The Impact and the Prevention of Burnout on Psychic activity and Nervous and Immune Systems

Keywords: Burnout, conflict, psychic activity, nervous and immune systems, prevention

The aim of my study is to analyze the causes and consequences of burnout on the dysfunction of psychic activity and weakening of the nervous and immune systems in their interaction. By presenting different research studies and my own analytical and practical results, this article traces the various orientations and identifies the symptoms, ranging from the quasi-normal state to the serious pathology in individuals suffering from burnout. The burnout methods and the psychotherapeutic techniques presented, which can generate the burnout cases, present a common backbone to the different approaches exposed in order to try to explain what this concept covers and the way it manifests itself.

Introduction

Currently the problems of mental, emotional and physiological exhaustion, the issues of depersonalization and the reduction of personal accomplishment among individuals overinvested professionally are of great interest to researchers in human sciences and medicine. The burn-in is the first phase of the exhaustion syndrome with presenteeism which refers to an abusive presence in the workplace and the burnout (burning to the end) is the ultimate phase of global exhaustion syndrome in which the coping mechanisms are consumed under the influence of tensions and professional overinvestment. But the persistent under-occupation, marked by routine and boredom at work, can also lead narcissistic, ambitious and very well qualified individuals to a burnout or boreout (exhaustion by ennui). The aim of my study is to explain above that burnout is a result of complex multi-factorial, additional and interactive interactions between an individual, his or her professional environment and society in general that interact and influence each other continually. Freudenberger has observed in a recurring way the links between psychic activity and the weakening of the nervous and immune systems of the overinvested individuals in their work. [START_REF] Freudenberger | Staff burn-out[END_REF] thinks that the process of burnout consists of four phases: 1) idealistic enthusiasm, 2) stagnation, 3) frustration and depression, 4) apathy and despair. Freudenberger not only described the symptoms of burnout but also listed personality factors that predispose people to suffer from burnout. It is primarily "the dedicated and the committed" who are most likely to burnout [START_REF] Freudenberger | Staff burn-out[END_REF]. Contrasting Freudenberger, which stresses the personal factors conducive to the appearance of a burnout, Maslach attributes them mainly to the professional environment and working conditions. According to the Maslach and Jackson three-dimensional model, of ''burnout syndrome includes: 1) professional exhaustion, 2) depersonalization, 3) diminution of personal achievement and the devaluation of the work itself''. [START_REF] Maslach | Burned-out/ Human Behavior[END_REF]. Based on the three dimensions of burnout, she developed the Maslach Burnout Inventory (MBI. [START_REF] Maslach | The measurement of experienced burnout[END_REF] Other questionnaires exist, such as the Copenhagen Burnout Inventory (CBI) and the Oldenburg Burnout Inventory to measure work-life and health data collected from clinical signs observed by the occupational health service team.

The essential methods of preventing burnout bring include the treatment of various somatic disorders, restoring the interactions of nervous and immune systems with effective relaxation exercises. In case of burnout syndrome, medicine intervention (use of antidepressants, tranquilizers treatments for musculoskeletal disorders, non-steroidal anti-inflammatory drugs, etc.) should be done in agreement with the doctor dealing. The controls of the working conditions by the doctors at work and psychologists must be systematic and effective. Human resources services, occupational health services in each society and organization have a preventive role to play in terms of screening for burnout, based on a set of both individual and collective signals (precisely, evaluating interpersonal conflicts, indicators related to the health and safety of workers, etc.). In the burnout prevention regimen, the psychotherapies (the Rogers helper relationship, Kernberg's empathy, Freudian psychoanalysis, Moreno psychodrama, Gestalt Therapy, etc.) are aimed at helping the person. As Freud points out in the psychoanalytic practice " For the work of analysis proceeds best if the patient's pathogenic experiences belong to the past, so that his ego can stand at a distance from them" [START_REF] Freud | Analysis Terminable and Interminable /In J Strachey. The Standard Edition of the Complete Psychological Works of Sigmund Freud[END_REF].

Materials and Methods

My analysis of the scientific burnout literature is based on a multiple approach. As I am interested in how burnout is investigated in the medical and psychologic field, I first identified all medical literature search for studies that investigate the burnout syndrome, using the PubMed database and others. This set of bibliographic databases references articles in the health sciences, particularly medicine, psychology, and neuroscience, and is a key resource for researches in these field. Specifically, I analyzed about 449 articles on burnout. According to this results and from the my point of view the burn-out is above all an ''interdependent consequence between the conflicts that exist in different levels '' [START_REF] Radtchenko-Draillard | L'étude psychanalytique de la résolution des conflits dans les interactions internationales et la négociation[END_REF]: 1) at the intrapersonal level (between antagonistic offensive and defensive drives, between the needs of his professional tasks and the means available to achieve them, etc.); 2) at the intersubjective level (between individuals who reveal differences in the satisfaction of each person's desires or the need to be recognized by his colleagues, etc.); 3) at the intragroupal level (between hierarchy and subordinates which manifest disagreements in decision-making, etc.); 4) at the intergroup level (between social groups in the situation where they show aggression in competition, precariousness, etc.); 5) at the international level (between opposing goals and the means of their realization of the different countries, aggravated by economic crises, etc.). In order to summarize the impacts of burnout on health in general and the nervous and immune systems in particular found in the various researches, I have synthesized in the table the main dysfunctions, disorders and diseases occurred with the development of burn-out. (See Annex 1, Table ).

Concerned to regain a positive, optimistic and constructive attitude towards oneself and life.

Emotional therapeutic stimulation and behavioral therapies also contribute to increasing the fixation of positive memories (satisfaction, happiness, pleasure) in individuals. Recall that the reaction to positive stress (eustress) or pleasure and evolution through its phases is often experienced as enjoyable and stimulating for the performance of individuals. In order to evaluate the effects of emotional therapeutic stimulation and analytical psychotherapy through the enhancement of pleasure I carried out a qualitative study (interviews) on the 119 managers (of the French companies and Foreigners located in Paris and its suburbs). According to my analysis, 75% of male managers and 69% women managers consider that pleasure is indispensable in the prevention of burnout. In addition, for 69.8% of managers the pleasure in the work increases the motivation and makes it easier to find problem solving, 81.3% of managers compare the pleasure with the satisfaction of being responsible for those of his team,

Conclusion

The analysis of burnout research has shown how a mental state has been socially and scientifically constructed and reproduced in the last years. Over time, the health sciences produced enough evidence to make burnout an important object of scientific enquiry and a challenge for public health. In my article I have shown that most studies focus on the causes and associated factors of burnout, in interactive link with the nervous and immune systems, and its prevention programs, but they do not adequately reflect without reflecting the psychological foundations of burnout or do not fully analyze the usefulness of this concept in a clinical context and in society at large. However, it is essential to also examine in this article all aspects of the onset of this syndrome, given the fact that many burnout studies find extremely high prevalence rates and that burnout is widely discussed in many modern and industrialized countries.

74. 3 %

 3 of managers the pleasure and the positive emotions necessary for the balance of the society. My study restores also the essential sources of pleasure in this professional work. (See Annex 2, Figure)

Annex 1 :

 1

5 .

 5 Loss of willingness to have contacts with colleagues, family and friends, resignation, psychic vacuum, depersonalization, etc. 5. Eating disorders, gastrointestinal disorders, gastric ulcer, binge-eating disorder, anorexia, bulimia, merycisme, sudden weight change, etc.

Table of essential dysfunctions, disorders and diseases observed during the burn-out Psychological and Psychoanalytic dysfunctions : Psychosomatic disorders and diseases:

	1. Mental and cognitive exhaustion, memory	1. Physical exhaustion, chronic fatigue
	impairment, speech impairment, perceptive	syndrome, lack of energy, bodily pain, vestiges,
	and sensitive difficulties, decreased attention	muscle tensions, headaches, sexual problems, ear
	and concentration, communication problems.	ringing, decreased hearing, sleep disturbances,
		sleep apnea, vomiting, etc.
	2. Negative Attitude towards oneself, inferior	2. Weakening of the immune system, increased
	feeling, repetitive masochism, worthlessness,	susceptibility to infections, colds and flu
	self-repentance, loss of self-esteem, feelings	infections, allergies, immune tolerance,
	of guilt, permanent doubts, demotivation, etc.	inflammatory reactions, general weakness of the
		immune system.
	3. Negative Attitude towards life, melancholy,	3. Heart palpitations, migraines, risk of
	post-traumatic stress, disgust, daily	developing high blood pressure,
	frustration, anxiety, suicidal thoughts, etc.	cardiomyopathies, cardio-renal disorders,
		tachycardia, myocardial infarction, stroke, etc.
	4. Negative Attitude towards others (scorn,	4. Neurasthenia , neurocognitive disorders,
	jealousy, conflicts, harassment, cynicism,	different neuroses, neuralgia, neurodermatitis,
	sadistic behaviors, humiliations, sarcasm, etc.	mental disorders, depression, addictions (drugs,
		alcohol, tobacco, medical self-treatment), etc.


Birgit Pepin

Sean Delaney
email: sean.delaney@mie.ie

Andreas Eckert
email: andreas.eckert@oru.se

Nataly Essonnier
email: nataly.essonnier@gmail.com

Andreas J Stylianides

Introduction to the papers of TWG22: Curriculum resources and task design in mathematics education

Introduction

The Thematic Working Group 22 (Curriculum Resources and Task Design in Mathematics Education, hereafter TWG22) was launched at CERME10 in recognition of the growing area of research in the field of curriculum resources [START_REF] Pepin | Curricular resources and textbooks[END_REF][START_REF] Stylianides | Curricular resources and classroom use: The case of mathematics[END_REF], their design and use by teachers and students, and the specific design of mathematical tasks, mostly by teachers acting as designers [START_REF] Laurillard | Teaching as a design science: Building pedagogical patterns for learning and technology[END_REF][START_REF] Jones | Research on mathematics teachers as partners in task design[END_REF] but also their place in textbooks (e.g., [START_REF] Stylianides | Special issue on "Reasoning-and-proving in mathematics textbooks: From the elementary to the university level[END_REF]. While this field shares approaches, methods and research topics found in other areas in mathematics education research (e.g. TWG15,16), it has its own distinctiveness. It focusses on the design and use of curriculum materials (digital and non-digital) for various educational purposes, including the design of classroom-based interventions to promote important but hard-to-teach and hard-to-learn learning goals (e.g., [START_REF] Komatsu | Fostering empirical examination after proof construction in secondary school geometry[END_REF][START_REF] Saxe | Towards curricular coherence in integers and fractions: A study of the efficacy of a lesson sequence that uses the number line as the principal representational context[END_REF][START_REF] Stylianides | Facilitating the transition from empirical arguments to proof[END_REF]. Moreover, leaning on work by [START_REF] Pepin | Digital curriculum resources in mathematics education: foundations for change[END_REF], we make here a clear distinction between (digital) curriculum resources and educational technology, as research on (digital) curriculum resources (in the area of mathematics) pays particular attention to  the aims and content of teaching and learning mathematics;  the teacher's role in the instructional design and task design process (i.e., how teachers select, revise, and appropriate curriculum materials; how teachers design tasks);  students' interactions with (digital) curriculum resources in terms of how they navigate learning experiences within a digital or non-digital environment;  the impact of (digital) curriculum resources in terms of how the scope and sequence of mathematical topics (and tasks) are navigated by teachers and students;  the educative potential of (digital) curriculum resources in terms of how teachers develop capacity to design pedagogic activities and tasks. (amended from Pepin, et al. 2017, p. 647) We acknowledge that "curriculum resources" is an elastic term: ranging from particular tasks and activities, over one-off worksheets or tests, to a full-blown curriculum program, and all of these notions were addressed in the different sessions at CERME10 and CERME11.

