When energy is deposited in some region of tissue, the pressures generated by thermoelastic expansion propagate away from that region into the rest of the tissue. Consider the time-resolved pressure (P) arriving at an observation point, for example a detector, in response to an impulse of energy deposition (W) distributed over some region of tissue.

The pressure (P) [Pa] is related to a quantity called the velocity potential (φ) which is proportional to the energy deposition (W):

where ρ is density [kg/m^{3}], **r** is the position of observation, and t is time [s]. The following figure illustrates the problem and give the solution for calculating the velocity potential from a W(r).

k | indicates the kth time point, in steps of Δt = dr/c_{s} . |

where c_{s} = sound velocity (for water: 1480 m/s), and dr [m] is voxel size of simulation | |

j | indicates the jth voxel. |

φ | is the velocity potential [m^{2}/s]. |

V[j] | is the jth voxel volume [m^{3}]. |

r[j] | is the jth voxel distance-from-observation-point [m]. |

W[j] | is the jth voxel energy deposition [J/m^{3}]. |

β | is the thermal expansivity [strain/degree C], (for water: 2.29x10^{-4} strain/degree C). |

ρ | is the density [kg/m^{3}] (for water: 1000 kg/m^{3}). |

C_{p} | is the specific heat [J/(kg degreeC)], (for water: 4180 J/(kg degreeC)). |

The velocity potential (φ) [m^{2}/s] observed at time t is proportional to the sum of all energy deposition (W [J/m^{3}]) at a distance r = c_{s}t, where c_{s} is the speed of sound in the medium. An isotropic acoustic detector would detect the pressure arriving from a spherical shell of energy deposition, radius = r. An acoustic detector with a preferred direction of detection would detect from a spherical arc, radius = r.

The expression for calculating the velocity potential seen at an observation point at time t due to a field of energy deposition W(r), the red region described by voxels j = 1 to N_{j}, is shown in the above figure.