Vascular injury triggers two intertwined processes, platelet deposition and coagulation, and can lead to the formation of a blood clot that may grow to occlude a vessel. Formation of the clot involves complex biochemical, biophysical, and biomechanical interactions that are also dynamic and spatially-distributed, and occur on multiple spatial and temporal scales. We previously developed a spatial-temporal mathematical model of these interactions and looked at the interplay between physical factors (flow, transport to the clot, platelet distribution within the blood) and biochemical ones in determining the growth of the clot. Recently, we extend this model to include reduction of the advection and diffusion of the coagulation proteins in regions of the clot with high platelet number density. The effect of this reduction, in conjunction with limitations on fluid and platelet transport through dense regions of the clot, can be profound. Our results suggest a possible physical mechanism for limiting clot growth.

Despite receiving 25% of the cardiac output, the mammalian kidney is susceptible to hypoxia. For example, hypoxia-induced acute kidney injury (AKI) is a prevalent complication of surgical procedures including cardiac surgeries that require cadiopulmonary bypass. How does AKI happen? What is a good biomarker for hospital-acquired AKI? We use an integrative model of kidney function and oxygenation to determine the extent to which the bypass procedure may cause mismatched changes in oxygen delivery and consumption, leading to AKI. We also use that model to assess the potential of urinary oxygen tension to be a biomarker for hospital-acquired AKI.

Recent advancements in computational fluid dynamics have enabled researchers to efficiently explore problems that involve moving elastic boundaries immersed in fluids for problems such as cardiac fluid dynamics, fish swimming, and the movement of bacteria. These advances have also made modeling the interaction between a fluid and an electromechanical model of an elastic organ feasible. The tubular hearts of some ascidians and vertebrate embryos offers a relatively simple model organ for such a study. Blood is driven through the heart by either peristaltic contractions or valveless suction pumping through localized periodic contractions. Models considering only the fluid-structure interaction aspects of these hearts are insufficient to resolve the actual pumping mechanism. The electromechanical model presented here will integrate feedback between the conduction of action potentials, the contraction of muscles, the movement of tissues, and the resulting fluid motion.

Myosin contraction and adhesion drag deform viscoelastic actin mesh in motile cells. Deformation of this mesh is crucial for pulling the cell body forward and constraining the cell sides. I will discuss the free boundary problem of cell mechanics and show how simulations of the mathematical model of the motile cell reproduce a number of modes of motility.

Multicellular organisms require groups of cells to function together as a unit. A common scenario involves the collective movement of cells. For example, when your skin gets cut, one of the first processes is re-epithelialization where epidermal cells crawl over the wounded region. Likewise, in cancer, tumor cells often move as a group to detach from the primary tumor and invade distal regions of the body. In this presentation, I will describe the work that we have been doing to develop a multiscale model for collective cell migration. This model is based in the fundamental biophysics of a single cell. We show that a combination of directed cell motility, dipole-distributed forces, and adhesion to neighboring cells and the environment is sufficient to explain in vitro wound healing dynamics and gives insight into the biophysical changes that occur when cancer cells become metastatic. This model provides testable predictions, such as that the rate of wound healing should be proportional to the contractile stress that the cells exert on their surroundings. I will conclude with some exciting new experimental results that confirm some of the model predictions.

Blood is a concentrated suspension of deformable cells, mainly red blood cells (RBCs), and does not possess a unique well-defined viscosity. The resistance to blood flow in narrow tubes is conveniently expressed in terms of the apparent viscosity, i.e., the viscosity of a Newtonian fluid that would give the same flow rate for the same pressure gradient. The apparent viscosity of blood flowing through narrow glass tubes decreases strongly with decreasing tube diameter over the range from 300 $\mu$m to 10 $\mu$m, a phenomenon known as the Fåhræus-Lindqvist effect. The main cause of this phenomenon is the presence of a cell-free or cell-depleted layer near the vessel wall. In capillaries with diameters up to about 8 $\mu$m, RBCs frequently flow in single file with narrow bullet-like shapes, and their mechanics can be analyzed by assuming axisymmetric geometry and using lubrication theory to describe the flow in the space between the cell and the vessel wall. In larger microvessels, the width of the cell-free layer is determined by two opposing effects. Individual flexible particles tend to migrate away from solid boundaries, but particle-particle interactions result in 'shear-induced diffusion' down the concentration gradient toward the wall. We used a two-dimensional model to simulate the motion of a single deformable RBC in a linear shear flow adjacent to a solid wall, subject to a lateral force that represents the effect of shear-induced diffusion. With increasing lateral force, we found a transition from tumbling to steady tank-treading behavior, with a hydrodynamic lift force generated as a result of the positive orientation of the long axis of the cell with respect to the wall. Under normal physiological hematocrit levels, the lateral force on cells at the edge of the central RBC column resulting from interactions with other cells is predicted to be sufficient to lead to tank-treading rather than tumbling, thereby stabilizing the cell-free layer.

