Introduction to Finite Element Modeling in Biomechanics

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Introduction to Finite Element Modeling in
Biomechanics

• Dr. N. Fatouraee
• Biomedical Engineering Faculty
• December, 2004

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Overview

• Introduction and Definitions
• Basic finite element methods
• 1-D model problem
• Application Examples

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Overview

• Finite Element Method
• numerical method to solve differential equations
• E.g.

Flow Problem u(r)
Heat Transfer Problem T(r,t)
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The Continuum Concept

• biomechanics example blood flow through aorta
• diameter of aorta ? 25 mm
• diameter of red blood cell ? 8 ?m (0.008 mm)
• treat blood as homogeneous and ignore cells

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The Continuum Concept

• biomechanics example blood flow through
  capillaries
• diameter of capillary can be 7 ?m
• diameter of red blood cell ? 8 ?m
• clearly must include individual blood cells in
  model

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Continuous vs. Discrete Solution

• What if the equation had no analytical solution
  (e.g., due to nonlinearities)?

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Continuous vs. Discrete Solution

• What if the equation had no analytical solution
  (e.g., due to nonlinearities)?
• How would you solve an ordinary differential
  equation on the computer?
• Numerical methods
• Runge-Kutta
• Euler method

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Discretization
0
1
0
1
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Discretization
in general, Euler method is given by

• Start with initial condition y(x0)y0
• Calculate f(x0,y0)
• Calculate y1y0 f(x0,y0) ?x
• Calculate f(x1,y1) ..

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Euler Example
ODE dy/dx (x,y) 0.05 yInitial Cond. y(0)100
Problem Use Euler with 2 steps Calculate y(x)
between at x20 and x40
Euler, 2 steps dy/dt(0,100) 5 ?x
20 y(20) y(0) ?xdy/dt(0,100) 100
205 200 y(40) y(20) ?xdy/dt(20,200)
200 20 10 400
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Discretization

• in general, the process by which a continuous,
  differential equation is transformed into a set
  of algebraic equations to be solved on a computer
• various forms of discretization
• finite element, finite difference, finite volume

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Finite Element Method

• discretization
• steps in finite element method
• weak form of differential equation
• interpolation functions within elements
• solution of resulting algebraic equations

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Basic Finite Element MethodsA 1-D Example
solve for u(x)
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Basic Finite Element MethodsA 1-D Example
Note that for a0, b1
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Basic Finite Element Method

• seek solution to allied formulation referred to
  as weak statement

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Basic Finite Element Method

• seek solution to allied formulation referred to
  as weak statement

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Basic Finite Element Method
The integral form is as valid as the original
differential equation.
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Basic Finite Element Method
note that by the chain rule
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Basic Finite Element Method
note that by the chain rule
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Basic Finite Element Method
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Basic Finite Element Method
recall w(x) is arbitrary?no loss in generality
to require w(a)w(b)0 i.e.,
subject w to same boundary conditions as u
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Basic Finite Element Method
weak statement
the above expression is continuous i.e., must
be evaluated for all x
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Discretization
0
1
0
1
elements
nodes
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Discretization
1
2
3
4
5
6
nodes
elements
1
2
3
4
5
u defined at nodes ? u1, u2 u(x1), u(x2)
goal ? solve for ui
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Discretization
1
2
3
4
5
6
nodes
elements
1
2
3
4
5
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Consider a Typical Element
e
x2
x1
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Interpolation Functions
Within the element we interpolate between u1 and
u2
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Interpolation Functions
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Interpolation Functions
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Interpolation Functions
at x x1 u u1 at x x2 u u2 x1 lt x lt x2
interpolation between u1 and u2
u1, u2 ? unknowns to be solved for
i.e., nodal values of u
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Approximation Functions
Now we have to choose functions for w
- referred to as Galerkin method
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We end up with a system of algebraic equations,
that canbe solved by the computer
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How many elements do we need?
0
1
1
2
3
4
5
6
nodes
elements
1
2
3
4
5
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5 elements
2 elements
20 elements
10 elements
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Practical Finite Element Analysis

• many commercial finite element codes exist for
  different disciplines
• FIDAP, FLUENT, FLOTRAN-ANSYS computational fluid
  mechanics (CFD)
• ANSYS, LS-Dyna, Abaqus solid mechanics
• FIDAP, ADINA, ANSYS fluid/solid interactions
  (FSI)

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Using a Commercial Code

• choose most appropriate software for problem at
  hand
• not always trivial
• can the code handle the key physical processes
• e.g., spatially varying material properties,
  nonlinearities

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Steps in Finite Element Method (FEM)

• Geometry Creation
• Material properties (e.g. mass density)
• Initial Conditions (e.g. temperature)
• Boundary Conditions
• Loads (e.g. forces)
• Mesh Generation
• Solution
• Time discretization (for transient problems)
• Adjustment of Loads and Boundary Conditions
• Visualization
• Contour plots (on cutting planes)
• Iso surfaces/lines
• Vector plots
• Animations
• Validation

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Model Validation

• most important part of the process, but hardest
  and often not done
• two types of validation
• code validation are the equations being solved
  correctly as written (i.e., grid resolution,
  etc.)
• model validation is the numerical model
  representative of the system being simulated
  (very difficult)