The role and importance assigned to curriculum resources and task design in the last decades has led to an enormous variety of approaches to research in this area. At CERME10, 17 papers and 6 posters were accepted, and at CERME11 the number of papers submitted to TWG22 has increased: 22 papers and 8 posters were accepted. In terms of organization, the paper contributions were presented (by the lead author) and discussed (by an 'independent' TWG colleague). The presentation/discussion sessions (see below) were followed by parallel table discussions (colleagues could choose which table to join), and finally a plenary, where amanuenses of each table/paper reported back on the main issues discussed.

The sessions were organized under the following six themes (distributed over six sessions):

1. International textbook studies 2. Teacher thinking and competences 3. Task design 4. Digital curriculum resources and task design 5. Particular task design and theories 6. School and workplace mathematics

Although there was considerable overlap between some of the sessions, for clarity we briefly discuss each theme separately below. During the first part of session 7 the eight posters were presented and discussed in a "poster walk". The second part of session 7 was devoted to a discussion of the issues raised during the six thematic sessions.

Within CERME, the 22 papers and 8 posters presented in TWG22 at CERME11 drew upon research from 25 countries across 4 continents, and offered a wide spectrum of perspectives.

Outside CERME, we saw an increase in handbook chapters, special issues, and whole books focusing on the design and use of curriculum resources (e.g. [START_REF] Pepin | E-textbooks in/for teaching and learning mathematics: A disruptive and potentially transformative educational technology[END_REF][START_REF] Pepin | Digital curriculum resources in mathematics education: foundations for change[END_REF][START_REF] Stylianides | Special issue on "Reasoning-and-proving in mathematics textbooks: From the elementary to the university level[END_REF][START_REF] Stylianides | Curricular resources and classroom use: The case of mathematics[END_REF]. At the same time, the work of TWG22 overlapped with research in the field of, for example, digital technology in mathematics education, or teacher education. New conferences have been initiated (e.g. ICMT; MEDA) with colleagues from TWG22 involved in their set-up and as IPC members.

Themes and paper contributions

In session 1 international and comparative research was presented, predominantly related to textbook studies. In two paper presentations (Memis; Ruwisch & Huang) a particular topic area (e.g. reasoning; length measurement) was compared in two countries' textbooks (Japanese & Turkish textbooks; German & Taiwanese textbooks). Results showed that, for example, in Taiwanese textbooks the emphasis was on concrete actions, whereas in the German textbooks it was on abstract and mental procedures; and the Japanese textbooks provided more "creative" mathematical reasoning opportunities (in the area of proportions) than the Turkish books. A third textbook study of Malawian primary textbooks (Lisnet) examined grade 1 textbooks with respect to mathematical opportunities (with respect to the number concept), whilst teachers' pedagogical design capacity with respect to using textbooks was investigated in another (Chowdhuri). A comparison of naming systems (of curriculum resources) by Chinese and Mexican teachers was conducted in Wang et al.'s study, which highlighted the problem of conceptual equivalence in international studies (see permeating themes).

In session 2 issues related to teachers' use of curriculum resources, their lesson planning and related instruction were discussed. Whilst in the paper by Siedel examined teachers' knowledge of their resource options in order to support and further their design of effective instruction, another group of researchers (Allsopp, et al.) explored teachers' use of a curriculum framework (involving mathematical competencies) for teaching activities. The team of Delaney and Gurhy investigated task characteristics and teacher practice that supported differentiated instruction and work on challenging tasks.

In session 3 core issues of task design was addressed. These related to the design of tasks with particular characteristics (e.g. self-explanation prompts In session 5 task designs related to particular theories were examined. These theories included the Hypothetical Learning Theory for the learning of rules for manipulating integers (Schumacher & Rezat) and for the learning of Calculus (Breen, et al.), and the Theory of Conceptual Fields for the analysis of textbooks on the concept of function (Sureda & Rossi).

In session 6 curriculum resource design related to school and workplace mathematics was explored.

Studies included those that investigated the 'building of bridges' between school and workplace mathematics (Herheim & Kacerja), and the integration of inquiry-based learning and workplace mathematics into mathematics teaching (Kalogeria & Psycharis).

In session 7 the eight posters were presented. The issues addressed in the posters ranged from those related to the design of the mathematics curriculum (e.g.

Issues raised across the six sessions

The sessions were organized under six themes, and naturally issues related to those themes were discussed in those sessions. However, there were also selected permeating strands/themes that ran across the themed sessions. The following themes permeated all six sessions: (1) terminology; (2) teacher learning and student learning;

(3) networking of theories; (4) particular theories: hypothetical and actual learning trajectories; (5) methodologies; (6) theory and practice; and (7) context.

Terminology

Questions such as the following were raised:

1. What constitutes a task? -An activity in a textbook? A lesson? A learning trajectory? 2. Where does task design finish? Should implementation be included in the design? 3. When is a task educative for students? 4. When is a task educative for teachers? E.g. in a teacher guide?

The issue of 'terminology' (and 'naming') was raised in the first session, initially because several comparative studies were presented (e.g. Memis; Ruwisch & Huang; Wang et al.). In the comparative studies the researcher/s have tried to establish 'equivalence of concepts', so that they could establish validity of concepts for the comparisons. This was linked to the translation from one language to another, and the meaning making of particular notions, so that the researchers could actually compare 'like with like'. It was also questioned what it means to design a 'task' (see also session 3 and 4): first, is a task similar, or equivalent, to an activity in a textbook, or do we mean a lesson, or even a whole lesson series, when we speak of a task? Second, what does 'task design' mean? Is it linked to task/lesson preparation, or is the enactment part of task design? Third, when is a task educative for students, and when for teachers? The issue of "educative materials" [START_REF] Davis | Designing educative curriculum materials to promote teacher learning[END_REF][START_REF] Pepin | Enhancing teacher learning with curriculum resources[END_REF] was raised, and educative for whom. Clearly, educative curriculum materials for students (e.g. in textbooks) are likely to be different from those for teachers (e.g. in teacher guides that accompany textbooks).

Teacher learning and student learning

Issues and questions such as the following were raised:

• It is often (naively) assumed that students "automatically" learn when using a particularly designed task -How do we investigate that? • How does teacher learning take place when working with tasks and materials?

-If tasks are developed in a community, how does this affect teacher learning? • What kind of teacher learning is required for choosing, designing, interpreting, and implementing tasks? -What is the possible role of educative curriculum materials?

The issue of 'teacher and student learning' links to the above-mentioned questions raised concerning educative curriculum materials. It was discussed in which ways teachers learn when working with curriculum materials (e.g. textbooks; digital curriculum materials) and mathematics tasks. How does one know that teachers learn when interacting with curriculum materials, and as importantly, what do they learn? And if they work in a collective, do they learn more, or less likely, or otherwise? Indeed, it seems to be necessary to establish under which conditions teachers learn best, and with which kinds of curriculum materials. Effective educative materials (for teachers) must have particular characteristics/heuristics.

Networking of theories

Issues and questions such as the following were raised:

• Abundance of theories -Activity Theory -Variation Theory

Theory and practice

Issues and questions such as the following were raised:

• Role of users -Teachers as designers • Researchers and teachers working together -Collaboration/cooperation? -Role of researcher/s and teacher/s • How can research findings inform practice and how can practice inform research?

Several studies presented in the TWG22 used a design-based research approach. This methodology was seen to bridge theory and practice, as theoretically underpinned designs/interventions are evaluated in practice, and subsequently re-designed and evaluated in iterative cycles to reach the desired results. One of the questions on design-based research links to the role/s of the designer and users: for example, when do teachers act as designers; when are they the 'implementers' of commercial designs (e.g. from textbooks), who are expected to appropriate the design/s, or even enact the design/s 'with fidelity'? What does 'teacher design' mean, and does design include creativity and novel design? Or is teacher design when teachers make small amendments to mathematical tasks during their lessons? Perhaps more importantly, it was questioned in which ways research findings can help teachers in their daily practice of 'task design', and how teachers' practice informs and stimulates research that addresses the issues relevant for teachers.

Context

Issues and questions such as the following were raised:

• Researchers are often not explicit about particular contexts, and societal influences are sometimes difficult to determine, so are particular teaching cultures (e.g. textbooksor lack of themplay different roles in different places).

-What might we be able to infer from a textbook analysis in terms of student learning opportunities in the classroom? -Affordances and limitations of textbook analysis • Exploratory studyare the systems stable? What has changed and what stays the same, when -Curricula change? -Resource systems vary? • Technology, and the changes it brings, can render some tasks redundant or unworkable One of the permeating strands throughout the six sessions related to the context of the design and enactment of curriculum resources. Mathematics teachers work under very different conditions with different kinds of curriculum resources, inside and outside the classroom. In some contexts, textbooks are crucial/mandatory resources, in others teachers prefer to 'pick and choose', or design their own materials. In some contexts, teachers can dedicate half of their time to the design of curriculum resources and learning trajectories, in others they are expected to 'teach by the book'. It became clear that what we measure as researchers might be valid for one particular school or region, or true for a particular time, but that parameters change with time and context, and hence research only provides glimpses of particular phenomena.

Possible directions for future research

Whilst there was a clear overlap of research themes (and directions) between CERME10 and CERME11 in TWG22 (e.g. task design, plurality of theoretical frames), at CERME11 issues related to international studies and international comparisons emerged as important points for discussion. This also related to theoretical frames, or those that were developed in one context and used in another. Moreover, teacher professional development practices and their work with curriculum resources were compared from one context to another, and this led the way to question the scalability of particular curricular practices and design activities.

