Numerical computation of an Evans function for travelling waves

We demonstrate a geometrically inspired technique for computing Evans functions for the linearised operators about travelling waves. Using the examples of the F-KPP equation and a Keller-Segel model of bacterial chemotaxis, we produce an Evans function which is computable through several orders of magnitude in the spectral parameter and show how such a function can naturally be extended into the continuous spectrum. In both examples, we use this function to numerically verify the absence of eigenvalues in a large region of the right half of the spectral plane. We also include a new proof of spectral stability in the appropriate weighted space of travelling waves of speed $c \geq 2 \sqrt{\delta}$ in the F-KPP equation.Comment: 37 pages, 11 figure

Application of the control volume method to a mathematical model of cell migration

In recent years, mathematical models of cell migration have become increasingly complex. These models have evolved from simple diffusion models to computationally troublesome reaction-diffusion-advection models. As such, the use of ``black box'' numerical solvers has become less appropriate. We discuss the application of the control volume technique for resolving a complicated nonlinear cell migration model. The nonlinearity is treated using an inexact Newton solver and flux limiting ensures that the cell migration fronts are captured adequately. Specifically, we analyse the model due to Perumpanani et al. (1999), comparing the numerical results of the proposed computational model developed in this research to previous results published by other researchers. We show that the finite volume computational model captures the physics of the processes with good accuracy using coarse meshes

Splittings of generalized Baumslag-Solitar groups

We study the structure of generalized Baumslag-Solitar groups from the point of view of their (usually non-unique) splittings as fundamental groups of graphs of infinite cyclic groups. We find and characterize certain decompositions of smallest complexity (`fully reduced' decompositions) and give a simplified proof of the existence of deformations. We also prove a finiteness theorem and solve the isomorphism problem for generalized Baumslag-Solitar groups with no non-trivial integral moduli.Comment: 20 pages; hyperlinked latex. Version 2: minor change

Quiescience as a mechanism for cyclical hypoxia and acidosis

Tumour tissue characteristically experiences fluctuations in substrate supply. This unstable microenvironment drives constitutive metabolic changes within cellular populations and, ultimately, leads to a more aggressive phenotype. Previously, variations in substrate levels were assumed to occur through oscillations in the hæmodynamics of nearby and distant blood vessels. In this paper we examine an alternative hypothesis, that cycles of metabolite concentrations are also driven by cycles of cellular quiescence and proliferation. Using a mathematical modelling approach, we show that the interdependence between cell cycle and the microenvironment will induce typical cycles with the period of order hours in tumour acidity and oxygenation. As a corollary, this means that the standard assumption of metabolites entering diffusive equilibrium around the tumour is not valid; instead temporal dynamics must be considered

A reaction-diffusion model for the growth of avascular tumor

A nutrient-limited model for avascular cancer growth including cell proliferation, motility and death is presented. The model qualitatively reproduces commonly observed morphologies for primary tumors, and the simulated patterns are characterized by its gyration radius, total number of cancer cells, and number of cells on tumor periphery. These very distinct morphological patterns follow Gompertz growth curves, but exhibit different scaling laws for their surfaces. Also, the simulated tumors incorporate a spatial structure composed of a central necrotic core, an inner rim of quiescent cells and a narrow outer shell of proliferating cells in agreement with biological data. Finally, our results indicate that the competition for nutrients among normal and cancer cells may be a determinant factor in generating papillary tumor morphology.Comment: 9 pages, 6 figures, to appear in PR

Establishment of Highly Tumorigenic Human Colorectal Cancer Cell Line (CR4) with Properties of Putative Cancer Stem Cells

BACKGROUND: Colorectal cancer (CRC) has the third highest mortality rates among the US population. According to the most recent concept of carcinogenesis, human tumors are organized hierarchically, and the top of it is occupied by malignant stem cells (cancer stem cells, CSCs, or cancer-initiating cells, CICs), which possess unlimited self-renewal and tumor-initiating capacities and high resistance to conventional therapies. To reflect the complexity and diversity of human tumors and to provide clinically and physiologically relevant cancer models, large banks of characterized patient-derived low-passage cell lines, and especially CIC-enriched cell lines, are urgently needed. PRINCIPAL FINDINGS: Here we report the establishment of a novel CIC-enriched, highly tumorigenic and clonogenic colon cancer cell line, CR4, derived from liver metastasis. This stable cell line was established by combining 3D culturing and 2D culturing in stem cell media, subcloning of cells with particular morphology, co-culture with carcinoma associated fibroblasts (CAFs) and serial transplantation to NOD/SCID mice. Using RNA-Seq complete transcriptome profiling of the tumorigenic fraction of the CR4 cells in comparison to the bulk tumor cells, we have identified about 360 differentially expressed transcripts, many of which represent stemness, pluripotency and resistance to treatment. Majority of the established CR4 cells express common markers of stemness, including CD133, CD44, CD166, EpCAM, CD24 and Lgr5. Using immunocytochemical, FACS and western blot analyses, we have shown that a significant ratio of the CR4 cells express key markers of pluripotency markers, including Sox-2, Oct3/4 and c-Myc. Constitutive overactivation of ABC transporters and NF-kB and absence of tumor suppressors p53 and p21 may partially explain exceptional drug resistance of the CR4 cells. CONCLUSIONS: The highly tumorigenic and clonogenic CIC-enriched CR4 cell line may provide an important new tool to support the discovery of novel diagnostic and/or prognostic biomarkers as well as the development of more effective therapeutic strategies

