For frequencies of 1 and 3 MHz with the same spatial domain in the pressure and intensity computation, the parameters shown in Table 3.1 were determined using Matlab script. These parameters were used in the Fortean code to simulate the 2D steady state pressure as a function of axial distance from the transducer. The steady state pressure along the transducer axis and the steady state focal intensity were obtained for a series of source pressures from 10-70 kPa in water for both 1 and 3 MHz frequencies respectively. Plots of the steady state pressure along the axial distance of transducer, the positive and negative focal pressures at both 1 and 3 MHz frequencies are shown in Figs. 3.1 and 3.2. For a frequency of 1MHz, these source pressures yielded peak positive focal pressures of 0.77 MPa to 5.81 MPa when the simulated space is water.
The steady state pressure for the complete signal can be seen in Fig. 3.1 (a) at source pressures ranging from 10 to 70 kPa. In order to make sure the pressure waveform reached to the steady state, different simulation time was applied and we tracked the wave trace , if the trace looks the same with the different time length, it has to be steady state , because it is not changing, otherwise it is not in steady state. In our case we ran our code with different simulation times, and we reached a steady state result at each time. So we made sure that the waveform looks good as can be seen in Fig (a). The windowed signal is shown in Fig (b) which indicates the expanded waveforms of fig (a), just to better track the signal’s behavior. It can be seen from the peaks, the focal pressures are increased as source pressures increase. Distortion of the waves increases with increasing source pressures as well.
The peak compressional pressure values ranging from 0.77 to 5.81 MPa and the peak rarefactional pressure values in range of 0.76 to 5.032 MPa can be seen at Figs (c) and (d) respectively for each source pressure. It is noticeable that the actual acoustic focal length is shorter (12.3cm) than the geometric focal zone 13 cm because of water refraction in the FDTD solution which is expected.
Similar results are given for 3 MHz in Fig. 3.2. For example, the peak positive focal pressures increased from 2.23 to 19.4 MPa in the water, while the corresponding values for peak rarefactional pressures range from 2.11 to 12.63 MPa. Herein it can be seen that the distortion of the waves increase with increasing frequency as we expected, because we know that as frequency grows, more harmonics and finally more distorted waves are appeared.
Table 3.1 Simulation Parameters for 2-D pressure code at 1and 3MHz frequencies in water
| Simulation Parameters | Values for 1MHz | Values for 3MHz |
| Max distance in z-direction- zmaxP | 20cm (0.2 m) | 20 cm (0.2 m) |
| Max distance in r-direction- rmaxP | 10cm (0.1 m) | 10 cm (0.1 m) |
| Max distance in z-direction- zmaxT ( temperature space) | 15cm (0.15m) | 15cm(0.15m) |
| Max distance in r-direction- zmaxT( temperature space) | 10cm (0.1m) | 2cm (0.02m) |
| Spatial step in z direction- dzp | m | m |
| Spatial step in r direction- drp | m | m |
| Time step- dtp | s = 0.05µs | s |
| Number of acoustic cycles- Nptspercycle | 20 | 20 |
| Max index in z-direction- Imaxp | 1334 | 4001 |
| Max index in r-direction – Jmaxp | 668 | 2001 |
| Max index in z-direction temp- ImaxT | 1001 | 3001 |
| Max index in r-direction temp- JmaxT | 668 | 401 |
| Distance from front face of transducer to front surface of tissue – Iskin | 68 | 201 |
| Coordinate of geometric focus of transducer in z axis – Ifocusp | 868 | 2601 |
| Coordinate of geometric focus of transducer in r axis-Jfocusp (symmetry) | 1 | 1 |
| Tend to end time index | s | s |
| Max index in time- Nmaxp | 3000 (each unit equivalent to 0.05µs) | 8000 (each unit equivalent to 0.016 µs) |
|
|
|
|
` |
|
| Fig.3.1: Characterization of 1MHz transducer showing (a) steady state pressure along axial distance of transducer (b) steady state pressure, windowed signal vs. axial distance (c) steady state compressional peak pressure vs. axial distance of transducer and (d) steady state rarefactional peak pressure vs. axial distance of transducer direction. | |
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|
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| Fig.3.2: Characterization of 3 MHz transducer in water showing (a) steady state pressure along axial distance of transducer (b) steady state pressure, windowed signal vs. axial distance (c) steady state compressional peak pressure vs. axial distance of transducer and (d) steady state rarefactional peak pressure vs. axial distance of transducer direction. | |
In order to confirm that our calculation results for 3MHz are still consistent, the FDTD code was applied for a smaller step size ( ) in our numerical model. The reason for this implementation was to obtain additional higher harmonics and the resulting ecxtra distortion of the waveforms beyond that which was observable from our calculation outputs and pressure profiles with the larger step size.
