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Über dieses Buch

This IMA Volume in Mathematics and its Applications COMPUTATIONAL MODELING IN BIOLOGICAL FLUID DYNAMICS is based on the proceedings of a very successful workshop with the same title. The workshop was an integral part of the September 1998 to June 1999 IMA program on "MATHEMATICS IN BIOLOGY." I would like to thank the organizing committee: Lisa J. Fauci of Tulane University and Shay Gueron of Technion - Israel Institute of Technology for their excellent work as organizers of the meeting and for editing the proceedings. I also take this opportunity to thank the National Science Founda­ tion (NSF), whose financial support of the IMA made the Mathematics in Biology program possible. Willard Miller, Jr., Professor and Director Institute for Mathematics and its Applications University of Minnesota 400 Lind Hall, 207 Church St. SE Minneapolis, MN 55455-0436 612-624-6066, FAX 612-626-7370 World Wide Web: v PREFACE A unifying theme in biological fluid dynamics is the interaction of moving, elastic boundaries with a surrounding fluid. A complex dynami­ cal system describes the motion of red blood cells through the circulatory system, the movement of spermatazoa in the reproductive tract, cilia of microorganisms, or a heart pumping blood. The revolution in computa­ tional technology has allowed tremendous progress in the study of these previously intractable fluid-structure interaction problems.



Fluid Mechanics of Ciliary Propulsion

Cilia have many functions in the animal kingdom, some of these being cleansing, feeding, excretion, locomotion and reproduction. They occur in all phyla of the animal kingdom with the possible exception of the class Nematoda. This lecture will discuss the development of fluid mechanical models and theories that help with our understanding and interpretation of locomotion of protozoa, mucous transport in the lung, filter feeding in bivalve molluscs and gamete transport.
The theoretical development for the fluid mechanics requires obtaining the fundamental singularities and image systems pertinent to the system under study, the physical interpretation of them and their constructive use to model the flow fields generated by fields of cilia.
These theories allow estimates of the flow fields in the cilia sublayer and for a greater understanding of propulsive mechanisms in both micro-organisms and mucous transport in the lower respiratory tract. The sophisticated models in turn allow us to develop better approximations for simplified models that provide an improved understanding of more complicated flows involving filter feeding in bivalve molluscs and with ovum transport in the oviduct.
Finally, studies of possible filter feeding strategies in the sessile organism, Vorticella, which alters the length of its stalk periodically, has led to the development of some interesting non-linear mathematics in a simplified ‘blinking stokeslet’ model of this filter feeding phenomenon. We shall demonstrate that this can lead to chaotic dynamics, which has been shown to enhance mixing and hence improve the efficiency of feeding currents. The continuous system is reduced to an area-preserving map, which allows for greater analytical progress to be made in this inertia-free system. Poincaré sections and Lyapunov exponents are used alongside other chaotic measures to determine the nature and extent of the chaos. Effects of molecular diffusion are mimicked via the incorporation of white noise in the map and enhanced feeding levels are predicted.
The author of this paper acknowledges, with gratitude, the enormous influence that Sir James Lighthill has had on his life and academic career. This paper is dedicated to his memory and is based on research work conducted by the author of this review over the last 30 years with the material in this article being taken from papers over this period.
John Blake

The Role of Cyclic Nucleotide Pathways and Calmodulin in Ciliary Stimulation

Cilia are tiny hairlike protrusions enveloped by a membrane contiguous with the cell membrane, which beat in a cooperative pseudo-periodic, spatial and temporal pattern called the metachronal wave. They are thin (0.25–0.3 µm), relatively long (6–50 µm) and densely packed on the cell surface (100–200 cilia per cell). In the mucociliary system, their primary function is transport of a mucus layer over the cell together with various objects that may be trapped in this layer. Highly cooperative beating of cilia at high frequencies enables the mucociliary system to carry relatively large objects, at remarkable velocities. Furthermore, high frequency of ciliary beating results in increased energy expenditure. Therefore, under normal conditions, cilia beat with either low frequency or may even be at rest. However, they can dramatically change their activity in response to a variety of receptor-mediated stimuli. For example, ciliary cells possess purinergic P1 and P2 [7], [12], [32], [33], cholinergic [1], [6], [24], adrenergic [19], [31], [34] receptors.
Alex Braiman, Natalya Uzlaner, Zvi Priel

