1 Principles of medical ultrasound imaging and measurements

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					Chapter 1.1                                                      1.1                                            April 30, 2000



                   1 Principles of medical ultrasound
                            imaging and measurements

1.1   Introduction.................................................................................................................... 3

1.2   Ultrasound waves and transducers
      A. Ultrasound compression waves in biological tissue.............................................. 3
      B. Spatial resolution and frequency........................................................................... 6
      C. Ultrasound transducers and beams........................................................................ 7
      D. Focusing and apodization.................................................................................. 13

1.3 Pulse echo amplitude imaging
    A. Reflection, refraction and scattering of ultrasound...............................................17
    B. A-mode and M-mode imaging- Power absorption and TGC........................... 21
    C. Two dimensional (2D) amplitude imaging...........................................................26
    D. Resolution, speckle and frequency in ultrasound imaging................................... 28
    E.    Signal compression and processing.....................................................................31
    F.    Three-dimenaional (3D) imaging.........................................................................31

1.4   Beam forming with electronic arrays
      A. Phased linear arrays.............................................................................................35
      B. Switched linear arrays..........................................................................................36
      C. Curvilinear arrays................................................................................................ 37
      D. Annular arrays.....................................................................................................40
      E.   Dynamic focus and aperture................................................................................ 40

1.5   Factors affecting image quality
      A. What is image quality?.........................................................................................44
      B. Effect of side lobes on image quality................................................................... 45
      C. Artifacts from inhomogeneities in the tissue........................................................46
      D. Reducing the effect of reverberations and wave front aberrations....................... 51

1.6   Blood velocity measurements using the Doppler effect of back-scattered ultrasound
      A. Basic principle..................................................................................................... 52
      B. PW and CW Doppler measurements................................................................... 53
      C. Spectral analysis.................................................................................................. 55
      D. Range ambiguity, frequency aliasing, and LPRF/HPRF PW Doppler.................57
      E.   Analysis of range ambiguity and velocity range with
           LPRF and HPRF PW Doppler............................................................................59

1.7   Multi range gated (MRG) Doppler and Color Flow Mapping (CFM)
      A. Measurement of velocity profile with MRG Doppler .........................................64
      B. 2D color flow mapping of blood velocities..........................................................66
      C. Methods of insonifying the image field ............................................................. 68
      D. Comparison between tissue and flow imaging.....................................................69
Chapter 1.1                                                       1.2                                           April 30, 2000


1.8     Combination of ultrasonic Doppler and amplitude imaging systems
        A. Need and problems with the combination...........................................................71
        B. Instrument optimization....................................................................................... 74
        C. The Missing Signal Estimator (MSE) method of simultaneous tissue imaging
            and Doppler measurements.................................................................................75

1.9 Ultrasound instrumentation
     A. Introduction ........................................................................................................ 77
     B. Instrument block diagram and display interface ..................................................77
     C. Interfacing to a computer..................................................................................... 80
     D. Ultrasound scanline processing for tissue imaging..............................................81
     E.    Dynamic range and digital representation of signals............................................83
     F.    Ultrasound scanline processing for PW/CW Doppler ........................................ 84
     G. Ultrasound scanline processing for color flow imaging ..................................... 86

1.10 Quick reference of important concepts
     Section 1.2.....................................................................................................................88
     Section 1.3.....................................................................................................................90
     Section 1.4.....................................................................................................................91
     Section 1.5.....................................................................................................................92
     Section 1.6.....................................................................................................................92
     Section 1.7.....................................................................................................................94
     Section 1.8.....................................................................................................................95
     Section 1.9.....................................................................................................................95

1.11 Further reading on basic principles and clinical applications.......................................... 99
Chapter 1.1                                   1.3                              April 30, 2000


