Biomechanical aspects of soft tissues

Benjamin Loret

Université de Grenoble, France

Fernando M.F. Simões

Instituto Superior Técnico

Universidade de Lisboa, Portugal

November 19, 2016

Crée comme Dieu Ordonne comme un roi Travaille comme un esclave Constantin Brancusi

Contents

Page

Contents 4

1 Biomechanical topics in soft tissues 23 1.1 Introduction...... 23 1.2 Diffusion, convection, osmosis: towards wearable artificialkidneys...... 24 1.3 Issuesindrugdelivery ...... 25 1.4 Macroscopic models of tissues, interstitium and membranes ...... 27 1.5 Energy couplings, passive and active transports ...... 28 1.6 Moregeneraldirectand reversecouplings ...... 29 1.7 Tissue engineering re-directed to tumor tissue exploration ...... 29 1.8 Frommechanicstobiomechanics ...... 30 1.9 Mechanisms of injury of the knee, ...... 31 1.10 Water and solid constituents of soft tissues ...... 33

I Solids and multi-species mixtures as open systems : a perspective 35

2 Elements of continuum mechanics 37 2.1 Algebraic relations and algebraic operators ...... 38 2.1.1 Notations and conventions for tensor calculus ...... 38 2.1.2 Algebraicoperators ...... 39 2.2 Characteristic polynomial, eigenvalues and eigenvectors ...... 42 2.2.1 Eigenvaluesandeigenvectors ...... 42 2.2.2 Characteristic polynomial of a second order tensor ...... 43 2.2.3 Scalarinvariantsof the stress deviator ...... 44 2.2.4 Spectral analysis of a symmetrized dyadic product of twovectors ...... 46 2.3 Afewusefultensorialrelations ...... 47 2.3.1 Cayley-Hamiltontheorem ...... 47 2.3.2 Application to the polar decomposition F = Q · U ...... 47 2.3.3 GeneralizedCayley-Hamiltonrelation ...... 48 2.3.4 The tensor equation X · A + A · X =2 S ...... 48 2.3.5 Rank one and rank two modifications of the identity ...... 49 2.3.6 Square-root of a positive definite symmetric second ordertensor ...... 50 2.4 The differential operators of continuum mechanics ...... 51 2.4.1 Definitions of the differential operators ...... 51 2.4.2 Applicationtokinematics ...... 55 2.4.3 Spherical, cylindrical and polar coordinates ...... 56 2.4.4 Convectiveacceleration ...... 61 2.5 Measuresofstrain ...... 63 2.5.1 Polar decomposition of the deformation gradient ...... 63 2.5.2 Spectral decomposition of the right and left Cauchy-Greentensors ...... 63 2.5.3 Lagrangianand Eulerian strain measures ...... 64

4 2.5.4 Rate of deformation and rates of strain measures ...... 64 2.5.5 Representationtheorems for rotations ...... 65 2.5.6 Infinitesimalstrains ...... 69 2.5.7 Volume change and incompressibility ...... 70 2.6 Transports between reference and current configurations...... 70 2.7 Time derivatives and Reynolds theorems ...... 73 2.8 Work-conjugatestress-strainpairs ...... 75 2.8.1 Stress conjugate to a given Lagrangian strain measure ...... 75 2.8.2 Issuesrelated to the logarithmicstrain ...... 76 2.9 Smalluponlarge ...... 77 2.10 Kinematical constraints and reaction stresses ...... 78 2.11 Invariance,objectivity,isotropy ...... 79 2.11.1 Tensorinvariance...... 80 2.11.2 Material frame indifference or material objectivity ...... 80 2.11.3 Materialsymmetries ...... 84 Exercises on Chapter 2 ...... 90 3 Thermodynamic properties of fluids 95 3.1 Basicthermodynamicdefinitions ...... 95 3.1.1 Open and closed systems. Intensive and extensive properties ...... 95 3.1.2 Energy, free energy, enthalpy, free enthalpy, Legendreduals ...... 96 3.1.3 Chemical equilibrium between two multi-species phases ...... 97 3.1.4 TheGibbs-Duhemrelation ...... 98 3.1.5 Chemical affinity associated with a chemical reaction ...... 99 3.1.6 Chemical equilibrium versus steady state ...... 100 3.1.7 Flux- and energy-driven chemical reactions ...... 100 3.2 Phasechange ...... 102 3.2.1 Phasechangeand the Clapeyronformulas ...... 102 3.2.2 The principle of Spite and the Kelvin-Helmholtz formula...... 105 3.2.3 Thetimecourseofadesiccationprocess ...... 107 3.2.4 Negative pressures, metastability and cavitation ...... 108 3.3 Thermodynamic functions of fluids ...... 109 3.3.1 Theperfectoridealgas ...... 110 3.3.2 Compressible and incompressible liquids ...... 111 3.3.3 Explicit thermodynamic functions for liquids ...... 112 3.3.4 Liquid-vapor equilibrium curve: IAPWS and simplified formulations ...... 118 3.4 Laplace’s law of mechanical equilibrium at the interface between two immiscible fluids . . 123 3.5 Henry’s law and Raoult’s law of chemical equilibrium betweenaliquidandagas . . . . . 124 3.6 Composition of fluids in plasma, interstitium, cell ...... 125 4 Multi-species mixtures as thermodynamically open systems 129 4.1 Thethermodynamicallyopensystem ...... 130 4.2 Chemomechanical behavior and growth of soft tissues ...... 130 4.3 Generalformofbalanceequations ...... 134 4.3.1 Partial, apparent and intrinsic properties ...... 134 4.3.2 Mass, geometry and kinematical descriptors ...... 134 4.3.3 Integralandlocalbalanceequations ...... 135 4.4 Balanceofmass...... 137 4.4.1 Balanceofmassforspecies ...... 137 4.4.2 Balanceofmassforthemixture ...... 138 4.4.3 Diffusion,masstransferandgrowth ...... 138 4.5 Balanceofmomentum ...... 139 4.5.1 Balanceofmomentumforspecies...... 139 4.5.2 Balanceofmomentumforthemixture ...... 140 4.5.3 Interactionmomenta...... 141 4.5.4 Interactionstresses...... 141 4.6 Balanceofenergy...... 142 4.6.1 Notation of thermodynamic functions in a mixture context ...... 142 4.6.2 First principle of thermodynamics ...... 143 4.6.3 Balanceofenergyforspecies ...... 143 4.6.4 Balance of energy for the mixture: direct derivation ...... 144 4.6.5 Thecaseofanelectrolyte ...... 146 4.7 Balance of entropy and Clausius-Duhem inequality ...... 146 4.7.1 Balanceofentropyforspecies...... 147 4.7.2 Internal and external entropy productions for a species...... 147 4.7.3 Second principle of thermodynamics ...... 149 4.7.4 TheClausius-Duheminequality ...... 150 4.8 Dissipationmechanisms ...... 153 4.8.1 Dissipation due to generalized diffusion and transfer ...... 154 4.8.2 Spontaneous reactions and stable configurations ...... 154 4.8.3 Coupledchemicalreactions ...... 156 4.8.4 Dissipation associated with a chemical reaction ...... 157 4.8.5 Are growth and structuration dissipative processes? ...... 158 4.8.6 Onsagerrelationsandratepotential ...... 158 4.9 General constitutive principles ...... 159 Appendices of Chapter 4 ...... 161

