Maria Pimpinella
email: maria.pimpinella@enea.it

Claudio Caporali

Antonio Stefano Guerra

Luca Silvi

Vanessa De Coste

Assunta Petrucci

Franck Delaunay

Stéphane Dufreneix

Jean Gouriou

Aimé Ostrowsky

Frank Delaunay

Benjamin Rapp

Jean-Marc Bordy

Josiane Daures

Maïwenn Le Roy

Line Sommier c , Didier Vermesse

Feasibility of using a dose-area product ratio as beam quality specifier for photon beams with small field sizes

Introduction

In recent years, the use of small beams (field sizes smaller than 2 cm × 2 cm) in routine radiotherapy techniques has increased steadily. Nevertheless, small beam dosimetry is still challenging. The determination of absorbed dose to water, D w , in narrow photon beams is particularly demanding, if the traditional approach of measuring D w with a point-like detector placed on the beam axis is followed. Even the use of high resolution detectors with sensitive volume of the order of tenths of mm 3 or less does not ensure a reliable D w measurement in the absence of lateral electronic equilibrium [START_REF] Benmakhlouf | Output correction factors for nine small field detectors in 6 MV radiation therapy photon beams: A PENELOPE Monte Carlo study[END_REF]. When used in small field sizes, point-like detectors can exhibit large response variations (even more than 10%) depending on the detector material and construction details near the sensitive volume. These variations generally result in underestimation (in the case of small volume ionization chambers) or overestimation (in the case of solid state detectors) of both D w and output factors [START_REF] Bouchard | Detector dose response in megavoltage small photon beams. I. Theoretical concepts[END_REF][START_REF] Scott | Characterizing the influence of detector density on dosimeter response in non-equilibrium small photon fields[END_REF]. As a consequence detector specific correction factors [START_REF] Alfonso | A new formalism for reference dosimetry of small and nonstandard fields[END_REF] are required for accurate D w measurement in small beams. Previous work in this field has focused on the evaluation of such correction factors, either using direct Monte Carlo calculation or, experimentally, taking a given detector as reference, and results for the most widely used detectors have been reported in literature [START_REF] Benmakhlouf | Output correction factors for nine small field detectors in 6 MV radiation therapy photon beams: A PENELOPE Monte Carlo study[END_REF][START_REF] Azangwe | Detector to detector corrections: a comprehensive experimental study of detector specific correction factors for beam output measurement for small radiotherapy beams[END_REF][START_REF] Czarnecki | Monte Carlo calculated correction factors for diodes and ion chambers in small photon fields[END_REF][START_REF] Francescon | Monte Carlo simulated corrections for beam commissioning measurements with circular and MLC shaped fields on the CyberKnife M6 System: a study including diode, microchamber, point scintillator, and synthetic microdiamond detectors[END_REF][START_REF] Lechner | Detector comparison for small field output factor measurements in flattening filter free photon beams[END_REF][START_REF] Masi | CyberKnife beam output factor measurements: a multi-site and multidetector study[END_REF][START_REF] O'brien | Small field detector correction factors for silicon-diode and diamond detectors with circular 6 MV fields derived using both empirical and numerical methods[END_REF][START_REF] Papaconstadopoulos | On the correction, perturbation and modification of small field detectors in relative dosimetry[END_REF][START_REF] Ralston | Over-response of synthetic microDiamond detectors in small radiation fields[END_REF][START_REF] Underwood | Application of the Exradin W1 scintillator to determine Ediode 60017 and microDiamond 60019 correction factors for relative dosimetry within small MV and FFF fields[END_REF][START_REF] Huet | Study of commercial detector responses in non-equilibrium small photon fields of a 1000 MU/min CyberKnife system[END_REF]. However, as no D w primary standards exist for field sizes smaller than 2 cm, discrepancies among published data are difficult to solve [START_REF] Andreo | On the Monte Carlo simulation of small-field micro-diamond detectors for megavoltage photon dosimetry[END_REF][START_REF] Marinelli | Experimental determination of the PTW 60019 microDiamond dosimeter active area and volume[END_REF]. Moreover, the positioning of point-like detectors in narrow beams is also very critical for the measurement accuracy, since errors larger than 1% can occur even for uncertainties on the detector position as low as a few tenths of mm [START_REF] Charles | A practical and theoretical definition of very small field size for radiotherapy output factor measurements[END_REF].

T

The above difficulties have raised interest in a different approach, based on measuring an integral quantity as reference quantity for small beam dosimetry, in analogy with the concept of dose-area product used for measurements in free space in diagnostic radiology [START_REF]Dosimetry in diagnostic radiology: an International Code of Practice[END_REF], and extending the concept to measurements performed in a material medium like in a water phantom [START_REF] Djouguela | The dosearea product, a new parameter for the dosimetry of narrow photon beams[END_REF][START_REF] Duane | International Symposium on Standards, Applications and Quality Assurance in Medical Radiation Dosimetry; Application of Dose Area Product and DAP Ratio to Dosimetry in IMRT and Small Field External Beam Radiotherapy[END_REF][START_REF] Roy | Establishment of dosimetric references in terms of dose-area product for small sizes MV X-ray beams[END_REF]. For radiotherapy dosimetry application, the dose-area product (DAP)i sd e fined as:

∫ = DAP D x y dxdy (,) A w ( 1
)
where A is an area, in a plane perpendicular to the beam central axis, larger than the beam size at that plane, and corresponding to the active area of the detector used for the DAP measurement. A proper DAP detector should have a flat sensitive volume with cross-sectional area larger than the radiation field. Large-area plane-parallel ionization chambers (LACs) have been shown to be adequate DAP detectors [START_REF] Sánchez-Doblado | A new method for output factor determination in MLC shaped narrow beams[END_REF][START_REF] Pimpinella | Measurements of Dose.area-product ratio in small radiotherapy photon beams using two types of large-area plane-parallel ionization chambers[END_REF]. In very small field sizes, positioning a DAP detector on the beam central axis is less critical than positioning a point-like detector.

In addition, measurement of the integral dose is expected to be less detector dependent, if compared with the point-dose measurement by point-like detectors [START_REF] Underwood | Detector density and small field dosimetry: integral versus point dose measurement schemes[END_REF]. In fact, the latter requires corrections that strongly dependent on beam shape, detector type, and off-axis position in the small field. Conversely, even in the case of a composite clinical field, Monte Carlo calculations in [START_REF] Underwood | Detector density and small field dosimetry: integral versus point dose measurement schemes[END_REF] showed that differences in response among point-like detectors substantially diminish, if the detector signal is integrated over the whole radiation field. The Laboratoire National Henri Becquerel (LNE-LNHB) recently developed a calorimeter for absolute measurement of DAP in small field sizes, making this quantity available for transfer from the calibration laboratory to the user's beam [START_REF] Dufreneix | Using a dose-area product for absolute measurements in small fields: a feasibility study[END_REF][START_REF] Dufreneix | Construction of a large graphite calorimeter for measurements in small fields used in radiotherapy[END_REF]. Then, in the scenario of a small beam dosimetry based on DAP references, defining the beam quality specifier in terms of DAP to link the calibration to the user's beam becomes an attractive possibility.

In the present paper, the feasibility of expressing the photon beam quality in terms of a DAP ratio is explored for field diameters below 2 cm. Specifically the ratio of DAP at 20 cm and 10 cm water depths

is considered in analogy with the traditional TPR 20,10 beam quality index [START_REF] Andreo | Absorbed dose determination in external beam radiotherapy: an international code of practice for dosimetry based on standards of absorbed dose to water[END_REF] based on point-dose measurement. We thoroughly investigated the properties of DAPR 20,10 in 6 MV and 10 MV small photon beams both by Monte Carlo calculation and experimentally, using LACs with different air-cavity diameters and characteristics. Two types of Linacs with different collimator systems were used. The aim of this work was: a) to verify the ability of DAPR 20,10 to discriminate between qualities of small beams for the purpose of selecting energy-dependent dosimetric data (i.e. ionization chamber calibration coefficient, correction factors); b) to establish appropriate measurement conditions and procedures for the experimental determination of DAPR 20,10 as a beam quality specifier.

Materials and methods

Monte Carlo simulations of 6 MV and 10 MV clinical photon beams were performed in order to investigate the dependence of DAPR 20,10 on beam energy and field size. Moreover, Monte Carlo calculation was applied to evaluate the influence on DAPR 20,10 of the area over which DAP is integrated (i.e. the detector active area). Measurement setups with fixed source-to-surface distance (SSD) or with fixed source-to-detector distance (SDD) were considered. Additionally, the water-to-air stopping power ratio, s w.air , the most important energy-dependent parameter affecting ionization chamber response, was calculated at reference depth as a function of the beam energy and field size, to assess whether the s w.air values are correlated with the corresponding DAPR 20,10 values. Finally, ratios of ionization signals at 20 cm and 10 cm water depths were measured under various experimental conditions (beam energies, field sizes and measurement setups) by means of LACs with different active areas, and experimental results were compared to those obtained by Monte Carlo calculation.

Accelerators and photon beams

Accelerators used in this work were a Varian DHX clinical accelerator available at San Filippo Neri Hospital in Rome and a General-Electric (GE) Saturne 43 clinical accelerator at LNE-LNHB. Since reproducibility of jaw positioning was not good enough for small beams (in some cases measurement reproducibility was larger than 1%), only beams shaped by fixed cones were considered for DAPR measurement.

