Ségolène Gaillard

Laurent Roche

Georges Deschênes

Denis E J Morin

Christine Vianey-Saban

Cécile Acquaviva-Bourdain

Patrice Nony

Fabien Subtil

Catherine Mercier

Pierre Cochat

Ségolene Gaillard
email: segolene.gaillard@chu-lyon.fr

Aurélia Bertholet-Thomas

Catherine Cornu

Behrouz Kassai

Collaboration between academics, small pharmaceutical company and patient organizations in the development of a new formulation of cysteamine in nephropathic cystinosis: A successful story

Keywords: Rare diseases, Multidisciplinary approach, Patient advocacy, Nephropathic cystinosis, Clinical research

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Rare diseases are defined as conditions affecting 1 out of 2000 people or less. The currently known rare diseases affect about 30 million people in Europe, a number probably underestimated [1,[START_REF] Eurordis Eofrd | Rare diseases: understanding this public health priority[END_REF].

Rare diseases usually concerned small and disseminated populations. Thus, implementing clinical research is challenging. Design and outcome measures should be adapted to small population and appropriate outcomes should be defined to meet patients 'expectations and competent authorities' requirements [START_REF] Abrahamyan | Alternative designs for clinical trials in rare diseases[END_REF]. Another issue is about patient recruitment which often requires multicenter, multinational trials and a complex logistic approach for the implementation of clinical trials [START_REF] Augustine | Clinical trials in rare disease: challenges and opportunities[END_REF]. In addition, there are few medical experts of one particular disease so not many potential investigators for clinical research. Funding remains also an issue as few budgets are dedicated to the development by pharmaceutical companies and few institutional budgets are available. Also coordination of the budgets should be at a larger scale than national level [START_REF] E-Rare | Rare disease research funding cross-sectional analysis[END_REF][START_REF] Joppi | Orphan drugs, orphan diseases. The first decade of orphan drug legislation in the EU[END_REF].

Collaborations are often needed at different levels: investigators experts of the disease, sponsors, trained clinical research staff and patients themselves [START_REF] Deward | Practical aspects of recruitment and retention in clinical trials of rare genetic diseases: the phenylketonuria (PKU) experience[END_REF][START_REF] Kempf | Challenges of developing and conducting clinical trials in rare disorders[END_REF]. A multidisciplinary approach is also required by the condition affecting the patients so implementation of research may require different categories of physicians or healthcare professionals on one site.

Objectives

The objective of this article is to provide an overview of a successful collaboration in the field of rare diseases: nephropathic cystinosis (NC). We present here the context and the way the development was performed and how studies were implemented in France trying to focus on what was the key of success and what could be improved.

Context

Development of a new formulation of cysteamine in nephropathic cystinosis

Cystinosis is an inherited autosomal recessive disease (Gahl et al [START_REF] Gahl | Cystinosis[END_REF]). It is considered as a very rare disease with a prevalence estimated around 1 case per 100 000 to 200 000 living births in developed nations [START_REF] Broyer | Cystinosis from childhood to adulthood[END_REF]. It has been identified as a generalized lysosomal storage disease linked to a molecular defect of a transport protein, cystinosin [START_REF] Gahl | Cystinosis[END_REF]. The diagnosis of NC is confirmed by increased cystine levels in leukocytes (above 2 nmol half-cystine/mg proteins, normal < 0.2) and genetic analyses [START_REF] Gahl | Cystinosis[END_REF].

Patients usually present during the first six months of their life with a Toni-Debré-Fanconi syndrome leading to a progressive chronic interstitial nephritis and then to end-stage renal disease (ESRD) in the first decade of life [START_REF] Lichtenberg-Geslin | Tubulopathies. EMC Pédiatrie maladies infectieuses[END_REF][START_REF] Vargas-Poussou | Molecular aspects of renal tubule diseases[END_REF]. Since the 1980's, cysteamine therapy has improved renal outcomes by delaying progression of ESRD and other extrarenal complications to the second and third decade of life [START_REF] Broyer | Cystinosis from childhood to adulthood[END_REF][START_REF] Brodin-Sartorius | Cysteamine therapy delays the progression of nephropathic cystinosis in late adolescents and adults[END_REF].

Until the 2010s, the only available treatment was a short acting formulation to be taken 4 times a day, every 6 hours strictly. The main side-effects of this treatment were breath and body odour, nausea, vomiting, drowsiness. Because of these side effects, the daily regimen (4 times a day) and the need for a life-long treatment, the patient's adherence to the treatment was not good [START_REF] Schneider | Treatment of cystinosis: simple in principle, difficult in practice[END_REF]. Levtchenko et al demonstrated that only about 25% (5 of 22) of patients were actually compliant with the 4 times a day regimen [START_REF] Levtchenko | Strict cysteamine dose regimen is required to prevent nocturnal cystine accumulation in cystinosis[END_REF]. Moreover, there is evidence that failure to adhere to this strict 4 times a day regimen results in rapid deterioration of kidney function [START_REF] Kleta | Longterm follow-up of well-treated nephropathic cystinosis patients[END_REF]. Thus, improving patient adherence should ultimately delay the time to kidney failure or the morbidity and mortality associated with renal replacement therapy such as dialysis or transplant, and hence greatly improves both the patient's quality and quantity of life.

In 2010, a small American company (Raptor pharmaceuticals) launched a program to develop a new formulation of cysteamine. The pharmaceutical company developed a delayedrelease formulation of cysteamine as encapsulated, beaded, enteric-coated, bitartrate salt of cysteamine for oral administration.

Seven clinical studies were part of the development plan discussed with the American Food drug Administration (FDA) to allow marketing authorization of this new formulation in the United States (US) and in Europe. The first study was performed on a restricted number of subjects in the US to determine pharmacokinetic (PK) and pharmacodynamic (PD) parameters as well as potential safety and tolerability of the treatment compared to the existing formulation [START_REF] Dohil | Twice-daily cysteamine bitartrate therapy for children with cystinosis[END_REF]. Four bioequivalence studies were performed in fasted and fed healthy volunteers in the US only to demonstrate bioequivalence of open capsules versus whole capsules and interactions of food or Proton Pump Inhibitors with the new formulation of cysteamine [START_REF] Pavloff | The effect of food and liquid pH on the integrity of enteric-coated beads from cysteamine bitartrate delayed-release capsules[END_REF][START_REF] Armas | A Phase 1 pharmacokinetic study of cysteamine bitartrate delayed-release capsules following oral administration with orange juice, water, or omeprazole in cystinosis[END_REF].

Two main studies were performed in NC patients worldwide, including France. The first one was an open-label randomized cross over trial to evaluate the non-inferiority of the delayed release formulation compared to the short acting cysteamine, for the maintenance of leukocyte cystine levels under 1 nmol half-cystine/mg protein. The second one was an open label, single-arm long term safety study [START_REF] Langman | A randomized controlled crossover trial with delayed-release cysteamine bitartrate in nephropathic cystinosis: effectiveness on white blood cell cystine levels and comparison of safety[END_REF][START_REF] Langman | Quality of life is improved and kidney function preserved in patients with nephropathic cystinosis treated for 2 years with delayed-release cysteamine bitartrate[END_REF].

Coordination of the study and collaboration between sponsor and academics

As few specialist physicians were available worldwide, the study needed to be multinational and multicenter. The site of Lyon was selected to be an investigation site among 8 other sites worldwide. A greater collaboration was decided based upon the experience of Lyon in NC and the lack of experience of the company in clinical research performed in Europe, at that time. The clinical investigation center (CIC) of Lyon was selected to coordinate the study in France, with the local physician (Pr Pierre Cochat). Three other sites were implemented in France (2 in Paris and 1 in Montpellier), all having a CIC as supporting research structure. A CIC is a structure which allows physicians and sponsors to perform research projects, providing resources to perform clinical research [START_REF]Clinical investigation centers[END_REF].

All sponsor study activities, except submissions to competent authorities, were delegated to CIC, including submissions to the French central ethic committee, adaptation of the study protocol and documents to local requirements, pharmacovigilance (through the hospital vigilance unit in the University hospital of Lyon), coordination and monitoring of the four French sites, ensuring the study was conducted in conformity with GCP and French law.

The recruitment rate in the first randomized controlled crossover trial was very good: 43 patients were randomized in one year. Under CIC coordination, France enrolled 13 patients i.e. 30% of the patients randomized in the study, in 2 months. Twenty-three (38%) out of 60 patients entered the long term safety study in France.

Discussion

Coordination was performed with CIC network procedures following local regulation and good clinical practice guidelines [START_REF] Chevassus | Good professional practices of French CICs -version # 2[END_REF]. The advantage of working with CIC of Lyon was: (1) its knowledge of the French regulation and experience in coordination; (2) its connections with other French CIC which participated as local investigation centers, helping investigators to perform the study on sites.

A good recruitment was possible through collaboration between the investigators and the clinical investigation centers (CIC) network offering access to research facilities to investigators and sponsors. As a result of these collaborations, the audit from the FDA which was performed during the course of the study resulted in no notes for the audited site in Paris. The CICs make it possible to carry out studies satisfying the quality standards required by the authorities. Also, two publications resulted from this collaboration. Moreover, coordinating these studies in France and collaborating with patients and investigators allowed the CIC team and the coordinating investigator to propose a research to better understand adherence to cysteamine and late complications of the disease: the CrYSTobs clinical research. This study enrolled 17 French patients, most of them already included in the studies on the new formulation of cysteamine. The study visits were proposed at the same timing in order not to add any constraints for the patients. Examinations specifically performed for the CrYSTobs study (adherence measurements, MRI, neuropsychological testing, blood samples, urinary samples…etc…) were proposed when patients were already scheduled at hospital for the main studies. This system allowed the participation of a rather important number of patients in our physiopathological study. Participation would have been more difficult in the CrYSTobs study which did not propose any therapeutic solution to the patients but only explorations to better understand the disease and its complications. Preliminary results of this study, particularly on neurological complications, were presented to patients and their families as well as the community.

Exploring neurological complications and providing new information was expected by the NC population who is currently facing this late complication of the disease (first results submitted).

Involvement of patients and collaboration with an academic coordination centre

The recruitment in the study was greatly facilitated because the expectations of the patients were very high. The twice daily formulation was highly demanded by patients and their families because of the difficulties in adherence and the reality of the progression of the disease. With the previous formulation, patients had to be awakened in the middle of the night to take their medication to maintain adequate white blood cell (WBC) cystine levels. Emerging from the University of California Los Angeles, the sponsor was directly collaborating with the physicians on the new formulation of cysteamine. Even though patients and patients' organizations worldwide were not involved in the design of the project, the recruitment rates in both studies were very good.

As there were few sites per country, patients wishing to participate had to travel sometimes more than 100 kilometers away from home. At the beginning of the cross-over study, patients had to come to hospital 8 times in the first month, staying up to 4 days in a row for some visits. After that, their visits were scheduled every three months until availability of the product in the patients 'country. There are several challenges with this type of organization: (1) Because of the distance, travel advances could be prohibitive for patients. A system had to be found to allow patients 'reimbursements in a timely manner. A research organization independent from the sponsor was chosen by the investigators and the coordination center; (2) Patients and their families had to engage to come every three months for study visits at hospital, on a long term. Psychological considerations had to be taken into account. Some patients did not have a regular follow-up before their enrollment in the study, some had left their standard of care follow-up for several years before volunteering for a research, others had high expectations related to this new formulation: better tolerance, greater quality of life…etc…This has been previously highlighted in the literature as an issue for the retention of patients in a study. In this case, the new formulation did not have a better tolerance profile nor resolved the side effects of cysteamine. Two French patients early-terminated the study however most of them remained with the new formulation until availability in France.

