Alexander Iomin

Migration and proliferation dichotomy in tumor cell invasion

We propose a two-component reaction-transport model for the migration-proliferation dichotomy in the spreading of tumor cells. By using a continuous time random walk (CTRW), we formulate a system of the balance equations for the cancer cells of two phenotypes with random switching between cell proliferation and migration. The transport process is formulated in terms of the CTRW with an arbitrary waiting-time distribution law. Proliferation is modeled by a standard logistic growth. We apply hyperbolic scaling and Hamilton-Jacobi formalism to determine the overall rate of tumor cell invasion. In particular, we take into account both normal diffusion and anomalous transport (subdiffusion) in order to show that the standard diffusion approximation for migration leads to overestimation of the overall cancer spreading rate.

Fractional-time quantum dynamics

Application of the fractional calculus to quantum processes is presented. In particular, the quantum dynamics is considered in the framework of the fractional time Schrödinger equation (SE), which differs from the standard SE by the fractional time derivative: partial differential/partial differentialt --> partial differential(alpha)/partial differentialt(alpha). It is shown that for alpha=1/2 the fractional SE is isospectral to a comb model. An analytical expression for the Greens functions of the systems are obtained. The semiclassical limit is discussed.

Probabilistic approach to a proliferation and migration dichotomy in tumor cell invasion

The proliferation and migration dichotomy of the tumor cell invasion is examined within a two-component continuous time random walk (CTRW) model. The balance equations for the cancer cells of two phenotypes with random switching between cell proliferation and migration are derived. The transport of tumor cells is formulated in terms of the CTRW with an arbitrary waiting time distribution law, while proliferation is modeled by a logistic growth. The overall rate of tumor cell invasion for normal diffusion and subdiffusion is determined.

Fractional-time Schrödinger equation: Fractional dynamics on a comb

The physical relevance of the fractional time derivative in quantum mechanics is discussed. It is shown that the introduction of the fractional time Schrodinger equation (FTSE) in quantum mechanics by analogy with the fractional diffusion ∂∂t→∂α∂tα can lead to an essential deficiency in the quantum mechanical description, and needs special care. To shed light on this situation, a quantum comb model is introduced. It is shown that for α = 1/2, the FTSE is a particular case of the quantum comb model. This exact example shows that the FTSE is insufficient to describe a quantum process, and the appearance of the fractional time derivative by a simple change ∂∂t→∂α∂tα in the Schrodinger equation leads to the loss of most of the information about quantum dynamics.

Comb-like models for transport along spiny dendrites

We suggest a modification of a comb model to describe anomalous transport in spiny dendrites. Geometry of the comb structure consisting of a one-dimensional backbone and lateral branches makes it possible to describe anomalous diffusion, where dynamics inside fingers corresponds to spines, while the backbone describes diffusion along dendrites. The presented analysis establishes that the fractional dynamics in spiny dendrites is controlled by fractal geometry of the comb structure and fractional kinetics inside the spines. Our results show that the transport along spiny dendrites is subdiffusive and depends on the density of spines in agreement with recent experiments.

Negative superdiffusion due to inhomogeneous convection

Fractional transport of particles on a comb structure in the presence of an inhomogeneous convection flow is studied [Baskin and Iomin, Phys. Rev. Lett. 93, 120603 (2004)]. The large scale asymptotics is considered. It is shown that a contaminant spreads superdiffusively in the direction opposite to the convection flow. Conditions for the realization of this effect are discussed in detail.

Subdiffusion in the nonlinear Schrödinger equation with disorder

The nonlinear Schrödinger equation in the presence of disorder is considered. The dynamics of an initially localized wave packet is studied. A subdiffusive spreading of the wave packet is explained in the framework of a continuous time random walk. A probabilistic description of subdiffusion is suggested, and a transport exponent of subdiffusion is obtained to be 2/5.

Electrostatics in fractal geometry: Fractional calculus approach

An electric field in a composite dielectric with a fractal charge distribution is obtained in the spherical symmetry case. The method is based on the splitting of a composite volume into a fractal volume Vd ∼ rd with the fractal dimension d and a complementary host volume Vh = V3 − Vd. Integrations over these fractal volumes correspond to the convolution integrals that eventually lead to the employment of the fractional integro-differentiation.

Subdiffusion on a fractal comb.

Subdiffusion on a fractal comb is considered. A mechanism of subdiffusion with a transport exponent different from 1/2 is suggested. It is shown that the transport exponent is determined by the fractal geometry of the comb.

Superdiffusion of cancer on a comb structure

The influence of cell fission on transport properties of the vessel network is studied. A simple mathematical model is proposed by virtue of heuristic arguments on tumor development. The constructed model is a modification of a so–called comb structure. In the framework of this model we are able to show that the tumor development corresponds to fractional transport of cells. A possible answer to the question how the malignant neoplasm cells appear at an arbitrary distance from the primary tumor is proposed. The model could also be a possible mechanism for diffusive cancers.