Biomech Model MechanobiolDOI 10.1007/s10237-016-0840-z

ORIGINAL PAPER

The breakup of intravascular microbubbles and its impacton the endothelium

Wolfgang Wiedemair1,3 · Zeljko Tukovic2 · Hrvoje Jasak2 ·Dimos Poulikakos1 · Vartan Kurtcuoglu3,4

Received: 28 November 2015 / Accepted: 27 September 2016© Springer-Verlag Berlin Heidelberg 2016

Abstract Encapsulated microbubbles (MBs) serve as endo-vascular agents in a wide range of medical ultrasoundapplications. The oscillatory response of these agents toultrasonic excitation is determined by MB size, gas content,viscoelastic shell properties and geometrical constraints. Theviscoelastic parameters of the MB capsule vary during anoscillation cycle and change irreversibly upon shell rupture.The latter results in marked stress changes on the endothe-lium of capillary blood vessels due to altered MB dynamics.Mechanical effects on microvessels are crucial for safety andefficacy in applications such as focused ultrasound-mediatedblood–brain barrier (BBB) opening. Since direct in vivoquantification of vascular stresses is currently not achievable,computational modelling has established itself as an alterna-tive. We have developed a novel computational frameworkcombining fluid–structure coupling and interface tracking tomodel the nonlinear dynamics of an encapsulatedMB in con-strained environments. This framework is used to investigatethe mechanical stresses at the endothelium resulting fromMB shell rupture in three microvessel setups of increasinglevels of geometric detail. All configurations predict substan-tial elevation of up to 150% for peak wall shear stress upon

B Vartan Kurtcuogluvartan.kurtcuoglu@uzh.ch

1 Laboratory of Thermodynamics in Emerging Technologies,Department of Mechanical and Process Engineering, ETHZurich, Sonneggstrasse 3, 8092 Zurich, Switzerland

2 Faculty of Mechanical Engineering and Naval Architecture,University of Zagreb, Ivana Lucica 5, 10000 Zagreb, Croatia

3 The Interface Group, Institute of Physiology, University ofZurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland

4 Zurich Center for Integrative Human Physiology, andNeuroscience Center Zurich, University of Zurich, Zurich,Switzerland

MB breakup, whereas global peak transmural pressure lev-els remain unaltered. The presence of red blood cells causesconfinement of pressure and shear gradients to the proximityof theMB, and the introduction of endothelial texture createslocal modulations of shear stress levels.With regard to safetyassessments, the mechanical impact ofMB breakup is shownto be more important than taking into account individual redblood cells and endothelial texture. The latter two may proveto be relevant to the actual, complex process of BBB openinginduced by MB oscillations.

Keywords Microbubbles · Shell breakup · Microvessel ·Wall shear stress · Geometrical complexity

1 Introduction

Microbubbles (MBs) with a viscoelastic encapsulation layerare used as systemic contrast-enhancing agents in med-ical ultrasound (US) radiology (Quaia 2007). They act asscattering centres for impinging ultrasound waves due tothe difference in acoustic impedance between gas contentand blood. These ultrasound contrast agents (UCAs) exhibitechogenic properties according to their size, which is typi-cally in the range of 1–10μm,gas content and shell properties(Stride and Saffari 2003a). More recently, the same UCAshave been employed in combination with focused ultrasoundfor the systemic transport and targeted delivery ofmedicationto specific tissues or locations in the body (Ferrara et al. 2007;Hernot and Klibanov 2008). Insonated UCAs are also usedto trigger an enhancement of the local permeability of theblood–brain barrier (BBB) (Abbott et al. 2010) for drug trans-fer (Abbott and Romero 1996; Hynynen 2008; Vykhodtsevaet al. 2008) and are employed in a range of other therapeuticapplications (Liu et al. 2006; Unger et al. 2004).

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The oscillatory dynamics of the endovascular MB dependon the excitation pressure amplitude, the shell characteris-tics, liquid properties and the geometrical setting. Typicalresonance frequencies are in the MHz range. OscillatingMBs produce microscopic pulsatile currents in the surround-ing liquid. This microstreaming projects oscillatory forcesonto nearby boundaries and can cause or augment mechan-ical or biological effects. The transient local opening of theBBB is associated with such processes (McDannold et al.2006), which need to be well controllable in order to pre-vent irreversible damage to the BBB (Hynynen et al. 2005).The intensity of mechanical surface effects depends on themicrostreaming characteristics which are a function of thevibrations of the MB, which in turn can be tuned by US set-tings and pre-determined by tailoring the shell parameters(Stride 2005).

The encapsulation serves to stabilize the MBs against dis-solution and coalescence. It is composed of a thin layer ofviscoelastic material that deforms with the expansion andcontraction of the MB, rendering the shell characteristicsa function of current bubble size. Upon MB compression,the shell is compacted, which can cause a wrinkling of thatlayer. Suchbuckling is observed in lipidmonolayers (Versluis2010) and attributed to the congestion of the constant num-ber of lipid molecules in a shell of decreasing size. The samephenomenon is observedwhen theMBshrinks due to gradualgas elusion (Borden and Longo 2002). The shell loses mostof its tension in the buckling state (Marmottant et al. 2005),causing a marked change in the dynamics of the MB, whichis otherwise typically dominated by surface tension. Duringbubble expansion in pressure rarefaction cycles, the vis-coelastic shell experiences strain and the tensionmay surpassa critical limit above which the shell ruptures. The breakupof the encapsulation (Stride and Saffari 2003b) exposes partsof the gas interface directly to the surrounding liquid, whileremnants of the shell may still float at the interface. Thiscauses an abrupt change in MB characteristics, leading to achange in oscillation patterns and subsequent dissolution ofthe MB (de Jong et al. 2009).

The rupture of MBs inside vessels of the circulatory sys-tem changes the dynamics of the involved MB. The alteredmicrostreaming can result in a marked change in mechan-ical stresses at the endothelium. The generated shear maytrigger biological responses, such as an increase in BBBpermeability (McDannold et al. 2006), or may damage cellmembranes (Wu 2002). Hence, the stress on the endothe-lial cell membranes exerted by MB breakup is of interest forthe assessment of potential tissue response or damage andultimately for a risk-benefit assessment.

