o.j cd - ntnufolk.ntnu.no/htorp/doktoravhandlinger_ultralyd/2004_runehansen.pdf · norway....
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8 2:~ NTNU Trondheim Norges teknisk-naturvitenskapelige universitet
Doktoravhandling 2004:5 Fakultet for informasjonsteknologi, matematikk og elektrotelmikk Institutt for teknisk kybernetikk
~ ~ ~ ~
New
Tec
hniq
ues
for
Det
ecti
on o
f U
ltra
soun
d C
ontr
ast A
gent
s
Run
e H
anse
n
Sub
mit
ted
to
The
Nor
weg
ian
Uni
vers
ity
of
Sci
ence
and
Tec
hnol
ogy
in p
arti
al f
ulfi
llm
ent o
f the
req
uire
men
ts
for
the
degr
ee o
f D
octo
r o
f Eng
inee
ring
D
okto
r In
geni
¢r
Nor
weg
ian
Uni
vers
ity
of
Sci
ence
and
Tec
hnol
ogy
Fac
ulty
of
Info
rmat
ion
Tec
hnol
ogy,
M
athe
mat
ics
and
Ele
ctri
cal E
ngin
eeri
ng
Dec
embe
r, 2
003
iii
Abs
trac
t
Thi
s th
esis
ana
lyse
s m
edic
al u
ltra
soun
d pu
lse
echo
det
ecti
on t
echn
ique
s o
f co
ntra
st b
ub
bles
em
bedd
ed in
sof
t tis
sue
and
thre
e ne
w d
etec
tion
tec
hniq
ues
are
desc
ribe
d.
In a
med
ical
ult
raso
und
imag
ing
situ
atio
n, t
he l
inea
rly
scat
tere
d ti
ssue
sig
nal
is s
tron
g an
d w
ill t
ypic
ally
mas
k th
e li
near
ly s
catt
ered
con
tras
t ag
ent
sign
al.
In c
ontr
ast
agen
t de
te
ctio
n te
chni
ques
, it i
s th
eref
ore
usua
lly
the
stro
ng lo
cal n
onli
near
bub
ble
resp
onse
whi
ch
is u
tili
zed
for
imag
e re
cons
truc
tion
.
In a
sm
all
tiss
ue v
olum
e el
emen
t un
derg
oing
com
pres
sion
and
exp
ansi
on d
ue t
o tr
ans
mit
ted
ultr
ason
ic w
aves
, th
ere
will
be
nonl
inea
r ef
fect
s in
trod
uced
due
to
defo
rmat
ion
of
the
volu
me
elem
ent
and
inte
rmol
ecul
ar f
orce
s. F
or ty
pica
l ul
tras
ound
bea
ms,
wit
h ph
ase
fron
ts t
hat a
re n
ot s
tron
gly
curv
ed, t
he d
omin
ant n
onli
near
eff
ects
are
due
to t
he n
onli
near
in
term
olec
ular
forc
es.
The
loca
l eff
ect o
f thi
s no
nlin
eari
ty is
low
but
the
nonl
inea
r di
stor
ti
on a
ccum
ulat
es in
the
for
war
d pr
opag
atio
n o
f the
wav
e an
d ca
n us
uall
y no
t be
negl
ecte
d in
med
ical
ult
raso
und
wav
e pr
opag
atio
n o
f fre
quen
cies
and
am
plit
udes
typi
call
y ap
plie
d.
Firs
t, th
e ef
fect
of
nonl
inea
r w
ave
prop
agat
ion
on n
onli
near
con
tras
t bub
ble
scat
teri
ng is
st
udie
d. T
he s
econ
d ha
rmon
ic c
ompo
nent
in th
e ul
tras
ound
tran
smit
fiel
d, i
ntro
duce
d du
e to
non
line
ar i
nter
mol
ecul
ar f
orce
s, i
s sh
own
to p
oten
tial
ly r
educ
e th
e no
nlin
ear
seco
nd,
thir
d, a
nd f
ourt
h ha
rmon
ic c
ompo
nent
s sc
atte
red
from
a c
ontr
ast b
ubbl
e. T
he d
imin
ishi
ng
effe
cts
on t
he s
catt
ered
thir
d an
d fo
urth
har
mon
ic c
ompo
nent
s ar
e es
peci
ally
sig
nifi
cant
.