5 Anisotropic and conewise 165 5.1 Hypoelasticity, elasticity and hyperelasticity ...... 166 5.1.1 Ahypoelasticmaterial...... 167 5.1.2 Anelasticmaterial...... 168 5.1.3 Aparticularhyperelasticmaterial ...... 168 5.2 Anisotropiclinearelasticity ...... 169 5.2.1 Matrixrepresentation ...... 169 5.2.2 Minorandmajorsymmetries ...... 170 5.2.3 Microstructureandsymmetrygroups...... 170 5.2.4 From anisotropic to isotropic linear hyperelasticity...... 173 5.2.5 Linearisotropicelasticmaterials ...... 176 5.2.6 Linearorthotropicelastic materials ...... 177 5.2.7 Prestress/prestrain in linear hyperelastic materials...... 180 5.2.8 Linear transversely isotropic elastic materials ...... 180 5.2.9 Linearcubicelasticity ...... 183 5.2.10 Saint-Venant’s relations for linear elasticity ...... 184 5.2.11 A particular elastic material with fabric anisotropy...... 184 5.3 Spectralanalysisoftheelasticmatrix ...... 186 5.3.1 Spectral analysis: linear elastic isotropic materials ...... 187 5.3.2 Spectral analysis: linear elastic transversely isotropicmaterials ...... 187 5.3.3 Spectral analysis: linear elastic orthotropic materials...... 188 5.4 Fiber-reinforced materials: extension-contraction response ...... 189 5.4.1 Basic remarks on the mechanical response of fiber-reinforcedtissues ...... 189 5.4.2 Conewise constitutive response: generalities ...... 190 5.4.3 Application to materials with one family of fibers ...... 191 5.4.4 Application to materials with two families of fibers ...... 191 5.4.5 Application to linear orthotropic materials: octantwiseresponse ...... 192 5.4.6 Application to transversely isotropic materials ...... 194 5.4.7 Application to isotropic materials ...... 194 5.4.8 Cartilageasa tissuewith cubic symmetry ...... 194 5.4.9 Cartilageasanorthotropictissue ...... 194 Exercises on Chapter 5 ...... 198