Breakup of different types of encapsulated MBs has beenexperimentally observed and characterized by Moran et al.(2000), and the influence of ultrasound parameters on theMB oscillations was assessed using acoustic and optical

feedback (Dayton et al. 1999; de Jonget al. 2009).Direct opti-cal observation employing ultrahigh-speed cameras (Chinet al. 2003) allowed to follow the transient kinetics of theMB (de Jong et al. 2000), while the mechanical damagecaused by MB oscillations and breakup in vessels was stud-ied in gel phantoms (Caskey et al. 2009). Direct observationof MB oscillation in microvessels was reported by Caskeyet al. (2007), and the resulting microvessel deformation in exvivo vessels was characterized by Chen et al. (2011). How-ever, experimental assessment is currently limited to kineticobservations and a posteriori damage assessment since thesmall time and length scales of the involved processes poseenormous challenges to in situ measurement of dynamicsquantities like shear and pressure at the vessel walls.

The dynamics of a singleMBwith clean surface in an infi-nite fluid space exposed to an ultrasound field is accuratelydescribed by the Rayleigh–Plesset equation (RPE) (Plessetand Prosperetti 1977). An analytical model including theeffect of a solid shell was proposed by Church (1995) andadapted for thin encapsulating layers by Hoff et al. (2000).Basedon experimental findings, Sarkar et al. (2005) proposeda model for MBs with infinitely thin viscoelastic shells char-acterized by size-dependent properties. Taking into accountshell buckling, a finite elastic range of the encapsulation andshell rupture, Marmottant et al. (2005) developed an exten-sion towards a three-statemodel categorized by the tension ofthe interface. Further refined rheological shell models incor-porating non linear shell viscosity were proposed by Li et al.(2013) and Doinikov et al. (2009). The modelling of MBsinside rigid tubes is achieved by introducing an effectivemassterm (Klotz and Hynynen 2010). However, most analyticalmodels are restricted to a spherically oscillating MB in sim-ple geometrical settings.

Numerical modelling permits greater flexibility in study-ing complex, coupled setups that are more relevant topractical, medical purposes. The oscillation of MBs in flexi-ble, plain tubes (Ye and Bull 2006) was studied, and lumpedparameter solid models allowed investigation of the interac-tion of MBs with compliant microvessels (MVs) of variablesize (Qin and Ferrara 2006). The mechanical stresses at theendothelial interface were assessed using coupled solid andfluid models (Hosseinkhah et al. 2013; Wiedemair et al.2012). The breakup of MBs near rigid walls (Hsiao andChahine 2013) and inside MVs (Hosseinkhah et al. 2015)was also studied numerically.

Currently available numerical models often simplifyMVsas liquid-filled straight tubeswith plainwalls. This geometricsimplification neglects many features of the microvascula-ture, such as the presence of red blood cells (RBCs) in closeproximity to the MB and the texture of the MV resultingfrom the endothelial cell morphology. RBCs may influencethe local microflow and thereby have an impact on mechani-cal stress at the endothelium. A more detailed representation

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of the endothelial wall morphology could cause alterationsof wall shear patterns, since this mechanical parameter isvery sensitive to the local surface configuration. Therefore,neglecting a subset of the geometric features may undulyreducemodel fidelity and ultimately limit the level of validityof the assessed mechanical wall parameters such as trans-mural pressure and wall shear stress.

The purpose of this study is to investigate the effect ofincluding additional geometric MV features that can be ofimportance, such as RBCs and endothelial texturing. Whilethe representations of the employed geometrical structuresare still idealized, they allow for general conclusions aboutthe value of increasing geometric complexity. Mechanicalparameters derived from models of enhanced complexitymay prove relevant to safety assessment of medical treat-ments involvingMBs.Moreover, a detailed andmore realisticrepresentation of the involvedmechanical force patterns maycontribute to elucidating the underlying mechanisms of BBBopening, a process that is currently not fully understood(Sheikov et al. 2004).

To enable this investigation, we introduce a novel fullygeneric numerical framework to treat the complex couplingof viscoelastic MB dynamics, blood microflow and vesselstructural dynamics in the event of MB shell breakup. Itcombines the ability to accuratelymodel non-spherical oscil-lations of MBs with size-dependent viscoelastic shell prop-erties and the coupling to arbitrarily shaped solid domains.This versatile tool allows the study of changes in the wallparameter patterns for increasingly detailed setups.

2 Methods

We introduce a computational framework to model thechanges in mechanical stresses on the endothelium causedby the breakup of a single encapsulated microbubble inside aMV at increasing levels of geometric detail. The interactionof multiple, physically distinct dynamic domains is imple-mented by means of iterative, bidirectional coupling. Thecoupling of intraluminal microflow and structural mechanicsof the vessel is achieved by employing a fluid–structure inter-action (FSI) algorithm. We translate the three-state approachby Marmottant et al. (2005) into an adaptive fluid boundarycondition for an interface tracking (IFT) algorithm, whichallows embedment of the MB dynamics into the model.This computational framework permits the modelling ofcompliant MVs hosting an encapsulated MB with variableviscoelastic properties immersed in blood plasma and sur-rounded by RBCs (Fig. 1). The RBC spacing of 5–8μmwascalculated based on the average concentration of RBCs inhuman blood (McHedlishvili and Varazashvili 1980) and themean capillary diameter (Cassot et al. 2006). This setup isexposed to oscillating pressure fields of varying amplitude,

Fig. 1 The geometrical setup comprises a single microbubble embed-ded in a train of RBCs inside a compliant MV. The interfaces betweenMB and blood plasma as well as the liquid–solid interface betweenblood plasma and the MV wall are deformable. The RBCs are motilein axial direction

while the bubble dynamics, microstreaming and endothelialinterface stresses are monitored.

Blood plasma is modelled as an incompressible, New-tonian fluid, and the structural component of the MV aswell as the RBCs is regarded as homogeneous, isotropic, lin-ear elastic solid (Tukovic and Jasak 2007). These domainsare coupled using an FSI algorithm (Kuttler et al. 2010)with Arbitrary Lagrangian–Eulerian (ALE) formulation ofthe fluid constitutive equations on a dynamically adapting,unstructured computational mesh (Jasak and Tukovic 2007).An interface quasi-Newton technique with inverse Jacobianfrom a least-squaresmodel (IQN–ILS) (Degroote et al. 2009)is employed tooptimize stability and convergenceof the com-putation. Both computational domains are discretized usinga second-order accurate cell-centred unstructured finite vol-ume scheme. The algebraic systems are solved iteratively,using a segregated solution procedure based on the PISOalgorithm (Issa 1986) for the pressure–velocity coupling inthe fluid and a segregated algorithm for determining thethree components of the displacement increment in the linearmomentum conservation law written in updated Lagrangianformulation for the solid (Tukovic and Jasak 2007). The tran-sient problem is treated in a time-marching manner withuniform time step size and an implicit, second-order accuratetime integration scheme (Ferziger and Peric 1995). Wiede-mair et al. (2012) provides details on the employed couplingalgorithm and a thorough description of the solid and fluidmodelling, including the formulations of constitutive equa-tions as well as references to model derivation.

