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Pilhwa Lee, University of Michigan
Charles Wolgemuth, University of Arizona
An immersed boundary method for two-phase fluids and gels and the swimming of Caenorhabditis elegans through viscoelastic fluids
The swimming of microorganisms typically involves the undulation or rotation of thin, filamentary objects in a fluid or other medium. While swimming in Newtonian fluids has been examined extensively, only recently have investigations into microorganism swimming through non-Newtonian fluids and gels been explored. The equations that govern these more complex media are often nonlinear and require computational algorithms to study moderate to large amplitude motions of the swimmer. Here we develop an immersed boundary method for handling fluid-structure interactions in a general two-phase medium, where one phase is a Newtonian fluid and the other phase is viscoelastic (e.g., a polymer melt or network). We use this algorithm to investigate the swimming of an undulating, filamentary swimmer in 2D (i.e., a sheet). A novel aspect of our method is that it allows one to specify how forces produced by the swimmer are distributed between the two phases of the fluid. The algorithm is validated by comparison to theoretical predictions for small amplitude swimming in gels and viscoelastic fluids. We show how the swimming velocity depends on material parameters of the fluid and the interaction between the fluid and swimmer. In addition, we simulate the swimming of Caenorhabditis elegans in viscoelastic fluids and find good agreement between the swimming speeds and fluid flows in our simulations and previous experimental measurements. These results suggest that our methodology provides an accurate means for exploring the physics of swimming through non-Newtonian fluids and gels.
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Cheryl Zapata-Allegro, University of Utah
Aaron L. Fogelson, University of Utah
Dynamic Fibrin Gelation Under Flow
Blood clots are composed of fibrin fibers and platelets. Fibrin fibers provide structural support and prevent emboli. The conditions during fibrin polymerization determine the structure of fibrin fibers. Static experiments have shown that fibrin polymerized in low thrombin concentration produce thick fibers with few branches. Conversely, fibrin polymerized in high thrombin concentration produce thin fibers with many branches. A mathematical model is created to understand fibrin gel formation under flow. The model describes the process of fibrin fiber formation beginning with an injury and a simplified coagulation model. Fibrin monomers are produced as a result of enzymatic reactions in which thrombin acts on fibrinogen. Fibrin monomers polymerize to form both linear and branching oligomers. Oligomers continue to polymerize to create a fibrin gel. The fibrin gel continues to develop because of reactions with oligomers. The growing fibrin gel forms a porous material that influences the local fluid motion and therefore affects the continued development of the gel.
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Brian Hong, University of Arizona
Timothy Secomb, University of Arizona
Michael Moulton, University of Nebraska Medical Center
Modeling the dynamics of the left ventricle: a low order approach
We discuss a low order model which describes the contraction of the left ventricle (LV). Using three parameters, the model describes the three essential modes of cardiac motion: contraction, elongation, and torsion. We compute the dynamics of the LV by incorporating three types of stress into the myocardium: active, elastic, and viscous. The active stress results from muscle fiber contraction during systole, which drives the overall periodic motion of the LV. The passive stresses constrain the motion of the LV in a manner consistent with the structure of the myocardium. A lumped parameter model for blood flow is coupled to the low order LV model to generate a system of ordinary differential equations. The resulting solution provides detailed results in terms of the principle attributes of cardiac function such as ejection fraction, flow rate, and relative pressures.
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Calina Copos, University of California, Davis
Robert Guy, University of California, Davis
Juan Carlos del Alamo, University of California, San Diego
Computational methods to study the coordination of mechanical forces involved in amoeboid cell migration
We present a computational model to study the interplay of cellular mechanics, substrate mechanics and cell-matrix interaction and the resulting migration. Our mathematical framework considers a porous viscoelastic cytoplasm, adhesion dynamics, and substrate mechanics. Our model introduces a novel way of simulating a viscoelastic deforming network. Using our methodology we present insight into the 3D cell-substrate forces for cells migrating on flat substrates.
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James Martindale, University of Nevada - Reno
Henry Fu, University of Nevada - Reno
A comparison of numerical methods in helical flagella-induced microscale flows and an application to microscale pumping
We are developing numerical tools to help design and create microscale pumps using magnetically actuated rotating bacterial flagella attached to a planar substrate. Since the interaction between flagella may play a key role in the function of the pump, the model must be accurate at both large and small scales, and also for a range of flagellar geometries to account for possible changes in polymorphic forms during experimentation.
