Over the past decade or so, there have been a large number of modelling approaches aimedat elucidating the most important mechanisms affecting the formation of new capillariesfrom parent blood vessels — a process known as angiogenesis. Most studies have focussedupon the way in which capillary sprouts are initiated and migrate in response todiffusible chemical stimuli supplied by hypoxic stromal cells and leukocytes in thecontexts of solid tumour growth and wound healing. However, relatively few studies haveexamined the important role played by blood perfusion during angiogenesis and fewer stillhave explored the ways in which a dynamically evolving vascular bed architecture canaffect the distribution of flow within it. From the perspective of solid tumour growthand, perhaps more importantly, its treatment (e.g. chemotherapy), it would clearly be ofsome benefit to understand this coupling between vascular structure and perfusion morefully. This paper focuses on the implications of such a coupling upon chemotherapeutic,anti-angiogenic, and anti-vascular treatments.

In an extension to previous work by the authors, the issue of pericyte recruitment duringvessel maturation is considered in order to study the effects of different anti-vascularand anti-angiogenic therapies from a more rigorous modelling standpoint. Pericytes are aprime target for new vascular disrupting agents (VDAs) currently in clinical trials.However, different compounds attack different components of the vascular network and theimplications of targeting only certain elements of the capillary bed are not immediatelyclear. In light of these uncertainties, the effects of anti-angiogenic and anti-vasculardrugs are re-examined by using an extended model that includes an interdependency betweenvessel remodelling potential and local pericyte density. Two- and three-dimensionalsimulation results are presented and suggest that it may be possible to identify aVDA-specific “plasticity window” (a time period corresponding to low pericyte density),within which a given VDA would be most effective.

In vitro transmesothelial migration assays of ovarian cancer cells, isolated oraggregated in multicellular spheroids, are reproduced deducing suitable Cellular PottsModels (CPM). We show that the simulations are in good agreement with the experimentalevidence and that the overall process is regulated by the activity of matrixmetalloproteinases (MMPs) and by the interplay of the adhesive properties of the cellswith the extracellular matrix and between cells, both of the same type and of differenttypes. In particular, the process depends on the ability of the cell to induce theloosening of cadherin-mediated junctions. Coherently with experiments, it is found thatsingle cell invasion is more conservative with a crucial role played by MMPs. A similarimportant role is played in cell spheroid invasion, which in comparison is moredisruptive. It achieves monofocal or multifocal characteristics according to the relativeadhesion affinity among cells or between them and the mesothelial layer.

In this paper, we will discuss the meshless polyharmonic reconstruction of vector fieldsfrom scattered data, possibly, contaminated by noise. We give an explicit solution of theproblem. After some theoretical framework, we discuss some numerical aspect arising in theproblems related to the reconstruction of vector fields

Tuberculosis (TB) is the leading cause of death among individuals infected with thehepatitis B virus (HBV). The study of the joint dynamics of HBV and TB present formidablemathematical challenges due to the fact that the models of transmission are quitedistinct. We formulate and analyze a deterministic mathematical model which incorporatesof the co-dynamics of hepatitis B and tuberculosis. Two sub-models, namely: HBV-only andTB-only sub-models are considered first of all. Unlike the HBV-only sub-model, which has aglobally-asymptotically stable disease-free equilibrium whenever the associatedreproduction number is less than unity, the TB-only sub-model undergoes the phenomenon ofbackward bifurcation, where a stable disease-free equilibrium co-exists with a stableendemic equilibrium, for a certain range of the associated reproduction number less thanunity. Thus, for TB, the classical requirement of having the associated reproductionnumber to be less than unity, although necessary, is not sufficient for its elimination.It is also shown, that the full HBV-TB co-infection model undergoes a backward bifurcationphenomenon. Through simulations, we mainly find that i) the two diseases will co-existwhenever their partial reproductive numbers exceed unity; (ii) the increased progressionrate due to exogenous reinfection from latent to active TB in co-infected individuals mayplay a significant role in the rising prevalence of TB; and (iii) the increasedprogression rates from acute stage to chronic stage of HBV infection have increased theprevalence levels of HBV and TB prevalences.