The dynamics of theMBare coupledwith the fluid domainvia an adaptable, deformable boundary at the MB–liquidinterface. The basic algorithm for tracking the MB interfaceis described by Tukovic and Jasak (2012) and was employedfor modelling the interaction of multiple MBs byWiedemairet al. (2014). This IFT algorithm is extended and generalizedby the introduction of viscoelastic shell properties that scalewith the dynamic size of the MB during oscillation.

2.1 Analytical MB model

A MB with a thin viscoelastic encapsulation can be charac-terized by its equilibrium radius a0 as well as the dynamicviscosity μs and the tension σs of its shell (Chatterjee

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and Sarkar 2003). When the MB radius changes duringUS-induced oscillation, the viscoelastic properties of theMBshell may be altered (Tu et al. 2009; van derMeer et al. 2007)due to the compaction or rarefaction of its constituent mole-cules. Marmottant et al. (2005) suggest that μs scales withthe inverse of the current bubble radius a according to

μs = κs

a(1)

with shell dilatational viscosity κs .The surface tension of the MB can be written in a piece-

wise form as

σ(A) =⎧⎨

⎩

0σs(A)

σ0

if A ≤ Aif A < A < Aif A ≥ A

, (2)

where A is the surface area below which buckling occursand A denotes the critical maximum surface area at whichthe shell ruptures. Experimental findings (Borden and Longo2002; Versluis 2010) indicate that the shell loses tension inthe buckling state. In the elastic regime, σs is proportional tothe surface area A of the MB according to

σs = χ

(A

A− 1

)

, (3)

where χ is the elastic shell compression modulus. κs and χ

are constants determined by the composition of the shell. Theelastic regime is limited by the buckling behaviour towardssmall surface areas and the shell rupture towards large surfaceareas.

Shell rupture occurs when surpassing the critical maxi-mumsurface area A. After rupture of the shell, the gas contentis directly exposed to the surrounding liquid, resulting in aconstant surface tension σ0 for the gas–liquid interface (Paulet al. 2010). The breakup surface area

A = A

(

1 + σc

χ

)

(4)

is determined from the maximum tension σc the shell cansustain. σc is usually of the same order of magnitude as σ0(Stride and Saffari 2003b) with proposed values for the upperend of the elastic regime ranging from σc = σ0 (Versluis2010) to several times σ0 (Marmottant et al. 2005).

Rupture changes the shell characteristics permanently toσ = σ0 and μs reduces to the residual viscosity μr . Thatquantity is a fraction of the original μs and accounts forthe effect of the shell remnants, of which large portions willstill reside at the interface. For spherical MBs, the equivalentbuckling radius a and the equivalent breakup radius a can beemployed as more intuitive limits of the linear regime.

The response of a MB with the described shell charac-teristics and polytropic gas content to a transient pressurevariation Pa(t) can be described by the modified RPE

ρl

(

aa + 3

2a2

)

=(

P0 + 2σ(A0)

a0

)(a0a

)3γ

−P0 − 2σ(A)

a− 4

aμt

a− Pa(t), (5)

where dots denote derivatives with respect to time, ρl is theliquid density, γ is the polytropic exponent, P0 denotes equi-librium pressure and μt is the total viscosity defined as

μt = μl + μs, (6)

where μl is the dynamic viscosity of the liquid.

2.2 Numerical MB model

A more general case of non-spherical MB oscillation can bemodelled by tracking theMB–liquid interface in a 3D numer-ical simulation employing the finite volume formulation. Weconsider a deformable liquid domain, representing the bloodcompartment of the setup, where the flow of incompress-ible fluid is governed by mass and momentum conservationaccording to theNavier–Stokes equationswritten inALE for-mulation. A portion of the boundary of this domain coincideswith the interface between blood and MB. At this interface,an adaptable boundary condition that is updated accordingto the dynamic conditions in the MB is applied. This MB–liquid interface Γb is iteratively deformed until normal stressequilibrium

P|�b= Pg − κσ (A) − 2μt (κ)∇s · v|�b

(7)

is reached. Here, P is the liquid pressure, κ denotes twice thelocal surface curvature calculated by κ = −∇s · n, where ∇s

is the surface gradient and n is the surface normal unit vector,v represents fluid velocity, and Pg is the transient internal gaspressure of the MB. Pg , considered homogenous throughoutthe bubble, is derived from the polytropic law

Pg = Pg,o

(Vb,oVb

)γ

, (8)

where Pg,0 is the equilibrium gas pressure, Vb,0 is the MBequilibrium volume, and Vb is the current MB volume.

The solution procedure relies on an interface trackingapproach (Tukovic and Jasak 2012) to determine the exactposition of the MB interface assuming zero mass transferacrossΓb. The sharp delineation of theMB interface is essen-tial for accurate evaluation of Eq. (7), which is dominated bythe surface tension term due to large κ values as a result of

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the small size of the MB. We developed a numerical solvertailored to address the combination of FSI at the endothelialinterface with IFT at the MB surface in the numerical plat-formOpenFOAM (Jasak et al. 2007;Weller et al. 1998). Thissolver relies on the iterative solution of the solid–fluid cou-pling problem using a segregated FSI algorithm with nestediteration loops addressing the IFT of the MB surface insidethe fluid domain.

With the application of transient pressure boundary condi-tions at the vessel inlets and at the outer MV surface, the MBimmersed in blood plasma at the centre of the vessel is sub-jected to a transient excitation signal. Pa is assumed spatiallyhomogenous throughout the considered domain because theconfiguration is much smaller than the USwavelength. Sincethe typical bloodflowvelocity of 1mm s−1 in capillaries (Fis-cher et al. 1996; Stucker et al. 2004) is about three orders ofmagnitude smaller than the radial MB interface velocity, theMBwill not be displaced significantly from its initial locationduring the small time window relevant to MB oscillation andbreakup, which is in the range of μs. Therefore, advectiondue to physiologic blood flow is neglected.

