Mec. Ind. (2001) 2, 283–287 2001 Éditions scientifiques et médicales Elsevier SAS. All rights reservedS1296-2139(01)01109-5/FLA

Microhydrodynamics in biological systems

O.E. JensenDivision of Theoretical Mechanics, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK

(Received 25 April 2001; accepted 4 May 2001)

Abstract—This article presents a brief review of three biological systems where microhydrodynamics plays a crucial role: the motionof swimming microorganisms; the motion of blood cells in narrow capillaries; and the flow of liquids in small lung airways. 2001Éditions scientifiques et médicales Elsevier SAS

Stokes flow / swimming microorganisms / microcirculation / surface tension / airway mechanics

Résumé—Microhydrodynamique dans les systèmes biologiques. Cet article présente une revue succincte de trois systèmesbiologiques dans lesquels la microhydrodynamique joue un rôle majeur : la nage de certains microorganismes ; le mouvement descellules rouges dans les petits capillaires : et l’écoulement de liquides dans les passages des structures pulmonaires. 2001 Éditionsscientifiques et médicales Elsevier SAS

écoulement de Stokes / nage de microorganismes / microcirculation / tension / superficielle / conduites pulmonaires

1. INTRODUCTION

There are numerous biological and medical phenom-ena where microhydrodynamics provides essential in-sights and understanding. Three topics are discussedhere that illustrate some of the diverse aspects of low-Reynolds-number flows in biological systems. These are(i) the swimming and feeding of microorganisms such asalgae or bacteria; (ii) the flow of blood in the microcircu-lation, where individual blood cells must squeeze throughnarrow capillaries; and (iii) the flow of air and other fluidsin the small airways of the lung, where surface tension isa dominant force.

2. SWIMMING AND FEEDING OFMICROORGANISMS

Consider a small organism of length a that swims atspeed U through a fluid of viscosity µ and density ρ,such that the Reynolds number

Re = ρUa

µ

is substantially less than unity. In this domain, viscousforces dominate inertia forces, a condition that imposes

E-mail address: Oliver.Jensen@nottingham.ac.uk (O.E. Jensen).

substantial constraints on mechanisms of propulsion.Two key features of such “creeping” flows are as follows.

(i) Drag is only weakly dependent on shape andorientation. This is clearly illustrated by the example of afinite-length cylinder of radius a and length l (� a). Thedrag on a cylinder moving parallel to itself with speed U

is [1]

2πµlU

log(l/a) − (3/2) + log 2whereas the drag on a cylinder moving perpendicular toitself with speed U is

4πµlU

log(l/a) − (1/2) + log 2

These drag forces differ by less than a factor of 2.This difference, though small, can be exploited to allowpropulsion. In both cases, the drag scales roughly withthe body’s largest dimension.

(ii) Creeping flows are reversible. This is nicely illus-trated by a bivalve scallop, an example suggested by Pur-cell [2] ( figure 1). This swimmer has only one degree offreedom (the opening and closing of its hinge), so thatas the scallop opens and closes it moves backwards andforwards, always returning to the same spot.

Swimmers must therefore employ cyclic but non-reversible motions to propel themselves forwards. Thusalgae such as Chlamydomonas Nivalis swim a “low-

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Figure 1. Purcell’s bivalve scallop, which cannot propel itselfany distance in Stokes flow.

Figure 2. Motion of cilia: time increases from left to right(after [31]).

Reynolds-number breast-stroke” using a pair of flagel-lae [3], and organisms such as Paramecium, which arecoated with arrays of cilia, move by generating coherentwave motion among the cilia. During the forward strokeeach cilium is straightened, generating maximum thrust,whereas it bends parallel to the body surface during therecovery stroke, as illustrated in figure 2.

Another important strategy is that adopted by the bac-terium Escherichia Coli, which has a number of flagellaemounted on tiny rotary motors in its membrane. Duringthe bacterium’s propulsive phase (the “run”), the flagellaeform bundles with helical shapes, with propulsion gener-ated by rotation of the helices. At the end of each runthe motors rotate in the opposite direction, the flagellaein each bundle fly apart, and the cell tumbles randomly toa new orientation. By controlling the length of each run,the cell can execute a biased random walk along nutri-ent gradients [4]. Diffusion is a powerful effect at thesesmall lengthscales, and so during each run the cell mustswim many body lengths to detect significant variationsof nutrient concentration [2].

