Biomechanics applies the laws and techniques of mechanics in the study of biological systems and related phenomena. Biomechanics uses mathematical and computational tools such as model construction of musclo-skeletal system, body fluid circulation, to aid medical diagnosis, therapeutics and surgery planning, designing of prostheses and implants or in tissue engineering. Present book targets specific topics pertaining to the biomechanics of soft tissues. Subjects addressed includes solids and multi-species mixtures as open systems: a continuum mechanics perspective; electro-chemo-mechanical couplings: tissues with a fixed electric charge and growth of biological tissues.
Table of Contents
1.Biomechanical topics in soft tissues. Introduction. Diffusion, convection, osmosis: towards wearable artificial kidneys. Issues in drug delivery. Macroscopic models of tissues, interstitium and membranes. Energy couplings, passive and active transports. More general direct and reverse couplings. Tissue engineering re-directed to tumor tissue exploration. From mechanics to biomechanics. Mechanisms of injury of the knee, fracture mechanics. Water and solid constituents of soft tissues. 2. I Solids and multi-species mixtures as open systems :a continuum mechanics perspective. 3. Elements of continuum mechanics. Algebraic relations and algebraic operators. Characteristic polynomial, eigenvalues and eigenvectors. A few useful tensorial relations. The differential operators of continuum mechanics. Measures of strain. Transports between reference and current configurations. Time derivatives and Reynolds theorems. Work-conjugate stress-strain pairs. Small upon large. Kinematical constraints and reaction stresses. Invariance, objectivity, isotropy. Exercises on Chapter. 4. Thermodynamic properties of fluids. Basic thermodynamic definitions. Phase change. Thermodynamic functions of fluids. Laplace’s law of mechanical equilibrium at the interface between two immiscible fluids. Henry’s law and Raoult’s law of chemicalequilibrium between a liquid and a gas. Composition of fluids in plasma, interstitium, cell. 5. Multi-species mixtures as thermodynamically open systems. The thermodynamically open system. Chemomechanical behavior and growth of soft tissues. General form of balance equations. Balance of mass. Balance of momentum. Balance of energy. Balance of entropy and Clausius-Duhem inequality. Dissipation mechanisms. General constitutive principles. Appendices of Chapter. 6. Anisotropic and conewise elasticity. Hypoelasticity, elasticity and hyperelasticity. Anisotropic linear elasticity. Spectral analysis of the elastic matrix. Fiber-reinforced materials: extension-contraction response. Exercises on Chapter. 7. Hyperelasticity, a purely mechanical point of view. Restrictions to the strain energy function. Properties of constitutive functions and partial differential equations. Total potential energy. Isotropic hyperelasticity. Anisotropic hyperelasticity and Fung’s strain energy. Elastic potentials with the elastic-growth multiplicative decomposition. Return to the conewise response in presence of fibers. Elasticity and workless constitutive stress. Use of right and left Cauchy-Green tensors. Exercises on Chapter. 8. Poroelasticity with a single porosity. Geometrical, kinematical and mechanical descriptors. Elements of mixture theory: the level/scale of the species. Anisotropic poroelasticity: a composite material format. Anisotropic poroelasticity: drained and undrained properties. Unconfined compression of a poroelastic cylinder. 9. Viscoelasticity and poro-viscoelasticity. Viscoelasticity, poroelasticity and poro-viscoelasticity. Intrinsic time dependent behavior of collagen fibrils. Complex modulus. Relaxation spectrum. The quasi-linear viscoelasticity QLV model of Fung. Overall viscoelastic properties: ECM and a contractile cell. An electrical analogy applied to the corneal endothelium. Fluid infusion in a viscoelastic polymer gel. Confined compression of a viscoelastic polymer gel. Compression of a poroelastic layer: displacement control. Visco-hyperelasticity and other memory effects. Exercises on Chapter. 10. Thermoelasticity and thermo-poroelasticity. Thermomechanical properties of elastic solids. Thermoelastic heat, stored energy and dissipated energy. Anisotropic thermo-poroelasticity : mechanics, transport, energy. Generalized compatibility conditions for the stress. Heating a thermo-poroelastic medium. Exercises on Chapter. 