The deformation of the MB according to the surroundingflow conditions is mediated by Eq. (7) and will in turn induceoscillatory flow to the surrounding liquid. The impact of thegenerated microstreaming on the displacement of the lumi-nal vessel interface Γs f can be determined through FSI. Theforces acting on this interface are evaluated in terms of wallshear stress

τwss = μl n · ∇ (v − (v · n) n)|�s f(9)

and transmural pressure

Ptm = P|�s f− (Pi + Pa) , (10)

where Pi represents the static pressure in the interstitial spaceoutside the vessel and Γs f indicates that the quantities areevaluated at the luminal vessel interface. These two quan-tities, termed mechanical wall parameters, characterize thenormal and transverse stresses on the MV endothelium. Thestresses at the endothelial vessel interface before and afterthe breakup of the shell of an encapsulated MB with size-dependent shell characteristics are investigated. They arerelevant to cavitation-induced damage and are hypothesizedto play a key role in BBB opening (Chen and Konofagou2014; McDannold et al. 2008; Nyborg 2001; Wamel et al.2006).

All studied setups comprise a single MB of equilibriumradius a0 = 1μm with a = 0.985μm and shell proper-ties κs = 15 nN sm−1, χ = 1.0Nm−1 and μr = 0.4μs

with a polytropic exponent of γ = 1.095, suitable for sulphurhexafluoride gas as employed in SonoVue� UCAs (Marmot-tant et al. 2005). A breakup tension of σc = 0.115Nm−1 is

assumed, corresponding to A = 13.6μm2 or a = 1.04μmfor an equivalent spherical shape. The MB is located at thecentre of a lt = 40μm long, compliant MV with luminalradius rt = 3μm (Cassot et al. 2006) and 0.5μm averagewall thickness (Bertossi et al. 1997), mimicking a portion ofa capillary blood vessel. The vessel wall has a mass densityρv of 1200 kgm−3, and Young’s modulus Ev is set to 8 MPa(Miao et al. 2008). The liquid properties are adjusted to rep-resent blood plasma with ρl = 1030 kgm−3 (Lowe 1987)and μl = 1.5 × 10−3 kgm−1 s−1 (Haidekker et al. 2002).

A configuration often employed for themodelling of oscil-lating endovascular MBs is a single MB placed in the centreof a blood plasma containing straight tube with plain luminalinterface.Weuse this as the basis of our study, referring to it asSetup I. The inherent rotational and central symmetries areexploited to accelerate computation, and the configurationis resolved with a hexahedral, block-structured computa-tional mesh. We evolve this into Setup II by adding pairsof RBCs upstream and downstream of the MB according toFig. 1 and investigate these cells’ impact on the wall parame-ters. While passing through a MV, the tips of the deformed,parachute-shaped RBCs (Secomb et al. 2006) slide along theendothelium (Pries et al. 2000). In the chosen axisymmetricconfiguration, the RBCs can move in axial direction drivenby the oscillation of the MB. To represent the textured MVsurface, endothelial cells lining the MV are approximated byhexagonal honeycomb structures arranged in nominal bloodflow direction in Setup III. They feature a bulge at the centrerepresenting the protruding cell nucleus region and indenta-tions at the cell junction. The geometries in Setups II & IIIare resolved using a polyhedral, unstructured computationalmesh. Average mesh densities in the three setups are 281–219 computational cells per μm3, corresponding to averageresolutions of 0.15– 0.17 μm, with interfaces being resolvedmuch finer than bulk regions.

The configurations are exposed to a sinusoidal pressurefield Pa(t) with transiently growing pressure amplitude Pex atan excitation frequency of fex = 2.6MHz (Marmottant et al.2005; Schlachetzki et al. 2002). MBs with thin viscoelasticshells can typically sustain acoustic pressures in the rangeof 100–150kPa (de Jong et al. 2009), depending on fex andthe exact shell configuration. Pex is linearly ramped up froman initial setting Pex,i = 130kPa to a final value Pex, f =180kPa during a ramp time interval tr = 5μs and staysconstant at Pex, f for t > tr . Theparameters are chosen suchthat the MB initially oscillates in the stable regime and thengradually approaches and surpasses the breakup threshold.

The employed computational framework is based on acombination of two algorithms, FSI and IFT with constantparameters. Validation studies for both methods have beenperformed and are provided in Wiedemair et al. (2012) andWiedemair et al. (2014), respectively. A validation of thenovel generalized IFT algorithm for scalable viscoelastic

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shell properties is provided in the “Appendix” section, wherewe established close correspondence of numerically deter-mined results with analytical predictions from Eq. (5) andalso find good agreement with a reference case.

3 Results

We shall increase the complexity of the geometric setup(Fig. 1) in three stages by introducing additional features.The impact of those model adjustments on τwss, Ptm andtheir derivatives, collectively referred to as wall parameters,is monitored. Starting from a straight tube with plain walls offixed thickness (Setup I), we shall assess the effect of intro-ducing RBCs into the MV (Setup II) and then of adding atexture that mimics the structure of endothelial cells at thevascular interface (Setup III).

3.1 Setup I: Plain vessel without RBCs

Figure 2 shows an arrangement of plots for the time windowaround the shell rupture in a straight tube with texture-freewalls (Setup I). The plots include the excitation pressure sig-nal Pa(t), the average MB radius a, the maximum of themagnitude of τwss(|τwss|max) and the maximum transmuralpressure, Ptm,max. To discard inflow and boundary effects atthe vessel ends, a sampling domain spanning 15 μm to bothsides of the location of the MB rather than the entire tube isconsidered for determining the maxima. A linear increase ofexcitation pressure amplitude is imposed over the first 5μs.

The MB shell breaks at a local minimum of Pa at tb =4.32μs (dashed line in Fig. 2), leading to an immediate sharpincrease in a. This growth of vibration amplitude is accompa-nied by an increase of |τwss|max to approximately 2.5 timesits pre-breakup value. Ptm,max shows a change in its tran-sient pattern after shell rupture, but no appreciable alterationin peak values. The occasionally observable abrupt changesof slope in the patterns of Ptm,max and |τwss|max are the resultof a switch of spatial location of the maximum value.

We observe a distinct preference for compression as evi-dent in the temporal pattern of the bubble radius curve, whichreaches values far below a0. This kinematic comportmentwas also pointed out by Marmottant et al. (2005) and exper-imentally confirmed (de Jong et al. 2007; Versluis 2010).The change of slope in the compression phase is caused bya decrease in σ (A) and a simultaneous increase of μs(κ). Itmarks the transition of the MB dynamics from the surface-tension-dominated regime to buckling comportment.

