Altered glymphatic function is observed for many neurological diseases. Glioma, one of the most common brain cancers, is known to have altered fluid dynamics in terms of edema and blood-brain barrier breakdown, both features potentially impacting the glymphatic function. To study glioma and its fluid dynamics, we propose a flexible mathematical model, including the tumor, the peri-tumoral edema and the healthy tissue. From a mechanical point of view, we consider the brain as a multicompartment porous medium and model both the fluid movement and the clearance of solutes within the brain. Our results indicate that the impairment of the glymphatic system due to glioma growth is two-fold. First, edema resulting from the leakage of fluid at the blood-brain barrier and/or the occlusion of the interstitial fluid exit routes (notably the perivascular spaces) due to migratory tumor cells result in a slight localized increase of pressure, consequently impairing negatively glymphatic clearance. Second, local changes of porosity (i.e. the volume fraction of certain compartments such as perivascular or extracellular spaces), result in a disruption of the transport of solutes in the brain. Our results indicate that an effect similar to the enhanced permeability and retention is obtained using biologically relevant changes of parameter values of our model. Our mathematical model is the first step towards a digital twin for drug or contrast product delivery within the cerebro-spinal fluid directly (e.g. from intrathecal injection) for patients suffering from gliomas.

One of the fundamental problems in shape analysis is to align curves or surfaces before computing geodesic distances between their shapes. Finding the optimal reparametrization realizing this alignment is a computationally demanding task, typically done by solving an optimization problem on the diffeomorphism group. In this paper, we propose an algorithm for constructing approximations of orientation-preserving diffeomorphisms by composition of elementary diffeomorphisms. The algorithm is implemented using PyTorch, and is applicable for both unparametrized curves and surfaces. Moreover, we show universal approximation properties for the constructed architectures, and obtain bounds for the Lipschitz constants of the resulting diffeomorphisms.

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.

AbstractThe tripartite synapse or “neural threesome” refers to the interplay in the synapse between neighbouring neurons, the synaptic cleft, and the surrounding glial cells. Despite extensive research, the effects of glial cells, such as astrocytes, on signal transduction between neurons are not fully understood. The Kirchhoff-Nernst- Planck (KNP) and Extracellular-Membrane-Intracellular (EMI) models constitute a promising framework for modeling these kinds of systems. However, they lack the neurotransmitter-related mechanisms that are necessary to bridge signal transduction across the synaptic cleft. Here, we propose an extension to the KNP-EMI model by a spatio-temporal diffusion-based description of the most prominent neurotransmitter, glutamate, that allows for investigation of the contribution of astrocytes to the functionality of the synapse. We validate our model by showing that the presence of an astrocyte in the domain affects the glutamate flux across the postsynaptic terminal, as observed physiologically. The proposed extension offers a sufficiently simple way of integrating synaptic glutamate dynamics into the KNP-EMI framework. It introduces the relevant interactions between electrical activity and diffusion processes at the tripartite synapse that are necessary to assess how astrocytes might contribute to the functionality of the synapse. This work has implications for future studies involving glial mechanisms and other charged species within the KNP-EMI framework.