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The 10 MHz ultrasonic near-field: calculations, hydrophone and optical diffraction tomography measurements
The 10 MHz ultrasonic near-field: calculations, hydrophone and optical diffraction tomography measurements Roger Eriksson, Anders Holm, Monica Landeborg, Hans W. Persson and Kjell Lindstr6m Department of Electrical Measurements, Lund Institute of Technology, PO Box 118, S-221 00 Lund, Sweden
Received 28 January 1993; accepted 22 April 1993 The near-field from a 10 MHz ultrasound transducer has been studied using three different techniques; theoretical calculations, miniature hydrophone measurements and optical diffraction tomography (ODT). The motive was twofold; to evaluate the measurement possibilities of ODT for high-frequency ultrasound and to verify the calculated complexity of near-fields from real ultrasound transducers designed for use in blood perfusion measurements. Calculations of the field from an ideal piston transducer were done with surface integrals of the Kirchhoff function and the measured transducer was designed to approximate this ideal. Contour maps, from the three methods, of pressure amplitude from 1 to 30 mm along the beam are presented with beam profiles at 5 and 45 mm. The results imply that ODT has an increased spatial resolution compared with the 0.5 and 1.0 mm hydrophones. However, even better spatial resolution is needed to draw definite conclusions regarding the complexity of the near-field of the 10 MHz transducer.
Keywords: near-field; hydrophones; optical diffraction tomography; blood perfusion
In this paper, results from numerical calculations and measurements of the ultrasonic near-field of a 10 MHz, 5 m m diameter piston transducer are presented. Two measurement methods were used, optical diffraction tomography and a miniature polyvinyldifluoride (PVDF) hydrophone. The two methods were chosen in order to achieve measurements with an adequate spatial resolution. The measurements were done in order to investigate the complexity of the ultrasonic near-field and to verify earlier assumptions concerning the effect this complexity has on blood perfusion measurements. The possibility of measuring blood perfusion in real-time in deep tissue has, for a long time, been desired by physicians and for an equally long time been a challenge for scientists. Hertz 1 suggested a method for blood perfusion measurement based on the ultrasonic Doppler method. The method has been examined further by others and found to be promising but there are also difficulties with the technique z 4. Among the problems are unexpected variations of the measured perfusion. Experimental studies suggest that these variations are related to ultrasonic interference combined with small transducer movements 5. It is essential for blood perfusion measurements to use high-frequency ultrasound owing to, among other things, 0041 624X/93/060439-08 © 1993 Butterworth-Heinemann Ltd
the low speed of the moving scatterers down to and below 0.1 m m s l. Also, scattering from small particles increases with frequency. The resulting interference in the near-field of an ultrasonic transducer, due to the wave nature of the ultrasound, causes spatial variations in the ultrasonic intensity. The feature size of these variations is of the order of a wavelength. If these differences are large and if the blood perfusion is not homogeneously distributed throughout the sample volume, then small movements of the transducer will lead to variations in the measured perfusion value. These variations are not related to changes in the blood perfusion and the extent of the variations has so far prevented clinical use of the measurement method. Measurements of the complex near-field, especially at frequencies above a few MHz, are difficult due to spatial resolution requirements. An established measurement method like the miniature hydrophone method has a few undesirable characteristics limiting its usefulness. Some of the more important characteristics are the size of the hydrophone, standing waves and hydrophone angle. A hydrophone integrates the pressure field on its active surface so that there is always a spatial averaging. Evidently, one can reduce the averaging area by reducing the size of the hydrophone, but this cannot be done
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The 10 M H z ultrasonic near-field." R. Eriksson et al. without decreasing the signal-to-noise ratio. Errors due to standing waves can be decreased by using burst excitation but will always be a problem in the very close near-field. However, hydrophones have advantages with their comparatively simple practical set-up and use. Optical diffraction tomography (ODT), on the other hand, promises high spatial resolution and potentially absolute measurements. The spatial resolution is limited by a combination of optical system resolution and mechanical scanning resolution. Standing waves are not a problem in O D T owing to its non-invasive nature. There are experimental difficulties with the laser beam angle that are similar to the hydrophone angle problems. However, O D T does not offer the experimental simplicity of the hydrophone. There are several variations of the theoretical calculations of the near-field and this paper does not introduce any new techniques nor does it include a review; instead, previously' published* mathematical models are used throughout. There are also several measurement systems based on miniature hydrophones with corresponding documentation, a recent description and discussion of such systems was edited by Preston ~. Diffraction of light by ultrasound is not a newly discovered phenomenon numerous papers have been published since the 1930s, among them the much cited paper by Raman and Nath s. ODT, as a measurement technique, was introduced by Reibold and Molkenstruck ~ who applied it to measurements of ultrasound transducers in the 3 MHz range. Reibold e l al. ~° 13 have published several papers describing measurements with O D T and other authors have also used the technique ~ ~'. Unfortunately, there is a lack of literature and data on measurements of high-frequency ultrasound near-fields so that any extensive comparison between O D T and other results is quite difficult. It is known from theory that the ultrasonic near-field is complex and, furthermore, at 10 MHz, the near-field is very sensitive to transducer design variables. It is thus useful to investigate it experimentally and compare experimental results with theory. As far as wc can ascertain, previous O D T measurements have been limited in frequency up to approximately 5 MHz, this paper is an attempt to extend the frequency range upwards. The blood perfusion measurements with 1 0 M H z ultrasound described earlier ~ have shown inconclusive results. It is believed that the interference pattern in the near-field causes these measurement problems and the O D T measurement results could eventually be taken into account in the design of a blood perfusion measurement system based on the Doppler technique. To summarize, the motives for this investigation are: (1) the extension of O D T to high-frequency near-field measurements and a comparison with the generally used miniature hydrophone measurement technique: and (2) to verify the intensity variations in the near-field of a real 10 MHz ultrasound transducer used for blood perfusion measurements.
M a t e r i a l s and m e t h o d s Numerical calculatious Analysing diffraction fields in front of a transducer involves surface integrals of the form
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U l t r a s o n i c s 1 9 9 3 Vol 31 No 6
t
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GdS
{1~
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where S is the surface of the transducer and ~/, is the scalar velocity potential. G is a Green's function to be chosen according to the boundary conditions of the transducer. The Kirchhoff function is an appropriate Green's function for a transducer in a thin case, totally immersed in an acoustic medium ~'. The rear wall of the transducer element must be insulated from the acoustic medium and not contribute to the acoustic field. In Equation (2) G is replaced with the Kirchhoff function =
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+
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(2)
where s is a field point to an element on the surface of the transducer and k = ~,~:c. A prediction of the scalar velocity field is best accomplished if this surface integral is rewritten in a single integral form. Archer-Hall and G e C used geometrical arguments to accomplish this transformation. Equation (3) shows the single integral form which, with reasonable computer power (a PC), can be numerically calculated to obtain the complete field of a plane piston radiator. ~bKlr,-)=
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and ~ is the radius of the transducer, r is the radial distance, _- is the distance from the transducer surface and ~, is the variable of integration. As can be seen, Equation (3) contains two parts. The first term represents a plane wave that contributes to the field inside the radius of the transducer and with half its value on the rim of the transducer. The second term is an edge wave term that contributes to the whole field of the transducer. For the numerical calculations, a personal computer (Macintosh Quadra 700) was used. The numerical integration was implemented with a commercial software package (Mathematica, Wolfram Research). Raman Nath diffraction Diffraction of light by sound has been studied extensively since the 1930s. Raman and Nath wrote a seminal paper s explaining acousto-optic interaction from a diffraction grating standpoint. They also introduced the Raman Nath equations relating diffracted light intensity to the pressure amplitude of an ultrasonic wave. There is no shortage of literature on the subject and a recent and excellent theoretical summary of ultrasonic light diffraction wits written by Korpel ~ The basis of the present experimental study is the optical diffraction method introduced by Reibold and Molkenstruck '~ in 1984. Consider a continuous (CWI progressive ultrasonic wave as illustrated in t:7:lm'C I a n d let the ultrasonic pressure-lield be described by P I x . v, z, t) - P,,,,,(.x-. y. z) sin ( K , z - - D - J
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where ~s is the angular frequency, K s is the wave-number of the ultrasound and ~ps(x, y, z) is the phase-angle related to some arbitrary reference. The index of refraction in the water is modulated by the ultrasound, and the elasto-optic coefficient that couples pressure and index of refraction is denoted 18 Pop ~ 1.51 x 10 -1° m 2 N -1 The details of the R a m a n Nath model of ultrasonic light diffraction are outside the scope of this paper but some definitions are needed. For instance, the Raman Nath parameter v, i.e. the integrated optical phase, is defined by
v(x, y, z, t)
= keoPo p
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--3C
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(5)
where keo is the vacuum wave-number of the probing light beam. If the ultrasound and optical parameters are within the Raman Nath model's limits 17 the light intensity detected by a photo-detector placed in the image plane includes a time-varying part as in Equation (4)
l(bO'), z, t) = Io2Jo(b)J t (b) cos(K~z - f2,t + 0s(Y))
(6)
where I 0 is the light intensity without sound and J,(b) are Bessel functions of the first kind. Two criteria must be met for the Raman Nath model to be valid for the current frequency, transducer size and pressure levels. These are Q = DK~k~ t < 1 and Qv < 1, where D is the active diameter of the transducer. With the 10 M H z transducer measured one calculates: Q ~ 0 . 6 7 and Qv ~ 0.3. For Equation (6), to be valid, all diffraction orders except the zeroth and first positive, or negative, must be removed by a spatial filter (Figure 1). If the amplitude and phase of the signal described by Equation (6) are measured, one can write two expressions that relate these quantities to the ultrasound pressure field at a specific z.
