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Ultrasonics the Acoustic Radiation Force of a Focused Ultrasound Beam on a Suspended Eukaryotic Cell

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Ultrasonics the Acoustic Radiation Force of a Focused Ultrasound Beam on a Suspended Eukaryotic Cell Ultrasonics 108 (2020) 106205 Contents lists available at ScienceDirect Ultrasonics journal homepage: www.elsevier.com/locate/ultras The acoustic radiation force of a focused ultrasound beam on a suspended eukaryotic cell T ⁎ ⁎ Xiangjun Penga,b,f, Wei Hea,b, Fengxian Xina,c, , Guy M. Genind,e,f, Tian Jian Lub,g, a State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an 710049, PR China b State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China c MOE Key Laboratory for Multifunctional Materials and Structures, Xi’an Jiaotong University, Xi’an 710049, PR China d Bioinspired Engineering and Biomechanics Center (BEBC), Xi’an Jiaotong University, Xi'an 710049, PR China e The Key Laboratory of Biomedical Information Engineering of Ministry of Education, Xi’an Jiaotong University, Xi'an 710049, PR China f U.S. National Science Foundation Science and Technology Center for Engineering Mechanobiology, and McKelvey School of Engineering, Washington University, St. Louis, MO 63130, USA g Nanjing Center for Multifunctional Lightweight Materials and Structures (MLMS), Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China ARTICLE INFO ABSTRACT Keywords: Although ultrasound tools for manipulating and permeabilizing suspended cells have been available for nearly a Acoustic radiation force century, accurate prediction of the distribution of acoustic radiation force (ARF) continues to be a challenge. We Gaussian wave therefore developed an analytical model of the acoustic radiation force (ARF) generated by a focused Gaussian Three layered shell ultrasound beam incident on a eukaryotic cell immersed in an ideal fluid. The model had three layers corre- Eukaryotic cell sponding to the nucleus, cytoplasm, and membrane, of a eukaryotic cell. We derived an exact expression for the Acoustofluidics ARF in relation to the geometrical and acoustic parameters of the model cell components. The mechanics of the cell membrane and nucleus, the relative width of the Gaussian beam, the size, position and aspect ratio of the cell had significant influence on the ARF. The model provides a theoretical basis for improved acoustic control of cell trapping, cell sorting, cell assembly, and drug delivery. 1. Introduction progressive sound field [19]. Yosioka and Hasegawa extended King’s work to compressible spherical particles [20], and extensive subsequent Ultrasound has been applied to manipulate [1] and lyse [2] cells theoretical and experimental works have shown the ARF exerted by a since the 1920s. These were amongst the first contactless particle planar acoustic field on a microsphere [21–23] to be very sensitive to trapping and manipulation methods in biology and continue to find the structure and acoustic properties of the micro-particle. application in bio-medical research [3,4] A particularly sensitive ap- Two theoretical approaches are commonly used to calculate the plication of ultrasound manipulation is acoustic tweezers, which has ARF: the partial-wave expansion method and the ray acoustics method. received particular attention due to its advantages in contamination- The ray acoustics method is limited to cases when the wavelength of the free and label-free cell handling [5–8]. Numerous experimental, theo- acoustic wave is far smaller than the radius of the sphere, but the retical and numerical studies have demonstrated that acoustic tweezers partial-wave expansion method is applicable to an arbitrary frequency can be used to align, move and sort microparticles and cells [3,6,9–11]. range [24]. The partial-wave expansion method has been used to ex- Higher energy versions of these focused ultrasound technologies can be plore a range of waves in spherical coordinates, including plane waves used to permeabilize membranes to ions and drugs [12–16]. [20], Bessel waves [25] and Gaussian waves [26]. Underlying all of these applications is fine control of acoustic ra- Gaussian waves are widely used to model optical and acoustical diation force (ARF). The study of ARF, which is the period-averaged wavefields converging to or diverging from focal regions [26]. Particles force caused by a sound wave, is just like the optical radiation force can become trapped by a Gaussian wave in the focal region [6]. Focused generated by electromagneticwaves striking on electrically or magne- Gaussian ultrasound waves have found utility in bioscience because tically responsive objects [17], therefore has a long history [18]. In- they can trap suspended cells for quantification of the cell’s mechanical vestigation of ARF on microparticles dates back to King’s theoretical properties [27]. study of ARF on a rigid sphere in an ideal fluid subjected a planar In existing theoretical studies of ARF in cell