Multi-species viscous models for tissue growth: incompressible limit and qualitative behaviour.

We introduce two 2D mechanical models reproducing the evolution of two viscous tissues in contact. Their main property is to model the swirling cell motions while keeping the tissues segregated, as observed during vertebrate embryo elongation. Segregation is encoded differently in the two models: by passive or active segregation (based on a mechanical repulsion pressure). We formally compute the incompressible limits of the two models, and obtain strictly segregated solutions. The two models thus obtained are compared. A striking feature in the active segregation model is the persistence of the repulsion pressure at the limit: a ghost effect is discussed and confronted to the biological data. Thanks to a transmission problem formulation at the incompressible limit, we show a pressure jump at the tissues' boundaries.

Analysis of the boundary integral equation for the growth of a parabolic/paraboloidal dendrite with convection.

The growth of a parabolic/paraboloidal dendrite streamlined by viscous and potential flows in an undercooled one-component melt is analyzed using the boundary integral equation. The total melt undercooling is found as a function of the Péclet, Reynolds, and Prandtl numbers in two- and three-dimensional cases. The solution obtained coincides with the modified Ivantsov solution known from previous theories of crystal growth. Varying Péclet and Reynolds numbers we show that the melt undercooling practically coincides in cases of viscous and potential flows for a small Prandtl number, which is typical for metals. In cases of water solutions and non-metallic alloys, the Prandtl number is not small enough and the melt undercooling is substantially different for viscous and potential flows. In other words, a simpler potential flow hydrodynamic model can be used instead of a more complicated viscous flow model when studying the solidification of undercooled metals with convection.

Mathematical modeling accurately predicts the dynamics and scaling of nuclear growth in discrete cytoplasmic volumes.

Scaling of nuclear size with cell size has been observed in many species and cell types. In this work we formulate a modeling framework based on the limiting component hypothesis. We derive a family of spatio-temporal mathematical models for nuclear size determination based on different transport and growth mechanisms. We analyse model properties and use in vitro experimental data to identify the most probable mechanism. This suggests that nuclear volume scales with cell volume and that a nucleus controls its import rate as it grows. We further test the model by comparing to data of early frog development, where rapid cell divisions set the relevant time scales.

Analysis of a mathematical model of immune response to fungal infection.

Fungi are cells found as commensal residents, on the skin, and on mucosal surfaces of the human body, including the digestive track and urogenital track, but some species are pathogenic. Fungal infection may spread into deep-seated organs causing life-threatening infection, especially in immune-compromised individuals. Effective defense against fungal infection requires a coordinated response by the innate and adaptive immune systems. In the present paper we introduce a simple mathematical model of immune response to fungal infection consisting of three partial differential equations, for the populations of fungi (F), neutrophils (N) and cytotoxic T cells (T), taking N and T to represent, respectively, the innate and adaptive immune cells. We denote by [Formula: see text] the aggressive proliferation rate of the fungi, by [Formula: see text] and [Formula: see text] the killing rates of fungi by neutrophils and T cells, and by [Formula: see text] and [Formula: see text] the immune strengths, respectively, of N and T of an infected individual. We take the expression [Formula: see text] to represent the coordinated defense of the immune system against fungal infection. We use mathematical analysis to prove the following: If [Formula: see text], then the infection is eventually stopped, and [Formula: see text] as [Formula: see text]; and (ii) if [Formula: see text] then the infection cannot be stopped and F converges to some positive constant as [Formula: see text]. Treatments of fungal infection include anti-fungal agents and immunotherapy drugs, and both cause the parameter I to increase.

The formation of spreading front: the singular limit of three-component reaction-diffusion models.

Understanding the invasion processes of biological species is a fundamental issue in ecology. Several mathematical models have been proposed to estimate the spreading speed of species. In recent decades, it was reported that some mathematical models of population dynamics have an explicit form of the evolution equations for the spreading front, which are represented by free boundary problems such as the Stefan-like problem (e.g., Mimura et al., Jpn J Appl Math 2:151-186, 1985; Du and Lin, SIAM J Math Anal 42:377-405, 2010). To understand the formation of the spreading front, in this paper, we will consider the singular limit of three-component reaction-diffusion models and give some interpretations for spreading front from the viewpoint of modeling. As an application, we revisit the issue of the spread of the grey squirrel in the UK and estimate the spreading speed of the grey squirrel based on our result. Also, we discuss the relation between some free boundary problems related to population dynamics and mathematical models describing Controlling Invasive Alien Species. Lastly, we numerically consider the traveling wave solutions, which give information on the spreading behavior of invasive species.

Geodesic Loewner paths with varying boundary conditions.

