O Beltramo-Martin
email: olivier.beltramo-martin@lam.fr

C M Correia

S Ragland

L Jolissaint

B Neichel

T Fusco

P L Wizinowich

PRIME: Psf Reconstruction and Identification for Multiple sources characterization Enhancement. Application to Keck NIRC2 imager

In order to enhance accuracy of astrophysical estimates obtained on Adaptive-optics (AO) images, such as photometry and astrometry, we investigate a new concept to constrain the Point Spread Function (PSF) model called PSF Reconstruction and Identification for Multi-sources characterization Enhancement (PRIME), that handles jointly the science image and the AO control loop data. We present in this paper the concept of PRIME and validate it on Keck II telescope NIRC2 images. We show that by calibrating the PSF model over the scientific image, PSF reconstruction achieves 1% and 3 mas of accuracy on respectively the Strehl-ratio and the PSF full width at half maximum. We show on NIRC2 binary images that PRIME is sufficiently robust to noise to retain photometry and astrometry precision below 0.005 mag and 100µas on a mH = 14 mag object. Finally, we also validate that PRIME performs a PSF calibration on the triple system Gl569BAB which provides a separation of 66.73±1.02 and a differential photometry of 0.538±0.048, compared to the reference values obtained with the extracted PSF which are 66.76 mas ± 0.94 and 0.532 mag ± 0.041.

Introduction

Estimation of key science quantities, such as astrometry and photometry are made possible by data-reduction techniques. For sources extraction or modeling, common pipelines are StarFinder ( [START_REF] Diolaiti | StarFinder: A code for stellar field analysis[END_REF]), SeXTRACTOR ( [START_REF] Bertin | SExtractor: Software for source extraction[END_REF]), or DAOPHOT ( [START_REF] Stetson | DAOPHOT -A computer program for crowded-field stellar photometry[END_REF]). Nonetheless, past analysis ( [START_REF] Fritz | What is limiting near-infrared astrometry in the Galactic Centre?[END_REF][START_REF] Yelda | Improving Galactic Center Astrometry by Reducing the Effects of Geometric Distortion[END_REF][START_REF] Schödel | Accurate photometry with adaptive optics in the presence of anisoplanatic effects with a sparsely sampled PSF. The Galactic center as an example of a challenging target for accurate AO photometry[END_REF][START_REF] Sheehy | Constraining the Adaptive Optics Point-Spread Function in Crowded Fields: Measuring Photometric Aperture Corrections[END_REF]) have assessed that 50% of the astrometry and photometry error breakdown in the Galactic center (GC) is induced by a problem of determination of the Point spread function (PSF) model. A more recent study ( [START_REF] Ascenso | PSF reconstruction for AO photometry and astrometry[END_REF]) has illustrated that photometry in the general case of globular clusters observations assisted by Adaptive Optics (AO) can be obtained at better than 1%-level by providing StarFinder with the exact PSF, advocating for pushing the PSF characterization further to reach more accurate science estimates. One may rapidly realize that the source confusion and measurement noise are important limitations to extract the PSF.

An alternative approach to identifying the AO PSF is PSF reconstruction (PSF-R) ( [START_REF] Veran | Estimation of the adaptive optics long-exposure point-spread function using control loop data[END_REF]) which estimates the PSF from real-time AO control loop data, i.e. wavefront sensor (WFS) measurements and deformable mirrors (DM) commands. Such a concept has already been demonstrated on-sky ( [START_REF] Ragland | Status of point spread function determination for Keck adaptive optics[END_REF][START_REF] Martin | Point spread function reconstruction validated using on-sky CANARY data in multiobject adaptive optics mode[END_REF][START_REF] Ragland | Point spread function determination for Keck adaptive optics[END_REF][START_REF] Jolissaint | Adaptive Optics Point Spread Function Reconstruction at W. M. Keck Observatory in Laser Natural Guide Star Modes : Final Developments[END_REF][START_REF] Veran | Estimation of the adaptive optics long-exposure point-spread function using control loop data[END_REF]) and gives the astronomer access to a PSF model without any extraction from the focal-plane, which has the great advantage of achieving a PSF determination insensible to science detector noise, potential source confusion or absence of PSF calibrators within the science field. However, PSF-R lacks operational use so far because it requires a large amount of specific AO skills, especially to capture comprehensively AO performance and behavior on-sky. For instance, in the case of AO-assisted observations led at the W.M. Keck Observatory with a quad-cells WFS, the determination of WFS optical gains plays a major role in the PSF reconstruction and is highly sensitive to seeing conditions and may vary unexpectedly ( [START_REF] Ragland | Point spread function determination for Keck adaptive optics[END_REF]). Moreover, the AO telemetry permits to reconstruct the PSF in the Guide Star (GS) direction only. To extrapolate the PSF across the Field of View (FOV), one must model the spatial variations due to anisoplanatism ( [START_REF] Beltramo-Martin | Offaxis point spread function characterization in laser guide star adaptive optics systems[END_REF][START_REF] Fried | Anisoplanatism in adaptive optics[END_REF]), whose model relies on atmospheric parameters such as the C 2 n (h). For single-conjugated AO (SCAO) systems that rely on a single Natural Guide Star (NGS) or a Laser Guide Star (LGS) plus a NGS to measure tip-tilt modes, the anisoplanatism model can not be calibrated from AO control loop data, which necessitates to obtain the C 2 n (h) from external instruments ( [START_REF] Osborn | Optical turbulence profiling with Stereo-SCIDAR for VLT and ELT[END_REF][START_REF] Butterley | Determination of the profile of atmospheric optical turbulence strength from SLODAR data[END_REF][START_REF] Wilson | SLODAR: measuring optical turbulence altitude with a Shack-Hartmann wavefront sensor[END_REF]) pointing at different sky directions than the telescope. Discrepancies at a level of 10% up to 20% on the C 2 n (h) estimation compared to AO control loop data-based technique on multiple NGS/LGS systems has been already observed ( [START_REF] Ono | Multi time-step wavefront reconstruction for tomographic adaptive-optics systems[END_REF]), which has an impact of the off-axis PSF model accuracy. On top of that, the PSF is affected by field-dependent static aberrations that must be characterized by using long-exposure phase diversity ( [START_REF] Mugnier | On-Line Long-Exposure Phase Diversity: a Powerful Tool for Sensing Quasi-Static Aberrations of Extreme Adaptive Optics Imaging Systems[END_REF]) or focal-plane sharpening ( [START_REF] Lamb | Exploring the operational effects of phase diversity for the calibration of non-common path errors on NFIRAOS[END_REF]) for instance, which potentially requires a large amount of telescope time ( [START_REF] Sitarski | Modeling instrumental field-dependent aberrations in the nirc2 instrument on the keck ii telescope[END_REF]). Finally, earlier studies have showed discrepancies between the reconstructed PSFs and the on-sky PSFs ( [START_REF] Ragland | Point spread function determination for Keck adaptive optics[END_REF][START_REF] Jolissaint | Adaptive Optics Point Spread Function Reconstruction at W. M. Keck Observatory in Laser Natural Guide Star Modes : Final Developments[END_REF]), that are attributed to the presence of instrumental effects, addressed at the system level ( [START_REF] Ragland | Status of point spread function determination for Keck adaptive optics[END_REF][START_REF] Ragland | A novel technique to measure residual systematic segment piston errors of large aperture optical telescopes[END_REF]) and telemetry issues.

