Biomedical Imaging I - Middle East Technical University

Transcript Biomedical Imaging I - Middle East Technical University

Ultrasound Introduction

History (Hendee and Ritenour, 2002)

• In 1880, French physicists Pierre and Jacques Curie discovered the piezoelectric effect. “Piezo” is Greek for pressure. Piezoelectricity refers to the generation of an electrical response to an applied pressure .

• French physicist Paul Langevin attempted to develop piezoelectric materials as senders and receivers of high frequency mechanical disturbances (ultrasound waves) through materials. His specific application was the use of ultrasound to detect submarines during Word War I. This technique, sound navigation and ranging (SONAR), finally become practical during World War II.

4/29/2020 10:24 AM .

• Industrial uses of ultasound began in 1928 with the suggestion of Soviet Physicist Sokolov that it could be used to detect hidden flaws in materials.

• Medical uses of ultrasound through the 1930s were confined to applications such as cancer treatments and physcial therapy for various ailments. therapeutic • Diagnostic applications of ultrasound began in the late 1940s through collaboration between the physicians and engineers with SONAR. 4/29/2020 10:24 AM .

Acoustic Wave Energy Ranges Infrasound Audible Ultrasound 20 Hz 20 kHz

• Just as there are infrared, visible, and ultraviolet ranges in the EM spectrum, so there are infrasound (“infra” = “below,” “beneath”), audible (i.e., sound) and ultrasound (“ultra” = “beyond,” “above”) ranges of acoustic wave frequencies • Note that the ratio of the highest to the lowest audible frequencies is 10 3 , or almost 10 octaves. On the other hand, the ratio of the highest to the lowest frequencies of visible light is a bit less than 2 (i.e., less than one octave).

4/29/2020 10:24 AM .

Different Forms of Energy

• Electromagnetic • Photons (quantum description), electromagnetic waves (classical description • Does not require a material medium through which to propagate – Mechanisms of propagation through material media are different from that of propagation through free space • Acoustic • Requires a material medium through which to propagate • Consists of oscillatory motions of the atoms/molecules of which a material is constituted.

• Oscillating particles have kinetic energy motions  square of amplitudes of their • Through action of intermolecular forces, particles transfer their energy to adjacent particles  energy wave traveling through material.

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Transfer/Transformation of Energy

• Light becomes sound — photoacoustic phenomena • Sound becomes light — sonoluminescence • Absorbed electromagnetic (EM) and acoustic energy both become heat • Nevertheless, EM and acoustic energy are FUNDAMENTALLY DISTINCT PHENOMENA!

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Ultrasound Intensity

• As an ultrasound wave passes through a medium, it transports through the medium. energy • The rate of energy transport is known as power.

• Medical ultrasound is produced in beams that are usuallys focused into a small area, and the beam is described in terms of the power per unit area, defined as the beam’s intensity .

• No universal standard reference intensity exist for ultrasound.

“ultrasound at 50 dB was used” is nonsensical.

“the returning echo was 50 dB below the transmitted signal” is informative. 4/29/2020 10:24 AM .

The power consumed by a force F İn time t is given by that has moved an object by a distance l





l t

An ultrasound is a pressure wave. Power P carried by an ultrasonic wave is

P



F



particle velocity

Thus the instantaneous intensity can be expressed as

i

(

t

) 

Force



Area particle velocity



p

(

t

) 

u

(

t

) 4/29/2020 10:24 AM .

Average intensity (Sinusoidal excitation) :

 1

T T

0 

P m Sin



t U m Sin



t dt



P m U m



Sin

2 

t dt

 1 2

P m U m

 1 2 

cU m

2  1 2

P m

2 

4/29/2020 10:24 AM .

Safety limits

Maximum ultrasound intensities (mW/cm 2 ) recommended by the US Food and Drug Administration for various diagnostic applications .

Use Cardiac Peripheral vessels Opthalmic Abdominal Fetal (Intensity) max 430 720 17 94 94 4/29/2020 10:24 AM .

Ultrasound velocity

• The velocity of ultrasound wave through a medium varies with the physical properties of the medium. • Low-density media (air and other gases): molecules may move over relatively large distances before they influence neighboring molecules.

 the velocity of ultrasound wave is low. • High-density media (solids): molecules are constrained in their motion.  the velocity of ultrasound wave is high.

• Liquids exhibit ultrasound velocities intermediate between those in gases and solids. • In different media, changes in velocity are reflected in changes in wavelength of the ultrasound waves, with the frequency remaining relatively constant. 4/29/2020 10:24 AM .