Equations of the Loewner class subject to non-constant boundary conditions along the real axis are formulated and solved giving the geodesic paths of slits growing in the upper half complex plane. The problem is motivated by Laplacian growth in which the slits represent thin fingers growing in a diffusion field. A single finger follows a curved path determined by the forcing function appearing in Loewner's equation. This function is found by solving an ordinary differential equation whose terms depend on curvature properties of the streamlines of the diffusive field in the conformally mapped 'mathematical' plane. The effect of boundary conditions specifying either piecewise constant values of the field variable along the real axis, or a dipole placed on the real axis, reveal a range of behaviours for the growing slit. These include regions along the real axis from which no slit growth is possible, regions where paths grow to infinity, or regions where paths curve back toward the real axis terminating in finite time. Symmetric pairs of paths subject to the piecewise constant boundary condition along the real axis are also computed, demonstrating that paths which grow to infinity evolve asymptotically toward an angle of bifurcation of π/5.

Mathematical modelling in cell migration: tackling biochemistry in changing geometries.

Directed cell migration poses a rich set of theoretical challenges. Broadly, these are concerned with (1) how cells sense external signal gradients and adapt; (2) how actin polymerisation is localised to drive the leading cell edge and Myosin-II molecular motors retract the cell rear; and (3) how the combined action of cellular forces and cell adhesion results in cell shape changes and net migration. Reaction-diffusion models for biological pattern formation going back to Turing have long been used to explain generic principles of gradient sensing and cell polarisation in simple, static geometries like a circle. In this minireview, we focus on recent research which aims at coupling the biochemistry with cellular mechanics and modelling cell shape changes. In particular, we want to contrast two principal modelling approaches: (1) interface tracking where the cell membrane, interfacing cell interior and exterior, is explicitly represented by a set of moving points in 2D or 3D space and (2) interface capturing. In interface capturing, the membrane is implicitly modelled analogously to a level line in a hilly landscape whose topology changes according to forces acting on the membrane. With the increased availability of high-quality 3D microscopy data of complex cell shapes, such methods will become increasingly important in data-driven, image-based modelling to better understand the mechanochemistry underpinning cell motion.

Spreading speed for a West Nile virus model with free boundary.

The purpose of this paper is to determine the precise asymptotic spreading speed of the virus for a West Nile virus model with free boundary, introduced recently in Lin and Zhu (J Math Biol 75:1381-1409, 2017), based on a model of Lewis et al. (Bull Math Biol 68:3-23, 2006). We show that this speed is uniquely defined by a semiwave solution associated with the West Nile virus model. To find such a semiwave solution, we firstly consider a general cooperative system over the half-line [Formula: see text], and prove the existence of a monotone solution by an upper and lower solution approach; we then establish the existence and uniqueness of the desired semiwave solution by applying this method together with some other techniques including the sliding method. Our result indicates that the asymptotic spreading speed of the West Nile virus model with free boundary is strictly less than that of the corresponding model in Lewis et al. (2006).

Mathematical modeling of dispersal phenomenon in biofilms.

A mathematical model for dispersal phenomenon in multispecies biofilm based on a continuum approach and mass conservation principles is presented. The formation of dispersed cells is modeled by considering a mass balance for the bulk liquid and the biofilm. Diffusion of these cells within the biofilm and in the bulk liquid is described using a diffusion-reaction equation. Diffusion supposes a random character of mobility. Notably, biofilm growth is modeled by a hyperbolic partial differential equation while the diffusion process of dispersed cells by a parabolic partial differential equation. The two are mutually connected but governed by different equations that are coupled by two growth rate terms. Three biological processes are discussed. The first is related to experimental observations on starvation induced dispersal [1]. The second considers diffusion of a non-lethal antibiofilm agent which induces dispersal of free cells. The third example considers dispersal induced by a self-produced biocide agent.

Numerical resolution of a model of tumour growth.

We consider and solve numerically a mathematical model of tumour growth based on cancer stem cells (CSC) hypothesis with the aim of gaining some insight into the relation of different processes leading to exponential growth in solid tumours and into the evolution of different subpopulations of cells. The model consists of four hyperbolic equations of first order to describe the evolution of four subpopulations of cells. A fifth equation is introduced to model the evolution of the moving boundary. The coefficients of the model represent the rates at which reactions occur. In order to integrate numerically the four hyperbolic equations, a formulation in terms of the total derivatives is posed. A finite element discretization is applied to integrate the model equations in space. Our numerical results suggest the existence of a pseudo-equilibrium state reached at the early stage of the tumour, for which the fraction of CSC remains small. We include the study of the behaviour of the solutions for longer times and we obtain that the solutions to the system of partial differential equations stabilize to homogeneous steady states whose values depend only on the values of the parameters. We show that CSC may comprise different proportions of the tumour, becoming, in some cases, the predominant type of cells within the tumour. We also obtain that possible effective measure to detain tumour progression should combine the targeting of CSC with the targeting of progenitor cells.