As for many rare diseases, the evaluation of the new formulation marketing authorization application by French health Authorities was considered quite long by the patients. One reason was that the studies were not designed to assess properly the impact of the new formulation on quality of life and patient related outcomes. Complementary analyses of adherence to treatment provided by an ancillary study performed only in French patients helped providing more information to resolve the issue. The patients' organizations also played an important role by pushing further the discussion to express the patient's point of view, the impact on quality of life, insufficiently taken into consideration into the studies [START_REF] Mamzer | How to strengthen the presence of patients in health technology assessments conducted by the health authorities[END_REF].

Keeping patients aware, informed and collaborating with them was the objective of the coordination center. Information on study progress were sent through specific newsletter, social media of the patients 'organizations and through several meetings with patients and patients 'organizations scheduled during the study. During these meetings dedicated to NC, patients were involved by presenting their experience with the new formulation or discussing other aspects of the disease or their participation in the study. Information on study progress and drug availability were given. When available, study results were presented [START_REF] Deward | Practical aspects of recruitment and retention in clinical trials of rare genetic diseases: the phenylketonuria (PKU) experience[END_REF].

Conclusion

This example of collaboration between sponsor, academics and patients was a success with the marketing authorization obtained for the new formulation of cysteamine in 2013 and most of all its availability in France in 2018. This was a long process during and after the studies. It could not have been a success without these collaborations. The availability of the new product in France could have been facilitated with patients' related outcomes such as quality of life, adherence to treatment more clearly defined and assessed in the studies. The patients and public themselves could have been of a great help in the conception of the protocol [START_REF] Mamzer | How to strengthen the presence of patients in health technology assessments conducted by the health authorities[END_REF]. Personally, this successful approach and collaboration contributed to the phD I'm currently doing on the involvement of young persons advisory groups in paediatric clinical research.

Involving patients and public early in the process, particularly in rare diseases as the patient is a great source of knowledge and has his own expectations is needed [START_REF] Gaasterland | The patient's view on rare disease trial design -a qualitative study[END_REF]. Priorities of research, design, conduct and reporting of clinical trials can be defined in collaboration with adults but also with young patients [START_REF] Gaasterland | The patient's view on rare disease trial design -a qualitative study[END_REF][START_REF] Gaillard | Involving children and young people in clinical research through the forum of a European Young Persons' Advisory Group: needs and challenges[END_REF] as they are the first to be concerned by rare diseases. Patients and public, whatever their age, can be involved as early as possible in clinical development: from priorization of research to dissemination of study results and Health authorities' evaluations.

Acknowledgements

We wish to thank the clinical investigation centers which participated to the recruitment and folloup of the patients and for their collaborative work: CIC of Robert Debré (Dr Florentia Kaguelidou, Mr François Luc, ,Mrs Christine Samy, Dr Ying Wang and Mrs Setareh Zarrabian), CIC of Montpellier (Mr Hugues Chevassus, Mrs Mirna Khalil and Mrs Cécile Rouge), CIC of Lyon (Fanny Abad, Béatrice Abel, Najet Belghali, Eleonore Brocard, Catherine Caire, Corine Di-Cara, Caroline Douchet, Hanane Gheit and Mélanie Romier).

We finally thank the patients' organizations (Cystinose France, Vaincre les Maladies Lysosomales and Association pour l'Information et la Recherche sur les maladies Rénales Génétiques) and the patients who agreed on participating to this study and gave their time for this project. Without their will to provide more knowledge in their disease, the researches would not have been possible.

Disclosure of interest

Authors have no competing interest to declare


Pierre-Louis Bourgeois
email: pierre-louis.bourgeois@cea.fr

Xavier Davoine

New mitigation approach to numerical Cherenkov radiation in PIC simulations of wakefield accelerators

Keywords: Plasma, Particle-In-Cell (PIC), Numerical Cherenkov Radiation, Laser WakeField Acceleration (LWFA)

Numerical Cherenkov radiation is a well-known problem of FDTD PIC codes caused by numerical dispersion errors creating spurious radiation around accelerated particles. It is usually dealt with by complex numerical schemes and low-pass filters. In this study we look at a simple way to circumvent this problem for wakefield accelerators simulations by only modifying the field interpolation algorithm. Taking advantage of the highly directional nature of the numerical Cherenkov radiation, we make sure that the resultant force created by the spurious radiation is as close to zero as possible for the accelerated electrons, negating its effects on the accelerated bunch without actually suppressing the numerical radiation. Applying the above to laser wakefield acceleration simulations, we observe both a reduction and a stabilization of the beam normalized emittance during the acceleration process as well as an improved regularity of the oscillating electrons trajectories which might help to further understand radiation processes such as betatron.

Introduction

Particle-in-cell (PIC) codes are known for their versatility and relative speed, they are used to simulate a wide range of phenomena in plasma physics. To solve Maxwell equations, many of them use the finitedifference time-domain (FDTD) method first proposed by K. Yee [START_REF] Yee | Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media[END_REF], a simple second-order method easily parallelizable but suffering from some numerical dispersion error which gives rise to what is known as numerical 5 Cherenkov radiation (NCR). This spurious radiation can couple with the plasma response and lead to the Numerical Cherenkov Instability (NCI). These effects have been studied since the 1970s [START_REF] Godfrey | Numerical cherenkov instabilities in electromagnetic particle codes[END_REF] and many ways to mitigate them have been proposed since [START_REF] Greenwood | On the elimination of numerical cerenkov radiation in pic simulations[END_REF][START_REF] Lehe | Numerical growth of emittance in simulations of laser-wakefield acceleration[END_REF][START_REF] Xu | Numerical instability due to relativistic plasma drift in em-pic simulations[END_REF][START_REF] Godfrey | Suppressing the numerical cherenkov instability in fdtd pic codes[END_REF][START_REF] Lehe | Elimination of numerical cherenkov instability in flowing-plasma particle-in-cell simulations by using galilean coordinates[END_REF][START_REF] Nuter | Suppressing the numerical cherenkov radiation in the yee numerical scheme[END_REF][START_REF] Blinne | A systematic approach to numerical dispersion in maxwell solvers[END_REF][START_REF] Li | Controlling the numerical cerenkov instability in pic simulations using a customized finite difference maxwell solver and a local fft based current correction[END_REF][START_REF] Pukhov | X-dispersionless maxwell solver for plasma-based particle acceleration[END_REF]. Those methods usually rely on complex modification of the computational stencil and/or heavy filtering which can themselves lead to new numerical limitations or reduce the code performance. The effects of NCR are especially important in simulations of wakefield acceleration, one of the most prominent applications of PIC simulations. The wakefield can be created by a beam of charged particles propagating in a plasma, which has given promising results recently [START_REF] Blumenfeld | Energy doubling of 42 gev electrons in a metre-scale plasma wakefield accelerator[END_REF][START_REF] Adli | Acceleration of electrons in the plasma wakefield of a proton bunch[END_REF], or by the propagation of a laser beam. Laser wakefield acceleration (LWFA) in plasmas is not a novel idea as it dates back to the 1970s [START_REF] Tajima | Laser electron accelerator[END_REF] but it has seen renewed interest for the last fifteen years thanks to advancements in experimentations and simulations [START_REF] Faure | A laser-plasma accelerator producing monoenergetic electron beams[END_REF][START_REF] Geddes | High-quality electron beams from a laser wakefield accelerator using plasma-channel guiding[END_REF][START_REF] Mangles | Monoenergetic beams of relativistic electrons from intense laser-plasma interactions[END_REF][START_REF] Kostyukov | Phenomenological theory of laser-plasma interaction in "bubble" regime[END_REF][START_REF] Lu | Nonlinear theory for relativistic plasma wakefields in the blowout regime[END_REF],

and energy up to a few GeV [START_REF] Leemans | Multi-gev electron beams from capillary-discharge-guided subpetawatt laser pulses in the self-trapping regime[END_REF] have been obtained experimentally.

In this paper we present a new and simple way to circumvent the effects of NCR in simulations of wakefield acceleration. In section 2 we describe in more detail the NCR phenomenon and its characteristics in laserwakefield simulations. Section 3 presents briefly existing methods to deal with NCR as well as their limitations and then describes our new method to limit its effects. We finally discuss in section 4 the results obtained with LWFA simulations using this new scheme and show it suppresses the spurious growth of the accelerated beam emittance as well as limiting the impact on the calculated electron motion.

Numerical Cherenkov Radiation

Numerical Cherenkov in PIC simulations

Cherenkov radiation is the result of a charged particle moving faster than the phase velocity v ϕ (ω) of an electromagnetic mode characterised by its angular frequency ω. It may physically occur with relativistic particles when the dispersion relation allows for a phase velocity lower than the speed of light (v ϕ < c) in a medium where the index of refraction is n > 1. The charged particle with a velocity v p will then generate radiation at all frequencies satisfying v p > v ϕ (ω).

For electromagnetic waves propagating in vacuum, the angular frequency ω and the wave number k = k 2

x + k 2 y + k 2 z are proportional:

ω 2 = c 2 k 2 ,
where k x , k y , k z are the wavevector components along the x, y, z directions. It ensues that no Cherenkov radiation should be possible as particles cannot be faster than v ϕ = c. However, in a PIC code using the Yee scheme [START_REF] Yee | Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media[END_REF], the spatial and temporal discretization leads to the following numerical dispersion relation in vacuum:

1 c 2 ∆t 2 sin 2 ω∆t 2 = 1 ∆x 2 sin 2 k x ∆x 2 + 1 ∆y 2 sin 2 k y ∆y 2 + 1 ∆z 2 sin 2 k z ∆z 2 (1)
where ∆t, ∆x, ∆y, ∆z refers to the time step and the cell size in each direction. This allows for some high frequency modes discretized on the grid to have v ϕ < c and thus for the creation of numerical Cherenkov radiation by fast-moving particles (β = v p /c ≈ 1) in the simulation when none should physically occur. In the remainder of this paper, x represents the longitudinal direction along which the propagation occurs and y and z are the two transverse directions. The excited modes satisfy the relation β x c = v ϕ,x = ω/k x with β x the particle normalized velocity along x and v ϕ,x the wave phase velocity along x. Thus with Eq. ( 1) we can characterise those modes by :

1 c 2 ∆t 2 sin 2 k x β x c∆t 2 = 1 ∆x 2 sin 2 k x ∆x 2 + 1 ∆y 2 sin 2 k y ∆y 2 + 1 ∆z 2 sin 2 k z ∆z 2 (2)
Wakefield acceleration is a typical case where numerical Cherenkov radiation can be observed and represents a substantial source of error. Reducing its impact in those simulations is thus especially important to obtain realistic results. In wakefield acceleration simulations, electromagnetic waves -whether physical (laser, beam self-field, wakefield) or numerical in origin (NCR) -propagate along the same direction as the accelerated electron beam. We focus in this paper on finding a solution to mitigate the effects of numerical Cherenkov radiation in the context of laser-wakefield simulations. However, the results could be applicable to other wakefield acceleration methods or even to any scenario where physical and numerical electromagnetic waves propagate at a velocity close to c mainly along the same direction as the particles generating NCR.

