Numerical Modeling of Non-Spherical Response of Therapeutic Encapsulated Microbubbles to Ultrasound

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1 Numerica Modeing of Non-Spherica Response of Therapeutic Encapsuated Microbubbes to Utrasound Chao-Tsung Hsiao and Georges L. Chahine DYNAFLOW, INC J Iron Bridge Road Jessup, MD 0794 Abstract. A 3-D zero-thickness she mode was deveoped to simuate the non-spherica response of therapeutic encapsuated microbubbes subjected to utrasound excitation. The 3-D mode was compared with previous deveoped modes for vaidation. This mode was then utiized to study the bubbe break-up during non-spherica deformations resuting from the presence of a nearby soid wa. The effect of she thickness and standoff distance to the soid wa on the bubbe break-up was studied parametricay. A diagram of bubbe shapes was derived to categorize bubbe forms at break-up, and four zones were identified. The diagram provides the good conditions for drug and gene deivery. Keywords: Therapeutic Utrasound; Encapsuated Bubbe Dynamics; Cavitation; Drug Deivery, Boundary Eement Method PACS: Ei, Sh, Hj, dd, dp INTRODUCTION Encapsuated microbubbes formed by a gas core and a viscous she have been used in utrasound medica appications for many years. The most estabished use of encapsuated microbubbes is acoustic contrast agents to enhance utrasonic images quaity [1,]. However, due to their inert properties under norma bioogica conditions and reactive character when subjected to utrasound excitation, there are rapidy emerging appications for microbubbes for targeted drug and gene deivery [3-5]. Severa ways of using encapsuated microbubbes for targeted drug and gene deivery have been studied. One approach is to inject reguar utrasound contrast agents and drug-containing vesices simutaneousy [5,6]. Then, a sufficienty strong utrasound puse is appied to the targeted area such that the contrast agents expand exposivey and coapse vioenty near ces and tissues to produce sonoporation; a temporary formation of pores on the ce membrane, which assist arge moecue drugs deivery from carriers to the interstitia tissue. An aternative approach is to incorporate the drug directy into the encapsuating ayer of the microbubbes [7-9]. When subjected to utrasound excitation, the she is forced to fragment and reease the payoad to the targeted area. Such drug-aden encapsuated microbubbes need a thick she to increase the payoad because the voume of the drug deivered by the microbubbes is reativey sma as compared to other means of drug deivery. Further improvement of drug-aden encapsuated microbubbes is to attach a igand that is

2 targeted to specific endotheia receptor [10]. This method aows the encapsuated microbubbes to adhere to the endotheia surface and thus increase oca microbubbe accumuation and targeted deivery. However, sonoporation is therapeutic ony at ow doses of utrasound. An increase in the utrasound intensity resuts in ces death and other detrimenta bioogica side effects. Therefore, characterization and understanding of the fragmentation mechanism of encapsuated microbubbes is pivota to its use for drug deivery. Athough many studies [11-14] have been dedicated to investigate she break-up of encapsuated microbubbes, they are imited to spherica modes and no rea ife deformations are incuded. From previous studies on bubbe dynamics [15-17], it is known that an osciating bubbe may form a re-entrant jet during its coapse depending on its distance to nearby boundaries. A simiar behavior is expected with the encapsuated microbubbes. Indeed, micro-jetting and micro-streaming associated with the vioent non-spherica breakup have been hypothesized as mechanisms to enhance drug deivery [18]. Understanding this phenomenon is particuary important for drug-aden microbubbes baring surface igands to adhere to the