The effect of acoustic velocity on phantom measurements

Ultrasound in Med. & Biol., Vol. 26, No. 7, pp. 1133–1143, 2000 Copyright © 2000 World Federation for Ultrasound in Medicine & Biology Printed in the USA. All rights reserved 0301-5629/00/$–see front matter

PII: S0301-5629(00)00248-9

● Original Contribution THE EFFECT OF ACOUSTIC VELOCITY ON PHANTOM MEASUREMENTS ALBERT GOLDSTEIN Department of Radiology, Wayne State University School of Medicine, Detroit, MI, USA (Received 4 October 1999; in final form 25 April 2000)

Abstract—Urethane rubber ultrasound (US) phantoms have a much lower acoustic velocity (1430 –1450 m/s) than the accepted soft tissue average of 1540 m/s. Two important questions arise: can the rod positions in these rubber phantoms be adjusted so that they may be used to test equipment distance measurement accuracy for all types of multielement transducers, and can they be used to measure beam focus (using the spread of the rod blur patterns)? These questions were addressed for linear-, phased-, convex- and vector-array transducers. Theoretical predictions for the different transducers’ distance measurement errors agreed with careful measurements obtained with a specially designed array of stainless-steel rods immersed in paraffin oil (1447 m/s). The conclusions of this study are that phantoms with acoustic velocities different from 1540 m/s cannot be used to check distance measurement accuracies of all the types of real-time transducers, nor to predict a transducer’s focusing performance in clinical scans. © 2000 World Federation for Ultrasound in Medicine & Biology. Key Words: Acoustic velocity, Ultrasound phantom measurements, Urethane rubber, Ultrasound quality assurance.

rods in these phantoms to compensate for velocity miscalibration when performing image distance measurement accuracy checks. However, there are four different types of multielement transducers presently in use and each has a different acquisition geometry. Can the urethane rubber phantom rod positions be adjusted to obtain “accurate” distance measurements for all of these transducer types? That is one of the questions addressed in this study. To answer this question, theoretical predictions for image misregistration due to velocity miscalibration have been derived for the four types of multielement transducers. These predictions then were tested using a liquid-based phantom whose acoustic velocity closely resembled that of the urethane rubber. To determine if beam focus measurements can be properly performed with urethane rubber phantoms, lateral beam profile measurements were performed using two commercially available phantoms: an agar gel/ graphite phantom and a urethane rubber phantom.

INTRODUCTION Ultrasound (US) phantoms are used for equipment acceptance and routine performance tests. Acceptance tests quantitatively determine equipment performance characteristics. Routine performance tests verify equipment operational stability over time. Most phantoms are tissuemimicking (their acoustic properties are identical to liver tissue), so that their measurement results are consistent with clinical performance. Recently, urethane rubber US phantoms have appeared in the market place. Although these phantoms have a longer useful life (and warranty period) than traditional agar gel/graphite phantoms, their acoustic velocities are much lower than the 1540 m/s design velocity of imaging equipment: 1450 m/s (ATS Laboratories Inc., Bridgeport, CT) and 1430 m/s (Computerized Imaging Reference Systems, Inc., Norfolk, VA). Due to this velocity miscalibration, the question arises as to whether or not image distance measurement accuracy checks or beam focus measurements can be properly performed using urethane rubber phantoms. It is possible to adjust deliberately the position of

THEORY A simple ray model was used, depicting the central ray of the transducer beam. Image lines are the sequential positions of the beam’s central ray when acquiring the

Address correspondence to: Albert Goldstein, Ph.D., Department of Radiology, Detroit Receiving Hospital, 4201 St. Antoine Blvd., Detroit, MI 48201 USA. E-mail: [email protected] 1133

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Fig. 1. Linear array. (F) the real-space location of the rod in the transducer scan plane; (E) the image location of the rod. crs ⬍ ccal.

data for each image frame. Due to cosmetic fill-in algorithms, they are usually not discernible in US images. The transducer scan plane is usually aligned perpendicular to the rods in phantoms. Because the intersection of a line and a plane is a point, the rods are effectively point targets. Linear array Most linear arrays do not beam steer.1 The image lines are formed by subgroups of array elements that step along the array length. The misregistration of a point target along a vertical US line in a linear array image is shown in Fig. 1. The echo time of flight is t tof ⫽

2D rs , c rs

(1)

where Drs is the real-space depth of the point target from the linear array front surface and crs is the acoustic velocity of the medium. The scanner is calibrated for tissue with c ⫽ ccal, so the echo is registered in the image along the image line at an image depth Dim given by: D im ⫽

c calt tof c cal ⫽ D . 2 c rs rs

(2)

when crs ⬍ ccal, Dim ⬎ Drs. 1 Some linear arrays do add a slight 20° beam steering called microsteering (Goldstein and Powis 1998) that will be neglected here.