In c
ontr
ast h
arm
onic
det
ecti
on te
chni
ques
, th
e no
ise
sign
al p
rese
nt in
a p
ulse
ech
o im
ag
ing
syst
em w
ill p
oten
tial
ly m
ask
the
rece
ived
har
mon
ic c
ontr
ast s
igna
l. T
he u
se o
f Bar
ker
code
s, w
hich
are
a ty
pe o
f pul
se c
ompr
essi
on c
odes
fam
ilia
r in
rada
r sy
stem
s an
d co
mm
uni
cati
on t
heor
y, i
s st
udie
d an
d th
e po
tent
ial
for
incr
easi
ng t
he C
ontr
ast
to N
oise
Rat
io o
f th
e re
ceiv
ed s
catt
ered
thir
d ha
rmon
ic c
ompo
nent
is i
nves
tiga
ted.
Gen
eral
ly, a
n in
crea
se o
f 6
to 9
dB
was
fou
nd b
oth
num
eric
ally
and
exp
erim
enta
lly
appl
ying
a fo
ur b
it B
arke
r cod
e.
The
Bar
ker
code
was
, ho
wev
er,
foun
d to
be
very
sen
siti
ve to
var
iati
ons
in a
cous
tic
prop
er
ties
of t
he b
ubbl
e an
d bu
bble
mov
emen
t dur
ing
inso
nifi
cati
on b
y th
e pu
lse
sequ
ence
.
The
Con
tras
t to
Noi
se R
atio
of
the
rece
ived
sca
tter
ed t
hird
or
four
th h
arm
onic
com
po
nent
s ca
n al
so b
e in
crea
sed
by tr
ansm
itti
ng a
dua
l fr
eque
ncy
band
pul
se w
here
in p
arti
cu
lar
a fu
ndam
enta
l ba
nd a
nd i
ts s
econ
d ha
rmon
ic c
ompo
nent
are
tra
nsm
itte
d, o
verl
appi
ng
in th
e ti
me
dom
ain.
The
rece
ived
thir
d or
four
th h
arm
onic
con
tras
t sig
nal a
nd ti
ssue
sig
nal
ampl
itud
es a
re t
hen
sign
ific
antl
y in
crea
sed
rela
tive
to
whe
n tr
ansm
itti
ng a
con
vent
iona
l fu
ndam
enta
l fr
eque
ncy
band
pul
se.
The
incr
ease
in
the
rece
ived
thir
d or
four
th h
arm
onic
ti
ssue
sig
nal
ampl
itud
e ca
n be
can
cele
d or
red
uced
by
tran
smit
ting
a s
econ
d du
al f
re
quen
cy b
and
puls
e, w
ith
inve
rted
pol
arit
y on
the
tra
nsm
itte
d se
cond
har
mon
ic b
and,
and
th
en c
ombi
ning
the
tw
o re
sult
ing
rece
ived
sig
nals
in
a ge
nera
l pu
lse
inve
rsio
n pr
oces
s.
iv
By
inve
rtin
g th
e po
lari
ty o
f th
e tr
ansm
itte
d se
cond
har
mon
ic c
ompo
nent
, on
e is
abl
e to
co
nstr
uct t
wo
asym
met
ric
puls
es w
ith
resp
ect t
o po
siti
ve a
nd n
egat
ive
tran
smit
am
plit
ude,
th
us p
reve
ntin
g th
e re
sult
ing
cont
rast
age
nt s
igna
l fr
om b
eing
sig
nifi
cant
ly r
educ
ed in
the
pu
lse
inve
rsio
n pr
oces
s.
Fin
ally
, a
cont
rast
age
nt d
etec
tion
tec
hniq
ue u
tili
zing
the
tot
al s
catt
ered
con
tras
t bu
bbl
e si
gnal
is
desc
ribe
d.
Har
mon
ic c
ontr
ast
dete
ctio
n te
chni
ques
typ
ical
ly i
mpo
se s
ome
impo
rtan
t li
mit
atio
ns o
n th
e ra
nge
reso
luti
on a
nd C
ontr
ast
to N
oise
Rat
io o
btai
nabl
e in
th
e fi
nal
ultr
asou
nd i
mag
e.
