Bioengineering 280A lecture 3

1

TT Liu, BE280A, UCSD, Fall 2006

Bioengineering 280APrinciples of Biomedical Imaging

Fall Quarter 2006Ultrasound Lecture 2


Depth of Penetration

!

Assume system can handle L dB of loss, then

L = 20log10

Az

A0

"

# $

%

& '

We also have the definition

( = -1

z20log10

Az

A0

"

# $

%

& '

and the approximation

( =(0 f

Total range a wave can travel before attenuation L is

z = L

(0 f

Depth of penetration is

dp =L

2(0 f

2


Depth of Penetration

22041085133202401

Depth ofPenetration(cm)

Frequency(MHz)

Assume L = 80 dB; α0= 1dB/cm/MHz


Pulse Repetition and Frame Rate

!

Need to wait for echoes to return before transmitting new pulse

Pulse repetition interval is

TR "2dp

c

Pulse repetition rate is

fR =1

TR

If N pulses are required to form an image, then the

frame rate is

F =1

NTR

3


Example

!

N = 256, L= 80dB, c =1540m /s, "0 =1dB /cm /MHz

What frequency should be used to achieve a frame rate of 15 frame/sec?

TR =1

FN= 0.26ms

TR #2dp

c=

L

"0 fc

f #L

acTR=1.99MHz


Huygen’s Principle

http://www.fink.com/thesis/chapter2.html http://www.cbem.imperial.ac.uk/ardan/diff/hfw.html

4


Huygen’s Principle

Anderson and Trahey 2000

!

Wavenumber

k =2"

#

!

Oliquity Factor

r01


Small-Angle (paraxial) Approximation


r01

5


Fresnel Approximation


Approximates spherical wavefront with a parabolic phaseprofile


Fresnel Approximation

!

U(x0,y0) =exp( jkz)

j"z## exp

jk

2zx1 $ x0( )

2+ y1 $ y0( )

2( )%

& '

(

) * s x1,y1( )dx1dy1

=exp( jkz)

j"zs(x0,y0)++exp

jk

2zx02 + y0

2( )%

& '

(

) *

%

& '

(

) *

6


Fraunhofer Approximation

!

kr01" kz 1+

1

2

x1# x

0

z

$

% &

'

( )

2

+1

2

y1# y

0

z

$

% &

'

( )

2$

% & &

'

( ) )

= kz 1+1

2z2x1

2 # 2x1x0

+ x0

2( ) +1

2z2y1

2 # 2y1y0

+ y0

2( )$

% &

'

( )

= kz +k

2zx1

2 + y1

2( ) +k

2zx0

2 + y0

2( ) #k

zx1x0

+ y1y0( )

" kz + +k

2zx0

2 + y0

2( ) #k

zx1x0

+ y1y0( )

Assume this termis negligible.


Fraunhofer Condition

!

k

2zx1

2 + y1

2( )Phase term due to position on transducer is

Far-field condition is

!

k

2zx1

2 + y1

2( ) <<1

z >>k

2x1

2 + y1

2( ) ="

#x1

2 + y1

2( )

For a square DxD transducer, x1

2 + y1

2 = D2 /4

z >>"D2

4#$D

2

#

7


Fraunhofer Approximation


Quadratic phase term

Fourier transform of thesource with

!

kx =x

0

"z ky =

y0

"z


Plane Wave (Fraunhofer) Approximationz

x0

x1

θ

θd

r01

!

d = "x1sin#

sin# =x0

r01

$x0

z

d $ "x0x1

z

8


Plane Wave Approximation

!

1

rexp( jkr) "

1

zexp jk z + d( )( ) =

1

zexp j

2#

$z %

x0x1

z

&

' (

)

* +

&

' (

)

* +

U(x0) = s(x1)%,

,

-1

rexp( jkr)dx1

" s(x1)%,

,

-1

zexp j

2#

$z %

x0x1

z

&

' (

)

* +

&

' (

)

* + dx1

=1

zexp j

2#z

$

&

' (

)

* + s(x1)%,

,

- exp %j2#x0x1

$z

&

' (

)

* + dx1

=1

zexp j

2#z

$

&

' (

)

* + s(x1)%,

,

- exp % j2#kxx1( )dx1

=1

zexp j

2#z

$

&

' (

)

* + F s(x)[ ]

k x=x0

$z


Plane Wave Approximation

!

