Evgeniy Khain

Dynamics and pattern formation in invasive tumor growth

We study the in vitro dynamics of the malignant brain tumor glioblastoma multiforme. The growing tumor consists of a dense proliferating zone and an outer less dense invasive region. Experiments with different types of cells show qualitatively different behavior: one cell line invades in a spherically symmetric manner, but another gives rise to branches. We formulate a model for this sort of growth using two coupled reaction-diffusion equations for the cell and nutrient concentrations. When the ratio of the nutrient and cell diffusion coefficients exceeds some critical value, the plane propagating front becomes unstable with respect to transversal perturbations. The instability threshold and the full phase-plane diagram in the parameter space are determined. The results are in a qualitative agreement with experimental findings for the two types of cells.

Onset of thermal convection in a horizontal layer of granular gas

The Navier-Stokes granular hydrodynamics is employed for determining the threshold of thermal convection in an infinite horizontal layer of granular gas. The dependence of the convection threshold, in terms of the inelasticity of particle collisions, on the Froude and Knudsen numbers is found. A simple necessary condition for convection is formulated in terms of the Schwarzschilds criterion, well known in thermal convection of (compressible) classical fluids. The morphology of convection cells at the onset is determined. At large Froude numbers, the Froude number drops out of the problem. As the Froude number goes to zero, the convection instability turns into a recently discovered phase-separation instability.

A model for glioma growth This paper was submitted as an invited paper resulting from “Understanding Complex Systems” conference held at University of Illinois-Urbana Champaign, May 2005

Glioblastoma Multiforme (GBM) is the most invasive form of primary brain tumor. We propose a mathematical model that describes such tumor growth and allows us to describe two different mechanisms of cell invasion: diffusion (random motion) and chemotaxis (directed motion along the gradient of the chemoattractant concentration). The results are in a quantitative agreement with recent in vitro experiments. It was observed in experiments that the outer invasive zone grows faster than the inner proliferative region. We argue that this feature indicates transient behavior, and that the growth velocities tend to the same constant value for larger times. A longer-time experiment is needed to verify this hypothesis and to choose between the two basic mechanisms for tumor growth.

Fluctuations and stability in front propagation

Propagating fronts arising from bistable reaction-diffusion equations are a purely deterministic effect. Stochastic reaction-diffusion processes also show front propagation which coincides with the deterministic effect in the limit of small fluctuations (usually, large populations). However, for larger fluctuations propagation can be affected. We give an example, based on the classic spruce budworm model, where the direction of wave propagation, i.e., the relative stability of two phases, can be reversed by fluctuations.

Minimizing the population extinction risk by migration.

Many populations in nature are fragmented: they consist of local populations occupying separate patches. A local population is prone to extinction due to the shot noise of birth and death processes. A migrating population from another patch can dramatically delay the extinction. What is the optimal migration rate that minimizes the extinction risk of the whole population? Here, we answer this question for a connected network of model habitat patches with different carrying capacities.

Pattern formation of glioma cells: Effects of adhesion

We investigate clustering of malignant glioma cells. In vitro experiments in collagen gels identified a cell line that formed clusters in a region of low cell density, whereas a very similar cell line (which lacks an important mutation) did not cluster significantly. We hypothesize that the mutation affects the strength of cell-cell adhesion. We investigate this effect in a new experiment, which follows the clustering dynamics of glioma cells on a surface. We interpret our results in terms of a stochastic model and identify two mechanisms of clustering. First, there is a critical value of the strength of adhesion; above the threshold, large clusters grow from a homogeneous suspension of cells; below it, the system remains homogeneous, similarly to the ordinary phase separation. Second, when cells form a cluster, we have evidence that they increase their proliferation rate. We have successfully reproduced the experimental findings and found that both mechanisms are crucial for cluster formation and growth.

Velocity fluctuations of noisy reaction fronts propagating into a metastable state

The position of a reaction front, propagating into a metastable state, fluctuates because of the shot noise of reactions and diffusion. A recent theory (Meerson et al 2011 Phys. Rev. E 84 011147) gave a closed analytic expression for the front diffusion coefficient in the weak noise limit. Here we test this theory in stochastic simulations involving reacting and diffusing particles on a one-dimensional lattice. We also investigate a noise-induced systematic shift of the front velocity compared to the prediction from the spatially continuous deterministic reaction?diffusion equation.

Noise induces rare events in granular media

The granular Leidenfrost effect [B. Meerson, et al., Phys. Rev. Lett. 91, 024301 (2003)PRLTAO0031-900710.1103/PhysRevLett.91.024301; P. Eshuis et al., Phys. Rev. Lett. 95, 258001 (2005)PRLTAO0031-900710.1103/PhysRevLett.95.258001] is the levitation of a mass of granular matter when a wall below the grains is vibrated, giving rise to a hot granular gas below the cluster. We find by simulation that for a range of parameters the system is bistable: the levitated cluster can occasionally break and give rise to two clusters and a hot granular gas above and below. We use techniques from the theory of rare events to compute the mean transition time for breaking to occur. This requires the introduction of a two-component reaction coordinate.

Clustering and phase separation in dense shear granular flow

We investigate various regimes of steady dense Couette flow of inelastically colliding hard disks in the absence of gravity. The two governing parameters in this two-dimensional system are the inelasticity of particle collisions and the average density of particles. The simplest steady state is the uniform shear flow (USF), where the temperature and the density profiles are homogeneous over the system and the velocity of the flow changes linearly between the two moving walls. The USF becomes unstable when the inelasticity of particle collisions exceeds a certain threshold, which depends on the average density of particles. Then the USF gives a way to a plug flow regime, where a solid-like cluster coexists with one or two fluid layers. These regimes are investigated using equations of granular hydrodynamics with constitutive relations that interpolate between low and high densities. The results are tested in event-driven molecular dynamics (MD) simulations, and a good agreement is observed.

Hydrodynamics of “Thermal” Granular Convection

We employ the Navier-Stokes granular hydrodynamics for determining the threshold of “thermal” convection in a horizontal layer of fluidized granular medium [1, 2]. A recent experiment with a highly fluidized three-dimensional granular flow [3] gives strong evidence for thermal convection. In the simplest model of inelastic hard spheres, the convection sets in when the restitution co-efficient becomes smaller than a critical value [4] . When gravity goes to zero, the convection instability turns into a recently discovered phase separation insta-bility [5]. A lower bound for the convection threshold is determined using the Schwarzschild criterion of stability of classical compressible fluid.