Research Interests

Most projects fall under the category of behavior of cells and tissues. Application areas include developmental biology, tissue engineering, and understanding of homeostasis and remodeling in tissues. Methods include continuum mechanics, mixture theory, and stochastic models. Most models are at the tissue level, but some take into account processes at the level of the individual cell or polymer. Tissues studied include cartilage, liver, blood vessels and networks, glands, and various in vitro systems. Students interested in this area of research should contact Dr. Lubkin; you should have some background in partial differential equations, fluid dynamics, and simulation. 

Branching development occurs in many organs such as the salivary gland, lung, and kidney. We have developed several alternative mechanical models of the tissue deformations of such branching development. We hope to answer many questions, including (1) Is it necessary to assume a contribution from morphogens? (2) Is active deformation of the epithelium or of the mesenchyme necessary for branching morphogenesis, or are both necessary in tandem? Are either or neither sufficient? (3) What is the mechanical role of the basal lamina separating the tissues? (4) What is the difference in mechanism which leads to the very different morphologies of lung, salivary gland, etc.? (5) How does the lumen form? This work is supported by the NSF, and is in collaboration with Prof. Yasuo Nakanishi of Osaka University, Prof. Zhilin Li of NC State University, Dr. David Warburton of Children's Hospital Los Angeles, and graduate students Xioahai Wan (now at Capital One), Qunlei Jiang, Oswaldo Lozoya, Erin Jerkins, and Megan Sawyer.

Past Research

Bacterial Patterns Lactic acid bacteria (LAB) are used in fermentation of food products, and may have an important future in preventing food poisoning through competitive exclusion of pathogens by inoculated LAB on fresh foods. We are studying several issues related to the growth, death, ecology, and metabolism of LAB: sensitivity to temperature, pH, and other factors, acid production, internal energy stores, life phases (lag, exponential, death), and interaction with other organisms. The mathematical modeling, parameter identification, and statistical analysis are leading to a much better understanding of the population dynamics of these important organisms. This work is in collaboration with Drs. Fred Breidt and Roger McFeeters of NC State, and has involved NCSU graduate students Dan Dougherty (now faculty at Michigan State), Prashant Mudgal, and Althea Smith.

A biomechanical question in cancer is how a capsule forms around a benign tumor, and why it fails to form around a malignant tumor. Capsules make diagnosis and surgical removal of tumors substantially easier, and are believed also to inhibit the growth of tumors. The existing competing theories of the mechanism of capsule formation are (1) expansion of the tumor compresses the passive normal tissue surrrounding it, (2) the tumor secretes copious amounts of fibrin, which stimulate an active wound healing response in the surrounding tissue. My two-phase model of the mechanics of tumor growth and tissue response provides a quantitative evaluation of the contribution which both phenomena provide to the formation of the capsule, applicable to the vast variety of types of cancer and types of host tissue and their associated mechanical parameter ranges.

Spectacular spatial patterns can be formed from the aggregation of microorganisms in an initially homogeneous environment, as they interact with each other and with their environment. We developed a chemotactic model of pattern formation in cultures of E. coli and Salmonella typhimurium, and a mechanical model of a non-chemotactic aggregating eukaryotic system which appears to generate pattern in vitro using only traction forces. Our simplest assumptions led to experimentally observed patterns in all of these systems under a wide range of conditions. The eukaryotic system is dependent on the cells dynamically changing the material properties of their extracellular matrix. Our full model included large-strain deformations of an anisotropic viscoelastic material, and anisotropic diffusion.

Urban sprawl is widely recognized as a widespread problem affecting quality of life, economic efficiency, and even environmental factors. Yet there is no consensus on the definition of sprawl, beyond "we know it when we see it." In particular, it is difficult to do any analysis of the phenomenon without a quantitative definition of sprawl. Our working group on quantifying sprawl, including George Hess (Forestry), Sharon Lubkin (Biomathematics), Fatih Rifki (Architecture), and several graduate students from NCSU and Duke, examined correlations between a large number of proposed measures of sprawl, and developed a combined principal-component measure, for use in studies of the causes and results of sprawl.

A persistent problem in cancer chemotherapy is the similar toxicity levels of the agents in tumor and healthy tissue. As a result, many potential cures are missed because sufficient drug levels are not achieved. A patented two-step process involving monoclonal antibody-enzyme conjugates followed by relatively high doses of a nontoxic prodrug appears to succeed at localizing high doses of the toxic drug in the tumor while minimizing toxicity in the blood, and has had dramatic results in animal models. The modeling and analysis of this promising two-step process was joint work with Prof. J.D. Murray of the University of Washington, and graduate student Trachette Jackson (now faculty at U of Michigan), in collaboration with the research group of Dr. Peter Senter at the Bristol-Myers Squibb Pharmaceutical Research Institute in Seattle, who developed the process in nude mouse models.

In 25+ years' psychophysiological study of 4500+ married couples, Prof. John Gottman and his colleagues in the Department of Psychology at the University of Washington have determined astonishingly accurate criteria predicting the success or failure of marriages. We collaborated on his 5-year NIMH study which uses our mathematical framework to help determine appropriate therapeutic interventions to change the phase portrait of a distressed marriage. The specific aim is to determine the Minimal Marital Therapy necessary to cross the bifurcation between the stable and unstable systems.

Feline Leukemia Virus (FeLV) is a slow retroviral autoimmune disease of cats which is transmitted primarily by allogrooming (shared saliva). In collaboration with Joseph Romatowski, DVM, of the Cat Clinic of NE Seattle, we developed an SIRS model of the endemic epidemiology of FeLV. Analysis and parameter estimation allowed us to make estimates of the potential impact of two control measures, vaccination and euthanasia. We concluded that in the low-density free-roaming population, where the incidence is low, very low levels of intervention are needed to control the disease, but in the high-risk subpopulation of cats in high-density living arrangements, no amount of intervention can control the disease once it has taken hold. Thus the high-risk population acts as a reservoir for infection of the low-risk population.

Twining vines climb supports by means of a flagellar motion called circumnutation. In Darwin's books on the subject it was observed that most vines twine only to the right and the rest to the left. Inspired by this chiral symmetry-breaking we studied what structural features could lead to plausible chiral symmetry-breaking in common biological models, including reaction-diffusion systems and coupled oscillators. Our 1992 results are considered by some to be the first published work leading to the now burgeoning field of Brownian ratchets.