In this paper, we present PRIME, which stands for PSF Reconstruction and Identification for Multiple sources characterization Enhancement, as a supplementary methodology to PSF-R that addresses the discrepancies in the post-processing stage. PRIME aims to calibrating a PSF model, preliminary instantiated from AO control loop data, by adjusting some key atmosphere and system parameters, such as the seeing, the C 2 n (h) and WFS optical gains, over PSF extracted out from the focal-plane images. The calibration process is made from a best-fitting process based on a non-linear leastsquared minimization. When this calibration done, one may extrapolate the PSF model at any science sky directions or wavelengths. PRIME gathers all relevant information to identify the PSF model as accurately as possible. It benefits the PSF-R framework to build a rich an accurate PSF model and tackle mis-identification of key parameters by retrieving them from the focal-plane image. PRIME has however the disadvantage to require the presence of one or several PSFs within the observed field, which suits only to a limited number of science cases. Therefore, our strategy is to deploy PRIME on engineering data and observations of globular clusters to learn how the parameters evolve with respect to observing conditions, such as the seeing or wind-shake for instance, and infer the PSF model without additional focal-plane calibration. In the future, we will also investigate for coupling the PSF-R facility at the Keck telescopes with the AIROPA package ( [START_REF] Ciurlo | Off-axis PSF reconstruction for integral field spectrograph: instrumental aberrations and application to Keck/OSIRIS data[END_REF][START_REF] Witzel | The AIROPA software package: milestones for testing general relativity in the strong gravity regime with AO[END_REF]) to enhance scientific exploitation of the near infra-red imager NIRC2 and the integral field spectrograph OSIRIS ( [START_REF] Larkin | OSIRIS: A diffraction limited integral field spectrograph for Keck[END_REF][START_REF] Mclean | Instrumentation for the Keck Observatory[END_REF]). In Sect. 2, we detail the PRIME framework that relies on the PSF-R model proposed by ([15]). We highlight the key model parameters and validate the approach performance in simulation and analyze its sensibility with respect to the star magnitude. We present a model validation over KECK II telescopes NIRC2 images in Sect. 3. We validate that the approach delivers PSF-based estimates, such as Strehl-ratio (SR) and Full Width at Half Maximum (FWHM), at 1%-level and compare the model outputs to external references. Sect. 4 illustrates an application of PRIME on tight-binaries characterization to asses accuracy and precision of astrometry and photometry.

PSF calibration with PRIME

PSF direct model

Classical PSF-R allows to model the system Optical transfer function (OTF) in closed loop from AO control loop data. The robustness and efficiency of this method is well established in the literature ( [START_REF] Ragland | Status of point spread function determination for Keck adaptive optics[END_REF][START_REF] Martin | Point spread function reconstruction validated using on-sky CANARY data in multiobject adaptive optics mode[END_REF][START_REF] Jolissaint | Adaptive Optics Point Spread Function Reconstruction at W. M. Keck Observatory in Laser Natural Guide Star Modes : Final Developments[END_REF][START_REF] Gilles | Simulation model based approach for long exposure atmospheric point spread function reconstruction for laser guide star multiconjugate adaptive optics[END_REF][START_REF] Flicker | PSF reconstruction for Keck AO[END_REF][START_REF] Veran | Estimation of the adaptive optics long-exposure point-spread function using control loop data[END_REF]). Our analysis focuses on the Keck SCAO system guided by either a NGS or LGS and relies on the PSF-R concept implemented by [START_REF] Jolissaint | Adaptive Optics Point Spread Function Reconstruction at W. M. Keck Observatory in Laser Natural Guide Star Modes : Final Developments[END_REF]. We have identified a small number of key parameters whose fitting over sky images makes the reconstruction more accurate. We define µ the meta-parameters vector that gathers the model parameters, which are C 2 n (h) the atmospheric profile, g DM and g TT the WFS and Tip-Tilt (TT) optical gains and a z the list of Zernike modes coefficients that allow to include an additional static phase. The model is passed to an iterative non-linear minimization algorithm that adjusts the parameters to match the observed PSF, as detailed in Sect. 2.2. The model expands as follows

OTF ε (ρ/λ, µ) = OTF Stat (ρ/λ).K ⊥ (ρ/λ, r 0 ).K TT (g TT ).K DM (ρ/λ, C 2 n (h), g DM ).K z (ρ/λ, a z ) (1)
where

• OTF Stat includes the pupil model and the residual static aberrations, such as Non common path aberrations (NCPA), telescope-dependent static-aberrations and segmented mirror cophasing errors. We assume in this paper that the model is fed by the calibration of those static terms using dedicated phase diversity ( [START_REF] Mugnier | On-Line Long-Exposure Phase Diversity: a Powerful Tool for Sensing Quasi-Static Aberrations of Extreme Adaptive Optics Imaging Systems[END_REF]) or focal sharpening ( [START_REF] Lamb | Exploring the operational effects of phase diversity for the calibration of non-common path errors on NFIRAOS[END_REF]) methods. For Keck observations, the calibration procedure is detailed in ([25, 33, 29]). If ϕ Stat is the static phase of the electric-field in the pupil during the observation, OTF Stat is given by OTF Stat (ρ/λ) = P P(r)P * (r + ρ). exp (i(ϕ Stat (r) + ϕ Stat (r + ρ))) dr

where P is the pupil function on which the static phase is defined, r the phase sample coordinates inside the pupil, ρ is the separation vector within the pupil and λ the wavelength. Calibration of static terms during the observations we have reduced were done right before the image acquisition.

• K ⊥ is a spatial filter that includes both the fitting and WFS aliasing error by following expressions given in ( [START_REF] Correia | Anti-aliasing Wiener filtering for wave-front reconstruction in the spatial-frequency domain for high-order astronomical adaptive-optics systems[END_REF]). K ⊥ is derived from the atmospheric phase Power spectrum density (PSD) that scales with r -5/3 0 .

• K TT is the TT spatial filter derived from the residual AO jitter. We name w TT (t) the time-dependent residual TT wavefront delivered by the tip-tilt sensor. OTF TT is readily obtained from the tip-tilt phase Structure function (SF)

D TT as K TT (ρ/λ, g TT ) = exp (-0.5 × D TT (ρ, g TT )) , (3)
where D TT (ρ) is derived from w TT based on

D TT (ρ, g TT ) = g 2 TT D 2 × ρ. w TT (t).w TT (t) t t -C TT η .ρ t (4)
where x t is the temporal average of vector x, C TT η the tip-tilt noise covariance matrix and D the telescope diameter. The scalar g TT is a tuning factor of the TT optical gain. Noise variance is obtained by adjusting a model of the AO loop rejection filter on temporal power spectrum density as proposed by [START_REF] Jolissaint | Adaptive Optics Point Spread Function Reconstruction at W. M. Keck Observatory in Laser Natural Guide Star Modes : Final Developments[END_REF].

• K DM corresponds to the TT excluded filter calculated from the AO residual error within the AO control radius. It is calculated from D θ (r, ρ) the residual phase SF in the science direction θ as follows K DM (ρ/λ, C 2 n (h), g DM ) = P P(r)P * (r + ρ) exp -0.5 × D θ (r, ρ, C 2 n (h), g DM ) dr P P(r)P * (r + ρ)dr [START_REF] Butterley | Determination of the profile of atmospheric optical turbulence strength from SLODAR data[END_REF] where D θ expands into two terms

D θ (r, ρ) = D 0 (r, ρ, g DM ) + D ∆ (r, ρ, C 2 n (h)), (6)
with D 0 as the TT excluded residual phase SF in the guide star direction and D ∆ the anisoplanatism SF. A complete methodology to derive D ∆ from the C 2 n (h) is given in ( [START_REF] Beltramo-Martin | Offaxis point spread function characterization in laser guide star adaptive optics systems[END_REF]). D 0 is derived from

D 0 (r, ρ, g DM ) = g 2 DM × F(r). w DM (t).w DM (t) t t -C DM η .F t (r + ρ), (7)
where w DM is the time-dependent residual wavefront in the DM commands space, C DM η is the noise covariance matrix on DM actuators, F the influence function matrix that converts the DM commands into a zonal description of the phase. Again, the scalar g DM is a tuning factor of the WFS optical gain.

• K z permits to incorporate n z additional static modes, presently Zernike modes, into the PSF model K z (ρ/λ, a z ) = P P(r)P * (r + ρ) exp (i(ϕ z (r) + ϕ z (r + ρ)) dr P P(r)P * (r + ρ)dr ,

with

ϕ z (r) = nz i a z .Z i (r). (9)
Including additional Zernike modes, like focus and astigmatism terms, will be particularly useful to compensate for any static terms that are not well identified. With the current formulation, we have the ability to describe an exact PSF for a given wavelength and field position with a minimal number of parameters: 3 for on-axis reconstruction (seeing, optical gains) up to 9 for off-axis reconstruction with a description of the C 2 n (h) profile over 7 layers. In the following, we will limit the number of reconstructed Zernike modes to focus and astigmatism terms only.

