Mathematical and computational modelling of skin biophysics: a review

The objective of this paper is to provide a review on some aspects of the mathematical and computational modelling of skin biophysics, with special focus on constitutive theories based on nonlinear continuum mechanics from elasticity, through anelasticity, including growth, to thermoelasticity. Microstructural and phenomenological approaches combining imaging techniques are also discussed. Finally, recent research applications on skin wrinkles will be presented to highlight the potential of physics-based modelling of skin in tackling global challenges such as ageing of the population and the associated skin degradation, diseases and traumas.

[1]  M Setaro,et al.  Irregularity skin index (ISI): a tool to evaluate skin surface texture , 2001, Skin research and technology : official journal of International Society for Bioengineering and the Skin (ISBS) [and] International Society for Digital Imaging of Skin (ISDIS) [and] International Society for Skin Imaging.

[2]  Georges Limbert,et al.  A transversely isotropic viscohyperelastic material Application to the modeling of biological soft connective tissues , 2004 .

[3]  R. Jedynak.  Approximation of the inverse Langevin function revisited , 2014, Rheologica Acta.

[4]  Y. Fung,et al.  Biorheology of soft tissues. , 1973, Biorheology.

[5]  Ellen Kuhl,et al.  Multi-view stereo analysis reveals anisotropy of prestrain, deformation, and growth in living skin , 2015, Biomechanics and modeling in mechanobiology.

[6]  Alberto Redaelli,et al.  Hierarchical structure and nanomechanics of collagen microfibrils from the atomistic scale up. , 2011, Nano letters.

[7]  Jay D. Humphrey,et al.  Review Paper: Continuum biomechanics of soft biological tissues , 2003, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[8]  M. Ben Amar,et al.  Morphogenesis of growing soft tissues. , 2007, Physical review letters.

[9]  P.M.F. Nielsen,et al.  Investigating stress-strain properties of in-vivo human skin using multiaxial loading experiments and finite element modeling , 2004, The 26th Annual International Conference of the IEEE Engineering in Medicine and Biology Society.

[10]  Boumediene Nedjar,et al.  An anisotropic viscoelastic fibre–matrix model at finite strains: Continuum formulation and computational aspects , 2007 .

[11]  Jia Lu,et al.  Physically motivated invariant formulation for transversely isotropic hyperelasticity , 2005 .

[12]  A. Gower.  Connecting the material parameters of soft fibre-reinforced solids with the formation of surface wrinkles , 2015 .

[13]  A. Cohen,et al.  A Padé approximant to the inverse Langevin function , 1991 .

[14]  Y. Lanir.  The rheological behavior of the skin: experimental results and a structural model. , 1979, Biorheology.

[15]  Gerhard A Holzapfel,et al.  Modelling non-symmetric collagen fibre dispersion in arterial walls , 2015, Journal of The Royal Society Interface.

[16]  C. Linder,et al.  Understanding geometric instabilities in thin films via a multi-layer model. , 2016, Soft matter.

[17]  R. Meijer,et al.  Characterisation of Anisotropic and Non-linear Behaviour of Human Skin In Vivo. , 1999, Computer methods in biomechanics and biomedical engineering.

[18]  L. Mahadevan,et al.  Nested self-similar wrinkling patterns in skins , 2005, Nature materials.

[19]  Ali Javili,et al.  Computational aspects of morphological instabilities using isogeometric analysis , 2017 .

[20]  Sidney Fels,et al.  Simulating the three-dimensional deformation of in vivo facial skin. , 2013, Journal of the mechanical behavior of biomedical materials.

[21]  Sophia Mã ¶ ller,et al.  Biomechanics — Mechanical properties of living tissue , 1982 .

[22]  J. Gosline,et al.  Elastic proteins: biological roles and mechanical properties. , 2002, Philosophical transactions of the Royal Society of London. Series B, Biological sciences.

[23]  Karl Grosh,et al.  A rheological network model for the continuum anisotropic and viscoelastic behavior of soft tissue , 2004, Biomechanics and modeling in mechanobiology.

[24]  Hung Nguyen-Xuan,et al.  Explicit finite deformation analysis of isogeometric membranes , 2014 .

[25]  K Y Volokh,et al.  Prediction of arterial failure based on a microstructural bi-layer fiber-matrix model with softening. , 2008, Journal of biomechanics.

[26]  E Mazza,et al.  Simulation of the aging face. , 2007, Journal of biomechanical engineering.

[27]  Benjamin C. Marchi,et al.  An error-minimizing approach to inverse Langevin approximations , 2015, Rheologica Acta.

[28]  Arthur Rook,et al.  Rook's Textbook of Dermatology , 2004 .

[29]  Jeffrey E. Bischoff,et al.  Finite element simulations of orthotropic hyperelasticity , 2002 .

[30]  Matthew J. Matava,et al.  Tattoo-Induced Skin “Burn” During Magnetic Resonance Imaging in a Professional Football Player , 2011, Sports health.