Attenuation of Ultrasound

As an ultrasound beam penetrates a medium, energy is removed from the beam by • absorption, • scattering, and • reflection As with x-rays, the term “attenuation” refers to any mechanism that removes energy from the ultrasound beam. Ultrasound is “absorbed” by the medium if part of the beam’s energy is converted into other forms of energy, such as an increase in the random motion of molecules.

If the obstacle’s size is large compared with the wavelength of sound then part of the beam may be “reflected” and the remainder “transmitted” through the obstacle as a beam of lower intensity.

If the size of the obstacle is comparable to or smaler than the wavelength of the ultraound, the obstacle will “scatter” energy in various directions. 4/29/2020 10:24 AM .

Attenuation coefficients



for 1 MHz Ultrasound

Material  (dB/cm) Material  (db/cm) Blood Fat Muscle (across fibers) Muscle a(along fibers) Aqueous and vitreous humor of eye Lens of eye Skull bone 0.18

0.6

3.3

1.2

0.1

2.0

20 Lung liver Brain Kidney Spinal cord water Caster oil 40 0.9

0.85

1 1 0.0022

2 4/29/2020 10:24 AM .

Clinical Potential of Attenuation Measurements

Note, overall attenuation coefficient β , not only absorption or only (back)scattering Infarcted myocardium Healthy myocardium That is, ultrasound attenuation and backscatter measurements can be used (among many other things) to assess extent of tissue death in myocardial infarction 4/29/2020 10:24 AM .

Reflection

• In most diagnostic applications of ultrasound, use is made of uultasound waves reflected from interfaces between different tissues in the patient. The fraction of the impringing energy reflected from an interface depends on the difference in acpustic impedance of the media on opposite sides of the interface. • The acoustic impedance Z of a medium is the product of the density of the medium and velocity of ultrasound in the medium.

 

An alternative definition : Acoustic impedance = pressure/particle velocity Compare electrical circuit analogue : impedance = voltage/current 4/29/2020 10:24 AM .

metal gas acrylic soft tissues hard tissue 4/29/2020 10:24 AM Notice how similar these values are to each other and to that for water, and how different they are from these.

Reflection and Refraction

• Behavior or ultrasound at an interface between materials of different refractive index.

Z is analogous to behavior of light at interface between materials of different • Fraction of pressure reflected = Reflection Coefficient, R ; fraction Z 1 , u 1 p i p r of pressure transmitted = Transmission Coefficient, T R  p p r i  Z Z 2 2

cos cos

  i i   Z Z 1 1

cos cos

  t t T  p t p i  Z 2

cos

Z  i  Z  i 1

cos

 t

.

,

Z 2 , u 2 p t • Intensity reflection and transmission coefficients are derived from the preceding equations and p = Zu and I = p 0 2 /(2 Z ): I I i r    Z Z 2 2

cos cos

  i i   Z Z 1 1

cos cos

  t t   2

,

I I t i   Z

4 cos

 i 2

cos

 Z 1 2  i

cos

 t  2

.

4/29/2020 10:24 AM .

Transmission through plates (normal incidence)

Z 1 Z 2 Z 3 incident wave Transmitted wave l 2 The coefficient for transmission of incident energy into medium 3 is given by:  

I t I i

 (

3 

1 ) 2

Cos

2  4

(

3 2

1 

1 /

2 ) 2

Sin

k

2  2 

l

2  2 4/29/2020 10:24 AM .

important cases: 1) Cos k 2 =1  l 2 =n  2 (where n is an integer)   (

Z

3 3 

Z

1 ) 2 Transmission through such a layer is independent of the layer material .

2) Sin k 2 =1  l 2 =(2n-1)  2 /4   (

2  4

Z Z

1 3 /

2 ) 2 If Z 2 =(Z 1 Z 3 ) 1/2 then  =1. This situation has considerable practical value in maximizing coupling between transducer materials and liquid media. 4/29/2020 10:24 AM .

Refraction

• As an ultrasound beam crosses an interface obliquely between two media, its direction is changed (i.e., the beam is bent). If the velocity of ultrasound is higher in the second medium, then the beam enters the medium at a more oblque (less steep) angle. This behavior of ultrasound transmitted obliquely across an interface is termed refraction . • The relationship between the incident and refraction angles is decribed by the Snell’s law:

Sin

i Sin



c c t i

• The incidence angle at which refraction causes no ultrasound to enter a medium is termed the critical angle  c.

