A concave annular array design, based on phasor summation—Part I: Design methodology

ULTRASONIC

IMAGING

10,

275-286

(1988)

A CONCAVE ANNULAR ARRAY DESIGN, BASED ON PHASOR SUMMATION -PART I: DESIGN METHODOLOGY Weiquan Yuanl, Steven A. Johnson17 2~3, Michael 3. Berggren*, Richard S. Eidens 1 @epartment of Electrical Engineering *Department of Bioengineering University of Utah Salt Lake City, UT 84112

A detailed method for the design of anular arrays based on phasor summation is developed. The net phase shift across each annulus obtained by phasor summation is a very important factor in the design of an annular array, and determines the strength of the signal received. The method presented in this paper allows the designer to evaluate the trade-offs in parameters such as the strength of the signal from each annulus, the efficiency of each annulus, and the depth of focus that would be achievable. A concave annular array has been fabricated according to this design method. A set of graphs are given for an actual design to illustrate how one may readily evaluate these trade-offs. 0 1988 Academic Press, Inc. Key words:

Annular array; dynamic focusing; focal range; imaging; phase shift; ultrasound; transducer.

I. INTRODUCTION In recent years, pulse-echo imaging has played a very important role in medical ultrasound diagnosis. In most case, a one-dimensional (or linear) anay is employed. The axial resolution is mainly determined by the transmitted pulse duration, and the lateral resolution is limited by the numerical aperture of the imaging system. A single ultrasonic piston transducer provides good axial resolution, but relatively poor lateral resolution because of its low numerical aperture and also because of diffraction and refraction effects. It has been shown that the large aperture of a phased array can be employed to obtain high lateral resolution in a plane. A two-dimensional array, such as an annular array, can provide a large aperture in both dimensions, and thus with mechanical motion can produce three-dimensional high resolution images. For these reasons, annular arrays have become the subject of investigation for improved dynamically focused clinical scanners.

3 Author to whom correspondence should be addressed. 0161-7346/88

275

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Copyright 0 1988 rights qf reproduction

$3.00

by Academic Press, Inc. in any form reserved.

WAN

ET AL

D. R. Dietz, S. I. Parks and M. Linzer developed a dynamically-focused

annular

array system for contact B-scanning, which is based on a constant f-number approach to increase the sensitivity of the system [ 11. M. Arditi et al. describe an annular array based system for breast echography [2]. They also describe general annular array design approaches and provide an excellent bibliography of papers dealing with annular array design and ultrasound breast echography [2]. M. S. Patterson and F. S. Foster built a hybrid ultrasound imaging system which combines spherical focusing on transmitting with axial focusing on receiving [3]. Similar annular array designs are appearing in new commercial clinnical scanners. Comparisons of theoretical calculations with experimental results for the wideband response of an annular array have been presented by D. R. Dietz, S. J. Norton and M. Linzer [43. The phase shift across each annulus is an important factor in the design of an annular array. In Section II of this paper a design method yielding a measure of efficiency is described which is based upon the phase shift across the annulus. The phase shift is chosen by making a compromise between the maximum signal received by the annulus and its efficiency. In Section III design formulas are developed to determine the geometry of the annular array in terms of the phase shift across the annulus and the desired depth of focus. The variables in these universal formulas are plotted in a series of graphs that provide complete information for the design of a variety of annular arrays and can be readily modified to model virtually any annular array. An example is given of an annular array that has been built based on this design method. Experimental results are given in Part II of this paper.

