Fahima Djelil
email: fahima.djelil@imt-atlantique.fr

Jean-Marie Gilliot
email: jm.gilliot@imt-atlantique.fr

Serge Garlatti
email: serge.garlatti@imt-atlantique.fr

Philippe Leray
email: philippe.leray@univ-nantes.fr

Supporting Self-Regulation Learning Using a Bayesian Approach. Some Preliminary Insights

Keywords: Self-Regulated Learning, Open Learner Models, Knowledge Tracing

Self-Regulated Learning (SRL) is actually challenging modern online environments (e-learning platforms, MOOCs, exercise-based platforms, ...). A large emergent literature points out the need for empirical studies on approaches that can help to build tools for measuring and scaffolding SRL. This paper is a state of the art aiming to identify a set of key avenues to conduct a future experimental research on this theme. More importantly, we present the approaches of Open Learner Models, and knowledge Tracing through Bayesian Networks that offer promising insights to model student knowledge, measure SRL levels and provide appropriate interventions to foster student SRL skills.

Introduction

Although self-regulation has a long-time existence in the Education literature [START_REF] Zimmerman | Development of a structured interview for assessing student use of selfregulated learning strategies[END_REF], it is a key challenge in modern online environments [START_REF] Araka | Research trends in measurement and intervention tools for self-regulated learning for e-learning environments-systematic review (2008-2018)[END_REF]. Indeed, with the advent of MOOCs, and the success of exercise-based platforms, large-scale online environments are becoming widespread, both in e-learning and in blended learning [START_REF] Light | Increasing student engagement in math: The use of khan academy in chilean classrooms[END_REF], and the success of such learning environments depends on the ability of the learners to take control on their own learning [START_REF] Araka | A conceptual model for measuring and supporting selfregulated learning using educational data mining on learning management systems[END_REF]. This learner's ability is referred to Self-Regulated Learning (SRL), which is known to have a good potential on autonomy development and maintaining motivation for learners [START_REF] Schunk | Motivation and self-regulated learning: Theory, research, and applications[END_REF]. SRL is also known to have a positive impact on academic achievements [START_REF] Zimmerman | Self-regulated learning and academic achievement: An overview[END_REF]. However, empirical studies aiming to support SRL at scale are lacking. There is a real need for analysing learner's activities for scaffolding SRL, especially in MOOCs [7; 8; 9]. Promoting SRL implies to provide students with interventions, defined as activities or events that can trigger student SRL development during a learning episode, for example during an online course [START_REF] Araka | Research trends in measurement and intervention tools for self-regulated learning for e-learning environments-systematic review (2008-2018)[END_REF].

Our main objective in this paper is to draw some insights on how to support successfully self-regulated learning at a large scale, and explore existing approaches that aim to estimate acquired skills levels and metacognitive levels about students to provide appropriate interventions. Our main research questions are the following:

1. What are the research requirements for SRL support at a large scale? 2. What are the main existing SRL supports that have been investigated? 3. What are the computational approaches for supporting SRL in online learning environments?

First of all, we introduce the concept of self-regulation and point out the need for empirical studies regarding SRL measurement and intervention. We present the approaches of Open Learner Models and Knowledge Tracing, which offer promising perspectives towards SRL. Finally, we discuss these avenues for a future research and we conclude on our interest in Bayesian modelling techniques to support personalization and self-regulated learning.

2 Self-Regulated Learning (SRL)

SRL Theory

Self-regulation is a well-established concept that includes the cognitive, metacognitive, behavioural, motivational and emotional/affective aspects of learning [START_REF] Panadero | A review of self-regulated learning: Six models and four directions for research[END_REF]. A typology of fourteen strategies of SRL that deal mainly with information processing, have been established, using a collection of qualitative data [START_REF] Zimmerman | Development of a structured interview for assessing student use of selfregulated learning strategies[END_REF]. This includes cognitive strategies (Reviewing courses, organising information, memorising), metacognitive strategies (self-assessment, planning) and strategies for seeking additional information (using documentary resources, asking others for help). Another facet of self-regulation is to protect the intention to learn by regulating motivation, emotion and attention [START_REF] Cosnefroy | Se mettre au travail et y rester: les tourments de l'autorégulation[END_REF].

Self-regulation development or learning requires giving a certain level of autonomy to learners [START_REF] Schunk | Motivation and self-regulated learning: Theory, research, and applications[END_REF]. Autonomy is apprehended here from a constructivist perspective as "construction" and as "constructed" at the same time. This comprises a certain balance between prior knowledge and new knowledge, resulting from exchanges between the learner and his environment, and may be a product of his representations and confrontation with others" [START_REF] Eneau | Un scénario collaboratif pour développer l'apprentissage d'adultes, en ligne et à distance[END_REF]. Self-Regulated learners have the ability to manage their own learning, and assume an active role in achieving their academic goals and controlling their learning process [START_REF] Araka | A conceptual model for measuring and supporting selfregulated learning using educational data mining on learning management systems[END_REF].

Research Requirements for Supporting SRL in Learning Environments

Self-regulated learning analysis has appeared as one of the main themes for future research in e-learning environments [START_REF] Araka | Research trends in measurement and intervention tools for self-regulated learning for e-learning environments-systematic review (2008-2018)[END_REF]. Many studies agree on the relevance of self-regulated learning in MOOCs, and point out the need for more empirical analysis of learners' activities [7; 8; 9]. Some of the main objectives of such research are to reduce the number of drop out students, to enhance learners' motivation and to increase learning success in MOOCs [START_REF] Alario-Hoyos | Understanding learners' motivation and learning strategies in moocs[END_REF]. Indeed, in such learning environments, the lack of direct guidance from an instructor makes self-regulation very relevant [START_REF] Alario-Hoyos | Understanding learners' motivation and learning strategies in moocs[END_REF]. Since learning environments supporting SRL place the learner as the controller of the learning process, these environments should provide tools for measuring and scaffolding SRL [START_REF] Araka | A conceptual model for measuring and supporting selfregulated learning using educational data mining on learning management systems[END_REF]. A scaffold or intervention is defined as an activity or event that can trigger SRL skills development of learners within an online course, while SRL strategy measures seek to establish SRL levels for learners [13; 4].

According to [START_REF] Jézégou | Se former à distance: regard sur les stratégies d'autorégulation environnementale d'étudiants adultes[END_REF], self-regulation is divided into three different types: internal, behavioural and environmental. Different approaches described in the literature deal with internal self-regulation, i.e., affects, emotions, and motivation of learners, and behavioural self-regulation, i.e., learning behaviours. As highlighted by [START_REF] Jézégou | Se former à distance: regard sur les stratégies d'autorégulation environnementale d'étudiants adultes[END_REF], there is a lack of works dedicated to analyse learning practices, leading to a kind of environmental self-regulation, i.e. to have an optimal educational environment for the success of learning.

A recent systematic review [START_REF] Araka | Research trends in measurement and intervention tools for self-regulated learning for e-learning environments-systematic review (2008-2018)[END_REF] makes a comprehensive synthesis, indicating that there is actually a noticeable shift from using tools that only measures SRL, to tools that go beyond the measurement to provide interventions. This review describes three trends of measurements that are used to promote SRL skills for learners in online learning environments: (i) SRL measured through Self-report tools, where learners reflect in a manual way, (ii) online measure of SRL based on learning data comprising a series of events, and (iii) measurement approaches that provide SRL interventions. This last trend is still emerging, and there is a lack of frameworks that help to integrate SRL measurement and interventions within data models [START_REF] Araka | Research trends in measurement and intervention tools for self-regulated learning for e-learning environments-systematic review (2008-2018)[END_REF].

Regarding the first research question "What are the research requirements for SRL support at a large scale?", we would say that more empirical work is needed for investigating how to provide personalised interventions and scaffolds to help learners in the SRL process. According to the literature, several approaches of interventions have been studied as shown in the next paragraph.

Existing Approaches for SRL Support and Interventions

Although a lack of empirical studies about how SRL can be supported in online learning environments, there is some evidence of different approaches that are used to provide SRL interventions. Three different approaches supporting learners to enhance their SRL skills were described [START_REF] Araka | A conceptual model for measuring and supporting selfregulated learning using educational data mining on learning management systems[END_REF]: (i) Learning Analytics (LA) through dashboard visualizations, enhancing learner activities through provision of suggested insights, (ii) Artificial Intelligent software agents, that offer assistive activity guidance to students, and (iii) learner feedback built from web-enabled prompts, facilitated through LA reports that assess learners' use of SRL strategies.

Other similar intervention approaches include [START_REF] Wong | Supporting self-regulated learning in online learning environments and moocs: A systematic review[END_REF]: (i) prompts such as questions, suggestions and short answer problems, (ii) feedbacks to trigger reflective activities and deepen learner understanding, and (iii) integrated support systems which are tools to support goals setting and SRL strategies. This also addresses the role of human factors including previous knowledge, cognitive and metacognitive abilities in adapting SRL approaches to best fit the individual learner needs.

According to [START_REF] Garcia | Systematic literature review: Self-regulated learning strategies using elearning tools for computer science[END_REF], some existing studies provide partial SRL support, and rarely combine different approaches. For example, self-evaluation is already well supported in programming courses thanks to automated assessment tools, as well as goal setting and planning, while other strategies are not yet experimented [START_REF] Garcia | Systematic literature review: Self-regulated learning strategies using elearning tools for computer science[END_REF].

Regarding the second research question "What are the main existing SRL supports that have been investigated ?", the most common approaches described in the literature, especially in online learning environments, include the use of Learning Analytics, prompts and feedback, and support systems for activity guidance, goal setting, planning and selfreflection.

Automatic Guidance VS Autonomy

Learning environments supporting SRL should provide students the ability to set up their learning goals, identify activities and strategies to achieve these goals, monitor and evaluate their progress [START_REF] Nussbaumer | A competence-based service for supporting selfregulated learning in virtual environments[END_REF]. Increasing students' autonomy supposes mixing self-reflection and explicit guidance when necessary. A standard solution to tailor the learning process to the learner needs include adaptive systems and Intelligent Tutoring Systems (ITS). While these systems are supposed to guide the learning process, SRL requires to give control to the learners. In other words, it is necessary to take a balance between automatic guidance and autonomy. Thus, adaptive learning systems and Intelligent Tutoring Systems should be modified to support SRL, that is to say, giving control to the learner and mixing self-reflection and explicit guidance when necessary. Moreover, adaptive systems and ITS led to the development of learner models [START_REF] Corbett | Intelligent tutoring systems[END_REF]. Some particular learner models appeared as a complementary way to support SRL: Open Learner Models (OLM) [START_REF] Nussbaumer | A competence-based service for supporting selfregulated learning in virtual environments[END_REF]. Indeed, Open Learner Models are scrutable and might support self-reflection and metacognition as well as efficient, learner-controlled student modelling.

