Dr Rudolf Schnee

Nathalie Kroichvili

Daniela Chrenko

Reiner Kriesten

Anforderungen der Elektromobilität an die Wertschöpfungsketten in der Automobilindustrie

Keywords: Digitalisierung, Klimawandel und alternative Antriebsformen, wie der Elektromobilität erzeugen neben technologischen Innovationen auch disruptive Veränderungen in den Wertschöpfungsketten der Automobilindustrie. Parallel entwickeln Automobilhersteller durch die globale und digitale Transformation neue Geschäftsmodelle, die nicht mehr ausschließlich auf das Produkt und nahestehende Dienstleistungen, wie Versicherungen oder Aftersales fokussiert sind. Es werden zusätzliche Leistungen, wie Mobilitätsangebote, umweltfreundlich erzeugter Strom, der Verkauf und die Installation von Wallboxen oder das Laden an eigenen Ladestationen angeboten. Diese Veränderungen führen dazu, dass existierende, überwiegend lineare Wertschöpfungsketten verändert und weiterentwickelt werden müssen. Bei der Transformation in die Elektromobilität ist die Bildung von Eco-Systemen unabdingbar, bei denen die Automobilhersteller mit anderen Unternehmen und Branchen ein Gesamtprodukt anbieten

Einführung

Die Automobilindustrie steht vor einem technologischen, wirtschaftlichen und gesellschaftlichen Wandel, welche auf mehrere einschneidenden Veränderungen beruhen: Erstens werden zunehmend fossil betriebene Fahrzeugantriebe durch Elektrofahrzeuge ersetzt. Diese Entwicklung wird primär durch politische Aktivitäten ausgelöst, um die Reduktion von Kohlendioxid (CO2) voranzutreiben, welches ein Haupttreiber für die globale Erwärmung ist. Obwohl die Emissionen der Treibhausgase in den letzten Jahren in Deutschland kontinuierlich gesunken ist, ist das Ziel der UN-Klimakonferenz von Paris, 2015 gefährdet, die menschengemachte globale Erderwärmung bis Ende des Jahrhunderts auf unter zwei Grad Celsius (möglichst 1.5 Grad Celsius) zu halten. Zusätzlich hat sich die Europäische Union in Ihrem Green Deal verpflichtet, bis 2050 klimaneutral zu werden und ab 2035 sollen nur noch lokal emissionsfreie Fahrzeuge neu zugelassen werden. Einen großen Anteil an den Treibhausgasen hat der Bereich Verkehr und Transport. So entfallen 19.85% der Treibhausgase in 2020 auf diesen Bereich [START_REF] Günther | umweltbundesamt[END_REF]. Neben den gesetzgeberischen Maßnahmen unterstützen die europäischen Staaten den Verkauf von elektrischen Fahrzeugen (BEV und Hybrid) mit hohen Subventionen. Zweitens wandeln sich die Anforderungen der Bevölkerung an die Mobilität hinsichtlich Vernetztheit und intermodale Transportmöglichkeiten, ohne den Wunsch auf individuelle Mobilität aufgeben zu müssen. Reisende möchten weiterhin die Flexibilität beibehalten, jedoch nicht auf Kosten der Umwelt oder sozialer Belange. Forciert wird dies durch Städte und Gemeinden, die versuchen, Fahrzeuge aus den Städten fernzuhalten. Geschwindigkeitsbeschränkungen und reduzierte Parkmöglichkeiten sollen die Bevölkerung zum Wechsel auf öffentliche Verkehrsmittel oder privaten Mobilitätsangebote motivieren. Drittens hat die Digitalisierung einen weiteren großen Einfluss auf die Automobilindustrie. Durch userfreundliche Anwendungen und Apps ist es einfach möglich, abhängig von Verkehrslage, Wetter und Verfügbarkeit von Verkehrsmitteln die individuelle Route zu planen und zu nutzen. Durch die ständige Erweiterung der Angebotspalette ist der Besitz des individuellen Fahrzeuges für den Einzelnen nicht mehr notwendig und kann Auswirkungen auf den Absatz der Automobilindustrie haben. [START_REF] Block | Zukunftsreport 2023[END_REF]. Dies ist besonders bemerkenswert, da die Automobilindustrie eines der wirtschaftsstärksten Branchen des Landes ist. Mit 2,2 Millionen Arbeitsplätzen sichert sie 7% der Arbeitsplätze in Deutschland und tragen ein Drittel der deutschen Investitionen in Forschung und Entwicklung bei [START_REF] Hagedorn | Automobile Wertschöpfung 2030/2050[END_REF].

Während Enthusiasten die hervorragenden Chancen für Kunden, Umwelt und Wirtschaft loben, sind die Automobilhersteller gezwungen, wichtige Entscheidungen zu treffen, die mittel-bis langfristig bindend sind und den Fortbestand des Unternehmens sichern oder gefährden können [START_REF] Clausen | Die erfolgreiche Transformation der Automobilbranche[END_REF] . Zusätzlich werden sie mit neuer Konkurrenz konfrontiert und der Verdrängungswettbewerb nimmt deutlich zu. Dazu gehören nicht nur junge Unternehmen und Startups, sondern insbesondere Technologieunternehmen, die über erhebliche finanzielle Ressourcen und hervorragende Kenntnisse in Software, Kundenakquise und Datenmanagement verfügen [START_REF] Tschiesner | Mastering new mobility[END_REF]. Infolgedessen kann das aktuelle fahrzeugzentrierte System durch ein radikal effizienteres, datengesteuertes und fahrerloses Ökosystem ersetzt werden, in dessen Mittelpunkt der Kunde steht. Benutzer können nahtlos zwischen öffentlichen, privaten, On-Demand-und planmäßigen Verkehrsmitteln wechseln, unterstützt durch dynamische Reiseinformationen [START_REF] Falck | Fahrzeugbau -wie verändert sich die Wertschöpfungskette?[END_REF].

Eines der größten Herausforderungen ist die Elektromobilität. Durch sie werden neue und einschneidende Anforderungen auf unterschiedlichen Ebenen an die Automobilindustrie gestellt. Neben der Veränderung des Antriebes beim Fahrzeug sind weitere Bereiche davon betroffen. So kann der weitere Ausbau und Nutzung von Mobilitätsdienstleistungen dazu führen, dass das Fahrzeug nur noch als reines Fortbewegungsmittel gesehen wird und nicht mehr als emotionalen Wertgegenstand mit Fahrfreude und sozialer Symbolik. Die Notwendigkeit von weiteren IT-Diensten führt dazu, dass zusätzliche Kompetenzen benötigt werden, die bisher bei den Automobilherstellern nicht notwendig waren und auch nicht vorhanden sind. Basierend auf diesen Herausforderungen ergibt sich eine hohe Transformationsleistung in der gesamten Automobilindustrie insbesondere in den Wertschöpfungsketten und Geschäftsmodellen. Diese sind aktuell auf Fahrzeuge mit Verbrennungsmotoren und deren Umfeldbedingungen optimiert und wurden über Jahrzehnte weiterentwickelt.

Die Frage ist, welche Anforderungen ergeben sich durch die Elektromobilität an die Weiterentwicklung der Wertschöpfungsketten? Sind es nur die Lieferketten, die durch einen veränderten Antrieb neu aufgestellt werden müssen oder hat die Elektromobilität einen weit höheren Einfluss auf das gesamte Umfeld der Automobilindustrie? Es ist anzunehmen, dass die Auswirkungen zu einer radikalen Umstrukturierung der Wertschöpfungsketten in der Automobilindustrie und im Mobilitätssektor führen werden, um einerseits die Kosten deutlich senken zu können, aber auch um neue Produkte kommerzialisieren zu können, um die Wünsche der Kunden erfüllen zu können. Dieser Wandel wird dazu führen, dass bestehende, überwiegend lineare Wertschöpfungsketten um zusätzliche Elemente erweitert werden müssen. Weitere Anforderungen des Umfeldes werden dazu führen, dass sich die Automobilindustrie zu einem Eco-System weiterentwickeln muss, welches auch zu Veränderungen der Geschäftsmodelle führen wird.

Elektromobilität

Elektromobilität, E-Mobilität oder Elektro-Mobilität ist eine Wortschöpfung, die in der Literatur nicht hinreichend definiert ist und mehrere Aussagen über ihre volle Bedeutung zulässt. Mehrheitlich wird sie im Sinne "elektrifiziert" und "Automobil" verwendet, wobei unterschiedliche Definitionen bzgl. der Fahrzeugkonzepte (batteriebetriebene Fahrzeuge (BEV, Hybride, die ihre Energie primär aus dem Stromnetz beziehen und extern nachgeladen werden können) zugelassen werden [START_REF] Reuter | Rohstoffe für innovative Fahrzeugtechnologien: Herausforderungen und Lösungsansätze[END_REF], [START_REF] Schönfelder | Elektromobilität[END_REF]. Wichtige Eigenschaften sind jedoch nicht nur der umweltfreundliche Betrieb, sondern auch eine umweltfreundliche Produktion incl. der Abbau der dazu benötigten Roh-und Hilfsstoffe. Zusätzlich sind die Möglichkeiten der europäischen Lieferkettengesetze auszuschöpfen, welches die Unternehmen verpflichtet, den Schutz der Menschenrechte zu wahren [START_REF] Kommission | DES RATES über die Sorgfaltspflichten von Unternehmen im Hinblick auf Nachhaltigkeit und zur Änderung der Richtlinie (EU[END_REF] Die Reduktion der Elektromobilität auf batteriebetriebene Fahrzeuge engt die zukünftigen Entwicklungen und Rahmenbedingungen der Automobil-und Mobilitätsindustrie jedoch zu sehr ein und würde deren Anforderungen an die Wertschöpfungsketten nur ungenügend darstellen. Deshalb ist eine Ausweitung auf die Elektromobilität als System sinnvoll und notwendig. Bonnema [START_REF] Bonnema | Electric mobility and charging: Systems of systems and infrastructure systems[END_REF] und das Projekt "Electric Mobility in Norway" [11, p. 4] Abbildung 2: Wertschöpfungskette nach M. Porter [START_REF] Porter | Creating and sustaining superior performance[END_REF] Diese Klassifizierung gilt jedoch nur für das Basismodell. Abhängig von der Branche und der Komplexität können weitere Wertschöpfungsketten vorhanden sein. Die Margin auf der rechten Seite des Grundmodells symbolisiert die Differenz zwischen dem Erlös und den eingesetzten Ressourcen bei der Herstellung des Produkts oder der Dienstleistung. Es vergleicht die angefallenen Kosten mit den Erlösen, die durch die geschaffenen Produkte generiert werden. Das Ziel eines Unternehmens ist die Maximierung der Wertschöpfung entlang der gesamten Wertschöpfungskette durch Kostensenkung und Umsatzsteigerung. Innerhalb einer Branche sind die Wertschöpfungsketten von Lieferanten und Kunden miteinander verknüpft und bilden ein gemeinsames Wertschöpfungskettensystem. Insbesondere in der Automobilbranche sind die Abhängigkeiten und die Komplexität der Wertschöpfungsketten sehr hoch und Störungen können zu hohen Einbußen in Umsatz und Produktion führen.

Zusammenfassung und Diskussion

Neue Gesetze und Kundenbedürfnisse zwingen die Automobilindustrie zu umwälzenden Veränderungen und Weiterentwicklungen. Neben den Wertschöpfungsketten hat die Elektromobilität auch erhebliche Auswirkungen auf die bestehenden Geschäftsmodelle. Diese stehen in einer engen Verbindung mit den Wertschöpfungsketten und sind in weiteren Forschungsaktivitäten zu untersuchen.

