Alfio Grillo

Remodelling in statistically oriented fibre-reinforced materials and biological tissues

We present a mathematical model of structural reorganisation in a fibre-reinforced composite material in which the fibres are oriented statistically, i.e. obey a probability distribution of orientation. Such a composite material exemplifies a biological tissue (e.g. articular cartilage or a blood vessel) whose soft matrix is reinforced by collagen fibres. The structural reorganisation of the composite takes place as fibres reorient, in response to mechanical stimuli, in order to optimise the stress distribution in the tissue. Our mathematical model is based on the Principle of Virtual Powers and the study of dissipation. Besides incompressibility, our main hypothesis is that the composite is characterised by a probability density distribution that measures the probability of finding a family of fibres aligned along a given direction at a fixed material point. Under this assumption, we describe the reorientation of fibres as the evolution of the most probable direction along which the fibres are aligned. To test our theory, we compare our simulations of a benchmark problem with selected results taken from the literature.

Do two-phase biogas plants separate anaerobic digestion phases? – A mathematical model for the distribution of anaerobic digestion phases among reactor stages

In this article a mathematical model is introduced, which estimates the distribution of the four anaerobic digestion phases (hydrolysis, acidogenesis, acetogenesis and methanogenesis) that occur among the leach bed reactor and the anaerobic filter of a biogas plant. It is shown that only the hydrolysis takes place in the first stage (leach bed reactor), while all other anaerobic digestion phases take place in both reactor stages. It turns out that, besides the usually measured raw materials of the acetogenesis and the methanogenesis phases (organic acids), it is also necessary to analyze the process liquid for raw materials of the acidogenesis phase, i.e., sugars, fatty acids, amino acids, etc. The introduced model can be used to monitor the inhibition of the anaerobic digestion phases in reactor stages and can, thus, help to improve the control system of biogas plants.

Mathematical modeling of process liquid flow and acetoclastic methanogenesis under mesophilic conditions in a two-phase biogas reactor

Acetoclastic methanogenesis in the second stage of a two-phase biogas reactor is investigated. A mathematical model coupling chemical reactions with transport of process liquid and with the variation of population of the microorganisms living on the plastic tower packing of the reactor is proposed. The evolution of the liquid is described by an advection-diffusion-reaction equation, while a monod-type kinetic is used for the reactions. Moreover, a new inhibition factor MO(max) is introduced, which hinders the growth of microorganisms when the plastic tower packing is overpopulated. After estimating the reaction parameters, the acetate outflow measured experimentally is in good agreement with that predicted by simulations. For coupling liquid transport with reaction processes, a spatial discretization of the reactor is performed. This yields essential information about the distribution of acetate and the production of methane in the reactor. This information allows for defining a measure of the effectiveness of the reactor.

Influence of nucleus deformability on cell entry into cylindrical structures

The mechanical properties of cell nuclei have been demonstrated to play a fundamental role in cell movement across extracellular networks and micro-channels. In this work, we focus on a mathematical description of a cell entering a cylindrical channel composed of extracellular matrix. An energetic approach is derived in order to obtain a necessary condition for which cells enter cylindrical structures. The nucleus of the cell is treated either (i) as an elastic membrane surrounding a liquid droplet or (ii) as an incompressible elastic material with Neo-Hookean constitutive equation. The results obtained highlight the importance of the interplay between mechanical deformability of the nucleus and the capability of the cell to establish adhesive bonds and generate active forces in the cytoskeleton due to myosin action.

Non-Linear Model for Compression Tests on Articular Cartilage

Hydrated soft tissues, such as articular cartilage, are often modeled as biphasic systems with individually incompressible solid and fluid phases, and biphasic models are employed to fit experimental data in order to determine the mechanical and hydraulic properties of the tissues. Two of the most common experimental setups are confined and unconfined compression. Analytical solutions exist for the unconfined case with the linear, isotropic, homogeneous model of articular cartilage, and for the confined case with the non-linear, isotropic, homogeneous model. The aim of this contribution is to provide an easily implementable numerical tool to determine a solution to the governing differential equations of (homogeneous and isotropic) unconfined and (inhomogeneous and isotropic) confined compression under large deformations. The large-deformation governing equations are reduced to equivalent diffusive equations, which are then solved by means of finite difference (FD) methods. The solution strategy proposed here could be used to generate benchmark tests for validating complex user-defined material models within finite element (FE) implementations, and for determining the tissues mechanical and hydraulic properties from experimental data.