The peak positive and negative pressures for the source pressures of 10, 40, and 70 kPa are shown in the water in figure 3.3.The positive peak amplitudes are 2.23, 9.81, and 19.4MPa respectively for 10, 40, and 70 kPa, while the negative peak pressures are 2.08, 7.50, and 11.81 MPa. Both compressional and rarefactional peak amplitudes are in a good agreement with previous calculations we made at 3MHz frequency. So we made sure that our results are stable.
| Figure 3.3: Smaller step size simulation for (a) positive amplitude and (b) negative amplitude at source pressures of 10, 40, and 70 kPa for 3MHz frequency in water. | |
| Fig.3.4: Acoustic intensity as a function of axial and radial location for FDTD solution to the wave equation. The peak intensity at the focus is 1194 for (a) 1MHz and 12338 for (b) 3MHz at 70 kPa source pressure. | |
The simulated spatial intensity profile as a function of axial and radial distance is plotted in Fig 3.4. .The acoustic intensity in water was computed in 2D in the simulation space as shown in the figure. In order to save computation time, the intensity space was mapped only in a subsection of the space containing the region of interest. This space is also axially symmetric. As can be seen from figures (a) and (b), the intensity is found at the real focus of the transducer which is before the geometric focus. Moreover, comparing the simulated intensity at 1 and 3 MHz, it is shown that the simulated intensity values for 3 MHz are much greater than 1MHz due to the sharper focus at the higher frequency. For instance, a source pressure of 10 kPa gives an intensity of 23.5 at 1MHz compared to 3 MHz having an intensity of 188.5 . These intensity levels are well below the in vivo cavitation threshold that has been reported in the literature (see Section 1.2).
RESULTS OBTAINED FROM 2D PRESSURE SIMULATION FOR 1 AND 3 MHZ FREQUENCIES IN TISSUE USING FDTD CODE
The pressure amplitude in tissue was calculated with the same process which used to calculate the pressure amplitude in water. Figure 3.5 shows the comparison of the pressure amplitude in water and tissue at both 1MHz drives at 10 kPa. Here also it can be seen that the actual acoustic focal length in tissue is shorter (12.21cm) than the actual focal length of water (12.3) cm. This is due to the fact that the degree of refraction of the beam from the transducer surface depends on the speed of sound in the medium. This leads to having the smaller focal pressure amplitudes in the case of tissue versus water. Moreover, it is clearly visible that a lower pressure is obtained in the focal region in tissue than water due to diffraction and also the higher attenuation coefficient of tissue.
The positive and negative peak pressures for 1 and 3 MHz frequencies are shown respectively in Figure 3.6. Figure (a) shows the compressional pressures as 0.72, 2.98, 5.40, and 11.81 MPa corresponding to the source pressures of 10, 40, 70, and 140 kPa respectively at a distance of 11cm from the transducer surface to the tissue edge 14cm, (Tissue thickness 3cm), while figure (b) shows the corresponding rarefactional pressure at the peak values of 0.70, 2.76, 4.70, and 8.80 MPa with the same source pressures and the same distance at 1MHz frequency. Similar results are given for 3 MHz in figures (c) and (d). Fig (c) shows the positive peak pressures as 1.72, 7.41, 14.21, and 43.74 MPa corresponding to the source pressures of 10. 40, 70 and 140 kPa respectively at a distance of 11 cm from the transducer surface to the tissue edge 14 cm, (Tissue thickness 3cm), while fig (d) illustrates the negative peak pressures at 1.64, 6.18, 10.14, and 18.38 MPa. Herein for 1 and 3 MHz, like the water media, the distortion of the waves increased with increasing driving level and frequency, and thus the peak pressures also became larger with increasing the source pressures.
