Validity of the Cauchy-Born rule applied to discrete cellular-scale models of biological tissues

The development of new models of biological tissues that consider cells in a discrete manner is becoming increasingly popular as an alternative to PDE-based continuum methods, although formal relationships between the discrete and continuum frameworks remain to be established. For crystal mechanics, the discrete-to-continuum bridge is often made by assuming that local atom displacements can be mapped homogeneously from the mesoscale deformation gradient, an assumption known as the Cauchy-Born rule (CBR). Although the CBR does not hold exactly for non-crystalline materials, it may still be used as a first order approximation for analytic calculations of effective stresses or strain energies. In this work, our goal is to investigate numerically the applicability of the CBR to 2-D cellular-scale models by assessing the mechanical behaviour of model biological tissues, including crystalline (honeycomb) and non-crystalline reference states. The numerical procedure consists in precribing an affine deformation on the boundary cells and computing the position of internal cells. The position of internal cells is then compared with the prediction of the CBR and an average deviation is calculated in the strain domain. For centre-based models, we show that the CBR holds exactly when the deformation gradient is relatively small and the reference stress-free configuration is defined by a honeycomb lattice. We show further that the CBR may be used approximately when the reference state is perturbed from the honeycomb configuration. By contrast, for vertex-based models, a similar analysis reveals that the CBR does not provide a good representation of the tissue mechanics, even when the reference configuration is defined by a honeycomb lattice. The paper concludes with a discussion of the implications of these results for concurrent discrete/continuous modelling, adaptation of atom-to-continuum (AtC) techniques to biological tissues and model classification

On the foundations of cancer modelling: selected topics, speculations, & perspectives

This paper presents a critical review of selected topics related to the modelling of cancer onset, evolution and growth, with the aim of illustrating, to a wide applied mathematical readership, some of the novel mathematical problems in the field. This review attempts to capture, from the appropriate literature, the main issues involved in the modelling of phenomena related to cancer dynamics at all scales which characterise this highly complex system: from the molecular scale up to that of tissue. The last part of the paper discusses the challenge of developing a mathematical biological theory of tumour onset and evolution

Multiscale modeling of granular flows with application to crowd dynamics

In this paper a new multiscale modeling technique is proposed. It relies on a recently introduced measure-theoretic approach, which allows to manage the microscopic and the macroscopic scale under a unique framework. In the resulting coupled model the two scales coexist and share information. This allows to perform numerical simulations in which the trajectories and the density of the particles affect each other. Crowd dynamics is the motivating application throughout the paper.Comment: 30 pages, 9 figure

Multiscale modeling of granular flows with application to crowd dynamics

In this paper a new multiscale modeling technique is proposed. It relies on a recently introduced measure-theoretic approach, which allows to manage the microscopic and the macroscopic scale under a unique framework. In the resulting coupled model the two scales coexist and share information. This allows to perform numerical simulations in which the trajectories and the density of the particles affect each other. Crowd dynamics is the motivating application throughout the paper.Comment: 30 pages, 9 figure

Error control for the FEM approximation of an upscaled thermo-diffusion system with Smoluchowski interactions

We analyze a coupled system of evolution equations that describes the effect of thermal gradients on the motion and deposition of $N$ populations of colloidal species diffusing and interacting together through Smoluchowski production terms. This class of systems is particularly useful in studying drug delivery, contaminant transportin complex media, as well as heat shocks thorough permeable media. The particularity lies in the modeling of the nonlinear and nonlocal coupling between diffusion and thermal conduction. We investigate the semidiscrete as well as the fully discrete em a priori error analysis of the finite elements approximation of the weak solution to a thermo-diffusion reaction system posed in a macroscopic domain. The mathematical techniques include energy-like estimates and compactness arguments