Problem fitting

We have deployed an iterative non-linear minimization to best-fit the PSF model over observed images. We present the PRIME architecture in Fig. 1. We remind that µ = [C 2 n (h), g DM , g TT , a z ] is a PSF meta-parameters that PRIME retrieves by minimizing the the following multiple point-sources criterion

ε(µ, p, α) = n psf i=1 I(θ i ) -I(θ i , µ, p i , α i ) + b 2 L2(Ω) , (10)
where I(θ i ) is the observed image at the sky location θ i and I(θ i , µ, p i , α i ) is the corresponding model that depends on PSF parameters µ = [C 2 n (h), g DM , g TT ], stellar photometry p i , sub-pixel astrometry α i around position θ i and b a scalar value to adjust the background level. We use a L 2 norm that integrates the residual over pixels within a square box of field of view given by Ω. We calculate the norm over at least twice the AO correction radius. The image model is then given by the following expression

I(θ i , µ, p i , α i ) = p i × F Ω OTF ε (θ i , µ) × exp -2iπ×αi.ρ/λ (11)
where OTF ε is the model OTF given by Eq. 1 that depends on the meta-parameters µ and is multiplied by the exponential term to introduce sub-pixel shifts. The operator F Ω applies the 2D Fourier transform and any required operations (zeropadding, cropping, interpolation) to get the model PSF at the desired resolution, position and pixel-scale. Practically, we compute the PSF model over n × Ω pixels, with 1 < n < 2 to capture the highest spatial frequency in the box. For a sole PSF, the PSF is centered and having n = 1 is enough, but when having multiple sources in the field, it can be necessary to get to n = 2 if stars are at image corners.

PRIME solves for this criterion iteratively using a Levenberg-Marquardt algorithm which derives at each iteration the PSF model gradient with respect to the adjusted parameters. The minimization process stops when reaching the stop conditions, which can be either a maximal number of iterations, or the variations on adjusted parameters or criterion are lower than a threshold given by the user. In practice, we have found that n max = 300, δ min = 10 -10 and ε min = 10 -10 provides a satisfactory convergence for Keck II telescope NIRC2 images. Eventually, PRIME delivers astrometry and photometry of stellar sources, the post-AO PSF best-fitted model and the corresponding parameters. So far, the method is only available to work over a limited number of stars [START_REF] Mugnier | On-Line Long-Exposure Phase Diversity: a Powerful Tool for Sensing Quasi-Static Aberrations of Extreme Adaptive Optics Imaging Systems[END_REF] for computing time reasons and must be fed with initial stellar astrometry at few pixels accuracy.

Also, Eq. 10 does include a data-based criteria only and no regularization term. To avoid the algorithm convergence to a local minimum, we pass to PRIME initial inputs that are assumed to be close to what expected, such as the MASS/DIMM C 2 n (h) and the optical gain calibration ( [START_REF] Ragland | Point spread function determination for Keck adaptive optics[END_REF]) and we bound accordingly the parameters space. By applying PRIME over more and more archive data, we will multiply the feedback on the system and make its performance more stable and predictable, as well as sampling the probability density function of the key parameters. Consequently, PRIME is going to be updated with respect to the what we will learn on future tests to make it as more robust and efficient as possible.

Image-assisted wavefront error breakdown

The wavefront error budget is calculated from the trace of covariance matrices that feed the PSF model described in Sect. 2.1. Thanks to the PSF calibration achieved by PRIME, the covariance matrix of the residual DM-compensated phase and tip-tilt are naturally scaled properly regarding the PSF through optical gains tuning factors g DM and g TT . This occurs similarly for the DM-fitting and anisoplanatism errors thanks to the atmospheric parameters adjustment process. The use of the sky PSF adds a supplementary constrain on the absolute level of each term that must deliver a overall Strehl-ratio (SR) very close to the image SR. The total wavefront error is given by

σ 2 ε = σ 2 stat + σ 2 fit + σ 2 dm + σ 2 jitter + σ 2 meas + σ 2 ang-aniso + σ 2 foc-aniso + σ 2 tilt-aniso (12)
where

• σ 2 stat refers to the static aberrations error that contains NCPA calibration residual, telescope co-phasing error if calibration available and additional Zernike terms adjusted by PRIME. It is derived from the variance of the static phase map ϕ stat + ϕ z over the pupil.

× μ k =μ k -1 +Δ μ k p k =p k -1 +Δ p k α k =α k -1 +Δ α k Get OTF ε (μ k , θ) Evaluate ε(μ k , p k , α k ) μ , p , α PSF(θ , μ ) ^I(θ , μ , p , α ) Stop conditions • k = n max ? • Δμ k & Δp k & Δα k <δ min ? • ε(μ k ,p k ,α k ) < ε min ? no yes Inputs Outputs w DM w TT Get Δ μ k , Δ p k , Δ α k I(θ) OTF STAT k=k+1 k =0 k >0 μ 0 PRIME Get ^I(θ ,μ k , p k , α k ) k =0 p 0 , α 0 k >0 Figure 1: Block-diagram of PRIME algorithm.
Increments on inputs are derived from image model gradient by the Levenberg-Marquardt algorithm. In the end of the procedure, we get the estimated PSF parameters vector µ, the stellar photometry p and astrometry α, the corresponding PSF and image model.

• σ 2 fit is the fitting error that corresponds to the uncompensated high-spatial frequencies and derived from the adjusted seeing estimation from PRIME by integrating the fitting PSD ( [START_REF] Jolissaint | Synthetic modeling of astronomical closed loop adaptive optics[END_REF]).

• σ 2 dm is the DM bandwidth error that computes from the trace of the matrix g 2 DM × w DM (t).w t DM (t) -C DM η that appears in the calculation of OTF DM in Eq. 5. It gives the AO correction residual excluded from high-spatial frequencies, tip-tilt modes and measurements errors, which mostly corresponds to the servo-lag error.

• σ 2 jitter is the jitter error given by the trace of the matrix g 2 TT × w TT (t).w t TT (t) -C TT η that gather the residual tip-tilt error.

• σ 2 meas corresponds to the measurement error that includes the WFS noise and aliasing filtered by the AO loop. The WFS noise error is estimated from the telemetry ( [START_REF] Jolissaint | Adaptive Optics Point Spread Function Reconstruction at W. M. Keck Observatory in Laser Natural Guide Star Modes : Final Developments[END_REF]) and the aliasing error is given by integrated the aliasing PSD across spatial frequencies ( [START_REF] Jolissaint | Synthetic modeling of astronomical closed loop adaptive optics[END_REF]) with the adjusted seeing value.

• σ 2 ang-aniso , σ 2 foc-aniso and σ 2 tilt-aniso refer respectively to the angular (off-axis operations), focal and tilt (LGS operations) anisoplanatism errors. For the present analysis, we have only considered focal anisoplanatism for LGS cases regarding the on-axis position of the NGS. Errors are derived from the trace of covariance matrices associated to each of these three terms ( [START_REF] Beltramo-Martin | Offaxis point spread function characterization in laser guide star adaptive optics systems[END_REF]) after adjustment of µ with PRIME.

Performance in simulation

We have simulated a AO-compensated H-band PSFs illustrated in Fig. 2 observed by NIRC2 and located at 10 arcsec from the guide star, which corresponds to the maximal separation between two stars observed with the NIRC2 detector in narrow field mode with 9.94 mas/pixel. We have contaminated the PSF with star photon-noise, electronic noise with 38 photo-events of standard-deviation, 0.08 e-/s of dark current and 13.6 mag/arcsec of sky brightness. The simulation includes NCPA residual and anisoplanatism, with a 2-layers atmospheric profile (64% at 180m and 36% at 7km) with r 0 = 16cm (500nm) and L 0 =25m. The simulated profile is defined by compressing with the mean-weight approach ( [START_REF] Robert | Tomographic wavefront error using multi-LGS constellation sensed with Shack-Hartmann wavefront sensors[END_REF]) the median Maunakea profile over two layers, which is sufficient to describe the anisoplanatism at this separation ( [START_REF] Beltramo-Martin | Offaxis point spread function characterization in laser guide star adaptive optics systems[END_REF]). Fig. 2 shows a slight PSF elongation due to anisoplanatism, which is drowned into the noise floor when including detector noise. We have verified that the algorithm converges towards the exact solution (ε = ε min ) when removing electronic and photon noise. Then, we have tuned the star magnitude and contaminated the simulated image with noise before calibrating the four parameters (layers strengths plus WFS and TT optical gains) in order to assess the algorithm robustness to noise. We report in Fig. 3 the accuracy on the adjusted parameters as a function of the calibration star magnitude. Fig. 4 reports the corresponding errors on PSF estimates, such as the SR, FWHM and Fraction of Variance Unexplained (FVU) that gives the overall PRIME PSF residual from the noise-free reference ( [START_REF] Beltramo-Martin | Offaxis point spread function characterization in laser guide star adaptive optics systems[END_REF]). SR is the most affected PSF estimate and reaches 1% of error for a magnitude 13, for which we get 1% on the altitude seeing estimation and 5% on the other parameters. Also, we learn that we can tolerate 5% of error on optical gains to ensure a PSF modeling to better than 1%-level of accuracy.