A Numerical Method for Simulating Fast-Swimming Motions

A numerical method for the simulation of thin, elastic immersed boundaries in a two-dimensional fluid is introduced. The method is Lagrangian and combines the use of vortices and impulse elements (vortex dipoles). Consequently, it is best suited for applications where the Reynolds number is high. The example presented here is the motion of an undulating filament, simulating the swimming of an organism in a slightly viscous fluid.
Ricardo Cortez

A Fluid-Structure Interaction Model of Ciliary Beating

The coupled system of a viscous, incompressible fluid and a single, force-generating organism is difficult to analyze, even when the kinetics of an organism’s waveform is taken as given. In the past decades, the efforts to describe quantitatively the fluid dynamics of spermatozoa and ciliary propulsion have been very successful. Since the Reynolds number is quite small and inertial effects can be neglected, the linear Stokes flow assumption has been used to investigate the hydrodynamic consequences of flagellar undulations [3]. These investigations have been both analytical and computational. Resistive-force theory, initially developed by Gray and Hancock [12], makes use of the linear Stokes flow assumption, and constructs the flow field by means of distributions of fundamental singularities. Lighthill later improved this theory by incorporating slender body approximations [19], since the diameter of a flagellum is much smaller than its length. More detailed hydrodynamic analysis, such as refined slender body theory and boundary element methods, have produced excellent simulations of both two- and three-dimensional flagellar propulsion in an infinite fluid domain or in a domain with a fixed wall [17], [18], [16], [22]. In addition, ciliary motion, where the effect of a single plane surface at the base of the cilia is included, is studied by Gueron and Liron [14], [15], wherein they also present a thorough survey of flagellar hydrodynamics.
Robert H. Dillon, Lisa J. Fauci

Energetic Considerations of Ciliary Beating

The internal mechanism of cilia which is responsible for their beating is among the most ancient biological motors on an evolutionary scale. Ciliary beat cycles consist of two phases: the effective stroke, where the cilium moves approximately as a straight rod, and the recovery stroke, where it bends and rolls close to the surface in a mostly tangential motion.
It is commonly agreed that for efficient functioning, the effective stroke is designed to encounter strong viscous resistance and to generate thrust, whereas the recovery stroke is designed to return to starting position while avoiding viscous resistance as much as possible. Metachronal coordination between cilia, which occurs when many of them beat close to each other, is believed to be an autonomous result of the hydrodynamical interactions in the multiciliary system. Qualitatively, metachronism is understood as a way for minimizing the energy expenditure required for beating.
This paper presents a quantitative investigation of the energetic advantages of metachronism. Using a new method for computing the work by a model cilium beating in a viscous fluid we demonstrate that the energy expenditure during the effective stroke for a single cilium is approximately five times the amount of work done during the recovery stroke. Investigation of multicilia configurations shows that having neighboring cilia beat metachronally is energetically advantageous and perhaps crucial for multiciliary functioning. Finally, the model is used to approximate the number of dynein arm attachments that are likely to occur during the effective and recovery strokes of a beat cycle.
Shay Gueron, Konstantin Levit-Gurevich

Fluid Dynamics of Animal Appendages that Capture Molecules: Arthropod Olfactory Antennae

Appendages bearing arrays of hair-like structures serve important biological functions in many animals from a variety of phyla. For example, feathery gills take up oxygen, hairy olfactory antennae capture odorant molecules, filamentous suspension-feeding structures catch single-celled algae, setulose appendages create ventilatory or feeding currents, and bristly legs or wings propel little animals through the surrounding water or air. To perform any of these functions, an array of hairs must interact with the fluid around it. Therefore, in order to elucidate basic rules governing how hairy appendages work, we have been studying the fluid dynamics of arrays of cylinders. The purpose of this paper is to provide a brief overview of what mathematical and physical models have taught us about fluid motion around and through arrays of hairs, and of how those insights have helped us unravel ways in which the function of hairy appendages depends on their structure and behavior. I will focus on examples of appendages that capture molecules: the olfactory antennae of various arthropods.
M. A. R. Koehl

Cartesian Grid Methods for Fluid Flow in Complex Geometries

Biological fluid dynamics typically involves geometrically complicated structures which are often deforming in time. We give a brief overview of some approaches based on using fixed Cartesian grids instead of attempting to use a grid which conforms to the boundary. Both finite-difference and finite-volume methods are discussed, as well as a combined approach which has recently been used for computing incompressible flow using the streamfunction-vorticity formulation of the incompressible Navier-Stokes equations.
Randall J. Leveque, Donna Calhoun

Computed Simulations of Ciliary and Flagellar Motility Using the Geometric Clutch Model can Replicate a Wide Variety of Experimental Conditions

