1,721,097 research outputs found
Asymptotic analysis of selection-mutation models in the presence of multiple fitness peaks
No full textWe study the long-time behaviour of phenotype-structured models describing the evolutionary dynamics of asexual populations whose phenotypic fitness landscape is characterised by multiple peaks. First we consider the case where phenotypic changes do not occur, and then we include the effect of heritable phenotypic changes. In the former case the model is formulated as an integrodifferential equation for the phenotype distribution of the individuals in the population, whereas in the latter case the evolution of the phenotype distribution is governed by a non-local parabolic equation whereby a linear diffusion operator captures the presence of phenotypic changes. We prove that the long-time limit of the solution to the integrodifferential equation is unique and given by a measure consisting of a weighted sum of Dirac masses centred at the peaks of the phenotypic fitness landscape. We also derive an explicit formula to compute the weights in front of the Dirac masses. Moreover, we demonstrate that the longtime solution of the non-local parabolic equation exhibits a qualitatively similar behaviour in the asymptotic regime where the diffusion coefficient modelling the rate of phenotypic change tends to zero. However, we show that the limit measure of the non-local parabolic equation may consist of less Dirac masses, and we provide a sufficient criterion to identify the positions of their centres. Finally, we carry out a detailed characterisation of the speed of convergence of the integral of the solution (i.e. the population size) to its long-time limit for both models. Taken together, our results support a more in-depth theoretical understanding of the conditions leading to the emergence of stable phenotypic polymorphism in asexual populations
Trade-offs between chemotaxis and proliferation shape the phenotypic structuring of invading waves
No full textChemotaxis-driven invasions have been proposed across a broad spectrum of biological processes, from cancer to ecology. The influential system of equations introduced by Keller and Segel has proven a popular choice in the modelling of such phenomena, but in its original form restricts to a homogeneous population. To account for the possibility of phenotypic heterogeneity, we extend to the case of a population continuously structured across space, time and phenotype, where the latter determines variation in chemotactic responsiveness, proliferation rate, and the level of chemical environment modulation. The extended model considered here comprises a non-local partial differential equation for the local phenotype distribution of cells which is coupled, through an integral term, with a differential equation for the concentration of an attractant, which is sensed and degraded by the cells. In the framework of this model, we concentrate on a chemotaxis/proliferation trade-off scenario, where the cell phenotypes span a spectrum of states from highly-chemotactic but minimally-proliferative to minimally-chemotactic but highly-proliferative. Using a combination of numerical simulation and formal asymptotic analysis, we explore the properties of travelling-wave solutions. The results of our study demonstrate how incorporating phenotypic heterogeneity may lead to a highly-structured wave profile, where cells in different phenotypic states dominate different spatial positions across the invading wave, and clarify how the phenotypic structuring of the wave can be shaped by trade-offs between chemotaxis and proliferation
Pattern formation within phenotype-structured chemotactic populations
Full text linkPopulations can become spatially organized through chemotaxis autoattraction, wherein population members release their own chemoattractant. Standard models of this process usually assume phenotypic homogeneity, but recent studies have shed illumination on the inherent heterogeneity within populations: in terms of chemotactic behaviour, trait heterogeneity can range from the sensitivity to attractant gradients to the rate at which attractants are produced. We propose a framework that accounts for this heterogeneity, extending the standard Keller-Segel model to a non-local formulation in which the population is continuously structured across some phenotype state space. Focusing on autoattraction, we allow both the chemotactic sensitivity and the rate of attractant secretion to vary across the population and suppose members can switch between different phenotype states. We extend classical Turing-type linear stability analyses to determine the impact of phenotypic structuring on pattern formation, showing that the rate of switching influences both the critical condition for self-organization and subsequent pattern dynamics. Scenarios in which the chemotactic sensitivity and attractant secretion are positively or negatively correlated are used to highlight the significance of these results
Asymptotic dynamics in continuous structured populations with mutations, competition and mutualism
No full textEffective interface conditions for continuum mechanical models describing the invasion of multiple cell populations through thin membranes
Full text linkWe consider a continuum mechanical model for the migration of multiple cell populations through parts of tissue separated by thin membranes. In this model, cells belonging to different populations may be characterised by different proliferative abilities and mobility, which may vary from part to part of the tissue, as well as by different invasion potentials within the membranes. The original transmission problem, consisting of a set of mass balance equations for the volume fraction of cells of every population complemented with continuity of stresses and mass flux across the surfaces of the membranes, is then reduced to a limiting transmission problem whereby each thin membrane is replaced by an effective interface. In order to close the limiting problem, a set of biophysically-consistent transmission conditions is derived through a formal asymptotic method. Models based on such a limiting transmission problem may find fruitful application in a variety of research areas in the biological and medical sciences, including developmental biology, immunology and cancer growth and invasion
Metal artefact reduction in computed tomography images by a fourth-order total variation flow
No full textPermanent metallic implants, such as dental fillings and cardiac devices, generate streaks-like artefacts in computed tomography (CT) images. In this article, we propose a strategy to perform metal artefact reduction (MAR) that relies on the total variation-H− 1 inpainting, a variational approach based on a fourth-order total variation (TV) flow. This approach has never been used to perform MAR, although it has been profitably employed in other branches of image processing. A systematic evaluation of the performance is carried out. Comparisons are made with the results obtained using classical linear interpolation and two other partial differential equation-based approaches relying, respectively, on the Fourier's heat equation and on a second order TV flow. Visual inspection of both synthetic and real CT images, as well as computation of similarity indexes, suggests that our strategy for MAR outperforms the others considered here, as it provides best image restoration, highest similarity indexes and for being the only one able to recover hidden structures, a task of primary importance in the medical field
Cell population heterogeneity and evolution towards drug resistance in cancer: Biological and mathematical assessment, theoretical treatment optimisation
Full text linkBackground Drug-induced drug resistance in cancer has been attributed to diverse biological mechanisms at the individual cell or cell population scale, relying on stochastically or epigenetically varying expression of phenotypes at the single cell level, and on the adaptability of tumours at the cell population level. Scope of review We focus on intra-tumour heterogeneity, namely between-cell variability within cancer cell populations, to account for drug resistance. To shed light on such heterogeneity, we review evolutionary mechanisms that encompass the great evolution that has designed multicellular organisms, as well as smaller windows of evolution on the time scale of human disease. We also present mathematical models used to predict drug resistance in cancer and optimal control methods that can circumvent it in combined therapeutic strategies. Major conclusions Plasticity in cancer cells, i.e., partial reversal to a stem-like status in individual cells and resulting adaptability of cancer cell populations, may be viewed as backward evolution making cancer cell populations resistant to drug insult. This reversible plasticity is captured by mathematical models that incorporate between-cell heterogeneity through continuous phenotypic variables. Such models have the benefit of being compatible with optimal control methods for the design of optimised therapeutic protocols involving combinations of cytotoxic and cytostatic treatments with epigenetic drugs and immunotherapies. General significance Gathering knowledge from cancer and evolutionary biology with physiologically based mathematical models of cell population dynamics should provide oncologists with a rationale to design optimised therapeutic strategies to circumvent drug resistance, that still remains a major pitfall of cancer therapeutics. This article is part of a Special Issue entitled “System Genetics” Guest Editor: Dr. Yudong Cai and Dr. Tao Huang
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