6 Hyperelasticity, a purely mechanical point of view 209 6.1 Restrictions to the strain energy function ...... 210 6.1.1 Mathematicalrequirements ...... 210 6.1.2 Constitutive requirements of mechanical nature ...... 211 6.2 Properties of constitutive functions and partial differentialequations ...... 211 6.2.1 Positivedefiniteness ...... 212 6.2.2 Ellipticity and Strong Ellipticity ...... 213 6.2.3 Convexity...... 215 6.2.4 Polyconvexity...... 216 6.3 Totalpotentialenergy ...... 217 6.4 Isotropichyperelasticity ...... 218 6.5 Anisotropic hyperelasticity and Fung’s strain energy ...... 220 6.5.1 Exponentialstrainenergy ...... 220 6.5.2 Locking versus softening materials: exponentials andenergylimiters ...... 221 6.5.3 Strong Ellipticity of Fung’s like strain energies (SE) ...... 222 6.5.4 Convexity of Fung’s like strain energies ...... 224 6.6 Elastic potentials with the elastic-growth multiplicativedecomposition ...... 224 6.6.1 Isotropic Fung’s like potentials ...... 225 6.6.2 Multiplicative volumetric-isochoric decomposition of the deformation gradient . . . 226 6.6.3 Additive decomposition of anisotropic Fung’s like potentials ...... 227 6.7 Return to the conewise response in presence of fibers ...... 229 6.8 Elasticity and workless constitutive stress ...... 229 6.9 UseofrightandleftCauchy-Greentensors ...... 230 Exercises on Chapter 6 ...... 232 7 Poroelasticity with a single porosity 235 7.1 Geometrical, kinematical and mechanical descriptors ...... 236 7.2 Elements of mixture theory: the level/scale of the species ...... 237 7.2.1 Intrinsicpropertiesofspecies ...... 237 7.2.2 Entropyproductionforaspecies ...... 242 7.3 Anisotropic poroelasticity: a composite material format ...... 243 7.3.1 Mechanical, viscous and inertial couplings ...... 243 7.3.2 supplies and Darcy’s law ...... 244 7.3.3 Temperature dependence of the viscosity coefficient of liquids and gases ...... 246 7.3.4 Momentum supplies, the viscous fluid and Brinkman’s law...... 246 7.3.5 Derivation of the momentum equations via Hamilton’s principle...... 247 7.3.6 Anisotropic poroelasticity in terms of apparent stressesandstrains ...... 247 7.4 Anisotropic poroelasticity: drained and undrained properties ...... 253 7.4.1 Constitutive equations in terms of drained properties...... 253 7.4.2 Constitutive equations in terms of undrained properties ...... 256 7.4.3 Independentmeasurements ...... 258 7.4.4 Compatibility conditions for isotropic porous media ...... 259 7.4.5 Compatibility conditions for transversely isotropicporousmedia ...... 260 7.4.6 Compatibility conditions for orthotropic porous media...... 263 7.4.7 Porousmediawithfabricanisotropy ...... 264 7.5 Unconfined compression of a poroelastic cylinder ...... 266 7.5.1 Poroelastic field and constitutive equations ...... 266 7.5.2 Axisymmetric deformations independent of the axial coordinate...... 267 7.5.3 SolutionintheLaplacedomain ...... 268 7.5.4 Relaxationtests: appliedaxialstrain ...... 270 7.5.5 Creeptests: appliedaxialload ...... 274 7.5.6 CompendiumofBesselfunctions ...... 276 8 Viscoelasticity and poro-viscoelasticity 281 8.1 Viscoelasticity, poroelasticity and poro-viscoelasticity...... 282 8.2 Intrinsic time dependent behavior of collagen fibrils ...... 282 8.3 Complexmodulus ...... 283 8.3.1 Storageandlossmoduli ...... 283 8.3.2 Timetranslationinvariance ...... 286 8.3.3 Thefadingmemoryprinciple ...... 289 8.3.4 BasicsofLaplace-Carsontransform...... 289 8.3.5 Losscoefficientandhysteresisloops ...... 290 8.3.6 Complex modulus of the standard linear solid model ...... 291 8.4 Relaxationspectrum ...... 292 8.4.1 Adiscretespectrum ...... 292 8.4.2 Acontinuousspectrum ...... 293 8.4.3 Aspecificrelaxationspectrum ...... 294 8.5 The quasi-linear viscoelasticity QLV model of Fung ...... 295 8.5.1 Elastic response and reduced relaxation function ...... 295 8.5.2 The three-dimensional reduced relaxation tensor ...... 296 8.6 Overall viscoelastic properties: ECM and a contractile cell...... 296 8.7 An electrical analogy applied to the corneal endothelium...... 297 8.8 Fluid infusion in a viscoelastic polymer gel ...... 299 8.8.1 Poro-viscoelastic field and constitutive equations ...... 299 8.8.2 The initial and boundary value problem ...... 300 8.8.3 The differential equation for the dilatation, or pressure...... 301 8.8.4 Radial displacement in a spherical setting ...... 301 8.8.5 Thesolutionintimedomain ...... 303 8.9 Confined compression of a viscoelastic polymer gel ...... 306 8.9.1 ThesolutionintheLaplacedomain ...... 307 8.9.2 Thesteadystatesolution ...... 307 8.9.3 Thetransientsolution ...... 307 8.10 Compression of a poroelastic layer: displacement control...... 308 8.10.1 Theanalyticalsolution...... 309 8.10.2 The elastic gel as a limit of the viscous gel with a zero relaxation time ...... 310 8.10.3 Inversion of the Laplace transform: the minimum numberofpoles ...... 310 8.11 Visco-hyperelasticity and other memory effects ...... 310 8.11.1 Linearviscousfluids ...... 311 8.11.2 Maxwell and Kelvin models in a dynamic context ...... 311 8.11.3 Rate effects as non-equilibrium thermodynamically consistent processes ...... 312 8.11.4 Visco-hyperelasticity ...... 314 8.11.5 Longrangememory ...... 314 8.11.6 Fadingmemoriesinamixturecontext ...... 314 8.11.7 Nonlinear viscoelastic models for collagen gels ...... 315 Exercises on Chapter 8 ...... 316 9 Thermoelasticity and thermo-poroelasticity 321 9.1 Thermomechanical properties of elastic solids ...... 322 9.1.1 Isothermalandisentropicmoduli ...... 322 9.1.2 Temperaturedependence of heat capacity ...... 324 9.1.3 The energy equationof thermoelasticity ...... 325 9.1.4 Cattaneo’sheatconductor...... 328 9.1.5 CauchystressversusEshelbystress...... 329 9.1.6 FreeenergyversusMassieufunction ...... 329 9.2 Thermoelastic heat, stored energy and dissipated energy...... 330 9.2.1 A thermomechanical model including an internal variable ...... 331 9.2.2 Plasticworkandinelasticheat ...... 332 9.2.3 Plastic work, recoverable and irrecoverable stored energies...... 333 9.3 Anisotropic thermo-poroelasticity : mechanics, transport,energy ...... 334 9.3.1 Thermodynamicderivation ...... 335 9.3.2 Linearthermo-poroelasticity ...... 335 9.3.3 Coupledflows: thermoosmosis ...... 338 9.3.4 Energy equation of a porous medium in local thermal equilibrium ...... 340 9.4 Generalized compatibility conditions for the stress ...... 341 9.4.1 Compatibility conditions for the stress for linear isotropic elasticity ...... 341 9.4.2 Compatibility conditions for the stress in presence of an inelastic strain ...... 342 9.4.3 Compatibility conditions for spherical problems withradialsymmetry ...... 342 9.5 Heatingathermo-poroelasticmedium ...... 343 9.5.1 Thermo-poroelastic field and constitutive equations ...... 343 9.5.2 Thermo-poroelastic field equations in a 1D context ...... 345 9.5.3 Prescribed temperature and drained boundary ...... 346 9.5.4 Prescribed temperature and impervious boundary ...... 348 9.5.5 Prescribed thermal flux and drained boundary ...... 348 9.5.6 Prescribed thermal flux and impervious boundary ...... 349 9.5.7 Mechanical output, strain, stress and displacement ...... 350 9.5.8 Simulations, drained and undrained limits ...... 350 9.5.9 Extension of the above model including thermoosmosis ...... 354 9.5.10 Chemo-poroelasticity versus thermo-poroelasticity ...... 354 Exercises on Chapter 9 ...... 355 10 Transfers of mass, momentum and energy 357 10.1 Summaryofbalanceequations ...... 358 10.2 Intercompartmentmasstransfers ...... 359 10.2.1 Linearmasstransfer ...... 359 10.2.2 Mass transfer of electrically charged species ...... 360 10.2.3 Nonlinear of mass transfer ...... 361 10.3 Momentum supplyin a poroelasticcontext ...... 362 10.4 Thebioheatequation...... 362 10.4.1 Thebioheatenergyequation ...... 362 10.4.2 The bioheat energy equation with a relaxation time ...... 363 10.4.3 Thermalboundaryconditions ...... 363 10.4.4 SignificanceoftheBiotnumber ...... 363 10.5 Mass and energy transfers between a liquid and a solid ...... 364 10.5.1 Constitutive equations for the solid and liquid ...... 364 10.5.2 Constitutive equations for mass and heat transfers ...... 365 10.6 A porous medium with two porosities and three temperatures...... 366 10.6.1 Thermomechanicalequations ...... 368 10.6.2 Generalizeddiffusion...... 374 10.6.3 Massandenergytransfers ...... 375 10.6.4 Linear constitutive equations of mass transfer ...... 375 10.6.5 Linear constitutive equations of energy transfer ...... 377 10.6.6 Theenergyequations ...... 377 10.6.7 Thecompletesetoffieldequations ...... 379 10.6.8 Increasing and decreasing the degree of sophistication ...... 379 10.7 A porous medium with a single porosity and two temperatures ...... 380 10.8 Interfacial transfer coefficients and specific surface areas ...... 381 10.8.1 Interfacial heat transfer coefficients ...... 381 10.8.2 Specificsurfacearea ...... 383 10.9 The coefficients of transfer: two scale derivation ...... 385 10.9.1 Smallscaleboundaryvalueproblems ...... 385 10.9.2 Heat transfer with a characteristic time ...... 387 10.9.3 Identification of the transfer coefficients ...... 388 11 Waves in thermoelastic solids and saturated porous media 391 11.1Basicdefinitions ...... 392 11.1.1 Phase wave and discontinuity wave ...... 392 11.1.2 LagrangianandEuleriananalyses ...... 393 11.1.3 Hadamard’s compatibility relations in the current configuration...... 394 11.1.4 Conservation laws in a solid domain with a crossing discontinuity ...... 395 11.1.5 Conservation laws in a mixture with a crossing discontinuity ...... 401 11.1.6 Propagation of discontinuities in a solid domain ...... 402 11.2 Wavesinelasticsolids ...... 405 11.2.1 Jump relations for accelerationwaves ...... 405 11.2.2 Acceleration waves in linear isotropic materials ...... 406 11.2.3 Acoustic tensor of a linear elastic ...... 406 11.2.4 Acoustic tensor of a linear transversely isotropic material ...... 407 11.2.5 Acceleration waves in materials with fabric anisotropy...... 407 11.2.6 Wave motions and Helmholtz potentials in isotropic elasticsolids ...... 409 11.3 Acceleration waves in thermoelastic solids ...... 410 11.3.1 Acceleration waves in solid conductors obeying Fourier’sheatlaw...... 410 11.3.2 Acceleration waves in solid conductors obeying Cattaneo’sheatlaw...... 411 11.3.3 Acceleration waves in non conductor thermoelastic solids ...... 411 11.4 Acceleration waves and acoustic waves in fluids ...... 412 11.4.1 Acceleration waves in a frictionless ideal fluid ...... 412 11.4.2 Acoustic waves in a frictionless ideal fluid ...... 413 11.4.3 Regimes in the steady irrotational flow in a barotropicfluid...... 413 11.4.4 ANewtonianviscousfluid ...... 414 11.5 Acceleration waves in saturated porous media ...... 414 11.5.1 The characteristic equation for the wave-speeds ...... 415 11.5.2 Acceleration wave-speeds in isotropic porous media ...... 417 11.5.3 Stationarydiscontinuity ...... 423 11.5.4 Interlacing and separation properties ...... 423 11.5.5 Inertial coupling: influence on acceleration waves ...... 426 11.5.6 Waves in fluid saturated porous media: pushing the windowajar ...... 428 11.6 Propagation of acceleration waves in saturated porous media ...... 429 11.6.1 Equationsofpropagation ...... 429 11.6.2 Thedecayingwave-front...... 430 11.7 Plane motions and Helmholtz potentials in poroelasticity ...... 431 11.7.1 Displacements and strains in terms of Helmholtz potentials ...... 431 11.7.2 Poroelastic stresses in terms of Helmholtz potentials ...... 432 11.7.3 Field equations in terms of Helmholtz potentials ...... 432 11.7.4 Boundary conditions in a dynamic non dissipative context...... 433 11.8 Silentporoelasticboundaries ...... 434 11.8.1 Impedance matrix, general case λsw =06 ...... 435 11.8.2 Impedance matrix: no poromechanical coupling, λsw =0 ...... 436 11.8.3 Impedance matrix: dry/drained medium ...... 436 11.8.4 Impedance matrix: undrained medium ...... 436 11.9 Surface waves in a saturated porous half-plane ...... 437 11.9.1 Time harmonic solutions decreasing with depth ...... 437 11.9.2 Thecompleteformalsolution ...... 438 11.9.3 Frequencydependentdissipation ...... 438 11.9.4 Determination of the coefficients of depth dependence Aj , j ∈ [1, 3] ...... 439 11.9.5 Permeableand tractionfreesurface...... 440 11.9.6 Impermeablefreesurface ...... 446 11.9.7 TheKelvinfunctionsberandbei ...... 449 11.9.8 Surface waves in an elastic half-plane ...... 451 11.9.9 Theprincipleoftheargument...... 453 11.9.10 Descartes’sruleofsigns...... 453 11.10 First order waves in saturated porous media ...... 454 11.10.1 Regimeofthefieldequations...... 455 11.10.2Formofthesolution...... 456 11.10.3 Discontinuities across the shear Mach line in regionsi)andii) ...... 458 11.10.4 Discontinuities across the longitudinal Mach line in regions i) and iii) ...... 460 11.10.5 Jumps of strains acrossfirst order waves ...... 460 11.10.6 Balances of momentum across first order waves ...... 460 Exercises on Chapter 11 ...... 461