Als
o, t
he n
onli
near
par
t o
f th
e co
ntra
st s
igna
l sc
atte
red
in
the
forw
ard
prop
agat
ion
dire
ctio
n ad
ds i
n ph
ase
wit
h th
e pr
opag
atin
g tr
ansm
it f
ield
and
m
ay i
ntro
duce
a s
igni
fica
nt p
robl
em w
hen
line
arly
bac
k-sc
atte
red
from
the
tis
sue.
In
the
new
det
ecti
on t
echn
ique
, ec
hoes
fro
m a
tra
nsm
itte
d du
al f
requ
ency
ban
d pu
lse
cons
ist
ing
of
a lo
w f
requ
ency
"pu
mpi
ng"
puls
e an
d a
high
fre
quen
cy i
mag
ing
puls
e ov
erla
ppin
g in
the
tim
e do
mai
n, a
re s
tore
d in
the
im
agin
g sy
stem
. A
sec
ond
dual
fre
quen
cy b
and
puls
e is
tra
nsm
itte
d, w
here
the
pol
arit
y o
f th
e tr
ansm
itte
d lo
w f
requ
ency
com
pone
nts
are
inve
rted
rel
ativ
e to
the
fir
st t
rans
mit
ted
puls
e, a
nd t
he r
esul
ting
new
ech
oes
are
line
arly
co
mbi
ned
wit
h th
e st
ored
ech
oes.
The
tra
nsm
itte
d lo
w f
requ
ency
pul
se m
anip
ulat
es t
he
acou
stic
sca
tter
ing
prop
erti
es o
f th
e co
ntra
st b
ubbl
es a
t th
e tr
ansm
itte
d hi
gh f
requ
ency
co
mpo
nent
s.
The
res
ulti
ng t
issu
e ec
hoes
wil
l be
can
cele
d in
the
lin
ear
com
bina
tion
of
the
echo
es w
hile
the
con
tras
t ag
ent
echo
es,
and
in p
arti
cula
r th
e li
near
ly s
catt
ered
hig
h fr
eque
ncy
com
pone
nts
of
thes
e, a
re p
rese
rved
and
may
be
used
for
im
age
reco
nstr
ucti
on.
v
Pre
face
Thi
s th
esis
is s
ubm
itte
d to
the
Fac
ulty
of I
nfor
mat
ion
Tec
hnol
ogy,
Mat
hem
atic
s an
d E
lec
tric
al E
ngin
eeri
ng a
t th
e N
orw
egia
n U
nive
rsit
y o
f S
cien
ce a
nd T
echn
olog
y, N
TN
U,
in
part
ial f
ulfi
llmen
t of
the
requ
irem
ents
for
the
deg
ree
of D
okto
r In
geni
pr.
The
wor
k ha
s be
en c
arri
ed o
ut a
t th
e D
epar
tmen
t of
Cir
cula
tion
and
Med
ical
Im
agin
g at
the
Fac
ulty
of
Med
icin
e w
here
I h
ave
been
em
ploy
ed a
nd w
here
my
supe
rvis
or h
as
been
Pro
fess
or B
j0m
A.
J. A
ngel
sen.
For
mal
ly,
I am
aff
iliat
ed to
the
Dep
artm
ent
of E
ngi
neer
ing
Cyb
erne
tics
. T
he th
esis
des
crib
es w
ork
done
in t
he p
erio
d fr
om S
prin
g 20
00 to
F
all
2003
reg
ardi
ng c
ontr
ast
agen
t de
tect
ion
tech
niqu
es i
n m
edic
al u
ltra
soun
d im
agin
g.
The
wor
k do
ne i
s m
ainl
y ba
sed
on t
heor
etic
al c
onsi
dera
tion
s an
d nu
mer
ical
sim
ulat
ions
w
here
as e
xper
imen
tal m
easu
rem
ents
are
car
ried
out
onl
y to
a s
mal
l ext
ent.
Fin
anci
ally
, the
wor
k is
sup
port
ed b
y th
e R
esea
rch
Cou
ncil
of N
orw
ay.
