Download BE112A Topic 1: Introduction to Biomechanics

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
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.
m
  lim
V 0 V
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.
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
MEASURE
MODEL INPUTS
PHYSIOLOGICAL
TESTING
THE CONTINUUM MODEL
Governing Equations
MODEL
IMPLEMENTATION
AND SOLUTION
CLINICAL AND
BIOENGINEERING
APPLICATIONS
Continuum Model of the Heart
MODEL INPUTS
anatomy
tissue properties
cellular properties
PHYSIOLOGICAL TESTING
myocardial ischemia
EP mapping
disease models
CONTINUUM MODEL
OF THE HEART
MODEL
IMPLEMENTATION
Computational
methods
supercomputing
visualization
CLINICAL APPLICATIONS
myocardial infarction
cardiac imaging
In-vivo devices
pacing and defibrillation
tissue engineering
Model Inputs
ANATOMY
Ventricular Anatomy Model
Model Inputs
TISSUE PROPERTIES
uniaxial stretch
oim
Stress (kPa)
Wildtype
WT
oim
8
6
4
2
0
0.05
0.1
0.15
Strain
0.2
Model Inputs
CELLULAR
PROPERTIES
Myocyte Contractile Mechanics
sarcomere
length 2.0
µm
2.15 µm
tension
1 µN
3 sec
3.5 min
Bluhm, McCulloch, Lew. J Biomech. 1995;28:1119-1122
Model Implementation
COMPUTATIONAL
METHODS
The Finite Element Method
14
12
28
29
26
27
15
13
Physiological Testing
MYOCARDIAL ISCHEMIA
Strains in Myocardial Ischemia
Occlusion
site
Radiopaque
beads
-0.05
0.00
0.05
Clinical Applications
CARDIAC IMAGING
End-diastole
End-systole
Before
ventricular
reduction
surgery
Cardiac MRI
After
ventricular
reduction
surgery
Clinical Applications
IN-VIVO DEVICES
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 ...
Conservation Laws
• Conservation of Mass
• Lagrangian
• Eulerian (continuity)
• Conservation of Momentum
• Linear
• Angular
• Conservation of Energy
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)”
 
0
dV    d v
 0 d X1 d X 2 d X 3    d x1 d x2 d x3   
Hence:
xi
d X1 d X 2 d X 3
X R
0 d v
xi


det F det U 123
 dV X R
Thus, for an incompressible solid:  =  0 detF = 1
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:

D


vi
div( v ) 
div v  vi
 0
t
Dt
t
xi
xi
Thus, for an incompressible fluid:
 = constant divv = trD = 0
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”
→
Dv

divT b
Dt
or
vi
vi Tji
 vk

bi
t
xk x j
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:
T T
T
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:
De
vi qi
 tr(T D ) div q Tji

Dt
x j xi
where: e is the internal energy density
q is the heat flux vector
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.