Finally, the FWHM estimation is more robust to noise compared to the SR evaluation. The main explanation of this is due to the PSF halo fitting that is strongly limited by the presence of noise, which affects the SR evaluation, compared to the brighter PSF core from which the FWHM is estimated. On top of that, the altitude seeing does impact the FWHM as well by introducing more or less anisoplanatism, which permits to enhance its estimation accuracy compared to the ground layer seeing that feeds the calculation of the PSF halo only. Although, this value modifies the PSF core morphology as well, this information is encoded into the telemetry and does not appear directly in the calculation of spatial filter K DM and K TT .

PSF model calibration on NIRC2 at Keck II

PRIME NGS/LGS results

In the following, we present PRIME results applied to the near infra-red imager NIRC2 at Keck II in narrow field mode with 9.94 mas/pixel sampling. We have processed 158 images collected simultaneously with the AO telemetry over three nights, in either NGS (August 1st 2013, March 14th 2017 and March 15th 2017, 80 images) ( [START_REF] Wizinowich | Performance of the W.M. Keck Observatory Natural Guide Star Adaptive Optic Facility: the first year at the telescope[END_REF]) or LGS (March 14th 2017, 78 images) ( [START_REF] Wizinowich | Observatory Laser Guide Star Adaptive Optics System: Overview[END_REF]) mode. We have systematically retrieved the seeing, optical gains, focus and astigmatism terms by using PRIME. In LGS mode, the observations were taken with the NGS on-axis, which creates only focal anisoplanatism. This latter was not strong enough to identify a full C 2 n (h) profile but does impact the SR and has been used in parallel with the PSF halo to constrain the seeing value. The sky image was cropped to 168 pixels to limit the noise propagation through the fitting process, which corresponds to 4 times the AO correction band. We report in Fig. 5 the reconstructed SR and FWHM versus measured values from images in NGS (1.65 µm) and LGS (2.12 µm) modes. PRIME reaches a vary good accuracy on SR and FWHM estimation of respectively 0.23% ±2.1% and -0.75 mas ± 3.5 mas in NGS mode and -0.71% ±0.27% and 3.2 mas ± 1.9 mas in LGS mode Furthermore, we provide in Fig. 6 a visual inspection of sky and PRIME-reconstructed PSFs, that illustrates that on top of PSF metrics, the PSF morphology is well retrieved for various conditions of performance, even with strong PSF elongation due to wind-shake (case n0070). Such results were expected in the sense that we do best-fit the PSF model over the sky image whose morphology must be retrieved accurately. However, such an excellent agreement can only be obtained thanks to a representative and rich enough PSF model, which points out that classical PSF-R performance is only limited by the determination of key parameters.

We have used equation 12 to derive the wavefront-error breakdown based on PRIME outputs. We report the decomposition of the total wavefront error in Fig. 8 in NGS and LGS modes with respect to the AO performance. Absolute values are verified by computing the Marchal SR from the total variance and compare it to the value estimated from the image. Results presented in in Fig. 9 shows an excellent agreement and provide confidence into the error breakdown estimation. Poor AO performance are explained by strong jitter effects that dominates the turbulence in terms of power and therefore residual error. On top of that wind-shake effects participate to boost the PSF residual jitter. To overcome this limitation as much as possible, Keck I AO system is equipped with TRICK, a Near infra-red tip-tilt sensor that is designed to improve the tip-tilt estimation ( [START_REF] Wizinowich | Tip-Tilt Sensing with Keck I Laser Guide Star Adaptive Optics: Sensor Selection and Performance Predictions[END_REF]).

LGS observations also suffered by very large low-order aberrations , i.e. focus and astigmatism, that explains the poor performance contrary to NGS cases for which static terms were dominated by NCPA residual. The Low-Bandwidth Wavefront Sensor (LBWFS) ( [START_REF] Wizinowich | Observatory Laser Guide Star Adaptive Optics System: Overview[END_REF]) is supposed to capture these low-order modes and update the AO loop at a low temporal frame rate to correct for them. However, the LBWFS telemetry is not yet available into the Keck Telemetry Recording System (TRS) and additional investigation must be led to verify if including the LBWFS measurements into Eq. 5 could enhance the PSF reconstruction. Nonetheless, if these modes are effectively perceived by the LBWFS, they must be compensated and not affect the PSF as strongly as we see on sky images, indicating that they are created by another effect, as potentially variations of static aberrations. We also notice that these static errors grow up while the atmospheric fitting error and the focal anisoplanatism decreases, i.e. for lower values of seeing. This observation indicates that static errors can be potentially introduced by variations of the WFS optical gain, which modify WFS reference slopes and may degrade the compensation of NCPA.

We observe a similar anti-correlation between the residual jitter and the seeing. When the wind blows faster, it shakes the telescope and the dome, which impacts the PSF as we saw in Fig. 6 with the case n0070. In LGS mode, the DM bandwidth error grows up despite the diminution of the seeing value, which pinpoints that this augmentation is due to a greater wind speed value. Observations lead to the conclusions that the good seeing conditions can be reached in presence of fast wind, but such good seeing conditions can not be optimally exploited because the telescope wind-shake.

PSF prediction

We have compared PRIME retrieved parameters to reference values to understand in such way we can calibrate the PSF model regarding the observing conditions when no point source is available in the field. We have focused on all the NGS data we have and a sub-sample of 40 LGS cases acquired on the 14th March 2017, in order to learn about the parameters evolution and predict the PSF from the telemetry on a test sample of the 38 remaining LGS data. All the errors bars in the following are obtained from the Levenberg-Marquardt best-fitting procedure that deliver the estimates precision from the PSF model gradients calculated empirically. We report in Fig. 10 the seeing retrieved with PRIME on both NGS and LGS images compared to the DIMM measurements at MaunaKea. It highlights a bias of 0.06" and a large discrepancy of 0.14" rms. We can consequently compensated for this bias in the PSF model to improve the PSF halo reconstruction but the large discrepancy will eventually limit the PSF prediction. Alternative seeing estimation must be therefore considered from the AO telemetry ( [START_REF] Jolissaint | Determination of the optical turbulence parameters from the adaptive optics telemetry: critical analysis and on-sky validation[END_REF][START_REF] Andrade | Estimation of atmospheric turbulence parameters from Shack-Hartmann wavefront sensor measurements[END_REF]). Our next step will be to combine seeing estimation method from AO control loop data ( [START_REF] Jolissaint | Determination of the optical turbulence parameters from the adaptive optics telemetry: critical analysis and on-sky validation[END_REF][START_REF] Andrade | Estimation of atmospheric turbulence parameters from Shack-Hartmann wavefront sensor measurements[END_REF]) and PRIME to constrain more efficiently the seeing determination and enhance the robustness to noise. We report in Fig. 11 the evolution of the retrieved optical gain versus the DIMM seeing. We observe a decreasing according to the following empirical law g DM = 1.03 -0.4243 × s DIMM , where s DIMM is the DIMM seeing. We observe a slighter decreasing for the TT optical gain for which we have g TT = 0.98 -0.037 × s DIMM . Once again, the samples cloud is quite dispersed, but the trend is clear enough to be calibrated and compensate for the reconstruction bias. Finally, we represent in Fig. 12 the three Zernike coefficients estimated by PRIME with respect to the DIMM seeing whose the dependency is modeled linearly with a 4 = -4.7 + 1.68 × s DIMM , a 5 = 48 + 110 × s DIMM and a 6 = 221 + 29.9 × s DIMM Furthermore, we have compared PRIME to AO telemetry-based only PSF-R on the LGS test case with two configurations: either we neglected system parameters variations with respect to the seeing (g DM = g DM = 1, a 4 =a 5 =a 6 = 0), or we use the empirical laws deduced on the learning set. We report in Fig. 13 the SR reconstruction error and the photometric accuracy with respect to the seeing. Photometry is obtained from the residual of the scaling of the reconstructed PSF over the sky image. We report in Tab. 1 the corresponding error statistics in terms of bias and standard-deviation. FWHM results are very similar between the three methods, but slightly better with PRIME and the calibrated PSF-R compared to the classical PSF-R, advocating that the PSF FWHM is already well estimated from the AO telemetry only.