endotheia surface because of their sma standoff from the tissue surface. Hsiao et a. [19] deveoped a 3-D finite-thickness she numerica mode based on a domain decomposition scheme to study the dynamics of encapsuated microbubbes. The computationa domain is composed of inner domains covering the shes, and an outer domain covering the host iquid medium. The inner domains are discretized and soved using an unsteady Navier-Stokes sover, 3DYNAFS-VIS, whie the fow fied in the outer domain is soved using a BEM-based potentia fow sover, 3DYNAFS- BEM. This method can best describe the dynamics of thick-sheed encapsuated microbubbes, but at high computationa cost. In this paper, to speed up computations, we present a 3-D zero-thickness she mode which repaces the she thickness with an equivaent diatationa viscosity term in the boundary condition. NUMERICAL MODELS The present 3-D zero-thickness she mode soves the fow fied ony in the outer domain, which is assumed to be irrotationa and incompressibe. This assumption aows the definition of a veocity potentia, φ, which satisfy the Lapace equation: φ = 0 and u = φ, (1) where u is the veocity vector. A boundary integra method is used to sove Equation (1), based on using Green s theorem, which can be written in the foowing form: ( φ G G φ ) d Ω= [ φ G G φ ] ds Ω n. () S Ω is the domain of integration, S is the boundary surface, n is oca norma unit vector, and G is Green's function. If G is seected to be harmonic everywhere but at some discrete points, such as G = 1/ x y, where x is a fixed point in Ω and y is a point on the boundary surface S, Equation () reduces to Green s formua with απ being the soid ange at x encosing the domain Ω: απφ( x) = n φ( y) G( x,y) G( x,y) φ( y ) ds, (3) S y [ ]

3 To sove Equation (3) either the veocity potentia, φ, or the norma veocity, φ/ n, to the boundary surface S needs to be provided. The boundary surface for the current probem (encapsuated microbubbe near a wa) incudes a rigid wa and the iquid/she interface. For the rigid wa, a no-penetration condition, φ/ n =0, is used to specify the norma veocity. For the iquid/she interface, the veocity potentia can be deduced from integration of the Bernoui equation: φ P( t) P 1 = φ φ, (4) t ρ where P(t) is the imposed acoustic pressure, ρ is the iquid density, and P is the iquid side pressure on the iquid/she interface. P can be determined through the norma stress baance across the iquid/she interface. However, in the zero-thickness she mode, the inner domain is degenerated and thus the iquid/she interface and the iquid/gas interface coincide. To express the norma stress boundary condition across the iquid/she/gas interface, we introduce an effective norma stress, τ n, to represent the norma stress due to the she. If the bubbe was spherica, then the norma stress baance across the interface woud be written as: γ R& P = pg 4μ + τ n, (5) R R where R is the bubbe radius, R is the bubbe wa veocity, μ is the iquid viscosity,γ is the surface tension and p g is the pressure of the gas core. In [1] Hsiao et a. found that if τn = 1 d0 μsr& / R, where μ s is the she viscosity and d 0 is the initia she thickness, then the spherica finite-thickness she mode recovers the spherica zerothickness she mode derived in [13], using a diatationa viscosity in the norma stress baance equation. To adapt Equation (5) for the current 3-D zero-thickness mode, a oca spherica motion approximation such that R ~/ C, where C is the interface curvature, and R& ~ φ n is appied. As a resut, the norma stress boundary condition is written as: φ P = pg Cγ C( μ + 3d0 μsc). (6) n RESULTS The resuts and computations in this study focus on thick oi-sheed microbubbes deveoped for drug deivery. For exampe, ImaRx Therapeutics (Tucson, AZ) has produced a prototype with a triacetin she. For a triacetin-sheed bubbe in water, we have the foowing fuid properties used in the numerica computations: 3 μ = 0.08kg ms, ρ = 1000 kg m, μ = 0.001kg ms, γ = 0.068kg s. s Vaidity of the 3-D Zero-Thickness Mode To vaidate the 3-D zero-thickness she mode we have compared its soutions with those obtained by a spherica finite-thickness she mode which has been vaidated with experimenta measurement by Hsiao et a. [19]. FIGURE 1(a) shows the comparison between these two modes for a bubbe with initia outer radius R 0 =1.7