L'(y)
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(7a)
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With Equations (6), (7a) and (7b) the relationship between the ultrasound pressure field and the measured light intensity is clarified. The end result comprises two ray integrals that can be used for separate tomographic r e c o n s t r u c t i o n s of Pmax(X,y) COS[q~s(x,y)] a n d Pmax(X,Y) sin[q~,(x,y)] respectively. These two reconstructions can be used to produce Pm,x(X, y) and q~s(x, y). In this paper, only Pmax(X, Y) is calculated and presented.
The basis of most transmission tomography algorithms is the ray-integral concept, i.e. a characteristic denoted f(x, y) of an object is integrated by a probing beam along a geometrically straight line. From sampled parallel collections of ray integrals ( = one projection) taken at angles covering 180 c' in a plane it is possible to reconstruct f(x, y). There are countless variations of tomography algorithms and we will not delve deeply into this field. A good introductory text as well as a source of algorithms is Kak and Slaney 19. In a practical ultrasound field measurement system there is always a trade-off between measurement time and desired accuracy/resolution. For instance, with hydrophone systems a doubling of the resolution in a plane gives a quadrupling of the required data points and thus also an increased measurement time. With diffraction tomography there is no simple relation between resolution and measurement time. This is due to the complex dependencies of resolution on the number of samples, the number of projections and the algorithm used. In many cases tomography algorithms are tested with simulated data intended to resemble some part of the human body, e.g. the Shepp and Logan head phantom; however, these test phantoms were not appropriate for the type of projection data measured in O D T . It was more appropriate to test algorithms with simulated projection data from ultrasound fields. The simulated projection data were calculated with the numerical techniques described above at distances of 5 m m and 23 m m from the transducer surface, i.e. in the near-field. A pressure field containing 101 × 101 values with a sample distance of 0.1 m m (each direction) was calculated and from this, field projection data were subsequently derived. Since we used an axisymmetric model all projections were the same, independent of projection angle. The algorithms tested were Filtered Backprojection (FBP) and Algebraic Reconstruction Technique (ART). These algorithms were implemented on a personal computer. Generally, the literature agrees that ART or other iterative algorithms (MART, SIRT) are better for reconstructions with a small number of projections, e.g. 1-50. Our reconstruction results indicated that this was not the case for the projection data expected in the present paper. On the contrary, our tests indicated that FBP is better even for a small number of projections. In Figure 2 two graphs illustrating combinations of simulated theoretical cross-sections and reconstructed cross-sections are presented. The sample distance in the simulated projections was 0.1 mm (101 samples (n)) and the number of projections 20 (m). In Figure 2a a comparison is presented between the theoretical cross-section, a reconstructed cross-section, and a reconstructed cross-section where 5% white noise was added independently to each projection. All
Ultrasonics 1993 Vol 31 No 6
441
The 10 MHz ultrasonic near-field. R. Eriksson et al. I
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reconstructions in this paper FBP has been used, motivated by the findings presented in Fixture 2.
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Figure2 Pressure cross-sections through transducer axis. 101 samples and 20 projections. (a) Comparison between reconstruc tion and simulation at 5 mm distance from transducer. Simulation (thin curve). Reconstruction with FB P (thick curve). Reconstruction (FBP) with 5% white noise added to each projection (thick dashed curve). (b) Comparison between simulation and different recon struction algorithms at 5 mm. Simulation (thin curve). Reconstruc tion with FBP (thick curve). Reconstruction with ART (thick dashed curve)
reconstruction planes were filtered with nearestneighbour averaging before the cross-sections were extracted. This filtering was done to equalize this situation to that of experimental data on which filtering has been done to smooth out noise. The amplitude values of the reconstructions are erroneous but the locations of the dips are correct and the variations in the theoretical cross-section are 'detected' both with noiseless and noisy projection data. In fact, the filtering procedure increases the amplitude difference between theory and reconstruction for projection data without noise, i.e. an unfiltered cross-section would appear more accurate. In Fiqure 2h a comparison between the theoretical cross-section, a reconstructed cross-section (FBP) and another reconstructed cross-section (ART) is presented. The ART algorithm 'misses' some of the theoretical variations and we judge it to be inferior to FBP in this case. With added noise the ART algorithm's accuracy is decreased. The reason for FBP's superiority over the iterative algorithm is most likely due to the symmetric nature of the expected projections and also the relatively limited spatial frequency content, i.e. there are no edges in the ultrasound field. One definite advantage with FBP compared with ART is speed, reconstruction time with ART is proportional to imn 3, where i is the number of iterations, and reconstruction time with FBP is proportional to mn. Judging from the graphs in Fiyure 2 and numerical comparisons, the numerical error from the reconstruction is roughly 4-10%. In all subsequent
442
Ultrasonics 1993 Vol 31 No 6
Transducer design The 10 MHz transducer used was made in-house and it is designed to have a radiating characteristic as close as possible to an ideal piston transducer. A piezoelectric ceramic was mounted in a plastic case and damped by backing material, Figure 3, to prevent internal reflections. In order not to distort the ultrasonic field the front surface was left uncovered, i.e. no matching or protective layer was used.
Hydrophone set-up The transducer was fixed in a water bath and excited with a 10 MHz, 8 Vpp, 15 cycle tone-burst signal from a quartz stabilized function generator (Hewlett Packard 3314A). The tone-burst signals as well as sound absorbing material on the walls ofthc water tank served to minimize noise echoes and standing waves. Two calibrated miniature (1 and 0.5 mm diameter) P V D F hydrophones (H.W. Persson) were used. During the measurements, the hydrophone was mounted in a test fixture controlled by three stepping motors enabling the transducer to be translated along three orthogonal axes. The signal from the hydrophone was amplified by a Panametrics Ultrasonic Preamplifier, captured by a Tektronix 2432A digital oscilloscope and retrieved by a computer. The hydrophones were calibrated by direct comparison with a P V D F bilaminar shielded membrane hydrophone manufactured by G E C - M a r c o n i Research Centre. This membrane hydrophone was calibrated at the National Physical Laboratory, UK, with uncertainties in the sensitivity value at 10 M H z of + 8 % . As the miniature hydrophone is similarly calibrated, a qualified assumption is that the uncertainty is in the same range, i.e. less than _+10%. The measuring oscilloscope has an uncertainty within + 3% (manufacturer's specificationt.
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In addition the oscilloscope was controlled by a computer yielding an estimate of the uncertainties of the actual signal measurements within _+5%. Assuming that all the uncertainties mentioned are random and independent they can be combined in quadrature producing a t o t a l uncertainty of _+ 14% in the hydrophone measurement results.
