Bone structure adaptation as a cellular automaton optimization process

1. Bone structure adaptation as a cellular automaton optimization process Andrés Tovar, Glen L. Niebur, Mihir Sen and John E. Renaud Department of Aerospace and Mechanical Engineering University of Notre Dame, Indiana Brian Sanders Air Force Research Laboratory Wright-Patterson AFB, Ohio Presentation at General Motors Corporation Detroit, Michigan 6 May 2004 Bone adaptation as a CA process

2. Bone adaptation as a CA process

3. Bone adaptation as a CA process

4. [Meyer and Culmann, 1867; Wolff, 1892] Bone adaptation as a CA process

5. Finite Element Method (FEM) Bone static models Cellular Automata (CAs) Biological dynamics Hybrid Cellular Automata (HCA) Bone adaptation dynamic model Bone adaptation as a CA process

6. Content 1. Bone Adaptation 2. Cellular Automata (CAs) 3. The Hybrid Cellular Automaton (HCA) method • Local Control Rule • Performance 4. Examples 5. Final remarks Bone adaptation as a CA process

7. 1. Bone Adaptation Bone adaptation as a CA process

8. Bone adaptation as a CA process

9. 100mm Bone adaptation as a CA process

10. BMU Basic multi-cellular unit Oscteoblasts form bone Osteoclasts resorb bone [Frost, 1964, 1969] Bone adaptation as a CA process

11. 10mm Bone adaptation as a CA process

12. Oscteocytes sense mechanical stimuli [Skerry et al., 1989; Cowin et al., 1991; Lanyon, 1993; Klein-Nulend et al., 1995] Bone adaptation as a CA process

13. [Mullender et al. 1994, Mullender and Huiskes, 1995] Bone adaptation as a CA process

14. [Ott, 2001] Bone adaptation as a CA process

15. 2. Cellular Automata 0.1mm The average density of osteocytes is 12,000 ~ 20,000 cells/mm3 [Frost, 1960; Bodyne, 1972] Bone adaptation as a CA process

16. 0.5mm 2. Cellular Automata CAs are dynamical systems that are discrete in space and time and operate on a uniform,regular lattice. Bone adaptation as a CA process

17. Local rule Neighborhood Empty N = 0 Von Neumann N = 4 Moore N = 8 Expanded Moore N = 24 Boundary 0 a . . . a a . . . b a b . . . z a . . . z Fixed Adiabatic Reflecting Periodic CAs are characterized by local interactions. Bone adaptation as a CA process

18. CAs have been used to simulate physical and biological phenomena since their creation by von Neumann in 1940s. [Wolfram, 2002] [Conway, 1970] [Chopard and Droz, 1998] [Tovar, 2003] Bone adaptation as a CA process

19. 3. Hybrid Cellular Automaton Model Mechanical set point Mechanical signal U* U [Hajela and Kim, 2001; Abdalla and Gürdal, 2002 ] Bone adaptation as a CA process

20. 3. Hybrid Cellular Automaton Model FEM U* U Bone adaptation as a CA process

21. Local control [Carter, 1977; Beaupre, 1990] [Bendsøe, 1989; Sigmund, 2001] 3. Hybrid Cellular Automaton Model FEM U* U Bone adaptation as a CA process

22. Start Local control 3. Hybrid Cellular Automaton Model FEM U* U no yes ? End Bone adaptation as a CA process

23. 3.1 Local control strategy a) Two-position control b) Proportional control c) Integral control d) Derivative control Bone adaptation as a CA process

24. 3.1 Local control strategyTwo-position control t:21 U:6.7170 M:0.539 Bone adaptation as a CA process [c.f. Sauter, 1992]

25. 3.1 Local control strategyProportional control t:23 U:6.3265 M:0.581 Bone adaptation as a CA process [c.f. Martin et al., 1998]

26. 3.1 Local control strategyProportional-Integral control t:16 U:6.4576 M:0.568 Bone adaptation as a CA process [c.f. Hazelwood et al., 2001]

27. 3.1 Local control strategyProportional-Derivative control t:23 U:6.2938 M:0.585 Bone adaptation as a CA process [c.f. Fyhrie and Schaffler, 1995]

28. 3.1 Local control strategyProportional-Integral-Derivative control t:15 U:6.4338 M:0.569 Bone adaptation as a CA process [c.f. Davidson et al., 2004]

29. 3.2 PerformanceInitial design M = 1.0 M = 0.5 t:15 U:6.4338 M:0.569 t:21 U:6.4502 M:0.568 M = 0.0 M≈ 0.5 t:21 U:6.4668 M:0.568 t:17 U:6.4350 M:0.568 Bone adaptation as a CA process

30. 3.2 PerformanceNeighborhood t:16 U:6.8073 M:0.529 t:15 U:6.4338 M:0.569 t:13 U:6.3511 M:0.574 t:16 U:6.2062 M:0.592 Bone adaptation as a CA process

31. 0 a . . . z a . . . z Fixed Periodic 3.2 PerformanceBoundary conditions t:15 U:6.4338 M:0.569 t:13 U:6.3905 M:0.584 Bone adaptation as a CA process

32. 3.2 PerformanceSize of the Design Domain 10x10 60x60 30x30 t:18 U:5.9944 M:0.598 t:20 U:6.8146 M:0.540 t:15 U:6.4338 M:0.569 120x120 90x90 t:17 U:7.1222 M:0.526 t:16 U:6.9805 M:0.533 Bone adaptation as a CA process

33. 3.2 PerformanceTarget mechanical stimulus U* U0 ≈ 0.005 U* = U0/5 U* = U0 t:15 U:6.4338 M:0.569 t:6 U:4.4033 M:0.920 U* = 5U0 U* = 10U0 t:19 U:13.1251 M:0.274 t:18 U:18.5379 M:0.193 Bone adaptation as a CA process

34. 3.2 PerformanceThe trade-off curve Bone adaptation as a CA process

35. 3.2 PerformanceThe trade-off curve [Sigmund, 2001] Bone adaptation as a CA process

36. 4. ExamplesStructures in cantilever 1:1 3:1 2:1 4:1 Bone adaptation as a CA process

37. 4. ExamplesStructures in cantilever t:10 U:10.3091 M:0.483 t:10 U:12.9910 M:0.189 t:18 U:11.0233 M:0.551 Bone adaptation as a CA process

38. 5. ExamplesTrabecular bone (one-load case) Bone adaptation as a CA process

39. 5. ExamplesTrabecular bone (two-load case) Bone adaptation as a CA process

40. 6. Final Remarks • HCA = CA + FEM, using local control rules. • HCA models are suitable to simulate biological structural optimization process. • HCA local control rules need to be “tuned” according to biological evidence. • Time effects, like mineralization of bone tissue, can be included in the model. • A probabilistic HCA model can be implemented to simulate non-deterministic process in bone remodeling. • The time scales are still a concern for HCA model. Bone adaptation as a CA process

41. Thanks Bone adaptation as a CA process