Stabilization of Chaos in a Cancer Model: The Effect of Oncotripsy

Year 2022, Volume: 10 Issue: 2, 139 - 149, 30.04.2022

Serpil Yılmaz

https://doi.org/10.17694/bajece.1039384

Abstract

There has been much interest in the development of therapies for the prevention and treatment of tumours. Recently, the method of oncotripsy has been proposed to destroy cancer cells by applying the ultrasound harmonic excitations at the resonant frequency of cancer cells. In this study, periodic disturbances whose frequency tuned to the fundamental frequency and the higher harmonics of the cancer cells are applied to a tumour growth model, respectively, and the appearance of periodic behaviors in a three-dimensional chaotic cancer model is investigated as a result of those harmonic excitations. The numerical results show that by choosing the appropriate values of the parameters of periodic disturbances, the chaotic cancer model induces periodic behaviors such as period-one and two limit cycles which may have important implications on cancer treatment. The results also provide a view to understanding the oncotripsy effect within the framework of stabilization of chaos.

Keywords

Bifurcation, Cancer model, Chaos, Stabilization

References

• [1] V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor, and A. S. Perelson, “Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis,” Bulletin of mathematical biology, vol. 56, no. 2, pp. 295–321, 1994.
• [2] D. Kirschner and J. C. Panetta, “Modeling immunotherapy of the tumor– immune interaction,” Journal of mathematical biology, vol. 37, no. 3, pp. 235–252, 1998.
• [3] V. A. Kuznetsov and G. D. Knott, “Modeling tumor regrowth and immunotherapy,” Mathematical and Computer Modelling, vol. 33, no. 12-13, pp. 1275–1287, 2001.
• [4] L. G. De Pillis and A. Radunskaya, “The dynamics of an optimally controlled tumor model: A case study,” Mathematical and computer modelling, vol. 37, no. 11, pp. 1221–1244, 2003.
• [5] A.d’Onofrio,“Ageneralframeworkformodelingtumor-immunesystem competition and immunotherapy: Mathematical analysis and biomedical inferences,” Physica D: Nonlinear Phenomena, vol. 208, no. 3-4, pp. 220–235, 2005.
• [6] L. G. de Pillis, W. Gu, and A. E. Radunskaya, “Mixed immunotherapy and chemotherapy of tumors: modeling, applications and biological interpretations,” Journal of theoretical biology, vol. 238, no. 4, pp. 841– 862, 2006.
• [7] M. Itik and S. P. Banks, “Chaos in a three-dimensional cancer model,” International Journal of Bifurcation and Chaos, vol. 20, no. 01, pp. 71–79, 2010.
• [8] H.-C.WeiandJ.-T.Lin,“Periodicallypulsedimmunotherapyinamath- ematical model of tumor-immune interaction,” International Journal of Bifurcation and Chaos, vol. 23, no. 04, p. 1350068, 2013.
• [9] J. Yang, S. Tang, and R. A. Cheke, “Modelling pulsed immunotherapy of tumour–immune interaction,” Mathematics and Computers in Simu- lation, vol. 109, pp. 92–112, 2015.
• [10] Y.Xu,J.Feng,J.Li,andH.Zhang,“Stochasticbifurcationforatumor– immune system with symmetric le ́vy noise,” Physica A: Statistical Mechanics and its Applications, vol. 392, no. 20, pp. 4739–4748, 2013.
• [11] M. Baar, L. Coquille, H. Mayer, M. Ho ̈lzel, M. Rogava, T. Tu ̈ting, and A. Bovier, “A stochastic model for immunotherapy of cancer,” Scientific reports, vol. 6, no. 1, pp. 1–10, 2016.
• [12] X. Liu, Q. Li, and J. Pan, “A deterministic and stochastic model for the system dynamics of tumor–immune responses to chemotherapy,” Physica A: Statistical Mechanics and its Applications, vol. 500, pp. 162– 176, 2018.
• [13] X. Li, G. Song, Y. Xia, and C. Yuan, “Dynamical behaviors of the tumor-immune system in a stochastic environment,” SIAM Journal on Applied Mathematics, vol. 79, no. 6, pp. 2193–2217, 2019.
• [14] J. Yang, Y. Tan, and R. A. Cheke, “Modelling effects of a chemother- apeutic dose response on a stochastic tumour-immune model,” Chaos, Solitons & Fractals, vol. 123, pp. 1–13, 2019.
• [15] P. Schulthess, V. Rottscha ̈fer, J. W. Yates, and P. H. van Der Graaf, “Optimization of cancer treatment in the frequency domain,” The AAPS journal, vol. 21, no. 6, p. 106, 2019.
• [16] S. Heyden and M. Ortiz, “Oncotripsy: Targeting cancer cells selectively via resonant harmonic excitation,” Journal of the Mechanics and Physics of Solids, vol. 92, pp. 164–175, 2016.
• [17] S. K. Jaganathan, A. P. Subramanian, M. V. Vellayappan, A. Balaji, A. A. John, A. K. Jaganathan, and E. Supriyanto, “Natural frequency of cancer cells as a starting point in cancer treatment,” Current Science, pp. 1828–1832, 2016.
• [18] M. Fraldi, A. Cugno, L. Deseri, K. Dayal, and N. Pugno, “A frequency- based hypothesis for mechanically targeting and selectively attacking cancer cells,” Journal of the Royal Society Interface, vol. 12, no. 111, p. 20150656, 2015.
• [19] S.HeydenandM.Ortiz,“Investigationoftheinfluenceofviscoelasticity on oncotripsy,” Computer Methods in Applied Mechanics and Engineer- ing, vol. 314, pp. 314–322, 2017.
• [20] E. Calabro` and S. Magazu`, “New perspectives in the treatment of tumor cells by electromagnetic radiation at resonance frequencies in cellular membrane channels,” The Open Biotechnology Journal, vol. 13, no. 1, 2019.
• [21] U. Lucia, G. Grisolia, A. Ponzetto, L. Bergandi, and F. Silvagno, “Thermomagnetic resonance affects cancer growth and motility,” Royal Society open science, vol. 7, no. 7, p. 200299, 2020.
• [22] D. R. Mittelstein, J. Ye, E. F. Schibber, A. Roychoudhury, L. T. Martinez, M. H. Fekrazad, M. Ortiz, P. P. Lee, M. G. Shapiro, and M. Gharib, “Selective ablation of cancer cells with low intensity pulsed ultrasound,” Applied Physics Letters, vol. 116, no. 1, p. 013701, 2020.
• [23] D. R. Mittelstein, “Modifying ultrasound waveform parameters to con- trol, influence, or disrupt cells,” Ph.D. dissertation, California Institute of Technology, 2020.
• [24] E. Schibber, D. Mittelstein, M. Gharib, M. Shapiro, P. Lee, and M. Ortiz, “A dynamical model of oncotripsy by mechanical cell fatigue: selective cancer cell ablation by low-intensity pulsed ultrasound,” Proceedings of the Royal Society A, vol. 476, no. 2236, p. 20190692, 2020.
• [25] E. Figueroa-Schibber, “High-cycle dynamic cell fatigue with applica- tions on oncotripsy,” Ph.D. dissertation, California Institute of Technol- ogy, 2020.
• [26] S. Abernethy and R. J. Gooding, “The importance of chaotic attractors in modelling tumour growth,” Physica A: Statistical Mechanics and its Applications, vol. 507, pp. 268–277, 2018.
• [27] M. Fahimi, K. Nouri, and L. Torkzadeh, “Chaos in a stochastic cancer model,” Physica A: Statistical Mechanics and its Applications, vol. 545, p. 123810, 2020.