Search

Motivated by the mathematical modeling of tumor invasion in healthy tissues, we propose a generalized compressible diphasic Navier-Stokes Cahn-Hilliard model that we name G-NSCH. We assume that the two phases of the fluid represent two different populations of cells: cancer cells and healthy tissue. We include in our model possible friction and proliferation effects. The model aims to be as general as possible to study the possible mechanical effects playing a role in the invasive growth of a tumor. In the present work, we focus on the analysis and numerical simulation of the G-NSCH model. Our G-NSCH system is derived rigorously and satisfies the basic mechanics of fluids and the thermodynamics of particles. Under simplifying assumptions, we prove the existence of global weak solutions. We also propose a structure-preserving numerical scheme based on the scalar auxiliary variable method to simulate our system and present some numerical simulations validating the properties of the numerical scheme and illustrating the solutions of the G-NSCH model., Comment: 29 pages + appendix + bibliography

Arsenic (As) and sulfate pollution are often found co-occurring as a result of smelting metal ores. Previous studies showed that sulfate reducing microbes (SRMs) can use As(V) as a terminal electron acceptor, while others reported that SRMs are the main mercury (Hg) methylators in freshwater systems. However, we have yet to fully explore how As(V) can affect methylmercury (MeHg) production. In this study, we examined whether additions of As(V) and sulfate in freshwater sediments collected near a major gold mine with a history of S and As emissions affect Hg methylation. First, we show that Hg methylation in lake sediments was primarily limited by carbon substrate availability rather than by that of sulfate as terminal electron acceptors. Then, under conditions where carbon is not limiting, sulfate addition to the system significantly increased Hg methylation rate constants. Finally, we show that MeHg production rates in sediments significantly decreased with increasing As(V) concentrations, regardless of the sulfate concentration amended to sediments. This work underscores the apparent antagonistic effects of As(V) on the one hand, and carbon and sulfate on the other hand on the kinetics of Hg methylation. Arsenic controls on Hg methylation are complex and a combination of direct impact on the methylators' fitness, the formation of As-bearing mineral phases affecting Hg bioavailability, or changes in the microbial community structures over increasing As concentrations should be the focus of additional investigations.

The degenerate Cahn–Hilliard equation is a standard model to describe living tissues. It takes into account cell populations undergoing short-range attraction and long-range repulsion effects. In this framework, we consider the usual Cahn–Hilliard equation with a singular single-well potential and degenerate mobility. These degeneracy and singularity induce numerous difficulties, in particular for its numerical simulation. To overcome these issues, we propose a relaxation system formed of two second-order equations which can be solved with standard packages. This system is endowed with an energy and an entropy structure compatible with the limiting equation. Here, we study the theoretical properties of this system: global existence and convergence of the relaxed system to the degenerate Cahn–Hilliard equation. We also study the long-time asymptotics which interest relies on the numerous possible steady states with given mass.

Supplementary text, tables, and figures.

The Keller-Segel model is a well-known system representing chemotaxis in living organisms. We study the convergence of a generalized nonlinear variant of the Keller-Segel to the degenerate Cahn-Hilliard system. This analysis is made possible from the observation that the Keller-Segel system is equivalent to a relaxed version of the Cahn-Hilliard system. Furthermore, this latter equivalent system has an interesting application in the modelling of living tissues. Indeed, compressible and incompressible porous medium type equations are widely used to describe the mechanical properties of living tissues. The relaxed degenerate Cahn-Hilliard system, can be viewed as a compressible living tissue model for which the movement is driven by Darcy's law and takes into account the effects of the viscosity as well as surface tension at the surface of the tissue. We study the convergence of the Keller-Segel system to the Cahn-Hilliard equation and some of the analytical properties of the model such as the incompressible limit of our model. Our analysis relies on a priori estimates, compactness properties, and on the equivalence between the Keller-Segel system and the relaxed degenerate Cahn-Hilliard system.