Ack
now
led
gmen
ts
Abo
ve a
ll, I
wou
ld l
ike
to t
hank
Pro
fess
or B
jpm
A.
J. A
ngel
sen
for
intr
oduc
ing
me
to t
he
fasc
inat
ing
fiel
ds o
f m
edic
al u
ltra
soun
d im
agin
g an
d ul
tras
ound
con
tras
t ag
ents
, an
d fo
r hi
s co
ntin
uous
sup
port
and
gui
danc
e du
ring
thi
s w
ork.
I w
ould
als
o li
ke t
o th
ank
all
my
othe
r co
llea
gues
at
the
Div
isio
n o
f Med
ical
Im
agin
g an
d at
the
GE
Vin
gmed
Ult
raso
und
grou
p in
Tro
ndhe
im fo
r an
enr
ichi
ng e
nvir
onm
ent b
oth
acad
emic
ally
and
soc
ially
. I w
ould
li
ke t
o gi
ve a
spe
cial
tha
nk to
Ton
ni F
. Joh
anse
n at
the
Div
isio
n of
Med
ical
Im
agin
g fo
r m
any
help
ful
disc
ussi
ons
and
for
read
ing
the
man
uscr
ipt.
Spe
cial
tha
nks
also
to
Sve
in
Eri
k M
aspy
at
the
Div
isio
n o
f M
edic
al I
mag
ing
and
Tro
nd V
arsl
ot a
t th
e D
epar
tmen
t of
M
athe
mat
ical
Sci
ence
s fo
r nu
mer
ous
frui
tful
dis
cuss
ions
. Fi
nally
, I
am g
rate
ful
to A
nja
Pat
rice
Auf
lem
, w
ith
who
m I
am
sha
ring
my
life,
for
alw
ays
bein
g so
sup
port
ive
and
cari
ng.
Con
tent
s
Abs
trac
t iii
Pre
face
v
Nom
encl
atur
e ix
1 In
trod
ucti
on
1 lo
l M
edic
al U
ltra
soun
d Im
agin
g 0
0 0
0 0
0 0
0 0
1 lo
2 U
ltras
ound
Con
tras
t Age
nts
0 0
0 0
0 o
o o
o o
3 1.
2.1
Con
tras
t Age
nt D
etec
tion
Tec
hniq
ues
4 1.
3 O
verv
iew
of t
he T
hesi
s 0
0 0
0 0
0 0
o o
o o
o 7
2 R
educ
tion
of N
onlin
ear
Con
tras
t Age
nt S
catt
erin
g du
e to
Non
linea
r In
cide
nt
Wav
e P
ropa
gati
on
11
2ol
Intr
oduc
tion
0 0
0 0
0 0
0 0
0 0
0 0
11
202
The
ory
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 12
20
2.1
Sing
le B
ubbl
e O
scil
lati
on
12
2020
2 Se
cond
Har
mon
ic C
ompo
nent
in T
rans
mit
Fie
ld
14
2.3
Res
ults
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
16
2.
3.1
Sim
ulat
ion
of T
rans
mit
ted
Wav
e F
ield
0 0
0 0
o o
16
20
302
Sim
ulat
ions
of B
ubbl
e O
scil
lati
on
0 0
o 0
o o
o o
16
2030
3 D
rivi
ng th
e B
ubbl
e w
ith th
e S
imul
ated
Tra
nsm
it F
ield
2.
4 C
oncl
usio
ns
0 0
0 20
5 A
know
ledg
emet
s 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0
3 U
sing
Bar
ker
Cod
es in
Con
tras
t Har
mon
ic I
mag
ing
301
Intr
oduc
tion
o o
o 3o
2 T
heor
y 0
0 0
0 0
0 0
0 3o
2.1
Ove
rvie
w
0 0
3o2o
2 B
arke
r C
odes
3o
2o3
Bub
ble
Osc
illa
tion
3.
3 N
umer
ical
Res
ults
0 0
0 0
0
3.3o
l N
oise
Fre
e S
eque
nce
wit
hout
Har
mon
ic C
ompo
nent
s o
3.30
2 Pr
esen
ce o
f Sev
eral
Har
mon
ics
in S
igna
l for
Pro
cess
ing
26
29
30
31
31
33
33
34
35
35
35
37
viii
3.3.
