## Optoacoustic imaging:## Velocity Potential related to energy deposition## Steven L. Jacques, Guenther Paltauf |

Now consider time-resolved velocity potential at a position distant from the site of energy depostion. The following equation is based on the thermoelastic wave equation:

where δ() indicates a Dirac delta function whose value is zero unless t = (|r' - r_{i}|/c_{s}), and c_{s} is speed of sound. The subscript _{i} refers to the i^{th} detector in an array of detectors. The term d^{3}r' refers to an incremental volume element dV in the volume integration.

W(r',t) directly contributes to φ(r_{i},t), attenuated by a 1/r dependence, 1/(|r' - r_{i}|). The summation of contributions (W)(dV)/(|r' - r_{i}|) from all positions whose contribution would arrive at r at time t yields the velocity potential at position r and time t. Such an integration can gather the contributions from various objects or from various parts of an extended irregular object. The φ(r_{i},t) is calculated as a function of time by repeated application of the above integral at different time points t. Finally, the time derivative of φ(r_{i},t) times -ρ yields the time-resolved pressure P(r,t).

Contributions to pressure P may be either positive or negative due to compressive and tensile components. The interference of positive and negative pressure waves arriving at a point of observation can be a difficult task to compute. However, by doing the bookkeeping in terms of the velocity potential, the problem is simple integration with a subsequent time derivative which is relatively simple to implement.