The following directions for future research emerged from the discussions in TWG22:

• Understanding better the appropriation of tasks by practitioners (teachers, lecturers) and the impact on students (and teachers) would be a valuable focus. Teacher design often includes appropriation. • There is additional potential to look at professional development around particular tasks.

• Students and their learning experiences should be included in considerations for task design.

They are typically included in design-based research, where selected aspects of student response (to tasks) are often included. • There is abundant potential to research the notion of 'task design', in particular when researchers or task designers let go of designs and hand them over to practitioners/teachers (e.g. in terms of the fidelity to the intended design of the task). Confrey and her team [START_REF] Sztajn | Learning trajectory-based instruction: toward a theory of teaching[END_REF] have developed learning trajectories for the common core standards, which have fidelity to the common core curriculum. • We discussed fidelity of resources that are passed on to teachers, but we have not sufficiently discussed fidelity of design to research on tasks or to the theory used and this may be addressed in future work. • Scalability of processes was not a strong focus of the working group. The mathematics education community is still more concerned about small groups of teachers implementing tasks. However, for example, Chinese mathematics education research has moved towards scalability of tasks/materials.

  -Dyrvold & Bergvall); to task design on particular mathematical topic areas (e.g. slope of a curve -Bos, et al.) or with focus on particular challenges (e.g. conceptual and/or creative challenges -Jaeder); to task design fostering students' engagement (e.g. construction of examples on a particular topic -Cusi & Olsher), and to the design of a task progression framework (e.g. Courtney & Glasnovic Gracin).In session 4 core issues of task design continued to be addressed, also in relation to digital curriculum resources. Whilst the study by Essonnier et al. addressed collaborative design (in an EU project) of a digital resource and the linked social creativity, the study by Eckert and Nilsson investigated the design of digitally enriched classroom talk. In her research Rafalska focused on the teaching and learning of 'algorithmics' in terms of resource design, whilst Geti and Ding examined the use of variation theory for problem-based task design.

-Theory of Conceptual Fields -Documental Approach to Didactics • What are their boundaries/ domains of application/ similarities & differences/ limitations? • Can they be combined/ networked?

-Is any combination viable? • How can they be linked to design principles?

In session 5 particular theories were emphasized as bases for the analysis and design of tasks (e.g. variation theory). Moreover, it was questioned whether particular theories could be combined or networked (e.g. Bikner-Ahsbahs & Prediger 2014), and under which circumstances this might be useful. For example, the theory of Hypothetical Learning Theory (HLT, Simon 1995) was seen as a theory supportive of teachers to design a coherent learning trajectory for their students. However, this particular theory might not suffice for defining particular design principles of a learning trajectory; for design principles particular curriculum design tools, or task design principles, would be also be needed (e.g. variation theory). Furthermore, issues related to the use of a theory outside of the context where it was 'born' were discussed, and whether a theory might be differently interpreted in different contexts.

Particular theories: Hypothetical and actual learning trajectories

Issues and questions such as the following were raised:

• What do they mean?

-Issue of singular (trajectory) vs. plural (trajectories) -Common/joint/collaborative learning path vs. individual learning path • What does the comparison of the two tell us?

-Implications for task design(ers) • How do they link (or how might they be linked) to design principles? Linked to the above, it was questioned whether the envisaged hypothetical learning theories would actually develop into actual student learning/study paths (e.g. Pepin & Kockthis conference;[START_REF] Weber | Expanding notions of "learning trajectories" in mathematics education[END_REF], and in which ways the comparison of the two would provide indications of design criteria/principles (for task/curriculum designers) or the efficacy of an intervention [START_REF] Stylianides | Facilitating the transition from empirical arguments to proof[END_REF].

Methodologies

Issues and questions such as the following were raised:

• Design (based) research as a methodology tailored to bridging theory and practice.

• How do theories inform methodologies for studying task design, implementation…?

• Are some methodologies better suited to study task design, implementation than others?

• How do we measure effectiveness of the task design, implementation, stability, for example? Is comparison of HLTs and ALTs a viable way to do this?

In terms of empirical research, questions were discussed linked to the 'measurement' of the design and implementation of tasks, the design and evaluation of processes expected to help students (and teachers) to develop deeper understandings of the mathematics and its learning (and teaching).


Rogier Bos

Michiel Doorman

Kristijan Cafuta

Selena Praprotnik

Sanja Antoliš

Matija Bašić

Supporting the reinvention of the slope of a curve in a point: A smooth slope to slide is a smooth slide to slope

Keywords: Mathematics education, reinvention, slope

This paper discusses a task, designing a slide, to support the reinvention of the notion of slope of a curve in a point. The field test show that the lesson scenario allows teachers to utilize students' informal models to introduce slope more formally.

Introduction

Introducing slope of a curve in a point and the derivative to students is a didactical challenge for teachers. It is tempting to choose for an instrumental approach swiftly progressing to differentiation techniques, both easier to teach and to learn. Without conceptual understanding this entails merely meaningless manipulation of symbols and execution of recipes. The meaning of limit, difference quotient and other concepts related to the slope of a curve at a point form serious obstacles for students (e.g. [START_REF] Tall | How humans learn to think mathematically: Exploring the three worlds of mathematics[END_REF][START_REF] Zandieh | A theoretical framework for analyzing student understanding of the concept of derivative[END_REF].

This paper discusses an attempt to deal with these obstacles inspired by design heuristics from the theory of Realistic Mathematics Education (RME) suggesting that a way for students to learn mathematics in a meaningful way is to involve them in a process of reinvention [START_REF] Freudenthal | Revisiting mathematics education: China lectures[END_REF]. A common approach in RME is to introduce a new concept through a task of which the context provides opportunities for students to focus on reasoning and representations related to the notion aimed at. As [START_REF] Gravemeijer | Context problems in realistic mathematics education: A calculus course as an example[END_REF]

state it

The students should first experience a qualitative, global, introduction of a mathematical concept. This qualitative introduction then should create the need for a more formal description of the concepts involved. (p. 113) This study reports initial experiences of an attempt to support students in (re)inventing the notion of slope of a curve in a point and related representations (e.g. [START_REF] Zandieh | A theoretical framework for analyzing student understanding of the concept of derivative[END_REF]. Students, collaborating in groups of three, are asked to design a slide consisting of a bended and a straight part joining without bumps. The desired outcomes are concrete equations describing a line and a curve that meet smoothly (imagine the purpose is to feed these equations into a 3D-printer to print the slide). The task aims for students to discuss what it means for the line and curve to meet in a not-bumpy, i.e. smooth, way. They should search for methods to design a slide and to decide to what extent smoothness is achieved. The hypothesis is that they will come up with ideas that are essential to the notion of slope of a curve in a point. Informal building blocks, meaningful to the student, that can be used by the teacher to afterwards introduce the slope of a curve in a point more formally.

The context of the task is a non-kinematic one. Previous research (e.g. [START_REF] Doorman | Modelling motion: From trace graphs to instantaneous change[END_REF] exploited kinematic contexts. Historically this approach makes sense, given the origin of the notion of derivative in mechanics. The presence of dimensions (like meters/second) helps to understand the meaning of slope and more generally to make a connection from derivative in mathematics to the physics curriculum. The slide-context offers opportunities that are not present in the kinematic approach. Students have a tactile and visual understanding of what it means for a surface (or better, a curve) to be smooth, as also pointed out by [START_REF] Tall | How humans learn to think mathematically: Exploring the three worlds of mathematics[END_REF]. He suggests students slide their hands along a curve to sense the changing slope. The slide task allows students to mathematize this tactile and visual experience. A main merit of the slide task is that it is formulated in a very open way: it allows for various approaches by students. We discern three options: the standard textbook approach through secants of the curve, the locally linear/linear approximation/"zooming in" approach [START_REF] Tall | How humans learn to think mathematically: Exploring the three worlds of mathematics[END_REF], and a (more algebraic) approach based on local bounding lines. The most important advantage of the slide task is that it allows the teacher connect the student's work to any or all of these approaches, depending on what students produce, possibly providing multiple views on the notion of slope.

The design of the lesson plan (scenario) for the task is based on the Theory of Didactical Situations (TDS) [START_REF] Brousseau | Theory of didactical situations in mathematics[END_REF]. We investigated whether combining RME and TDS as frameworks for the design of an inquiry based mathematics lesson can lead to a successful scenario. This approach of combining ideas from RME and TDS to task design for inquiry based mathematics teaching (IBMT) is pioneered in the Erasmus+ project Meria (Winsløw, 2017).

The research question addressed in this paper is: How do RME-inspired task characteristics and a TDS-inspired teaching scenario support students' invention of the notion of slope of a curve in a point?

RME perspective on the slide task

The design of the task originates from the RME-principle of a didactical phenomenology [START_REF] Freudenthal | Didactical phenomenology of mathematical structures[END_REF] of the slide context: What a didactical phenomenology can do is … starting from those phenomena that beg to be organized, and from that starting point teaching the learner to manipulate these means of organising. (Didactical phenomenology of mathematical structures)

The challenge of realizing a smooth connection in the slide context is a phenomenon that begs to be organized by the tangent line. Visually at first, but then it begs for symbolic means to decide whether the candidate line and curve really join in a smooth way. Some questions that arise are: how good is the slide at begging? Does it lead students to mathematize in direction of the learning goal, or even reinvent the slope of a curve at a point? What can be expected from students can be explained using the emergent models design heuristic [START_REF] Doorman | Emergent modeling: Discrete graphs to support the understanding of change and velocity[END_REF]. The context-problem is expected to invite students to develop their own situation-specific solution methods. These methods will have to be discussed and compared. A (situational) model expected to emerge from the slide task is that of the slope of a curve at a point as the slope of a tangent line. This is a model of, produced after one episode of mathematizing. At best, this is a starting point for further mathematizing towards symbolic and computational aspects of the slope at a point and the derivative.

The slide task as a TDS-scenario

How to organize the classroom for a reinvention activity? [START_REF] Freudenthal | Revisiting mathematics education: China lectures[END_REF] claims that "guiding means striking a delicate balance between the force of teaching and the freedom of learning" (p. 55). TDS might provide a suitable framework to balance the two. Central to the theory is the difference between didactical and adidactical situations [START_REF] Brousseau | Theory of didactical situations in mathematics[END_REF]. In a didactical situation the teacher acts intentionally to share his knowledge. In an adidactical situation the teacher purposefully withdraws, leaving space for students to develop their own activities. Students need this space to invoke their own meaningful strategies to address the slide problem.