  Abbildung 1: Elemente der Elektromobilität als System[START_REF] Schnee | Towards a Conceptual Framework of Sustainable Business Models for Electromobility[END_REF]

3. 1 4 . 1 : 4 . 2

 14142 Abbildung 3: Zuliefererpyramide der Automobilindustrie[17, p. 8]

4. 5

 5 Digitalisierung Die fortschreitende Digitalisierung unterstützt die Elektromobilität in unterschiedlichen Bereichen und stellt neue Anforderungen an die Automobilindustrie. Digitale Technologien ermöglichen neue Mobilitätsstrukturen, bei denen das Fahrzeug zwar als Transportmittel noch benötigt wird, jedoch seine singuläre Stellung verlieren könnte. Es wird zum Bestandteil eines datenbasierten Verkehrsökosystems. Durch digitale Kommunikation ist es möglich, das geeignete Verkehrsmittel zu wählen und zu nutzen. Durch Echtzeit-Datenaustausch zwischen den Verkehrsteilnehmern und den Transportanbietern ist diese Verknüpfung möglich und ermöglicht einen geplanten sowie ungeplanten Transport. So können die Bahn, e-car-sharing und e-scooter für die letzte Meile für eine Reise genutzt werden. Aber auch weitere Dienstleistungen sind mit der Digitalisierung möglich. Das Auffinden und Bezahlen von Parkplätzen oder Ladestationen für Elektrofahrzeuge ist mit der Nutzung der geeigneten Apps userfreundlich möglich. Elektrofahrzeuge werden diese Anforderungen weiter erhöhen, da sich durch die Leistungselektronik der Batterien zusätzliche Information abbilden lassen. Diese werden die Automobilhersteller in die Kommunikation mit unterschiedlichen Devices aufnehmen und erfüllen, da sie ein wettbewerbsdifferenzierender Faktor, insbesondere bei jüngeren Kunden, darstellen können.

  Abbildung 7: Neues Ökosystem der Elektromobilität umfasst Akteure aus verschiedenen Bereichen

4.3 Forderung nach Umweltfreundlichkeit und sozialer Ausgeglichenheit

  Neben der Reduzierung des CO2 fokussiert die Elektromobilität auf weitere Maßnahmen in Bezug auf Umwelt und soziale Gleichberechtigung. Der Wunsch der Bevölkerung nach umweltfreundlichem Transport wird immer größer und der Staat unterstützt dieses Vorhaben durch strengere Gesetze und hohen Subventionen. So ist es den Käufern von Elektrofahrzeugen wichtig, dass die Rohstoffe, insbesondere Lithium, Blei und Kobalt für die Antriebsbatterie, auf eine möglichst umweltfreundliche Art und Weise abgebaut wird. Ebenso wird Elektrizität nicht nur für die Nutzung der Elektrofahrzeuge benötigt, sondern auch für den Abbau der Rohmaterialen, die Produktion und Wartung der Fahrzeuge sowie für das Recycling. Diese soll umweltfreundlich ohne fossile Brennstoffe erzeugt werden. Die Europäische Union untermauert diese Anforderungen mit einem Vorschlag für die Richtlinie zur nachhaltigen Unternehmensführung mit dem Ziel, dass negative Auswirkungen der Geschäftstätigkeiten von Unternehmen in Europa auf die Menschenrechte und die Umwelt in ihren Wertschöpfungsketten zu vermeiden sind. Diese Sorgfaltspflicht bezieht sich auf die vorgelagerten (umfasst alle Aktivitäten eines Unternehmens bei der Herstellung eines Produktes bzw. der Erbringung von Dienstleistungen) als auch nachgelagerten Wertschöpfungsketten (eigen Aktivitäten oder der Geschäftspartner in Bezug auf Vertrieb, Lagerung, Transport oder Entsorgung) und geht über den heuten Status Quo der Gesetzgebung weit hinaus[START_REF] Kommission | DES RATES über die Sorgfaltspflichten von Unternehmen im Hinblick auf Nachhaltigkeit und zur Änderung der Richtlinie (EU[END_REF].4.4 Nachfrage nach IntermodalitätDas Mobilitätsverhalten der Bevölkerung hat sich in den letzten Jahren stark verändert. Der Bedarf an komfortabler und nachhaltiger Mobilität wächst, und der Wunsch, ein Auto zu besitzen, ist bei jüngeren Menschen nicht mehr so stark ausgeprägt[START_REF] Johansson | Mobility as a Service: Exploring Young People's Mobility Demands and Travel Behavior[END_REF]. Je nach Entfernung, Anzahl der zu befördernden Personen, Verfügbarkeit der Verkehrsmittel, Preis und verfügbarer Zeit können verschiedene Verkehrsmittel genutzt werden. Das Angebot an Mobilitätsdienstleistungen wächst ständig, und herkömmliche Verkehrsmittel wie der öffentliche Nahverkehr werden durch innovative Transportmittel wie E-Scooter, Uber oder Ride-Sharing ergänzt. Die Forderung nach Intermodalität erfordert ein Umdenken in der Automobilindustrie und kann zu einer Erweiterung der Wertschöpfung führen. So startet Mercedes-Benz sehr früh sein Angebot für das Car-Sharing mit Car2Go und wechselte seine Verbrennerflotte später in BEVs aus. Auch andere Automobilhersteller gründeten eigene Unternehmen in den Mobilitätsbereichen, wie Moia von VW als Anbieter für Ridepooling[START_REF]MOIA: Sieben Fragen und Antworten zum Mobilitätsunternehmen[END_REF] oder die Park-App PayByPHone von Ford[START_REF] Pertschy | MSR: Daimler und BMW dominieren bei Mobilitätsdiensten[END_REF]. Der Markt für Intermodalität ist jedoch sehr volatil und stark abhängig von Gesetzgebung und Bedürfnissen der Kunden. So haben bereits mehrere Automobilhersteller ihre Aktivitäten in diesen Bereich wieder eingechränkt.

  Technologien, die seit Jahrzehnten die Basis für das Fahrzeug gebildet haben, sind nicht mehr zukunftsfähig und werden durch neue Antriebssysteme ersetzt. Aber nicht nur die Technologie hin zu alternativen Antrieben verändert sich, sondern auch die Kundenbedürfnisse sind nicht mehr auf den Besitz eines Fahrzeuges fokussiert. In Zeiten der Urbanisierung, langer Staus und fehlender Parkplätze ist es wichtiger, möglichst komfortabel seine tägliche Wegstrecke zur Arbeit oder für Reisen am Wochenende bewältigen zu können. Und dies sollte möglichst umweltfreundlich und mit sozial gerecht gefertigten und betriebenen Fortbewegungsmitteln stattfinden. Auch die zunehmende Digitalisierung stellte neue Anforderungen an die Automobilindustrie, die diese bisher outgesourced hatte und sich primär auf die Fahrzeugtechnologie und Emotionen konzentriert hat. Die Folgen sind erheblich und führen zu veränderten und neuen Ansätzen in allen Funktionen und Bereichen der Automobilindustrie. Während die technologischen Veränderungen bereits stattfinden und viele Automobilhersteller sich darauf eingestellt haben, gibt es bei den Wertschöpfungsketten und den Geschäftsmodellen noch erheblichen Nachholbedarf. Die Transformation von Verbrennerfahrzeuge auf Elektrofahrzeuge hat erhebliche Auswirkungen auf die Lieferketten. Neue Lieferanten müssen gesucht werden und die Verantwortung über Umwelt und soziale Behandlung sollen weit über die eigenen Lieferketten hinaus verpflichtend werden. Auch wenn das Gesetz diese Maßnahmen nur für Unternehmen mit mehr als 3.000 Mitarbeitern verpflichtend vorsieht, sind die nächsten Staffelungen für Unternehmen mit weniger Mitarbeiter bereits beschlossen. Während bisher das Fahrzeug im Fokus der Wertschöpfungsketten stand, welches um fahrzeugnahe Dienstleistungen, wie Finanzierung oder Aftersales ergänzt wurden, gehen die Anforderungen heute weit darüber hinaus. Mobilität bedeutet nicht mehr nur, mit dem eigenen Fahrzeug von A nach B zu fahren, sondern es geht um die Befriedigung der individuellen Transportwünsche, die möglichst umweltfreundlich, sozial gleichberechtigt, komfortabel, spontan und userfreundlich umgesetzt werden sollen. Der dadurch erweiterte Begriff der Elektromobilität geht über die heutigen Wertschöpfungsketten der Unternehmen weit hinaus. Primär lineare Wertschöpfungsketten müssen aufgesplittet werden und durch Kooperationen oder eigene Funktionen erweitert werden. Das führt dazu, dass neue Partner angebunden werden müssen, mit denen bisher keine oder nur geringe Kontakte bestanden haben. Auch die Digitalisierung führt dazu, dass neue Funktionen und Kompetenzen benötigt werden, die sich heute in den Wertschöpfungsketten nicht oder nur in geringem Maße widerspiegeln.


Aflah Elouneg

Jérôme Chambert

Arnaud Lejeune

Quentin Lucot

Emmanuelle Jacquet

Stéphane P A Bordas
email: stephane.bordas@uni.lu

Anisotropic mechanical characterization of human skin by in vivo multi-axial ring suction test

Keywords: Skin Mechanical Characterization, Multi-Axial Extension, Anisotropy, Langer's line, CutiScan ® CS

come

Anisotropic mechanical characterization of human skin by in vivo multi-axial ring suction test

Introduction

It is essential to study the mechanics of the skin since its behavior under different loads is linked to several skin developments and regeneration dysfunctions in both cellular and tissular scales, including wound healing and scar formation [START_REF] Corr | Biomechanics of scar tissue and uninjured skin[END_REF][START_REF] Buganza Tepole | Systems-based approaches toward wound healing[END_REF]. A thorough in vivo analysis of the mechanisms underlying these phenomena combined with clinical trial data may help develop novel therapeutic strategies. The skin, considered a soft tissue, is composed of three primary layers: the epidermis, the dermis, and the hypodermis. Each layer is made of several components. The methodology followed to characterize the mechanical properties of the skin lies in essential steps ordered as follows, measurement, modeling, and parameter identification. There are no general standards for measuring human skin's elasticity in vivo. The choice of methods and materials depends on the targeted material parameters and site accessibility. The epidermis, a superficial layer of the skin, has a thickness of approximately 75-150 µm [START_REF] Odland | Physiology, biochemistry and molecular biology of the skin[END_REF] and is composed principally of flattened cells that overlie a base sublayer consisting of columnar cells arranged perpendicularly. The dermis, which has a thickness of 0.87-2.65 mm (combined with the epidermis) [START_REF] Tan | Skin thickness measurement by pulsed ultrasound; its reproducibility, validation and variability[END_REF], is a dense connective tissue consisting of fibrous proteins, collagen, and elastin. Procedures used for in vivo testing of the skin include tensile, suction, indentation, and torsion. The mechanical behavior of the dermis, believed to dominate the mechanics of the skin, is dictated by elastin fibers at small strains and stress and by collagen fibers at large strains [START_REF] Oxlund | The role of elastin in the mechanical properties of skin[END_REF].

