Abstract
After an initial phase of growth and development, bone undergoes a continuous cycle of repair, renewal and optimisation by a process called remodelling. This paper describes a novel mathematical model of the trabecular bone remodelling cycle. It is essentially formulated to simulate a remodelling event at a fixed position in the bone, integrating bone removal by osteoclasts and formation by osteoblasts. The model is developed to construct the variation in bone thickness at a particular point during the remodelling event, derived from standard bone histomorphometric analyses. The novelties of the approach are the adoption of a predator–prey model to describe the dynamic interaction between osteoclasts and osteoblasts, using a genetic algorithm–based solution; quantitative reconstruction of the bone remodelling cycle; and the introduction of a feedback mechanism in the bone formation activity to co-regulate bone thickness. The application of the model is first demonstrated by using experimental data recorded for normal (healthy) bone remodelling to predict the temporal variation in the number of osteoblasts and osteoclasts. The simulated histomorphometric data and remodelling cycle characteristics compare well with the specified input data. Sensitivity studies then reveal how variations in the model’s parameters affect its output; it is hoped that these parameters can be linked to specific biochemical factors in the future. Two sample pathological conditions, hypothyroidism and primary hyperparathyroidism, are examined to demonstrate how the model could be applied more broadly, and, for the first time, the osteoblast and osteoclast populations are predicted for these conditions. Further data are required to fully validate the model’s predictive capacity, but this work shows it has potential, especially in the modelling of pathological conditions and the optimisation of the treatment of those conditions.
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References
Agerbaek M, Eriksen E, Kragstrup J, Mosekilde L, Melsen F (1991) A reconstruction of the remodelling cycle in normal human cortical iliac bone. Bone Miner 12: 101–112
Allori A, Sailon A, Warren S (2008) Biological basis of bone formation, remodelling, and repair—Part I: Biochemical signaling molecules. Tissue Eng Part B 14(3): 259–273
Aubin JE (1998) Advances in the osteoblast lineage. Biochem Cell Biol 76(6): 899–910
Boyce BF, Xing L (2008) Functions of RANKL/RANK/OPG in bone modelling and remodelling. Arch Biochem Biophys 473: 139–146
Buenzli PR, Pivonka P, Smith DW (2011) Spatio-temporal structure of cell distribution in cortical bone multicellular units: a mathematical model. Bone 48(4): 918–926
Canalis E (1993) Regulation of bone remodelling. In: Favus M, Christakos S, Gagel R, Kleerekoper M, Langman C, Shane E, Stewart A, Whyte M (eds) Primer on the metabolic bone diseases and disorders of mineral metabolism. Lippincott-Raven, New York, pp 33–37
Christiansen P (2001) The skeleton in primary hyperparathyroidism: a review focusing on bone remodelling, structure, mass, and fracture. APMIS Suppl 102: 1–52
Einhorn TA (1996) The bone organ system: form and function. In: Marcus R, Feldman D, Kelsey J (eds) Osteoporosis, Chap 1. Academic Press, New York, pp 3–21
Eriksen E, Gundersen H, Melsen F, Mosekilde L (1984a) Reconstruction of the resorptive site in iliac trabecular bone: a kinetic model for bone resorption in 20 normal individuals. Metab Bone Dis Rel Res 5: 235–242
Eriksen E, Gundersen H, Melsen F, Mosekilde L (1984b) Reconstruction of the formative site in iliac trabecular bone in 20 normal individuals employing a kinetic model for matrix and mineral apposition. Metab Bone Dis Rel Res 5: 243–252
Eriksen E, Mosekilde L, Melsen F (1985) Trabecular bone remodelling and bone balance in hyperthyroidism. Bone 6: 421–428
Eriksen E, Mosekilde L, Melsen F (1986a) Kinetics of trabecular bone resorption and formation in hypothyroidism: evidence for a positive balance per remodelling cycle. Bone 7: 101–108
Eriksen E, Mosekilde L, Melsen F (1986b) Trabecular bone remodelling and balance in primary hyperparathyroidism. Bone 7: 213–221
