Use of mathematical modeling to study pressure regimes in normal and Fontan blood flow circulations

We develop two mathematical lumped parameter models for blood pressure distribution in the Fontan blood flow circulation: an ODE based spatially homogeneous model and a PDE based spatially inhomogeneous model. We present numerical simulations of the cardiac pressure-volume cycle and study the effect of pulmonary resistance on cardiac output. We analyze solutions of two initial-boundary value problems for a non-linear parabolic partial differential equation (PDE models) with switching in the time dynamic boundary conditions which model blood pressure distribution in the cardiovascular system with and without Fontan surgery. We also obtain necessary conditions for parameter values of the PDE models for existence and uniqueness of non-negative bounded periodic solutions.Comment: 32 pages, 6 figures, 1 tabl

In Vitro Biological Characterization of Silver-Doped Anodic Oxide Coating on Titanium

Despite the high biocompatibility and clinical effectiveness of Ti-based implants, surface functionalization (with complex osteointegrative/antibacterial strategies) is still required. To enhance the dental implant surface and to provide additional osteoinductive and antibacterial properties, plasma electrolytic oxidation of a pure Ti was performed using a nitrilotriacetic acid (NTA)-based Ag nanoparticles (AgNP)-loaded calcium–phosphate solution. Chemical and structural properties of the surface-modified titanium were assessed using scanning electron microscopy (SEM) with energy dispersive X-ray (EDX) and contact angle measurement. A bacterial adhesion test and cell culture biocompatibility with collagen production were performed to evaluate biological effectiveness of the Ti after the plasma electrolytic process. The NTA-based calcium–phosphate solution with Ag nanoparticles (AgNPs) can provide formation of a thick, porous plasma electrolytic oxidation (PEO) layer enriched in silver oxide. Voltage elevation leads to increased porosity and a hydrophilic nature of the newly formed ceramic coating. The silver-enriched PEO layer exhibits an effective antibacterial effect with high biocompatibility and increased collagen production that could be an effective complex strategy for dental and orthopedic implant development

Asymptotic decay and non-rupture of viscous sheets

For a nonlinear system of coupled PDEs, that describes evolution of a viscous thin liquid sheet and takes account of surface tension at the free surface, we show exponential (H1, L2) asymptotic decay to the flat profile of its solutions considered with general initial data. Additionally, by transforming the system to Lagrangian coordinates we show that the minimal thickness of the sheet stays positive for all times. This result proves the conjecture formally accepted in the physical literature (cf. Eggers and Fontelos in Singularities: formation, structure, and propagation. Cambridge Texts in Applied Mathematics, Cambridge, 2015), that a viscous sheet cannot rupture in finite time in the absence of external forcing. Moreover, in the absence of surface tension we find a special class of initial data for which the Lagrangian solution exhibits L2-exponential decay to the flat profile

Stationary Solutions for the Cahn - Hilliard Equation Coupled with Neumann Boundary Conditions

The structure of stationary states of the one-dimensional Cahn - Hilliard equation coupled with the Neumann boundary conditions has been studied. Here the free energy is given by a fourth order polynomial. The bifurcation diagram for existence and uniqueness of monotone solutions for this problem has been constructed. Namely, we find the length of the interval on which the solution monotonically increases or decreases and has one zero for some fixed values of physical parameters. Under the non-uniqueness we understand a possibility of existence of more than one monotone solutions for the same values of physical parameters.Исследована структура стационарного состояния одномерного уравнения Кана - Хилларда в сочетании с граничными условиями Неймана. Здесь свободная энергия задается полиномом четвертого порядка. Была построена диаграмма бифуркации существования и единственности монотонных решений этой задачи. А именно, найдена длина интервала, на котором решение монотонно возрастает или убывает и имеет один нуль для некоторых фиксированных значений физических параметров. Под неоднозначностью понимается возможность существования более чем одного монотонного решения для некоторых значений физических параметров