manipulation, cells were ⁎ Corresponding authors. E-mail addresses: [email protected] (F. Xin), [email protected] (T.J. Lu). https://doi.org/10.1016/j.ultras.2020.106205 Received 29 July 2019; Received in revised form 2 March 2020; Accepted 6 June 2020 Available online 18 June 2020 0041-624X/ © 2020 Elsevier B.V. All rights reserved. X. Peng, et al. Ultrasonics 108 (2020) 106205 Fig. 1. Schematic of a Gaussian beam incident upon a triple-layered shell (three-layer model) model of a eukaryotic cell. modeled as homogeneous microspheres [28]. However, eukaryotic cells circular frequency of the Gaussian wave. are heterogeneous, and the nucleus has been reported to affect wave In a progressive focused Gaussian ultrasound wave field, the in- propagation significantly. Thus, the simple homogenous sphere model cident wave pressure is expressed by: does not accurately represent eukaryotic cells. p (,xyzt , , ) As a first step towards understanding how cell shape and hetero- i geneity affect ARF, we studied an ellipsoidal cell consisting of a mem- pW 22 22 0 ⎡ ()xy+ ⎤ ⎧ ⎧ ⎡ ()xy+ ⎤ −1⎛ z ⎞⎫⎫ = exp − exp −ik + z − tan ⎜⎟ brane, cytoplasm, and nucleus. This three-layered model was embedded wz() ⎢ wz2 () ⎥ ⎨ ⎨ ⎢ 2()Rz ⎥ f ⎬⎬ ⎣ ⎦ ⎩ ⎣ ⎦ ⎝ C ⎠⎭ in an ideal fluid that was subjected to a focused Gaussian ultrasound ⎩ ⎭ wave. The partial wave expansion method was employed to calculate exp(−iωt ) (1) the ARF on the cell. Results show that the nucleus and membrane play 2 where w ()zW=+ 1 (/ zfC ) is the beam width, R ()zfzffz=+CCC (/ /) an important role in determining the ARF, along with the aspect ratio of −1 is the radius of curvature of the isophase surface, tan (zf / C ) is the the cell and the size of the cell relative to the Gaussian beam waist. 2 phase factor, and fC = kW /2 is the confocal factor. Although the phase front of the fundamental mode of the incident 2. Theoretical model Gaussian wave is not planar in general, it is very nearly planar in the neighborhood of the beam waist and can be approximated as an With reference to Fig. 1, a focused Gaussian ultrasound wave is acoustic wave with Gaussian amplitude distribution [26]: incident on a eukaryotic cell immersed in an inviscid fluid, with z 222 0 pi (xyzt , , , )≈−+ p0 exp( ( x y )/ W ) exp( ikz1 ) exp(− iωt ) (2) being the location of the center of the cell relative to the origin of the fi Cartesian coordination system, which is also the beam waist center. The We de ne the wavelength in a particular medium as wave with beam waist radius W propagates along the +z direction. The λ ==2/πk11 2 πc /ω and skW= 1/( ). In a spherical coordinate system, with xr= sin θ cos φ, yr= sin θ sin φ, zr= cos θ, the incident acoustic eukaryotic cell consists of an outer cell membrane with radius r1,a wave pressure may be expanded into a generalized Rayleigh wave middle layer (cytoplasm) with radius r2, and an inner core (cell nucleus) series, as: with radius r3. Let the mass densities and acoustic velocities of the surrounding medium, the cell membrane, the cytoplasm and the nu- ∞ n cleus be denoted by (ρc, ), (ρc, ), (ρc, ) and (ρc,), respectively. pi (,rθt , )=+ p0 ∑ Λn i (2 n 1)( jn kr11 )P(cos)exp(n θ − iωt ) 1 1 2 2 3 3 4 4 (3) Corresponding acoustic impedances and wave numbers are n=0 Zi ==ρci i ( i 1,2,3,4) and kωciii==/ ( 1, 2, 3, 4), ω being the where: 2 X. Peng, et al. Ultrasonics 108 (2020) 106205 p fi Γ(p + 1) Γ(pj++ 1/2) 2 j generated by the sound eld over the instantaneous surface St()of the Λ2p = ∑ QQs00(4−− )exp( iαz10) Γ(p + 1/2) ()!!pjj− sphere, as: j=0 (4) Fn()tp=− ((,,))rθtp− dS Λ21p+ ∫∫St() 0 (13) p Γ(p + 1) Γ(pj++ 3/2) 2 j = ∑ ()QQjQQs01−− 1( −4) 0 where n is the outward normal to St(). To evaluate the ARF, the excess Γ(p + 3/2) ()!!pjj− j=0 of pressure should be taken up to second-order terms in the velocity fi exp(−iα10 z ) (5) potential. For a periodic wave, the ARF is de ned as a time-averaged quantity over period of the sound field. The time-averaged force acting 2 2 Here, Q00=+1/(1 2iz / l), Q11=−2/[αl ( i 2 z 0 / l ) ], lαW= 1 , jn (·) is on a sphere immersed in an infinite ideal fluid is: the spherical Bessel function of the first kind, Pn(·) is the Legendre polynomial of order n, and Γ(·) is the Gamma function. Fnpp dS 〈〉=−∫∫St() 〈() −0 〉 The scattered wave field can be expressed as: =−∫∫ ρv 〈()ntnntn + v v 〉 dS ∞ S0 n (1) ps (,rθt , )=+ p0 ∑ Λn i (2 n 1) Ahn n ( kr11 )P(cos)exp(n θ− iωt) 2 ⎡ 1 ρ ∂ψ 1 2 ⎤ n=0 (6) + ∫∫ 2 −〈∇〉ρψ|| n dS S0 ⎣ 2 c ()∂t 2 ⎦ (14) in which An is the scattering coefficient to be determined by the boundary condition. Therefore, the total wave field outside the three- where 〈·〉 represents the time average, t is an in-plane unit tangential layer model (eukaryotic cell) takes the form: vector of St(), S0 is the surface of the target at its equilibrium position, ∂ψ 1 ∂ψ dSrdrd= θ, and the parameters vnr| = r =− and vtr| = r =− are the p rθt 1 ∂r 1 r ∂θ 1 (,1 ,) radial and tangential components of the velocity at the surface, re- ∞ n (1) spectively.
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