Mass Transport in Porous Media With Variable Mass

We present a theoretical and numerical study of mass transport in a porous medium saturated with a fluid and characterised by an evolving internal structure. The dynamics of the porous medium and the fluid as well as their reciprocal interactions are described at a coarse scale, so that the fundamental tools of Mixture Theory and Continuum Mechanics can be used. The evolution of the internal structure of the porous medium, which is here primarily imputed either to growth or to mass exchange with the fluid, is investigated by enriching the space of kinematic variables of the mixture with a set of structural descriptors, each of which is power-conjugate to generalised forces satisfying a balance law. Establishing the influence of the structural change of the porous medium on the transport properties of the mixture and, thus, on the quantities characterising fluid flow is the crux of our contribution.

Three-dimensional simulation of the thermohaline-driven buoyancy of a brine parcel

We provide three-dimensional numerical simulations of the thermohaline-driven buoyancy of a brine “parcel” immersed in an initially homogeneous porous medium of hydrological interest. Our purpose is to improve our understanding of the thermohaline flow through the 3D visualization of the evolving patterns generated by the distributions of brine, temperature, and fluid density in the porous medium. We propose a possible physical interpretation of our results, which are obtained within the approximations usually employed in the context of density- and temperature-driven flow.

A generalised algorithm for anelastic processes in elastoplasticity and biomechanics

A computational algorithm for solving anelastic problems in finite deformations is introduced. The presented procedure, termed the Generalised Plasticity Algorithm (GPA) hereafter, takes inspiration from the Return Mapping Algorithm (RMA), which is typically employed to solve the Karush–Kuhn–Tucker (KKT) system arising in finite elastoplasticity, but aims to modify and extend the RMA to the case of more general flow rules and strain energy density functions as well as to non-classical formulations of elastoplasticity, in which the plastic variables are not treated as internal variables. To assess its reliability, the GPA is tested in two different contexts. First, it is used for solving two classical problems (a shear-compression test and the necking of a circular bar). In both cases, the GPA is compared with the RMA in terms of structural set-up, computational effort and flexibility, and its convergence is evaluated by solving several benchmarks. Some restrictions of the classical form of the RMA are pointed out, and it is shown how these can be overcome by adopting the proposed algorithm. Second, the GPA is applied to characterise the mechanical response of a biological tissue that undergoes large deformations and remodelling of its internal structure.

An avascular tumor growth model based on porous media mechanics and evolving natural states

Mechanical factors play a major role in tumor development and response to treatment. This is more evident for tumors grown in vivo, where cancer cells interact with the different components of the host tissue. Mathematical models are able to characterize the mechanical response of the tumor and can provide a better understanding of these interactions. In this work, we present a biphasic model for tumor growth based on the mechanics of fluid-saturated porous media. In our model, the porous medium is identified with the tumor cells and the extracellular matrix, and represents the system’s solid phase, whereas the interstitial fluid constitutes the liquid phase. A nutrient is transported by the fluid phase, thereby supporting the growth of the tumor. The internal reorganization of the tissue in response to mechanical and chemical stimuli is described by enforcing the multiplicative decomposition of the deformation gradient tensor associated with the solid phase motion. In this way, we are able to distinguish the contributions of growth, rearrangement of cellular bonds, and elastic distortions which occur during tumor evolution. Results are shown for three cases of biological interest, addressing (i) the growth of a tumor spheroid in the culture medium, and (ii) the evolution of an avascular tumor growing first in a soft host tissue and then (iii) in a three-dimensional heterogeneous region. We analyze the dependence of tumor development on the mechanical environment, with particular focus on cell reorganization and its role in stress relaxation.

Synaptic bouton properties are tuned to best fit the prevailing firing pattern

The morphology of presynaptic specializations can vary greatly ranging from classical single-release-site boutons in the central nervous system to boutons of various sizes harboring multiple vesicle release sites. Multi-release-site boutons can be found in several neural contexts, for example at the neuromuscular junction (NMJ) of body wall muscles of Drosophila larvae. These NMJs are built by two motor neurons forming two types of glutamatergic multi-release-site boutons with two typical diameters. However, it is unknown why these distinct nerve terminal configurations are used on the same postsynaptic muscle fiber. To systematically dissect the biophysical properties of these boutons we developed a full three-dimensional model of such boutons, their release sites and transmitter-harboring vesicles and analyzed the local vesicle dynamics of various configurations during stimulation. Here we show that the rate of transmission of a bouton is primarily limited by diffusion-based vesicle movements and that the probability of vesicle release and the size of a bouton affect bouton-performance in distinct temporal domains allowing for an optimal transmission of the neural signals at different time scales. A comparison of our in silico simulations with in vivo recordings of the natural motor pattern of both neurons revealed that the bouton properties resemble a well-tuned cooperation of the parameters release probability and bouton size, enabling a reliable transmission of the prevailing firing-pattern at diffusion-limited boutons. Our findings indicate that the prevailing firing-pattern of a neuron may determine the physiological and morphological parameters required for its synaptic terminals.