Lightning Problem Presentation: Groups 1 - 5 (15 min. each)
Monday, September 9, 2013 - 9:15am - 10:30am
- Modeling Fluid Flow Induced by Bacterial Carpets
Lisa J. Fauci (Tulane University)Karin Leiderman (University of California)
Recent developments in microfluidic devices have enabled controlled studies and manipulation of fluid flows with length scales at the micron level . At this length scale, viscous forces are very important and processes such as diffusion and surface tension dominate. In many microfluidic experiments that measure chemical and biological processes, mixing of the fluid within the chamber is desirable. Turbulent mixing and pumping do not work in this microscale world where inertia is negligible. For example, two laminar streams flowing in contact with each other will not mix except by diffusion. Strategies for generating mixing at the microscale are therefore an important component of experiment design. A novel approach to microfluidic mixing introduced by N. Darnton et al  is the use of flagellated bacteria as fluidic actuators. In this approach, large numbers of bacteria are made to adhere to a substrate. The adherent bacteria on this “bacterial carpet” freely rotate their flagella, which move the fluid near them and act as microscopic propellers. Furthermore,
they can live on small amounts of simple nutrients (e.g., sugars) and can even maintain mobility for several hours without food . Since no external power source is needed, the use of bacteria can be advantageous over conventional micro- or nano-fabricated devices.
The objective of this project is to study the flow induced by the collective flagellar motion of bacterial carpets. In particular, we will model a doubly-periodic array of rotating helical flagella stuck to a surface immersed in a viscous, incompressible fluid. We will examine the transport of fluid particles above the carpet as a function of flagellar distribution and geometries. In addition, since many applications in microfluidics involve suspended macromolecules and biopolymers, we will examine the transport of particles of non-zero volume as well as fibers. Our mathematical model and computational framework will be based upon the method of regularized Stokeslets [1, 2], which
was designed to study the coupling of elastic structures with fluid at zero Reynolds number, and a more recent extension of this method to incorporate periodicity .
 R Cortez. The method of regularized stokeslets. SIAM J. Sci. Comput., 23:1204, 2001.
 R Cortez, L Fauci, and A Medovikov. The method of regularized stokeslets in three dimensions:
analysis, validation, and application to helical swimming. Phys. Fluids, 17:031504, 2005.
 N Darnton, L Turner, K Breuer, and HC Berg. Moving fluid with bacterial carpets. Biophys.
J., 86:1863–1870, 2004.
 M Kim and K Breuer. Use of bacterial carpets to enhance mixing in microfluidic systems. J.
Fluids Engr., 129:319, 2007.
 K Leiderman, EL Bouzarth, R Cortez, and AT Layton. A regularization method for the nu-
merical solution of periodic stokes flow. J. Comput. Phys., 236:187–202, 2013.
 HA Stone, AD Stroock, and A Ajdari. Engineering flows in small devices: Microfluidics toward
a lab-on-a-chip. Annu. Rev. Fluid Mech., 36:381–411, 2004.
- Anti-Cancer Drug Resistance: A Pre‐existing or Emerging Phenomenon?
Jana Gevertz (The College of New Jersey)Katarzyna Anna Rejniak (Moffitt Cancer Center)
Practically all chemotherapeutic agents and potentially many targeted therapies that are used in the clinical treatment of cancer lead to drug resistance. There is, however, no consensus on whether drug resistance is pre-existing or acquired. Pre-existing drug resistance means the cancer contains a subpopulation of drug resistant cells at the initiation of treatment, and that these cells become activated or selected for during the course of therapy. On the other hand, acquired resistance involves the tumor gradually developing drug resistance due to drug action and other factors, such as microenvironmental or metabolic conditions.
Mechanisms of drug resistance are currently being studied in cell culture. Biologists produce drug-resistant cell lines by exposing the cells to the drug, collecting the surviving cell subpopulation, and repeating this process through several passages until the remaining subpopulation of cells no longer responds to the treatment. While this is an effective way to generate a resistant cell population, this in vitro process does not reveal whether the surviving cells become more resistant to the chemotherapeutic treatment with each cell passage, or if a small population of susceptible cells was present from the beginning and
simply has overgrown the other cells during the course of the experiment.
This poses several questions. If the hypothesis of a pre-existing population of resistant cells is true, what mechanisms enable those cells to resist the drug action of the often multiple chemotherapeutic treatments that may be given to a patient sequentially or in parallel? If the hypothesis of gradual emergence of drug resistance is true, what factors contribute to the development of acquired drug resistance?
(a) Develop a hybrid discrete-continuous model of cancer response to a single drug or drug combinations based on tumor histology and immunohistology.