Results highlight that the parameters calibration allows to correct for the bias on estimates with a similar level compared to PRIME and decreases the discrepancy by 50% compared to the uncalibrated PSF-R. If point source are available in the field to calibrate the PSF model using PRIME, we can expect a photometric accuracy at the level of 0.004 mag, that degrades to 0.06 when relying on parameters calibration only with the present status of the system and the PRIME method. Such results are already promising to deploy PRIME for constraining the PSF model in the image post-processing pipeline as we plan to do. On top of that, by deploying PRIME or more and more archive data, we will be able to improve our understanding of the AO system and enhance its performance, make it more stable and more predictable, which will also make the system parameters prediction more robust and reliable.

4 Application of PRIME to tight-binaries

Impact of noise on stellar parameters estimation

We handle in this section images of binary data 1732-0319-1 and 1742-1793-1 acquired with NIRC2 in narrow field band by using the FeII filter (1.6455 µm), with a good Signal-to-noise ratio (S/N) over 5 frames of 50s exposure each. We have measured the stability of the photometry and astrometry measurements by using either PRIME, or predicted PSF-R or a Moffat model. We present in Figs. 14 and 15 a 2D visual comparison of best-fitted models as well as azimuthal profile Thanks to PRIME, we gain by having more degree of freedom to achieve better stellar characterization, but we do potentially increase the noise propagation though the minimization loop. Because the S/N was very good on the present images, we can artificially play on the object digital intensity to change its apparent magnitude regarding NIRC2 photometric specifications and simulate the observation we would have with a different S/N. The goal of our approach is to capture how the metrics stability degrades with respect to the equivalent magnitude and the PSF model. We illustrate in Fig. 16 the photometry and astrometry precision with respect to the artificial object magnitude. This plot shows two distinct regimes below and beyond magnitude 14 mag that respectively correspond to PSF-model and noise limitation regimes. For stars with m H > 14 mag, the noise propagation through the fitting process dominates the estimation precision, while for stars with m H ≤ 14 mag, we clearly see the huge gain obtained thanks to PRIME. In this PSF-model limited regime, for a given object magnitude, we improve the precision by a factor 2 and 1.5 on respectively the photometry and astrometry; or in other words, we get the same precision for objects one to two magnitudes fainter. Also, because PRIME does estimate PSF parameters in addition to photometry and astrometry, we were expecting more sensitivity to noise and see the metrics precision getting worse faster than the classical PSF-model, which does not occur. Finally, we understand that the present implementation of PRIME is efficient when feeding it with image of stars of magnitude m H ≤ 14 mag stars with 50s of exposure time, which is accessible in the Galactic center ( [START_REF] Yelda | Improving Galactic Center Astrometry by Reducing the Effects of Geometric Distortion[END_REF]) and makes it feasible to deploy PRIME on such a science case. In practice, we can provide several pieces of the field to PRIME to make sure we gather enough S/N for the PSF calibration.

Astrometry and photometry accuracy versus PSF model

We have applied PRIME on 82 observations taken with NGS AO on 2017 Mar 15 of the triple system Gl 569 formed by a low-mass binary component Gl 569B located 4.9 arcsec from the primary Gl 569A. The AO loop was closed on Gl 569A and unsaturated images of all three components were obtained in the K cont filter (λ eff = 2.2705µm) over 20 coadds of 0.2s exposure each. Thanks to the presence of the third companion Gl 569A, we have extracted out the PSF from the scientific image and use it to estimate reference values for characterizing the separation and differential photometry of Gl 569B by using the non-linear minimization process implemented into PRIME. In other words, PRIME has solves the criterion given in Eq. 10 by playing on [p 1 , p 2 ] and [α 1 , α 2 ] with OTF ε given by the 2D Fourier transform of the extracted on-axis PSF. Furthermore, we have played the exact same game by providing to PRIME three different PSF models. Firstly, we have predicted the PSF from the AO control loop data and the predictive laws for parameters proposed in Sect. 3.2. Secondly, we have best-fitted a asymmetric Moffat model on the Gl 569A image and use it as our PSF calibration. Finally, we have calibrated the PSF model described in Sect. 2.1 on the Gl 569A image by retrieving seeing, optical gains. For the three situation, we have also convolve the on-axis PSF model to a angular anisoplanatism filter obtained from MASS/DIMM C 2 n (h) values. We report in Fig. 17 results of the fitting process for a particular acquisition compared to the original image of Gl 569B. The figures highlight that the binary observation is very well reproduced when relying on the extracted PSF; regarding the position of Gl 569A to the binary and the imaging wavelength, the anisoplanatism effect is extremely weak and can be neglected, in a way that the PSF extraction does not suffer substantially from any PSF model error. The reconstruction stands very close to the real PSF but miss its asymmetric shape, which is certainly introduced by a particular pattern of co-phasing errors of the primary mirror segments that produces low-order modes aberrations ( [START_REF] Ragland | A novel technique to measure residual systematic segment piston errors of large aperture optical telescopes[END_REF]). PRIME succeeds partially restoring the asymmetric shape while the Moffat model is not accurate enough to mimic the complex PSF structure. We report in Tab. 3 estimation results averaged over the 82 acquisitions which confirm the visual inspection on Fig. 17: PRIME allows to catch up reconstruction errors to achieve stellar parameters retrieval similar to the PSF extraction approach technique. It estimates 66.73" ± 1.02 of separation compared to the reference value of 66.76" ± 0.94 and 0.538 mag ±0.048 of differential photometry compared to 0.532 mag ±0.041 obtained from the exact PSF model. We were expecting the prediction process to mitigate the photometric bias as observed in Fig. 13, but the PSF was perturbed by co-phasing errors, which we can not predict so far, and strong telescope oscillations outside the regime of observing Also, we have played the game of estimating directly the PSF and stellar parameters on the binary image itself by using PRIME. We reached very same results, differences are within the uncertainty bars, although Gl 569B is 3.7 mag fainter than Gl 569A, which suggests that the joint estimation should be more efficient to the sole PSF calibration at equivalent S/N and points out that providing multiple sources PSF to PRIME allows effectively to strengthen the best-fitting process and mitigate the noise contamination.

Conclusion

We have introduced PRIME as a new approach that couples AO telemetry and focal plane images to provide an accurate PSF model across field locations and wavelengths. PRIME has the ability to calibrate the atmospheric and system parameters that the AO telemetry-based PSF model depends on, such as the C 2 n (h) profile and WFS optical gains, using a non-linear least-square minimization routine that allows to mitigate temporal variations and poor-knowledge of model inputs. In addition to providing the PSF model, PRIME yields an assessment of those input variations regarding the observation conditions and allows to scale up the wavefront error breakdown absolute value with respect to both the AO telemetry and the image.

We have deployed PRIME on Keck II on-sky PSF in engineering mode when guiding the system either on a NGS or a LGS on-axis. We have validated that only few parameters must be estimated carefully (seeing, optical gains plus additional focus and astigmatism terms) to reach -0.7% ± 0.27 of accuracy on SR, 3.2 mas ± 1.2 on FWHM and 0.001 ± 0.004 on photometry. Over a sub-sample of data, we have identified empirical laws of these parameters with respect to the DIMM seeing that have been used to predict the PSF from the direct model. We have shown that this calibration permits to increase the absolute photometry accuracy from 0.22 mag ± 0.12 down to -0.03 mag ± 0.06.

We have deployed PRIME over tight-binaries observations and analyzed, on the one hand the robustness to noise, on the other hand the PSF model accuracy. We have shown on NIRC2 binary images that PRIME is sufficiently robust to noise to retain photometry and astrometry precision below 0.005 mag and 100µas on a m H = 14 mag object. Finally, we have also validated that PRIME can perform a PSF calibration on the triple system Gl569BAB which provides a separation of 66.73±1.02 and a differential photometry of 0.538±0.048, compared to the reference values obtained with the extracted PSF which are 66.76 mas ± 0.94 and 0.532 mag ± 0.041. PRIME is capable of calibrating the PSF model in the case of observations of the globular clusters and future application to engineering data will permit to make the PSF prediction more robust and efficient. The next step of this work will consist in coupling existing sources extraction softwares with PRIME to increase the measurements accuracy of astrometry and photometry on crowded fields observations. Moreover, we will deploy the PSF-based AO diagnostic to improve a posteriori the AO running operations. The algorithm is currently implemented in Matlab and will be migrated into Python and made accessible to the community.