Laser wakefield acceleration simulations

Laser-wakefield accelerators are created by focusing an ultra-intense laser beam on a gas jet. The laser beam ionizes the gas, propagates inside the created low density plasma and excites a plasma wave in its wake. If the intensity is high enough, the plasma wave takes the form of an ionic cavity devoid of electron. Electrons injected in this cavity -or bubble -can be accelerated to high energies in a short distance thanks to the high amplitude fields created inside the bubble by the charge separation. an accelerating effect on electrons (red zone where E x < 0) while the front is decelerating (blue zone where

E x > 0)
. Electrons injected at the rear of the bubble thus gain energy while travelling within the bubble until they dephase and reach the decelerating zone. This energy gain can be as high as a few GeV [START_REF] Leemans | Multi-gev electron beams from capillary-discharge-guided subpetawatt laser pulses in the self-trapping regime[END_REF] with current laser technology. While the longitudinal fields are accelerating, the transverse ones are focusing and will thus tend to keep the electron beam on axis and create transverse oscillations of the accelerated electrons.

The numerical simulations presented in this paper have all been made in 3D using the PIC code calder [START_REF] Lefebvre | Electron and photon production from relativistic laser-plasma interactions[END_REF] in a moving window with a 3220 × 200 × 200 grid and ∆x = 0.019 µm, ∆y = ∆z = 0.38 µm and c∆t = 0.993 ∆x.

We consider typical laser-wakefield acceleration parameters: A 4 J laser pulse (λ 0 = 0.8µm) with a FWHM duration τ 0 = 25fs, a waist w 0 = 18 µm giving a normalised peak vector potential a 0 = 4.35 with polarisation along the y direction. a 0 is normalised to mec e with m e and e the mass and charge of an electron. The electronic density is n e = 6.22 × 10 18 cm -3 = 3.57 × 10 -3 n c where n c =

ω 2 0 me 0 e 2
is the critical density at this wavelength with ω 0 = 2π λ0 and 0 the vacuum permittivity. Those parameters have been selected to allow for an important self-injected charge so as to make sure the effects of numerical Cherenkov are easily visible. and 2.b) the low frequency field of the bubble and the beam self-field but also some higher frequency radiation surrounding the injected electron beam. This is even clearer looking in Fourier space (Fig. 2.c and 2.d). The physically relevant frequencies are confined on the left where k x < 2 ω 0 /c with notably the laser field visible in Fig. 2.c around k x = 1 ω 0 /c. Then there is some higher frequency radiation covering quite a large range (2.5 ω 0 /c < k x < 15 ω 0 /c) corresponding to the higher frequency fields surrounding the electron beam as can be seen in Fig. 2.e and 2.f where only fields with k x > 2.5 ω 0 /c have been plotted.

To ensure these are indeed NCR, we can, using Eq. ( 2), characterize theoretical NCR in either plane in Fourier space by:

k y = 2 ∆y arcsin ∆y 1 c 2 ∆t 2 sin 2 k x β x c∆t 2 - 1 ∆x 2 sin 2 k x ∆x 2 (3)
k z = 2 ∆z arcsin ∆z 1 c 2 ∆t 2 sin 2 k x β x c∆t 2 - 1 ∆x 2 sin 2 k x ∆x 2 (4)
These equations are represented by the black dashed line in Fig. 2.c and 2.d where we have used β x = 0.99995, corresponding to a 50 MeV beam (p x ≈ 100 m e c). This theoretical prediction correspond pretty well to the observed radiation that is thus conclusively identified as NCR.

It is worth pointing out that the numerical Cherenkov observed here does not couple with the plasma or the beam itself to give rise to NCI so that this work deals only with the direct impact of the generated NCR.

Note also that the numerical Cherenkov radiation does have a longitudinal component which we can observe on E x in the simulations. However its amplitude is lower by more than an order of magnitude compared to the physical simulated fields making it far less impactful on the simulation. We focus therefore in this paper exclusively on the transverse components of NCR.

Electron motion in LWFA

The NCR around the electron beam modify the electromagnetic fields inside the ionic cavity and may hence influence the transverse motion of the electrons inside the bubble. For simplicity's sake, we will mainly consider in the rest of this paper the transverse motion of electrons in the (xy) plane. The effects and equations are quite similar in the (xz) plane. For a particle of charge e moving along the x-axis at v x ≈ c, as is the case of the electrons in the accelerated beam, the transverse force in the y direction is given by: F y = e (E y -v x B z ) ∼ e (E y -cB z ). only the contribution from the cavity fields. In the bubble regime this transverse force is supposed to be linear with the transverse position as F y = 1 2 m e ω 2 p y [START_REF] Kostyukov | Phenomenological theory of laser-plasma interaction in "bubble" regime[END_REF][START_REF] Lu | Nonlinear theory for relativistic plasma wakefields in the blowout regime[END_REF] where ω p = nee 2 0me is the plasma frequency. The strong correlation between F y and y can indeed be observed in Fig. 3 where E y -cB z is stronger off axis and decreasing linearly to become zero on the axis of the bubble. against the position (blue line), to be compared with the theoretical Fy = 1 2 meω 2 p y (red dashed line) (respectively Fz = 1 2 meω 2 p z).

Suppressing Numerical Cherenkov

Existing methods

The simplest method to eliminate the numerical Cherenkov radiation is to remove it through filtering. As the radiation is not confined to a small range of frequencies near the Nyquist frequency, simple binomial filters

115
are not enough and more sophisticated low-pass filters are required [START_REF] Greenwood | On the elimination of numerical cerenkov radiation in pic simulations[END_REF][START_REF] Xu | Numerical instability due to relativistic plasma drift in em-pic simulations[END_REF][START_REF] Godfrey | Suppressing the numerical cherenkov instability in fdtd pic codes[END_REF][START_REF] Friedman | A second-order implicit particle mover with adjustable damping[END_REF] Various modified numerical schemes have also been proposed so that the generation of numerical Cherenkov radiation cannot arise at all. The basic principle is to modify the dispersion relation so as to ensure that v ϕ > c.

This can be achieve through modifying the computational stencil in the Maxwell-solver [START_REF] Greenwood | On the elimination of numerical cerenkov radiation in pic simulations[END_REF][START_REF] Lehe | Numerical growth of emittance in simulations of laser-wakefield acceleration[END_REF][START_REF] Xu | Numerical instability due to relativistic plasma drift in em-pic simulations[END_REF][START_REF] Lehe | Elimination of numerical cherenkov instability in flowing-plasma particle-in-cell simulations by using galilean coordinates[END_REF][START_REF] Nuter | Suppressing the numerical cherenkov radiation in the yee numerical scheme[END_REF][START_REF] Blinne | A systematic approach to numerical dispersion in maxwell solvers[END_REF].

However, some of these methods are not straightforward to implement, while other may induce some unreal-120 istic behaviour, such as light propagating faster than c in vacuum. Our proposed solution, besides being easy to implement, relies on a different approach: only the field interpolation is modified, so the Maxwell-solver remains unchanged and no smoothing is used. Our method can thus be used in addition of some of the previously published methods, in order to reduce even further the NCR effects.

Standard Field interpolation with the Yee scheme

In a PIC code with the Yee FDTD scheme [START_REF] Yee | Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media[END_REF], Electric and Magnetic fields are calculated using a leapfrog method. This implies they are defined on different grids which are offset spatially by half a cell but also temporally by ∆t/2. As the fields are only known on specific grid points while the particles move freely in the whole simulated space, it is necessary to interpolate the EM fields at the charged particles positions in order to calculate their motion.

Fig. 5 shows a simple situation with the fields amplitude values known on the different points of the grid and an electron moving freely between those points. To simplify the notations we consider only the longitudinal (a) spatial dimension x, A n i then refers to the value of the field A at the n th time step and the i th point of the spatial grid or in other terms A n i = A (t = n∆t, x = i∆x). Due to the leap-frog nature of the Yee scheme, values of the E y and E z fields are known at integer time-steps and grid points while values for B y and B z are known at half-integer time-steps and grid points. In the remainder of this section, we will only consider E y and B z and then omit the y and z indices when it is not absolutely necessary so as not to clutter the notation.

t ∆t n + 1 2 n n -1 2 x ∆x i -1 i -1 2 i i + 1 2 i + 1 B n+ 1 2 i-1 2 B n+ 1 2 i+ 1 2 E n i-1 E n i E n i+1 B n-1 2 i-1 2 B n-1 2 i+ 1 2 e - (b) t ∆t n + 1 2 n n -1 2 x ∆x i -1 i -1 2 i i + 1 2 i + 1 E n i-1 E n i E n i+1 Bn i-1 2 Bn i+ 1 2
Before the spatial interpolation, a temporal one is necessary in order to get Bn and have the fields known at the same time step. The simplest and most common way being the linear time interpolation Bn = 1 2 (B n-

1 2 + B n+ 1 2
) as is shown in the Fig. 5.a but quadratic time interpolation -with Bn

i+ 1 2 = 3 8 B n+ 1 2 i+ 1 2 + 3 4 B n-1 2 i+ 1 2 -1 8 B n-3 2 i+ 1 2
-can sometimes be used as well to get higher accuracy. Once the E and B fields are known both at the same time step, they are interpolated spatially on the particles. This step is shown in Fig. 5.b. Once again different kinds of spatial interpolation can be used with varying orders of interpolation. The one depicted here is, for simplicity's sake, a 1 st order method (linear interpolation).

Proposed B-Translated Interpolation Scheme

NCR is highly directional with propagation mainly along the axis of propagation of the electron beam, as can be seen in Fig. 2 where k y < 1ω 0 /c k x . Therefore with a particle moving along the x-axis at v x ∼ c, as is the case of electrons in the accelerated beam, the fields should satisfy:

E NCR x ∼ cB NCR x ≈ 0 → F NCR x = e (E NCR x ) ≈ 0 E NCR y ∼ cB NCR z → F NCR y = e E NCR y -v x B NCR z ≈ 0 E NCR z ∼ -cB NCR y → F NCR z = e E NCR z + v x B NCR y ≈ 0
where E NCR , B NCR are the electromagnetic fields due solely to the NCR and F NCR is the force resulting from those fields. As all components of F NCR are close to zero, we should expect NCR to generate a negligible force and thus have a negligible effect in LWFA simulations. Yet, as we have seen, its effect is clearly noticeable and the generated force is far from negligible.

In fact the relation E NCR at the particle position. The idea of our new approach is to modify the interpolation of the EM fields so as to make sure the impact of numerical Cherenkov radiation through its generated force

F NCR is indeed negligible.
Since all the main electromagnetic fields in our LWFA case -the wakefield itself, the laser field, the beam's self field and the NCR -propagate approximately at c along the x direction, then we can assume that B (n∆t, i∆x) ≈ B (n + 1 2 )∆t, (i + 1 2 )∆x , as in half a time step the fields have propagated and advanced of half a cell. This works because while still satisfying the CFL condition, c∆t is very close to ∆x in the simulations. The idea is then to interpolate the fields at the particle position using (

(a) t ∆t n + 1 2 n n -1 2 x ∆x i -1 i -1 2 i i + 1 2 i + 1 B n+ 1 2 i-1 2 B n+ 1 2 i+ 1 2 E n i-1 E n i E n i+1 B n-1 2 i-1 2 B n-1 2 i+ 1 2 (b) t ∆t n + 1 2 n n -1 2 x ∆x i -1 i -1 2 i i + 1 2 i + 1 E n i-1 E n i E n i+1 Bn i-1 Bn i Bn i+1
E n i , Bn i = B n+ 1 2 i+ 1
2

) instead of (E n i , Bn

i+ 1 2
) as previously shown. This new method is described in Fig. 6.

