Albane Gril -Lium

Le

Valérie

A COMPARATIVE ANALYSIS OF APPROACHES TO DESIGN AND CAPITALIZE DATA INDICATORS

Keywords: Learning analytics, data indicator capitalization, user-centered design, comparative analysis

In Technology Enhanced Learning field, learning analytics cover multiple research challenges, among which tracking data analysis, data indicators design and visualization and their adoption by end-users. Part of our research effort is dedicated to changing their design and use process, in order to capitalize them. This would allow cost saving on design workflow and encourage the adoption by teachers and students throughout their implications with a better understanding of the capitalization mechanism. A study of the state of the art has been made to explore various solutions which could be exploited in the capitalization of indicators on both levels, design and use. In this paper, a proposition of capitalization is detailed through measurable and accessible functionalities. The list of those functionalities is being used as criteria in a comparative grid to analyze how existing design approaches enable data indicators capitalization process. This analysis points out the purpose of our work to reach a larger scope of users, thus helping them understand and work on data indicators capitalization. It sets a direction toward a user-centered approach, aiming at a better control of users over their own observation needs and the use of their data.

INTRODUCTION

In a context where digital environments play an important role in education, practitioners use TEL (Technology Enhanced Learning) systems on a daily basis, generating at the same time a massive quantity of traces. Numerous works exploit these traces with data analysis and visualization mechanisms, transforming raw traces into data indicators as actionable knowledge [START_REF] Van Leeuwen | Teachers' perceptions of the usability of learning analytics reports in a flipped university course: When and how does information become actionable knowledge?[END_REF] that are much more comprehensible to users. Our research effort falls into this research area and has been carried out along with a multi-party university project écri+ (ecriplus.fr) since 2018. The main objective of the latter is to promote students' success via a learning platform DESCRIPTIVE AND CLOSE TO THE PROPOSED TITLE 3 A study on the state of the art of data indicators design approaches has helped us consolidate the definition of capitalization, before isolating specific challenges to focus on. By using low level criteria to identify data indicators capitalization, we can get closer to the user's pragmatic needs. A higher level of interpretation of those requirements will, then, allow us to confront our primary conception of capitalization in a process of consolidating the definition of the notion.

This paper is structured as follows. We start by presenting a number of research efforts dedicated to data analysis and data indicators visualization. We continue by pointing out the need for data indicators capitalization and making an attempt to provide a definition of the latter. Then we specify how we established a list of criteria used to compare different approaches for data indicators design. The last section is dedicated to the description of the approaches chosen for this state of the art. The comparative study is made later in the same section.

RELATED WORKS

Our work is part of the Learning Analytics (LA) field, which aims at understanding and improving learning, more specifically the one carried out through TEL systems (Siemense & Long, 2011). LA life cycle is commonly divided into three phases of traces collection, analysis and visualization of learning traces [START_REF] Clow | The learning analytics cycle: closing the loop effectively[END_REF]. illustrates an example of a life cycle of learning traces in a TEL system. In this example, both learners and teachers use a Web platform to perform learning activities. A learner can take an academic test or view a video, a teacher can monitor learners' progress. These activities generate a large quantity but also diversity of learning traces, captured by tracking systems. This data, called traces, can include activity traces, survey's answers, logs, and so on.

The data collection phases can take many forms due to the diversity of learning activities and tools currently available. Various studies are seeking to store and structure it in a coherent way, beyond its inherent diversity. In Settouti et al. (2007), modeled traces (m-trace) are considered as the association between a collection of temporally situated events and an explicit model of this collection. That formalism allows transformations, requests and even computes indicators while handling heterogeneous data from multiple sources. On the other hand, in UTL (Usage Tracking Language) [START_REF] Choquet | Modeling tracks for the model driven re-engineering of a tel system[END_REF], a trace is not necessarily temporally situated, but a result of a transformation described by the language. UTL uses a model for all types of data, defining them according to three facets: The Defining (D) facet defines the observation needs; The Getting (G) facet describes the observation means to implement the data acquisition; and the Using (U) facet defines the use of data once they are computed. Each proposal aims at offering a relevant model to support a specific type of information. It will then significantly influence the way this data can be processed and analyzed.

The collected traces are commonly processed by analysis tools. As found in Laalys [START_REF] Muratet | Understanding learners' behaviors in serious games[END_REF], a statistical tool that uses a semi-automatically process to analyze learner's behavior by generating pedagogical labels in serious games. UnderTracks [START_REF] Bouhineau | Conception et mise en place d'un entrepôt de traces et processus de traitement EIAH: UnderTracks[END_REF] assists the analysis process with its visual construction. UTL4CL [START_REF] Ngoc | Modeling of Indicators using UTL: A study case with Hop3x system[END_REF] is based on the UTL language to generate data indicators with SQL requests on traces.

The transformation of raw data into usable information through an analysis process, thus enabling the creation of learning data indicators. These can then be displayed to users via a learning dashboard, created to suit different pedagogical practices. For example, during a course, in a synchronous way, teachers can use a dedicated dashboard to keep track of students' activities and their progress. In an asynchronous way, teachers can assess the use of a resource or adjust the structure of a course. Some work aims at creating data indicators for a specific pedagogical and digital context. For instance, Travis [START_REF] May | TrAVis to enhance online tutoring and learning activities: Real-time visualization of students tracking data[END_REF] displays data indicators for students' activities in discussion forums. Lab4CE [START_REF] Broisin | Lab4CE: a remote laboratory for computer education[END_REF] implements visualization tools in a remote laboratory for computer education intended for instructors and learners. Radar and Reflector [START_REF] Phielix | Group awareness of social and cognitive performance in a CSCL environment: Effects of a peer feedback and reflection tool[END_REF] is a reflection tool for learners on group assessment.

Thus far, data indicators are seen as a tool of Learning Analytics that inherited its challenges in adoption by teachers as claimed by [START_REF] Kaliisa | Overcoming challenges to the adoption of learning analytics at the practitioner level: A critical analysis of 18 learning analytics frameworks[END_REF]. Their work studied main challenges in teacher's adoption of LA, including the integration of technical and pedagogical expertise, the lack of connection with educational theories or pedagogies, and the failure to align with teachers' practice.

It is worth mentioning that learning indicators designate indicators focused on the study of learning, whereas data indicators designate from a general perspective the object computed throughout the analysis process of traces, which are not necessarily acquired within a learning setting. In our work, we focus on data indicators on learning, designed and used in a learning context.

The approach based on indicators capitalization is not easily highlighted among the existing comparative works in LA [START_REF] Cherigny | L'analytique des apprentissages avec le numérique Groupes thématiques de la Direction du numérique pour l'Éducation[END_REF][START_REF] Croon | A systematic review of the effect of gamification on adherence across disciplines[END_REF][START_REF] Basak | A framework on the critical success factors of e-learning implementation in higher education: A review of the literature[END_REF], Alowayr and Badii, 2014[START_REF] Corbi | Review of Current Student-Monitoring Techniques used in eLearning-Focused recommender Systems and Learning analytics. The Experience API & LIME model Case Study[END_REF][START_REF] Choquet | Modeling tracks for the model driven re-engineering of a tel system[END_REF], Dimitrakoupoulou, 2004). For that, we take a closer look at how each approach, formalism as implementation, enables users to interact with data indicators, leading to their capitalization. Indeed, in the idea of supporting digital learning, tools were developed to provide users (student as teacher, experts as non experts) with assistance in accessing and feeding learning resources and environments. The involvement of users in all LA phases is therefore essential to the analysis outcomes and their relevance. But it is also crucial to the way indicators are used after the completion of their design and evaluation. To analyze indicators' life cycle beyond those phases, as the capitalization process suggests, a new set of requirements is needed.

CAPITALIZATION: DEFINITION AND CONCEPTS

Defining data indicators capitalization

The concept of capitalization is usually tied to monetary valuation. Here, the capitalization is considered as a valuation of the data and the value of the information they provide. When it comes to capitalized data indicators, their value can be determined by several factors including how relevant an indicator is, or how easy to use and reuse the existing indicators are, according to users and contexts.

For example, a teacher creates, for a particular course, an indicator for students' progression and grading based on a series of short quizzes. He can easily give it to another colleague working in the same course. But the colleague, while using a different pedagogical scenario and evaluation system, can not directly use the given data indicator. He will need to alter the core definition of the indicator's analysis process, which requires time and resources.