There are several popular numerical methods, each with various advantages and disadvantages in terms of accuracy and computational cost, used to model the dynamics of cilia- and flagella-induced flows in swimming and pumping for both biological and artificial systems. We compare the accuracy and cost of slender body theory and the method of regularized Stokeslets distributed both on the surface and centerline of the flagellum. By examining the pumping due to a helical flagellum attached to a no-slip plane as well as the swimming of a bacterium due to a single helical flagellum for a range of geometries specified by total length, helical radius, helical pitch, and filament thickness, we ascertain which methods provide the optimal balance between accuracy and computational efficiency, not only for the various polymorphic forms of bacterial flagella but across a range of biological and artificial scenarios.
Based on the above results, we apply the method of regularized Stokeslets using a centerline distribution to obtain the flux through a vertical plane due to a tilted bacterial flagellum rotating above a no-slip plane in its various polymorphic configurations. In order to more accurately model the experimental parameters, current work is under way to find an analytical representation of the flux due to this flagellum through a rectangular microchannel.
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Nicholas Battista, University of North Carolina at Chapel Hill
Laura Miller, University of North Carolina at Chapel Hill
Hematocrit and Trabeculation: It takes two to tango, or something like that
Proper cardiogenesis requires a delicate balance between genetic and environmental (epigenetic) signals, and mechanical forces. Hemodynamic processes, such as vortex formation, are important in the generation of shear at the endothelial surface layer and strains at the epithelial layer, which aid in proper morphology and functionality. Hematocrit first appears in embryonic zebrafish hearts around 25 hpf, while ventricular trabeculae form around 72 hpf, for Womersley number (Wo) on the order of 0.1. Effects of hematocrit and trabeculation in this flow regime is not well understood. In this study, computational fluid dynamics is used to quantify the effects of Wo, idealized trabeculae morphology, and hematocrit on intracardial flows of embryonic zebrafish hearts.
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Ben Fogelson, Courant Institute
Alex Mogilner, Courant Institute
Force scaling behavior of actomyosin stress fibers
Actin stress fibers are important mechanical elements of many cells, but for typical cells the geometry and interconnectedness of the cytoskeleton makes it hard to get a quantitative understanding of how an individual stress fiber behaves and, in particular, how much force it generates. Growing cells on a micropatterned substrate constrains their shape and can simplify the stress fiber geometry. By using data from cells grown on such a substrate, we construct a simple 1-D model of actomyosin force production to explain a puzzling peak in force production at intermediate stress fiber lengths. As an additional result, we present at least one cute baby picture of Aaron Fogelson.
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Alexander Hoover, University of North Carolina at Chapel Hill
Boyce E. Griffith, University of North Carolina at Chapel Hill
Laura Miller, University of North Carolina at Chapel Hill
An active tension model for jellyfish propulsion and turning
Jellyfish represent one of the earliest and simplest examples of swimming by a macroscopic organism. Through a process of elastic deformation and recoil, jellyfish propulsion is generated via the coordinated contraction of its elastic bell by its coronal swimming muscles and a complementary re-expansion that is passively driven by stored elastic energy. With this in mind, we present a model constructed a hybrid immersed boundary/finite element framework that examines the mechanics of swimming by incorporating material models that are informed by the musculature present in jellyfish and the bell's passive properties into a model of the elastic jellyfish bell in three dimensions. We will apply our model to examine onto how the underlying acephalic neuromuscular organization of their bell allows for complicated swimming behaviors, such as steering and maneuvering. Scaling and morphological properties in jellyfish propulsion will be examined.
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Hoang-Ngan Nguyen, University of California, Merced
Karin Leiderman, University of California, Merced
Computation of Brinkman Flows due to Triply-Periodic Arrays of Regularized Forces
A fast summation method of Ewald type is developed for Brinkman flow due to triply-periodic arrays of regularized forces. The flow is decomposed into two sums, one in physical space and one in reciprocal space. A class of suitable regularized forces are determined such that both sums have summands decaying fast. And an FFT based is applied to the sum in reciprocal space to speed up computation time. Initial simulation results for fluid-cilia interaction problems are also presented.