The third layer is the hypodermis, a fatty tissue composed mainly of adipose cells. Its thickness varies all over the body and functions as an insulator, constituting about 10% of the body mass [START_REF] Burton | Subcutaneous Fat: text book of dermatology[END_REF]. In in vivo extension tests, the skin is stretched parallel to its surface. The load can be uni-axial [START_REF] Sodeman | A direct method for the estimation of skin distensibility with its application to the study of vascular states[END_REF][START_REF] Wijn | Mechanical properties of the human skin in vivo for small deformations; a comparison of uniaxial strain and torsion measurements[END_REF][START_REF] Burlin | A method of in vivo measurement of the elastic properties of skin in radiotherapy patients[END_REF][START_REF] Manschot | The measurement and modelling of the mechanical properties of human skin in vivo-i. the measurement[END_REF][START_REF] Manschot | The measurement and modelling of the mechanical properties of human skin in vivo-ii. the model[END_REF][START_REF] Meijer | Characterisation of anisotropic and nonlinear behaviour of human skin in vivo[END_REF][START_REF] Khatyr | Model of the viscoelastic behaviour of skin in vivo and study of anisotropy[END_REF][START_REF] Jacquet | A new experimental method for measuring skin's natural tension[END_REF][START_REF] Evans | Measuring the mechanical properties of human skin in vivo using digital image correlation and finite element modelling[END_REF][START_REF] Flynn | Mechanical characterisation of in vivo human skin using a 3d force-sensitive micro-robot and finite element analysis[END_REF][START_REF] Boyer | Assessment of the in-plane biomechanical properties of human skin using a finite element model updating approach combined with an optical full-field measurement on a new tensile device[END_REF][START_REF] Jacquet | Ultra-light extensometer for the assessment of the mechanical properties of the human skin in vivo[END_REF][START_REF] Chambert | Multimodal investigation of a keloid scar by combining mechanical tests in vivo with diverse imaging techniques[END_REF], bi-axial [START_REF] Wa | Biaxial tension test of human skin in vivo[END_REF] or multi-axial [START_REF] Alexander | Accounting for natural tension in the mechanical testing of human skin[END_REF][START_REF] Reihsner | Two-dimensional elastic properties of human skin in terms of an incremental model at the in vivo configuration[END_REF][START_REF] Reihsner | On the orthogonal anisotropy of human skin as a function of anatomical region[END_REF][START_REF] Laiacona | Non-invasive in vivo quantification of human skin tension lines[END_REF]. This experiment quantifies the skin's stiffness and the elastic parameters along the stretched direction(s). Those parameters can also be assessed through other techniques, suction [START_REF] Grahame | The influence of ageing on the in vivo elasticity of human skin[END_REF][START_REF] Elsner | Mechanical function of the skin: state of the art[END_REF][START_REF] Diridollou | In vivo model of the mechanical properties of the human skin under suction[END_REF][START_REF] Hendriks | A numerical-experimental method to characterize the non-linear mechanical behaviour of human skin[END_REF][START_REF] Barel | Suction chamber method for measurement of skin mechanics: The new digital version of the cutometer[END_REF][START_REF] Delalleau | A nonlinear elastic behavior to identify the mechanical parameters of human skin in vivo[END_REF][START_REF] Müller | A novel ultra-light suction device for mechanical characterization of skin[END_REF], indentation [START_REF] Delalleau | A nonlinear elastic behavior to identify the mechanical parameters of human skin in vivo[END_REF][START_REF] Bader | Mechanical characteristics of skin and underlying tissues in vivo[END_REF][START_REF] Zheng | Effective elastic properties for lower limb soft tissues from manual indentation experiment[END_REF][START_REF] Khaothong | In vivo measurements of the mechanical properties of human skin and muscle by inverse finite element method combined with the indentation test[END_REF][START_REF] Iivarinen | Experimental and computational analysis of soft tissue stiffness in forearm using a manual indentation device[END_REF][START_REF] Zhou | In vivo measurement of the anisotropic mechanical properties of human skin by indentation test[END_REF], and torsion [START_REF] Wijn | Mechanical properties of the human skin in vivo for small deformations; a comparison of uniaxial strain and torsion measurements[END_REF][START_REF] Sanders | Torsional elasticity of human skin in vivo[END_REF][START_REF] Agache | Mechanical properties and young's modulus of human skin in vivo[END_REF][START_REF] Escoffier | Age-related mechanical properties of human skin: an in vivo study[END_REF]. The mechanical response of human skin undergoing large deformation exhibits a J-shaped stress-strain curve comprising at least two different regimes. Their respective tangents represent skin stiffness, commonly introduced as the elastic moduli. Their identified values in the earlier cited studies range from 1 kPa to 57 MPa [START_REF] Agache | Measuring the Skin[END_REF][START_REF] Kalra | Mechanical behaviour of skin: a review[END_REF][START_REF] Joodaki | Skin mechanical properties and modeling: A review[END_REF]. This high variability is due to the choice of the hypotheses involved in developing theoretical models to determine elastic properties. Every model was built to mimic one or many aspects, such as viscoelasticity, hyperelasticity, anisotropy, multi-layer scaling, etc. Using a complex model with all the biomechanical aspects is computationally and mathematically challenging [START_REF] Limbert | Mathematical and computational modelling of skin biophysics: a review[END_REF] and makes the model prone to overfitting.

In the light of recent studies about the mechanical characterization of bi-material keloid-skin [START_REF] Chambert | Multimodal investigation of a keloid scar by combining mechanical tests in vivo with diverse imaging techniques[END_REF][START_REF] Sutula | An open source pipeline for design of experiments for hyperelastic models of the skin with applications to keloids[END_REF][START_REF] Elouneg | An open-source fenics-based framework for hyperelastic parameter estimation from noisy full-field data: Application to heterogeneous soft tissues[END_REF], we address the anisotropic characteristics of the healthy skin surrounding the keloid scar. It has been asserted in [START_REF] Yannas | Principles of skin regeneration[END_REF][START_REF] Yannas | Tissue and Organ Regeneration in Adults: Extension of the Paradigm to Several Organs[END_REF] that during the skin wound healing process, a mechanical stretch field develops in the wound plane. Also, quantitative measurements by laser light scattering showed that the collagen fibers are oriented along the direction of the major contraction vector of the wound rather than quasirandomly oriented [START_REF] Ferdman | Scattering of light from histologic sections: a new method for the analysis of connective tissue[END_REF]. According to Hendriks [START_REF] Hendriks | Mechanical behaviour of human skin in vivo-a literature review[END_REF], the biomechanical properties determined from testing the whole skin are seemingly driven by the dermal collagen [START_REF] Edwards | Evaluation of biomechanical properties of human skin[END_REF] since those collagen fibers are the main constituent of the dermis (77% of the fat-free dry weight) running almost parallel with the skin surface [START_REF] Finlay | Scanning electron microscopy of the human dermis under uni-axial strain[END_REF]. The density distribution of collagen fibers shows a preferential direction of lower extensibility correlated with Langer's lines [START_REF] Langer | Zue anatomie und physiologie de haut. ueber der spaltbarkeit der cutis[END_REF][START_REF] Langer | On the anatomy and physiology of the skin: I. the cleavability of the cutis[END_REF] in numerous studies [START_REF] Ridge | The directional effects of skin: A bio-engineering study of skin with particular reference to Langer's lines[END_REF][START_REF] Gibson | Directional variation in extensibility of human skin in vivo[END_REF][START_REF] Finlay | Biodynamic studies of human skin: torsional characteristics in relation to structure[END_REF][START_REF] Markenscoff | On the stress-strain relation for skin[END_REF][START_REF] Annaidh | Characterization of the anisotropic mechanical properties of excised human skin[END_REF][START_REF] Ní Annaidh | Automated estimation of collagen fibre dispersion in the dermis and its contribution to the anisotropic behaviour of skin[END_REF]. Therefore, it is vital to identify, in vivo, the privileged direction of the collagen bundles in order to prevent the formation of undesired scars after incision [START_REF] Aarabi | Mechanical load initiates hypertrophic scar formation through decreased cellular apoptosis[END_REF][START_REF] Son | Overview of surgical scar prevention and management[END_REF].

The anisotropy of the skin has been characterized in vivo through different non-invasive methods in the last decades, with optical approaches [START_REF] Nickell | Anisotropy of light propagation in human skin[END_REF][START_REF] Sakai | In vivo evaluation of human skin anisotropy by polarization-sensitive optical coherence tomography[END_REF], elastography [START_REF] Gahagnon | Skin anisotropy in vivo and initial natural stress effect: A quantitative study using high-frequency static elastography[END_REF][START_REF] Kirby | Probing elastic anisotropy of human skin in vivo with light using non-contact acoustic micro-tapping oce and polarization sensitive oct[END_REF], suction tests [START_REF] Tonge | Full-field bulge test for planar anisotropic tissues: Part i -experimental methods applied to human skin tissue[END_REF][START_REF] Lakhani | Non-invasive in vivo quantification of directional dependent variation in mechanical properties for human skin[END_REF], and multi-axial tests [START_REF] Khatyr | Model of the viscoelastic behaviour of skin in vivo and study of anisotropy[END_REF][START_REF] Flynn | Mechanical characterisation of in vivo human skin using a 3d force-sensitive micro-robot and finite element analysis[END_REF][START_REF] Boyer | Assessment of the in-plane biomechanical properties of human skin using a finite element model updating approach combined with an optical full-field measurement on a new tensile device[END_REF][START_REF] Reihsner | Two-dimensional elastic properties of human skin in terms of an incremental model at the in vivo configuration[END_REF][START_REF] Reihsner | On the orthogonal anisotropy of human skin as a function of anatomical region[END_REF][START_REF] Gibson | Directional variation in extensibility of human skin in vivo[END_REF][START_REF] Stark | Directional variations in the extensibility of human skin[END_REF][START_REF] Kvistedal | Estimating material parameters of human skin in vivo[END_REF][START_REF] Verhaegen | A reliable, non-invasive measurement tool for anisotropy in normal skin and scar tissue[END_REF][START_REF] Rosicka | Skin anisotropy: Finding the optimal incision line for volar forearm in males and females[END_REF]. The latter method is in the form of multiple simple tensile tests carried out in many directions. This method seems to be the most reliable as the deformations are in-plane, corresponding to the alignment of collagen fibers parallel to the dermis layer. However, it has several drawbacks. Conducting multiple tests to obtain a load-extension curve in each direction is time-consuming. Conversely, reducing the number of angles with which the direction of the minimum extensibility can be defined will affect the accuracy [START_REF] Serup | Handbook of Non-Invasive Methods and the Skin[END_REF].

A novel technique was recently used to investigate the anisotropy of soft tissue in all directions simultaneously, continuously, and locally. By applying a suction load within an annular surface, the central zone is subjected to multi-axial stretch. The preferential direction of collagen fibers is precisely quantified from the resulting full-field displacements. For that purpose, Laiacona et al. [START_REF] Laiacona | Non-invasive in vivo quantification of human skin tension lines[END_REF] have developed a home-made device with negative pressure of 170 mbar and inner and outer diameters, respectively, 30 mm and 49 mm. A commercial device CutiScan ® CS 1001, utilized in [START_REF] Rosado | Cutiscan®-a new system of biomechanical evaluation of the skin in vivo-comparative study of use depending on the anatomical site[END_REF][START_REF] Rosado | About the in vivo quantitation of skin anisotropy[END_REF][START_REF] Silva | Texture analysis is a useful tool to characterize the cutaneous biomechanical profile: A análise de textura é uma ferramenta útil para caracterizar o perfil biomecânico cutâneo[END_REF][START_REF] Kim | Evaluation of anisotropic properties of striae distensae with regard to skin surface texture and viscoelasticity[END_REF][START_REF] Elouneg | Biomechanical characterization of earlobe keloid by ring suction test[END_REF][START_REF] Anthonissen | Can the CutiScan CS 100® measure anisotropy and viscoelasticity in scar tissue after mastectomy? a reliability and validity study[END_REF], has been introduced in the market by Courage + Khazaka electronic GmbH, with an inner diameter of 5 mm (respectively outer diameter of 14 mm), granting measurement of multi-axial stretch on a small region, such as the earlobe [START_REF] Elouneg | Biomechanical characterization of earlobe keloid by ring suction test[END_REF].