For Brousseau it served another purpose: to change the expectations the students and the teacher have of each other. He calls it finding a new didactical contract. During this adidactical situation the teacher cannot be expected to be involved in the mathematizing process. Instead (groups of) students interact with the milieu. For the slide task the milieu consists of the problem itself together with the artifacts that may be needed to tackle it, for example GeoGebra.

The slide task lesson plan is set up according to phases from TDS and starts with the teacher explaining the problem and introducing artifacts that can be used to work on it. Then the teacher symbolically hands over the milieu to the students and withdraws. What follows is an adidactical action phase of 20 to 30 minutes. In this phase students work on the problem in groups of three. They may apply any approach they think is useful. The teacher, even though not interacting with the students, is not inactive: she registers solution strategies from the students and identifies examples that might be used in the following formulation phase. The teacher makes sure that for groups that have different strategies one student explains their approach on the blackboard. Then follows a validation phase. The teacher asks: "are some solutions better than others? Is there a best solution? How do you know?". The classroom discussion that these questions provoke form the input for the last phase: the institutionalization phase. In this phase the teacher is expected to be able to organize the ideas and strategies presented by students into solution models of the slide problem. The teacher makes a start with transforming the emerging model of the situation produced by students into a model for mathematical reasoning.

Method

Context. This task was conceived during a project meeting of the Erasmus+ project Meria on inquiry based mathematics teaching. A working group designed a detailed scenario describing teacher and student activities and possible solution strategies.

Procedure. The data for this pilot include self-reports, including students work, of pilot lessons by three Dutch teachers (from now on referred to as teacher A, B and C). Teacher B is involved with the whole Meria project as a teacher who performs pilots; the other two teachers signed up for the course after reading about it in a national newsletter. They were trained to give the lesson in a three hour course taught by the authors. In the course we had a try-out of the scenario and discussed the scenario in detail. The potential of the adidactical phase(s) was stressed as this is an uncommon position for teachers to be in. At some point in the following four weeks they taught the lesson. After the lesson teachers assembled the work of the students and wrote a report including their observations and experiences during the lesson. In a second meeting, after those four weeks, we discussed the experiences of the pilot and their reports with the teachers.

Other data is the self-report on a pilot conducted by the first author.

Analysis.

In an a priori analysis we discerned three hypothetical student approaches, based on three formal approaches to the tangent line. In the a posteriori analysis, we consider signs or traces of each approach as an indicator of a situational student model, which may later be vertically mathematized into a more formal mathematical model given by the definition.

1. (Bounding line approach) Students choose a free line and then move (translate and rotate) it until there seems to be just one intersection point in the area of focus. This informal approach relates to a formal definition of the tangent line as a unique local bounding line (this we will explain more precisely in the continuation of this research project). 2. (Secant lines approach) Students choose one point on the curve, the intended point of tangency. Then they choose another point on the curve, draw the line between the two points, and move the second point closer to the first (this works best in Geogebra or similar) to obtain a smoother fit. This informal approach relates, of course, to the standard school book approach of the tangent line and slope. 3. (Linear approximation approach) Students choose one point on the curve, draw a line and then try to adjust the slope so that it fits best against the curve (perhaps by wiggling their ruler). In GeoGebra or similar this can be done by moving the second point needed to draw the line. This informal approach relates to the more formal approach of the tangent line as the best linear approximation at a point of the curve (this will also be explained more precisely in future work).

The main distinction between the approaches is in the way the smoothness of the slide designs is evaluated or controlled by the students in the slider context: (1) by looking for intersection points, (2) by moving the point closer and closer and (3) by looking for intuitive smoothness. If students allow GeoGebra to compute the intersection point of the line and the curve, then approach 1 and 3 will resemble approach 2, since they will see the two intersection points approaching each other as they move closer to the desired line. In approach 1 students fix the slope and search for corresponding point, whereas in approach 2 they fix the point and find the corresponding slope. In approach 3 both options are present.

After students have drawn some lines and curves and written corresponding equations, they will have to find out whether their solution is 'smooth' enough. In a second a priori analysis, of what might be discussed by the students and/or the teacher in the validation phase, we expect three possible ways for students to evaluate their own design (in line with [START_REF] Zandieh | A theoretical framework for analyzing student understanding of the concept of derivative[END_REF]:

I. (Visual). Some will rely on their visual evaluation of the design: if it looks goods, then it is good. Students may also choose to zoom in on the curve (work on a smaller scale), for example using GeoGebra or similar, to check whether their fit is smooth. Zooming in the curve should more or less coincide with line (see the Locally Straight Approach in Tall 2013). II. (Algebraic). The students may compute whether their system of equation has the intended intersection point as a solution. Moreover, students could try and compute that there is precisely one solution to the system of two equations (at least locally). For example, if the curve is described by and the line is translated, so . Finding the value of for which the discriminant of the equation is zero, gives the desired line. For general algebraic curves one could study for which value of the intersection point has multiplicity . III. (Numerical). As a continuation of the second and third approach it would seem natural to choose the second point very close to the first and see how the slope of the obtained line compares to the suggested line. For example, if and , the line through and has a slope very close to 2. Students may even study a sequence of points ever closer to , which would bring them very close to the definition of a derivative through the difference quotient.

We collected the work of the students. In some cases that was individual paper work, but in most cases groups handed in one A4 with their notes and solutions. Furthermore, the teachers reported about the lessons, what they observed during the action phase, to what extent they were able to follow the scenario and what they saw as main results of the lesson. We analyzed the data for occurrences (of traces) of approaches 1, 2 and 3 during the action phase. In addition, these resources were used to find occurrences of the evaluation methods I, II and III during the action phase and validation phase (possibly the teacher introducing them during a didactical validation phase). These traces were used to hypothesize what informal situational mathematical models emerged from the students' work.

Finally these observed approaches were analyzed for their connection to what the teachers reported to have institutionalized in the final phase of the lessons. In particular, we tried to decide whether the potential in the students' work was seen and used by the teacher.

Results

We discuss the work of some groups that represent various observed approaches. A group of grade 8 pre-university school students from teacher A used approach 1: in GeoGebra they translated the line with fixed slope until it seemed to be tangent to the hyperbola (see Figure 1). The suggested line has equation . This looks acceptable (evaluation method I), but actually the two intersection points are approximately 2 apart. Clearly these students have neither The teacher reports that students mention that they are looking for the line such that there is one intersection point. So these students' informal model of the situation relates to the bounding line definition of tangent line. The teacher writes in her report that she is disappointed that the students did not arrive at the notion of slope of a curve at a point. But, in this case, in the institutionalization phase the teacher could have used the students' explanation of their strategy and the fact that students mention that they are looking for one intersection point to discuss the tangent line as a bounding line. Next, with the students, she could discuss evaluation method II as a step towards the target knowledge. This makes clear that there is a difficulty for teachers to realize the potential in the informal solutions students present. The challenge is to relate students' models to the target knowledge.

A group of grade 10 vocational school students from teacher B draws a tangent line to at the point (see Figure 2). First they seem to have drawn a secant line through and , but this is then dismissed for the sequel. They somehow come up with a for this candidate tangent line on the interval . They compute the slope of the line. What follows is unfortunately nonsense, but nevertheless the teacher was very excited about their approach and let them present it at the blackboard during the formulation phase. Rightfully so, since these students' informal solution method is precisely the model one needs to introduce the slope of a curve at a point. The teacher writes in her self-report: "The notion of … slope in a point became meaningful".

Another group from teacher B goes through a sequence of improved slide designs (see Figure 3). The first one has an asymptote; students write "does not have a straight bit". The second one is correct, but is dismissed, because the straight bit is horizontal instead of going up. The third is still with decimal coefficients, but the next four all are clearly attempts at exact solution (5 and 6 the same). Their approach can be classified as type 3. Teacher B told the class that zooming in was a good idea, so all students had a means of evaluating their solutions (method I). What happens in the last attempt is a "nice number wonder": by choosing nice numbers and the students hit the bull's eye. this curve". As a consequence, most students did not work with a line and a non-linear curve and the teacher got stuck trying to institutionalize the target knowledge as aimed for in the scenario.

The first author performed a pilot study with twelve grade 9 pre-university students equally divided in three groups. One group made drawings of circle segments and approximate tangent lines, using that the radius is perpendicular to the tangent. Within a few minutes they say: "the slope of the tangent line is important". Then they solve the problem by choosing a circle which goes through the point . For this point they infer that the slope of the tangent line is -1 (also visually obvious), leads them to the equation . This group's approach seems to be a case of vertical mathematization. Tangents to circles were already meaningful to them and for them solving the problem easily changes into symbolic work: doing some computations. After they showed the teacher their solution, he asked them to find a solution using a parabola. Their quick solution was a vertical line (

) and a parabola ( ). Then they asked: "How can you decide the right slope for a different point on the parabola?". This was a very welcome question, later referred to by the teacher in the institutionalization phase.

Another group quickly decided they wanted to use a tangent line with a parabola and a line design and surprisingly knew the algebraic way (II) to evaluate it: using the discriminant to decide whether the intersection point is unique. It turned out that the class made an exercise about exactly this a few weeks ago. Still the group took a long time to decide on a suitable line, finally settling for a correct one. According to the students this was obtained by guessing! In the institutionalization phase the teacher used the students' solution strategies and remarks to stress the importance of the slope of the tangent line (both in the circle solution and parabola solution). He explained this was called the slope of the curve at a point. With the students they decided that for a circle they could compute the slope in every point, but for the parabola new techniques would be necessary. This would be the subject for later lessons.

Discussion

Did the task and scenario provide opportunities for students to "experience a qualitative, global, introduction" to the slope of a curve in a point, as was the goal of the task? More concretely: did students come up with ideas that are essential to the notion of slope of a curve in a point? Based on the observations above one could conclude the students did.

Two ideas that are essential to the notion of slope did not come up naturally for students in the pilot. Firstly, the secant lines approach (2) did not occur in any of the pilot lessons, neither did any numerical evaluation of the design (III). This indicates that for the students in the pilot this approach does not come naturally (given the current Dutch curriculum and the task). Many students do have a dynamical approach, in the sense that they try to improve their initial and following attempts by varying parameters of the curve or the line to end up with a (visually) seemingly smoother result. But using two points on the curve to construct such a sequence of better and better fitting lines seems not a natural idea to come up with. On the positive side: the dynamical approach of "improving on your previous line" is a first step toward understanding the tangent line as the result of limit procedure. Secondly, the idea of zooming in or changing scale as an evaluation method has not occurred naturally to students. Once told by teacher B, students immediately grasped the idea and could use it, but they did not reinvent it. Students seem to prefer to stick to the scale at which they drew their first picture and work with reasonably nice numbers within that scale.

