On the relationship between power mode and pressure amplitude decorrelation

Estimation of mean transit time, along with tissue blood volume, are important factors in determining soft tissue perfusion. Recently, power mode decorrelation techniques have been successfully used to estimate mean transit time of red blood cells or contrast material through a region-of-interest (ROI) both in laminar flow phantoms and in vivo. The previously described theory for power mode decorrelation derives from a phenomenological stochastic differential equation (Langevin equation) based on conservation of matter, relating the detected signal power to the measured rate of decorrelation. Given the experimental support for power mode decorrelation as a method to estimate mean transit time, it becomes important to determine the relationship between the phenomenological parameters that appear in the corresponding stochastic equation and system parameters, such as the transducer point response function. With this equation as a starting point, and using the fact that the pressure amplitude is a Gaussianly distributed random process, the following stochastic differential equation for the pressure amplitude p(t) is derived, a necessary first step in establishing the relationship between the measured decorrelation rate and system parameters (i.e., point response function): dp(t)/dt = -(v/2+2ik x v)p(t)+f(t), where v/2 represents the rate of decorrelation, 2k x v is the Doppler shift for an insonating wave vector k and particle velocity v.f(t) is a stationary, white noise Gaussian random process.