Final work
“Computer simulation of ultrasound imaging”
Course: “Selected parts from medical physics: Ultrasound”
Submitted by: Yulia Preezant (Belougin)
ID322071986
Spring 2002
Computer simulation of ultrasound imaging
Preezant Yulia
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Contents
Introduction ______________________________________________________________ 3
The problem posing ________________________________________________________ 4
Spatial Resolution __________________________________________________________ 7
Axial resolution ___________________________________________________________ 7
Lateral resolution __________________________________________________________ 8
Depth of focus ___________________________________________________________ 11
Apodization _____________________________________________________________ 14
Computer simulation of ultrasound imaging
Preezant Yulia
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Introduction
This summary has been prepared as a final work in frame of the course “Selected parts from
medical physics: Ultrasound”.
The summary performs the computer simulation of an ultrasound imaging. We studied the
case of an ultrasound field attained from multi-elements array of transducers for different methods
of ultrasound beam forming and apodization. Also we tested dependence of image functional
characteristics (as resolution of image, depth of field and e.g.) on base system parameters (length of
ultrasound wave, size of aperture, size of elementary transducer, focal distance). In the simulation
we worked in “far field” and used a linear description of ultrasound phenomena.
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The problem posing
We have a one-dimensional array of 64 elementary transducers with aperture width (D) 18
mm, which transmits Gaussian pulse1 with frequency 2.5MHz to set of points.
Figure 1 Array of transducers.
The probe operated in the phased-array mode. It means that an ultrasonic field was formed
on the step of transmitting; we set time delays for beginning pulse transmission for each elementary
transducer so that all impulses from transducers were focused. And on step of receiving we formed
an image; we set time delays for beginning signal receiving for each elementary transducer so that
we wished receive signals only from point of interest.
Figure 2 Phased array mode. Appropriately delayed pulses are transmitted form an array to achieve
steering and focusing at the point of interest (top). The echoes returned are likewise delayed before they are
summed together to form of strong echo signal from the region of interest (bottom).
1
 1  t 2 
f (t )  sin t  exp      , where  - characteristics pulse parameter
 2   


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The pressure in j elementary transducer, which was created by ultrasonic wave, which was
emitted from element i, and reflected from point on distance rp(x) (from central element):
 ka sin  i
P(i, j, x, t )  Const  sin c
2

where
 ka sin  j

  sin c
2



 f t   ij  k ri ( x p )  r j ( x p ) ,




 ij -time delays for beam forming, which are chosen in accordance with beam
forming technique,  i -angle of view for i transducer element, k-wave vector, Const-constant, which
involve parameters of transducer elements, tissue and ultrasonic wave.
Figure 3 Illustration to transducer’s geometry
For received pressures we set pick detection for time dependent P(i, j, x), and maximal on
time value summed:
P( x)   Pmax (i, j , rp ( x))
i
j
The Intensity of pressure field in logarithmic units is a Beam Profile, which determines
characteristics of a system:
 P( x) 
 I ( x) 
  10 log 
 ,
I dB  20 log 
 P( focus) 
 I ( Focus) 
Using simulator for this system:
1.
Define axial and lateral resolution
a)
For distance 80mm from the center of probe, if we are focusing on 60, 80, 100mm
b)
Change  (characteristics pulse parameter) so that 10dB resolution has improved by a
factor of 2.
c)
What is the dependence of resolutions on wavelength?
d)
What is the dependence of resolutions on probe aperture?
Computer simulation of ultrasound imaging
Preezant Yulia
2.
Define depth of focus
a)
For focus that have range between 25 and 100 mm
e)
What is the dependence of depth focus on wavelength?
b)
What is the dependence of depth focus on probe aperture?
3.
Apodization
4.
The system was deteriorated
6
Computer simulation of ultrasound imaging
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Spatial Resolution
Spatial resolution is defined as the minimum distance at which two bright point reflectors
can be resolved. Its value is a key parameter in quantitating the ability to spatially resolve small
structures and to provide good edge definition within the ultrasound image.
Axial resolution
Definition: Axial resolution is a measurement of an instrument’s ability to resolve two
reflecting boundaries that are closely spaced in the axial direction of the instrument
If we determine the effective pulse time to be t ' , we get that axial resolution (AR):
AR 
t 'c
2
Figure 4 The form of the simulated pressure pulse.
For our pulse function:
f (t )  sin  t  e
 1  t 2 
   
 2  


,
approximately we can estimate the effective pulse time by decreasing exponential envelope
of the function. It can be said that:
1
t '10 dB   2ln    2.146  4.292
 10 
It leads that axial resolution for our type of signal is:
AR10 dB  2.146  c
It’s clear that a shorter transmitted pulse will lead to a more precise measurement.
Computer simulation of ultrasound imaging
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Higher transducer frequency producing shorter pulse results in an improved axial resolution.
On the other hand, higher frequency results in an increase of attenuation of ultrasound in tissue.
Therefore, efforts to improve resolution by increasing the frequency will decrease the maximum
penetration depth.
A
R
mm
  1107 sec
  4 107 sec
  8 107 sec
0.33
1.32m
2.64
m
mm
We changed exponential parameter of the pulse in range (1  10 7  1) sec what leads to range of
axial resolutions (0.33  ) mm. It means that in continuous wave regime (   1 ) the probe cannot
resolve between points in longitudinal direction.
Lateral resolution
Definition: The spacing in the transverse plane at which the points are just separately
resolvable is known as the lateral resolution (LR). A reasonable measure of LR is a width (-XdB
beam width) of the focused beam, as shown on figure:
Figure 5 The –10dB lateral resolution.
It is also governed by the shape of the sound, which, in general, is characterized by the
aperture, the transmitting frequency and by transducers focusing.
a)
Lateral resolution (-10dB) for different focusing methods:
Computer simulation of ultrasound imaging
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An ultrasound beam generated by a linear phased array can be focused and steered by
properly delaying the transmitted and received signal. In ultrasound imaging system focusing is
required, since spherical waves propagate in the near field region.
The primary function of focusing is to improve the lateral resolution. The focusing delay is
dependent on the point’s range r(xp). Therefore, focusing can be fixed and dynamic.