If students discover that some of their solutions have two intersection points, in the institutionalization phase the teacher could try and use this discovery to form a bridge to approach 2 and the difference quotient. As soon as you have two intersection points the idea of moving one of them to find a new line is a reasonable step.

A more radical conclusion in line with RME principles would be to go with the students' informal models all the way, instead of trying to facilitate a bridge towards the traditional way to introduce the slope. If the students' approach suggests the bounding line definition of a tangent line is the more meaningful for them, then the teaching sequence should be based on it. One can design instructional sequences following on approach 1 and 3 (in addition to existing sequences based on approach 2). These questions will be topic of study during the next steps in the context of this project.

Finally, in this project we combined principles from TDS and RME. This experience showed that the TDS emphasis on organizing didactic and adidactic phases by a teacher, in connection with the RME principle of reinvention guided by emergent models during these phases, supported the design of an inquiry-based scenario for students.

Figure 1 .

 1 Figure 1. Sample student work

  Teacher C did not formulate the task properly. She said: "Figure out in groups which functions could describe

Figure 2 :Figure 3 :

 23 Figure 2: Sample student work

Acknowledgment

The authors would like to thank the members of the Meria project-team for their contributions to the development and evaluation of the slide task.


Gaspard Jankowiak
email: gaspard@math.janko.fr.

Cécile Écile Taing

Clair Poignard

Annabelle Collin

Comparison and calibration of different electroporation models -Application to rabbit livers experiments

come

Electroporation (EP) is a complex phenomenon that occurs when biological tissues are subjected to short, high intensity electric pulses. The physical rationale of EP at the cell scale consists of the increase of membrane permeability when the cell is subjected to an electric field strong enough. More precisely, when the cell transmembrane voltage (the difference of the electric potential across the cell membrane) reaches around 1V, then defects are created in the membrane, which thus becomes permeant to extracellular molecules [START_REF] Leguèbe | Conducting and permeable states of cell membrane submitted to high voltage pulses: Mathematical and numerical studies validated by the experiments[END_REF][START_REF] Leguèbe | Mathematical model for transport of DNA plasmids from the external medium up to the nucleus by electroporation[END_REF].

The clinical interest for the phenomenon has constantly increased for the last decades. Indeed, EP makes it possible to either kill directly the cells in the target region (tumor) by a nonthermal mechanism named irreversible electroporation (IRE) [START_REF] Davalos | Tissue Ablation with Irreversible Electroporation[END_REF][START_REF] Edd | In Vivo Results of a New Focal Tissue Ablation Technique: Irreversible Electroporation[END_REF][START_REF] Davalos | Theoretical analysis of the thermal effects during in vivo tissue electroporation[END_REF][START_REF] Golberg | A statistical model for multidimensional irreversible electroporation cell death in tissue[END_REF], or to introduce non permeant molecules (ions, cytotoxic drugs like bleomycin, DNA plasmids, etc.) into living cells, which is referred to as reversible electroporation [START_REF] Mir | Electrochemotherapy potentiation of antitumour effect of bleomycin by local electric pulses[END_REF][START_REF] Gehl | Vascular reactions to in vivo electroporation: characterization and consequences for drug and gene delivery[END_REF][START_REF] Gehl | Investigational treatment of cancer using electrochemotherapy, electrochemoimmunotherapy and electro-gene transfer[END_REF]. The clinical interests in electroporation-based ablation therapies (EPAs) -either IRE alone or electroporation combined with drug delivery, which is referred to as electrochemotherapy (ECT) -have dramatically increased recently, after publications showing that EPAs provide interesting alternatives to standard non surgical ablative techniques -radiation therapy, radiofrequency ablation (RFA), or cryoablation (CrA)-to treat deep-seated non resectable tumors.

However EPAs are currently mainly limited to cutaneous and subcutaneous tumors. One the main reason of the limitation in the clinical routine use of EPAs comes from the technical difficulties raised by such therapies. Unlike standard ablative techniques which mainly deal with one needle, EPAs need at least two and usually three to four needles (even more for complex tumor shapes): the a priori determination of the treated zone is thus trickier than for standard ablative techniques. Moreover unlike thermal based ablation therapies, as RFA or CrA, for which the monitoring of the temperature in the tumor gives to physician an instantaneous assessment of the treatment, for EPAs the treated region is linked to the amplitude of the electric field, which cannot be monitored instantaneously since the pulse durations are very short (several tens of microseconds per pulse).

Electroporation is tightly linked to the electroquasistatic description of the electric field in the tissue. Therefore numerical modeling of the electric field magnitude provides a powerful strategy to assess the treatment efficacy: thanks to well-designed simulations, the numerical distribution of the electric field is achievable, providing a numerical evaluation of the treatment. However the choice of the model and the patient-dependent calibration, and particularly the patient-dependent organ conductivity, are crucial challenges which are still open.

The aims of the project is to compare the solutions of the different electroporation models found in the literature and to investigate their calibrations for electrostatic descriptions of a tissue from bioelectrical measurements on rabbit livers performed by Sel et al. [START_REF] Sel | Sequential finite element model of tissue electropermeabilization[END_REF]. After a presentation of 3 different static and 1 dynamical models of electroporation performed in Section 2, Section 3 presents the experimental set-up of Sel et al. and shows how to link the models outputs to the data. After the presentation of numerical schemes used to solve the models, Section 4 proposes a comparison of the solutions in the framework of Sel et al.. Section 5 is devoted to the investigation of parameters estimation from data measurements.

Electroporation models

In this section, we present three static models and one dynamical model, which provide the electrical description of tissue. To compare the different models, the following simple configuration is considered (see Figure 1). The domain Ω is a connected, open smooth subset of R d , d ∈ {2, 3}, and consists of two non intersecting subsets: Ω, which corresponds to the tissue under consideration (rabbit liver for instance in the experiment of Sel et al.) and the needles, represented by two parallel cylinders E ± . The needles are set at the isopotential ±g respectively. The outer boundary is denoted by Γ out := ∂Ω \ E ± .

2.1.

The standard static model. The most used model to describe tissue electroporation consists of the phenomenological electrostatic problem. The tissue is described as a conductive medium, whose conductivity tensor σ depends on the amplitude of the electric field -∇u. The model reads then

-∇ • (σ( ∇u )∇u) = 0, in Ω, (1a) ∂ n u| Γout = 0, u| E ± = g ± , (1b) Ω Γ out E + E - Figure 1.
The typical geometrical configuration consists of the tissue domain Ω in gray, deprived of the two needles E ± , in white. The outer boundary Γ out is represented in bold.

The tissue conductivity consists of a 4 parameters sigmoid function. Typically [START_REF] Breton | Non-Linear Steady-State Electrical Current Modeling for the Electropermeabilization of Biological Tissue[END_REF] ∀λ ≥ 0,

σ(λ) = σ 0 + σ 1 -σ 0 2 (1 + erf(k ep (λ -E th ))) ,
where σ 0 is the conductivity of the non electroporated tissue, σ 1 is the tissue conductivity of the fully porated tissue, E th is the threshold amplitude for electroporation, and k ep is the slope of the nonlinearity. Here, erf is the Gauss error function. The qualitative behaviour of σ is depicted in Figure 2. It should be noted that this choice of σ is largely phenomenological, and as shown in [START_REF] Ivorra | Electrical modeling of the influence of medium conductivity on electroporation[END_REF], the available experimental data does not seem sufficient to characterize the dependence of the conductivity on the electric field.

E th σ 1 σ 0 λ σ(λ)

2.2.

The electric circuit approach of Voyer et al. In [START_REF] Voyer | Dynamical modeling of tissue electroporation[END_REF], Voyer et al. proposed a biphasic dynamical model based on the description of an individual cell and surrounding matrix as an electric circuit. The ODEs at the cell level are formally generalized to PDEs at the tissue level. It describes the electric potential outside cells φ e and the electric field inside cells J c . The parameters are the extracellular and intracellular electric conductivities, respectively σ e and σ c . The conductivity of the cell membrane σ m depends on time in a way which mimics the effects of poration, i.e. the appearance of holes on the membrane, and permeabilisation, that is the degradation of membrane molecules. Both phenomena increase the conductivity. The resulting system reads ∇ • (σ e ∇φ e + J c ) = 0, (3a)

ε m ∂ t J c + (σ m (t, E m ) + σ c )J c = σ c σ m ∇φ e , (3b)
∂ n φ e | Γout = 0, φ e | E ± = g ± , J| Γout = 0 , (3c)
where E m = ∇φ eσ -1 c J c , and

σ m (t, E m ) = σ m 0 + σ m 1 X 1 (t, E m ) + σ m e X 2 (t, E m ). (3d)
The functions X 1 and X 2 are the respective degrees of poration and degrees of permeabilisation. They satisfy

Ẋ1 = β 1 ( E m ) -X 1 τ 1 , and Ẋ2 = β 2 (X 1 ) -X 2 τ 2 , (4)
where β 1 and β 2 are 2-parameters sigmoid functions of the form

β i (x) = 1 2 1 + erf(k i x -x trans i ) ,
where k i is the stiffness of the sigmoid and x trans i the transition threshold. A simplified static version of this model allows to compare it with the standard model. Indeed, the static version of this model writes

∇ • (σ e ∇φ e + J c ) = 0, (σ m ( E m ) + σ c )J c = σ c σ m ( E m )∇φ e , ∂ n φ e | Γout = 0, φ e | E ± = g ± , J| Γout = 0.
where σ m is a sigmoid function similar to [START_REF] Breton | Non-Linear Steady-State Electrical Current Modeling for the Electropermeabilization of Biological Tissue[END_REF]. Written in φ e and E m , the equations become

∇ • (σ e + σ c )∇φ e -σ c E m = 0, in Ω, (5a) (σ m ( E m ) + σ c )E m = σ c ∇φ e ,
in Ω, (5b)

∂ n φ e | Γout = 0, φ e | E ± = g ± , E m | Γout = 0. (5c)
Problem (5) makes it possible to define the equivalent tissue conductivity. Actually, thanks to (5b), the potential φ e satisfies

∇ • σ e + σ c σ m ( E m ) σ c + σ m ( E m ) ∇φ e = 0.
The equivalent tissue conductivity can thus be defined as

σ eq ( E m ) = σ e + σ c σ m ( E m ) σ c + σ m ( E m ) . (6)
A simplified version of the problem consists in assuming that σ m depends on ∇φ e instead of E m . Then the simplified problem reads

∇ • σ e + σ c σ m ( ∇φ e ) σ c + σ m ( ∇φ e )
∇φ e = 0, in Ω, (7a)

∂ n φ e | Γout = 0, φ e | E ± = g ± , (7b)
which is similar to the standard model (1) since the function σ eq defined by

∀λ ≥ 0, σ eq (λ) = σ e + σ c σ m (λ) σ c + σ m (λ)
, is a sigmoid function.