Capitalizing indicators would be about keeping and multiplying their value in time, contexts and users, while maintaining the reusability. Our approach considers data indicators as objects with challenges to overcome, like production cost or user validation. Indeed, they have a complex design process, highly dependent on context. Despite numerous attempts to simplify the design workflow for analysts, or the identification of observation needs for users [START_REF] Laforcade | Specification of observation needs in an instructional design context-A Model-Driven Engineering approach[END_REF], the adoption of data indicators in learning practice for both teacher and student remains challenging [START_REF] Kaliisa | Overcoming challenges to the adoption of learning analytics at the practitioner level: A critical analysis of 18 learning analytics frameworks[END_REF]. On top of that, while the usefulness of dashboards has been demonstrated in [START_REF] Verbert | Learning analytics dashboard applications[END_REF] to back up educational practitioners in any type of pedagogical scenario, many barriers remain to fully assimilate indicators as a common pedagogical resource [START_REF] Basak | A framework on the critical success factors of e-learning implementation in higher education: A review of the literature[END_REF]. The inadequacy of user's implication and expertise with the current solutions can # -will be assigend by editors. TITLE -SHOULD BE SHORT, DESCRIPTIVE AND CLOSE TO THE PROPOSED TITLE 7 be considered as a design workflow issue. The commitment to create an adequate solution has led to the rise of co-design processes with users [START_REF] Dollinger | Working together in learning analytics towards the co-creation of value[END_REF]. These challenges are those addressed through data indicator capitalization.

The approach of capitalization is therefore the combination of several multidisciplinary concepts, as illustrated in Figure 2. The first concept is "appropriability", often used in human sciences (Ollagnier-Beldame, 2010) and in the field of HCI (Human-Computer Interaction). For instance, when adopting the indicators, users can apprehend and customize them without the intervention of experts. Then, comes the idea of "reusability", that is, indicators must be designed to work with similar context. Next, "adaptability" that makes modifications possible to match other contexts or meet different needs. Finally, "shareability" must be ensured between users, thus, allowing users to access and share capitalized indicators within the system.

The association of all those concepts under the idea of data indicator capitalization aims to tackle the identified challenges and encourage a broader use of capitalized data indicators in e-learning practices. In order to verify the achievement of those four concepts, each was broken down into concrete technical properties as validation criteria, as presented hereafter.

Criteria to compare approaches in data indicator design

The criteria of this comparative analysis are intended to specify how an existing approach can achieve indicators capitalization. Several iterations have been made to better define each criterion.

In order to come up with a solution suitable to the platform's users, communication with the project's stakeholders has been maintained right from the start of the design process. Working groups, held primarily with teachers, allowed us to identify some of their needs and expectations. This observation work did not include any presentation of the data indicators or their capitalization process. The latter was tackled through the prism of pedagogical information, their use and challenges, creating a real involvement of the teachers, even without specific knowledge in learning analytics. This will allow us to create a relevant protocol for further experimentation on the relation between teachers or learners and data indicators. The preliminary list of needs that emerged was used as a basis to compose a series of technical properties, necessary for the verification of capitalization's concepts (Table 1). The existing criteria are meant to be fed from the experience of different users' context and profiles. The multiple iteration of user needs gathering does not affect the existing criteria, but rather help create a hierarchy among the identified needs.

The properties formulated are low level to get closer to technical functionalities and make the validation process easier. Indeed, an abstract concept is too open to interpretation to be used in a comparative approach. In the meantime, formulating a set of technical properties generic enough to apply to different kinds of approaches is no less tedious. Here after a detailed list of all currently selected technical properties classified between the four concepts:

2.2.1 Appropriability

Users need to regain control over the information available to them, which means that anyone can use the system, regardless of specific technical skills:

• Granularity of entities: the idea that each complex entity can be broken down into smaller and less complex entities. The more levels of decomposition, the better. This gives the opportunity to consider a nesting architecture to simplify the notions being handled. • Visualization choices: it is an important part of data exploitation, a formalism should be able to specify how an indicator will be visualized, and an implementation should give the possibility to change it. A tool should have customizable visualization options.

• Categorization of entities: by classifying all entities, users will be able to access them more easily. It is even more efficient on multiple levels or with a tag system. • Search in existing indicators: users can access and select existing indicators for their own use. • Filtering of indicators: a variety of data indicators can be specified and narrowed down, allowing a better choice for the most relevant ones for each user's need.

2.2.2 Reusability

Any entity created has to be valued within the system, without losing its value through time:

• Description of entities: any information is useful. It can be considered as meta-data of any sort. It is important for meta-data not to be mandatory, but its availability, updatable by any user, is a plus. This co-construction raises the question of community regulations to maintain a high level of reliability. • Knowledge evolution: evolution of the existing entities has to be eased. An existing indicator can be modified to create multiple new ones, without losing any previously created entities.

• Versioning system: the idea of keeping track of data indicator construction through the different steps, as well as its evolution as it passes from user to user. The functionality will allow "Knowledge evolution" without losing value through time. • Interlinking of indicators: the possibility to create more complex indicators not only from traces but also from other indicators, all entities should be linkable and those links modifiable. It is an easier way to promote "Knowledge evolution" by using some kind of "Granularity of entities". • Storage of entities: it is necessary to store the existing and created entities to ensure their future reuse. This is not necessarily handled by the tool itself, but needs to be considered when designing a system.

2.2.3 Shareability

To promote community and co-construction, communication between the user of the system can be achieved through different means:

• User access management: to create a system open to many types of user, being able to handle role and permission on data access or edition is necessary. Also, from an ethical perspective, an open system has to consider the questions of data anonymization and protection.

• Circulation of indicators to users: in an effort to co-build knowledge, spreading of the created and existing entities will allow different levels of user commitment and real cost saving. Ideally, any entity or set of entities can be shared to any related group of users. • Rating of indicators: to promote reuse of existing entities and help newcomers navigate the potentially great quantity of entities. It can be done in many ways, including assessing different aspects of the indicators, such as their usefulness and their relevance. • Comments can be made on any data indicator. Users participate through the commenting system in describing indicators and their properties.

2.2.4 Adaptability

The system has to provide a strong support to the expansion of indicators use:

• List of existing entities: an entity includes data indicators but also traces, operators, analysis processes or even users. It is required for many previous properties to have an available list of all existing entities. • Import of external traces: in a new context, traces and data in general, won't necessarily be the same, adding new information has to be facilitated by design. • Context of the indicator: if the context changes, the one in which an indicator has been created has to be specifically detailed in its description. That context includes the pedagogical situation and the environment setting of the TEL system. • Abstraction of indicators: regardless of the context of use, each indicator can be created independently. • Modular data computing: the calculation process needs to be easily adaptable to a new context, if possible automatically by the system, requiring a rigorous formalism set in "Abstraction of indicators".

These technical properties are classified between the four pillar concepts that define the capitalization design workflow and use. However, most of this classification can be questioned. For instance, the possibility to search and filter indicators can both enable appropriation, but are related to properties classified in other concepts. In order to search among existing entities, it is necessary to have an available list of the existing ones. This related property "List of existing entities" is placed in the "Adaptability" category. In the meantime, an implementation of a formalism allowing access to existing entities does not necessarily have a "search functionality" either. Likewise, if the filtering property is strayed forward in a tool, for a formalism it means that indicators Considering the relation of some criteria (Figure . 3), their classification has been adjusted throughout multiple iterations of the literature review. The lack of information and inadequate wording was also considered to make criteria relevant to the contexts encountered.

With this list of capitalization properties, we seek to experiment how design workflow responds to the identified requirements. Comparing existing solutions through this new perspective and the present grid of criteria will allow us to consolidate the definition of data indicators capitalization within the current context.

COMPARING DATA INDICATOR DESIGN APPROACHES

Selection of indicators design approaches

Approaches and tools for designing indicators described in various research works have been selected. The starting point was a study of the state of the art carried out in the Thematic Groups of the Digital Education Directorate by [START_REF] Cherigny | L'analytique des apprentissages avec le numérique Groupes thématiques de la Direction du numérique pour l'Éducation[END_REF], completed by additional research. Once this initial list of tools had been compiled, those closest to the criteria described above were chosen, distinguishing between formalisms of various kinds and tangible implementations.

Our project's platform is meant for students' online training before a certification in accordance to their achievement and the levels they reached. It is also meant for teachers to follow the class's progress and help them through available resources. All the selected approaches are relevant in the specific context of our project. This helps us narrow down our selection on many other tools designed exclusively for interaction analyses, as well as lab sessions and practical works. Likewise, some work specialized on innovative hardware interfaces (tablets, surface tables, VR glasses, …) were discarded as irrelevant to our work.

Each of these approaches were selected for their relevance in a capitalization of indicators design, but also for the diversity of their nature and solution provided.

Rule-based systems [START_REF] Schauer | Playing by the rules: A philosophical examination of rule-based decisionmaking in law and in life[END_REF] reproduce the cognitive mechanisms of an expert in a particular domain. They are based on facts and modifiable rules, which offer flexibility in data computing and the opportunity for knowledge evolution. It has been implemented in GINDIC (Generator of INDICators). Designers can use this tool to define indicators, the calculations performed on the traces and their visualization [START_REF] Gendron | Towards a structured approach to the definition of indicators for collaborative activities in engineering design[END_REF].