To analyse the spatial stress distribution, we sampled wallshear stress in axial direction, τwss,z, and transmural pres-sure along a line, Ξ , parallel to the longitudinal vessel axis zon the luminal interface. Since the setup is axially symmet-ric, this sampling represents the stress patterns on the entire

Fig. 2 Temporal evolution around the time point of shell rupture (dot-ted line) of Pa, a, |τwss|max and Ptm,max patterns for Pex,i = 130kPa,Pex, f = 180kPa and tr = 5μs in a vessel with plain luminal wallsand without RBCs (Setup I). As Pa increases linearly from Pex,i toPex, f , shell rupture takes place at tb = 4.32μs, causing a markedchange in MB oscillation pattern and amplitude, and a substantialincrease in |τwss|max. The Ptm,max pattern changes shape, but peak val-ues do not deviate significantly from pre-rupture levels

luminal surface. We stack the data sampled along Ξ at con-secutive time steps to allow for a spatio-temporal analysis ofthe wall parameters and their axial gradients (Fig. 3). τwss,zand Ptm show oscillatory temporal patterns that change afterMB shell breakup. The initially weak transient variation ofPtm in the region around the location of the MB (z = 0)increases in amplitude after tb, and its shape changes froma single pressure peak to that of a pressure peak pair dur-ing each excitation pressure oscillation cycle. This can alsobe seen in the Ptm,max pattern in Fig. 2. At the same time,the pressure oscillation in distal regions continues with pre-rupture amplitude.

τwss,z increases substantially along Ξ following shellrupture, and isolated local extrema become more apparent.Pronounced peaks of τwss,z establish immediately lateral tothe flow stagnation point located at z = 0. The gradients ofτwss,z and Ptm along z show wave-like patterns and increasemarkedly after tb.

3.2 Setup II: Plain vessel containing RBCs

The assessment of Pa(t), a, |τwss|max and Ptm,max for atexture-free vessel containing RBCs, performed in the samefashion as for Setup I, reveals that theMB comportment doesnot change considerably in the presence of RBCs, with MBshell breakup again taking place at tb = 4.32μs (Fig. 4).While the peak values of the wall parameters stay in the

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Fig. 3 Spatio-temporal behaviour of wall parameters along the axialsampling lineΞ for Setup I. Ptm shows a layered temporal comportmentwith ripples in z direction and a marked change in pattern at the MBlocation (z = 0) after shell rupture at tb = 4.32μs (white horizontalline). The peak Ptm values do not change appreciably due to the rup-ture. τwss,z increases substantially after shell rupture, and isolated localextrema become more apparent. The axial gradients of both parametersincrease after tb and exhibit a wave-like pattern

same range as without RBCs, their transient patterns havenoticeably different shapes.

A study of the spatio-temporal comportment of Ptm andτwss,z (Fig. 5) along Ξ reveals the impact of RBCs on thewall parameters. The Ptm pattern is generally similar to SetupI, but shows clear damping in the RBC regions (marked bydashed lines). The peak pressure amplitude in the vicinity ofthe MB is greatly enhanced after shell rupture, but surpassespre-rupture Ptm levels only marginally. Moreover, a transientpressure peak pair establishes in this region after tb. Themajor variations in axial gradients are contained between theinnermost RBCs. Substantially lower gradients are observedin the space between the, with respect to the MB, proximaland distal RBCs.

τwss,z increases substantially after tb to similar values asin Setup I, but exhibits a different, more layered pattern withjumps in the downstream (z > 0) RBC locations and localmaxima in their upstream positions. This loss of symmetryof the upstream and downstream regions, as compared toSetup I, is caused by the parachute shape of the RBCs. Onlythe symmetry about the longitudinal axis is still preserved inSetup II. Similar to the Ptm gradients, large values of the axialτwss,z gradients are contained between the innermost RBCs,while lower gradients are observed in the inter-RBC spacebetween the two upstream and the two downstream RBCs.

Overall, the introduction of RBCs causes a confinementof the major stress variations in the vicinity of the MB

Fig. 4 Temporal evolution around the timeof shell rupture (dotted line)of Pa, a, |τwss|max and Ptm,max with Pex,i = 130 kPa, Pex, f = 180kPa and tr = 5μs in a vessel with plain luminal walls containingRBCs (Setup II). The peak values of the wall parameter are similar toSetup I, but the transient comportment shows noticeable differences inthe observed fluctuation patterns

and creates a distinction between patterns in upstream anddownstream regions, which is more pronounced in the τwss,zpatterns and its respective gradients than in the Ptm distribu-tions.

3.3 Setup III: Texturized vessel containing RBCs

Figure 6 shows temporal patterns of Pa, a, |τwss|max andPtm,max for a textured MV containing RBCs. The MB shellruptures at tb = 4.7μs, which is later than in the previous con-figurations. The peak values of Ptm and |τwss| are elevatedby 20% and 30%, respectively, but the qualitative transientcomportment of the wall parameters is similar to the previ-ous setups with substantial increase in |τwss|max after shellrupture and a change in the pattern of Ptm,max without anincrease in overall peak values.

The spatio-temporal comportment of Ptm and τwss,z alongΞ , whichwas placed such that it does not run along a cell–celljunction or directly across a cell nucleus bulge, is displayedin Fig. 7. Qualitatively, little difference to the patterns foundfor Setup II is visible, except for a damping of τwss,z at thelocation where Ξ crosses a cell junction (black arrow). Theincrease in peak values at a location where Ξ passes nextto a cell nucleus bulge (white arrow) is not discernible dueto overlay with already present regions of high τwss,z. How-ever, it can be assessed from slightly higher peak values inthe τwss,z islands immediately left of the MB location ascompared to their right counterparts.

