Biological invasions and epidemics with nonlocal diffusion along a line.

The goal of this work is to understand and quantify how a line with nonlocal diffusion given by an integral enhances a reaction-diffusion process occurring in the surrounding plane. This is part of a long term programme where we aim at modelling, in a mathematically rigorous way, the effect of transportation networks on the speed of biological invasions or propagation of epidemics. We prove the existence of a global propagation speed and characterise in terms of the parameters of the system the situations where such a speed is boosted by the presence of the line. In the course of the study we also uncover unexpected regularity properties of the model. On the quantitative side, the two main parameters are the intensity of the diffusion kernel and the characteristic size of its support. One outcome of this work is that the propagation speed will significantly be enhanced even if only one of the two is large, thus broadening the picture that we have already drawn in our previous works on the subject, with local diffusion modelled by a standard Laplacian. We further investigate the role of the other parameters, enlightening some subtle effects due to the interplay between the diffusion in the half plane and that on the line. Lastly, in the context of propagation of epidemics, we also discuss the model where, instead of a diffusion, displacement on the line comes from a pure transport term.

Controlling release from encapsulated drug-loaded devices: insights from modeling the dissolution front propagation.

Dissolution of drug from its solid form to a dissolved form is an important consideration in the design and optimization of drug delivery devices, particularly owing to the abundance of emerging compounds that are extremely poorly soluble. When the solid dosage form is encapsulated, for example by the porous walls of an implant, the impact of the encapsulant drug transport properties is a further confounding issue. In such a case, dissolution and diffusion work in tandem to control the release of drug. However, the interplay between these two competing processes in the context of drug delivery is not as well understood as it is for other mass transfer problems, particularly for practical controlled-release considerations such as an encapsulant layer around the drug delivery device. To address this gap, this work presents a mathematical model that describes controlled release from a drug-loaded device surrounded by a passive porous layer. A solution for the drug concentration distribution is derived using the method of eigenfunction expansion. The model is able to track the dissolution front propagation, and predict the drug release curve during the dissolution process. The utility of the model is demonstrated through comparison against experimental data representing drug release from a cylindrical drug-loaded orthopedic fixation pin, where the model is shown to capture the data very well. Analysis presented here reveals how the various geometrical and physicochemical parameters influence drug dissolution and, ultimately, the drug release profile. It is found that the non-dimensional initial concentration plays a key role in determining whether the problem is diffusion-limited or dissolution-limited, whereas the nature of the problem is largely independent of other parameters including diffusion coefficient and encapsulant thickness. We expect the model will prove to be a useful tool for those designing encapsulated drug delivery devices, in terms of optimizing the design of the device to achieve a desired drug release profile.

The crucial role of elasticity in regulating liquid-liquid phase separation in cells.

Liquid-liquid phase separation has emerged as a fundamental mechanism underlying intracellular organization, with evidence for it being reported in numerous different systems. However, there is a growing concern regarding the lack of quantitative rigor in the techniques employed to study phase separation, and their ability to account for the complex nature of the cellular milieu, which affects key experimentally observable measures, such as the shape, size and transport dynamics of liquid droplets. Here, we bridge this gap by combining recent experimental data with theoretical predictions that capture the subtleties of nonlinear elasticity and fluid transport. We show that within a biologically accessible range of material parameters, phase separation is highly sensitive to elastic properties and can thus be used as a mechanical switch to rapidly transition between different states in cellular systems. Furthermore, we show that this active mechanically mediated mechanism can drive transport across cells at biologically relevant timescales and could play a crucial role in promoting spatial localization of condensates; whether cells exploit such mechanisms for transport of their constituents remains an open question.

A traveling-wave solution for bacterial chemotaxis with growth.

Bacterial cells navigate their environment by directing their movement along chemical gradients. This process, known as chemotaxis, can promote the rapid expansion of bacterial populations into previously unoccupied territories. However, despite numerous experimental and theoretical studies on this classical topic, chemotaxis-driven population expansion is not understood in quantitative terms. Building on recent experimental progress, we here present a detailed analytical study that provides a quantitative understanding of how chemotaxis and cell growth lead to rapid and stable expansion of bacterial populations. We provide analytical relations that accurately describe the dependence of the expansion speed and density profile of the expanding population on important molecular, cellular, and environmental parameters. In particular, expansion speeds can be boosted by orders of magnitude when the environmental availability of chemicals relative to the cellular limits of chemical sensing is high. Analytical understanding of such complex spatiotemporal dynamic processes is rare. Our analytical results and the methods employed to attain them provide a mathematical framework for investigations of the roles of taxis in diverse ecological contexts across broad parameter regimes.

