# Calculations of pressures and intensities for radiation force modeling with FOCUS

## Abstract

Rapid calculations of the pressure-squared (as an approximation to the intensity) are required for models of the radiation force generated by linear ultrasound phased arrays. To determine the computation times and numerical errors produced by FOCUS and Field II, comparisons of the calculated complex pressures are performed using a simulated Vermon L5 phased array probe. The simulated phased array is populated with 128 elements. In the simulated array, the element pitch is 0.245mm, the kerf is 0.03mm wide, the element height is 7mm, and the fixed elevation focus is located at 30mm. For simulations of the radiation force, the array generates an electronically steered focus at 25mm. The array is excited with a 5MHz sinudoidal input, and the pressure field is computed on a 133 by 133 by 261 point rectilinear grid that is approximately centered at the focus. Half wavelength sampling is employed in all three directions. The results obtained with each method are then compared to a reference pressure field. In Field II calculations that employ a sampling frequency of 100MHz, this calculation takes about 4 hours on a 32 bit desktop running Windows XP, and in FOCUS, this calculation takes about 3 minutes on the same computer. The numerical errors generated by each model are also evaluated for each method, where the maximum error with Field II is just over 3% and the maximum error with FOCUS is just under 1%. To achieve a maximum error below 1% for this particular array and computational grid combination, a sampling frequency of 500MHz is required in Field II. The computation time for this transducer and grid geometry in Field II with 500MHz sampling is about 20hours. [Note: this is an updated version of the abstract that was presented at the upcoming Acoustical Society of America meeting held in Baltimore, MD in April, 2010].

## A. Array geometry

The array defined for these simulations is shown in Figure 1. Curved elements with a geometric focus at z=30mm are approximated with thin elements partitioned into 5 flat rectangular sections in FOCUS and in Field II. Both programs use the exact same coordinates for the element vertices and for the subelement vertices.**Figure 1:** 128 element linear array, where each element is modeled as five smaller segments that approximate a larger curved element that is geometrically focused in the elevation direction. The elevation focus is located 30mm from the array, the width of each element is 0.245mm, the kerf between elements is 0.030mm, and the height of each element is 7mm. For pressure and pressure-squared calculations, the excitation frequency is 5MHz.

## B. Pressure calculations

The array is electronically focused at z=25mm. The same values for the time delays (which are then converted into phases via θ=ωt for calculations in FOCUS) are used in Field II and FOCUS. The 3D pressure field is then calculated with FOCUS and Field II. The pressure field is computed with both programs on the same 133 by 133 by 261 point rectilinear grid that is approximately centered at the focus. Half wavelength sampling is emplayed in all three directions. The excitation frequency for these calculations is 5MHz.

The time required for the full 3D pressure field calculation with FOCUS is about 3minutes. This value includes the time required for FOCUS to calculate the source pressure plane with 7 abscissas and then compute the pressure in the 3D volume with the angular spectrum approach. The time required for the full 3D pressure field calculation with Field II, using a sampling frequency of 100MHz, is about 4hours.

Cross-sections of the 3D pressures obtained with FOCUS are shown in Figures 2, 3, and 4. Figure 2 contains the pressure in the xy plane at z=25mm (the location of the electronically steered focus), Figure 3 shows the pressure in the xz plane at y = 0mm, and Figure 4 displays the pressure in the yz plane at z = 0mm.

**Figure 2:** Pressure calculated by FOCUS in the xy plane at z = 25mm. As shown in Figure 1, the aperture is much larger in azimuth (the x direction) than in elevation (the y direction). Likewise, the scales in the x and y directions differ by a factor of 5.

**Figure 3:** Pressure calculated by FOCUS in the xz plane at y = 0mm. The array is electronically focused at z = 25mm.

**Figure 4:** Pressure calculated by FOCUS in the yz plane at z = 0mm. The elevation focus is located at z = 30mm, and the array is also electronically focused at z = 25mm.

## C. On-axis evaluations of the pressure-squared

The errors achieved with each program are evaluated in a comparison with the on-axis pressure-squared reference values calculated with the Rayleigh-Sommerfeld integral. The on-axis pressure-squared reference is shown in Figure 5. The Rayleigh-Sommerfeld reference is calculated with 9 abscissas in each direction for each subelement. The comparison is only shown here for the on-axis case (i.e., the 1D case) due to the very long computation times that are required for 3D Rayleigh-Sommerfeld calculations.

Figure 6 shows the on-axis errors in the pressure-squared values calculated with FOCUS and Field II, where the temporal sampling rate is 100MHz for Field II calculations. Figure 6 indicates that the on-axis errors are consistently equal or better with FOCUS. The peak error is slightly over 3% with Field II (sampling frequency of 100MHz), whereas the peak error is just under 1% with FOCUS. The error is also smaller in the nearfield region with FOCUS.

Figure 7 shows the on-axis errors in the pressure-squared values calculated with FOCUS and Field II, where the temporal sampling rate is increased to 500MHz for Field II calculations. In Figure 7, the peak error is just under 1% with Field II (sampling frequency of 500MHz) and with FOCUS. Therefore, to achieve roughly equivalent peak errors in the focal zone for this array and grid geometry, Field II requires a temporal sampling rate of 500MHz. However, this increases the computation time with Field II to about 20 hours, whereas the computation time with FOCUS is still only 3 minutes. The error with FOCUS is again smaller in the nearfield region, as in Figure 6.

**Figure 5:** Reference pressure-squared calculations evaluated on-axis with the Rayleigh-Sommerfeld integral.

**Figure 6:** On-axis errors in the pressure-squared obtained with Field II (100MHz sampling rate) and with FOCUS.

**Figure 7: **On-axis errors in the pressure-squared obtained with Field II (500MHz sampling rate) and with FOCUS.

## D. Contributors

### Simulations

- Robert J. McGough, Michigan State University
- Don VanderLaan, Michigan State University
- Matt Urban, Mayo Clinic

### Web Site

- Chris Johnson, Michigan State University
- Alec Dutch, Michigan State University
- Leslie Thomas, Michigan State University
- Peter Beard, Michigan State University