Although the CutiScan device is reliable and promising [START_REF] Anthonissen | Can the CutiScan CS 100® measure anisotropy and viscoelasticity in scar tissue after mastectomy? a reliability and validity study[END_REF], its original software has some limitations, as reported in [START_REF] Elouneg | Biomechanical characterization of earlobe keloid by ring suction test[END_REF]. Indeed, the angle of the minimum displacement (equivalent to Langer's line) is given in the associated software, but access to the data of the displacement full-field is not provided. Furthermore, the anisotropy axis is hard to estimate as the displacement field shows a symmetry axis barely. And the displacement values given in pixels are multiplied by an unknown factor, which makes it unable to convert the displacements into millimeters to assess the strains. Thus, we had the motivation to adapt the Device's output treatment according to our research context and popularize it through this article.

In this work, we present an open-source methodology to identify the privileged direction of collagen fibers in the skin from a multi-axial ring suction test performed by the CutiScan ® CS 100 probe. The displacement fields are computed via the DIC technique starting from a video file recorded by the commercial device and divided into frames. From a field matching the quasi-static configuration, displacement data are gathered on a specific circle and fitted with an analytical orthotropic linear model. The anisotropy direction and the elastic material parameters are identified through an optimization method. We present the application of this numerical pipeline2 on a latex material and the forearm of a 28-years-old Caucasian volunteer, whose public dataset 3 was described in the data paper [START_REF] Elouneg | In vivo skin anisotropy dataset from annular suction test[END_REF]. The reader can refer to the repository https://figshare.com/s/0f3612238200cc98fb27 [START_REF] Elouneg | Anisotropic mechanical characterization of human skin by in vivo multi-axial ring suction test[END_REF] to reproduce the results of the paper.

Material and methods

Experiments

Multi-axial ring suction test

The in vivo experimental design used to characterize the anisotropic behavior of human skin in the present study is built around the commercial apparatus CutiScan ® CS 100 (referenced CutiScan ® thereafter), developed by Courage + Khazaka electronic GmbH. It comprises two main parts (Fig. 1): a suction probe and a central device. The probe is equipped with a black-and-white CCD camera (charge-coupled device), including UV light (Fig. 2a). It applies a constant negative pressure 𝑝 within an annular ring area (inner diameter 5 mm and outer diameter 14 mm) to draw a uniformly soft layer for a predefined phase 𝑡 creep (Fig. 2b). A double-sided adhesive prevents sliding between the probe and the skin surface. Then, the pressure is completely and instantaneously released during another phase 𝑡 relax . For the whole creep-relaxation cycle, the in-plane displacement of the material studied is recorded with a µEye ® camera inside a zone of interest (ZOI) of 5 mm diameter. The pre-cycle can be skipped if unnecessary deformations are not desired (soft materials may take longer to recover their initial configuration due to viscous effects). However, we recommend maintaining the pre-cycle to regulate brightness inside the probe. To minimize unnecessary tensions, we set 𝑡 creep = 1 s and 𝑡 relax = 1 s in the pre-cycle, then 𝑡 standby = 10 s. Whilst in the principal cycle, 𝑡 creep = 3 s, and 𝑡 relax = 3 s.

Displacement field measurement

A closed-source framework is provided with the device to convert the videos into steady displacement plots for every 360 angle through the DIC. However, the exact position of the point where the displacement-time curve is built is unknown, and the user has no access to the full-field data. To overcome this limitation, an open-source Python-based program, PyDIC [START_REF] André | PyDIC, a python suite for local digital image correlation[END_REF], has been adapted (https://github.com/aflahelouneg/MARSAC/blob/main/pydic.py) to split the recorded video extracted from the commercial software into frames and compute the displacement vectors at the mapped nodes. The Lucas-Kanade method was employed to track motions [START_REF] Lucas | An iterative image registration technique with an application to stereo vision[END_REF]. The DIC technique in this framework is implemented based on OpenCV (Computer Vision), a graphic open-source Python library [START_REF] Solem | Programming Computer Vision with Python: Tools and Algorithms for Analyzing Images[END_REF]. On the basis of a calibration study, the correlation parameters, grid resolution, and correlation window size were fixed to be 20 pixels and 40 pixels, respectively. The coordinate grid is mapped with 49 × 49 nodes (Fig. 3).

Reference material

The validation of the method was carried out on the basis of a calibration study, where we checked the presented method's ability to characterize the soft materials reliably. Thus, a thin homogeneous layer of rubber latex of 220-µm thickness was subjected a the multi-axial ring suction test at constant pressure 𝑝 = 300 mbar. From the recorded video of 6 s, 81 frames were extracted and converted into displacement fields. We retained only the frame corresponding to the quasi-static configuration at the end of the creep phase (around 3 s). The output displacement field was expected to be purely radial. In a parallel experiment, by means of the uni-axial tensile test, 3 rectangular samples of the same latex tissue were cut out along the directions 0 • , 45 • , and 90 • . By calculating the initial tangent of the stress-stretch curve, one may obtain the elastic modulus 𝐸 latex = 2.7 ± 0.42 MPa. The results from the multi-axial and uni-axial experiments were combined to calibrate a linear orthotropic elastic model in Section 3.2.1.

Subject forearm test

30 series of multi-axial ring suction tests were conducted on the volar left forearm of a 28-years-old Caucasian male (Fig. 5), following the principles outlined in the Helsinki Declaration of 1975, as revised in 2000. The site has been chosen for its flatness and easy access. The measured area is located at a distance of 15 cm from the wrist. The epidermis-dermis layer has a thickness of 𝑡ℎ h = 1.47 mm and was quantified by ultrasound echography. A single series consists of 21 cycles with pressure set-point values varying from 100 mbar to 500 mbar with a step of 20 mbar. A delay of 2 minutes was arbitrarily set between each complete suction cycle by considering that a weak load has a non-significant impact on skin conditioning. It is a common practice in dermatology that a waiting period between successive loadings is brief. In Takema et al.'s study [START_REF] Takema | Age-related changes in the elastic properties and thickness of human facial skin[END_REF], the relaxation period was 2 seconds for 3 pressure levels, 200, 300, and 400 mbar, with a Cutometer device. In addition, the CutiScan ® applies a pre-load cycle of 10 s before each recordable load cycle, making the influence of the delay period less significant.

The video files recorded with CutiScan ® software were stored as primary data in a public repository https://search-data.ubfc.fr/femto/FR-18008901306731-2021-08-25_In-vivo-skin-a nisotropy-dataset-for-a-young-man.html, and from which displacement fields were computed as secondary data using our numerical DIC tool. The latter is available in the same repository, as well as Python codes used to generate them (for more technical details about the experiment, see data paper [START_REF] Elouneg | In vivo skin anisotropy dataset from annular suction test[END_REF]). As the present study focuses on the characterization of the anisotropy in healthy skin independently of viscoelastic effects, a data pre-selection was done. The displacement field resulting from the frame captured at the quasi-static state was selected for a pressure 𝑝 = 300 mbar. According to the observations, the results are more reproducible at that pressure level. The mean displacement over 30 tests is then computed at every node of the DIC coordinate grid. As a result, the mean-displacement field will be used as one set of data for the primary analysis.

Modeling

Problem description

The annular suction test can be modeled as a simplified 2D problem with Neumann boundary conditions characterized by radial traction force 𝒕 applied to the external frontier and locking conditions (or zero displacements) at the center of the domain to avoid rigid body displacements (Fig. 6a). The Cartesian basis {𝒆 1 , 𝒆 2 } (with Cartesian coordinates 𝑥 1 and 𝑥 2 ) on which the model is built is not to be confused with the basis {𝒆 ′ 1 , 𝒆 ′ 2 } used for the experimental displacement fields. They differ by an angle 𝜙 = (𝒆 ′ 1 , 𝒆 1 ), representing the orientation of the Langer's line (Fig. 7).

We assume in our approach that:

• The considered problem can be treated by plane stress conditions in the (𝑥 1 , 𝑥 2 ) plane.

• The friction effects between the skin and CutiScan ® probe are not considered (Fig. 2b).

• A uniform in-plane radial stress 𝜎 𝑟 taking place over all the observable zone is linearly correlated with the supplied negative pressure.

• The skin within the observable zone is subjected to small strains and behaves as a linear orthotropic material. Consider the deformable body occupying a disk corresponding to the observable zone defined by Cuti-Scan ® (Fig. 6a). Knowing that the axes (𝑥 Similarly, as in [START_REF] Doghri | Mechanics of deformable solids: linear, nonlinear, analytical and computational aspects[END_REF], the local formulation of the static problem consists in finding the tensor fields of displacements 𝒖, strains 𝜺 and stresses 𝝈 which satisfy the following equations:

Radial traction 𝒕 force

Prescribed displacement 𝒖(0, 0) = {0, 0} T 𝒆 1 𝒆 2 (a) Global problem 𝜕Ω 𝐷 𝜕Ω 𝐷 Ω 𝜕Ω 𝑁 𝒆 1 𝒆 2 (b) Reduced problem
∇ • 𝝈 = 0 in Ω 𝝈 • 𝒏 = 𝜎 𝑟 𝒏 on 𝜕Ω N 𝑢 1 (0, 𝑥 2 ) = 0 and 𝑢 2 (𝑥 1 , 0) = 0 on 𝜕Ω D 𝜺 = 1 2 ∇𝒖 + ∇ T 𝒖 𝝈 = C : 𝜺 (1)
In the absence of body forces, Eq. (1a) stands for the equilibrium equations within the deformable domain Ω. Eq. (1b) represents the Neumann boundary conditions with 𝒏 the normal vector to the external boundary domain 𝜕Ω N . 𝜎 𝑟 is the radial stress resulted from the applied 𝒕. Eq. (1c) describes the Dirichlet boundary conditions. Eq. (1d) represents the infinitesimal strain-displacement relations.

Eq. (1e) constitutes the stress-strain equations where C is the fourth-order tensor, also called the elasticity operator. In matrix form, Eq. (1e) for an orthotropic linear elastic material under plane-stress conditions [START_REF] Jones | Mechanics of composite materials[END_REF] is given by:

         𝜎 11 𝜎 22 𝜎 12          = 1 1 -𝜈 12 𝜈 21        𝐸 1 𝜈 21 𝐸 1 0 𝜈 12 𝐸 2 𝐸 2 0 0 0 (1 -𝜈 12 𝜈 21 )𝐺 12                 𝜀 11 𝜀 22 2𝜀 12          (2)
where 𝜎 11 , 𝜎 22 and 𝜎 12 (respectively, 𝜀 11 , 𝜀 22 and 𝜀 12 ) are the components of the stress tensor 𝝈 (respectively, the strain tensor 𝜺). 𝐸 1 and 𝐸 2 are the Young's moduli, 𝐺 12 is the shear modulus, and 𝜈 12 and 𝜈 21 are the Poisson's ratios. In the observable zone, the in-plane radial stress is presupposed to be uniform, as 𝜎 11 = 𝜎 22 = 𝜎 𝑟 .