The Geometric Clutch hypothesis proposes that the strain that develops between the outer doublets of the flagellar/ciliary axoneme acts as the principal control to regulate the function of the dynein motor proteins. In this hypothetical scheme, the forces that develop transverse to the axis of the doublets (t-forces) act as a clutch to engage or disengage the dynein arms from their binding sites on adjacent doublets. These forces can be easily computed from the longitudinal tension, or compression, on a doublet and the local curvature. A computer model has been developed based on the Geometric Clutch principle. When the model is scaled as closely as possible to the physical dimensions and mechanical properties that have been measured in real cilia and flagella, the computed simulations successfully replicate the basic patterns of motility of the biological systems. Observed phenomena, such as the effective and recovery stroke of cilia, can be readily reproduced; and mechanical-sensitivity, a known property of cilia and flagella, is intrinsic to the computer simulation. Recently, the model has been further tested by comparing computed behavior and real behavior of bull sperm under identical conditions of mechanical restraint and dissection. The results of the real and computed experiments are in good agreement. The simulation accurately predicts the observed changes in the beating pattern, and the conditions that cause the beat to arrest.
Charles B. Lindemann

A One-Dimensional Fluid Dynamic Model of the Systemic Arteries

The systemic arteries are modeled as a bifurcating tree of compliant and tapering vessels. Blood flow and pressure in the vessels are determined by solving the axisymmetric Navier-Stokes equations. The arterial tree ranging from the aorta to the arterioles consists of a tree with more than 20 generations. Computing blood flow and pressure for all vessels requires a prohibitive amount of time. To avoid using too much time, we have truncated the arterial tree after a limited number of generations and applied a suitable outflow boundary condition. To this end we propose a structured tree model in which a root impedance is determined using a semi-analytical approach. The structured tree is a binary asymmetric tree in which the radii of the daughter vessels are scaled linearly with the radius of the parent vessel. The root impedance of the structured tree is found by propagating solutions of a wave equation from the terminals to the root of the structured tree. The wave equation is derived by linearizing the axisymmetric Navier-Stokes equations together with applying a long-wave approximation. The root impedance of the structured tree provides a dynamical outflow boundary condition, which is computationally feasible. The structured tree outflow boundary condition is based on physiologic principles and it accommodates wave propagation effects for the entire systemic arterial tree. Blood flow in the large systemic arteries is verified by comparing simulations with data obtained from magnetic resonance measurements. The outflow boundary condition is verified by comparisons with literature data and with a standard model (the three-element windkessel model).
Mette S. Olufsen

Hydrodynamics of Liquid Capsules Enclosed by Elastic Membranes

The flow-induced deformation of liquid capsules enclosed by elastic membranes is discussed with reference to the biofluid-dynamics of vesicles and biological cells. The deformation of a membrane from a reference configuration causes the development of non-isotropic in-plane elastic tensions, transverse shear tensions, and bending moments according to constitutive laws that reflect the membrane constitution. The type and degree of capsule deformation, the internal and external structure of the flow, and the macroscopic rheological properties of a suspension are determined by the magnitude of the developing elastic tensions and bending moments relative to the strength of the imposed flow. Integral representations of the flow past a collection of capsules are reviewed, and integral equations for the velocity distribution over the membranes are presented for two-dimensional, three-dimensional, and axisymmetric flow at vanishing Reynolds number. The velocity field due to the presence of the capsules is expressed in terms of distributions of Stokes flow singularities over the membranes, represented by the single-layer and double-layer hydrodynamic potential of Stokes flow. The strength of the distribution of the single-layer potential expressing interfacial distributions of point forces is identified with jump in the hydrodynamic traction across a membrane. Expressions for this jump are reviewed and derived in terms of the developing elastic tensions and bending moments under the formalism of the theory of thin shells.
C. Pozrikidis

Unsteady Aerodynamics of Two Dimensional Insect Flight

Motivated by our interest in unsteady aerodynamics of insect flight, we compute the Navier-Stokes equation around a two dimensional flapping wing. The analysis of unsteady flows in forward flight reveals a mechanism of frequency selection, which results from the two intrinsic time scales associated with the shedding of leading and trailing edge vortices. The predicted preferred frequency scales inversely with the size of the wing, which is consistent with the zoological observation. The computation of hovering flight uncovers an intrinsic mechanism of generating lift by creating a downward jet of counter-rotating vortex pairs. The average computed forces in a generic hovering flight are shown to be sufficient to support typical insect weight.
Lisa J. Fauci, Shay Gueron


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