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Fig. 2. Beam steering. There is a constant time delay of ⌬t between the sequential transmissions of the three elements shown. The elements transmit from left to right. At the instant shown the right element is transmitting. The solid arcs are the wave-fronts from the previous element transmissions. The dotted line is the resultant wave-front from all the transmissions and the dashed line (perpendicular to the combined wave-front) is the beam steered direction of the US ray.

The Y and X orthogonal directions are defined vertically down and horizontal to the beam center line, respectively. The image misregistration in the directions Y and X are defined as ⌬Y ⫽ Yim ⫺ Yrs and ⌬X ⫽ Xim ⫺ Xrs, and we have: ⌬Y ⫽

冉

冊

c cal ⫺ 1 D rs c rs

(3)

and ⌬X ⫽ 0,

(4)

where Yrs and Xrs are the real-space coordinates of the point target and Yim and Xim are its image coordinates. For linear arrays, the misregistration is one-dimensional (1-D), with the point target image shifted only in the vertical direction. Misregistration, ⌬Y, is positive in the vertical down direction. Phased array Phased arrays employ beam steering, so image misregistration due to beam steering errors must also be considered. All of the array elements contribute to each image line. Figure 2 demonstrates beam steering, where there is a constant time delay ⌬t between the transmissions (and receptions) of adjacent array elements. Only three array elements are shown. The element on the left transmitted first, the middle one next and, at the instant shown, the right element is firing. The cylindrical wavefronts (seen as circular in the scan plane) have progressed outward from the element centers the distances 2crs⌬t

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and ⌬Y ⫽ D im cos ␪cal ⫺ D rs cos ␪rs

冉 冑 冉冊

冊

2

ccal ccal ⫽ Drs 1⫺ sin2 ␪rs ⫺ 冑1 ⫺ sin2 ␪rs . (8) crs crs

Fig. 3. Phased array. (F) the real-space location of the rod in the transducer scan plane; (E) the image location of the rod; D ⫽ the real-space depth of the rod in the phantom; and x ⫽ its distance from the sector image center line.

Here, and for all other transducer acquisition geometries, misregistration in the vertical direction, ⌬Y, is positive vertically down, misregistration in the horizontal direction, ⌬X, is positive away from the image center line (Y axis), D is the real-space depth of a rod from the transducer front surface, and x is the rod’s real-space lateral distance from the image center line. Then eqns (7) and (8) can be rewritten as: ⌬X ⫽ x

and crs⌬t, respectively. A resultant wave-front, tangentual to the individual element circular wave-fronts, will form along the dotted line in Fig. 2. In an isotropic homogeneous medium, the wave propagation direction is perpendicular to the wave front along the dashed lines in Fig. 2. With an interelement distance d, the real space beam steering angle ␪rs is given by: c rs⌬t sin ␪rs ⫽ . d

(5)

However, in the image, the US line is positioned at an angle ␪cal derived from the assumed ccal of tissue. The relationship between ␪cal and ␪rs is: sin ␪cal ⫽

c cal⌬t c cal ⫽ sin ␪rs. d c rs

(6)

This relationship resembles Snell’s law, where a US wave traveling at an angle ␪cal in a medium with c ⫽ ccal is refracted at a flat interface with a medium that has c ⫽ crs to an angle ␪rs. Figure 3 shows the real-space path of the US line in the phantom and the misregistration of this line (beam steering error) in the image when crs ⬍ ccal. The point target is misregistered at a greater depth along the image line due to eqn (1). The X and Y image misregistrations of the point target are found to be: ⌬X ⫽ D im sin ␪cal ⫺ D rs sin ␪rs ⫽ D rs sin ␪rs

冋冉 冊 册 c cal c rs

2

⫺1

(7)

冋冉 冊 册 2

c cal c rs

⫺1

(9)

and ⌬Y ⫽ 冑D2 ⫹ x2

冋 冑 冉冊

册

2

D ccal ccal x2 1⫺ . 2 2⫺ 2 crs crs D ⫹ x 冑D ⫹ x2 (10)

⌬X is independent of D, so a vertical set of rods in a phantom will remain vertical in the image regardless of the value of c. Because ⌬Y depends on both x and D, a horizontal set of rods in a phantom at a constant depth will present in the image along a curve when crs ⫽ ccal. The horizontal, constant depth rods in a phantom are usually equally spaced, xL, on both sides of the phantom vertical center line. Each set of two rods equidistant from the phantom center line are called co-lateral rods. When the phantom center line is centered in the phased array image, the co-lateral rods have the same xL and D values and will be shifted vertically in the image by the same amount. So the measured error in the spatial distance between them, 2⌬XL in the image can be computed using eqn (9), that is: 2⌬X L ⫽ 2x L

冋冉 冊 册 c cal c rs

2

⫺1 .