The UTL 2 language [START_REF] Choquet | Modeling tracks for the model driven re-engineering of a tel system[END_REF]) is an XML-based metalanguage describing the traces of a scenario. It uses the description of observation analysis methods to facilitate their capitalization. It can be used for indicators modeling with a strong attachment to pedagogical situations.

The trace-based systems, TBS, [START_REF] Settouti | A trace-based system for technologyenhanced learning systems personalisation[END_REF] consider traces as a temporal sequence of observed events along with a capitalizable model. Traces in heterogeneous formats can thus be manipulated to create different indicators abstracted from environment settings. TBS-IM [START_REF] Dimitrakopoulou | A new approach based on modelled traces to compute collaborative and individual indicators human interaction[END_REF] is a TBS implemented for indicators calculation in the Moodle learning platform. Another TBS is DDART [START_REF] Michel | Supporting awareness and self-regulation in project-based learning through personalised dashboards[END_REF], a dynamic dashboard # -will be assigend by editors. TITLE -SHOULD BE SHORT, DESCRIPTIVE AND CLOSE TO THE PROPOSED TITLE 13 based on activity traces and reports in the form of a Moodle plug-in. Learners can create customizable indicators related to their activities.

The Reusable Indicator Template of [START_REF] David | Design patterns for recording and analysing usage of learning systems: State of art of tracking and analysing usage[END_REF] is composed of a computational function of traces and metadata explaining the reuse conditions, including the function's domain of definition, the learning situation and a description. The EM-AGIIR tool [START_REF] David | Design patterns for recording and analysing usage of learning systems: State of art of tracking and analysing usage[END_REF] proposes an implementation of the RIT and an open multi-agent architecture to apply the indicators with new traces.

The ontological framework of [START_REF] Lebis | Capitalisation of analysis processes: Enabling reproducibility, openness and adaptability thanks to narration[END_REF] is used to capitalize the analysis processes of learning traces in a so-called narrative way. It is not meant for indicator capitalization, but its originality motivated us to compare it with the rest. It is part of a larger set of tools built for analysis process capitalization.

The Academic Analytics Tool [START_REF] Ross | The academic analytics tool: Workflow and use cases[END_REF], was developed for the Moodle Analytics project. AAT is a software allowing teachers and course managers to access and analyze student behavior data in learning systems. Course Insights is a Learning Analytics Dashboard that provides filterable and comparative visualizations of aggregated students' daily activity and learning events to teachers. The flexibility of those tools, which are not based on a studied formalism, is interesting from a user perspective.

We looked into indicators design approaches from the Learning Analytics domain, but it seems relevant to expand this research scope into other domains where context is equally important. Overall, few tools offer a solution for capitalization either from system or user standpoints. Nonetheless, each tool presents a solution or part of a solution to the issues we have pointed out above. A closer look at the existing approaches is therefore interesting for the design of our own indicator capitalization approach, while selecting the criteria that match our design choice based on the four pillar concepts, presented earlier in section 3.

Literature review grid

The findings from our study on the state of the art is synthesized in Table 1. Our proposal is based on four levels of validation of a criterion: [V] validated with respect to our work and in line with our requirements on capitalization, [PV] partially validated, [NV] not validated and [NR] not relevant, in case of inadequacy with a criterion. The number of "not relevant" criteria is due to the effort to compare objects of different nature with a common grid. The formulation of the criteria as technical properties is interpreted in a distinct way depending on whether the analyzed object is a theoretical formalism or a tangible implementation. For instance, "Visualization options" would be the information stored for a language, the display handled for an architecture and real visualizations for implemented tools. In some cases, the property does not apply for some level of abstraction, hence the use of "not relevant".

Validation of each criterion was done through a careful analysis of the literature. Nonetheless, the collected information for our analysis can be limited, as each tool we studied is not always meant for data indicator capitalization. The grid's filling considers interconnection of criteria in the implementation level to fill the information gaps. It may occur that specific information contradicts the interconnection as seen in Figure 3, for instance no filtering implemented even where categorization is permitted. 1, it can be seen that a rule-based system does not offer any specific means to categorize entities, whereas GINDIC does include the dimension and the nature of the created indicators. On its part, the Reusable Indicator Patron provides three types, Cognitive, Social and Affective, which are here considered as categorization. If we take a look at another example, "User access management" may seem a stray forward functionality for an implemented tool, therefore, it can be seen that three out of five of them have it implemented. Then for the formalisms, both rule-based system and tracebased system are considered as partially validating the criterion by suggesting a system architecture in which the property could be easily added. But this concern has not been found in the studied documentation and therefore not fully validated. For the rest of the formalisms, their nature is too disconnected from such a functionality to be considered relevant. A similar reflection has been followed for each criterion in the grid, while studying the available documentation.

Regarding the criteria suggested for the capitalization of indicators, some limits in the design solutions can be noticed. It would be interesting to look for an approach which facilitates sharing abilities of existing indicators without losing the flexibility of their design. Indeed, the criteria for appropriation and sharing of indicators are the least developed.

This involvement of the end-user is key in a successful capitalization process. But the current indicators design approach focuses mostly on the feasibility and the technical aspect of indicators, rather than the plus value and the experience the user will take from using a specific indicator. It is worth taking note that the "real use" or the "effectiveness" of the capitalized indicators is not the goal of our approach and so cannot be considered a success measure for our work. But considering the way individuals will use the created object influences how it is designed. The use of each object can evolve in time and practice behavior, thus requiring a versatile and evolving approach.

The technical usability of an approach, as an ergonomic criterion, is not directly included in this grid, but this important factor of user adoption can be partially assessed through the present criteria.

It can also be noted that an implementation does not necessarily preserve all the technical properties of the corresponding formalism. Some other properties do not seem to be related to the implemented indicators design formalism. However, the chosen formalism clearly has an impact on the implemented functionality. Thus, the conception model for an indicator design approach should not be taken lightly.

To sum up, the current grid will be used in the continuation of our work on designing an approach for capitalization, as a guideline and validation criteria. The computerization process includes consideration of the formalism as much as the implementation and the user's interactions with it. Users will find the presented functionalities to enable interesting actions in the design process but the possibility of an interaction alone, as the collection of data without an efficient indexation, will still be limited. It is therefore our main focus to enable the whole potential of users' interactions and their design activities within the idea of data indicator capitalization.

CONCLUSION

The democratization of data indicators in learning practices faces challenges with high design cost and limited user adoption. Capitalization as a solution to those challenges requires a closer look at the user's needs regarding data collection, transformation, computation, exploration, and visualization. The multiplicity of users' level of interaction with the TEL system, pedagogical information and data indicators make the design of a common approach more complex.

The comparative analysis developed here allows us to point out a range of concrete technical points to enhance data indicators design formalism and to support their implementations. By using this custom grid to compare a wide range of approaches, we saw how the formalism used can have an impact on the available functionalities. This initial list of criteria for a tool aiming to support capitalization is also how we choose to start building our approach. A number of specialized tools have successfully offered solutions for data analysts with highly efficient data processing and visualization, or for teachers and learners with turnkey solutions for dashboard. But the co-design of data indicators by heterogeneous users is still a challenge.

It is important for us to point out that the work presented here has some limits. First, the domain and project context in which were made the choices of design approaches remain narrow. This choice was also due to size constraints. For that reason, extending this analysis to new design solutions has already been set among our research perspectives. The lack of empirical studies to validate the grid has also been pointed out, and an analysis carried out exclusively through the literature would need to be further investigated. Finally, the criteria developed here are based on a user-centered approach that has only been initiated.

Data indicator design approach focusing on its capitalization considers the indicator's entire life cycle. Throughout this process, tools will be developed for users to operate and specific skills are needed to effectively carry out a capitalization process. Our focus is neither on a particular tool nor a user, but on observing and studying how users interact with a tool during the capitalization process. As we have already mentioned, the capitalization is the result of a constant interaction between users and tools. The facilitation of these interactions is valuable and pertinent to help users make decisions in data indicator design and use. The provided solution needs to be modular, so each task can be, or not, achieved by using adequate formalism and implementation. The analysis we have made so far shows the necessity to accentuate the accessibility of the entities handled by the users to ensure the durability of our approach.

Capitalization also requires the active involvement of users. They need to be proactive for successful indicators' assimilation in their learning practices. This proactivity has to be supported and promoted by the process itself. With this perspective, our future work will implicate users as they will help us validate the initial conclusions. Focus groups will be organized and get involved in our design approach. Moreover, a co-design of the capitalization process has already been initiated with working groups to identify the users' actions when creating and using data indicators. This first attempt will allow us to detect the key interactions when users are involved in a capitalization process. The grid that we have presented here will serve as the base of an evolutive approach where users and their interactions will be observed and studied in each iteration of data design and capitalization.