[31]  S Derler,et al.  Influence of epidermal hydration on the friction of human skin against textiles , 2008, Journal of The Royal Society Interface.

[32]  K. Nouri,et al.  Lasers for tattoo removal: a review , 2010, Lasers in Medical Science.

[33]  Sergio Pellegrino,et al.  WRINKLED MEMBRANES PART III: NUMERICAL SIMULATIONS , 2006 .

[34]  Manuel Doblaré,et al.  On finite‐strain damage of viscoelastic‐fibred materials. Application to soft biological tissues , 2008 .

[35]  Sandy L. Jones,et al.  Human Anatomy and Physiology. 6th ed , 1970 .

[36]  Yanping Cao,et al.  From wrinkles to creases in elastomers: the instability and imperfection-sensitivity of wrinkling , 2012, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[37]  Walter Noll,et al.  Foundations of Linear Viscoelasticity , 1961 .

[38]  Paul Ackermans,et al.  Linear shear response of the upper skin layers. , 2011, Biorheology.

[39]  G. Limbert,et al.  On the mechanics of bacterial biofilms on non-dissolvable surgical sutures: a laser scanning confocal microscopy-based finite element study. , 2013, Acta biomaterialia.

[40]  Ellen Kuhl,et al.  Growing skin: tissue expansion in pediatric forehead reconstruction , 2012, Biomechanics and modeling in mechanobiology.

[41]  J. L. Lévêque.  EEMCO guidance for the assessment of skin topography 1 , 1999 .

[42]  Balasundar I Raju,et al.  3-D finite-element models of human and monkey fingertips to investigate the mechanics of tactile sense. , 2003, Journal of biomechanical engineering.

[43]  M. Biot.  Surface instability of rubber in compression , 1963 .

[44]  M Jabareen,et al.  Suction based mechanical characterization of superficial facial soft tissues. , 2015, Journal of biomechanics.

[45]  S. Persechino,et al.  Laser Removal of Tattoos , 2012, International journal of immunopathology and pharmacology.

[46]  V. Langer,et al.  Zur Anatomie und Physiologie der Haut. I. Über die Spaltbarkeit der Cutis , 1861 .

[47]  Brad Lee Boyce,et al.  Modeling the anisotropic finite-deformation viscoelastic behavior of soft fiber-reinforced composites , 2007 .

[48]  遠山 正彌,et al.  人体の解剖生理学 = Human anatomy & physiology , 2010 .

[49]  G. Porter.  Cause and effect , 2010, Nature Methods.

[50]  M. Carfagna,et al.  Oblique wrinkles , 2020, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[51]  G. Piérard,et al.  The riddle of genuine skin microrelief and wrinkles , 2006, International journal of cosmetic science.

[52]  Brian J. Briscoe,et al.  Friction and lubrication of human skin , 2007 .

[53]  Paul Steinmann,et al.  Theory and numerics of geometrically non‐linear open system mechanics , 2003 .

[54]  K. Grosh,et al.  Remodeling of biological tissue: Mechanically induced reorientation of a transversely isotropic chain network , 2004, q-bio/0411037.

[55]  T J Joyce,et al.  Resolving the viscoelasticity and anisotropy dependence of the mechanical properties of skin from a porcine model , 2016, Biomechanics and modeling in mechanobiology.

[56]  H. Zahouani,et al.  In vivo characterization of viscoelastic properties of human skin using dynamic micro-indentation , 2007, 2007 29th Annual International Conference of the IEEE Engineering in Medicine and Biology Society.

[57]  Yewang Su,et al.  Mechanics of Epidermal Electronics , 2012 .

[58]  James C Birchall,et al.  An anisotropic, hyperelastic model for skin: experimental measurements, finite element modelling and identification of parameters for human and murine skin. , 2013, Journal of the mechanical behavior of biomedical materials.

[59]  A. Khunger,et al.  Complications of Tattoos and Tattoo Removal: Stop and Think Before you ink , 2015, Journal of cutaneous and aesthetic surgery.

[60]  A. Le Bot,et al.  Dynamic indentation on human skin in vivo: ageing effects , 2009, Skin research and technology : official journal of International Society for Bioengineering and the Skin (ISBS) [and] International Society for Digital Imaging of Skin (ISDIS) [and] International Society for Skin Imaging.

[61]  E. A. de Souza Neto,et al.  On the determination of the path direction for arc-length methods in the presence of bifurcations and `snap-backs' , 1999 .

[62]  Georges Limbert,et al.  Implications of multi‐asperity contact for shear stress distribution in the viable epidermis – An image‐based finite element study , 2017 .

[63]  A. McCulloch,et al.  Stress-dependent finite growth in soft elastic tissues. , 1994, Journal of biomechanics.

[64]  G. Piérard,et al.  From skin microrelief to wrinkles. An area ripe for investigation , 2003, Journal of cosmetic dermatology.