The focal peak pressures and intensities were calculated for 1MHz in tissue of different thicknesses of 5, 4, 3, and 2 cm, when the tissue width ends up to 13 cm which is the geometric focus. Likewise, the focal peak pressures and intensities were calculated for 3MHz in tissue of various depths of 4, 3 and 2 cm, when the tissue edge ends up to 14 cm which is the total simulation space for temperature in axial distance from transducer. The results of compressional and rarefactional peak pressure and intensities are indicated for 1 and 3 MHz frequencies in Table 3.3 (a) and (b) respectively. From the results it can be seen that less tissue thickness shows more pressure amplitudes at both positive and negative peaks.
Table 3.2 Simulation Parameters for 2-D pressure code at 1and 3 MHz frequencies in tissue
| Simulation Parameters | Values for 1MHz | Values for 3MHz |
| Max distance in z-direction- zmaxP | 15cm (0.15m) | 15cm (0.15m) |
| Max distance in r-direction- rmaxP | 6.5 cm (0.65m) | 6.5 cm (0.65m) |
| Max distance in z-direction- zmaxT ( temperature space) | 14cm (0.14m) | 14cm (0.14m) |
| Max distance in r-direction- zmaxT( temperature space) | 6.35cm (0.635m) | 6.35cm (0.635m) |
| Distance from front face of transducer to front surface of tissue – Iskin | 11 cm (0.11m) | 11 cm (0.11m) |
| Spatial step in z direction- dzp | m | m |
| Spatial step in r direction- drp | m | m |
| Time step- dtp | s = 0.05µs | s = 0.0016µs |
| Number of acoustic cycles- Nptspercycle | 20 | 20 |
| Max index in z-direction- Imaxp | 1001 | 3001 |
| Max index in r-direction – Jmaxp | 434 | 1301 |
| Max index in z-direction temp- ImaxT | 934 | 2801 |
| Max index in r-direction temp- JmaxT | 424 | 1271 |
| Distance from front face of transducer to front surface of tissue – Iskin | 734 | 2201 |
| Coordinate of geometric focus of transducer in z axis – Ifocusp | 868 | 2601 |
| Coordinate of geometric focus of transducer in r axis-Jfocusp (symmetry) | 1 | 1 |
| Tend to end time index | s | s |
| Max index in time- Nmaxp | 2500 (each unit equivalent to 0.05µs) | 6000 (each unit equivalent to 0.0016µs) |
Figure3.5: Comparison of tissue and water pressure amplitudes at the focus at 1 and 3 MHz
Table 3.3(a): Peak focal pressures and intensities in tissues of various thicknesses at 1 MHz
Tissue thickness 5 cm (8-13 cm)
| Source
Pressure (kPa) |
Positive
Pressure (MPa) |
Negative
Pressure (MPa) |
Intensity in tissue | |
| Calculated
(W/cm2) |
by derating
(W/cm2) |
|||
| 10 | 0.58 | 0.57 | 10.7 | 7.42 |
| 40 | 2.40 | 2.25 | 174.92 | 126.06 |
| 70 | 4.30 | 3.85 | 548.06 | 414.71 |
| 140 | 9.20 | 7.31 | 2330 | 2004.3 |
Tissue thickness 4 cm (9-13 cm)
| Source
Pressure (kPa) |
Positive
Pressure (MPa) |
Negative
Pressure (MPa) |
Intensity in tissue | |
| Calculated
(W/cm2) |
by derating
(W/cm2) |
|||
| 10 | 0.63 | 0.61 | 12.12 | 9.06 |
| 40 | 2.55 | 2.40 | 199.07 | 154.23 |
| 70 | 4.65 | 4.11 | 626.18 | 563.9 |
| 140 | 10.00 | 7.78 | 2683.2 | 2452.12 |
Tissue thickness 3 cm (10-13 cm)
| Source
Pressure (kPa) |
Positive
Pressure (MPa) |
Negative
Pressure (MPa) |
Intensity in tissue | |
| Calculated
(W/cm2) |
by derating
(W/cm2) |
|||
| 10 | 0.67 | 0.66 | 13.42 | 11.06 |
| 40 | 2.76 | 2.58 | 223.61 | 188.5 |
| 70 | 4.97 | 4.42 | 686.87 | 620.14 |
| 140 | 10.74 | 8.35 | 2622.6 | 2997.16 |
Tissue thickness 2 cm (11-13 cm)
| Source
Pressure (kPa) |
Positive
Pressure (MPa) |
Negative
Pressure (MPa) |
Intensity in tissue | |
| Calculated
(W/cm2) |
by derating
(W/cm2) |
|||
| 10 | 0.72 | 0.70 | 15.80 | 13.51 |
| 40 | 2.98 | 2.76 | 237.11 | 230.25 |
| 70 | 5.40 | 4.70 | 778.5 | 757.1 |
| 140 | 11.81 | 8.80 | 3172.9 | 3658.87 |
“By derating” means use equation 2 from Dunn et al, 1975 [I = I0 exp (-md)] with I0 equal to the computed values for water, m = 0.20f cm-1, and d = 1, 2, 3, 4, or 5 cm.
