Skip Navigation Links
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


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:

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

Download Full Text PDF



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.


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).


  1. [1]  Raggatt, L.J. and Partridge, N.C. (2010), Cellular and molecular mechanisms of bone remodeling, The Journal of Biological Chemistry, 285, 25103-25108.
  2. [2]  Komarova, S.V., Smith, R.J., Dixon, S.J., Sims, S.M., and Wahlb, L.M. (2003), Mathematical model predicts a critical role for osteoclast autocrine regulation in the control of bone remodeling, Bone, 33, 206-215.
  3. [3]  Ayati, B.P., Edwards, C.M., Webb, G.F., and Wikswo, J.P. A mathematical model of bone remodeling dynamics for normal bone cell populations and myeloma bone disease, Biology Direct, 5(28).
  4. [4]  Suva, L.J., Washam, C., Nicholas, R.W., and Griffin, R.J. (2011), Bone metastasis: mechanisms and therapeutic opportunities, Nature Reviews Endocrinology, 7, 208-218.
  5. [5]  Magin, R.L. (2004), Fractional Calculus in Bioengineering, Begell House.
  6. [6]  Lenzi, E.K., Neto, R.M., Tateishi, A.A., Lenzi, M.K., and Ribeiro, H.V. (2016), Fractional diffusion equations coupled by reaction terms, Physica A, 458, 9-16.
  7. [7]  Prodanov, D. and Delbeke, J. (2016), A model of space-fractional-order diffusion in the glial scar, Journal of Theoretical Biology, 403, 97-109.
  8. [8]  Iyiola, O.S. and Zaman, F.D. (2014), A fractional diffusion equation model for cancer tumor, AIP Advances, 4, 107121.
  9. [9]  Iomin, A. (2005), Superdiffusion of cancer on a comb structure, Journal of Physics: Conference Series, 7, 57-67.
  10. [10]  Benhamou, C.L., Poupon, S., Lespessailles, E., Loiseau, S., Jennane, R., Siroux, V., Ohley, W., and Pothuaud, L. (2001), Fractal analysis of radiographic trabecular bone texture and bone mineral density: Two complementary parameters related to osteoporotic fractures, Journal of Bone and Mineral Research, 16(4), 697-704.
  11. [11]  Chen, J., Liu, F., and Anh, V. (2008), Analytical solution for the time-fractional telegraph equation by the method of separating variables, Journal of Mathematical Analysis and Applications, 338(2), 1364-1377.
  12. [12]  Velasco, M.P. and Vázquez, L. (2014), On the fractional newton and wave equation in one space dimension, Applied Mathematical Modelling, 38(13), 3314-3324.
  13. [13]  Langlands, T.A.M., Henry, B.I., and Wearne, S.L. (2009), Fractional cable equation models for anomalous electrodiffusion in nerve cells: infinite domain solutions, Journal of Mathematical Biology, 59(6), 761-808.
  14. [14]  Magdziarz, M., Gajda, J., and Zorawik, T. (2014), Comment on fractional fokker lanck equation with space and time dependent drift and diffusion, Journal of Statistical Physics, 154(5), 1241-1250.
  15. [15]  Barkai, E. and Silbey, R.J. (2000), Fractional kramers equation, The Journal of Physical Chemistry B, 104, 3866-3874.
  16. [16]  Ahmad, W.M. and El-Khazali, R. (2007), Fractional-order dynamical models of love, Chaos, Solitons and Fractals, 33, 1367-1375.
  17. [17]  M.A. Savageau, Introduction to S–systems and the underlying power-law formalism, Mathematical and Computer Modelling 11 (1988) 546–551.
  18. [18]  Casimiro, S., Guise, T.A., and Chirgwin, J. (2009), The critical role of the bone microenvironment in cancer metastases, Molecular and Cellular Endocrinology, 310(1-2), 71-81.
  19. [19]  Makatsoris, T. and Kalofonos, H.P. (2009), Bone Metastases, Springer, Ch. The Role of Chemotherapy in the Treatment of Bone Metastases, pp. 287-298.
  20. [20]  Pinheiro, J.V., Lemos, J.M., and Vinga, S. (2011), Nonlinear MPC of HIV-1 infection with periodic inputs, in: IEEE Conference on Decision and Control and European Control Conference, 65-70.
  21. [21]  Valério, D. and Sá da Costa, J. (2013), An Introduction to Fractional Control, IET, Stevenage, iSBN 978-1- 84919-545-4.
  22. [22]  Birkhold, A., Razi, H., Weinkamer, R., Duda, G.N., Checa, S., and Willie, B.M. (2015), Monitoring in vivo (re)modeling: a computational approach using 4D microCT data to quantify bone surface movements, Bone, 75, 210-221.
  23. [23]  Coelho, R., Lemos, J.M., Valério, D., Alho, I., Costa, L., and Vinga, S. (2016), Dynamic modeling of bone metastasis, microenvironment and therapy — integrating parathyroid hormone (PTH) effect, antiresorptive treatment and chemotherapy, Journal of Theoretical Biology, 391, 1-12.
  24. [24]  Pivonka, P., Buenzli, P.R., Scheiner, S., Hellmich, C., and Dunstan, C.R. (2013), The influence of bone surface availability in bone remodelling — a mathematical model including coupled geometrical and biomechanical regulations of bone cells, Engineering Structures, 47, 134-147.
  25. [25]  Scheiner, S., Pivonka, P., and Hellmich, C. (2013), Coupling systems biology with multiscale mechanics, for computer simulations of bone remodeling, Computer Methods in Applied Mechanics and Engineering, 254, 181-196.
  26. [26]  Valério, D. and Sá da Costa, J. (2011), Introduction to single-input, single-output Fractional Control, IET Control Theory & Applications, 5(8), 1033-1057.
  27. [27]  Samko, S.G., Kilbas, A.A., and Marichev, O.I. (1993), Fractional integrals and derivatives, Gordon and Breach, Yverdon.
  28. [28]  Miller, K.S. and Ross, B. (1993), An introduction to the fractional calculus and fractional differential equations, John Wiley and Sons, New York.
  29. [29]  Podlubny, I. (2000), Matrix approach to discrete fractional calculus, Fractional Calculus and Applied Analysis,3(4), 359-386.
  30. [30]  Podlubny, I., Chechkin, A., Skovranek, T., Chen, Y., and Jara, B.M.V. (2009), Matrix approach to discrete fractional calculus II: Partial fractional differential equations, Journal of Computational Physics, 228, 3137- 3153.
  31. [31]  Podlubny, I., Skovranek, T., Jara, B.M.V., Petras, I., Verbitsky, V., and Chen, Y. (1990), Matrix approach to discrete fractional calculus III: non-equidistant grids, variable step length and distributed orders, Philosophical Transactions of the Royal Society A, 371.
  32. [32]  Podlubny, I. (2012), Matrix approach to distributed-order ODEs and PDEs, matlabcentral/fileexchange/36570-matrix-approach-to-distributed-order-odes-and-pdes.
  33. [33]  Hristov, J. (2015), Approximate solutions to time-fractional models by integral-balance approach, in: Cattani, C., Srivastava, H.M., and Yang, X.-J. (Eds.), Fractional Dynamics, De Gruyter, Ch.5, 78-109.