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Piezoelectric Effect

• The piezoelectric effect is exhibited by certain crystals that, in response to applied pressure, develop a voltage across oppsite surfaces. This effect is used to produce an electrical signal in response to incident ultrasound waves. • Similarly, application of voltage across the crystal casues deformation of the crystal. This deforming effect, termed the converse piezoelectric effect , is used to produce an ultrasound beam from a transducer. • Many crystals exhibit the piezoelectric effect at low temperatures, but are unsuitable aas ultrasound transducers because their piezoelectric properties do not exist at room temperature. The temperature above which a crystals’s piezoelectric properties disappear isa known as Curie point of the crystal. 4/29/2020 10:24 AM .

Piezoelectric Properties

• Efficiency of the transducer is the fraction of applied energy that is converted to the desired energy mode. For an ultrasound transducer, this definition of efficiency is dexribed as the electromechanical coupling coeffcient k c . • If mechanical energy (i.e., pressure) is applied, we obtain

k c

2 

mechanical energy converted to electrical energy applied mechanical energy

• If electrical energy is applied, we obtain

k c

2 

electrical energy converted to mechanical energy applied electrical energy

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Properties of selected piezoelectric crystals

Materials Electromagnetic coupling coefficient (k c ) Curie point (° C) Quartz (occur in nature) Rochelle salt (occur in nature) Barium titanate (man-made) Lead zirconate titanate (PZT-4) (man-made) Lead zirconate titanate (PZT-5) (man-made) 4/29/2020 10:24 AM 0.11

0.78

0.30

0.70

550 45 120 328 365 .

Transducer design

• The piezoelectric crystal is the functional component of an ultasound transducer. A crystal exhibits its greatest response at the resonance frequency .

• The resonance frequency is determined by the thickness dimension of the crystal along the axis of the ultrasound beam). A crystal of half-wavelength thickness resonates at a frequency v: t of the crystal (the



 

Example: a 1.5 mm thick quartz disk (c =5740 m/sec in quartz) has a resonance frequency of v=5740/2 (0.0015)=1.91 MHz. 4/29/2020 10:24 AM .

Transducer Q-factor

• Disc of piezoelectric material (usually PZT), mechanical resonance frequencies f res  Resonance curve (Q-factor, Q = f res /D f ; D f is -3 dB width of curve  D f lo-Q • High Q: strong resonance (narrow curve) Amplitude • Low Q: A ( f res ) = 0 dB strongly damped, weak resonance (broad curve) - 3 dB D f hi-Q Frequency • Tradeoff of high Q: + – Efficient at f res (high signal-to-noise ratio) Pulse distortion (ringing effect) 4/29/2020 10:24 AM .

Typical Ultrasound Transducer

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Trasducer Backing

• With only air behind the crystal, ultrasound transmitted back into the cylinder from the crystal is reflected from the cylinder’s opposite end. The reflected ultrasound reinforces the ultrasound propagated in the forward direction from the transducer. This reverberation of ultrasound in the transducer itself contributes energy to the ultrasound beam (i.e., it increases the efficiency). It also extends the time over which the ultrasound pulse is produced. Extension of the pulse duration (decreases bandwidth, increases Q) is no problem in some clinical uses of ultrasound such as continuous wave applications. • However, most ultrasound imaging applications utilize short pulses of ultrasound, and suppression of ultrasound reverberation is desirable. Backing of transducer with an absorbing material (tungsten powder embedded in epoxy resin) reduces reflections from back, causes damping at resonance frequency • Reduces efficiency • Increases Bandwidth (lowers Q) 4/29/2020 10:24 AM .

Transducers in Pulsed / C.W. Mode

• Low bandwidth: • No backing • High efficiency (SNR) • High-Q • Strong “Pulse ringing”  applications c.w. • Large Bandwidth: • Pulsed applications • Backing, matching • Low-Q • Lowered efficiency 4/29/2020 10:24 AM The characteristics of a 5MHz transducer for pulsed applications .