Fixed focusing on transmitting =Fixed focus on receiving
Fig.6 shows the results of the simulation for the fixed focusing case for different foci. For all
those numerical experiments we used a parameters:   0.616mm ,   4 107 sec.
Such focused beams have the smallest LR at ranges equal focal distances. Away from this
ranges, the LR becomes worse (the lateral beam width increases).
60mm - LR-10dB=3mm
60mm – no focusing
60mm – no focusing
80mm- no focusing
80mm - LR-10dB=4.2mm
80mm - LR-10dB=4.2mm
100mm – no focusing
100mm- LR-10dB=6mm
100mm- LR-10dB=4.9mm
Figure 6 Dependence of LR on focal distances (fixed focusing)
Such a focused beams have the smallest beam width at the range of L. Away from this
range, the lateral beam width drastically increase.

Fixed focusing on transmitting – Dynamic focusing on receiving
Dynamically focused reception increases the focal length of the receiving beam
electronically with time. On fig. 7 shown that the lateral resolution of images can be further
improved and side lobes can be further suppressed if dynamic focusing is used.
Computer simulation of ultrasound imaging
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60mm - LR-10dB=3mm
60mm - LR-10dB=3.3mm
60mm - LR-10dB=3.2mm
80mm - LR-10dB=4.3mm
80mm - LR-10dB=4.1mm
80mm - LR-10dB=3.8mm
100mm - LR-10dB=5.3mm
100mm - LR-10dB=4.9mm
100mm - LR-10dB=4.6mm
Figure 7 Dependence of LR on focal distances (dynamic focusing)
Dynamic focusing on receiving improves the lateral resolution in circumvent of focus. Also
for both focusing methods we can obtain that:
LR  L
b)
Dependence of LR on frequency (on wave length) of pulse.
Figure 8 Dependence of LR wavelength    c  for the case of dynamic focusing (on receiving)
f

It can be noted that lateral resolution is proportional to wavelength:
LR 
c)
1
, LR  
f
Dependence of LR on transducer’s aperture
Computer simulation of ultrasound imaging
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Figure 9 Dependence of LR on transducer’s aperture (D) for the case of dynamic focusing (on receiving)
We can see that the larger the aperture of the transducer, the smaller is the tendency of
divergence to expand the beam, and the smaller the spot of focus.
LR 
1
D
Approximately expression for lateral resolution is:
L
LR10 dB  1.5   
 D
L
An expression for a single rectangular unfocused element is: LR  2   
 D
Now we can make an important observation: We obtained well-focused beam in near field
region.
Depth of focus
Definition: the axial distance over which the beam maintains its approximate focused size is
termed the depth of focus (DF), as shown on fig.10.
Figure 10 Illustration for pressure field profile and depth of focus
a) Dependence on focusing
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Figure 11 Diagrams for calculation depth of focus. Difference between fixed focusing and dynamic
focusing
Dynamic focus on reception is commonly used so that depth of field is extended without any
reduction of frame rate. Dynamically focused reception increases the focal length.
DF2 dB  7.5mm
DF2 dB  25mm
DF2 dB  44mm
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DF2 dB  29mm
Figure 12 Depth of focus for different focal distances
We can make the interest observation: The depth of focus as a function of focal distance for
dynamic receiving does not obey the law for unfocused beam (see fig.13):
DF  ( L)2
Figure 13 Dependence of depth of focus on range
b) Dependence of DF on frequency (on wave length) of pulse
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Figure 14 Dependence of DF on wavelength
As we can see on fig.14:
DF 
1
, DF  
f
c) Dependence of DF on transducer’s aperture
DF(-2dB)=25mm
DF(-2dB)=18mm
Figure 15 Dependence of DF on transducer’s aperture
For most of data we can determine the depth of focus as:
2
Depth2 dB
L
 4,76   
D
Apodization
There are some methods that serve to smooth out pressure variation, resulting in beam with
low amplitude modulation in the near field, and low acoustic side lobe amplitudes, such as pulsing,
using of rectangular elements and apodization.
Computer simulation of ultrasound imaging
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The beam profile can be manipulated by apodizing the aperture, e.g. by setting for each
element the certain weight.

On fig.16 demonstrated the effect of apodization for a Gaussian type apodization
with  =2 for a points at distance of 25mm from the transducer face.
For apodization we used Gaussian distribution function :
a( x)  e
 x 2
 2  
 D
Figure 16 Gaussian type of apodization
In general, it’s easy to find that apodization contributes in increasing LR but improve image
smopth out side lobes.

The system has deteriorated, and only 48 central elements remain operative. The
effect of changing is shown on fig.17
Figure 17 The results of deterioration
Such manipulation makes LR and quality of image worse.
Computer simulation of ultrasound imaging
Preezant Yulia
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16
The system has deteriorated, and only 48 elements remain operative. The operative
elements are evenly distributed through the aperture.
Figure 18 The results of evenly deteriorated aperture

The system has deteriorated, and only 48 central elements remain operative. The
probability of an element “stay alive” is a Gaussian function centered
Figure 19 Comparison results of a Gaussian type of apodization and deterioration with Gaussian
probability