II. DETERMINATION

OF THE PHASE SHIFT ACROSS AN ANNULUS

In pulse-echo imaging systems the signal received by each annulus of the concave annular array depends upon a function of the phase shift across the annulus and the area of the annulus. It is obvious that, if there were no phase shifts, the strength of the received signal would increase as the area of the annulus increases. However, increasing the area of the annulus will lead to an increase of the phase shift for ethos from all points except the geometrical focal point and thus lead to cancellations in the resultant signal. Thus for every range interval of focal depth, there is an optimum width for each annulus for maximum power output. Since the total output of the atmular array is the summation of the output signals of each annulus of the array, the phase shifts must be optimized for each annulus. The remainder of this section discusses how the phase shift of each annulus and its efficiency may be determined from simple geometrical considerations. Figure l(a) shows an isometric side view of a concave annular array and figure 1(b) shows a front view. Note that each concentric annulus can be either a transmitter or a

276

CONCAVE

i-th annulus

-,

i-th transducer ( antylus)

ANNULAR

ARRAY-PART

element

Fig. 1 the ce&al

(4

I

element

0))

(a) Side view of a concave annular array. (b) The front view of an annular array.

receiver. The basic procedure used to determine the phase shift across an annulus begins by dividing the i-th annulus into N subelements of equal areas as shown in figure 2. Because the subelements are chosen to have equal area, the output signal of each subelement not only has the same magnitude, but also nearly the same differential phase shift at any chosen point along the central axis [5]. Let the signal received by the k-th subelement of the i-th annulus be represented by the phasor Pik , then the composite signal received by the i-th annulus , phasor Pi , is the vector summation of PIk , k = l,....., N , Pi =

C Pi,

k=l,......, N

(1)

Let @ik be the mean phase of the k-th subelement of the i-th annulus, each of which has approximately the same resultant phase; that is, $il = $iz = @i3= ...... = $iN . Let [Pikl = JPua] for all k. Then, the determination of the resultant phase from the i-th annulus, $i , depends on the vector summation as shown in figure 3. Such a summation defines a circle. When +i = ~12, PI reaches its maximum, which by inspection is the diameter of the circle. Note that the resultant phase, $i , from the i-th annulus is related to the maximum differential phase shij? across that annufus, wi , by the reIation \yi = 2+i . See figure 4(a) and (b) for the relation between these different phasors. There are several factors that must be considered in selecting the resultant phase, oi, from the i-th annulus. As shown above, $i = n/2 is the choice which produces the strongest signal received from the i-th annulus. But a large resultant phase will result in a large width of the annulus, and increasing the width of each annulus will decrease the number of annuli that fit within a given maximum outer diameter of the array. In practice we observe that the number of annuli should be 5 or more in order to perform the dynamic focusing efficiently over a large range. Furthermore, as the resultant phase from an annulus becomes smaller, its efficiency increases. The definition of efficiency, rl, is the net output signal strength (as seen at a point on the axis) divided by the sum of the signal

277

YUANETAL

k-th

subelement

0

I 2.---------

2

Fig. 2

The i-th annulartransducerelementis divided into N subelements of equalarea.

Fig. 3

The output of i-th annulus,Pi , is the sumor compositionof all Pk , k = 1, N.

strengthsof the individual subelements(asseenat this samepoint). Let arC(Pi)be the arc facing Pi . Then note that atC(PI ) = 2 R Qi and 1Pi 1= 2 R sin($i) , where R is the radiusof the circle in figure 4. Thus, assumingthe phasorcomponentsPI, lie on a circle asshown in figure 4, the efficiency can be easily calculatedfrom the ratio of 1Pi 1and the XC facing Pi , that is TJ =

[SiIl($i)

/ $i ]

l

100%

(2)

The relation between q and @iis shown in figure 5. For $i = ~12 and ~14, the respective efficiency is q = 64% and 90%. In practice, a resultantphaseof 7~14 is often a

(b) Fig. 4

(a) The definition of efficiency for Cpi< n/2. Note that kril is a normalized vector representingthe signalfrom the inner radiusof the annulusand kri2 is the correspondingvector representingthe signalfrom the outer radius. (b) The definition of efficient y for $1> n/2.

278

CONCAVE

05

0.0

1.0

2.0

ANNULAR

ARRAY-PART

I

3.0

Phase $ i Fig. 5

Relation between efficiency and phase shift from the i-th annulus.