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Fig. 5 Spatio-temporal behaviour of wall parameters along Ξ forSetup II. Ptm shows a layered temporal comportment with noticeabledamping in areas where the distance between vessel wall and RBCis below 1μm (narrow regions between vertical dashed lines) and amarked change in pattern at the MB location (z = 0) after rupture. Thepeak values of Ptm do not change appreciably due to shell rupture, butits axial gradients increase after tb. Areas of high spatial transmuralpressure gradient are visible between the innermost RBCs, followedby low gradient regions in the gaps between the RBCs. τwss,z risesstrongly after shell rupture along with its axial gradients, where againconfinement effects due to RBCs are observed

Fig. 6 Temporal evolution around the time point of shell rupture (dot-ted line) of Pa, a, |τwss|max and Ptm,max for Setup III. Shell breakupoccurs later than in previous configurations, and the peak values ofthe wall parameters are elevated by 20–30%. The transient patterns of|τwss|max and Ptm,max are similar to those seen in Setup II

Axial gradients along a longitudinal sample line cannot beconsidered fully representative for this setup because the sur-face texture breaks rotational symmetry. Significant changesinwall parameterswill appear in directions other than the ves-

Fig. 7 Spatio-temporal behaviour of wall parameters along Ξ for aconfiguration with RBCs and vessel wall texture. No major differencesin patterns compared to Setup II can be observed. A discernible reduc-tion of τwss,z is seen around a cell junction (black arrow), while anincrease at the site of a cell nucleus bulge (white arrow) is barely visi-ble due to overlap with an also otherwise present region of strong shear

Fig. 8 Cutaway of a textured vessel containing RBCs (Setup III) withthe luminal vessel face coloured to represent |τwss|. The largest shearvalues are contained between the innermost RBCs. Maxima are visibleat the central cell nucleus bulges, and cell–cell interfaces have slightlylower |τwss| values

sel axis at cell nucleus bulges and cell–cell junctions, wherethe latter could be of particular relevance with respect topossible endothelial damage or changes in permeability. Theshear stress magnitude distribution on the vessel interface att = 5.6μs is shown in a cutaway in Fig. 8. Shear stress levelsincrease around the protruding cell cores and are diminishedat cell–cell junctions. This distribution gives rise to strong,highly localized gradients.

Overall, the introduction of MV wall morphology dampsthe amplitude ofMB oscillation, delayingMB shell breakup.While τwss and Ptm patterns do not differ appreciably fromSetup II, the peak values increase due to variations in localMV cross section. These peak values are observed highlylocalized at the protruding bulges and receding cell–cell junc-tions of the endothelial cell pattern.

4 Discussion

We have analysed a series of three increasingly detailedsetups for the modelling of ultrasound-induced MB vibra-tions and shell breakup inside a MV to assess the influenceof geometric simplifications on the prediction of wall para-meters. System dynamics were calculated using a novelcomputational framework featuring interface tracking capa-bility combinedwith fluid–structure interaction.We assessedthe spatial and temporal comportment of wall parameters

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for all setups and identified newly emerging features whenincreasing the level of geometric detail. The assessed wallparameters have an impact on theMV endothelium, and theirmagnitude and spatial distribution are relevant with respectto inducing cell damage (Wu 2002) or evoking biologicalresponses (McDannold et al. 2006).

The rupture of theMB shell was achieved by exposing thesetup to transient pressure variations of increasing amplitudeuntil the surface stresses surpassed a critical σc. The pres-ence of RBCs did not noticeably change the dynamics ofthe MB (Stride and Saffari 2004) and the time tb until shellbreakup was unaltered compared to Setup I. However, shellbreakup was delayed in Setup III, where the cross sectionof the vessel lumen is locally narrowed by portions of theendothelial cells protruding from the wall. This imposes anincreased damping effect on the forced MB oscillations dueto confinement (Klotz and Hynynen 2010; Wiedemair et al.2014). Consequently, the MB oscillates with reduced ampli-tude and surpasses the rupture threshold at a later time andat higher Pex.

A substantial increase of peak τwss values after shell rup-turewasobserved in all three investigated setups (Figs. 2, 4, 6).The wall shear scales with the surface normal gradient of theflow velocity (9) of the microstreaming caused by the MBvibrations. This velocity is considerably increased after shellbreakup due to the rise inMB oscillation amplitude observedin all setups. This translates to a considerable increase in liq-uid volumedisplacement.Hence, elevatedpost-rupture levelsof τwss are a direct consequence of the increased MB oscil-lation amplitude after shell breakup and should therefore beexpected in all setups.

Unaltered overall peak levels of Ptm before and after shellrupture were found for all investigated setups (Figs. 2, 4, 6).However, the local pressure levels in the vicinity of the MBrise, and a double-peak morphology establishes in the post-rupture patterns. The pressure levels at the MB interface aredetermined according to Eq. (7) by shell tension σ , surfacecurvature κ , viscous forces and internal gas pressure,which isa function ofMB volume according to Eq. (8). The change oftension fromσs toσ0 and the reduction ofμs toμr due to shellbreakup are compensated by an increase in surface velocityand a larger variation in MB volume and, consequently, ininternal gas pressure. This equilibration effect can leave theliquid pressure levels atΓb and hence Ptm basically unaltered.

Overall, shell breakup in MVs is characterized by animmediate substantial increase in oscillation amplitude andwall shear stress accompanied by no appreciable changein transmural pressure. This qualitative comportment isobserved irrespective of the level of geometric complexity.It implies that any consideration on mechanical MV load-ing by encapsulated MBs should include assessment of thedynamics of the ruptured bubble state, because this generallycreates the highest τwss.

The introduction of RBCs (Setup II) leads to a changein shape of the spatio-temporal τwss,z patterns. The axialdisplacement of the RBCs with the flow causes the fluidin the upstream and downstream inter-RBC gaps to movelike a plug. This creates homogenous τwss,z strips in theseregions (Fig. 5), as compared to the observed island struc-tures in Setup I (Fig. 3). In contrast, spatially confined andlocally varying τwss,z patterns are observed in proximity tothe MB. Taken together, these patterns cause strong localgradients, which are confined to the vicinity of the MB, andlow gradient values beyond the proximal RBCs. Therefore,the endothelium directly adjacent to the MB location expe-riences maximum levels of strain, while more distal regionsare less exposed.

The spatio-temporal patterns of Ptm are less affected by thepresence ofRBCs than those ofτwss. Because of the plugflowin the inter-RBC space, homogeneous pressure layers areobserved. These layers are disrupted by faster flow throughthe narrow annular passage between RBC tips and vesselinterface, causing locally reduced pressure levels. Due to thislayering and the increased pressure levels close to the MB,the high-intensity regions of the Ptm gradients are confinedbetween the innermost RBCs. Low levels are visible beyondthe proximal RBCs (Fig. 5). Consequently, the maximumlevels of transverse strain are imposed on the endotheliumdirectly adjacent to the MB location.

The introduction of RBCs into the MV (Setup II) addsthe aspect of confinement of wall parameter variations to thevicinity of the MB. This setup indicates that neighbouringblood cells have substantial impact on local microstreamingand hence on the local MV loading. The fully motile, per-fectly parachute-shaped RBCs aligned along the vessel axisused in our model may in fact underestimate confinementeffects caused by stacked, tilted and deformed RBCs. Bloodcells should thus be taken into consideration when determin-ing the location of maximum vessel straining.