3 Learner Models to support SRL

Open Learner Models

A self-regulated learning system can be seen as a modified adaptive learning systems or ITS that supports SRL [START_REF] Nussbaumer | A competence-based service for supporting selfregulated learning in virtual environments[END_REF]. Providing a higher level of information could help learners to refine critical reflection and to make informed decisions. Indeed, the system has to provide personalized learning (rec-ommending a next exercise, giving a simple list of relevant exercises, etc.), but also to encourage self-reflection, and to be able to justify interventions according to learners' goals and achievements. This is in relation to the concept of Open Learner Model (OLM), that goes beyond the learner model used in ITS and limited to a personalized learning perspective. While adaptive systems make use of learner information (e.g., knowledge state) by using a learner model to adapt a course or a system behaviour, the OLM approach pursues the idea of displaying learner information by making the learner model open to the learner and letting him choose the next steps [START_REF] Bull | Open learner models[END_REF]. In traditional ITS, control is given to the system which guides the learner in the learning process, but a self-regulated learner should take over the control on his own [START_REF] Nussbaumer | A competence-based service for supporting selfregulated learning in virtual environments[END_REF]. This is why the OLM approach is used to support the learner's reflection process by providing formative feedback on the learning process or even engaging negotiation with the learner [START_REF] Bull | Open learner models[END_REF]. Previous works on OLMs showed interesting effects on engagement with the learning content and the system [START_REF] Guerra | An intelligent interface for learning content: Combining an open learner model and social comparison to support self-regulated learning and engagement[END_REF]. For example, [START_REF] Mitrovic | Evaluating the effect of open student models on self-assessment[END_REF] showed that OLM helps learners to select better problems. Another example is reported in on the Leaning Tracker tool [START_REF] Davis | Retrieval practice and study planning in moocs: Exploring classroom-based self-regulated learning strategies at scale[END_REF] which provides a significant increased students' success, comprising a dashboard with visualisation comparing learners' own behaviour with successful learners' behaviour, in order to promote metacognition.

Personalised learning also comprises recommender systems that aim behavioural development by proposing specific activities and providing a structured environment. However, as noticed in the UNESCO report (2019) [START_REF]Rethinking pedagogy: exploring the potential of digital technology in achieving quality education[END_REF], it is important not to delegate decision-making to Artificial Intelligence (AI), but rather to serve the aims of humans. Learners should be able to make informed decisions by themselves. OLM allows the user (learner, teacher, peers and/or other stakeholders in the education process) to view the content of the learner model, in a human understandable form [START_REF] Bull | Smili: A framework for interfaces to learning data in open learner models, learning analytics and related fields[END_REF]. This focus on understandability is necessary if users are to be able to act appropriately on the learner model information [START_REF] Bull | Smili: A framework for interfaces to learning data in open learner models, learning analytics and related fields[END_REF]. To ensure the support is effective, the learner models must be interpretable by users [START_REF] Conati | Ai in education needs interpretable machine learning: Lessons from open learner modelling[END_REF], i.e., based on explainable models. As claimed in [START_REF] Rosé | Explanatory learner models: Why machine learning (alone) is not the answer[END_REF]: "the recommender system will not be transparent enough to adapt in an insightful way to the individual needs of students nor to the context of use". It is necessary to bring interdisciplinary expertise to develop explanatory learner models that provide interpretable and actionable insights in addition to accurate prediction, rather than relying on AI expertise alone [START_REF] Rosé | Explanatory learner models: Why machine learning (alone) is not the answer[END_REF].

A great variety of student modelling techniques have emerged based on student interactions in learning activities and knowledge measuring [START_REF] Bodily | Open learner models and learning analytics dashboards: a systematic review[END_REF]. Knowledge Tracing is a popular technique used in ITS, in which modelling and predicting student knowledge is a fundamental task.

Knowledge Tracing

Knowledge Tracing is a standard for inferring student knowledge [START_REF] Donghua | Deep knowledge tracing based on bayesian neural network[END_REF]. It is the task of modelling student knowledge over time, so that we can accurately predict how students will perform on future interactions. This means that resources can be suggested to students based on their individual needs, and content which is predicted to be too easy or too hard can be skipped or delayed. The problem of knowledge tracing is inherently difficult since human learning is complex, and has been heavily studied within the intelligent tutoring community to find appropriate and rich models [START_REF] Piech | Deep knowledge tracing[END_REF]. In this section we describe two main approaches that are used to build student models, with their related pros and cons: Bayesian Knowledge Tracing (BKT) [START_REF] Corbett | Knowledge tracing: Modeling the acquisition of procedural knowledge[END_REF], which is the most popular one and Deep Knowledge Tracing (DKT) [START_REF] Piech | Deep knowledge tracing[END_REF], which emerges as an alternative to BKT.

Bayesian Knowledge Tracing

A popular approach for student modelling is Bayesian Knowledge Tracing (BKT), using simple Bayesian Networks [START_REF] Millán | Bayesian networks for student model engineering[END_REF], or mostly Dynamic Bayesian Networks, having the potential to increase the representational power of the student model and improve prediction accuracy [33; 34; 35; 36]. BKT models a learner's knowledge state as a set of binary variables [START_REF] Corbett | Knowledge tracing: Modeling the acquisition of procedural knowledge[END_REF], each of which represents understanding or non-understanding of a single concept. For instance, when making a prediction, the model is provided with the tag of the exercise being answered, and must predict whether the student will get the exercise correct, or false. This binary representation of students' understanding or skills is seen as the main limitation of BKT [START_REF] Piech | Deep knowledge tracing[END_REF]. However, this limitation can easily be lifted, since Bayesian networks are not restricted to binary random variables [START_REF] Alpaydin | Introduction to machine learning[END_REF].

Knowledge representation. The selection of Bayesian

Networks to model students' learning is justified by several reasons in the literature. First, such methods are used in different domains in decision making [START_REF] Alpaydin | Introduction to machine learning[END_REF], and in a wide variety phenomenon in ITS including models of students' knowledge [START_REF] Baker | More accurate student modeling through contextual estimation of slip and guess probabilities in bayesian knowledge tracing[END_REF]. Bayesian Networks can accommodate both empirical and theoretical knowledge [START_REF] Conati | Empirically building and evaluating a probabilistic model of user affect[END_REF]. More specifically, this modelling approach can take advantage of both empirical and theoretical knowledge of SRL in large-scale online environments [START_REF] Sabourin | Utilizing dynamic bayes nets to improve early prediction models of self-regulated learning[END_REF]. Bayesian Networks are defined with a directed acyclic graph between random variables that encode the probabilistic dependencies between variables, and a set of conditional probability distributions (the parameters) that encode the strength of these dependencies. Both the structure and the conditional dependencies can be learned using a variety of possible algorithms [START_REF] Alpaydin | Introduction to machine learning[END_REF] or specified by hand. At the intersection between Knowledge representation and Machine Learning in Artificial Intelligence, these models can be iteratively built from knowledge elicitation [41; 42] to learning from data through successive refinements [START_REF] Kjaerulff | Probabilistic networks for practitioners-a guide to construction and analysis of bayesian networks and influence diagrams[END_REF].

Interpretable models. Bayesian Networks also have other important advantages derived from their typology as probabilistic graphical models. One is that we can visualize the interaction of variables and have a better "understanding" of the process, for example, when the dependencies between the random variables have been defined in a causal way [START_REF] Van Der Gaag | Probabilities for a probabilistic network: a case study in oesophageal cancer[END_REF]. In fact, thanks to their graphical dimension and their causality expressiveness, Bayesian Networks are de-facto "interpretable" [START_REF] Lacave | A review of explanation methods for bayesian networks[END_REF], i.e., can be easily "inspected" and "understood" by experts or users.

Deep Knowledge Tracing

Recent studies showed improvement over state of the art performance on dynamic probabilistic models. Models supported by Recurrent Neural Networks (RNNs) have been applied to the Knowledge Tracing Task, leading to the concept of Deep Knowledge Tracing (DKT), showing predictive power superior to BKT [30; 45]. RNNs were applied to data from Khan Academy's online courses to predict students' performance [START_REF] Piech | Deep knowledge tracing[END_REF]. An improvement of this work is performed later on a dataset from the MOOC Hour of Code, in which data is structured in the form of program submissions, instead of a binary form [START_REF] Wang | Deep knowledge tracing on programming exercises[END_REF]. The deep learning model trains on a student's history of past code submissions and predicts the student's performance on the next exercise [START_REF] Wang | Deep knowledge tracing on programming exercises[END_REF]. These studies highlighted a rich property of RNNs, such as their ability to use information from an input in a prediction at a much later point in time. The learned model does not need expert annotations (it can learn concept patterns on its own) and it can operate on any student input that can be vectorized. However, one disadvantage of RNNs over the previous Bayesian Network based approaches, which was highlighted [START_REF] Piech | Deep knowledge tracing[END_REF], is that RNNs require large amounts of training data, and so are not well suited to a small classroom environment. A more recent work [START_REF] Donghua | Deep knowledge tracing based on bayesian neural network[END_REF] explored a novel method to improve the prediction process by combining Bayesian Neural Network with DKT, in order to better model students' learning and track the process of students' knowledge acquisition. This approach showed performances on modelling feature-rich students' behaviour data, preventing overfitting, enhancing the generalisation ability of the model, and accelerating the convergence speed of the model [START_REF] Donghua | Deep knowledge tracing based on bayesian neural network[END_REF].

Discussion and Conclusions

SRL aims both increasing students' autonomy and guiding their own learning while providing personalised and necessary support. Based on that, one approach to face the challenges of measuring SRL in Open Online Environments and building personalised interventions is to combine the OLM and Knowledge Tracing approaches. To provide effective SRL support, student models have to be interpretable by users [START_REF] Conati | Ai in education needs interpretable machine learning: Lessons from open learner modelling[END_REF]. Providing explanations on learning process stimulate reflection on metacognition, beyond activity recommendations.

From the presented student modelling approaches, we would say that DKT seems to be an improvement over BKT with regard to the prediction performance [30; 45], however BKT offers more valuable insights on how explanations are communicated to the end user to be understandable [START_REF] Lacave | A review of explanation methods for bayesian networks[END_REF]. DKT is also empirically well suited in case of large amounts of data, having high semantic complexity and manually choosing features is tedious or insufficient [START_REF] Wang | Deep knowledge tracing on programming exercises[END_REF]. In addition, from the approach that combines Bayesian neural network with DKT [START_REF] Donghua | Deep knowledge tracing based on bayesian neural network[END_REF], we would say that DKT may not be sufficient on its own and needs to take advantage from Bayesian Networks, although it provides a high prediction performance. Since SRL needs theoretical knowledge to be tracked and measured, we could expect a set of human input and expert knowledge needed to feed the learner model, and so Bayesian Networks would be more suited.

Moreover, existing work [START_REF] Sabourin | Utilizing dynamic bayes nets to improve early prediction models of self-regulated learning[END_REF] has shown promise in being able to predict SRL behaviours early into interaction, using Bayesian Networks. This may include features related to the self-regulation process, such as planning and monitoring behaviours [START_REF] Sabourin | Utilizing dynamic bayes nets to improve early prediction models of self-regulated learning[END_REF].