Using this approximation, there is no need for a temporal interpolation anymore and it also simplifies the spatial interpolation as we now know the magnetic field on the same grid points as the electric field. As the above effectively results in a translation of the calculated B field before the spatial interpolation, we will in the remainder of this paper refer to this new process as the B-Translated Interpolation Scheme (B-TIS). We will also refer to the previous temporal interpolation method as LTI when using the linear time interpolation or QTI for the quadratic one.

3.4. Accuracy comparison of the two methods

Theoretical calculation

Let us discuss in further details the accuracy of the two methods on the computed magnetic fields. We first consider a scalar field B of the form :

B(x, y, z, t) = B 0 cos (k x x + k y y + k z z -ωt) (5)
Discretized in the simulation, this gives us:

B n i,j,k = B 0 cos (k x i∆x + k y j∆y + k z kδz -ωn∆t) = B 0 cos ϕ n i,j,j .
The linear time interpolation used is defined by: Bn

i,j,k = 1 2 B n+ 1 2 i,j,k + B n-1 2 i,j,k
which we can rewrite with

δϕ = ω∆t 2 as Bn i,j,k = B 0 cos ϕ n i,j,k cos (δϕ) = B n i,j,k cos (δϕ) (6)
This shows that this temporal interpolation is a good approximation as long as δϕ 1 when cos (δϕ) 1. To highlight the error introduced by this interpolations we will rewrite Bn i,j,k as Bn i,j,k = B n i,j,k + εB with εB being εB = B 0 cos ϕ n i,j,k (cos (δϕ) -1)

Then at the lowest order we have:

εB B 0 1 2 cos ϕ n i,j,k δϕ 2 < 1 2 δϕ 2 (8)
Similarly, we can show that the quadratic time interpolation defined by Bn i,j,k = 3 8 B

n+ 1 2 i,j,k + 3 4 B n-1 2 i,j,k -1 8 B n-3 2 i,j,k
gives an error εB :

εB = B 0 cos ϕ n i,j,k 9 8 cos (δϕ) - 1 8 cos (3δϕ) -1 + B 0 sin ϕ n i,j,k 1 8 sin (3δϕ) - 3 8 sin (δϕ) (9)
And at the lowest order: εB

B 0 1 2 sin ϕ n i,j,k δϕ 3 < 1 2 δϕ 3 (10)
Looking at the accuracy of our new B-TIS, we consider

B n i,j,k = B n+ 1 2 i+ 1 2 ,j,k = B 0 cos ϕ n i,j,k + δϕ (11)
with δϕ = 1 2 (k x ∆x -ω∆t). As before we will rewrite this so as to have Bn i,j,k expressed as

B n i,j,k = B n i,j,k + ε B with this time ε B being ε B = B 0 cos ϕ n i,j,k cos δϕ -1 -B 0 sin ϕ n i,j,k sin δϕ (12)
which at the lowest order finally gives us:

ε B B 0 sin ϕ n i,j,k δϕ < δϕ (13)
So, as long as we can ensure that δϕ < 1 2 δϕ 2 , B-TIS should provide more accurate computation of the B field. For longitudinally propagating waves with ω/k x ≈ c and c∆t ≈ ∆x, as is usually the case for LWFA simulations, δϕ can become very small and the gain should be significant.

If we consider the transverse forces applied to an electron:

F y = -e (E y + v z B x -v x B z ) (14)
F z = -e (E z + v x B y -v y B x ) (15)
then the translation or temporal interpolation of B introduce an error on the transverse forces such as:

ε Fy = -e (v z ε Bx -v x ε Bz ) (16)
ε Fz = -e v x ε By -v y ε Bx [START_REF] Mangles | Monoenergetic beams of relativistic electrons from intense laser-plasma interactions[END_REF] Note that most of the fields in the simulations propagate mainly longitudinally so B 0 , x ≈ 0 making ε Bx really small. Furthermore for most electrons v y v x and v z v x . We can thus safely approximate the error to:

ε Fy ≈ e v x ε Bz (18)
ε Fz ≈ -e v x ε By [START_REF] Lu | Nonlinear theory for relativistic plasma wakefields in the blowout regime[END_REF] Minimising the error on the transverse magnetic fields will thus minimise the error on the the transverse forces and help reduce the impact of NCR.

Error on Electromagnetic waves in vacuum

The presence of charged particles or a low density plasma may affect the phase velocity of an electromagnetic wave but for the sake of simplicity, we neglect this effect in this section and consider waves propagating in the vacuum. For ease of visualisation, we also consider waves propagating in the (x, y) plane only (k z = 0) but the results can easily be extended to a more general 3D case.

As shown previously, with the Yee scheme, propagating electromagnetic waves abide to the numerical dispersion equation [START_REF] Yee | Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media[END_REF]. We can thus express δϕ and δϕ as :

δϕ = arcsin c∆t ∆x sin 2 k x ∆x 2 + c∆t ∆y sin 2 k y ∆y 2 (20) δϕ = 1 2 k x ∆x -arcsin c∆t ∆x sin 2 k x ∆x 2 + c∆t ∆y sin 2 k y ∆y 2 (21)
From there it is quite clear that the results will depend heavily on the numerical parameters chosen for the simulation, namely : ∆x, ∆y and ∆t.

Assuming ∆y = ∆z, the choice of the numerical time step ∆t is limited by the CFL condition ∆t < ∆t CFL with:

c∆t CFL = ∆x 1 + 2 ∆x 2 ∆y 2 (22)
Thus, if we express ∆t as a fraction of ∆t CFL , ∆x and ∆y are the only free numerical parameters and we can test the accuracy of our two methods depending on (k x , k y ) by comparing :

˜ = εB B 0 max = max 0<θ<2π |cos (θ) (cos (δϕ) -1)| = |cos (δϕ) -1| (23) = ε B B 0 max = max 0<θ<2π cos (θ) cos δϕ -1 -sin (θ) sin δϕ (24)
When there is no laser to model, as is the case for beam-driven acceleration, the grid size must be small enough so as to accurately describe plasma oscillations. It is thus common to use ∆x = ∆y chosen between λ p /25 and λ p /50. For LWFA simulations, it is in addition necessary to accurately describe the laser pulse longitudinally. To that end ∆x is usually chosen between λ 0 /25 and λ 0 /50 and, as λ p is usually 10 times bigger than λ 0 , it is more efficient to use ∆x ∆y.

In order to look at the impact of the cell size on accuracy, we compare the error values of LTI and B-TIS Note that if we take ∆x = λ 0 /40 then k 0 ∆x/π = 0.05, so everything above 0.01 can be considered as high frequency, which is mostly some noise and not some physically relevant component. Knowing that, it is clear that LTI ensures good accuracy for low frequency fields (which are phisically relevant) but its precision decreases rapidly outside of that range. Increasing the ratio ∆y/∆x increase the accuracy for transverse high frequency fields but reduces it for those with high longitudinal frequency. On the other hand, the accuracy of B-TIS is maximum when v ϕ = ω/k ≈ c and increasing the ratio ∆y/∆x improves significantly the accuracy for the whole spectrum as c∆t becomes increasingly closer to the value of ∆x.

LTI is better with ∆y = ∆x, but as soon as ∆y is at least four times bigger than ∆x, B-TIS gives similar or better accuracy to LTI for the laser field and lower frequency fields and much better accuracy for the high The precision with B-TIS is then better than with LTI and on the same order of magnitude as with QTI.

As stated in Section 2, the frequency of NCR satisfy ω = k x β x c with β x being the normalized velocity of the particle generating the NCR along the x direction. This means, with the same notation as previously that δϕ = 1 2 β x k x ∆t and δϕ = 1 2 k x (∆x -β x c∆t). Using our particular numerical parameters and β x = 0.99995 (p x ≈ 100m e c) we get, for all possible values of k x : 0.011 ≤˜ (k x ) ≤ 0.99 (28)

0.0016 ≤ ˜ (k x ) ≤ 1.10 (29) 0.0010 ≤ (k x ) ≤ 0.011 (30)
As expected the results are significantly better for B-TIS for all the possible spectrum of NCR but especially for higher frequencies where the difference can be as big as two orders of magnitude. Note that for those highest frequencies QTI is no better than LTI.

Error on Wakefield and beam self-field

Beam self-field and wakefield are not purely electromagnetic fields so they do not satisfy the numerical dispersion relation given by Eq. 1. Instead, as long as the quasistatic approximation can be applied [START_REF] Sprangle | Nonlinear theory of intense laser-plasma interactions[END_REF], these fields are characterised by ω = vk x where v = v b is the beam velocity in the case of the beam self-field or v = v wf is the wakefield velocity in the wakefield case, both moving along the x axis.

Thus, in that case, δϕ = kxv∆t 2 and δϕ = 1 2 k x (∆x -v∆t) and we can show similarly to previous cases that: The value of v has a very small impact on the precision as in both cases v ≈ c though its effect is more noticeable with B-TIS. For very low frequencies such as the wakefield frequency k p (k p ∆x/π ≈ 0.005 if ∆x ≈ λ 0 /40 ≈ λ p /400), LTI ensures a better precision than B-TIS but both methods give very small errors in that case. The error of LTI increases very rapidly with the longitudinal frequency so that for higher frequencies B-TIS again gives an error at least an order of magnitude smaller.

˜ (k x ) 1 2 δϕ 2 (k x ) (31) ˜ (k x ) 1 2 δϕ 3 (k x ) (32) (k x ) δϕ(k x ) (33)
Using again the numerical parameters of the simulation introduced in section 2, we can compute the error for typical values of k x for the wakefield and the beam self-field. Choosing k x = k p for the wakefield we get: For those typical values, LTI and B-TIS give pretty similar values resulting in good accuracy for both methods.

However higher frequencies may arise due to inhomogeneities in the electron beam or high electron density gradients in the wake, those would benefit from the improved accuracy of B-TIS at higher frequencies. Beam 245 hosing or transverse evolution of the bubble may also occur but would result in very low longitudinal frequency oscillations where both method give good results.

Implementation of B-TIS in Calder

We show in Fig. 11

F LTI y (x, y, z) = e i,j,k S i,j,k (x, y, z) E n i,j,k -S i+ 1 2 ,j,k (x, y, z) c Bn i+ 1 2 ,j,k (36)
F B-TIS y (x, y, z) = e i,j,k S i,j,k (x, y, z) E n i,j,k -S i,j,k (x, y, z) c Bn i,j,k (37)
where i,j,k are the indices on the spatial grid along the x,y,z directions respectively, and S i,j,k (x, y, z) is the interpolation form factor. A third order form factor (cubic interpolation) was used here as in the simulations 250 for spatial interpolation. When calculating F y = e (E y -cB z ), the laser field, the beam self field and NCR are expected to cancel out leaving only the transverse force created by the bubble fields. Yet we can clearly see in Fig. 11 on the first row that F y has some higher frequency components. At the front of the bubble we can observe oscillations clearly due to the laser beam while there are even higher frequency perturbations all along the beam position.

Looking at the right-most picture, we can estimate that λ ≈ 4∆x or k x ≈ 10ω 0 /c which is coherent with NCR characteristics as seen in Fig 2 , we can therefore conclude that the aforementioned perturbations are indeed caused by NCR.

The spurious rapid oscillations of the field are effectively suppressed when using B-TIS instead of LTI and only the expected transverse bubble field are left on the bottom row of Fig. 11. Coincidentally, the error due to the laser fields in front of the bubble is reduced as expected according to the error levels reported earlier.

From the above observations, we conclude that B-TIS provides better accuracy than the other above-mentioned methods with regards to the motion of particles in LWFA simulations.