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OD T set- up A test fixture controlled by four stepping motors held the transducer under measurement. These stepping motors controlled the transducer's movement along three orthogonal axes and the rotation with respect to the transducer axis. Tomographic algorithms assume that projections taken at 180° separation are equal except for mirroring, and if the mechanical mounting of the transducer is not correct this will not be the case. To dispose of potential experimental errors due to incorrect mounting, great care was taken in mounting the transducer case so that the angle between the transducer surface and laser is 90 °, and also the rotation axis was aligned with the centre sample point of a projection. The alignment was performed by cross-correlating two projections with 180 ° separation. The importance of these experimental 'tricks' increases with frequency and resolution requirements. Figure 4 shows a schematic block diagram of the experimental set-up. The light source was a 20 mW HeNe single-mode laser, the laser beam was not expanded. For good diffraction resolution the light beam should 'cover' several wavelengths of the measured ultrasound. With a 0.7 mm diameter beam almost five wavelengths are covered and this was experimentally satisfactory. The optical phase grating formed by the ultrasound was imaged by a lens onto the photodetector surface and all negative diffraction orders were removed by a spatial filter located one focal distance away from the lens (1000 mm). A high frequency photodiode with a 100/~m pinhole was connected to a two-stage amplifier, the amplifying electronics were incorporated with the photodiode in a small metal tube. The imaging process enlarges the phase grating approximately five times and, combined with the pinhole, this yields a sample size in the sound field of roughly 20#m. Two signals were output from the photodetector assembly; one proportional to a 'DC' light level and one high-frequency signal. The DC-part was used for calibration purposes and the high-frequency signal was measured by a quadrature detector producing
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two DC voltage levels proportional to the integrals in Equations 7a and 7b. The excitation signal and reference signals were controlled by temperature controlled quartz-stabilized function generators (HP 3325) and the voltages were measured by digital multimeters (H P 3478). All aspects of the measuring system were controlled by a personal computer. The transducer was scanned across the probing light beam, yielding one projection, and thereafter rotated. At each sampling point, five voltage values, for each quadrature component, were averaged. In all the presented data 20 projections of 101 samples with 0.1 mm sample distance have been used. Considering all the aspects of the measurement system a rigorous error analysis is difficult. The measuring multimeters have an error of less than + 1 % , the calibration error is of the order of _+4%, and the reconstruction algorithm error is of the order of _+ 10%. Several other error sources exist, among them the positioning of the transducer in the x-direction, electronic noise, and so on; consequently, a total error of approximately _+15% is a reasonable and qualified estimate. However, several aspects of the measurement system can be improved, thus reducing the error.
Results Preliminary measurements indicated that the 5 mm transducer had an actual 'active' diameter of 4.3 mm. Field calculations were performed with this diameter in a central plane parallel to the transducer axis. Calculations with both 0.25 mm and 0.1 mm sampling distances were performed. The smaller sampling distance data were used for cross-sectional comparisons with O D T and hydrophone measurements and the larger sampling distance data were used for the contour-map presentations. The area 'covered' by the field calculations was z = {1, 30} mm and x = { - 5 , 5} mm. The normalized results are shown in Fioure 5. A sinusoidal CW signal with an amplitude of 8 Vpp at 10 MHz excited the transducer during the measurements. Tomographic scans with 20 projections each with 101 samples and a sampling distance of 0.1 mm were carried out. For each scanned plane a reconstruction of the pressure amplitude (101 x 101 values) was calculated with the FBP algorithm. A total of 117 planes were measured and reconstructed with a z-sampling distance of 0.25 mm. The compilation of the 117 reconstruction planes forms a reconstruction volume, and from this volume one can extract, in theory, any desired reconstruction plane. A central plane parallel to the transducer axis was extracted and is shown in Fioure 6.
Ultrasonics 1993 Vol 31 No 6
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The I 0 M H z ultrasonic near-field." R. Eriksson et al. P, , , m , H ]plilud,. ~1'~
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different spatial locations. The hydrophone measurements were performed with the 0 . 5 m m diameter hydrophone and the O D T measurements were performed with the same number of samples and projections as above (101 x 20). Four curves are shown in Figure 9h. T w o O D T cross-sections are included due to the reconstruction error noticed in one (sharp dip). The second O D T cross-section is rotated 45 from the first. Discussion Generally, the measurement of the near-field from a l0 M H z ultrasound transducer is a s o m e w h a t complicated procedure due to the small feature size in the pressure field. In this paper we investigated two different techniques that offer g o o d spatial resolution. The two m e t h o d s are compared with a calculated theoretical pressure field from an ideal piston transducer. The transducer measured had a physical diameter of 5 m m , a microscope m e a s u r e m e n t of the conductive layer on the transducer surface gave a smaller diameter (4.7 m m ) . H o w e v e r , preliminary c o m p a r i s o n s of field calculations with O D T measurements indicated that a proper "acoustic" diameter was 4.3 m m . This choice was based on the location of the last major pressure dip. Consequently, the diameter used in the theoretical field calculations presented in Fiqure 5 was 4.3 m m instead of the physical diameter, 5 m m . It is clear from Fiqurc 5 that the pressure field contains large variations, especially in the extreme near-field.
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The same transducer was measured with a l m m miniature P V D F hydrophone. The transducer was excited with a 10 M H z , 8 Vpp, 15 cycle tone-burst signal. The hydrophone was scanned in a measurement pattern covering z - ~ l , 3 0 1 m m and x = [ - 5 , 5 1 m m with a sampling distance of 0.3 ram. The scanning plane was parallel to the transducer axis and symmetrically located. Measurement results are shown in Figure 7. Unfortunately, the transducer in the measurements presented above fractured forcing us to use a replacement. The new transducer was identical in design and as near identical in fabrication as possible. In Figure A' hydrophone m e a s u r e m e n t s from the second transducer are shown. The m e a s u r e m e n t s were performed with a 0.5 m m hydrophone in the same m e a s u r e m e n t pattern as with the 1 m m hydrophone. A consequence of this replacement is that a direct c o m p a r i s o n of pressure levels between Fiqures 6 and 7 and Figure 8 is impossible. Graphs of calculated, O D T and hydrophone crosssections at 5 m m along and at 45 m m distance from the transducer surface are presented in Figures 9a and 9h. The c o m p a r i s o n m e a s u r e m e n t s were done on the second transducer with careful consideration to alignment so as to eliminate any discrepancies due to measurements in
444
Ultrasonics 1993 Vol 31 No 6
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Figure 9 The scale to the left applies to the ODT and hydrophone measurements and the scale to the right applies to the calculated data. (a) Comparison between cross-sections from calculated data, ODT- and 0 . 5 m m hydrophone-measurements at z 5ram. (b) Comparison between cross sections from calculated data, ODT, ODT rotated 45 and 0 . 5 m m hydrophone-measurements at • 45 mm
The 10 MHz ultrasonic near-field: R. Eriksson et al. At a glance, it appears there is a discrepancy between the ODT-measurement results (Figure 6) and the field calculations. The large variations visible in Figure 5 in the extreme near-field are detected by ODT-measurement except for the geometrically small pressure dips along the symmetry axis. Real transducers only approximate ideal ones and, consequently, it might be possible that these variations do not exist in the near-field of the transducer measured. For instance, the 'pressure ridge' in the centre of the field close to the transducer surface indicates that the transducer is not ideally symmetric. If the near-field dips exist, with an axial extent of roughly 0.2 m m at z = 3 mm, then they could have been missed due to the restricted spatial resolution of the measurements. The axial-sampling distance in the O D T measurement was 0.25 mm. Also, the reconstruction program, whose resolution is determined by the number of projections, number of samples and sample density (both z and y), could be at fault and not compute reconstructed fields correctly. An increase in the number of projections and/or samples would enhance the spatial resolution of a reconstruction, none the less the discussion about algorithm accuracy above implies that the number of projections and projection samples are adequate even in the extreme near-field. Even though the optical set-up is not an interferometric one an unexpected vibration sensitivity was found and this can be seen in those measurements performed during an extended period of