11. Transfers of mass, momentum and energy. Summary of balance equations. Intercompartment mass transfers. Momentum supply in a poroelastic context. The bioheat equation. Mass and energy transfers between a liquid and a solid. A porous medium with two porosities and three temperatures. A porous medium with a single porosity and two temperatures. Interfacial transfer coefficients and specific surface areas. The coefficients of transfer: two scale derivation. 12. Waves in thermoelastic solids and saturated porous media. Basic definitions. Waves in elastic solids. Acceleration waves in thermoelastic solids. Acceleration waves and acoustic waves in fluids. Acceleration waves in saturated porous media. Propagation of acceleration waves in saturated porous media. Plane motions and Helmholtz potentials in poroelasticity. Silent poroelastic boundaries. Surface waves in a saturated porous half-plane. First order waves in saturated porous media. Exercises on Chapter. 13. II Electro-chemomechanical couplings in tissues with a fixed electrical charge. 14. Directional averaging for fiber-reinforced tissues. Directional analysis. A simple fiber model for the collagen network. Directional models of tissues. The mechanical response of individual fibers. Fabric tensors. Spatial homogenization over cells and ECM. 15. Electro-chemomechanical couplings. Chemomechanical couplings in engineering and biology. Molar volumes, electrostriction. Electrokinetic processes. Nanoscopic aspects. Hydration, hydrolysis, complexation and solubility. Some basic notions of electrostatics. Semi-permeable membrane and osmotic effect. Reverse or Inverse osmosis. Electrical repulsion and electrical shielding. The pore composition in materials with fixed charge. The heart muscle and cell electrophysiology. Reverse couplings. A partially coupled chemo-poroelastic model. Exercises on Chapter. 16. Chemomechanical couplings in articular cartilages. Overview. Histological aspects of articular cartilages. Pathologies, osteoarthritis, rheumatoid arthritis. A brief review of modeling aspects. Interpretation of laboratory experiments. Partition of the tissue into phases. The constitutive structure: deformation, mass transfer, diffusion. Chemoelastic energy of the tissue. Features of the constitutive framework. Remarks on constitutive frameworks and constraints. Exercises on Chapter. 17. Passive transport in the interstitium and circulation. Diffusion as a mode of passive transport. Coupled flows. Propagation, diffusion, convection. Physico-chemical processes involving diffusion, convection and reaction. Diffusion versus convection. Newtonian viscous fluids and Reynolds number. Hydraulic conductivity. The steps of drug delivery, extravasation, transport in the interstitium. Blood circulation. Mechanics of vessels. Transport of oxygen and carbon dioxide in blood. Basics of enzymatic kinetics. Acid-base equilibrium. Exercises on Chapter. 18. Coupled transports in tissues with a fixed electrical charge. Macroscopic transports in tissues with a fixed charge. Generalized and coupled diffusion: structure of constitutive equations. Generalized diffusion: the membrane effect. Generalized diffusion: ranges of the coefficients and refinements. Generalized diffusion with thermal effects. Exercises on Chapter. 19. Effects of the pH on transport and mechanics. Overview and laboratory observations. pH agents embedded in the mixture framework. Acid-base reactions in a thermodynamic framework. Calcium binding in a thermodynamic framework. Variation of the fixed electrical charge with pH. Biochemical composition of articular cartilages. Concentrations of ions, sites and charges. Chemical equilibrium at the cartilage-bath interface. Constitutive equations of generalized diffusion. Simulations of bath-cartilage equilibria. Constitutive mechanical framework: effects of pH and calcium binding. Chemically induced stiffening/softening by fiber recruitment/deactivation. Mechanical and chemical tests at varying pH. The limit case of a conewise linear collagen response. Comments on the mechanical model. 