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, and Prof. Zhilin Li of NC State
University, and graduate students Xioahai Wan (now at Capital One),
Qunlei Jiang, and Oswaldo Lozoya.
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.
Past Research
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.