The peak levels of τwss at feature locations of the endothe-lial wall texture (Setup III) are substantially higher than fortexture-free setups (Setups I and II). τwss is increased at cellnucleus bulges and decreased in cell–cell indentations. Bothphenomena can be explained by the, respectively, enhancedand reduced exposure of these structures to flow and hence toshearing. The local surface configuration thus significantlymodulates τwss. The surface features introduced in Setup IIIare rather shallow and could be much more pronounced inreal MVs. Hence, the impact of the real endothelial morphol-ogy can be expected to exceed the levels encountered in thisstudy.

The introduction of endothelial cell texturing results ina substantial increase in overall Ptm levels. However, Ptm,unlike τwss, does not depend directly on the local surfaceshape. Therefore, we cannot infer that more pronounced tex-turingwill necessarily lead to significantly higher Ptm. In this

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context, the overall reduction in vessel lumen cross section,whichmay result frommore prominent endothelial texturing,is relevant.

Vessel wall texture causes a substantial increase in peakwall parameters and creates localized τwss peaks at featurelocations of the endothelial cells, causing punctual loadingof the MV. This implies that the strain on the MV wall is notonly confined to the close proximity of the MB, but is alsofocused on some critical subcellular features.

Themagnitude of Ptm and τwss observed prior toMB shellbreakup agrees with values previously derived with differentmodels (Hosseinkhah and Hynynen 2012; Wiedemair et al.2012). Post-rupture wall parameter levels are in the samerange as reported by (Hosseinkhah et al. 2015) using a differ-ent numerical approach and amodel equivalent in complexityto Setup I. To our knowledge, no experimental data on thedynamic loading of MVs in the presented parameter rangeare currently available.

Beyond the surface texture introduced in Setup III, theinner surface of a real blood vessel is covered by an endothe-lial surface layer (ESL). This approximately 0.5μm thicklayer consists ofmacromolecules that are, in part, anchored inthe cellmembrane (Pries et al. 2000). The ESL could bemod-elled as a porous medium, partially impeding liquid motionand hence impacting local flow patterns near the endothe-lium. However, limited availability of mechanical materialproperties precludes propermodelling of the ESL. Taking theESL into account would likely result in a reduction of shearstress on the endothelium itself. However, shear stress wouldbe transmitted from theESL surface by the torque of glycoca-lyx protein strands to their anchor points at the cellmembrane(Pries et al. 2000). Themembrane-boundmacromolecules ofthe ESL are known to act as mechanotransducers and playa role in the regulation of cell function (Davies et al. 1997).Therefore, the ESL could act as a stress mediator betweenthe flow and the cell membrane.

Sliding of RBCs past the ESL should reduce flow throughthe annular gaps around the RBCs substantially, increasingthe level of confinement created by the presence of RBCs.Hence, taking the ESL into account would likely furtherincrease the reported compartmentalization of shear gradi-ents and pressure gradients as well as the bulk flow characterin the distal intra-RBC gaps. Finally, the ESL reduces theeffective luminal cross section of the vessel with respectto liquid motion. Such reduction increases the damping ofMB oscillations (Klotz and Hynynen 2010; Wiedemair et al.2014), which in turn leads to shell breakup at higher Pex.

The introduced geometrical refinements to the modelsetup discussed in this study are not exhaustive. Parame-ters like RBC spacing, MV size and shape of the endothelialcells could be varied. Also, non-symmetric setups with off-axis MBs, tilted RBCs or a bent MV could offer additionalinsight. While the exact distribution of Ptm and τwss would

change with altered flow patterns in a modified geometricalsetting, the qualitative comportment of these parameters inthe event ofMB shell rupture is expected to be preserved. Thepresented computational framework can handle these modi-fied geometries provided the MB remains entirely immersedin plasma. In contrast, themodelling ofMBs that are in directcontact with the endothelium or with RBCs would requireadditional treatment of contact and wetting, which is notimplemented in the present framework.

The MB shell fracture process itself was not modelled inthis study, since an additional coupling to a two-dimensionaldynamic system at a different time scale was beyond thescope of this work. The fracture was assumed to happeninstantaneously because the fracture time is much smallerthan a MB oscillation cycle. Short-term dynamic phenom-ena that might cause instantaneous peaks of wall parametersare therefore not covered by this approach. Subsequent to theshell fracture, molecular species such as proteins dissolvedin the plasma might adsorb on the liquid–gas interface (Fangand Szleifer 2001), thereby changing its viscoelastic prop-erties. However, the time scale of the diffusion processesrequired for this (Putnam 1975), and in particular the timerequired for proteins to reach the interface by passing throughthe MB’s momentum boundary layer, is substantially largerthan multiple periods of MB oscillation. Adsorption is there-fore not considered for the short time interval investigatedin this study. Nonlinear viscous terms were omitted in theshell model because appropriate choice of initial viscoelasticproperties allows for a comparable fidelity of themodelwhilemaintaining the computational complexity at an acceptablelevel.

Experimental investigation of the modelled processeswould pose great technical challenges due to the small timeand length scales involved. It is, indeed, likely beyond thereach of current technology. Nevertheless, the followingideas may prove useful as starting points for experimentalexamination of at least parts of the presented phenomena:In vitro setups based on cultured endothelial layers mim-icking the BBB have been produced for various purposes(Patabendige et al. 2013; Zhang et al. 2006). MBs immersedin liquid could be positioned close to such a cell layer grownon a flat substrate and excited by US. Employing a high-speed camera (Chin et al. 2003) would allow the recordingof MB kinematics and a comparison to the computationalpredictions. However, this approach would not provide val-ues of local shear stress and pressure. It is, furthermore, notclear how the MBs can be steadied close enough to the celllayer throughout the entire process and how the response ofthe flat in vitro BBB would compared to the real, tubularbarrier. Alternatively, a high-frequency oscillator, e.g. piezo-electric actuator, positioned in close proximity to the in vitrocell layer could be employed to generate controlled shear andpressure conditions similar to the ones reported by ourmodel.

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The BBB response could then be assessed by tracking thepassage of solutes across the endothelium. Yet it is unclearhow exactly the shear and pressure gradients predicted byour model could be achieved, and the precise monitoring ofhighly localized shear stresses and pressure poses a greatchallenge.

Although direct experimental validation is currently notpossible and despite the discussed limitations in scope,this study demonstrates the effects of geometric simplifica-tions on the computational prediction of microvascular wallstresses during MB oscillation and breakup.