A striking macroscopic manifestation of microscopicswimming motions is bioconvection [3]. If a suspensionof chemotactic bacteria such as Bacillus Subtilis, whichswim up oxygen gradients, is placed in a shallow dish, the

bacteria swim up to the layer’s free surface, where theyaccumulate and concentrate in a thin layer that is slightlydenser than the rest of the suspension. Eventually thislayer becomes convectively unstable and dense plumes ofbacteria fall downwards. Similar bioconvection patternsare seen in suspensions of Chl. Nivalis: in this case thealgal cells are bottom heavy, and so a gravitational torqueacts on the cells that causes them to swim upwards, onaverage, which again leads to convective instabilities.

Finally, many small crustacea and small beetles havearrays of hairs on their limbs, which they use for feedingor propulsion. When moved slowly, so that Re � 1, anarray of hairs acts as a paddle — an efficient way ofpushing a lot of fluid around. If moved more rapidly,with Re = O(1), the array operates as a sieve that maybe used for filter feeding [5]. The organism can thereforeuse its arrays of hairs for these two different purposes, byvarying the speed of motion.

3. THE FLOW OF BLOOD IN THEMICROCIRCULATION

The microcirculation is comprised of small arteries(arterioles), capillaries and small veins (venules) havingdiameters less than 100 µm. This part of the circulatorysystem is important because it provides the dominantresistance to blood flow, and therefore controls the loadplaced on the heart. At these small dimensions, inertiaand flow pulsatility are unimportant, but the rheologyof blood becomes complex. Blood is a concentratedsuspension of deformable cells, mostly red blood cells,suspended in plasma (which is 90 % water). Each redblood cell has a membrane that resists area changes butwhich bends easily, and under normal conditions eachcell has a familiar biconcave shape and a width of about8 µm. However, these cells can deform substantially tosqueeze through capillaries as narrow as 3 µm, assuminga range of exotic shapes in the process [6]. The cells alsoexperience a hydrodynamic lift away from the walls ofblood vessels, so that the cells accumulate at the centre ofvessels, supported by a lubricating plasma layer. Thus thevolume fraction of red blood cells (the hematocrit) fallsfrom 40–45 % (its value in large blood vessels) down to25 % in capillaries, and at the same time the resistanceto flow in small capillaries (as measured by an apparentviscosity) can fall to less than half of its normal value.

This behaviour has been successfully modelled by anumber of workers (including Lighthill [7], Skalak, Sec-omb and coworkers [8–10]), since the mechanics of thered blood cell membrane has been well characterised, andthe thin plasma layer between each cell and the capil-

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Figure 3. Airway closure: surface tension causes the initially uniform liquid lining of an airway (a) to become non-uniform and possiblyto occlude the airway (b), which then leads to airway collapse (c).

lary wall is well described using lubrication theory. Sec-omb [11] provides a detailed review of this and relatedwork. He and Hsu have shown, in particular, that non-axisymmetry and tank-treading motions — while presentexperimentally — do not significantly contribute to flowresistance [12]. However, it has recently been recognisedthat the very thin layer of glycocalyx lining each capil-lary, the “sugar-coat” on the surface of endothelial cells,has an unexpectedly large effect on flow resistance [13,14].

A topic which has received considerable attentionover the last decade concerns the adhesion of whiteblood cells (leukocytes, and particularly neutrophils) tothe walls of blood vessels in the presence of injury orinflammation. Specific receptor and ligand molecules onthe surface of the cells, and on the endothelial cells liningsmall venules, can form transient adhesive bonds. Thisallows neutrophils to “roll” over the vessel wall, withindividual bonds forming and breaking in the process.Hydrodynamic forces cause the cell to pivot about thetethers formed by each molecular bond. Rolling is aprecursor to full adhesion and subsequent migration ofthe cell through the vessel wall. Remarkably, the rollingspeed (of a few µm·s−1) is almost independent of theimposed shear stress [15], for reasons that remain to befully explained.

4. FLOWS IN SMALL AIRWAYS OF THELUNG

The human lung consists of a network of bifurcating,deformable airways, terminating after roughly 20 gener-ations of branching in terminal air-sacs known as alveoli.Gas exchange occurs across the walls of alveoli, throughwhich the blood of the pulmonary circulation flows. Airflow in the smallest airways, which occurs on length-scales of a few hundred microns, is in the creeping-flowregime (Re � 1). While most gas exchange here occursby diffusion, inhaled aerosol particles (for which dif-fusive effects are very weak) exhibit mixing propertiesdominated by Lagrangian chaos [16, 17]. Small asym-metries between the airway wall shapes on inspiration

(wall expansion) and expiration (wall contraction) gener-ate stirring flows that stretch and fold material fluid par-ticles, giving rise over many breaths to effective mixingof the inspired air. A key feature of these flows is that astagnation saddle point in the primary flow is perturbedby geometric asymmetries, a characteristic signature ofmixing by chaotic advection [18].