Table 3.3 (b): Peak focal pressures and intensities in tissue of various thicknesses at 3 MHz
Tissue thickness 4 cm (10-14 cm)
| Source
Pressure (kPa) |
Positive
Pressure (MPa) |
Negative
Pressure (MPa) |
Intensity in tissue | |
| Calculated
(W/cm2) |
by derating
(W/cm2) |
|||
| 10 | 1.40 | 1.35 | 55.30 | 15.15 |
| 40 | 5.90 | 5.11 | 886.32 | 289.70 |
| 70 | 11.00 | 8.51 | 2642.3 | 1036.97 |
| 140 | 27.5 | 14.90 | 12423 | |
Tissue thickness 3 cm (11-14 cm)
| Source
Pressure (kPa) |
Positive
Pressure (MPa) |
Negative
Pressure (MPa) |
Intensity in tissue | |
| Calculated
(W/cm2) |
by derating
(W/cm2) |
|||
| 10 | 1.72 | 1.65 | 82.10 | 27.60 |
| 40 | 7.41 | 6.18 | 527.6 | |
| 70 | 14.21 | 10.14 | 4934 | 1889.86 |
| 140 | 43.74 | 18.40 | 2434.9 | |
Tissue thickness 2 cm (12-14 cm)
| Source
Pressure (kPa) |
Positive
Pressure (MPa) |
Negative
Pressure (MPa) |
Intensity in tissue | |
| Calculated
(W/cm2) |
by derating
(W/cm2) |
|||
| 10 | 2.13 | 2.02 | 50.30 | |
| 40 | 9.48 | 7.37 | 961.20 | |
| 70 | 19.06 | 11.81 | 3441.33 | |
| 140 | ||||
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| Fig 3.6: Characterization of 1 and 3 MHz transducer in tissue (depth 11-14cm) showing (a) steady state compressional pressure vs. axial distance of transducer at 1MHz (b) steady state rarefactional peak pressure vs. axial distance of transducer direction at 1MHz (c) steady state compressional pressure vs. axial distance of transducer at 3MHz (d) steady state rarefactional peak pressure vs. axial distance of transducer direction at 3MHz. | |
RESULTS OBTAINED FROM 2D PRESSURE SIMULATION FOR TRANSDUCER LENS ASSEMBLY BOTH CALCULATION AND PHYSICAL MEASUREMENT
As explained before, we need to verify that our sound propagation model accurately predicts the spatial pressure distribution generated by focused transducer. Pressure is the most conveniently measurable quantity and thus will be used for comparison. As described in Section 2.6, the pressure field was calculated using the FORTRAN code to compere our calculations with the measurements we made in the scan tank. This can be achieved in pure water with the hydrophone fixed in place. Results for the simulation parameters using the code are presented at Table 3.3.
Figure 3.7 shows the pressure profile as a function of the radial and axial distance from the acoustic axis of x, y, and z for the 1.21 MHz transducer in water, the peak voltages (pressure) are 277, 279, and 275 mV, respectively.