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Example 1 Liver Cancer Treatment
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Radiofrequency Ablation forLiver Cancer

• Surgical Resection is currently the
  gold-standard, and offers 5-year survival of
  around 30
• Surgical Resection only possible in 10-20 of the
  cases
• Radiofrequency Ablation heats up tissue by
  application of electrical current
• Once tumor tissue reaches 50C, cancer cells die

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Effects of RF energy on tissue
Na
K
Cl-
Cl-
Cl-
Na

• Electrical Current is applied to tissue
• Electrical current causes heating by ionic
  friction
• Temperatures above 50 C result in cell death
  (necrosis)

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Clinical procedure
Insertion
Probe Extension

• Ground pad placed on patients back or thighs
• Patient under local anesthesia and conscious
  sedation, or light general anesthesia

Application of RF power(12-25 min)
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Current RF Devices
200W RF-generator (Radionics / Tyco)
9-prong probe, 5 cm diameter, (Rita Medical)
Cool-Tip probe, 17-gauge needle, (Radionics /
Tyco)
12-prong probe,4 cm diameter,(Boston Scientific)
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RF Lesion Pathology
Coagulation Zone ( RF lesion, gt50 C)
Hyperemic Zone (increased perfusion)
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Finite Element Modeling for Radiofrequency
Ablation

• Purpose of Models
• Investigate shortcomings of current devices
• Simulate improved devices
• Estimate RF lesion dimensions for treatment
  planning
• Thermo-Electrically Coupled Model
• Solve Electric Field problem (Where is heat
  generated)
• Solve thermal problem (Heat Conduction in Tissue,
  Perfusion, Vessels)

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Electric Field Problem (Where is heat being
generated?)
Laplaces Equation
Boundary Conditions
P
? M
Electric Field
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Thermal ProblemConservation of Energy
rate of change of energy in a body
rate of energy generation
rate of energy addition
- rate of energy lost
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Model Geometry
2-D axisymmetric model
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Animations
Electrical Current Density(Where is heat being
generated?)
Temperature
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Model Results
Temperature at end of ablation
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Ex-vivo Validation in Animal Tissue

• Verify Temperature, Impedance and Lesion Diameter
• We applied same power as in computer model

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Experimental Setup
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Comparison Model ? Experiment
Impedance
Temperature
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Conclusion

• Lesion Diameter
• Model 33 mm
• Experiment 29 3 mm
• RF Lesion in model 14 larger
• Information on Electrical Tissue Conductivity vs.
  Temperature needed

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Impact of large vessels
Computer Model Geometry12-prong probe next to
10mm-vessel (e.g. portal vein)Flow rate 23 cm/s

• Vessel cooling simulated by estimating
  convective heat transfer coefficient

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Model Results

• Cancer cells next to vessel could survive

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Improved Configuration

• Improved configuration heats from both sides, and
  may create lesions closer to vessel

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Temperatureat End of Ablation
Bipolar

• Improved configuration creates lesion up to
  vessel
• Next Step Experimental Validation

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Example 2 Simulation of Artificial Heart Valve
Phantom I
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MR Imaging Bioprosthetic Valve
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Comparison between Experiment and Simulation
MRI
simulation
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Example 3 Artificial Heart Valve II
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J. De Hart et al. / Journal of Biomechanics 36
(2003) 699712 703
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Configurations of the fiber-reinforced stentless
valve and corresponding velocityvector fields
taken at six successive points in time. The
leftand right diagram at the bottom of each
frame denote the applied velocityand pressure
curves, respectively.
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Configurations of the fiber-reinforced stentless
valve and corresponding velocityvector fields
taken at six successive points in time. The
leftand right diagram at the bottom of each
frame denote the applied velocityand pressure
curves, respectively.
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Maximum principle Cauchystresses in the leaflet
matrix material during systole. In all frames the
right leaflet is taken from the nonreinforced
model for comparison. MPSr denotes the maximum
principle stress ratio of the reinforced and
non-reinforced leaflets. The stress scale on the
bottom is given in kPa.
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Maximum principle Cauchystresses in the leaflet
matrix material during systole. In all frames the
right leaflet is taken from the nonreinforced
model for comparison. MPSr denotes the maximum
principle stress ratio of the reinforced and
non-reinforced leaflets. The stress scale on the
bottom is given in kPa.
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Other Examples in Biomedical Engineering
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from Shirazi-Adl et al., J. Biomech. Engr.
123391 2001
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Pressure on vertebrae disks
from Miga et al., J. Biomech. Engr. 123354 2001
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Blood flow in Vessel Aneurism
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Blood flow in Vessel Aneurism
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Blood flow in Vessel Aneurism
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Strain in Knee Ligaments
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Electric Heart Activity
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Turbulence modelingPulsatile transitional flows

• Blood Flow in arteries is crucial in the
  development and prevention of cardiovascular
  diseases.
• Flow patterns in the previous Carotid Bifurcation
  during two different instants of the heart beat.

• Blood Flow Paterns

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Blood Flow Analysis Improves Stent-Grafts
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Blood Flow Analysis Improves Stent-Grafts