A flowgraph model for bladder carcinoma

Background: Superficial bladder cancer has been the subject of numerous studies for many years, but the evolution of the disease still remains not well understood. After the tumor has been surgically removed, it may reappear at a similar level of malignancy or progress to a higher level. The process may be reasonably modeled by means of a Markov process. However, in order to more completely model the evolution of the disease, this approach is insufficient. The semi-Markov framework allows a more realistic approach, but calculations become frequently intractable. In this context, flowgraph models provide an efficient approach to successfully manage the evolution of superficial bladder carcinoma. Our aim is to test this methodology in this particular case. Results: We have built a successful model for a simple but representative case. Conclusion: The flowgraph approach is suitable for modeling of superficial bladder cancer.Rubio Navarro, G.; García Mora, MB.; Santamaria Navarro, C.; Pontones Moreno, JL. (2014). A flowgraph model for bladder carcinoma. Theoretical Biology and Medical Modelling. 11(1):1-11. doi:10.1186/1742-4682-11-S1-S3S111111van Rhijn BW, Burger M, Lotan Y, Solsona E, Stief CG, Sylvester RJ, Witjes JA, Zlotta AR: Recurrence and progression of disease in non-muscle-invasive bladder cancer: from epidemiology to treatment strategy. Eur Urol. 2009, 56: 430-42. 10.1016/j.eururo.2009.06.028.Sylvester RJ, van der Meijden AP, Oosterlinck W, Witjes JA, Bouffioux C, Denis L, Newling DW, Kurth K: Predicting recurrence and progression in individual patients with stage Ta T1 bladder cancer using EORTC risk tables: a combined analysis of 2596 patients from seven EORTC trials. Eur Urol. 2006, 49: 475-7.Fernández-Gómez J, Madero R, Solsona E, Unda M, neiro LMP, González M, Portillo J, Ojea A, Pertusa C, Rodríguez-Molina J, Camacho J, Rabadan M, Astobieta A, Montesinos M, Isorna S, nola PM, Gimeno A, Blas M, neiro JAMP: The EORTC Tables Overestimate the Risk of Recurrence and Progression in Patients with Non-Muscle-Invasive Bladder Cancer Treated with Bacillus Calmette-Guerin: External Validation of the EORTC Risk Tables. Eur Urol. 2011, 60: 423-30. 10.1016/j.eururo.2011.05.033.Butler RW, Huzurbazar AV: Stochastic network models for survival analysis. J Am Statist Assoc. 1997, 92: 246-57. 10.1080/01621459.1997.10473622.Klein JP, Moeschberger ML: Suvival Analysis Techniques for Censored and Truncated Data. 2003, Springer, segundaNeuts MF: Matrix Geometric Solutions in Stocastic Models An Algoritmic Approach. 1981, Baltimore: The Johns Hopkins University PressLatouche G, Ramaswami V: Introduction to Matrix Analytic Methods in Stochastic Modeling. 1999, Philadelphia: SIAMPérez-Ocón R, Segovia MC: Modeling lifetimes using phase-type distributions. Risk, Reliability and Societal Safety, Proceedings of the European Safety and Reliability Conference 2007 (ESREL 2007). Edited by: Taylor & Francis re. 2007Huzurbazar A, Williams B: Incorporating Covariates in Flowgraph Models: Applications to Recurrent Event Data. Thecnometrics. 2010, 52: 198-208. 10.1198/TECH.2010.08044.Collins DH, Huzurbazar AV: System reliability and safety assessment using non-parametric flowgraph models. Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability December 1, 2008 vol 222 no 4. 2008, 667-664.Huzurbazar A: Multistate Models, Flowgraph Models, and Semi-Markov Processes. Communications in Statistics - Theory and Methods. 2004, 33: 457-474. 10.1081/STA-120028678.Huzurbazar A: Flowgraph Models for Multistate Time-To-Event Data. 2005, New York: WileyMullen KM, van Stokkum IHM: nnls: The Lawson-Hanson algorithm for non-negative least squares (NNLS). 2012, [R package version 1.4], http://CRAN.R-project.org/package=nnlsAbate J, Whitt W: The Fourier-Series Method For Inverting Transforms Of Probability Distributions. Queueing Syst. 1992, 5-88.Collins DH, Huzurbazar AV: Prognostic models based on statistical flowgraphs. Appl Stochastic Models Bus Ind. 2012, 28: 141-51. 10.1002/asmb.884.OMS: International Classification of Tumours. 1999, 2™, World Health Organization, Histological typing of urinary bladder tumours, Volumen 10, GenevaLujan S: Modelización matemática de la multirrecidiva y heterogeneidad individual para el cálculo del riesgo biológico de recidiva y progresión del tumor vesical no músculo invasivo. PhD thesis. 2012, Universitat de ValènciaTeam RDC: R: A Language and Environment for Statistical Computing. 2010, R Foundation for Statistical Computing, Vienna, Austria,Goulet V, Dutang C, Maechler M, Firth D, Shapira M, Stadelmann M, expm-developers@listsR-forgeR-projectorg: expm: Matrix exponential. 2011, [R package version 0.98-5], http://CRAN.R-project.org/package=expmBates D, Maechler M: Matrix: Sparse and Dense Matrix Classes and Methods. 2011, R package version 1.0-1.Therneau T: survival: Survival analysis, including penalised likelihood. 2011, original Splus: R port by Thomas Lumley, [R package version 2.36-10], http://CRAN.R-project.org/package=survivalJackson CH: Multi-State Models for Panel Data: The msm Package for R. Journal of Statistical Software. 2011, 38 (8): 1-29. http://www.jstatsoft.org/v38/i08