2.3.

The static bidomain model. In electrocardiology, the so-called bidomain model has been proven to be the homogenisation limit of the cell scale electric potential. It consists of 2 electric potentials, u e and v, which satisfy the following PDE system proposed in [START_REF] Deville | Mathematical Modeling of enhanced drug delivery by mean of Electroporation or Enzymatic treatment[END_REF]:

∇ • (σ e + σ c )∇u e -σ c ∇v = 0, (8a) S m ( ∇u e )v -∇ • (σ c ∇v) + ∇ • (σ c ∇u e ) = 0, (8b) σ e ∇u e • n| Γout = 0, u e | E ± = g ± , σ c ∇v • n| ∂Ω = σ c ∇u e • n| ∂Ω , (8c)
where S m = A m Sm , with A m (m -1 ) the ratio of membrane area by unit volume and Sm (S.m -2 ), is the conductance of the cell membranes. This model enables to link rigorously the membrane conductance S m to the electric potential thanks to an homogenization procedure presented in [START_REF] Colli Franzone | Degenerate evolution systems modeling the cardiac electric field at micro and macroscopic level[END_REF][START_REF] Collin | Mathematical analysis and 2-scale convergence of a heterogeneous microscopic bidomain model[END_REF]. In this sense, it is more physiological than the previous phenomenological models (see Remark 1). To account for the electroporation phenomenon -that is the increase of the membrane conductance -we assume that Sm follows a sigmoid function of the norm of the outer electric field -∇u e

Sm ( ∇u

e ) = S0 m + 1 2 ( S1 m -S0 m ) [1 + erf(k ep ( ∇u e -E th ))] .
Remark 2 (On the nonlinearity). In cardiac electrophysiology, the nonlinearity usually depends on the homogenised transmembrane voltage u eu c . However such a nonlinearity is inconsistent with the electroporation phenomenon. Actually, assuming that g -= -g + is constant, the symmetry along the axis (Oy) implies that u e , u c and the transmembrane voltage vanishes along (Oy), which is inconsistent with the fact that electroporation occurs between the needles. For this reason, a nonlinearity depending on ∇u e is chosen, which is relevant with the nonlinearity of the static nonlinear model.

Remark 3 (On the boundary conditions on v). The Neumann boundary condition satisfied by v states the electric field does not go out of the cell phase, and thus u c given by

u c = u e -v satisfies ∂ n u c | ∂Ω = 0.

Data measurements

The experimental set-up of Sel et al. consists of an ex-vivo cubic piece of rabbit liver in which two needles are inserted (see Figure 3). Square pulses of different amplitudes are applied on the needles and the ents have shown that the extent of cell ation depends on electric field intensity, raction with surrounding cells [START_REF] Julier | Unscented filtering and nonlinear estimation[END_REF], [START_REF] Leguèbe | Mathematical model for transport of DNA plasmids from the external medium up to the nucleus by electroporation[END_REF]. o permeabilized cells such as cells volume of medium, membrane conductivity, cell l transmembrane potential (TMP) affect ctivity. The relationship between these e conductivity of cells in suspension has usly [START_REF] Leguèbe | Conducting and permeable states of cell membrane submitted to high voltage pulses: Mathematical and numerical studies validated by the experiments[END_REF]. Besides effective conductivity permeabilization on a tissue level is the nsity. As the field results from a voltage electrodes, the electrode configuration stribution in tissue [START_REF] Mir | Electrochemotherapy potentiation of antitumour effect of bleomycin by local electric pulses[END_REF]- [START_REF] Sel | Sequential finite element model of tissue electropermeabilization[END_REF]. Electrode or tissue permeabilization are parallel act plate electrodes as well as needle arrays [START_REF] Sel | Sequential finite element model of tissue electropermeabilization[END_REF]- [21]. ents have shown that the extent of cell ation depends on electric field intensity, raction with surrounding cells [START_REF] Julier | Unscented filtering and nonlinear estimation[END_REF], [START_REF] Leguèbe | Mathematical model for transport of DNA plasmids from the external medium up to the nucleus by electroporation[END_REF]. o permeabilized cells such as cells volume of medium, membrane conductivity, cell l transmembrane potential (TMP) affect ctivity. The relationship between these e conductivity of cells in suspension has usly [START_REF] Leguèbe | Conducting and permeable states of cell membrane submitted to high voltage pulses: Mathematical and numerical studies validated by the experiments[END_REF]. Besides effective conductivity permeabilization on a tissue level is the nsity. As the field results from a voltage electrodes, the electrode configuration stribution in tissue [START_REF] Mir | Electrochemotherapy potentiation of antitumour effect of bleomycin by local electric pulses[END_REF]- [START_REF] Sel | Sequential finite element model of tissue electropermeabilization[END_REF]. Electrode or tissue permeabilization are parallel act plate electrodes as well as needle

8 > > > > < > > > > : r • ( (x)rV(x)) = 0 x 2 rV • n = 0 x 2 out V = g ±
x 2 E ± I V: electric potential (rV electric field) I : conductivity The expression of the numerical electric intensity depends on the choice of the models. Denoting by E + the length of the electrode in the perpendicular plane to the 2D simulation plane, one has

I I ± = R E ± rV • n ds: intensity 7/38
I 1 E = E + E + σ ( ∇V ) ∇V • n ds,
where V is the solution to (1),

I 2 E = E + E + σ eq ( E m
)∇φ e • n ds, where (φ e , E m ) is the solution to (5), or to (3)

I 2,simp E = E + E + σ eq ( ∇φ e
)∇φ e • n ds, where φ e is the solution to [START_REF] Deville | Mathematical Modeling of enhanced drug delivery by mean of Electroporation or Enzymatic treatment[END_REF],

I 3 E = E + E +
σ e ∇u e • n ds, where (u e , v) is the solution to [START_REF] Edd | In Vivo Results of a New Focal Tissue Ablation Technique: Irreversible Electroporation[END_REF].

The above definition of the electrical intensities of the models involve the gradient of the solution along the needle. This is numerically unstable since the electric field is the most intense nearby the electrodes, and thus numerical instabilities may appear. To smoothen this behavior, it is possible to introduce a specific function w such that the surface integral is replaced by a volume integral. More precisely, let w be defined by

-∆w = 0, in Ω, ( 9a
)
w| E ± = ±1, ∇w • n| Γout = 0. (9b)
Then, thanks to appropriate integration by parts one has

I 1 E = E + 2 Ω σ ( ∇V ) ∇V • ∇w dx,
where V is the solution to (1), (10a)

I 2 E = E + 2 Ω σ eq ( E m
)∇φ e • ∇w dx, where (φ e , E m ) is the solution to (5), (10b)

I 2,simp E = E + 2 Ω
σ eq ( ∇φ e )∇φ e • ∇w dx, where φ e is the solution to [START_REF] Deville | Mathematical Modeling of enhanced drug delivery by mean of Electroporation or Enzymatic treatment[END_REF], or to (3) (10c)

I 3 E = E + 2 Ω ((σ e + σ c
)∇u eσ c ∇v) • ∇w dx, where (u e , v) is the solution to (8). (10d) 4. Numerical discretization and qualitative comparison between the models 4.1. Discretization and numerical approximation. In this section, we present the discretizations that have been used to solve the different models. All these strategies have been implemented using the Python finite element library FEniCS [START_REF] Alnaes | The fenics project version 1.5[END_REF]. 4.1.1. Static models: illustration on the standard static model. We want to solve (1) which we recall here:

-∇ • (σ( ∇u )∇u) = 0, ∂ n u| Γout = 0, u| E ± = g ± , (1b)
which is a non-linear problem since σ depends on ∇u. In Breton et al. [START_REF] Breton | Non-Linear Steady-State Electrical Current Modeling for the Electropermeabilization of Biological Tissue[END_REF] a modified fixed point has been presented and the numerical convergence of this scheme has been shown. In this work, we compare two other strategies. The first one consists in using iteration schemes coming from the time-discretization of the following non-linear evolution equation

∂ s u -∇ • (σ( ∇u )∇u) = 0 in Ω ,
with the same boundary conditions as above, and where s acts as the pseudo time variable. We follow the same strategy for the static version of the biphasic model and for the bidomain model. Depending on the eigenvalue of the linearised problem, one can expect that u behaves asymptotically as the static solution. We explicit the strategy (which can be easily extended to the other static models) for the standard model. First, we consider the decomposition u = v + u lin , where u lin solves the linear static problem associated to [START_REF] Alnaes | The fenics project version 1.5[END_REF], that is the Laplace equation on Ω with boundary conditions (1b). Thus, we obtain the following equation on v

∂ s v -∇ • (σ( ∇(v + u lin ) )∇(v + u lin )) = 0, (11)
with homogeneous Dirichlet boundary conditions, since u lin carries the boundary conditions of [START_REF] Alnaes | The fenics project version 1.5[END_REF]. By discretizing the s derivative by finite differences with step ∆s, multiplying this equation by a test function ϕ ∈ C ∞ c (Ω) and integrating over Ω, we get

Ω v n+1 -v n ∆s ϕ + Ω σ( ∇(v n + u lin ) )∇v n+1 • ∇ϕ + Ω σ( ∇(v n + u lin ) )∇u lin • ∇ϕ - E ± ∪Γout σ( ∇(v n + u lin ) )∇v n+1 • n ϕ = 0.
The term on Γ out vanishes because of the homogeneous Neumann boundary conditions, and we can restrict to test functions with zero trace on E ± by integrating the Dirichlet data in the function space, see below. Hence, the variational problem to solve reads A(v n+1 , ϕ) = L(ϕ) where

A(v n+1 , ϕ) := Ω v n+1 ϕ + ∆s Ω σ( ∇(v n + u lin ) )∇v n+1 • ∇ϕ, L(ϕ) := Ω v n ϕ -∆s Ω σ( ∇(v n + u lin ) )∇u lin • ∇ϕ.
This iterative method is initialized by taking v 0 = u lin . We then proceed by approximating v, u lin using P 2 finite elements, i.e. we have

v h , u h lin ∈ V h with V h = w ∈ C 0 (Ω) : w| T ∈ P 2 ∀ T ∈ T h ,
where P 2 is the space of polynomials in two variables, of degree at most 2 and T h is a triangulation of Ω. We also introduce

V h ψ := w ∈ V h : w| E ± = ψ . We are left with the problem of finding v h n+1 ∈ V h g ± such that A(v h n+1 , ϕ h ) = L(ϕ h ) , ∀ϕ h ∈ V h 0 , for n > 0, with v h 0 = u h lin .
This problem is well-posed according to standard finite element analysis arguments.