The Varian DHX accelerator produces 6 MV and 10 MV photon beams and it is equipped with Radionics stereotactic collimators. These are tapered conical collimators using Cerrobend (27% lead, 50% bismuth, 13% tin and 10% cadmium) as collimating material. A Cerrobend cylinder with central conical opening is inserted into a stainless steel cylindrical housing with length of 12.5 cm and outer diameter of 7.5 cm. Using such cones and a constant 7 cm × 7 cm secondary collimator (e.g. linac jaws) setting, circular beams with diameters of 2.00 cm, 1.50 cm and 1.25 cm at the isocenter are produced. The beams shaped by the stereotactic collimators were used for investigating the properties of DAPR 20,10 by Monte Carlo simulations and measurements.

For the Saturne 43 accelerator, specifically designed external collimators made of tungsten alloy (D185) with a length of 10 cm and a conically-shaped hole were added to produce 6 MV beams with diameters of 2 cm, 1 cm and 0.75 cm at the reference plane. The alignment was checked optically with a telescope. The entry and exit apertures of the collimator had to be centred on the same axis and the radial dose distribution was checked with EBT3 films. Two additional monitor ionization chambers were mounted on the external collimator in front of the beam defined by the linac jaws. For the three small beams, DAPR 20,10 was experimentally determined setting an SDD of 100 cm and integrating the absorbed dose over a detector surface of 3 cm diameter.

Monte Carlo simulations

Varian DHX accelerator

The BEAMnrc code [START_REF] Rogers | BEAM: a Monte Carlo code to simulate radiotherapy treatment units[END_REF] of the EGSnrc Monte Carlo system version V4-r2-4 [START_REF] Kawrakow | The EGSnrc Code System: Monte Carlo simulation of electron and photon transport[END_REF] was used for simulating the 6 MV and 10 MV photon beams produced by the Varian DHX accelerator (Fig. 1). First, square photon beams shaped by the jaws were simulated with the purpose of validating the accelerator model by comparing the calculated and measured dose distributions. The mean energy of the initial electron beam was tuned by comparing calculated and measured percentage depth dose (PDD) curves for the 10 cm × 10 cm field size. A circular, Gaussian spatial distribution was assumed for the electron beam incident on the target. The full width at half maximum (FWHM) of the Gaussian distribution was determined by comparing simulated and measured total scatter factors for the 1 cm × 1 cm field size, according to the procedure proposed by Francescon et al. [START_REF] Francescon | Total scatter factors of small beams: a multidetector and Monte Carlo study[END_REF]. Then, the 6 MV and 10 MV stereotactic beams shaped by the Radionics cones were simulated and the corresponding phase-space files (PSFs) generated. To extend the range of beam sizes studied, square fields (side 1.6 cm, 1.0 cm, 0.8 cm and 0.5 cm at the isocenter) defined by the linac jaws were also simulated. The BEAMDP program [START_REF] Ma | BEAMDP as a General-Purpose Utility[END_REF] was used to analyse the phase-space files and to derive the photon beam energy spectra.

The values of the EGSnrc simulation parameters are summarized in table 1. For the BEAMnrc simulations, cutoff energies were set to 10 keV for photons (PCUT) and 700 keV for electrons (ECUT, electron rest mass included). The range rejection (RR) and the directional bremsstrahlung splitting (DBS) techniques were applied to improve the calculation efficiency of photon beam simulations [START_REF] Rogers | BEAMnrc user manual[END_REF][START_REF] Kawrakow | Large efficiency improvements in BEAMnrc using directional bremsstrahlung splitting[END_REF]. The RR parameter ESAVE, defined as the threshold energy below which secondary electrons having ranges shorter than the nearest region boundary are stopped, was set to 2 MeV, but RR was not applied in the accelerator target. For the DBS technique, a splitting value of 1000 was used with a splitting field size extending at least 2 cm beyond the beam edge at 100 cm source distance. PSFs were scored in air at 80 cm, 90 cm and 100 cm distance from the target and used as source inputs into the EGSnrc/DOSXYZnrc, EGSnrc/DOSRZnrc and EGSnrc/SPRRZnrc user codes [START_REF] Rogers | NRCC user codes for EGSnrc[END_REF] for calculating dose profiles (PDD curves, cross-plane and inplane profiles), DAP and s w.air values, respectively. For these calculations, the simulation parameters were the same as for the BEAMnrc simulations except for the electron cutoff energy which was set to 521 keV (electron rest mass included). To investigate the field size dependency of the beam energy distribution in water, a water phantom was included in the BEAMnrc simulations at a source distance of 90 cm. Phase-space files were generated at 10 cm and 20 cm depths in water, and analysed.

Saturne 43 accelerator

For the Monte Carlo simulation of the GE Saturne 43 beams, two different Monte Carlo codes, EGSnrc [START_REF] Kawrakow | The EGSnrc Code System: Monte Carlo simulation of electron and photon transport[END_REF] and PENELOPE [START_REF] Salvat | A code system for Monte Carlo Simulation of Electron and Photon Transport[END_REF], were used. The beam radial anisotropy was measured using EBT3 films and the PDD curves, to be used for verification of the simulation results, were measured using ionization chambers. A PTW 31014, an Exradin A1SL and a homemade plane-parallel ionization chamber were used. The Monte Carlo beam parameters (energy and spot size of the initial electron beam) were defined for the 2 cm beam diameter. Then, the same beam parameters were applied for the 1 cm and 0.75 cm beam diameters and the simulation results compared with measurements. The beam parameters chosen for a parallelized version of PENELOPE 2006 with the MPI library [START_REF] Salvat | A code system for Monte Carlo Simulation of Electron and Photon Transport[END_REF][START_REF] Tola | Optimization of Monte Carlo codes PENELOPE 2006 and PENFAST by parallelization and reduction variance implementation[END_REF] and EGSnrc version V4-r2-4 simulations were rather different. For the PENELOPE calculations, the electron energy spectrum was Gaussian (6.25 MeV, standard deviation of 450 keV) and the electrons reached the target uniformly on a discshaped focal spot of 1 mm diameter. For the EGSnrc calculations, the electron energy spectrum was not Gaussian but based on a more representative spectrum [START_REF] Constantin | Modeling the truebeam linac using a CAD to Geant4 geometry implementation: dose and IAEAcompliant phase space calculations[END_REF] and the electrons reached the target with a Gaussian spatial distribution (FWHM of 1 mm) and a maximum angular spread around the Z axis of 0.2°. The EGSnrc parameters were almost identical to those described in the previous section, except for the DBS splitting parameter set to 1500. The PENELOPE simulation parameters are summarized in Table 2. The C1 parameter is linked to the average angular deflection produced by multiple elastic scattering along a path length equal to the mean free path between two hard elastic events. The C2 parameter is the maximum average fractional energy loss between two consecutive hard elastic events. The WCC and WCR parameters are the cutoff energy losses for hard inelastic collisions and hard bremsstrahlung emission respectively. DSMAX is the maximum mean free path between two hard elastic events. For variance reduction, oriented splitting for bremsstrahlung events in the target (i.e. when a particle exits a specific area, the particle is split into several equivalent particles distributed around a circle of radius defined by current particle position) and an additional splitting using the cylindrical symmetry before reaching the jaws were used. This is based on the activation of forced bremsstrahlung emission for primary electrons whose direction is within a cone with a specific half angle. When this occurs, the number of secondary bremsstrahlung photons that are emitted from primary electrons is increased. For dose calculations, another splitting is done just before the water-phantom entrance surface.

Calculation of water-to-air stopping power ratios

The EGSnrc/SPRRZnrc user code [START_REF] Rogers | NRCC user codes for EGSnrc[END_REF] was applied to calculate s w,air for all the beams used in this work. The SPRRZnrc code calculates s w,air using an on-the-fly technique to score the energy deposition in a cavity filled with the transport medium (i.e. water) and then deriving the energy deposition in the cavity when filled with air [START_REF] Rogers | NRCC user codes for EGSnrc[END_REF]. Compared to off-line s w,air calculation based on scoring the electron fluence spectrum in water, the on-the-fly technique avoids possible effects on the calculation results due to the actual number of energy bins used in the spectrum.

A water phantom with 20 cm radius and 30 cm height was considered and s w,air values were calculated at 10 cm and 20 cm depths in water, using cylindrical scoring regions centred on the beam axis with radii in the range 0.1-5 cm, and a height of 2 mm. Values of s w,air obtained with a scoring region radius of 1 mm are related to D w point measurement. Scoring regions larger than the stereotactic field sizes were considered more appropriate to evaluate s w,air values related to the DAP quantity as defined in this work.

DAPR 20,10 calculations

Using the EGSnrc/DOSRZnrc user code [START_REF] Rogers | NRCC user codes for EGSnrc[END_REF], DAPR 20,10 values were determined for all the simulated Varian DHX beams as a function of the radius of the surface over which the absorbed dose is integrated. To this aim, DAP values were calculated at 10 cm and 20 cm depths in water as a function of the radius of the scoring region. Since the Monte Carlo scoring region corresponds to the detector active area, radii were varied from 1.5 cm to 6 cm, to match the range of radii of the available LACs. Both the SSD setup (i.e. DAP at 10 and 20 cm water depths determined with a fixed SSD, moving the chamber) and the SDD setup (i.e. DAP at 10 and 20 cm water depths determined with the detector at a fixed distance from the source, moving the phantom) were simulated. SSD values of 80 cm, 90 cm and 100 cm as well as an SDD of 100 cm were considered. DAPR 20,10 values referring to an integration surface with radius 1.5 cm were also calculated for the GE Saturne 43 6 MV beams with diameters 2 cm, 1 cm and 0.75 cm using both the EGSnrc and PENELOPE codes.