time as an increase in the noise level of a reconstructed cross-section. In Figure 6 this is visible at z-distances ~ {9 17 mm} and {25 mm}. These locations correspond to specific times of day, it was noted that an increase in noise always began at approximately 06.30 (morning) and ended at approximately 17.00 (afternoon); the total scan time, all 117 z-planes, was 45 hours. These times fit well to the activity levels of the building in which the lab is located. The 1 m m hydrophone data shown in Figure 7 do not show the detail of the calculated field. This is expected, mainly because of the large measuring area of the hydrophone. The details close to the transducer surface are not resolved but the pressure 'ridge' mentioned above is visible and similar in direction to that seen in Figure 6. Another distinct feature that is seen in all three contour maps is the pressure dip at 16 mm. The pressure levels measured with O D T appear to be in agreement with the pressure levels measured with the hydrophone although the hydrophone generally shows slightly larger pressure values. The 0.5 m m hydrophone data shown in Figure 8 show somewhat increased resolution around the 5 m m and 11 mm distances but there is also some loss of detail close to the transducer surface. The new transducer does not seem to have the angled pressure 'ridge' seen with the first transducer. A casual look at Figures 6 8 reveals that the spatial resolution of O D T is greater than that with the hydrophone. All the contour maps presented suffer from a general scaling problem, which is not due to the data but the presentation method, i.e. the number of grey levels used and the interpolation method. A better comparison of pressure values and resolution is shown by cross-sectional beam profiles instead. Figure 9a shows a comparison between calculations, O D T measurements and 0.5 m m hydrophone measurements. The measurement curves have been shifted in the x-direction to facilitate a direct comparison with
calculated values. The detail in the theoretical field is better resolved with O D T than with the 0 . 5 m m hydrophone. Although great care was taken to make the measurements at the same locations this cannot be confirmed. The hydrophone was positioned manually and, as can be seen, the size of the field details is of the order of 0.5 mm. The peak pressure values agree well but the locations of the peaks differ. The decreasing trend in pressure values to the left is depicted by both methods. This trend could be due to a small asymmetry of the transducer surface. The larger noise in the O D T values is partly due to the reconstruction algorithm. As before, the reconstruction from which the O D T cross-section was extracted was filtered with a nearest-neighbour averaging. Other filters could possibly reduce the noise further. In Figure 9b (45 ram) four three-measurement and one calculated, curves are shown. As in Figure 9a the peak pressure values agree well. The O D T and the hydrophone data represent measurements at the same locations. The sharp feature seen with O D T is a typical reconstruction error probably due to an outlier in the projection data. Because of the tomographic reconstruction 'erroneous' data can affect the entire reconstructed data set. An O D T cross-section extracted at 45 ° relative to the other two is also shown in Figure 9b, this cross-section better represents O D T ' s ability to measure pressure values at this distance (45 mm). A numerically calculated cross-section is included as well. With both the hydrophone and O D T measurements the expected sidelobes are visible as inclined plateaus on both sides of the profiles. The width of the calculated beam agrees well with both measurement methods. The agreement between hydrophone data and calculated data is excellent and superior to O D T . From the curves presented in Figure 9b it could be argued that the variations in pressure amplitude in the O D T cross-section from 5 r a m are not due to real pressure variations but instead are reconstruction errors. However, it is believed that the regularity and positions of the pressure dips, in comparison with the calculated data, are confirmations of the validity of the O D T data. The O D T data from 45 m m also show small pressure 'bumps' not expected from the calculations and not detected by the hydrophone. These bumps are most likely due to reconstruction noise. As a summary, a few items can be pointed out. It appears that O D T has a potentially better spatial resolution compared with 1 m m and 0.5 m m hydrophones. The finer details of a 10 M H z near-field are resolved better with O D T . The noise in O D T measurements can be a problem and is generally greater than that in hydrophone measurements. The experimental difficulties with O D T limits the method's widespread use in a non-specialist setting in contrast to hydrophones which are comparatively simple to use. In principle O D T is a self-calibrating method and does not need any external calibration. Consequently, O D T can be useful for reference measurements and/or measurements where great spatial detail is required. Careful design of an O D T system could possibly offer an alternative to other reference measurement methods such as the laser interferometer. The results imply that O D T has an increased spatial resolution compared with the 0.5 and 1.0 mm hydrophones. However, even better spatial resolution is needed to draw definite conclusions regarding the complexity of
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The 10 M H z ultrasonic near-field." R. Eriksson et al. the near-field of the l0 MHz transducer. A cautious conclusion would be that the pressure field complexity for the investigated transducer is different from an ideal model even though the transducer was designed to approximate the pressure field. The hypothesis that interference patterns in the near-field are a factor behind the problems with Doppler blood perfusion measurements is not refuted.
7 8 9 10 11
References 1 Hertz, C.H. Fourth European Congress on U'/trasound in Medicim' and Bioloqy Dubrovnik, 17 24 May (1981) 2 Dymling, S.O. Measurements of blood perfusion in tissue using Doppler ultrasound, PhD Thesis, Department of Electrical Measurements, Lund Institute of Technology, LUTEDX:(TEEM1027)/1 4 (1985) 3 Basler, S., Vieli, A. and Aoliker, M. Measurement of tissue blood flow by high frequency Doppler ultrasound. In A,h'am'cs iH E.vperimental Medicine and Bioh~,qy, Vol. 220 Continuous 71anscutaneous Monitormo (Ed. Huch, A., Huch, R. and Rooth, G.) Plenum Press, New York, p. 223 11987) 4 Erikssoo, R., Persson, H.W., Dymling, S.O. and Lindstr6m, K. Evaluation of Doppler ultrasound for blood perfusion measurements Ultrasound in Med. & Biol. (1991) 17 445 5 Eriksson, R., Persson, H.W., Dymling, S.O. and Lindstr6m, K. Blood perfusion measurement with multi-frequency Doppler ultrasound. Submitted to Uhrasound in Med & Biol 6 Archer-Hall, J.A. and Gee, D. A single integral computer method
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12 13 14
15 I(~ 17 lS 19
for axisymmetric transducers with wtrious boundary conditions N D T hlternationa111980) 13 95 Preston, R.C. (Ed) Output Measurement.s/m" Medical (/Itrclsound Springer-Verlag, London (19911 Raman, C.V. and Nath, N.S.N. t h e diffraction of light by high frequency sound waves: Part I 11 Proc lnd A cad A'ci ( 1936 ) IIA 4(16 Reibold, R. and Molkenstruck, W. Light diffraction tomography applied to the investigation of ultrasonic tields. Part I: Continuous waves Acustica 11984) 56 180 Reibold, R. and Holzer, F. Complete mapping of ultrasonic fields from optically measured data in a single cross-section Acustica 11985) 58 11 Reihold, R. Light diffraction tomography applied to the investigation of ultrasonic fields. Part 1I: Standing waves ,4custica 11987) 63 283 Reibold, R. and Kwiek, P. Optical near-field investigation into the Raman Nath and KML regimes of diffraction by ultrasonic wave,,, A cuxtica (1990) 70 223 Richter, K.P., Reibold, R. and Molkenstruck, W. Sound lield characteristics of ultrasonic composite pulse transducers U/:ra.~onic.s (1991) 29 76 Larsen, P.N. and Bj6rn6, L. Transducer defect studies using light diffraction tomography Ultra,sonics (1987) 27 86 Holm, A., Persson, H.W. and Lindstr6m, K. Measurements of uhrasonic fields with optical diffraction tomography ~:ltrasound in Med & Biol 11991) 17 505 Holm, A. and Persson, tt.W. Optical diffraction tomograph 3 applied to airborne ultrasotmd l/trasoni~s 11993) 31 259 Korpel, A. Acousto-Optics Marcel Dckker. New York 11988) Riley, W.A. and Klein, W.R. Piezo-optic coefficients of liquids ,I 4cou.~t Soc Am (19671 42 1258 Kak, A.C. and Slaney, M. Prim.iph,.s O/('omPu:criccd Tomo:lral;hi, hml#in~t IEEE Press, New York II9881
Received 28 January 1993; accepted 22 April 1993 The near-field from a 10 MHz ultrasound transducer has been studied using three different techniques; theoretical calculations, miniature hydrophone measurements and optical diffraction tomography (ODT). The motive was twofold; to evaluate the measurement possibilities of ODT for high-frequency ultrasound and to verify the calculated complexity of near-fields from real ultrasound transducers designed for use in blood perfusion measurements. Calculations of the field from an ideal piston transducer were done with surface integrals of the Kirchhoff function and the measured transducer was designed to approximate this ideal. Contour maps, from the three methods, of pressure amplitude from 1 to 30 mm along the beam are presented with beam profiles at 5 and 45 mm. The results imply that ODT has an increased spatial resolution compared with the 0.5 and 1.0 mm hydrophones. However, even better spatial resolution is needed to draw definite conclusions regarding the complexity of the near-field of the 10 MHz transducer.
