Maria Hadjinicolaou

An analytic solution for low-frequency scattering by two soft spheres

A plane wave is scattered by two small spheres of not necessarily equal radii. Low-frequency theory reduces this scattering problem to a sequence of potential problems which can be solved iteratively. It is shown that there exists exactly one bispherical coordinate system that fits the given geometry. Then R-separation is utilized to solve analytically the potential problems governing the leading two low-frequency approximations. It is shown that the Rayleigh approximation is azimuthal independent, while the first-order approximation involves the azimuthal angle explicitly. The leading two nonvanishing approximations of the normalized scattering amplitude as well as the scattering cross-section are also provided. The Rayleigh approximations for the amplitude and for the cross-section involve only a monopole term, while their next order approximations are expressed in terms of a monopole as well as a dipole term. The dipole term disappears whenever the two spheres become equal, and this observation provide...

A Mathematical Model for the Blood Plasma Flow Around Two Aggregated Low-Density Lipoproteins

The rheological behaviour of low-density lipoprotein (LDL) particles within the blood plasma and their role in atherogenesis, as well as their ability to aggregate under certain circumstances, is the subject of many clinical tests and theoretical studies aiming at the prevention of atherosclerosis. In the present study we develop a mathematical model that describes the flow of the blood plasma around two aggregated LDLs. We consider the flow as a creeping steady incompressible axisymmetric one, while the two aggregated LDLs are described by an inverted oblate spheroid. The mathematical methods of Kelvin inversion and the semi-separation of variables are employed and analytical expressions for the stream function are derived. These expressions are expected to be useful for further model developing and screening as well as the theoretical justification and validation of laboratory results.

Translation of Two Aggregated Low-Density Lipoproteins Within Blood Plasma: A Mathematical Model

Arteriosclerosis is a disease in which the artery walls get thicker and harder. Atherosclerosis is a specific form of arteriosclerosis which allows less blood to travel through the artery and increases blood pressure. Low-density lipoproteins (LDLs) and their ability to aggregate are important in atherosclerosis. In the present study we develop a mathematical model that describes the translation of two aggregated LDSs through blood plasma. We model the two aggregated LDLs as an inverted oblate spheroid and the flow as a creeping steady incompressible axisymmetric one. The mathematical tools that we used are the Kelvin inversion and the semi-separation of variables in the spheroidal coordinate systems. The stream function is given as a series expansion of even order terms of combinations of Gegenbauer functions of angular and radial dependence. The analytical solution is expected to give insight into the study of the various chemical precipitation methods used for the precipitation of lipoproteins, as this is the first step for the measurement of their concentration within blood plasma.

Studying the correlation between the extracellular environment and the diffusion processes in tumor growth

The tumor behavior is understood as a complex dynamical system encountering many different scales. Following the principles of Jiang et.al, we also employ a multiscale model, where the environment of a tumor, at the extracellular level is described by reaction diffusion, while at the cellular level an agent based model is applied. We further extend this model by employing a health function, which describes at every time step the health state of any tumor cell. This health function takes into account the biological and biochemical micro environment. A stochastic function is applied to model the mitosis process of proliferating tumor cells.

Eigenfunction Expansions for the Stokes Flow Operators in the Inverted Oblate Coordinate System

When studying axisymmetric particle fluid flows, a scalar function, , is usually employed, which is called a stream function. It serves as a velocity potential and it can be used for the derivation of significant hydrodynamic quantities. The governing equation is a fourth-order partial differential equation; namely, , where is the Stokes irrotational operator and is the Stokes bistream operator. As it is already known, in some axisymmetric coordinate systems, such as the cylindrical, spherical, and spheroidal ones, separates variables, while in the inverted prolate spheroidal coordinate system, this equation accepts -separable solutions, as it was shown recently by the authors. Notably, the kernel space of the operator does not decompose in a similar way, since it accepts separable solutions in cylindrical and spherical system of coordinates, while semiseparates variables in the spheroidal coordinate systems and it -semiseparates variables in the inverted prolate spheroidal coordinates. In addition to these results, we show in the present work that in the inverted oblate spheroidal coordinates, the equation also -separates variables and we derive the eigenfunctions of the Stokes operator in this particular coordinate system. Furthermore, we demonstrate that the equation -semiseparates variables. Since the generalized eigenfunctions of cannot be obtained in a closed form, we present a methodology through which we can derive the complete set of the generalized eigenfunctions of in the modified inverted oblate spheroidal coordinate system.

A chemical energy approach of avascular tumor growth: multiscale modeling and qualitative results.

AbstractIn the present manuscript we propose a lattice free multiscale model for avascular tumor growth that takes into account the biochemical environment, mitosis, necrosis, cellular signaling and cellular mechanics. This model extends analogous approaches by assuming a function that incorporates the biochemical energy level of the tumor cells and a mechanism that simulates the behavior of cancer stem cells. Numerical simulations of the model are used to investigate the morphology of the tumor at the avascular phase. The obtained results show similar characteristics with those observed in clinical data in the case of the Ductal Carcinoma In Situ (DCIS) of the breast.

Virtual Class – An Appropriate Environment for Distance Learning Mathematics at an Open University

Abstract Distance learning at an Open University usually engages technology mediated teaching tools and learning environments that have been designed and developed during the last decades. Unlike the apparently governing social learning pedagogical point of view and the benefits of asynchronous e-learning platforms, the cognitive perspectives of education and the need for synchronous interaction between the tutor and the students as well as the students themselves, is decisive for the quality of attained knowledge. Open universities, traditionally, used to organize consultative group meetings aiming at the better understanding of concepts, the development of skills, the gaining of positive attitudes and finally the enhancement of knowledge. Tutors in mathematics courses face significant difficulties due to the students’ heterogeneous mathematical background and thus problems arise that may impede a lot the acquisition of knowledge. These problems must be also addressed in the group meetings and therefore the reconstruction of the pre-existing mathematical knowledge under a common basis and the acquisition of cognitive flexibility in handling mathematical concepts in different fields are two fundamental objectives of such meetings. Virtual classes provide the appropriate environment for achieving these aims, since they allow a qualitative dynamical, on-line communication among the members of a group.

Studying the blood plasma flow past a red blood cell, with the mathematical method of Kelvin's transformation

A mathematical tool, namely the Kelvin transformation, has been employed in order to derive analytical expressions for important hydrodynamic quantities, aiming to the understanding and the study of the blood plasma flow past a Red Blood Cell (RBC). These quantities are the fluid velocity, the drag force exerted on the cell and the drag coefficient. They are obtained by employing the stream function ψ which describes the Stokes flow past a fixed cell. The RBC, being a biconcave disk, has been modelled as an inverted prolate spheroid. The stream function is given as a series expansion in terms of Gegenbauer functions, which converge fast. Therefore the first term of the series suffices for the derivation of simple and ready to use expressions.