Measurements

LACs with collecting electrode diameters larger than the field size were used for measuring the ratios of ionization signals at 20 cm and 10 cm depths in water (M M (20)/ [START_REF] O'brien | Small field detector correction factors for silicon-diode and diamond detectors with circular 6 MV fields derived using both empirical and numerical methods[END_REF]). The M M (20)/ (10) ratio re- presents the DAPR 20,10 integrated over the chamber active area, if the ionization chamber response in terms of DAP does not change with the water depth.

Measurements at the Varian DHX accelerator were made using two ionization chambers: a PTW type 34070 (BP_IC) and a PTW type 7862 (TC_IC) (PTW, Freiburg, Germany). The BP_IC chamber is waterproof and has a collecting electrode diameter of 8.16 cm, an inter-electrode spacing of 2 mm and PMMA entrance and exit windows of 0.4 g cm -2 water equivalent thickness. The T_IC chamber is a circular plane-parallel transmission chamber with thin entrance and exit windows (0.028 g cm -2 water equivalent thickness). The chamber sensitive volume has a diameter of 9.65 cm and an inter-electrode spacing of 2.4 mm. A PTW Unidos E Universal Dosimeter was used for DAPR measurements and a PTW Tandem Dual Channel Electrometer was used for scanning measurements (i.e. lateral beam profiles and depth ionization distributions).

Measurements at the GE Saturne 43 accelerator were made using the following ionization chambers: a BP_IC, a PTW type 34073 (collecting diameter of 3.96 cm) and a homemade plane-parallel ionization chamber (SV-PMMA: collecting diameter of 3 cm, inter-electrode spacing of 2 mm) [START_REF] Dufreneix | Etablissement de références dosimétriques dans les faisceaux de rayons × de hautes énergies et de très petites sections (< 1 cm 2 ) pour la radiothérapie[END_REF].

The chambers were positioned in a water phantom with their reference point (i.e. the centre of the chamber at the air cavity entrance) at the measurement depth. Since both the TC_IC and the SV-PMMA chambers are not waterproof, a PMMA envelope (0.5 mm thick in front of the chamber entrance window) was used for measurements in water. Signal stability, measurement reproducibility and saturation curves were measured for all the chambers at different depths in water. Corrections for ion recombination and polarity effects were determined and applied to the chamber signal at each measurement depth.

Concerning ion recombination effects, it should be mentioned that when an ionization chamber larger than the beam is used in a linac, volume recombination is dominant in the direct irradiation field, but near the collecting electrode edge, initial recombination prevails. Examination of the Boag's theory [START_REF] Boag | Ionization chambers[END_REF] shows that the two-voltage method [START_REF] Andreo | Absorbed dose determination in external beam radiotherapy: an international code of practice for dosimetry based on standards of absorbed dose to water[END_REF] can also be used to calculate the saturation correction for partially irradiated ionization chambers as long as the number of created charges per volume unit is homogeneous in each electric field tube of force between electrodes (homogeneous irradiation in the tube of force). This is not difficult to achieve for plane-parallel ionization chambers with a small gap between the electrodes. However, measurements showed that among the different plane-parallel ionization chambers tested, the charges measured (M) for different inter-electrode voltages (V) do not always follow Boag's theory in beams smaller than the ionization chamber. Thus the saturation correction factor was calculated using the tangent of the best fit curve 1/M = f(1/V) at the point corresponding to the usually applied polarizing voltage, but the twovoltage method was applied whenever measurements showed linearity of 1/M versus 1/V.

Uncertainties

Uncertainties were estimated according to the guidelines of the Guide to the Expression of Uncertainties in Measurement [START_REF] Jcgm | Evaluation of measurement data -Guide to the expression of uncertainty in measurement[END_REF] and are expressed in terms of standard uncertainties (coverage factor k = 1). According to [START_REF] Jcgm | Evaluation of measurement data -Guide to the expression of uncertainty in measurement[END_REF] the number in parenthesis after a value is the numerical value of the uncertainty referred to the last digit(s) of the reported result.

Results

Monte Carlo results

Water-to-air stopping power ratios

Table 3 shows s w,air values calculated at a depth of 10 cm in water for the 6 MV and 10 MV Varian DHX beams, together with the corresponding photon fluence-weighted mean energies. Type A relative standard uncertainties were typically below 0.1%.

Significant differences in mean energy values were observed between the reference (10 cm × 10 cm) and the stereotactic beams. On the contrary, mean energy variations between small beams were always below 4% and the corresponding differences in s w,air values were within 0.2%. The s w,air values specifically calculated for the 10 MV and 6 MV stereotactic beams were up to about 0.6% and 0.3% lower than the corresponding values pertaining to the 10 cm × 10 cm field size (i.e. the

Table 3

Water-to-air stopping-power ratios, s w,air , calculated by means of the EGSnrc/SPRRZnrc user code for the Varian DHX 6 MV and 10 MV beams at 10 cm water depth for various field sizes. Radius of the scoring region was 0.1 cm or 1.5 cm for the s w,air values in brackets. The standard uncertainty of s w,air is typically below 0.1%. The photon fluenceweighted mean energies are also reported in the 2nd and 4th column. values currently used for radiotherapy reference dosimetry based on point measurement of D w ). These results confirm previous findings in the literature reporting differences that are smaller than 0.5% between broad (10 cm × 10 cm) and narrow 6 MV beams at reference depth [START_REF] Eklund | Fast modelling of spectra and stopping power ratio using differentiated fluence pencil kernels[END_REF][START_REF] Araki | Monte Carlo study of a CyberKnife stereotactic radiosurgery system[END_REF][START_REF] Sánchez-Doblado | Ionization chamber dosimetry of small photon fields: a Monte Carlo study on stopping-power ratios for radiosurgery and IMRT beams[END_REF][START_REF] Verhaegen | Monte Carlo dosimetry study of a 6 MV stereotactic radiosurgery unit[END_REF][START_REF] Andreo | Stopping power data for high-energy photon beams[END_REF]. Previous studies also indicate that differences become larger at higher energies and values up to 1.1% are reported in case of a 24 MV beam [START_REF] Sánchez-Doblado | Ionization chamber dosimetry of small photon fields: a Monte Carlo study on stopping-power ratios for radiosurgery and IMRT beams[END_REF]. Mean energy and s w,air data in table 3 refer to the beam axis and it should be noted that the photon spectra vary with the radial distance from the central axis. At a given depth in water, an energy decrease is generally observed when the radial distance increases. This is especially true beyond the beam edge where contributions to the photon energy spectra come from low-energy photons generated by radiation interactions in water. For the Varian stereotactic beams, the photon fluenceweighted mean energy calculated for scoring regions with radius ranging from 0.5 cm to 5 cm varied up to about 10%. The associated effects on the s w,air values were up to 0.2% for the 6 MV beams and up to 0.5% for the 10 MV beams. Differences between s w,air values at 10 cm and 20 cm were always within 0.2%.

For the GE Saturne 43 accelerator, s w,air values calculated for the 6 MV beam with diameter of 2 cm, 1 cm and 0.75 cm and disc-shaped (radius 1.5 cm) scoring region at 10 cm depth in water were very close to each other (from 1.1170 to 1.1171) as well as the photon fluenceweighted mean energies (from 1.977 to 1.979 MeV). Differences from the corresponding s w,air values for the Varian DHX linac (table 3, values in brackets) were about -0.2%, in line with the slightly lower (around -5%) mean energies of the Varian simulated beams.

Calculation of DAPR 20,10

Table 4 shows DAPR 20,10 values calculated for the Varian DHX 6 MV and 10 MV small circular beams of diameters ϕ, using dose scoring regions with different radii (R). Type A relative standard uncertainties were in the range 0.15-0.3%.

The DAPR 20,10 values reported in table 4 refer to the SSD setup with SSD = 100 cm and the SDD setup with SDD = 100 cm. For both setups, the DAPR 20,10 values relevant to the same integration area were almost independent of the beam diameter, in line with the results of Monte Carlo calculations showing that the mean beam energies were almost the same (section 3.1.1). Even if an increase in the DAPR 20,10 value can be noted when the field diameter decreases, especially for the SDD setup, differences among DAPR 20,10 values were generally within the expanded type A uncertainty (coverage factor k = 3). On the other hand, data in table 4 clearly show a variation of DAPR 20,10 value with both setup and scoring region radius. Due to the different distances from the source in the SSD configuration, the scoring regions at 120 cm source distance (20 cm depth) receive less of the scattered radiation outside the beam than the equivalent scoring regions at 110 cm source distance (10 cm depth). As a consequence DAPR 20,10 values calculated for the SSD setup were always lower than those calculated for the SDD setup. Deviations about 1-2% were observed with no clear dependence on energy, field diameter, or scoring region radius.

