30 In this study, we investigated the excitation of guided waves within a 6.3-mm thick brass plate and a 31 6.5-mm thick bovine bone plate using an ultrasound phased array system with tw
Trang 15
6
7Q1 Kim-Cuong T Nguyena,b, Lawrence H Lea,⇑, Tho N.H.T Trana, Edmond H.M Louc
Department of Radiology and Diagnostic Imaging, University of Alberta, Edmonton, Alberta T6G 2B7, Canada
Department of Biomedical Engineering, Ho Chi Minh City University of Technology, Ho Chi Minh City, Viet Nam
10 c
Department of Surgery, University of Alberta, Edmonton, Alberta T6G 2B7, Canada
11
1 4
a r t i c l e i n f o
15 Article history:
16 Available online xxxx
17 Keywords:
18Q3 Ultr asound
19 Phased array
20 Beam steering
21 Osteoporosis
22 Cortical bone
23
2 4
a b s t r a c t
25 Long bones are good waveguides to support the propagation of ultrasonic guided waves The low-order
26 guided waves have been consistently observed in quantitative ultrasound bone studies Selective
excita-27 tion of these low-order guided modes requires oblique incidence of the ultrasound beam using a
trans-28 ducer-wedge system It is generally assumed that an angle of incidence, hi, generates a specific phase
29 velocity of interest, co, via Snell’s law, hi= sin1(vw/co) wherevwis the velocity of the coupling medium
30
In this study, we investigated the excitation of guided waves within a 6.3-mm thick brass plate and a
31 6.5-mm thick bovine bone plate using an ultrasound phased array system with two 0.75-mm-pitch array
32 probes Arranging five elements as a group, the first group of a 16-element probe was used as a
transmit-33 ter and a 64-element probe was a receiver array The beam was steered for six angles (0°, 20°, 30°, 40°,
34 50°, and 60°) with a 1.6 MHz source signal An adjoint Radon transform algorithm mapped the time-offset
35 matrix into the frequency-phase velocity dispersion panels The imaged Lamb plate modes were
identi-36 fied by the theoretical dispersion curves The results show that the 0° excitation generated many modes
37 with no modal discrimination and the oblique beam excited a spectrum of phase velocities spread
asym-38 metrically about co The width of the excitation region decreased as the steering angle increased,
render-39 ing modal selectivity at large angles The phenomena were well predicted by the excitation function of
40 the source influence theory The low-order modes were better imaged at steering angle P30° for both
41 plates The study has also demonstrated the feasibility of using the two-probe phased array system for
42 future in vivo study
43
Ó 2013 Elsevier B.V All rights reserved
44 45
1 Introduction
47 Osteoporosis is a systemic skeletal disease characterized by
48 gradual loss of bone density, micro-architectural deterioration of
49 bone tissue, and thinning of the cortex, leading to bone fragility
50 and an enhanced risk of fractures Cortical thickness of long bone
51 measurement has been investigated for the incidence of
osteopo-52 rosis Loss of cortical bone involves an increase of intracortical
53 porosity due to trabecularization of cortical bone[1,2]and cortical
54 thinning due to the expansion of marrow cavity on the endosteal
55 surface[3] The cortical thicknesses at distal radius and tibia in
56 postmenopausal women with osteopenia were found to be thinner
57 than those of normal women in an in vivo study using
high-58 resolution peripheral quantitative computed tomography[4]
Re-59 cently, a high correlation was demonstrated between proximal
60 humeral cortical bone thickness measured from anteroposterior
61 shoulder radiographs and bone mineral density measured by
62 Dual-energy X-ray absorptiometry in an in vivo study for
osteopo-63 rosis diagnosis[5]
64 Ultrasound has been exploited to study long bones using the
so-65 called axial transmission technique, where the transmitter and the
66 receiver are deployed as a pitch–catch configuration with the
re-67 ceiver moving away from the transmitter Since the acoustic
68 impedance (density velocity) of the cortex is much higher than
69 those of the surrounding soft-tissue materials, the cortex is a
70 strong ultrasound waveguide The propagation of ultrasound is
71 guided by the cortical boundaries and its propagation
characteris-72 tics depend on the geometry (thickness) and material properties
73 (elasticities and density) of the cortex and the surrounding tissues
74 Ultrasonic guided waves (GWs) propagate within long bone in
75 their natural vibrational modes, known as guided modes at
differ-76 ent phase velocities, which depend on frequency The GWs travel
77 longer distance and suffer less energy loss than the bulk waves
be-78 cause the boundaries keep most of the GW energies within the
79 waveguide
80 The application of GWs to study long bones is quite recent but
81 the results so far are quite interesting Nicholson et al found the
82 velocity of the fundamental Lamb mode A0 differed by 15% 0041-624X/$ - see front matter Ó 2013 Elsevier B.V All rights reserved.
