When invading the tissue, malignant tumour cells (i.e. cancer cells) need to detach fromneighbouring cells, degrade the basement membrane, and migrate through the extracellularmatrix. These processes require loss of cell-cell adhesion and enhancement of cell-matrixadhesion. In this paper we present a mathematical model of an intracellular pathway forthe interactions between a cancer cell and the extracellular matrix. Cancer cells usesimilar mechanisms as with normal cells for their interactions with the extracellularmatrix. We develop a model of cell-matrix adhesion that accounts for reactions between thecell surface receptor integrins, the matrix glycoprotein fibronectin, and the actinfilaments in the cytoskeleton. Each represents components for an intermediate compartment,the extracellular compartment, and the intracellular compartment, respectively. Binding offibronectin with integrins triggers a clustering of protein complexes, which thenactivates and phosphorylates regulatory proteins that are involved in actin reorganisationcausing actin polymerization and stress fibre assembly. Rearrangement of actin filamentswith integrin/fibronectin complexes near adhesion sites and interaction with fibrillarfibronectin produces the force necessary for cell migration, accounting for cell-matrixadhesion.

The introduction of the ragweed leaf beetle in the South of Russia in 1978–1989 wasaccompanied by a number of spectacular phenomena that determined the general success ofthe ragweed control and further dispersal and acclimatization of the beetles:(i) formation of solitary population waves (SPW), characterized by anextremely high density of the phytophage population at the narrow band of the front of amoving wave defoliating nearly all ragweed plants, and (ii) rapid, within5-6 generations, development of flight in the leaf beetle species that in its homelandlost the ability to fly. We present here a demogenetic model capable of reproducing boththese phenomena, assuming that the flight ability of a phytophage population is governedby a single diallelic locus with flight and flightless alleles that determine threegenotypes of the ragweed leaf beetle. Simulation results agree well with the practicalrecommendation of retaining a high density of common ragweed in the release area in orderto provide the necessary conditions for the initial increase of the leaf beetle populationand the formation of the wave. The model confirms the earlier hypothesis that the SPW isthe key factor that determines efficiency of weed biocontrol program. We demonstrate alsothat the formation of the wave has crucially accelerated the development of the beetles’ability to fly.

Epidermal wound healing is a complex process that repairs injured tissue. The complexityof this process increases when bacteria are present in a wound; the bacteria interactiondetermines whether infection sets in. Because of underlying physiological problemsinfected wounds do not follow the normal healing pattern. In this paper we present amathematical model of the healing of both infected and uninfected wounds. At the core ofour model is an account of the initiation of angiogenesis by macrophage-derived growthfactors. We express the model as a system of reaction-diffusion equations, and we presentresults of computations for a version of the model with one spatial dimension.

In this work we propose a method for analysis of postsurgical haemodynamics after femoralartery treatment of occlusive vascular disease. Patient specific reconstruction algorithmof 1D core network based on MRI data is proposed as a tool for such analysis. Along withpresurgical ultrasound data fitting it provides effective personalizing predictive methodthat is validated with clinical observations.

The inner knowledge of volumes from imagesis an ancient problem. This question becomes complicated when itconcerns quantization, as the case of any measurement and inparticular the calculation of fractal dimensions. Trabecular bonetissues have, like many natural elements, an architecture whichshows a fractal aspect. Many studies have already been developedaccording to this approach. The question which arises however is toknow to which extent it is possible to get an exact determination ofthe fractal dimension of a given volume only from the fractalmeasurement made on the projections or slice images given by medicalimaging. This paper gives general results about the Minkowskidimensions and contents of projections and sections of a set. Wealso show with examples that they depend essentially on thedirections of the planes and so there is - in general case - norelation between 3D and 2D fractal dimensions. This consideration isthen illustrated with examples from synthetic models and from CTscan images of wrists. In conclusion, this study reveals that thequantitative characterization of an organic volume (in particularosseous) requires taking into account the whole volume, and not onlysome of its slices or projections.

Atherosclerosis always develops in plaques, and the reasons are not clear. We test the hypothesis that plaque morphology results from a self-perpetuating propagating process driven by macrophages (Mphs). A computer model of atherogenesis was written in which the computer screen represents a surface view of a flattened area of an arterial wall on which greatly accelerated atherogenesis is depicted. Rate of Mph recruitment from blood monocytes is set as a steeply rising function of the number of Mphs locally present. Smooth muscle accumulation depends on Mph number, Mphs have a probability of death/loss, and lipid accumulation results directly from the death of Mphs. The program runs in reiterative cycles. From an initially normal wall, fatty streak-like foci of Mphs form at random sites, which may progress or regress. Some develop into progressive focal lesions resembling advanced plaques, which are Mph-rich and have a central fibrous cap-like central region of smooth muscle cells. Lipid accumulates centrally in them. To investigate a fetal origin of atherosclerosis, the simulation was initially loaded with Mphs: lesion development was greatly enhanced. These results strongly resemble atherosclerosis in vivo, and support the Mph-dependent hypothesis of spreading plaque growth. 