Keywords: near-field; hydrophones; optical diffraction tomography; blood perfusion
In this paper, results from numerical calculations and measurements of the ultrasonic near-field of a 10 MHz, 5 m m diameter piston transducer are presented. Two measurement methods were used, optical diffraction tomography and a miniature polyvinyldifluoride (PVDF) hydrophone. The two methods were chosen in order to achieve measurements with an adequate spatial resolution. The measurements were done in order to investigate the complexity of the ultrasonic near-field and to verify earlier assumptions concerning the effect this complexity has on blood perfusion measurements. The possibility of measuring blood perfusion in real-time in deep tissue has, for a long time, been desired by physicians and for an equally long time been a challenge for scientists. Hertz 1 suggested a method for blood perfusion measurement based on the ultrasonic Doppler method. The method has been examined further by others and found to be promising but there are also difficulties with the technique z 4. Among the problems are unexpected variations of the measured perfusion. Experimental studies suggest that these variations are related to ultrasonic interference combined with small transducer movements 5. It is essential for blood perfusion measurements to use high-frequency ultrasound owing to, among other things, 0041 624X/93/060439-08 © 1993 Butterworth-Heinemann Ltd
the low speed of the moving scatterers down to and below 0.1 m m s l. Also, scattering from small particles increases with frequency. The resulting interference in the near-field of an ultrasonic transducer, due to the wave nature of the ultrasound, causes spatial variations in the ultrasonic intensity. The feature size of these variations is of the order of a wavelength. If these differences are large and if the blood perfusion is not homogeneously distributed throughout the sample volume, then small movements of the transducer will lead to variations in the measured perfusion value. These variations are not related to changes in the blood perfusion and the extent of the variations has so far prevented clinical use of the measurement method. Measurements of the complex near-field, especially at frequencies above a few MHz, are difficult due to spatial resolution requirements. An established measurement method like the miniature hydrophone method has a few undesirable characteristics limiting its usefulness. Some of the more important characteristics are the size of the hydrophone, standing waves and hydrophone angle. A hydrophone integrates the pressure field on its active surface so that there is always a spatial averaging. Evidently, one can reduce the averaging area by reducing the size of the hydrophone, but this cannot be done
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439
The 10 M H z ultrasonic near-field." R. Eriksson et al. without decreasing the signal-to-noise ratio. Errors due to standing waves can be decreased by using burst excitation but will always be a problem in the very close near-field. However, hydrophones have advantages with their comparatively simple practical set-up and use. Optical diffraction tomography (ODT), on the other hand, promises high spatial resolution and potentially absolute measurements. The spatial resolution is limited by a combination of optical system resolution and mechanical scanning resolution. Standing waves are not a problem in O D T owing to its non-invasive nature. There are experimental difficulties with the laser beam angle that are similar to the hydrophone angle problems. However, O D T does not offer the experimental simplicity of the hydrophone. There are several variations of the theoretical calculations of the near-field and this paper does not introduce any new techniques nor does it include a review; instead, previously' published* mathematical models are used throughout. There are also several measurement systems based on miniature hydrophones with corresponding documentation, a recent description and discussion of such systems was edited by Preston ~. Diffraction of light by ultrasound is not a newly discovered phenomenon numerous papers have been published since the 1930s, among them the much cited paper by Raman and Nath s. ODT, as a measurement technique, was introduced by Reibold and Molkenstruck ~ who applied it to measurements of ultrasound transducers in the 3 MHz range. Reibold e l al. ~° 13 have published several papers describing measurements with O D T and other authors have also used the technique ~ ~'. Unfortunately, there is a lack of literature and data on measurements of high-frequency ultrasound near-fields so that any extensive comparison between O D T and other results is quite difficult. It is known from theory that the ultrasonic near-field is complex and, furthermore, at 10 MHz, the near-field is very sensitive to transducer design variables. It is thus useful to investigate it experimentally and compare experimental results with theory. As far as wc can ascertain, previous O D T measurements have been limited in frequency up to approximately 5 MHz, this paper is an attempt to extend the frequency range upwards. The blood perfusion measurements with 1 0 M H z ultrasound described earlier ~ have shown inconclusive results. It is believed that the interference pattern in the near-field causes these measurement problems and the O D T measurement results could eventually be taken into account in the design of a blood perfusion measurement system based on the Doppler technique. To summarize, the motives for this investigation are: (1) the extension of O D T to high-frequency near-field measurements and a comparison with the generally used miniature hydrophone measurement technique: and (2) to verify the intensity variations in the near-field of a real 10 MHz ultrasound transducer used for blood perfusion measurements.
M a t e r i a l s and m e t h o d s Numerical calculatious Analysing diffraction fields in front of a transducer involves surface integrals of the form
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U l t r a s o n i c s 1 9 9 3 Vol 31 No 6
t
(./)=,
GdS
{1~
!
where S is the surface of the transducer and ~/, is the scalar velocity potential. G is a Green's function to be chosen according to the boundary conditions of the transducer. The Kirchhoff function is an appropriate Green's function for a transducer in a thin case, totally immersed in an acoustic medium ~'. The rear wall of the transducer element must be insulated from the acoustic medium and not contribute to the acoustic field. In Equation (2) G is replaced with the Kirchhoff function =
k 1+
+
dS
(2)
where s is a field point to an element on the surface of the transducer and k = ~,~:c. A prediction of the scalar velocity field is best accomplished if this surface integral is rewritten in a single integral form. Archer-Hall and G e C used geometrical arguments to accomplish this transformation. Equation (3) shows the single integral form which, with reasonable computer power (a PC), can be numerically calculated to obtain the complete field of a plane piston radiator. ~bKlr,-)=
1 :~:- r = .
0 + 2~
e
ik~
r > ~/3 ~
e ik~ 1 +
. r cos i/, ,~, a 2 + r - '
,l~'
2arcos~k.
d~k (3l
Now ,s
1-2 + a 2 + 1.2
2 a t c o s ~/J)~ 2
and ~ is the radius of the transducer, r is the radial distance, _- is the distance from the transducer surface and ~, is the variable of integration. As can be seen, Equation (3) contains two parts. The first term represents a plane wave that contributes to the field inside the radius of the transducer and with half its value on the rim of the transducer. The second term is an edge wave term that contributes to the whole field of the transducer. For the numerical calculations, a personal computer (Macintosh Quadra 700) was used. The numerical integration was implemented with a commercial software package (Mathematica, Wolfram Research). Raman Nath diffraction Diffraction of light by sound has been studied extensively since the 1930s. Raman and Nath wrote a seminal paper s explaining acousto-optic interaction from a diffraction grating standpoint. They also introduced the Raman Nath equations relating diffracted light intensity to the pressure amplitude of an ultrasonic wave. There is no shortage of literature on the subject and a recent and excellent theoretical summary of ultrasonic light diffraction wits written by Korpel ~ The basis of the present experimental study is the optical diffraction method introduced by Reibold and Molkenstruck '~ in 1984. Consider a continuous (CWI progressive ultrasonic wave as illustrated in t:7:lm'C I a n d let the ultrasonic pressure-lield be described by P I x . v, z, t) - P,,,,,(.x-. y. z) sin ( K , z - - D - J
+~p, Ix, y, .:)) 14)
The 10 MHz ultrasonic near-field. R. Eriksson et al. Transducer rotational axis
Transducer ~ -
•
i!!i
Presented measurement plane
Scanning direction ~-~ z
ix
Y _
Photodetector
.I,,Spatial filter Lens
Tomography algorithm
~_._(reconstruction surface) Scanning plane
Laser /
Figure 1
/.'-;
Optical diffraction tomography schematic
where ~s is the angular frequency, K s is the wave-number of the ultrasound and ~ps(x, y, z) is the phase-angle related to some arbitrary reference. The index of refraction in the water is modulated by the ultrasound, and the elasto-optic coefficient that couples pressure and index of refraction is denoted 18 Pop ~ 1.51 x 10 -1° m 2 N -1 The details of the R a m a n Nath model of ultrasonic light diffraction are outside the scope of this paper but some definitions are needed. For instance, the Raman Nath parameter v, i.e. the integrated optical phase, is defined by
v(x, y, z, t)
= keoPo p
Pmax(X, y, Z)
--3C
x sin(Ksz - f~,t + q~,(x, y)) dx = f~(y) sin(K,z - f2d + 0,(Y))
(5)
where keo is the vacuum wave-number of the probing light beam. If the ultrasound and optical parameters are within the Raman Nath model's limits 17 the light intensity detected by a photo-detector placed in the image plane includes a time-varying part as in Equation (4)
l(bO'), z, t) = Io2Jo(b)J t (b) cos(K~z - f2,t + 0s(Y))
(6)
where I 0 is the light intensity without sound and J,(b) are Bessel functions of the first kind. Two criteria must be met for the Raman Nath model to be valid for the current frequency, transducer size and pressure levels. These are Q = DK~k~ t < 1 and Qv < 1, where D is the active diameter of the transducer. With the 10 M H z transducer measured one calculates: Q ~ 0 . 6 7 and Qv ~ 0.3. For Equation (6), to be valid, all diffraction orders except the zeroth and first positive, or negative, must be removed by a spatial filter (Figure 1). If the amplitude and phase of the signal described by Equation (6) are measured, one can write two expressions that relate these quantities to the ultrasound pressure field at a specific z.
