Modeling of Trabecular Architecture as Result of an Optimal Control Procedure

Part of the Lecture Notes in Computational Vision and Biomechanics book series (LNCVB, volume 4)


The primary mechanical function of bones is to provide rigid levers for muscles to pull against and to remain as light as possible to allow for efficient locomotion. To accomplish this function, bones must adapt their shape and architecture efficiently. Bone tissue during skeletal growth and development continuously adjusts its mass and architecture to changing mechanical environments.

The trabecular structure in bone is the result of a dynamic remodeling process controlled by mechanical loads.

In this study, the process of adaptive bone remodeling is investigated mathematically and simulated by a finite element model. Bone tissue is described as continuous material with variable mass density. Topology optimization and cellular automata models are adopted with the aim to characterize the osteocytes located within the bone as sensors of mechanical signals.

An implemented optimization process provides structures resembling actual trabecular architectures and aligning them with the actual principal stress orientation. Two cost indices are derived from the structural optimization task of simultaneously minimizing the weight and maximizing the stiffness of the investigated domain. A trabecular architecture characterized by the highest structural efficiency is showed as result of the proposed optimal control process.


Optimal control PID control Finite element method (FEM) Bone functional adaptation Minimum mass Maximum stiffness 



This research was partially funded by Sapienza University of Rome under Progetto di Ateneo 2004 grant no. C26A044385, Progetto di Ateneo 2005 grant no. C26A059503 and Progetto di Università 2008 grant no. C26A08E7B3.