http://dx.doi.org/10.1016/j.ultras.2013.08.004
⇑ Corre
Q2 sponding author Tel.: +1 (780)4071153; fax: +1 (780)4077280.
E-mail address: lawrence.le@ualberta.ca (L.H Le).
Contents lists available atScienceDirect
Ultrasonics
j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / u l t r a s
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Q1 et al., Excitation of ultrasonic Lamb waves using a phased array system with two array probes: Phantom
Trang 283 between eight healthy and eight osteoporotic subjects (1615 m/s
84 versus 1300 m/s) [6] The same group studied a population of
85 106 pubertal girls and also found the velocity of a slow-traveling
86 wave (1500–2300 m/s) consistent with that of the fundamental
87 A0 mode[7] Protopappas et al identified four low-order modes,
88 S0, S1, S2, and A1in an ex vivo study of an intact sheep tibia[8]
89 Lee et al found a strong correlation between the phase velocities
90 of A0and S0modes with cortical thicknesses in bovine tibiae[9]
91 Ta et al found that the L(0, 2) mode was quite sensitive to the
92 thickness change in the cortex[10] Basically in most studies, the
93 first few low-order guided modes have been consistently observed
94 and further studied for their potential to characterize long bones
95 Guided modes are dispersive and might come close together,
96 posing a challenge for their identification The ability to isolate
97 the guided modes of interest is the key for a successful analysis
98 of ultrasound data Post-acquisition signal processing techniques
99 such as singular value decomposition[11],s p transform[12],
100 group velocity filtering [13], dispersion compensation [14], and
101 the joint approximate diagonalization of eigen-matrices algorithm
102 (JADE) [15] are viable methods to separate wavefields Guided
103 modes can also be selectively excited by using angle beam
104 Preferential modal excitation and selectivity using angle beam
105 is widely used in ultrasonic non-destructive testing and material
106 characterization It is generally assumed that given the
compres-107 sional wave velocity of the angle wedge and an incident angle, only
108 a phase velocity is generated via Snell’s law However in practice,
109 the ultrasound beam has a finite beam size and does not generate
110 just a single phase velocity for a given wedge angle The element
111 size of the transducer and the incident angle influence the
excita-112 tion of the GWs within the structure, which is generally known as
113 the source influence[16–18] Instead of being excited with a
defin-114 itive phase velocity (single excitation), GWs with a spectrum of
115 phase velocities are generated at oblique incidence For normal
116 incidence, the phase velocity spectrum is very broad and
disper-117 sive, which implies infinite phase velocities to be excited, thus
118 making mode isolation difficult For a fixed size transducer,
119 increasing the beam angle decreases the width of the phase
veloc-120 ity spectrum, thus generating fewer guided modes
121 The use of angle beam to study long bone is very limited Le et al
122 used a 51° angle beam to study bulk waves at receivers deployed
123 downstream from the point of excitation[19] Ta et al used various
124 angle beams to excite low-order longitudinal modes and was the
125 first to mention briefly the concept of phase velocity spectrum in
126 the bone community without much details[10,20] Although a pair
127 of transducers is still the most common means to acquire bone data,
128 ultrasound array system has been used in axial transmission bone
129 study[21] The array system or multi-transmitter–multi-receiver
130 system has many advantages over
single-transmitter–single-recei-131 ver system The former has better resolution because of the smaller
132 element footprint, fast acquisition speed, accurate coordination of
133 the receivers, and less motion-related problems In case the system
134 is a phased array (PA) system, beam steering is possible
135 The objective of this work is to investigate the use of a PA system
136 to excite guided waves in brass and bone plates The system has two
137 multi-element array probes with one acting as a transmitter and the
138 other as a receiver The acquired data are processed and transformed
139 to the dispersion maps via an adjoint Radon transform The
theoret-140 ical dispersion curves based on plate models are used for modal
141 identification We attempt to explain the variation of guided-wave
142 excitation with the incident angle using the source influence theory
143 (SIT) The novelties of our work lie in our employment of two phased
144 array probes and the use of Radon transform to estimate dispersion