II Electro-chemomechanical couplings in tissues with a fixed electrical charge 463

12 Directional averaging for fiber-reinforced tissues 465 12.1 Directionalanalysis...... 466 12.1.1 Spatialanddirectionalaveraging ...... 466 12.1.2 Directionalintegration ...... 466 12.2 A simple fiber model for the collagen network ...... 470 12.2.1 Asinglefamilyofcollagenfibers ...... 470 12.2.2 Random directional distribution of collagen fibers ...... 471 12.2.3 Conewise linear stress-strain response of collagen fibers ...... 473 12.2.4 Conewise nonlinear stress-strain response of collagenfibers ...... 476 12.2.5 Random planar directional distribution of collagen fibers...... 476 12.3 Directionalmodelsoftissues...... 478 12.3.1 Directional distribution of collagen fibers in cornea...... 478 12.3.2 Continuous planar directional distributions of collagenfibers ...... 479 12.3.3 A directional model of active rod-like cells ...... 481 12.4 The mechanical response of individual fibers ...... 482 12.4.1 The toe region, nonlinearity and fiber uncrimping ...... 482 12.4.2 Intrinsic viscous behavior of collagen fibrils ...... 484 12.5Fabrictensors...... 484 12.5.1 Elementary examplesof fabric tensors ...... 484 12.5.2 Higherorderfabrictensors ...... 486 12.6 Spatial homogenization over cells and ECM ...... 489 13 Electro-chemomechanical couplings 491 13.1 Chemomechanical couplings in engineering and biology ...... 492 13.2 Molarvolumes,electrostriction ...... 492 13.2.1 Molarmassesandmolarvolumes ...... 492 13.2.2 Themolarvolumeofdissolvedsalts ...... 493 13.2.3 Themolarvolumeofdissolvedions ...... 494 13.3 Electrokineticprocesses ...... 495 13.3.1 Laboratorysetups ...... 496 13.3.2 Developmentofanacidfront ...... 497 13.3.3 Electroinjection...... 497 13.4 Nanoscopicaspects...... 497 13.4.1 Thefixedelectricalcharge...... 497 13.4.2 MechanismsduetopH...... 499 13.5 Hydration, hydrolysis, complexation and solubility ...... 499 13.5.1 Solvationandhydration ...... 500 13.5.2 Hydrolysisandcomplexation ...... 501 13.5.3 Individual and simultaneous solubilities of NaCl andKCl ...... 504 13.6 Somebasicnotionsofelectrostatics ...... 505 13.6.1 Permittivity and relative permittivity ...... 505 13.6.2 Gauss’slawandPoisson’sequation ...... 505 13.6.3 Characteristic time of electric relaxation ...... 506 13.7 Semi-permeable membrane and osmotic effect ...... 507 13.8 ReverseorInverseosmosis...... 509 13.9 Electrical repulsion and electrical shielding ...... 510 13.10 The pore composition in materials with fixed charge ...... 510 13.10.1 Chemical consolidation and swelling in clays ...... 511 13.10.2 Modification of the electrolyte circulating an articularcartilage ...... 514 13.11 The heart muscle and cell electrophysiology ...... 522 13.12Reversecouplings ...... 522 13.12.1 Mechanoelectric feedback in the heart muscle ...... 523 13.12.2 Mechanobiology and engineered cartilages ...... 524 13.13 A partially coupled chemo-poroelastic model ...... 525 13.13.1 The chemo-poroelastic field and constitutive equations ...... 525 13.13.2 Uniaxial deformation under chemical loading ...... 526 13.13.3 Diffusionofthechemical ...... 527 13.13.4 Thesolutionforthedisplacement ...... 528 13.13.5 Thesolutionfortheporepressure ...... 529 13.13.6Theaxialforce...... 530 13.13.7 Simulations of chemical loadings ...... 530 Exercises on Chapter 13 ...... 533 14 Chemomechanical couplings in articular cartilages 541 14.1Overview ...... 542 14.2 Histological aspects of articular cartilages ...... 543 14.2.1 The three major types of cartilage in the human body ...... 543 14.2.2 Alayeredstructure...... 545 14.2.3 Compositionofcollagen ...... 546 14.2.4 Composition and structure of proteoglycans ...... 551 14.2.5 Physicalproperties...... 553 14.2.6 Electroneutrality in the two compartment context ...... 554 14.2.7 Methods of measurement of the fixed charge...... 554 14.2.8 Mechanicalproperties ...... 554 14.2.9 Typicallaboratoryexperiments ...... 555 14.2.10Cell density and spatial organization ...... 555 14.2.11Depth heterogeneities, split-lines and anisotropy ...... 556 14.2.12 Natural repair and transport of nutrients ...... 556 14.2.13 In vitro engineeredcartilage ...... 557 14.2.14 Interspecies, age and spatial heterogeneity ...... 557 14.3 Pathologies, osteoarthritis, rheumatoid arthritis ...... 557 14.4 Abriefreviewofmodelingaspects ...... 559 14.4.1 Intrafibrillar (IF) and extrafibrillar (EF) compartments ...... 559 14.4.2 Tensileexperiments ...... 561 14.4.3 The triphasic model of the Columbia University group ...... 561 14.4.4 The quadriphasic model of the Eindhoven University group ...... 562 14.4.5 TheexperimentsoftheMITgroup ...... 563 14.4.6 Molecular and microstructural models ...... 564 14.5 Interpretation of laboratory experiments ...... 564 14.5.1 Difficultiesofmethodology ...... 564 14.5.2 The fictitious bath: a tool for heterogeneous and transientprocesses ...... 564 14.6 Partitionofthetissueintophases...... 566 14.6.1 Definition of the phases: a measure of spatial organization...... 566 14.6.2 The fictitious membrane surrounding a cartilage specimen...... 567 14.6.3 Astronglyinteractingmodel ...... 568 14.6.4 Measuresofthe chemicalcomposition ...... 569 14.6.5 Workand electrochemicalpotentials ...... 570 14.6.6 Balanceequations ...... 575 14.6.7 Incompressibilityofspecies ...... 577 14.7 The constitutive structure: deformation, mass transfer,diffusion ...... 577 14.8 Chemoelasticenergyofthetissue ...... 578 14.8.1 The Gibbs-Duhem relation for the intrafibrillar fluid phase ...... 578 14.8.2 Electrochemical energy of the intrafibrillar phase ...... 579 14.8.3 Chemoelastic energy in terms of generalized strains ...... 579 14.8.4 Chemoelastic energy in terms of mixed generalized variables...... 582 14.9 Features of the constitutive framework ...... 583 14.9.1 Theeffectivestressformat...... 583 14.9.2 Equilibrium of the extrafibrillar phase with a sodium chloride bath ...... 584 14.9.3 Equilibrium between the two fluid phases ...... 585 14.9.4 Thereferencehypertonicstate ...... 585 14.9.5 Thelimitcaseofafreshbath ...... 586 14.9.6 Exhaustion and recharge of the intrafibrillar phase ...... 586 14.10 Remarks on constitutive frameworks and constraints ...... 586 Exercises on Chapter 14 ...... 588 15 Passive transport in the interstitium and circulation 593 15.1 Diffusionasamodeofpassivetransport ...... 594 15.2Coupledflows...... 595 15.3 Propagation,diffusion, convection...... 597 15.3.1 An example of hyperbolic PDE: propagation of a shock wave ...... 597 15.3.2 AparabolicPDE:diffusionofheat ...... 600 15.3.3 Diffusion of a solute in a medium of finite length ...... 605 15.3.4 A parabolic PDE displaying diffusion and convection ...... 606 15.3.5 Hyperbolicheatequation ...... 610 15.3.6 Moleculardiffusion...... 610 15.4 Physico-chemical processes involving diffusion, convectionandreaction ...... 610 15.4.1 Reaction-diffusion in a porous medium context ...... 