Variable order 3D models of bone remodelling

• Autorzy: Valério D. , Neto J. , Vinga S.

Identyfikatory

DOI

10.24425/bpasts.2019.129649

• Języki publikacji: EN

Abstrakty

EN

This paper presents simulations of a three-dimensional model of the bone remodelling process. The model consists of a set of variable order partial differential equations, in which the varying order depends on the presence of tumour cells. The simulations are of a two-dimensional bone, to make visualisation simpler. They show that this model corresponds to the known evolution of bone remodelling, and is simpler than integer order models found in the literature.

Słowa kluczowe

EN

variable-order derivative; bone remodelling; cancer

PL

przebudowa kości; rak

• Wydawca: Polska Akademia Nauk, Wydział IV Nauk Technicznych
• Czasopismo: Bulletin of the Polish Academy of Sciences. Technical Sciences
• Rocznik: 2019
• Tom: Vol. 67, nr 3
• Strony: 501--508
• Opis fizyczny: Bibliogr. 19 poz., wykr., rys.

Twórcy

autor

Valério D.

• IDMEC, Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal

autor

Neto J.

• IDMEC, Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal

autor

Vinga S.

• IDMEC, Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal

Bibliografia

• [1] B.P.Ayati, C.M. Edwards, G.F.Webb, and J.P.Wikswo, “Amathematical model of bone remodeling dynamics for normal bone cell populations and myeloma bone disease”, Biology Direct 5 (28), (2010).
• [2] J. Belinha, L.M.J.S. Dinis, and R.M. Natal Jorge, “The mandible remodelling induced by dental implants: a meshless approach”, Journal of Mechanics in Medicine and Biology 15 (4), 1550059 (2015).
• [3] J. Belinha, L.M.J.S. Dinis, and R.M. Natal Jorge, “The meshless methods in the bone tissue remodelling analysis”, Procedia Engineering, 110, 51–58 (2015).
• [4] A.I. Birkhold, H. Razi, R. Weinkamer, G.N. Duda, S. Checa, and B.M. Willie, “Monitoring in vivo (re)modeling: a computational approach using 4D microCT data to quantify bone surface movements”, Bone 75, 210–221 (2015).
• [5] L.F. Christ, D. Valério, R.M. Coelho, and S. Vinga, “Models of bone metastases and therapy using fractional derivatives”, Journal of Applied Nonlinear Dynamics 7 (1), 81–94 (2017).
• [6] R. Coelho, J.M. Lemos, D. Valério, I. ,Alho, L. Costa, and S. Vinga, “Dynamic modeling of bone metastasis, microenvironment and therapy – integrating parathyroid hormone (PTH) effect, antiresorptive treatment and chemotherapy”, Journal of Theoretical Biology 391, 1–12 (2016).
• [7] S.V. Komarova, R.J. Smith, S.J. Dixon, S.M. Sims, and L.M. Wahlb, “Mathematical model predicts a critical role for osteoclast autocrine regulation in the control of bone remodeling”, Bone 33, 206–215 (2003).
• [8] M. Macias and D. Sierociuk, “An alternative recursive fractional variable-order derivative definition and its analog validation”, in Proceedings of International Conference on Fractional Differentiation and its Applications, Catania, Itally, 2014.
• [9] R.L. Magin, Fractional Calculus in Bioengineering. Begell House, 2004.
• [10] W. Malesza and M. Macias, “Numerical solution of fractional variable order linear control system in state-space form”, Bull. Pol. Ac.: Tech. 65 (5), 715–724 (2017).
• [11] K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, JohnWiley and Sons, New York, 1993.
• [12] J. Neto, R.M. Coelho, D. Valério, S. Vinga, D. Sierociuk, W. Malesza, M. Macias, and A. Dzielinski, “Variable order differential models of bone remodelling”, in IFAC World Congress, 2017.
• [13] S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach, Yverdon, 1993.
• [14] M.A. Savageau, “Introduction to S-systems and the underlying power-law formalism”, Mathematical and Computer Modelling 11, 546–551 (1988).
• [15] D. Sierociuk,W. Malesza, and M. Macias, “Derivation, interpretation, and analog modelling of fractional variable order derivative definition”, Applied Mathematical Modelling 39 (13), 3876–3888 (2015). http://dx.doi.org/10.1016/j.apm.2014.12.009.
• [16] F. Teles, “Cancer therapy optimization based on unsupervised learning and multiple model adaptive control”, Master’s thesis, University of Lisbon, IST, 2017.
• [17] D. Valério, R. Coelho, and S. Vinga, “Fractional dynamic modelling of bone metastasis, microenvironment and therapy”, in International Conference on Fractional Differentiation and its Applications, 2016.
• [18] D. Valério and J. Sá da Costa, “Introduction to single-input, single-output Fractional Control”, IET Control Theory & Applications 5 (8), 1033–1057 (2011).
• [19] D. Valério and J. Sá da Costa, An Introduction to Fractional Control, IET, Stevenage, 2013, ISBN 978-1-84919-545-4.

Uwagi

PL

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).