for , where the difference can be explained by the modified formulation of the saturation effect used in Kase et al calculation.

•

  The radiosensitive cell line: SCC61,established from squamous cell carcinomas of the pharynx.• The radioresistant cell line: SQ20B,established from squamous cell carcinomas of the neck.We used as input parameters :The experimental dose effect relation obtained with 250 KV X rays.Experimental data obtained at GANIL with 72 MeV/n carbon ions and 85 MeV/n argon ions

•

  The low value of indicates that the contribution of the monomolecular transformation of the type II lesions is weak SQ20B•The high value of indicates that the most lethal lesions are produced by the monomolecular transformation of the type II lesions SCC61

D. DABLI

, G. MONTAROU (a) , M. BEUVE (b) and C. RODRIGUEZ-LAFRASSE (c)