[65]  K. Kwiatkowski,et al.  Friction and deformation behaviour of human skin , 2009 .

[66]  T J Walters,et al.  INTER-SPECIES EXTRAPOLATION OF SKIN HEATING RESULTING FROM MILLIMETER WAVE IRRADIATION: MODELING AND EXPERIMENTAL RESULTS , 2003, Health physics.

[67]  S. Stupkiewicz,et al.  Micromechanical analysis of friction anisotropy in rough elastic contacts , 2014 .

[68]  Thao D Nguyen,et al.  Full-field bulge test for planar anisotropic tissues: part I--experimental methods applied to human skin tissue. , 2013, Acta biomaterialia.

[69]  L. Mahadevan,et al.  Geometry and physics of wrinkling. , 2003, Physical review letters.

[70]  Jan Genzer,et al.  Soft matter with hard skin: From skin wrinkles to templating and material characterization. , 2006, Soft matter.

[71]  Incremental elastic coefficients of an isotropic medium in finite strain , 1963 .

[72]  Georges Limbert,et al.  Skin Microstructure is a Key Contributor to Its Friction Behaviour , 2016, Tribology Letters.

[73]  Y. Lanir.  Constitutive equations for fibrous connective tissues. , 1983, Journal of biomechanics.

[74]  M. Rubin,et al.  Modeling a smooth elastic–inelastic transition with a strongly objective numerical integrator needing no iteration , 2013 .

[75]  A. N. Annaidh,et al.  Non‐invasive evaluation of skin tension lines with elastic waves , 2016, Skin research and technology : official journal of International Society for Bioengineering and the Skin (ISBS) [and] International Society for Digital Imaging of Skin (ISDIS) [and] International Society for Skin Imaging.

[76]  F. Silver,et al.  Mechanobiology of force transduction in dermal tissue , 2003, Skin research and technology : official journal of International Society for Bioengineering and the Skin (ISBS) [and] International Society for Digital Imaging of Skin (ISDIS) [and] International Society for Skin Imaging.

[77]  Michael D. Gilchrist,et al.  Automated Estimation of Collagen Fibre Dispersion in the Dermis and its Contribution to the Anisotropic Behaviour of Skin , 2012, Annals of Biomedical Engineering.

[78]  A. E. Ehret,et al.  A micromechanically motivated model for the viscoelastic behaviour of soft biological tissues at large strains , 2009 .

[79]  M. Boyce,et al.  A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials , 1993 .

[80]  Peter Wriggers,et al.  An improved EAS brick element for finite deformation , 2010 .

[81]  David Barber,et al.  Bayesian reasoning and machine learning , 2012 .

[82]  K. Volokh.  On Irreversibility and Dissipation in Hyperelasticity With Softening , 2014 .

[83]  Reinhold H Dauskardt,et al.  Solar UV radiation reduces the barrier function of human skin , 2012, Proceedings of the National Academy of Sciences.

[84]  Y Lanir,et al.  Two-dimensional mechanical properties of rabbit skin. I. Experimental system. , 1974, Journal of biomechanics.

[85]  Mikhail Itskov,et al.  Modeling of anisotropic softening phenomena: Application to soft biological tissues , 2009 .

[86]  Urša Šolinc,et al.  A simple way to improved formulation of $$\hbox {FE}^2$$FE2 analysis , 2015 .

[87]  R E Baier,et al.  Mechanical characteristics of human skin subjected to static versus cyclic normal pressures. , 1999, Journal of rehabilitation research and development.

[88]  A. E. Ehret,et al.  Porcine dermis in uniaxial cyclic loading: Sample preparation, experimental results and modeling , 2011 .

[89]  Maria A. Holland,et al.  Growth on demand: reviewing the mechanobiology of stretched skin. , 2013, Journal of the mechanical behavior of biomedical materials.

[90]  Adrian Buganza Tepole,et al.  Computational systems mechanobiology of wound healing , 2017 .

[91]  Ellen Kuhl,et al.  Frontiers in growth and remodeling. , 2012, Mechanics research communications.

[92]  Cormac Flynn,et al.  Finite element modelling of forearm skin wrinkling , 2008, Skin research and technology : official journal of International Society for Bioengineering and the Skin (ISBS) [and] International Society for Digital Imaging of Skin (ISDIS) [and] International Society for Skin Imaging.

[93]  P. Nielsen,et al.  Estimating material parameters of human skin in vivo , 2009, Biomechanics and modeling in mechanobiology.

[94]  J. Lloyd,et al.  A FINITE ELEMENT MODEL OF THE FACE INCLUDING AN ORTHOTROPIC SKIN MODEL UNDER IN VIVO TENSION , 2012 .

[95]  Mikhail Itskov,et al.  A class of orthotropic and transversely isotropic hyperelastic constitutive models based on a polyconvex strain energy function , 2004 .