145 energy To our knowledge, these have never been done in the bone
146 community While the SIT has been studied for a circular disk
147 transducer, we find it interesting to apply the theory to our data
148 acquired by a PA system
149
2 Materials and methods
150 2.1 Preparation of samples
151
We performed experiments on a brass plate and a bovine bone
152 plate The brass plate was 6.3 mm thick with a 255 mm 115 mm
153 surface dimension We prepared a bone plate from a fresh bovine
154 tibia The skin and soft tissue were removed Using a table
band-155 saw, both ends of the tibia were cut and then the diaphysis was
156 cut along the axial direction to make a plate Both surfaces of the
157 plate were sanded and smoothed by a disk sander The resultant
158 bone plate had a relatively flat (190 mm 48 mm) surface area
159 with a thickness of approximately 6.5 mm The top face was
pol-160 ished further to prepare the surface ready for the placement of
161 the probes
162 2.2 Ultrasound phased array system
163
We used an Olympus TomoScan FOCUS LTTM Ultrasound PA
164 system (Olympus NDT Inc., Canada) with two array probes as
165 shown in Fig 1a The system has the following specifications:
166 0.5–20 MHz bandwidth, 20 kHz pulsing rate, 10-bit A/D converter,
167 and up to 100 MHz sampling frequency Real-time data
compres-168 sion and signal averaging are also available The scanner has a
169 high-speed data acquisition rate of 4 MB/s with maximum 1 GB file
170 size and 8192 data point per A-scan (or time series) The unit is
171 connected to a computer via Ethernet port The Windows XP-based
172 computer was loaded with the TomoviewTM software (Version 2.9
173 R6) to control the data acquisition process and to modify the
174 parameters of the ultrasound beam such as scanning mode, beam
175 angle, focal position, and active aperture The acquired data can
176
be exported to the computer for further post-acquisition analysis
177 The scanner also supports multi-probe operations such as
sin-178 gle-transducer-element probe combination or two
multi-Fig 1 The ultrasound phased array system: (a) The TomoScan FOCUS LTTM phased array acquisition system (1), the Windows XP-based computer with the Tomo-ViewTM software to control the acquisition process (2), and the probe unit (b) The housing with the 16-element and 64-element probes The P16 was the transmitter array while the P64 was the receiver array.
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Trang 3179 element probes up to 128 elements Beam steering and focusing
180 (transmit focus) at oblique angle can be achieved by electronically
181 delaying the firing of the elements without mechanical movement
182 Receive-focusing is also possible The recorded echoes are stored,
183 delayed, and then summed to produce an ultrasound signal The
184 scanner was previously used to study scoliosis[22]
185 The two array probes used are the 16-element (2.25L16) and
186 64-element (2.25L64) array probes with a central frequency of
187 2.25 MHz (Fig 1b) Here we denote them as P16 and P64
respec-188 tively The two probes sat tightly within a housing, which was
de-189 signed and built in-house to ensure the probes were stabilized and
190 the relative distance between them were fixed during data
acqui-191 sition The active areas of the P16 and the P64 probes are
respec-192 tively 12 mm 12 mm and 48 mm 12 mm Both have the same
193 pitch of 0.75 mm Pitch is defined as the distance between the
cen-194 ters of two adjacent elements
195 2.3 Data acquisition
196 The data were acquired using an axial transmission
configura-197 tion The experiment setup is schematically shown inFig 2 The
198 setup shows the arrangement of two probes (within housing) on
199 ultrasound coupling media, which were in contact with the
under-200 lying plate The plate was a brass plate or a bone plate in our case
201 Two pieces of 5-mm thick ultrasound gel pad, acting as coupling
202 medium, were cut from a commercial ultrasound gel pad
(Aqua-203 flex, Parker Laboratories, Inc., USA) with surface areas slightly
lar-204 ger so that the probes rested comfortably on the pads The whole
205 set up was held in place by the 3MTM transpose medical tape
206 The ultrasound gel (Aquasonic 100, Parker Laboratories, Inc.,
207 USA) was applied to all contact surfaces to ensure good coupling
208 The experiments were performed at room temperature of 22 °C
209 We chose to use five transducer elements as a group due to the
210 compromise between maximum steering angle and frame size
211 (number of acquired A-scan) The first five elements of P16 probe
212 were used as the transmitter For the receivers, five elements