(11)

The fractional error of the digital caliper measurement in the horizontal direction is the term in square brackets. For a vertical set of rods, x is constant. For two rods at depths D2 and D1, where D2 ⬎ D1 a vertical digital caliper measurement will have an error of ⌬Y(D2,x) ⫺ ⌬Y(D1,x) with ⌬Y given by eqn (10).

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solved either graphically or numerically. A mathematical computational computer program (Mathcad, Mathsoft Corp. Cambridge, MA) was used to obtain that relation here when necessary. From inspection of Fig. 4: D rs ⫽

D , cos ␪rs

(15)

so the equations for ⌬X and ⌬Y become: ⌬X ⫽ D tan ␪rs Fig. 4. Vector array. (F) the real-space location of the rod in the transducer scan plane; (E) the image location of the rod; D ⫽ the real-space depth of the rod in the phantom; and x ⫽ its distance from the sector image center line; L ⫽ the height in the image of the virtual vertex above the transducer front face.

Vector array Vector arrays have a truncated sector image format. The image lines are formed by a subgroup of array elements that steps along the array length. Each line is beam steered. Figure 4 shows the generation of one image line when crs ⬍ ccal. L is the vertical distance from the transducer face to the virtual vertex (convergent point of all image lines) of the image. Then: ⌬X ⫽

冉

冊

L ⫹ D im sin ␪cal cos ␪cal ⫺ 共L tan ␪cal ⫹ D rs sin ␪rs兲

(12)

and ⌬Y ⫽

冉

冊

L ⫹ Dim cos ␪cal ⫺ 共L ⫹ Drs cos ␪rs兲. (13) cos ␪cal

Equation (6) relates ␪cal and ␪rs and eqn (2) relates Drs and Dim. When these two equations are substituted into eqns (12) and (13), the resulting misregistration equations are found to be identical in appearance to eqns (7) and (8). From inspection of Fig. 4: tan ␪rs ⫽

x ⫺ L tan ␪cal . D

(14)

For a rod with real-space locations x and D, the relation between ␪rs and ␪cal can be obtained from the solution of the two transcendental eqns (6) and (14), which must be

冋冉 冊 册 2

c cal c rs

⫺1

(16)

and ⌬Y ⫽

冉 冑 冉 冊

c cal D cos ␪rs c rs

1⫺

c cal c rs

2

sin2 ␪rs

冊

⫺ 冑1 ⫺ sin2 ␪rs .

(17)

A vertical set of rods in a phantom with crs ⫽ ccal will not be vertical in the image. For a set of rods equally spaced about the image center line (and the transducer properly centered in the phantom), the horizontal distance measurement error will be double that given by eqn (16). Convex array Convex arrays are “linear” arrays whose front faces are curved (with a radius of curvature R). The image lines are formed by a subgroup of array elements that steps along the array length. Beam steering is not needed, due to the curvature of the array front face. Convex arrays have a truncated sector image format. Figure 5 shows the generation of one image line when crs ⬍ ccal. Then, employing eqn (2): ⌬X ⫽ 共R ⫹ D im兲sin ␪ ⫺ 共R ⫹ D rs兲sin ␪ ⫽ D rs

冉

冊

(18)

冊

(19)

c cal ⫺ 1 sin ␪ c rs

and ⌬Y ⫽ 共R ⫹ D im兲cos ␪ ⫺ 共R ⫹ D rs兲cos ␪ ⫽ D rs

冉

c cal ⫺ 1 cos ␪ c rs

These equations accurately predict image misregistration only when the convex array front face is pushed down on the phantom so that there is continuous contact between

Effect of acoustic velocity ● A. GOLDSTEIN

Fig. 5. Convex array. (F) the real-space location of the rod in the transducer scan plane; (E) the image location of the rod; D ⫽ the real space depth of the rod relative to the “lowest” point on the array front face; and x ⫽ its distance from the sector image center line; R ⫽ the radius of curvature of the transducer front face.

the transducer and the phantom. Then, no refraction will occur at the coupling gel interfaces. From inspection of Fig. 5: x tan ␪ ⫽ D⫹R

(20)

D rs ⫽ 冑共D ⫹ R兲 2 ⫹ x 2 ⫺ R,

(21)

and

where D here is the real-space depth of the target relative to the “lowest” point of the array front face. Equations (18) and (19) then can be put in the form: ⌬X ⫽ x

冉

c cal ⫺1 c rs

冊冉

1⫺

R

冑共D ⫹ R兲 2 ⫹ x 2

冊

(22)

and

冉 冊冉

⌬Y ⫽ 共D ⫹ R兲

冊

ccal R ⫺1 1⫺ . (23) crs 冑共D ⫹ R兲2 ⫹ x2

Equations (22) and (23) can be used to obtain the digital caliper measurement error for horizontal and vertical sets