[96]  Wenguang Li,et al.  Modelling methods for In Vitro biomechanical properties of the skin: A review , 2015 .

[97]  Roland Bazin,et al.  Influence of age on the wrinkling capacities of skin , 2002, Skin research and technology : official journal of International Society for Bioengineering and the Skin (ISBS) [and] International Society for Digital Imaging of Skin (ISDIS) [and] International Society for Skin Imaging.

[98]  A.J.M. Spencer,et al.  Constitutive Theory for Strongly Anisotropic Solids , 1984 .

[99]  Gerhard A. Holzapfel,et al.  A viscoelastic model for fiber-reinforced composites at finite strains: Continuum basis, computational aspects and applications , 2001 .

[100]  N. Veijgen,et al.  Variables influencing the frictional behaviour of in vivo human skin. , 2013, Journal of the mechanical behavior of biomedical materials.

[101]  Michel Destrade,et al.  Characterization of the anisotropic mechanical properties of excised human skin. , 2013, Journal of the mechanical behavior of biomedical materials.

[102]  Manuel Doblaré,et al.  An anisotropic visco-hyperelastic model for ligaments at finite strains. Formulation and computational aspects , 2007 .

[103]  P. Ciarletta,et al.  Shear instability in skin tissue , 2013, 1306.0198.

[104]  Susanne Hertz,et al.  Statistical Mechanics Of Chain Molecules , 2016 .

[105]  Michele Marino,et al.  Molecular and intermolecular effects in collagen fibril mechanics: a multiscale analytical model compared with atomistic and experimental studies , 2016, Biomechanics and modeling in mechanobiology.

[106]  Cormac Flynn,et al.  A simplified model of scar contraction. , 2008, Journal of biomechanics.

[107]  Miguel Ángel Martínez,et al.  Anisotropic microsphere-based approach to damage in soft fibered tissue , 2012, Biomechanics and modeling in mechanobiology.

[108]  K. Grosh,et al.  A continuum treatment of growth in biological tissue: the coupling of mass transport and mechanics , 2003, q-bio/0312001.

[109]  J. C. Simo,et al.  On a fully three-dimensional finite-strain viscoelastic damage model: Formulation and computational aspects , 1987 .

[110]  P Vanezis,et al.  Puncture resistance and tensile strength of skin simulants , 1999, Proceedings of the Institution of Mechanical Engineers. Part H, Journal of engineering in medicine.

[111]  R. Ogden,et al.  A New Constitutive Framework for Arterial Wall Mechanics and a Comparative Study of Material Models , 2000 .

[112]  Georges Limbert,et al.  A mesostructurally-based anisotropic continuum model for biological soft tissues--decoupled invariant formulation. , 2011, Journal of the mechanical behavior of biomedical materials.

[113]  Xiaoyu Luo,et al.  An Invariant-Based Damage Model for Human and Animal Skins , 2016, Annals of Biomedical Engineering.

[114]  Ellen Kuhl,et al.  Isogeometric Kirchhoff-Love shell formulations for biological membranes. , 2015, Computer methods in applied mechanics and engineering.

[115]  Peter J. Hunter,et al.  Estimating material parameters of a structurally based constitutive relation for skin mechanics , 2011, Biomechanics and modeling in mechanobiology.

[116]  T. Christian Gasser,et al.  The numerical implementation of invariant-based viscoelastic formulations at finite strains. An anisotropic model for the passive myocardium , 2011 .

[117]  P. Dokládal,et al.  Ex vivo multiscale quantitation of skin biomechanics in wild-type and genetically-modified mice using multiphoton microscopy , 2015, Scientific Reports.

[118]  K. Langer,et al.  On the anatomy and physiology of the skin: II. Skin Tension (With 1 Figure) , 1978 .

[119]  P Ciarletta,et al.  Contour instabilities in early tumor growth models. , 2011, Physical review letters.

[120]  J. Mansour,et al.  Analysis of shear wave propagation in skin; application to an experimental procedure. , 1990, Journal of biomechanics.

[121]  Mark Taylor,et al.  On the constitutive modeling of biological soft connective tissues: A general theoretical framework and explicit forms of the tensors of elasticity for strongly anisotropic continuum fiber-reinforced composites at finite strain , 2002 .

[122]  A. Forrester.  Black-box calibration for complex-system simulation , 2010, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[123]  Pasquale Ciarletta,et al.  Papillary networks in the dermal–epidermal junction of skin: A biomechanical model , 2012 .

[124]  Reinhold H Dauskardt,et al.  Mechanical properties of human stratum corneum: effects of temperature, hydration, and chemical treatment. , 2006, Biomaterials.

[125]  Amit Gefen,et al.  How do microclimate factors affect the risk for superficial pressure ulcers: a mathematical modeling study. , 2011, Journal of tissue viability.

