Numerical analysis for transverse microbead trapping using 30 MHz focused ultrasound in ray acoustics regime

Abstract

We have recently devised a remote acoustic manipulation method with high frequency focused ultrasonic beam of 30-200 MHz, and experimentally realized it by the intensity gradient near the beam's focus. A two-dimensional (or transverse) acoustic trapping was demonstrated by directly applying the acoustic radiation force on lipid spheres and leukemia cells that were individually moved towards the focus. Only longitudinal waves were then considered because both target and propagation media involved were fluid e.g., water or phosphate buffer saline. In order for our current technique to be applicable to bead-based assay approaches using micron-sized polystyrene spheres as in optical tweezers, the possibility of microbead trapping must first be investigated from theoretical perspective. In this paper, a simulation study in the ray acoustics regime (bead diameter D>ultrasonic wavelength λ of trapping beam) is thus undertaken by calculating the acoustic radiation force on a polystyrene bead generated from 30 MHz focused beam of Gaussian intensity profile. Analytical trapping models for a bead located in the near-/far fields and on the focal plane are derived by incorporating both longitudinal- and shear force terms into our existing ray acoustics model for liquid targets. The net radiation force is computed by adding the two terms, and the resultant trapping force is defined as a negative net radiation force in the positive transverse direction (y>0). The magnitude of the trapping force and its spatial range are evaluated in the same direction by varying bead size (D=2 λ=100 μm or 3 λ=150 μm), location, and transducer's f-number (= 1 or 2). When the bead size is increased, all force components exerted on the bead is increased in the near field of ultrasound for both f-numbers. With f-number=1 being used, the peak longitudinal-, shear- , and net forces are -3.1 nN, -9.8 nN, and -12.7 nN for D=2 λ, whereas the forces are increased to -5.3 nN, -21.0 nN, and -25.7 nN for D=3 λ. In case of f-number=2, the peak magnitudes of the forces are 1.2 nN, -7.8 nN, and -6.6 nN for D=2 λ, whereas they are increased to 5.9 nN, -17.1 nN, and -12.0 nN for D=3 λ. With f-number=1, the net trapping forces at (0, y, -2 λ) can be reached to -39.8 nN for D=2 λ and -65.2 nN for D=3 λ, and -7.8 nN for D=2 λ and -15.2 nN for D=3 λ at (0, y, -14 λ). When f-number=2 is used, the peak trapping forces at (0, y, -2 λ) can be -3.4 nN for D=2 λ and -5.9 n N for D=3 λ, while they are -6.3 nN for D=2 λ and -12.0 nN for D=3 λ at (0, y, -14 λ). In the near filed, the bead can be trapped in the range from 0 to 340 μm for D=2 λ, and from 0 to 380 μm for D=3 λ. The trapping range Rtrap with f-number=2 lies from 0 to 295 μm for D=2 λ, and from 0 to 340 μm for D=3 λ. As either a larger bead or a lower f-numbered trapping beam is used, a stronger trapping force can be produced in the region. When a bead is more closely positioned to the focus, the trapping occurs in multiple locations and the net force variation becomes more complicated. In the far field, with f-number=1 being used, the peak longitudinal-, shear- , and net forces are 4.6 nN, 6.8 nN, and 11.4 nN for D=2 λ, whereas the forces are increased to 11.4 nN, 12.1 nN, and 23.6 nN for D=3 λ. In case of f-number=2, the maximum value of each force is 4.4 nN, 1.8 nN, and 5.0 nN for D=2 λ, respectively, whereas it becomes 12.3 nN, -0.7 nN, and 10.6 nN for D=3 λ. The bead is forced to move away from the beam axis by a positive net force for y>0 and a negative net force for y<0. With f-number=1, the peak repulsive forces at (0, y, 5 λ) can be 25.8 nN for D=2 λ and 49.9 nN for D=3 λ, and 3.4 nN for D=2 λ and 7.5 nN for D=3 λ at (0, y, 20 λ). When f-number=2 is used, the forces at (0, y, 5 λ) can be 3.9 nN for D=2 λ and 9.5 nN for D=3 λ, while they are 3.7 nN for D=2 λ and 7.8 nN for D=3 λ at (0, y, 20 λ). As the bead is placed farther away from the focus, the net repulsive force is reduced and yet the bead trapping is difficult throughout the far-field region. On the focal plane, with f-number=1, the peak longitudinal-, shear- , and net trapping forces are 31.8 nN, -36.2 nN, and -16.5 nN for D=2 λ, whereas the forces are changed to 73.9 nN, -58.2 nN, and -42.7 nN for D=3 λ. In case of f-number=2, the peak magnitudes of the forces are 6.4 nN, -7.0 nN, and -1.6 nN for D=2 λ, whereas they are increased to 18.1 nN, -15.8 nN, and -3.9 nN for D=3 λ. The Rtrap ranges from 33 to 131 μm for D=2 λ, and from 52 to 170 μm for D=3 λ when f-number=1. The Rtrap with f-number=2 is then located from 0 to 238 μm for D=2 λ, and from 73 to 288 μm for D=3 λ. Hence, the results suggest that microbeads such as polystyrene spheres may acoustically be controlled as remote handles with focused sound beam for bead-bioassay applications, where trapped beads can be used to induce cellular response change by exerting mechanical stress on single cells.

Keywords: High frequency focused ultrasonic beam; Longitudinal component; Polystyrene bead; Ray acoustics regime; Shear component.