Regarding our ultimate research question "What are the computational approaches for supporting SRL in online learning environments?", applying Bayesian modelling techniques would be a sound approach towards supporting SRL in Open Online Environments, and so at scale. This can include SRL measures and personalised interventions. Bayesian Networks (or Dynamic ones) offer interesting insights for acquiring and updating learners' models as OLMs, about cognitive and metacognitive skills and progression of learners. In a future work, we will attempt to test this approach on data from large populations, to conduct experimental research. This will include data corpus resulting from available large scale learning platforms comprising MOOCs series (available on FUN and Edx platforms) and an exercise-based platform (France IOI).

Adopting a design-based research approach is necessary to tackle empirical analysis of learner activities [START_REF] Štemberger | Design based research: The way of developing and implementing[END_REF]. This will enable to combine two complementary research focuses with feedback from real experiments (technical and social approach). In such context, conducting an experimental study necessitate an interdisciplinary research.

Acknowledgements

Based on this preliminary work, future research experiments will be conducted as part of the Xcale project (2021-2024), funded by the French National Research Agency (ANR)/AAPG 2020.


domain: chim sdv

text:
Xiaoxi Yu

Hélène Sotin

Nathalie Marnet

Sylvain Guyot

Characterization of complex phenolic compounds in rapeseed and sunflower biomass generated during biorefinery

In addition to lipids, proteins and polysaccharides, rapeseed and sunflower seeds are known to contain significant amounts of phenolic compounds. They are located both in the kernels and in their fibrous hulls. After oil production, valorization of those phenolic compounds can be considered due to their putative health benefits. Previous studies reported that the concentration of some complex polyphenols such as condensed tannins in rapeseed hulls tends to decrease with advanced plant maturity. They may be oxidized or form strong interactions with other polymers in the plant (proteins, polysaccharides). Our work aims at evaluating the applicability of acidolysis methods (i.e. direct phloroglucinolysis and butanol-HCl acidolysis) for determining complex phenolic compounds in rapeseed and sunflower biomass (oil cake and hull).

Introduction

Materials and methods

Bio2actives -From biomass &biorefinery to actives and ingredients 6th & 7th July, 2017, Centre des Congrès, Quimper (France)

Main phenolic compounds in the rapeseed

 After delipidation, phloroglucinolysis or butanol-HCl reaction can be applied directly on rapeseed or sunflower samples for the determination of complex phenolic compounds (e.g. condensed tannins). Contrary to rapeseed hulls, few condensed tannins were detected in rapeseed oil cakes and in sunflower samples.  Further work will be done to improve the estimation of oxidized phenolic compounds.

Conclusions Acknowledgement

  : adduct (extention unit + phloro) CAT: catechin after reaction (terminal units) EC: epicatechin after reaction (terminal units


Dima El

Mohammed Al-Kharaz

Bouchra Ananou

Guillaume Graton

Mustapha Ouladsine

Jacques Pinaton

Combining Approaches of Brownian Motion and Similarity Principle to Improve the Remaining Useful Life Prediction

This paper proposes a data-driven framework for Remaining Useful Life (RUL) prediction, based on the Brownian Motion model (BM) and the similarity principle, for an operating system given its health indicator. It addresses the issues of noisy and limited run-to-failure (R2F) data. The Percentile filtering is used to extract, from the R2F data, 100 monotonic profiles used as references in the modeling and the RUL prediction. Then, the similarity is computed between these references and the Health Indicator (HI) of the operating system. Fitting the most similar reference into the BM improves the RUL prediction. A numerical application using simulated data justifies the accuracy of this approach.

I. INTRODUCTION

Nowadays, Condition-Based Maintenance (CBM) [START_REF] Jardine | A review on machinery diagnostics and prognostics implementing condition-based maintenance[END_REF] is a maintenance strategy that the manufacturing industry tries to adopt more and more in order to reduce the life cycle cost of the equipment units and to extend their availability. CBM uses real-time information to optimize the maintenance moment by restoring the functional properties of the equipment units. It is based on the current health monitoring of the equipment units, thus it is important to add prognosis tools to predict the future state and to anticipate the maintenance.

Fault prognosis is one of the major tasks in CBM. It estimates the RUL of the equipment units based on the condition monitoring information. Generally, prognosis approaches can be classified into three main categories depending on the type of the used information. These categories [START_REF] Mrugalska | Remaining useful life as prognostic approach: a review[END_REF], [START_REF] Javed | State of the art and taxonomy of prognostics approaches, trends of prognostics applications and open issues towards maturity at different technology readiness levels[END_REF] are defined as physical modelbased approaches, data-driven approaches and fusion-based approaches.

The physical model-based approaches [START_REF] Luo | Model-based prognostic techniques applied to a suspension system[END_REF] use an explicit mathematical model to represent a dynamical system's degradation. Data-driven approaches are based on condition mon-itoring data for analyzing and predicting the current and the future health conditions. Fusion-based approaches [START_REF] Liao | A hybrid framework combining data-driven and model-based methods for system remaining useful life prediction[END_REF] are a combination of these two approaches. In the real case, understanding and modeling the degradation of complex systems are difficult especially when taking into account all possible degradation effects. Thus, the physical model-based approaches are complicated to use even though they give the most accurate RUL estimation. On the other hand, data-driven approaches do not require understanding the physical behavior and the degradation process which makes them faster to deploy compared to the other approaches. Hence, for these reasons, this category of approaches is adopted in this paper.

Data-driven approaches include mainly statistical and Artificial Intelligence (AI) methods. The statistical methods estimate the RUL based on the expected health trend under known operating conditions (e.g. Wiener Process [START_REF] Wang | An adaptive remaining useful life estimation approach for newly developed system based on nonlinear degradation model[END_REF], Gamma Process [START_REF] Wu | Online prognostication of remaining useful life for random discharge lithiumion batteries using a gamma process model[END_REF], autoregressive model [START_REF] Ahsan | Remaining useful life prediction of gas turbine engine using autoregressive model[END_REF], Hidden Markov / Semi-Markov models [START_REF] Fan | Fatigue crack fault diagnosis and prognosis based on hidden semi-markov model[END_REF]). The AI methods estimate the RUL by mimicking human brain structure using several methods like neural networks (e.g. Long short-term memory [START_REF] Huang | Tool remaining useful life prediction based on edge data processing and lstm recurrent neural network[END_REF]), support vector machine [START_REF] Huang | Support vector machine based estimation of remaining useful life: current research status and future trends[END_REF], etc.

The Brownian Motion [START_REF] Whitmore | Modelling accelerated degradation data using wiener diffusion with a time scale transformation[END_REF] is one of the statistical methods widely used to model degradation processes. In [START_REF] Nguyen | Fault prognosis for discrete manufacturing processes[END_REF], it is used with an adaptive drift parameter to model the HI of the operating device and to predict its future state for RUL prediction, based on its past degradation information and R2F indicator. In [START_REF] Zhang | A model for residual life prediction based on brownian motion in framework of similarity[END_REF], the BM is introduced in the similaritybased approach. This category of approach employs all the R2F profiles generated in the same operating conditions of the operating device to predict its RUL [START_REF] You | A framework of similarity-based residual life prediction approaches using degradation histories with failure, preventive maintenance, and suspension events[END_REF]. These R2F profiles similar to the operating device's HI provide additional contributions to the prediction. Thus, in our work we combine the strengths of these two methods by using the adaptive BM model proposed in [START_REF] Nguyen | Fault prognosis for discrete manufacturing processes[END_REF] that introduces the references's information and the operating device's HI after a similarity computation as in [START_REF] Zhang | A model for residual life prediction based on brownian motion in framework of similarity[END_REF]. However, the problem is that the R2F data are sometimes limited as the devices are not allowed to be used until failure for finance and safety issues. Hence, the objective of this paper is to deal with the lack of R2F data. The proposed solution consists in generating references for the RUL prediction in the case of the availability of just one noisy R2F indicator. This can be done using the Percentile filtering suggested in [START_REF] Nguyen | Fault prognosis for batch production based on percentile measure and gamma process: Application to semiconductor manufacturing[END_REF]. Then, the similarity principle is introduced in the BM model in a new way in order to predict the RUL.

The remainder of this paper is structured as follows: the proposed approach is detailed in section II. Section III presents the approach's application on simulated data and a comparison with a non-similarity-based approach [START_REF] Nguyen | Fault prognosis for discrete manufacturing processes[END_REF]. Finally, section IV concludes the work and presents some perspectives.

II. APPROACH FRAMEWORK

The proposed approach is illustrated in Fig. 1. Given the R2F indicator of a system, the objective is to predict the RUL of an operating device sharing similar operating conditions. Generally, health indicators are highly noisy, hence a preprocessing step is required. The filtering method proposed in [START_REF] Nguyen | Fault prognosis for batch production based on percentile measure and gamma process: Application to semiconductor manufacturing[END_REF] and based on the notion of percentile is applied for this purpose. This filter extracts from the R2F indicator, 100 monotonic profiles that cover it and are considered as references in this study. Similarly, the HI of the operating device is also filtered but only the 50 th extracted profile is used for RUL prediction as it is the median profile. At each prediction time, the similarity between the references and the 50 th extracted profile from the operating device's HI is calculated and the most similar reference to it is chosen and modeled by BM as in [START_REF] Nguyen | Fault prognosis for discrete manufacturing processes[END_REF]. Its parameters are used to predict the RUL of the operating device. The different parts of this approach are detailed in the following subsections.

A. HI filtering

The Percentile filtering is used since it minimizes information loss and uncertainty in the data compared to other existing methods, and it is suitable for finding a solution to our issues [START_REF] Nguyen | Fault prognosis for batch production based on percentile measure and gamma process: Application to semiconductor manufacturing[END_REF]. This method is briefly described in the following steps.

Let Y t be the raw HI that measures the degradation over time t, V N and V F be respectively the normal and the failure threshold values set by the experts.

• First, the interval [V N , V F ] is divided into several subintervals with step s, where s is chosen in such a way to make a compromise between the preservation of information and the complexity of calculation:

s = min Median(|Y t -Y t-1 |) 10 , 0.01 (1)
where,

Median(|Y t -Y t-1 |) is the median value of the variability (Y t -Y t-1 ). (Y ti < v) & (Y ti+1 ≥ v) (2)
This selection is based on the fact that the HI is noisy and it can cross the value v many times. • In the following, for each v ∈ V, the goal is to calculate the percentiles denoted as t p v for p ∈ {1, 2, . . . , 100} of the set T v . They are used next to construct the monotonic profiles (t p v , v) relative to each p. These percentiles are computed in two steps:

1) After placing the values t i of the set T v in ascending order, the n corresponding percentiles to these values are computed according to relative position calculation as: p = 100(j -0.5)/n, j ∈ {1, 2, . . . , n}.

