Mathematical Modeling of a Brain-on-a-Chip: A Study of the Neuronal Nitric Oxide Role in Cerebral Microaneurysms

Brain tissue is a complex material made of interconnected neural, glial, and vascular networks. While the physics and biochemistry of brain’s cell types and their interactions within their networks have been studied extensively, only recently the interactions of and feedback among the networks have started to capture the attention of the research community. Thus, a good understanding of the coupled mechano-electrochemical processes that either provide or diminish brain’s functions is still lacking. One way to increase the knowledge on how the brain yields its functions is by developing a robust controlled feedback engineering system that uses fundamental science concepts to guide and interpret experiments investigating brain’s response to various stimuli, aging, trauma, diseases, treatment and recovery processes. Recently, a mathematical model for an implantable neuro-glial-vascular unit, named brain-on-a-chip, was proposed that can be optimized to perform some fundamental cellular processes that could facilitate monitoring and supporting brain’s functions, and highlight basic brain mechanisms. In this paper we use coupled elastic, viscoelastic and mass elements to model a brain-on-a-chip made of a neuron and its membrane, and astrocyte’s endfeet connected to an arteriole’s wall. We propose two constrained Lagrangian formulations that link the Hodgkin-Huxley model of the neuronal membrane, and the mechanics of the neuron, neuronal membrane, and the glia’s endfeet. The effects of the nitric oxide produced by neurons and endothelial cells on the proposed brain-on-a-chip are investigated through numerical simulations. Our numerical simulations suggest that a non-decaying synthesis of nitric oxide may contribute to the onset of a cerebral microaneurysm

Constitutive Models for Tumour Classification

The aim of this paper is to formulate new mathematical models that will be able to differentiate not only between normal and abnormal tissues, but, more importantly, between benign and malignant tumours. We present preliminary results of a tri-phasic model and numerical simulations of the effect of cellular adhesion forces on the mechanical properties of biological tissues. We pursued the following three approaches: (i) the simulation of the time-harmonic linear elastic models to examine coarse scale effects and adhesion properties, (ii) the investigation of a tri-phasic model, with the intent of upscaling this model to determine effects of electro-mechanical coupling between cells, and (iii) the upscaling of a simple cell model as a framework for studying interface conditions at malignant cells. Each of these approaches has opened exciting new directions of research that we plan to study in the future

A Multiscale Pressure-Volume Model of Cerebrospinal Fluid Dynamics: Application to Hydrocephalus

ABSTRACT Hydrocephalus is a brain disease characterized by abnormalities in the cerebrospinal fluid (CSF) circulation. The treatment is surgical in nature and continues to suffer of poor outcomes. The first mathematical model for studying the CSF pressure-volume relationship in hydrocephalus was proposed by Marmarou in the 1970s. However, the model fails to fully capture the complex CSF dynamics controlled by CSF-brain tissue interactions. In this paper we use fractional calculus to introduce multiscaling effects in Marmarou's model. We solve our fractional order non-linear differential equation analytically using a modified Adomian decomposition method

Modeling the Non-linear Viscoelastic Response of High Temperature Polyimides

A constitutive model is developed to predict the viscoelastic response of polyimide resins that are used in high temperature applications. This model is based on a thermodynamic framework that uses the notion that the `natural configuration' of a body evolves as the body undergoes a process and the evolution is determined by maximizing the rate of entropy production in general and the rate of dissipation within purely mechanical considerations. We constitutively prescribe forms for the specific Helmholtz potential and the rate of dissipation (which is the product of density, temperature and the rate of entropy production), and the model is derived by maximizing the rate of dissipation with the constraint of incompressibility, and the reduced energy dissipation equation is also regarded as a constraint in that it is required to be met in every process that the body undergoes. The efficacy of the model is ascertained by comparing the predictions of the model with the experimental data for PMR-15 and HFPE-II-52 polyimide resins.Comment: 16 pages, 4 figures, submitted to Mechanics of Material

Direct and Inverse Solutions for Thermal-and Stress-Transients and the Analytical Determination of Boundary Conditions Using Remote Temperature or Strain Data

From an analytical standpoint, a majority of calculations use known boundary conditions (temperature or flux) and the so-called direct route to determine internal temperatures, strains, and/or stresses. For such problems where the thermal boundary condition is known a priori, the analytical procedure and solutions are tractable for the linear case where the thermophysical properties are independent of temperature. On the other hand, the inverse route where the boundary conditions must be determined from remotely determined temperature and/or flux data is much more difficult mathematically, as well as inherently sensitive to data errors (i.e., ill-posed). When solutions are available, they are often restricted to a harsh, albeit unrealistic step change in temperature or flux and/or are only valid for relatively short time frames before temperature changes occur at the far boundary. While the two approaches may seem to be at odds with each other, a generalized direct solution based on polynomial temperature or strain-histories can also be used to determine unknown boundary conditions via least-squares determination of coefficients. Once the inverse problem (and unknown boundary condition) is solved via these coefficients, the resulting polynomial can then be used with the generalized direct solution to determine the thermal-and stress-states as a function of time and position. When used for both thick slabs and tubes, excellent agreement was seen for various test cases. In fact, the derived solutions appear to be well suited for many thermal scenarios, provided the analysis is restricted to the time interval used to determine the polynomial and the thermophysical properties that do not vary with temperature. Since temperature dependent properties can certainly be an issue that affects accuracy in these types of calculations, some recent analytical procedures for both direct and inverse solutions are also discussed