All the airways of the lung are wet, being lined witheither mucus (in larger airways) or a thin aqueous liq-uid lining (in terminal airways and alveoli). Thus withinthe lung is an air-liquid interface that has both enormoussurface area (80 m2 in an adult) and very high curvature(alveolar diameters are roughly 250 µm). Surface tensionis therefore a dominant force in respiratory mechanics.Lung inflation requires expansion of the air–liquid inter-face, so the work done against surface tension is a ma-jor factor in determining lung compliance. Since surfacetension creates subatmospheric capillary pressures in theliquid lining, there is the potential for these low pres-sures to draw fluid into airways across permeable airwaywalls, or to cause deformable airways to collapse. Ex-cessively high surface tension thus leads to stiff lungshaving a high proportion of flooded and collapsed air-ways. Healthy lungs maintain high lung compliance andnearly dry, patent airways by manufacturing pulmonarysurfactant, a complex mixture of phospholipids and pro-teins that reduces the surface tension of the airway liningfluid. Infants born very prematurely lack the capacity toproduce adequate quantities of their own surfactant, andso an artificial surfactant must be delivered to their lungsto help them through their first weeks of life.

A number of microhydrodynamical problems havebeen investigated recently that help us understand someof the fundamental effects of surface tension on pul-monary mechanics [19].

(i) Airway closure occurs when surface tension de-stabilises the liquid lining of a cylindrical airway, caus-ing the spontaneous formation of a liquid bridge that oc-cludes the airway ( figure 3a, b). While the presence ofsurfactant slows down this instability, it cannot prevent itoccurring [20]; weak gravitational forces either stabiliseor destabilise the primary capillary instability, depend-

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Figure 4. Airway reopening: the model problem.

ing on the airway’s orientation [21, 22]. Perhaps mostsignificantly, since airways are deformable they can col-lapse under the compressive load induced by low capil-lary pressures in a liquid bridge, buckling in the processto complex, non-axisymmetric configurations ( figure 3c)[23, 24].

(ii) Airway reopening. Once a flexible-walled airwayhas flooded and collapsed, it is important to considerhow it may be reopened so that gas exchange can berestored. This issue has been investigated through themodel problem illustrated in figure 4. A semi-infinitebubble of air is blown at pressure pb into a fluid-filled,elastic-walled channel formed by membranes supportedby external springs and held under longitudinal tension.Experiments [25] and theory [26, 27] ( figure 5) haveshown that pb must exceed a critical threshold beforethe bubble propagates steadily into the channel, peelingopen the channel walls. The membranes are held togetherahead of the advancing bubble tip by very low fluidpressures generated by the combined effects of largemembrane tension and locally large membrane curvature.Understanding the mechanical origins of such stresses isimportant because artificial ventilation can easily harmdelicate airway walls.

(iii) Surfactant transport. When an artificial surfac-tant is delivered to the lungs of a premature infant,for example through an endotracheal tube, it spreadsspontaneously along liquid-lined lung airways throughflows driven by gradients of surface tension [19, 28](a so-called Marangoni flow). The surface tension of thesurfactant-deficient airway liquid lining deep in the lungis higher than that of the proximal surfactant-rich lining,and this difference drives both liquid and surfactant to theperiphery of the lung. The spreading process is of a non-linear diffusive character, and exhibits a rich range of dy-namical structures, including kinematic shock formation[29] and stress singularities [30]. Models of fundamentalfluid-mechanical aspects of surfactant-driven flows havebeen used to provide a foundation for more general mod-els of the entire delivery process during Surfactant Re-placement Therapy [28].

Figure 5. Airway reopening: steady flow computations showhow the bubble pressure pb depends on the bubble speed Ca.The right-hand solution branch, representing peeling motion,is stable when pb is held fixed; the left-hand branch is unstable(from [26]).

5. CONCLUSIONS

The examples presented above cover a broad class ofmicrohydrodynamical flows, from motion of individualobjects in Stokes flow (Section 2) to fluid-structure inter-actions (Section 3) and flows driven by interfacial forces(Section 4). It is particularly instructive to see how evolu-tion has helped organisms overcome the mechanical con-straints of their environment, or exploit mechanical phe-nomena to their advantage. These natural solutions maybe valuable guides as we strive to make new technologi-cal advances involving small-scale fluid mechanics.

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