Natural immunity to breast and prostate cancers is predicted by a novel, saturatedordered mutation model fitted to USA (SEER) incidence data, a prediction consistent withthe latest ideas in immunosurveillance. For example, the prevalence of natural immunity tobreast cancer in the white female risk population is predicted to be 76.5%; this immunitymay be genetic and, therefore, inherited. The modeling also predicts that 6.9% of WhiteFemales are born with a mutation necessary to cause breast cancer (the hereditaryform) and, therefore, are at the highest risk of developing it. By contrast,16.6% of White Females are born without any such mutation but are nonetheless susceptibleto developing breast cancer (the sporadic form). The modeling determinesthe required number of ordered mutations for a cell to become cancerous as well as themean time between consecutive mutations for both the sporadic and hereditary forms of thedisease. The mean time between consecutive breast cancer mutations was found to varybetween 2.59 - 2.97 years, suggesting that such mutations are rare events and establishingan upper bound on the lifetime of a breast cell. The prevalence of immunity to breastcancer is predicted to be 79.7% in Blacks, 86.5% in Asians, and 85.8% in Indians.Similarly, the prevalence of immunity to prostate cancer is predicted to be 67.4% forWhites, 50.5% for Blacks, 77.7% for Asians, and 78.6% for Indians. It is of paramountimportance to delineate the mechanism underlying immunity to these cancers.

We discuss Laplacians on graphs in a framework of regular Dirichlet forms. We focus onphenomena related to unboundedness of the Laplacians. This includes (failure of) essentialselfadjointness, absence of essential spectrum and stochastic incompleteness.

Real-world medical decisions rarely involve binary Ðsole condition present or absent-patterns of patient pathophysiology. Similarly, provider interventions are rarely unitaryin nature: the clinician often undertakes multiple interventions simultaneously.Conventional approaches towards complex physiologic derangements and their associatedmanagement focus on the frequencies of joint appearances, treating the individualderangements of physiology or elements of intervention as conceptually isolated. Thisframework is ill suited to capture either the integrated patterns of derangement displayedby a particular patient or the integrated patterns of provider intervention. Here weillustrate the application of a different approach-that of symbolic dynamics-in which theintegrated pattern of each patients derangement, and the associated provider response, arecaptured by defining words based on the elements of the pattern offailure. We will use as an example provider practices in the context of mechanicalventilation- a common, potentially harmful, and complex life support technology. We alsodelineate other domains in which symbolic dynamics approaches might aid in quantitatingpractice patterns, assessing quality of care, and identifying best practices.

In this article a novel model framework to simulate cells and their internal structure is described. The model is agent-based and suitable to simulate single cells with a detailed internal structure as well as multi-cellular compounds. Cells are simulated as a set of many interacting particles, with neighborhood relations defined via a Delaunay triangulation. The interacting sub-particles of a cell can assume specific roles – i.e., membrane sub-particle, internal sub-particle, organelles, etc –, distinguished by specific interaction potentials and, eventually, also by the use of modified interaction criteria. For example, membrane sub-particles may interact only on a two-dimensional surface embedded on three-dimensional space, described via a restricted Delaunay triangulation. The model can be used not only to study cell shape and movement, but also has the potential to investigate the coupling between internal space-resolved movement of molecules and determined cell behaviors.

We have carried out a polysystem analysis of the population dynamics by using a varietyof hypotheses and their respective models based on different system interpretations of thephenomenon under investigation. Each of the models supplements standard dynamic equationsfor explaining the effects observed. A qualitative model-based analysis is made of theage-specific male mortality for a Siberian industrial city. The study revealed thetendencies for background mortality to increase with age and over time, whichcharacterizes in an integral manner the influence of socio-ecological factors on thedecline in population viability. It is shown that these tendencies are similar fordifferent years and for different population age groups.

We present an iterative method based on an infinite dimensional adaptation of thesuccessive overrelaxation (SOR) algorithm for solving the 2-D neutron transport equation.In a wide range of application, the neutron transport operator admits a Self-Adjoint andm-Accretive Splitting (SAS). This splitting leads to an ADI-like iterative method whichconverges unconditionally and is equivalent to a fixed point problem where the operator isa 2 by 2 matrix of operators. An infinite dimensional adaptation of a SOR algorithm isthen applied to solve the matrix operator equation. Theoretical and numerical results ofconvergence are given

We review some recent results for a class of fluid mechanics equations called activescalars, with fractional dissipation. Our main examples are the surface quasi-geostrophicequation, the Burgers equation, and the Cordoba-Cordoba-Fontelos model. We discussnonlocal maximum principle methods which allow to prove existence of global regularsolutions for the critical dissipation. We also recall what is known about the possibilityof finite time blow up in the supercritical regime.