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Fig. 6. Schematic representation of the experimental apparatus. Both the tank and transducer holder are made from 6.35-mm thick Plexiglas. The rods are 0.86-mm diameter, type 304 stainless steel. The transducer holder rests on top of the tank with its rear vertical runner pressed against the back of the tank. The transducer (not shown) is fastened on the vertical pivoting platform. The arrows indicate the degrees of freedom that are used to position properly the transducer over the rod pattern.

of rods in the same manner as with the phased array above. A set of vertical rods may be mispositioned to compensate for range equation misregistration due to velocity mismatch. If these vertical rods are imaged in the center of the transducer’s field-of-view, where beam steering is not utilized, then they will be “properly” registered when scanned by all of the four types of multielement transducers. However, the question remains whether there is a way to misposition a phantom’s horizontal rods to compensate for velocity mismatch for all the different transducer types. METHODS AND MATERIALS To test experimentally the theoretical predictions, a specialized apparatus shown schematically in Fig. 6 was constructed. It consisted of two components: a pattern of parallel rods in a liquid-filled tank and a transducer holder that permitted accurate and repeatable placement of the transducer relative to the rod pattern. Two equally spaced sets of parallel 0.86-mm diameter, type 304 stainless-steel rods were used. The vertical set consisted of 13 rods spaced at 1-cm intervals. (Note

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that the large rod diameter does not compromise measurement accuracy because the received specular echoes originate from the rods’ top surfaces, which are spaced at 1 cm intervals.) The horizontal set consisted of seven rods spaced at 3-cm intervals. The horizontal set of rods was placed at the level of the eleventh vertical rod down in the vertical column (10 cm deeper than the first rod), so that the eleventh rod of the vertical column is also the center rod of the horizontal set. After fabrication, the real-space center-to– center distances between the rods to be used in the digital caliper measurements were accurately determined with a vernier caliper, taking into consideration the rod diameters. The transducer was fastened on the vertical pivoting platform of the transducer holder with two rubber bands. Its cable was supported from above, so that its weight did not shift the transducer position. The transducer could be positioned up and down and also tilted sideways until its central ray was parallel to the vertical set of rods. The vertical pivoting platform permited alignment of the transducer scan plane perpendicular to the rods. It has a slight force fit in its rectangular hole so that it could maintain its position without mechanical fasteners. A search of the ultrasound literature (Selfridge 1985) found paraffin oil that has an acoustic velocity similar to that of the urethane rubbers used in US phantoms. The experiments were performed using an Acuson 128 XP/10 clinical scanner. Four Acuson transducers were used: a phased-array (S228) transducer with a 2.5MHz center frequency, a vector-array (V4) with a 4-MHz center frequency, a convex-array (C5) with a 5-MHz center frequency and a linear-array (L558) with a 5-MHz center frequency. From the derived theoretical formulas, it is evident that the misregistration of the rod images is dependent on the exact position of the rods in the transducer field of view. So it was crucial to align accurately the transducer scan plane with respect to the rod pattern. To accomplish this task, the following experimental protocol was adopted. The transducer was mounted on the vertical pivoting platform with two rubber bands. It was moved down until its front surface just touched the top vertical rod (this ensured that the horizontal set of rods is exactly 10 cm deep in its scan plane). Any air bubbles seen on the transducer front face were removed. The abdominal tissue-specific mode was chosen along with an image depth of 140 mm. The multizone focusing scheme that had the most contiguous focal zones was chosen. The vertical pivoting platform was angled until a maximum amplitude echo was received from the rods (ensuring that the scan plane was perpendicular to the rods). The transducer was then tilted (in its scan plane)

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Table 1. Measured rod-to–rod distances in apparatus Distance Co-lateral horizontal rods 3–3 cm 6–6 cm 9–9 cm Vertical rods Rod 2 to rod 13 Rod 3 to rod 13