The direction we are taking, where user involvement is continually present, leads us to acknowledge other important issues of data privacy and ethics. Their impact on users' perception and even performance is not negligible [START_REF] Kaliisa | Overcoming challenges to the adoption of learning analytics at the practitioner level: A critical analysis of 18 learning analytics frameworks[END_REF][START_REF] May | The side effect of learning analytics: An empirical study on e-learning technologies and user privacy[END_REF]. Thus, this consideration must be assimilated in the design process to reach a thorough user-adapted proposition.

• ACKNOWLEDGEMENT

Our work is carried out within the French project PIA écri+ (n° ANR-17-NCUN-0015) <http://ecriplus.fr/>.

• REFERENCES

Alowayr, A., Badii, A., (2014). Review of Monitoring Tools for E-Learning Platforms.

International Journal of Computer Science and Information Technology, 6, 79-86.

Figure 1 .

 1 Figure 1. An example of Learning Analytics' life cycle

Figure 2 .

 2 Figure 2. Pillar concepts of data indicator capitalization

Figure 3 .

 3 Figure 3. Related criteria for capitalization

Table 1 .

 1 Comparison of formalisms and implementations of data indicator capitalization

	# -will be assigend by editors. TITLE -SHOULD BE SHORT,	15
	DESCRIPTIVE AND CLOSE TO THE PROPOSED TITLE
	In Table


Sándor Bilicz
email: biliczs@sch.bme.hu

Emmanuel Vazquez

József Pávó

Marc Lambert

Szabolcs Gyimóthy

Eddy-current testing with the Expected Improvement optimization algorithm

Keywords: eddy current testing, inversion, global optimization, Gaussian processes, Expected Improvement algorithm

This paper presents an inverse problem methodology in the domain of non-destructive testing, and more precisely eddy-current testing. Our objective is to use a precise but expensive-to-evaluate model of the electromagnetic induction phenomenon in a conductive material and to estimate the characteristics of a flaw by minimization of a regularized criterion with the Expected Improvement (EI) global optimization algorithm. The EI algorithm is designed to estimate a global optimum of a function with a restricted budget of function evaluations. Thus, we expect to be able to estimate the characteristics of a flaw with a relatively low cost despite resorting to an expensive model of the induction phenomenon. The efficiency of the approach is discussed in the light of preliminary numerical examples obtained using synthetic data.

INTRODUCTION

This paper presents an inverse problem methodology in the domain of non-destructive testing, and more precisely eddycurrent testing (ECT). As illustrated by a large number of contributions in the last ten years, the estimation of the characteristics of scattering objects remains quite challenging from a theoretical, computational and/or experimental perspective, even in the case of the simplest configurations (e.g., a homogeneous embedding space). In this paper, our objective is to use a precise but expensive-to-evaluate model of the electromagnetic induction phenomenon in a conductive material. Then, to estimate the characteristics of a flaw, a regularized criterion is minimized using the Expected Improvement (EI) global optimization algorithm. The EI algorithm is designed to estimate a global optimum of a function with a restricted budget of function evaluations. Thus, we expect to be able to estimate the characteristics a flaw with a relatively low cost despite resorting to an expensive model of the induction phenomenon.

The EI algorithm is well-known in the domain of global optimization, but to the best of our knowledge, using the EI algorithm for the ECT problem remains largely unexplored. Preliminary results for the characterization of a single volumetric defect have been presented by the authors in [START_REF] Bilicz | Characterization of a 3D defect using the Expected Improvement algorithm[END_REF]. In this paper, a more challenging case is studied. We consider two infinitesimally thin material defects close to each other. Our contribution is twofold. First, from the point of view of ECT, we would like to show that the methodology we propose is promising. More generally, from the point of view of the inverse problem community, we would like to bring attention to an efficient optimization algorithm that renders practical the use of precise but expensive direct models.

The paper is organized as follows. First, we present the theoretical formulation of the direct problem and some aspects of its numerical implementation ( § 2). Then the inverse problem -along with our regularizing assumptions -will be presented ( § 3). Next the new stochastic approach and the optimization task will be in the focus ( § 4). Finally, numerical examples will illustrate the proposed method and the conclusions of the experiments will be drawn ( § 5).

THE FORWARD PROBLEM

Assume the arrangement shown in Fig. 1 where two infinitesimally thin cracks located in the planar surfaces S 1 and S 2 inside a non-ferromagnetic plate are searched by a pancake type ECT coil. The normal vectors of the S 1 and S 2 surfaces are n1 and n2 , respectively. In this Subsection the method used for the solution of the forward problem is discussed. In other words, the method used for the calculation of the change of the impedance of the ECT coil due to the presence of the cracks (∆Z) is described when the location and the orientation of S 1 and S 2 are arbitrary. During the solution of the inverse problem, however it will be assumed that the cracks are outer defects (OD) and they are parallel to each other, consequently n1 = n2 = x ( x is the unit vector of the x co-ordinate axis). Fig. 1 shows the particular arrangement that is actually assumed for the inverse problem.

The electromagnetic interaction of the exciting coil with nonconducting cracks whose thickness are considerably smaller than their other dimensions might be calculated based on the so-called infinitesimally thin crack approximation described by [START_REF] Bowler | Eddy-current interaction with an ideal crack. I. The forward problem[END_REF]. This model assumes the cracks in the otherwise homogeneous conductor as mathematical surfaces that block the flow of the eddy currents across their surfaces, in other

x x y z A B D 1 D 2 L 1 L 2 σ 0 , µ 0 h l d S 1 S 2 r 1 r 2
Fig. 1. Schematic drawing of the geometry of the investigated forward and inverse problems words the normal component of the electric field must be zero on the surfaces representing the cracks. In the same time, the magnetic field is assumed to be continuous on the two sides of these surfaces.

The above boundary conditions might be satisfied by replacing the S 1 and S 2 mathematical surfaces with current dipole distributions, p 1 (r) n1 and p 2 (r) n2 , having the same effect as the presence of the crack (the function p 1 (r) and p 2 (r) are non-zero only on the surfaces S 1 and S 2 , respectively). Thus, the sum of the electromagnetic field generated by the exciting coil and the secondary sources in the homogeneous conductor will provide the field generated by the crack-probe interaction. Consequently this field must satisfy the above boundary conditions. In the case of cracks with planar surfaces, the described consideration leads to the following integral equation reported by [START_REF] Pávó | Reconstruction of group of cracks in plate specimens using ECT impedance data[END_REF]:

0 = E i n1 (r 1 ) -jωµ 0 S 2 g 21 (r 1 |r ′ )p 2 (r ′ ) dr ′ - -jωµ 0 lim r→r 1± S 1 g 11 (r|r ′ )p 1 (r ′ ) dr ′ , r 1 ∈ S 1 , (1) 0 = E i n2 (r 2 ) -jωµ 0 S 1 g 12 (r 2 |r ′ )p 1 (r ′ ) dr ′ - -jωµ 0 lim r→r 2± S 2 g 22 (r|r ′ )p 2 (r ′ ) dr ′ , r 2 ∈ S 2 , ( 2
)
where r k± (k = 1, 2) denotes the limiting values of the approaches towards the point r k from the nk ornk directions, ω is the angular frequency of the sinusoidal excitation of the probe coil and µ 0 is the permeability of the vacuum.

E i nk = nk • E i (k = 1, 2)
, where E i is the so-called incident field that is the electric field generated by the current of the exciting coil in the defect-free plate. g kl (r|r ′ ) = nl • G(r|r ′ ) • nk (k, l = 1, 2) where G(r|r ′ ) is the Green's dyad transforming the current excitation into the electric field.

The surfaces of the cracks are discretized on a regular lattice where the unknown dipole density function is approximated by a piecewise linear function. The integral equation is tested by first order testing functions and the kernel of the integral equation is evaluated by the formulae published by [START_REF] Pávó | Reconstruction of crack shape by optimization using eddy current field measurement[END_REF]. After solving the linear system of equations obtained by the described discretization of the integral equations (1)-( 2), the dipole density function can be calculated. Knowing p 1 (r) and p 2 (r), the impedance variation of the probe, ∆Z, due to the presence of the cracks can be calculated based on the reciprocity theorem [START_REF] Bowler | Eddy-current interaction with an ideal crack. I. The forward problem[END_REF]:

∆Z = - 1 I 2 2 k=1 S k E i nk (r)p k (r) dr, ( 3
)
where I is the current of the exciting coil.

The results of the described solution of the forward problem are tested against measurements available for single artificial cracks (unfortunately measured results on two parallel cracks are not available in the literature). The comparison verified the validity of the results.