Fig. 6

Geometry of an annular array. Note that this drawing serves equally well to represent a slice through a conical array or for a set of linear elements arranged along a circular arc.

good compromise choice for which both the signal strength and efficiency may be considered near optimal. Of course, other choices of $i between 0 and x/2 can also be considered. For our choice, ~14 I CpIn12 it follows that 7~122 w 6 n, which is consistent with that given by Eq. (13) in Arditi et al , where the phase shift Y across each element is chosen to be 2.33 radians or 133.5 degrees [2]. The ultimate choice depends on fixed transducer characteristics (e.g. bandwidth), or design compromises between side lobe level and spatial resolution, and on the imaging environment. The following section will develop the formulas for determining the geometry of the annular array from the resultant phase, +i , from each annuli.

III. GEOMETRY OF THE ANNULAR

ARRAY

Basic Geometrical Formulas When the phase shift $i across the i-th annulus has been chosen, an optimization of the geometrical parameters can be readily done. Figure 6 shows the geometry of a concave annular array, where Wi is the width (chord) of the i-th annulus, R is the radius of curvature of the array, 0 is the geometrical focal point of the array, 01 and 02 are respectively the nearest and farthest required dynamic focal points, and ai1 and aa are the inner and outer radii of the i-th annulus. The selected maximum phase shift occurs at the

279

WANETAL

nearest focal point 01 . Thus, from figures 4 and 6, the maximum phase shift, vi, across the i-th annulus is vi = 2+i = k(lri21-IrilI)

(3)

where k = 2n / h is the wave number, h is the wavelength,

and

1ril( 2 = ai12 + [ ( R2 - ait )I’2 - Lt 12

(4)

( fi212 = ai

(5)

+ [ ( R2 - ai

)1/z - Lt 12

Formulas and graphical procedures for optimizing these parameters are given in the next section. Also from the geometry of figure 6, the width of the i-th annulus, wi , is given by Wi = { (ai - ai1 )’ + [ ( R’ - ai12 )t”

- ( R2 - ai

)l” 12 }I”

(6)

In these discussions, the only approximation used was that the arc containing the width of the annulus was replaced by its corresponding chord. This approximation is quite accurate for wi I R << 1. The relative error is given by ]AtC(Wi)-Wi]/]ArC(Wi)]=[l-2*sin(8/2)/8]@100%

(7)

In most practical cases, 8 530” , that is, the relative error I 1.2 %, is quite acceptable. The gap between two adjacent annuli is another design consideration.

It is

important to avoid cross-talk coupling between adjacent annuli. Thus, this space should be large enough to provide good isolation, but as small as possible for maximum efficiency.

A Graphical Design Procedure We will

now describe graphical

procedures

for optimizing

a design.

The

procedures developed in this section can be equally applied to an annular ring or linear elements arranged on a circular arc, either of which can be represented with figure 6. Let us first consider a situation for which we have chosen the radius of curvature, R, and wish

280

CONCAVE

ANNULAR

ARRAY-PART

I

to find an optimal trade-off between the nearest focal range, L1 , and the height of the central element, a,-,~. Let 6 represent a path distance normalized by the wavelength. Then we may express the maximum phase shift of Eq.(3) in terms of the equivalent path difference, (h)(6), so that a 6 = 0.5 corresponds to Y = K or @I= 1112. (Note also that the axes of all graphs in figures 7 to 12 have been normalized in units of wavelength.) From Eqs. (3), (4) and (5) we find that the maximum near focal range of the central ring, L, , is given by the following equation:

h&(R+h6/2)

L, =

(8)

[R+h6-

Figure 7 plots a family of curves relating the distance from the center of the array to the nearest point in the focal range, (R - Lr) , to the height of the central element, a02 , for selected radii, R, from 50 to 380 wavelengths for 6 = .25. The design of our annular array used R = 340 h and a02 = 26.5 h for which (R - L1) = 273.9 h . Once these parameters have been selected, the calculation of the radii of the outer elements is quite straightforward. For a path difference (6)(h) , with an inner radius ail, the maximum outer radius is given by the following equation: ai

(9)

= JZ3

400

e z 2 i5 8

IL ‘m P

300

200

100

or.

i. 0

10

;. Height

Fig.7

Cm%

20

;.