Figure 2 :

 2 Figure 2: Simulated NIRC2 PSFs in H-band narrow field mode with 30s of exposure time including electronic and photon noise. The PSF is affected by NCPA residual and is located at 10 arcsec from the guide star. The isoplanatic angle was set to θ 0 = 25 arcsec and r 0 = 16cm. Left: No noise Middle: magnitude 9 Right: magnitude 14.

Figure 3 :

 3 Figure 3: Accuracy on the PSF parameters estimation delivered by PRIME with respect to the calibration star magnitude with 30s of exposure time. Envelopes give the 1 -σ standard-deviation.

Figure 4 :

 4 Figure 4: Accuracy on PSF estimates derived from the PRIME PSF best-fit with respect to the calibration star magnitude. Envelopes give the 1 -σ standard-deviation.

Figure 5 :

 5 Figure 5: Top : Reconstructed SR Bottom : FWHM versus measurements over images obtained through classical PSF-R or PRIME in NGS mode. Values of Fe correspond to the WFS temporal frequency that only affects the PSF-R results only. Dashed line is the y=x line.

Figure 6 :

 6 Figure 6: Visual inspection of PRIME reconstruction results (left) on-sky PSFs (middle) PRIME best-fit (right) reconstruction residual. (1st row) n0004 case in NGS mode in H-band (August 1st 2013) (2nd row) n0089 case, NGS mode, H-band (August 1st 2013) (3rd row) n0070 case, NGS mode, H-band (August 1st 2013) (4th row) n0086 case, LGS modes, K-band (March 14th 2017)

Figure 7 :Figure 8 :

 78 Figure 7: Cuts of sky PSF compared to PRIME best-fit (Top-left) n0004 case in NGS mode H-band (August 1st 2013) (Top-right) n0089 case, NGS mode, H-band (August 1st 2013) (Bottom-left) n0070 case, NGS mode, H-band (August 1st 2013) (Bottom-right) n0086 case, LGS modes, K-band (March 14th 2017)

Figure 9 :

 9 Figure 9: Marchal SR derived from the error breakdown versus the sky SR.

Figure 10 :

 10 Figure 10: Seeing estimates from PRIME with regard to DIMM measurements. Dash line corresponds to y=x and the dotted line is the linear regression showing a bias of 0.06 arcsec.

Figure 11 :

 11 Figure 11: Optical gain estimates with PRIME versus the DIMM seeing over the 40 LGS learning sets.

Figure 12 :

 12 Figure 12: Static terms retrieved with PRIME versus the DIMM seeing over the 40 LGS learning sets.

Figure 13 :

 13 Figure 13: Top Strehl ratio error Bottom photometric error versus the seeing for the 40 LGS test samples. The figure compares the classical PSF-R (optical gains set to one and use of the DIMM seeing), calibrated PSF-R (use of predictive parameters evolution laws) or PRIME (focal-plane best-fit).

Figure 14 :

 14 Figure 14: 1732-03910-1 case. Left panel: Top images are respectively PRIME and Moffat fitting; bottom figures give the fitting residual over sky images. Right panel: Azimuthal average of the binary image averaged over the five acquisition with overplotting of best-fit obtained with PRIME, PSF-R and a PSF Moffat model.

Figure 15 :

 15 Figure 15: 1742-1793-1 case. Left panel: Top images are respectively PRIME and Moffat fitting; bottom figures give the fitting residual over sky images. Right panel: Azimuthal average of the mean binary image over the five acquisition with overplotting of best-fit obtained with PRIME, PSF-R and a PSF Moffat model.

Figure 16 :

 16 Figure 16: Top : Photometry Bottom : astrometry precision on binary object regarding the object magnitude (50s of exposure time) and the PSF model. Results are averaged out over the 5 frames of each of the two observed binary.

Figure 17 :

 17 Figure 17: From left to right 1) H-band NIRC2 image of Gl589B acquired over 20 coadds of 0.2s integration each 2) Best-fit using the the on-axis NGS image 3) Best-fit using the PSF-R model 4) Best-fit using the myopic PRIME estimation 5) Best-fit using the myopic estimation based on a Moffat PSF model.

Figure 18 :

 18 Figure 18: Estimated Top : differential photometry Bottom : binary separation obtained by modeling the PSF with a Moffat, PSF-R and PRIME with respect to the Gl569A-based retrieval. Each sample corresponds to one of the 82 acquisitions. The dash line corresponds to y=x and the dashed-dotted lines refer to linear regression for each model.

Table 1 :

 1 Statistics on SR and photometric error reconstruction using either the classical PSF-R (optical gains set to one and use of the DIMM seeing), calibrated PSF-R (use of predictive parameters evolution laws) or PRIME (focal-plane best-fit) over the LGS test images.

		∆ SR (%)	∆ mag (mag)
		Mean RMS	Mean RMS
	Classical PSF-R	8	3.2	0.22	0.12
	Calibrated PSF-R	-0.8	2.0	-0.03	0.06
	PRIME	0.7	0.3	0.0009 0.004

Table 2 :

 2 Statistics on estimation stability of the binary separation and differential photometry regarding the PSF model. Values are averaged over the two objects. the five frames for the 1732-03910-1 and 1742-1793-1 cases. Present results confirm that PRIME can estimate jointly PSF and stellar parameters efficiently. A single Moffat function is not accurate enough to reproduce the entire PSF shape and the PSF-R direct model inputs were not sufficiently well identified to reach the same residual level than PRIME. We particularly notice that PRIME has played on the seeing value to adjust the PSF wings level.

	FVU (%)	∆α 2 1 + ∆α 2 2 (µas)	∆p 2 1 + ∆p 2 2 (mag)
	Moffat 1.67±0.34	473 ± 18	0.014 ± 0.001
	PSF-R 0.77±0.16	333 ± 63	0.001 ± 0.001
	PRIME 0.49±0.05	263 ± 31	0.008 ± 0.0005
	averaged out over

Table 3 :

 3 Statistics on estimation of the binary separation and differential photometry regarding the PSF model. Error bars are obtained from an average over 82 successive acquisitions. PSF model Separation (mas) m A -m B (mag) test data sets, which can be mitigated by extending the number of test samples. PRIME shows already excellent agreements with the extracted PSF which gives evidence that PSF-R can be coupled efficiently with PSF extraction tools.

	Gl 569A	66.76 +/-0.94	0.532 +/-0.041
	PRIME	66.73 +/-1.02	0.538 +/-0.048
	PSF-R	64.50 +/-1.00	0.628 +/-0.051
	Moffat	68.17 +/-1.35	0.689 +/-0.061
	conditions of the

Acknowledgments

The research leading to these results received the support of the A*MIDEX project (no. ANR-11-IDEX-0001-02) funded by the Investissements d'Avenir French Government program, managed by the French National Research Agency (ANR). This work has received partial funding from the European Unions Horizon 2020 research and innovation programme under grant agreement No 730890. This work was also supported by the Action Spcifique Haute Rsolution Angulaire (ASHRA) of CNRS/INSU co-funded by CNES. This material reflects only the authors views and the Commission is not liable for any use that may be made of the information contained therein. The data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Maunakea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain.


Julien Taïeb

Léonor Benhaim

MD-PhD Pierre Laurent Puig

Karine Le Malicot

Jean François

PhD Flore Geillon

David Tougeron

MD-PhD Sylvain Manfredi

MD-PhD Marion Chauvenet

Valérie Taly

Julien Taïeb Md-Phd

Léonor Benhaim Md-Phd

Karine Le Malicot Master2

Emile Md-Phd

Valerie Taly

PhD Côme Lepage

MD-PhD Thierry André

Julien Pr

Taieb
email: jtaieb75@gmail.com

Decision for adjuvant treatment in stage II colon cancer based on circulating tumor DNA:The CIRCULATE-PRODIGE 70 trial

Keywords: Colon cancer, adjuvant, circulating tumor DNA, methylated biomarker

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Abstract

Background

Adjuvant treatment for stage II colon cancer remains debated. Finding a tool to select patients at risk for disease recurrence may help the clinical decision. Circulating tumor DNA (ctDNA) has been reported recently as a potential predictive marker for disease recurrence. We thus aim to test its ability to better select stage II colon cancer patients for adjuvant therapy.

