Abstract
In this paper, an in vitro model of HER2+ breast cancer cells dynamics resulting from various dosages and timings of paclitaxel and trastuzumab combination regimens is considered. Since, the combined in vitro results and development of dynamics of drug synergy has a potential to evaluate and improve standard of care, then combination therapies in timings of paclitaxel and trastuzumab combination regimens, thus, HER2+ breast cancer cells dynamics are extended to a system of fractal fractional partial differential equations in order to enable one to capture the dynamics of the deadly breast cancer in terms of combination of the two therapies. Moreover, the well-posedness of solutions is presented and the extended dynamics are analysed to that effect. Since it is not that easy to obtain the analytic solution a novel numerical method based on fractal fractional derivatives is design, implemented and the results with respect to the stability conditions are presented.
Similar content being viewed by others
References
Atangana, A.: Fractal-fractional differentiation and integration: connecting fractal calculus and fractional calculus to predict complex system. Chaos, Solitons Fractals 102, 396–406 (2017)
Atangana, A., Qureshi, S.: Modeling attractors of chaotic dynamical systems with fractal-fractional operators. Chaos, Solitons Fractals 123, 320–337 (2019)
Atangana, A., Baleanu, D.: New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model. Thermal Sci. 20, 763–769 (2016)
Araz, S.I.: Numerical analysis of a new volterra integro-differential equation involving fractal-fractional operators. Chaos, Solitons Fractals 130, 109396 (2020)
Atangana, A., Owolabi, K.M.: New numerical approach for fractional differential equations, Math. Model. Natural Phenomena, 13 (2018) 21 pages. https://doi.org/10.1051/mmnp/2018010
Bartel, C.A., Jackson, M.W.: HER2+ breast cancer cells expressing elevated FAM83A are sensitive to FAM83A loss. PLoS ONE 12(5), e0176778 (2017)
Burden, R.L., Faires, J.D.: Numerical Analysis. Brooks/Cole, USA (2011)
Caputo, M.: Elasticità e dissipazione. Zanichelli, Bologna (1969)
Carslaw, H.S., Jaeger, J.C.: Conduction of Heat in Solids. Oxford University Press, London (1956)
Chen, W., Sun, G., Zhanga, X., Koroãk, D.: Anomalous diffusion modeling by fractal and fractional derivatives. Comput. Math. Appl. 59, 1754–1758 (2010)
Goufo, E.F.D.: Fractal and fractional dynamics for a 3D autonomous and two-wing smooth chaotic system. Alexandria Eng. J. 59, 2469–2476 (2020)
Hassett, M.J., Li, H., Burstein, H.J.: Neoadjuvant treatment strategies for HER2-positive breast cancer: cost-effectiveness and quality of life outcomes. Breast Cancer Res. Treat. 181, 43–51 (2020)
Heydari, M.H.: Numerical solution of nonlinear 2D optimal control problems generated by Atangana-Riemann-Liouville fractal-fractional derivative. Appl. Numer. Math. 150, 507–518 (2020)
Imran, M.A.: Application of fractal fractional derivative of power law kernel to MHD viscous fluid flow between two plates. Chaos, Solitons Fractals 134, 109691 (2020)
Jarrett, A.M., Shah, A., Bloom, M.J., McKenna, M.T., Hormuth, D.A., Yankeelov, T.E., Sorace, A.G.: Experimentally-driven mathematical modeling to improve combination targeted and cytotoxic therapy for HER2+ breast cancer. Sci. Rep. 9, 12830 (2019)
Jordan, D.W., Smith, P.: Nonlinear Ordinary Differential Equations. Clarendon Press, Oxford (1987)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Netherlands (2006)
Lakhtakia, R., Burney, I.: A brief history of breast cancer. Sultan Qaboos Univ. Med. J. 15(1), e34–e38 (2015)
Luchko, Y.: Maximum principle for the generalized time-fractional diffusion equation. J. Math. Anal. Appl. 351, 218–223 (2009)
Metzler, R., Barkai, E., Klafter, J.: Deriving fractional Fokker-Planck equations from a generalized master equation. Europhys. Lett. 46(4), 431–436 (1999)
Owolabi K.M., Patidar K.C., Shikongo A.: Mathematical analysis and numerical simulation of a tumor-host model with chemotherapy application. Commun. Math. Biol. Neurosci. 2018, Article ID 21 (2018)
Ortigueira, M.D.: Riesz potential operators and inverses via fractional centred derivatives. Int. J. Math. Math. Sci. 2006(48391), 1–12 (2006)
Owolabi, K.M.: Mathematical analysis and numerical simulation of patterns in fractional and classical reaction-diffusion systems. Chaos, Solitons Fractals 93, 89–98 (2016)
Owolabi, K.M.: Mathematical modelling and analysis of two-component system with Caputo fractional derivative order. Chaos, Solitons Fractals 103, 544–554 (2017)
Owolabi, K.M.: Numerical patterns in reaction-diffusion system with the Caputo and Atangana-Baleanu fractional derivatives. Chaos, Solitons Fractals 115, 160–169 (2018)
Palle, J., Rochand, A., Pernot, S.: Human epidermal growth factor receptor 2 (HER2) in advanced gastric cancer: Current knowledge and future perspectives. Drugs 80, 401–415 (2020)
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
Podlubny, I., Chechkin, A.V., Skovranek, T., Chen, Y., Jara, B.M.V.: Matrix approach to discrete fractional calculus II: partial fractional differential equations. J. Comput. Phys. 228(8), 3137–3153 (2009)
Saichev, A., Zaslavsky, G.: Fractional kinetic equations: solutions and applications. Chaos 7(4), 753–764 (1997)
Xie, B., Zhu, L., Ma, C.: A network meta-analysis on the efficacy of HER2-targeted agents in combination with taxane-containing regimens for treatment of HER2-positive metastatic breast cancer. Breast Cancer 27, 186–196 (2020). https://doi.org/10.1007/s12282-019-01007-9
Acknowledgements
The authors are grateful to all of the anonymous reviewers for their valuable suggestions
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Competing interests
The authors have no competing interests
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Owolabi, K.M., Shikongo, A. Fractal Fractional Operator Method on HER2+ Breast Cancer Dynamics. Int. J. Appl. Comput. Math 7, 85 (2021). https://doi.org/10.1007/s40819-021-01030-5
Accepted:
Published:
DOI: https://doi.org/10.1007/s40819-021-01030-5
Keywords
- Paclitaxel
- Trastuzumab
- BT474 HER2+
- Fractal fractional operator
- Well-posed
- Stability analysis
- Numerical method