As shown in Table 4, DAPR 20,10 increases with the scoring region radius for both SSD and SDD setups and for all of the field sizes considered. When the radius of the scoring region increases, the relative contribution to DAP from the scattered radiation increases more at 20 cm depth than at 10 cm depth. It could be reasonably contended that if the radius of the scoring region increases, more and more of the scattered radiation will be seen and DAPR 20,10 will tend to an upper limit. However, the calculated DAPR 20,10 value increased all over the R range (1.5 cm to 6 cm) considered, with an approximately linear relation. It could be expected that, for even higher R values, DAPR 20,10 deviates from linearity to reach a limit, but this is likely to occur for R values far larger than the radius of the available LACs. As an example, DAPR 20,10 for the SDD setup is shown in Fig. 2 as a function of R for the 6 MV and 10 MV beams with field diameter 1.25 cm. The increase in the DAPR 20,10 value in the R range from 1.5 cm to 6 cm is about 1.9% per cm and 1.5% per cm for the 6 MV and 10 MV beams, respectively. These figures slightly decrease when the beam radius increases, their values being 1.7% per cm and 1.4% per cm for the beam with a diameter of Table 4 Ratios of DAP at 20 cm and 10 cm in water calculated by Monte Carlo with different scoring region radii (R) for Varian DHX 6 MV and 10 MV circular beams with field diameter (ϕ) 2.00 cm, 1.50 cm and 1.25 cm for the SSD and SDD setups. The reported uncertainties are type A standard uncertainties (between 0.15% and 0.3%). 2 cm. Calculations performed for the square beams defined by the linac jaws confirmed that DAPR 20,10 value referred to a given integration area is almost independent of the beam size (differences were within 1% in the range of field side from 1.6 cm to 0.5 cm). On the other hand, DAPR 20,10 for the square beams was less sensitive to the radius of the scoring region than for the stereotactic beams. The DAPR 20,10 variation with R was typically 1.2% per cm. When comparing results obtained for beams with equivalent sections (i.e. circular beam with diameter 1.25 cm and square beam with side 1.1 cm) differences between DAPR 20,10 values were within the statistical uncertainty (0.1%) for a scoring region radius of 1.5 cm but around 1.5% for a scoring region radius of 6 cm.

A possible influence of the size of the water phantom on the calculated DAPR 20,10 was also considered. No appreciable effects were found for phantom radii larger than 20 cm and phantom heights above 30 cm. The maximum variations of DAPR 20,10 above these phantom size limits were lower than 0.15% and, therefore, within the statistical uncertainties.

Finally effects of electron spot size and energy of the initial electron beam on DAPR 20,10 calculation were evaluated by modifying those simulation parameters for the 6 MV beam. The FWHM of the electron spatial distribution was varied from 1.0 mm to 2.5 mm and the corresponding variations in DAPR 20,10 values were smaller than 0.3% even for the smallest beam simulated (square beam with 0.5 cm side). The initial electron beam energy was varied in the range from 5 MeV to 7 MeV for the 6 MV beam with diameter of 1.25 cm. As illustrated in Fig. 3, the calculated DAPR 20,10 values show high correlation with the initial electron beam energy whatever the scoring region radius in the range 1.5-5 cm.

Table 5 shows DAPR 20,10 values for the GE Saturne 43 accelerator calculated in the SDD setup with EGSnrc and PENELOPE using a scoring region radius of 1.5 cm. Except for the value corresponding to the beam diameter of 1 cm, results from both codes agree within one statistical standard deviation. Results are also rather close for the different beam diameters with very similar energy spectra. On the other hand, DAPR 20,10 values referring to the GE Saturne 43 accelerator are larger than those calculated for the Varian DHX accelerator (6 MV, SDD = 100 cm, R = 1.5 cm) by about 1.7%. This could be partially ascribed to the different mean energies of the simulated beams, as mentioned in Section 3.1.1.

Experimental results

All the tested ionization chambers showed signal drifts in water in a 60 Co beam. The drift was of 0.06% per hour for the SV-PMMA, 0.1% per hour for the BP_IC, 0.2% per hour for the T_IC and 0.4% per hour for the PTW 34073. Given the large drift, the PTW 34073 chamber was rejected for this study. The drift for the SV-PMMA ionization chamber was attributed to deformations of the PMMA waterproof box which is in contact with the front graphite electrode.

In the Varian DHX beams, the repeatability of the ionization chamber signal was better than 0.1% and the short-term reproducibility, evaluated during a measurement session, was 0.3%. The signal variability during a measurement session was mostly ascribed to the reproducibility of the beam profile, since the uncertainty component due to the chamber positioning at the measurement depth was 0.05% (positioning uncertainty was 0.1 mm). The saturation correction factor determined at various depths in water ranged from 1.001 to 1.003 for the BP_IC chamber and was around 1.001 for the TC_IC chamber, with no dependence on depth or field size. The polarity correction factor for both chambers ranged from 1.001 to 1.002. Fig. 4 shows the normalized depth ionizing curves measured using the BP_IC chamber in the Varian 6 MV beams with reference and stereotactic field sizes. Differences along the curves referring to stereotactic beams were generally within ± 1%. Similar results were obtained for the 10 MV beams. Differences in M M (20)/ (10) values obtained with SSD values of 80 cm, 90 cm and 100 cm were well within 0.5% for both photon energies with no evidence of SSD dependence. However, M M (20)/ [START_REF] O'brien | Small field detector correction factors for silicon-diode and diamond detectors with circular 6 MV fields derived using both empirical and numerical methods[END_REF] values obtained in the SSD setups using the BP_IC chamber were systematically lower than those obtained by the T_IC chamber. Differences were typically around 1.5%.M M (20)/ (10) measurements were also made in the SDD setup (SDD = 100 cm) and results are shown in table 6.D i fferences between M M (20)/ (10) values obtained by the BP_IC and TC_IC chambers were typically around 1%. These differences, as well as those observed for the SSD setup, can be ascribed to the different collecting electrode diameters of the two ionization chambers, in agreement with the Monte Carlo calculations which showed a dependence of DAPR 20,10 on the radius of the dose scoring region (for the SDD setup, Monte Carlo differences were in the range 0.9% to 1.2% using R = 4.08 cm and 4.825 cm).

Data in table 6 also show a tendency of M M (20)/ (10) ratio to in- crease when the field diameter decreases. At 6 MV, the values referring to 2 cm and 1.25 cm beam diameters differ from each other by 1.3% and 1.6% for the BP_IC and the T_IC chambers, respectively. The corresponding differences for the 10 MV beams are 0.7% and 1.2%. The Monte Carlo calculated differences were 1.0% and 1.2% for 6 MV, and 0.3% and 0.4% for 10 MV.

For the GE Saturne 43 accelerator, M M (20)/ (10) ratios measured in the SDD setup for three beam diameters (2 cm, 1 cm and 0.75 cm) with the SV-PMMA and the BP_IC chambers are shown in table 7.Differences among the measured ratios were 0.43% for the 3 cm sensitive diameter and 1.4% for the 8.16 cm sensitive diameter. The differences between the M M (20)/ (10) ratios measured by the two ionization chambers were larger (around 4%) than data shown in table 6 as the size difference of the collecting surfaces is larger. The results obtained with the BP_IC chamber are consistent between the two accelerators, the M M (20)/ (10) difference being 0.36% for the 6 MV beams with 2 cm diameter.

In Fig. 5 the M M (20)/ (10) ratios measured using the SV-PMMA chamber are compared to the Monte Carlo DAPR 20,10 values calculated with a scoring region of 1.5 cm radius. While PENELOPE and EGSnrc calculations gave comparable results, the differences between the calculated and the corresponding measured DAPR 20,10 were around 0.5% and the calculated values were systematically smaller than the measured ones. The 0.5% difference is consistent with the type B uncertainty (0.55%, k = 1) associated to the ratio of DAP values calculated by Monte Carlo using the same integration area (see details in Appendix A).

Discussion

Reference integration area

In this work, the properties of DAPR 20,10 in water were studied both by Monte Carlo simulations and by measurements using large-area ionization chambers in 6 MV and 10 MV photon beams with field sizes below 2 cm. Using the Monte Carlo phase-space files describing stereotactic photon beams, DAPR 20,10 was calculated for different values of the integration area (always larger than the beam size), using both SSD and SDD setups. Calculation results showed a dependence of DAPR 20,10 on both the setup and the size of the integration area. Such dependences were experimentally confirmed by comparing ratios of ionization signals at 20 cm and 10 cm water depths obtained by means of LACs with different collecting electrode diameters. Taken together, Fig. 4. Normalized depth ionization curves (SSD 100 cm) measured with a PTW 34070 ionization chamber with collecting electrode diameter of 8.16 cm at 6 MV for the reference field size (10 cm × 10 cm) and three stereotactic beams with diameter 2.00 cm, 1.50 cm and 1.25 cm at the phantom surface.

Table 6

Ratios of ionization chamber readings at 20 cm and 10 cm depths in water obtained at a fixed source to detector distance of 100 cm (SDD) for 6 MV and 10 MV stereotactic beams using a transmission chamber (T_IC) and a PTW 34070 ionization chamber (BP_IC) with collecting electrode diameter of 9.65 cm and 8.16 cm, respectively. The combined relative standard uncertainty of M( 20 these results indicate that a reference setup and a reference integration area must be defined to obtain a DAPR 20,10 parameter useful as beam quality specifier. Then, to allow measurement of such beam quality specifier by means of a generic LAC, correction factors are required to convert the M M (20)/ ( 10) measured values to the values 'as they would be' if they were measured, under the same experimental conditions, by a chamber with collecting electrode area equal to the reference area. Different computational or experimental methods can be used for determining such correction factors. In this work, a method based on Monte Carlo simulations was applied. Accordingly, for a given ionization chamber with collecting electrode radius R IC , the correction factor for the small field size fs is defined as

= [ () () ]
where R ref is the radius of the reference integration area. Then the value of DAPR 20,10 to be used as beam quality index is obtained as

= ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ DAPR M M k (20) (10) R R fs 20,10 R R, R fs IC IC ref ref IC (4)
Using the DAP values calculated by Monte Carlo at 20 cm and 10 cm depths as a function of the radius of the integration area, k R, R fs ref IC correction factors can be determined for any ionization chamber radius. It should be noted that Eq. ( 4) is derived under the assumption that the ionization chamber response in terms of DAP does not change with water depth (i.e. ). Although results in section 3.1.1 showed that the s w,air variation with depth is marginal, changes in the chamber response due to perturbation effects cannot be excluded in principle. The agreement between the calculated DAPR 20,10 and the measured M M (20)/ (10) ratios shown in Fig. 5 supports the above assumption for the SV-PMMA chamber. The assumption was further verified for the BP_IC chamber by Monte Carlo simulation. The ionization chamber was modelled according to the manufacturer's drawings and the chamber response in terms of DAP was calculated as

= rd DAP d DAP d () () () air IC w , ( 5
)
where d is the water depth, DAP air,IC (d) is the average absorbed dose in the air cavity integrated over the ionization chamber (IC) active area and DAP w (d) is the integral of D w , in homogeneous water, over the cross-sectional area of a water voxel equal to the IC active volume. Circular beams with diameter 2.0 cm and 1.25 cm and square beams with side 1.0 cm and 0.5 cm were considered. Calculation results showed variations below 0.2% in r(d) with depth, thereby supporting the validity of Eq. ( 4).