Comparison of results for LWFA simulations

Influence on the bubble and the beam

We implemented the modified interpolation scheme B-TIS into calder, applying it for every particle with p x > 50 m e c. Indeed, the underlying approximation works well inside the cavity for the injected electrons with v x ≈ c but would give poor results with slower electrons. It shall therefore only be applied to particles satisfying β x > 0.9998 (p x > 50 m e c) while other particles shall be treated using the previously existing temporal interpolation method.

A 3D simulation was run with the exact same parameters as presented in Sec.2 but using our newly implemented B-TIS instead of the previous LTI method. Comparison of their results reveal a high degree of similarity, with almost identical bubbles -same shape, same fields -the differences coming from the accelerated beam itself and its generated fields. Fig. 12 shows electron density maps in the two different cases, the shape of the bubble appears identical in both instances but the injected beam is much more narrow with the modified scheme. In fact, only the transverse spatial distribution of the beam seems to be really affected by the change, with higher local electron density because of the beam focusing.

Looking at Fig. 13, the injection process appears unaffected. The injected charge is almost identical for the whole duration of the simulation and the linear density of the beam is also extremely similar. Four autoinjection events can clearly be identified in Fig. 13.a when the injected charge jumps from one plateau to another.

The third event is the most important and occurring for a relatively long time. The energy spectrum of the accelerated electrons is also quite comparable although differences are more noticeable there, highlighting how NCR can impact the acceleration process of the electrons.

The above results seem to confirm that our modified scheme does not impact the physics of wakefield creation, injection or acceleration process but merely corrects the spurious effect of NCR on the accelerated beam without actually suppressing the radiation.

Looking at the electromagnetic fields in the simulation with B-TIS, the numerical Cherenkov radiation is indeed still present as expected. Fig. 14 presents maps of the electric fields E y and E z and their respective Fourier transform, to be compared with Fig. 2. Once again the bubbles and their associated fields are extremely similar in both cases, the numerical Cherenkov radiation is the main observable difference between the two 290 simulations. The NCR is much more intense around the electron bunch in the new simulation, especially at the front of the beam, and seems more focused and clearly defined in Fourier space, this is likely due to the beam being much more narrow and more uniform in motion. The differences in terms of transverse force, though not visible when looking at the fields, translate into important difference in terms of beam emittance. Fig. 15 and Fig. 16 present the evolution with time of the 295 normalized emittance of sections of the electron beam.

The colored bunches were defined by the number of injection events occurring during the simulation as can be seen on Fig 13.a. We can observe the spurious growth of emittance described by Lehe [START_REF] Lehe | Numerical growth of emittance in simulations of laser-wakefield acceleration[END_REF] for all sections of the beam and in both directions for the simulations using LTI. Instead, with B-TIS, the emittance of the beam is smaller, but in addition it also tends to stay constant during the acceleration of the beam. This is in line with 300 what is expected according to theoretical models: in the case of a fully evacuated plasma bubble, the transverse

Influence on electrons trajectories

Looking in more details into the individual electron trajectories, the effect of the modification is quite remarkable. Fig. 17 presents an example of such trajectories to be compared to those from Fig. 4. The oscillations of the electrons are far more regular and there is a much stronger correlation between the position of the particle and the transverse force especially in the (xz)-plane. There are still however some remaining disparities along the y direction which we will discuss further on.

Fig. 18 shows a comparison between the two simulations of the average difference between the recorded transverse force and its theoretical value for every recorded particles. The difference between the theoretical values and the recorded values are quite significant when using the LTI scheme revealing that the electrons are influenced by something other than the ideal EM fields of a perfect bubble. These effects are much less visible when B-TIS is used, especially in the (xz)-plane.

The difference of behaviour between the two planes can be explain by the polarisation of the laser beam along the y-axis. A 'wiggling' of the bubble in that plane is frequently observed in LWFA simulations and is indeed visible in Fig. 12. It is a well-known physical effect due to the increasing dissymmetry of the laser beam as it is depleted. This 'wiggling' may affect the electromagnetic fields inside the cavity leading to significant disparity with the theoretical fields inside the bubble. This effect is actually visible on Fig. 17. Note how the trajectory along the y direction is still somewhat irregular and the correlation between F y and y -although much better than previously -is not as good as between F z and z. This is the consequence of the asymmetry of the bubble in the y plane due to its 'wiggling'. The axis of the bubble where F y ≈ 0 is effectively constantly displaced thus impacting the correlation between F y and the particle position along y.

This effect may be as important as the error introduced by NCR which would explain the small improvement in the (xy)-plane relative to the one in the (xz)-plane. Using B-TIS allows us to clearly see the impact of this effect while it was previously masked by the effects of NCR. Our new improved scheme thus enables us to study rather subtle physical effects that might otherwise be lost due to numerical errors. is the force recorded by the particle (ie computed using Eq. 36 and 37) and F theor y = 1 2 meω 2 p y . Blue dots represents results using the LTI while the red squares are obtained with B-TIS. The picture on the right shows the same for the z direction.

Conclusion

We showed that the simple Yee scheme used by most PIC codes, though quite robust, has some intrinsic problems when simulating wakefield accelerators leading to an overestimation of the transverse size and emittance of the accelerated beam as well as a misrepresentation of the particle trajectories due to non physical numerical Cherenkov radiation. By using a simple modification of the interpolation scheme, we were able to significantly reduce the impact of this spurious radiation and improve the accuracy of our LWFA simulations. The improved accuracy on particle trajectories might especially help better understand electron motion and thus subtle beam evolution or even betatron radiation sources which are one very promising application of LWFA.

Though this paper is focused on LWFA, this work could be extended to study other wakefield acceleration methods, or even any simulation of charged particles propagating relativistically in a plasma as long as all the main EM fields propagate at a velocity close to c in the beam propagation direction. In addition, as it does not modify the Maxwell solver, our B-TIS could potentially be used with any FDTD scheme, especially with a dispersion free scheme that can allow use of ∆t = ∆x, making it a simple yet versatile tool to deal with NCR.

Finally, B-TIS might prove useful to improve the simulated motion of the particles for any simulation of relativistic particles co-propagating with electromagnetic fields such as direct laser acceleration or vacuum laser accelerations cases where electrons are accelerated directly by the co-propagating laser field, in order to improve the accuracy on the interaction.

10

 10

Fig. 1 Figure 1 :

 11 Fig.1shows a snapshot of a numerical simulation with an electron bunch accelerated in the wake of a laser pulse. The longitudinal field E x presents two distinct regions inside the cavity: the rear of the bubble has

70

  In LWFA simulations, numerical Cherenkov radiation can appear around the accelerated electron beam. It creates high frequency radiation inside the ionic cavity which leads to a non-physical growth of the transverse size, divergence and beam emittance[START_REF] Lehe | Numerical growth of emittance in simulations of laser-wakefield acceleration[END_REF]. Fig. 2 illustrates such occurrence of NCR by showing the transverse electric fields of the wakefield accelerator. The laser pulse is clearly visible on Fig. 2.a as well as (on both Fig. 2.a

Figure 2 :

 2 Figure 2: Numerical Cherenkov radiation in a LWFA simulation: (a) shows a map of the transverse field Ey in the (xy) plane while (b) shows a map of Ez in the (xz) plane. (c) and (d) show their respective spatial Fourier transform with the black dashed lines corresponding to Eq. (3) and (4) respectively for a particle with px ≈ 100 mec. (e) and (f) show maps of the electric transverse fields for kx > 2.5 ω 0 /c only.

Fig. 3

 3 Fig. 3 shows a map of the difference E y -cB z . Note that, since the laser and the beam self-field both propagates at v ϕ ≈ c along the x direction, their contributions cancel out as E laser y ≈ cB laser z and E self-field y

Figure 3 :

 3 Figure 3: Map of Ey -cBz, slicecut in the (xy)-plane. Only the contribution from the bubble fields is still apparent.

Figure 4 :

 4 Figure 4: Examples of particle trajectories in presence of Numerical Cherenkov radiation. The top row is along the y-axis and the bottom one along the z-axis. On the left is the y (respectively z) position of the electron with time, the color showing the transverse force Fy = Ey -vxBz (respectively Fz = Ez + vxBy) recorded by the particle. On the right the transverse force has been plotted

Figure 5 :

 5 Figure 5: Usual interpolation process of EM fields on a 1D spatial grid at the n th time step in order to get the transverse force F n y applied on the particle represented by the yellow dot. (a) shows the initial configuration where the fields values are known at different times and spatial points. The temporal interpolation to get Bn is shown with dashed lined arrows. Then (b) shows the subsequent spatial interpolation of E n and Bn on the particle (again with dashed lined arrows and first order).

y

  ∼ cB NCR z holds only when the fields are not interpolated. Even with small grid spacings and time steps, high frequency fields such as the numerical Cherenkov radiation may vary noticeably in one time step and over one cell size. Using the average Bn then introduces an error on the magnetic field which might be small in absolute value but becomes quite important when one considers the difference E NCR y -cB NCR z . Spatial interpolation of E and B from different points as shown in Fig. 5 may also increase the error on the computed E NCR y -cB NCR z

Figure 6 : 2 i+ 1 2

 621 Figure 6: Modified interpolation process of EM fields: B-TIS on a 1D grid at the n th time step. (a) shows the same initial configuration as Fig. 5 (a) but instead of the temporal interpolation of B, the relation Bn i = B n+ 1 2 i+ 1 2

  across the field of possible k x and k y for different sets of numerical parameters. We present next the results of

Figure 7 :

 7 Figure 7: Absolute error on B field using (a) LTI ˜ (kx, ky) or (b) B-TIS (kx, ky) with ∆y = ∆x and ∆t = 0.99 ∆tCFL. The black line describe possible NCR modes according to eq. 3 with βx = 0.99995 (px ≈ 100mec).

Figure 8 :

 8 Figure 8: Absolute error on B field using (a) LTI ˜ (kx, ky) or (b) B-TIS (kx, ky) with ∆y = 4∆x and ∆t = 0.99 ∆tCFL. The black line describe possible NCR modes (and aliases) according to eq. 3 with βx = 0.99995 (px ≈ 100mec).

Figure 9 :Fig. 9 , 1 2 1 2

 9911 Figure 9: Absolute error on B field using (a) LTI ˜ (kx, ky) or (b) B-TIS (kx, ky) with ∆y = 20∆x and ∆t = 0.99 ∆tCFL. The black line describe possible NCR modes (and aliases) according to eq. 3 with βx = 0.99995 (px ≈ 100mec).

Figure 10

 10 Figure 10 compares the errors ˜ (k x ) of LTI and (k x ) of B-TIS, both for v = v wf = 0.99822 (case of laser-driven wakefield in a plasma with n e = 0.00357n c ) and v = v b = 0.99995 (case of a 50 MeV electron beam).

˜ ≈ 1 .

 1 10 -5 , ˜ ≈ 4. 10 -8 , ≈ 4. 10 -5 (34) and choosing k x = 4k p for the beam self-field (assuming a electron beam as long as a quarter of the bubble length) we get: ˜ ≈ 2. 10 -4 , ˜ ≈ 3. 10 -6 , ≈ 1. 10 -4

Figure 10 :

 10 Figure 10: Absolute error on B field using LTI or B-TIS with ∆t = 0.99 ∆tCFL.

  a comparison of the resulting transverse force F y = e (E y -cB z ) calculated with either LTI or B-TIS. The pictures are successive zooms in the (xy) plane of the simulation. The first one shows the bubble in its entirety while the second focuses on fine details around the axis of the bubble and in the rightmost one grid points used to calculate the EM fields are visible. Between those grid points, the fields are interpolated in the same way as they would in calder to calculate fields at the particles positions. The force F y that would apply to an electron is the difference between the interpolated E y and interpolated B z , F LTI y is shown on the first row and F B-TIS y on the second, with:

Figure 11 :

 11 Figure11: Successive zooms on a map of the interpolated transverse force Fy that would apply to an electron.The first row when using LTI, the second row when using B-TIS. For each on the rightmost picture, + and × materialize the grid points used for the spatial interpolation: + for E n , × for Bn and × for Bn .