Dynamic Computational Model Suggests That Cellular Citizenship Is Fundamental for Selective Tumor Apoptosis

Computational models in the field of cancer research have focused primarily on estimates of biological events based on laboratory generated data. We introduce a novel in-silico technology that takes us to the next level of prediction models and facilitates innovative solutions through the mathematical system. The model's building blocks are cells defined phenotypically as normal or tumor, with biological processes translated into equations describing the life protocols of the cells in a quantitative and stochastic manner. The essentials of communication in a society composed of normal and tumor cells are explored to reveal “protocols” for selective tumor eradication. Results consistently identify “citizenship properties” among cells that are essential for the induction of healing processes in a healthy system invaded by cancer. These properties act via inter-cellular communication protocols that can be optimized to induce tumor eradication along with system recovery. Within the computational systems, the protocols universally succeed in removing a wide variety of tumors defined by proliferation rates, initial volumes, and apoptosis resistant phenotypes; they show high adaptability for biological details and allow incorporation of population heterogeneity. These protocols work as long as at least 32% of cells obey extra-cellular commands and at least 28% of cancer cells report their deaths. This low percentage implies that the protocols are resilient to the suboptimal situations often seen in biological systems. We conclude that our in-silico model is a powerful tool to investigate, to propose, and to exercise logical anti-cancer solutions. Functional results should be confirmed in a biological system and molecular findings should be loaded into the computational model for the next level of directed experiments

A Three Species Model to Simulate Application of Hyperbaric Oxygen Therapy to Chronic Wounds

Chronic wounds are a significant socioeconomic problem for governments worldwide. Approximately 15% of people who suffer from diabetes will experience a lower-limb ulcer at some stage of their lives, and 24% of these wounds will ultimately result in amputation of the lower limb. Hyperbaric Oxygen Therapy (HBOT) has been shown to aid the healing of chronic wounds; however, the causal reasons for the improved healing remain unclear and hence current HBOT protocols remain empirical. Here we develop a three-species mathematical model of wound healing that is used to simulate the application of hyperbaric oxygen therapy in the treatment of wounds. Based on our modelling, we predict that intermittent HBOT will assist chronic wound healing while normobaric oxygen is ineffective in treating such wounds. Furthermore, treatment should continue until healing is complete, and HBOT will not stimulate healing under all circumstances, leading us to conclude that finding the right protocol for an individual patient is crucial if HBOT is to be effective. We provide constraints that depend on the model parameters for the range of HBOT protocols that will stimulate healing. More specifically, we predict that patients with a poor arterial supply of oxygen, high consumption of oxygen by the wound tissue, chronically hypoxic wounds, and/or a dysfunctional endothelial cell response to oxygen are at risk of nonresponsiveness to HBOT. The work of this paper can, in some way, highlight which patients are most likely to respond well to HBOT (for example, those with a good arterial supply), and thus has the potential to assist in improving both the success rate and hence the cost-effectiveness of this therapy

Population Receptive Field Dynamics in Human Visual Cortex

Seminal work in the early nineties revealed that the visual receptive field of neurons in cat primary visual cortex can change in location and size when artificial scotomas are applied. Recent work now suggests that these single neuron receptive field dynamics also pertain to the neuronal population receptive field (pRF) that can be measured in humans with functional magnetic resonance imaging (fMRI). To examine this further, we estimated the pRF in twelve healthy participants while masking the central portion of the visual field. We found that the pRF changes in location and size for two differently sized artificial scotomas, and that these pRF dynamics are most likely due to a combination of the neuronal receptive field position and size scatter as well as modulatory feedback signals from extrastriate visual areas