Considering that the absolute measurement of DAP at LNE-LNHB refers to an integration area with radius 1.5 cm [START_REF] Dufreneix | Using a dose-area product for absolute measurements in small fields: a feasibility study[END_REF][START_REF] Dufreneix | Construction of a large graphite calorimeter for measurements in small fields used in radiotherapy[END_REF] R ref values obtained by the two ionization chambers do not exceed 0.4%, while differences between the measured M M (20)/ (10) values were up to 1.3% (see data in Table 8 and Table 6).

The ratio between the M M (20)/ ( 10) measured with a BP_IC chamber and the SV-PMMA chamber (whose area corresponds to the area chosen as reference) is equal to 1.041 for the Saturne 43 accelerator (data in table 7 -6 MV, 2 cm field diameter). The equivalent calculated ratio for the Varian DHX beam, obtained from data in tables 6 and 8, is very close (1.043) indicating that the correction factor is nearly independent of those linac types and those collimating systems. [START_REF] O'brien | Small field detector correction factors for silicon-diode and diamond detectors with circular 6 MV fields derived using both empirical and numerical methods[END_REF] for a scoring region corresponding to a disc of 3 cm diameter. The uncertainties correspond to one standard deviation including the statistical and estimated (0.55%) uncertainties. therefore set to 0.8% (k = 1). Concerning dosimetric data, results in section 3.1.1 showed that changes in photon spectra among small fields have negligible effects (around 0.2%) on s w,air value. Moreover, simulations of the BP_IC chamber response in terms of DAP revealed that effects due to the field size are within 0.5%. On the basis of these results, for a given nominal energy, the same beam quality index can be associated to the small beams, for any field diameter from 2 cm to 0.75 cm.

The sensitivity of DAPR 20,10 to the beam energy was demonstrated by Monte Carlo results in section 3.1.2 (data in Fig. 3). Additionally, data in table 8 indicate a change of about 10% in the experimental DAPR 20,10 R ref value from 6 MV to 10 MV beams. Such a change is similar to that obtained for the beam quality specifiers currently used for radiotherapy photon beams (i.e. TPR 20,10 defined for the 10 cm × 10 cm reference field). This is also shown in Fig. 6 where the s w,air calculated for the beams used in this work are reported as a function of both TPR 20,10 and DAPR 20,10 R ref . Thus, given a reference integration area, DAPR 20,10 determined in SDD setup appears suitable for specifying the photon beam quality for fields with diameter from 2 cm to 0.75 cm, since it combines field size independence with good sensitivity to beam energy. Further investigation would be desirable to extend these results to even smaller beams.

Conclusions

Results of this work show that a ratio of DAP that is useful as beam quality specifier for small beams is obtained for the reference condition that combines the SDD setup (SDD = 100 cm) and the integration surface with radius R ref = 1.5 cm (DAPR 20,10 R ref ). In beams with diameter below 2 cm, DAPR 20,10 R ref is independent of field size and its sensitivity to the beam energy is similar to that of currently used beam quality specifiers defined for the 10 cm × 10 cm field size (i.e. TPR 20,10 ). Measuring DAPR 20,10 R ref in small beams can be achieved using large-area plane-parallel ionization chambers with cross-sectional area larger than the beam size. If the air cavity radius differs from R ref , Monte Carlo correction factors accounting for effects due to the chamber size are required. Specific correction factors should be calculated for any nominal beam energy and collimator system. Since Monte Carlo calculations could be problematic at clinical level, an investigation on alternative practical methods for determining the above correction factors is worthwhile.

were used [START_REF] Dufreneix | Construction of a large graphite calorimeter for measurements in small fields used in radiotherapy[END_REF][START_REF] Daures | Small section graphite calorimeter (GR-10) at LNE-LNHB for measurements in small beams for IMRT[END_REF]. The GR11 calorimeter was designed for absolute measurements of DAP in field sizes smaller than 2 cm [START_REF] Dufreneix | Construction of a large graphite calorimeter for measurements in small fields used in radiotherapy[END_REF] and the size of the calorimeter core is the result of a compromise between two conflicting options: enlarging it to cover the greatest possible part of the scattered beam and reducing it to increase the measured temperature elevation in the core. On the other hand, the size of the GR10 calorimeter ensures that the core is widely within the field size for the largest circular beam (2 cm diameter) considered in this work. The GR10 and GR11 calorimeters were modelled and irradiations in a graphite phantom were simulated with both PENELOPE and EGSnrc codes.

For the beam with 2 cm diameter, the mean absorbed dose in the core (D core ) was calculated for both the GR10 and GR11 calorimeters and the ratio D core (GR10)/D core (GR11) obtained by means of PENELOPE and EGSnrc codes are compared to the measured value in Fig. A1.Ad i fference of 0.9% was found between the value calculated using PENELOPE and the measured value with a corresponding type A uncertainty (k = 1) of 0.2%. The corresponding difference was of 1.4% for EGSnrc calculations with a corresponding type A uncertainty (k = 1) of 0.15%. To explain such large differences between calculation and measurement, additional simulations were done to calculate, in a water phantom, the mean absorbed dose in water in a volume υ around the reference point corresponding to the calorimeter core volumes (D w (υ ≡ core)). The D w (υ)/D core ratios pertaining to the GR11 and GR10 calorimeter cores calculated by PENELOPE and EGSnrc are compared in A2 for the beam with diameter of 2 cm. Results obtained by means of the two Monte Carlo codes are in agreement (differences of 0.1% and 0.2% with type A uncertainties (k = 1) of 0.25% and 0.1%, respectively) for the ratios D w (υ ≡ GR10)/D core (GR10) and D w (υ ≡ GR11)/D core (GR11), that is when the scoring region is the same in both water and graphite. For D w (υ ≡ GR10)/D core (GR11) or D w (υ ≡ GR11)/D core (GR10), the differences are much larger, i.e. 0.4% and 0.8% with type A uncertainties (k = 1) of 0.3% and 0.2% respectively.

The above results indicate that Monte Carlo calculation of absorbed dose is more critical outside the beam field (out-scattered part) than inside and that differences between results obtained using different Monte Carlo codes can be significant. However, when the volume of water (υ) for the mean absorbed dose to water corresponds to the volume of the calorimeter core, the two codes are in agreement when calculating D w (υ ≡ core)/ D core . In this case, the increase of correlation in the calculation of the ratios seems to reduce the problem scale, although the related type B uncertainty should be enlarged (0.55%, k = 1), compared to the value of 0.2% that is typically adopted when the irradiated surface is well inside the beam. When the volumes involved in the dose ratio are really different (as in the case of the GR10 and GR11 calorimeter core volumes), a type B uncertainty of 0.8% (k = 1) should be used. This figure reflects the 1.4% maximum difference observed between measured and calculated dose ratios in terms of standard uncertainty of a rectangular distribution.

Fig. 1 .

 1 Fig. 1. Sketch of the Varian DHX accelerator model used for the simulation of stereotactic photon beams shaped by Radionics cones used as tertiary collimators: 1-target, 2static collimators, 3-vacuum window, 4flattening filter, 5monitor chamber, 6-jaws, 7-stereotactic cone holder (Aluminum), 8-stereotactic Cerrobend cone. Features of the stereotactic collimators (holder and cones) are shown by the picture on the right.

Fig. 3 .

 3 Fig. 3. Calculated DAPR 20,10 with scoring region radii (R) of 1.5 cm, 3 cm, 4 cm and 5 cm vs the initial electron beam energy used for the Monte Carlo simulation of the Varian 6 MV beam with diameter of 1.25 cm.

  )/M(10) is 0.4%. Field diameter M(20)/M(10), SDD =

  the same area is tentatively assumed in this work as reference for the DAPR 20,10 R ref measurement. Accordingly, k R, R fs ref IC correction factors with R ref = 1.5 cm were determined for the BP_IC and T_IC ionization chambers and applied to the ratios M M (20)/ (10) measured in the Varian DHX beams with SDD 100 cm. Differences between DAPR 20,10

Fig. 5 .

 5 Fig.5. Ratios of calculated DAPR 20,10 to measured M(20)/ M[START_REF] O'brien | Small field detector correction factors for silicon-diode and diamond detectors with circular 6 MV fields derived using both empirical and numerical methods[END_REF] for a scoring region corresponding to a disc of 3 cm diameter. The uncertainties correspond to one standard deviation including the statistical and estimated (0.55%) uncertainties.

Fig. A1 .

 A1 Fig. A1. Left hand side, representation of the core diameters for the GR11 and GR10 calorimeters (yellow circles) in a 2 cm diameter beam (measured with EBT3 films) and right hand side, D core (GR10)/D core (GR11) ratios measured and calculated with EGSnrc and PENELOPE (error bars are type A uncertainties at one standard deviation).

