Time series forecasting using ARIMA for modeling of glioma growth in response to radiotherapy
DOI:
https://doi.org/10.5433/1679-0375.2021v42n1p3Keywords:
Mathematical models, Tumor growth, Glioblastoma, Time series forecast, RadiotherapyAbstract
In present days, the growing number of people suffering from cancer has been a major cause for concern worldwide. Glioblastoma in particular, are primary tumors in glial cells located in the central nervous system. Because of this sensitive location, mathematical models have been studied and developed as alternative tools for analyzing tumor growth rates, assisting on the decision-making process for treatment dosage, without exposing the patient’s life. This paper presents two time series models to estimate the growth rate of glioblastoma in response to ionizing radiotherapy treatment. The results obtained indicate that the proposed time series methods attain predictions with a Mean Absolute Percentual Error (MAPE) of approximately 1% to 4%, and simulations show that the Autoregressive Integrated Moving Average (ARIMA) method surpasses the Holt method based on the Mean Square Error (MSE) and MAPE values obtained. Furthermore, the results show that the time series method is applicable to data from two different mathematical models for glioblastoma growth.Downloads
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References
BARBOSA, O. X.; ASSIS, W. L. S.; GARCIA, V. S.; ALVAREZ, G. B. Computational simulation of gliomas using stochastic methods. Pesquisa e Ensino em Ciências Exatas e da Natureza, Cajazeiras, v. 3, p. 199–215, 2019.
BARNETT, G. H. High-grade gliomas: diagnosis and treatment. New Jersey: Humana Press, 2007.
BELLOMO, N.; CHAPLAIN, M.; ANGELIS, E. Selected topics in cancer modeling: genesis, evolution, immune competition, and therapy. Boston: Birkhäuser, 2008.
BOX, G. E.; JENKINS, G. M.; REINSEL, G. C.; LJUNG, G. M. Time series analysis: forecasting and control. 5th ed. New Jersey: John Wiley & Sons, 2015.
CANCER INSTITUTE NSW. What is cancer. Alexandria: NSW, 2018. Available from: https://www.cancerinstitute.org.au/understandingcancer/ what-is-cancer. Access in: jun. 2018.
DOLGIN, E. The mathematician versus the malignancy. Nature Medicine, New York, v. 20, p. 460–463, 2014.
DOMINGOS, S. D. O.; OLIVEIRA, J. F. D.; MATTOS NETO, P. S. G. An intelligent hybridization of arima with machine learning models for time series forecasting. Knowledge- Based Systems, Netherlands, v. 175, p. 72–86, 2019.
FERNÁNDEZ-CARA, E.;PROUVÉE, L. Optimal control of mathematical models for the radiotherapy of gliomas: the scalar case. Computational and Applied Mathematics, Petrópolis, v. 37, p. 745–762, 2018.
HOLLAND, E. C. Glioblastoma multiforme: the terminator. Proceedings of the National Academy of Sciences, Washington, v. 97, n. 12, p. 6242–6244, 2000.
HYNDMAN, R.; KOEHLER, A. B.; ORD, J. K.; SNYDER, R. D. Forecasting with exponential smoothing: the state space approach. Berlin: Springer, 2008.
HYNDMAN, R. J.; AKRAM, M.; ARCHIBALD, B. C. The admissible parameter space for exponential smoothing models. Annals of the Institute of Statistical Mathematics, Tokyo, v. 60, n. 2, p. 407–426, 2008.
HYNDMAN, R. J.; KOEHLER, A. B.; SNYDER, R. D.; GROSE, S. A state space framework for automatic forecasting using exponential smoothing methods. International Journal of Forecasting, Amsterdam, v. 18, n. 3, p. 439–454, 2002.
IARC - INTERNATIONAL AGENCY FOR RESEARCH ON CANCER. Latest global cancer data: Cancer burden rises to 18.1 million new cases and 9.6 million cancer deaths in 2018. França: IARC, 2018. Available from: https://www.iarc.fr/. Access in: dec. 2018.
INCA - INSTITUTO NACIONAL DE CÂNCER. Estimativas 2018: Incidência de câncer no Brasil. Rio de Janeiro: INCA, 2018. Available from: . Access in: jun. 2018.
JESUS, J. C. D.; CHRISTO, E. S.; GARCIA, V. S.; ALVAREZ, G. B. Time series analysis for modeling of glioma growth in response to radiotherapy. IEEE Latin America Transactions, Canadá, v. 14, n. 3, p. 1532–1537, 2014.