The collective effect of finite-sized inhomogeneities on the spatial spread of populations in two dimensions.

The dynamics of a population expanding into unoccupied habitat has been primarily studied for situations in which growth and dispersal parameters are uniform in space or vary in one dimension. Here, we study the influence of finite-sized individual inhomogeneities and their collective effect on front speed if randomly placed in a two-dimensional habitat. We use an individual-based model to investigate the front dynamics for a region in which dispersal or growth of individuals is reduced to zero (obstacles) or increased above the background (hotspots), respectively. In a regime where front dynamics is determined by a local front speed only, a principle of least time can be employed to predict front speed and shape. The resulting analytical solutions motivate an event-based algorithm illustrating the effects of several obstacles or hotspots. We finally apply the principle of least time to large heterogeneous environments by solving the Eikonal equation numerically. Obstacles lead to a slow-down that is dominated by the number density and width of obstacles, but not by their precise shape. Hotspots result in a speed-up, which we characterize as function of hotspot strength and density. Our findings emphasize the importance of taking the dimensionality of the environment into account.

Ionotropic Gelation Fronts in Sodium Carboxymethyl Cellulose for Hydrogel Particle Formation.

Hydrogel microparticles (HMPs) find numerous practical applications, ranging from drug delivery to tissue engineering. Designing HMPs from the molecular to macroscopic scales is required to exploit their full potential as functional materials. Here, we explore the gelation of sodium carboxymethyl cellulose (NaCMC), a model anionic polyelectrolyte, with Fe3+ cations in water. Gelation front kinetics are first established using 1D microfluidic experiments, and effective diffusive coefficients are found to increase with Fe3+ concentration and decrease with NaCMC concentrations. We use Fourier Transform Infrared Spectroscopy (FTIR) to elucidate the Fe3+-NaCMC gelation mechanism and small angle neutron scattering (SANS) to spatio-temporally resolve the solution-to-network structure during front propagation. We find that the polyelectrolyte chain cross-section remains largely unperturbed by gelation and identify three hierarchical structural features at larger length scales. Equipped with the understanding of gelation mechanism and kinetics, using microfluidics, we illustrate the fabrication of range of HMP particles with prescribed morphologies.

Hydrodynamic descriptions for surface roughness in fracture front propagation.

Fracture is ubiquitous in a crystalline material. Inspired by the observed phenomenological similarities between the spatial profile of a fractured surface and velocities in hydrodynamic turbulence, we set up a hydrodynamic description for the dynamics of fracture surface propagation mode I or opening fracture front. We consider several related continuum hydrodynamic models and use them to extract the similarities between the profile of a fractured surface and velocities in hydrodynamic turbulence. We conclude that a fractured surface should be generically self-similar with an underlying multifractal behaviour.This article is part of the theme issue 'Statistical physics of fracture and earthquakes'.

Dissipative structures induced by photoisomerization in a dye-doped nematic liquid crystal layer.

Order-disorder phase transitions driven by temperature or light in soft matter materials exhibit complex dissipative structures. Here, we investigate the spatio-temporal phenomena induced by light in a dye-doped nematic liquid crystal layer. Experimentally, for planar anchoring of the nematic layer and high enough input power, photoisomerization processes induce a nematic-isotropic phase transition mediated by interface propagation between the two phases. In the case of a twisted nematic layer and for intermediate input power, the light induces a spatially modulated phase, which exhibits stripe patterns. The pattern originates as an instability mediated by interface propagation between the modulated and the homogeneous nematic states. Theoretically, the phase transition, emergence of stripe patterns and front dynamics are described on the basis of a proposed model for the dopant concentration coupled with the nematic order parameter. Numerical simulations show quite a fair agreement with the experimental observations.This article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)'.

Tree crowns grow into self-similar shapes controlled by gravity and light sensing.

