A nonlinear model of cell interaction with an acoustic field

A theoretical and experimental nonlinear analysis of cellular response/displacement to ultrasound excitations is presented. Linear cell models can predict the resonant frequency (f_R∼5MHz), but only a nonlinear analysis can reveal the amount of mechanical energy that couples into the cell and the bifurcation behavior of the cell when it is excited near resonance. The cell dynamics is described by the nonlinear viscoelastic constitutive behavior of the cytoplasm, nucleus and their respective membranes, in the presence of a fluid with an oscillating pressure field. The method of multiple scales is used to derive the amplitude of oscillation of the cytoplasm and nucleus as a function of frequency. A major finding is the existence of multiple solutions for a range of sub-resonant frequencies. At positive detuning (f>f_R), the mechanical energy that couples into the cell is small, it is higher at resonance but significantly higher at sub-resonant frequencies in the multiplicity range. Experimentally it was shown when 3.5 MHz is approached sub- and supra-resonance and 6.5 MHz is approached sub-resonance, gene expression was statistically higher than that when stimulated directly. Thus, there exists an optimal range of frequencies for ultrasound treatment – in the region of multiplicity where deformation and thus mechanical energy coupling is maximized. The ultrasound protocol must be designed to operate at the solution associated with the higher mechanical energy – thus the start-up conditions should be in the domain of attraction of the high energy solution.