For rectangular shaped cracks being parallel to each other and being perpendicular to the surface of the plate, the integral equations ( 1)-( 2) can be solved by approximating p 1 (r) and p 2 (r) with the function series introduced in [START_REF] Pávó | Calculation of eddy current probe signal with global approximation[END_REF]. Using this method, the results on several test configurations are very close to those obtained by the method described above. The good agreement supports the validity of both solution methods.

THE INVERSE PROBLEM

When the probe coil scans the zone containing the cracks, the variations of the coil impedance are measured at each point of a regular rectangular grid, above the supposedly damaged zone of the plate. In other words, a so-called surface scan is performed. The scan consists of N coil locations where the measured impedance variations are ∆Z k , k = 1, 2, . . . , N.

The problem is to characterize the flaw from this set of measured impedance variations, which will be viewed as the input data of the inverse problem.

Regularizing assumptions

It is customary to introduce some regularizing assumptions to reduce the ill-posedness of the inverse problem. In this work, two rectangular shaped thin cracks are assumed in the plate specimen, under (or near) the surface scan measurement (see Fig. 1). Both cracks are of OD type. Their orientation is supposedly known, together with the "geometrical center" of the two-cracks-system (i.e. the distances of the crack midpoints from the origin are the same for both cracks in the x (A/2) and in the y (B/2) directions as well). As a result, only 6 geometrical parameters are needed to describe the configuration: the length of the cracks (L 1 , L 2 ), their depth (D 1 , D 2 ) and the x-y distances of their centers (A, B), respectively (see Fig. 1). To simplify the notations, the unknown defect parameters are collected in a vector t:

t = [A, B, L 1 , L 2 , D 1 , D 2 ]
. The definition domain of t will be denoted by T, and will be referred to as the parameter space.

The forward solver ( § 2) is used to compute the impedance variation due to a fault characterized by its parameter vector t. A simulated surface scan consists of such computed impedance variations, denoted as ∆Z k (t), k = 1, 2, . . . , N. Obviously, the simulation is performed at exactly the same coil positions as the real measurement, thus both the real and the simulated surface scan consists of N coil locations.

Optimization task

The next step for solving the inverse problem is to achieve the strongest resemblance between the impedance signal obtained by simulation, {∆Z k (t), k = 1, . . . , N}, and the measurement, {∆Z k , k = 1, . . . , N}, by tuning the parameter vector t.

The applied inversion method requires a finite parameter space T, meaning that one has to set lower and upper bounds for each parameter and, moreover, the parameter axes are needed to be discretized. Consequently, the space T has to consist of a finite number of parameter points t. The bounds and discretization of each parameter are summarized in Table 1. The parameters are independent from each other, thus, the parameter space T can be imagined as a 6 dimensional hypercube, spanned by the 6 parameter axes. The hypercube includes 5 2 • 7 4 = 60025 parameter points. To give a mathematical form to the resemblance between two surface-scan impedance signals, we define the similarity function

Q(t) = N k=1 |∆Z k (t) -∆Z k | 2 N k=1 |∆Z k | 2 . ( 4
)
Our objective is thus to minimize Q(t), i.e. to find t = arg min t∈T Q(t) .

STOCHASTIC MODELING AND OPTIMIZATION OF THE SIMILARITY FUNCTION

Our objective is to implement the Expected Improvement algorithm to minimize Q(t). One iteration of such an algorithm involves mainly two steps:

(1) the construction of an approximation of the similarity function from a set of past evaluations of the function obtained at previous iterations. To this end, a random process is chosen as a model of the similarity function and an interpolation by kriging is performed. (2) the search of the maximum of the Expected Improvement over the parameter space T then the computation of the similarity at that parameter point.

This two-step procedure is repeated iteratively until a stopping criterion is met, i.e. a sequential optimization algorithm is obtained.

Kriging interpolation

Let us assume that we have already observed the multivariate scalar function Q(t) at n points t 1 , t 2 , . . . , t n of T. In this way,

n function values Q 1 = Q(t 1 ), Q 2 = Q(t 2 ), . . . , Q n = Q(t n )
are known. We would like to predict the function value at unobserved sites. One method to achieve this goal is to use kriging, a random process approach developed in the 60s in geostatistics [START_REF] Chiles | Geostatistics, Modeling Spatial Uncertainty[END_REF]. The method is also well-known for modeling computer simulations [START_REF] Sacks | Design and analysis of computer experiments[END_REF].

Let ξ(t) be a Gaussian random process that models the function Q(t). Thus, each observation Q k is considered as the realization of the Gaussian random variable ξ(t k ) (k = 1, 2, . . . , n). Kriging computes the best linear unbiased predictor (BLUP) of ξ(t). Let us denote this prediction by ξ(t). The predictor is linear in the sense that it is a linear combination of the observed random variables ξ(t k ), k = 1, 2, . . . , n, which can therefore be written as

ξ(t) = n k=1 λ k (t)ξ(t k ). ( 5
)
Unbiasedness relates to the fact that the mean of ξ(t) is equal to the mean of ξ(t), i.e. the mean prediction error is zero:

E[ε(t)] = E[ ξ(t) -ξ(t)] = 0. ( 6
)
The term "best" means that the prediction error ε(t) of the kriging predictor has the smallest variance among all unbiased predictors. This variance (also called kriging error) may be written as

σ2 (t) = var[ ξ(t) -ξ(t)] = E[( ξ(t) -ξ(t)) 2 ] , (7)
using the unbiasedness condition (6).

The objective is to find the coefficients λ k (t) in ( 5) that provide the BLUP. The kriging error can be written using the covariance function, which describes the dependence between two random variables of the process at different points. Let us denote the covariance function by k(t a , t b ) = cov[ξ(t a ), ξ(t b )], where t a and t b are two points in T. Let us denote by K, the covariance matrix whose entries correspond to the covariances of the random process between the observation points t 1 , t 2 , . . . , t n :

K =                 k(t 1 , t 1 ) k(t 1 , t 2 ) . . . k(t 1 , t n ) k(t 2 , t 1 ) k(t 2 , t 2 ) . . . k(t 2 , t n ) . . . . . . . . . . . . k(t n , t 1 ) k(t n , t 2 ) . . . k(t n , t n )                 (8)
If one has some prior knowledge on the function to be modeled it can be reflected by giving a prior mean to the predictor. Since in our case no information is available but the observed function values, a constant (but unknown) mean E[ξ(t)] = C is assumed.

To simplify the notations let us collect the coefficients λ k (t) into a vector λ λ λ(t) = [λ 1 (t) λ 2 (t) . . . λ n (t)] T , and denote by k(t) the vector whose elements are the values of the covariance between t and the observation points:

k(t) = [k(t, t 1 ) k(t, t 2 ) . . . k(t, t n )] T .
It can be shown that the determination of λ λ λ(t) boils down to computing the solution of the linear system of equations (see, e.g., [START_REF] Villemonteix | An informational approach to the global optimization of expensive-to-evaluate functions[END_REF]):

             1 K : 1 1 . . . 1 0                           λ λ λ(t) µ(t)              =              k(t) 1              (9)
where µ(t) is the Lagrange multiplier, which corresponds to the enforcement of the unbiasedness condition.

Once the vector λ λ λ(t) has been computed, a predicted value of the similarity function Q(t) can be written as

Q(t) = n k=1 λ k (t)Q k . ( 10
)
It is easy to show that the function t → Q(t) interpolates Q(t) at observed points. An interesting property of kriging, is that an estimate of the uncertainty of the prediction is available via the kriging error, which can be written as

σ2 (t) = k(t, t) -λ λ λ(t) T k(t) -µ(t).
(11) This feature will be essential in the adaptive sampling strategy of the Expected Improvement algorithm.

Covariance model

Before focusing on the optimization algorithm, we mention very briefly how the covariance function is chosen in practice (see also [START_REF] Villemonteix | An informational approach to the global optimization of expensive-to-evaluate functions[END_REF]).

First, a simplifying assumption is made -as usually done in geostatistics (see, e.g., [START_REF] Chiles | Geostatistics, Modeling Spatial Uncertainty[END_REF]) -namely that the random process is stationary. Then, the covariance function is a univariate function k(h) where h is a distance between two points t a , t b ∈ T. This distance needs not to be the classical Euclidean distance. In our case, the components of t ∈ T are of different kinds. Thus, it is reasonable to use some anisotropic distance, which may be written as

h = D d=1 t a,d -t b,d ρ d 2 . ( 12
)
where t a,d and t b,d are the d th components of the vectors t a and t b , respectively, and D the dimension of T. The parameters ρ d , d = 1, 2, . . . , D, represent the range of the covariance, or the typical correlation distance, in the direction of the d th component.