2) The value of the rest (100 -n) percentiles is calculated as follows:

t p v =      t 1 for p < 100(1 -0.5)/n t n for p > 100(n -0.5)/n LI for (1 -0.5) < np 100 < (n -0.5) (3)
where LI stands for Linear Interpolation: given (p 1 , t p1 v ) and (p 2 , t p2 v ) two pairs of a percentile and its corresponding value respectively, the percentiles p i between p 1 and p 2 (with p 1 < p 2 ) are computed by LI as follows:

t pi v = t p1 v + (p i -p 1 ) t p2 v -t p1 v p 2 -p 1 (4)
• 100 profiles are generated, each one corresponding to a percentile p (see Fig. 3). These profiles Y p t are constructed by assimilating to each splitting value v of V , the time t p v corresponding to the percentile p of T v . The t p v that breaks the monotony of the profile is omitted.

B. HI modeling

The Brownian Motion with an adaptive drift parameter is used to model the HI as in [START_REF] Nguyen | Fault prognosis for discrete manufacturing processes[END_REF] and [START_REF] Wang | A model for residual life prediction based on brownian motion with an adaptive drift[END_REF]. The evolution of the filtered HI is modeled iteratively over time. The strength of this method relies on the introduction of the current and the historical degradation information in the model to forecast the future trend of the health indicator. This is done by updating the drift parameter whenever a new observation is available. The degradation model is given as follows:

y ti = y ti-1 + µ ti-1 (t i -t i-1 ) + σε ti-1,ti (5)
where, y ti is the degradation information at time t i , µ ti-1 is the updated drift parameter at t i-1 after observing y ti-1 , σ is a constant and σε ti-1,ti is the error such that ε ti-1,ti ∼ N (0, t i -t i-1 ) by the BM. The variance t i -t i-1 makes sense as the random quantity should somehow increases when t i -t i-1 increases. The Kalman Filter (KF) [START_REF] Harvey | Forecasting, structural time series models and the Kalman filter[END_REF] is a recursive procedure used in this case to estimate and update the drift parameter. It computes the optimal estimator of µ t at time t, based on the available observations up to y t and including it. In order to apply the KF, the system and the observation equations are given respectively in the following:

µ ti = µ ti-1 + ν (6)
y ti -y ti-1 = µ ti-1 (t i -t i-1 ) + σε ti-1,ti (7)
where, ν is the system error normally distributed such as ν ∼ N (0, Q). Under the normality assumption, the initial µ t0 has a normal distribution with mean μt0 and variance P 0 . Thus, µ t>0 is a linear combination of two random variables, both with normal distributions, hence it is itself normally distributed and its mean (8) and variance ( 9) are updated by KF as follows:

μti = μti-1 +P i|i-1 (t i -t i-1 )F -1 i (y ti -y ti-1 -μ ti-1 (t i -t i-1 )) (8)
P i = P i|i-1 -P i|i-1 (t i -t i-1 ) 2 F -1 i P i|i-1 (9)
where,

P i|i-1 = P i-1 + Q F i = (t i -t i-1 ) 2 P i|i-1 + σ 2 (t i -t i-1 ) (10)
Thus, the model of the future behavior of the HI after observing y ti is:

y t = y ti + μti (t i -t i-1 ) + σε ti-1,ti (11)
where μti is a function of (y 1 , . . . , y ti ) for t 1 < t 2 < . . . < t i ≤ t.

C. Similarity

In this subsection, the key elements of the similarity principle are presented. Let Y op and Y r k be respectively the health indicator of the operating device and the reference k considered both in the same operating conditions. Thus, the references can improve the device's RUL prediction.

As in [START_REF] Zhang | A model for residual life prediction based on brownian motion in framework of similarity[END_REF] and [START_REF] You | A framework of similarity-based residual life prediction approaches using degradation histories with failure, preventive maintenance, and suspension events[END_REF], the similarity between the operating device's HI and a reference is measured over a time interval I by a similarity function S. For the similarity computation, (N + 1) consecutive monitoring points are considered. In particularly, the last (N + 1) observations of the operating device's HI are chosen as they reflect the current state of the device. Thus, the time interval can be represented as I = (N + 1)∆t, where ∆t is the condition monitoring time interval.

In this paper, S is the Euclidean distance and S op↔k (t i , N ) denotes the similarity calculated between the operating device and the reference k at the time t i for a determined duration I:

S op↔k (t i , N ) = N n=0 [Y op (t i -n∆t) -Y r k (t i -n∆t)] 2 (12)
Note that the following condition should be satisfied:

N + 1 ≤ min(N op , N r k ) (13)
where N op and N r k are respectively the number of condition monitoring points of Y op and Y r k . At each prediction time t i , the most similar reference to Y op is determined and used as a support for the RUL prediction. It is chosen in such a way to minimize the similarity function S op↔k (t i , N ) and is denoted as r sim :

r sim = arg min 1<k<N T S op↔k (t i , N ) (14)
where, N T is the total number of references.

D. RUL prediction

The objective of our work is to predict accurately the time required for the system to fail. It is done by forecasting the path of the degradation indicator Y op up to the failure threshold. For the RUL prediction, only the profile Y 50 op of the operating device (p = 50) is considered after filtering as it is the median of the generated profiles. At each prediction time, the degradation information is available until t i (Y 50 op (t 1 -→ t i )) and when the profile Y 50 op exceeds the normal operating threshold V N , the prognosis is triggered.

In order to have an accurate prediction, the degradation information of both Y 50 op and Y rsim are used in the BM model [START_REF] Nguyen | Fault prognosis for discrete manufacturing processes[END_REF]: the behavior of Y rsim is taken into account in order to forecast Y 50 op until failure. This is done by predicting the future evolution of the drift parameter of Y 50 op based on the evolution of Y rsim as detailed just below. At each prediction time t i :

• The corresponding time t c on which Y rsim has almost the same degradation state as Y 50 op is determined as :

Y rsim (t c ) ≤ Y 50 op (t i ) ≤ Y rsim (t c+1 ) (15)
• The drift parameter μop of Y 50 op and μsim of Y rsim are calculated using the KF (equations 8-10).

• The residual is computed as:

r i = μop ti -μsim tc (16)
• The predicted values of the drift parameter μop to consider for t > t i are:

P ti =
• The predicted RUL at t i is equal to k∆t unit of time, such that:

k = inf{k : E(Y 50 op (t i+k )) ≥ V F } (20)

III. APPLICATION

The proposed approach is tested on simulated data on Matlab. Its evaluation is then performed using metrics dedicated to fault prognosis.

In order to highlight the effectiveness of our approach, it is compared to a non-similarity-based approach [START_REF] Nguyen | Fault prognosis for discrete manufacturing processes[END_REF]. For that, the RUL is computed in two ways:

• With similarity (approach presented in this paper): the RUL is computed by picking at each prediction time t i , the suitable reference to Y 50 op from the entire available collection.

• Without similarity (non-similarity-based approach): the RUL is predicted using the same reference during all the prediction times t i as in [START_REF] Nguyen | Fault prognosis for discrete manufacturing processes[END_REF]. For this computation, the 50 th profile of the operating device and the reference Y r50 are used.

The two approaches are compared on two different scenarios: in the first one, the HI of the operating device and the reference present almost the same trend, but their tendencies are different in the second one.

In the remainder of this section, the given data and the Percentile filtering are presented in (III-A), the RUL prediction based on the similarity principle and the BM is detailed in (III-B). Finally, the approaches's evaluation is given in (III-C).

A. Data & Filtering

The R2F indicator and its 100 extracted profiles by the Percentile filtering are displayed in Fig. 3. The normal and the failure threshold are predefined as V N = -10 and V F = 28 respectively. As shown, this indicator (in blue) is noisy. Thus, the filtering is required as the quality of the HI affects the RUL prediction. The extracted profiles are monotonic and each one corresponds to a percentile p. They represent the useful information drowned in the noise of the HI. These extracted profiles are the references in the two cases. op of the operating device is used. This profile, in addition to the references, are illustrated in Fig. 5.

For the first scenario, the profiles Y 50 op and Y r50 have almost the same degradation dynamics but a notable difference is especially shown in the second scenario. In fact, the degradation of Y 50 op is slow at first in comparison with the reference Y r50 , and becomes faster after the 1000 th instant. Therefore, this scenario is considered to show the impact of the similarity on RUL prediction.

Fig. 1 :

 1 Fig. 1: Framework of the proposed approach

Fig. 2 :

 2 Fig. 2: Y t passes through a value v many times

Fig. 3 :

 3 Fig. 3: Profiles extracted from the R2F indicator

2 Fig. 4 :

 24 Fig. 4: Health indicators of the operating device and the reference

  {μ sim tc + |r i |, μsim tc+1 + |r i |, . . . μsim t M ax + |r i |} (17) where t M ax is the time of the last monitoring point of Y rsim . The absolute value of r i is taken in order to keep the index monotone and subsequently to avoid its divergence from the failure threshold V F . i+k+1 ) = E Y 50 op (t i+k )+ μsim t M ax +|r i |+σ 0,1 = E Y 50 op (t i+k ) + μsim t M ax + |r i |

	• The expected value of Y 50 op is predicted iteratively over t
	as:
	for 0 ≤ k < t M ax -t c
	E Y 50 op (t i+k+1 ) = E Y 50 op (t i+k )+ μsim tc+k +|r i |+σ 0,1
	= E Y 50 op (t i+k ) + μsim tc+k + |r i |	(18)
	for [ ]k ≥ t M ax -t c
	E (Y 50 op (t

• The splitting values v of [V N , V F ] are summarized in V = {V N , V N +s , V N +2s , . . . V F }. A set T v = {t 1 , t 2 , . . . , t n },with n the number of its elements, is constructed for each value v, by selecting the time t i verifying the following condition (see Fig.2):

B. RUL Prediction

When the profile Y 50 op exceeds the phase of normal functioning (V N = -10), RUL prediction is launched. Thus, the prediction starts at T init = 500 in the first scenario, and T init = 600 in second one (Fig. 5). In this application, N is set to 50 and ∆t 1 unit of time.

The real failure time denoted as T end is the time when the 50 th filtered profile of the operating system reaches the failure threshold as:

In the first scenario, T end = 1907, and the real RUL at each prediction time t i is RUL real (t i ) = T end -t i . This RUL is computed in order to evaluate our predictions. Idem for the second case where T end = 1807.

The comparison between the two predicted RUL (with and without similarity) is performed for the two scenarios and illustrated respectively in Fig. 6 and Fig. 7. The result shows that the proposed approach outperforms the non-similarity-based approach as the difference between the real and the predicted RUL is more important when fixing the reference than in our approach. It can also be noticed that the RUL prediction is improved during the time as more data become available. This is especially notable on the last 200 prediction times of our approach as shown on the zoom in the figures. The improvement of the RUL prediction by the proposed approach is explained by the fact that using the most similar reference to Y 50 op at each prediction time t i is more effective especially when the two HI do not have exactly the same trend. In Fig. 8, the similar references chosen by our approach during the prediction are presented. op since the two HI have the same trend all along the time interval. However, in the second scenario, a greater variety of similar profiles is chosen: 67 profiles are particularly picked by the proposed approach between p = 1 and p = 75 following the same evolution of Y 50 op . Thus, using only Y r50 as the reference for the RUL prediction is not sufficient specially when Y r50 and Y 50

op have different trends.