213 worked as a group and each group was spaced by one pitch
214 (0.75 mm) increment, that is, 1-2-3-4-5, 2-3-4-5-6, etc The offset
215 spanned from 22.75 mm to 67 mm with an aperture of
216 44.25 mm The scanner had an option to select source pulse of
dif-217 ferent dominant period, thus controlling the central frequency of
218 the incident pulse We chose a pulse with 1.6 MHz central
fre-219 quency The calculated near field length L, was around 3.7 mm,
220 as given by L = kA2f/4v [23], where the aspect ratio constant k,
221 which is the ratio between the short and long dimensions of the
222 transmitter, is 0.99; the transmitter aperture, A is 3.75 mm for a
223 five-element source; the frequency, f, is 1.6 MHz;v is 1500 m/s,
224 the sound velocity in the ultrasound gel pad The axial resolution
225 of the beam was 0.24 mm based on one-half of the pulse length
226 There were 60 A-scans and each A-scan was 2500-point long with
227
a sampling interval of 0.02ls The data filled a 60 2500
time-228 offset (t x) matrix of amplitudes In our experiments, we steered
229 both the transmitter and receiver in sync at six angles: 0° (normal
230 incidence), 20°, 30°, 40°, 50°, and 60° The synchronization at the
231 same inclination enhanced the sensitivity of the receivers to record
232 the guided waves traveling at the phase velocity of interest[24]
233 Depending on the steering angle, the calculated lateral resolution
234 ranged from 0.23 mm to 0.78 mm [25] In this paper, beam was
235 steered at an incident angle and thus we use the terms, steering
236 angle and incident angle, interchangeably
237 2.4 Beam steering
238 When an ultrasound beam incidents on the bone surface at an
239 angle, hi, a guided wave traveling with a phase velocity, co, between
240 the transmitter and receiver and along the bone structure (parallel
241
to the interface within the bone structure) will be generated
242 according to Snell’s law (Fig 2):
243
hi¼ sin1 vw
co
ð1Þ 245 246 or
247
ko¼ kwsin hi¼xsin hi
250 where vw is the ultrasonic velocity of the coupling medium,
251
kw=x/vw is the incident wavenumber in the coupling medium,
252
ko=x/cois the horizontal wavenumber of the guided wave in the
253 cortex, andxis the radial frequency Based on Eqs.(1)or(2), only
254 phases with a single phase velocity or wavenumber are generated,
255 which corresponds to a horizontal excitation line at cofor all
fre-256 quencies in the f c panel However, we observed more phase
257 velocities in our experimental data and thus Snell’s law was not
258 adequate to explain the phenomenon
259 Based on the SIT[16,17], there exists an excitation zone where
260 guided waves traveling with phase velocities around coare excited
261 The excited phase velocity spectrum is mainly governed by the size
262
of the transducer element and the incident angle and can be
263 approximated by the excitation function F for a piston-type source
264
of width, A[17],
265 Fðf ; cÞ ¼rojRðhiÞj
2ðk koÞsin
Aðk koÞ
2 cos hi
ð3Þ 267 268 where k =x/c androis the uniform pressure on the source surface
269 The factor jR(hi)j accounts for the change in traction at the interface
270 and more detail about this factor is referred to[16,17] In our work,
271
we assumeroand jR(hi)j take the values of unity and A = 3.75 mm
272 for a five-element source For the six steering angles we considered
273 and using 1.6 MHz as the central frequency of the incident pulse,
274 their excitation spectra are shown in Fig 3 We also follow Ditri
275 and Rose[16]to define a 9 dB phase velocity bandwidth,r9dB,
276 for the source array:
277
r9dB¼c
þ
o c o
co ¼ðDcoÞ9dB
280 where c
o and cþ
o are the phase velocities smaller and larger than co
281 respectively when jFj drops by 9 dB of the maximum
282 2.5 Adjoint radon transform
283 Following[12], we consider a series of ultrasonic time signals
284 d(t, xn) acquired at different offsets, x0, x1, , xN1where t denotes
285 time and the x-axis is not necessarily evenly sampled We write the
286 time signals as a superposition of Radon signals, m(s, p):
Fig 2 A cross-section of the experiment setup The housing hosted two ultrasound
probes in place: a 16-element (P16) probe as the transmitter and a 64-element
probe as the receiver The probes rested on the ultrasound gel pads, which acted as
coupling media The pads then overlaid the plate Only one group (five elements) in
P16 was used as source generator and 60 groups in P64 as receivers The receivers
were steered at the same inclination as the transmitting beam to enhance the
receiving sensitivities to propagating guided waves with phase velocity, c o related
to the inclination, h i by Snell’s law, sin h i =vw /c o wherevw is the velocity of the
coupling medium.