Methods

This national, phase III trial (NCT 2019-000935-15) conducted in more than 100 centers in France, plans to screen around 2640 patients in order to randomize (2:1; minimization method) 198 ctDNA positive patients. Patients aged 18 to 75 years with ECOG performance status ≤1 with R0 surgical resection of a pT3-T4aN0 colon or high rectum adenocarcinoma will be randomized within 63 days after curative-intent surgery, to adjuvant mFOLFOX6 (oxaliplatin 85 mg/m², leucovorin 400 mg/m², and 5-FU bolus 400 mg/m2 then 5FU Continuous infusion 2.4 g/m²) every two weeks for 12 cycles or observation. Patients will be followed for maximum 7 years. A gain of 17.5% in 3-yr disease free survival (DFS) is expected (42.5% in the experimental arm vs. 25% in the control arm; HR:0.62; α, 5% [two-sided log-rank test]; 1-β, 80%).

Secondary endpoints include 2-yr DFS, overall survival, and toxicity.

The study was opened to inclusion in the end of January 2020.

Rational

With an estimated number of 446,000 new patients per year in Europe and 44,000 new patients registered each year in France, colorectal cancer (CRC) is the second most common cancer in Europe and the third in France. It leads to death around 214,000 patients per year in Europe (2012)accounting for 13% of all cancer deaths [START_REF] Ferlay | Cancer incidence and mortality patterns in Europe: Estimates for 40 countries in 2012[END_REF].

Patients with colon cancer (CC) represent approximately 68% of all CRC patients including 36% of stage II [START_REF] Taieb | Refining adjuvant therapy for non-metastatic colon cancer, new standards and perspectives[END_REF]. Death after recurrence following stage II colon cancer contributes approximately to 16% of the mortality from colon cancer [START_REF] Guyot | Time trends in the treatment and survival of recurrences from colorectal cancer[END_REF]. This represents about 1,600 deaths per year in France-similar to all deaths from melanoma-and 23,000 deaths per year in Europe.

In stage II colon cancer, adjuvant therapy with 5-Fluorouracil and leucovorin-if administered to all patients -results in merely 3% improvement in 5-year survival [START_REF] Group | Adjuvant chemotherapy versus observation in patients with colorectal cancer: a randomised study[END_REF].

Although various histo-prognostic factors may influence the prognosis, the only widely accepted method to drive the treatment decision is the screening for MSI status which predicts a good prognosis and potentially no benefit from adjuvant chemotherapy [START_REF] Sargent | Defective mismatch repair as a predictive marker for lack of efficacy of fluorouracil-based adjuvant therapy in colon cancer[END_REF].

A Cochrane review has demonstrated that although associated with a small benefit in disease free survival (hazard ratio 0.83 [95% CI 0.75-0.91], adjuvant systemic chemotherapy does not result into a significant improvement in overall survival (hazard ratio 0.96 [0.87-1.05]). The authors confirmed the need to define which risk features should be used to select patients with stage II colon cancer for adjuvant therapy [START_REF] Figueredo | Adjuvant therapy for completely resected stage II colon cancer[END_REF]. A fluoropyrimidine based adjuvant therapy is therefore not the standard of care in stage II colon cancer according to the TNCD (French guidelines, http://www.tncd.org) nor ESMO guidelines but can be considered in patients with a higher risk of disease recurrence defined by clinical and pathological factors. In proficient MMR (pMMR) tumors, adjuvant chemotherapy is currently discussed individually for patients with T4-stage, bowel perforation, number of nodes examined < 10 or 12, and less consuensually for patients with vascular, perineural or lymphatic invasion, undifferentiated tumor and bowel obstruction. Six months of oxaliplatinbased treatment is an option as adjuvant chemotherapy for patients with high risk stage II CC especially T4, inadequate nodal harvest and perforation [START_REF] Taieb | Refining adjuvant therapy for non-metastatic colon cancer, new standards and perspectives[END_REF]. Overall, the T4bN0 tumor stage (tumor directly invades or is adherent to other organs or structures) is one of the strongest prognostic factor, associated with reported survivals of 58.4% (5-year relative survival) and 45.7% (5 year overall survival) [START_REF] Tournigand | Adjuvant therapy with fluorouracil and oxaliplatin in stage II and elderly patients (between ages 70 and 75 years) with colon cancer: subgroup analyses of the Multicenter International Study of Oxaliplatin, Fluorouracil, and Leucovorin in the Adjuvant Treatment of Colon Cancer trial[END_REF][START_REF]American Joint Committee on Cancer[END_REF].

Furthermore, beyond prognostic, the value of these factors in predicting the benefit of adjuvant chemotherapy is not proven.

With the increased sensitivity of current next generation sequencing (NGS) techniques, ctDNA can be detected by sequencing tumor specific mutations as i.e. TP53, APC, KRAS or NRAS that are present in the vast majority of patients with colorectal cancer. It can also be detected with simpler techniques such as the detection of 2 methylated markers WIF1 and NPY by digital droplet PCR after bisulfitation of the DNA extracted from plasma accordingly to a method developed and validated in colorectal cancer patients [START_REF] Roperch | Aberrant methylation of NPY, PENK, and WIF1 as a promising marker for blood-based diagnosis of colorectal cancer[END_REF][START_REF] Garrigou | A Study of Hypermethylated Circulating Tumor DNA as a Universal Colorectal Cancer Biomarker[END_REF][START_REF] Garlan | Early Evaluation of Circulating Tumor DNA as Marker of Therapeutic Efficacy in Metastatic Colorectal Cancer Patients (PLACOL Study)[END_REF][START_REF] Bachet | RAS mutation analysis in circulating tumor DNA from patients with metastatic colorectal cancer: the AGEO RASANC prospective multicenter study[END_REF]. Briefly, it has been demonstrated that these markers are present in most tumor samples independently of the disease stage and that the monitoring of methylated ctDNA (Met-ctDNA) using these markers is as efficient as when using patient tumor mutation-specific assays [START_REF] Garrigou | A Study of Hypermethylated Circulating Tumor DNA as a Universal Colorectal Cancer Biomarker[END_REF][START_REF] Garlan | Early Evaluation of Circulating Tumor DNA as Marker of Therapeutic Efficacy in Metastatic Colorectal Cancer Patients (PLACOL Study)[END_REF] and that it could also be used as a control of presence of ctDNA in advanced cancer [START_REF] Bachet | RAS mutation analysis in circulating tumor DNA from patients with metastatic colorectal cancer: the AGEO RASANC prospective multicenter study[END_REF]. Moreover, Met-ctDNA concentration, measured by these markers, was shown to be prognostic of disease progression and patient survival, both in advanced and localized disease. Finally, early decrease of ctDNA concentration bas been demonstrated to be predictive of response to chemotherapy in CC patients in the metastatic setting [START_REF] Garlan | Early Evaluation of Circulating Tumor DNA as Marker of Therapeutic Efficacy in Metastatic Colorectal Cancer Patients (PLACOL Study)[END_REF].

In a recent observational study in stage II colon cancer, the detection of residual circulating tumor DNA (ctDNA) after complete surgical resection was highly predictive for tumor recurrence. The estimated recurrence free survival in ctDNA positive patients not treated with chemotherapy was 22% compared to 90% in ctDNA negative [START_REF] Tie | Circulating tumor DNA analysis detects minimal residual disease and predicts recurrence in patients with stage II colon cancer[END_REF]. Comparable results have been observed for stage III colon cancer patients [START_REF] Reinert | Analysis of Plasma Cell-Free DNA by Ultradeep Sequencing in Patients With Stages I to III Colorectal Cancer[END_REF]. More recently, our group has shown that ctDNA was also highly predictive of recurrence in stage III patients coming from a large phase III prospective trial. We also observed that 3 months of adjuvant FOLFOX seemed insufficient in ctDNA+ patients, compared to 6 months treatment underlining the potential guidance that ctDNA might bring to our therapeutic strategy in the adjuvant setting [START_REF] Taieb | Analysis of circulating tumour DNA (ctDNA) from patients enrolled in the IDEA-FRANCE phase III trial: Prognostic and predictive value for adjuvant treatment duration[END_REF] .

