A mixture-like model for tumor-immune system interactions.

We introduce a mathematical model based on mixture theory intended to describe the tumor-immune system interactions within the tumor microenvironment. The equations account for the geometry of the tumor expansion, and the displacement of the immune cells, driven by diffusion and chemotactic mechanisms. They also take into account the constraints in terms of nutrient and oxygen supply. The numerical investigations analyze the impact of the different modeling assumptions and parameters. Depending on the parameters, the model can reproduce elimination, equilibrium or escape phases and it identifies a critical role of oxygen/nutrient supply in shaping the tumor growth. In addition, antitumor immune cells are key factors in controlling tumor growth, maintaining an equilibrium while protumor cells favor escape and tumor expansion.

A size and space structured model describing interactions of tumor cells with immune cells reveals cancer persistent equilibrium states in tumorigenesis.

The recent success of immunotherapies for the treatment of cancer has highlighted the importance of the interactions between tumor and immune cells. Mathematical models of tumor growth are needed to faithfully reproduce and predict the spatiotemporal dynamics of tumor growth. We introduce a mathematical model intended to describe by means of a system of partial differential equations the early stages of the interactions between effector immune cells and tumor cells. The model is structured in size and space, and it takes into account the migration of the tumor antigen-specific cytotoxic effector cells towards the tumor micro-environment by a chemotactic mechanism. We investigate on numerical grounds the role of the key parameters of the model such as the division and growth rates of the tumor cells, and the conversion and death rates of the immune cells. Our main findings are two-fold. Firstly, the model exhibits a possible control of the tumor growth by the immune response; nevertheless, the control is not complete in the sense that the asymptotic equilibrium states keep residual tumors and activated immune cells. Secondly, space heterogeneities of the source of immune cells can significantly reduce the efficiency of the control dynamics, making patterns of remission-recurrence appear.

A mathematical model to investigate the key drivers of the biogeography of the colon microbiota.

The gut microbiota, mainly located in the colon, is engaged in a complex dialogue with the large intestinal epithelium through which important regulatory processes for the health and well-being of the host take place. Imbalances of the microbial populations, called dysbiosis, are related to several pathological status, emphasizing the importance of understanding the gut bacterial ecology. Among the ecological drivers of the microbiota, the spatial structure of the colon is of special interest: spatio-temporal mechanisms can lead to the constitution of spatial interactions among the bacterial populations and of environmental niches that impact the overall colonization of the colon. In the present study, we introduce a mathematical model of the colon microbiota in its fluid environment, based on the explicit coupling of a population dynamics model of microbial populations involved in fibre degradation with a fluid dynamics model of the luminal content. This modeling framework is used to study the main drivers of the spatial structure of the microbiota, specially focusing on the dietary fibre inflow, the epithelial motility, the microbial active swimming and viscosity gradients in the digestive track. We found 1) that the viscosity gradients allow the creation of favorable niches in the vicinity of the mucus layer; 2) that very low microbial active swimming in the radial direction is enough to promote bacterial growth, which sheds a new light on microbial motility in the colon and 3) that dietary fibres are the main driver of the spatial structure of the microbiota in the distal bowel whereas epithelial motility is preponderant for the colonization of the proximal colon; in the transverse colon, fibre levels and chemotaxis have the strongest impact on the distribution of the microbial communities.

An ant navigation model based on Weber's law.

We analyze an ant navigation model based on Weber's law, where the ants move across a pheromone landscape sensing the area using two antennae. The key parameter of the model is the angle [Formula: see text] representing the span of the ant's sensing area. We show that when [Formula: see text] ants are able to follow (straight) pheromone trails proving that for initial conditions close to the trail, there exists a Lyapunov function that ensures ant trajectories converge on and follow the pheromone trail, with these solutions being locally asymptotically stable. Furthermore, we indicate that the features of the ant trajectories such as convergence speed or oscillation wave length are controlled by the angle [Formula: see text]. For [Formula: see text], we present numerical evidence that indicates that ants are unable to follow pheromone trails. We also assess our model by comparing it to previous experimental results, showing that the solutions' behavior falls into biologically meaningful ranges. Our work provides solid mathematical support for experimental studies where it was found that ant perception follows a Weber's law, by proving that such models lead to the desired robust and stable trail following.

Signal propagation of the MAPK cascade in Xenopus oocytes: role of bistability and ultrasensitivity for a mixed problem.

The MAPK signaling cascade is nowadays understood as a network module highly conserved across species. Its main function is to transfer a signal arriving at the plasma membrane to the cellular interior. Current understanding of 'how' this is achieved involves the notions of ultrasensitivity and bistability which relate to the nonlinear dynamics of the biochemical network, ignoring spatial aspects. Much less, indeed, is so far known about the propagation of the signal through the cytoplasm. In this work we formulate, starting from a Michaelis-Menten model for the MAPK cascade in Xenopus oocytes, a reaction-diffusion model of the cascade. We study this model in one space dimension. Basing ourselves on previous general results on reaction diffusion models, we particularly study for our model the conditions for signal propagation. We show that the existence of a propagating front depends sensitively on the initial and boundary conditions at the plasma membrane. Possible biological consequences of this finding are discussed.

Analysis of the Equilibrium Phase in Immune-Controlled Tumors Provides Hints for Designing Better Strategies for Cancer Treatment.

When it comes to improving cancer therapies, one challenge is to identify key biological parameters that prevent immune escape and maintain an equilibrium state characterized by a stable subclinical tumor mass, controlled by the immune cells. Based on a space and size structured partial differential equation model, we developed numerical methods that allow us to predict the shape of the equilibrium at low cost, without running simulations of the initial-boundary value problem. In turn, the computation of the equilibrium state allowed us to apply global sensitivity analysis methods that assess which and how parameters influence the residual tumor mass. This analysis reveals that the elimination rate of tumor cells by immune cells far exceeds the influence of the other parameters on the equilibrium size of the tumor. Moreover, combining parameters that sustain and strengthen the antitumor immune response also proves more efficient at maintaining the tumor in a long-lasting equilibrium state. Applied to the biological parameters that define each type of cancer, such numerical investigations can provide hints for the design and optimization of cancer treatments.

A size and space structured model of tumor growth describes a key role for protumor immune cells in breaking equilibrium states in tumorigenesis.

Switching from the healthy stage to the uncontrolled development of tumors relies on complicated mechanisms and the activation of antagonistic immune responses, that can ultimately favor the tumor growth. We introduce here a mathematical model intended to describe the interactions between the immune system and tumors. The model is based on partial differential equations, describing the displacement of immune cells subjected to both diffusion and chemotactic mechanisms, the strength of which is driven by the development of the tumors. The model takes into account the dual nature of the immune response, with the activation of both antitumor and protumor mechanisms. The competition between these antagonistic effects leads to either equilibrium or escape phases, which reproduces features of tumor development observed in experimental and clinical settings. Next, we consider on numerical grounds the efficacy of treatments: the numerical study brings out interesting hints on immunotherapy strategies, concerning the role of the administered dose, the role of the administration time and the interest in combining treatments acting on different aspects of the immune response. Such mathematical model can shed light on the conditions where the tumor can be maintained in a viable state and also provide useful hints for personalized, efficient, therapeutic strategies, boosting the antitumor immune response, and reducing the protumor actions.