611 15.4.2 Diffusionofoxygeninthetissue ...... 612 15.4.3 The reaction-diffusion equations of Turing [1952]...... 614 15.5 Diffusionversusconvection ...... 615 15.6 Newtonian viscous fluids and Reynolds number ...... 617 15.6.1 Acompressibleviscousfluid...... 617 15.6.2 Anincompressibleviscousfluid ...... 617 15.6.3 Hagen-Poiseuille flows along rigid and elastic walls ...... 618 15.7 Hydraulicconductivity...... 621 15.7.1 Circulation, theorems of Kelvin and Helmholtz ...... 622 15.7.2 Bernoulli equation for steady irrotational flow ...... 623 15.7.3 Hubbertpotential ...... 623 15.7.4 Porosity-permeabilityrelation ...... 624 15.7.5 Planarnetwork ...... 626 15.7.6 Three-dimensionalnetwork ...... 627 15.7.7 Dependence of permeability with respect to a characteristiclength ...... 627 15.7.8 The drag coefficient kSd for spherical and cylindrical particles ...... 628 15.7.9 ModificationsofDarcy’slaw ...... 629 15.7.10Measurement of pressure and tissue hydraulic conductivity via a manometer . . . 630 15.8 The steps of drug delivery, extravasation, transport in the interstitium ...... 631 15.8.1 Deliveryofmoleculesandparticles ...... 631 15.8.2 Thesuccessivesteps in drugdelivery ...... 632 15.8.3 Extravasationacross the vascular wall ...... 633 15.8.4 Transport within the interstitium ...... 634 15.8.5 Transportofcellsandadhesion ...... 635 15.9 Bloodcirculation ...... 636 15.9.1 The overallblood circulationnetwork ...... 636 15.9.2 Thevascularnetwork ...... 637 15.9.3 Capillaries: filtration and re-absorption ...... 638 15.10Mechanicsofvessels ...... 639 15.10.1 Constraints in the formation of vessels ...... 639 15.10.2Collapseoftheveins ...... 640 15.11 Transport of oxygen and carbon dioxide in blood ...... 641 15.11.1 Dissolutionandbindingofoxygen ...... 641 15.11.2 Dissolution and combination of carbon dioxide ...... 642 15.11.3 Respiration and circulation of oxygen ...... 643 15.12Basicsofenzymatickinetics ...... 644 15.12.1 Catalysisandenzymaticcatalysis ...... 644 15.12.2 Reversible and irreversible enzymatic kinetics ...... 646 15.12.3 The irreversible Michaelis-Menten model ...... 646 15.12.4 Two reversible Michaelis-Menten models ...... 647 15.12.5 Catalysisinhibition ...... 649 15.13Acid-baseequilibrium ...... 650 15.13.1 pHs in plasma, interstitium and cytoplasm ...... 650 15.13.2Buffersinplasma ...... 650 Exercises on Chapter 15 ...... 652 16 Coupled transports in tissues with a fixed electrical charge 657 16.1 Macroscopic transports in tissues with a fixed charge ...... 657 16.1.1 Overviewoftransportissues ...... 658 16.1.2 Theconstitutiveframework ...... 659 16.1.3 Thermodynamic arguments governing mass transfer anddiffusion...... 663 16.2 Generalized and coupled diffusion: structure of constitutive equations ...... 664 16.2.1 Diffusion in terms of fluxes relative to the solid ...... 664 16.2.2 Diffusion in terms of diffusive fluxes and current density...... 665 16.2.3 Relations between the diffusion matrices κ and K ...... 666 16.2.4 Procedures of identification of the constitutive functions...... 667 16.2.5 A general form of diffusion with mobile and non-mobile ions...... 667 16.2.6 The particular case of arrowhead diffusion matrices ...... 669 16.2.7 Summary of constitutive equations of transport ...... 670 16.2.8 Special cases: electrically and hydraulically open circuits...... 671 16.3 Generalized diffusion: the membrane effect ...... 672 16.3.1 Equilibrium relations between a gel and a bath ...... 673 16.3.2 The osmotic, or reflection, coefficient/matrix ...... 674 16.3.3 Appropriate interpretation of diffusion tests ...... 677 16.4 Generalized diffusion: ranges of the coefficients and refinements...... 679 16.4.1 Hydraulicconductivity...... 679 16.4.2 Diffusion coefficients and ionic mobilities ...... 680 16.4.3 Tortuosityfactor ...... 680 16.4.4 Electricalconductivity ...... 682 16.4.5 Electroosmoticcoefficient ...... 682 16.4.6 Variations and general trends of coefficients of generalized diffusion ...... 683 16.4.7 Perspective ...... 684 16.5 Generalized diffusion with thermal effects ...... 685 Exercises on Chapter 16 ...... 689 17 Effects of the pH on transport and mechanics 695 17.1 Overviewandlaboratoryobservations ...... 697 17.1.1 Indications on the biochemical composition of collagenandPGs ...... 697 17.1.2 Laboratoryobservations ...... 698 17.1.3 Scope ...... 699 17.2 pH agents embedded in the mixture framework ...... 700 17.2.1 Theconstituentsandphases...... 700 17.2.2 Macroscopic descriptors of the geometry and mass of themixture...... 701 17.3 Acid-base reactions in a thermodynamic framework ...... 702 17.3.1 Electrochemicalpotentials ...... 702 17.3.2 Spontaneousdissociationofwater ...... 703 17.3.3 The chemical affinities associated with the acid-base reactions...... 703 17.4 Calcium binding in a thermodynamic framework ...... 706 17.4.1 Bindingofmetallicions ...... 706 17.4.2 Calcium binding to carboxyl sites of PGs ...... 707 17.5 Variation of the fixed electrical charge with pH ...... 708 17.6 Biochemical composition of articular cartilages ...... 709 17.6.1 Molarmassesandvolumes...... 709 17.6.2 pKsoftheacid-basereactions...... 709 17.6.3 CompositionofGAGs ...... 709 17.6.4 Composition and partition of the charge of collagen ...... 710 17.6.5 Additional relative proportions ...... 711 17.6.6 Overallmassandvolumecomposition ...... 711 17.7 Concentrations of ions, sites and charges ...... 711 17.7.1 Physiologicalionic concentrations...... 711 17.7.2 Evolution of sites and charges as a function of the extrafibrillar pH ...... 712 17.8 Chemical equilibrium at the cartilage-bath interface ...... 712 17.8.1 Abinaryelectrolyte ...... 713 17.8.2 Aternaryelectrolyte...... 714 17.8.3 A ternary electrolyte and enthalpies of formation ...... 714 17.8.4 Sodium chloride electrolyte and enthalpies of formation ...... 715 17.8.5 Sodium chloride electrolyte and enthalpies of formation in absence of pH effect . . 716 17.9 Constitutive equations of generalized diffusion ...... 716 17.9.1 Fluxes...... 717 17.9.2 Generalized diffusion in presence of hydrogen and hydroxylions...... 717 17.10 Simulations of bath-cartilage equilibria ...... 718 17.10.1Materialdata ...... 719 17.10.2 Variationofionicstrength ...... 719 17.10.3 VariationofpHofthebath...... 721 17.10.4 Commentsonthetransportmodel...... 724 17.11 Constitutive mechanical framework: effects of pH and calcium binding ...... 725 17.11.1 Structure of the constitutive equations ...... 726 17.11.2Equilibria...... 728 17.11.3 A word on the kinetics of acid-base reactions and calcium binding ...... 728 17.12 Chemically induced stiffening/softening by fiber recruitment/deactivation ...... 729 17.12.1 Asinglefamilyofcollagenfibers ...... 729 17.12.2 Thestressofthecollagennetwork ...... 730 17.12.3 Chemoelastic energy with electro-chemomechanicalcouplings...... 731 17.12.4 Constitutiveparameters ...... 732 17.13 Mechanical and chemical tests at varying pH ...... 