For symmetry conditions on stiffness and compliance matrices, we get:

𝜈 12 𝐸 1 = 𝜈 21 𝐸 2 (3)
Consequently, only 4 material parameters are independent: 𝐸 1 , 𝐸 2 , 𝐺 12 , and 𝜈 12 .

Analytical model

As Fig. 7 shows, the initial circular domain transforms into an elliptical deformed shape due to multiaxial loading and orthotropic behavior. Accordingly, we assume the following analytical form for the Cartesian displacement field:

𝑢 1 (𝑟, 𝜃) = 𝐴 1 𝑟 cos 𝜃 𝑢 2 (𝑟, 𝜃) = 𝐴 2 𝑟 sin 𝜃 ; 𝑟 ∈ [0, 𝑅] ; 𝜃 ∈ [0, 360 • ] , (4)
where 𝑟 and 𝜃 are the cylindrical coordinates defined by 𝑟 = ∥ ---→ 𝑂 𝑀 ∥ and 𝜃 = (𝒆 1 , ---→ 𝑂 𝑀), in which ---→ 𝑂 𝑀 is the position vector of a material point in the Cartesian system {𝑂; 𝒆 1 , 𝒆 2 }. The radius 𝑅 corresponds to the frontier of the observable zone. 𝐴 1 and 𝐴 2 are constants.

From the infinitesimal strain-displacement relations (1d) and the constitutive equations of orthotropic linear elasticity (2), we obtain uniform strain and stress fields. Consequently, the equilibrium equations (1a) are always satisfied. Using Neumann boundary conditions (1b) and after computations, we express 𝐴 1 and 𝐴 2 as:

           𝐴 1 = 𝜎 𝑟 (1 -𝜈 12 ) 𝐸 1 𝐴 2 = 𝜎 𝑟 (1 -𝜈 21 ) 𝐸 2 (5)
Finally, the analytical solution for the Cartesian displacement field in terms of Cartesian coordinates 𝑥 1 and 𝑥 2 is given by:

           𝑢 1 (𝑥 1 ) = 𝜎 𝑟 (1 -𝜈 12 ) 𝐸 1 𝑥 1 𝑢 2 (𝑥 2 ) = 𝜎 𝑟 (1 -𝜈 21 ) 𝐸 2 𝑥 2 (6)

FEM model

Subjected to the Dirichlet and Neumann conditions (Fig. 6), the problem can be easily written in a weak form over skin domain Ω (observable disk-zone) with boundary 𝜕Ω, and solved with Finite Element Method (FEM) by using FEniCS software/framework [START_REF] Logg | Automated solution of differential equations by the finite element method[END_REF]. In the absence of body force, its weak formulation writes:

Find 𝒖 ∈ U such that ∀𝒗 ∈ V 0 , ∫ Ω 𝝈(𝒖) : ∇ sym 𝒗 dΩ = ∫ 𝜕Ω ( 𝑝𝒏) • 𝒗 dΓ (7)
Assuming U, a space of sufficiently smooth functions and V 0 , its counterpart satisfying homogeneous boundary condition on the Dirichlet boundary 𝜕Ω D . The domain Ω, taking the shape of a quarter of a disk, is meshed into quadratic triangular finite elements, and the size of each element is approximately equal to 50 µm (55174 degrees of freedom).

Methodology for material parameters characterization

In the case of 2D orthotropic linear elastic material under plane-stress conditions, we present in this section a determination methodology for the material parameters of the skin by using CutiScan ® apparatus: Poisson's ratio 𝜈 12 , shear modulus 𝐺 12 , and Young's moduli 𝐸 1 & 𝐸 2 .

Determination of Poisson's ratio and shear modulus

CutiScan ® apparatus imposes a hydrostatic stress state with in-plane pressure 𝑝 set at the boundary of the observable zone. This loading implies a multi-axial stretch that does not allow to identify the Poisson's ratio 𝜈 12 (or 𝜈 21 ) for skin orthotropic material. Indeed, Ting and Chen [START_REF] Ting | Poisson's ratio for anisotropic elastic materials can have no bounds[END_REF] have shown that Poisson's ratio for anisotropic materials can have no theoretical bounds. Though, it had been fixed arbitrarily for human skin between 0.3 and 0.5 because of its rubber-like mechanical properties [START_REF] Khatyr | Model of the viscoelastic behaviour of skin in vivo and study of anisotropy[END_REF][START_REF] Agache | Measuring the Skin[END_REF][START_REF] Pailler-Mattei | In vivo measurements of the elastic mechanical properties of human skin by indentation tests[END_REF][START_REF] Levy | The biomechanical efficacy of dressings in preventing heel ulcers[END_REF]. Destrade et al. [START_REF] Destrade | Slight compressibility and sensitivity to changes in poisson's ratio[END_REF] pointed out that the usual practice of arbitrarily choosing a value of Poisson's ratio when numerically modeling rubbers and soft tissue will, almost certainly, lead to a significant difference between the simulated and actual normal stresses in a sheared block. Moreover, the collagen fiber orientation affects Poisson's ratio variation [START_REF] Dwivedi | Effect of collagen fibre orientation on the poisson's ratio and stress relaxation of skin: an ex vivo and in vivo study[END_REF]. The modeler would need a robust hypothesis to associate the classical Poisson's ratio with the parameter 𝜈 12 . Hence, based on an multi-axial study conducted by Reihsner et al. [START_REF] Reihsner | Two-dimensional elastic properties of human skin in terms of an incremental model at the in vivo configuration[END_REF], we define: 𝜈 12 = 𝐶 12 /𝐶 22 = 0.43 ± 0.09 with 𝐶 12 = 𝜈 12 𝐸 2 1-𝜈 12 𝜈 21 = 4.0 ± 0.6 Nm/g and 𝐶 22 = 𝐸 2 1-𝜈 12 𝜈 21 = 9.4 ± 1.4 Nm/g are elastic coefficient determined by the experiment on 16 sites of a 30-years-old subject skin [START_REF] Reihsner | Two-dimensional elastic properties of human skin in terms of an incremental model at the in vivo configuration[END_REF]. Considering that no shear strain and no shear stress are activated by CutiScan ® loading within the skin observable zone, the shear modulus 𝐺 12 can be arbitrarily chosen: thereafter, 𝐺 12 = 1 MPa.

Identification of geometrical parameters and Young's moduli

The strategy adopted to identify the material parameters consists of two steps.

First, by means of the non-linear Newton-Raphson method, the geometrical parameters 𝒎 𝑔 = {𝑎, 𝑏, 𝜙} are calibrated from the elliptic deformation configuration associated with a circle with radius 𝑟 in the undeformed state (Fig. 7).

x1 (𝑟, 𝜃) = 𝑎(𝑟) cos 𝜃 x2 (𝑟, 𝜃) = 𝑏(𝑟) sin 𝜃 ; 𝑟 ∈ [0, 𝑅] ; 𝜃 ∈ [0, 360 • ] (8)
Second, the material parameters 𝒎 𝑚 = {𝐸 1 , 𝐸 2 } are derived from the principal minor and major semi-axes of the ellipse, respectively, 𝑎 and 𝑏, via the model as

           𝑎(𝑟) = 𝑟 + 𝑢 1 (𝑟) = 𝑟 1 + 𝜎 𝑟 (1 -𝜈 12 ) 𝐸1 𝑏(𝑟) = 𝑟 + 𝑢 2 (𝑟) = 𝑟 1 + 𝜎 𝑟 (1 -𝜈 12 𝐸 2 /𝐸 1 ) 𝐸2 (9)
The material parameters can be assessed from Eqs. ( 6) and ( 9) given 𝑟 as

           𝐸 1 (𝑟) = 𝜎 𝑟 (1 -𝜈 12 )𝑟 𝑎 -𝑟 𝐸 2 (𝑟) = 𝜎 𝑟 (1 -𝜈 12 )𝑟 𝑏(1 -𝜈 12 ) + 𝜈 12 𝑎 -𝑟 (10)
The inverse identification of the geometrical parameter set 𝒎 𝑔 is processed according to a least-squares non-linear optimization method. By a similar method as in [START_REF] Zhou | In vivo measurement of the anisotropic mechanical properties of human skin by indentation test[END_REF], the error function on one point {𝑥 1 ′ 𝑘 , 𝑥 2 ′ 𝑘 } T is the mismatch of ellipses radii between the model -in the rotating system -and the data from [START_REF] Elouneg | In vivo skin anisotropy dataset from annular suction test[END_REF]. The optimization problem considers the rigid body motion, which is assessed by identifying the coordinates 𝑥 10 and 𝑥 20 of the center after deformation. Hence, the total parameter set to be estimated is m𝑔 = 𝒎 𝑔 ∪ {𝑥 10 , 𝑥 20 }. The objective function reads 12)) result from the rotation of {𝒆 1 , 𝒆 2 } with respect to {𝒆 ′ 1 , 𝒆 ′ 2 } by the angle 𝜙 (Fig. 7).

𝐽 𝑘 (𝑟, 𝜃 𝑘 , m𝑔 ) = 1 2 √ xR 1 (𝑟,
Since the displacement vectors are independent of each other, the total error function for each ellipse 𝐽 (𝑟) is quantified for all 𝑁 𝑘 chosen nodes as

𝐽 (𝑟, m𝑔 ) = 𝑁 𝑘 ∑ 𝑘=1 𝐽 𝑘 (𝑟, 𝜃 𝑘 , m𝑔 ) (13)
Starting from an initial guess m (0) 𝑔 , we compute iteratively the change on the parameters 𝛿 m𝑔 by way of the Newton-Raphson method (see Eq. ( 14)) until reaching the absolute convergence criteria of 10 -9 on every parameter. The detailed expression of the Gradient vector 𝜕𝐽 𝜕 m𝑔 and the symmetrical Hessian matrix 𝜕 2 𝐽 𝜕 m2 𝑔 can be found in the framework code https://github.com/aflahelouneg/MARSAC/b lob/main/newton.py.

m (𝑛+1) 𝑔 = m (𝑛) 𝑔 - 𝜕 2 𝐽 𝜕 m2 𝑔 -1 • 𝜕𝐽 𝜕 m𝑔 (14)

Statistical analysis

The interquartile range method was used in this study to discard the outlier values among the results of 30 tests. Therefore, 12 values were removed for 𝐸 1 , 4 values for 𝐸 2 , and 3 values for 𝜙. We denote Ê1 , Ê2 , and φ, the average values over the tests of 𝐸 1 , 𝐸 2 , and 𝜙, respectively. The global uncertainty of Ê 𝑗 ( 𝑗 = {1, 2}) takes into account the day-to-day variation of mechanical response and Δ 𝜈 12 (uncertainty of 𝜈 12 ). They were computed according to the weighted standard deviation method.