(b) Simulate and compare outcomes of possible resistance mechanisms, such as pre-existing populations of resistant cells; role of cancer stem cells; drug penetration of tumor tissue architecture; dynamics of drug absorption and efflux; role of irregular metabolite gradients.
(c) Determine a hierarchy of factors that contribute to tumor resistance to anti-cancer drugs.
We will use the agent-based techniques (such as cellular automata or particle-spring models) and partial differential equations to develop a hybrid model of tumor response to chemotherapy. We will employ image analysis and classification methods to analyze histology images and define model initial conditions. Computational simulations will be conducted to investigate model outcomes under various drug resistance mechanisms.
Tredan O, Galmarini CM, Patel K, Tannpck IF, Drug resistance and the solid tumor
microenvironment, J. Natl Cancer Inst, 2007, 99:1441-1454
Dean M, Fojo T, Bates S, Tumor stem cells and drug resistance, Nature Reviews Cancer, 2005, 5:275-284
Lambert G, Estevez-Salmeron L, Oh S, Liao D, Emerson BM, Tlsty TD, Austin RH, An
analogy between the evolution of drug resistance in bacterial communities and malignant tissues, Nature Reviews Cancer, 2011, 11:375-382
- Modeling Autoregulation in the Kidney
Julia C. Arciero (Sparks) (Indiana University-Purdue University)Anita Layton (Duke University)
What is autoregulation?
Autoregulation is a biological process in which an internal adaptive mechanism works to adjust (or miti-gate) an animal’s response to stimuli. For example, the autoregulation process results in the maintenance of blood flow to tissues at a certain level despite variations in blood pressure or metabolism. Autoregulation is most prominent in the kidney, the heart, and the brain, inasmuch as appropriate perfusion of these organs is essential for life, and through autoregulation the body can divert blood (and thus, oxygen) where it is most needed.
Why study the kidney?
Autoregulation of renal blood flow and glomerular fil-tration rate in the kidney is critical, since about 25% Pressure of one’s cardiac output passes through the kidney. The kidney is responsible for excreting a small (but appro-
priate) fraction of the filtrate depending on the condi-tions of the body. Impaired renal autoregulation is a symptom of and a contributing factor to the progress of diseases such as hypertension and diabetes. The frequencies of these diseases have skyrocketed among the US and overseas population in recent decades. By gaining a better understanding of renal autoregulation, in physiological and pathophysiological conditions, we might have a better idea of how to control the progres-sion of hypertension and diabetes.
What are the autoregulation mechanisms in the kidney?
The two major autoregulation mechanisms in the kidney are the tubuloglomerular feedback (TGF) and myo-genic response. TGF attempts to balance glomerular filtration with tubular reabsorptive capacity, whereas the myogenic mechanism induces vasoconstriction when blood pressure is increased. These two mechanisms
share a common effector in the afferent arteriole, which is the vessel that delivers blood to the kidney’s filter (glomerulus).
What will this team do?
We will develop a model that represents the TGF and myogenic mechanisms. The model will include mass transport along a loop of Henle (PDE) and a circuit-based representation of the afferent arteriole (algebraic equations). There are a lot of interesting things we can do with the model:
- Bifurcation analysis. TGF is a negative feedback system with a nonzero feedback delay. One can ask the question: when the system is given a transient perturbation, will it return to a steady state, or will it evolve into a sustained oscillation (i.e., limit-cycle oscillation)? How does the answer depend on model parameters (i.e., feedback gain and delay, myogenic response strength, etc.)? One can study this question by analyzing a linearized version of the model equations.
- Numerical simulations. The full nonlinear model can be run, for different sets of parameters, to validate predictions of the linear model above.
- Stochastic PDE with feedback. We can introduce stochasticity into some of the model parameters and see how the stability of the system is affected.
- We can use the model to study pathophysiological mechanisms in diabetes, including salt-handling and glomerular hyperfiltration.
What do you need to know to be part of this project?
You should be interested in the physiology, but you do not need to be an expert on the kidney. You should know how to code in matlab.
- Modeling Anti‐coagulation Therapy
Lisette de Pillis (Harvey Mudd College)Erica Graham (North Carolina State University)Ami Radunskaya (Pomona College)
Anti-coagulation therapy is often prescribed after surgery in order to prevent
the formation of dangerous clots that could cause strokes or respiratory obstructions. However, the clotting properties of blood are naturally tightly regulated by the healthy body, and these treatments can upset these innate regulatory processes. Furthermore, anti-coagulants have negative side effects such as excessive bleeding. For example, people taking anti-coagulants can experience dangerous clotting after airplane flights, despite a lack of clinical evidence that flying adversely affects clotting activity. In this project, we hope to build mathematical models to address several key questions about the administration of these anti-coagulant drugs. Some of these questions are:
- What should the dosing be: how much drug should be given, and how
- How do current monitoring practices reflect actual anti-coagulation activity (for example, how good of a proxy is the INR)?