               1 Principles of medical ultrasound
                     imaging and measurements
    1.1 Introduction
    This chapter gives an overview of the basic principles of medical ultrasound imaging, as an
introduction to the more elaborate mathematical treatment of wave propagation, scattering, and
signal processing presented in the following chapters. Further in depth presentations are also
found in the reference list at the end of this Chapter.
    The amplitude of the back-scattered ultrasound from a transmitted pulse was first used to
identify organ structures along a fixed beam direction using the A-mode display described in
Section 1.3B. The development continued by scanning the beam in a plane to obtain two
dimensional (2D) images of tissue structures. This was first done by manual scanning of the
non-moving organs. With the moving structures of the heart, the manual scanning was to
slow, and the M-mode display described in Section 1.3B was invented. This allowed for
detailed examination of the fast movement of the cardiac valves and the cardiac wall. The need
for real time imaging of the heart spurred the development of real time ultrasound beam
scanning, first by fast mechanical rotation of the transducer, and later by beam steering with
electronic arrays. Over the years, the real time scanning has proven to have so many practical
advantages that it has also taken over from the manual scanning of non-moving structures.
    Ideas of three-dimensional (3D) imaging of organs by beam scanning in 3D directions,
was first introduced in the 50'ies, but severe practical problems arouse from the lack of
adequate 3D display technology. The power of todays computers and displays now has
spurred new life into 3D imaging, but many practical problems is still left due to the high noise
level in clinical clinical ultrasound images. Real time 3D imaging is also a problem due to the
low speed of sound in tissue and the large number of beam directions that must be used.
Processing to visualize volumetric data is also time consuming.
    When the scatterer is moving, the back-scattered signal will also have a change in
frequency in accordance with the Doppler effect. This can be used for the measurement of the
velocity of moving targets, for example the blood. By pulsing the beam, blood velocities in a
specific area can be measured. However, there is a problem with the pulsed beam since the
Doppler signal is sampled for each pulse only, which limits the maximum Doppler frequency
that can be analyzed by the sampling theorem. This limits the maximum velocity that can be
measured with pulsed wave (PW) Doppler. To avoid this limitation a continuously transmitted
wave (CW) must be used. Then the range resolution in the measurement is lost.
    With scanning of the ultrasound beam, the Doppler effect can be used to image the spatial
variation of the blood velocities. In the display, the blood velocities are usually shown in a
color scale, which has given the name color flow mapping (CFM) to the technique.
    1.2 Ultrasound waves and transducers
    A. Ultrasound compression waves in biological tissue

     Ultrasound is a term used to describe sound waves that have frequencies above the audible
range, that means any sound wave with a frequency in excess of 15-20 kHz. Medical
diagnostic ultrasound usually operates in the range 2-10 MHz for transcutaneous
measurements, but frequencies up to 40 MHz have been used intraoperatively and with
intraarterial imaging with ultrasound catheters.
Chapter 1.2                                     1.4                              April 30, 2000




                                                                                          distance



                     l              compression         decompression
    pressure




                                                                                          distance




                                   Moves with wave velocity, c

    Figure 1-1. Schematic illustration of a longitudinal compression wave. The oscillatory
particle movement is along the propagation direction of the wave.
    Figure 1-1 illustrates a plane, compression type acoustic wave. The medium is compressed
and decompressed with a defined spatial period along the propagation direction, produced by
an increase and a decrease in the pressure, respectively. The compression/decompression
pattern moves with the wave velocity along the propagation direction.
    For a frozen instant in time, the spatial period (e.g. the distance between neighboring
pressure maxima) of the wave is called the wavelength, Â. At any fixed point along the wave,
the spatial variations in the pressure will pass by with the wave propagation velocity, c, and an
oscillation in the pressure with frequency f can be observed. The relationship between the
wavelength and the frequency is

    Â = c/f                                                                              (1.1)

    At each spatial position, the material points are oscillating around their equilibrium position,
producing deformation of the material. Acoustic waves can be of the compression type or of
the shear type. For the compression type, termed compression or longitudinal waves, the
particle movement is along the direction of wave propagation. For the shear type, the particle
movement is transverse to the wave propagation, termed shear or transverse waves. In soft
tissue, the shear waves have low propagation velocity (100m/s) and are heavily attenuated
(f>1MHz) and can therefore be neglected. However, at interfaces conversion of longitudinal to
shear waves introduces extra energy losses.
    Soft tissue is mainly water with some solids added. Therefore, the wave propagation
velocity varies little between different types of soft tissue, and is only slightly above that of
water, except for bone where the solid structure is predominant. Table 1-1 shows the sound
velocity, mass density, compressibility, and acoustic impedance of some biological and non-
biological materials.
Chapter 1.2                                     1.5                             April 30, 2000