In this paper, a new a posteriori error estimator for nonconforming convection diffusionapproximation problem, which relies on the small discrete problems solution in stars, hasbeen established. It is equivalent to the energy error up to data oscillation without anysaturation assumption nor comparison with residual estimator

In this note, I wish to describe the first order semiclassical approximation to thespectrum of one frequency quasi-periodic operators. In the case of a sampling functionwith two critical points, the spectrum exhibits two gaps in the leading orderapproximation. Furthermore, I will give an example of a two frequency quasi-periodicoperator, which has no gaps in the leading order of the semiclassical approximation.

This paper extends the volume filling chemotaxis model [18, 26] by taking into account the cell population interactions. Theextended chemotaxis models have nonlinear diffusion and chemotactic sensitivity dependingon cell population density, which is a modification of the classical Keller-Segel model inwhich the diffusion and chemotactic sensitivity are constants (linear). The existence andboundedness of global solutions of these models are discussed and the numerical patternformations are shown. The further improvement is proposed in the end.

This paper is related to the spectral stability of traveling wave solutions of partialdifferential equations. In the first part of the paper we use the Gohberg-Rouche Theoremto prove equality of the algebraic multiplicity of an isolated eigenvalue of an abstractoperator on a Hilbert space, and the algebraic multiplicity of the eigenvalue of thecorresponding Birman-Schwinger type operator pencil. In the second part of the paper weapply this result to discuss three particular classes of problems: the Schrödingeroperator, the operator obtained by linearizing a degenerate system of reaction diffusionequations about a pulse, and a general high order differential operator. We studyrelations between the algebraic multiplicity of an isolated eigenvalue for the respectiveoperators, and the order of the eigenvalue as the zero of the Evans function for thecorresponding first order system.

We derive a residual a posteriori error estimates for the subscales stabilization ofconvection diffusion equation. The estimator yields upper bound on the error which isglobal and lower bound that is local

Early studies of the novel swine-origin 2009 influenza A (H1N1) epidemic indicateclinical attack rates in children much higher than in adults. Non-medical interventionssuch as school closings are constrained by their large socio-economic costs. Here wedevelop a mathematical model to ascertain the roles of pre-symptomatic influenzatransmission as well as symptoms surveillance of children to assess the utility of schoolclosures. Our model analysis indicates that school closings are advisable whenpre-symptomatic transmission is significant or when removal of symptomatic children isinefficient. Our objective is to provide a rational basis for school closings decisionsdependent on virulence characteristics and local surveillance implementation, applicableto the current epidemic and future epidemics.

We study the number of lattice points in ℝd ,d ≥ 2, lying inside an annulus as a function of the centre of theannulus. The average number of lattice points there equals the volume of the annulus, andwe study the L 1 and L 2 norms ofthe remainder. We say that a dimension is critical, if these norms do not have upper andlower bounds of the same order as the radius goes to infinity. In [Duke Math. J., 107 (2001), No. 2, 209–238], it was proved that in the case of the ball (instead of an annulus)the critical dimensions are d ≡ 1 mod 4. We show that the behaviour ofthe width of an annulus as a function of the radius determines which dimensions arecritical now. In particular, if the width is bounded away from zero and infinity, thecritical dimensions are d ≡ 3 mod 4; if the width goes to infinity, butslower than the radius, then all dimensions are critical, and if the width tends to zeroas a power of the radius, then there are no critical dimensions.

In this paper we study the error rate of RNA synthesis in the look-ahead model for therandom walk of RNA polymerase along DNA during transcription. The model’s centralassumption is the existence of a window of activity in whichribonucleoside triphosphates (rNTPs) bind reversibly to the template DNA strand beforebeing hydrolyzed and linked covalently to the nascent RNA chain. An unknown, butimportant, integer parameter of this model is the window size w. Here, weuse mathematical analysis and computer simulation to study the rate at whichtranscriptional errors occur as a function of w. We find dramaticreduction in the error rate of transcription as w increases, especiallyfor small values of w. The error reduction method provided by look-aheadoccurs before hydrolysis and covalent linkage of rNTP to the nascent RNAchain, and is therefore distinct from error correction mechanisms that have previouslybeen considered.

Anisotropic adaptive methods based on a metric related to the Hessian of the solution areconsidered. We propose a metric targeted to the minimization of interpolation errorgradient for a nonconforming linear finite element approximation of a given piecewiseregular function on a polyhedral domain Ω ofℝ d , d ≥ 2. We alsopresent an algorithm generating a sequence of asymptotically quasi-optimal meshes relativeto such a nonconforming discretization and give numerical asymptotic behavior of the errorreduction produced by the generated mesh