We study a nonlinear diffusion equationut = uxx + f(u)with Robin boundary condition at x = 0 and with a free boundary conditionat x = h(t), whereh(t) > 0 is a moving boundary representing theexpanding front in ecology models. For anyf ∈ C1 with f(0) = 0, weprove that every bounded positive solution of this problem converges to a stationary one.As applications, we use this convergence result to study diffusion equations withmonostable and combustion types of nonlinearities. We obtain dichotomy results and sharpthresholds for the asymptotic behavior of the solutions.

There is evidence that cancer develops when cells acquire a sequence of mutations thatalter normal cell characteristics. This sequence determines a hierarchy among the cells,based on how many more mutations they need to accumulate in order to become cancerous.When cells divide, they exhibit telomere loss and differentiate, which defines anothercell hierarchy, on top of which is the stem cell. We propose a mutation-generation model,which combines the mutation-accumulation hierarchy with the differentiation hierarchy ofthe cells, allowing us to take a step further in examining cancer development and growth.The results of the model support the hypothesis of the cancer stem cell’s role in cancerpathogenesis: a very small fraction of the cancer cell population is responsible for thecancer growth and development. Also, according to the model, the nature of mutationaccumulation is sufficient to explain the faster growth of the cancer cell population.However, numerical results show that in order for a cancer to develop within a reasonabletime frame, cancer cells need to exhibit a higher proliferation rate than normalcells.

In this paper we explore the potential of the pairwise-type modelling approach to beextended to weighted networks where nodal degree and weights are not independent. As abaseline or null model for weighted networks, we consider undirected, heterogenousnetworks where edge weights are randomly distributed. We show that the pairwise modelsuccessfully captures the extra complexity of the network, but does this at the cost oflimited analytical tractability due the high number of equations. To circumvent thisproblem, we employ the edge-based modelling approach to derive models corresponding to twodifferent cases, namely for degree-dependent and randomly distributed weights. Thesemodels are more amenable to compute important epidemic descriptors, such as early growthrate and final epidemic size, and produce similarly excellent agreement with simulation.Using a branching process approach we compute the basic reproductive ratio for both modelsand discuss the implication of random and correlated weight distributions on this as wellas on the time evolution and final outcome of epidemics. Finally, we illustrate that thetwo seemingly different modelling approaches, pairwise and edge-based, operate on similarassumptions and it is possible to formally link the two.

We investigate the origin of deterministic chaos in the Belousov–Zhabotinsky (BZ) reaction carried out in closed and unstirred reactors (CURs). In detail, we develop a model on the idea that hydrodynamic instabilities play a driving role in the transition to chaotic dynamics. A set of partial differential equations were derived by coupling the two variable Oregonator–diffusion system to the Navier–Stokes equations. This approach allows us to shed light on the correlation between chemical oscillations and spatial–temporal dynamics. In particular, numerical solutions to the corresponding reaction-diffusion-convection (RDC) problem show that natural convection can change the evolution of the concentration distribution as well as oscillation patterns. The results suggest a new way of perceiving the BZ reaction when it is conducted in CURs. In conflict with the common experience, chemical oscillations are no longer a mere chemical process. Within this framework the evolution of all dynamical observables are demonstrated to converge to the regime imposed by the RDC coupling: chemical and spatial–temporal chaos are genuine manifestations of the same phenomenon.

We analyze a stochastic neuronal network model which corresponds to an all-to-all networkof discretized integrate-and-fire neurons where the synapses are failure-prone. Thisnetwork exhibits different phases of behavior corresponding to synchrony and asynchrony,and we show that this is due to the limiting mean-field system possessing multipleattractors. We also show that this mean-field limit exhibits a first-order phasetransition as a function of the connection strength — as the synapses are made morereliable, there is a sudden onset of synchronous behavior. A detailed understanding of thedynamics involves both a characterization of the size of the giant component in a certainrandom graph process, and control of the pathwise dynamics of the system by obtainingexponential bounds for the probabilities of events far from the mean.

