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

Correlation Between Animal and Mathematical Models for Prostate Cancer Progression

This work demonstrates that prostate tumour progression in vivo can be analysed by using solutions of a mathematical model supplemented by initial conditions chosen according to growth rates of cell lines in vitro. The mathematical model is investigated and solved numerically. Its numerical solutions are compared with experimental data from animal models. The numerical results confirm the experimental results with the growth rates in vivo

A Fast Parallel Algorithm for Delay Partial Differential Equations Modeling the Cell Cycle in Cell Lines Derived from Human Tumors

We present a fast numerical algorithm for solving delay partial differential equations that model the growth of human tumor cells. The undetermined model parameters need to be estimated according to experimental data and it is desired to shorten the computational time needed in estimating them. To speed up the computations, we present an algorithm invoking parallelization designed for arbitrary numbers of available processors. The presented numerical results demonstrate the efficiency of the algorithm

Monotone iterative method for Caratheodory solutions of differential-functional equations

The paper deals with initial problem for  differential-functional equations. The sufficient conditions for the existence of some monotone function sequences are given.The error extimation of the approximate solutions is given. The Caratheodory solutions of the  differential-functional equations are considered. The differential inequalities technique is applied

Numerical Algorithm for the Growth of Human Tumor Cells and Their Responses to Therapy

We investigate a system of delay partial differential equations that models the growth of human tumor cells and their responses to therapy. The model includes unknown parameters that need to be estimated according to experimental data. We introduce a numerical algorithm, which shortens the computational time for solving the model equations and estimating their parameters. Numerical results demonstrate the efficiency of our algorithm and show correspondence between predicted and experimental data

Propagation of errors in dynamic iterative schemes

summary:We consider iterative schemes applied to systems of linear ordinary differential equations and investigate their convergence in terms of magnitudes of the coefficients given in the systems. We address the question of whether the reordering of equations in a given system improves the convergence of an iterative scheme

Numerical Versus Experimental Data for Prostate Tumour Growth

The goal of this paper is to solve mathematical model equations on solid tumour growth and compute their parameter values by applying growth rates of prostate cancer cell lines in vivo. For these computations, we investigate previously developed C3(1)/Tag transgenic models of prostate cancer. To make the computations fast, we have constructed an algorithm, which is based on small amounts of spatial grid-points and obtained a correspondence between the in vivo growth of tumours and the solutions of the model equations

Numerical Experiments with Model Equations of Cancer Invasion of Tissue

In this paper we investigate a mathematical model of cancer invasion of tissue, which incorporates haptotaxis, chemotaxis, proliferation and degradation rates for cancer cells and the extracellular matrix, kinetics of urokinase receptor, and urokinase plasminogen activator cycle. We solve the model using spectrally accurate approximations and compare its numerical solutions with laboratory data. The spectral accuracy allows to use low-dimensional matrices and vectors, which speeds up the computations of the numerical solutions and thus to estimate the parameter values for the model equations. Our numerical results demonstrate correlations between numerical data computed from the mathematical model and in vivo tumour growth rates from prostate cell lines

Parallel Computations and Numerical Simulations for Nonlinear Systems of Volterra Integro-Differential Equations

We investigate thalamo-cortical systems that are modeled by nonlinear Volterra integro-differential equations of convolution type. We divide the systems into smaller subsystems in such a way that each of them is solved separately by a processor working independently of other processors results of which are shared only once in the process of computations. We solve the subsystems concurrently in a parallel computing environment and present results of numerical experiments, which show savings in the run time and therefore efficiency of our approach. For our numerical simulations, we apply different numbers np of processors and each case shows that the run time decreases with increasing np. The optimal speed-up is obtained with np=N, where N is the (moderate) number of equations in the thalamo-cortical model