The introduction of the similarity principle in the RUL prediction is suitable for cases like this scenario where the degradation information of the operating device and the R2F data are not exactly the same as in the first one (ideal scenario). In addition, this approach is more robust and more suitable for real cases as it improves RUL prediction when R2F data are limited.

C. Evaluation

To evaluate numerically the two presented approaches, the Root Mean Square Error (RMSE) and the (α-λ) accuracy are computed. These metrics [START_REF] Lei | Machinery health prognostics: A systematic review from data acquisition to rul prediction[END_REF] can only be used when the real RUL is available. They evaluate the prognosis's results from different aspects and allow us to compare these approaches. 1) RMSE is defined by:

This metric is computed for the two approaches throughout the prediction interval and over the last 200 predictions time (see Table I). It can be noticed that the prediction error is smaller when the similarity principle is introduced in the BM model to predict the RUL.

2) (α-λ) accuracy is a binary metric that determines wether a prediction result falls or not within the accuracy zone at a specific time t λ , where λ ∈ [0, 1]. The accuracy zone is defined with respect to a percentage α of accuracy related to the real RUL. This zone becomes smaller as time increases by reflecting that the prediction accuracy increases as more data are available. In this application, α is fixed at 20% for precision.

The time t λ is computed between the beginning time of the prediction (λ = 0) and the real failure time (λ = 1) as in the following:

It can be noticed as shown in Fig. 9 and Fig. 10, that the predicted RUL corresponding to t λ for λ ∈ [0, 0.1, 0.2, . . . , 1] falls more in the accuracy zone in our approach compared to the non-similarity-based approach. This result is also valid for λ calculated in a step of 1/100 between 0 and 1. When computing the RUL at each of these t λ , we found that 70 predictions in our approach fall in the accuracy zone compared to only 53 in the non-similarity-based approach. Similarly, for the second scenario where 29 more predictions in the proposed approach belong to the accuracy zone especially in the last 200 prediction times as seen in the figures.

According to these metrics, the proposed approach outperforms the non similarity-based approach confirming the graphic results. This paper deals with the problem of RUL prediction when R2F data are limited. It proposes a new approach that combines the Brownian Motion model and the similarity principle after a pre-processing step of the health indicators by the Percentile filtering. An implementation with experiments on simulated data is carried out. The obtained results show the effectiveness of our approach compared to existing methods particularly to the non-similarity-based approaches.

In the future, we intend to extend this work in at least two directions: testing another method for HI modeling and applying the approach to real data.


Louis Dutheil

Maïmouna Bocoum

Mathias Fink

Sébastien M Popoff

François Ramaz

Jean-Michel Tualle
email: tualle@univ-paris13.fr

Fourier-Transform Acousto-Optic Imaging with Off-axis Holographic Detection

imaging is an in-depth optical imaging technique of highly scattering media. One challenging end-application for this technique is to perform imaging of living biological tissues. Indeed, because it relies on coherent illumination, AO imaging is sensitive to speckle decorrelation occurring on the millisecond timescale. Camera-based detections are well suited for in vivo imaging provided their integration time be lower than those decorrelation time scales. In this work, we present Fourier transform acousto-optic imaging combined with off-axis holography which relies on plane-waves and long-duration pulses. We demonstrate, for the first time, a two-dimensional imaging system fully compatible with in vivo imaging prerequisites. The method is validated experimentally by performing in-depth imaging inside a multiple scattering sample.

Introduction

Imaging biological tissues at large depths is a challenge inherent to multiple medical applications such as early tumour detection. Indeed, after a few millimetres of propagation, predominance of the multiple scattering regime is detrimental to optical imaging relying on ballistic light in the visible to near-infrared range [START_REF] Cheong | A review of the optical properties of biological tissues[END_REF]. Optical coherence tomography [START_REF] Huang | Optical coherence tomography[END_REF][START_REF] Drexler | Optical coherence tomography: technology and applications[END_REF] is for instance limited to about 100 µm in depth, even when standard wavefront correction [START_REF] Badon | Smart optical coherence tomography for ultra-deep imaging through highly scattering media[END_REF] is used to compensate for strong aberrations. As a way around, diffuse optical tomography [START_REF] Boas | Imaging the body with diffuse optical tomography[END_REF] has emerged to infer millimeter-to-centimeter depth optical properties of the medium, based on prior knowledge about scattering properties of the medium [START_REF] Arridge | Nonuniqueness in diffusion-based optical tomography[END_REF][START_REF] Brendel | Selection of optimal wavelengths for spectral reconstruction in diffuse optical tomography[END_REF].

In this context, Photo-Acoustic Imaging (PAI) [START_REF] Xu | Photoacoustic imaging in biomedicine[END_REF][START_REF] Zhang | Functional photoacoustic microscopy for high-resolution and noninvasive in vivo imaging[END_REF][START_REF] Bossy | Time reversal of photoacoustic waves[END_REF][START_REF] Wang | Photoacoustic tomography: in vivo imaging from organelles to organs[END_REF][START_REF] Chaigne | Controlling light in scattering media non-invasively using the photoacoustic transmission matrix[END_REF] and Acousto-Optic Imaging (AOI) [START_REF] Gunther | Review of current methods of acousto-optical tomography for biomedical applications[END_REF][START_REF] Bengtsson | Characterization and modeling of acousto-optic signal strengths in highly scattering media[END_REF] have drawn increasing interest over the past decades. They both provide a direct mapping of optical absorption properties of scattering tissues. In PAI, an intense Q-switched nanosecond laser beam is used to heat optical absorbers inside the medium. As they undergo thermal relaxation, those absorbers emit ultrasound (US) waves in the MHz range which are detected to reconstruct an optically contrasted image. In AOI, the medium is illuminated with a laser of high temporal coherence, while insonified with an auxiliary US pulse in the MHz range. As a result of the AO effect [START_REF] Leutz | Ultrasonic modulation of multiply scattered light[END_REF], the spectrum of light propagating through the insonified region shows two side-bands which are both shifted from the optical carrier by the US frequency. Those so-called tagged photons which leave the medium are then selectively detected. Due to the local character of the tagging, the absorption properties of the medium can be monitored during US ballistic propagation. Up until now, the development of PAI has been faster than that of AOI, as it relies on incoherent illumination. In fact, for clinical applications, coherent illumination is challenging because of the speckle decorrelation caused by Brownian motion of scatterers, blood circulation, breathing. This decorrelation occurs on time scales in the order or below the millisecond-range [START_REF] Gross | Heterodyne detection of multiply scattered monochromatic light with a multipixel detector[END_REF][START_REF] Lev | In vivo demonstration of the ultrasound-modulated light technique[END_REF][START_REF] Liu | Optical focusing deep inside dynamic scattering media with near-infrared time-reversed ultrasonically encoded (TRUE) light[END_REF][START_REF] Qureshi | In vivo study of optical speckle decorrelation time across depths in the mouse brain[END_REF]. In the case of AOI, this sensitivity is highly dependent on the technique used to detect the tagged photons. In addition, because only a few percent of the light is tagged by the US, weak signal-to-noise ratio (SNR) is another major challenge to address before considering in vivo applications.

The most basic scheme to perform AOI is to use temporally-pulsed spatially-focused US for insonification. Using a detection fast enough to follow US pulse propagation, a direct space-to-time mapping allows to reconstruct an in-depth image. Fast detection is for instance performed using photorefractive holography [START_REF] Ramaz | Photorefractive detection of tagged photons in ultrasound modulated optical tomography of thick biological tissues[END_REF] or spectral hole burning [START_REF] Bengtsson | Characterization and modeling of acousto-optic signal strengths in highly scattering media[END_REF][START_REF] Venet | Ultrasound-modulated optical tomography in scattering media: flux filtering based on persistent spectral hole burning in the optical diagnosis window[END_REF], both relying on high-sampling-rate photodiode. Robustness against speckle decorrelation makes spectral hole burning promising, but these systems require cryogenic equipment and are not tunable in wavelength.

Camera-based detection [START_REF] Guillaume | Acousto-optical coherence tomography with a digital holographic detection scheme[END_REF] of tagged light is an alternative to those detections, provided the camera-integration time remains lower than that of speckle decorrelation. In the particular case of off-axis holography [START_REF] Gross | Shot-noise detection of ultrasound-tagged photons in ultrasound-modulated optical imaging[END_REF], the US frequency-shifted light interferes on the camera with a reference beam of the same frequency. This gives rise to spatial fringes on the detector with a spacing dependent upon the off-axis angle between the two beams. By digitally processing this signal [START_REF] Kronrod | Reconstruction of a hologram with a computer[END_REF][START_REF] Onural | Digital decoding of in-line holograms[END_REF], we can retrieve the total number of tagged photons recorded during the camera exposure. Since the camera frames rate (<kHz) is orders of magnitude lower than the sampling rate of a fast photodiode (MHz), it is no longer possible to resolve the propagation of the US pulse which typically progresses at 1.5 mm µs -1 . Furthermore, to maximize the number of tagged photons recorded during one camera exposure, temporally-long unfocused US pulses impose as a natural choice because they maximize the insonification volume within the sample. The problem arising with such US wave is a complete loss in spatial resolution, since tagged photons now come from the entire insonification plane.

The problem of recovering spatial resolution using plane and/or continuous wave and been extensively reported in the literature [START_REF] Bocoum | Structured ultrasound-modulated optical tomography[END_REF][START_REF] Marks | Comprehensive approach to breast cancer detection using light: photon localization by ultrasound modulation and tissue characterization by spectral discrimination[END_REF]. In the temporal (i.e. axial) direction, one solution is to increase the spectral bandwidth of the tagging US, and thereby, that of tagged light. This is achieved for instance by imposing a frequency chirp [START_REF] Forget | High-contrast fast Fourier transform acousto-optical tomography of phantom tissues with a frequency-chirp modulation of the ultrasound[END_REF], or random phase jumps [START_REF] Lesaffre | Acousto-optical coherence tomography using random phase jumps on ultrasound and light[END_REF] on the US carrier frequency. Imposing the same phase modulation on the reference beam when performing off-axis holography, interferences on the camera only build up when the relative delay between the two beams does not exceed the temporal coherence of these modulations. As a result, the tagged photons seem to been emanating from a confined volume along US propagation axis, yielding a recovery of axial resolution. The volume of tagged light however reduces as we increase the spectral bandwidth, which ultimately also degrades the SNR.

