2. The basic idea

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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).

kindicates 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.
φis 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.

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