Fig. A2 .

 A2 Fig. A2. (D w (υ)/D core ) PENELOPE /(D w (υ)/D core ) EGSnrc for cores and volumes of water υ corresponding to the GR10 and GR11 calorimeter cores (error bars are type A uncertainties at one standard deviation).

Table 1

 1 Settings of radiation transport parameters for EGSnrc simulations.

	Parameter	Setting value
	PCUT (keV)	10
	ECUT (keV)	700 (BEAMnrc simulations)
		521 (DOSRZnrc, DOSXYZnrc, SPRRZnrc
		simulations)
	Electron step algorithm	PRESTA-II
	Boundary crossing algorithm	EXACT
	Photon cross sections	XCOM

Table 2

 2 PENELOPE simulation parameters in the accelerator head and in water.

	Parameter	Setting value
	C1/C2	0.05/0.05
	Cutoff energies (keV): electrons /	500/10/500 (accelerator head)
	photons / positrons	50/10/50 (water phantom)
	Wcc/Wcr (keV)	10/10 (accelerator head)
		5/5 (water phantom)
	DSMAX	1/20th of the smallest dimension for
		the target
		1/10th of the thickness for other
		elements
	Oriented splitting	Splitting number 100
		Cone half-angle 4°C
	ylindrical symmetry splitting	Splitting number 20
	Phantom splitting	Splitting number 20

  Monte Carlo calculated DAPR 20,10 with SDD 100 cm as a function of the scoring region radius, R, for the 6 MV and 10 MV stereotactic beams with diameter 1.25 cm produced by the Varian DHX accelerator.

					0.80
						y = 0.0101x + 0.684
					0.75	R² = 0.996
				DAPR 20,10	0.65 0.70	y = 0.0118x + 0.6126 R² = 0.999
					0.60	10 MV -1.25 cm
						6 MV -1.25 cm
					0.55
					01234567
						Scoring region radius (cm)
				Fig. 2.
		.617 (1)	0.621 (1)	0.622 (1)	0.628 (1)	0.629 (1)	0.630 (1)
	4.0	0.646 (1)	0.644 (1)	0.643 (1)	0.654 (1)	0.657 (1)	0.661 (1)
	5.0	0.658 (1)	0.657 (1)	0.658 (1)	0.664 (1)	0.668 (1)	0.672 (1)
				DAPR 20,10 -10 MV-Varian DHX
			SSD setup (SSD = 100 cm)			SDD setup (SDD = 100 cm)
	R (cm)	ϕ = 2.00 cm	ϕ = 1.50 cm	ϕ = 1.25 cm	ϕ = 2.00 cm	ϕ = 1.50 cm	ϕ = 1.25 cm
	1.5	0.682 (2)	0.688 (2)	0.691 (2)	0.697 (2)	0.698 (2)	0.698 (2)
	4.0	0.713 (2)	0.709 (2)	0.708 (2)	0.723 (1)	0.724 (1)	0.725 (2)
	5.0	0.724 (2)	0.722 (2)	0.722 (2)	0.732 (1)	0.734 (1)	0.735 (2)

Table 5

 5 Ratios of DAP at 20 cm and 10 cm in water calculated by PENELOPE and EGSnrc Monte Carlo codes with a scoring region radius (R) of 1.5 cm for GE Saturne 43 6 MV circular beams with field diameter (ϕ) 2 cm, 1 cm and 0.75 cm for the SDD setup (SDD = 100 cm). The last line gives the ratios between the DAP at 20 cm and 10 cm obtained by PENELOPE and EGSnrc. The reported uncertainties are type A standard uncertainties.

		DAPR 20,10 -6MV-GE Saturne 43
		ϕ =2cm	ϕ =1cm	ϕ = 0.75 cm
	PENELOPE	0.639 (1)	0.642 (2)	0.641 (3)
	EGSnrc	0.6385 (2)	0.6392 (4)	0.6404 (2)
	Ratio	1.0009 (14)	1.0038 (24)	1.0011 (39)

Table 7 M

 7

			Field diameter	6 MV-SDD = 100 cm
				SV-PMMA	BP_IC	BP_IC/SV-PMMA
			2.00 cm	0.6414 (6)	0.6676 (16)	1.041
			1.00 cm	0.6433 (4)	0.6706 (16)	1.042
			0.75 cm	0.6442 (6)	0.6767 (19)	1.051
	2.00 cm	0.6652 0.6723 0.990	0.7332 0.7379 0.994
	1.50 cm	0.6719 0.6804 0.988	0.7386 0.7449 0.992
	1.25 cm	0.6741 0.6832 0.987	0.7379 0.7469 0.988

[START_REF] Duane | International Symposium on Standards, Applications and Quality Assurance in Medical Radiation Dosimetry; Application of Dose Area Product and DAP Ratio to Dosimetry in IMRT and Small Field External Beam Radiotherapy[END_REF]

/M

[START_REF] O'brien | Small field detector correction factors for silicon-diode and diamond detectors with circular 6 MV fields derived using both empirical and numerical methods[END_REF]

in water obtained at a fixed source to detector distance of 100 cm for 6 MV produced by the GE Saturne 43 accelerator using a home-made chamber (SV-PMMA) and a PTW 34070 chamber (BP_IC) with sensitive air cavity diameter of 3 cm and 8.16 cm, respectively.

Table 8 DAPR 20

 820 Data in Table8show that the maximum variation in the DAPR 20,10 R ref value for the Varian DHX stereotactic beams is below 0.8% and 1% for 6 MV and 10 MV, respectively. Similarly, data in Table7show that theDAPR 20,10R ref values measured by means of the SV-PMMA chamber in the GE Saturne 43 6 MV beams with diameters 2 cm, 1 cm and 0.75 cm differ from one another by less than 0.5%. Data summarized in Table7and Table8indicate a tendency of the DAPR 20,10 R ref value to increase when the field size decreases, in line with the slight increase of the mean photon energy shown by Monte Carlo simulations. However, the differences observed among field sizes are not statistically significant, particularly for the data in Table8obtained applying k R, R

	R ref values referred to a reference area with diameter 3 cm (R ref = 1.5 cm) obtained ,10 applying Monte Carlo correction factors to the ratio of ionization signals at 20 cm and
	10 cm water depths.
	Field diameter	6 MV			10 MV
		BP_IC	T_IC	BP_IC/T_IC	BP_IC	T_IC	BP_IC/T_IC
	2.00 cm	0.638	0.637	1.001	0.706	0.703	1.004
	1.50 cm	0.643	0.641	1.002	0.712	0.710	1.003
	1.25 cm	0.642	0.642	1.002	0.710	0.711	0.999

Acknowledgments

This work is supported by the EMRP joint research project MetrExtRT which has received funding from the European Union on the basis of Decision No 912/2009/EC. The EMRP is jointly funded by the EMRP participating countries within EURAMET and the European Union.

The computing resources and the related technical support used for this work at ENEA have been provided by CRESCO/ENEAGRID High Performance Computing infrastructure and its staff [46]. CRESCO/ ENEAGRID High Performance Computing infrastructure is funded by ENEA, the Italian National Agency for New Technologies, Energy and Sustainable Economic Development and by Italian and European research programs, see http://www.cresco.enea.it/english for information.

Conflicts of interest

None.

Appendix A. Accuracy in DAPR calculation

The accuracy of Monte Carlo calculation of absorbed dose in regions extending laterally beyond the edge of the radiation was evaluated by comparing the results of simulations with the PENELOPE and EGSnrc codes, as well as via calculations and measurements of the mean absorbed doses in the core of two graphite calorimeters in small 6 MV beams produced by the GE Saturne 43 accelerator. Specifically a graphite calorimeter (GR11) with a core of 30 mm diameter and 3 mm thickness and a graphite calorimeter (GR10) with a core diameter of 6 mm and 6 mm thickness


Jean-Christophe Cexus
email: jean-christophe.cexus@ensta-bretagne.fr

Ali Khenchaf
email: ali.khenchaf@ensta-bretagne.fr

2D Teager-Kaiser Energy Operator for the translation motion compensation in ISAR signatures

Keywords: Teager-Kaiser energy operator, Inverse Synthetic Aperture Radar, Translation motion compensation, Focal quality indicator

In this paper, a new focal quality indicator, the 2D-TKEO indicator, for the translational motion compensation is proposed. This indicator is based on the 2D Teager-Kaiser Energy Operator which represents the 'energy' of an image. In Inverse Synthetic Aperture Radar (ISAR) imaging of a non-cooperative target, the relative motion is unknown and it is therefore necessary to implement an autofocus approach to obtain high resolution imaging. This step is based on an optimization problem via the minimization of a cost function, thus making it possible to estimate the translational kinematic (speed, acceleration, jerk. . . ) of the moving target. This paper describes the use of a new quadratic focal quality indicator to estimate the quality of an ISAR image. To validate our approach is presented on simulation results of a target constituted by ideal point scatterers models. The effectiveness of 2D-TKEO indicator is demonstrated on both noise-free and noisy simulations and results compared to current methods.