[126]  H. H. Pennes.  Analysis of tissue and arterial blood temperatures in the resting human forearm. , 1948, Journal of applied physiology.

[127]  G Pennati,et al.  An Axisymmetric Computational Model of Skin Expansion and Growth , 2007, Biomechanics and modeling in mechanobiology.

[128]  B Notarberardino,et al.  An efficient approach to converting three-dimensional image data into highly accurate computational models , 2008, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[129]  J. Sanders,et al.  Skin response to repetitive mechanical stress: a new experimental model in pig. , 1998, Archives of physical medicine and rehabilitation.

[130]  S. Reese,et al.  A theory of finite viscoelasticity and numerical aspects , 1998 .

[131]  Yanping Cao,et al.  Wrinkling Phenomena in Neo-Hookean Film/Substrate Bilayers , 2012 .

[132]  Nicole Propst,et al.  Mathematical Foundations Of Elasticity , 2016 .

[133]  Ellen Kuhl,et al.  Stretching skin: The physiological limit and beyond. , 2012, International journal of non-linear mechanics.

[134]  J. Sherratt,et al.  Theoretical models of wound healing: past successes and future challenges. , 2002, Comptes rendus biologies.

[135]  Louise Reips,et al.  A variational framework for fiber‐reinforced viscoelastic soft tissues , 2012 .

[136]  Ellen Kuhl,et al.  On the biomechanics and mechanobiology of growing skin. , 2012, Journal of theoretical biology.

[137]  Soo-Young Kang,et al.  An elastic second skin. , 2016, Nature materials.

[138]  P. Fuchs,et al.  Burn injuries caused by air bag deployment. , 2001, Burns : journal of the International Society for Burn Injuries.

[139]  Yoram Lanir,et al.  Micro and macro rheology of planar tissues. , 2009, Biomaterials.

[140]  Pasquale Ciarletta,et al.  Morphogenesis of thin hyperelastic plates: A constitutive theory of biological growth in the Föppl-von Kármán limit , 2009 .

[141]  Martyn P Nash,et al.  Computational and experimental characterization of skin mechanics: identifying current challenges and future directions , 2013, Wiley interdisciplinary reviews. Systems biology and medicine.

[142]  Lamers E,et al.  Large amplitude oscillatory shear properties of human skin. , 2013, Journal of the mechanical behavior of biomedical materials.

[143]  Zoujun Dai,et al.  Dynamic viscoelastic models of human skin using optical elastography , 2015, Physics in medicine and biology.

[144]  S. Pellegrino,et al.  Wrinkled membranes III: numerical simulations , 2006 .

[145]  J. A. Bea,et al.  An experimental study of the mouse skin behaviour: damage and inelastic aspects. , 2008, Journal of biomechanics.

[146]  R. Lavker,et al.  The anatomy and pathogenesis of wrinkles , 1985, The British journal of dermatology.

[147]  K. Langer,et al.  On the anatomy and physiology of the skin: I. The cleavability of the cutis , 1978 .

[148]  D. R. Veronda,et al.  Mechanical characterization of skin-finite deformations. , 1970, Journal of biomechanics.

[149]  Ren G Dong,et al.  Estimation of the viscous properties of skin and subcutaneous tissue in uniaxial stress relaxation tests. , 2006, Bio-medical materials and engineering.

[150]  M L Kreidstein,et al.  MRI interaction with tattoo pigments: case report, pathophysiology, and management. , 1997, Plastic and reconstructive surgery.

[151]  Jeffrey E. Bischoff,et al.  A microstructurally based orthotropic hyperelastic constitutive law , 2002 .

[152]  A. E. Ehret,et al.  A polyconvex anisotropic strain–energy function for soft collagenous tissues , 2006, Biomechanics and modeling in mechanobiology.

[153]  E M Arruda,et al.  Finite element modeling of human skin using an isotropic, nonlinear elastic constitutive model. , 2000, Journal of biomechanics.

[154]  M. Rubin,et al.  An anisotropic discrete fiber model with dissipation for soft biological tissues , 2014 .

[155]  D. Béchet,et al.  Hierarchical mechanics of connective tissues: integrating insights from nano to macroscopic studies. , 2014, Journal of biomedical nanotechnology.

[156]  Y. Itzchak,et al.  Assessment of mechanical conditions in sub-dermal tissues during sitting: a combined experimental-MRI and finite element approach. , 2007, Journal of biomechanics.

[157]  M. Röckner,et al.  Stochastic Partial Differential Equations: An Introduction , 2015 .

[158]  V. Lubarda.  Constitutive theories based on the multiplicative decomposition of deformation gradient: Thermoelasticity, elastoplasticity, and biomechanics , 2004 .

[159]  清水 宏,et al.  Shimizu's textbook of dermatology , 2007 .

[160]  E. Kuhl,et al.  On skin microrelief and the emergence of expression micro-wrinkles. , 2018, Soft matter.