60.4 ⫾ 0.2 mm 119.7 ⫾ 0.2 mm 180.0 ⫾ 0.2 mm 109.8 ⫾ 0.2 mm 100.1 ⫾ 0.2 mm

until the line of vertical rods was presented vertically in the image. The scan plane angulation and transducer tilting was repeated several times to ensure the correct orientation of the rod pattern in the transducer scan plane. The final positioning task was to place the vertical set of rods along the image center line. This was accomplished by sliding the transducer holder left and right along the tank top, making certain that its lower vertical surface was pressed firmly against the back of the tank (see Fig. 6). The correct position was verified in the image by dropping two digital caliper cursors from the transducer front surface to the most lateral horizontal rods visible in the image. These two measurements were equal when the vertical set of rods was along the image center line. After this position was found, the vertical runner was clamped to the back of the tank with office binder clips (Universal Office Products, Des Plains, IL). The paraffin oil acoustic velocity was obtained using a digital caliper measurement of the image distance between the second or third and thirteenth vertical rods (spanning a real-space distance of approximately 10 or 11 cm). The second or third rods were chosen to eliminate the possibility of image misregistration at very shallow depths due to excessive transient diffraction (Goldstein and Langrill 1979). The acoustic velocity was determined using eqn (2) and the measured real-space distance between these two rods (Table 1). For the vector and convex arrays, the image parameters L and R had to be determined (Figs. 4 and 5, respectively). They were obtained from hardcopy images in which the overall gain was increased so that the image plane filled with noise signals, sharply delineating the sector image sides. On the film hard copy, the sector image sides were extended back (with a pointed scribe) until they intersected at the image vertex. The image distance scale provided a convenient ruler for measuring L and R. Many images were obtained with different magnifications. The ones that proved the most useful had the higher magnifications (smaller fields of view). After the transducer alignment was assured, the theory was tested by digital caliper measurements of the horizontal image distances between co-lateral horizontal rods. For highest accuracy, the equipment gain was re-

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Table 2. Nominal phantom parameters

Material Speed of sound Attenuation coefficient Line target material Line target diameter

RMI 403*

ATS 539†

Agar gel with graphite 1540 m/s at 22°C 0.5 dB/(cm-MHz) Monofilament nylon 0.12 mm

Urethane rubber 1450 m/s at 23°C 0.5 dB/(cm-MHz) Monofilament nylon 0.10 mm

* Gammex RMI, Middleton, WI; † ATS Laboratories Inc., Bridgeport, CT.

duced until the rod blur patterns2 were as small as possible laterally, but still being visible in the image. After the distance measurement was performed, one cursor was slowly moved horizontally in the image. The measured distance changed incrementally, indicating the image pixel width. The effect of acoustic velocity on beam defocusing was tested using two commercial US phantoms: a urethane rubber phantom and an agar gel/graphite phantom. Their nominal parameters, as reported by their manufacturers, are listed in Table 2. They were essentially equivalent except for their acoustic velocities. Using the vertical set of rods in each phantom, an attempt was made to measure the phantom acoustic velocity similar to the measurement of the paraffin oil acoustic velocity. A nylon rod 10-cm deep was chosen in each. The C5 transducer and standard coupling gel were used. The abdominal tissue-specific mode was chosen with an image depth of 120 mm. The multizone focusing scheme with the most contiguous focal zones was implemented. Using standard laboratory clamps, the C5 transducer was held mechanically to insure that it remained motionless during the measurement. The transducer was carefully positioned to place the chosen rod in the vertical center of its field of view and to maximize its echo amplitude (thus, aligning the transducer scan plane perpendicular to the rod). Finally, to obtain the highest measurement accuracy, the write zoom mode (RES mode) on the Acuson scanner was implemented (with a pixel width of 0.1 mm). The equipment echo amplitude dynamic range was reduced to 30 dB to increase image contrast to its maximum value. Then the equipment gains were reduced until the rod blur pattern was just visible. Hard copy was obtained of this image and succeeding images in which the overall gain was increased in 2- or 5-dB increments. Using a precision optical comparator (Edmund Scientific, Barrington, NJ) on the film hard copy images, the 2 Image blurring occurs because the US pulse has a nonzero pulse length and the beam pattern has a nonzero width. The detection of a target echo at multiple positions along an image line or on adjacent image lines causes the image of the target to be larger than its actual spatial dimensions.

Fig. 7. Convex array dual distance measurement proof that the transducer is centered horizontally. The two digital caliper measurements are equal for both rods 6 cm from the vertical column of rods.

full lateral width of each blur pattern was measured. When plotted against the overall gain changes, these full lateral blur width measurements present a good approximation to the lateral beam profile at the rod depth (with the assumption that the beam profile is symmetrical). RESULTS Table 1 contains the results of real-space distance measurements between the relevant rods in the rod pattern obtained with a vernier caliper. The measurement uncertainty is estimated to be ⫾ 0.2 mm. Figure 7 demonstrates the double digital caliper measurements used to verify the proper left-right alignment of the transducer holder. The equipment gain was increased to clearly demonstrate the rods. The horizontal rod blur patterns demonstrate comet tail artefacts caused by the rods’ mechanical resonances (Hefner and Goldstein 1981). The digital caliper measurements were made from the center of the proximal portion of the horizontal rod blur patterns. Because the measurement distance was perpendicular to this proximal surface, any uncertainty in locating the center of this surface resulted in negligible measurement error. The multiple focal zone scheme selected is indicated by the carets on the left edge of the sector field of view. The vertical rods closest to the transducer are barely seen, due to the acoustic shadowing caused by the top rod touching the transducer surface and the small focusing apertures used at shallow image depths. Figure 8 demonstrates one of the digital caliper distance measurements between the second and thirteenth vertical rods that was used to calculate the paraffin