To maximize the tagging volume and maintain spatial resolution, Fourier-Transform Acousto-Optic Imaging (FT-AOI) was first introduced using spatially focused US [START_REF] Barjean | Theoretical study of Fourier-transform acousto-optic imaging[END_REF][START_REF] Barjean | Fourier transform acousto-optic imaging with a custom-designed cmos smart-pixels array[END_REF]. There, a long US pulse is periodically modulated either in amplitude or phase. When the same modulation is reproduced on the reference arm, tagged photons appear as if generated by several static points periodically distributed along the direction of propagation. Consequently, one acquisition gives access to one component of the image Fourier Transform. In a recent article [START_REF] Bocoum | Reconstruction of bi-dimensional images in Fourier-transform acousto-optic imaging[END_REF], we have shown that using a transducer array, the same idea could be extended to fetch the two-dimension Fourier components of the image. However, this work relied on photorefractive-based holographic detection, and therefore sensitive to speckle decorrelation. In this paper we present, for the first time, FT-AOI combined with off-axis heterodyne holographic detection.

Theory

The theoretical framework of the method is described in [START_REF] Barjean | Theoretical study of Fourier-transform acousto-optic imaging[END_REF]. The present section focuses on true implementation of the method using a transducer array [START_REF] Bocoum | Reconstruction of bi-dimensional images in Fourier-transform acousto-optic imaging[END_REF]. In FT-AOI, the pressure field is periodically structured along the 𝑥 and 𝑧-axis, corresponding respectively to the direction along the transducer array and the direction of acoustic propagation. Reconstructing an image requires to perform multiple acquisitions so as to fetch all of its Fourier components. For each acquisition, the structuring frequencies 𝜈 𝑥 along x-axis is an increasing harmonic of fundamental frequencies

𝜈 0 𝑥 = 𝐿 -1
where 𝐿 is the width of the US probe, such that 𝜈 𝑥 = 𝑚 𝜈 0 𝑥 with 𝑚 integer. The modulation along 𝑧 is done by imposing a periodic temporal emission on each transducer with a relative phase imposed by the periodicity set along x. At any given time during propagation, the field thereby appears spatially modulated along 𝑧 at a frequency 𝜈 𝑧 = 𝑛 𝜈 0 𝑧 , where 𝑛 is an integer corresponding to the harmonic order and 𝜈 0 𝑧 the fundamental modulation frequency. By definition, we have that 𝜈 0 𝑧 = (𝑐 s 𝑇 0 ) -1 with 𝑇 0 the user-defined fundamental temporal period and 𝑐 s the sound velocity inside the medium. In addition, by controlling the emission time on all transducers simultaneously, we can vary the relative phase 𝜙 of the structuring pattern. We note 𝑃

𝐴 𝜙 𝑚,𝑛 (𝑥, 𝑧, 𝑡) = 𝑃 0 ℎ 𝜙 𝑚,𝑛 (𝑡 -𝑐 -1 s 𝑧, 𝑥), (2)
with 𝜔 us the US carrier pulsation and 𝑃 0 the nominal pressure. As US propagate, some photons are frequency-shifted from the incident optical carrier 𝜔 L at 𝜔 L ± 𝜔 us with a phase and amplitude inherited from 𝐴 𝜙 𝑚,𝑛 . In the holographic digital detection scheme, the collected field 𝐸 interferes with a reference field 𝐸 ref 𝑛 centered at 𝜔 L + 𝜔 us . The resulting image 𝐼 𝜙 𝑚,𝑛 (𝜂, 𝜉) obtained over an integration time 𝜏 e writes:

𝐼 𝜙 𝑚,𝑛 (𝜂, 𝜉) = 𝜖 0 ∫ 𝜏 e d𝑡 |𝐸 ref 𝑛 (𝜂, 𝜉, 𝑡) + 𝐸 (𝜂, 𝜉, 𝑡)| 2 (4)
where (𝜂, 𝜉) are the pixels coordinates on the camera sensor. By imposing that 𝐸 ref 𝑛 also be 𝑇0 /𝑛 periodic in amplitude, constructive interference over the integration time 𝜏 e will only build up for tagged light coming from specific locations in the interaction plane. These locations are defined by our so called "tagging function". The temporal expression of the reference field incident on the camera writes: where 𝑐 is the velocity of light, 𝐸 0 is the field amplitude, and 𝛼 𝜂 and 𝛼 𝜉 the off-axis angles.

𝐸 ref 𝑛 (𝜂, 𝜉, 𝑡) = 𝐸 0 sin 2 𝜋 𝑛 𝑇 0 𝑡 𝑒 -𝑖 𝜔 L +𝜔us 𝑐 ( 𝛼 𝜂 𝜂+𝛼 𝜉 𝜉 ) 𝑒 𝑖 ( 𝜔 L +𝜔 us )𝑡 , (5)
Following the basic principle of off-axis holography, the recorded image is numerically filtered in the Fourier domain so as to only keep the tilted component. This operation leads to the following hologram resulting from the interferences between the scattered light and the reference beam:

𝐻 𝜙 𝑚,𝑛 (𝜂, 𝜉) = 𝜖 0 𝑒 𝑖 𝜔 L +𝜔us 𝑐 ( 𝛼 𝜂 𝜂+𝛼 𝜉 𝜉 ) ∫ 𝜏 e d𝑡 𝐸 * 0 𝐸 (𝜂, 𝜉, 𝑡) sin 2 𝜋 𝑛 𝑇 0 𝑡 𝑒 -𝑖 ( 𝜔 L +𝜔 us )𝑡 (6)
The averaged square modulus of this hologram on all of the pixels, 𝑠 𝜙 𝑚,𝑛 ≡ |𝐻 𝜙 𝑚,𝑛 | 2 , is proportional to the average intensity of the tagged photons selectively detected in the US interaction plane [START_REF] Barjean | Theoretical study of Fourier-transform acousto-optic imaging[END_REF]. We write it in the following form:

𝑠 𝜙 𝑚,𝑛 ∝ ∫ 𝑥,𝑧 d𝑥 d𝑧 𝐼 (𝑥, 𝑧) 𝑀 𝜙 𝑚,𝑛 (𝑥, 𝑧) 2 . ( 7
)
where 𝐼 (𝑥, 𝑧) is the image we wish to reconstruct, which depends on the local average intensity of the diffuse light, on the tagging efficiency, and on the propagation from the medium to the camera [START_REF] Barjean | Theoretical study of Fourier-transform acousto-optic imaging[END_REF]. 𝑀 𝜙 𝑚,𝑛 (𝑥, 𝑧) is the tagging function which ensures spatial selectivity. As illustrated in Fig 2, the acoustic field modulation pattern has an arrival phase which varies along x and z. As a result, tagged photons will only be fully correlated with the reference wave at specific positions (x,z). The expression of tagged photons is easily derived from Eq 2, such that the tagging function becomes the following cross-correlation [START_REF] Barjean | Theoretical study of Fourier-transform acousto-optic imaging[END_REF]:

𝑀 𝜙 𝑚,𝑛 (𝑥, 𝑧) = 𝑃 0 𝜏 e ∫ 𝜏 e d𝑡 sin 2 𝜋 𝑛 𝑇 0 𝑡 ℎ 𝜙 𝑚,𝑛 (𝑡 -𝑐 -1 s 𝑧, 𝑥) = 𝑃 0 2 1 + 1 𝜋 cos 2𝜋 𝑚 𝜈 0 𝑥 𝑥 + 2𝜋 𝑛 𝜈 0 𝑧 𝑧 + 𝜙 (8)
derivation of Eq 8 is performed in Supplement 1. This expression shows that the sinusoidal modulation applied on the reference beam leads to periodically distributed tagged photons both in 𝑥 and 𝑧-axis. By combining measurements obtained using four different phase shifts, namely 𝜙 ∈ {0, 𝜋 /2, 𝜋, 3 𝜋 /2}, we access the complex Fourier components Ĩ of the acousto-optic image:

𝑠 3 𝜋 /2 𝑚,𝑛 -𝑠 𝜋 /2 𝑚,𝑛 + 𝑖 𝑠 𝜋 𝑚,𝑛 -𝑠 0 𝑚,𝑛 ∝ Ĩ 𝑚 𝜈 0 𝑥 , 𝑛 𝜈 0 𝑧 , (9)
Image 𝐼 (𝑥, 𝑧) is thus simply reconstructed by performing an inverse Fourier Transform (iFT) of Ĩ. In our model, we however recall that diffraction is neglected (Eqs 2 and 8), an hypothesis which might not always verify experimentally. Thus, to investigate potential effect of diffraction, we have performed a simulation of the actual pressure emitted by the probe we used in our later described experiment. Simulations were performed using Field II open-source software as already described in [START_REF] Jensen | Simulation of advanced ultrasound systems using field ii[END_REF]. A simulation grid is placed in the US plane, 35mm below the probe emission surface with -15 ≤ 𝑥 ≤ 15 and 20 ≤ 𝑧 ≤ 50. On each point of a simulation grid, we have calculated the tagging function using the simulated pressure and the expression of our theoretical optical reference field. As a result, we conclude that diffraction effects can indeed be neglected for all 𝑛 (tested up to 20). As 𝑚 value increases however, second harmonic artefacts start to appear in the tagging function, together with 𝜋-phase shifts of the sin wave. This effect is to be expected as a result of Talbot near-field diffraction effect and will be fully explained in future work. Example of resulting 𝑀 𝜙 𝑚,𝑛 functions are shown Fig. 3 for 𝑚 = 3, 5, 6 and 𝑛 = 8. In our simulated experimental conditions, diffraction artefacts start to appear typically for |𝑚| ≥ 6 with a noticeable phase shift at ∼ 38 mm. For this reason, we constrained our experiment to |𝑚| < 6 so as to remain with the scope of validity of our model.

Experiment

The sample is an hydrogel mimicking both the optical and mechanical properties of soft biological tissues [START_REF] Jacques | Optical properties of biological tissues: a review[END_REF]. The dimensions of the hydrogel along 𝑥, 𝑦 and 𝑧 axes are respectively 5 × 1 × 5 cm 3 . The 50 kPa Young modulus is set by the 2 % concentration of Agar [START_REF] Hall | Phantom materials for elastography[END_REF] whereas the concentration of Intralipid 10 % is adjusted to obtain a reduced scattering coefficient 𝜇 s = 1 cm -1 (transport mean free path 𝑙 * = 1 cm) [START_REF] Van Staveren | Light scattering in Intralipid-10% in the wavelength range of 400-1 100 nm[END_REF]. Two ink inclusions are inserted at a 5 mm depth. The dimensions of their cylindrical shapes are 4 mm in height and 2 mm of diameter. Their centers are separated by 4 mm. A photograph showing the two inclusions taken during the molding is shown on Fig. 4. The sample is illuminated with a single-longitudinal mode laser centered at 780 nm. After the tapered amplifier (MOPA, Sacher Lasertchnik GmBH), the incident power is about 370 mW for a beam diameter of 13 mm. A transmitted photons flux of 3.6 µW is collected by the CMOS camera (Ximea xiB-64) located 34 cm behind the sample output surface. This distance corresponds to a trade-off between proper sampling of the interference fringes and a maximization of the optical collection. A scheme of the experimental setup is presented on Fig. 5. Fig. 5. Experimental setup. Light emitted by a laser diode at 780 nm is split between the probe beam and the reference arm with a half-wave plate (HWP) followed by a polarizing beam-splitter (PBS). Both arms are then modulated in frequency by two acousto-optic modulators (AOM#) around 80 MHz with a relative difference of 3 MHz corresponding to the carrier frequency of the acoustic waves emitted by the transducer array (SL10-2 -Aixplorer SL10-2 US probe, SuperSonic Imagine, Aix-en-Provence, France) through the scattering hydrogel. Lenses (L#) and irises (I#) widen and filter the beams before they meet the camera (Ximea) sensor.