L'(y)
COS (Ps -keoPop
k~oPop
Pmax(X, y) COS[qgs(X, y ) ] dx
(7a)
Pmax(X,y) sin[~o~(x, y)] dx
(7b)
- x
~'0') sin O~ = ,,
With Equations (6), (7a) and (7b) the relationship between the ultrasound pressure field and the measured light intensity is clarified. The end result comprises two ray integrals that can be used for separate tomographic r e c o n s t r u c t i o n s of Pmax(X,y) COS[q~s(x,y)] a n d Pmax(X,Y) sin[q~,(x,y)] respectively. These two reconstructions can be used to produce Pm,x(X, y) and q~s(x, y). In this paper, only Pmax(X, Y) is calculated and presented.
The basis of most transmission tomography algorithms is the ray-integral concept, i.e. a characteristic denoted f(x, y) of an object is integrated by a probing beam along a geometrically straight line. From sampled parallel collections of ray integrals ( = one projection) taken at angles covering 180 c' in a plane it is possible to reconstruct f(x, y). There are countless variations of tomography algorithms and we will not delve deeply into this field. A good introductory text as well as a source of algorithms is Kak and Slaney 19. In a practical ultrasound field measurement system there is always a trade-off between measurement time and desired accuracy/resolution. For instance, with hydrophone systems a doubling of the resolution in a plane gives a quadrupling of the required data points and thus also an increased measurement time. With diffraction tomography there is no simple relation between resolution and measurement time. This is due to the complex dependencies of resolution on the number of samples, the number of projections and the algorithm used. In many cases tomography algorithms are tested with simulated data intended to resemble some part of the human body, e.g. the Shepp and Logan head phantom; however, these test phantoms were not appropriate for the type of projection data measured in O D T . It was more appropriate to test algorithms with simulated projection data from ultrasound fields. The simulated projection data were calculated with the numerical techniques described above at distances of 5 m m and 23 m m from the transducer surface, i.e. in the near-field. A pressure field containing 101 × 101 values with a sample distance of 0.1 m m (each direction) was calculated and from this, field projection data were subsequently derived. Since we used an axisymmetric model all projections were the same, independent of projection angle. The algorithms tested were Filtered Backprojection (FBP) and Algebraic Reconstruction Technique (ART). These algorithms were implemented on a personal computer. Generally, the literature agrees that ART or other iterative algorithms (MART, SIRT) are better for reconstructions with a small number of projections, e.g. 1-50. Our reconstruction results indicated that this was not the case for the projection data expected in the present paper. On the contrary, our tests indicated that FBP is better even for a small number of projections. In Figure 2 two graphs illustrating combinations of simulated theoretical cross-sections and reconstructed cross-sections are presented. The sample distance in the simulated projections was 0.1 mm (101 samples (n)) and the number of projections 20 (m). In Figure 2a a comparison is presented between the theoretical cross-section, a reconstructed cross-section, and a reconstructed cross-section where 5% white noise was added independently to each projection. All
Ultrasonics 1993 Vol 31 No 6
441
The 10 MHz ultrasonic near-field. R. Eriksson et al. I
,, 0.8
,,
/
~
,
,
,, ~
reconstructions in this paper FBP has been used, motivated by the findings presented in Fixture 2.
,, ', i
,
{I.7 '~
0.6-
0,4-
z
03 0,2 0J
a
ii 5
4
~
2
-I
II
I
2
1
4
Position (mm)
I
:', ',
,'~ ,'; i', d' ' ':
;,'
O.8
"¢
,
>
O.4
0,2
b
o
4
~
2
1
I
2
Posilion ( m m )
Figure2 Pressure cross-sections through transducer axis. 101 samples and 20 projections. (a) Comparison between reconstruc tion and simulation at 5 mm distance from transducer. Simulation (thin curve). Reconstruction with FB P (thick curve). Reconstruction (FBP) with 5% white noise added to each projection (thick dashed curve). (b) Comparison between simulation and different recon struction algorithms at 5 mm. Simulation (thin curve). Reconstruc tion with FBP (thick curve). Reconstruction with ART (thick dashed curve)
reconstruction planes were filtered with nearestneighbour averaging before the cross-sections were extracted. This filtering was done to equalize this situation to that of experimental data on which filtering has been done to smooth out noise. The amplitude values of the reconstructions are erroneous but the locations of the dips are correct and the variations in the theoretical cross-section are 'detected' both with noiseless and noisy projection data. In fact, the filtering procedure increases the amplitude difference between theory and reconstruction for projection data without noise, i.e. an unfiltered cross-section would appear more accurate. In Fiqure 2h a comparison between the theoretical cross-section, a reconstructed cross-section (FBP) and another reconstructed cross-section (ART) is presented. The ART algorithm 'misses' some of the theoretical variations and we judge it to be inferior to FBP in this case. With added noise the ART algorithm's accuracy is decreased. The reason for FBP's superiority over the iterative algorithm is most likely due to the symmetric nature of the expected projections and also the relatively limited spatial frequency content, i.e. there are no edges in the ultrasound field. One definite advantage with FBP compared with ART is speed, reconstruction time with ART is proportional to imn 3, where i is the number of iterations, and reconstruction time with FBP is proportional to mn. Judging from the graphs in Fiyure 2 and numerical comparisons, the numerical error from the reconstruction is roughly 4-10%. In all subsequent
442
Ultrasonics 1993 Vol 31 No 6
Transducer design The 10 MHz transducer used was made in-house and it is designed to have a radiating characteristic as close as possible to an ideal piston transducer. A piezoelectric ceramic was mounted in a plastic case and damped by backing material, Figure 3, to prevent internal reflections. In order not to distort the ultrasonic field the front surface was left uncovered, i.e. no matching or protective layer was used.
Hydrophone set-up The transducer was fixed in a water bath and excited with a 10 MHz, 8 Vpp, 15 cycle tone-burst signal from a quartz stabilized function generator (Hewlett Packard 3314A). The tone-burst signals as well as sound absorbing material on the walls ofthc water tank served to minimize noise echoes and standing waves. Two calibrated miniature (1 and 0.5 mm diameter) P V D F hydrophones (H.W. Persson) were used. During the measurements, the hydrophone was mounted in a test fixture controlled by three stepping motors enabling the transducer to be translated along three orthogonal axes. The signal from the hydrophone was amplified by a Panametrics Ultrasonic Preamplifier, captured by a Tektronix 2432A digital oscilloscope and retrieved by a computer. The hydrophones were calibrated by direct comparison with a P V D F bilaminar shielded membrane hydrophone manufactured by G E C - M a r c o n i Research Centre. This membrane hydrophone was calibrated at the National Physical Laboratory, UK, with uncertainties in the sensitivity value at 10 M H z of + 8 % . As the miniature hydrophone is similarly calibrated, a qualified assumption is that the uncertainty is in the same range, i.e. less than _+10%. The measuring oscilloscope has an uncertainty within + 3% (manufacturer's specificationt.
Smm j
z
•
- - Piezoelectric
.......
Backing
4
il
'~
~
Plastic
4
J
,
~---Metal
case
I
Ill
Figure3
Ii
Basic transducer design schematic
material
material
The 10 MHz ultrasonic near-field. R. Eriksson et al.