- How do we model pressure changes that reflect airplane flight (ground
level air pressure versus the pressure in an airplane that is flying at 30,000
- How would blood vessel damage affect clotting? Can scar tissue in the
blood vessels or damage to the vascular pumps have a significant effect?
Currently, mathematical models of blood clotting exist. As a starting point, we
propose to add a PK-PD model of a common anti-cogulant, such as Warfarin or Coumadin, to exiting models of blood clotting, such as the spatial model of
Choose a common anticoagulation therapy (such as Warfarin or Coumadin). Add the associated PK/PD representing Coumadin action to the Fogelson-Leiderman spatial blood clot formation model. Heuristic optimization procedures could be used to answer the doing question. Simulations could illuminate the relationship between commonly used proxies for anit-coagulation activity and what is actually happening. Finally, model parameters can be adjusted to represent changes in the host’s environment, such as the change in pressure experienced during an airplane flight, or the changes in tissue properties that follow major surgery or trauma.
 A. J. M. dePont, J. H. Hofstra, D. R. Pik., J. C. M. Meijers and M. J. Schultz. Pharmacokinetics and pharmacodynamics of danaparoid during continuous venovenous hemofiltration: a pilot study. Critical Care 11:R102 (2007). (doi:10.1186/cc6119)
 K. Leiderman and A. Fogelson. Grow with the flow: a spatialtemporal model of platelet deposition and blood coagulation under flow. Mathematical Medicine and Biology 28: 47?84 (2011) (doi:10.1093/imammb/dqq005).
 E. Pasterkamp, C. J. Kruithof, F. J. M. Van der Meer, F. R. Rosendaal and J. P. M. Vanderschoot. A model-based algorithm for the monitoring of long-term anticoagulation therapy. Journal of Thrombosis and Haemostasis, 3:915921 (2005).
 S. Thijssen, A. Kruse, J. Raimann, V. Bhalan, N. W. Levin and P. Kotanko
. A Mathematical Model of Regional Citrate Anticoagulation in Hemodial-
ysis. Blood Purif. 29:197203 (2009). (doi: 10.1159/000245647)
 R. Vink, R. A. Kraaijenhagen, M. Lei and H. R. Büller. Individualized
duration of oral anticoagulant therapy for deep vein thrombosis based on a decision model. Journal of Thrombosis and Haemostasis, 1: 2423–2530 (2003)
 F. F. Weller. A free boundary problem modeling thrombus growth. Model
development and numerical simulation using the level set method. J. Math.
Biol., 61: 805–818 (2010) (doi:10.1007/s00285-009-0324-1).
Ami Radunskaya, (Pomona College, Claremont CA 91711)
Lisette de Pillis (Harvey Mudd College, Claremont CA 91711)
Erica Graham (North Carolina State, Raleigh NC )
- What should the dosing be: how much drug should be given, and how
- Mathematical Modeling of Evolutionary Diversification
Jasmine Foo (University of Minnesota, Twin Cities)Natalia L Komarova (University of California)
Evolutionary diversification is one of the central concepts in ecology. The mechanisms leading to diversification of species in geographic isolation are well understood. There is however another scenario of diversification (sympatric diversification) whereby speciation occurs without the geographical separation of diverging populations. While it is becoming increasingly apparent that sympatric diversification is an important source of biological diversity, its underlying mechanisms are poorly characterized.
In two recent papers [1,2], experimental results are reported where sympatric diversification was observed in several E. coli bacterial systems. The observed evolutionary dynamics were driven at least in part by a co-‐evolutionary process, in which mutations causing one type of physiology changed the ecological environment, which in turn allowed the invasion of mutations causing an alternate physiology. The parallel genetic changes underlying similar phenotypes in independently evolved lineages provided the first ever empirical evidence of adaptive diversification as a predictable evolutionary process 
In this project, we will work on creating a mathematical model that can describe the observed data. In particular, we will work on an explanation of the observed coexistence of two different types of bacteria in the same spatial location. As a reference point, we will use existing mathematical models of adaptive diversification due to frequency-‐dependent ecological interactions.
The project will involve the parts: (1) Exploration/model construction, (2) Model validation by comparing with the experimental data in [1,2], (3) Explanation of the observed process of diversification and coexistence. The work will include both analytical and numerical components.
 Herron, Matthew D., and Michael Doebeli. "Parallel evolutionary dynamics of adaptive diversification in Escherichia coli." PLoS biology 11.2 (2013): e1001490.
 Burgess, Darren J. "Evolution: Experimental evolution probes neighbourly niches." Nature Reviews Genetics (2013).