                                               Table 1-1.
             Acoustic parameters for typical isotropic, homogeneous, elastic materials [6].
    Material         Mass density     Compressibility      Sound velocity Acoustic impedance
                       kg/m3           10-12 m2 /Nt             m/s         106 kg/(m2 s)

    Biological material:
    Fat                   950                508                 1 440              1.37
    Neurons              1 030               410                 1 540              1.59
    Blood                1 025               396                 1 570              1.61
    Kidney               1 040               396                 1 557              1.62
    Liver                1 060             375-394            1 547-1 585        1.64-1.68
    Spleen               1 060             380-389            1 556-1 575        1.65-1.67
    Muscles              1 070             353-393            1 542-1 626        1.65-1.74
    Bone              1 380-1 810          25-100             2 700-4 100         3.75-7.4
    Non biological material:
    Air (0¼ C)               1.2            8³10-6                 330             0.0004
    Rubber               950                 438                  1550             1.472
    Fresh Water (25¼ C) 988                   452                 1 497            1.48
    Salt Water         1 025                 416                  1 531            1.569
    Lucite             1 198                 117                  2 670            3.2
    Polystyrene        1 120                 143                  2 500            2.8
    Hard PVC           1 350                 175                  2 060            2.78
    Typ. Araldit       1 200                 160                  2 300            2.8
    Silicon, RTV-11 1 260                    793                  1 000            1.26
    Quartz             2 650                 11.4                 5 750           15.2
    Fused quartz       2 200                  13                  5 900           13
    PZT-5A             7 750                 5.65                 4 350           33.71
    Silver           10 410                   7.2                 3 650           38
    Gold             19 290                  4.93                 3 240           62.5
    Aluminium          2 875                   9                  6 260           18
    Brass             8 578                  5.94                 4 430           38

    The sound velocity is related to the mass density ¨ and the volume compressibility û
(defined i Section 2.2A) of the material as

    c =1      rk                                                                        (1.2)

    A typical biological material is inhomogeneous , i.e. composed of different types of soft
tissues in defined small regions. However, by excluding fat, the variation of the sound velocity
is only .2 -.3% so that the wave propagates approximately like in a homogeneous and isotropic
elastic material. By homogeneous we mean that the acoustic parameters are independent of the
location and by isotropic we mean that they are independent of the direction in the material.
However the material is inhomogeneous enough to scatter the ultrasound from the bulk of all
biological materials, as is further discussed in Chapter 7. Although the wave velocity is
isotropic, one have found absorption to depend on direction in fibrous and muscular tissue, as
discussed in Section 1.3B. The acoustic or characteristic impedance is
    Z = ¨c                                                                              (1.3)

    Variations in acoustic impedance between two materials causes a reflection of the
ultrasound as discussed in Section 1.3A. The wave velocity and impedance of fat is so much
lower than for muscular tissue, 9% for the velocity and 19% for the impedance, that it causes a
strong scattering of the ultrasound together with a refraction (bending) of the beam. This causes
strong artifacts in the image and is discussed in Section 1.5.
    Figure 1-2 shows the compressibility of the soft tissues as a function of the mass density.
The Figure demonstrates the compressibility is reduced as the materials get denser.
Chapter 1.2                                                     1.6                            April 30, 2000

                                       600



                                       500
        Compressibility 10-12 m2 /Nt


                                       400



                                       300



                                       200



                                       100



                                         0
                                          940   960   980      1000    1020      1040   1060       1080

                                                            Mass density kg/m3

    Figure 1-2. Compressibility of soft tissues as a function of their mass density.