733 17.14 The limit case of a conewise linear collagen response ...... 737 17.14.1 The apparent Poisson’s ratio of the solid skeleton ...... 737 17.14.2 Isotropicfreeswelling ...... 737 17.14.3 Uniaxialtractionalongaxis1 ...... 738 17.15Commentsonthemechanicalmodel ...... 739 18 Finite element analysis of ECM couplings 741 18.1 Overviewofthefiniteelementanalysis ...... 742 18.1.1 Theconstitutiveframework ...... 742 18.1.2 IBVPs including chemomechanical couplings in articularcartilages ...... 743 18.1.3 Scope ...... 743 18.2 Field and constitutive equations ...... 744 18.2.1 Balanceequations ...... 744 18.2.2 Theelectrochemicalpotentials ...... 745 18.2.3 Chemoelasticrelations ...... 746 18.2.4 Equations of generalized diffusion in the extrafibrillarphase...... 747 18.2.5 Equations of mass transfer between the two fluid phases...... 747 18.3 Finiteelementformulation...... 748 18.3.1 Theprimaryvariables ...... 748 18.3.2 Thesemi-discreteequations ...... 748 18.3.3 Time integrationand equation solving ...... 750 18.3.4 Dependentvariables ...... 752 18.3.5 Aslightlydifferentapproach ...... 752 18.4 Testingsetupandmaterialdata...... 753 18.4.1 Specimengeometry...... 753 18.4.2 Initialdata ...... 753 18.4.3 Mechanicalandtransportparameters ...... 754 18.4.4 Characteristictimes ...... 755 18.5 Mechanical and chemical loadings with NaCl ...... 755 18.5.1 Mechanical consolidation at different bath chemistries ...... 756 18.5.2 Intermingled mechanical and chemical consolidation ...... 757 18.5.3 Reversibility ...... 758 18.5.4 Uniaxialfreeswelling ...... 759 18.5.5 Influence of the presence of the intrafibrillar compartment...... 759 18.5.6 Influence of the presence of a porous layer at the boundaries ...... 760 18.5.7 Evolution in time of spatial profiles ...... 761 18.5.8 The membrane surrounding the cartilage specimen ...... 764 18.6 CyclicsubstitutionofNaClandCaCl2 ...... 765 18.7 Improvements of the chemomechanical model ...... 767 Exercises on Chapter 18 ...... 768 19 Cornea and annulus fibrosus 775 19.1 Function,structureandcomposition ...... 776 19.1.1 Alayeredstructure...... 778 19.1.2 Histologyofthecornea ...... 779 19.1.3 Keratoconus ...... 783 19.1.4 Variationsoverspecies ...... 784 19.2 Biomechanicalaspects ...... 784 19.2.1 Osmoticswelling ...... 784 19.2.2 Bindingofchlorideions ...... 785 19.2.3 Electricalshielding ...... 785 19.2.4 EffectsofpH ...... 786 19.2.5 Dependence of material properties on PG and collagen content ...... 786 19.2.6 Partitionofwater ...... 786 19.3 Active transport in the endothelium: the cell and organ scales...... 788 19.3.1 Activetransport: theorganscale ...... 788 19.3.2 Activetransport: thecellscale ...... 789 19.3.3 Active transport: energetic coupling ...... 791 19.4 Physicaldata: literaturereview ...... 793 19.4.1 Measuresofthe fixed electricalcharge ...... 793 19.4.2 Standardchemicalcontent ofstroma ...... 794 19.4.3 Standard mechanical properties of stroma ...... 795 19.4.4 Pressures ...... 796 19.4.5 Standard transport properties of endothelium, epithelium and stroma ...... 798 19.4.6 Acquisitionofmechanicaldata ...... 801 19.5 Scattering of light by the corneal stroma and transmittance ...... 803 19.6Cornealsurgery...... 806 19.7 Corneaengineering...... 807 19.8 Theconstitutiveframework ...... 808 19.8.1 Definitionofthephases ...... 809 19.8.2 Geometry,volumesandmasses ...... 809 19.8.3 Electrochemicalpotentials ...... 810 19.8.4 Electroneutrality ...... 810 19.9 Biochemicalcompositionofstroma ...... 811 19.9.1 Molarmassesandmassdensities ...... 811 19.9.2 Composition: mole numbers, masses and volumes ...... 811 19.10 The fixed electrical charge: negatively charged PGs and chloride binding ...... 812 19.10.1 Chloride binding: partition of species and sites ...... 812 19.10.2 Chloride binding: the fixed electrical charge ...... 814 19.10.3 Chemical equilibrium at the cornea-bath interface ...... 814 19.11 Effects of pH and chloride binding on the fixed charge ...... 817 19.11.1 Contributions to the fixed electrical charge by GAGs ...... 818 19.11.2 Collagen contributions to the fixed electrical charge...... 819 19.11.3 Acid-base reactions, chloride binding and fixed charge...... 820 19.11.4 The mechanochemical effect: experimental data ...... 822 19.12 Global constitutive structure: mechanics and transport ...... 825 19.12.1Balanceequations ...... 825 19.12.2 Clausius-Duheminequality ...... 828 19.13 Generalized diffusion in the extrafibrillar phase ...... 829 19.14 Rest potential and active fluxes at the cell scale ...... 831 19.14.1 Diffusivefluxesacrossamembrane ...... 832 19.14.2 Reverse Nernst potential associated with an ion ...... 832 19.14.3 Reverse Nernst potential in presence of active transport...... 832 19.14.4 Huxley-Hodgkin-Katz (HHK) formulas in presence of severalions ...... 833 19.14.5 Thechordconductanceequation ...... 834 19.15 Coupled diffusion across a membrane and active transport ...... 835 19.15.1 Work-conjugate pairs for diffusion across a membrane...... 835 19.15.2 Dissipation in terms of absolute and diffusive fluxes ...... 836 19.15.3 Equivalence of the formulations based on absolute and diffusive fluxes ...... 838 19.15.4 Absolute and diffusive fluxes : charged membrane ...... 840 19.15.5 Absolute and diffusive fluxes: membrane without fixed charge ...... 842 19.15.6 Ion-dependent osmoticcoefficients ...... 846 19.16 Chemomechanical framework including chloride binding...... 846 19.16.1 Incompressible constituents and electroneutrality ...... 847 19.16.2 Structure of the constitutive equations ...... 847 19.16.3 Equilibria forpassiveexchanges ...... 848 19.17 Constitutive equations of chemo-hyperelasticity ...... 849 19.17.1 Chemoelastic energy with electrical and chemical couplings...... 849 19.17.2Matrixenergy ...... 850 19.17.3 Twofamiliesofcollagenfibrils ...... 851 19.18Boundaryvalueproblems ...... 851 19.18.1 Summaryofbalanceequations ...... 851 19.18.2 Semi-discreteequations ...... 852 19.18.3 The element effective capacity matrix ...... 853 19.18.4 The contributions of the endothelial and epithelialfluxes ...... 854 19.18.5 Extraction of dependent variables from primary variables...... 855 19.18.6 Derivation of the element effective capacity matrices...... 857 19.18.7Initialconditions...... 860 19.18.8Boundaryconditions ...... 860 19.19 The purely mechanical contribution: nonlinear elasticity ...... 860 19.19.1 Asinglefamilyoffibers...... 861 19.19.2 Two families of fibers : clinotropic materials ...... 865 19.19.3 Additive decomposition of the strain energy for the cornealstroma ...... 875 19.20 Annulus fibrosus: another lamellar tissue with two familiesoffibers ...... 876 19.20.1 Histologyoftheannulusfibrosus...... 876 19.20.2 Single lamellar and multi-lamellar points of view ...... 877 19.20.3 A short review of mechanical models for the annulus fibrosus...... 879 19.20.4 A short review of chemomechanical models ...... 880