Ē 𝑗 = 𝑛 𝑗 𝑖 𝑤 𝑖 𝐸 (𝑖) 𝑗 𝑛 𝑗 𝑖 𝑤 𝑖 Δ𝐸 𝑗 = 𝑛 𝑗 𝑖 𝑤 𝑖 (𝐸 (𝑖) 𝑗 -Ē 𝑗 ) 2 (𝑛 𝑗 -1) 𝑛 𝑗 𝑖 𝑤 𝑖 𝑛 𝑗 (15)
with 𝑛 𝑗 the number of admissible values within the interquartile range, and the weights

𝑤 𝑖 = 1/Δ (𝑖) 𝐸 𝑗 2 , such that Δ (𝑖)
𝐸 𝑗 is the uncertainty of 𝐸 (𝑖) 𝑗 for each test 𝑖. Therefore, Ê 𝑗 = Ē 𝑗 ± Δ𝐸 𝑗 3 Results

FEM versus analytical model

The solution of the linear elastic problem defined by Eq. ( 1) was computed by FEM, and the analytical model with ranges of material parameter set coming from skin's literature ( [8, 10-14, 17, 19, 41]). For different orders of 𝐸 1 -magnitude, 𝐸 1 = {0.01, 0.1, 1, 10} MPa, we set the following ranges: 𝐸 1 /𝐸 2 = {1, 2, . . . , 19, 20}, 𝜈 12 = {0, 0.05, . . . , 0.90, 0.95} and 𝐺 12 = 1 MPa. Thus, (4𝐸 1 ) × (20𝐸 2 ) × (20𝜈 12 ) = 1600 FE simulations were performed. The minimum and maximum absolute errors between FEM and analytical solutions evaluated on nodes are 2.81 × 10 -6 and 3.0 × 10 -15 , respectively. For the arbitrary choice of parameter set (𝐸 1 = 10 kPa, 𝐸 2 = 0.5 kPa, 𝜈 12 = 0.5, 𝐺 12 = 1 MPa), the displacement vector fields for analytical (𝒖) and FEM (𝒖 FEM ) models have been carried out along circles 𝑟 = {0.5, 1.0, 1.5, 2.0, 2.5} mm in Fig. 8. A full-field displacement map obtained by FEM is presented in Fig. 9 for the same parameter set.

Material characterization

Calibration study

The presented method was validated based on two significant features: radial displacement and stiffness. As the radial in-plane stress 𝜎 𝑟 is uniform, the anisotropy is due only to the directional variation of the stiffness. Thus, using an isotropic latex material for validation of an orthotropic model (Eq. ( 6)) is plausible for the condition 𝐸 1 = 𝐸 2 . The data acquired from the experiments are described in Section 2.1.3.

Firstly, we check that the deformation of the reference material captured by the camera is radially regular. In Fig. 10, we plotted the mechanical response of the latex material in the quasi-static state. It consists of a displacement field captured by DIC and the deformation of the circle with 1 mm radius in the undeformed state. By using the linear interpolation all over the grid (provided by SciPy library), the values of the radial component of the displacements (𝑢 𝑟 ) were computed and displayed in Fig. 11 as a form of contour lines, as well as along the radial lines with the respective orientations {0 • ; 45 • ; 90 • ; 135 • }.

Starting from an initial guess corresponding to an anisotropic situation m (0) 𝑔 = {𝑎 = 1.0 mm; 𝑏 = 1.1 mm; 𝜙 = 0 • ; 𝑥 10 = 0 mm; 𝑥 20 = 0 mm} and by choosing 𝑁 𝑘 = 100, the inverse identification performed on each test data over the circle (𝑟 = 1 mm) results in the following converging parameter set Consequently, the ratio 𝑢 1 /𝑢 2 used to quantify the isotropicity of the latex material over the 3 tests is 0.9875 ± 0.0097, whereas the uncertainty on the captured displacement was evaluated to be Δ exp = 2.7 µm.

(a) 𝑢 1 (𝑥 1 -component) (b) 𝑢 2 (𝑥 2 -component) (c) 𝑢 𝑟 (radial component) (d) 𝑢 𝜃 (ortho-radial component)
Secondly, we investigate the correlation between the applied pressure on the annular section and the radial stress in the observable zone. This study was performed on the basis of a frictionless contact-based 2D axisymmetric simulation model built in COMSOL Multiphysics ® software (Fig. 13a). The details about the mesh and the boundary conditions are reported in Appendix A.1. Before exploiting the model to establish a link between 𝑝 and σ𝑟 , we validate it using the data of the first multi-axial test conducted on the latex material, whose elastic modulus is 𝐸 latex = 2.7 ± 0.42 MPa. σ𝑟 represents the radial stress in the 2D axisymmetric FE model, while 𝜎 𝑟 is associated with the analytical plane model (Eq. ( 6)). A corrective coefficient 𝜂 is used to guarantee that the elastic moduli in both models are equivalent. Thus, 𝜎 𝑟 = 𝜂 σ𝑟 . To simulate incompressibility, the Poisson's ratio was set to 𝜈 latex = 0.495. Once the rigid body motion was removed, the displacement of the latex was confronted against the simulation in Fig. 13b.

The simulation was run for different values of pressure 𝑝 ∈ {50, 100, 150, 200, 250, 300} mbar, where σ𝑟 was evaluated on the top surface, for each step. Fig. 14 exhibits a linear correlation between 𝑝 and σ𝑟 with a correlation coefficient 𝑅 cor = 0.995 and a slope of 𝜉 = 633. Consequently, we can establish the function 𝜎 𝑟 ( 𝑝) = 𝜂 𝑝/𝜉 to assess the radial stress for any applied pressure between 0 and 300 mbar at least. Given the stress value σ𝑟 = 0.47 MPa (corresponding to 𝑝 = 300 mbar), the measured displacements along one principal axis 𝑎 -𝑟 = 71 µm, and 𝜂 = 0.81, the elastic modulus was evaluated via Eq. ( 10) as 𝐸 latex = 2.7 MPa.

Application to human forearm

Analogically, the application of our methodology to the data from a human forearm introduction in Section 2.1.4 is presented in this part. The mechanical response consists of displacement vectors (Fig. 15) and their relative radial component evaluated on circles and radial cutlines (Fig. 16). The parameter identification was performed on the 1 mm radius circle to avoid edge effects. As an example, the best-fit solution for the data set of the test 1 is shown in Fig. 17, for a converging parameter, set m (1) 𝑔 = {𝑎 = 1.0029 mm; 𝑏 = 1.0632 mm; 𝜙 = 60.24 • ; 𝑥 10 = -17.8 µm; 𝑥 20 = 14.9 µm} with 𝑅 cor = 0.9849. The latter was juxtaposed with a simulation-based study carried out for the thickness 𝑡ℎ h = 1.47 mm and for different elastic modulus values to calibrate the coefficients 𝜉 = 24300 and 𝜂 = 1.03. By applying Eq. ( 10) (where 𝜎 𝑟 = 0.0123 MPa), Young's moduli along and across Langer's line are 𝐸 (1) 1 = 2.42 ± 0.38 MPa and 𝐸 (1) 2 = 0.188 ± 0.002 MPa, respectively. Skin material is known to exhibit a non-linear stress-strain curve; therefore, Eq. ( 6) can be applied only under a certain restriction: the correspondence between the incremental stress 𝑑𝜎 and strain 𝑑𝜀 is described by an elastic constant [START_REF] Reihsner | Two-dimensional elastic properties of human skin in terms of an incremental model at the in vivo configuration[END_REF]. In other words, the applied stresses and their respective strains must have linear dependence, which can occur within the first and second linear regimes. However, in a non-unidirectional stretch, the slope 𝑑𝜎/𝑑𝜀 = 𝑐𝑡𝑒 does not represent the real value of the elastic moduli 𝐸 1 and 𝐸 2 but ensures their constancy. In our case, the stress-strain curve for pressures between 100 and 500 mbar revealed that the elastic behavior is associated with the second linear regime, where the collagen fiber bundles are no longer crimped.

The identification results of m𝑔 for all tests are listed in Table 1. Once outliers had been discarded based on the interquartile range [START_REF] Dekking | A Modern Introduction to Probability and Statistics: Understanding why and how[END_REF], the resulting material parameters were similarly calculated (Fig. 18). Subsequently, φ = 40.9 ± 8.2 • . Using the formula (15), we obtained Ê1 = 1.35 ± 0.65 MPa, and Ê2 = 0.43 ± 0.07 MPa. Through Eq., we obtained Δ𝐸 1 = 0.86 MPa and Δ𝐸 2 = 0.03 MPa.

Discussion

The solutions of the static problem (1) given the boundary conditions on 𝜕Ω 𝐷 and 𝜕Ω 𝑁 were calculated numerically and analytically for numerous sets of material parameters (Section 3.1). Fig. 8 shows that the analytical solutions fit the FE data for a given parameter set. Similar results were obtained for the other parameter sets. Therefore, the proposed analytical model [START_REF] Burton | Subcutaneous Fat: text book of dermatology[END_REF] has been validated in an unclassical approach. One of its relevant benefits is its application as a forward solver in stochastic optimization algorithms requiring thousands of computations, such as the Bayesian calibration [START_REF] Zeraatpisheh | Bayesian model uncertainty quantification for hyperelastic soft tissue models[END_REF][START_REF] Han | Bayesian calibration of a computational model of tissue expansion based on a porcine animal model[END_REF].

In Section 3.2.1, the model parameters of two soft materials were identified using an inverse solver using the Newton-Raphson algorithm. A sheet of homogeneous latex material was analyzed in the context of a calibration study to verify the developed framework's capability to accurately characterize the mechanical behavior of soft tissue with data issued from the CutiScan ® apparatus. Figs. 11a and11b demonstrate the material's isotropy and linear elasticity spatial linearity behaviors but within a limited area. Indeed, the displacement data captured between the circles of 𝑟 = 1.3 mm and 𝑟 = 2 mm (zone 𝑍 𝑎 ) are highly perturbed. While between 𝑟 = 2 mm and 𝑟 = 2.5 mm (zone 𝑍 𝑏 ), the displacements decrease drastically until they reach 0 mm. This edge effect occurs because the material slips under the observable limit during the ring suction process. The disappearing zone is then interpreted as zero displacements with the DIC reader. This abnormality spreads toward the center and impacts the data in zones 𝑍 𝑎 and 𝑍 𝑏 .

If the multi-axis ring suction experiment was somehow identical to Laiacona et al.'s [START_REF] Laiacona | Non-invasive in vivo quantification of human skin tension lines[END_REF], the analysis approaches were quite different. the anisotropy angle is determined by fitting a deformed-and-rotated circle instead of fitting the radial strain, and the directional variations in Poisson's ratio were considered. In the given study, the data subjected to model optimization are gathered on a circle of radius 𝑟 = 1 mm. The consistency of the mechanical response within the circular zone 𝑟 < 1.3 mm and the deterministic nature of the inverse problem justify the choice of data quality over quantity. However, it is not recommended to select data near the center 𝑟 < 0.5, where the experimental uncertainty Δ exp is not neglected with respect to small displacements.

Identifying the offset rigid body motion parameters, {𝑥 10 , 𝑥 20 } is of major concern. The consequence of the offset is exhibited in Fig. 12, where the model and experimental data expressed with respect to the new origin are seemingly isotropic. The relative positions of the identified center shift are of the order of magnitude of the displacements. Thus, they cannot be neglected. Two technical reasons may describe the necessity of recentering. First, the origin of the DIC grid was fixed such that the width matches the diameter of the observable zone of the CutiScan ® probe (Fig. 3). The real center of the deformed skin then flexible. Second, even the pressure is uniformly applied around the zone, factors may lead to imperfect load distribution. Among them are the curvature of the forearm and the non-perpendicularity of the probe. In the latter situation, the device is not perfectly tangent to the skin surface.