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Trang 4dðt; xnÞ ¼XK1
k¼0
mðs¼ t pkxn;pkÞ; n ¼ 0; ; N 1 ð5Þ 289
290 where the ray parameter or slowness, p is sampled at p0, p1, , pK1,
291 and the intercept,s, also known as the arrival time at zero-offset, is
292 also discretized ass= gDt where g = 0, 1, , G 1 Taking the
tem-293 poral Fourier transform of Eq.(5)yields
294
Dðf ; xnÞ ¼XK1
k¼0
Mðf ; pkÞei2pfp k x n: ð6Þ 296
297 Rewriting Eq.(6)using matrix notation for each frequency gives
298
300
301 and
302
L ¼
ei2pfp 0 x 0 ei2pfpK1x 0
ei2pfp 0 xN1 ei2pfpK1xN1
2
6
4
3 7
304
305 The adjoint Radon solution is given by
306
308
309 where LHis the adjoint of the operator L and H denotes the complex
310 conjugate transpose The adjoint Radon solution is slightly better
311 than the Fourier transform, yields better results at fixed number
312 of records, and is less susceptible to aliasing problem Finally, the
313 dispersion curve or f c panel is obtained by replacing p = 1/c in
314 M To avoid aliasing, sampling in slowness, Dp, must be smaller
315 than 1=ðraperfmaxÞ where raperis the acquisition aperture and fmaxis
316 the maximum frequency of the data[26]
317 3 Results and discussion
318 3.1 Brass plate
319 Prior to dispersion analysis, the acquired t x data underwent
320 some simple but essential signal processing steps First, the
trig-321 gers were muted Second, the mean amplitude was subtracted
322 from the data to remove the background There was insignificant
323 energy beyond 1.0 MHz and the data were band-pass filtered with
324
a trapezoidal window (0.1, 0.15, 0.9, 1.0 MHz)
325
Fig 4shows the Radon panels or dispersion panels of the brass
326 plate for the six steering angles: 0° (normal incidence), 20°, 30°,
327 40°, 50°, and 60° Also indicated on the figures are the excited
328 phase velocities, co, as predicted by Snell’s law (Eq.(1)) given the
329 corresponding steering angles We also superimposed the
theoret-330 ical dispersion curves of the Lamb modes on the figure We used
331 the commercial DISPERSE software[27]to simulate the dispersion
332 curves based on an elastic plate model The material properties of
333 the brass plate are listed inTable 1 Before we discuss each case in
334 detail, several general observations can be made when the steering
335 angle changes from normal incidence to 60° First, the excitation
336 does not generate GWs of mono phase velocity as predicted by
337 Snell’s law Instead a spectrum of phase velocities is excited
Sec-338 ond, as the steering angle increases, less high-velocity GWs are
ex-339 cited and more low-velocity GWs are generated and focused Third,
340 the phase velocity spectrum becomes smaller and more selective
341
as the steering angle increases
342
As shown inFig 4, the normal beam (Fig 4a) lacks the focusing
343 power and gives rise to a wide spectrum of GW energies of all
fre-344 quencies The normal beam excites more energetic high-velocity
345 GWs than the low-velocity GWs The first three antisymmetric
346 A-modes (A0, A1, and A2) and the first four symmetric S-modes
347 (S0, S1, S2, and S3) can be identified The majority of the energies lies
348 above 4 km/s and between 0.5 MHz and 0.9 MHz The three
stron-349 gest modes are A2, S2and S3 The low-velocity GW energies are very
350 weak The beam excites the higher frequency portion of the
low-351 order modes A0and S0at 0.8 MHz where they converge into a small
352 energy cluster around 1.95 km/s At 0°, the Snell’s law-predicted
353 phase velocity is a very large value or, theoretically speaking,
infin-354 ity, indicating that the high-velocity GWs are favorably excited It
355
is quite challenging to identify and isolate guided modes using
356 normal beam excitation At 20° incidence (Fig 4b), the beam
ex-357 cites guided waves around the predicted phase velocity of
358 4.38 km/s All seven previously-identified guided modes are
