Solutions for the Cell Cycle in Cell Lines Derived from Human Tumors

The goal of the paper is to compute efficiently solutions for model equations that have the potential to describe the growth of human tumor cells and their responses to radiotherapy or chemotherapy. The mathematical model involves four unknown functions of two independent variables: the time variable t and dimensionless relative DNA content x. The unknown functions can be thought of as the number density of cells and are solutions of a system of four partial differential equations. We construct solutions of the system, which allow us to observe the number density of cells for different t and x values. We present results of our experiments which simulate population kinetics of human cancer cells in vitro. Our results show a correspondence between predicted and experimental data

Reversible Superconductivity in Electrochromic Indium-Tin Oxide Films

Transparent conductive indium tin oxide (ITO) thin films, electrochemically intercalated with sodium or other cations, show tunable superconducting transitions with a maximum $T_c$ at 5 K. The transition temperature and the density of states, $D(E_F)$ (extracted from the measured Pauli susceptibility $\chi_p$ exhibit the same dome shaped behavior as a function of electron density. Optimally intercalated samples have an upper critical field $\approx 4$ T and $\Delta/{k_BT_c} \approx 2.0$. Accompanying the development of superconductivity, the films show a reversible electrochromic change from transparent to colored and are partially transparent (orange) at the peak of the superconducting dome. This reversible intercalation of alkali and alkali earth ions into thin ITO films opens diverse opportunities for tunable, optically transparent superconductors

An Iterated Pseudospectral Method for Functional Partial Differential Equations

Chebyshev pseudospectral spatial discretization preconditioned by the Kosloff and Tal-Ezer transformation [10] is applied to hyperbolic and parabolic functional equations. A Jacobi waveform relaxation method is then applied to the resulting semi-discrete functional systems, and the result is a simple system of ordinary differential equations d/dtUk+1(t) = MαUk+1(t)+f(t,U kt). Here Mα is a diagonal matrix, k is the index of waveform relaxation iterations, U kt is a functional argument computed from the previous iterate and the function f, like the matrix Mα, depends on the process of semi-discretization. This waveform relaxation splitting has the advantage of straight forward, direct application of implicit numerical methods for time integration (which allow use of large time steps than explicit methods). Another advantage of Jacobi waveform relaxation is that the resulting systems of ordinary differential equation can be efficiently integrated in a parallel computing environment. The Kosloff and Tal-Ezer transformation preconditions the matrix Mα, and this speeds up the convergence of waveform relaxation. This transformation is based on a parameter α∈ (0, 1], thus we study the relationship between this parameter and the convergence of waveform relaxation with error bounds derived here for the iteration process. We find that convergence of waveform relaxation improves as α increases, with the greatest improvement at α=1 if the spatial derivative of the solution at the boundaries is near zero. These results are confirmed by numerical experiments, and they hold for hyperbolic, parabolic and mixed hyperbolic-parabolic problems with and without delay terms

Discrete variable methods for delay-differential equations with threshold-type delays

AbstractWe study numerical solution of systems of delay-differential equations in which the delay function, which depends on the unknown solution, is defined implicitly by the threshold condition. We study discrete variable numerical methods for these problems and present error analysis. The global error is composed of the error of solving the differential systems, the error from the threshold conditions and the errors in delay arguments. Our theoretical analysis is confirmed by numerical experiments on threshold problems from the theory of epidemics and from population dynamics

Numerical Solutions for a Model of Tissue Invasion and Migration of Tumour Cells

The goal of this paper is to construct a new algorithm for the numerical simulations of the evolution of tumour invasion and metastasis. By means of mathematical model equations and their numerical solutions we investigate how cancer cells can produce and secrete matrix degradative enzymes, degrade extracellular matrix, and invade due to diffusion and haptotactic migration. For the numerical simulations of the interactions between the tumour cells and the surrounding tissue, we apply numerical approximations, which are spectrally accurate and based on small amounts of grid-points. Our numerical experiments illustrate the metastatic ability of tumour cells

Particle Acceleration in Turbulence and Weakly Stochastic Reconnection

Fast particles are accelerated in astrophysical environments by a variety of processes. Acceleration in reconnection sites has attracted the attention of researchers recently. In this letter we analyze the energy distribution evolution of test particles injected in three dimensional (3D) magnetohydrodynamic (MHD) simulations of different magnetic reconnection configurations. When considering a single Sweet-Parker topology, the particles accelerate predominantly through a first-order Fermi process, as predicted in previous work (de Gouveia Dal Pino & Lazarian, 2005) and demonstrated numerically in Kowal, de Gouveia Dal Pino & Lazarian (2011). When turbulence is included within the current sheet, the acceleration rate, which depends on the reconnection rate, is highly enhanced. This is because reconnection in the presence of turbulence becomes fast and independent of resistivity (Lazarian & Vishniac, 1999; Kowal et al., 2009) and allows the formation of a thick volume filled with multiple simultaneously reconnecting magnetic fluxes. Charged particles trapped within this volume suffer several head-on scatterings with the contracting magnetic fluctuations, which significantly increase the acceleration rate and results in a first-order Fermi process. For comparison, we also tested acceleration in MHD turbulence, where particles suffer collisions with approaching and receding magnetic irregularities, resulting in a reduced acceleration rate. We argue that the dominant acceleration mechanism approaches a second order Fermi process in this case.Comment: 6 pages, 1 figur

Finite-Difference and Pseudo-Sprectral Methods for the Numerical Simulations of In Vitro Human Tumor Cell Population Kinetics

Pseudo-spectral approximations are constructed for the model equations which describe the population kinetics of human tumor cells in vitro and their responses to radiotherapy or chemotherapy. These approximations are more efficient than finite-difference approximations. The spectral accuracy of the pseudo-spectral method allows us to resolve the model with a much smaller number of spatial grid-points than required for the finite-difference method to achieve comparable accuracy. This is demonstrated by numerical experiments which show a good agreement between predicted and experimental data