One of the main challenges in cancer modelling is to improve the knowledge of tumor progression in areas related to tumor growth, tumor-induced angiogenesis and targeted therapies efficacy. For this purpose, incorporate the expertise from applied mathematicians, biologists and physicians is highly desirable. Despite the existence of a very wide range of models, involving many stages in cancer progression, few models have been proposed to take into account all relevant processes in tumor progression, in particular the effect of systemic treatments and angiogenesis. Composite biological experiments, both in vitro and in vivo, in addition with mathematical modelling can provide a better understanding of theses aspects. In this work we proposed that a rational experimental design associated with mathematical modelling could provide new insights into cancer progression. To accomplish this task, we reviewed mathematical models and cancer biology literature, describing in detail the basic principles of mathematical modelling. We also analyze how experimental data regarding tumor cells proliferation and angiogenesis in vitro may fit with mathematical modelling in order to reconstruct in vivo tumor evolution. Additionally, we explained the mathematical methodology in a comprehensible way in order to facilitate its future use by the scientific community.

The future challenges in oncology imaging are to assess the response to treatment even earlier. As an addition to functional imaging, mathematical modeling based on the imaging is an alternative, cross-disciplinary area of development. Modeling was developed in oncology not only in order to understand and predict tumor growth, but also to anticipate the effects of targeted and untargeted therapies. A very wide range of these models exist, involving many stages in the progression of tumors. Few models, however, have been proposed to reproduce in vivo tumor growth because of the complexity of the mechanisms involved. Morphological imaging combined with "spatial" models appears to perform well although functioning imaging could still provide further information on metabolism and the micro-architecture. The combination of imaging and modeling can resolve complex problems and describe many facets of tumor growth or response to treatment. It is now possible to consider its clinical use in the medium term. This review describes the basic principles of mathematical modeling and describes the advantages, limitations and future prospects for this in vivo approach based on imaging data.

The possibility that fundamental discreteness implicit in a quantum gravity theory may act as a natural regulator for ultraviolet singularities arising in quantum field theory has been intensively studied. Here, along the same expectations, we investigate whether a nonstandard representation, called polymer representation can smooth away the large amount of negative energy that afflicts the Hamiltonians of higher-order time derivative theories; rendering the theory unstable when interactions come into play. We focus on the fourthorder Pais-Uhlenbeck model which can be reexpressed as the sum of two decoupled harmonic oscillators one producing positive energy and the other negative energy. As expected, the Schrödinger quantization of such model leads to the stability problem or to negative norm states called ghosts. Within the framework of polymer quantization we show the existence of new regions where the Hamiltonian can be defined well bounded from below.