pres-359 ent but their cluster peaks show up at different (higher)
frequen-360 cies The low-velocity GWs below 4.38 km/s are energetic The
361 beam excites strong S0energy at low 0.2 MHz and beyond
How-362 ever above 0.4 MHz, the dispersion curves of the A0or S0come
to-363 gether and it is difficult to tell which mode the energy clusters
Fig 3 The normalized e
Q5 xcitation spectra for six different steering angles The velocity value shown above each figure is the phase velocity determined by Snell’s law (Eq (1)
in the text) The phase velocity determined by Snell’s law is denoted by c o ; The phase velocities, c
o and c þ
o , are defined at the values of jFj equal to 9 dB of the maximum; The
c
p1 and c
p2 refer to the phase velocities (<c o ) of the peaks of the first and second sidelobes.
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Trang 5364 between 0.6 and 0.8 MHz belong to When the steering angle
in-365 creases to 30° (Fig 4c), the predicted phase velocity is 3.0 km/s
366 The number of GW modes, especially the high-velocity modes
367 above 4.0 km/s, reduce and the low-velocity modes around
368 1.95 km/s are more enhanced At this angle, we only identify five
369 significant modes, A0, S0, A1, S1, and A2with the majority of the
370 GW energies lying between 0.4 and 0.9 MHz For the A0, S0, A1,
371 and S1, the simulated dispersion curves match very well with the
372 corresponding modal clusters with the curves going through the
373 first two low-order modal clusters on their tracks It is interesting
374 to note from these dispersion curves that the same modal energies
375 are not continuous on their respective dispersion curves This
376 might imply that the modes experience strong attenuation at
cer-377 tain frequencies The first two low-order modes consistently show
378 their presence between 0.5 MHz and 0.9 MHz with progressively
379 stronger energies than those at smaller steering angles The last
380 three modes start to lose their strength as the steering angle
in-381 creases The weakening of the A1mode is obvious When the
steer-382 ing angle increases from 40° (Fig 4d) to 60° (Fig 4f), the predicted
383 imaged phase velocities drops from 2.33 km/s to 1.73 km/s and all
384 high-velocity GWs become less visible Only two low-velocity A0
385 and S0modes exist and are imaged with enhanced focus and
reso-386 lution In addition, more energy is excited at frequencies lower
387 than 0.4 MHz At 60° incidence, there is some 0.4 MHz energy
trav-388 eling around 1.8 km/s, which seems to belong to the A0 mode
389 Overall, the dispersion curves match the excited Lamb modes
rea-390 sonably well
391 3.2 Bovine bone plate
392
We investigated further the guided mode selectivity and
focus-393 ing by beam-steering on a bovine bone plate The data was
pro-394 cessed using the same procedures and parameters as those for
395 the brass plate data The dispersion panels (Fig 5) show the same
396 general observations as those in the brass plate Notably, the
ultra-397 sound beam energizes a spectrum of phase velocities and at
smal-398 ler steering angles, the beam favors the excitation of high-velocity
399 GWs As the steering angle is more oblique, the spectrum is
400 narrower and the beam is more selective toward low-velocity
401 excitation
402 The predicted phase velocities, co, for the six steering angles
re-403 main the same as those for brass plate (Table 2) The normal beam
404 excites a wide spectrum of phase velocities with an emphasis on
405 high-velocity GWs (Fig 5a) The energetic GWs travel at velocities
406 above 3.5 km/s There are a few weak and small energy clusters
407
We calculated the dispersion curves using a plate model with
Fig 4 The dispersion panels of the brass plate data for the six different steering angles Superimposed are the theoretical dispersion curves The c o ;c
o ;c þ
o ;c p1 , and c p2 are referred to Fig 3 for their definitions.