Discrete-to-continuum modelling of cells to tissues

Constitutive models for the mechanics of soft tissues are typically constructed by fitting phenomenological models to in vitro experimental measurements. However, a significant challenge is to construct macroscale soft tissue models which directly encode the properties of the constituent cells and their extracellular matrix in a rational manner. In this work we present a general framework to derive multiscale soft tissue models which incorporate the properties of individual cells without necessarily assuming homogeneity or periodicity at the cell level. The aim of this thesis is to derive a new model for cardiac soft tissue which we approach by forming an individual based model. First, we consider a reduced viscoelastic model for each individual cell and couple this to a network description of a one-dimensional line of cells. We utilise a discrete-to-continuum approach to upscale this array to form new (nonlinear) continuum partial- differential equation (PDE) models for the tissue which allows for gradients in the cell properties along the line. This system is implemented for a test problem inducing a prescribed displacement at one end of the array (while remaining fixed at the other) for both uniform and non-uniform stiffness of cells. A cluster of stiffer cells in the centre of the domain (mimicking a cluster of dead cells in myocardium after an infarction) is investigated and results show that the majority of the deformation is taken on by the more flexible cells while the stiff cells undergo a minimal deformation. We extend this model to include the effects of active contraction, to simulate myocardium behaviour in a periodic domain and we observe a travelling wave of contraction moving through the domain. For all formulations, the discrete and continuum results agree well. For the test problem, these systems also agree well with analytical results of the linearised continuum PDE. We further extend this model to incorporate cell growth and proliferation to consider the dy- namics of a proliferating array, examining how assumptions about cell dissipation translate into different global behaviour. Utilising the theory of morphoelasticity, we introduce cell growth into the system by multiplicative decomposition of the deformation tensor for each cell into an unstressed growth phase and an elastic deformation phase. We investigate stress-driven growth, where a cell grows fastest when it is unstressed and the growth rate reduces under compression (the set up does not allow the cells to be in tension). In order to assess the effect of cell dis- sipation on the system, we compare two cases: first, that the dissipation is independent of cell surface area; and second, that the dissipation coefficient is linearly proportional to the current cell surface area. We observe that in the latter case, cells pay an extra penalty for enlarging and overall growth of the array is decreased. We further consider cell proliferation in this system, with cells dividing when they reach double their initial size. In this case we can predict changes in the number of cells with time showing that the growth eventually attains a constant rate. Sub- strate dissipation results in division events becoming localised to the free end of the domain, replicating the behaviour of a proliferating rim. We also observe that cell proliferation generally leads to slower growth of the array (except in cases with very small substrate dissipation). We then extend the approach to a two-dimensional rectangular array of cells atop a fixed substrate and the upper boundary of cells parallel to this is subject to zero stress, again utilis- ing a discrete-to-continuum approach to form new (nonlinear) two-dimensional continuum PDE models. We specify the general formulation where each cell’s deformation must (in general) be solved numerically, and then focus on two simpler cases where the cell deformation is ap- proximated as either a uniaxial deformation or a simple shear. For cells undergoing uniaxial deformation, we consider a time-dependent prescribed deformation along one edge of the rect- angular domain (while keeping the edge parallel to this fixed) with two different cases for the boundaries normal to the moving edge. First, we consider zero external stress where the re- sulting deformation is in all three dimensions and the cell area in contact with the substrate decreases. Second, we consider the two boundaries normal to the moving edge to be periodic. In this case, there is no deformation normal to the periodic boundaries, and the prescribed com- pression on the array is in the out-of-plane direction alone. For a simple shear deformation, we apply a constant shearing force on one edge of the rectangular array (with the opposite edge held fixed) and periodic boundary conditions on the remaining two edges. In this case, we prohibit motion normal to the periodic boundaries, allowing motion only in the direction of the shearing force. Dissipation in the system results in a transient delay in the transmission of the shearing force to all the cells in the array. Cells closer to the sheared boundary move ahead of those closer to the fixed boundary. In this case we show that this deformation can be solved analytically. We conclude this thesis with an overview of how the approaches developed within can be extended to produce new models of soft tissue mechanics

Enlarging the possibility space for scientific model-based explanation

Two prominent views in the scientific explanation literature are: (1) that scientific explanations should be ontic or track causal or constitutive relations between the explanans and explanandum; (2) Idealizations in scientific models can be either epistemically dispensable or indispensable in principle. (1) manifests in the requirements which proponents of that view hold for scientific models to be deemed explanatory. Per these advocates, scientific models must not only track causal or constitutive relations but must include some mapping from the model components to the target system. (2) represents something like the current state of play for understanding the place of idealizations in scientific models and involves the longstanding issue of intertheoretic reduction. Idealizations can either be epistemically indispensable (that is not derivable from or reducible to) the relevant micro-level theory or epistemically dispensable in principle. The following project aims to rebut both of these views, thereby seeking to enlarge the possibility space for scientific explanation. For this reason, this project gestures towards and develops new dimensions for scientific model-based explanation. Pace (1), there are many scientific models which do not track ontic or causal relations but are nevertheless explanatory. The first chapter considers a cognitive dynamical model --the HKB model of bimanual coordination-- which fails these requirements for explanation but is one which I claim can still be shown to be explanatory. This represents a promising bit of evidence which can be marshalled and directed against this commitment. Along the lines of (1), proponents of this requirement claim that scientific models must be ontic or risk facing a problematic "directionality problem." The second chapter provides a route of response for the advocate of non-ontic scientific explanations, demonstrating how this problem can be resolved along pragmatic lines. Finally, the partition of the possibility space for understanding the role of idealizations in scientific models encapsulated in (2) is challenged in the third chapter. Therein, a certain species of idealization -continuum idealizations- are discussed and a pragmatic and deflationary approach to the issue of intertheoretic reduction is argued for. These chapters all serve to demonstrate countervailing considerations which, if successful, act as important challenges for the veracity of both (1) and (2). Rather than achieving a mere refutation of these commitments, the success of this project calls for a re-imagining and enlargement of the possibility space for scientific model-based explanations.Includes bibliographical references