The maintenance and reorganization of neuronal connections in the brain are fundamental questions whose understanding is being revolutionized through the development of new experimental and computational techniques. The branches of neuronal axons often terminate along dendrites by forming synapses onto micron-sized structures known as dendritic spines. These spines are characterized by their thin necks and bulbous heads, a geometry which is thought to allow them to function as separate biochemical compartments despite sharing a common dendritic shaft. In this way, one spine might be associated with a strong synaptic connection while a neighboring spine may be associated with a weak connection, and these differences can persist for long times. Membrane receptors responsible for sensing the neurotransmitters released into the synaptic cleft are critical constituents of the post-synaptic densities on spines. These membrane receptors are actively trafficked into spines by vesicles that can be larger than the spine necks. Such vesicles must deform significantly to squeeze into the bulbous heads of the spines. However, the precise mechanisms of this process, including quantitative estimates of the force and energy required, are still unknown. In this talk, I will describe our attempts to use three-dimensional immersed boundary method simulations to realistically capture the fluid dynamics of vesicle transport into spines. The spines are represented in our simulations as triangulated surfaces obtained by stitching together a cylinder with a sphere. We vary the applied force and neck geometry to identify the region in phase space in which the vesicle can squeeze into the spine. These results are compared to pass-stuck diagrams computed previously in the case of vesicles squeezing through narrow open channels. The resulting estimates for the force are found to be consistent with the physiological density of motor proteins. Resolving the thin lubricating layers between the vesicles and spine poses significant numerical challenges, and we have begun using lubrication theory as an alternative to fully resolving these boundary layers. This is joint work with Remy Kusters and Chris Rycroft.

The study of shock waves propagating in biological tissue and bone has several biomedical applications. In lithotripsy, focused shock waves are used to pulverize kidney stones without surgery, while in Extracorporeal Shock Wave Therapy (ESWT), focused shock waves of smaller amplitude are used to stimulate healing and bone growth. On the negative side, blast-induced traumatic brain injury (TBI) affects countless veterans and civilians who have survived nearby explosions. In this talk I will describe some of these applications and efforts to obtain a better understanding of the affect of wave propagation on biological media. High-resolution wave propagation algorithms can robustly handle interfaces between different materials, while methods recently developed for poroelasticity may be valuable in studying wave propagation in bone.

The immersed boundary (IB) method is a general approach to modeling systems in which a structure is immersed in and interacts with a viscous incompressible fluid. A key feature of this approach is that it uses an Eulerian description of the momentum, viscosity, and incompressibility of the coupled fluid-structure system and a Lagrangian description of the structural deformation and force generation. Integral equations couple the Eulerian and Lagrangian frames. The "conventional" IB method treats the particular case in which the structure is described using systems of elastic fibers immersed in fluid. However, the IB method is not restricted to structures described by fiber-based elasticity models. This talk will describe ongoing work to develop extensions of the IB method that enable the effective use of more general material models, including elastic bodies described by finite-strain constitutive models and immersed rigid bodies, as well as work to develop efficient solution strategies for such formulations.

We will discuss a Stokesian fluid model that incorporates forces due to elastic structures in the fluid environment of the actuated flagellum. We will present recent computational investigations of hyperactivated sperm detachment from oviductal epithelium as well as swimming through viscoelastic networks.

The fluid media surrounding many microorganisms are often mixtures of multiple materials with very different physical properties. The composition and rheology of the mixture may strongly affect the related locomotive behaviors. We study the classical Taylor’s swimming sheet problem within a two-fluid model, which describes a mixture of a viscous fluid solvent and a viscoelastic polymer network. Our results indicate that depending on the interactions between the swimming surface and the network, elasticity may have drastically different effects on the swimming speed.

Microswimmers or microrobots have recently received much attention due to their possible applications in microscale sensing and actuation, including many biomedical applications such as drug delivery, in vivo diagnostics, and microsurgery. One class of these are rigid microswimmers that can be propelled through bulk fluid when rotated by an external magnetic field. I describe an analysis of the rotational dynamics and stability of magnetic microscale objects in a rotating magnetic field. We are using this analysis to understand a number of features of these types of microswimmers observed experimentally. First, I describe how to understand the minimal geometries needed for effective propulsion. Second, I discuss the tumbling, wobbling, and propulsion of helical microswimmers reported in the literature. Third, I analyze bistability -- how for a given set of experimental conditions, specified by the strength and rotation rate of the magnetic field, more than one steady rotational orbit can be stable. I discuss the implications of bistability and its connections to observations for both achiral and helical microswimmers.