Table 1
Parameters used to simulate dispersion curves for the brass plate and bone plate The
compressional wave velocity (vp ) and shear wave velocity (vs ) of the brass plate were
taken from Table A-2 of [23] while the density (q) was measured Thevp ,vs , andqof
the bone plate were taken from [28] while the attenuation coefficients,ap andap were
from [19] We also measured the vp of the brass and bone plates and the
measurements were 4.56 km/s and 4.09 km/s respectively, which are very close to
the reported values in the literature [23,28]
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Trang 6408 absorption (Table 1) The bone plate model is different from the
409 brass plate model in shear wave velocity and density The match
410 is not perfect with some clusters lying between dispersion curves
411 We managed to identify eight Lamb modes: 4 antisymmetric
412 modes, A0 A3 and 4 symmetric modes, S0 S3 The two
high-413 intensity clusters are the fast-traveling A2 and S2 modes The A1
414 curve goes through a low-frequency cluster (0.32 MHz) and a
415 high-frequency cluster (0.75 MHz) In the neighborhood of
416 1.9 km/s, there is a noticeable fragmented band of weak energies
417 at 0.25 MHz and between 0.7 and 0.8 MHz, which is better imaged
418 in the 20° panel (Fig 5b) Except the A3mode, which is absent in
419 the brass plate dispersion panel, the modes in the brass plate
420 and bone panel are very similar with A2and S2being the two
stron-421 gest modes As the steering increases, this group of low-velocity
422 energies receive enhanced focusing and imaging resolution while
423 the high-velocity phases become weaker, say the A2and S2modes
424 for example At 30° (Fig 5c), only GWs traveling below 4.2 km/s are
425 energized As the steering angle increases from 30° (Fig 5c) to 60°
426 (Fig 5f), only the bundle of GW energies bounded between A0and
427 S1and over 0.5 MHz are imaged while the rest of the GWs lose
428 their intensity and become hardly visible At 50° (Fig 5e) and 60°
429 (Fig 5f), this band of low-order energies experience enhanced
430 excitation It is worth mentioning here that the adjoint Radon
431 transform is able to resolve these closely packed energy clusters
432 The steered beam greatly enhances and focuses the slow-traveling
433 (around 1.75 km/s) small energy cluster around 0.3 MHz, which lie
434 between the A0and S0 Quite similar to the brass plate, the A0and
435
S0modes persist in all steering angles
436 3.3 Excitation function
437 The SIT predicts that the loading size of the transducer
influ-438 ences the range of phase velocities generated by the transmitting
439 source[16,17] For a given angle of incidence, the beam does not
440 generate a single phase velocity (based on Snell’s law) but a
spec-441 trum of phase velocities The GW modes with dispersion curves
442 passing through the phase velocity zones for a given frequency
443 has greater ‘‘chance’’ to be excited as compared to the portion of
444 the curves, which are far away from the zone This excitation
prob-445 ability is provided by the excitation function defined by Eq.(3)
446 However, the strength of the excitation at that particular frequency
447
is governed by the excitability function of the theory As shown in
Fig 5 The dispersion panels of the bone plate data for six different steering angles Superimposed are the theoretical dispersion curves The c o ;c
o ; c þ
o ;c p1 , and c p2 are referred
to Fig 3 for their definitions.