The structured US plane waves are emitted with a 192-element linear transducer array (Aixplorer SL10-2 US probe, SuperSonic Imagine, Aix-en-Provence, France) with a fixed focal elevation of 35 mm. The excitation voltage of the transducer is set to 15 V such that the nominal pressure of the US wave was measured to be 𝑃 0 ≈ 300 kPa. Such a pressure corresponds to a US peak intensity of ≈ 3 W cm -2 . Accounting for the acquisition repetition rate, the average acoustic intensity is ≈ 150 mW cm -2 , far below biomedical norms of 720 mW cm -2 [START_REF] Phillips | Information for manufacturers seeking marketing clearance of diagnostic ultrasound systems and transducers[END_REF]. We set the fundamental temporal window of our periodic pattern to 𝑇 0 = 20 µs. The pattern is then repeated five times so as to match the exposure time of the camera 𝜏 e = 100 µs. The camera is triggered at a 300 Hz repetition rate and each component (given 𝑛,𝑚,𝜙 value) is measured with a single-shot image. Here, we only use a 1024 × 1024 pixel area of the 16 Mpixel chip, to prevent frame dropping which occurs when our frame grabber gets filled up. In fact, our image processing performed using Labview software runs on multi-core CPU unit, but remains too slow to prevent this buffer saturation at full chip capacity. A full frame acquisition would improve the SNR as the sensor surface is increased but would require Graphics Processing Unit (GPU) for the reconstruction process. The total acquisition to image a complete field of view of 30 × 30 mm 2 is 1.5 s.

We image the two ink inclusions hidden within hydrogels of two different reduced scattering coefficient 1 cm -1 and 10 cm -1 . The image corresponding to the 𝜇 s = 1 cm -1 hydrogel featured on Fig. 6 shows two dark spots in the halo of the tagged beam. with a diameter of 2 mm, and separated by 4 mm, as expected. This image was reconstructed from the 110 complex Fourier components, 𝑚 ∈ -5, 5 , 𝑛 ∈ 1, 10 which modulus is shown in Fig. 6(c). The resolution of FT-AOI is given by the spatial period of the higher harmonics used in the structuring pattern ℎ 𝑚,𝑛 with the wavelength of the US carrier as the ultimate limit. As the maximum values for 𝑚 and 𝑛 are respectively 𝑚 = 5 and 𝑛 = 10, the resolutions along 𝑥 and 𝑧 are respectively (𝑚 𝜈 0 𝑥 ) -1 = 7.7 mm and (𝑛 𝜈 0 𝑧 ) -1 = 3 mm, which discriminates our two inclusions. To outline time-efficiency of FT-AOI, we draw a comparison with AO imaging in real space as for instance performed using spatially-focused US with random phase jumps modulation [START_REF] Lesaffre | Acousto-optical coherence tomography using random phase jumps on ultrasound and light[END_REF][START_REF] Benoit A La Guillaume | Acousto-optical coherence tomography with a digital holographic detection scheme[END_REF]. In this respect, lets assume such an image be sampled every ∼ 2 mm over a 30 × 30 mm 2 field of view. This corresponds to 15 × 15 = 225 points to acquire. For each sampled point, the AO detected signal in experimental conditions similar as in present work is nearly one to two orders of magnitude lower [START_REF] Benoit A La Guillaume | Acousto-optical coherence tomography with a digital holographic detection scheme[END_REF] than for FT-AOI. In fact, because the tagging volume is reduced to few cubic millimeters, it requires a proportional increase in the number of averaging. As a result, the acquisition time for an image acquired in real space is on the order of a minute. In FT-AOI where acquisition is performed in reciprocal space, the tagging volume is significantly increased such that averaging is no longer required. As a result, the method is typically one order of magnitude more time efficient. For the stronger-diffusive hydrogel 𝜇 s = 10 cm -1 , the transmitted intensity through the sample is 6 times weaker than in previous case. This decreases the SNR to ≈ 1.6. Fourier components are here again fetched without averaging, and modulus of the noisy Fourier components are shown in Fig. 7 (c). The two inclusions are still discriminated. The noise however affects the image reconstruction as we see on Fig. 7 (a-b). We observe a phase offset between high and low frequency components such that several inclusions of different sizes seem to appear, which shows the limitation of the method at low SNR. We nevertheless emphasize that this result is obtained without averaging, thereby demonstrating the robustness of our imaging strategy.

In conclusion, we have shown that the FT-AOI combined with an off-axis detection can achieve averaging-free imaging over a 3 × 3 cm 2 field-of-view with mm resolution. This camera-based detection integrates the energy of multiple speckle grains impinging on the detector. A reference beam is used to extract and demodulate the tagged signal, and as a result of autocorrelation, only photons allowed by the tagging function are detected. This allows to perform imaging in the Fourier domain of the object while still tagging a large volume inside the sample. The camera integration time is kept below a millisecond such as to be insensitive to speckle decorrelation. In addition, the overall acquisition time of an image was 1.5 s, making this method adapted for in vivo monitoring.

This short acquisition time is only possible because of the large volume of tagged photon inherent to the method, even with limited effective collection on the camera and low pressure peak powers. We plan to further increase the image SNR by using the full camera chip along with GPU stream analysis. In this article, image reconstruction was performed using a straightforward twodimensional inverse Fourier transform. In the future, we plan to investigate other reconstruction strategies, in particular one which can account for diffraction effects visible on the tagging function. This can be done for instance by measuring an acousto-optic tagging matrix, thereby opening the way to wide variety of tagging strategy for optimal imaging.

  Fig. 1. Examples of the amplitude of modulation ℎ 𝜙 𝑚,𝑛

Fig. 2 .

 2 Fig. 2. Principle of the method. two-dimensional probe sends a temporally modulated acoustic beam (top caption) specified by ℎ 𝜙 𝑚,𝑛 . The acoustic field which propagates inside the scattering medium is represented at two different arbitrary positions inside the sample. The tagging function will result in the cross-correlation of this local field with the reference wave (lower caption).

Fig. 3 .

 3 Fig. 3. Simulated tagging functions for 𝑚 = 3 (a), 5 (b) and 6 (c), 𝑛 = 8 on a 20 ≤ 𝑧 ≤ 50 mm and -15 ≤ 𝑥 ≤ 15 mm simulation grid.

Fig. 4 .

 4 Fig. 4. Photograph of the two inclusions taken above the gel cast. The two inclusions stands at 35 mm from the top surface the gel. This distance corresponds to the elevation focus of the probe.

Fig. 6 .

 6 Fig. 6. FT-AOI of two ink inclusions buried within a 1 cm-thick scattering hydrogel with a reduced scattering coefficient of 1 cm -1 . (a) The image is obtained by gathering the 110 experimentally measured Fourier components for -5 ≤ 𝑚 ≤ 5 and 1 ≤ 𝑛 ≤ 10. The profile (b) is drawn following the blue line on the acousto-optic image crossing the two dark spots. (c) Modulus of the 4-phase acquired Fourier components.

Fig. 7 .

 7 Fig. 7. FT-AO images of two ink inclusions embedded in 1 cm-thick scattering hydrogel with a reduced scattering coefficient of 10 cm -1 . (a) The image is obtained by gathering the 110 experimentally measured Fourier components for -5 ≤ 𝑚 ≤ 5 and 1 ≤ 𝑛 ≤ 10. The profile (b) is drawn following the blue line on the acousto-optic image crossing the two dark spots. (c) Modulus of the 4-phase acquired Fourier components.

Funding. Plan Cancer ITMO-INSERM 2014-2019 (MALT-C16027HS); Défi Imag'In-CNRS (MALT); French LABEX WIFI (ANR-10-LABX-24 and ANR-10-IDEX-0001-02 PSL).

Disclosures. The authors declare no conflicts of interest.

Data availability. Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request. Supplemental document. See Supplement 1 for supporting content.


Anh Le

Quoc Anh Nguyen

Pascal Ngo

Thanh Binh Retailleau

Nguyen

Anh Quoc

Ngo

Pascal Retailleau

Thanh Binh Nguyen
email: thanh-binh.nguyen@cnrs.fr

Break-and-Build Strategy

Keywords: sulfur, 2-benzoylbenzoxazole, DMSO

published or not. The documents may come

Benzoxazole derivatives are prevalent in biologically and pharmacologically important molecules as well as in functional materials. [START_REF] Demmer | For a selected review on benzoxazoles[END_REF] A subgroup of this family is 2-benzoylbenzoxazoles, which are well-known for their significant biological activities such as antiproliferative activity (compound I), [2] inhibition of fatty acid amide hydrolase FAAH [3] (compound II) (Scheme 1). Consequently, several methods have been developed for the synthesis of 2benzoylbenzoxazoles, [4] most of them were based on functionalization of simpler benzoxazole cores IX [5] or using complex starting materials which required multistep syntheses. [START_REF] Aljaar | For an example of redox condensation between onitrosophenols and phenacyl bromides[END_REF] The intuitive approach based on the construction of the benzoxazole ring by condensation of oaminophenols 1 were described with phenylglyoxalic derivatives such as dithioester III [7] or imidoyl cyanide IV [8] via non redox condensation. Alternatively, when oxidation conditions were applied, condensation of 1 with derivatives of lower oxidation states such as α,αdihaloacetophenones V [9] or 2-bromophenylacetylenes VI [10] were known. In this context, a direct use of acetophenones 2, which are inexpensive and readily available in a wide range of structure, in a selective oxidative condensation with o-aminophenols 1 would provide a convenient and cost-effective approach to a library of 2-benzoylbenzoxazoles for drug discovery study. Scheme 1. 2-Benzoylbenzoxazoles: Bioactivity and Syntheses We have recently reported a Willgerodt rearrangement-type benzoazolation of acetophenones 2 with o-aminophenols 1 promoted by sulfur [START_REF] Nguyen | For recent reviews on the organic transformations involving elemental sulfur[END_REF] and Nmethylpiperidine (Scheme 2). [12] In this reaction leading to 4aa, while the methyl group of acetophenone 2a was oxidized and benzoxazolized with 2-aminophenol 1a, the carbonyl group was reduced to a methylene group. Since this methylene group is surrounded by the phenyl group and the newly installed benzoxazole moiety, this new situation could open up opportunities for oxidation of 4aa into a carbonyl in 3aa. Such operation was previously performed on separated 2-benzylbenzoxazole derivatives using relatively complex reaction conditions involving strong oxidizing agents in the presence of transition metal catalysts. [5m] We reason that a direct transformation from 1a and 2a to 3a could be performed in one-pot in the presence of a suitable oxidant in the reaction medium. As DMSO is well known as an excellent solvent and a mild and selective oxidizing agent, especially in the presence of sulfur, [13] we focused our attention to develop this strategy.