[161]  J. Lagarde,et al.  Model of the viscoelastic behaviour of skin in vivo and study of anisotropy , 2004, Skin research and technology : official journal of International Society for Bioengineering and the Skin (ISBS) [and] International Society for Digital Imaging of Skin (ISDIS) [and] International Society for Skin Imaging.

[162]  R. Burgeson,et al.  The dermal-epidermal junction. , 1997, Current opinion in cell biology.

[163]  A. Gefen,et al.  In Vitro “Wound” Healing: Experimentally Based Phenomenological Modeling , 2012 .

[164]  M. B. Rubin,et al.  An anisotropic discrete fibre model based on a generalised strain invariant with application to soft biological tissues , 2012 .

[165]  Katia Bertoldi,et al.  Spatial resolution of wrinkle patterns in thin elastic sheets at finite strain , 2014 .

[166]  Gerhard A. Holzapfel,et al.  Nonlinear Solid Mechanics: A Continuum Approach for Engineering Science , 2000 .

[167]  A. Douglas,et al.  Physically based strain invariant set for materials exhibiting transversely isotropic behavior , 2001 .

[168]  K. Hashimoto,et al.  NEW METHODS FOR SURFACE ULTRASTRUCTURE: COMPARATIVE STUDIES OF SCANNING ELECTRON MICROSCOPY, TRANSMISSION ELECTRON MICROSCOPY AND REPLICA METHOD , 1974, International journal of dermatology.

[169]  G. Limbert.  State-of-the-Art Constitutive Models of Skin Biomechanics , 2014 .

[170]  Kai-Nan An,et al.  Direct quantification of the flexibility of type I collagen monomer. , 2002, Biochemical and biophysical research communications.

[171]  Vlado A. Lubarda,et al.  On the mechanics of solids with a growing mass , 2002 .

[172]  W A Wan Abas.  Biaxial tension test of human skin in vivo. , 1994, Bio-medical materials and engineering.

[173]  Jean Krutmann,et al.  Environmental influences on skin aging and ethnic-specific manifestations , 2012, Dermato-endocrinology.

[174]  H S Wiley,et al.  Studies on blisters produced by friction. I. Results of linear rubbing and twisting technics. , 1966, The Journal of investigative dermatology.

[175]  Peter M. Pinsky,et al.  Multiscale Modeling Framework of Transdermal Drug Delivery , 2009, Annals of Biomedical Engineering.

[176]  Wan Abas Wa.  Biaxial tension test of human skin in vivo. , 1994 .

[177]  M. Jabareen,et al.  Elastic–viscoplastic modeling of soft biological tissues using a mixed finite element formulation based on the relative deformation gradient , 2014, International journal for numerical methods in biomedical engineering.

[178]  Wen Yang,et al.  On the tear resistance of skin , 2015, Nature Communications.

[179]  Jeffrey E. Bischoff,et al.  Reduced Parameter Formulation for Incorporating Fiber Level Viscoelasticity into Tissue Level Biomechanical Models , 2006, Annals of Biomedical Engineering.

[180]  Wing Kam Liu,et al.  Nonlinear Finite Elements for Continua and Structures , 2000 .

[181]  Serge Winitzki,et al.  Uniform Approximations for Transcendental Functions , 2003, ICCSA.

[182]  J. Middleton,et al.  An Anisotropic Viscous Hyperelastic Constitutive Model of the Posterior Cruciate Ligament Suitable for High Loading Rate Situations , 2005 .

[183]  Michael J Sherratt,et al.  Age-Related Tissue Stiffening: Cause and Effect. , 2013, Advances in wound care.

[184]  B. Audoly,et al.  Influence of Stratum Corneum on the entire skin mechanical properties, as predicted by a computational skin model , 2013, Skin research and technology : official journal of International Society for Bioengineering and the Skin (ISBS) [and] International Society for Digital Imaging of Skin (ISDIS) [and] International Society for Skin Imaging.

[185]  N. Yoshikawa,et al.  Microrelief suppresses large wrinkling appearance: an in silico study , 2015, Skin research and technology : official journal of International Society for Bioengineering and the Skin (ISBS) [and] International Society for Digital Imaging of Skin (ISDIS) [and] International Society for Skin Imaging.

[186]  Alain Goriely,et al.  Differential growth and instability in elastic shells. , 2005, Physical review letters.

[187]  Mélanie Ottenio,et al.  Strain rate and anisotropy effects on the tensile failure characteristics of human skin. , 2015, Journal of the mechanical behavior of biomedical materials.

[188]  J. Lagarde,et al.  In vivo model of the mechanical properties of the human skin under suction , 2000, Skin research and technology : official journal of International Society for Bioengineering and the Skin (ISBS) [and] International Society for Digital Imaging of Skin (ISDIS) [and] International Society for Skin Imaging.

[189]  T. Holliday.  :Skin: A Natural History , 2007 .