A simulation FE model was developed in COMSOL Multiphysics ® software to support the calibration study, where the ring suction experiment was modeled in a 2D axisymmetric geometry with a frictionless contact between the probe and the material surface (Figs. 13a and19). The elastic modulus assigned 1 and Eq. [START_REF] Manschot | The measurement and modelling of the mechanical properties of human skin in vivo-i. the measurement[END_REF]. Based on the interquartile range (IQR) method, the positive values inside the grey stripe are selected for calculating the dad-to-day deviation in (b) and (d). IQR is the interval [median -1.5×IQ, median + 1.5×IQ], such that IQ is the difference between the 3 rd and the 1 st quartiles. The median was computed regarding all tests, while the mean was computed after discarding the outliers.

to the material was identified from a parallel uni-axial tensile test according to 3 directions with 𝐸 latex = 2.7 ± 0.4 MPa. In the first step, the simulation was validated by comparing the calculated displacements along 4 directions of the deformed latex rubber versus the experimental data, which can be observed in Fig. 13b. A series of computations were performed for different negative pressure 𝑝 values, where the radial stress in the observable zone was evaluated in post-treatment. Therefore, a linear correlation between 𝑝 and 𝜎 𝑟 was noticed, which validates the third hypothesis formulated in Section 2.2.1. Such a result would help simplify the full configuration into a radial stretch since the stress can be deduced directly from the applied pressure (Fig. 6a). Calibrating only the slope, 𝜉, would not be enough to evaluate the elastic moduli accurately. We introduced a corrective coefficient to compensate for the discrepancy resulting from the transition from a 2D axisymmetric to a plane model. We recommend that values 𝜉 and 𝜂 should be calibrated according to the material's thickness and stiffness by virtue of the simulation.

Once the methodology had been explored and verified based on the latex material, it was applied to 30 series of data acquired from the multi-axial ring suction test (CutiScan ® ) conducted on the forearm of one patient. Figs. 15, 16a, and 16b show that the human skin behaves as anisotropic material with two principal directions neighboring 45 • and 135 • , and that the validation limit of the spatial linearity is around 𝑟 = 1 mm. Directions 0 • and 90 • with mostly identical mechanical responses indicate that the angle between the two principal axes is about 90 • . The material is intended to be orthotropic. Since collagen fibers are stiffer than the ground substance, their axis of minimum, respectively maximum displacement is aligned along Langer's line, respectively across Langer's line [START_REF] Manschot | The measurement and modelling of the mechanical properties of human skin in vivo-ii. the model[END_REF][START_REF] Khatyr | Model of the viscoelastic behaviour of skin in vivo and study of anisotropy[END_REF][START_REF] Reihsner | Two-dimensional elastic properties of human skin in terms of an incremental model at the in vivo configuration[END_REF][START_REF] Delalleau | A nonlinear elastic behavior to identify the mechanical parameters of human skin in vivo[END_REF][START_REF] Sanders | Torsional elasticity of human skin in vivo[END_REF]. Therefore, the Langer's line on the studied subject is oriented obliquely at a mean angle φ = 40.9 ± 8.2 • relative to the forearm axis (Fig. 17). This result is in accordance with the literature [START_REF] Boyer | Assessment of the in-plane biomechanical properties of human skin using a finite element model updating approach combined with an optical full-field measurement on a new tensile device[END_REF][START_REF] Rosicka | Skin anisotropy: Finding the optimal incision line for volar forearm in males and females[END_REF][START_REF] Zahouani | Contribution of human skin topography to the characterization of dynamic skin tension during senescence: morpho-mechanical approach[END_REF][START_REF] Cha | Noninvasive determination of fiber orientation and tracking 2dimensional deformation of human skin utilizing spatially resolved reflectance of infrared light measurement in vivo[END_REF] but sensibly different from Khatyr et al. [START_REF] Khatyr | Model of the viscoelastic behaviour of skin in vivo and study of anisotropy[END_REF] (0-10 • ) . If the symmetry is admitted in the contralateral study on the right forearm, our result is similar to one reported by Lakhani et al. [START_REF] Lakhani | Non-invasive in vivo quantification of directional dependent variation in mechanical properties for human skin[END_REF] and Kirby et al. [START_REF] Kirby | Probing elastic anisotropy of human skin in vivo with light using non-contact acoustic micro-tapping oce and polarization sensitive oct[END_REF]. Nevertheless, most recent studies, including ours, described efficient tools and methodology to determine the Langer's line orientation, historically defined on the forearm to be parallel to the forearm's long axis of the forearm [START_REF] Langer | Zue anatomie und physiologie de haut. ueber der spaltbarkeit der cutis[END_REF][START_REF] Langer | On the anatomy and physiology of the skin: I. the cleavability of the cutis[END_REF][START_REF] Liang | Biomechanical properties of in vivo human skin from dynamic optical coherence elastography[END_REF].

The model parameters were identified with MARSAC for every test to investigate day-to-day variation material parameters and explore the proposed method's sensitivity (Table 1). The values of the correlation coefficient witness an accurate model fit for 28 tests (𝑅 cor > 0.95). The parameters were retrieved from m𝑔 through Eq. ( 10) and have been projected in Fig. 18 for better visualization. As a result, Young's moduli along and across Langer's line are, respectively, Ê1 = 1.35 ± 0.65 MPa and Ê2 = 0.43±0.07 MPa. The plots reveal that the deviations on φ, Δ𝜙 = 8.2 • , and on Ê2 , Δ𝐸 2 = 0.07 MPa, whose removed outliers were no more than 4, are relatively low, unlike Ê1 ( Δ𝐸 1 = 0.65 MPa), whose 12 outliers were discarded, as occurring in one of the two following situations: (i) 𝑢 1 is almost zero along Langer's line, and thus 𝐸 1 reaches higher stiffness, (ii) 𝐸 1 takes a negative value. In the latter scenario, negative 𝐸 1 indicates that the skin is compressed along Langer's line instead of being stretched. A dome inside the observable zone takes shape by pressing the probe strongly against the skin, and the mechanical response becomes unpredictable. The high sensitivity of 𝐸 1 assessment can be related to the fact that 𝑢 1 values are within the measurement uncertainty evaluated earlier in Section 3.2.1, Δ exp = 2.7 µm.

On the other hand, the displacement 𝑢 2 across the higher stiffness direction is substantial compared to Δ exp , which may justify the reliability of its assessment. As for 𝜙, completely dissociated with 𝐸 1 and 𝐸 2 , regarding the deviation Δ𝜙 = 8.2 • , the process of its identification can be considered robust.

To elucidate the variability of the material parameters from one day to another, we display in Ap-pendix A.3 a series of skin pictures recorded at the first 6 tests. A change in the topography of the skin surface was visually noticed. A correlation between the skin lines' main direction and the anisotropy axis may appear. If we look at the tests {3, 4, 5, 6}, the furrows are mostly oriented along the same Langer's line identified in Table 1 with the respective angles 30.77 . This result confirms the observations made in [START_REF] Zahouani | Contribution of human skin topography to the characterization of dynamic skin tension during senescence: morpho-mechanical approach[END_REF][START_REF] Ayadh | Methods for characterizing the anisotropic behavior of the human skin's relief and its mechanical properties in vivo linked to age effects[END_REF]. Nevertheless, the correlation cannot be established on test 2, witnessing an absence of skin lines, and the relative 𝜙 is the most extreme with respect to the median. The skin microrelief network may depend on hydration conditions: the wrinkles are more pronounced in a dry stratum corneum [START_REF] Limbert | On skin microrelief and the emergence of expression micro-wrinkles[END_REF]. The experimental protocol should consider particular attention to skin hydration to reduce the fluctuations of the material parameters, for instance, avoiding exposing the studied site to any liquid. Despite that, the day-today variation of skin wrinkles observed in Appendix A.3 could be associated with the change in the anisotropy axis. Considering that the uppermost layer's deformation is directly correlated to the radial stretch, any change in its texture would have an impact on the radial displacement distribution, and thus on 𝜙. As for the rigid body motion, the identified values of 𝑥 10 and 𝑥 20 in Table 1 witness randomness over the 30 tests. Two causes can be suggested for this center shifting: the probe alignment with the skin surface and the friction between them. For the former, a manual setup of tangential contact would definitely be imperfect, and the skin slippage under the probe could not be uniform. As for the latter, a thorough tribology investigation is suggested to study the friction's effect on the center shifting.

Regarding the recent observations in our experiments, we recommend carrying out the multi-axial ring suction test enclosed by a strictly standardized protocol. The protocol would include the following steps, reproducing the hydration conditions (e.g., drying the measured site many hours before the test to ensure the appearance of the furrows), and holding the probe in a perfect perpendicular position to the skin surface with zero-load contact.

Since the value of the parameter 𝜈 12 = 0.43 ± 0.09 were fixed, based on the literature, the impact of its uncertainty on the identification accuracy had been studied. By averaging the standard deviations of the calibrated 𝐸 1 and 𝐸 2 for each test, we determined Δ𝐸 1 = 0.86 MPa and Δ𝐸 2 = 0.03 MPa. By comparing the uncertainties and the average values, one could notice that a 21% change in 𝜈 12 with regard to the displacement occurs in a change of 26% in 𝐸 1 and 3% in 𝐸 2 . The uncertainty on 𝐸 2 is weakly correlated to 𝜈 12 because of the predominant displacement 𝑢 2 against 𝑢 1 . However, as 𝑢 1 is low, 𝐸 1 is highly affected by Δ 𝜈 12 . It is greatly suggested that particular attention must be given to the precision of 𝜈 12 measurement. To do this, a uni-axial test should be performed along the anisotropy axis 𝜙 identified by the MARSAC method.

Two topics regarding the material parameters are addressed hereby, their comparison to the previous studies in the literature and their physical interpretation. The ratio Ê1 / Ê2 = 3.14 ± 1.60 is conform to the results in [START_REF] Wijn | Mechanical properties of the human skin in vivo for small deformations; a comparison of uniaxial strain and torsion measurements[END_REF][START_REF] Manschot | The measurement and modelling of the mechanical properties of human skin in vivo-ii. the model[END_REF][START_REF] Khatyr | Model of the viscoelastic behaviour of skin in vivo and study of anisotropy[END_REF][START_REF] Flynn | Mechanical characterisation of in vivo human skin using a 3d force-sensitive micro-robot and finite element analysis[END_REF][START_REF] Boyer | Assessment of the in-plane biomechanical properties of human skin using a finite element model updating approach combined with an optical full-field measurement on a new tensile device[END_REF][START_REF] Liang | Biomechanical properties of in vivo human skin from dynamic optical coherence elastography[END_REF]. However, Ê1 = 1.35 ± 0.65 MPa and Ê2 = 0.43 ± 0.07 MPa are in good agreement with Young's modulus in some of the earlier cited papers [START_REF] Wijn | Mechanical properties of the human skin in vivo for small deformations; a comparison of uniaxial strain and torsion measurements[END_REF][START_REF] Khatyr | Model of the viscoelastic behaviour of skin in vivo and study of anisotropy[END_REF] but higher compared to [START_REF] Flynn | Mechanical characterisation of in vivo human skin using a 3d force-sensitive micro-robot and finite element analysis[END_REF][START_REF] Boyer | Assessment of the in-plane biomechanical properties of human skin using a finite element model updating approach combined with an optical full-field measurement on a new tensile device[END_REF]. This latter behavior may occur when the effect of the epidermis layer (known to be stiffer than the dermis [START_REF] Agache | Measuring the Skin[END_REF][START_REF] Ferdman | Scattering of light from histologic sections: a new method for the analysis of connective tissue[END_REF][START_REF] Feng | In vivo stiffness measurement of epidermis, dermis, and hypodermis using broadband rayleigh-wave optical coherence elastography[END_REF]) on the mechanical response is not neglected. This assumption is adequate with the CutiScan ® device's function, which applies out-plane suction to cause an in-plane stretch in the superficial layers. Although the epidermis under superficial stretch dominates the skin stiffness, the dermis has a major effect on the mechanical response because the collagen and elastin fibers are present in the dermis, not in the epidermis. As an attempt to associate the material parameters with the micro-structure components, we state the following. Along Langer's line, 𝐸 1 describes the stiffness of the combination epidermis-collagen. While across Langer's line, 𝐸 2 describes the stiffness of the combination epidermis-elastin. On the bright side, the anisotropy direction angle 𝜙 is insensitive to skin stiffness and purely driven by collagen's highest distribution angle.

