Mechanical Model of Biological Cell

Since the time of my PhD research, I have been interested in mechanical properties of cell and particularly cancel cells.  What is the difference in mechanical properties of normal and cancer cell? Can the cancer cell be killed by ultrasound. These are the questions I want to find answers in my research. During work on my PhD thesis, together my colleague Vadim Levin developed a mechanical model of a biological cell which we called a "shell model" (Zinin, Levin, Biophysics (USSR),  32, 202, 1987; Zinin, Allen, Levin, Phys. Rev. E. 72, 61907, 2005). Within the shell model, a cells assumed to have a spherical shape of radius a. A spherical shape of the cell is assumed for two reasons. First, it is possible to obtain an analytical solution for spherical objects. Second, many bacteria indeed have a spherical shape (cocci). Within the shell model, the motion of the cell is composed of three components: the motions of the internal fluid and the surrounding fluid, and the deformation of cell shell.  The fluid within and outside the cell is characterized by a density, a velocity of sound, a compressional or bulk viscosity and a shear viscosity.

For the cells considered in our study, the thickness of the shell h is much less than the characteristic size of the cell a: h << a. For thin shells the equations of motion include the total values of the internal forces distributed over the thickness of the shell. Primarily two forces resist deformation of the shell of the cell: constant tension, To, and the force of surface elasticity. The resistance to the change in the surface area is characterized by the area compression modulus K_A, and the resistance to the shear deformation by modulus µ . The Hook's law  a  cellular shell have the following form :

The Mechanical Resonances of Cells

If a mechanical model of the biological cell is developed then the first question that strikes your mind is "Does such a system has mechanical resonances?". The question of resonance in mechanical oscillations of cells was originally investigated by E. Ackerman (E. Ackerman, Bull. Math. Biophys. 13, 93 1951). Based on early works of Rayleigh and Lamb (Landau, Lifshitz, Fluid Mechanics, vol. 6 (Pergamon Press, Oxford, 1959)  Ackerman estimated resonance frequencies and qualities of red blood cells modeling the cells as a spherical, isotropic elastic shells filled with and surrounded by viscous fluids. However, his simplified cell model and the mathematical errors in the derivation of the quality of the natural cell's oscillations impose limitations on potential applicability of his work. His experimental results have never been reproduced; nevertheless, they cannot be completely disregarded. A more rigorous theory of the natural oscillations of biological cells based on a more complete understanding of the elasticity of cellular materials was subsequently developed (Zinin, Levin,  Maev. Biophysics (USSR),  32, 202, 1987). The obtained dispersion equation has a complex form and only simple approximations were obtained for red blood cells (RBC). It was found that due to small values of the shear elastic modulus of the RBC and high viscosity of the internal fluid, the natural oscillations of the RBC were always aperiodic relaxation movements.

We also applied a shell model for a biological cell to estimate quality of the natural vibrations of the specific types of bacteria (Zinin, AllenI, Levin, Phys. Rev. E. 72, 61907, 2005). We found that high quality resonances were possible for several types of bacteria, which have radii greater than 5 µm. It is more likely that Gram positive bacteria would have resonances than Gram negative bacteria because the cell wall (shell) of the Gram-positive bacteria is much stiffer than that of Gran negative bacteria. It may be possible to achieve optimized ultrasound destruction of specific bacteria with the modulated acoustic radiation pressure technique developed for exciting shape resonances in drops by Marston and Apfel.

Left: Resonance oscillations of the cells were observed by Miller who imaged standing wave (high-quality resonance oscillation) in the cell wall of algae in a 1 MHz ultrasonic wave (Miller, IEEE Trans. Ultrasonic. Ferroelect. Freq. Contr. 33, 165, 1986).

It is interesting to note that the frequencies of the normal modes of vibration for a spherical virus particle were estimated by Ford in 2003 (Ford, Phys. Rev. E 67, 051924, 2003). Subsequent theoretical studies led to the successful detection of the normal oscillations for low-frequency vibrational modes of bacteriophage M13 in water by Raman spectroscopy (Tsen et al., J. Biomed. Opt. 12, 024009, 2007).

Deformation of Biological Cells in the Acoustical Field of an Oscillating Bubble

Is it possible to kill cell using ultrasonic contrast agents? I think the answer is yes! Recently, we develop a theoretical framework of the interaction of microbubbles with bacteria in the ultrasound field using a shell model of the bacteria (Zinin,  Phys Rev. E. 79 021910, 2009). It was predicted that deformation of the cell wall at the frequency is high enough to rupture small bacteria such as E. coli.

What about cancer cell such as HeLa cells? That is the question I would like to answer in of my current research.