R=380

h

-

R&IO

k

-

R=3oOh

I -

R=250 R=200 R=150 R=lOO R&O

h h A h 1

1

Elern~~t

of

1

50

A

Relation of the near focal range ( R-L1 ) to the height of the central element for selected values of R with a path difference of 6 = .25.

281

YUANETAL

where

D=C+ 1

and where

C=Jq.

Figure 8 shows a plot of the height of the outer rings, ai , versus the height of the central element, a02 , for 6 = .25 and a radius of curvature R = 340 h. The gap between each annulus was 6 1. The design choices for our S-ring, 3 MHz annular array were ai2 = 26.52, 41.94, 54.74, 66.22, and 76.86 wavelengths for i = 0 to 4 respectively. Note that as long as the ratio (a02 / R)
One could again solve for I+ in terms of 6, R, and am and

obtain the following equation: h6(R+h6/2) (10)

L2 = [R-1&d-]

However, it can occur that for a given choice of parameters that the normalized path length difference, 6, is less than a chosen maximum for all L;! out to infinity. In these cases one

annulus

no. i=4 i=3 id i=i

Fig. 8

Relation of the height of the outer four elements to the height of the ten ter element for R = 340 h.

CONCAVE

ANNULAR

ARRAY--PART

I

0.6

a a2 = 35 h

a ,=30X a o2 =25X Jz iii a

a,=20h a ,=151

0

50

100

150

200

Far Focal Distance

Fig. 9

Variation of the normalized path difference, 6 , as a function of the distance from the geometric focal point for R = 340 h for selected heights of the central element (a&

can obtain extraneous negative roots for b. This can be readily seen if we use the following expression for fl = (R - L1) and f2 = (R + L2) given by Arditi [3]: 2 -= R

1 ( -+f1

1 fi

>

which can readily be verified from Eqs. (8) and (10) as long as R/h >> 6. From this expression, it is clear that if fl < R/2, then IQ will have negative roots. We have found it more useful to pick the radius of curvature, R, and then to plot the normalized path difference, 6 , versus the the far focal length, L2, for a set of choices for the height of the central element, a02. The design procedure is then to make certain than 6 remains less than a chosen value for the selected range of parameters. Figure 9 gives a typical plot the far focal distance, L2, for R = 340 wavelengths and selected a02 values annular array. For our Sring, 3 MHz annular array, when a(~ = 26.52 h and f, = then the path difference, 6 , remains less than .25 out to La = 108.2 h , which

of 6 vs. for the 273.9 h can be

verified on figure 9. We also designed a smaller linear array for which a relatively large focal region was a primary concern. The linear array was designed for a frequency of 1 MHz and we chose to use 6 = 0.5. A plot of the near focal distance (R - L1) vs. the height of the center element (ao2) for selected values of R is given in figure 10. We selected a radius of curvature, R = 16.67 h and a02 = 2.25 h for which (R - L1) = 3.7 h. Note that for small values of (am / R), all the curves are nearly identical - i.e. in this region the distance from