Second, a parameterized covariance function is chosen and its parameters are estimated using the data with a maximum likelihood method (see, e.g, [START_REF] Villemonteix | An informational approach to the global optimization of expensive-to-evaluate functions[END_REF]). We use the Matérn covariance function, which can be written as

k(h) = σ 2 2 ν-1 Γ(ν) 2 √ ν h ν K ν 2 √ ν h , (13)
where K ν is the modified Bessel function of the second kind of order ν. The parameter ν controls the regularity of the random process -the higher the ν, the more regular the process is. The parameter σ 2 is the variance of the process (k(0) = σ 2 ). The parameters ρ d in ( 12) are also estimated by maximum likelihood (the covariance function has indeed D + 2 parameters).

Expected Improvement

Since the similarity function Q (to be minimized) requires to compute the solution of a forward problem, the computational cost of evaluating Q is non-negligible. Moreover, for an inversion method to be useful in practice, the computational burden of the method must not be too high. Thus, we wish to limit the number of evaluations of Q in the inversion procedure, which means that the optimization method used to minimize Q must be efficient.

The expected improvement (EI) algorithm is an iterative method to find the global minimizers of an expensive-to-evaluate function [START_REF] Jones | A taxonomy of global optimization methods based on response surfaces[END_REF]. The method is based on interpolation by kriging of the function to be optimized. Let us assume that

mm

Step in the y dir. 0.5 mm

Q has been evaluated at n points Q 1 = Q(t 1 ), Q 2 = Q(t 2 ), . . . , Q n = Q(t n
). An iteration of the EI algorithm provides the location of the next evaluation.

First, an interpolation Q of Q is computed by kriging from the set of past evaluations, along with the variance of the kriging error σ2 (t). Denote the current minimal value by

Q min = min k=1,...,n Q k . Define the improvement over Q min at a point t ∈ T by I(t) = max(0, Q min -Q(t)
). However, Q(t) is unknown except at the n evaluation points. Since Q(t) is modeled by the Gaussian random process ξ, a natural idea is to express the expected value of I(t), which is called the expected improvement and appears to have a very convenient analytical form:

EI(t) ≡ E[I(t)] = σ(t) uΦ(u) + ϕ(u) , (14)
where Φ(•) is the normal cumulative distribution function, ϕ(•) is the normal density function, and u is defined by

u = Q min -Q(t) σ(t) .
The next evaluation point is chosen according to the highest value of the expected improvement. Since ( 14) is straightforward to compute, the maximization of the EI over T is not a problem in practice. A natural stopping criterion is when the EI is smaller than a small positive number.

Note that the EI algorithm is known to be consistent, i.e. the algorithm converges to the location of the global minimizer (under some assumptions) [Locatelli, 1997, Vazquez and[START_REF] Vazquez | On the convergence of the expected improvement algorithm[END_REF]. In theory, the convergence rate is unknown but it appears in practice that this type of algorithm converges very rapidly, as illustrated in our numerical studies (see next section).

NUMERICAL EXAMPLES

In this section a couple of numerical examples are selected to illustrate the presented inversion method. The parameters of the ECT configuration are shown in Table 2 (see also Fig. 1).

The surface scan consists of 451 measured impedance valuesobserved at the nodes of a rectangular grid characterized by the parameters given in Table 2. The center of the surface scan is at the origin of the x-y coordinate system and the edges of the grid are parallel to the related coordinate axes.

Since no experimental measurements are available at that time, the so-called "measured impedance variation" (∆Z k , k = 1, . . . , N in (4)) of the defects in the assumed test cases are obtained by numerical simulation.

The necessary initialization of our iterative process implies the evaluation of the similarity function at some well-chosen points of the parameters space. There is no "best" way to choose such initial points without complementary a priori informations, Table 3. Defect parameters, in case of Initialization "A". (exact/reconstructed)

No. A mm B mm L 1 mm L 2 mm D 1 % D 2 % It. Q min 10 -3
#1 0.50/0.50 3/3 3/3 5/5 40/40 60/60 57 0 #2 0.75/0.75 1/1 4/4 4/3 30/40 50/50 112 21.6 #3 0.75/0.75 2/3 7/5 3/3 50/50 30/50 95 28.6 #4 1.00/0.75 0/0 6/7 6/3 60/60 40/40 60 8.56

Table 4. Defect parameters, in case of Initialization "B". (exact/reconstructed)

No. A mm B mm L 1 mm L 2 mm D 1 % D 2 % It. Q min 10 -3
#1 0.50/0.50 3/3 3/3 5/5 40/40 60/60 81 0 #2 0.75/0.75 1/1 4/2 4/4 30/40 50/50 118 6.98 #3 0.75/0.75 2/2 7/7 3/3 50/50 30/30 79 0 #4 1.00/0.75 0/0 6/6 6/5 60/60 40/40 109 4.73 therefore it seems reasonable to spread them more or less uniformly on the parameter space. Too few points can give misleading information on the similarity function whereas too many points may be unnecessary (thus "uneconomical" in the sense of computation time). Two types of initialization are presented in the paper:

• Initialization "A": all the vertices of the 6D parameter hypercube (2 6 = 64 points) and the midpoint of the hypercube (1 point): Σ65 points; • Initialization "B": the same points as in Init. "A", and all the midpoints of the sides of the hypercube (2 • 6 = 12 points): Σ77 points.

The parameters of the cracks to be retrieved and the obtained results are reported in Table 3 andTable 4. The depths (D 1 , D 2 ) are given in percents of the plate thickness d. After the initialization (65 or 77 similarity function evaluations) a fixed number of iteration cycles were performed (120 in all the cases). In the column "It." the number of cycle, in which the final best function value (Q min ) was found, is presented; that minimum value (related to the obtained solution) is also reported in the tables.

Table 3 and Table 4 illustrate the efficiency of EI algorithm with which an "acceptable" solution -in the sense of a low similarity function -is obtained in a few number of iterations.

The performance of the algorithm is illustrated in Fig. 3. On the bottom diagrams in Fig. 3(a) and Fig. 3(b), the log 10 of the maximal expected improvement (i.e. over the total discrete parameter space) at the related iteration cycle is presented.

One can see that it is decreasing as the iterations are being performed. However, the decrease is not monotonous. As it was mentioned, in our experiments a fixed number of iterations were performed. However, it would be better to construct a stopping criterion which stops the iterations automatically right after finding the solution -to reduce the number of "unnecessary" function calls at the end. By now, no really good stopping rule has been developed. One can rely on the maximal EI value, stating that if it is lower than a small limit, the iterations can be stopped. But the choice of this limit is not obvious -as it is illustrated in Fig. 3. In Fig. 3(b), one can see that a small Q value (quite close to the "final" solution in iteration No. 109) was obtained in iteration No. 60. The parameter vector related to the observation in iteration No. 60 is t = [1. 0,0,6,4,60,40] (in the same units as in Table 4). This parameter point is close to the real solution and to the reconstructed one (see Table 4) Line scan along the x = 0 line. Solid line: real crack configuration, dashed line: retrieved configuration (case Initialization "B"). The measured data is almost the same in both cases, thus it is extremely difficult to make difference between the configurations.

as well. However, the maximal EI value in iteration No. 60 is almost 100 times higher than the one in iteration No. 109! Note that the two presented performances are related to the same physical problem, only the initialization of the inversion procedure differs. In spite of this, significant differences can be found between the behaviour of the algorithm in the two cases. Thus, one can see that the choice of initial points strongly influences the latter performance. As it was mentioned above, there is no general best choice of initial points, moreover, it may also depend on the problem to be solved.

It is also important to note that the similarity function must be complicated. This is illustrated in Fig. 5: the function Q(t) is probably quite flat around its global minimizer, or physically, different configurations can cause very similar measurable output signal.

The computations were performed on a PC with 16 Gb RAM and a 64 bits CPU at 3 GHz. The computation of the Green's function and of the incident field was made in advance, separately. One average iteration cycle of the optimization loop took approximately 106 sec. Almost all of that time is needed to perform the kriging prediction and EI computation, the computation of the objective function takes just a few seconds. (At first sight, one can think that the applied inversion method is more expensive-to-evaluate than the objective function itself -but note that by using this inversion method, at most some hundred function calls are needed, whereas that the parameter space T consists of more than 60 thousand points!)

CONCLUSION

A method for ECT inversion was presented. The stochasticbased optimization algorithm was coupled with a surface integral-based forward solver -the developed inversion method seems to be efficient in the light of the presented test cases.

The kriging interpolation of the similarity function provides a cheap surrogate model of the electromagnetic "black-box". Since this "black-box" includes electromagnetic field computation tasks, it is usually expensive-to-evaluate. This is why the 3 andTable 4). Top: current best objective function value (the minimum of all the past observations). The found minimum is marked by a circle. Bottom: log 10 of the maximal EI in the current cycle.

use of such surrogate models seems to be fruitful. This kind of surrogate modeling of electromagnetic phenomena has been a popular method for years and now seems to be having its "second honeymoon" [START_REF] Sykulski | New trends in optimization in electromagnetics[END_REF]. Several methods exist for the solution of electromagnetic optimization problems. The main novelty of this paper is the application of such an existing (and in other domains widely used) optimization method for the inverse problem of double-crack characterization in ECT.

