Molecular, Cellular & Tissue Biomechanics Goal: Develop a fundamental understanding of biomechanics over a wide range of length scales. 20.310, 2.793, 6.024 Fall, 2006 20.310, 2.793, 6.024 Biomolecules and intermolecular forces Single molecule biopolymer mechanics Formation and dissolution of bonds Motion at the molecular/macromolecular level MOLECULAR MECHANICS Structure/function/properties of the cell Biomembranes The cytoskeleton Cell adhesion and aggregation Cell migration Mechanotransduction CELLULAR MECHANICS TISSUE MECHANICS Molecular structure --> physical properties Continuum, elastic models (stress, strain, constitutive laws) Viscoelasticity Poroelasticity Electrochemical effects on tissue properties Fall, 2006 1Some Learning Objectives 1. To understand the fundamental concepts of mechanics and be able to apply them to simple problems in the deformation of continuous media 2. To understand the underlying basis for the mechanical properties of molecules, cells and tissues 3. To be able to model biological materials using methods appropriate over diverse length scales 4. To be familiar with the wide spectrum of measurement techniques that are currently used to determine mechanical properties 5. To appreciate the close interconnections between mechanics and biology/chemistry of living systems 20.310, 2.793, 6.024 Fall, 2006 Biomechanics of tissues Mechanics Biology I. Linear elastic behavior I. Biochemical and molecular biology ofII. Viscoelasticity ECM molecules III. Poroelasticity A. Collagen IV. Electrochemical and superfamily physicochemical properties B. Proteoglycan superfamily C. Other glycoproteins II. Nanomolecular structures <--> tissue III. Mechanobiology 20.310, 2.793, 6.024 Fall, 2006 2Some preliminaries Equilibrium -- balance of forces concept of a stress tensor Compatibility -- relations between displacements and strains or deformation normal strains; shear strains; strain tensor Constitutive laws stress -- strain Young’s modulus; Poisson’s ratio, shear modulus linear, isotropic, elastic materials 20.310, 2.793, 6.024 Fall, 2006 20.310, 2.793, 6.024 Fall, 2006 Force balance in a single cell Desprat et al., Biophys J., 2005. Figure by MIT OCW.3Stress distributions along the basal surface of a resting cell 20.310, 2.793, 6.024 Fall, 2006 Several material testing methods Linear extension Linear shear Biaxial tension Cone-and-plate rheometer Confined compression 20.310, 2.793, 6.024 Fall, 2006 4 Graphical image of stress distributions on a cell surface removed due to copyright restrictions.Hu, et al., AJP Cell, 2003Tissue properties: common simplifying assumptions Linear -- the elastic modulus is constant, indpendent of strain amplitude Homogeneous -- the material is spatially uniform Isotropic -- the material exhibits the same elastic properties in all directions Time-independent -- stresses and strains are uniquely related, independent of rate of strain 20.310, 2.793, 6.024 Fall, 2006 Constitutive laws for a linear elastic, isotropic material = )+ 2G1 11 ( 11 + 22 + 33 11 (11 = 11  22 + 33 )E 22 = 11 + 22 + 33 )+ 2G22( 1 ( ) = )+ 2G22 = E 22  11 + 33  33 ( 11 + 22 + 33 33 1 12 = 2G12 33 = E 33  11 + 22 ) 13 = 2G13 ( 12 (1 + )12 23 = 2G2312 = = 2G E 13 (1 + )13 2G E = =  = = 13 2G E 1  2 (1 + )(1  2 )23 (1 + )2323 = = 2G E E = Young’s modulus  = Lame’ constant  = Poisson’s ratio G = shear modulus 20.310, 2.793, 6.024 Fall, 2006 520.310, 2.793, 6.024 Fall, 2006 1.E+00 1.E+02 1.E+04 1.E+06 1.E+08 1.E+10 1.E+12 DIAMOND STEEL BONE CONCRETE SILK F-ACTIN TENDON WOOD TUBULIN CONTRACTED SKELETAL MUSCLE ELASTIN RELAXED SKELETAL MUSCLE COLLAGEN GELS LUNG PARENCHYMA FIBROBLAST CELLS ENDOTHELIAL CELLS NEUTROPHILS LYMPHOCYTES Values of the elastic or Young’s modulus (E) for various materials 20.310, 2.793, 6.024 Fall, 2006 Linear? Unidirectional tensile tests Stress-strain behavior of a peptide hydrogel Linear behavior up to fracture Relatively low toughness due to small fracture straina bneedle needle peptide matrix peptide matrix plastic mesh 6 Figure by MIT OCW. Adapted from Leon, et. al., 1998Constant Es11e1100.0050.010.0150.020.0250.030.0350.04102030506040 Stress(N/m 2)>Strain7 Linear? Elastin is one of main structural components of tissue Linear; little hysteresis. Provides the “stretchiness” of tissues. Combination of single-molecule characteristics and microscale structure. 20.310, 2.793, 6.024 Fall, 2006 Linear? ACL -- different strain rates 20.310, 2.793, 6.024 Fall, 2006 Figure by MIT OCW.Figure by MIT OCW.Slow Medium FastRange of linearityE = ds/de = 109Pa2 4 6 8 10 12080604020Strain(%)Strass (MPa)Control1008060402005101520 Specimen fixed at zero stretch in 10% formalinELASTINLig. Nuchae denatured% Strain = (∆L/L0)*100Stress (kPa)The stress-strain curve of elastin. Data from Fung and Sobin (1981).Linear and Isotropic? Right tibia 20.310, 2.793, 6.024 Fall, 2006 20.310, 2.793, 6.024 Fall, 2006 Canine aorta showing elastic fiber content 8 Figure by MIT OCW.Scanning electron micrographs showing a low-power view of dog'saorta and a high-power view of the dense network of longitudinallyoriented elastic fibers in the outer layer of the same blood vessel.Images removed due to copyright restrictions. See Haas, K. S., S. J. Phillips, A. J. Comerota, and J. W. White. Anat. Rec. 230 (1991): 86-96.25020015010050001 234Stress (MPa)Strain (%)RA-directionSample no.Sample no.BA-direction66113224455LongitudinaldirectionRadial direction20.310, 2.793, 6.024 Fall, 2006 Collagen fiber arrangement in skin and cornea with alternating directions 20.310, 2.793, 6.024 Fall, 2006 Homogeneous? Only rarely 9 Electron micrograph of a cross-section of tadpole skin. The arrangement of collagen fibrils is plywoodlike, with successive layers of fibrils laid down nearly at right angles to each other. Image removed due to copyright restrictions.Electron micrographs of parallelcollagen fibrils in a tendon and themesh work of fibrils in skin removed due to copyright restrictions.Figure by MIT OCW.TendonFascicleTendon HierarchyX ray EM X ray EM X ray EMSEMEM SEMOMSEMOMEvidence:X- rayReticularmembraneFascicularmembraneWaveform orcrimp structureFibroblastsTropocollagen35