III Growth of biological tissues 883

20 Tissue Engineering 885 20.1 Biomechanical perspectives of tissue engineering ...... 886 20.1.1 Issues ...... 887 20.1.2 Composition ofarticularcartilages ...... 889 20.2 Basicnotionsofbiochemistry ...... 890 20.2.1 Macromolecules...... 890 20.2.2 ATPandenergeticcoupling ...... 891 20.2.3 Glycolysis...... 893 20.2.4 Hormones...... 895 20.2.5 Growthfactors ...... 897 20.2.6 Cytokines ...... 898 20.2.7 Genesandgeneexpression...... 898 20.3 Effectsofhormonesandgrowthfactors...... 900 20.3.1 Insulin-likegrowthfactorIGF-1 ...... 900 20.3.2 BonemorphogeneticproteinsBMPs ...... 901 20.3.3 Responsetoinjury ...... 901 20.4Cellcultures...... 902 20.4.1 Sourceofcells...... 902 20.4.2 Celllayersandscaffolds ...... 902 20.4.3 Experiencegainedfromcultures ...... 904 20.5 Mechanobiology: experimentaldata ...... 905 20.5.1 The evolutions of mass and mechanical properties correlate ...... 905 20.5.2 Mechanosensitivecandidates ...... 906 20.5.3 Experimentalobservations...... 907 20.6 Mechanobiology: models ofgrowthof mass ...... 913 20.6.1 Regularturnoverofcartilage ...... 913 20.6.2 The model of Bonassar and coworkers: scaffold degradation ...... 913 20.6.3 The chemical engineering model of Galban and Locke [1997] [1999]ab...... 914 20.6.4 The chemical engineering approach of the MIT and Cagliarigroups...... 915 20.6.5 A two phase mixture model for articular cartilages ...... 917 20.7 Growth and structuration of the collagen network ...... 918 20.7.1 Chemical engineering aspects, the experiments by Tranquillo and coworkers . . . . 918 20.7.2 Cellandcollagenfibrilsalignment ...... 919 20.7.3 The partial model of Hunter and Levenston [2002]a...... 921 20.7.4 TheapproachoftheTexasA&Mgroup ...... 921 20.7.5 TheapproachoftheUCSDgroup ...... 922 20.8 Useoflightfortissuefabrication ...... 926 20.9 Biochemical and mechanical factors: a synthesis ...... 927 21 Growth of soft tissues 929 21.1 Naturalgrowthandtissue engineering ...... 929 21.1.1 Mechanobiology and other points of view on biologicalgrowth ...... 930 21.1.2 Cartilage engineering versus tumor growth ...... 931 21.1.3 Scope ...... 932 21.2 Residual stresses in elastic solids and material symmetries...... 932 21.3 Kinematicsofgrowth...... 934 21.3.1 Kinematics of growth in a solid and residual stresses ...... 934 21.3.2 Kinematicsofgrowthinamixture ...... 941 21.3.3 Thepropertiesofthedepositedmass ...... 943 21.3.4 Consequences of mass balance for growing species ...... 944 21.3.5 Growth, remodeling, structuration and dissipation ...... 946 21.4 Constitutive assumptions and restrictions: the growthlaw...... 947 21.4.1 Tentativeassumptions ...... 948 21.4.2 A growth law consistent with the entropy production ...... 949 21.4.3 Retrospect ...... 952 21.5 Internal entropy production for a single solid ...... 952 21.5.1 The relations for a single solid in the current configuration ...... 952 21.5.2 The relations for a single solid in the reference configuration...... 954 21.6 Some models and their thermodynamic structure ...... 954 21.6.1 Homeostatic states and stimuli of growth ...... 955 21.6.2 Featuresofmodelsofgrowth ...... 957 21.6.3 Growthmodelswithoutgrowthstrain ...... 957 21.6.4 Growth models including growth strain ...... 960 21.7 Boundary value problems for elastic-growing solids ...... 964 21.7.1 Aortic growth, cardiac growth and development ...... 964 21.7.2 Residual stresses due to the growth of a spherical shell...... 975 21.8 Incompatible strains and residual stresses ...... 976 21.8.1 Incompatiblestrains ...... 977 21.8.2 Compatible strains in an infinitesimal strain context...... 977 21.8.3 Compatible strain in a finite strain context ...... 980 Exercises on Chapter 21 ...... 982 22 Elastic-growing solids 985 22.1 The tools: dissipation, homeostatic domain and convexity ...... 986 22.2 Relations relative to the intermediate configurations ...... 986 22.2.1 Entities relative to the intermediate configurations...... 986 22.2.2 Dissipation inequality in the intermediate configurationsforasolid ...... 988 22.3 Growth law based on quadratic dissipation ...... 989 22.4 Bounded structure variables via convexity ...... 991 22.4.1 Rates of structure variables through heuristic differentialequations ...... 991 22.4.2 Constitutive prototypes: free energy and homeostaticsurface ...... 992 22.4.3 Thegeneralizednormalityrule ...... 993 22.4.4 Heuristic differential equations versus generalized normality rule ...... 994 22.5 A thermodynamically consistent growth law ...... 996 22.5.1 The state of the deposited mass and the free energy of growth ...... 997 22.5.2 Dissipation inequality for the rates of growth and structure variables ...... 997 22.5.3 A growth law based on a convex homeostatic surface ...... 999 22.5.4 The ellipsoidal homeostatic surface ...... 1001 22.5.5 Rate laws for the structure variables ...... 1004 22.5.6 Application to volumetric growth ...... 1005 22.6 Modes of mass deposition and mechanical response ...... 1011 22.6.1 The mechanical state of the newly deposited mass ...... 1011 22.6.2 Free energy of growth for an elastic-growing solid ...... 1012 22.7 Constitutive equations for purely elastic solids ...... 1015 22.7.1 The deposition modes and their free energies ...... 1016 22.7.2 The constitutive equations for the generalized stressesformode(a)...... 1017 22.7.3 The constitutive equations for the generalized stressesformode(b)...... 1018 22.7.4 Thegrowthratelawforthemass...... 1018 22.7.5 Rate equations for the structure variables ...... 1019 22.8 Growth of a bar under unconfined tension/compression ...... 1020 22.8.1 Unconfined tension/compression of the bar ...... 1020 22.8.2 Thecriticalloadforthebar ...... 1023 22.8.3 The elastic-growing bar versus the elastic bar under tension ...... 1026 22.9 Growth of a hollow cylinder under internal pressure ...... 1029 22.9.1 Thethinwallapproximation ...... 1029 22.9.2 The hollow cylinder under internal pressure ...... 1029 22.9.3 Thecriticalloadforthecylinder ...... 1030 22.9.4 The elastic-growing cylinder versus the elastic cylinder...... 1030 Appendices on Chapter 22 ...... 1032