A couple of numerical examples illustrate the performance of the developed inversion method. The most important conclusion to be drawn is that quite few number of similarity function evaluations were enough to find an acceptable solution -even if the similarity function is "badly behaved" from the point of view of optimization. It is also important to emphasize that the performance of the algorithm is strongly influenced by the choice of initial observation points -as it was pointed out via our numerical examples.

A pitfall of the EI algorithm was also highlighted. The choice of an adequate stopping criterion is not straightforward and no general solution has been found yet. Further investigations are needed concerning this problem.

In this paper, only the brief introduction to the stochastic tools and the first results were presented -the research is far from being finished. In the authors' opinion, the use of surrogate models in ECT inversion is a hopeful idea.

Fig. 2 .

 2 Fig. 2. Magnitude of the measured impedances in case No. #4.Line scan along the x = 0 line. Solid line: real crack configuration, dashed line: retrieved configuration (case Initialization "B"). The measured data is almost the same in both cases, thus it is extremely difficult to make difference between the configurations.

Fig. 3 .

 3 Fig. 3. Typical performance of the algorithm. The curves are realted to Case No. #4, (see Table3 and Table 4). Top: current best objective function value (the minimum of all the past observations). The found minimum is marked by a circle. Bottom: log 10 of the maximal EI in the current cycle.

Table 1 .

 1 Discretization of the parameter axes.

	Parameter	A	B	L 1	L 2	D 1	D 2
	Minimum value (mm)	0.25	0	2	2	0.1250	0.1250
	Maximum value (mm)	1.25	4	8	8	0.8750	0.8750
	Step size (mm)	0.25	1	1	1	0.0125	0.0125
	No. of points	5	5	7	7	7	7

Table 2 .

 2 Parameters of the ECT configuration.

		Metal plate
	Thickness (d)	1.25 mm Conductivity (σ 0 )	10 6 S/m
		Probe coil
	Inner radius (r 1 )	0.6 mm	Outer radius (r 2 )	1.6 mm
	Height (l)	0.8 mm	Lift-off (h)	0.5 mm
	No. of turns	140	Frequency	150 kHz
		Surface scan
	Points in the x dir.	11	Points in the y dir. 41
	Step in the x dir.


Laurent Maïmoun
email: l-maimoun@chu-montpellier.fr

Sandrine Alonso

Krishna Kunal Mahadea

Vincent Boudousq

Thibault Mura

Denis Mariano-Goulart

Krishna Kunal Mahadea

Denis Mariano-Goulart1

Cross-calibration study of the Stratos and Hologic QDR 4500A dual-energy X-ray

Keywords: Cross-calibration, Dual X-ray absorptiometry (DXA), Stratos, QDR4500A, Areal bone mineral density (aBMD), Body composition, Body mass index

Introduction

Dual-energy X-ray absorptiometry (DXA) is the reference technique for the evaluation of areal bone mineral density (aBMD). It is noninvasive and characterized by low radiation and high precision and reproducibility, making it useful to monitor changes in aBMD over time and to determine treatment efficacy (1). However, a difference in aBMD calibrations between manufacturers has been well documented (234). Typically, it has been reported that a measurement of spine aBMD on a system from Hologic is lower than on the GE Healthcare Lunar system (23). Moreover, aBMD variations have also been observed for different scanners within the same manufacturer's product lines (56). The official guidelines of the International Society for Clinical Densitometry (ISCD) (7) therefore recommend that patient follow-up be performed with the same device. However, in situations where an old DXA is replaced by a newer one, or to meet the needs of a multicenter study, the ISCD recommends a cross-calibration study between devices to ensure consistent results (7).

In addition to aBMD, there is growing interest in evaluating body composition with DXA, which is today the gold standard in clinic for determining WB, regional lean tissue mass (LTM) and fat mass (FM) (8). Indeed, DXA use for patients with various conditions was advocated in the last ISCD recommendations, published in 2013 9 . These included patients living with HIV and taking antiretroviral agents, obese patients undergoing bariatric surgery (1011), patients with other conditions that induce significant weight loss, and patients with weakness or poor physical functioning. For example, when LTM is evaluated with DXA in weak or poorly functioning patients, it contributes to the sarcopenia diagnosis by adding the concomitant evaluation of muscle quantity or mass to the measurements of muscle strength by handgrip strength and physical performance by gait speed (12). Thus, low LTM can be defined by different cut-offs using appendicular lean mass evaluation 12, 13, 14, which requires the capacity of the device to perform segmental analysis - only possible with DXA technology. As for aBMD, the ISCD recommends a cross-calibration study between devices to ensure consistent results for body composition analysis (7).

To date, no data have been published on the cross-calibration between the Stratos (DMS®) and the QDR 4500A (Hologic Inc.®). To our knowledge, Stratos was only cross-calibrated with Discovery A, another device from Hologic®, but only lumbar spine and hip regions were analyzed, with no information on body composition(15).

The purpose of this study was to perform an in vivo calibration of total and regional body scans between two DXA devices: Stratos (DMS-APELEM; pencil-beam technology) and QDR 4500A (Hologic, Inc.; fan-beam technology), with a focus on aBMD and body composition.

Section snippets

Subjects

Fifty-five adults were recruited via an intra-hospital email invitation or from among patients referred to the Department of Nuclear Medicine, Lapeyronie Hospital (Centre Hospitalier Universitaire de Montpellier, CHRU Montpellier, France) for evaluation of aBMD and body composition. All participants were Caucasian and, to be eligible, they had to be over 18 years old, able to lie supine for 10-15 minutes, and have body weight <160 kg (table weight limit restriction of the QDR 4500A system). Participants were excluded from the study if they had a history of lumbar spine or proximal femur fracture, hip replacement or osteosynthesis and cementoplasty at the lumbar spine, or were pregnant or breast feeding. In accordance with the ISCD recommendations (7), these participants were representative of our patients and represented the full spectrum of scans performed in our Department.

Study procedures

Ethics approval for the study was obtained from the Ethics Review Committee of Nîmes, France (Commission de Protection des Personnes, Sud Méditerranée III on 9 July 2015: reference 2015-06.02 bis), and permission for the clinical trials was granted by the French Medicine and Health Care Products Regulatory Agency (ID RCB: 2015-A00596-43, on 21 September 2015). All study participants were volunteers and gave specific signed consent before the scans. All activities performed in this study were in accordance with the Declaration of Helsinki.

Standing height was measured with a stadiumeter to the nearest 0.1cm and recorded as the mean of two consecutive measurements. Body weight was measured with a calibrated scale with a precision of 0.1kg. Body mass index (BMI) was calculated as weight (kg) divided by the square of height (m).

Dual-energy X-ray absorptiometry measurements

As the two systems were located at the same site, each participant was scanned consecutively on the QDR 4500A (Hologic, Inc.) and immediately after on the Stratos (DMS, Mauguio, France). They wore the same light clothing with all metal and plastic artifacts removed. The QDR 4500A was used as the reference device. The lumbar spine (L1-L4) in the posterioranterior projection, the left proximal femur, the radius and the whole body were scanned on both DXA scanners. This was done after following the procedures described in each densitometer's operating manual. For aBMD analysis, the regions of interest (ROIs) were defined for the lumbar spine, femoral neck, total hip, radius and whole body, while for body composition [i.e., LTM and FM], the ROIs were defined for legs, arms and whole body. Moreover, appendicular lean tissue mass (ALM) was defined as the sum of LTM of the arms and legs ,as described by Heymsfield et al.(13).

Specificity of the DXA

-Stratos employs pencil-beam DXA technology. The generator operates on a 90-kV power supply and a 0.1-to 2-mA current depending on the type of scan and the patient's morphology. The X-ray source has a continuous spectrum with two energy peaks (35 and 65 keV) filtered by a k-edge filter (samarium). A pencil-shaped beam is obtained when the raw X-rays pass through by a circular collimator. Photon counting is done with a scintillation crystal (LaCl 3 ) coupled to a photomultiplier. The X-ray tube is placed beneath the patient's table and the detector is lodged in the arm. Scattered rays are stopped by a brasscollimator placed in front of the detector. The X-ray source and the detector move. Simultaneously along the x and y axes and the maximum scan area is 200 x 65cm 2 . The scanned image is a projection of the object with no magnifying effects and independent of the position of the object along the source/detector axis. Software version V5.2.1.2 was used. Other details and dosimetry were previously published (15). The least significant change (LSC) for Stratos was previously reported, with values of 3.39% for lumbar spine (L1-L4), 5.84% for femoral neck, and 2.38% for total hip (15).