A hybrid model of red blood cell production, where cells are considered as discreteobjects while intra-cellular proteins and extra-cellular biochemical substances aredescribed with continuous models, is proposed. Spatial organization and regulation of redblood cell production (erythropoiesis) are investigated. Normal erythropoiesis issimulated in two dimensions, and the influence on the output of the model of someparameters involved in cell fate (differentiation, self-renewal, and death by apoptosis)is studied.

The paper deals with an iterative method for numerical solving frictionlesscontact problems for two elastic bodies. Each iterative step consists of aDirichlet problem for the one body, a contact problem for the other one and twoNeumann problems to coordinate contact stresses. Convergence is proved by theBanach fixed point theorem in both continuous and discrete case. Numericalexperiments indicate scalability of the algorithm for some choices of therelaxation parameter.

During vertebrate development cells acquire different fates depending largely ontheir location in the embryo. The definition of a cell's developmentalfate relies on extensive intercellular communication that produces positionalinformation and ultimately generates an appropriately proportioned anatomy. Herewe place reaction-diffusion mechanisms in the context of general conceptsregarding the generation of positional information during development and thenfocus on these mechanisms as parsimonious systems for positioning anatomicalstructures relative to one another. In particular, we discuss the evidence forreaction-diffusion systems operating in the developing skin to yield theperiodic arrangements of hairs and feathers and discuss how best tobring together experimental molecular biology and numerical simulations to yielda more complete understanding of the mechanisms of development and naturalvariation.

The Generalized Elastic Model is a linear stochastic model which accounts for the behaviour of many physical systems in nature, ranging from polymeric chains to single-file systems. If an external perturbation is exerted only on a single point x⋆ (tagged probe), it propagates throughout the entire system. Within the fractional Langevin equation framework, we study the effect of such a perturbation, in cases of a constant force applied. We report most of the results arising from our previous analysis and, in the present work, we show that the Fox H-functions formalism provides a compact, elegant and useful tool for the study of the scaling properties of any observable. In particular we show how the generalized Kubo fluctuation relations can be expressed in terms of H-functions.

We investigate the structure oftravelling waves for a model of a fungal disease propagating overa vineyard. This model is based on a set of ODEs of the SIR-typecoupled with two reaction-diffusion equations describing thedispersal of the spores produced by the fungus inside and over thevineyard. An estimate of the biological parameters in the modelsuggests to use a singular perturbation analysis. It allows us tocompute the speed and the profile of the travelling waves. Theanalytical results are compared with numerical simulations.

We illustrate the appearance of oscillating solutions in delay differential equationsmodeling hematopoietic stem cell dynamics. We focus on autonomous oscillations, arising asconsequences of a destabilization of the system, for instance through a Hopf bifurcation.Models of hematopoietic stem cell dynamics are considered for their abilities to describeperiodic hematological diseases, such as chronic myelogenous leukemia and cyclicalneutropenia. After a review of delay models exhibiting oscillations, we focus on threeexamples, describing different delays: a discrete delay, a continuous distributed delay,and a state-dependent delay. In each case, we show how the system can have oscillatingsolutions, and we characterize these solutions in terms of periods and amplitudes.

Impairment of left ventricular contractility due tocardiovascular diseases is reflected in left ventricle (LV) motionpatterns. An abnormal change of torsion or long axis shortening LVvalues can help with the diagnosis and follow-up of LVdysfunction. Tagged Magnetic Resonance (TMR) is a widely spreadmedical imaging modality that allows estimation of the myocardialtissue local deformation. In this work, we introduce a novelvariational framework for extracting the left ventricle dynamicsfrom TMR sequences. A bi-dimensional representation space of TMRimages given by Gabor filter banks is defined. Tracking of thephases of the Gabor response is combined using a variationalframework which regularizes the deformation field just at areaswhere the Gabor amplitude drops, while restoring the underlyingmotion otherwise. The clinical applicability of the proposedmethod is illustrated by extracting normality models of theventricular torsion from 19 healthy subjects.

We present a model for describing the spread of an infectious disease with publicscreening measures to control the spread. We want to address the problem of determining anoptimal screening strategy for a disease characterized by appreciable duration of theinfectiveness period and by variability of the transmission risk. The specific disease wehave in mind is the HIV infection. However the model will apply to a disease for whichclass-age structure is significant and should not be disregarded.

A growing body of literature testifies to the importance of quantitative reasoning skillsin the 21st-century biology curriculum, and to the learning benefits associated withactive pedagogies. The process of modeling a biological system provides an approach thatintegrates mathematical skills and higher-order thinking with existing course contentknowledge. We describe a general strategy for teaching model-building in an introductorybiology course, using the example of a model of an infectious disease outbreak.Preliminary assessment data suggest that working through the formal process of modelconstruction may help students develop their scientific reasoning and communicationskills.