LEDER, K. et al. Mathematical modeling of PDGF-driven glioblastoma reveals optimized radiation dosing schedules. Cell, Cambridge, v. 156, n. 3, p. 603–616, 2014.
MONTGOMERY, D. C.; JENNINGS, C. L.; KULAHCI, M. C. Introduction to time series analysis and forecasting. 2th ed. New Jersey: John Wiley & Sons, 2015.
MORETTIN, P. A.; TOLOI, C. M. C. Modelos para previsão de séries temporais. Rio de Janeiro: IMPA, 1981. v. 1.
MURRAY, J. D. Vignettes from the field of mathematical biology: the application of mathematics to biology and medicine. Interface Focus, London, v. 2, n. 4, p. 397–406, 2012.
ROCKNE, R.; ALVORD, E. C.; ROCKHILL, J. K.; SWANSON, K. R. A mathematical model for brain tumor response to radiation therapy. Journal of Mathematical Biology, New York, v. 58, n. 4/5, p. 561–578, 2008.
SILVA, J. J. da; ALVAREZ, G. B.; GARCIA, V. S.; LOBÃO, D. C. Modelagem computacional do crescimento de glioma via diferenças finitas em resposta à radioterapia. In: CONGRESSO NACIONAL DE MATEMÁTICA APLICADA E COMPUTACIONAL, 36, 2016, Gramado. Proccedings [...]. Gramado: Sbmac Publishing, 2016. p. 1-7.
SOUZA, E. B.; NEVES, T. A.; ALVAREZ, G. B. Estudo sobre otimização da radioterapia em pacientes com gliomas. In: SIMPOSIO BRASILEIRO DE PESQUISA OPERACIONAL, 47., 2015, Porto de Galinhas. Proccedings [...]. Porto de Galinhas: Sbpo Publishing, 2015. p. 1-12.
SWANSON, K. R.; ALVORD JUNIOR, E. C.; MURRAY, J. D. A quantitative model for differential motility of gliomas in gray and white matter. Cell proliferation, Cambridge, v. 33, n. 5, p. 317– 329, 2000.
TEIXEIRA, J. G.; IKEHARA, H. C.; BRANDT, S. A. Estrutura de oferta e desenvolvimento agrícola: um modelo de equações recursivas. Semina: Ciências Exatas e Tecnológicas, Londrina, v. 2, n. 8, p. 183–188, 1981.
WALKER, M. D.; STRIKE, T. A.; SHELINE, G. E. An analysis of dose-effect relationship in the radiotherapy of malignant gliomas. International Journal of Radiation Oncology, Biology, Physics, New York, v. 5, n. 10, p. 1725–1731, 1979.
ZHANG, G. P. Time series forecasting using a hybrid arima and neural network model. Neurocomputing, Amsterdam, v. 50, p. 159–175, 2003.
BARNETT, G. H. High-grade gliomas: diagnosis and treatment. New Jersey: Humana Press, 2007.
BELLOMO, N.; CHAPLAIN, M.; ANGELIS, E. Selected topics in cancer modeling: genesis, evolution, immune competition, and therapy. Boston: Birkhäuser, 2008.
BOX, G. E.; JENKINS, G. M.; REINSEL, G. C.; LJUNG, G. M. Time series analysis: forecasting and control. 5th ed. New Jersey: John Wiley & Sons, 2015.
CANCER INSTITUTE NSW. What is cancer. Alexandria: NSW, 2018. Available from: https://www.cancerinstitute.org.au/understandingcancer/ what-is-cancer. Access in: jun. 2018.
DOLGIN, E. The mathematician versus the malignancy. Nature Medicine, New York, v. 20, p. 460–463, 2014.
DOMINGOS, S. D. O.; OLIVEIRA, J. F. D.; MATTOS NETO, P. S. G. An intelligent hybridization of arima with machine learning models for time series forecasting. Knowledge- Based Systems, Netherlands, v. 175, p. 72–86, 2019.
FERNÁNDEZ-CARA, E.;PROUVÉE, L. Optimal control of mathematical models for the radiotherapy of gliomas: the scalar case. Computational and Applied Mathematics, Petrópolis, v. 37, p. 745–762, 2018.
HOLLAND, E. C. Glioblastoma multiforme: the terminator. Proceedings of the National Academy of Sciences, Washington, v. 97, n. 12, p. 6242–6244, 2000.
HYNDMAN, R.; KOEHLER, A. B.; ORD, J. K.; SNYDER, R. D. Forecasting with exponential smoothing: the state space approach. Berlin: Springer, 2008.