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Chris Vogl, University of Washington
Mike Miksis, Northwestern University
Steve Davis, Northwestern University
Dave Salac, University at Buffalo
The Effect of Concentration-Dependent Viscosity on Vesicle Membrane Dynamics
Lyopreservation is a potential alternative to cryopreservation for biological material storage. Unfortunately, current techniques for inducing the highly viscous state required for lyopreservation have not been successful, despite the ability of certain organisms to do this naturally. This work investigates the hypothesis that large cell membrane deformation, during the drying process, is to blame. A vesicle membrane is simulated under a flow containing a viscosity-inducing solute. The results focus on the effect of the various material parameters on membrane deformation.
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David Stein, University of California, Davis
Robert Guy, University of California, Davis
Becca Thomases, University of California, Davis
High order solutions to the Stokes Oldroyd-B equations using FFT-based spectral methods
We present a method for simulating complex fluid flow in domains with arbitrary smooth boundaries using simple FFT-based spectral methods. Dirichlet conditions for the velocity are imposed by solving Stokes' equations using a novel high order Immersed Boundary like method. Our numerical scheme automatically generates a smooth extension for the velocity field over the non-physical regions of the computational domain, allowing for straightforward coupling with the stress equation. We demonstrate high order convergence of solutions to several test problems, and explore applications to the study of elastic turbulence.
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Chuanbin Li, University of California, Davis
Robert Guy , University of California, Davis
Becca Thomases, University of California, Davis
A Constraint Approach to Simulating the Swimming Behavior of Alga Cells in Complex Flows
In this project, we model and simulate the motility behavior of the microscopic alga Chlamydomonas Reinhardtii in Stokes fluids based on the shape data taken from experiments. We set up a constrained system to determine the forces on the swimmer and its swimming velocity, using the shape of the swimmer in one complete stroke. Our approach to simulating the swimming behavior demonstrates low computational costs even in three dimensions. Simulations also yield results reasonably close to those from experiments. We can further adapt our formulation to investigating the dynamics of the alga cell swimming in viscoelastic fluids.
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Gregory Herschlag, Duke University
Jian-Guo Liu, Duke University
Anita Layton, Duke University
Material transport across channels at low Reynolds number
All living organisms of a sufficient size rely on complex systems of tubular networks to efficiently collect, transport and distribute nutrients or waste. These networks exchange material with the interstitium via embedded channels leading to effective permeabilities across the wall separating the channel interior from the interstitium. For example, in many invertebrates, respiratory systems are made of complex tracheal systems that branch out through the entire body allowing for passive exchange of oxygen and carbon dioxide. Certain of these animals utilize various pumping mechanisms that alter the flow of the air or fluid being transported. Although the net effect of the averaged rates of flow is typically well understood, it is still a largely open problem to understand how, and in what circumstances, pumping enables and enhances the exchange of material across channel walls. It has been demonstrated experimentally that when certain insects flap their wings, compression of the trachea allow for more efficient oxygen extraction, however it is unclear if this pumping is optimized for flight, oxygen uptake or neither, and understanding this problem quantitatively will shed insight on this biological process as well as potentially inspire novel material extraction techniques in engineering applications. Insect respiration typically occurs at low Reynolds number and this regime will be the focus of the presentation. In this talk, I will discuss several new results in fluid flows through channels with permeable membranes, both with static walls and simple pumping dynamics. The results with be placed in the context of a model for insect respiration, however are be applicable to a broader classification of problems. The results will then be used to examine the efficacy of pumping as a mechanism for material exchange across channel walls.
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Nguyenho Ho, Worcester Polytechnic Institute
Sarah Olson, Worcester Polytechnic Institute
Rods with Bend and Twist in a Brinkman Fluid
We develop a Lagrangian algorithm to model an elastic rod in a porous
medium. The 3-D fluid is governed by the incompressible Brinkman equation and the Kirchhoff rod model captures bend and twist of the rod.
Regularized solutions are derived and we compare numerical results to
asymptotics for swimming speeds in a Brinkman fluid.