4 μm and an initia thickness of d * 0 =10-4 μm suspended in water at 1 atm ambient pressure driven by a acoustic wave of ampitude, P a =0.Mpa, and frequency, f =.5MHz. It can be seen that the soution of the zero-thickness mode matches very we with the thick she case. FIGURE 1(b) compares the maximum radius of the osciation versus normaized with the initia she thickness after periodic osciations are reached in both modes. It can be seen that the zero-thickness mode deviates significanty from the finite-thickness mode when the she thickness exceeds more than 10% of the initia outer radius, but matches very we with it, otherwise. (a) Zero thickness mode Finite thickness mode (b) Finite thickness mode Zero thickness mode R Rmax/R t d/r0 FIGURE 1. Comparison between spherica finite-thickness she mode and 3-D zero-thickness she mode of (a) the time history of the sheed bubbe radius for d0*=10-4 μm and (b) the normaized maximum radius versus normaized initia she thickness. FIGURE. Comparison of the bubbe shape variations between the 3-D finite-thickness mode and the zero-thickness she mode at severa time steps during the first bubbe osciation cyce. The 3-D zero-thickness she mode was aso appied to simuate the dynamics of a thick-sheed microbubbes near a rigid wa under utrasound acoustic excitation with P a =1Mpa, and f =.5MHz. The thick-sheed microbubbe had an initia inner radius of 1. μm and an initia outer radius of 1.7μm and was initiay ocated 3.6 μm away from the soid wa. The resuts of the zero-thickness mode are compared to those of the finite-thickness she mode for sma standoff conditions. During the comparison we found that using the average across the thickness initia radius of R 0 =1.45 μm for the zero-thickness she mode best match the finite-thickness she mode resuts. FIGURE shows a comparison of the bubbe shape variations between the two modes at severa time steps during the first bubbe osciation cyce. It is seen that the bubbe shape of the zero-thickness she mode amost fas between the shape of the inner and outer she interfaces obtained from the finite-thickness she mode

5 when she thickness is think. The soutions start to deviate when the thickness become very thick near the end of bubbe coapse. Effect of She Thickness and Standoff on She Break-up It has been shown by Chahine [] that an exposion bubbe may become pinched off and cut into two instead of forming a reentrant jet during its coapse depending on its initia distance to a soid boundary. For the current probem other parameters such as she thickness, she materia, initia bubbe radius, acoustic pressure ampitude, frequency may aso infuence the form of the breakup. In this study we ony focus on the effect of standoff and she thickness on the she breakup. Simuations were conducted for a triacetin-sheed bubbe with an initia mi-thickness radius of R 0 =1.45 μm under utrasound acoustic excitation with P a =1 Mpa, and f =.5 MHz. To cover a wide range of parametric space, we conducted simuations for five different standoff distances, i.e. X = 1.45,.85, 4.35, 5.80 and 7.5 μm, and five different she thicknesses, i.e. d 0 = 0.1, 0.3, 0.5, 0.7 and 0.9 μm. The simuation of each case was continued for more than one osciation cyce uness breakup occurred earier. FIGURE 3. Diagram of bubbe shapes within the parametric space studied. Within the parametric space