Biomechanics

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Biomechanics

• Mechanics applied to biology
• the interface of two large fields
• includes varied subjects such as
• sport mechanics gait analysis
• rehabilitation plant growth
• flight of birds marine organism swimming
• surgical devices prosthesis design
• biomaterials invertebrate mechanics
• Our focus continuum mechanics applied to
  mammalian physiology
• Objective to solve problems in physiology with
  mathematical accuracy

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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.

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Continuum MechanicsFundamentals

• 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.

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Biomechanics Mechanics?Physiology

• Continuum Mechanics Physiology
• Geometry and structure Anatomy and morphology
• Boundary conditions Environmental influences
• Conservation laws Biological principles
• mass mass transport, growth
• energy metabolism and energetics
• momentum motion, flow, equilibrium
• Constitutive equations Structure-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
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Continuum Model of the Heart
MODEL INPUTS
PHYSIOLOGICAL TESTING
anatomy
myocardial ischemia
tissue properties
EP mapping
cellular properties
disease models
CONTINUUM MODEL
OF THE HEART
MODEL
CLINICAL APPLICATIONS
IMPLEMENTATION
myocardial infarction
Computational methods
cardiac imaging
In-vivo devices
supercomputing
pacing and defibrillation
visualization
tissue engineering
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Model InputsANATOMY
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Ventricular Anatomy Model
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Model InputsTISSUE PROPERTIES
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Model InputsCELLULAR PROPERTIES
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Myocyte Contractile Mechanics
Bluhm, McCulloch, Lew. J Biomech.
1995281119-1122
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Model ImplementationCOMPUTATIONAL METHODS
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The Finite Element Method
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Physiological TestingMYOCARDIAL ISCHEMIA
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Strains in Myocardial Ischemia
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Clinical ApplicationsCARDIAC IMAGING
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Cardiac MRI
End-diastole
End-systole
Before ventricular reduction surgery
After ventricular reduction surgery
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Clinical ApplicationsIN-VIVO DEVICES
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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
...
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Conservation Laws

• Conservation of Mass
• Lagrangian
• Eulerian (continuity)
• Conservation of Momentum
• Linear
• Angular
• Conservation of Energy

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Conservation of Mass Lagrangian
The mass dm (?0dV) of the material in the
initial material volume element dV remains
constant as the element deforms to volume dv with
density ?, and this must hold everywhere (i.e.
for dV arbitrarily small)
Hence
Thus, for an incompressible solid ? ?0 ? detF
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Conservation of Mass EulerianThe 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
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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
?
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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
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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
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Topic 1 Summary of Key Points

• Biomechanics is mechanics applied to biology our
  specific focus is continuum mechanics applied to
  physiology.
• 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.
• 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.
• Biomechanics involves the interplay of
  experimental measurement in living tissues and
  theoretical analysis based on physical
  foundations
• Biomechanics has numerous applications in
  biomedical engineering, biophysics, medicine, and
  other fields.
• Knowledge of the fundamental conservation laws of
  continuum mechanics is essential.