Mathematical modeling of neoplasms odes and statistical analysis of medical data

Resumen

Although cancer is not the leading cause of death in developed countries, still it's the first as far as mortality rates are concerned. Population aging will result globally in significant increases in the rankings for most noncommunicable diseases, particularly cancers. Also increasing levels of tobacco smoking in many middle- and low-income countries will contribute to increase deaths from cardiovascular disease, chronic obstructive pulmonary disease and some cancers. Despite of the enormous efforts taken to find a method which could solve the problem definitively or at least turn cancer into chronic not lethal disease, we still have not an answer. The main reason is complexity of the problem and that cancer can not be generalized, in fact this term that encompasses more than 200 types of malignancies, each of them with some special features, its causes, its evolution and its specific treatment. Mathematical modeling can be a great help on this field because it provides a powerful tool for simulations "in silico", which nowadays are an element of any investigation. There exist thousands of models of tumor development, simpler and more complicated ones, none of them is the perfect one. That is why the main objective should be usefulness of a model as far as applications are concerned, not its complexity. The same rule might be applied to different elements of mathematics involved in medical problems, for example statistical methods. The thesis consists of three parts: analysis of three models of cancer-immune system competition, a stochastic model of lymphoma and statistical analysis of clinical data. In the first chapter three models of competition between cancer and immune system are presented and analyzed. Each of them is a simple system of ODEs, and the motivation are mathematical models of competing species in an ecosystem. For each of the models their strong and week points are discussed in the terms of agreement with biological reality and stability analysis and numerical simulations are performed. The final model is a model of competition between an artificially induced tumor and the adaptive immune system. The aim of this work was to reproduce experimental, found two possible outcomes depending on the initial quantities of tumor and adaptive immune cells. We came to the conclusion that the hypothesis of an equilibrium state before the treatment possibly refers to a small solution that tends to zero, but it has not disappeared yet. So, the final model is a known model of two species competition with finite carrying capacities and it has two groups of solutions depending on the initial conditions. In the first one, the immune system wins against the tumor cells, so the cancer disappears (elimination). In the second one, cancer keeps on growing. The second chapter is dedicated to the model of precursor T-lymphoblastic lymphoma, so one particular type of leukemia, based on the model of the competitive exclusion between different clonotypes in the maintenance of the native T cell repertoire developed by E. Stirk . We consider a situation of lack of any foreign antigen in the organism (healthy individual), so the only activation signal for T-cells is self antigen presented by APCs. Therefore, the main mass of the tumor are native T-cells about which we know that they live relatively shortly and divide poorly. Thus, we can assume that they belong to the clonotype which is very competitive, so very similar to many other clonotypes in terms of the specific survival stimuli which it is able to recognize. We also introduce a term for a constant influx of new native T cells from the thymus, which originate from mutated lymphoid stem cells. This way, despite of the fact that the tumoral clonotype of T cells is highly competitive, it does not become extinct and it keeps on proliferating, which is the main and most important difference between the results of E.Stirk and this one. We model the competition between two clonotypes of native T cells as a continuous time bivariate Markov process, for which later we formulate a deterministic approximation (systems of ODEs), using Van Kampens. We consider four cases, for each of them we perform stability analysis and numerical simulations. In the third chapter a statistical analysis made for clinical data in the form of mammograms is presented. It consists of two parts. In the first one, on the images, microcalcifications as individual objects and in the form of clusters were detected and described by means of 16 variables. The objective of this work was to determine the most significant parameters for cancer recognition in order to improve the accuracy of diagnosis. Factorial analysis provided factors in the form of linear combinations of the original variables, which are able to distinguish between the two groups of data, the malignant and benignant ones, for both, objects and clusters. In discriminant analysis, linear discriminant functions were determined. They classify the data into two groups, benignant and malignant, with the efficiency of 71.8% for objects and 100% for clusters. It shows that from statistical point of view is more reliable to consider the clusters of microcalcifications rather than particular objects. In the second part statistical analysis is performed for textural features of clinical data in the form of 322 digitalized mammographic images. The data are assigned into four types of mammary tissue, based on the classification introduced by J. N. Wolfe. For the images the coefficients of Haralick are calculated for 1, 3 and 5 pixels of distance and for angles 0, 45, 90 and 135 degrees. In factorial analysis we determine one factor which distinguishes significantly between the four groups. Euclidean norm calculated for 19-dimensional vectors defined by the Haralick coefficients is shown to be significant for type of parenchymal mammary tissue recognition.