Conclusions

MARSAC, an open-source framework coded in Python language, was built to characterize the anisotropic mechanical behavior of soft tissues. From an output video file gathered with the CutiScan ® device during a ring suction test, MARSAC computes the displacement field using DIC and identifies the Langer's line direction 𝜙, as well as the elastic moduli 𝐸 1 and 𝐸 2 , along and across Langer's line, for a given Poisson's ratio value. An analytical orthotropic linear model was built and used to solve the displacement for applied pressure. In the first instance, the pipeline was applied to a homogeneous latex material as a verification and calibration step. Then, it was applied to public data consisting in a series of 30 multiaxial ring suction tests, at pressure 𝑝 = 300 mbar, performed on a forearm of a 28-year-old Caucasian subject. The identification of Langer's line, φ = 40.9 ± 8.2 • , interpreted as collagen fiber preferred direction, and the anisotropy ratio, Ê2 / Ê1 = 3.14 ± 1.60, were in good agreement with the literature. The presented method showed a precise assessment of 𝜙 and 𝐸 2 , but not 𝐸 1 . The measured displacement along the stiffest direction is most of the time smaller than measurement error Δ exp = 2.7 µm; hence 𝐸 1 is less reliable. To conclude, the developed numerical method combined with the CutiScan ® is a fast and accurate means to determine Langer's line locally in small areas. It would help the experimentalists efficiently use the commercial tool whose associated original software showed a lack of reliability [START_REF] Anthonissen | Can the CutiScan CS 100® measure anisotropy and viscoelasticity in scar tissue after mastectomy? a reliability and validity study[END_REF].

The presented methodology will be applied to analyze clinical trial data of 30 volunteers to study the inter-subject variability. Also, It will also be integrated into a bi-material characterization software [START_REF] Elouneg | An open-source fenics-based framework for hyperelastic parameter estimation from noisy full-field data: Application to heterogeneous soft tissues[END_REF] to enhance its accuracy by adding the anisotropy aspect to the healthy skin around the abnormal scar.

Figure 1 :

 1 Figure 1: CutiScan ® CS 100.

Figure 2 :

 2 Figure 2: (a) A cross-sectional view through a CAD model of the probe. The suction fits the circular opening between the blue plastic part and the yellow aluminum reflector. The UV light is emitted from the purple rectangles distributed around the camera lens in the center. Permission was granted from Courage + Khazaka GmbH to publish the drawing. (b) Suction probe cross-section [77]. The aspiration uniformly applied between the cone and the wall results in a multi-axial extension of the soft layer.

Figure 3 :

 3 Figure 3: DIC coordinate grid in the initial configuration.

Figure 4 :

 4 Figure 4: Ring suction operation performed on a latex tissue: (a) Bottom view before loading; (b) Bottom view after loading; (c) Top view of the observable zone obtained by the camera before loading. In the central observable zone (1), the UV camera captures the displacement of the stretched layer subjected to suction applied to the annular zone (2). A double-sided annular sticker was settled on the annular zone (3).

Figure 5 :

 5 Figure 5: Experimental set-up of the ring suction test performed on the volar forearm and illustration of displacement field vectors. Source: Obtained by DIC method [79].

Figure 6 :

 6 Figure 6: Boundary conditions of the global (a) and reduced (b) problems.

  1 = 0) and (𝑥 2 = 0) are symmetry axes from geometrical and material points of view, only a quarter of the problem is taken into account. The external boundary 𝜕Ω of the domain Ω (a quarter of disk) can be split into a Dirichlet part 𝜕Ω D and a Neumann part 𝜕Ω N such as 𝜕Ω = 𝜕Ω D ∪ 𝜕Ω N and 𝜕Ω D ∩ 𝜕Ω N = ∅ (Fig. 6b).

Figure 7 :

 7 Figure 7: Initial and deformed configurations for 𝑟 = 1 mm. {𝒆 ′1 , 𝒆 ′ 2 } coordinate system corresponds to the probe referential (with 𝒆 ′ 1 oriented along the forearm toward the hand). The model is expressed in the system {𝒆 1 , 𝒆 2 } to fit the targeted data.

xR 1 (

 1 𝑟, 𝜃, 𝑎, 𝑏, 𝜙) = 𝑎(𝑟) cos (𝜃 -𝜙) cos 𝜙 -𝑏(𝑟) sin (𝜃 -𝜙) sin 𝜙 xR 2 (𝑟, 𝜃, 𝑎, 𝑏, 𝜙) = 𝑎(𝑟) cos (𝜃 -𝜙) sin 𝜙 + 𝑏(𝑟) sin (𝜃 -𝜙) cos 𝜙

Figure 8 :

 8 Figure 8: Components of analytical and FEM displacement vector fields on circles with various radius 𝑟 for the parameter set (𝐸 1 = 10 kPa, 𝐸 2 = 0.5 kPa, 𝜈 12 = 0.5, 𝐺 12 = 1 MPa).

Figure 9 :Figure 10 :

 910 Figure 9: Displacement vector field 𝒖 FEM for the parameter set (𝐸 1 = 10 kPa, 𝐸 2 = 0.5 kPa, 𝜈 12 = 0.5, 𝐺 12 = 1 MPa). The norm ∥𝒖 FEM ∥ is given by the color bar. The directions of the vector field 𝒖 FEM are represented by arrows without taking into account the magnitude scale.

( a )

 a Contour lines of 𝑢 𝑟 -component (b) Radial cutlines

Figure 11 :

 11 Figure 11: Radial displacement component of the latex material under multi-axial ring suction load. Data on circles and cutlines are extracted from the displacement field.

Figure 12 : 1 𝐼0 , 𝑥 2 𝐼 0 )

 12120 Figure 12: (a) Model fitting with the identified parameters of the latex material 𝒎 I 𝑔 . The center position shifted from the black to the green dot. (b) Radius change 𝑟 deformed of the deformed ellipse before and after removing the rigid body motion (recentring). 𝑟 deformed equals to 𝑟 + 𝑢 𝑟 , with 𝑢 𝑟 the radial component of the displacement vector. 𝜃 ′ = (𝒆 ′ 1 , ---→ 𝑂 𝑀), with ---→ 𝑂 𝑀 the position vector of a material point in the coordinate system {𝒆 ′ 1 , 𝒆 ′ 2 } related to the camera. For recentred ellipses, the shifting center (𝑥 1 𝐼 0 , 𝑥 2 𝐼 0 ) was retrieved from each point on the measured deformed ellipse.

Figure 13 :

 13 Figure 13: (a) Deformed configuration of a 2D axisymmetric model subjected to a multi-axial ring suction test. (b) Comparison of radial displacement obtained with the FE model and the CutiScan. The filled grey area corresponds to 95% confidence interval of radial obtained with the range 𝐸 = 2.7 ± 0.42 MPa.

Figure 14 :

 14 Figure 14: Correlation between the suction pressure 𝑝 and the radial stress 𝜎 𝑟 , with the slope 𝜉 = 633.

Figure 15 :

 15 Figure 15: Displacement vector field for test 1 evaluated at the quasi-static state on DIC nodes for 𝑝 = 300 mbar.

Figure 16 :

 16 Figure 16: Mechanical response of the forearm skin subjected to multi-axial ring suction load. In (16a), the deformation is scaled by a factor of 2.

Figure 17 :

 17 Figure 17: (a) Model fitting with the identified parameters of the skin on the day 1, m (1)𝑔 . The center position shifted from the black to the green dot. (b) Radius change 𝑟 deformed of the deformed ellipse before and after removing the rigid body motion (recentring). More details are available in Fig.12.

( a )Figure 18 :

 a18 Figure18: Illustration of the skin material parameters 𝐸 1 (a), 𝐸 2 (c), and 𝜙 (e) for every test deducted from Table1and Eq.[START_REF] Manschot | The measurement and modelling of the mechanical properties of human skin in vivo-i. the measurement[END_REF]. Based on the interquartile range (IQR) method, the positive values inside the grey stripe are selected for calculating the dad-to-day deviation in (b) and (d). IQR is the interval [median -1.5×IQ, median + 1.5×IQ], such that IQ is the difference between the 3 rd and the 1 st quartiles. The median was computed regarding all tests, while the mean was computed after discarding the outliers.

Figure 19 :

 19 Figure 19: 2D axisymmetric model boundary conditions.

6 Figure 20 :

 620 Figure 20: Model fitting along the circle of 1 mm radius for some tests.

  𝜃 𝑘 , 𝑎, 𝑏, 𝜙) -𝑥 10 2 + xR 2 (𝑟, 𝜃 𝑘 , 𝑎, 𝑏, 𝜙) -𝑥 20 2 -

			√ 𝑥 1	′ 𝑘	2 + 𝑥 2	′ 𝑘	2	2	(11)
	such that 𝜃 𝑘 = arctan(𝑥 2	′ 𝑘 /𝑥 1	′ 𝑘 ). xR 1 , and xR

2 (see Eq. (

Table 1 :

 1 Numerical results of the inverse identification of m𝑔 . The initial parameter values were fixed to be {𝜙 = 45°; 𝑥 10 = 0 mm; 𝑥 20 = 0 mm}, whereas 𝑎 and 𝑏 are chosen according to the circle used for identification such that 𝑎 < 𝑏. In our case, for 𝑟 = 1 mm, we choose 𝑎 = 1 mm and 𝑏 = 1.01 mm. The best-fit curves of the 6 first test are available in Appendix A.2 . Test 𝑎 (mm) 𝑏 (mm) 𝜙(°) 𝑥 10 (µm) 𝑥 20 (µm)

	𝑅 cor

  • , 27.61 • , 31.25 • , and 50.26 • , but not on test 1 with 𝜙 = 60.24

Acknowledgments

The authors acknowledge the financial support of Region de Bourgogne Franche-Comté, France (grants n • 2018-Y-07543), the Luxembourg National Research Fund (OTP S-STR-8001-00-B), and French Agence Nationale de Recherche (ANR) grant (No. ANR-21-CE45-0025: https://anr.fr/Project -ANR-21-CE45-0025) for the project "(S-KELOID) Understanding Keloid Disorders: A multi-scale in vitro/in vivo/in silico approach towards digital twins of skin organoids on the chip". They thank Tiguida Kadiakhe for her help with the experiments.

A Appendix

A.1 Calibration study: Simulation model

A FE model was constructed in COMSOL Multiphysics ® that simulates the multi-axial annular suction test. It was used to validate and calibrate the radial stress in the othrotopic linear elastic model (Eq. ( 6)) for assessing accurate values of the elastic moduli. In this 2D axisymmetric problem a thickness of 0.22 mm, the linear material elastic was adopted with Young's modulus 𝐸 = 2.7 MPa and Poisson's ratio 𝜈 = 0.495. The boundary conditions are described in Fig. 19.

A.2 Model fitting

See Fig. 20.

A.3 Test snapshots

See Fig. 21.