[190]  C. Schär.  Climate extremes: The worst heat waves to come , 2016 .

[191]  R. Ogden,et al.  Hyperelastic modelling of arterial layers with distributed collagen fibre orientations , 2006, Journal of The Royal Society Interface.

[192]  F. Silver,et al.  Viscoelastic properties of human skin and processed dermis , 2001, Skin research and technology : official journal of International Society for Bioengineering and the Skin (ISBS) [and] International Society for Digital Imaging of Skin (ISDIS) [and] International Society for Skin Imaging.

[193]  Yoram Lanir,et al.  Viscoelasticity and preconditioning of rat skin under uniaxial stretch: microstructural constitutive characterization. , 2009, Journal of biomechanical engineering.

[194]  Alain Goriely,et al.  Instabilities in elastomers and in soft tissues , 2006, 0711.4664.

[195]  S. R. Bodner,et al.  A three-dimensional nonlinear model for dissipative response of soft tissue , 2002 .

[196]  M. B. Rubin,et al.  A model for the anisotropic response of fibrous soft tissues using six discrete fibre bundles , 2011 .

[197]  C. G. Neumann.  THE EXPANSION OF AN AREA OF SKIN BY PROGRESSIVE DISTENTION OF A SUBCUTANEOUS BALLOON: Use of the Method for Securing Skin for Subtotal Reconstruction of the Ear , 1957, Plastic and reconstructive surgery.

[198]  R. Ogden.  Nonlinear Continuum Mechanics and Modeling the Elasticity of Soft Biological Tissues with a Focus on Artery Walls , 2017 .

[199]  E Kuhl,et al.  Computational modeling of chemo-bio-mechanical coupling: a systems-biology approach toward wound healing , 2016, Computer methods in biomechanics and biomedical engineering.

[200]  F. Peyraut,et al.  A new based error approach to approximate the inverse langevin function , 2014, Rheologica Acta.

[201]  Georges Limbert,et al.  A constitutive model of the posterior cruciate ligament. , 2006, Medical engineering & physics.

[202]  Jay D. Humphrey,et al.  An invariant basis for natural strain which yields orthogonal stress response terms in isotropic hyperelasticity , 2000 .

[203]  E. Kuhl,et al.  A novel strategy to identify the critical conditions for growth-induced instabilities. , 2014, Journal of the mechanical behavior of biomedical materials.

[204]  Paul Steinmann,et al.  Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound , 2016 .

[205]  S. Derler,et al.  Friction of human skin against smooth and rough glass as a function of the contact pressure , 2009 .

[206]  Paul Ackermans,et al.  In vitro indentation to determine the mechanical properties of epidermis. , 2011, Journal of biomechanics.

[207]  Brendan A O McCormack,et al.  A three-layer model of skin and its application in simulating wrinkling , 2009, Computer methods in biomechanics and biomedical engineering.

[208]  B. de Campos Vidal,et al.  Skin Collagen Fiber Molecular Order: A Pattern of Distributional Fiber Orientation as Assessed by Optical Anisotropy and Image Analysis , 2013, PloS one.

[209]  O. Kratky,et al.  Röntgenuntersuchung gelöster Fadenmoleküle , 1949 .

[210]  H. G. Allen.  WRINKLING AND OTHER FORMS OF LOCAL INSTABILITY , 1969 .

[211]  Georges Limbert,et al.  Trabecular bone strains around a dental implant and associated micromotions--a micro-CT-based three-dimensional finite element study. , 2010, Journal of biomechanics.

[212]  Markus J. Buehler,et al.  Large-Scale Hierarchical Molecular Modeling of Nanostructured Biological Materials , 2006 .

[213]  David W. DeGroot,et al.  Extremes of human heat tolerance: life at the precipice of thermoregulatory failure , 2004 .

[214]  C. Linder,et al.  An algorithmic approach to multi-layer wrinkling , 2016 .

[215]  H Alexander,et al.  Accounting for natural tension in the mechanical testing of human skin. , 1977, The Journal of investigative dermatology.

[216]  Georges Limbert,et al.  A mechanistic insight into the mechanical role of the stratum corneum during stretching and compression of the skin. , 2015, Journal of the mechanical behavior of biomedical materials.

[217]  H. Dobrev.  Use of Cutometer to assess epidermal hydration , 2000, Skin research and technology : official journal of International Society for Bioengineering and the Skin (ISBS) [and] International Society for Digital Imaging of Skin (ISDIS) [and] International Society for Skin Imaging.

[218]  N. Yoshikawa,et al.  Skin wrinkling morphology changes suddenly in the early 30s , 2012, Skin research and technology : official journal of International Society for Bioengineering and the Skin (ISBS) [and] International Society for Digital Imaging of Skin (ISDIS) [and] International Society for Skin Imaging.

[219]  J. Weiss,et al.  Finite element implementation of incompressible, transversely isotropic hyperelasticity , 1996 .