    The variability of the data in Table 1-1 is demonstrated together with the line showing the
linear least square approximation. This line gives an approximate linear relationship between
the compressibility and the mass density

             dû
    û ò û0 + ®® ¨ ò {1545 - 1.1 ¨}³10-12 m2 /Nt
             d¨
                                                                                                      (1.4)
             d¨
    ¨ ò ¨0 + ®® û ò 1405 - 0.9³101 2 û kg/m3
             dû

    This relationship between compressibility and mass density has implications for the angular
variation of the scattered intensity from tissue, as further analysed in Chapter 7.
    Typical frequencies used in ultrasound imaging together with the wavelengths for a wave
velocity c = 1580 m/s are shown in the Table 1-2.

    B. Spatial resolution and frequency
    The resolution in an ultrasound image is proportional to the wavelength, i.e. inversely
proportional to the frequency. Therefore to get good resolution, one should use as high a
frequency as possible. Unfortunately the attenuation of the ultrasound also increases with
frequency as discussed in Section 1.3B.
    To get adequate penetration, it is necessary to lower the frequency and therefore
frequencies in the range 2.5 - 5 MHz are used in adult cardiology and imaging of deep organs.
For pediatric cardiology, frequencies up to 7.5 MHz are used and for imaging of peripheral
vessels and during surgery, frequencies up to 10 - 15 MHz have been used. For intravascular
imaging of atherosclerosis, frequencies up to 40 MHz (Â = 39 µm) have been used producing
a spatial resolution around 100 µm.
Chapter 1.2                                   1.7                                April 30, 2000


                                           Table 1-2.
                   Typical wavelengths with diagnostic ultrasound [6,11,16,21].
    Frequency, MHz 2           2.5      3       3.5       5        7.5      10        20      40
    Wavelength, µm 790        632     527      451      316       211      158        79      39
    Tissue image: ®®®®®®®®®®®®®®®®®®
                   Deep organs, Adult heart
                                            ®®®®®®®®®®®®®
                                                Pediatric heart
                                                                ®®®®®®®®®
                                                                Periph. vessels
                                                                               ®®®®®®®®
                                                                               Intravascular
    Doppler:      ®®®®®®®®®®®®®
                  Deep vess., Ad. heart
                                        ®®®®®®®®®
                                          Ped.heart
                                                     ®®®®®®®®®®®
                                                       Periph. vessels
                                                                         ®®®®®®®
                                                                          Intravasc.

     At audible frequencies, we are used to sound waves bending around corners. We can hear
the train before it comes around the bend. However, as the frequency increases and the
wavelength reduces, the sound tends to move along straight lines. With loud-speakers we find
that the treble sound is more directive than the base. How well the treble sound is heard often
depends upon where in the room one is standing, while the base bends around all corners.
     The bending of waves around corners is called diffraction . The critical factor for how much
the wave bends, is the size of the transducer face in number of wavelengths. If this number is
large, the sound tends to move in straight lines. Typical non invasive diagnostic transducers are
15-25 mm in diameter, and from Table 1-2, we see that these transducers are many
wavelengths wide. For example, a circular transducer of 15 mm diameter will be approximately
20Â at 2 MHz and 48Â at 5 MHz. This makes it possible to generate beams of ultrasound. The
size of the transducer is called its aperture.

    C. Ultrasound transducers and beams
     An ultrasonic transducer can be made of a plate of piezoelectric material with thin metal
electrodes on each face as illustrated in Figure 1-3. The transducer is thus an electrical
capacitor. When a voltage source is coupled to the electrodes, the plate either increases or
decreases its thickness depending on the polarity of the voltage, as illustrated in Figure 1-3b
and c. The transducer can also be used to transfer energy from acoustic vibrations into an
electric voltage and in this mode, can be used to receive ultrasonic waves. When an incoming
wave hits the transducer surface, it causes the plate to vibrate and the piezoelectric effect
produces a voltage between the metal electrodes.
       If we connect an oscillating voltage source to the electrodes, the plate thickness will
vibrate. When the plate thickness is one half wavelength, we get a resonance in the thickness
vibration with increased vibration amplitude. The middle of the plate is then a vibration node,
i.e. not moving, while the surfaces of the plate are vibration antinodes moving to and from each
other with large amplitude. The pressure in the plate is zero on the surfaces when the plate is
vibrating freely in air, and maximum in the center of the plate as illustrated in Figure 1-3d.
Thus the pressure has antinodes where the material vibration has nodes, and nodes where the
vibration has antinodes.
Chapter 1.2                                     1.8                             April 30, 2000



piezo-electric                                                                      metal
material                                                                            electrodes


