Tools and Methods
Mathematical and computational models of biological systems often contain a large degree of uncertainty, stemming both from random or unaccounted for dynamics (aleatory uncertainty) and from unknown values of input parameters (epistemic uncertainty). Quantifying these sources of uncertainty and their effects on model predictions is critical to interpreting model results. My work in this area has primarily focused on quantifying epistemic uncertainty through parameter sensitivity and identifiability analysis. I have applied these analyses in various contexts, ranging from cell biology to within-host disease dynamics to epidemiology. For more information on these applications, see Applications below. For more on parameter sensitivity analysis, see our recent review paper on global sensitivity analysis of biological multi-scale models.
I am currently working on methodological advances in structural parameter identifiability for PDE models. Methods for structural identifiability analysis are relatively well-established for ODE models, but the same cannot be said for PDEs. Our work aims to extend differential algebra based methods for structural parameter identifiability to the PDE setting. In particular, we have considered age-structured epidemic models (manuscript in preparation). However, we plan to extend this methodology to other PDE systems such as reaction-diffusion systems. This work is in collaboration with Dr. Marisa Eisenberg at University of Michigan.
I am interested in applying machine learning and topological data analysis to agent-based biological models and biological images. I aim to use these analyses to better understand model dynamics and experimental results, and for model calibration and validation with experimental data. I have focused on the analysis of simulated and experimental tuberculosis granulomas in the lung (see Applications below).
My most recent work in the Kirschner Lab at University of Michigan focuses on combining a network modeling framework with synthetic population datasets to realistically model disease spread at the county level. In the Kirschner lab, we are currently using this model to study tuberculosis (TB) in Arkansas through collaboration with the Arkansas Department of Health. See our preprint on medRxiv for detailed information on the modeling framework using Washtenaw County, MI as a test population. We are also using this model to study COVID-19 dynamics in Washtenaw County. For more information on that project, visit my COVID-19 page.
Previously, we applied an age-structured PDE model to study TB at the national level for the United States and Cambodia. We utilized this model to evaluate and optimize age-targeted vaccination strategies for theoretical pre- and post-exposure vaccines. We found that the optimal age groups tended to be younger for Cambodia than for the US, and that post-exposure vaccination had a significantly larger effect than pre-exposure vaccination in the US.
Within-host disease dynamics
The focus of the Kirschner lab is on within-host modeling of tuberculosis infection. One of the major efforts of the Kirschner lab over the last several years has been multi-scale hybrid modeling of TB granuloma formation (for more information, see the lab webpage on GranSim). Together with graduate student Louis Joslyn and undergraduate Caleb Weissman, I applied methods for data analysis to understand and classify granulomas based on their temporal and spatial characteristics. This work applies machine learning, correlation analysis, topological data analysis, and geographic information systems to study simulated granulomas (GranSim) and experimental immunohistochemistry (IHC) images of granulomas from non-human primates.
Previously, as part of a group led by Dr. Helen Moore, I worked as part of an interdisciplinary team of mathematicians, clinicians, and immunologists to build and parameterize a model of tumor-immune interactions in multiple myeloma. We then applied a variety of methods for parameter sensitivity and identifiability analysis to identify the key components of the model which could potentially serve as future treatment targets. This work was initiated during the Women Advancing Mathematical Biology workshop at the Mathematical Biosciences Institute in Columbus, OH.
During my PhD, I worked on two projects in cell biology under the direction of my advisor Dr. Ching-Shan Chou at The Ohio State University. The first was to analyze models of stem cell lineages under different assumptions about negative feedback control. We found that control of cell division rates as well as the probability of stem cell self-renewal is necessary to keep the fractions of stem cells to differentiated cells in the total population as robust as possible to variations in cell division parameters, and to minimize the time for tissue recovery in a non-oscillatory manner. This work was in collaboration with Dr. Leili Shahriyari and Dr. Alexandra Jilkine.
The second project focused on parameter uncertainty quantification for a spatial model of yeast mating polarization. The model consisted of a large system of reaction-diffusion equations simulating various protein species on the cell membrane. With many unknown parameters, the computational cost of traditional sampling-based uncertainty quantification was prohibitive. We thus utilized polynomial surrogate models to significantly reduce the cost of derivative-based sensitivity analysis and Bayesian parameter estimation. We demonstrated this methodology on a simpler ODE model to show its accuracy and efficiency before applying it to the full PDE model. This work was in collaboration with Dr. Tau-Mu Yi and Dr. Dongbin Xiu.