I. INTRODUCTION

In the last decades, the use of electromagnetic to image a moving target has been extensively considered in order to identify or to discriminate various objects (cars, aircrafts, ships, and so on). While the recent progress of ISAR imaging has been described in [START_REF] Ozdemir | Inverse Synthetic Aperture Radar imaging with MATLAB algorithms[END_REF], [START_REF] Chen | Inverse Synthetic Aperture Radar Imaging -Principles, Algorithms and Applications[END_REF]. The ISAR image formation revealed its limitation as it is related to the motion of a noncooperative target. Indeed, generally unknown, it is necessary to estimate the relative movement of an object to obtain high quality ISAR images (in other words, to obtain focused ISAR images). This estimation will be all the more necessary if the movements of the target are complicated such as pitch, roll, or yaw motions [START_REF] Chen | Inverse Synthetic Aperture Radar Imaging -Principles, Algorithms and Applications[END_REF], [START_REF] Chen | Time-Frequency Transforms for Radar Imaging and Signal Analysis[END_REF]. The kinematic of a target can be decomposed as translational and rotational motion. While the rotational kinematic of a target is necessary to obtain an ISAR image formation, the phase term associated to the translational kinematic (velocity, acceleration, and jerk) is the principal source of motion error and must be removed to obtain a highly focused ISAR image. The current methods to estimate the kinematics of a target are based on the motion compensation methods or autofocus algorithms. In the literature and various fields (biological microscopy, video, camera, radar. . . ), wide of autofocus algorithms have been proposed to measure the degree of focus [START_REF] Ozdemir | Inverse Synthetic Aperture Radar imaging with MATLAB algorithms[END_REF], [START_REF] Chen | Inverse Synthetic Aperture Radar Imaging -Principles, Algorithms and Applications[END_REF], [START_REF] Groen | A comparison of different focus functions for use in autofocus algorithms[END_REF].

Indeed, the translational motion can be approximated with the Taylor polynomial model in radar imaging and the issue can be resolved as an optimization problem. Several techniques are therefore based on the use of image focus indicators such as contrast approach [START_REF] Berizzi | Focusing of two dimensional ISAR images by contrast maximization[END_REF], entropy approach [START_REF] Xi | Autofocusing of ISAR images based on entropy minimization[END_REF] and many others works. In this context, the main goal is understand how the new focal quality indicator, 2D-TKEO indicator, can be used in translational motion compensation.

We will perform such description as follow: in section II, we will first introduce the ISAR geometry, and describe the formation of the unprocessed ISAR spatial frequency spectrum of a target. We also present the translational motion compensation approach based on an optimization problem related to a definition of a focal quality indicator (or cost function). In the 3 rd section is described a new focal quality indicator based on the 2D Teager-Kaiser Energy Operator. Finally, we present numerical results and comments in section IV before conclusions and perspectives are addressed in the last section.

II. SIGNAL MODEL

Based on the equations of Maxwell, the electromagnetic back-scattering mechanisms can be complicated for complex targets. However, the point-scatterer model can be used to simply characterize the received Radar Cross-Section (RCS) signal of a moving target [START_REF] Ozdemir | Inverse Synthetic Aperture Radar imaging with MATLAB algorithms[END_REF]- [START_REF] Chen | Time-Frequency Transforms for Radar Imaging and Signal Analysis[END_REF]. In this model, the target is modeled as a set of localized point-scatterers, the total number is K (such as equation 2). Furthermore, the target is at the far field of the radar and the distance between the radar and the target is bigger than the dimensions of the object. The movement of the target are described by a translational and rotational motion relative to the radar. Thus, r(t) describes the translational position of the target's center of mass relative to the radar, and θ(t) is the rotational position of the target relative to xy (Fig. 1). The transmitted signal of radar is a stepped-frequency continuous waveform (SFCW) consisting of M bursts; with each burst containing N pulses (Fig. 2). It should be noted that the radar Line-Of-Sight (LOS) is fixed during each burst. The backscattered signal received by the radar receiver can be represented as the sum of each localized point-scatterers of the object as follow:

S(n, m) = e -4iπfn rn,m c H(n, m) + η(n, m) , (1)
where n = 0, ..., N -1, m = 0, ..., M -1, c is the speed of light, and η(n, m) is the additive complex white Gaussian noise of zero mean. Here, the two-dimensional (N by M matrix) complex array, S(n, m), represents the unprocessed (uncompensated) ISAR spatial frequency spectrum of the target. The H(n, m) target echo transfer function is the ideal translational motion compensated ISAR spatial frequency spectrum, and is given by:

H(n, m) = K k=1 ρ k e -4iπ fn c [x k cosθn,m+y k sinθn,m] , (2)
where (x k , y k ) is the position of the k th point-scatterer and ρ k is the backscattered field amplitude of this point. The frequency, f n , of the n th pulse in a m th burst is given by:

f n = f 0 + nδf , (3)
where f 0 is the frequency of the first pulse and δf is the frequency step from pulse to pulse. The equations for translational and rotational movements of the object at a specific time point is illuminated by the (n, m) th pulse and can be defined as:

r n,m = r 0 + v 0 t n,m + 1 2 a 0 t 2 n,m + 1 6 j 0 t 3 n,m , (4)
where r n,m is the instantaneous target range, and (r 0 , v 0 , a 0 , j 0 ) are the initial values of slant-range, velocity, acceleration, and jerk respectively. Similarly:

θ n,m = θ 0 + ω 0 t n,m + 1 2 α 0 t 2 n,m , (5)
where θ n,m is the instantaneous target angular displacement, and (θ 0 , ω 0 , α 0 ) the initial values of the angular displacement, velocity, and acceleration respectively. The sampling time, t n,m , is given by:

t n,m = (n + mN )δt , ( 6
)
where δt is the time interval between adjacent pulses.

In principle, to reconstruct the ISAR image of the target, the conventional algorithms are based on the two-dimensional Fourier Transform (FT):

I(p, q) = F 2D {S(n, m)} , (7)
where F 2D denotes the two-dimensional Fourier Transform of S(n, m), and p = 0, ..., N -1, q = 0, ..., M -1, and |I(p, q)| is a range-Doppler matrix (ISAR image) that represents the reflectivity of the target.

It should be noted that the phase term r n,m in (Eq. 7) with (Eq. 1) and (Eq. 4) contained the quantities (r 0 , v 0 , a 0 , j 0 ). The constant value r 0 has not really impacting on the imaging procedure. It assigns the centering of the target in the image without introducing any defocusing. An estimated value of r 0 can be selected easily to adjust the center of the target. By contrast, v 0 , a 0 , and j 0 give additional components called Doppler-shift error. This motion parameters cause degradation; distortions and blur that alter the appearance of the ISAR image (Fig. 9 and 10).

To reduce the distortions in the ISAR image, the principal objective of the translation motion compensation algorithm is to estimate the translational kinematic quantities (ṽ, ã, j) to obtain an estimate H(n, m; ṽ, ã, j) in (Eq. 2):

H(n, m; ṽ, ã, j) = S(n, m)F (n, m; ṽ, ã, j) ,

where F (n, m) function is obtained by:

F (n, m; ṽ, ã, j) = e + 4iπfn c [ rm,n ṽt n,m + 1 2 ãt 2 n,m + 1 6 jt 3 n,m ] , (9)
with ṽ, ã, and j the estimated motion parameters of a moving target: v 0 , a 0 , and j 0 respectively. After correction of the translation motion, the quality of the ISAR image is drastically improved, and the expression (Eq. 7) can be replaced by:

I(p, q) = F 2D H(n, m; ṽ, ã, j) . ( 10
)
As a result, the ISAR image estimated from (Eq. 10) will be more localized with less blurring. In fact, the precision of the translational kinematic quantities estimation is essential to enhance the quality of ISAR image formation. As such, the next paragraph introduces a classical scheme to estimate the motion parameters (ṽ, ã, j).

Various ISAR autofocus approaches have been proposed [START_REF] Chen | Inverse Synthetic Aperture Radar Imaging -Principles, Algorithms and Applications[END_REF], a few are based on the notion of image focus indicators such as matching pursuit [START_REF] Ozdemir | Inverse Synthetic Aperture Radar imaging with MATLAB algorithms[END_REF], [START_REF] Li | ISAR 2-D imaging of uniformly rotating targets via matching pursuit[END_REF], contrast [START_REF] Berizzi | Focusing of two dimensional ISAR images by contrast maximization[END_REF] and, entropy approaches [START_REF] Xi | Autofocusing of ISAR images based on entropy minimization[END_REF]. These methods suppose that the translational kinematic of the target can be approximated by a polynomial model (see Eq. 4). The estimation of the parameters is then reduced to a relatively simple optimization. In our case, it is equivalent to a problem of minimizing (or maximizing 1 ) a cost function, called focal quality indicator, including the three motion parameters as follows:

(ṽ, ã, j) = arg min

(v,a,j)∈Ω {f cost (v, a, j)} , (11)
with f cost (.) the cost function, Ω = {V, A, J}, and (V, A, J) the search spaces of (ṽ, ã, j) values respectively with V = [ṽ min , ṽmax ], A = [ã min , ãmax ], and J = [ jmin , jmax ].

It should be noted that the variables are discrete. The expression (Eq. 11) therefore describes a discrete optimization problem and can be resolved using an exhaustive search (brute force approach). Of course, there are various optimization techniques to effectively resolve this problem such as algorithms based on Gradient approaches, Nelder-Mead algorithm [START_REF] Nelder | A simplex method for function minimization[END_REF], or Genetic algorithm [START_REF] Martorella | Use of genetic algorithms for contrast and entropy optimization in ISAR autofocusing[END_REF].