Global stability of steady states in the classical Stefan problem for general boundary shapes.

The classical one-phase Stefan problem (without surface tension) allows for a continuum of steady-state solutions, given by an arbitrary (but sufficiently smooth) domain together with zero temperature. We prove global-in-time stability of such steady states, assuming a sufficient degree of smoothness on the initial domain, but without any a priori restriction on the convexity properties of the initial shape. This is an extension of our previous result (Hadzic & Shkoller 2014 Commun. Pure Appl. Math. 68, 689-757 (doi:10.1002/cpa.21522)) in which we studied nearly spherical shapes.

A free boundary approach to shape optimization problems.

The analysis of shape optimization problems involving the spectrum of the Laplace operator, such as isoperimetric inequalities, has known in recent years a series of interesting developments essentially as a consequence of the infusion of free boundary techniques. The main focus of this paper is to show how the analysis of a general shape optimization problem of spectral type can be reduced to the analysis of particular free boundary problems. In this survey article, we give an overview of some very recent technical tools, the so-called shape sub- and supersolutions, and show how to use them for the minimization of spectral functionals involving the eigenvalues of the Dirichlet Laplacian, under a volume constraint.

A Stefan problem on an evolving surface.

We formulate a Stefan problem on an evolving hypersurface and study the well posedness of weak solutions given L(1) data. To do this, we first develop function spaces and results to handle equations on evolving surfaces in order to give a natural treatment of the problem. Then, we consider the existence of solutions for L(∞) data; this is done by regularization of the nonlinearity. The regularized problem is solved by a fixed point theorem and then uniform estimates are obtained in order to pass to the limit. By using a duality method, we show continuous dependence, which allows us to extend the results to L(1) data.

Incompressible limit of a mechanical model of tumour growth with viscosity.

Various models of tumour growth are available in the literature. The first type describe the evolution of the cell number density when considered as a continuous visco-elastic material with growth. The second type describe the tumour as a set, and rules for the free boundary are given related to the classical Hele-Shaw model of fluid dynamics. Following previous papers where the material is described by a purely elastic material, or when active cell motion is included, we make the link between the two types of description considering the 'stiff pressure law' limit. Even though viscosity is a regularizing effect, new mathematical difficulties arise in the visco-elastic case because estimates on the pressure field are weaker and do not immediately imply compactness. For instance, travelling wave solutions and numerical simulations show that the pressure is discontinuous in space, which is not the case for an elastic material.

An overview of unconstrained free boundary problems.

In this paper, we present a survey concerning unconstrained free boundary problems of type [Formula: see text] where B1 is the unit ball, Ω is an unknown open set, F(1) and F(2) are elliptic operators (admitting regular solutions), and [Formula: see text] is a functions space to be specified in each case. Our main objective is to discuss a unifying approach to the optimal regularity of solutions to the above matching problems, and list several open problems in this direction.

Modeling multispecies biofilms including new bacterial species invasion.

A mathematical model for multispecies biofilm evolution based on continuum approach and mass conservation principles is presented. The model can describe biofilm growth dynamics including spatial distribution of microbial species, substrate concentrations, attachment, and detachment, and, in particular, is able to predict the biological process of colonization of new species and transport from bulk liquid to biofilm (or vice-versa). From a mathematical point of view, a significant feature is the boundary condition related to biofilm species concentrations on the biofilm free boundary. These data, either for new or for already existing species, are not required by this model, but rather can be predicted as results. Numerical solutions for representative examples are obtained by the method of characteristics. Results indicate that colonizing bacteria diffuse into biofilm and grow only where favorable environmental conditions exist for their development.

Multiple steadily translating bubbles in a Hele-Shaw channel.

Analytical solutions are constructed for an assembly of any finite number of bubbles in steady motion in a Hele-Shaw channel. The solutions are given in the form of a conformal mapping from a bounded multiply connected circular domain to the flow region exterior to the bubbles. The mapping is written as the sum of two analytic functions-corresponding to the complex potentials in the laboratory and co-moving frames-that map the circular domain onto respective degenerate polygonal domains. These functions are obtained using the generalized Schwarz-Christoffel formula for multiply connected domains in terms of the Schottky-Klein prime function. Our solutions are very general in that no symmetry assumption concerning the geometrical disposition of the bubbles is made. Several examples for various bubble configurations are discussed.