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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Models of Bone Metastases and Therapy using Fractional Derivatives

Journal of Applied Nonlinear Dynamics 7(1) (2018) 81--94 | DOI:10.5890/JAND.2018.03.007

Luiz Filipe Christ, Duarte Valério, Rui Moura Coelho, Susana Vinga

IDMEC, Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal

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Abstract

The adult skeleton is a highly specialised organ that undergoes constant remodelling over time. Bone metastasis affect the dynamic of this process. For this reason, the study of this dynamic model is essential to the development of better therapies for the disease. Anomalous diffusion phenomena are often found in biological systems, and can be modelled using fractional order derivatives. Consequently, this paper modifies models presented in the literature, consisting of differential equations of order one, checking how their behaviour changes when fractional derivatives are used instead. Results for both local and one-dimensional models of healthy bone tissue and of tumourous bone tissue have the characteristics expected from the literature, with higher fractional orders leading to a faster, more oscillatory system whereas lower orders have a more damped behaviour.

Acknowledgments

This work was supported by FCT, through IDMEC, under LAETA, project UID/EMS/50022/2013, joint Polish–Portuguese project “Modelling and controlling cancer evolution using fractional calculus”, and PERSEIDS (PTDC/EMS-SIS/0642/2014). R. Moura Coelho acknowledges support by grant ZEUGMA-BiNOVA, n AD0075. S. Vinga acknowledges support by Program Investigador FCT (IF/006-53/2012) from FCT, co-funded by the European Social Fund (ESF) through the Operational Program Human Potential (POPH).

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