Functional adaptation of bone as an optimal control problem

• Autorzy: Lekszycki T.

Warianty tytułu

PL

Funkcjonalna adaptacja kości jako zagadnienie optymalnego sterowania

• Języki publikacji: EN

Abstrakty

EN

The functional adaptation of bone is a process of bone tissue remodeling induced by variable in time mechanical demands that the skeleton has to satisfy. It is a very complex but highly organized process composed of events at micro-level (molecular and cellular) but having effects in macro-scale (variation of bone internal structure and external shape). Mathematical models of this phenomenon proposed in the literature represent formulas postulated on the basis of the results of medical observations or laboratory investigations and describe locally the evolution of a material in space and time. In the present paper a use is made of the hypothesis of optimal response of bone, proposed earlier by the author, what enables derivation (instead of postulation) the remodeling rules from a very general and global assumption. It turns out that such a formulation has many similarities to engineering optimal control problems. The link between the postulated local adaptation rules and those derived from the global assumption is also discussed.

PL

Funkcjonalna adaptacja kości jest procesem polegającym na przebudowie tkanki kostnej wywołanej zmieniającymi się w czasie wymaganiami mechanicznymi, jakie musi spełniać szkielet kostny. Proces ten jest niezwykle złożony, ale doskonale zorganizowany i składa się z szeregu zjawisk zachodzących na poziomie mikro (molekularnym i komórkowym) lecz mających efekt na poziomie makro (zmiana zewnętrznego kształtu kości oraz jej struktury wewnętrznej). Matematyczne modele tego zjawiska, postulowane w oparciu o obserwacje medyczne i badania laboratoryjne, opisują lokalną ewolucję materiału w czasie i przestrzeni. W tej pracy zastosowano hipotezę optymalnej odpowiedzi kości zaproponowaną wcześniej przez autora w celu wyprowadzenia (zamiast postulowania) związków rządzących przebudową kości w oparciu o bardzo ogólne założenia. Okazuje się, że takie sformułowanie ma wiele wspólnego z zagadnieniami optymalnego sterowania. W pracy zaprezentowano przykład zastosowania omawianego podejścia oraz przeprowadzono krótką dyskusję na temat związków między postulowanymi modelami i wyprowadzonymi w oparciu o przyjętą hipotezę.

Słowa kluczowe

EN

adaptation; bone; objective functional; optimal control; remodeling; tissue

• Wydawca: Polskie Towarzystwo Mechaniki Teoretycznej i Stosowanej
• Czasopismo: Journal of Theoretical and Applied Mechanics
• Rocznik: 2005
• Tom: Vol. 43 nr 3
• Strony: 555--574
• Opis fizyczny: Bibliogr. 44 poz., rys.

Twórcy

autor

Lekszycki T.

• Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw,

Bibliografia

• 1. Baud C.A., 1968, Submicroscopic structure and functional aspects of the osteocyte, Clinical Orthopaedics and Related Research, 56, 227-236
• 2. Bendsoe M.P., Kikuchi N., 1988, Generating optimal topologies in structural design using a homogenisation method, Comput. Methods Appl. Mech. Eng., 71, 197-224
• 3. Bronckers A.L.J.J., Goei W., Luo G., Karsenty G., D'Souza R.N., Lyaruu D.M., Burger E.H., 1996, DNA fragmentation during bone formation in neonatal rodents assessed by transferasemediated end labeling, Journal of Bone and Mineral Research, 11, 1281-1291
• 4. Burr D.B., Martin R.B., 1992, Mechanisms of bone adaptation to the mechanical environment, Triangle, 31, 2/3, 59-76
• 5. Burger E.H., Klein-Nulend J., 1999, Mechanotransduction in bone { role of the lacuno-canalicular network, FASEB J., 13S, S101-112
• 6. Burger E.H., Klein-Nulend J., Smit T.H., 2003, Strain-derived canalicular uid ow regulates osteoclast activity in a remodelling osteon { a proposal, J. Biomechanics, 36, 1453-1459
• 7. Carter D.R., Orr T.E., 1992, Skeletal development and bone functional adaptation, J. of Bone and Mineral Research, 7, S389-S395
• 8. Cowin S.C., 1995, On the minimization and maximization of the strain energy density in cortical bone tissue, J. of Biomech., 28, 4, 445-447
• 9. Cowin S.C., Hegedus D.H., 1976, Bone remodeling i: theory of adaptive elasticity, J. Elascity, 6, 3, 313-326
• 10. Cowin S.C., Moss-Salentijn L., Moss M.L., 1991, Candidates for the mechanosensory system in bone, J. Biomech. Eng., 113, 191-197
• 11. Cowin S.C., Weinbaum S., 1998, Strain amplification in the bone mechanosensory system, Am. J. Med. Sci., 316, 184-188
• 12. Cowin S.C., Weinbaum S., Zeng Y., 1995, A case for bone canaliculi as the anatomical site of strain generated potentials, J. Biomech., 28, 1281-1296
• 13. Doblare M., Garcia J.M., 2002, Anisotropic bone remodeling model based on a continuum damage-repair theory, J. Biomechanics, 35, 1-17
• 14. Doty S.B., 1981, Morphological evidence of gap junctions between bone cells, Calcif. Tissue Int., 33, 509-512
• 15. Dudley H.R., Spiro D., 1961, The fine structure of bone cells, The Journal of Biophysical and Biochemical Cytology, 11, 627-649
• 16. Fernandes P., Rodrigues H.C., Jacobs C.R., 2000, A model of bone adaptation using a global optimization criterion based on the trajectorial theory of Wolff, Mechanics in Biology, ASME 2000, AMD-Vol.242/BED-Vol.46, 173-184
• 17. Folgado J., Fernandes P.R., Guedes J.M., Rodrigues H.C., 2004, Evaluation of osteoporotic bone quality by a computational model for bone remodeling, Computers and Structures, 82, 1381-1388
• 18. Frost H.M., 1964, Dynamics of bone remodeling, In: Frost H.M. (Edit.), Bone Biodynamics, Little, Brown, Boston, MA, 315-333
• 19. Harringan T.P., Hamilton J.J., 1993, Bone strain sensation via transmembrane potential changes in surface osteoblasts: loading rate and microstructural implications, J. of Biomech., 26, 183-200
• 20. Hart R.T., Davy D.T., 1989, Theories of bone modeling and remodeling, In S.C. Cowin (Edit.), Bone Mechanics, CRC Press, Boca Raton, FL, 253-277
• 21. Hegedus D.H., Cowin S.C., 1996, Bone remodeling ii: small strain adaptive elasticity, J. of Elasticity, 6, 337-355
• 22. Klein-Nulend J., Van der Plas A., Semeins C.M., Ajubi N.E., Frangos J.A., Nijweide P.J., Burger E.H., 1995, Sensitivity of osteocytes to biomechanical stress in vitro, FASEB J., 9, 441-445
• 23. Knothe T.M.L., Adamson J.R., Tami A.E., Bauer T.W., 2004, The osteocyte, The International Journal of Biochemistry and Cell Biology, 36, 1-8
• 24. Lekszycki T., 1999, Optimality conditions in modeling of bone adaptation phenomenon, J. Theoret. Appl. Mech., 37, 3, 607-623
• 25. Lekszycki T., 2002, Modelling of bone adaptation based on an optimal response hypothesis, Meccanica, 37, 343-354
• 26. Lemairea V., Tobina F.L., Grellera L.D., Choa C.R., Suvab L.J., 2004, Modeling the interactions between osteoblast and osteoclast activities in bone remodeling, J. of Theoretical Biology, 229, 293-309
• 27. Levenston M.E., Carter D.R., 1998, An energy dissipation-based model for damage stimulated bone adaptation, J. of Biomech., 31, 7, 579-586
• 28. Luo G., Cowin S.C., Sadegh A.M., Arramon Y.P., 1995, Implementation of strain rate as a bone remodeling stimulus, J. of Biomechanical Engineering, 117, 3, 329-338
• 29. Mullender M.G., Huiskes R., 1995, A proposal for the regulatory mechanizm of Wolff's law, J. of Orthopaedic Research, 13, 503-512
• 30. Nefussi J.R., Sautier J.M., Nicolas V., Forest N., 1991, How osteoblasts become osteocytes: a decreasing matrix forming process, J. Biol. Buccale, 19, 75-82
• 31. Noble B.S., Stevens H., Loveridge N., Reeve J., 1997, Identification of apoptotic changes in osteocytes in normal and pathological human bone, Bone, 20, 182-273
• 32. Palumbo C.S., Palazzini, Zaffe D., Marotti G., 1990, Osteocyte differentiation in the tibia of newborn rabbit: an ultrastructural study of the formation of cytoplasmic processes, Acta Anat. (Basel), 137, 350-358
• 33. Parfitt A.M., 1977, The cellular basis of bone turnover and bone loss: a rebuttal of the osteocytic resorption { Bone ow theory, Clin. Orthop., 236-247
• 34. Prendergast P.J., Huiskes R., Soballe K., 1997, ESB Research Award 1996. Biophysical stimuli on cells during tissue differentiation at implant interfaces, J. of Biomechanics, 30, 6, 539-548
• 35. Prendergast P.J., Taylor D., 1994, Prediction of bone adaptation using damage accumulation, J. Biomech., 27, 1067-1076
• 36. Rodan G.A., 1991, Mechanical loading, estrogen deficiency, and coupling of bone formation to bone resorption, Journal of Bone and Mineral Research, 6, 527-530
• 37. Rodrigues H.C., Jacobs C.R., Guedes J.M., Bendsoe M.P., 1999, Global and local material optimization models applied to anisotropic bone adaptation, In P. Pedersen and M.P. Bendsoe (Edit.), Synthesis in Bio Solid Mechanics, Kluwer Academic Publishers, 209-220
• 38. Ruimerman R., Hilbers P., Van Rietbergen B., Huiskes R., 2005, A theoretical framework for strain-related trabecular bone maintenance and adaptation, J. of Biomechanics, 38, 931-941
• 39. Taber L.A., 1995, Biomechanics of growth, remodeling, and morphogenesis, Appl. Mech. Rev., 48, 8, 487-545
• 40. Tanaka N., Adachi T., 1999, Lattice continuum model for bone remodeling considering microstructural optimality of trabecular architecture, In P. Pedersen and M.P. Bendsoe (Edit.), Synthesis in Bio Solid Mechanics, Kluwer Academic Publishers, 43-54
• 41. Tezuka K., Wada Y., Takahashi A., Kikuchi M., 2005, Computersimulated bone architecture in a simple bone-remodeling model based on a reaction-diffusion system, J. Bone Miner. Metab., 23, 1-7
• 42. Verborgt O.G.J., Gibson, Schaffler M.B., 2000, Loss of osteocyte integrity in association with microdamage and bone remodeling after fatigue in vivo, J. of Bone and Mineral Research, 15, 60-67
• 43. Weinbaum S., Cowin S.C., Zeng Y., 1994, A model for the excitation of osteocytes by mechanical loading-induced bone uid shear stresses, J. Biomech., 27, 339-360
• 44. Wolff J., 1892, Das Gesetz der Transformation der Knochen, A. Hirchwild, Berlin (Translated by Maquet P., Furlong R., The Law of Bone Remodeling, 1986, Springer, Berlin)