283

YUANETAL,

2

0

4

Height

Fig. 10

8

6

of Center

Element

*

R=4Q

* * * * *

I?=35 a R=3Q a R=25 X R=20 a R=15 h

a

10

A

Relation of the near focal range ( R-L1 ) to the height of the center element for selected values of R for the smaller array with a path difference of 6 = 5

the array to the nearest focal point is nearly independent of the radius of curvature, R. From figure 11 we can see that the height of the next outer element should be 3.27 h . Figure 12 gives a plot of of the normalized path difference, 6 , vs. the far focal distance, L2 , for the 2-D linear array where R = 16.67 wavelengths. From figure 12 we see that the path difference will be less than .5 h out the the farthest distance plotted (actually out to infinity), giving the desired large focal depth for this atray. Design Example and Actual Array We have designed a S-ring, 3 MHz (h = 0.5 mm) concave annular array with a radius of curvature, R, of 170 mm and an outer diameter of 76.9 mm. The space between adjacent annuli is 3.0 mm. Each annulus has approximately the same area so that the phase shift across each annulus is nearly equal [5]. The array was designed so that the maximum differential phase shift across any annulus, Yi, for an echo from any point along the central

14 12 10 a 6

E .cn P

4 2

Fig. 11

0

0

2

Height

4

of Center

6

Element

6

10

A

284

Variation of the height of the 1st annulus with the height of the central element for R = 16.67 h.

CONCAVE

-

ARRAY-PART

I

1.0 0.6 0.6 0.4 0.2 0.0 0

10

20

Far Focal Distance

Fig. 12

30

0.

Variation of the normalized path difference, 6 , as a function of the distance from the geometric focal point for R = 16.67 h for selected heights of the central element (a&.

axis was less than .5n for focal ranges from 137 mm to 224 mm. Table I gives a more detailed description of the array’s design specifications (in mm). The atray used in the experiments described in the following sections was manufactured for us by Panametrics, Inc. of Waltham, Mass., according to a previous, nearly identical, design. The actual dimensions of this array are listed in Table II. IV CONCLUSIONS This paper presents methods for designing concave annular arrays using phasor summation. The phase shift across each annulus, which determines the strength of the signal received, is a very important factor in the design of the annular array. These

Table I. Design specifications for a king center

1St anlllus

2nd annulus

annular array. (dimensions in mm) 3rd annulus 4th annulus

inner radius

0.0

16.26

23.97

30.37

36.11

outer radius

13.26 13.27

20.97 4.74

27.37

33.11 2.79

38.43

width area (mm2)

552.8

3.44

554.0

554.9

285

555.8

2.38 557.0

YUANETAL

Table II. Dimensions of actual 5ring annular array. (dimensions in mm) center

1st anulus

2nd annulus

3rd annulus

4th annulus

inner radius

0.0

16.31

23.93

30.43

36.55

outer radius

13.26

20.88

27.38

33.5

38.71

methods allow one to optimize the resolution and sensitivity of an array. Graphs are given which alow one to readily evaluate these trade-offs.

V. ACKNOWLEDGEMENTS We thank Mr. Michael Tracy and Mr. Steve Jacobs for assistance with the design of the circuits used in these experiments. We also thank Dr. James F. Greenleaf of the Mayo Clinic, Rochester, Minnesota, for assistance and encouragement in the original design of the annular array. This work was supported in part by grants CA 29728 and HL 34995 from the National Institutes of Health.

VI. REFERENCES Expanding-aperture

annular array,

[l]

Dietz, D. R., Parks, S. I. and Linzer, M, Ultrasonic Imaging I, 56-75 (1979).

[2]

Arditi, M., Taylor, W. B., Foster, F. S.,. and Hunt, J. W., An annular array System for high resolution breast echography, Ultrasonic Imaging 4, 1-31 (1982).

[3]

Patterson, M. S. and Foster, F. S., The improvement and quantitative assessment of B-Mode imaging produced by annular array/cone hybrid, Ultrasonic Imaging 5, 195213 (1983).

[4]

Dietz, D. r., Norton, S. J. and Linzer, M., Wideband annular Array Response, in 1978 IEEEE Ultrasonics Symposium Proceedings, pp. 206211 (IEEE Cat. No. 78CH 1344-1 SU).

[5]

Arditi, M., Foster, F. S. and Hunt, J. W., Transient field of concave annular array, Ultrasonic Imaging 3, 37-61 (198 1).

286