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Fig. 8. Convex array vertical distance measurement to measure paraffin oil acoustic velocity. Note that the equipment gain had to be set high to observe all the vertical rods in the image so the horizontal rod blur pattern is large and a comet tail artefact is seen due to mechanical resonances in the rod.

oil acoustic velocity. The digital caliper cursors were positioned to one side of the rod blur patterns, with no loss of vertical distance measurement accuracy. Similar measurements were made for all transducers tested. The paraffin oil acoustic velocity mean value was 1447 m/s, with an SD of 0.67 m/s. From six vector-array high-gain images, an L value of 1.42 cm with an SD of 0.0075 cm was obtained for the V4 transducer. From four convex-array high-gain images, an R value of 5.08 cm with an SD of 0.029 cm was obtained for the C5 transducer. The expected error in the digital caliper measurements between pairs of co-lateral horizontal rods for different transducers may be calculated using the known real-space rod positions, the experimentally determined L, R (when necessary) and paraffin oil acoustic velocity in the appropriate theoretical expressions and then subtracting the known real-space distance between these horizontal rods from these theoretical results. The final results of these calculations are presented in Table 3. When using the above equations to predict the mis-

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Fig. 9. Convex array equipment gains set low for horizontal distance measurement. The horizontal rod blur patterns are still quite wide.

registration of the rods in phantom images, the uncertainties in the measured values of crs, x, D, R and L, must be taken into account. These uncertainties were either known directly or were estimated by assuming them to be the SD of their measured values. The values of the high and low errors of the predicted misregistrations were found numerically by trial and error and are included in Table 3. The theoretically predicted horizontal rod digital caliper measurement errors are almost the same for the phased and vector arrays. This is because of the similarity of their acquisition geometries, especially for small values of L. The predicted horizontal rod digital caliper measurement errors for the convex array are much smaller due to its different acquisition geometry (no beam steering). Figure 9 demonstrates a typical convex-array horizontal digital caliper measurement between the two 6-cm lateral rods. The equipment gain was set as low as possible to minimize the rod blur patterns and reduce the potential cursor placement error (Goldstein 2000). The digital caliper cursors were place directly over the faint rod blur patterns for maximum measurement accuracy. Table 4 presents the experimental results. They

Table 3. Calculated digital caliper error for horizontal co-lateral rods 10 cm in depth, c ⫽ 1447 m/s 0–3 cm Phased array Vector array Convex array Linear array

– – – 0 mm

3–3 cm

6–6 cm

9–9 cm

⫹7.96 ⫾ 0.12 mm ⫹6.91 ⫾ 0.10 mm ⫹2.58 ⫾ 0.04 mm

⫹15.92 ⫾ 0.18 mm ⫹13.79 ⫾ 0.15 mm ⫹5.30 ⫾ 0.07 mm –

⫹23.88 ⫾ 0.24 mm ⫹20.63 ⫾ 0.20 mm ⫹8.22 ⫾ 0.10 mm –

The uncertainties in the theoretical predictions arise from the uncertainties in the experimentally measured parameters. See text for details.

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Table 4. Measured digital caliper error for horizontal co-lateral rods 10 cm in depth with paraffin oil.

Phased array Vector array Convex array Linear array

0–3 cm

3–3 cm

6–6 cm

9–9 cm

– – – ⫹0.75 ⫾ 0.49 mm

⫹7.3 ⫾ 0.63 mm ⫹7.63 ⫾ 0.63 mm ⫹2.7 ⫾ 0.63 mm –

⫹15.3 ⫾ 0.63 mm ⫹14.9 ⫾ 0.63 mm ⫹8.5 ⫾ 0.63 mm –

– ⫹21.7 ⫾ 0.63 mm – –

Missing values indicate that the rods were not in the image field of view. The ⫾0.63 mm is due to a 0.4-mm pixelation error uncertainty (Goldstein 2000) and 0.2-mm real-space rod-to–rod distance uncertainty. The ⫾0.49 mm is due to a 0.3-mm pixelation error uncertainty (Goldstein 2000) and 0.2-mm real-space rod-to–rod distance uncertainty.

were obtained by subtracting the known rod-to–rod realspace distances from the digital caliper measurement results. Missing values indicate that the rods were not in the image field of view. The linear array had a limited horizontal field of view, so that only two of the horizontal rods were present in the image at the same time. There were two sources of measurement uncertainty; a pixelation error uncertainty of 1.5 times the image pixel width (Goldstein 2000) and the ⫾ 0.2-mm uncertainty of the real space rod-to–rod distance measurements. All sector images had a 0.4-mm pixel width. The linear array pixel width was estimated to be 0.3 mm.

