Research Interests
Glioblastoma Multiforme Modeling
Glioblastoma Multiforme (GBM) is an aggressive form of brain cancer which has survival time, with treatment, of approximately 12-15 months. GBM is of interest for clinicians and mathematicians alike because it is inherently heterogeneous. This complicates both mathematically modeling tumor growth and diffusion and devising treatment strategies.
In a large collaborative effort, I recently published a paper examining the ability of the basic reaction-diffusion equation, which assumes cellular homogeneity with respect to growth and diffusion, to fit in vivo data of GBM growth and diffusion. Results from this work are shown below with actual tumor in green and simulated tumor in red:
This poor fit inspired me to investigate methods of incorporating cellular heterogeneity into my models. I, along with Tracy Stepien and Yang Kuang, investigated a both a reaction-convection-diffusion equation and a two-equation model to explain in vitro experimental data. For the reaction-convection-diffusion model, we proposed a density-dependent diffusion term in which cells in high-density areas exhibited lower motility and cells in low-density areas exhibited higher motility. We proved existence and stability of traveling wave solutions. For the two-equation model, deemed 'go-or-grow' we assumed two subpopulations: one that grows and one that moves. Fits to experimental data for both models, along with the reaction-diffusion equation are displayed below.
Most recently, with Kevin Flores and H. T. Banks, we proposed a random differential equation version of the reaction-diffusion equation for modeling GBM. This model assumes that growth and diffusion are random variables rather than static parameters. We have shown that, using synthetic data, we are able to recover the probability distributions of the growth and diffusion parameters. We hope to continue our work in the area by quantifying uncertainties in this model and fitting to experimental data. Results of the inverse problems for synthetic data are portrayed below:
Machine Learning for Biomedical Image Analysis
Cell segmentation is an important task in the biomedical field, but it can be laborious and time consuming. Machine learning has emerged as a powerful tool that can automate image segmentation tasks with accuracy. We recently proposed a convolutional neural network that traces the cell boundary, ensuring a contiguously segmented region. The video below displays the cell tracer in action. The white represents the ground truth segmentation of the cell, the green is the machine learning predicted cell trace. The 'predicted moves' shows the next 20 pixels predicted by our network and the direction choice shows the direction the tracer moves in next.
Toxicological Modeling of Daphnia magna
Daphnia magna is a species of water flea used for toxicity testing because of their sensitivity to exogenous chemicals. They are also considered a model organism for ecological theory. Toxicity tests are often performed at the individual level, however, we really want to know what the effect on an entire population is, so we use mathematics to infer population-level effects from individual-level effects.
Recently, Kevin Flores, H. T. Banks, Gerald LeBlanc, and I formulated a continuous structured population model incorporating density-independent birth, survival and fecundity rates (derived from individual-level data) multiplied with density-dependent birth, survival, and fecundity rates. We performed parameter estimation and uncertainty quantification of the model with respect to data collected in a population-level microcosm. Resulting fits are presented below.
Future work will extend this framework to animals treated with toxicological substances.
One reason toxicity tests are not performed at the population level is that counting populations of daphnids is time-consuming. We recently devised a method of collecting daphnid population counts using convolutional neural networks and grey-scale images obtained with a document scanner. We used this software to perform a toxicological assessment of pendimethalin, an herbicide, on Daphnia magna populations. Currently, we are re-training our convolutional neural network to segment and classify daphnids in color images. In the future, we plan to do video classification in order to distinguish between female and male daphnids.
Prostate Cancer Modeling
As part of my thesis, I worked with Yang Kuang to generalize an existing model combining androgen deprivation therapy with immunotherapy (in the form of dendritic cell vaccines) for treating late-stage prostate cancer. We showed that, assuming the same total dosage of dendritic cells, more frequent injections resulted in delayed treatment resistance. We analyzed the 6D system and determined a critical dendritic cell vaccine dosage that resulted in stability of the disease-free equilibrium. Applying a quasi-steady state argument, we reduced the dimensionality of the system in order to determine the global dynamics.