Figure 3.8 shows computed and measured results obtained in water at 30°C for the sound sources. The pressures shown at (a) and (b) are all peak positive quantities that have been normalized to the spatial maximum value present at the real focal point. The axial distance in plot (b) shows the shifted axis by some amounts in order to have the same range and unit with (a). The plot (c) shows the solid red and blue lines which correspond to calculation and measurement respectively. Except for the peripheral regions of the field, we see that our model is able to predict all the detailed structure, and good agreement was found in the focal region (real focus of 5.20 cm) and beyond.
Table 3.4: Simulation parameters for 1.21-MHz transducer-lens assembly
| FDTD Simulation Parameters | Value |
| Max distance in z-direction- zmaxp | 10 cm (0.1 m) |
| Max distance in r-direction- rmaxp | 5 cm (0.05 m) |
| Spatial step in z direction- dzp | m |
| Spatial step in r direction- drp | m |
| Time step- dtp | s = 0.05µs |
| Number of acoustic cycles- Nptspercycle | 17 |
| Max index in z-direction- Imaxp | 668 |
| Max index in r-direction – Jmaxp | 334 |
| Max index in z-direction temp- ImaxT | 401 |
| Max index in r-direction temp- JmaxT | 201 |
| Distance from front face of transducer to tissue front surface – Iskin | 68 |
| Actual focal length | 5.20 cm |
| Geometric focal length | 5.5 cm |
| Coordinate of geometric focus of transducer – Ifocusp | 386 |
| Coordinate of focus of transducer-Jfocusp (symmetry) | 1 |
| Tend to end time index | s |
| Max index in time- Nmaxp | 1500 (each unit equivalent to 0.05µs) |
|
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Figure 3.7: Characterization of the 1.21 MHz Sonostat transducer showing, (a) voltage amplitude in radial direction x, (b) voltage amplitude in radial direction y, (c) voltage amplitude in axial direction z. |
Figure.3.8: Characterization of the 1.21-MHz transducer-lens combination showing normalized calculated (a) and measured (b) positive peak pressure profiles in axial direction , and (c) a comparison of the normalized pressure profiles verses axial distance for the transducer obtained from calculation (using the code) and measurement (using the scan-tank).
In the following section, the experimental data collected during the experiment are presented. Figure. 3.9 contains four images; the top two images (a) and (b) marked focal zone are the experimental data from the plane scan using the hydrophone. The bottom images (c) and (d) marked with the positive values which indicate relative depths beyond the focal zone of 1 and 10 mm, respectively. Images (c) and (d) indicate results for far field plane scans of transducer.
Another sets of data are show in figure 3.10 which are the near field measurements of the transducer with a negative distance meaning relative distance from the focal zone to the transducer.
The images show millimeters on both axes and the color adds a third dimension to the graph, the relative amplitude of the pressure field. The dark blue indicates lowest pressure and the dark red indicates the highest pressure.
By comparing the color distribution in the experimental data and also the pressure amplitude for each graph, it is noticeable that whenever the hydrophone is further from the center of the scan plane, in general, the amplitude of the pressure field (and therefore the electrical signal) captured by the oscilloscope decreases. The size of the signal can vary greatly over the scan plane.
| Fig.3.9: Pressure field pattern of transducer-lens, peak-to-peak voltage of: a) 587.8 mV at focus b) 275 mV at focus ; c) 270.3 mV far field; and d) 143.8 mV far field (furthest distance) | |
| Fig.3.10: Pressure field pattern of transducer-lens, peak-to-peak voltage of: a) 257.8 mV near field b) 248.5 mV near field ; and c) 218.18 mV near field (closest distance) | |
RESULTS OBTAINED FROM INTENSITY CALCULATIONS FOR 1 and 3 MHZ IN WATER
Tables 3.5 and 3.6 illustrate the results from the intensity calculations in water at 1 and 3 MHz. As mentioned in section 2.7, the goal of these intensity calculations was to compare them with the published numbers in the papers.