The second approach consists in using Newton's method directly on the finite element system. Consider the problem consisting in finding u h ∈ V h g ± such that

A(u h , ϕ h ) = 0 , ∀ϕ h ∈ V h 0 ,
where A is given by

A(u h , ϕ h ) = Ω σ( ∇u h )∇u h • ∇ϕ h ,
which is the discretized weak formulation of [START_REF] Alnaes | The fenics project version 1.5[END_REF]. Assuming that some u h 0 satisfying the boundary conditions is known, one can try to solve this system of non linear equations using the Newton's method. Such a u h 0 is given by solving Laplace's equation on Ω with boundary conditions (1b). In FEniCS, the computation of the corresponding Jacobian can be performed automatically. For the models we consider, this approach is faster and more stable compared to the evolution-based iterative method presented above, although it requires a good guess for initialization. In practice, such a good guess can be found by starting from the solution to the linear problem and increasing gradually the value of k ep until the desired value is reached. The Newton's approach is also simpler, in the sense that is does not require a choice of a step size (∆s in the evolution approach).

A qualitative comparison of the two methods can be seen in Figure 5. σ( ∇u ) is projected on piecewise constant discontinuous elements. On the right of the domain, the profile given by the Newton's method is noticeably smoother.

Figure 5. Comparison of σ( ∇u ) for typical solutions of 1 given by the evolution-based iterative method (left) and the Newton's approach (right) for the standard model. 4.1.2. The dynamical biphase model. Concerning the dynamical biphase model, we use an iterative time discretization method. We denote the time step by ∆t, so that t n = n∆t. At the n th step, we compute the membrane electric field, and its amplitude

E n m = ∇u n e - 1 σ c J n c , A n m = d c |E n m |,
where d c is the mean of the cell diameters. From there, we solve the differential equations for the degrees of poration and permeabilization X 1 and X 2 using an explicit Euler scheme

X n 1 = X n-1 1 + ∆t τ 1 β 1 (A n m ) -X n-1 1 , X n 2 = X n-1 2 + ∆t τ 2 β 1 (X n-1 1 ) -X n-1 2 .
Then, we compute the membrane conductivity

σ n m = σ m 0 + σ 1 X n 1 + σ 2 X n 2 .
Similarly to the static models, we multiply (3) by the test functions ϕ 1 ∈ C ∞ (Ω) and ϕ 2 ∈ C ∞ 0 (Ω) 2 respectively and integrate by parts to obtain

-σ e Ω ∇φ n+1 e • ∇ϕ 1 -σ c Ω J n+1 c • ∇ϕ 1 + σ e E ± ∪Γout ∇φ n+1 e • n ϕ 1 + σ c E ± ∪Γout J n+1 c • n ϕ 1 = 0, ε m Ω J n+1 c -J n c • ϕ 2 + ∆t Ω (σ n m + σ c )J n+1 c • ϕ 2 - Ω σ n m σ c ∇φ n+1 e • ϕ 2 = 0.
We discretize again on P 2 finite elements and integrate the Dirichlet boundary conditions in the finite element space. We then have to find

φ n+1,h e ∈ V h g ± , J n+1 c ∈ (V h 0 ) 2 such that A(Φ n+1,h , Ψ h ) = L n (Ψ h ) ,
where

Φ h = (φ n+1,h e , J n+1,h c ) and Ψ h = (ϕ h 1 , ϕ h 2 ) ∈ V h 0 × (V h 0 ) 2 .
The functionals A and L are given by:

A(Φ n+1,h , Ψ h ) := -σ e Ω ∇φ n+1,h e • ∇ϕ h 1 -σ c Ω ∇J n+1,h c • ∇ϕ h 1 + m Ω J n+1,h c • ϕ h 2 + ∆t Ω (σ n m + σ c )J n+1,h c • ϕ h 2 - Ω σ n m σ c ∇φ n+1,h e • ϕ h 2 , L n (ϕ) := m Ω J n c • ϕ h 2 .

4.2.

Qualitative comparison between the models. In this section we show the characteristic solution of all three models presented in Section 2. We consider an electrode diameter of 0.7mm, a voltage of 800V.

Results for the standard, the bidomain and the dynamical biphase models are illustrated in Figures 6, 7 and 8 respectively. In all cases, the contour lines of u and u e look similar.

Estimation strategy: illustrations on the standard static model

As we have seen in the previous section, the solutions of the different models are realistic and can be qualitatively compared. They allow to represent the electroporation phenomena. The aim of this section is to investigate numerical calibration strategies of these models. We will focus for this work on the standard static model.

First, the estimation strategy is introduced. Then, it is illustrated with synthetic data and finally, it is applied on real data [START_REF] Sel | Sequential finite element model of tissue electropermeabilization[END_REF] presented in Section 3. 1], with a 800V voltage step. It is easy to see that σ m is mainly driven by the value of X 1 , i.e. the poration process. At the lower left is the evolution of the current intensity flowing through the electrodes, the spike qualitatively matches experiments, see [START_REF] Voyer | Dynamical modeling of tissue electroporation[END_REF]. 5.1. Parameters estimation algorithm. We denote by θ ∈ Θ the vector which concatenates all the parameters which have to be estimated. We denote by y ∈ Y the solution and by z ∈ Z the observations. We denote by • Θ,(P ) -1 and by • Z,R the norms defined by

Sm( ue ) in S/m²

• 2 Θ,(P ) -1 = • , (P ) -1 • and • 2 Z,R = • , R • .
The static problem can be rewritten as A(y, θ) = 0, where A corresponds to the model operator. We denote by C the observation operator which relies y to z z = C(y). and we denote by O the following operator O : θ → y, such that A(y, θ) = 0.

Our objective is to inverse the operator Ψ = C • O. In order to do that, the criterium that we want to minimize is given by

J (θ) = 1 2 θ -θ 2 Θ,(P ) -1 + 1 2 z -Ψ(θ) 2 Z,R
, where θ corresponds to an a priori value of θ. This criterium corresponds to a likelihood functional where all disturbances are assumed to be Gaussian

θ ∼ N (θ , P 1/2 ) and z ∼ N (Ψ(θ), R -1/2 ).
If we want to use N z measurements denoted by (z k ) 1≤k≤Nz associated to (R k ) 1≤k≤Nz , we have to consider the following criterium

J (θ) = 1 2 θ -θ 2 Θ,(P ) -1 + 1 2 Nz k=1 z k -Ψ(θ) 2 Z,R k ,
which can be rewritten as

J (θ) = 1 2 θ -θ 2 Θ,(P ) -1 + 1 2 Nz k=1 z -Ψ(θ) 2 Z, R , with z =    z 1 . . . z Nz    , Ψ =    Ψ . . . Ψ    , Z = Z × • • • × Z and R =    R 1 . . . R Nz    .
It exists many methods to solve this kind of problem as for example as a non-exhaustive list: gradient descent based methods ; stochastic strategies as for example Monte-Carlo based strategies or expectationmaximization algorithm. In this work, we decided to use a stochastic algorithm based on an unscented transform [START_REF] Julier | Unscented filtering and nonlinear estimation[END_REF]. Let (ω i ) 1≤i≤p and (e (i) ) 1≤i≤p be some weight parameters and directions in Θ such that

1≤i≤p ω i e (i) (e (i) ) T = 1 ρ .
This is called a set of sigma points and many different sets of sigma points have been proposed in the literature. In this work, we will consider the simplex sigma points, which allow to consider only dim Θ +1 points [START_REF] Julier | The spherical simplex unscented transformation[END_REF]. The algorithm reads as follows:

• Initialisation: θ0 = θ and P 0 = P .

• Generation of p particles of mean θk and covariance P k :

θ (i) = θk + ρP k e (i) , 1 ≤ i ≤ p.
• Computation of the observations generated by the solution of the system for each particle:

Z i = Ψ(θ (i) ).
• Computation of the mean and of the covariance of Z i :

Z * = E( Z i ) def = ω i Z i and P z = Cov( Z i ) def = ω i ( Z i -Z * )( Z i -Z * ) T .
• Computation of the covariance between θ (i) and J(θ (i) ) (sensitivity of the observations to the parameters):

P θz = Cov(θ (i) , J(θ (i) )) def = ω i (θ (i) -θk )( Z i -Z * ) T .
• Computation of the new value of the parameters and of their covariances θk+1 = θk + P θz (P z + R -1 ) -1 ( z -Z * ), Table 2. Coupled estimation of E th , σ 0 and σ 1 (2 digits of precision). The true and estimated values are given after the log-transformation. The third (resp. fourth) column corresponds to the case where the 3 (resp. 1) electrode(s) are (resp. is) considered.

P k+1 = P k -P θz (P z + R -1 ) -1 (P θz ) T .

Estimations on synthetic data.

As said previously, we focus on the standard static model in this section. The parameters are then the parameters of the sigmoid function: E th , σ 0 , σ 1 and k ep . We want to validate on synthetic data the estimation strategy. To do that, we consider a framework close from the real data. Following [START_REF] Sel | Sequential finite element model of tissue electropermeabilization[END_REF], we build synthetic data with intensity measurements for 5 different voltages and 3 electrode sizes (the same of the article). The corresponding observation space has then dimension N z = 15. The synthetic data are obtained using E th = 5.75 × 10 4 V /m, σ 0 = 0.065 S/m, σ 1 = 0.1483 S/m and k ep = 10 -3 . The parameters that we want to estimate are all positive which will be constrained by a log-transformation during the estimation procedure. We assume that log E th ∼ N (log(4.5 × 10 4 ), √ 0.2), log σ 0 ∼ N (log(0.2), √ 0.2), log σ 1 ∼ N (log(0.09), √ 0.2) and log k ep ∼ N (log(10 -4 ), √ 0.2).

Concerning the uncertainties quantification of the observations, we assume that R -1/2 = 0.025 Id Nz,Nz which corresponds to a noise standard deviation of approximately 20%: R -1/2 ≈ 20% 1 Nz Nz k=1 |z k |Id Nz,Nz ≈ 0.025. First of all, we start by an individual estimation of the parameters and the synthetic data are not noisy. Table 1 presents the results (with 2 digits of precision). The algorithm stops when the convergence is reached.