studied, FIGURE 3 shows a diagram of the bubbe shapes either at the end of the simuation due to bubbe becoming muti-connected or at the end of the first bubbe osciation cyce. According to the ast shape of the bubbe surface we can divide the diagram into four zones: In Zone A, a reentering jet is found to punch through the bubbe. In Zone B, a ring type of jet is found at touch down. In Zone C, the bubbe is found to pinch off into two. In Zone D, the bubbe surface does not become muti-connected during the first cyce. Based on this diagram, the bubbe break-up in Zones A and B coud be the best conditions for drug or gene deivery. That is because the break-up of bubbe in the first cyce aows ower utrasound dose and the jet directed to the wa can hep drug or gene partices

6 penetrate the ce membrane. Athough the bubbe aso breaks up due to pinch-off in Zone C within the first cyce, there is no wa-heading jet to enhance the sonoporation. CONCLUSIONS A 3-D zero-thickness she mode was deveoped to study she break-up mechanisms for a triacetin-sheed microbubbe under utrasound excitation. It is shown that the zero-thickness she mode can predict we the dynamics of encapsuated microbubbes if the she thickness is ess than 10% of the initia she outer radius. A parametric study on she thickness and standoff shows a variety of bubbe shapes at break-up. Four zones are identified in the parametric space. Based on the diagram, the best condition for drug and gene deivery enhancement can be provided. ACKNOWLEDGMENTS This work was supported by the Nationa Institute of Biomedica Imaging and Bioengineering under SBIR Phase I and Phase II programs. REFERENCES 1. H. Becher, P. N. Burns, Handbook of Contrast Echocardiography: Left Ventricuar Function and Myocardia Perfusion, New York, Spring Verag, 000. pp B. B. Godberg, J. S. Raichen and F. Forsberg (Eds.), Utrasound Contrast Agents: Basic Principaes and Cinica Appications, London, Martin Dunitz, K. W. Ferrara, R. Poard and M. Annu Rev. Biomed. Eng., 9, (007). 4. C. R. Mayer and R. Bekeredjian. Advanced Drug Deivery Reviews, 60, (008). 5. J. Song, J. C. Chappe, Q. Ming, E. J. VanGieson, S. Kau and R. J. Price. J. Am. Co. Radio. 39, (00). 6. R. J. Price, D. M. Skyba, S. Kau, and T. C. Skaak. Circuation 98, (1998). 7. E. C. Unger, T. O. Matsunaga, T. McCreery, P. Schumann, R. Switzer and R. Quigey. Eur. J. Radio. 4, (00). 8. S. Y. Chen, J. H. Ding, R. Bekeredjian, B. Z. Yang and R. V. Shohet. Proc. Nat. Acad. Sci. USA, 103, (006) 9. P. Hauff, S. Seemann, R. Reszka, M. Schutze-Mosgau and M. Reinhardt. Radioogy 36, (005). 10. M. S. Tartis, J. McCaan, A. F. Lum, R. Labe, and S. M. Stieger. Utrasound Med. Bio. 3, (006). 11. C. C. Church. J. Acoust. Soc. Am. 97(3), (1995). 1. P. Marmottant P, S. van der Meer, M. Emmer, M. Versuis, N. de Jong, S. Higenfedt and D. Lohse. J. Acoust. Soc. Am. 118(6), (005). 13. K. Sarkar, W. T. Shi, D. Chatterjee and F. Forsberg. J. Acoust. Soc. Am. 118(1), (005). 14. J. S. Aen, D. J. May and K. W. Ferrara. Utrasound in Med. & Bio. 9(6), (00). 15. L. A. Crum. Journa de Physique. cooque c8, suppement au N 11, tome 40: c8-85 (1979). 16. G. L. Chahine. Appied Scientific Research 38, (198). 17. S. Zhang, J. H. Duncan, G. L. Chahine. J. Fuid Mech. 57, (1993). 18. M. W. Mier. Utrasound in Med. & Bio. 6, S59-S6 (000). 19. C.-T. Hsiao, X. Lu, G. L. Chahine. To appear in Utrasound in Med. & Bio. (010) 0. G. L. Chahine, T. O. Perdue, C. B. Interaction between underwater exposion bubbe and a soid submerged body. DYNAFLOW, INC. Technica Report , August, C.-T. Hsiao, X. Lu, G. L. Chahine. 3D modeing of the dynamics of therapeutic utrasound contrast agents SBIR Phase I fina Report. DYNAFLOW, INC. Technica Report M6018-NIH-1, May, G. L. Chahine. Numerica and experimenta study of exposion bubbe crown jetting behavior.. DYNAFLOW, INC. Technica Report ONR, Juy, 1997.