                                               a)


                       +      + +          +        +     +      + +




                                               b)




                       +       + +          +         +   +       + +
                                               c)




                 velocity                                  pressure
                 ampl.                                     ampl.




                                               d)


    Figure 1-3. a) Cross-section of schematic piezoelectric transducer plate with thin metal
electrodes. b) Expansion of the plate thickness by applying a voltage across the plate, and c)
compression of the plate thickness by applying a voltage of opposite polarity to the plate. The
lowest panel, d), shows the amplitude of the particle velocity and the pressure in the plate for a
free vibrating plate in air.
Chapter 1.2                                       1.9                          April 30, 2000


   Pressure




                                     time
                                                                        Bw
                   Tp

                                                                                   frequency
   Pressure                                  a)
                                                                                       1
                                                                                  Tp ~
                                                                                       Bw


                                      time
                                                                        Bw

              Tp
                                             b)                                     frequency

     Figure 1-4. a) Examples of shortest possible pulses obtained from a transducer with air
backing and no quarter wave matching, and b) a transducer with air backing and quarter wave
matching. The panels to the right show the Fourier transform of the pulses, which gives the
distribution of frequencies in the pulses. The width of these distributions is the bandwidth of
the transducer.
    Connecting a vibrating transducer surface to the tissue, will cause the tissue surface to
vibrate and thus radiate an ultrasound beam into the bulk tissue. The vibration of the surface
will be somewhat reduced by the contact loading of the tissue, and a pressure will develop on
the surface. However, the plate is much stiffer than the tissue so that the reduction in the
vibration at the surface of the transducer due to the load of the tissue is very low.
    Since the transducer is at resonance, it will ring for quite a few oscillations after we have
removed the voltage source. This is called the ring-down . Figure 1-4a shows the velocity on
                                                                                          i
the transducer surface when the transducer is driven with a very short electric pulse (impulse
response), and illustrates the minimum length of the ultrasound pulse which can typically be
transmitted with this design. This is undesirably long for pulse echo applications where we
want a short pulse for high range resolution. There are two ways to reduce the ringing:

    i)   By mounting the transducer plate on a heavy, stiff and absorbing backing , the ringing
         is dampened because energy is transmitted into the backing. The disadvantage of this
         method is that it introduces acoustic power losses into the backing, and reduces the
         amplitude of the pulse transmitted into the tissue as well as the sensitivity of the
         transducer as a receiver.

    ii) By using a thin plate between the transducer and the tissue which has a stiffness and
        mass density between that of the transducer and that of the tissue, the energy coupling
        between the transducer and the tissue will be stronger, and the ringing will die out
        more quickly. The plate should have a thickness of Â/4, where  is the wavelength in
        the plate. Such a plate transforms the mechanical impedance of the tissue and is often
        referred to as a quarter wave impedance transformer . The effect on the ring-down by
        such a quarter wave impedance matching is illustrated in Figure 1-4b.
Chapter 1.2                                     1.10                            April 30, 2000