Transducer – Tissue Mismatch

• Impedance mismatch causes reflection, inefficient coupling of acoustical energy from transducer into tissue:

Z T Z L

  30 MRayl 1.5 MRayl 

I t

I i

= 0.18

I t I i

 

Z T

Z Z T l



Z l

 2 • Solution: Matching layer(s) • increases coupling efficiency • damps crystal oscillations, increases bandwidth (reduces efficiency) 4/29/2020 10:24 AM

Z L Z T

Transducer Load (tissue)

I r I i I t

Matching Layers

• A layer between transducer and tissue with

Z T

Z l

Z L

creates stepwise transition • Ideally , 100 % coupling efficiency across a matching layer is possible because of destructive interference of back reflections if • layer thickness = •

Z l

 chosen so that

r,1

I r,2

Z l



• Problems: Finding material with exact

Z l

value (~6.7 MRayl) • Dual-layer:

Z l

,1 

Z T

3/ 4

1/ 4

;

Z l

,2 

1/ 4

T Z

3/ 4

4/29/2020 10:24 AM

I i I r,1 I r,2 Z T I t I r,l Z l Z L I t,L

Axial beam profile

• Ultrasound sources may be considered to be a collection of point sources, each radiating spherical wavelets into the medium.

• Interference of the spherical wavelets establishes a characteristic pattern for the resulting wavefronts. • The reinforcement and cancellation of individual wavelets are most noticable in the region near the source of ultrasound. They are progressively less dramatic with increasing distance from the ultrasound source.

• The region near the source where the interference of wavelets is most apparent is termed the Fresnel (or near) zone . For a disk shape transducer of radius r, the length Z 0 of the Fresnel zone is

0 

2  4/29/2020 10:24 AM .

4/29/2020 10:24 AM Fresnel zone Fraunhofer zone .

• Within the Fresnel zone, most of the ultrasound energy is confined to a beam width no greater than the transducer diameter. • Beyond the Fresnel zone, some of the energy escapes along the preriphery of the beam to produce a gradual divergence of the ultrasound beam that is described by sin   0 .

6 

where  is the Fraunhofer divergence angle in degrees. The region beyond the Fresnel zone is termed the Fraunhofer (or far) zone .

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Rules for Transducer design

• For a given transducer diameter, • the near field length increases with increasing frequency, • beam divergence in the far field decreases with incresing frequency, • For a giver transducer frequency, • the near field length increases with increasng transducer diameter, • beam divergence in the far field decreases with increasing transducer diameter.

Example: What is the length of the Fresnel zone for a 10-mm diameter, 2MHz unfocused ultrasound transducer?

 Z 0 = 1540 m/sec / 2x10 6 /sec =0.77 mm.

= (5mm) 2 /0.77 mm = 32.5 mm 4/29/2020 10:24 AM .

Transducer radius and ultrasound frequency and their relationship to Fresnel zone and beam divergence Frequency (Mhz) Wavelength (cm) Fresnel zone (cm) Fraunhofer divergence angle (degrees) Transducer radius constant at 0.5 cm 0.5

0.30

1.0

0.15

2.0

4.0

8.0

Radius(cm) 0.075

0.0325

0.0163

0.82

1.63

3.25

6.5

13.0

Fresnel zone (cm) 21.5

10.5

5.2

2.3

1.1

Fraunhofer divergence angle (degrees) Frequency constant at 2 MHz Radius (cm) 0.25

0.5

1.0

4/29/2020 10:24 AM 2.0

0.83

3.33

13.33

53.33

10.6

5.3

2.6

1.3

Lateral Beam Profile

• Isoecho contours: each contour depicts the locations of equal echo intensity for the ultrasound beam. At each of these locations, a reflecting object ( small steel ball) will be detected with equal sensitivity. Connecting these locations with lines yields isoecho contours. • Isoecho contours help depict the lateral resolution of a transducer, as well as variations in lateral resolution with depth. • For disc (radius r , piston source): 4/29/2020 10:24 AM sin   0.61



Axial and Lateral Resolution

• Axial resolution determined by spatial pulse length  c (  = pulse duration). Pulse length determined by location of -3 dB point.

• Lateral resolution determined by beam width (-3 dB beam width or - 6 dB width) 4/29/2020 10:24 AM .

Focusing of Ultrasound

• Increased spatial resolution at specific depth • Self-focusing radiator or acoustic lens 4/29/2020 10:24 AM .

Transducer Arrays

• Switched Array  lateral scan • Phased Array for beam steering, focusing 4/29/2020 10:24 AM .

Array Types

a) Linear Sequential (switched) ~1 cm  10-15 cm, up to 512 elements b) Curvilinear similar to (a), wider field of view c) Linear Phased up to 128 elements, small footprint  cardiac imaging d) 1.5D Array 3-9 elements in elevation allow for focusing e) 2D Phased Focusing, steering in both dimensions 4/29/2020 10:24 AM .