We study two classes of mathematical models currently used for the modeling in time and space of tumors: the Cahn-Hilliard equation for living tissues and the Keller-Segel model. The numerical methods we propose aim to represent these equations efficiently and accurately while preserving their properties. For the Cahn-Hilliard equation, our study is based on a relaxation method for which we prove the convergence to the original model. Even though the physical effects modeled by these equations are close to the ones studied in fluid dynamics, the equations used to model living tissues are different in order to represent the active behavior of cells. The resulting equations contain numerous singularities and degeneracies, which result in technical difficulties to analyze and simulate them efficiently.Our relaxation method has been introduced to facilitate the implementation of our numerical schemes. Hence, we propose numerical schemes that are easy to implement in already existing finite element software. In order to preserve the properties of the equations during numerical simulations, we design numerical schemes based on the Scalar Auxiliary Variable method. Based on these numerical works, we present two studies of biological phenomena.; Nous étudions deux types de modèles couramment utilisés pour la représentation en temps et en espace des tumeurs: l’équation de Cahn-Hilliard pour les tissus vivants et le modèle de Keller-Segel. Les méthodes numériques que nous développons cherchent à représenter de manière précise et efficace ces équations tout en préservant leurs propriétés. Pour l'équation de Cahn-Hilliard, notre étude s’appuie sur une méthode de relaxation dont nous prouvons la convergence vers le modèle initial. Même si elles représentent mathématiquement des phénomènes physiques proches de ceux étudiés en dynamique des fluides, les équations utilisées pour les tissus vivants sont souvent différentes pour rendre compte du caractère actif des cellules. Les équations résultantes contiennent de nombreuses singularités et dégénérescences qui sont difficiles à analyser théoriquement et simuler numériquement de manière efficace. La méthode de relaxation a été introduite pour faciliter l’implémentation de nos schémas numériques; nous proposons ainsi des schémas numériques éléments finis simples à adapter dans les codes pré-existants. Afin de préserver les propriétés des équations continues lors des simulations numériques, nous proposons des schémas numériques basés sur la Méthode de Variable Auxiliaire. Sur la base de ces travaux numériques, nous présentons l’étude de deux phénomènes biologiques.

Motivated by biological applications on tumour invasion through thin membranes, we study a porous-medium type equation where the density of the cell population evolves under Darcy's law, assuming continuity of both the density and flux velocity on the thin membrane which separates two domains. The drastically different scales and mobility rates between the membrane and the adjacent tissues lead to consider the limit as the thickness of the membrane approaches zero. We are interested in recovering the effective interface problem and the transmission conditions on the limiting zero-thickness surface, formally derived by Chaplain et al. (2019), which are compatible with nonlinear generalized Kedem-Katchalsky ones. Our analysis relies on a priori estimates and compactness arguments as well as on the construction of a suitable extension operator which allows to deal with the degeneracy of the mobility rate in the membrane, as its thickness tends to zero.

Motivated by biological applications on tumour invasion through thin membranes, we study a porous-medium type equation where the density of the cell population evolves under Darcy's law, assuming continuity of both the density and flux velocity on the thin membrane which separates two domains. The drastically different scales and mobility rates between the membrane and the adjacent tissues lead to consider the limit as the thickness of the membrane approaches zero. We are interested in recovering the effective interface problemand the transmission conditions on the limiting zero-thickness surface, formally derived by Chaplain et al., (2019), which are compatible with nonlinear generalized Kedem-Katchalsky ones. Our analysis relies on a priori estimates and compactness arguments as well as on the construction of a suitable extension operator which allows to deal with the degeneracy of the mobility rate in the membrane, as its thickness tends to zero.

We describe and analyze a finite element numerical scheme for the parabolic-parabolic Keller-Segel model. The scalar auxiliary variable method is used to retrieve the monotonic decay of the energy associated with the system at the discrete level. This method relies on the interpretation of the Keller-Segel model as a gradient flow. The resulting numerical scheme is efficient and easy to implement. We show the existence of a unique non-negative solution and that a modified discrete energy is obtained due to the use of the SAV method. We also prove the convergence of the discrete solutions to the ones of the weak form of the continuous Keller-Segel model.

We propose and analyse a finite element approximation of the Cahn-Hilliard equation regularised in space with single-well potential of Lennard-Jones type and degenerate mobility. The Cahn-Hilliard model has recently been applied to model evolution and growth for living tissues: although the choices of degenerate mobility and singular potential are biologically relevant, they induce difficulties regarding the design of a numerical scheme. We propose a finite element scheme in one and two dimensions and we show that it preserves the physical bounds of the solutions thanks to an upwind approach adapted to the finite elements method. Moreover, we show well-posedness, energy stability properties and convergence of solutions to the numerical scheme. Finally, numerical simulations in one and two dimensions are presented.

Temperatures and precipitations in the Arctic are expected to increase dramatically over the next century, yet little is known about how microbial communities and their underlying metabolic processes will be affected by these environmental changes. To address this knowledge gap, we adopted a unique space-for-time design to analyse sediments sampled from Lake Hazen, NU Canada. Over the past decade, in this world’s largest high Arctic lake by volume, climate change has enhanced glacial melt, resulting in increased annual runoff from the watershed. We exploit the spatial heterogeneity created by varying runoff regimes, conducting metagenomic analyses of lake sediments along these spatial gradients to study how a transition from low to high runoff, used as a proxy for environmental change, affects microbial community structure and functional potential. Here we show that increasing runoff leads to a decrease in taxonomic and functional diversity. While our data suggest that rising Arctic temperatures will negatively impact microbial diversity and nutrient cycling in Arctic lakes, it is still unclear how these losses at the microbial community level will affect biogeochemical cycles, or whether this will lead to feedback loops of uncertain direction and magnitude.

International audience

Thèse numérisée par la Direction des bibliothèques de l'Université de Montréal.

