Nick Freeman^{1}, Henrik Odéen^{1}, and Dennis L Parker^{1}

In-vivo determination of tissue thermal and acoustic properties, such as the specific absorption rate (SAR) for focus ultrasound, are important for accurate thermal modeling in treatment planning, monitoring, and control using, e.g., Pennes bioheat transfer equation. In this work we derive and present a numerical method using the Green’s function and MR thermometry data as input to estimate SAR non-invasively. The method is as accurate but substantially faster (on the order of seconds, compared to minutes) compared to two other SAR determination methods. Simulation and phantom experiments in a tissue mimicking gel phantom are performed.

**Introduction**

*Numerical Method* The
equations describing the method are given in Figure 1. The Green’s
function solution for PBTE (Equation 1) is given in Equation 2 where T_{0} and G are the respective solutions of Equation 3 and 4. These
solutions, shown in Equation 5 and 6, are obtained using the spatial Fourier transform,
where the heat kernel K is given in
equation 7. Using Equations 5 and 6, Equation 2 can be written in the form of a
sum of convolutions as shown in Equation 8, assuming that the temporal part of
the integral in Equation 2 only spans 0-t. Equation 8 can be solved for Q (the heat
deposition) using the convolution theorem. Further applying the Fourier
transform and re-arranging gives Equation 9. SAR is calculated as Q/ρ, which
can be simplified in the presented case since the simulations and phantoms do
not have any perfusion, giving Equation 10.

*Linear and Analytical
methods* The described numerical method of estimating Q was compared to two previously published methods^{2–6}. The linear method assumes there
is no heat conduction or perfusion during the heating, and MRTI data is then
fitted to this model. The analytical method fits a 2D radial Gaussian to an
analytical solution of the FUS heating.

*Simulation experiment*
“True” HIFU Q was simulated using the hybrid angular spectrum method^{7,8}, and a FDTD implementation of
the PBTE was used to create temperature maps using known phantom values^{9}. To mimic the phantom study
the temperature maps were converted to phase using the PRF equation^{10}, combined with a complex
image of a cylinder, and finally different amounts of zero-mean Gaussian noise
was added in k-space (to create temperature noise-standard deviation (SD)
corresponding to 0.0-0.4 in steps of 0.1°C). New (noisy) temperature maps were
created from the resulting complex image, and used as input for the three
different methods of determining Q.

*Phantom experiment*
FUS sonications at 4 different power levels (6,9,19,40W) were performed in tissue-mimicking
gelatin phantoms^{9}. MRTI was performed with a 3D
GRE segmented-EPI sequence (TR/TE=22/11ms, FOV=147x96x30mm,
resolution=1.15x1.15x2.5mm, flip-angle=15°, bandwidth=752Hz/pixel, ETL=7, Tacq=4.8s)
on a 3T scanner (Siemens Trio). MRTI
maps from the 4 different sonications were used as input for the three methods
of determining Q.

*Error comparison* For
all experiments, accuracy was assessed by comparing the “true” input temperature
distributions with temperature predictions created by inserting the determined Q
distributions in the PBTE-solver. RMSE was determined for all voxels where the
original temperature rise was above a chosen threshold (6 or 2.5°C). For
consistency the Q values presented are
in terms of Q_{rel}, defined
as the obtained Q normalized by input-power.

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2 R.B. Roemer, A.M. Fletcher, and T.C. Cetas, “Obtaining local SAR and blood perfusion data from temperature measurements: steady state and transient techniques compared.,” Int. J. Radiat. Oncol. Biol. Phys. 11(8), 1539–50 (1985).

3 C.R. Dillon, U. Vyas, A. Payne, D.A. Christensen, and R.B. Roemer, “An analytical solution for improved HIFU SAR estimation.,” Phys. Med. Biol. 57(14), 4527–44 (2012).

4 C.R. Dillon, N. Todd, A. Payne, D.L. Parker, D.A. Christensen, and R.B. Roemer, “Effects of MRTI sampling characteristics on estimation of HIFU SAR and tissue thermal diffusivity.,” Phys. Med. Biol. 58(20), 7291–307 (2013).

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6 Y.C. Shi, D.L. Parker, and C.R. Dillon, “Sensitivity of tissue properties derived from MRgFUS temperature data to input errors and data inclusion criteria: ex vivo study in porcine muscle,” Phys. Med. Biol. 61(15), N373–N385 (2016).

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9 A.I. Farrer et al., “Characterization and evaluation of tissue-mimicking gelatin phantoms for use with MRgFUS,” J. Ther. Ultrasound 3(1), 9 (2015).

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