In general

U(x0,y0) =1

zexp j

2"z

#

$

% &

'

( ) F s(x,y)[ ]

k x=x0

#z,ky =

y0

#z,

Example

s(x,y) = rect(x /D)rect(y /D)

U(x0,y0) =1

zexp( jkz)D2 sinc Dkx( )sinc Dky( )

=1

zexp( jkz)D2 sinc D

x0

#z

$

% &

'

( ) sinc Dky

y0

#z

$

% &

'

( )

Zeros occur at x0 =n#z

D and y0 =

n#z

D

Beamwidth of the sinc function is #z

D

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!

"z

D1

!

D1

!

"z

D2

!

D2

2

"

!

D1

2

"

!

D2


Example


!

rectx

D

"

# $

%

& ' rect

x

d

"

# $ %

& ' (1

dcomb

x

d

"

# $ %

& '

)

* +

,

- . /Dsinc(Dk

x)( d sinc(dk

x)comb(dk

x)[ ]

Sidelobes

Question: What should we do to reduce the sidelobes?

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Transducer Dimension

Anderson and Trahey 2000!

Goal : Operate in the Fresnel Zone

z < D2 /"

Dopt # "zmax

Example

zmax = 20 cm

" = 0.5 mm

Dopt =1 cm


Focusing in Fresnel Zone

!

U(x0,y0) =exp( jkz)

j"z## exp

jk

2zx1 $ x0( )

2+ y1 $ y0( )

2( )%

& '

(

) * s x1,y1( )dx1dy1

=exp( jkz)

j"z## exp

jk

2zx12 + y1

2( ) + x02 + y0

2( ) $ 2 x1x0 + y1y0( )( )%

& '

(

) * s x1,y1( )dx1dy1

=exp( jkz)

j"zexp

jk

2zx02 + y0

2( )%

& '

(

) *

exp( jkz)

j"z## exp

jk

2zx12 + y1

2( )%

& '

(

) * exp $

jk

zx1x0 + y1y0( )

%

& '

(

) * s x1,y1( )dx1dy1

!

Use time delays to compensate for this phase term

!

U(x0,y0) =exp( jkz)

j"zexp

jk

2zx02 + y0

2( )#

$ %

&

' ( F exp

jk

2zx12 + y1

2( )#

$ %

&

' ( s x1,y1( )

)

* +

,

- .

11


Focusing in Fresnel Zone

!

U(x0,y0) =exp( jkz)

j"zexp

jk

2zx02 + y0

2( )#

$ %

&

' ( F exp

jk

2zx12 + y1

2( )#

$ %

&

' ( s x1,y1( )

)

* +

,

- .

Make

!

s x1,y1( ) = s0 x1,y1( )exp "jk

2z0x12 + y1

2( )#

$ %

&

' (

!

U(x0,y0) =exp( jkz0)

j"z0exp

jk

2z0x02 + y0

2( )#

$ %

&

' ( F s x1,y1( )[ ]

At the focal depth z =z0

Beamwidth at the focal depth is:

!

"z0

D


Acoustic Lens

!

"z0

D= "F

D

c > c0

z0

12


Depth of Focus

!

When z " z0, the phase term is #$ = exp %jk

2z0

x1

2 + y1

2( )&

' (

)

* + exp %

jk

2zx1

2 + y1

2( )&

' (

)

* +

and the lens is not perfectly focused.

Consider variation in the x - direction.

#$ =kx

2

2

1

z%

1

z0

&

' (

)

* +

For transducer of size D, x

2

2=D

2

4

If we want #$ =,D2

2-

1

z%

1

z0

&

' (

)

* + <1 radian then

1

z%

1

z0

<2-

,D2

The larger the D, the smaller the depth of focus.

Bioengineering 280A lecture 3

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