1. Biomechanics Mechanics applied to biology –the interface of two large fields –includes varied subjects such as: sport mechanicsgait analysis rehabilitationplant growth flight of birdsmarine organism swimming surgical devicesprosthesis design biomaterials invertebrate mechanics Our focus: continuum mechanics applied to mammalian physiology Objective: to solve problems in physiology with mathematical accuracy

2. Continuum Mechanics is concerned with: the mechanical behavior of solids and fluids … on a scale in which their physical properties (mass, momentum, energy etc) can be defined by continuous or piecewise continuous functions i.e. the scale of interest is “large” compared with the characteristic dimension of the discrete constituents (e.g. cells in tissue, proteins in cells) in a material continuum, the densities of mass, momentum and energy can be defined at a point, e.g.

3. Continuum Mechanics Fundamentals The key words of continuum mechanics are tensors such as stress, strain, and rate-of-deformation The rules are the conservation laws of mechanics – mass, momentum and energy. Stress, strain, and rate of deformation vary with position and time. The relation between them is the constitutive law. The constitutive law must generally be determined by experiment but it is constrained by thermodynamic and other physical conditions. The language of continuum mechanics is tensor analysis.

4. Biomechanics: Mechanics↔Physiology Continuum MechanicsPhysiology Geometry and structureAnatomy and morphology Boundary conditionsEnvironmental influences Conservation lawsBiological principles massmass transport, growth energymetabolism and energetics momentummotion, flow, equilibrium Constitutive equationsStructure-function relations Therefore, continuum mechanics provides a mathematical framework for integrating the structure of the cell and tissue to the mechanical function of the whole organ

5. MEASURE MODEL INPUTS PHYSIOLOGICAL TESTING CLINICAL AND BIOENGINEERING APPLICATIONS THE CONTINUUM MODEL Governing Equations THE CONTINUUM MODEL Governing Equations MODEL IMPLEMENTATION AND SOLUTION

6. MODEL INPUTS anatomy tissue properties cellular properties PHYSIOLOGICAL TESTING myocardial ischemia EP mapping disease models MODEL IMPLEMENTATION Computational methods supercomputing visualization CLINICAL APPLICATIONS myocardial infarction cardiac imaging In-vivo devices pacing and defibrillation tissue engineering CONTINUUM MODEL OF THE HEART Continuum Model of the Heart

7. Model Inputs ANATOMY

8. Ventricular Anatomy Model

9. Model Inputs TISSUE PROPERTIES

10. Model Inputs CELLULAR PROPERTIES

11. Myocyte Contractile Mechanics Bluhm, McCulloch, Lew. J Biomech. 1995;28:1119-1122 1 µN 3 sec 3.5 min 2.0 µm 2.15 µm sarcomere length tension 1 µN 3 sec 3.5 min 2.0 µm 2.15 µm sarcomere length tension

12. Model Implementation COMPUTATIONAL METHODS

13. The Finite Element Method

14. Physiological Testing MYOCARDIAL ISCHEMIA

15. Strains in Myocardial Ischemia Radiopaque beads Occlusion site -0.05 0.00 0.05

16. Clinical Applications CARDIAC IMAGING

17. Cardiac MRI End-diastoleEnd-systole Before ventricular reduction surgery After ventricular reduction surgery

18. Clinical Applications IN-VIVO DEVICES

19. Bioengineering Design Applications prosthetic heart valves orthopedic implants tissue engineered vascular grafts surgical techniques and devices clinical image analysis software catheters pacemaker leads wheel chairs stents crash helmets airbags infusion pumps athletic shoes etc...

20. Conservation Laws Conservation of Mass Lagrangian Eulerian (continuity) Conservation of Momentum Linear Angular Conservation of Energy

21. Conservation of Mass: Lagrangian “The mass  m (=  0  V) of the material in the initial material volume element  V remains constant as the element deforms to volume  v with density , and this must hold everywhere (i.e. for  V arbitrarily small)” Hence: Thus, for an incompressible solid:  =  0  detF = 1

22. Conservation of Mass: Eulerian The Continuity Equation “The rate of increase of the mass contained in a fixed spatial region R equals the rate at which mass flows into the region across its bounding surface S” Hence by the divergence theorem and the usual approach, we get: Thus, for an incompressible fluid:  = constant  divv = trD = 0

23. Conservation of Linear Momentum “The rate of change of linear momentum of the particles that instantaneously lie within a fixed region R equals the resultant of the body forces b per unit mass acting on the particles in R plus the resultant of the surface tractions t (n) acting on the surface S” →

24. Conservation of Angular Momentum “The rate of change of angular momentum of the particles that instantaneously lie within a fixed region R equals the resultant couple about the origin of the body forces b per unit mass acting on the particles in R plus the resultant couple of the surface tractions t (n) acting on S”. Subject to the assumption that no distributed body or surface couples act on the material in the region, this law leads simply to the symmetry of the stress tensor:

25. Conservation of Energy “The rate of change of kinetic plus internal energy in the region R equals the rate at which mechanical work is done by the body forces b and surface tractions t (n) acting on the region plus the rate at which heat enters R across S”. With some manipulation, this leads to: where:e is the internal energy density q is the heat flux vector

26. Topic 1: Summary of Key Points Biomechanics is mechanics applied to biology; our specific focus is continuum mechanics applied to physiology.Biomechanics Continuum mechanics is based on the conservation of mass, momentum and energy at a spatial scale where these quantities can be approximated as continuous functions.Continuum mechanics The constitutive law describes the properties of a particular material. Therefore, a major objective of biomechanics is identifying the constitutive law for biological cells and tissues.constitutive law Biomechanics involves the interplay of experimental measurement in living tissues and theoretical analysis based on physical foundationsinterplay Biomechanics has numerous applications in biomedical engineering, biophysics, medicine, and other fields.applications Knowledge of the fundamental conservation laws of continuum mechanics is essential.fundamental conservation laws