III. RADIAL MOTION COMPENSATION USING 2D TEAGER-KAISER ENERGY OPERATOR

In the 1990s, Kaiser and al. proposed to compute the energy of a system using Teager-Kaiser Energy Operator (TKEO) [START_REF] Kaiser | On a simple algorithm to calculate the 'energy' of a signal[END_REF]. Despite its simple definition (Eq. 15), this quadratic operator is used in many field of signal processing such as Time-Frequency analysis [START_REF] Cexus | Nonstationary signals analysis by Teager-Huang Transform (THT)[END_REF] and, demodulation of AM-FM signals [START_REF] Maragos | Energy separation in signal modulations with application to speech analysis[END_REF] . . . Some various generalizations have also been recently proposed such as for complex-valued signals [START_REF] Cexus | Link between cross-Wigner distribution and cross-Teager energy operator[END_REF], [START_REF] Boudraa | Some useful properties of cross-ψ B -energy operator[END_REF], and also for two dimensions image both real-valued [15]- [START_REF] Boudraa | Teager-Kaiser energy bi-level thresholding[END_REF] or complex-valued [START_REF] Cexus | 2D cross-ψ B -energy operator for images analysis[END_REF].

The 2D Teager-Kaiser Energy Operator TKEO, 2D-TKEO, is close to a local mean weighted 2D-Laplacian-filter and is defined as:

Ψ 2R [I] ∇I 2 -I∆I , (12)
where I ≡ I(x, y) is a real-valued 2D image, (x, y) are the spatial coordinates of the pixel. ∇ and ∆=∇ 2 stand for the gradient and the Laplacian respectively:

∇I 2 = ∂I ∂x 2 + ∂I ∂y 2 , ∆I = ∂ 2 I ∂x 2 + ∂ 2 I ∂y 2 (13)
From this definition, it follows directly that:

Ψ 2R [I] = Ψ R [I](x) + Ψ R [I](y) , (14)
1 to maximize fcost(.) is equivalent to minimize the negation of fcost(.)

where

Ψ R [I](x) (or Ψ R [I](y)) is a classical form of Teager- Kaiser Energy Operator of 1D signal f (t): Ψ R [f ](t) ḟ (t) 2 -f (t) f (t) , (15)
Ψ 2R [I] is a sum of each 1D energy component applied along the two directions x and y. By extending few of its properties:

Ψ 2R [I + J] = Ψ 2R [I] + Ψ 2R [J] -J∆I -I∆J + 2∇I.∇J Ψ 2R [IJ] = I 2 Ψ 2R [J] + J 2 Ψ 2R [I] Ψ 2R [cI] = c 2 Ψ 2R [I] Ψ 2R [I + c] = Ψ 2R [I] -c∆I Ψ 2R [c] = 0
where c is a constant image, and I, J real images. The symbol . denotes inner product. The next paragraph introduces a common method to estimate a discrete version of

Ψ 2R [I].
For a discrete image I(p, q) 0≤p≤(N -1), 0≤q≤(M -1) , the expression (Eq. 12) becomes:

Ψ A 2R [I(p, q)] = 2I 2 (p, q) -I(p -1, q)I(p + 1, q) -I(p, q -1)I(p, q + 1) . ( 16
)
This equation (Eq. 16) is obtained by applying the one-sample difference operation along both the vertical and the horizontal directions. Note that this is only one among many suitable techniques used to obtain a discrete version of Ψ 2R [I]. Another extension can be obtained along diagonal directions:

Ψ B 2R [I(p, q)] = 2I 2 (p, q) -I(p -1, q + 1)I(p + 1, q -1) -I(p -1, q -1)I(p + 1, q + 1) . (17)
As a result, the energy of an image I at location (p, q) can be determined by:

Ψ 2R [I(p, q)] = max Ψ A 2R [I(p, q)], Ψ B 2R [I(p, q)] . (18
) This max(.) operator reflects more precisely the local activity of the pixel rather than, for example, the amplitude of the gradient [START_REF] Boudraa | Teager-Kaiser energy bi-level thresholding[END_REF], [START_REF] Cexus | 2D cross-ψ B -energy operator for images analysis[END_REF]. (ṽ, ã, j) = arg min

(v,a,j)∈Ω - N -1 p=0 M -1 q=0 Ψ 2R Ī(p, q; v, a, j) , (19)
where the power normalized ISAR image is defined as:

Ī(p, q; v, a, j) = |I(p, q; v, a, j)| 2 N -1 p=0 M -1 q=0 |I(p, q; v, a, j)| 2 , (20)
and I(p, q; v, a, j) is the ISAR spectrum using the 2D-Fourier Transform of H(n, m; v, a, j) (Eq. 10). Note that this is only one among other possible expressions and another extension can be obtained based on a maximum or even a standard deviation form.

IV. RESULTS

In this section, several simulations are presented in order to evaluate the performances of ISAR imaging after compensation of translation motion based on 2D-TKEO indicator (Eq. [START_REF] Cexus | 2D cross-ψ B -energy operator for images analysis[END_REF]. In all our simulations, the geometry of the airplane is depicted Fig. 3. This synthetic target is the MIG25 dataset described in [START_REF] Chen | Inverse Synthetic Aperture Radar Imaging -Principles, Algorithms and Applications[END_REF] which consists of 120 point scatterers of equal reflectivity. For clarity, the simulation parameters of radar and target are reported table I. The values of N and M are set at 256 to ensure that figures (Fig. 9, 10, and 11) better reflect the potentials of the method. We use an exhaustive optimization procedure over a three dimensional (v, a, j) search space confined to the region V = [-6.02, Figure 4 shows the results of the exhaustive search based on the cost function of 2D-TKEO indicator. The convexity near the center of the search space corresponds to the position of the minimum. Figure 5 shows translational velocity, acceleration, and jerk slices through the 2D-TKEO indicator surface described in equation [START_REF] Cexus | 2D cross-ψ B -energy operator for images analysis[END_REF]. The green vertical curve in these subplots show the true values. Clearly, optimization techniques can be used to find the translation motion parameters that correspond to the global minimum. For comparing the 2D-TKEO indicator approach with matching pursuit, contrast and entropy methods, the simulations have been corrupted by additive complex white Gaussian noise (Eq. 1). To characterize the performances, a total of 500 trials have been used for each SNR value, and associated Root Mean Square Error (RMSE) and Mean Absolute Error (MAE) are computed. RMSE and MAE are defined as:

RM SE = 1 W W w=1 |x w -x w | 2 , (21)
M AE = 1 W W w=1 |x w -x w | , ( 22
)
where W is the number of Monte-Carlo simulation (here W is set as 500), x is the prediction and x is the true value. From figures 6, 7, and 8, the proposed approach performs better than matching pursuit method. When comparing to the two others approaches (contrast and entropy), the new method is nearly equivalent except that the 2D-TKEO indicator computes less outliers values (Fig. 6). From both figures 7 and 8, it should be noted that the computed curves (RMSE and MAE) based on entropy and contrast methods are superimposed.

Although the problematic of the rotational motion compensation is not studied in this article, it is interesting to note that there are also many algorithms such as interpolation methods [2] and, Time-Frequency Transforms [START_REF] Chen | Time-Frequency Transforms for Radar Imaging and Signal Analysis[END_REF], [START_REF] Ahmed | ISAR image formation with a combined Empirical Mode Decomposition and time-frequency representation[END_REF]. Figures 11(a

Fig. 1 .

 1 Fig. 1. ISAR Geometry of a rotating target.

Fig. 2 .

 2 Fig. 2. Representation of SFCW radar signal.

Finally, it

  is possible to define a new focal quality indicator, 2D-TKEO indicator, based on Ψ 2R [.], applied to the ISAR image:

Fig. 3 .

 3 Fig. 3. Geometry of the target.

Fig. 4 .

 4 Fig. 4. Cost function of 2D-TKEO indicator with translational jerk j = 0.05m/s 3 .

Fig. 5 .

 5 Fig. 5. Translational velocity, acceleration, and jerk surface slices.

  (a) Velocity estimation. (b) Acceleration estimation. (c) Jerk estimation.

Fig. 6 .

 6 Fig. 6. Boxplots of the translational kinematic estimations.

Fig. 7 .

 7 Fig. 7. RMSE comparison between various methods for the estimation of v(left), a(middle), and j(right).

Fig. 8 .

 8 Fig. 8. MAE comparison between various methods for the estimation of v(left), a(middle), and j(right).

  ) and11(b) show one frame ISAR image formation using Spectrogram and NSBEMD-TFD algorithm respectively. The NSBEMD-TFD is a new method based on Non uniformly Sampled Bivariate Empirical Mode Decomposition Time-Frequency Distribution[START_REF] Couderc | ISAR imaging based on the Empirical Mode Decomposition Time-Frequency Representation[END_REF]. From these simulation results, we can see that these approaches allow for the compensation of the rotational motion of a target. ISAR using 2D-FT.

Fig. 9 .

 9 Fig. 9. Range profile and ISAR image of uncompensated target.

TABLE I RADAR

 I AND TARGET PARAMETERS. Rotational kinematic [rad, rad/s, rad/s 2 ] [θ 0 , ω 0 , α 0 ] [1.58, 0.057, 1.8e -4 ]

	Parameter name	Symbol	Value
	Number of pulses	N	64 (256)
	Number of Burst	M	64 (256)
	Step frequency	δf	2.13 MHz
	Initial frequency	f 0	0.372 GHz
	Pulse Repetition Interval	δt	1.59 ms
	Translational kinematic [m, m/s, m/s 2 , m/s 3 ]	[r 0 , v 0 , a 0 , j 0 ]	[4400, -4.02, 14.2, 0.05]

V. CONCLUSIONS

In this paper, a new focal quality indicator based on Teager-Kaiser Energy

Operator is used to estimate the translational kinematic of a moving target. The results show the validity of the approach proposed. The efficiency of ISAR imaging after the translational motion compensation is very interesting and is well adapted to non-stationary signals. To confirm the presented results, more simulations must be studied, preferentially using real dataset, and the results compared to other published algorithms.