Figure 12 :

 12 Figure 12: Comparison of the bubble and beam shape. Those are slice-cuts showing the electronic density normalised to the critical density ne/nc. (a) and (c) are slices in the (xy) and (xz) planes respectively in the case using LTI, while (b) and (d) are their counterparts with B-TIS.

Figure 13 :

 13 Figure 13: Comparison of beam properties when using the LTI scheme or the new B-TIS: (a) Total injected charge (px > 50mec) with the length of propagation in the plasma. (b) Beam distribution along the x axis and (c) energy spectrum of the accelerated electrons, both after 2.25 mm propagation in the plasma.

Figure 14 :

 14 Figure 14: Similar results to Fig. 2 but using the B-TIS for electrons with px > 50mec: (a) shows a map of the transverse field Ey in the (xy) plane while (b) shows a map of Ez in the (xz) plane. (c) and (d) show their respective spatial Fourier transform with the black dashed lines corresponding to Eq. (3) and (4) respectively for a particle with px ≈ 100 mec. (e) and (f) show maps of the transverse field for kx > 2.5 ω 0 /c only.

Figure 15 :

 15 Figure 15: Beam transverse size and emittance in the (xy) plane. (a) and (c) : map of the longitudinal electric field Ex and distribution of the beam particles (colored circles). (b) and (d) : Evolution of the normalized emittance εy of the different electron bunches defined in Fig. 13. The results in (a) and (b) are obtained using LTI, while (c) and (d) used B-TIS.

Figure 16 :

 16 Figure 16: Beam transverse size and emittance in the (xz) plane. (a) and (c) : map of the longitudinal electric field Ex and distribution of the beam particles (colored circles). (b) and (d) : Evolution of the normalized emittance εz of the different electron bunches defined in Fig. 13. The results in (a) and (b) are obtained using LTI, while (c) and (d) used B-TIS.

21 t

 21

Figure 17 :

 17 Figure 17: Trajectories of a electron from the simulation with the modified scheme B-TIS. The top row is along the y-axis, the bottom one along the z-axis. On the left is the y (respectively z) position of the electron with time, the color showing the transverse force Fy = Ey -vxBz (respectively Fz = Ez + vxBy) recorded by the particle. On the right the transverse force has been plotted against the position (blue line), to be compared with the theoretical Fy = 1 2 meω 2 p y (red dashed line) (respectively Fz = 1 2 meω 2 p z).

Figure 18 :

 18 Figure 18: The picture on the left presents the temporal average |F rec y -F theor y

Acknowledgements

This work has been supported by Laserlab-Europe (EU-H2020 654148). We also acknowledge GENCI for granting us access to the supercomputer Irene under the grants No. A0050506129 and No. A0060507594.


Germain Homerin

Adrian Sorin Nica

Amaury Farce

Joëlle Dubois

Alina Ghinet
email: alina.ghinet@yncrea.fr

Cl Cl

Ultrasounds-mediated 10-seconds synthesis of chalcones as potential farnesyltransferase inhibitors

Keywords: Farnesyltransferase Inhibitor Chalcone Claisen-Schmidt Ultrasounds Ar, 2-thiophene, ferrocene Ar, 2-thiophene, ferrocene

A broad range of chalcones and derivatives were easily and rapidly synthesized, following Claisen-Schmidt condensation of (hetero)aryl ketones and (hetero)aryl aldehydes using a ultrasound probe. A comparison was made with classical magnetic stirring experiments, and an optimization study was realized, showing lithium hydroxide to be the best basic catalyst of the studied condensations. By-products of the reactions (β-hydroxy-ketone, diketones, and cyclohexanols) were also isolated. All compounds were evaluated in vitro for their ability to inhibit human farnesyltransferase, a protein implicated in cancer and rare diseases and on the NCI-60 cancer cell lines panel. Molecules showed inhibitory activity on the target protein and cytostatic effect on different cell lines with particular activity against MCF7, breast cancer cells.

Chalcones are particularly easily synthesized and of great interest in medicinal chemistry. With a broad spectrum of pharmacological activities, chalcones were found to be mainly active against inflammation, 1,2 tuberculosis, 3 HIV, 4 malaria 5 and cancer. [6][7][8][9] A literature review dedicated to chalcones over the past ten years reveals almost 13,000 references with more than 1,500 references that describe the antitumor activity of reported chalcones. This biological activity seems the most investigated for such derivatives. The reported mechanisms of antitumor action are numerous, chalcones acting as inhibitors of: tubulin polymerization, [10][11][12][13][14][15] HDAC, 16 EGFR-tyrosine kinase (EGFR TK) phosphorylation, 17,18 thioredoxin reductases (TrxRs), 19 PI3K/Akt/mTOR pathway, 20 farnesyltransferase, 9 etc. The most documented is the antimitotic activity of chalcones involving the inhibition of tubulin polymerization. On the contrary, the activity of chalcones on human farnesyltransferase was very little explored and one study has been described by our group. 9 The best farnesyltransferase inhibitor identified in the study was compound A (Figure 1). 9 Farnesyltransferase (FTase), heterodimeric metalloenzyme, is a prenyltransferase involved in the thiolinkage of a farnesyl (C15) isoprenoid coming from the biosynthetic pathway of cholesterol, a critical metabolite for the prenylation of proteins. This process is a critical mediator in several diseases. Indeed, preventing the farnesylation process can constitute an approach in the treatment of cancers, acquired immunodeficiency syndrome (AIDS) or progeria. 21 Progeria or Hutchinson-Gilford syndrome is an extremely rare disease and has a prevalence of one case in five million births. The appearance of the newborn baby is normal at birth; an accelerated aging is noted a year after. Currently, only farnesyltransferase inhibitors were efficient on progeria models by blocking the progerin prenylation. Five phases I/II and II clinical trials using Lonafarnib (molecule B, Figure 1), known farnesyltransferase inhibitor previously developed for oncology with moderate results, alone or in combination with zoledronic acid, pravastatin or with everolimus are currently ongoing in the hope of finding a treatment. 22 First results showed additional bone mineral density benefit for patients but likely no added cardiovascular benefit with the addition of pravastatin and zoledronic acid. 23 The discovery of new efficient farnesyltransferase inhibitors remains a significant challenge in the fight against this extremely rare disease but also other diseases. 21 In this study, the investigation of chalcones bearing different heterocyclic units (thiophene, pyridine and ferrocene) than phenothiazine investigated previously (e.g. compound A) as potential human farnesyltransferase inhibitors is reported (Figure 1). Moreover, it has been described that chalcones were powerful chelators of zinc cation (Zn 2+ ). 24 This property is important in the inactivation of farnesyltransferase because the enzyme needs Zn 2+ to perform its functions. Chalcone derivatives could inhibit Zn 2+ metalloenzymes such as human FTase. Chalcones of this study were prepared by Claisen-Schmidt condensation of corresponding (hetero)aryl ketones and (hetero)aryl aldehydes (Scheme 1). 25 According to the literature, bis(thienyl) chalcone 1a can be obtained from 1-thiophen-2-ylethanone and thiophene-2-carbaldehyde using strong bases such as alkali hydroxides. 2 However, this method suffers from drawbacks: it is very sensitive to reaction conditions such as dilution of the medium or base quantity. As an example, βhydroxy-ketone 2a was obtained when water was used as the solvent, 26,27 and cyclohexanol 3a was isolated in presence of a large amount of base in low diluted medium. 28,29 In our hands, using a classical magnetic stirring and sodium hydroxide in tertbutanol, only 13% yield was achieved in the best case for the synthesis of chalcone 1a, which was hard to separate from byproducts such as cyclohexanol 3a (see supplementary material section). Moreover, using 1-(6-chloro-3-pyridinyl)-1-ethanone and thiophene-2-carbaldehyde in ethanol and water, the substitution of the chlorine by an ethoxy group led to 1b' and 3b' in very low yield, and as the only isolated compounds of a complex mixture.

In this work, the sonication of the mixture was tested instead of classical magnetic steering. Globally, the results were comparable except in the case of chalcones 1a and 1b, but the reaction time was decreased from hours to seconds. Different bases were next selected and tested for the Claisen-Schmidt condensation of 1-thiophen-2-yl-ethanone and thiophene-2carbaldehyde (Table 1). The blank gave no reaction. In the case of cesium carbonate, the reaction was not complete even after 60 seconds of sonication, and the use of cesium fluoride led to a complex mixture. Alkali hydroxides (NaOH and LiOH) were more suitable than other tested bases and led to the desired chalcone 1a in good yield after 10 seconds of sonication. However, for the synthesis of 1b, the use of NaOH provided only ethoxy-pyridine 1b' in poor yield. The same reaction performed using LiOH afforded the desired chalcone 1b. Consequently, the following experiments were realized using LiOH as the base. A screening of different solvents was next performed (Table 2). Mixtures of water and ethanol provided only the desired chalcone 1a (Table 2, entries 6-8), and a 10 time up-scaled experiment was realized with success (Table 2, entry 9). Water alone does not enable the dehydration of 2a, ethanol alone led to the formation of by-product 3a, while non-polar solvents are not suitable for the reaction. A second screening was realized to determine the relative amounts of water, ethanol and lithium hydroxide needed, and to determine the settings of sonication like amplitude, duration, and temperature (Table 3). The amount of lithium hydroxide, the amplitude and duration of the sonication, and the temperature of the medium do not seem to play a role in the reaction progress. The dilution and the relative quantity of ethanol in water, on the contrary, are of great importance. β-Hydroxy-ketone 2a was the major product when water alone was used as the solvent (Table 3, entries 1-7 and 16), except in diluted conditions (Table 3, entry 9), where chalcone 1a was obtained as the major product. The dilution factor was raised and chalcone 1a was obtained in good yields when water was used (Table 3, entries 9-11, 13 and 15). With less than 50% of water in the mixture, the conversion was medium (Table 3, entries 8, 12, and 14), and undesired products were formed. Finally, conditions were chosen as follows: i) sonication of the medium was preferred to classical magnetic stirring; ii) the solvent was a mixture of water and ethanol (1:1); iii) one equivalent of lithium hydroxide was optimal; iv) a concentration of 0.02 mol.L -1 was preferred. In these conditions, chalcone 1a was isolated in 91% yield (10-15% was obtained using classical magnetic stirring). The above described parameters were applied to corresponding ketones and aldehydes, and chalcones 1a-1q were obtained in medium to high yields (see Scheme 1). Interestingly, 6-chloropyridine chalcone 1b was obtained in 31% yield only under sonication while it was not possible to isolate it when realizing the reaction under magnetic stirring. Also, lower amount of by-product 6-ethoxypyridine 1b' was formed (6% yield under magnetic stirring and 2% yield under sonication). Consequently, sonication induced less substitution of the chlorine, however, di-keto-aryl compound 4b and cyclohexanol 3b were also isolated.