Scheme 2. Break-and Build Strategy for the Synthesis of 3

For this purpose, we first performed the benzoxazolation of acetophenone 2a with 2aminophenol 1a under the standard conditions (80 ºC, 16h) using sulfur equiv), N-methylpiperidine (NMP) (1 equiv) in the presence of DMSO (3 equiv) (Table 1, entry 1).

To our delight, this initial test resulted in a 80% conversion of 2-aminophenol 1a into a nearly 1:1 mixture of 4aa:3aa. Oxidation of the methylene moiety was significantly accelerated by heating at higher temperature (110 ºC, entry 2). At this stage, we realized that DMSO could act as oxidant to regenerate sulfur from H2S byproduct of the first step of Willgerodt-type benzoxazolation, the amount of sulfur could be lowered to stoichiometric amount, i.e. 1 equiv (entry 3).

Moreover, we noticed that the amount of NMP, which acted as a basic sulfur activator, could be further reduced to 0.5 equiv without lowering the reactivity (entries 4-5). Interestingly, other nitrogen bases such as pyridine, N-methylmorpholine (NMM), diisopropylethylamine (DIPEA), tri-n-propylamine or 1,4-diazabicyclo[2.2.2]octane (DABCO) (entries 6-10) could be used in place of NMP. It is noteworthy that the expected product 3aa could be formed even in the absence of a base, despite in lower yield (entry 11). The reaction in the absence of sulfur resulted in recovery of unchanged starting materials, which confirmed clearly the crucial role of this element in promoting the cascade transformation (entry 12).

Finally, lowering the DMSO amount (entry 13) of replacing DMSO by another polar aprotic solvent such as DMF (entry 14) prevented partially or totally the formation of 3aa, which confirmed the importance of DMSO in this oxidative condensation.

Table 1. Screening of the Reaction Conditions Entry [a] x additive y yield (%) [b] 1 [c] 3 N-methylpiperidine N-methylmorpholine 0.5 0 13 e 1 N-methylmorpholine 0.5 36 14 f 1 N-methylmorpholine 0.5 0 [d] [a] Reaction conditions: o-aminophenol 1a (1 mmol), acetophenone 2a, sulfur (x mmol, 32 mg.mmol -1 ), additive (y mmol), DMSO (0.5 mL), 110 °C, 16 h unless otherwise noted. [b] isolated yield unless otherwise noted. [c] Reaction temperature 80 ºC. [d] Determined by 1 H NMR of the crude mixture. [e] DMSO (0.3 mL) was used. [f] DMF (0.5 mL) was used in place of DMSO.

1 30 2 3 N-methylpiperidine 1 65 3 1 N-methylpiperidine 1 68 4 1 N-methylpiperidine 0.
With this set of optimized reaction conditions, we next studied the scope and limitations of our strategy. In general, starting from 2-aminophenol 1a and acetophenones 2 bearing a wide range of substituents of different electronic nature or positions, the reactions led to the expected 2-benzoylbenzoxazole products 3 in reasonable yields (Scheme 3). Acetophenones bearing electron donating groups such as methyl, methoxy, acetamido displayed good reactivity, provide the expected benzoxazoles 3ab-3aj in moderate to good yields. We emphasized that the product 3ai (compound I in Scheme 1) was previous demonstrated to be biologically active. [2] Interestingly, the reaction conditions could be applied to acetophenones bearing unprotected hydroxy groups in good to excellent yields. It should be noted that 3al could serve as a direct precursor to provide a wide range of bioactive derivative via acylation of its hydroxy group (compound II in Scheme 1). Acetophenones bearing halogen groups in different positions such as p-Cl, m-Cl, o-F and m-Br (2m-2p) displayed good reactivities and gave the products 3am-3ap in good yields. Gratifyingly, acetophenones substrates having an electron withdrawing group such as CF3, CN, CO2Me or NO2, showed also to be competent substrates. The structure of nitro 3aw was unambiguously confirmed by X-ray diffraction analysis. Although CN and NO2 groups were known to react with H2S, [14] which is a possible intermediate/byproduct of this transformation, via nucleophilic addition and reduction, respectively, the expected benzoxazole products 3as, 3at, 3av and 3ax were obtained in reasonable yields. In case of pnitroacetophenone 2v, the superiority of NMM compared to other basic additive such as NMP and pyridine or without base in assisting the benzoxazolation process was clearly demonstrated as NMM was the only base that could afford the expected benzoxazole 3av, despite in low yield.

2-Acetonapthone as well as heterocyclic substrates derived from pyridine, thiophene and benzofuran gave moderate to good yields of the corresponding benzoxazole products 3ax-3aaa. The twofold reaction could be performed when p-diacetylbenzene (0.6 equiv) was used.

The reaction conditions could also be applied successfully to substituted 2-aminophenols (Scheme 4). Benzoxazoles bearing a methyl group 3ba and 3ca were obtained in yields of 63% and 73%, respectively. Chloro substituents were also compatible with the reaction conditions: the oxidative coupling products 3da and 3ea were formed in moderate yields (63% and 57%, respectively). The presence of an electron withdrawing groups such as sulfone, sulfonamide or nitro on the phenyl ring of 2-aminophenol could reduce the nucleophilicity of the hydroxy group, they had little effect on the yields. In the case of 4-nitro-2aminophenol 1g, thanks to low solubility in methanol of product 3ga, it was easily isolated from the reaction mixture by a simple trituration/filtration operation with methanol. The structure of sulfone 3fa obtained in 61% yield was confirmed by X-ray diffraction analysis. o-Aminophenol substrate bearing an electron donating group MeO 1g showed also good reactivity under our conditions, leading to benzoxazole 3ga in good yield. Our reaction conditions were however not applicable to aliphatic methyl such as pinacolone, 3-methyl-2-butanone of 2-heptanone (Scheme 5, eq. 1). In these cases, the reactions mixtures were found to be complex, possibly because both step of benzoxazolation of methyl ketones and methylene oxidation occurred more difficultly in the absence of an aryl group on the methyl ketone substrates. Indeed, the presence of the products issued from the first step could be detected in the crude mixtures, the subsequent oxidation of their methylene moiety failed. The reaction conditions applied to homologous of acetophenones such as propiophenone, butyrophenone or valerophenone led to complex mixtures since the intermediates of Willgerodt rearrangement of the longer chains underwent uncontrollable oxidation by DMSO (Scheme 5, eq. 2).

Scheme 5. Control Experiments

In order to support the proposed reaction mechanism, the primary Willgerodt-type product 4aa was prepared separately and allowed to react with DMSO (0.5 equiv) (Scheme 5, eq. 3). The product 3aa was formed in high yield under the standard conditions. On the other hand, performing the similar reaction in the absence of sulfur resulted in recovery of 4aa. This confirmed unambiguously that the oxygen atom of 2benzoylbenzoxazole 3aa product comes from DMSO.

Additionally, we found that diphenylmethane could not be oxidized into benzophenone and remained unchanged under these oxidizing conditions (eq. 4). These results suggested the important role of both aryl and 2-benzoxazolyl substituents as well as the participation of sulfur to the oxidation of methylene moiety. Scheme 6. Proposed Reaction Mechanism On the basis of these results and our previous works, [13] a plausible mechanism is proposed in Scheme 6. The reaction commenced with a Willgerodttype reaction as described previously to provide benzoxazole 4aa.

An easily tautomerization 4aa 4aa' facilitated by the presence of both benzoxazole and phenyl moieties would favor the formation of polysulfide salt 5. Sulfur extrusion from 5 followed by a sulfur-promoted oxidation of thiol 6 to thione 8 would proceed via polysulfide 7 and its tautomer 7'. Hydrolysis of thione 8 would lead to 3aa. [15] Since the oxygen atom of a carbonyl function of a ketone was known to be exchanged rapidly with water, it is not conclusive to use labelled water to confirm if the ketone oxygen atom 3aa was issued from water or not via the hydrolysis step 8 ® 3aa. [16] Water involving in this step is generated from the recycling of sulfur by oxidation of hydrogen (poly)sulfide with DMSO. [17] At the present stage, we cannot exclude the pathway involving direct functionalization of the methyl group of acetophenone 1a without the intermediacy of a Willgerodt-type reaction.

In conclusion, we have reported an efficient method for a direct access to 2-benzoylbenzoxazoles from 2aminophenols and acetophenones via a cascade of sulfur-promoted Willgerodt type benzoxazolation and methylene oxidation in DMSO. The reactions conditions are compatible with many functional groups and heterocycles. The method is highlighted by the fact that both starting materials o-aminophenols and acetophenones are inexpensive and readily available in a wide range of structures. Moreover, since sulfur and DMSO were used as oxidant in the presence of substoichiometric amount of N-methylmorpholine as an additive, our method is inarguably the cheapest and most convenient method to provide a library of 2benzoylbenzoxazoles.

Experimental Section

A mixture of 2-aminophenol 1 (1 mmol), acetophenone (1.2 mmol), elemental sulfur (32 mg, 1 mmol), Nmethylmorpholine (56 mg, 0.5 mmol), and DMSO (0.5 mL) was heated under an argon atmosphere in a 7-mL test tube at 110 °C for 16 h. The product was purified by column chromatography on silica gel (heptane:EtOAc 1:0 to 5:1 or dichloromethane:heptane 2:1 to 1:0). For experimental details, full characterization, and copies of NMR spectra of all compounds, see Supporting Information.

CCDC 2045870 and 2042799 contain the supplementary crystallographic data for 3aw and 3fa, respectively in this paper. These data can be obtained free of charge from The Cambridge Crystallographic

Scheme 3 .

 3 Scheme 3. Scope of Methyl Aryl Ketones 2

Scheme 4 .

 4 Scheme 4. Scope of o-Aminophenols 1

  -Build Strategy for the Synthesis of 2-Benzoylbenzoxazoles from o-Aminophenols and Acetophenones Adv. Synth. Catal. Year, Volume, Page -Page Le Anh Nguyen, b,c Quoc Anh Ngo,* b,c Pascal Retailleau, a and Thanh Binh Nguyen* a

Acknowledgements

We thank Vietnam Academy of Science and Technology for valuable financial support (KHCBHH.01/18-20). We thank Dr. A. Baron, C. Moriou, S. Beaupierre and S. Thoret for their valuable assistance.

Data Centre via www.ccdc.cam.ac.uk/data_request/cif.