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Tyler Skorczewski, Cornell College
Boyce E. Griffith, University of North Carolina at Chapel Hill
Aaron L. Fogelson, University of Utah
Multi-bond Models for Platelet Adhesion and Cohesion
The initial response to blood vessel injury is formation of a
platelet aggregate to seal off the damage to the
vascular wall. To form the aggregate, platelets adhere to the
vascular wall and cohere to one another. Both of these processes
involve the interplay of multiple types of receptor-ligand bonds with
different force-dependent binding kinetics. The local fluid dynamics
affects the bond dynamics by exerting shear stresses on the platelets.
We present a mesoscale stochastic binding model based on recent
experimental data about platelet receptor-ligand interactions and
incorporate it into an immersed-boundary-based platelet aggregation
model. Multiple bond types and activation of platelets in response to
binding are parts of the model. Simulation results illustrate that
the model can capture the stop-start motion of a platelet along the
vessel wall as well as the activation-dependent firm adhesion that has
been observed experimentally.
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Lingxing Yao, Case Western Reserve University
Carme Calderer, University of Minnesota
Yoichiro Mori, University of Minnesota
Ronald Siegel, University of Minnesota
Rhythmomimetic drug delivery using volume phase transition in gels
It is well known that hydrogels are made of cross linked polymer
networks that are capable of absorbing large amounts of water, they can
exist in swollen or collapsed states. Under proper physical and chemical conditions, we could observe volume phase transitions in polyelectrolyte hydrogels between those states, sometimes repeatedly. In this presentation, we present modeling, analysis and numerical simulation of a prototype glucose driven drug delivery device based on chemomechanical interactions and volume phase transitions in polyelectrolyte gels. The device consists of two fluid compartments, an external cell (I) mimicking the physiological environment, and a closed chamber (II), separated by a hydrogel membrane. We combine the theory of the elastic, electrostatic and mixing energy in the model membrane, and based on these energy principles, we will propose a mathematical model system, which is capable of explaining the oscillatory volume phase transition seen in the experiments. The predictions of the model are in good agreement with earlier experimental results obtained with a laboratory device.
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Shigeru Kuroda, Hokkaido University
Toshiyuki Nakagaki, Hokkaido University
A mathematical model for adaptive crawling locomotion
Crawling with locomotory wave is a fundamental method of biological locomotion in invertebrate including limbless and legged animals. We conducted observations of crawling locomotion in largely different conditions. In particular, it was found that centipedes can largely change their leg-density waves which represent spatio-temporal coordination pattern of ground friction. Here, we present a simple mechano-mathematical model for crawler so that it provides observed mode transition depending on the external / internal conditions.
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Longhua Zhao, Case Western Reserve University
Roberto Camassa, University of North Carolina at Chapel Hill
James D. Martindale, University of Nevada, Reno
Richard M. McLaughlin, University of North Carolina at Chapel Hill
Leandra Vicci, University of North Carolina at Chapel Hill
Fluid dynamics of flows induced by a precessing rod
Theoretical study and experiments are developed to emulate dynamics biological interest such as primary cilia in developing embryos, where primary cilia are the main agent for the embryonic forms of nutrient circulation. Experiments are performed using high viscosity silicon oil with magnetically actuated precessing rod in a table-top setup. Stereoscopic Lagrangian tracking show quantified long-time agreement with an appropriately imaged slender body theory to enforce the no-slip condition at the floor. In contrast, breaking symmetry by a bent rod creates additional flow components which destroy quantitative short time agreement with the theory while maintaining its qualitative features including the creation of large scale Lagrangian tori.
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Toshiyuki Nakagaki, Hokkaido University
Itsuki Kunita, Hokkaido University
Shigeru Kuroda, Hokkaido University
Kaito Ooki, Hokkaido University
Tatsuya Yamaguchi, Kyushu University
Atsushi Tero, Kyushu University
Capacity for learning the shape of arena in the single-celled swimmer, viewed from slow dynamics of membrane potential.
We have studied for some years the learning capacity for time and space in unicellular organisms. Here we present that protozoan ciliates, Paramecium and Tetrahymena, show the swimming trajectory adaptive to the shape of swimming arena in which they experienced just before (e.g. capillary and droplet). As the swimming activity is regulated by membrane potential in the ciliates, a possible mechanism is considered according to the equations of Hodgkin-Huxley type with the additional slower variable.