20. Finite element analysis of ECM couplings. Overview of the finite element analysis. Field and constitutive equations. Finite element formulation. Testing setup and material data. Mechanical and chemical loadings with NaCl. Cyclic substitution of NaCl and CaCl2. Improvements of the chemomechanical model. Exercises on Chapter. 21. Cornea and annulus fibrosus. Function, structure and composition. Biomechanical aspects. Active transport in the endothelium: the cell and organ scales. Physical data: literature review. Scattering of light by the corneal stroma and transmittance. Corneal surgery. Cornea engineering. The constitutive framework. Biochemical composition of stroma. The fixed electrical charge: negatively charged PGs and chloride binding. Effects of pH and chloride binding on the fixed charge. Global constitutive structure: mechanics and transport. Generalized diffusion in the extrafibrillar phase. Rest potential and active fluxes at the cell scale. Coupled diffusion across a membrane and active transport. Chemomechanical framework including chloride binding. Constitutive equations of chemo-hyperelasticity. Boundary value problems. The purely mechanical contribution: nonlinear elasticity. .Annulus fibrosus: another lamellar tissue with two families of fibers. 22. III Growth of biological tissues. 23. Tissue Engineering. Biomechanical perspectives of tissue engineering. Basic notions of biochemistry. Effects of hormones and growth factors. Cell cultures. Mechanobiology: experimental data. Mechanobiology: models of growth of mass. Growth and structuration of the collagen network. Use of light for tissue fabrication. Biochemical and mechanical factors: a synthesis. 24. Growth of soft tissues. Natural growth and tissue engineering. Residual stresses in elastic solids and material symmetries. Kinematics of growth. Constitutive assumptions and restrictions: the growth law. Internal entropy production for a single solid. Some models and their thermodynamic structure. Boundary value problems for elastic-growing solids. Incompatible strains and residual stresses. Exercises on Chapter. 25. Elastic-growing solids. The tools: dissipation, homeostatic domain and convexity. Relations relative to the intermediate configurations. Growth law based on quadratic dissipation. Bounded structure variables via convexity. A thermodynamically consistent growth law. Modes of mass deposition and mechanical response. Constitutive equations for purely elastic solids. Growth of a bar under unconfined tension/compression. Growth of a hollow cylinder under internal pressure. Appendices on Chapter. 26. Elastic-growing mixtures. Thermodynamically consistent growth laws in a mixture context. Modes of mass deposition in an elastic-growing mixture. The fixed charge: electro-chemomechanical couplings. Growth equations with an evolving microstructure. Growth equations for proteoglycans. Summary of constitutive equations and residual. Growth laws with evolving microstructure: simulations. Self-healing in engineering materials. 27. Solid tumors. Solid tumors: the recent intrusion of mechanics. From DNA alteration to metastatic migration. Oncology and molecular basis of cancer. Mechanobiology: molecular basis and observations. Early mathematical models of tumor growth. Models of transport of fluids, nutrients and drugs. Mechanical boundary value problems. Poroelastic mixture models without growth strain. Poroelastic mixture models with a growth strain. Methods of imaging transport and mechanical properties. 28. Units and physical constants. Units. Physical constants. 29. Bibliography. 30. Index.
Benjamin Loret is professor of mechanics and civil engineering at the University of Grenoble, France. His research addresses the constitutive responses of engineering and biological materials to static and dynamic loadings. He has especially focused on the couplings of thermal, hydraulical, electrical, chemical and mechanical natures that are ubiquitous in fluid saturated porous media. Applications target innovative energy production systems and biomechanics of soft tissues.
Fernando. M.F. Simões was born on July 23rd, 1964, in Lisboa, Portugal. He has received is Ph.D. degree in civil engineering from the Instituto Superior Técnico of the Technical University of Lisboa, Portugal, in 1997. Presently, he is Assistant Professor at the Department of Civil Engineering, Architecture and GeoResources (University of Lisbon). His research interests include structural mechanics and biomechanics.