arXiv subjects

Leonard M. Sander

Publications and source records attributed to Leonard M. Sander.

41 records · Page 3Linked to original sources

arXiv2002

Growth Patterns of Microscopic Brain Tumors

Highly malignant brain tumors such as Glioblastoma Multiforme (GBM) form complex growth patterns in vitro in which invasive cells organize in tenuous branches. Here, we formulate a chemotaxis model for this sort of growth. A key element controlling the pattern is homotype attraction, i.e., the tendency for invasive cells to follow pathways previously explored. We investigate this in two ways: we show that there is an intrinsic instability in the model, which leads to branch formation. We also give a discrete description for the expansion of the invasive zone, and a continuum model for the nutrient supply. The results indicate that both, strong heterotype chemotaxis and strong homotype chemo-attraction are required for branch formation within the invasive zone. Our model thus can give a way to assess the importance of the various processes, and a way to explore and analyze transitions between different growth regimes.

physics.bio-ph↗

arXiv2002

Off-lattice noise reduction and the ultimate scaling of DLA in two dimensions

Off-lattice DLA clusters grown with different levels of noise reduction are found to be consistent with a simple fractal fixed point. Cluster shapes and their ensemble variation exhibit a dominant slowest correction to scaling, and this also accounts for the apparent ``multiscaling'' in the DLA mass distribution. We interpret the correction to scaling in terms of renormalized noise. The limiting value of this variable is strikingly small and is dominated by fluctuations in cluster shape. Earlier claims of anomalous scaling in DLA were misled by the slow approach to this small fixed point value.

cond-mat.stat-mech↗

arXiv1998

Diffusion Limited Aggregation and Iterated Conformal Maps

The creation of fractal clusters by diffusion limited aggregation (DLA) is studied by using iterated stochastic conformal maps following the method proposed recently by Hastings and Levitov. The object of interest is the function $Φ^{(n)}$ which conformally maps the exterior of the unit circle to the exterior of an $n$-particle DLA. The map $Φ^{(n)}$ is obtained from $n$ stochastic iterations of a function $ϕ$ that maps the unit circle to the unit circle with a bump. The scaling properties usually studied in the literature on DLA appear in a new light using this language. The dimension of the cluster is determined by the linear coefficient in the Laurent expansion of $Φ^{(n)}$, which asymptotically becomes a deterministic function of $n$. We find new relationships between the generalized dimensions of the harmonic measure and the scaling behavior of the Laurent coefficients.

chao-dyn↗

arXiv1998

Front Propagation: Precursors, Cutoffs and Structural Stability

We discuss the problem of fronts propagating into metastable and unstable states. We examine the time development of the leading edge, discovering a precursor which in the metastable case propagates out ahead of the front at a velocity more than double that of the front and establishes the characteristic exponential behavior of the steady-state leading edge. We also study the dependence of the velocity on the imposition of a cutoff in the reaction term. These studies shed new light on the problem of velocity selection in the case of propagation into an unstable state. We also examine how discreteness in a particle simulation acts as an effective cutoff in this case.

patt-sol↗

arXiv1996

Anomalous Dimension and Spatial Correlations in a Point-Island Model

We examine the island size distribution function and spatial correlation function of a model for island growth in the submonolayer regime in both 1 and 2 dimensions. In our model the islands do not grow in shape, and a fixed number of adatoms are added, nucleate, and are trapped at islands as they diffuse. We study the cases of various critical island sizes $i$ for nucleation as a function of initial coverage. We found anomalous scaling of the island size distribution for large $i$ . Using scaling, random walk theory, a version of mean-field theory we obtain a closed form for the spatial correlation function. Our analytic results are verified by Monte Carlo simulations.

cond-mat↗