Because these two sources of error are statistically independent, the total measurement error was estimated by taking the square root of the sum of their squares. The measurement of the two phantoms’ acoustic velocities was analyzed assuming a 1-cm center-to– center separation of the vertical set of rods in each phantom. The agar gel/graphite phantom had an acoustic velocity about 1% lower than 1540 m/s. The urethane rubber phantom had an acoustic velocity of 1540 m/s (demonstrating that the manufacturer had shifted the rods’ positions to yield this result). These measurements are not relevant in this study because it was not possible to measure independently the real-space separation of the vertical rods. So the manufacturer’s reported acoustic velocity values are assumed here. However, a result of this measurement is a demonstration of the long-term stability of the urethane rubber’s acoustic velocity. The beam profile measurements were performed carefully and repeatedly for the two commercial phantoms (4 times for the agar gel/graphite phantom and 6 times for the urethane rubber phantom). Figure 10 a and b presents typical results of the two beam lateral profiles obtained using the same 10-cm depth nylon rod, tissuespecific mode and multizone focusing scheme with the two commercial US phantoms. The lateral beam width obtained with the urethane rubber phantom was roughly twice as large as that obtained with the agar gel/graphite phantom and exhibits a double peak. Note that the data in these two figures overestimate the lateral beam width by one pixel width (0.1 mm). This is because the optical comparator measurement spanned the full width of all the pixels in the rod blur pattern, but the desired distance was the center-to– center distance between the two pixels at the lateral edges of the blur pattern (Goldstein 2000). Because this systematic error was small compared to the other measurement errors and the measured distance, it was neglected. DISCUSSION

Fig. 10. Beam width measurements of a 10-cm deep nylon rod in two commercial phantoms (see Table 2). (a) Agar gel/ graphite phantom; (b) urethane rubber phantom.

The measured co-lateral rod distance errors for the different transducers agree quite well with the theoretical predictions (Tables 3 and 4). In many cases, the two

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overlap. For the vector array, the two larger co-lateral rod distance errors differ by 0.33 mm and 0.24 mm (including errors), respectively, which are within the error margin in this work. The linear array distance errors differ by 0.26 mm (including errors), again a small amount (and negligible in clinical measurements). But the larger co-lateral rod distance errors for the convex array (6-6 cm) differ by 2.5 mm (including errors). The explanation for this large difference between theory and experiment for the convex array came from the image positions of the two 6-cm lateral rods. In Fig. 9, they are seen to be positioned near the image sides. Beam patterns generated near image sides are not generated (or received) by symmetrical transducer apertures, and might not have symmetrical lateral beam profiles. This hypothesis was tested by remeasuring the 6-6 cm co-lateral rod distance with a 9.1% by volume mixture of ethylene gylcol and water (c ⫽ 1540 m/s) in the tank (Goldstein and Langrill 1979). The same 2.5 mm disagreement between measurement and theory was obtained. When the equipment gain was increased, the rods’ image blur patterns widened and this disagreement was gone. Careful investigation of the 6-6 cm co-lateral rods’ lateral beam profiles revealed that they were doublepeaked with the exterior peak (nearer the image side) being the larger. Figure 11a and b demonstrates the measurement situation. When the equipment overall gain was set low, only this large peak was seen in the image and the rod position shifted toward the image side. When the overall gain was increased so that both peaks were present in the image (but not perceived as separate peaks), the larger blur pattern was laterally centered on the true rod position. Even with high-resolution equipment, engineering design trade-offs cause the transducer beam patterns generated at the image sides to be not as well focused as those beam patterns in the central portion of the image (with symmetrical transducer apertures). Beam focus parameters (time delays, dynamic aperture, dynamic apodization, etc.) are different in each tissue-specific mode available on modern high-resolution equipment, to obtain optimal images in different clinical situations. Thus, when scanning phantoms for routine QA equipment tests, it is essential always to use the same standard transducer in a standard tissue-specific mode with the same multifocus beam pattern. The above-derived theoretical predictions of image misregistration due to velocity miscalibration for various transducer acquisition geometries have been shown to be true and accurate. They indicate that: linear-array images have point target images shifted only in depth; phased and vector arrays have almost similar shifts of point target images due to their almost similar acquisition geometries; and convex arrays have vastly different point target misregistrations than the other transducer types.

Volume 26, Number 7, 2000

Fig. 11. Schematic representation of convex array 6-6 cm co-lateral rod distance measurement when the rods are close to the image sides. The resulting asymmetric beam apertures produce double-peaked lateral beam profiles, with the larger peak nearer the image side. The dashed line represents the echo display threshold. All echo amplitudes above this threshold value are displayed, others not. (a) Overall gain set so that only the larger of the two peaks is displayed; (b) increasing the overall gain displays both peaks (not resolved in the image). The measured co-lateral rod separation will be larger in (a).