-QDR 4500A employs fan-beam DXA technology. Briefly, the X-ray source has a continuous spectrum with two energy peaks (100 and 140 keV).The detector system is composed of a 216-multichannel detector consisting of CdWO4 scintillators coupled to silicon diodes. Software version 12.6 was used. Other details and dosimetry were previously published (16).

Procedure for analysis of femoral regions

Shared procedure: For the two DXAs, the left hip that is scanned is internally rotated using a hip positioner. Hip is internally rotated because it will open up the femoral neck area, putting the femoral shaft in line with the scanner bed and minimizing the lesser trochanter. Specific procedures: For QDR-4500A, the bottom of the global ROI should be 10 lines below the bottom of the lesser trochanter, 5 lines above the top of the head of the femoral neck, 5 lines laterally to the head of the 5 lines above the top of the head of the femoral neck, 5 lines laterally to the head of the femoral neck, and 5 lines laterally to the great trochanter. When the global ROI is defined, the software automatically places the different ROIs (i.e., head of the femur, greater trochanter and bottom of the lesser trochanter) and the femoral midline. By default, the width of the femoral neck box is a rectangular region of 1.5 * 5 cm and is placed tangent to the crest of the greater trochanter. For Stratos, the bottom of the global ROI is automatically positioned 6 cm below the insertion between the femoral neck axis and the lower limit of the femoral neck ROI (yellow line). The red line defines the femoral neck's axis and cuts the femoral neck in half. The femoral neck box in blue is positioned tangent to the crest of the greater trochanter. Last, the green line should be placed from the inflection point of the trochanter edge up to the boundary of the trochanter. Fig. 1 presents the left proximal femur images of both bone densitometers, including their respective ROIs.

Statistical analysis

Characteristics are expressed as mean ± standard deviation (SD) and frequency with percentage. Correlations between the QDR 4500A and Stratos measurements were calculated using Pearson correlation coefficients (with 95% CI). Inter-scanner differences were evaluated by calculating the systematic difference between the QDR 4500A and Stratos measurements (systematic bias) and tested against the 0 value using a paired Student's t-test. Agreement was analyzed using a Bland-Altman plot.

Cross-calibration equations were estimated using linear regression. Systematic bias and agreement between the Stratos and QDR-4500A measurements converted using the crosscalibration equations were assessed using paired Student's t-tests and Bland-Altman plots.

The BMI effect and effect modification on the cross-calibration model were evaluated by adding an effect and an interaction term with a "Severe-Obesity" variable to the linear regression model. This dichotomization was chosen because it best fit the data for the interaction effects under consideration. All the analyses were performed using SAS software version 9.4 (SAS Institute Inc., Cary, NC).

Results

The characteristics of the participants are described in Table 1. Fifty-five individuals (46 women: 84%) with a mean age of 41 ± 13.0 years (range: 20 to 64) and a mean BMI of 31.9 ± 10 kg/m² (range: 12.2 to 49.5) were included in this study. The repartition of participants according to BMI class is presented in Table 2.

Cross-calibration

Results of the comparisons of mean bone mass parameters and body composition parameters measured by the Stratos and QDR 4500A are given in Table 3. Values of aBMD on the two DXAs were highly correlated (p<0.001 for all) with a coefficient of correlation (r) ranging from 0.89 to 0.99 (r=0.89 for whole body, r=0.92 for radius, r=0.95 for femoral neck, r=0.96 for total hip, and r=0.99 for L1-L4 aBMD). For whole body composition, r ranged from 0.96 for ALM to 1.0 for FM (all p<0.001). All these correlations are presented in Table3 and Figs. 2 and3.

However, although the two measures were highly correlated, the paired t-test indicated a statistically significant bias between the Stratos and QDR 4500A for most of the measurements. Compared to QDR 4500A, Stratos under estimated whole body aBMD with a difference of 0.116 g/cm 2 , corresponding to a mean relative change of 10.4%. Specifically, the mean femoral neck and total hip aBMD were, respectively, a mean 0.07 g/cm 2 and 0.04 g/cm 2 lower with QDR 4500A than Stratos, corresponding to mean relative changes of 9% and 4.5. No significant bias was demonstrated for mean aBMD at L1-L4 and radius between the two DXAs .Compared to QDR 4500A ,Stratos over estimated FM (kg and %) and underestimated LTM (p< 0.001 for the three parameters) .The systematic differences for bone parameters and body composition parameters are depicted in Figs. 4a and5a with Bland-Altman plots (Stratos versus QDR 4500A).

As the subjects included in this study presented a large dispersion of BMI (range 12.2 to 49.5 kg/m 2 ), the concordance of measurements between QDR 4500A and Stratos was analyzed according to BMI class (see Supplementary Table1). For most parameters, the bias between the two measurements obtained by the two DXAs was not influenced by BMI. Only limited significant interactions of BMI with whole body aBMD, whole body BMC and whole body FM (kg and %) were found (see Supplementary Table1 and Supplementary Fig. 1).

Due to the observed differences between the two densitometers, linear cross-calibration equations were derived for bone mass parameters and body composition parameters. These equations are presented in Table4. The agreement between the Stratos measurements and the QDR 4500A measurements after conversion using the cross-calibration equations is depicted in Figs. 4b and 5b with Bland-Altman plots. Specific cross-calibration equations for parameters influenced by BMI are presented for the two classes of BMI (>/=35kg/m 2 or >35kg/m 2 ) in Table 5.

Discussion

For a group of participants presenting a large range of BMI values, in vivo cross-calibration demonstrated good linear agreement between the Stratos and the QDR 4500A for both bone and body composition parameters. However, the regression slope in most of the ROIs was significantly different from unity, demonstrating the need to cross-calibrate these two DXAs from different manufacturers. aBMD Although all the aBMD measurements were significantly correlated between the two densitometers, the aBMD values with the Stratos were significantly lower for whole body and higher for femoral neck and total hip compared to QDR 4500A. For the latter two skeletal sites, mean 0.07 g/cm² and 0.04 g/cm² differences were, respectively, observed, which amounted to percentage variations of 9% and 4.5%. In cases of differences exceeding 1%, the ISCD recommends (7) the development of cross-calibration equations to ensure comparability between the two densitometers; for example, to ensure correct patient management when a currently commercialized Stratos replaces an older QDR 4500A, or to ensure the consistency of measurements when a multicenter study is being conducted. Various technological reasons have been proposed to explain the aBMD offsets ,including differences in the manufacturers'calibration processes, computing algorithms, and detection technologies (17)(18)(19). In the current study, the type of beam (i.e., fanorpencil) may also be implicated. However, in a recent meta-analysis based on 14 studies, Yoon et al. (20) concluded that here was no significant difference in aBMD at the hip region between the two modes of DXA scanners, although pencil-beam underestimated aBMD in whole body compared to fan-beam, as was found in our study. It is never the less interesting to note that only studies comparing DXAs from the same manufacturer (i.e., Prodigy vs iDXA from Lunar or QDR 2000W vs QDR 4500A from Hologic) had been compiled (20). This methodology may have introduced a selective bias because an intra-device cross-calibration may have been performed by the manufacturers to limit aBMD differences within their range of densitometers. Other studies comparing pencil-beam (DPX-NT) and fan-beam (QDR 4500C) DXAs from two different manufacturers reported that, although the two DXAs were highly correlated, pencil-beam overestimated aBMD at the femur (0.082 g/cm 2 ) (21). Our results at the femoral region confirmed these findings.

Body composition

DXA analysis of total and regional body composition has progressively become important in the diagnosis and follow-up of patients with a range of pathologies or clinical condition s(9), including patients with HIV, obese patients undergoing bariatric surgery (1011), and patients with muscle weakness or poor physical function (1213) , (30). However, as DXA findings for body composition differ across the scanning systems produced by different manufacturers (3), the development of cross-calibration equations based on in vivo analysis, instead of in vitro phantoms that may be misleading (31), is required and recommended by the ISCD (9).

As expected, although both DXAs showed excellent reliability for whole body LTM and FM (r=0.98 to 1), there were significant differences between the Stratos and QDR 4500A for body composition evaluation. The whole body LTM determined by Stratos was lower (bias of 4.2 kg corresponding to a mean relative change of 7.8%)

Conclusion

To summarize, the study showed good linear agreement between a currently commercialized pencil-beam Stratos and an older fan-beam QDR 4500A. Using the equations generated from a population presenting a large range of BMI, an accurate transformation of the Stratos values was obtained, removing all significant differences between the two DXAs. These results can now be used to assess aBMD or LTM and FM in multicenter studies or when a change from QDR 4500A to Stratos is necessary.

Acknowledgments

The authors would like to thank all the participants.