Plants have developed different tropisms: in particular, they reorient the growth of their branches towards the light (phototropism) or upwards (gravitropism). How these tropisms affect the shape of a tree crown remains unanswered. We address this question by developing a propagating front model of tree growth. Being length-free, this model leads to self-similar solutions after a long period of time, which are independent of the initial conditions. Varying the intensities of each tropism, different self-similar shapes emerge, including singular ones. Interestingly, these shapes bear similarities to existing tree species. It is concluded that the core of specific crown shapes in trees relies on the balance between tropisms.

Proliferation of cells with aggregation and communication.

Cell proliferation is often considered to occur via front propagation with constant velocity. This scenario proposed by Fisher, Kolmogorov, Petrovsky, and Piskunov is based on the solution of the corresponding mean-field reaction-diffusion equations and does not take into account that due to adhesion the cells have tendency to aggregate and that the rate of cell division may depend on the cell-cell communication. Herein, the author presents extensive Monte Carlo simulations taking both these factors into account and illustrating that the former factor can dramatically modify the spatio-temporal kinetics of cell proliferation. In particular, the conventional relation between the front velocity and diffusion coefficient may fail, the front velocity may appreciably increase with increasing time, and/or the front may be partly or fully smeared on the realistic length scales.

Hindrances to bistable front propagation: application to Wolbachia invasion.

We study the biological situation when an invading population propagates and replaces an existing population with different characteristics. For instance, this may occur in the presence of a vertically transmitted infection causing a cytoplasmic effect similar to the Allee effect (e.g. Wolbachia in Aedes mosquitoes): the invading dynamics we model is bistable. We aim at quantifying the propagules (what does it take for an invasion to start?) and the invasive power (how far can an invading front go, and what can stop it?). We rigorously show that a heterogeneous environment inducing a strong enough population gradient can stop an invading front, which will converge in this case to a stable front. We characterize the critical population jump, and also prove the existence of unstable fronts above the stable (blocking) fronts. Being above the maximal unstable front enables an invading front to clear the obstacle and propagate further. We are particularly interested in the case of artificial Wolbachia infection, used as a tool to fight arboviruses.

A time-space model for the growth of microalgae biofilms for biofuel production.

We present in this paper a spatial model describing the growth of a photosynthetic microalgae biofilm. In this model we consider photosynthesis, extracellular matrix excretion, and mortality. These mechanisms are described precisely using kinetic laws that take into account some saturation effects which limit the reaction rates and involve different components that we treat individually. In particular, to obtain a more detailed description of the microalgae growth, we consider separately the lipids they contain and the functional part of microalgae (proteins, RNA, etc ...), the latter playing a leading role in photosynthesis. We also consider the components dissolved in liquid phase as CO2. The model is based on mixture theory and the behaviour of each component is described on the one hand by mass conservation, which takes into account biological features of the system, and on the other hand by conservation of momentum, which describes the physical properties of the components. Some numerical simulations are displayed in the one-dimensional case and show that the model is able to estimate accurately the biofilm productivity.

A Fully-Automatic Method to Segment the Carotid Artery Layers in Ultrasound Imaging: Application to Quantify the Compression-Decompression Pattern of the Intima-Media Complex During the Cardiac Cycle.

The aim of this study was to introduce and evaluate a contour segmentation method to extract the interfaces of the intima-media complex in carotid B-mode ultrasound images. The method was applied to assess the temporal variation of intima-media thickness during the cardiac cycle. The main methodological contribution of the proposed approach is the introduction of an augmented dimension to process 2-D images in a 3-D space. The third dimension, which is added to the two spatial dimensions of the image, corresponds to the tentative local thickness of the intima-media complex. The method is based on a dynamic programming scheme that runs in a 3-D space generated with a shape-adapted filter bank. The optimal solution corresponds to a single medial axis representation that fully describes the two anatomical interfaces of the arterial wall. The method is fully automatic and does not require any input from the user. The method was trained on 60 subjects and validated on 184 other subjects from six different cohorts and four different medical centers. The arterial wall was successfully segmented in all analyzed images (average pixel size = 57 ± 20 mm), with average segmentation errors of 47 ± 70 mm for the lumen-intima interface, 55 ± 68 mm for the media-adventitia interface and 66 ± 90 mm for the intima-media thickness. The amplitude of the temporal variations in IMT during the cardiac cycle was significantly higher in the diseased population than in healthy volunteers (106 ± 48 vs. 86 ± 34 mm, p = 0.001). The introduced framework is a promising approach to investigate an emerging functional parameter of the arterial wall by assessing the cyclic compression-decompression pattern of the tissues.