5 Conclusions

The most significant change in mechanical wall parametersfound in this study is the substantial rise in τwss caused bythe rupture of the MB shell. This phenomenon is common toall three investigated setups and is locally modulated by theintroduction of RBCs and endothelial cell patterning. Withregard to the safety assessment of treatments includingMBs,it appears to be sufficient to account for MB shell rupturein the basic geometric setup treated here. The application ofconservative safety margins in clinical practice would mostlikely supersede any local modulations caused by RBCs andcell texturing.

On the other hand, the observed confinement and localiza-tion effects may prove to be highly relevant to advancing theknowledge on the mechanisms of BBB opening through MBoscillations. Here, the differences in stress exposure alongthe endothelial cell membrane as well as the compartmental-ization caused byRBCs could be of critical importance.Mostprominently, the confinement of stress gradients to the vicin-ity of the MB and the pronounced local modulation of τwssat the endothelial cell features offer detailed insight into themechanical vessel loading on a microscopic scale. Correlat-ing the predicted locations of maximum mechanical stresseswith sites of increased trans-endothelial substance transportcould help to further elucidate the chain of processes involvedin BBB opening. A controlled reproduction of numericallyattained shear levels, patterns and gradients on endotheliallayers in vitro could be used to find such correlation.

Acknowledgments The authors gratefully acknowledge the funding ofthis research by the Swiss National Science Foundation through NCCRCo-Me and NCCR Kidney.CH.

Appendix: Validation of the IFT algorithm

An IFT algorithm with a modified pressure boundary condi-tion according to Eq. (7) is employed in this study to modelthe dynamics of a non-spherically oscillating encapsulated

MB immersed in liquid. It allows for variations of σs and μs

as a function of MB size, including abrupt changes at shellbreakup or during buckling. This algorithm is validated withrespect to analytical predictions from Eq. (5).

The validation setup comprises a single MB of equilib-rium radius a0 = 0.975μm with variable settings for κsand χ . This MB is enclosed in a fluid sphere of radiusRl = 100μm, mimicking an infinitely large liquid envelope.The liquid is considered to be water with ρl = 1000 kgm−3

and μl = 10−3 kgm−1 s−1. This setup is exposed to a tran-sient pressure field at various excitation frequencies fex andeither constant or linearly growing excitation amplitude Pex.

In a first step, the conformity with predictions in the elas-tic regime and for buckling is established. The maximumtension that the shell can sustain is set to σc = 1Nm−1,which is deliberately high to preclude shell rupture at thisstage of the validation process. Figure 9 shows a compar-ison between the transient radius plots acquired with IFTand predicted by the modified RPE in Eq. (5) for the oscil-lation of a MB driven by an US field of fex = 2.9MHzand Pex = 130kPa. The shell properties were chosen to fita SonoVue� UCA with κs = 15 nN sm−1, χ = 1Nm−1

and a = a0 = 0.975μm (Marmottant et al. 2005). We quan-tify the conformity between the numerical and the analyticalresults using themaximumof the transient relative difference

ε(t) =∣∣∣∣aa(t) − an(t)

aa,max − aa,min

∣∣∣∣ , (11)

where aa(t) represents the transient analytically determinedMB radius and an(t) represents the transient numericallydetermined MB radius, while aa,max and aa,min denote theoverall maximum and minimum of aa(t). The analyticallypredicted and the numerically modelled comportments showexcellent agreement with εmax = 0.45% in a simulation

Fig. 9 Comparison of radius response curves from theoretical predic-tions and numerical results for a configuration with fex = 2.9MHz,Pex = 130kPa, κs = 15 nN sm−1, χ = 1Nm−1 and a = a0 =0.975μm. The numerically determined a(t) shows excellent agreementwith the analytical prediction, having a relative deviation of less than0.45%

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Fig. 10 Comparison between numerical results and analytical predic-tion for shell break up of two MBs with viscoelastic encapsulationof a0 = 0.975μm and a = 0.95μm. An encapsulated MB ofκs = 7.2 nN sm−1, χ = 0.55Nm−1, σc = 0.0615Nm−1 andμr = 0.4μs (top) exposed to fex = 1.5MHz, Pex,i = 100kPa andPex, f = 130kPa with tr = 5μs shows breakup at 3.48 μs. AnotherMB of κs = 15 nN sm−1, χ = 1Nm−1, σc = 0.148Nm−1 andμr = 0.4μs breaks up at 4.92μs when excited with fex = 2.9MHz,Pex,i = 150kPa and Pex, f = 210kPa with tr = 5μs. Both casesshow close agreement between analytical prediction and numericalresults

interval of 10 μs. The numerically acquired data also closelycorrespond to the results based on simulations using thesame parameters published by Marmottant et al. (2005). Asexpected, the radius pattern exhibits compression preference(de Jong et al. 2007; Versluis 2010) and a change in slopeduring contraction.

After checking the predictive capability of the numericalmodel for the elastic regime and the buckling state, we exam-ine its ability to model shell rupture. To achieve a transitionfrom oscillation with encapsulation to oscillation withoutencapsulation, Pex is ramped up from an initial Pex,i to afinal value Pex, f over a time interval tr and stays constantat Pex, f for t > tr . Figure 10 displays a portion of theradius–time curves for two configurations with the instantsof breakup indicated by arrows. Both encapsulatedMBs havea buckling radius of a = 0.95μm.

The top panel of Fig. 10 shows results obtained for fex =1.5MHz, pressure amplitude buildup from Pex,i = 100kPato Pex, f = 130kPa over tr = 5μs and shell properties ofκs = 7.2 nN sm−1, χ = 0.55Nm−1 (Gorce et al. 2000),

σc = 0.0615Nm−1 and μr = 0.4μs . The shell’s breakuptension is surpassed at Pex = 121 kPa and tb = 3.48μs,after which the oscillation amplitude still keeps growingwithincreasing Pex. εmax in the elastic regime does not surpass0.6% and peaks at 1.9% after shell breakup.

Figure 10 (bottom) displays findings for the oscillationof a MB with fex = 2.9MHz, Pex,i = 150 kPa, Pex, f =210 kPa, tr = 5μs, κs = 15nN sm−1, χ = 1Nm−1,σc = 0.148Nm−1 and μr = 0.4μs . Shell breakup occursat 4.92 μs and Pex = 209 kPa. Subsequently, a(t) quicklystabilizes to a new pattern with constant amplitude. εmax inthe elastic regime does not surpass 0.4% and increases to2.4% after shell breakup.

All three presented setups show an agreement betweennumerical results and analytical predictions with a differencebelow 1% before shell rupture and lower than 2.5% afterrupture.

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