In addition the oscilloscope was controlled by a computer yielding an estimate of the uncertainties of the actual signal measurements within _+5%. Assuming that all the uncertainties mentioned are random and independent they can be combined in quadrature producing a t o t a l uncertainty of _+ 14% in the hydrophone measurement results.
_~ E 5 3
t 0.4 0.2 0 ~-5"1
OD T set- up A test fixture controlled by four stepping motors held the transducer under measurement. These stepping motors controlled the transducer's movement along three orthogonal axes and the rotation with respect to the transducer axis. Tomographic algorithms assume that projections taken at 180° separation are equal except for mirroring, and if the mechanical mounting of the transducer is not correct this will not be the case. To dispose of potential experimental errors due to incorrect mounting, great care was taken in mounting the transducer case so that the angle between the transducer surface and laser is 90 °, and also the rotation axis was aligned with the centre sample point of a projection. The alignment was performed by cross-correlating two projections with 180 ° separation. The importance of these experimental 'tricks' increases with frequency and resolution requirements. Figure 4 shows a schematic block diagram of the experimental set-up. The light source was a 20 mW HeNe single-mode laser, the laser beam was not expanded. For good diffraction resolution the light beam should 'cover' several wavelengths of the measured ultrasound. With a 0.7 mm diameter beam almost five wavelengths are covered and this was experimentally satisfactory. The optical phase grating formed by the ultrasound was imaged by a lens onto the photodetector surface and all negative diffraction orders were removed by a spatial filter located one focal distance away from the lens (1000 mm). A high frequency photodiode with a 100/~m pinhole was connected to a two-stage amplifier, the amplifying electronics were incorporated with the photodiode in a small metal tube. The imaging process enlarges the phase grating approximately five times and, combined with the pinhole, this yields a sample size in the sound field of roughly 20#m. Two signals were output from the photodetector assembly; one proportional to a 'DC' light level and one high-frequency signal. The DC-part was used for calibration purposes and the high-frequency signal was measured by a quadrature detector producing
I
tTl /°c-","
Istepping
I..... [ - - 3 1 Transd.... .~
Laser beam
l
~ l
~~;
,o
:
"?
~
Iz
/I
x •
Photodetector
Approx. 6 m
Figure4
ODT measurement system. Thick curves indicate IEEE-488 communications. Thin curves indicate signal lines and stepper motor control. The 6 m optical path was folded with mirrors (not shown)
N~I realized pyessur~~ mnplhudc
S i m u l a t e d data
3
5
7
9 I[ 13 15 17 19 2i 23 Distance frora transducer surl~a.'e ( r a m )
25
27
29
Figure 5 Field calculations. 10 M Hz 4.3 mm diameter transducer, sampling distance 0.25 ram. The darker colour iudicates higher pressure
two DC voltage levels proportional to the integrals in Equations 7a and 7b. The excitation signal and reference signals were controlled by temperature controlled quartz-stabilized function generators (HP 3325) and the voltages were measured by digital multimeters (H P 3478). All aspects of the measuring system were controlled by a personal computer. The transducer was scanned across the probing light beam, yielding one projection, and thereafter rotated. At each sampling point, five voltage values, for each quadrature component, were averaged. In all the presented data 20 projections of 101 samples with 0.1 mm sample distance have been used. Considering all the aspects of the measurement system a rigorous error analysis is difficult. The measuring multimeters have an error of less than + 1 % , the calibration error is of the order of _+4%, and the reconstruction algorithm error is of the order of _+ 10%. Several other error sources exist, among them the positioning of the transducer in the x-direction, electronic noise, and so on; consequently, a total error of approximately _+15% is a reasonable and qualified estimate. However, several aspects of the measurement system can be improved, thus reducing the error.
Results Preliminary measurements indicated that the 5 mm transducer had an actual 'active' diameter of 4.3 mm. Field calculations were performed with this diameter in a central plane parallel to the transducer axis. Calculations with both 0.25 mm and 0.1 mm sampling distances were performed. The smaller sampling distance data were used for cross-sectional comparisons with O D T and hydrophone measurements and the larger sampling distance data were used for the contour-map presentations. The area 'covered' by the field calculations was z = {1, 30} mm and x = { - 5 , 5} mm. The normalized results are shown in Fioure 5. A sinusoidal CW signal with an amplitude of 8 Vpp at 10 MHz excited the transducer during the measurements. Tomographic scans with 20 projections each with 101 samples and a sampling distance of 0.1 mm were carried out. For each scanned plane a reconstruction of the pressure amplitude (101 x 101 values) was calculated with the FBP algorithm. A total of 117 planes were measured and reconstructed with a z-sampling distance of 0.25 mm. The compilation of the 117 reconstruction planes forms a reconstruction volume, and from this volume one can extract, in theory, any desired reconstruction plane. A central plane parallel to the transducer axis was extracted and is shown in Fioure 6.
Ultrasonics 1993 Vol 31 No 6
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different spatial locations. The hydrophone measurements were performed with the 0 . 5 m m diameter hydrophone and the O D T measurements were performed with the same number of samples and projections as above (101 x 20). Four curves are shown in Figure 9h. T w o O D T cross-sections are included due to the reconstruction error noticed in one (sharp dip). The second O D T cross-section is rotated 45 from the first. Discussion Generally, the measurement of the near-field from a l0 M H z ultrasound transducer is a s o m e w h a t complicated procedure due to the small feature size in the pressure field. In this paper we investigated two different techniques that offer g o o d spatial resolution. The two m e t h o d s are compared with a calculated theoretical pressure field from an ideal piston transducer. The transducer measured had a physical diameter of 5 m m , a microscope m e a s u r e m e n t of the conductive layer on the transducer surface gave a smaller diameter (4.7 m m ) . H o w e v e r , preliminary c o m p a r i s o n s of field calculations with O D T measurements indicated that a proper "acoustic" diameter was 4.3 m m . This choice was based on the location of the last major pressure dip. Consequently, the diameter used in the theoretical field calculations presented in Fiqure 5 was 4.3 m m instead of the physical diameter, 5 m m . It is clear from Fiqurc 5 that the pressure field contains large variations, especially in the extreme near-field.