Table 2
Parameters for the 9-dB phase velocity bandwidth for five steering angles The c
p1 and c p2 refer to the phase velocities (<c o ) of the peaks of the first and second sidelobes.
h i (°) c o (km/s) c þ
o (km/s) c
o (km/s) (Dc o ) 9dB (km/s) r9dB c
p1 (km/s) c
p2 (km/s)
Please cite this article in press as: K.-C.T Nguyen
Q1 et al., Excitation of ultrasonic Lamb waves using a phased array system with two array probes: Phantom
Trang 7448 the results, even though the dispersion curves of the modes run
449 through the 9 dB phase velocity bandwidth, the modal energies
450 are sporadic and not continuous along the dispersion tracks It is
451 uncertain that the absence of some frequency components is due
452 to the attenuation of those components or the preferential modal
453 excitation Studying the excitability function is beyond the scope
454 of this study Here, we attempt to use the excitation function of
455 the source influence theory to explain the observed behaviors of
456 the dispersion energies with steering angles
457 The 9 dB bandwidth defines a range of phase velocities within
458 which significant excitation may happen The bandwidth
parame-459 ters are tabulated inTable 2 An example of how to delineate the
460 bandwidth of a spectrum and other relevant parameters is
pro-461 vided by the 20° case inFig 3b As shown inFig 3, the normal
462 beam excitation function exceeds the 9 dB value over an
‘‘infi-463 nitely’’ wide phase velocity range and there is basically no
selectiv-464 ity to phase velocity This is evident in both data sets
465 (Figs 4 and 5a) The selectivity to phase velocities improves when
466 the steering angle increases The bandwidth decreases from 128%
467 to 21% of cowhen the incident angle increases from 20° to 60°,
468 offering enhanced modal selectivity The bandwidth-narrowing,
469 as predicted by the excitation function, is supported by our
exper-470 imental data While the upper boundary of the observed GW
re-471 gion is well defined by cþ
o, the lower boundary of the region is
472 less so by the c
o For the two largest steering angles, 50° (Figs 4
473 and 5e) and 60° (Figs 4 and 5f), the c
o and cþ
o are the lower and
474 upper boundaries of the observed phase velocity spectra For the
475 brass plate, the c
o and cþ
o are sufficient to define the 40° phase
476 velocity spectrum (Fig 4d) However the c
p1and c p2are required
477 to define the lower bounds of the observed phase velocity regions
478 for the 30° (Fig 4c) and 20° (Fig 4b) respectively For the bone
479 plate, the c
p1fixes the lower bounds of the 30° (Fig 5c) and 40°
480 (Fig 5d) while the c
p2 outlines the lower bound for the 20°
481 (Fig 5b) Although the 9 dB bandwidth defines the dominant
482 phase velocity region where the excitation may happen, the
excita-483 tion can happen in the sidelobes of the excitation function as
illus-484 trated by the dispersion panels of the experiment data
485 4 Concluding remarks
486 In this study, we investigated the use of a commercial
non-487 medical PA system to excite GWs within a brass plate and a bone
488 plate with two array probes Acquisition with two probes not only
489 eliminated the crosstalk between transducer elements but also
490 allowed adjustable long offset for GW buildup By using a fixed
491 five-element of a 16-element probe as the loading and a
64-492 element receiver probe, many energetic fast and slow GWs of a
493 wide frequency range were excited and observed in both plates
494 We also studied the effects of modal selectivity within the plates
495 by beam-steering By varying the angle of the steering beam, the
496 excited bandwidth of the phase velocity spectrum changed, in
497 good agreement with the prediction by the excitation function of
498 the source influence theory Consequently, modal selectivity is
499 possible by choosing larger steering angle The results of this study
500 allow us to consider a PA system for clinical work The acquisition
501 time by a PA system is reduced significantly by 100 fold without
502 any mechanical probe movement as compared to a
single-trans-503 mitter-single-receiver system and problems relating to inaccurate
504 recording transducer’s coordination, patient discomfort, and
505 patient motion are minimized if used in clinical settings The
506 low-order low-velocity GW modes have been consistently
507 observed in both plates The adjoint Radon transform has
success-508 fully extracted the dispersion energies with good resolution Using
509 the PA system in combination of low frequency toneburst for
510 in vivo study will be the next avenue for our future studies
511 Acknowledgment
512
KC Nguyen wants to thank Ministry of Education and Training
513
of Vietnam for the award of a 322 scholarship to make the study
514 possible We also thank Devlin Morrison for building the
trans-515 ducer housing
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