Can a linguistic serial founder effect originating in Africa explain the worldwide phonemic cline?

It has been proposed that a serial founder effect could have caused the present observed pattern of global phonemic diversity. Here we present a model that simulates the human range expansion out of Africa and the subsequent spatial linguistic dynamics until today. It does not assume copying errors, Darwinian competition, reduced contrastive possibilities or any other specific linguistic mechanism. We show that the decrease of linguistic diversity with distance (from the presumed origin of the expansion) arises under three assumptions, previously introduced by other authors: (i) an accumulation rate for phonemes; (ii) small phonemic inventories for the languages spoken before the out-of-Africa dispersal; (iii) an increase in the phonemic accumulation rate with the number of speakers per unit area. Numerical simulations show that the predictions of the model agree with the observed decrease of linguistic diversity with increasing distance from the most likely origin of the out-of-Africa dispersal. Thus, the proposal that a serial founder effect could have caused the present observed pattern of global phonemic diversity is viable, if three strong assumptions are satisfied.

Front propagation speeds of T7 virus mutants.

We propose a new reaction-diffusion model with an eclipse time to study the spread of viruses on bacterial populations. This new model is both biologically and physically sound, unlike previous ones. We determine important parameter values from experimental data, such as the one-step growth. We verify the proposed model by comparing theoretical and experimental data of the front propagation speed for several T7 virus strains.

Regulating the cryo-freezing region of biological tissue with a controlled thermal device.

Cryoablation is a minimally invasive technique that kills tissue in situ through freezing and is used to destroy unresectable benign and malignant tumors. A key objective of this therapeutic technique is to kill cells within a closely defined malignant region while inflicting minimal thermal injury to the surrounding healthy tissue. The extremely low temperatures used in cryoablation inevitably cause varying degrees of damage to the surrounding healthy tissue. Thus, we proposed a simple, effective, and non-invasive heating device that can be easily incorporated into the existing cryosurgical technology. The chief aim of this device is to reduce the over-freezing of neighboring healthy tissue. A model was developed to study the performance of the proposed device during cryo-freezing of a biological tissue, for example porcine liver. The model, validated with the in vitro experimental data, demonstrated good agreement of up to 6.3%. The performance of the proposed device was evaluated using a dimensionless parameter termed the heating coil coefficient. Results demonstrated that the implementation of a heating coil is instrumental for reducing the size of undesired boundary lesions. The adoption of a 50-mm-diameter heating coil reduced the freezing of neighboring tissue by up to 56% within a freezing time of 10 min. This study establishes a framework for the selection of a correctly sized heating device to reduce the over-freezing of neighboring healthy tissue optimally.

Improved segmentation of white matter tracts with adaptive Riemannian metrics.

We present a novel geodesic approach to segmentation of white matter tracts from diffusion tensor imaging (DTI). Compared to deterministic and stochastic tractography, geodesic approaches treat the geometry of the brain white matter as a manifold, often using the inverse tensor field as a Riemannian metric. The white matter pathways are then inferred from the resulting geodesics, which have the desirable property that they tend to follow the main eigenvectors of the tensors, yet still have the flexibility to deviate from these directions when it results in lower costs. While this makes such methods more robust to noise, the choice of Riemannian metric in these methods is ad hoc. A serious drawback of current geodesic methods is that geodesics tend to deviate from the major eigenvectors in high-curvature areas in order to achieve the shortest path. In this paper we propose a method for learning an adaptive Riemannian metric from the DTI data, where the resulting geodesics more closely follow the principal eigenvector of the diffusion tensors even in high-curvature regions. We also develop a way to automatically segment the white matter tracts based on the computed geodesics. We show the robustness of our method on simulated data with different noise levels. We also compare our method with tractography methods and geodesic approaches using other Riemannian metrics and demonstrate that the proposed method results in improved geodesics and segmentations using both synthetic and real DTI data.