Aplicación de los Métodos Analíticos Matriciales en el estudio de la supervivencia al cáncer de vejiga

Abstract

This thesis presents two state-space models for analyzing the evolution of the bladder cancer in two groups of patients. This cancer is one of the most extended in the population in many countries, it is highly aggressive, the treatment and following up of the patients is long and, consequently, the associated costs are very high in comparison with other types of cancer. The two state-space models applied to study the disease are Markovian processes: a multidimensional Markov process, based on the matrix-analytical methods and a semi-Markov process. Both procedures have different methodologies and allow to obtain information about the disease in two different and complementary ways. The matrix-analytic methods are applied to a dataset. Based on the data and the characteristics of the disease, the states are defined and the staying empirical times in states are calculated. Phase-type distributions are fitted to these times with an adequate goodness of fit. Under these conditions, the Markovian structure is introduced to analyze the model, though the staying times did not follow exponential distributions. This shows the versatility and power of these methods. A multidimensional Markov process is constructed based on the phase-type distributions. The performance measures are calculated by using matrix calculations and presented in a algorithmic form that can be performed using computational algorithm. In the analysis of the data, different groups of patients are distinguished in terms of the disease. The analysis and comparison of these groups is carried out introducing covariates in the model. This is a structure that has proved to be very useful in survival, and it has been applied when a Markov process governs the evolution of the system. We introduce them in the multidimensional case, extending the Cox model. In the multidimensional case, the covariates are affected by matrix coefficients. These parameters have been estimated using an adaptation of the Nelder-Mead algorithm to optimize the partial likelihood, which has allowed the entry of censored data. The procedure used allows to carry out later extensions of the study, with the estimation of the parameters from the optimization of more complex matrix functions. The analysis of the disease is completed and the groups are compared. The model is applicable even in the case of few data. The results contribute to a better knowledge of the disease. The semi-Markov model uses a different methodology. Based on the data, the states are defined and the transition probabilities among states are estimated. In this case, the future evolution of the system depends on the occupied state at a certain instant and on the time since the last transition. The sojourn times in states can have arbitrary distributions. The model presented in an adequate goodness of fit to the real process, as a consequence of the strong consistency properties of the used estimators. The estimations have involved procedures that have been generalized and can be easily adapted to be used in other applications. The proposed models and the estimation of the main survival measures, stand out relevant achievements and provide useful information to health care experts who will be able to plan preventive actions, specially when the survival time is large as we observe in this disease. They are constructed starting from a dataset, that is analyzed in detail, and then, two ways are applied to contribute to a better knowledge of the disease. In each case, the algorithms are easily adaptable in other domains of applications, such as reliability, ensuring their applicability even despite the limitations of the available information.