  Subharmonics emitted from microbubble-based ultrasound contrast agents (UCA)
are considered as a noninvasive indicator for diagnostic applications in
medical fields. Contrast ultrasound imaging using the modality of subharmonics
could be also better than that of using harmonics. Unlike the harmonics of an
UCA that increase with increasing the amplitude of excitation pressure,
subharmonics occur only when the excitation pressure exceeds a threshold value.
The excitation pressure inducing the onset of subharmonic components from the
UCA during nonlinear oscillations is known as the subharmonic threshold.
Although numerous studies on the subharmonics of free bubbles or UCAs have been
carried out for decades, several experimental observations, such as the shift
of the subharmonic resonance and subharmonic threshold at low excitation
pressures, cannot be well characterized by existing theories. An analytical
method for resolving these inconsistencies remains incomplete. The present
article theoretically investigates the subharmonic threshold of an UCA through
the techniques of energy balancing in the phase plane and instability analysis
of subharmonic-free oscillations. The influence of shell parameters of the UCA
on the subharmonic threshold is studied. Those experimental observations are
explained by the amplitude-dependent nonlinear resonance and damping. The
effect of nonspherical oscillations on subharmonic emissions is also discussed.