[220]  Cwj Cees Oomens,et al.  A numerical‐experimental method to characterize the non‐linear mechanical behaviour of human skin , 2003, Skin research and technology : official journal of International Society for Bioengineering and the Skin (ISBS) [and] International Society for Digital Imaging of Skin (ISDIS) [and] International Society for Skin Imaging.

[221]  S Mantero,et al.  Possible role of decorin glycosaminoglycans in fibril to fibril force transfer in relative mature tendons--a computational study from molecular to microstructural level. , 2003, Journal of biomechanics.

[222]  Zvi Hashin,et al.  Continuum Theory of the Mechanics of Fibre-Reinforced Composites , 1984 .

[223]  W. Kuhn,et al.  Beziehungen zwischen Molekülgröße, statistischer Molekülgestalt und elastischen Eigenschaften hochpolymerer Stoffe , 1936 .

[224]  L. Chan,et al.  Human skin basement membrane in health and in autoimmune diseases. , 1997, Frontiers in bioscience : a journal and virtual library.

[225]  E. Kuhl,et al.  A continuum model for remodeling in living structures , 2007 .

[226]  J. Humphrey.  Continuum biomechanics of soft biological tissues , 2003 .

[227]  Alain Goriely,et al.  Growth and instability in elastic tissues , 2005 .

[228]  E. Kuhl,et al.  Period-doubling and period-tripling in growing bilayered systems , 2015, Philosophical magazine.

[229]  John Williams,et al.  Human anatomy and physiology , 2019, Human anatomy and physiology.

[230]  J C Barbenel,et al.  The time-dependent mechanical properties of skin. , 1977, The Journal of investigative dermatology.

[231]  G. Limbert,et al.  Thermoelastic modelling of the skin at finite deformations. , 2016, Journal of thermal biology.

[232]  Andreas Menzel,et al.  Anisotropic micro-sphere-based finite elasticity applied to blood vessel modelling , 2009 .

[233]  G. Piérard,et al.  Assessment of Aging and Actinic Damages by Cyanoacrylate Skin Surface Strippings , 1987, The American Journal of dermatopathology.

[234]  Xi-Qiao Feng,et al.  Instabilities of soft films on compliant substrates , 2017 .

[235]  Y Lanir,et al.  Two-dimensional mechanical properties of rabbit skin. II. Experimental results. , 1974, Journal of Biomechanics.

[236]  Yoram Lanir,et al.  Multi-scale Structural Modeling of Soft Tissues Mechanics and Mechanobiology , 2017 .

[237]  Qi-Chang He,et al.  Conewise linear elastic materials , 1994 .

[238]  L. Taber.  Biomechanics of Growth, Remodeling, and Morphogenesis , 1995 .

[239]  Gerhard A. Holzapfel,et al.  On Fiber Dispersion Models: Exclusion of Compressed Fibers and Spurious Model Comparisons , 2017 .

[240]  Ellen Kuhl,et al.  Computational modeling of skin: Using stress profiles as predictor for tissue necrosis in reconstructive surgery. , 2014, Computers & structures.

[241]  Mikhail Itskov,et al.  A simple and accurate approximation of the inverse Langevin function , 2015, Rheologica Acta.

[242]  Brendan A O McCormack,et al.  Simulating the wrinkling and aging of skin with a multi-layer finite element model. , 2010, Journal of biomechanics.

[243]  Ellen Kuhl,et al.  Characterization of living skin using multi-view stereo and isogeometric analysis. , 2014, Acta biomaterialia.

[244]  E Mazza,et al.  Nonlinear elastic-viscoplastic constitutive equations for aging facial tissues , 2005, Biomechanics and modeling in mechanobiology.

[245]  K Y Volokh,et al.  Modeling failure of soft anisotropic materials with application to arteries. , 2011, Journal of the mechanical behavior of biomedical materials.

[246]  M. Epstein.  Mathematical characterization and identification of remodeling, growth, aging and morphogenesis , 2015 .

[247]  Alexandre Delalleau,et al.  Characterization of the mechanical properties of skin by inverse analysis combined with the indentation test. , 2006, Journal of biomechanics.

[248]  D P Pioletti,et al.  Viscoelastic constitutive law in large deformations: application to human knee ligaments and tendons. , 1998, Journal of biomechanics.

[249]  Christopher J. Ploch,et al.  Author ' s personal copy Growing skin : A computational model for skin expansion in reconstructive surgery , 2011 .

[250]  Feng Xu,et al.  Introduction to Skin Biothermomechanics and Thermal Pain , 2011 .

[251]  J M Mansour,et al.  Dynamic measurement of the viscoelastic properties of skin. , 1991, Journal of biomechanics.

[252]  Thao D Nguyen,et al.  Full-field bulge test for planar anisotropic tissues: part II--a thin shell method for determining material parameters and comparison of two distributed fiber modeling approaches. , 2013, Acta biomaterialia.