So, it must be concluded that it is not possible to misplace the rods in US phantoms to compensate for phantom velocities different from 1540 m/s for all the different transducer types. In Fig. 10, the measured beam widths greater than 18 dB down from peak signal are roughly constant, due to the nylon rod/fill contrast in the phantom. This is illustrated schematically in Fig. 12. The phantom rod subject contrast (echo amplitude difference between the rod and the background fill echoes) is slightly greater than 18 dB in both phantoms. When the equipment overall gain is set so that only the rod echo is displayed, the width of the rod blur pattern is a measure of beam lateral width at a certain number of decibels down from peak. But, when the overall gain is increased so that the fill echoes are also present in the image, then the displayed width of the rod blur pattern is a measure of the

Effect of acoustic velocity ● A. GOLDSTEIN

Fig. 12. Beam width measurement using US tissue-mimicking phantoms. The dashed line represents the echo display threshold. The dotted line represents the average echo amplitude of the fill echoes. The double arrow indicates the perceived blur pattern lateral dimension. (a) Only the rod echo is displayed and its image blur width is a measure of the lateral beam profile at the level of the echo display threshold; (b) the equipment gain has been increased to display also the phantom fill echoes. Now, the rod blur pattern width is determined by the lateral beam profile at the level of the average fill echoes. Whenever the fill echoes are displayed, their magnitude determines the rod blur pattern dimensions.

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toms also may be due to its attenuation’s stronger frequency dependence (f1.6 vs. f1.1 for liver tissue) (Zagzebski and Madsen 1995). The pulse average frequency would then be lower in urethane rubber due to the increased attenuation of high frequencies. Lower pulse average frequencies lead to wider beam widths. Due to the difficulty of finding appropriate transmission media, this effect could not be studied here. But it remains as a further reason not to use urethane rubber phantoms for beam width measurements. The large increase in blur pattern width (lateral beam profile) of rods in urethane rubber phantoms is presumably due to beam defocusing caused by the urethane rubber’s low acoustic velocity. The measured beam lateral profile in urethane phantoms is seen to bear no relation to the beam lateral profile in tissue (or in phantom materials with c ⫽ 1540 m/s). One QA measurement that is not dependent on phantom acoustic velocity is a routine check of image penetration. Image penetration depends upon transducer sensitivity, transducer frequency, receiver gain and phantom attenuation. As long as these parameters remain stable over time, the observed image penetration distance will not change. When penetration is seen to change, additional tests must be performed to isolate the specific parameter that has changed. The long-term stability of urethane rubber phantoms make them advantageous for this measurement. CONCLUSIONS US phantoms with acoustic velocities different from 1540 m/s cannot be used to check the distance measurement accuracies of all transducer types. US phantoms with acoustic velocities different from 1540 m/s cannot be used to measure or predict a transducer’s focusing performance in clinical scans. REFERENCES

rod blur pattern width at the level of the average fill echoes. Further increases in overall gain should yield a constant width of the rod blur pattern, with some variance due to measurement uncertainty and a varying thickness of the grey-level slices. The data presented here demonstrate that the measurement of lateral beam profile is difficult with commercial US phantoms. These measurements require a mechanical support for the transducer that also permits accurate alignment of the beam central ray. As seen here, phantom fill echoes limit the dynamic range of the beam profile measurement. More complete measurements of beam profile could be made in an anechoeic tissuemimicking liquid (Madsen et al. 1998). The larger beam widths in urethane rubber phan-

Goldstein A. Errors in ultrasound digital image distance measurements. Ultrasound Med Biol 2000;26:1125–1132. Goldstein A, Langrill LN. Ethylene glycol-water mixtures for use in ultrasound test objects. J Clin Ultrasound 1979;7:465– 470. Goldstein A, Powis RL. Medical ultrasonic diagnostics In: Papadakis EP, ed. Ultrasonic instruments and devices I: Reference for modern instrumentation, techniques and technology. Physical acoustics series. Thurston RN, Pierce AD, eds. Vol. 23. New York: Academic Press, 1998:43–191. Hefner LV, Goldstein A. Resonance by rod-shaped reflectors in ultrasound test objects. Radiology 1981;139:189 –193. Madsen EL, Frank GR, Dong F. Liquid or solid ultrasonically tissuemimicking materials with very low scatter. Ultrasound Med Biol 1998;24:535–542. Selfridge A. Approximate material properties in isotropic materials. IEEE Trans Sonics Ultrason 1985;SU-32:381–394. Zagzebski JA, Masen EL. Ultrasound Phantoms—Concepts and Construction. In: Goldman LW, Fowlkes JB, eds. Medical CT and Ultrasound: Current Technology and Applications. College Park, MD: Advanced Medical Publishing, 1995:121–142.