Ultrasound Imaging

A Mode (Amplitude Mode)

• Oldest, simplest type • Display of the envelope of pulse-echoes vs. time, depth d • Pulse repetition rate ~ kHz (limited by penetration depth, c  1.5 mm/  s  20 cm  additional wait time for reverberation and echoes) = ct /2 270  s, plus 4/29/2020 10:24 AM .

B Mode (“Brightness Mode”)

• The location of echo-producing interfaces is displayed in two-dimensions (x,y) on a video screen. The amplitude of each echo is represented by the brightness value at the xy location. • Lateral scan across tissue surface 4/29/2020 10:24 AM .

Real-Time B Scanners

• Frame rate R f ~30 Hz R f  d  N = c/2 d: depth, N: no. of lines 4/29/2020 10:24 AM .

M-Mode (“Motion Mode”)

• Recording of variation in A scan over time (cardiac imaging: wall thickness, valve function) 4/29/2020 10:24 AM .

Doppler Effect

• When there is relative motion between a source and a detector of ultrasound, the frequency of the detected ultrasound differes from t he emitted by the source. detector detector • An ultrasound source is moving with velocity time v s toward the detector. After t , following the production of any wavefront, the distance between the wave front and the source is (c-v s )t , where c is the velocity of the ultrasound in the medium. The wavelength motion is shortened from the source.  =(c-v s )/ f 0  of the ultrasound in the direction of where f 0 is the frequency of ultrasound 4/29/2020 10:24 AM .

• With the shortened wavelength, the ultrasound reaches the detector with an increased frequency :



   (

c c



v s

)

0  

c c



v s

 

0 • That is, the frequency of the detected ultrasound shifts to a higher value when the ultrasound source is moving toward the detector. The shift in the frequency D





0 

0  

c c



v s

  

0 

0  

c v



s v s

  4/29/2020 10:24 AM .

• If the velocity c of ultrasound in the medium is much greater than the velocity v s of the ultrasound source, then c-v c ~ c and D



v s c

• A similar expression is applicable to the case in which the ultrasound source is stationary and the detector is moving toward the source with velocity v d . In this case, the Doppler shift frequency is approximately D



v d c

where c>>v d . 4/29/2020 10:24 AM .

• If the ultrasound source is moving away from the detector, then the distance between the source and a wavefront is ct+v elapsed since the production of the wavefront. The wavelength ultrasound is  =(c+v s )/ f 0 s t = (c+v s )t, where t is the time and the apparent frequency f is  of the

 

  (

c c



v s

)

0  

c c



v s

 

0 • That is, the frequency shifts to a lower value when the ultrasound source is moving away from the detector. The shift in frequency is D

 



0 

0  

 

v s v s

 

0  



c v s

  

0 4/29/2020 10:24 AM .

• If the velocity c of ultrasound in the medium is much greater than the velocity v s of the ultrasound source, then c+v S ~ c and D



c v s



c v d

where c>>v d . 4/29/2020 10:24 AM .

• If the source and detector are at the same location and ultrasound is reflected from an object moving toward the location with velocity v, the object first acts as a moving detector and receives the ultrasound signal, and then as a moving source as it reflects the signal. v Source /detector • As a results the ultrasound signal received by the detector exhibits a frequency shift (when c>>v) D

 2

v c

4/29/2020 10:24 AM .

• Similarly, for an object moving away from the source and detector, the shift in frequency is D

 2

c v

where the negative sign indicates that the frequency of the detectedultrasound is lower than that emitted by the source. • For the more general case where the ultrasound beam strikes a moving object at an angle  , D

 2

v c

cos   v 4/29/2020 10:24 AM .

CW Doppler

• Doppler shift in detected frequency

f d





f v

: blood flow velocity c: speed of sound  : angle between direction of blood flow and US beam 4/29/2020 10:24 AM .

CW Doppler Clinical Images

• CW ultrasonic flowmeter measurement (radial artery)

[10cm/s] • Spectrasonogram: Time-variation of Doppler Spectrum

[0.2 s]

f t

4/29/2020 10:24 AM .

ULTRASONIC COMPUTED TOMOGRAPHY

• Ultrasound CT is very similar to X-ray computerized tomography. In both cases, a transmtter illuminates the object and a line integral of the attenuation can be estimated by measuring the energy on the far side of the object. • Ultrasound sifferes from x-rays because the propagation speed is much lower; it is possible to measure the exact pressure of the wave as a function of time. From the pressure waveform it is possible to measure • The attenuation of the pressure field, • The delay in the signal indıced by the object.

