Introduction to Finite Element Modeling in Biomechanics

1. Introduction to Finite Element Modeling in Biomechanics Dr. N. Fatouraee Biomedical Engineering Faculty December, 2004

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

3. Overview • Finite Element Method • numerical method to solve differential equations E.g.: Flow Problem u(r) Heat Transfer Problem T(r,t)

4. 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

5. 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

6. Continuous vs. Discrete Solution • What if the equation had no “analytical solution” (e.g., due to nonlinearities)?

7. 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

8. Discretization 0 1 0 1

9. 0 1 Discretization in general, Euler method is given by: • Start with initial condition: y(x0)=y0 • Calculate f(x0,y0) • Calculate y1=y0 + f(x0,y0) x • Calculate f(x1,y1) …………..

10. y Exact Solution 8 Steps 4 Steps 2 Steps x 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 x=20 and x=40 Euler, 2 steps: dy/dt(0,100) = 5 ; x = 20 y(20) = y(0) + x*dy/dt(0,100) = 100 + 20*5= 200 y(40) = y(20) + x*dy/dt(20,200) = 200 + 20* 10 = 400

11. 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

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

13. Basic Finite Element Methods:A 1-D Example solve for u(x)

14. Basic Finite Element Methods:A 1-D Example Note that for a=0, b=1:

15. Basic Finite Element Method • seek solution to allied formulation referred to as “weak” statement

16. Basic Finite Element Method • seek solution to allied formulation referred to as “weak” statement

17. Basic Finite Element Method The integral form is as valid as the original differential equation.

18. Basic Finite Element Method note that by the chain rule:

19. Basic Finite Element Method note that by the chain rule:

20. Basic Finite Element Method

21. 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

22. Basic Finite Element Method “weak statement”: the above expression is “continuous” i.e., must be evaluated for all x

23. Discretization 0 1 0 1 “elements” “nodes”

24. 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

25. Discretization 1 2 3 4 5 6 “nodes” “elements” 1 2 3 4 5

26. Consider a Typical Element e x2 x1

27. Interpolation Functions Within the element we interpolate between u1 and u2:

28. Interpolation Functions

29. Interpolation Functions

30. e x2 x1 Interpolation Functions at x = x1: u = u1 at x = x2: u = u2 x1 < x < x2: interpolation between u1 and u2 u1, u2 unknowns to be solved for i.e., nodal values of u

31. Approximation Functions Now we have to choose functions for w: - referred to as “Galerkin” method

32. We end up with a system of algebraic equations, that canbe solved by the computer

33. How many elements do we need? 0 1 1 2 3 4 5 6 “nodes” “elements” 1 2 3 4 5

34. 5 elements 2 elements 20 elements 10 elements

35. Practical Finite Element Analysis • many commercial finite element codes exist for different disciplines • FIDAP, FLUENT: fluid mechanics • ANSYS, LS-Dyna, Abaqus: solid mechanics

36. 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

37. 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

38. 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)

39. Example 1: Liver Cancer Treatment

40. 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 50°C, cancer cells die

41. Electric Field Effects of RF energy on tissue • Electrical Current is applied to tissue • Electrical current causes heating by ionic friction • Temperatures above ~50 °C result in cell death (necrosis) Na+ K+ Cl- Cl- Cl- Na+

42. 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)

43. 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)

44. RF Lesion Pathology Coagulation Zone (= RF lesion, >50 °C) Hyperemic Zone (increased perfusion)

45. 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)

46. Electric Field Problem (Where is heat being generated?) Laplace’s Equation Boundary Conditions P  M Electric Field

47. Thermal Problem:Conservation of Energy rate of change of energy in a body = + rate of energy generation + rate of energy addition - rate of energy lost

48. energy transfer (“conducted”) to surrounding tissue energy transfer to blood flow carrying heat away (“convected”) energy transferred (“conducted”) back to electrode energy storage by tissue energy added due to metabolism energy added by electric current (Power = current*voltage)

49. 1 cm Model Geometry 2-D axisymmetric model

50. Animations Electrical Current Density(Where is heat being generated?) Temperature