Helen M. Byrne

Heather A. Harrington, Dhagash Mehta, Helen M. Byrne et al.

Many systems in biology, physics and engineering can be described by systems of ordinary differential equation containing many parameters. When studying the dynamic behavior of these large, nonlinear systems, it is useful to identify and characterize the steady-state solutions as the model parameters vary, a technically challenging problem in a high-dimensional parameter landscape. Rather than simply determining the number and stability of steady-states at distinct points in parameter space, we decompose the parameter space into finitely many regions, the steady-state solutions being consistent within each distinct region. From a computational algebraic viewpoint, the boundary of these regions is contained in the discriminant locus. We develop global and local numerical algorithms for constructing the discriminant locus and classifying the parameter landscape. We showcase our numerical approaches by applying them to molecular and cell-network models.

John T. Nardini, Charles W. J. Pugh, Helen M. Byrne

Disease complications can alter vascular network morphology and disrupt tissue functioning. Diabetic retinopathy, for example, is a complication of types 1 and 2 diabetes mellitus that can cause blindness. Microvascular diseases are assessed by visual inspection of retinal images, but this can be challenging when diseases exhibit silent symptoms or patients cannot attend in-person meetings. We examine the performance of machine learning algorithms in detecting microvascular disease when trained on statistical and topological summaries of segmented retinal vascular images. We apply our methods to three publicly-available datasets and find that, among the 13 total descriptor vectors we consider, either a statistical Box-counting descriptor vector or a topological Flooding descriptor vector achieves the highest accuracy levels on these datasets. We then created a fourth dataset by merging several datasets: the Box-counting vector outperforms all descriptors on this dataset, including the topological Flooding vector which is sensitive to differences in the annotation styles within the combined dataset. Our work represents a first step to establishing which computational methods are most suitable for identifying microvascular disease as well as some of their current limitations. In the longer term, these methods could be incorporated into automated disease assessment tools.