Sources of pesticide losses to surface waters and groundwater at field and landscape scales

Pesticide residues in groundwater and surface waters may harm aquatic ecosystems and result in a deterioration of drinking water quality. EU legislation and policy emphasize risk management and risk reduction for pesticides to ensure long-term, sustainable use of water across Europe. Different tools applicable at scales ranging from farm to national and EU scales are required to meet the needs of the various managers engaged with the task of protecting water resources. The use of computer-based pesticide fate and transport models at such large scales is challenging since models are scale-specific and generally developed for the soil pedon or plot scale. Modelling at larger scales is further complicated by the spatial and temporal variability of agro-environmental conditions and the uncertainty in predictions. The objective of this thesis was to identify the soil processes that dominate diffuse pesticide losses at field and landscape scales and to develop methods that can help identify 'high risk' areas for leaching. The underlying idea was that pesticide pollution of groundwater and surface waters can be mitigated if pesticide application on such areas is reduced. Macropore flow increases the risk of pesticide leaching and was identified as the most important process responsible for spatial variation of diffuse pesticide losses from a 30 ha field and a 9 km² catchment in the south of Sweden. Point-sources caused by careless handling of pesticides when filling or cleaning spraying equipment were also a significant source of contamination at the landscape scale. The research presented in this thesis suggests that the strength of macropore flow due to earthworm burrows and soil aggregation can be predicted from widely available soil survey information such as texture, management practices etc. Thus, a simple classification of soils according to their susceptibility to macropore flow may facilitate the use of process-based models at the landscape scale. Predictions of a meta-model of the MACRO model suggested that, at the field scale, fine-textured soils are high-risk areas for pesticide leaching. Uncertainty in pesticide degradation and sorption did not significantly affect predictions of the spatial extent of these high-risk areas. Thus, site-specific pesticide application seems to be a promising method for mitigating groundwater contamination at this scale

How to scale up from animal movement decisions to spatiotemporal patterns: An approach via step selection

Uncovering the mechanisms behind animal space use patterns is of vital importance for predictive ecology, thus conservation and management of ecosystems. Movement is a core driver of those patterns so understanding how movement mechanisms give rise to space use patterns has become an increasingly active area of research.This study focuses on a particular strand of research in this area, based around step selection analysis (SSA). SSA is a popular way of inferring drivers of movement decisions, but, perhaps less well appreciated, it also parametrises a model of animal movement. Of key interest is that this model can be propagated forwards in time to predict the space use patterns over broader spatial and temporal scales than those that pertain to the proximate movement decisions of animals.Here, we provide a guide for understanding and using the various existing techniques for scaling up step selection models to predict broad-scale space use patterns. We give practical guidance on when to use which technique, as well as specific examples together with code in R and Python.By pulling together various disparate techniques into one place, and providing code and instructions in simple examples, we hope to highlight the importance of these techniques and make them accessible to a wider range of ecologists, ultimately helping expand the usefulness of SSA

State Of the Art Report in the fields of numerical analysis and scientific computing. Final version as of 16/02/2020 deliverable D4.1 of the HORIZON 2020 project EURAD.: European Joint Programme on Radioactive Waste Management

Document information Project Acronym EURAD Project Title European Joint Programme on Radioactive Waste Management Project Type European Joint Programme (EJP) EC grant agreement No. 847593 Project starting / end date 1 st June 2019-30 May 2024 Work Package No. 4 Work Package Title Development and Improvement Of NUmerical methods and Tools for modelling coupled processes Work Package Acronym DONUT Deliverable No. 4.