Sperm have been observed to form sperm trains and self organize into vortices. What is the relative role of biochemistry, flagellar waveform, and hydrodynamics in these interactions? In this talk, we will present recent computational results on attraction, synchronization, and the formation of sperm trains in 2-d and 3-d using the method of regularized Stokeslets. This corresponds to studying finite length sheets in 2-d and finite filaments in 3-d, leading to different behaviors and flow fields. These results will be studied for planar waveforms that are symmetrical and asymmetrical (varying amplitude).

In this talk we present two different strategies for the numerical discretization of an embedded two-dimensional manifold: high-order continuous Galerkin spectral/hp elements and Radial Basis Function (RBF)-Finite Differences (FD). The first of these methods is designed to provide high-order solutions (both in terms of function and geometric fidelity) based upon a spectral element tessellation of the surface. The latter relaxes the need for a tessellation, and allows for high-order solutions given scattered node representations of the surface. We will demonstrate the use of these methods in bioengineering applications — specifically electrochemical propagation across a human left atrium and surface chemical modeling over platelet-like objects.

We consider the numerical description of the coupled Stokes-Darcy flow, i.e., of flow where one part of the domain is governed by a Stokes flow, and the other corresponds to porous media flow, along with coupling conditions on the interface. We propose the use of a constraint preconditioner for this problem. We provide spectral bounds for the preconditioned problems, which are independent of the size of the finite element mesh. We use both standard (continuous) finite elements for both flows, and also consider the case where the porous media flow is modeled with Discontinuous Galerkin methods. Numerical experiments in two and three dimensions, illustrate our results, and comparisons with other saddle-point preconditioners found in the literature demonstrate the advantage of our approach.

We propose a new projection method based on radial basis functions (RBFs) for discretizing the incompressible unsteady Stokes equations. The novelty of the method comes from the application of a new technique for computing the Helmholtz-Hodge decomposition of a vector field, i.e., the decomposition into a sum of divergence-free and curl-free fields, which forms the foundation for projection methods. This approach only requires samples of the field at (possibly scattered) nodes over the domain and allows one to impose boundary conditions on the vector field, not its potentials, distinguishing it from many current methods. We apply this decomposition to the incompressible Stokes equations in such a way as to avoid any time-splitting errors. Numerical results will be presented demonstrating high-order convergence in both space (between 5th and 6th order) and time (up to 4th order) for some model problems in irregular geometries.

A new field of biomedical research, the structure and biomechanics of hemostasis and thrombosis, has been quickly developing over the past few years. The structure and mechanical properties of blood clots are essential in vivo for their ability to stop bleeding in flowing blood but also determine the likelihood of obstructive thrombi that cause heart attacks and strokes. Despite such critical importance, the structural basis of clot formation and mechanics is not well understood. This talk will focus on recent biophysical research on both platelet aggregation and fibrin polymerization and mechanics. An optical trap system has been developed to study protein-protein binding/unbinding at the single molecule level, and used to characterize fibrinogen-integrin interactions that are responsible for platelet aggregation and are relevant to the behavior of platelets in flowing blood. Furthermore, optical trap studies of interactions of individual fibrin molecules in clots demonstrate the existence of catch bonds. Experimental investigation of the forced elongation and compression of fibrin provide important qualitative and quantitative characteristics of the molecular mechanisms underlying fibrin mechanical properties at the microscopic and macroscopic scales. Contraction of blood clots via platelets and fibrin results in redistribution of platelets and fibrin to the exterior and compressed red blood cells in the interior forming a tessellated array of polyhedra. Through research on the structure and mechanical behavior of clots at the macroscopic, network, fiber and molecular levels, we know that their remarkable extensibility and compressibility can only be understood by integration of their material properties at all these levels. The goal of this talk is to introduce these experimental results to encourage collaborations with mathematicians to model and understand these phenomena.

We present a stochastic multiscale model of fibrinolysis, the enzymatic degradation of the fibrin fibers that stabilize blood clots. Detailed biochemistry is considered in a microscale model of a single fibrin fiber. Data from the microscale model are used in a macroscale model of a full 3-dimensional fibrin clot. We present turbidimetric and scanning electron microscopy data that validate the model over a range of experimental conditions and we discuss the clinical implications of model results.

In this talk, recent advances in the study of the dynamics of fibrin clot formation will be described. In particular, I will derive and discuss features of a new partial differential equation model that describes the growth of fibrin clots as a polymerization/gelation reaction. The solution of this PDE model gives insight into the branching structure of clots that are formed under various physiological conditions.