Research
My research can be divided into the following categories.
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• Brain Tumor Invasion
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• Collagen Micromechanics
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• Cell Motility
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• Clinical Applications of Cancer Modeling
Each is described in more detail below. At the bottom of this page is my modeling philosophy.
Brain Tumor Invasion
Glioblastoma is the most malignant form of brain cancer and the median survival time is one year past diagnosis. One reason these tumors are so difficult to treat is their extreme invasiveness. The mechanisms that govern invasion are not well understood, but seem to involve cell-cell adhesion, cell-matrix interactions, chemotaxis, random motility, and cell proliferation. Our collaborators implant tumor spheroid in collagen gel and track the paths of individual cells using confocal microscopy. We develop PDE and SDE models, along with image processing tools to better quantify and understand invasion.
Looking at fluctuations in the image intensity, we have developed AIRE, an algorithm for automatically estimating the radii of tumor spheroids. A Matlab GUI for this algorithm can be obtained by contacting me.
When biologists study the growth of tumor spheroids and the responsiveness to different drugs, it is frequently only a radius of invasion that is reported. However, these images contain significantly more information. We use a PDE model to break invasion into four components, random motility, directed motility, cell shedding from the surface of the core, and proliferation and quantitative comparisons are made between the model and experiment. From experiments like the one pictured below, we learned that U87WT cells are shed faster and move in a more directed fashion than the U87dEGFR cells.
By tracking individual cells, we observe a significant radial bias of the invasive cells away from the tumor spheroid. A radially biased Orstein-Uhlenbeck model captures some features of cell behavior.
Collagen-I Micromechanics
Recent work suggests that understanding cell-gel interactions is critical to understanding to understanding tumor invasion. It is also important in understanding cell motility and in the design of tissue engineering scaffolds. While continuum, viscoelastic models of collagen are valid up to small strains, at larger strains, there is considerable strain stiffening, as well as plastic deformations. To understand these phenomena, we are developing micromechanical models for collagen.
FIbeR Extraction (FIRE)
The first step in understanding collagen-I micromechanics is to understand the network architecture of the collagen-I gel itself. We have developed image processing algorithms for extracting the gel architecture from a set of 3d confocal microscope images. Below are images of the 3d collagen-I gel (left), the extracted the 3d network (middle), and a reduced, interpolated network that is used for mechanical modeling (right).
flattened 3d image extracted 3d network reduced 3d network
Finite Element Models of Collagen
Once extracted, each fiber is treated as an elastic beam that can resist stretching and bending. The cross-links are modeled as torsional springs. This model is able to both qualitatively and quantitatively reproduce experimental results, such as the strain stiffening.
Cell Motility
In collaboration with Hans Othmer and Victor Barocas, I am developing a multiphase mixture model for cell motility atop a 2d substrate. The idea is to model how actin polymerization and depolymerization, myosin contraction, and cell-matrix adhesion lead to cell velocity and see how changes in these factors can alter cell speed.
Clinical Applications of Cancer Modeling
I am working with Arkadiusz Dudek on predicting time to progression of non-small-cell lung carcinoma patients by combining information from both PET and CT scans. The staging of tumors and assessment of therapeutic response is traditionally done by measuring the volumetric change of the tumor from Computed Tomography (CT). However, tracking tumor volume alone does not provide enough information because CT does not indicate whether cells in tumor that are still viable. Positron Emission Tomography (PET) may be used to overcome this issue. Using a radiolabeled tracer, PET images indicate the glucose metabolic activity of tissue. Active tumor tissue is typically more metabolically active than healthy tissue. However, if damaged tumor tissue becomes inflamed, it will appear to be metabolically active even if the tumor cells are not viable. My goal is to investigate the use of a mathematical model for integrating both CT and PET measurements in order to make better predictions of patient prognosis.
I have also interned at Novartis for two summers where I developed two models. One was for predicting gleevec resistance in Chronic Myeloid Leukemia patients using white blood cell counts, cytogenic response and molecular response data. The other model was for predicting the time of maximum caspase expression after a murine tumor is treated with an apoptosis inhibitor.
Modeling Philosophy
It is my belief that for a mathematical model to be of scientific interest, it must either quantitatively reproduce experimental observations, or make novel predictions that can be validated by experiment. Therefore, the type of model used to describe a system must depend on both the system itself and the type of observations that can be made. When describing in vitro experiments, it is reasonable to use complex models because a great deal of information can be measured. For clinical applications, however, the system of interest is typically much more complex while at the same time, the types of measurements that can be made are restricted Thus the most useful clinical models tend to be quite simple, such as Alan Perelson’s early models of HIV, Kristin Swanson’s models of Glioblastoma invasion, and Franziska Michor’s models of Chronic Myeloid Leukemia. Models such as these are understandable by anyone with a basic understanding of ODEs or PDEs, while also providing important insight into very serious diseases.
