BE280A 07 US2 - University of California, San...

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TT Liu, BE280A, UCSD, Fall 2007

Bioengineering 280APrinciples of Biomedical Imaging

Fall Quarter 2007Ultrasound Lecture 2


Single-slit Diffraction

Source:wikipedia

!

Beamwidth "z

D


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


Huygen’s Principle

Anderson and Trahey 2000

!

Wavenumber

k =2"

#

!

Oliquity Factor

r01

2


Small-Angle (paraxial) Approximation


r01


Fresnel Approximation


Approximates spherical wavefront with aparabolic phase profile


Fresnel Approximation

!

U(x0,y0) =1

j"r01## exp jkr01( )s x1,y1( )dx1dy1

$1

j"z## exp jk z

2 + x1 % x0( )2

+ y1 % y0( )2( )( )s x1,y1( )dx1dy1

$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( )&

' (

)

* +

&

' (

)

* +


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( )&

' (

)

* + s x1,y1( )exp %

jk

zx1x0 + y1y0( )

&

' (

)

* + dx1dy1

=exp( jkz)

j#zexp

jk

2zx02 + y0

2( )&

' (

)

* + F exp

jk

2zx12 + y1

2( )&

' (

)

* + s x1,y1( )

-

. /

0

1 2

Phase across transducer face

3


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 /2

z >>"D2

2#$D

2

#


!

"z

D1

!

D1

!

"z

D2

!

D2

2

"

!

D1

2

"

!

D2

!

"z

Dz=D

2/"

= D


Fresnel Zone

Anderson and Trahey 2000TT Liu, BE280A, UCSD, Fall 2007

c > c0

Focusing in Fresnel Zone

!

U(x0,y0) =exp( jkz)

j"zexp

jk

2zx02 + y0

2( )#

$ %

&

' ( F exp

jk

2zx12 + y1

2( )#

$ %

&

' ( s x1,y1( )

)

* +

,

- .

D

z0

!

expjk

2z0x12 + y1

2( )"

# $

%

& '

!

exp "jk

2z0x12 + y1

2( )#

$ %

&

' (

4


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

!

U(x0,y0) =exp( jkz0)

j"z0exp

jk

2z0x02 + y0

2( )#

$ %

&

' ( F exp

jk

2z0x12 + y1

2( )#

$ %

&

' ( exp )

jk

2z0x12 + y1

2( )#

$ %

&

' ( s x1,y1( )

*

+ ,

-

. /

=exp( jkz0)

j"z0exp

jk

2z0x02 + y0

2( )#

$ %

&

' ( F s x1,y1( )[ ]

At the focal depth z =z0

TT Liu, BE280A, UCSD, Fall 2007 TT Liu, BE280A, UCSD, Fall 2007

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

8

If we want #$ =,D2

4-

1

z%

1

z0

&

' (

)

* + <1 radian then

1

z%

1

z0

<4-

,D2.-

D2/

#z

z0

2<-

D2

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

5


!

"z0

D= "F

D

z0

D


Focusing with Phased Array



Focusing and Steering

Prince and Links 2005TT Liu, BE280A, UCSD, Fall 2007

Dynamic Focusing

Prince and Links 2005

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Doppler Effect

http://www.centrus.com.br/DiplomaFMF/SeriesFMF/doppler/capitulos-htmłchapter_01.htm


Doppler Effect

!

"f =2vf

0

c # v$2vf

0

c

Examplev = 50 cm/sc = 1500 m/sf0 = 5 MHz

2vf0

c= 3333 Hz


Doppler



CW Doppler


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CW Doppler

f0 f0+Δf-f0-f0-Δf

*Effective width = 1/Tobs


CW Doppler

!

Resolution "f =1/Tobs

"v =c"f

2 f0

=c

2Tobs f0

ExampleDesign goal : "v = 5 cm /s; f

0= 5 MHz

Tobs =c

2"vf0

=1500m /s

2 0.05m /s( ) 5 #106( )= 3ms

Note that for a depth of 15 cm, it takes only200 usec for echos to return.


PW Doppler



PW Doppler


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TT Liu, BE280A, UCSD, Fall 2007 Siemens Medical Systems; jnormal common carotid artery TT Liu, BE280A, UCSD, Fall 2007

Aliasing

!

Measure Doppler shifts at a specified range

For unambiguous range, one pulse at a time.

TPR =2rmax

c (e.g. 200 usec for 15 cm depth)

To avoid aliasing require

1

TPR

> 2"fmax

Δf-Δf*

1/TPR

=Δf-Δf

AliasedΔf-Δf


Aliasing



Aliasing


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Aliasing



PW Doppler

!

Velocity Resolution (same as with CW)

Tobs >1

"f=

c

2"vf0

Range Resolution

Want to interrogate velocities from a small region "z =cTpulse

2

We also need to make sure that particles remain within this region

over the observation time Tobs

vmaxTobs < "z# Tobs <"z

vmax

=cTpulse

2vmax


PW Doppler

!

Design Example

Rmax = 6 cm " TPR =2(0.06m)

1500m /s= 80 µsec

1

TPR

> 2#fmax =4vmax f0

c

c

4TPR f0

> vmax " for f0 = 5MHz we find that vmax < 93.75cm /s

If we choose #v =1cm /s then Tobs =c

2#vmax f0

=15ms

Range resolution : #z > vmaxTobs =1.4cm

Tpulse =2#z

c=18.8usec

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