Gabriel F. Calvo

Hypoxic Cell Waves Around Necrotic Cores in Glioblastoma: A Biomathematical Model and Its Therapeutic Implications

Glioblastoma is a rapidly evolving high-grade astrocytoma that is distinguished pathologically from lower grade gliomas by the presence of necrosis and microvascular hyperplasia. Necrotic areas are typically surrounded by hypercellular regions known as “pseudopalisades” originated by local tumor vessel occlusions that induce collective cellular migration events. This leads to the formation of waves of tumor cells actively migrating away from central hypoxia. We present a mathematical model that incorporates the interplay among two tumor cell phenotypes, a necrotic core and the oxygen distribution. Our simulations reveal the formation of a traveling wave of tumor cells that reproduces the observed histologic patterns of pseudopalisades. Additional simulations of the model equations show that preventing the collapse of tumor microvessels leads to slower glioma invasion, a fact that might be exploited for therapeutic purposes.

Bright solitary waves in malignant gliomas.

We put forward a nonlinear wave model describing the fundamental dynamical features of an aggressive type of brain tumors. Our model accounts for the invasion of normal tissue by a proliferating and propagating rim of active glioma cancer cells in the tumor boundary and the subsequent formation of a necrotic core. By resorting to numerical simulations, phase space analysis, and exact solutions we prove that bright solitary tumor waves develop in such systems. Possible implications of our model as a tool to extract relevant patient specific tumor parameters in combination with standard preoperative clinical imaging are also discussed.

A Mathematical Model for the Glucose-Lactate Metabolism of in Vitro Cancer Cells

We propose a mathematical model of tumor cell nutrient uptake governed by the presence of two key biomolecular fuels: glucose and lactate. The model allows us to describe, in a remarkably simple way, different in vitro scenarios previously reported in experiments of tumor cell metabolism using distinct energy sources. The predictions of our model show good agreement with all the examined tumor cell lines (cervix, colon, and glioma) and provide a first step toward the development of more comprehensive frameworks accounting for in vivo cancer dynamics under complex spatial heterogeneities.

Effective particle methods for Fisher–Kolmogorov equations: Theory and applications to brain tumor dynamics

Abstract Extended systems governed by partial differential equations can, under suitable conditions, be approximated by means of sets of ordinary differential equations for global quantities capturing the essential features of the systems dynamics. Here we obtain a small number of effective equations describing the dynamics of single-front and localized solutions of Fisher–Kolmogorov type equations. These solutions are parametrized by means of a minimal set of time-dependent quantities for which ordinary differential equations ruling their dynamics are found. A comparison of the finite dimensional equations and the dynamics of the full partial differential equation is made showing a very good quantitative agreement with the dynamics of the partial differential equation. We also discuss some implications of our findings for the understanding of the growth progression of certain types of primary brain tumors and discuss possible extensions of our results to related equations arising in different modeling scenarios.

Combined therapies of antithrombotics and antioxidants delay in silico brain tumour progression

Glioblastoma multiforme (GBM), the most frequent type of primary brain tumour, is a rapidly evolving and spatially heterogeneous high-grade astrocytoma that presents areas of necrosis, hypercellularity and microvascular hyperplasia. The aberrant vasculature leads to hypoxic areas and results in an increase in oxidative stress, selecting for more invasive tumour cell phenotypes. In our study, we assay in silico different therapeutic approaches which combine antithrombotics (ATs), antioxidants and standard radiotherapy (RT). To do so, we have developed a biocomputational model of GBM that incorporates the spatio-temporal interplay among two glioma cell phenotypes corresponding to oxygenated and hypoxic cells, a necrotic core and the local vasculature whose response evolves with tumour progression. Our numerical simulations predict that suitable combinations of ATs and antioxidants may diminish, in a synergistic way, oxidative stress and the subsequent hypoxic response. This novel therapeutical strategy, with potentially low or no toxicity, might reduce tumour invasion and further sensitize GBM to conventional RT or other cytotoxic agents, hopefully increasing median patient overall survival time.

Modeling the connection between primary and metastatic tumors

We put forward a model for cancer metastasis as a migration phenomenon between tumor cell populations coexisting and evolving in two different habitats. One of them is a primary tumor and the other one is a secondary or metastatic tumor. The evolution of the different cell phenotype populations in each habitat is described by means of a simple quasispecies model allowing for a cascade of mutations between the different phenotypes in each habitat. The cell migration event is supposed to be unidirectional and take place continuously in time. The possible clinical outcomes of the model depending on the parameter space are analyzed and the effect of the resection of the primary tumor is studied.

Exact bright and dark spatial soliton solutions in saturable nonlinear media

Abstract We present exact analytical bright and dark (black and grey) solitary wave solutions of a nonlinear Schrodinger-type equation describing the propagation of spatial beams in media exhibiting a saturable nonlinearity (such as centrosymmetric photorefractive materials). A qualitative study of the stationary equation is carried out together with a discussion of the stability of the solutions.

Linear Canonical Transforms on Quantum States of Light

Many quantum information and quantum computation protocols exploit high-dimensional Hilbert spaces. Photons, which constitute the main carrier of information between nodes of quantum networks, can store high-dimensional quantum bits in their spatial degrees of freedom. These degrees of freedom can be tailored by resorting to the symplectic invariant approach based on lossless linear canonical transformations. These transformations enable one to manipulate the transverse structure of a single photon prepared in superpositions of paraxial modes. We present a basic introduction of these transformations acting on photons and discuss some of their applications for elementary quantum information processing.

Hypoxia in Gliomas: Opening Therapeutical Opportunities Using a Mathematical-Based Approach

This chapter explores the use of mathematical models as promising and powerful tools to understand the complexity of tumors and their, frequently, hypoxic environment. We focus on gliomas, which are primary brain tumors derived from glial cells, mainly astrocytes and/or oligodendrocytes. A variety of mathematical models, based on ordinary and/or partial differential equations, have been developed both at the micro and macroscopic levels. The aim here is to describe in a quantitative way key physiopathological mechanisms relevant in these types of malignancies and to suggest optimal therapeutical strategies. More specifically, we consider novel therapies targeting thromboembolic phenomena to decrease cell invasion in high grade glioma or to delay the malignant transformation in low grade gliomas. This study has been the basis of a multidisciplinary collaboration involving, among others, neuro-oncologists, radiation oncologists, pathologists, cancer biologists, surgeons and mathematicians.

A transfer integral technique for solving a class of linear integral equations

An eigenvalue problem, the convergence difficulties that arise and a mathematical solution are considered. The eigenvalue problem is motivated by simplified models for the dissociation equilibrium between double-stranded and single-stranded DNA chains induced by temperature (thermal denaturation), and by the application of the so-called transfer integral technique. Namely, we extend the Peyrard-Bishop model for DNA melting from the original one-dimensional model to a three-dimensional one, which gives rise to an eigenvalue problem defined by a linear integral equation whose kernel is not in L 2 . For the one-dimensional model, the corresponding kernel is not in L 2 either, which is related to certain convergence difficulties noticed by previous researchers. Inspired by methods from quantum scattering theory, we transform the three-dimensional eigenvalue problem, obtaining a new L 2 kernel which has improved convergence properties.