HYNDMAN, R. J.; AKRAM, M.; ARCHIBALD, B. C. The admissible parameter space for exponential smoothing models. Annals of the Institute of Statistical Mathematics, Tokyo, v. 60, n. 2, p. 407–426, 2008.
HYNDMAN, R. J.; KOEHLER, A. B.; SNYDER, R. D.; GROSE, S. A state space framework for automatic forecasting using exponential smoothing methods. International Journal of Forecasting, Amsterdam, v. 18, n. 3, p. 439–454, 2002.
IARC - INTERNATIONAL AGENCY FOR RESEARCH ON CANCER. Latest global cancer data: Cancer burden rises to 18.1 million new cases and 9.6 million cancer deaths in 2018. França: IARC, 2018. Available from: https://www.iarc.fr/. Access in: dec. 2018.
INCA - INSTITUTO NACIONAL DE CÂNCER. Estimativas 2018: Incidência de câncer no Brasil. Rio de Janeiro: INCA, 2018. Available from: . Access in: jun. 2018.
JESUS, J. C. D.; CHRISTO, E. S.; GARCIA, V. S.; ALVAREZ, G. B. Time series analysis for modeling of glioma growth in response to radiotherapy. IEEE Latin America Transactions, Canadá, v. 14, n. 3, p. 1532–1537, 2014.
LEDER, K. et al. Mathematical modeling of PDGF-driven glioblastoma reveals optimized radiation dosing schedules. Cell, Cambridge, v. 156, n. 3, p. 603–616, 2014.
MONTGOMERY, D. C.; JENNINGS, C. L.; KULAHCI, M. C. Introduction to time series analysis and forecasting. 2th ed. New Jersey: John Wiley & Sons, 2015.
MORETTIN, P. A.; TOLOI, C. M. C. Modelos para previsão de séries temporais. Rio de Janeiro: IMPA, 1981. v. 1.
MURRAY, J. D. Vignettes from the field of mathematical biology: the application of mathematics to biology and medicine. Interface Focus, London, v. 2, n. 4, p. 397–406, 2012.
ROCKNE, R.; ALVORD, E. C.; ROCKHILL, J. K.; SWANSON, K. R. A mathematical model for brain tumor response to radiation therapy. Journal of Mathematical Biology, New York, v. 58, n. 4/5, p. 561–578, 2008.
SILVA, J. J. da; ALVAREZ, G. B.; GARCIA, V. S.; LOBÃO, D. C. Modelagem computacional do crescimento de glioma via diferenças finitas em resposta à radioterapia. In: CONGRESSO NACIONAL DE MATEMÁTICA APLICADA E COMPUTACIONAL, 36, 2016, Gramado. Proccedings [...]. Gramado: Sbmac Publishing, 2016. p. 1-7.
SOUZA, E. B.; NEVES, T. A.; ALVAREZ, G. B. Estudo sobre otimização da radioterapia em pacientes com gliomas. In: SIMPOSIO BRASILEIRO DE PESQUISA OPERACIONAL, 47., 2015, Porto de Galinhas. Proccedings [...]. Porto de Galinhas: Sbpo Publishing, 2015. p. 1-12.
SWANSON, K. R.; ALVORD JUNIOR, E. C.; MURRAY, J. D. A quantitative model for differential motility of gliomas in gray and white matter. Cell proliferation, Cambridge, v. 33, n. 5, p. 317– 329, 2000.
TEIXEIRA, J. G.; IKEHARA, H. C.; BRANDT, S. A. Estrutura de oferta e desenvolvimento agrícola: um modelo de equações recursivas. Semina: Ciências Exatas e Tecnológicas, Londrina, v. 2, n. 8, p. 183–188, 1981.
WALKER, M. D.; STRIKE, T. A.; SHELINE, G. E. An analysis of dose-effect relationship in the radiotherapy of malignant gliomas. International Journal of Radiation Oncology, Biology, Physics, New York, v. 5, n. 10, p. 1725–1731, 1979.
ZHANG, G. P. Time series forecasting using a hybrid arima and neural network model. Neurocomputing, Amsterdam, v. 50, p. 159–175, 2003.
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Published
2021-04-07
How to Cite
Silva, L. M. da, Alvarez, G. B., Christo, E. da S., Pelén Sierra, G. A., & Garcia, V. da S. (2021). Time series forecasting using ARIMA for modeling of glioma growth in response to radiotherapy. Semina: Ciências Exatas E Tecnológicas, 42(1), 3–12. https://doi.org/10.5433/1679-0375.2021v42n1p3
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