    The right panel in Figure 1-4 shows the Fourier transform of the pulses which shows the
distribution of frequencies in each pulse. The width of these distributions is called the
bandwidth of the transducer, and we see that the bandwidth is inversely proportional to the
pulse length, i.e. the wider the bandwidth, the shorter the pulse. Since the backing introduces a
reduction in receiver sensitivity, it is avoided if possible. In some cases the backing is required
for mechanical support, for example with linear phased arrays, and cannot be avoided. This
accounts for a less than optimal sensitivity noted with the linear phased array compared to some
other types of transducers. With the second method, the ring-down is reduced through a better
coupling of the energy into the tissue. The best transducer is then obtained with air backing and
quarter wave matching of the transducer to the tissue load. Two or more Â/4 layers are also
used to improve the bandwidth at the cost of more difficult machining.
    To analyze the radiated beam from such a transducer plate, we can use Huygens' principle
where each point on the surface acts as a source of a spherical wave. According to Huygens'
principle, these partial waves will interfere and generate the resulting beam as illustrated in
Figure 1-5. The beam is composed of three regions: The farfield region where, due to
diffraction the beam expands with a fixed opening angle. In the farfield, the beam is composed
of a central main lobe with side lobes as skirts around. The beam amplitude falls gradually off
from the axis, and we can define the width of the main lobe as where the amplitude has fallen X
dB off from the axial value.



                                         2
                                       D /2Â
                         D2/4Â
                          2
                    0.8 D /4Â                                                  Side lobe
  D=2a




                                                           Main lobe                       Q12dB




                                                                               Side lobe

            Extreme near field               Transition region

                                   Near field                                    Far field




    Figure 1-5. Schematic illustration of the formation of an ultrasonic beam from a circular
vibrating plate. According to Huygens' principle, the beam is formed by the interference
between spherical waves from each point on the transducer surface. The resulting beam has a
characteristic farfield region where it expands with a defined opening angle. In the farfield, the
beam is composed of a typical main lobe with surrounding side lobes. Between the transducer
and the farfield, is the nearfield region. This is composed of the extreme nearfield where the
beam is a cylindrical extension of the transducer, and a transition zone between the extreme
nearfield and the farfield.
Chapter 1.2                                        1.11                             April 30, 2000


                                              Table 1-3.
                       Values of kX for various definitions of the beam width.
              X = ‚ is obtained at the zero between the main lobe and the first side lobe.
    X dB                      3                10                12           ‚
    kX                        1                1.8               2           2.44

    The dual sided opening angle of the main lobe for a plane circular transducer is then

    ¯XdB = kX Â/D                                                                            (1.5)

    where D = 2a is the diameter of the transducer. Values of kX for different reductions X in
beam amplitude are shown in Table 1-3. Typical values of 12dB opening angles are shown in
Table 1-4. The distance x3 along the transducer axis to the start of the farfield region is for a
plane, circular transducer

    x3 = 2a2 /Â = D2 /2Â                                                                     (1.6)

    This can be defined geometrically as the intersection between the cylindrical extension of
the transducer and the 12dB opening angle of the beam, as illustrated in Figure 1-5. This is
discussed at length in Sections 5.4-6. Between the transducer and this limit is the nearfield
region which is composed of the extreme nearfield for

    x3 < 0.8 a2 /Â = 0.2 D2 /Â                                                               (1.7)

     where the beam is approximately the cylindrical extension of the transducer. The region
between the extreme nearfield and the farfield is called the transition region . In this region the
beam contracts before it starts to expand in the farfield, causing an apparent focusing that is
referred to as diffraction focusing.
     The transition between the near field and the far field is not sharp, and the whole concept of
dividing the beam into a near field and a far field region is a simplified means of enhancing
some aspects of the character of the beam rather than describing well defined regions. Also, the
shape of the far field side lobes depends upon the length of the transmitted pulse. For
continuous wave (CW) excitation, we have directions of zero field between each lobe with an
amplitude variation as illustrated in Figure 1-6a, while with a short p u l s e d w a v e (PW)
excitation the zeros disappear and we get an angular variation of the field amplitude as
illustrated in Figure 1-6b. This is further analyzed in Section 5.7.

                                        Table 1-4.
      Dual sided 12 dB opening angle in degrees for practical circular transducers in the range
                                        2-50 MHz.