The current CIRCULATE-PRODIGE 70 trial aims to evaluate the efficacy of risk stratification in stage II patient based on the post-operative ctDNA detection and the efficacy of adjuvant therapy in patients with positive detection of ctDNA. The patients with negative ctDNA detection will be randomized to be followed within the trial or not, as following all negative patients within the trial will not be possible with the budget allocated. The generated knowledge on this patients population may enable to tailor the surveillance programs accordingly.

Patients and study design

The CIRCULATE-PRODIGE 70 study is an open-label randomized (2:1) phase III trial conducted in France, sponsored by Dijon Bourgogne University Hospital and coordinated operationally by the FFCD data center, comparing adjuvant mFOLFOX6 (oxaliplatin 85 mg/m², leucovorin 400 mg/m², and 5-FU bolus 400 mg/m 2 then 2.4 g/m² continuous infusion over 46 h) every two weeks for 24 weeks (12 cycles) to observation.

Eligible patients, ≥18 years and < 76 years, ECOG ≤1, curatively R0 resected within 63 days before randomization will have to be positive for ctDNA detection 2 weeks or more after the surgical procedure and before week 8 to allow the randomization in due course.

The ctDNA status will be assessed by digital droplet PCR after bisulfitation of the DNA extracted from the plasma, by the detection of 2 methylated markers (WIF1 and NPY) according to a method developed and validated for colorectal cancer [START_REF] Roperch | Aberrant methylation of NPY, PENK, and WIF1 as a promising marker for blood-based diagnosis of colorectal cancer[END_REF][START_REF] Garrigou | A Study of Hypermethylated Circulating Tumor DNA as a Universal Colorectal Cancer Biomarker[END_REF][START_REF] Garlan | Early Evaluation of Circulating Tumor DNA as Marker of Therapeutic Efficacy in Metastatic Colorectal Cancer Patients (PLACOL Study)[END_REF][START_REF] Bachet | RAS mutation analysis in circulating tumor DNA from patients with metastatic colorectal cancer: the AGEO RASANC prospective multicenter study[END_REF].

Eligibility criteria are shown in table 1.

TABLE 1: main eligibility criteria

Main inclusion criteria

-Colon and high rectum cancer stage II excluding low and medium rectal cancers (tumor location ≥ 12 cm from the anal verge by endoscopy and/or above the peritoneal reflection at surgery), -without gross or microscopic evidence of residual disease after surgery with curative intent Randomization planned up to 8 weeks after curative resection -Age ≥ 18 years and ≤ 75 years -ECOG performance Status (ECOG-PS) <2 -Signed written informed consent obtained prior to any study specific procedures -Available ctDNA analysis -Available preoperative CT-scan within 21 days before surgery

Main non-inclusion criteria

-Known DPD deficiency -T4b tumors -Radio or radiochemotherapy for rectal tumors -Contraindication to chemotherapy (inadequate bone marrow, hepatic, renal functions) -Comorbidity influencing the prognosis of the patients (i.e. active secondary cancer or others) -Participation to another interventional study for postoperative therapy

The randomization is adjusted for the following stratification factors: emergency surgery (yes vs no); T4a stage (yes vs no) and study site.

Eligible ctDNA positive patients will be randomized using a 2:1 ratio ("Chemotherapy" or "Follow-up within the trial"). Eligible ctDNA negative patients will be randomized using a 1:4 ratio ("Follow-up within the trial" or "Follow-up outside the trial"). In total 2640 patients will be screened for inclusion and exclusion criteria. Assuming 25% of patients not filling eligibility criteria, around 2640 patients should be screened for ctDNA assessment, 554 will be followed up within the study and 198 ctDNA+ will be randomized to receive or not adjuvant chemotherapy (Figure 1). This study (clinicaltrials.gov NCT 2019-000935-15) is performed in accordance with the Declaration of Helsinki and Good Clinical Practice Guidelines. A French ethics committee approved the study. All patients have to provide written informed consent before starting the study.

Endpoints and Assessments

The primary objective is DFS rate at 3 years defined as the time from the date of randomization up to the date of first local, regional or distant relapse, second CRC, or death from any cause including treatment-related death. Secondary objectives include DFS at 2 years, time to recurrence (TTR), Overall Survival (OS), and toxicities according to the NCI-CTCAE version 4.03 comparisons between patients assigned to FOLFOX or to observation.

Tumor assessment is recommended to be performed before surgery and every 6 months, with thoraco-abdominal and pelvic CT-scan after surgery for 5 years. The CEA level will be assessed according to the same schedule. After 5 years postsurgery, a thoraco-abdominal and pelvic CT-scan is recommended every year. DFS and OS in ctDNA positive versus ctDNA negative patients, without adjuvant treatment will also be compared.

Finally, the added value of ctDNA on classical prognostic factors for stage II colon cancer will be tested together with the predictive value of clinical/pathological and biological factors for adjuvant treatment efficacy in ctDNA+ stage II colon cancer.

Statistical analysis plan

The study will include patients during 48 months and the follow-up will continue until 36 months after the last randomized patient (for evaluation of DFS at 3 years). Two interim efficacy analyses for futility and efficacy purposes are planned after 33% of events and 60% of events. To reach an overall alpha of 5% , the alpha spend at first stage is 0.00019 and 0.00762 at second stage. We expect a 17.5 % gain for DFS at 3-year in the experimental arm (FOLFOX) compared with the reference arm (observation), 42.5 % versus 25 %, respectively; HR:0.617). In order to significantly conclude with 80 % power and 5 % alpha level in a two-sided log-rank test, we need to obtain at least 154 events and 198 patients have to be analyzed.

Ancillary study: biomarker analysis

All patients enrolled in this trial will be proposed to consent for their participation to a biological ancillary study. Prognostic and predictive biomarkers can be studied both on blood and on tumor-derived samples.

In addition to ctDNA status, this study will allow us to better assess in a large series of patients the prognostication of stage II colon cancer for which little is known compared to stage III.

We aim to define a nomogram using different prognostic factors focused on the characterization of the different tumor molecular phenotype including (but not limited to):

• after DNA extraction from formalin fixed paraffin embedded tumor the Mismatch repair deficiency phenotype (dMMR), the CpG island methylator phenotype (CIMP) and the most frequent mutations that occur in colon cancer. Theses molecular characteristics will be correlated to ctDNA status, time to recurrence, DFS and OS. The univariate and multivariate analyses will be performed in order to determine the best model to predict DFS in this homogenous group of stage II colorectal tumor including or not the ctDNA status.

Conclusion

CIRCULATE is a phase III randomized trial evaluating, in patients with detectable ctDNA after the curative surgical resection of a stage II colon or high rectum cancer, the benefit of mFOLFOX6 adjuvant therapy compared to surveillance alone. We plan to enroll a total of 554 ctDNA patients. The 198 ctDNA positive patients will be randomized (2:1) between mFOLFOX6 or observation and the remaining 356 ctDNApatients will be followed up to improve our knowledge on the prognosis value of ctDNA in stage II colon cancer patients.

The study was opened to inclusion in the end of January 2020. Overall, 100 centers will be opened in France. The end of inclusions is scheduled for December 2024.

•

  The molecular subtype (CMS) by using the nanostring technology after RNA extraction from the same sample tumor • And finally the tumor microenvironment characterized by immunochemistry technics with a focus on Immunoscore™ which will be determined according to HalioDX test plus other scores under development taking into account the stromal reaction observed in the tumor (DGmune).

Acknowledgements

This work was partially supported by the A.R.C.A.D (Aide et Recherche en Cancérologie Digestive) foundation and a national grant from the French national cancer institute (INCa, PHRC 2019).

board member for Merck Serono, Amgen, Boehringer Ingelheim, Biocartis, Roche, Bristol-Myers Squibb and MSD.

CL has served in a consulting/advisory role and or received honoraria for, Amgen, Pierre Fabre, Novartis, AAA.

TA has served in a consulting/advisory role and or received honoraria for, Amgen, Bristol-Myers Squibb, Chugai, Clovis, Halliodx, MSD Oncology, Pierre Fabre, Roche/Ventana, Sanofi, Servier and has received travel, accommodations, and expenses from Roche/Ventana, MSD Oncology, and Bristol-Myers Squibb. LB, This work was partially supported by the A.R.C.A.D (Aide et Recherche en Cancérologie Digestive) foundation and a national grant from the French national cancer institute (INCa, PHRC 2019).

Conflict of interest: JT has received honoraria for speaker or advisory role from Sanofi, Roche, Merck, Amgen, Sirtex, Servier, Lilly, Celgene and MSD.