  1. Andreaus U, Colloca M, Iacoviello D, Pignataro M (2011) Optimal-tuning PID control of adaptive materials for structural efficiency. Struct Multidiscip Optim 43:43–59CrossRefGoogle Scholar
  2. Andreaus U, Colloca M, Iacoviello D (2012) An optimal control procedure for bone adaptation under mechanical stimulus. Control Eng Pract 20:575–583Google Scholar
  3. Bagge M (2000) A model of bone adaptation as an optimization process. J Biomech 33:1349–1357CrossRefGoogle Scholar
  4. Battiti R (1992) First and second order methods for learning: between steepest descent and Newton’s method. Neural Comput 4:141–166CrossRefGoogle Scholar
  5. Beaupré GS, Orr TE, Carter DR (1990a) An approach for time-dependent bone modeling and remodeling-theoretical development. J Orthop Res 8:651–661CrossRefGoogle Scholar
  6. Beaupré GS, Orr TE, Carter DR (1990b) An approach for time-dependent bone modeling and remodeling-application: a preliminary remodeling simulation. J Orthop Res 8:662–670CrossRefGoogle Scholar
  7. Bourgery JM (1832) Traité complet de l’anatomie de l’homme. Delaunay, ParisGoogle Scholar
  8. Carter DR (1984) Mechanical loading histories and cortical bone remodeling. Calcif Tissue Int 36:519–524CrossRefGoogle Scholar
  9. Carter DR (1987) Mechanical loading history in skeletal biology. J Biomech 20:1095–1105CrossRefGoogle Scholar
  10. Carter DR, Hayes WC (1977) The compressive behaviour of bone as a two-phase porous structure. J Bone Joint Surg Am 59:954–962Google Scholar
  11. Carter DR, Fyhrie DP, Whalen RT (1987) Trabecular bone density and loading history: regulation of connective tissue biology by mechanical energy. J Biomech 20:785–794CrossRefGoogle Scholar
  12. Carter DR, Orr TE, Fyhrie DP (1989) Relationships between loading history and femoral cancellous bone architecture. J Biomech 22:231–244CrossRefGoogle Scholar
  13. Cowin SC, Doty SB (2007) Tissue mechanics. Springer, New YorkMATHCrossRefGoogle Scholar
  14. Cowin SC, Hegedus DH (1976) Bone remodeling I: a theory of adaptive elasticity. J Elast 6:313–326MathSciNetMATHCrossRefGoogle Scholar
  15. Currey JD (1988) The effect of porosity and mineral content on the Young’s modulus of elasticity of compact bone. J Biomech 21:131–139CrossRefGoogle Scholar
  16. Curry J (1984) The mechanical adaptations of bones. Princeton University Press, PrincetonGoogle Scholar
  17. Fletcher R (1987) Practical methods of optimization, 2nd edn. John Wiley and Sons, ChichesterMATHGoogle Scholar
  18. Frost HM (1983) A determinant of bone architecture. The minimum effective strain. Clin Orthop Relat Res 175:286–292Google Scholar
  19. Fyhrie DP, Carter DR (1986) A unifying principle relating stress to trabecular bone morphology. J Orthop Res 4:304–317CrossRefGoogle Scholar
  20. Gill PE, Murray W, Wright MH (1981) Practical optimization. Academic, LondonMATHGoogle Scholar
  21. Huiskes R, Weinans H, Grootenboer J, Dalstra M, Fudala M, Slooff TJ (1987) Adaptive bone remodeling theory applied to prosthetic-design analysis. J Biomech 20:1135–1150CrossRefGoogle Scholar
  22. Huiskes R, Ruimerman R, van Lenthe GH, Janssen JD (2000) Effects of mechanical forces on maintenance and adaptation of form in trabecular bone. Nature 405:704–706CrossRefGoogle Scholar
  23. Jacobs CR, Simo JC, Beaupré GS, Carter DR (1997) Adaptive bone remodeling incorporating simultaneous density and anisotropy considerations. J Biomech 30:603–613CrossRefGoogle Scholar
  24. Lekszycki T (2005) Functional adaptation of bone as an optimal control problem. J Theor Appl Mech 43:555–574Google Scholar
  25. Lennartson B, Kristiansson B (2006) Evaluation and tuning of robust PID controllers. IET Control Theor Appl 3:294–302CrossRefGoogle Scholar
  26. Liu GP, Daley S (1999) Optimal-tuning PID controller design in a frequency domain with application to a rotary hydraulic system. Control Eng Pract 7:821–830CrossRefGoogle Scholar
  27. Liu GP, Daley S (2001) Optimal-tuning PID control for industrial system. Control Eng Pract 9:1185–1194CrossRefGoogle Scholar
  28. Mullender MG, Huiskes R (1995) Proposal for the regulatory mechanism of Wolff’s law. J Orthop Res 13:503–512CrossRefGoogle Scholar
  29. Mullender MG, Huiskes R, Weinans H (1994) A physiological approach to the simulation of bone remodeling as a self-organizational control process. J Biomech 27:1389–1394CrossRefGoogle Scholar
  30. Patel NM, Tillotson D, Renaud JE, Tovar A, Izui K (2008) Comparative study of topology optimization techniques. AIAA J 46:1963–1975CrossRefGoogle Scholar
  31. Penninger CL, Patel NM, Niebur GL, Tovar A, Renaud JE (2008) A fully anisotropic hierarchical hybrid cellular automaton algorithm to simulate bone remodeling. Mech Res Commun 35:32–42CrossRefGoogle Scholar
  32. Roesler H (1987) The history of some fundamental concepts in bone biomechanics. J Biomech 20:1025–1034CrossRefGoogle Scholar
  33. Roux W (1881) Der ziichtende kampf der teile, oder die ‘teilauslese’ im organismus (theorie der ‘finktionellen unpassung’). Wilhelm Engelmann, LeipzigGoogle Scholar
  34. Rubin CT (1984) Skeletal strain and the functional significance of bone architecture. Calcif Tissue Int 36:11–18CrossRefGoogle Scholar
  35. Rubin CT, Lanyon LE (1987) Osteo-regulatory nature of mechanical stimuli: function as a determinant for adaptive remodeling in bone. J Orthop Res 5:300–310CrossRefGoogle Scholar
  36. Subbarayan G, Bartel DL (1989) Fully stressed and minimum mass structures in bone remodeling. In: Friedman MH (ed) Biomechanics symposium, vol 98. American Society of Mechanical Engineers, New York, pp 173–176Google Scholar
  37. Tovar A (2004) Bone remodeling as a hybrid cellular automaton optimization process. Dissertation, University of Notre DameGoogle Scholar
  38. Tovar A, Patel NM, Niebur GL, Sen M, Renaud JE (2006) Topology optimization using a hybrid cellular automaton method with local control rules. ASME J Mech Des 128:1205–1216CrossRefGoogle Scholar
  39. Tovar A, Patel NM, Kaushik AK, Renaud JE (2007) Optimality conditions of the hybrid cellular automata for structural optimization. AIAA J 45:673–683CrossRefGoogle Scholar
  40. Turner CH (1991) Homeostatic control of bone structure: an application of feedback theory. Bone 12:203–217CrossRefGoogle Scholar
  41. Weinans H, Huiskes R, Grootenboer HJ (1992) The behaviour of adaptive bone-remodeling simulations models. J Biomech 25:1425–1441CrossRefGoogle Scholar
  42. Wolff J (1892) Das Gesetz der transformation der knochem. Hirschwald, BerlinGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of Structural and Geotechnical Engineering, Faculty of Civil and Industrial EngineeringSapienza University of RomeRomeItaly
  2. 2.Department of Mechanical and Aerospace EngineeringPolytechnic Institute of New York UniversityBrooklynUSA
  3. 3.Department of Computer, Control and Management Engineering Antonio RubertiSapienza University of RomeRomeItaly

Personalised recommendations