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The same transducer was measured with a l m m miniature P V D F hydrophone. The transducer was excited with a 10 M H z , 8 Vpp, 15 cycle tone-burst signal. The hydrophone was scanned in a measurement pattern covering z - ~ l , 3 0 1 m m and x = [ - 5 , 5 1 m m with a sampling distance of 0.3 ram. The scanning plane was parallel to the transducer axis and symmetrically located. Measurement results are shown in Figure 7. Unfortunately, the transducer in the measurements presented above fractured forcing us to use a replacement. The new transducer was identical in design and as near identical in fabrication as possible. In Figure A' hydrophone m e a s u r e m e n t s from the second transducer are shown. The m e a s u r e m e n t s were performed with a 0.5 m m hydrophone in the same m e a s u r e m e n t pattern as with the 1 m m hydrophone. A consequence of this replacement is that a direct c o m p a r i s o n of pressure levels between Fiqures 6 and 7 and Figure 8 is impossible. Graphs of calculated, O D T and hydrophone crosssections at 5 m m along and at 45 m m distance from the transducer surface are presented in Figures 9a and 9h. The c o m p a r i s o n m e a s u r e m e n t s were done on the second transducer with careful consideration to alignment so as to eliminate any discrepancies due to measurements in
444
Ultrasonics 1993 Vol 31 No 6
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The 10 MHz ultrasonic near-field: R. Eriksson et al. At a glance, it appears there is a discrepancy between the ODT-measurement results (Figure 6) and the field calculations. The large variations visible in Figure 5 in the extreme near-field are detected by ODT-measurement except for the geometrically small pressure dips along the symmetry axis. Real transducers only approximate ideal ones and, consequently, it might be possible that these variations do not exist in the near-field of the transducer measured. For instance, the 'pressure ridge' in the centre of the field close to the transducer surface indicates that the transducer is not ideally symmetric. If the near-field dips exist, with an axial extent of roughly 0.2 m m at z = 3 mm, then they could have been missed due to the restricted spatial resolution of the measurements. The axial-sampling distance in the O D T measurement was 0.25 mm. Also, the reconstruction program, whose resolution is determined by the number of projections, number of samples and sample density (both z and y), could be at fault and not compute reconstructed fields correctly. An increase in the number of projections and/or samples would enhance the spatial resolution of a reconstruction, none the less the discussion about algorithm accuracy above implies that the number of projections and projection samples are adequate even in the extreme near-field. Even though the optical set-up is not an interferometric one an unexpected vibration sensitivity was found and this can be seen in those measurements performed during an extended period of time as an increase in the noise level of a reconstructed cross-section. In Figure 6 this is visible at z-distances ~ {9 17 mm} and {25 mm}. These locations correspond to specific times of day, it was noted that an increase in noise always began at approximately 06.30 (morning) and ended at approximately 17.00 (afternoon); the total scan time, all 117 z-planes, was 45 hours. These times fit well to the activity levels of the building in which the lab is located. The 1 m m hydrophone data shown in Figure 7 do not show the detail of the calculated field. This is expected, mainly because of the large measuring area of the hydrophone. The details close to the transducer surface are not resolved but the pressure 'ridge' mentioned above is visible and similar in direction to that seen in Figure 6. Another distinct feature that is seen in all three contour maps is the pressure dip at 16 mm. The pressure levels measured with O D T appear to be in agreement with the pressure levels measured with the hydrophone although the hydrophone generally shows slightly larger pressure values. The 0.5 m m hydrophone data shown in Figure 8 show somewhat increased resolution around the 5 m m and 11 mm distances but there is also some loss of detail close to the transducer surface. The new transducer does not seem to have the angled pressure 'ridge' seen with the first transducer. A casual look at Figures 6 8 reveals that the spatial resolution of O D T is greater than that with the hydrophone. All the contour maps presented suffer from a general scaling problem, which is not due to the data but the presentation method, i.e. the number of grey levels used and the interpolation method. A better comparison of pressure values and resolution is shown by cross-sectional beam profiles instead. Figure 9a shows a comparison between calculations, O D T measurements and 0.5 m m hydrophone measurements. The measurement curves have been shifted in the x-direction to facilitate a direct comparison with
calculated values. The detail in the theoretical field is better resolved with O D T than with the 0 . 5 m m hydrophone. Although great care was taken to make the measurements at the same locations this cannot be confirmed. The hydrophone was positioned manually and, as can be seen, the size of the field details is of the order of 0.5 mm. The peak pressure values agree well but the locations of the peaks differ. The decreasing trend in pressure values to the left is depicted by both methods. This trend could be due to a small asymmetry of the transducer surface. The larger noise in the O D T values is partly due to the reconstruction algorithm. As before, the reconstruction from which the O D T cross-section was extracted was filtered with a nearest-neighbour averaging. Other filters could possibly reduce the noise further. In Figure 9b (45 ram) four three-measurement and one calculated, curves are shown. As in Figure 9a the peak pressure values agree well. The O D T and the hydrophone data represent measurements at the same locations. The sharp feature seen with O D T is a typical reconstruction error probably due to an outlier in the projection data. Because of the tomographic reconstruction 'erroneous' data can affect the entire reconstructed data set. An O D T cross-section extracted at 45 ° relative to the other two is also shown in Figure 9b, this cross-section better represents O D T ' s ability to measure pressure values at this distance (45 mm). A numerically calculated cross-section is included as well. With both the hydrophone and O D T measurements the expected sidelobes are visible as inclined plateaus on both sides of the profiles. The width of the calculated beam agrees well with both measurement methods. The agreement between hydrophone data and calculated data is excellent and superior to O D T . From the curves presented in Figure 9b it could be argued that the variations in pressure amplitude in the O D T cross-section from 5 r a m are not due to real pressure variations but instead are reconstruction errors. However, it is believed that the regularity and positions of the pressure dips, in comparison with the calculated data, are confirmations of the validity of the O D T data. The O D T data from 45 m m also show small pressure 'bumps' not expected from the calculations and not detected by the hydrophone. These bumps are most likely due to reconstruction noise. As a summary, a few items can be pointed out. It appears that O D T has a potentially better spatial resolution compared with 1 m m and 0.5 m m hydrophones. The finer details of a 10 M H z near-field are resolved better with O D T . The noise in O D T measurements can be a problem and is generally greater than that in hydrophone measurements. The experimental difficulties with O D T limits the method's widespread use in a non-specialist setting in contrast to hydrophones which are comparatively simple to use. In principle O D T is a self-calibrating method and does not need any external calibration. Consequently, O D T can be useful for reference measurements and/or measurements where great spatial detail is required. Careful design of an O D T system could possibly offer an alternative to other reference measurement methods such as the laser interferometer. The results imply that O D T has an increased spatial resolution compared with the 0.5 and 1.0 mm hydrophones. However, even better spatial resolution is needed to draw definite conclusions regarding the complexity of
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The 10 M H z ultrasonic near-field." R. Eriksson et al. the near-field of the l0 MHz transducer. A cautious conclusion would be that the pressure field complexity for the investigated transducer is different from an ideal model even though the transducer was designed to approximate the pressure field. The hypothesis that interference patterns in the near-field are a factor behind the problems with Doppler blood perfusion measurements is not refuted.
7 8 9 10 11
References 1 Hertz, C.H. Fourth European Congress on U'/trasound in Medicim' and Bioloqy Dubrovnik, 17 24 May (1981) 2 Dymling, S.O. Measurements of blood perfusion in tissue using Doppler ultrasound, PhD Thesis, Department of Electrical Measurements, Lund Institute of Technology, LUTEDX:(TEEM1027)/1 4 (1985) 3 Basler, S., Vieli, A. and Aoliker, M. Measurement of tissue blood flow by high frequency Doppler ultrasound. In A,h'am'cs iH E.vperimental Medicine and Bioh~,qy, Vol. 220 Continuous 71anscutaneous Monitormo (Ed. Huch, A., Huch, R. and Rooth, G.) Plenum Press, New York, p. 223 11987) 4 Erikssoo, R., Persson, H.W., Dymling, S.O. and Lindstr6m, K. Evaluation of Doppler ultrasound for blood perfusion measurements Ultrasound in Med. & Biol. (1991) 17 445 5 Eriksson, R., Persson, H.W., Dymling, S.O. and Lindstr6m, K. Blood perfusion measurement with multi-frequency Doppler ultrasound. Submitted to Uhrasound in Med & Biol 6 Archer-Hall, J.A. and Gee, D. A single integral computer method
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15 I(~ 17 lS 19
for axisymmetric transducers with wtrious boundary conditions N D T hlternationa111980) 13 95 Preston, R.C. (Ed) Output Measurement.s/m" Medical (/Itrclsound Springer-Verlag, London (19911 Raman, C.V. and Nath, N.S.N. t h e diffraction of light by high frequency sound waves: Part I 11 Proc lnd A cad A'ci ( 1936 ) IIA 4(16 Reibold, R. and Molkenstruck, W. Light diffraction tomography applied to the investigation of ultrasonic tields. Part I: Continuous waves Acustica 11984) 56 180 Reibold, R. and Holzer, F. Complete mapping of ultrasonic fields from optically measured data in a single cross-section Acustica 11985) 58 11 Reihold, R. Light diffraction tomography applied to the investigation of ultrasonic fields. Part 1I: Standing waves ,4custica 11987) 63 283 Reibold, R. and Kwiek, P. Optical near-field investigation into the Raman Nath and KML regimes of diffraction by ultrasonic wave,,, A cuxtica (1990) 70 223 Richter, K.P., Reibold, R. and Molkenstruck, W. Sound lield characteristics of ultrasonic composite pulse transducers U/:ra.~onic.s (1991) 29 76 Larsen, P.N. and Bj6rn6, L. Transducer defect studies using light diffraction tomography Ultra,sonics (1987) 27 86 Holm, A., Persson, H.W. and Lindstr6m, K. Measurements of uhrasonic fields with optical diffraction tomography ~:ltrasound in Med & Biol 11991) 17 505 Holm, A. and Persson, tt.W. Optical diffraction tomograph 3 applied to airborne ultrasotmd l/trasoni~s 11993) 31 259 Korpel, A. Acousto-Optics Marcel Dckker. New York 11988) Riley, W.A. and Klein, W.R. Piezo-optic coefficients of liquids ,I 4cou.~t Soc Am (19671 42 1258 Kak, A.C. and Slaney, M. Prim.iph,.s O/('omPu:criccd Tomo:lral;hi, hml#in~t IEEE Press, New York II9881