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Priscilla Elizondo, University of Utah
Aaron L. Fogelson, University of Utah
Toward a Model of Venous Thrombosis Initiation in Venous Valve Pockets
A previously validated mathematical model of intravascular platelet deposition and the tissue factor pathway of blood coagulation is modified to study the initiation of venous thrombosis, the formation of blood clots within deep veins usually in the legs. The model takes into account plasma-phase and surface-bound enzymes and zymogens, coagulation inhibitors, and activated and unactivated platelets. Reactions are initiated by the appearance of tissue factor on activated endothelial cells. Subsequent reactions take place on the endothelium, in the plasma, and on platelet surfaces. The model also allows for platelet deposition on the activated endothelial cells. A small fraction of incoming platelets is regarded as partly activated. Additional platelet activation can occur when thrombin or ADP are present. Low shear rates are considered (below 20/sec) to take into account the fact that venous thrombosis frequently is initiated near the bottom of venous valve pockets. The production of activated protein C by thrombomodulin takes place on the same endothelial cell surfaces on which tissue factor reactions occur. Fibrin is accounted for in terms of its role in buffering thrombin. Different shear rates and densities of tissue factor and thrombomodulin are examined, along with the role of chemical inhibitors. The model responds slowly and in a threshold manner to changes in TF density. The chemical inhibitor activated protein C plays a significant role in determining the tissue factor threshold at low shear rates and heparan-sulfate accelerated antithrombin and fibrin buffering both significantly influence the final thrombin production for above threshold tissue factor levels.
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Qiang Yang, Tulane University
Lisa J. Fauci, Tulane University
Macroscopic fiber motion in a polymeric fluid driven by a four roll mill
We study the dynamics of an elastic fiber in a viscoelastic fluid described by the FENE-P model. We examine the effects of Weissenberg number, fiber rigidity and maximum distention parameter on fiber motion and the evolution of polymer stress. We also look at the ability of fiber to escape closed streamlines in Newtonian fluids and viscoelastic fluids.
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Amneet Pal Singh Bhalla, University of North Carolina at Chapel Hill
Bakytzhan Kallemov, Courant Institute
Boyce E. Griffith, University of North Carolina at Chapel Hill
Aleksandar Donev, Courant Institute
An Immersed Boundary Method for Rigid Bodies
The traditional immersed boundary (IB) method is a very flexible method for coupling elastic structures to fluid flow. When rigid bodies are modeled using an IB approach, a penalty method is usually employed to approximately enforce the rigidity of the body; this requires small time step sizes and leads to difficult-to-control errors in the solution. We develop a method that exactly enforces a rigidity constraint by solving a linear system coupling a standard semi-implicit discretization of the fluid equations with a rigidity constraint. We develop a preconditioned iterative solver that combines an approximate multigrid solver for the fluid problem with an approximate direct solver for the Schur complement system. We demonstrate the efficiency and study the accuracy of the method on several test problems for both zero and finite Reynolds numbers.
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Nicholas A. Danes, University of California, Merced
Karin Leiderman, University of California, Merced
Sriram Krishnaswamy, University of Pennsylvania Cardiovascular Institute
A Mathematical Model of Factor Xa Regulation by Rivaroxaban
Rivaroxaban is an oral anticoagulant drug (blood thinner) used to prevent clots from forming. There is a black box warning stating that premature discontinuation of the drug increases the risk of clotting events. This risk, which is higher than the one prior to treatment, is not fully understood. Rivaroxaban works by reversibly binding to factor Xa, a protein involved in the biochemical reactions of coagulation. Regulation of factor Xa is crucial for the production of thrombin, the final enzyme in the coagulation pathway. Here, we consider a mathematical model of a reduced coagulation pathway including the reactions up to the production Xa, in the presence of rivaroxaban. Using kinetic parameters from the literature, we find that the competition for Xa between plasma inhibitors and rivaroxaban reduces the
negative feedback of Xa, enhancing Xa production. In addition, we find conditions under which the coagulation initiator, TF, and the plasma inhibitor, TFPI, enhance the peak concentration of Xa in the presence of rivaroxaban, compared to the case without. Finally, we find that there is always a time in which Xa concentration becomes higher in the presence of rivaroxaban, compared to the case without. These studies suggest that further investigation of the kinetics of rivaroxaban are warranted.
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