D=2a mm              1     1.5 2   3     4     5     6     8     10 12 15 17        20    24 28      33
Â=0.75 (2MHz)                                                                5.9    4.3   3.6 3.1    2.6
Â=0.5 (3MHz)                                                         4.8 3.8 3.4    2.9   2.4 2.0    1.7
Â=0.3 (5MHz)                                               4.3   3.4 2.9 2.3 2.0    1.7   1.4 1.2    1.0
Â=0.2 (7.5MHz)                                 4.6   3.8   2.9   2.3 1.9 1.5 1.3    1.1   0.95
Â=0.15 (10MHz)                           1.2   3.4   2.9   2.1   1.7 1.4 1.2 1.0    0.9
Â= 75µ (20MHz)               4.3   2.9   2.1   1.7   1.4   1.1   0.9 0.7 0.6
Â= 50µ (30MHz)       5.7 3.8 2.9   1.9   1.4   1.1   0.9   0.7   0.6
Â= 37µ (40MHz)       4.3 2.9 2.1   1.4   1.1   0.9   0.7   0.5
Â= 30µ (50MHz)       3.4 2.3 1.7   1.1   0.9   0.7   0.6
Chapter 1.2                                    1.12                             April 30, 2000


                    dB
                0



              -10



              -20



              -30
                 -10                            0                             10
                                                a)                          degrees
                    dB
                0



              -10



              -20



              -30
                -10                            0                              10
                                               b)                           degrees

     Figure 1-6. Far field angular variation of the amplitude of the transmitted wave for a plane,
circular 3MHz transducer with 15 mm diameter. CW excitation of the transducer is shown in
a), and PW excitation in b).


                            F

                                                      side lobe

                                                      main lobe
     D=2a




                                                                            QF =Q12dB    Qg
                                              DF                     x3




                                            LF (1dB)


    Figure 1-7. Focusing of an ultrasound beam by shaping the transducer disc as a spherical
shell.
Chapter 1.2                                    1.13                              April 30, 2000


1.0                                                1.0


0.5                                                0.5


  0                                                   0
   20         10        0           10      20            20    10         0        10         20
                                            mm                                                mm
                        a)                                                b)


1.0                                                1.0


0.5                                                0.5


  0                                                   0
              transducer diameter                      20       10        0         10         20
                                                                                              mm
                        c)                                                d)

     Figure 1-8. Focal field distribution from a 3 MHz spherical shell with diameter 15 mm and
focal radius 75 mm. CW excitation with uniform vibration amplitude over the disc is shown in
a), and PW excitation with uniform vibration amplitude over the disc is shown in b). With an
apodization of the vibration amplitude over the disc as in c), the PW excitation produces the
field shown in d).
      D. Focusing and apodization
     The beam can be focused in a simple way by forming the transducer as part of a spherical
shell as illustrated in Figure 1-7. The geometrical focus will be in the center of curvature of the
shell. However, because the transducer disc is a limited number of wavelengths in diameter, we
get diffraction of the sound and the focus will not be as sharp as the focus obtained with
geometrical beams. The field distribution will be as shown in Figure 1-7, with some variations
depending on how the transducer is excited.
     Figure 1-8a shows the cross sectional distribution of the amplitude at the focal point for
CW excitation of the beam with equal vibration amplitude over the disc. The field is composed
of a main lobe and side lobes with clear zeros in between. Figure 1-8b shows the amplitude
variation with PW excitation of the disc producing a transmitted pulse as in Figure 1-4b with
uniform vibration amplitude over the whole disc. The zeros between the sidelobes with CW
excitation have disappeared. It can be advantageous to let the amplitude of the plate vibration
degrade from the center to the edges. This is called apodization . With a variation of the
vibration amplitude as shown in Figure 1-8c, we get a variation of the field in the focus with
PW as shown in Figure 1-8d.
     When apodization of the excitation amplitude over the transducer surface is done, the width
of the main lobe increases, but the amplitude of the sidelobes decreases. One reason to do this
is that the sidelobes generate acoustic noise since they detect targets that are not along the beam
direction. This reduces the contrast resolution of the instrument, which is the ability to
discriminate a weak target that is close to a strong one. The effect of the sidelobes is further
discussed in Section 1.5. Although the amplitude of the sidelobes is much lower than for the
main lobe, the sidelobes contain a fair amount of energy because they form a skirt around the
main lobe.