As a conclusion, it is possible to estimate three parameters: E th , σ 0 and σ 1 . The sensitivity of σ 0 is weaker and more iterations are needed (compared to E th and σ 1 ). Then, k ep is not identifiable: there is no sensitivity associated to this parameter. As k ep is not identifiable, we fix it at its true value and we make a coupled estimation of the three other parameters, see the third column of Table 2. We consider also a case where only one size of electrodes is considered. The results are very encouraging and show that it is possible to estimate the three parameters together. The results are slightly better when three electrodes are considered. Finally, we add a gaussian noise of standard deviation σ n = 0.005, 0.01 (resp. 5% and 10%) on the synthetic data to see the sensitivity of the estimation to measurement noise. In the case of σ n = 0.01, we also try to estimate using only one size of electrodes. The results are given in Table 3. As expected, the more the noise on the data is important, the more it is difficult to well estimate the parameters but the results are still encouraging. More precisely, with noisy data, it is crucial to consider measurements performed with different size of electrodes.

( ∇V ) = σ 0 + (σ 1 -σ 0 ) exp E 2 th ∇V 2 .
The remaining three parameters, E th , σ 0 and σ 1 have to be estimated jointly, since we have no a priori value for them. We assume that the parameters are centered around the values given in Section 5.2, i.e.:

(12) log E th ∼ N (log(5.75 × 10 4 ), 2), log σ 0 ∼ N (log(0.065), 0.2), log σ 1 ∼ N (log(0.1483), 0.5) .

The noise on the measurements is modeled by R -1/2 = 0.2, which corresponds to a noise of 25%.

When all electrodiameters are considered together, the convergence of the estimation procedure is shown in Figure 9. As is the synthetic data case, the convergence of σ 0 is much slower than the other two parameters, which requires a large number of iterations. For each individual electrode size, the convergence of the estimation procedure is shown in Figure 10. The estimated values differ significantly from one electrode diameter to the next, and σ 0 converges again slower than E th and σ 1 . As we can see, the estimated parameters differ significantly between electrode diameters, especially for 0.3mm. We solve the direct problem with parameters estimated for each diameter individually ; and for the parameters estimated when considering all diameters simultaneously, see Figure 11. We can see the resulting distribution of σ( ∇u ) for the three electrode diameters and for voltages of 200V and 1000V . The distribution is very different, especially for 1000V . Table 6 presents the maximum relative error on the intensity, and shows that the best results are obtained when estimating the parameters only on the electrode of diameter 1.1mm. This is not surprising regarding to the relatively low values for the current for a diameter of 0.3mm, the quasi-equivalent values for diameters of 0.7 and 1.1mm and also the relatively low values for low voltages (see Table 4).

Maybe, better results could be achieved by understanding better the impact on the measurement noise using for example synthetic data. One strategy could be for example to consider different values of the standard deviations of the observations in order to give less importance to the most noisy data. Table 6. Maximum relative error on the intensity across voltages, for each set of estimated parameters (one for each electrode diameter plus one for the estimated on all three simultaneously).

Conclusion and perspectives

In this paper, we have compared different models of electroporation: the standard static model, the dynamical biphase model and the bidomain model in the same framework built using bioelectrical measurements on rabbit livers proposed by Sel et al. [START_REF] Sel | Sequential finite element model of tissue electropermeabilization[END_REF]. The solutions of the different models are realistic and can be qualitatively compared. Then we investigate a numerical calibration of the parameters only on the standard model. The strategy is based on a stochastic algorithm using an unscented transform for estimating parameters of static systems. The results on synthetic data are convincing. Finally, the estimation strategy is applied on the rabbit measurements and the attempt is promising even if the best fit is not achieved using all geometries simultaneously. As explained previously, this could be due to the measurements noise. This work allows to lay the ground of the calibration of electroporation models but many improvements have to be studied. The work on synthetic data has to be continued in order to better understand the estimation difficulties due to the measurements noise. Furthermore, the parameters estimation has to be applied on the bidomain model. Finally, concerning the dynamical biphase model, a dynamical version of the estimation strategy has to be written. The main difficulty lies in the fact that we need an efficient strategy which does not involve prohibitive computational times. To address this issue, Reduced-order Kalman filter method, introduced by Moireau and Chapelle [START_REF] Moireau | Reduced-order unscented Kalman filtering with application to parameter identification in large-dimensional systems[END_REF], will be investigated in a near future.

1 .

 1 The standard static model 2.2. The electric circuit approach of Voyer et al. 2.3. The static bidomain model 3. Data measurements 4. Numerical discretization and qualitative comparison between the models 4.1. Discretization and numerical approximation 4.2. Qualitative comparison between the models 5. Estimation strategy: illustrations on the standard static model 5.1. Parameters estimation algorithm 5.2. Estimations on synthetic data 5.3. Estimations on the data set of Sel et al. 6. Conclusion and perspectives 1. Introduction

Figure 2 .

 2 Figure 2. The shape of the conductivity σ

Fig. 1 .

 1 Fig. 1. Needle electrodes used in experiments: (a) side view; (b) top view.

Figure 3 .

 3 Figure 3. Experimental set-up for electroporation measurements as performed by Sel et al. 3 electrodes diameter are considered. The piece of rabbit liver is located below the plexiglass on a thickness of 7 mm [19].

I

  Starting point: classical electrostatic equation:

Figure 2 . 4 :

 24 Figure 2.4: 2D full domain and final computational mesh obtained considering the symmetry of the problem.

Figure 4 .

 4 Figure 4. (Left): Symmetrical domain corresponding to the horizontal section of the experimental set-up. (Right): Computational domain restricted to a quadrant. Homogeneous Dirichlet condition is set on the left border, homogeneous Neumann conditions are set on the top, bottom and right borders, non homogeneous Dirichlet condition is set on the needle boundary.

Figure 6 .

 6 Figure 6. Solution of the static standard model, using parameters E th = 5.75 × 10 4 V /m, k ep = 10 -3 V -1 , σ 0 = 0.065S/m and σ 1 = 0.1483S/m.

Figure 7 .

 7 Figure 7. Solution of the static bidomain model, using the parameters of [7, Table 3.2]

Figure 8 .

 8 Figure 8. Solution to the biphase model at around 8ms, using the parameters of [20, Table1], with a 800V voltage step. It is easy to see that σ m is mainly driven by the value of X 1 , i.e. the poration process. At the lower left is the evolution of the current intensity flowing through the electrodes, the spike qualitatively matches experiments, see[START_REF] Voyer | Dynamical modeling of tissue electroporation[END_REF].

Figure 9 .

 9 Figure 9. Estimation of E th , σ 0 and σ 1 , after log-transformation. All voltages and all electrodes diameters are considered simultaneously. The thin dashed green and blue lines denote the standard deviation of the uncertainty on the estimation.

Figure 10 .

 10 Figure 10. Estimation of E th , σ 0 and σ 1 , after log-transformation. All voltages are considered, but electrode diameters are considered separately. The thin lines denote the standard deviation of the uncertainty on the estimation.

Figure 11 .

 11 Figure 11. σ( ∇u ) in S/m for applied voltages 200V and 1000V and the three geometries, for the parameters estimated separately on each geometry (top 2 lines) and for the parameters estimated on all geometries simultaneously (bottom 2 lines).

Table 1 .

 1 Parameter Real value Estimated value Number of iterations Individual estimation of the parameters (2 digits of precision). The true and estimated values are given after the log-transformation.

	E th	10.96	10.95	13
	σ 0	-2.73	-2.74	20
	σ 1	-1.91	-1.91	3
	k ep	-6.91	-9.20	∞
	Parameter Real value Estim. Estim., 1 elec.
	E th	10.96	10.95	10.95
	σ 0	-2.73	-2.76	-2.77
	σ 1	-1.91	-1.90	-1.88

Table 3 .

 3 5.3. Estimations on the data set ofSel et al. We now turn to the problem of parameter estimation for real data.Parameter Real value Estim. σ n = 0.005 Estim. σ n = 0.01 Estim. σ n = 0.01, 1 elec. Coupled estimation of E th , σ 0 and σ 1 (2 digits of precision). The true and estimated values are given after the log-transformation. The third (resp. fourth) column corresponds to σ n = 0.005 (resp. σ n = 0.01). The last column is for σ n = 0.01 and only one electrode is considered.

	E th	10.96	10.94	10.83	10.86
	σ 0	-2.73	-2.75	-2.72	-2.59
	σ 1	-1.91	-1.91	-1.94	-1.98
	Parameter Initial value Estim. 0.3mm Estim. 0.7mm Estim 1.1mm Estim all 3
	E th	10.96	10.9	9.59	9.60	9.75
	σ 0	-2.73	-1.88	-5.78	-4.23	-9.06
	σ 1	-1.909	-0.37	-0.58	-0.77	-0.65

Table 5 .

 5 Initial values and results of the 4 estimations, one for each electrode diameter, and one with all diameters combined. 5.3.1. Dataset. The dataset used is the one provided by[START_REF] Sel | Sequential finite element model of tissue electropermeabilization[END_REF], the corresponding experimental setup is recalled in Figure3. It provides intensity measurements for 5 different voltages and 3 electrode sizes, which are collected in Table4. As for the synthetic data, the corresponding observation space has then dimension N z = 15. Estimation and validation. As was made clear in Section 5.2, the sensivity on k ep in the case of a 4-parameter sigmoid is very low. Trying to estimate this parameter would most likely result in an unreliable value. To avoid having to choose an arbitrary value for k ep , we consider a 3-parameter sigmoid instead, of the form

	Electrode diameter	Applied voltage		1.5		0.3mm 0.7mm
		200V 400V 600V 800V 1000V				1.1mm
	0.3mm 0.7mm 1.1mm	0.104 0.275 0.571 0.877 1.260 0.170 0.412 0.850 1.389 1.663 0.168 0.437 0.776 1.315 1.533	Intensity/A	1
				0.5
	Table 4. Intensity (A) of the cur-
	rent flowing through the electrodes
	measured by [19]		0	200	400	600	800	1,000
						Applied voltage/V
	5.3.2. σ

  max V |I computed (V ) -I measured (V )|/I measured (V )

	Geometry for estimation Geometry for validation
		0.3mm 0.7mm 1.1mm
	0.3mm	3%	33%	26%
	0.7mm	52%	24%	31%
	1.1mm	27%	24%	15%
	all	31%	34%	26%

Acknowledgements References

Acknowledgements

The authors would like to thank very warmly Philippe Moireau and Sébastien Impériale from the Inria team MEDISIM and Olivier Gallinato from the SophiaGenetics company for their help and their fruitful discussions. This research was granted by the plan Cancer NUMEP, PC201615, led by C. Poignard.