Electromagnetic treatment heads. The −3 dB focal zone is shown for the 30 mm (top) and 15 mm (bottom) standoff. Axial focal length from the tip depends on which standoff is used (15 and 30 mm, respectively). The focal dimensions shown were obtained from measurements.
Illustration of experimental setup. The therapy head can be either ballistic (no standoff used) or electromagnetic (as shown). Coupling of the treatment head to the water was facilitated by a Tegaderm window and coupling gel.
Typical averaged pressure waveform (25 averages) at the focus (15 mm from the long standoff) at the maximum output setting of the Duolith SD1 T-Top. The main pulse has a sharp front followed by a longer negative tail. Here, maximum and minimum pressure amplitudes are p+ = 42.3 MPa and p− = 9.9 MPa. There is a repeatable trailing pulse that is particular to electromagnetic sources. Inset: Close up of the pulse front of the waveform (top) and its time derivative (bottom). These are used to measure the rise time (trt ) of the main pulse, in this case trt = 8 ns (see text).
Selected radial scan averaged waveforms collected from the measurements in a plane 5 mm from the short standoff, along with simulation fit from model. Radial scan had a total length of 14 mm and waveforms along the way were used as a boundary condition for the model.
Boundary condition map for the modeling algorithm. Experimental pressure waveforms were measured 5 mm from the short standoff along the radial (transverse) coordinate out to 14 mm. Numerically extrapolated waveforms extends the map out to 20 mm. In addition, 36 additional numerically interpolated waveforms were added between each two experimental waveforms, giving the map a total of 3701 waveforms. The low pressure tail described by Eq. (3) is not shown. Waveforms are represented by vertical lines; pressure is shown as in gray scale; the actual computational windows were 65.5 μs in time and 43.2 mm in the radial (transverse) direction. The time axis is relative to the time in Fig. 4 where the “0” corresponds to a time shift by 32 μs from the data in Fig. 4 .
Waterfall plot for all machine output settings from the long standoff. Individually labeled waveforms correspond to the machine setting shown in Table I . Inset displays averaged waveforms for three different machine settings indicated as dark lines in the waterfall plot. (a) Setting no. 7 of 62 MPa nominal peak positive pressure; (b) setting no. 3 of 31 MPa; (c) setting no. 2 of 10 MPa.
Axial scans along the beam axis for the focused source using 30 mm and 15 mm standoffs with a 1 mm step-size and 20 averages per location. Measurements include p+ and p− (“x” symbols for 30 mm standoff and black circles for 15 mm standoff). “Short” and “Long” correspond to the position of the tips of the short and long standoffs, respectively. The distance between standoff tips is 15 mm.
Axial distribution of the averaged peak positive and peak negative pressures measured for the short standoff (black circles) and compared to the modeling results (solid lines) at the highest machine output setting (no. 7, 62 MPa). “Short” corresponds to the position of the therapy head edge of the short standoff.
Axial waveform measured and modeled for the short standoff at the distances −20 mm (a), −10 mm (b), 0 mm (c, the focus), and 10 mm (d) away from focus at the highest machine output setting (no. 7, 62 MPa). The experimental waveform was averaged over 25 individual waveforms in order to reduce the noise level.
Radial (transverse) scans at the focus for both long and short standoffs. The labels and legends apply for all of the plots: (a) and (c) - x and y scan for the 30 mm standoff; (b) and (d) - x and y scan for the 15 mm standoff. The coordinate x corresponds to the vertical direction and y - to in and out of the page from geometry of Fig. 2 . Each figure represents the peak positive pressure and absolute value of the peak negative pressure. Modeling results are presented for the short standoff experiments (b) and (d).
(Color online) Two-dimensional spatial distributions of the peak positive (left) and peak negative (middle) pressures, and energy density (right) in the field generated with the short standoff obtained in the modeling.
(a) p + and p − pressures measured at each machine output setting for both standoffs. (b) Calculated energy density (E.D.) at each machine setting for both standoffs. 1:1 line plotted for comparison on each plot. Measurements were collected at the measured focal point starting with the highest energy level and recording 25 waveforms independently for each energy level. Means and standard deviations (error bars) are shown.
Change in p + (solid symbols) and p − (open symbols) pressure with respect to a change in machine PRF for three selected machine output settings: 62 MPa (no. 7), 51 MPa (no. 5), and 36 MPa (no. 4). The long 30 mm standoff was used for these measurements. Data acquired shows averaged pulses (50 pulses averaged) at a specific PRF. Means and STDs are less than 2% in all the cases.
Rise time versus machine setting pressure levels for most of the machine settings using the long standoff. The rise time of the shock front of the electromagnetic source at a range of machine output settings is measured from averaged waveforms obtained at the beam focus. Rise time called “derivative definition” was calculated at the 36% level of the derivative of the pressure wave based on Fig. 3 . The “traditional definition” is the standard rise time from 10% to 90% of p +. Settings between no. 2 and no. 1 (Table I ) are difficult to detect since the signal is very noisy and the noise level is comparable to the amplitude when the derivative is calculated. A more powerful derivative method could be implemented to calculate rise times in these low amplitude sinusoidal-like waveforms.
Simulations of the axial peak pressure distributions. Inset: The focusing gain for the peak positive pressure (inset). The dashed lines correspond to the results simulating the experimental conditions for the short standoff at the highest machine output (no. 7, Table I ). The inset shows the ratio of the peak positive pressure at the focus to its initial value at the boundary as a function of the source pressure output. The dashed circle in the inset indicates the experimental point corresponding to that gain curve.
The pressure waveform (a) and corresponding spectrum (b) measured at a distance of 5 mm from the metal applicator for the ballistic source operated at 5 bar setting level.
Selected output settings of the Duolith SD1 discussed throughout the results and discussion sections.
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