We carry out the convergence analysis of the scalar auxiliary variable (SAV) method applied to the nonlinear Schrödinger equation, which preserves a modified Hamiltonian on the discrete level. We derive a weak and strong convergence result, establish second-order global error bounds and present longtime error estimates on the modified Hamiltonian. In addition, we illustrate the favorable energy conservation of the SAV method compared to classical splitting schemes in certain applications.

Improper design and maintenance of stormwater ponds (SWPs) may lead to hypoxic conditions, poor water quality and the production of hydrogen sulfide (H2S). The objective of this study is to develop a comprehensive understanding of hypoxic conditions of SWPs, with a focus on the potential for H2S production and emission. This study was conducted at two retention SWPs in Ottawa, Canada; a problematic pond with the propensity for H2S emission and a reference pond that did not demonstrate H2S emission. The investigation illustrated a significant impact of low dissolved oxygen (DO) concentrations, hypoxic conditions, on the concentration of total sulfides in the water column. Both ponds were shown to periodically experience hypoxic conditions at depth, especially during summer periods with less precipitation and across longer periods of winter, ice-covered conditions. The problem pond, however, was shown to experience lower DO and longer hypoxic conditions than the reference pond in both non-ice-covered and ice-covered conditions due to greater depth and a longer hydraulic retention time. Hypoxic conditions were initiated at the deepest locations in the problem pond and subsequently were spread across the entirety of the pond under winter, ice-covered conditions. Algal biomass (Chlorophyll-a) and soluble biochemical oxygen demand concentrations were shown to not likely be significant factors in the development of hypoxia in the H2S-generating pond. Algal blooms of colonial Chrysophyceae, Synura, a known mixotroph, were observed during ice-covered conditions in the problem pond possibly due to stress-coping mechanisms of algae.

Improper design and maintenance of stormwater ponds (SWPs) may lead to hypoxic conditions, poor water quality and the production of hydrogen sulfide (H2S). The objective of this study is to develop a comprehensive understanding of hypoxic conditions of SWPs, with a focus on the potential for H2S production and emission. This study was conducted at two retention SWPs in Ottawa, Canada; a problematic pond with the propensity for H2S emission and a reference pond that did not demonstrate H2S emission. The investigation illustrated a significant impact of low dissolved oxygen (DO) concentrations, hypoxic conditions, on the concentration of total sulfides in the water column. Both ponds were shown to periodically experience hypoxic conditions at depth, especially during summer periods with less precipitation and across longer periods of winter, ice-covered conditions. The problem pond, however, was shown to experience lower DO and longer hypoxic conditions than the reference pond in both non-ice-covered and ice-covered conditions due to greater depth and a longer hydraulic retention time. Hypoxic conditions were initiated at the deepest locations in the problem pond and subsequently were spread across the entirety of the pond under winter, ice-covered conditions. Algal biomass (Chlorophyll-a) and soluble biochemical oxygen demand concentrations were shown to not likely be significant factors in the development of hypoxia in the H2S-generating pond. Algal blooms of colonial Chrysophyceae, Synura, a known mixotroph, were observed during ice-covered conditions in the problem pond possibly due to stress-coping mechanisms of algae.

The glymphatic system is the subject of numerous pieces of research in biology. Mathematical modelling plays a considerable role in this field since it can indicate the possible physical effects of this system and validate the biologists’ hypotheses. The available mathematical models that describe the system at the scale of the brain (i.e. the macroscopic scale) are often solely based on the diffusion equation and do not consider the fine structures formed by the perivascular spaces. We therefore propose a mathematical model representing the time and space evolution of a mixture flowing through multiple compartments of the brain. We adopt a macroscopic point of view in which the compartments are all present at any point in space. The equations system is composed of two coupled equations for each compartment: One equation for the pressure of a fluid and one for the mass concentration of a solute. The fluid and solute can move from one compartment to another according to certain membrane conditions modelled by transfer functions. We propose to apply this new modelling framework to the clearance of 14C-inulin from the rat brain.

International audience; Motivated by experimental observations in 3D/organoid cultures derived from glioblastoma, we develop a mathematical model where tumour aggregate formation is obtained as the result of nutrient-limited cell proliferation coupled with chemotaxis-based cell movement. The introduction of a chemotherapeutic treatment induces mechanical changes at the cell level, with cells undergoing a transition from rigid bodies to semi-elastic entities. We analyse the influence of these individual mechanical changes on the properties of the aggregates obtained at the population level by introducing a nonlinear volume-filling chemotactic system of partial differential equations. The elastic properties of the cells are taken into account through the so-called squeezing probability, which allows us to change the packing capacity of the aggregates, depending on the concentration of the treatment in the extracellular microenvironment. We explore two scenarios for the effect of the treatment: firstly, the treatment acts only on the mechanical properties of the cells and, secondly, we assume it also prevents cell proliferation. A linear stability analysis enables us to study the ability of the system to create patterns. We provide numerical simulations in 1D and 2D that illustrate the shrinking of the aggregates due to the presence of the treatment.