B A R 1 O R 1 O R 2 R 1 O 4 1 R 1 OH R 2 O R 1 R 1 O R 2 3 R 1 O R 2 OH 2 R 1 O R 2 1 R 1 O R 2 O H + -H 2 O LiOH H 2 O/EtOH O S S 1a (91%) (13%) O N Cl S 1c (
Fe 2+ C C C C C N O C C C C C Cl Fe 2+ C C C C C N C C C C C Cl O Scheme 1. Claisen-Schmidt
If 6-chloropyridyl chalcones 1b-1j were obtained in medium yields (31-43%) due to purification issues, other aryl and thienyl chalcones 1a and 1k-1r were obtained in very good yields (91-99%). Most of the time, by-products of general formula 3 and 4 were observed in very low quantity in the crude. Using 1-(6chloro-pyridin-3-yl)-ethanone and 6-chloro-pyridine-3carbaldehyde, the isolation in very low yield (0.1%) of compound 4d succeeded but not compound 3d. In the same light, using 1-thiophen-2-yl-ethanone and 2,4-dichloro-benzaldehyde, compound 4q (14% yield) was isolated but not compound 3q (which was isolated from the corresponding experiment realized under magnetic stirring). For the other examples, only the chalcone was isolated, except for the synthesis of chalcone 1j, where enone 5 was isolated (Scheme 2). This compound was also observed as traces in the crude of the syntheses of chalcones 1c-1f (from 6-chloro-pyridine-3-carbaldehyde), but was not isolated despite efforts. In order to understand the formation of enone 5, the ultrasonication of 6-chloro-pyridine-3-carbaldehyde alone with lithium hydroxide was performed. The Cannizzaro reaction products 30 alcohol 6 31 and acid 7 were obtained (Scheme 3). The formation of compound 5 is thus difficult to explain however oxidations and/or reductions of the reagents and/or the solvents would be a track to follow. The geometry of cyclohexanols 3 was then investigated. Spectroscopic studies for 3a, 3b and 3q ( 1 H, 13 C, 2D-NMR) were undertaken. Results were in accordance with Gezegen's cyclohexanol 28 (see Supplementary Data for spectra), not with Shan's product. 29 It confirmed that 3b has the same geometry as 3a and 3q, and it showed the position of the ethyl moiety, which is located on the pyridine next to the hydroxyl group (Figure 3). It is to be noted that a degradation of compound 1o was observed over time (the efforts to isolate the degradation product did not succeed). Finally, desired chalcones were all obtained in better yields using ultra-sonication and lithium hydroxide compared to the classical procedure.

Chalcones 1a-r, β-hydroxy-ketone 2a, cyclohexanol 3 and diketones 4 were evaluated in vitro, according to a previously described protocol, 32 for their ability to inhibit human FTase (Table 4). DMSO and chaetomellic acid A were used as negative and positive controls, respectively. First, it is to be noted that the determination of chalcone 1b activity was not possible due to intrinsic fluorescence at the test wavelength. Ferrocenyl compounds 1i and 1j have a similar activity with IC50 values in the low micromolar range and were the most active chalcones in the current study. Chalcones bearing a thienyl unit, except inactive chalcones 1c and 1m, were all from medium to highly active. Chalcone 1l was the most active in the thienyl-containing series (IC50 = 7.4 µM, Table 4). All molecules decorated with two chloropyridine units (e. g. chalcone 1d) or a chloropyridine and a phenyl unit (e. g. chalcones 1e-h) were completely inactive against human FTase. By-products (cyclohexanols 3a, 3b and 3q and diketones 4b, 4d and 4q) were also unable to inhibit the protein probably due to their very bulky structure preventing them to reach the binding site and interact with farnesyltransferase. Symmetric tetrachloro-substituted chalcones 1o and 1p displayed modest inhibitory activity, the 2,4dichlorophenyl substitution being preferred for the activity over the 3,4-dichlorophenyl ring. (1o: IC50 = 38.7 µM; 1p: IC50 = 96.6 µM, Table 4). Dissymmetric tetrachloro-substituted chalcones 1n and 1r interacted completely different with FTase. The 2,4dichlorophenylcarbonyl unit and the 3,4-dichlorophenyl ring placed next to the ethylenic bridge in chalcone 1n were not tolerated to inhibit the protein while the switching of the two rings (placement of the 2,4-dichlorophenyl next to the ethylenic bridge and the 3,4-dichlorophenyl next to the carbonyl function in chalcone 1r) resulted in favorable interaction (1r: IC50 = 8.7 µM, Table 4).

Finally, the most active compound in this study was not a chalcone but the β-hydroxy-ketone 2a (IC50 = 564 nM, Table 4) obtained as intermediate in the synthesis of chalcone 1a. Compared to structurally closed chalcone 1a, β-hydroxy-ketone 2a was 17.8 times more active than its congener (1a: IC50 = 12 µM, Table 4).

Fourteen molecules 1b-h, 1l-m, 1p, 1r, 3a, 3q and 4q have been selected by the National Cancer Institute (NCI), Germantown, for evaluation of their antiproliferative potential against the NCI-60 cancer cell lines, including multidrugresistant (MDR) tumor cell lines (HCT-15: colorectal adenocarcinoma; NCI/ADR-RES: human ovary adenocarcinoma; RXF 393: human kidney poorly differentiated hypernephroma; MCF7: human breast adenocarcinoma and SF-539: human CNS glioblastoma). Resumed representative biological efficacy is described in Table 5. All selected molecules underwent the NCI 60 cell one-dose screen and were tested at a 10 µM concentration.

Farnesyltransferase inhibitors generally display a cytostatic effect rather than cytotoxic activity. This behavior was also registered in this study. Unfortunately, the most active chalcones against FTase 1i and 1j and β-hydroxy-ketone 2a were not selected by the NCI for biological evaluation. Nevertheless, the other selected molecules displayed selective cytostatic effect. It is to be noted that no molecule was active against any SNC, renal and prostate cancer cells from the NCI-60 panel (see Supplementary information section for full one dose graphs for tested molecules). The most sensitive cancer cell line to chalcones was MCF7 (breast cancer). Seven chalcones (1c, 1e-h, 1l and 1m) inhibited the growth of multi-drug resistant MCF7 breast cancer cells. Chalcones 1c and 1m were the most active compounds on MCF7 cells displaying cell growth inhibition of 84.3% and 87.1% at 10 µM concentration. Leukemia K-562, colon cancer HCT-116 and melanoma LOX IMVI cells were also sensitive after treatment with chalcones. Chalcone 1f bearing a 2,4dichlorophenyl and a chloropyridine rings was the most cytostatic agent in the current study (Table 5). The chemical agent induced excellent cell growth inhibition on K-562 and CCRF-CEM leukemia cells (92.3% and 93% inhibition, respectively), on LOX IMVI melanoma cells (93.5% inhibition) and on MCF7 breast cancer cells (82.9% inhibition). A cytotoxic effect on SW-620 colon cancer cells was also registered for chalcone 1f (33.2% reduction of pre-existing tumor cells). Chalcones 1c, 1g and 1m were less active than chalcone 1f on the NCI-60 panel but have demonstrated growth inhibition greater than 50% against four different cell lines. It is interesting to note that chalcones 1f and 1g have similar structures the only difference being the positioning of the chloropyridine and 2,4dichlorophenyl units. The positioning of the chloropyridine next to the carbonyl group in chalcone 1f is more favorable for antiproliferative activity while the positioning of the same motif next to the ethylenic bridge in chalcone 1g greatly decreased the activity. The same observation applies to chalcones 1l and 1m having the left and the right units switched. When the heterocyclic unit (in this case 2-thienyl) was placed next to the carbonyl in chalcone 1m, the antiproliferative activity was stronger than that of the analogue chalcone 1l where the thienyl was placed next to the ethylenic linkage (Table 5). Chalcone 1l has thus completely lost the activity against the leukemia cells and induced 72.5% inhibition only on MCF7 cells. Differences between IC50 on human FTase in the (sub)micromolar range and IC50 on cancer cell growth in the micromolar or ten micromolar range is characteristic to potent FTIs and was reported previously on different series of compounds. 33 In the case of the presence of two different heterocyclic units in the structure of chalcones (e.g. chalcones 1b and 1c), the best cell growth inhibition was induced by chalcone 1c bearing the thienyl unit next to the carbonyl instead of chloropyridine in the structure of chalcone 1b (Table 5). The presence of two identical chloropyridines in chalcone 1d completely abolished the biological activity.

The absence of a heterocyclic ring in chalcones 1p and 1r bearing two chloro-substituted phenyl units was detrimental for the antiproliferative activity. Moreover, bulkier derivatives 3a, 3q and 4q obtained as by-products in the synthesis of target chalcones 1a and 1q, respectively, were not active on the NCI-60 panel (Table 5). Chalcones 1i-l, 1p and 1q and β-hydroxy-ketone 2a have been docked in the binding site of human farnesyltransferase in order to understand the binding mode and highlight the structural elements necessary for interaction with the protein (Figure 4).

Chalcone 1i has two-thirds of the solutions that superimposed with the highest score, as represented in fig. 4(a). The chloropyridine unit of chalcone 1i is positioned in front of the zinc of the protein while the ferrocene unit occupies the pocket formed by amino acids Tyr861, Arg702 and Tyr166, also known as the A2 pocket. 34 Chalcone 1j, reverse analogue of chalcone 1i, also interacts with the zinc via the chloropyridine unit while the ferrocene occupies the same pocket as in chalcone 1i (Fig. 4(b)). The positioning of the ferrocene from chalcones 1i and 1j agrees with our previous studies on ketones bearing bulky lipophilic groups such as paracyclophane, adamantane, noradamantane or ferrocene located in the A2 binding site of farnesyltransferase. 34 Chalcone 1k has twelve solutions superimposed with the best score (Fig. 4(c)). In this configuration, it attaches to the Tyr166 via a hydrogen bond between the carbonyl of the chalcone and the phenol of the amino acid. The other solutions for chalcone 1k are placed in the opposite direction with the thienyl unit in place of the 2,4-dichlorophenyl ring and do not make any single interaction. This may explain the moderate IC50 value of 20.8 µM obtained for this compound on FTase. Chalcone 1l, structurally very closed to chalcone 1k, interacts in similar way (Fig. 4(d)). The hydrogen bond is conserved between the carbonyl of the chalcone linkage and the phenol of the Tyr166 and the chlorine atoms are slightly closer for interaction with the zinc atom compared to chalcone 1k.

Tetrachloro-substituted chalcone 1p places less well, with two solutions (Fig. 4(e) and 4(f)) and at least two other smaller groups that are less well defined. The conformation with the highest score does not form hydrogen bond but is correctly placed to interact with the zinc cation (Fig. 4(e)). The second conformation is placed conversely and forms a hydrogen bond with Tyr166 (Fig. 4(f)). However, the very small number of solutions can justify the modest activity of this molecule on the FTase (IC50 = 96.6 µM).

Chalcone 1q behaves differently and forms a hydrogen bond via its carbonyl group but with the Tyr861 (Fig. 4(g)) and is fixed in the active site by stacking with the farnesyl moiety.

The best farnesyltransferase inhibitor identified in the current study was β-hydroxy-ketone 2a synthesized as racemate. The both enantiomers R and S of 2a were investigated in the binding site of the protein (Fig. 4(h) and 4(i)). The enantiomer R has two conformations which are positioned in the same place and which maintain a hydrogen bond with Tyr861 via the carbonyl unit (Fig. 4(h)). The difference between the two conformations retrieved for (R)-2a is the change of side of the 2-thienyl rings. The enantiomer S is even better positioned and has a single solution for 26 of the 30 conformations. The hydrogen bond with Tyr166 is still present and there is a second interaction with Arg702.

