Deformation of biological cells in the acoustic field of an oscillating bubble

In this work we develop a theoretical framework of the interaction of microbubbles with bacteria in the ultrasound field using a shell model of the bacteria, following an approach developed previously [P. V. Zinin et al., Phys. Rev. E 72, 61907 (2005)]. Within the shell model, the motion of the cell in an ultrasonic field is determined by the motion of three components: the internal viscous fluid, a thin elastic shell, and the surrounding viscous fluid. Several conclusions can be drawn from the modeling of sound interaction with a biological cell: (a) the characteristics of a cell’s oscillations in an ultrasonic field are determined both by the elastic properties of the shell the viscosities of all components of the system, (b) for dipole quadrupole oscillations the cell’s shell deforms due to a change in the shell area this oscillation depends on the surface area modulus $K_A$, (c) the relative change in the area has a maximum at frequency $f_K\sim \frac{1}{2\pi }\sqrt{K_A∕\left(\rho a^3\right)}$, where $a$ is the cell’s radius and $\rho$ is its density. It was predicted that deformation of the cell wall at the frequency $f_K$ is high enough to rupture small bacteria such as $E$. coli in which the quality factor of natural vibrations is less than 1 $(Q<1)$. For bacteria with high value quality factors $(Q>1)$, the area deformation has a strong peak near a resonance frequency $f_K$; however, the value of the deformation near the resonance frequency is not high enough to produce sufficient mechanical effect. The theoretical framework developed in this work can be extended for describing the deformation of a biological cell under any arbitrary, external periodic force including radiation forces unduced by acoustical (acoustical levitation) or optical waves (optical tweezers).

Figure 1

Stresses on the element of the spherical shell in the spherical coordinates $(r,\theta ,\phi )$.

Figure 2

(Color online) Diagram of the interaction of a vibrating bubble with bacteria in ultrasound field. Following notations were introduced in the text and shown in the figure: $xyz$ are Cartesian coordinates; $\theta$ is the zenith angle in the spherical coordinate system; $a$ is the radius of the cell; $R_o$ is the equilibrium radius of the bubble; $L$ is the distance between center of the cell $\left(O\right)$ and center of the bubble $\left(Q\right)$; $P$ is a point on the cell surface; $R$ is the distance between center of the bubble and point $P$; $k_o$ is the wave vector of the incident plane sound wave; $c$ and $\eta$ are sound velocity and viscosity of the corresponding liquids.

Figure 3

(Color online) Radius of a bubble at its linear resonance as a function of the frequency from Minnaert’s Eq. (67).

Figure 4

(Color online) Frequency dependence of the area deformation $(n=2)$ of the B. emersonii in the vicinity of an oscillating bubble calculated with the parameters from Table .

Figure 5

(Color online) Frequency dependence of area deformation of the E. coli in the vicinity of oscillating bubble calculated for two radii of the cell with the parameters taken from Table .

Figure 6

(Color online) Frequency dependence of the area deformation $(n=2)$ of the E. coli in the vicinity of oscillating bubble calculated for different internal and external viscosities.