| Source Pressure (Pa) | Calculated focal pressure at the real focus using the code | Calculated intensity using the equation: | Calculated waveform vs. time at the position of the peak pressure | Calculated intensity using the equation: | Simulated intensity using the code:
|
| 10,000 | 7.776 | 20.15 | 7.777 | 20.16 | 23.5 |
| 20000 | 1.571 | 82.26 | 1.572 | 82.26 | 94.5 |
| 30,000 | 2.383 | 189.3 | 2.383 | 189.3 | 213 |
| 40,000 | 3.212 | 343 | 3.21 | 343.5 | 381 |
| 50,000 | 4.06 | 550 | 4.06 | 550 | 559 |
| 60,000 | 4.929 | 810 | 4.93 | 810 | 871 |
| 70,000 | 5.819 | 1126 | 5.823 | 1,130 | 1194 |
| 140,000 | 12.8 | 5461.3 | 12.8 | 5461.3 | 5360 |
Table 3.5: Intensity calculations for 1MHz frequency in water
Table 3.6: Intensity calculations for 3MHz frequency in water
| Source Pressure (Pa) | Calculated focal pressure at the real focus using the code | Calculated intensity using the equation: | Calculated waveform vs. time at the position of the peak pressure | Calculated intensity using the equation: | Simulated intensity using the code:
|
| 10,000 | 2.237 | 167 | 2.234 | 167 | 188.61 |
| 20000 | 4.599 | 705.02 | 4.65 | 720.75 | 777.72 |
| 30,000 | 7.093 | 1677 | 7.108 | 1684.12 | 1758.9 |
| 40,000 | 9.817 | 3212.33 | 9.788 | 3193.33 | 3238.7 |
| 50,000 | 12.67 | 5350.9 | 12.21 | 4969.47 | 4918.9 |
| 60,000 | 15.9 | 8427 | 15.75 | 8400.52 | 8236.2 |
| 70,000 | 19.4 | 12545 | 18.5 | 11433 | 12338 |
Figure 3.11 shows the time-varying pressure waveforms at the location of the peak pressure for series of source pressures from 10-70 kPa .The averaged peak positive pressure over several positions was found in the range of 0.77-5.82 MPa at 1MHz.
Fig.3.11: Time varying pressure waveforms for a series of source pressure in water for (a) 1MHz and (b) 3MHz frequency
RESULTS OBTAINED FROM INTENSITY CALCULATION IN THE PUBLISHED PAPER FOR 1 AND 3 MHZ FREQUNECIS
In this section, the intensity numbers published in paper [25] were calculated at different exposure times using the slope and intercepts given in the paper. It has been suggested empirically that on a log-log plot (Figure 3.12), intensity as a function of exposure time can produce lesions at three collinear regions [25].
Table 3.7 indicates the intensities which we calculated manually for both 1 and 3 MHz frequencies from the results given by Dunn and coworkers. Then the square root of focal intensities (Tables 3.5, 3.6) versus source pressure numbers were plotted to find a proper linear relationship between intensity and source pressure to specify the exact source pressure numbers in the study using the published intensity numbers. Figure 3.13 illustrates this linear relation.
In addition, the calculated focal intensity using the simulation code was compared to the intensity numbers in the paper, except some different numbers we found in the calculation due to lack of data and information from the intensity plot in the paper, our simulation satisfies pretty well the published data.
Figure 3.12: Threshold acoustic intensity at the irradiation site versus duration required for a single pulse to produce a lesion in mammalian brain, with relevant data for the figure. [25]
Table 3.7: Calculated intensity numbers using slope and intercept of published plot
| Exposure time | Calculated intensity 1MHz | Calculated Exact Source Pressure
1MHz (a) |
Calculated intensity 3MHz | Calculated Exact Source Pressure 3MHz (b) |
| 10119.3 | 190731 | 9486.83 | 65000 | |
| 3200 | 109201 | 2999.9 | 38357.6 | |
| 1011.9 | 63352.5 | 948.7 | 23375.4 | |
| 1 | 320 | 37571.3 | 300 | 14950.3 |
- P= + 2.4 /0.00054
- P= + 6.6 /0.0016
Fig. 3.13. The square root of calculated focal intensity vs source pressure at (a) 1 and (b) 3MHz