The cyclohexanol derivatives (e.g. compound 3q) narrowly enter in the pocket of farnesyltransferase. There is no solution that stands out from the different loose conformations (data not shown) which may explain the absence of the biological activity of these bulky derivatives.

To conclude, an ultra-fast green Claisen-Schmidt synthesis of a series of chalcones has been developed under sonication in a mixture of ethanol/water and using lithium hydroxide as base. The chemical transformations were compared with corresponding classical magnetic stirring reactions and allowed to reduce the reaction duration from hours to 10 seconds with similar to superior results. In some cases, by-products were detected in the medium and were purified and identified as cyclohexanol or bisketone derivatives.

Synthesized molecules 1-4 have been evaluated on human farnesyltransferase inhibitors in vitro. Chalcones 1 were globally able to inhibit the protein and allowed to establish additional structure-activity relationships in the series: i) the ferrocenyl moiety was tolerated and provided the most active chalcones in this study with IC50 in the low micromolar range (e.g. 1i and 1j); ii) the presence of two pyridine units (e. g. chalcone 1d) or a chloropyridine and a phenyl unit (e. g. chalcones 1e-h) in the structure of chalcones abolished the activity; iii) thienylsubstituted chalcones conserved an inhibitory potential; iv) symmetric tetrachloro-substituted chalcones 1o and 1p displayed modest activity; v) placement of the 2,4-dichlorophenyl next to the ethylenic bridge and the 3,4-dichlorophenyl next to the carbonyl function in chalcone 1r resulted in favorable interaction; vi) bulky by-products 3 and 4 were not active on FTase; vii) the β-hydroxy-ketone 2a was the most active compound in the current study.

Compounds were also tested for their ability to prevent cancer cell growth and underwent the one-dose screen at the National Cancer Institute (NCI) at 10 µM concentration. Molecules showed a cytostatic effect on different cell lines with particular activity against MCF7, breast cancer cells.

Tyr166

Tyr166 Tyr166 Zn Zn Zn FPP FPP FPP Tyr861 Tyr861

Arg702

The β-hydroxy-ketone 2a deserves further biological investigation on progeria cells. Additional pharmacomodulations are expected on this scaffold and could lead to a new generation of potent farnesyltransferase inhibitors.

Figure 1 .

 1 Figure 1. Structure of reported human FTase inhibitors (indolizine-chalcone (A) and Lonafarnib (B)) and of target chalcones 1.

Scheme 2 .

 2 Scheme 2. Synthesis of enone 5 as a by-product of the Claisen-Schmidt condensation of 1-(ferrocenyl)-ethanone and 6-chloropyridine-3-carbaldehyde.

Scheme 3 .

 3 Scheme 3. Cannizzaro reaction of 6-chloro-pyridine-3carbaldehyde.

Figure 3 .

 3 Figure 3. Structure of cyclohexanols 3a, 3b' and 3q compared to literature equivalents.

2 a

 2 Data obtained from NCI's in vitro 60-cell one dose screen (experiments conducted with 10 µM concentration of tested molecule). b GI% is the percentage of growth inhibition of tumor cells. c No inhibition effect. d Not tested. e Cytotoxic effect: growth percent of -33.2% on SW-620 cells induced by chalcone 1f.

Figure 4 .

 4 Figure 4. Docking of synthesized molecules in the active site of human farnesyltransferase: chalcone 1i (a), chalcone 1j (b), chalcone 1k (c), chalcone 1l (d), chalcone 1p (e-f), chalcone 1q (g) and β-hydroxy-ketone 2a (enantiomer R (h); enantiomer S (i)). Yellow dashed lines represent hydrogen bonds between investigated molecules and aminoacids present in the active site of human farnesyltransferase.

Table 1 .

 1 Influence of the base on the Claisen-

	condensation of (hetero-)aryl ketones
	and (hetero-)aryl aldehydes, and compounds synthesized in this
	study (yields are given within brackets, relatively to aldehyde and
	ketone reagents; regular writing correspond to ultrasounds mediated
	experiments, and italic writing correspond to classical stirring
	experiments).

b

Complex mixture.

Table 2 .

 2 Influence of the solvent on the Claisen-Schmidt condensation of 1-thiophen-2-yl-ethanone and thiophene-2carbaldehyde (1 equiv LiOH, 0.32 mmol, 0.25 mol/L). a

					2a/1a/3a
	Entry	Solvent	Ti -Tf (°C)	Energy (J)	( 1 H NMR yield relatively to ketone reagent)
	1	H2O	22-26	181	>90/0/0
	2	EtOH	23-28	288	0/67/15
	3	DMF	23-32	306	0/0/0
	4	THF	23-30	264	0/0/0
	5	2-Me-THF	23-30	256	0/0/0
	6	H2O:EtOH (1/1)	26-32	356	0/>90/0
	7	H2O:EtOH (1/3)	25-31	290	13/77/0
	8	H2O:EtOH (3/1)	27-32	354	0/>90/0
	9	Upscale of entry 6 (10X)	28-30	377	0/>90/0

a 1 H NMR of the crude was realized immediately after ultrasonic-probe mediated condensation of 1-thiophen-2-yl-ethanone and thiophene-2carbaldehyde. Ti = initial medium temperature; Tf = final medium temperature.

Table 3 .

 3 Screening of the optimal volumes of water and EtOH, and number of equivalents of LiOH on the Claisen-Schmidt condensation of 1-thiophen-2-yl-ethanone and thiophene-2-carbaldehyde. a

	Entry	Water (mL)	EtOH (mL)	Quantities of reagents (mmol)	LiOH (equiv)	Amplitude	Duration (s)	Ti -Tf (°C)	Energy (J)	2a/1a/4a/3a ( 1 H NMR yield relatively to ketone reagent)
	1	12.5	0	3.2	1	0.3	10	25-28	350	>90/0/0/0
	2	12.5	0	3.2	0.1	0.3	10	23-27	340	>90/0/0/0
	3	12.5	0	3.2	0.01	0.3	10	24-28	362	>90/0/0/0
	4	12.5	0	3.2	1	0.3	5	21-26	173	>90/0/0/0
	5	12.5	0	3.2	1	0.3	20	25-38	699	>90/0/0/0
	6	12.5	0	3.2	1	0.2	10	23-29	283	>90/0/0/0
	7	12.5	0	3.2	1	0.4	10	23-29	279	>90/0/0/0
	8	0	12.5	0.32	1	0.3	10	23-28	288	0/67/0/15
	9	12.5	0	0.32	1	0.3	10	22-26	181	0/>90/0/0
	10	6.25	6.25	0.32	1	0.3	10	26-32	356	0/>90/0/0
	11	62.5	62.5	3.2	1	0.3	10	28-30	377	0/>90/0/0
	12	3.1	9.4	0.32	1	0.3	10	25-31	290	13/77/0/0
	13	9.4	3.1	0.32	1	0.3	10	27-32	354	0/>90/0/0
	14	0	12.5	0.32	0.1	0.3	10	23-31	310	0/67/0/15
	15	6.25	6.25	0.32	0.1	0.3	10	27-32	342	0/>90/0/0
	16	12.5	0	0.32	0.1	0.3	10	23-27	354	>90/0/0/0

a 1 H NMR of the crude was realized immediately after ultrasonic-probe mediated condensation of 1-thiophen-2-yl-ethanone and thiophene-2-carbaldehyde. Ti = initial medium temperature; Tf = final medium temperature.

Table 4 .

 4 Inhibitory activity of studied compounds on human FTase. a

	Entry Compound	% Inhibition of FTase at 100µM	IC50 (µM) b	R 2
	1	1a	100	12.0	0.9946
	2	1b	__ c	__ c	__ c
	3	1b'	86	24.9	0.9228
	4	1c	28	-	-
	5	1d	26	-	-
	6	1e	54	-	-
	7	1f	53	-	-
	8	1g	39	-	-
	9	1h	22	-	-
	10	1i	64	1.0	0.9464
	11	1j	97	4.6	0.9818
	12	1k	70	20.8	0.9823
	13	1l	98	7.4	0.9493
	14	1m	47	-	-
	15	1n	53	-	-
	16	1o	69	38.7	0.9815
	17	1p	60	96.6	0.9090
	18	1q	84	25.8	0.9415
	19	1r	74	8.7	0.9867
	20	2a	86	0.6	0.8356
	21	3a	2	-	-
	22	3b'	24	-	-
	23	3q	11	-	-
	24	4b	19	-	-
	25	4d	4	-	-
	26	4q	0	-	-
	27	Chaetomellic acid A	100	0.18	0.9898

a Inhibition of human farnesyltransferase in vitro at 100 µM concentration of tested compound.

b IC50 values are indicated as the mean of two independent experiments each realized in duplicate.

c Intrinsic fluorescence at the test wavelength (data not reliable).

Table 5 .

 5 Results of the in vitro human cancer cell growth inhibition for selected compounds 1b

-h, 1l-m, 1p-r, 3a, 3q and

  4q a,b

		Compound	1b	1c	1d	1e	1f	1g	1h	1l	1m	1p	1r	3a	3q	4q
	Cell type	Cell line								GI% a,b
		SR	17.5	n.t. d	n.t.	n.t.	n.t.	55.7	n.t.	-	78.0	6.1	2.8	7.9	10.0	9.7
		K-562	2.9	65.8	6.1	48.3	92.3	18.9	25.3	1.7	71.6	9.7	2.4	14.5	16.5	25.0
	Leukemia	CCRF-CEM		38.6	3.1	26.6	93.0	6.5	15.2	-	31.3	3.7	-	5.0	22.3	7.9
		RPMI-8226 95.8	n.t.	n.t.	n.t.	n.t.	8.9	n.t.	-	66.9	7.2	5.4	17.7	26.8	17.3
	Non-Small Cell	NCI-H522	18.0	37.2	5.3	45.8	61.3	24.2	17.0	8.6	24.7	8.1	11.4	15.3	18.5	17.3
	Lung Cancer
		KM12	-c	57.8	1.0	44.4	23.5	3.6	32.4	2.8	15.4	-	0.3	3.4	4.7	4.8
	Colon Cancer	SW-620	-	47.6	-	40.1	100.0 e	7.9	17.6	5.5	29.6	-	-	-	12.7	4.4
		HCT-116	8.9	35.5	-	55.1	63.3	56.6		2.2	39.8	6.1	13.9	6.1	21.2	34.7
	Melanoma	LOX IMVI	-	85.0	-	37.6	93.5	65.1	21.9	-	45.7	-	-	7.8	3.1	10.9
	Ovarian Cancer	OVCAR-3	-	45.8	-	43.5	21.5	-	11.2	-	6.8	-	-	-	11.5	-
	Breast cancer	MCF7	32.3	84.3	8.8	66.4	82.9	63.4	69.8	72.5	87.1	14.2	28.3	12.2	18.3	10.

Acknowledgments

The authors gratefully acknowledge Interreg NWE for financial support of this work which is part of the RIVER project. Acknowledgements are also addressed to Yncréa Hauts-de-France, HEI, for financial support of this work, especially for the investments allocated to our team.

The authors also acknowledge the National Cancer Institute (NCI) for the biological evaluation of compounds on their 60-cell panel: the testing was performed by the Developmental Therapeutics Program, Division of Cancer Treatment and Diagnosis (the URL to the Program's website: http://dtp.cancer.gov).