Markovian modeling for dependent interrecurrence times in bladder cancer

[EN] A methodology to model a process in which repeated events occur is presented. The context is the evolution of non-muscle-invasive bladder carcinoma (NMIBC), characterized by recurrent relapses. It is based on the statistical flowgraph approach, a technique specifically suited for semi-Markov processes. A very useful feature of the flowgraph framework is that it naturally incorporates the management of censored data. However, this approach presents two difficulties with the process to be modeled. On one hand, the management of covariates is not straightforward. However, it is of great interest to know how the characteristics of a certain patient influence the evolution of the disease. On the other hand, repeated events on the same subject are generally not independent, in which case the semi-Markov framework is not sufficient because the semi-Markov assumption implies independence among waiting time distributions. We solve this issue by extending the flowgraph methodology using the Markovian arrival process (MAP), which does successfully model the dependence between events. Along the way, we provide a procedure to consider covariates and censored times in MAPs, a pending task needed in this field. In short, we have managed to extend the flowgraph methodology beyond the semi-Markovian framework, simplifying the incorporation of covariates and keeping the management of censored times. All of which has allowed us to build a multistate model of the evolution of NMIBC. The developed model allows us to compute the Survival function for any evolution of a patient with specific clinic-pathological characteristics in this primary tumor.García Mora, MB.; Santamaria Navarro, C.; Rubio Navarro, G. (2020). Markovian modeling for dependent interrecurrence times in bladder cancer. Mathematical Methods in the Applied Sciences. 43(14):8302-8310. https://doi.org/10.1002/mma.6593S830283104314Meszáros, A., Papp, J., & Telek, M. (2014). Fitting traffic traces with discrete canonical phase type distributions and Markov arrival processes. International Journal of Applied Mathematics and Computer Science, 24(3), 453-470. doi:10.2478/amcs-2014-0034García-Mora, B., Santamaría, C., & Rubio, G. (2018). Modeling dependence in the inter-failure times. An analysis in Reliability models by Markovian Arrival Processes. Journal of Computational and Applied Mathematics, 343, 762-770. doi:10.1016/j.cam.2017.12.022Montoro-Cazorla, D., & Pérez-Ocón, R. (2011). A shock and wear system with memory of the phase of failure. Mathematical and Computer Modelling, 54(9-10), 2155-2164. doi:10.1016/j.mcm.2011.05.024Neuts, M. F., & Bhattacharjee, M. C. (1981). Shock models with phase type survival and shock resistance. Naval Research Logistics Quarterly, 28(2), 213-219. doi:10.1002/nav.3800280204Kiemeney, L. A. L. M., Witjes, J. A., Heijbroek, R. P., Verbeek, A. L. M., & Debruyne, F. M. J. (1993). Predictability of Recurrent and Progressive Disease in Individual Patients with Primary Superficial Bladder Cancer. Journal of Urology, 150(1), 60-64. doi:10.1016/s0022-5347(17)35397-1Butler, R. W., & Huzurbazar, A. V. (1997). Stochastic Network Models for Survival Analysis. Journal of the American Statistical Association, 92(437), 246-257. doi:10.1080/01621459.1997.10473622Huzurbazar, A. V. (2000). Modeling and Analysis of Engineering Systems Data Using Flowgraph Models. Technometrics, 42(3), 300-306. doi:10.1080/00401706.2000.10486050Collins, D. H., & Huzurbazar, A. V. (2008). System reliability and safety assessment using non-parametric flowgraph models. Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability, 222(4), 667-674. doi:10.1243/1748006xjrr165Collins, D. H., & Huzurbazar, A. V. (2011). Prognostic models based on statistical flowgraphs. Applied Stochastic Models in Business and Industry, 28(2), 141-151. doi:10.1002/asmb.884Yau, C. L., & Huzurbazar, A. V. (2002). Analysis of censored and incomplete survival data using flowgraph models. Statistics in Medicine, 21(23), 3727-3743. doi:10.1002/sim.1237Rubio, G., García-Mora, B., Santamaría, C., & Pontones, J. L. (2014). A flowgraph model for bladder carcinoma. Theoretical Biology and Medical Modelling, 11(S1). doi:10.1186/1742-4682-11-s1-s3RubioG García–MoraB SantamaríaC PontonesJL.Incorporating multiple recurrences in a flowgraph model for bladder carcinoma. In: International Work–Conference on Bioinformatics and Biomedical Engineering IWBBIO.Granada Spain:2015;61.Mason, S. (1956). Feedback Theory-Further Properties of Signal Flow Graphs. Proceedings of the IRE, 44(7), 920-926. doi:10.1109/jrproc.1956.275147Abate, J., & Whitt, W. (1992). The Fourier-series method for inverting transforms of probability distributions. Queueing Systems, 10(1-2), 5-87. doi:10.1007/bf01158520McClureT.Numerical Inverse Laplace Transform(https://www.mathworks.com/matlabcentral/fileexchange/39035-numerical-inverse-laplace-transform) MATLAB Central File Exchange. Retrieved January 7 2019;2020.Santamaría, C., García-Mora, B., Rubio, G., & Luján, S. (2011). An analysis of the recurrence–progression process in bladder carcinoma by means of joint frailty models. Mathematical and Computer Modelling, 54(7-8), 1671-1675. doi:10.1016/j.mcm.2010.11.004Huzurbazar, A. V., & Williams, B. J. (2010). Incorporating Covariates in Flowgraph Models: Applications to Recurrent Event Data. Technometrics, 52(2), 198-208. doi:10.1198/tech.2010.08044García-Mora, B., Santamaría, C., Rubio, G., & Pontones, J. L. (2016). Bayesian prediction for flowgraph models with covariates. An application to bladder carcinoma. Journal of Computational and Applied Mathematics, 291, 85-93. doi:10.1016/j.cam.2015.03.04