Frost H (1979) Treatment of osteoporoses by manipulation of coherent bone cell populations. Clin Orthop Relat Res 143: 227–244
Frost H (1986) Intermediary organization of the skeleton. CRC press, Boca Raton
Gause GF, Smaragdova N, Witt A (1936) Further studies of interaction between predator and prey. J Anim Ecol 5: 1–18
Gilsanz V, Gibbens D, Carlson M, Boechat I, Cann C, Schulz E (1988) Peak trabecular bone density: a comparison of adolescent and adult females. Calcif Tissue Int 43: 260–262
Jaworski Z, Hooper C (1980) Study of cell kinetics within evolving secondary haversian systems. J Anat 131: 91–102
Jaworski Z, Duck B, Sekaly G (1981) Kinetics of osteoclasts and their nuclei in evolving secondary haversian systems. J Anat 133: 397–405
Kanatani M, Sugimoto T, Sowa H, Kobayashi T, Kanzawa M, Chihara K (2004) Thyroid hormone stimulates osteoclast differentiation by a mechanism independent of RANKL–RANK interaction. J Cell Physiol 201(1): 17–25
Komarova S, Smith R, Dixon S, Sims S, Wahl L (2003) Mathematical model predicts a critical role for osteoclast autocrine regulation in the control of bone remodelling. Bone 33: 206–215
Kroll M (2000) Parathyroid hormone temporal effects on bone formation and resorption. Bull Math Biol 62: 163–188
Kuang Y (1990) Global stability of Gause-type predator-prey systems. J Math Biol 28: 463–474
Lemaire V, Tobin F, Greller L, Cho C, Suva L (2004) Modeling the interactions between osteoblast and osteoclast activities in bone remodelling. J Theor Biol 229: 293–309
Manolagas SC (2000) Birth and death of bone cells: basic regulatory mechanisms and implications for the pathogenesis and treatment of osteoporosis. Endocr Rev 21(2): 115–137
Miura M, Tanaka K, Komatsu Y, Suda M, Yasoda A, Sakuma Y, Ozasa A, Nakao K (2002) A novel interaction between thyroid hormone and 1,25(OH)(OH)2D3 in osteoclast formation. Biochem Biophys Res Commun 291: 987–994
Moroz A, Crane M, Smith G, Wimpenny D (2006) Phenomenological model of bone remodelling cycle containing osteocyte regulation loop. Biosystems 84: 183–190
Parfitt A (1994) Osteonal and hemi-osteonal remodelling: the spatial and temporal framework for signal traffic in adult human bone. J Cell Biochem 55: 273–286
Parfitt A (2000) The mechanism of coupling: a role for the vasculature. Bone 26: 319–323
Pivonka P, Zimak J, Smith D, Gardiner B, Dunstan C, Sims N, Martin T, Mundy G (2008) Model structure and control of bone remodelling: a theoretical study. Bone 43: 249–263
Pivonka P, Zimak J, Smith D, Gardiner B, Dunstan C, Sims N, Martin T, Mundy G (2010) Theoretical investigation of the role of the RANK–RANKL–OPG system in bone remodelling. J Theor Biol 262: 306–316
Rattanakul C, Lenbury Y, Krishnamara N, Wollkind D (2003) Modelling of bone formation and resorption mediated by parathyroid hormone: response to oestrogen/PTH therapy. Biosystems 70: 55–72
Rodan GA (1998) Control of bone formation and resorption: biological and clinical perspective. J Cell Biochem 30–31(Suppl): 55–61
Rodan GA, Martin TJ (2000) Therapeutic approaches to bone diseases. Science 289: 1508–1514
Roodman GD (1999) Cell biology of the osteoclast. Exp Hematol 27(8): 1229–1241
Ryser MD, Komarova SV, Nigam N (2010) The cellular dynamics of bone remodelling: a mathematical model. SIAM J Appl Math 70(6): 1899–1921
Seeman E, Delmas P (2006) Bone quality the material and structural basis of bone strength and fragility. N Engl J Med 354: 2250–2261
Sommerfeldt DW, Rubin CT (2001) Biology of bone and how it orchestrates the form and function of the skeleton. J Eur Spine 10: S86–S95
Teitelbaum SL (2000) Bone resorption by osteoclasts. Science 289: 1504–1508
Udagawa N, Nakamura M, Sato N, Takahashi N (2006) The mechanism of coupling between bone resorption and formation. J Oral Biosci 48: 185–197
Zaidi M (2007) Skeletal remodelling in health and disease. Nat Med 13: 791–801
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Ji, B., Genever, P.G., Patton, R.J. et al. A novel mathematical model of bone remodelling cycles for trabecular bone at the cellular level. Biomech Model Mechanobiol 11, 973–982 (2012). https://doi.org/10.1007/s10237-011-0366-3
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DOI: https://doi.org/10.1007/s10237-011-0366-3