23 Elastic-growing mixtures 1035 23.1 Thermodynamically consistent growth laws in a mixture context ...... 1036 23.1.1 Themixtureasawhole ...... 1036 23.1.2 Growth laws for constituents based on quadratic dissipation...... 1040 23.1.3 The dissipation inequalities in the intermediate configurations...... 1041 23.1.4 Constituents with evolving microstructure: options...... 1042 23.2 Modes of mass deposition in an elastic-growing mixture ...... 1046 23.2.1 The constituent is deposited in a natural configuration...... 1047 23.2.2 The constituent is deposited with the properties at the deposition site ...... 1047 23.2.3 Theexpressionforthestress ...... 1047 23.3 The fixed charge: electro-chemomechanical couplings ...... 1048 23.4 Growth equations with an evolving microstructure ...... 1050 23.4.1 The free energy in terms of director vectors ...... 1050 23.4.2 Rateofgrowth ...... 1052 23.4.3 Rateofstructurevariables...... 1054 23.4.4 Evolution of the fiber directions mκi, for i ∈ [1,ne] ...... 1055 23.4.5 Evolution of the relative volume fractions n˜κi, for i ∈ [1,ne] ...... 1057 23.4.6 Evolution of the pyridinoline crosslink density ...... 1058 23.4.7 Evolutionofthecellenergysource ...... 1059 23.4.8 Gathering the contributions to the dissipation inequality...... 1061 23.5 Growthequationsfor proteoglycans...... 1061 23.5.1 Thefreeenergy...... 1061 23.5.2 Rate of growth and rates of structure variables ...... 1062 23.5.3 Evolutionofthecellenergysource ...... 1063 23.5.4 Gathering the contributions to the Clausius-Duhem inequality ...... 1064 23.6 Summary of constitutive equations and residual ...... 1064 23.7 Growth laws with evolving microstructure: simulations...... 1066 23.7.1 Physical and constitutive parameters ...... 1066 23.7.2 The reference isotropic configuration ...... 1067 23.7.3 Initialdata ...... 1069 23.7.4 Boundaryconditions ...... 1069 23.7.5 Asymptotic properties of the growth law ...... 1071 23.7.6 SequentialgrowthofPGs andcollagen ...... 1072 23.7.7 Stiffening of collagen by pyridinoline ...... 1074 23.7.8 Directional distribution of collagen network ...... 1074 23.7.9 Growthandresorption ...... 1075 23.7.10 Dissipations and structuration damping ...... 1078 23.7.11 Influence of the initial cell energy density ...... 1079 23.8 Self-healing in engineering materials ...... 1080

24 Solid tumors 1083 24.1 Solid tumors: the recent intrusion of mechanics ...... 1084 24.2 From DNA alteration to metastatic migration ...... 1086 24.3 Oncologyandmolecularbasisofcancer ...... 1089 24.3.1 Tissuecomposition...... 1089 24.3.2 Normalcellularcycle...... 1091 24.3.3 Formationoftumorcells...... 1093 24.3.4 Physiological consequences of the presence of tumor cells ...... 1094 24.3.5 Tissue modification and capsule formation ...... 1095 24.3.6 Angiogenesis and lymphangiogenesis ...... 1095 24.3.7 Sometherapies ...... 1097 24.4 Mechanobiology: molecular basis and observations ...... 1100 24.4.1 Mechanosensitivecandidates ...... 1100 24.4.2 Effects of fluid pressure and mechanical stress ...... 1101 24.5 Early mathematical models of tumor growth ...... 1102 24.5.1 Residualstresses ...... 1102 24.5.2 Sphericalmodels ...... 1103 24.5.3 Criterionfornecrosis...... 1104 24.5.4 Mixture models of mathematical flavor ...... 1104 24.6 Models of transport of fluids, nutrients and drugs ...... 1108 24.6.1 Influenceofthemicrostructure ...... 1108 24.6.2 Hydraulic permeability and diffusion coefficients in tumors ...... 1108 24.6.3 Models that include nutrients ...... 1109 24.6.4 Models that include drug delivery aspects ...... 1109 24.6.5 Models that include pH influence ...... 1111 24.6.6 Specialeffects...... 1112 24.7 Mechanicalboundaryvalueproblems ...... 1112 24.7.1 Cylindricalproblems ...... 1112 24.7.2 A sphere or spherical shell under purely radial displacement...... 1115 24.8 Poroelastic mixture models without growth strain ...... 1118 24.8.1 A two phase poroelastic tissue with external mass exchanges ...... 1118 24.8.2 A three phase poroelastic tissue with internal mass exchanges...... 1126 24.8.3 Infusion length in a three phase poroelastic tissue ...... 1130 24.9 Poroelastic mixture models with a growth strain ...... 1132 24.9.1 A two phase model including a growth strain and nutrients ...... 1132 24.9.2 A two phase model including the effects of lymphatics ...... 1135 24.10 Methods of imaging transport and mechanical properties ...... 1137 24.10.1Basicideas...... 1137 24.10.2 Tumor tissues have distinct elastic and transport properties ...... 1138 24.10.3 Ultrasonic techniques provide local strain ...... 1138 24.10.4 MR elastography (MRE) provides wave-length ...... 1138 24.10.5 Ultrasound and Computer Tomography (CT) for blood flow imaging ...... 1138 24.10.6 Inverse analysis based on surface measurements ...... 1139 24.10.7 Imagingporoelasticproperties ...... 1139 24.10.8 Extracting information by minimization ...... 1140

25 Units and physical constants 1143 25.1Units...... 1143 25.1.1 Basicunits ...... 1143 25.1.2 Standardunits ...... 1143 25.1.3 Nonstandardunits...... 1144 25.2 Physicalconstants ...... 1144 Bibliography 1147

Index 1199