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/m3], 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/cs . |
| where cs = 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 [m2/s]. |
| V[j] | is the jth voxel volume [m3]. |
| r[j] | is the jth voxel distance-from-observation-point [m]. |
| W[j] | is the jth voxel energy deposition [J/m3]. |
| β | is the thermal expansivity [strain/degree C], (for water: 2.29x10-4 strain/degree C). |
| ρ | is the density [kg/m3] (for water: 1000 kg/m3). |
| Cp | is the specific heat [J/(kg degreeC)], (for water: 4180 J/(kg degreeC)). |
The velocity potential (φ) [m2/s] observed at time t is proportional to the sum of all energy deposition (W [J/m3]) at a distance r = cst, where cs 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 Nj, is shown in the above figure.