• Thus from these measurements, it is possible to estimate • the attenution coefficient, • refractive index of the object • It is clear that in computerized tomography, it is essential to know the path that a ray traverses from the source to the detector. In x-ray and emission tomography, the paths are straight lines. But in ultrasound, this is not always the case. 4/29/2020 10:24 AM .

Fundamental considerations

• Ultrasonic waves in the range of 1 10MHz are highly attenuated by air, thus the tissue is immersed in water. Water – serves to couple the energy of the transducer into the object, – provides a good refractive index match with the tissue.

• If an electrical signal, x(t) is applied to the trasmitting transducer, a number of effects can be identified that determine the electrical signal produced by the receiving signal.

• We can write an expression for the received signal y(t), by considering each of these effects in the frequency domain.

l w1 l l w2 tissue water 4/29/2020 10:24 AM y(t) Receiving Transducer, T 2 y a (t) x a (t) x(t) Transmitting Transducer, T 1 .

The Fourier Transform of the received signal Y(f), is given by a simple multiplication of the following factors: 1) the transmitter transfer function relating the electrical signal to the resulting pressure wave, T 1 (f); 2) the attenuation exp[  w (f)l w1 ], and phase change exp [-j  w (f)l w1 ], caused by the near side of the tissue, 3) the transmittence of the front surface of the tissue or the percentage of energy in the water that is coupled into the tissue,  1.

4/29/2020 10:24 AM .

4) the attenuation exp[  (f)l], and phase change exp [-j  (f)l], caused by the near side of the tissue, 5) the transmittence of the rear surface of the tissue,  1.

6) the attenuation exp[  w (f)l w2 ], and phase change exp [-j  w (f)l w2 ], caused by the near side of the tissue, 7) the receiver transfer function relating the pressure to the resulting electrical signal, T 2 (f);

Y

(

f

) 

X

(

f

)

T

1 (

f

)

T

2 (

f

)  

e

 (  (

) 

 (

))

e

 ( 

(

) 



(

))

l w A

 4/29/2020 10:24 AM .

• For the direct water path signal, it is also possible to write a similar expression:

Y

(

f

) 

X

(

f

)

T

1 (

f

)

T

2 (

f

)

e

 ( 

(

) 



(

))

l t

y w (t) Receiving Transducer, T 2 l t water x(t) Transmitting Transducer, T 1 4/29/2020 10:24 AM .

(

f f

) 

Y w

( ) 

Y w

(

f f

)



 (  (

) 

 (

))

l e

( 

(

) 



(

))

)



 [(  (

)  

(

))



(  (

)  

(

))

] Extending this rationale to multilayered objects,

(

) 

Y w

(

)



 0 

(  (

)  

(

))



0 

(  (

)  

(

))

Attenuation in water is negligible, i.e,  w (f)  0 4/29/2020 10:24 AM .

 (

, 

(

f f

) )  2 



(

) 2 

f c w

 (

)  

(

)   2 

c w f

2 

  

c w c w c

(

)   (

)  1     1  Refraction index 4/29/2020 10:24 AM .

T d

 1

c w Y

(

)  0 

(  (

)

  1 )

(

)

 0

  (

)

dx e



2 

fT d Y w

' The corresponding signal can be obtained by taking the Inverse Fourier Transform:

(



T d

) Attenuated water path signal (It is a hypothetical signal that would be received if it underwent the same loss as the actual signal going through.) 4/29/2020 10:24 AM .

Reconstructing the attenuation coefficient



(x,y)

• For soft tissues the coefficient A  is negligible. The time delay in the measured signal may not be taken into account. Thus a line integral about the attenuation coefficient can be obtained from the amplitudes of the water path signal and the signal transmitted from the object :

y w y

 0 

 (

)

• The same approach can be applied for different view angles and projection data can be obtained for each view. • The reconstruction algorithms established for x-ray computerized tomography can be used to reconstruct  (x,y).

4/29/2020 10:24 AM .

Ultrasonic Reflection Tomography

• Here the aim is to make cross sectional images for refractive index coefficient of the soft tissue. Remember the expression about the time delay of the wave propagating in x direction:

T d

 1

c w

0 

(  (

)  1 )

• This can be generalized for waves propagating in any direction. Thus measurement of T d provides projection data of 1  (x,y) for a general view angle. • Well known image reconstruction algorithms can be used to reconstruct  (x,y) from time delay measurements.

4/29/2020 10:24 AM .