Tracy L. Stepien

Mathematical Analysis of Glioma Growth in a Murine Model

Five immunocompetent C57BL/6-cBrd/cBrd/Cr (albino C57BL/6) mice were injected with GL261-luc2 cells, a cell line sharing characteristics of human glioblastoma multiforme (GBM). The mice were imaged using magnetic resonance (MR) at five separate time points to characterize growth and development of the tumor. After 25 days, the final tumor volumes of the mice varied from 12 mm3 to 62 mm3, even though mice were inoculated from the same tumor cell line under carefully controlled conditions. We generated hypotheses to explore large variances in final tumor size and tested them with our simple reaction-diffusion model in both a 3-dimensional (3D) finite difference method and a 2-dimensional (2D) level set method. The parameters obtained from a best-fit procedure, designed to yield simulated tumors as close as possible to the observed ones, vary by an order of magnitude between the three mice analyzed in detail. These differences may reflect morphological and biological variability in tumor growth, as well as errors in the mathematical model, perhaps from an oversimplification of the tumor dynamics or nonidentifiability of parameters. Our results generate parameters that match other experimental in vitro and in vivo measurements. Additionally, we calculate wave speed, which matches with other rat and human measurements.

Traveling Waves in a One-Dimensional Elastic Continuum Model of Cell Layer Migration with Stretch-Dependent Proliferation ∗

Collective cell migration plays a substantial role in maintaining the cohesion of epithelial cell layers and in wound healing. A number of mathematical models of this process have been developed, all of which reduce to essentially a reaction-diffusion equation with diffusion and proliferation terms that depend on material assumptions about the cell layer. In this paper we extend a one-dimensional mathematical model of cell layer migration of Mi et al. (Biophys. J., 93 (2007), pp. 3745-3752) to incorporate stretch-dependent proliferation, and show that this formulation reduces to a generalized Stefan problem for the density of the layer. We solve numerically the resulting partial differential equation system using an adaptive finite difference method and show that the solutions converge to self-similar or traveling wave solutions. We analyze self-similar solutions for cases with no prolifera- tion, and necessary and sufficient conditions for existence and uniqueness of traveling solutions for a wide range of material assumptions about the cell layer.

Tubuloglomerular Feedback-Mediated Dynamics in Three Coupled Nephrons

A model of three coupled nephrons branching from a common cortical radial artery is developed to further understand the effects of equal and unequal coupling on tubuloglomerular feedback. The integral model of Pitman et al. (2002), which describes the fluid flow up the thick ascending limb of a single, short-looped nephron of the mammalian kidney, is extended to a system of three nephrons through a model of coupling proposed by Pitman et al. (2004). Analysis of the system, verified by numerical results, indicates that stable limit-cycle oscillations emerge for sufficiently large feedback gain magnitude and time delay through a Hopf bifurcation, similar to the single nephron model, yet generally at lower values. Previous work has demonstrated that coupling induces oscillations at lower values of gain, relative to uncoupled nephrons. The current analysis extends this earlier finding by showing that asymmetric coupling among nephrons further increases the likelihood of the model nephron system being in an oscillatory state.

Traveling Waves of a Go-or-Grow Model of Glioma Growth

Glioblastoma multiforme is a deadly brain cancer in which tumor cells excessively proliferate and migrate. The first mathematical models of the spread of gliomas featured reaction-diffusion equatio...

Modeling Autoregulation of the Afferent Arteriole of the Rat Kidney

One of the key autoregulatory mechanisms that control blood flow in the kidney is the myogenic response. Subject to increased pressure, the renal afferent arteriole responds with an increase in muscle tone and a decrease in diameter. To investigate the myogenic response of an afferent arteriole segment of the rat kidney, we extend a mathematical model of an afferent arteriole cell. For each cell, we include detailed Ca2+ signaling, transmembrane transport of major ions, the kinetics of myosin light chain phosphorylation, as well as cellular contraction and wall mechanics. To model an afferent arteriole segment, a number of cell models are connected in series by gap junctions, which link the cytoplasm of neighboring cells. Blood flow through the afferent arteriole is modeled using Poiseuille flow. Simulation of an inflow pressure up-step leads to a decrease in the diameter for the proximal part of the vessel (vasoconstriction) and to an increase in proximal vessel diameter (vasodilation) for an inflow pressure down-step. Through its myogenic response, the afferent arteriole segment model yields approximately stable outflow pressure for a physiological range of inflow pressures (100–160 mmHg), consistent with experimental observations. The present model can be incorporated as a key component into models of integrated renal hemodynamic regulation.

Discovering News Frames: Exploring Text, Content, and Concepts in Online News Sources to Address Water Insecurity in the Southwest Region

The Internet is a major source of online news content. Current efforts to evaluate online news content, including text, story line and sources is limited by the use of small-scale manual techniques that are time consuming and dependent on human judgments. This article explores the use of machine learning algorithms and mathematical techniques for Internet-scale data mining and semantic discovery of news content that will enable researchers to mine, analyze and visualize large-scale datasets. This research has the potential to inform the integration and application of data mining to address real-world socio-environmental issues, including water insecurity in the Southwestern United States. This paper establishes a formal definition of framing and proposes an approach for the discovery of distinct patterns that characterize prominent frames. Our experimental evaluation shows that the proposed process is an effective and efficient semi-supervised machine learning method to inform data mining for inferring classification.