3 E
ffec
t of A
cous
tic P
ower
Abs
orpt
ion
3.3.
4 B
ubbl
e Si
gnal
with
Inf
inite
CN
R ...
3.3.
5 T
rans
mit
Puls
e B
andw
idth
.
. .
. .
. 3.
3.6
Var
iabl
e A
cous
tic B
ubbl
e Pa
ram
eter
s 3.
3.7
Bub
ble
Mov
emen
t ...... .
3.
3.8
Eff
ect o
f Hav
ing
a Fi
nite
CN
R .
3.
4 E
xper
imen
tal R
esul
ts
3.5
Con
clus
ions
.
. .
. 3.
6 A
ckno
wle
dgm
ents
.
Con
tent
s
38
40
43
46
48
50
51
55
56
4 A
New
Dua
l Fre
quen
cy B
and
Con
tras
t Age
nt D
etec
tion
Tec
hniq
ue
57
57
59
59
60
61
63
63
67
72
78
79
79
4.1
Intr
oduc
tion ........................ .
4.
2 M
etho
d .......................... .
4.
2.1
Wav
e Pr
opag
atio
n an
d Sc
atte
ring
fro
m S
oft T
issu
e 4.
2.2
Scat
teri
ng f
rom
Con
tras
t Age
nts
. 4.
2.3
A N
ew P
ulse
Inv
ersi
on T
echn
ique
4.
3 N
umer
ical
Sim
ulat
ions
.
. .
. 4.
3.1
The
Tra
nsm
it F
ield
....... .
4.
3.2
Scat
teri
ng f
rom
Tis
sue
. .
. .
. .
4.3.
3 Sc
atte
ring
fro
m a
Con
tras
t Bub
ble
4.4
Con
clus
ions
. .
. .
4.
5 Fu
rthe
r Wor
k .
. .
4.6
Ack
now
ledg
men
ts
5 L
inea
r C
ontr
ast
Age
nt D
etec
tion
thr
ough
Low
Fre
quen
cy M
anip
ulat
ion
of
Hig
h F
requ
ency
Sca
tter
ing
Pro
pert
ies
81
5.1
Intr
oduc
tion
. .
. .
. .
. .
. .
. .
. .
. .
. .
. .
. .
. .
. 81
5.
2 T
heor
y.
. .
. .
. .
. .
. .
. .
. .
. .
. .
. .
. .
. .
. .
. 83
5.
2.1
Wav
e Pr
opag
atio
n an
d Sc
atte
ring
fro
m S
oft T
issu
e 83
5.
2.2
Con
tras
t Age
nt S
catte
ring
84
5.
3 M
etho
d .
. .
. .
. .
. .
. .
. .
. 87
5.
4 B
ubbl
e O
scill
atio
ns
. .
. .
. .
. .
89
5.5
Prop
agat
ion
of T
rans
mitt
ed P
ulse
s 94
5.
6 C
oncl
usio
ns .
. .
.
98
5.7
Furt
her W
ork
. .
. 98
5.
8 A
ckno
wle
dgm
ents
99
Bib
liogr
aphy
10
1
Nom
encl
atur
e L
atin
lett
ers
a po
siti
ve a
mpl
itud
e fu
ncti
on
a bu
bble
rad
ius
a ra
dius
vel
ocit
y i:i
radi
us a
ccel
erat
ion
A
bulk
mod
ulus
b
dam
ping
fac
tor
of r
eson
ant
syst
em
b po
siti
ve a
mpl
itud
e fu
ncti
on
b bi
nary
Bar
ker
sequ
ence
B
F
ouri
er tr
ansf
orm
of B
arke
r se
quen
ce
B
nonl
inea
rity
par
amet
er
c po
siti
ve a
mpl
itud
e fu
ncti
on
c sp
eed
of s
ound
d
loss
fac
tor
of r
eson
ant
syst
em
e ba
se o
f nat
ural
log
arit
hm
f fr
eque
ncy
Fr
radi
atio
n fo
rce
h ca
usal
low
-pas
s fi
lter
H
tran
sfer
fun
ctio
n im
agin
ary
unit,
i =
A
I in
tens
ity
of w
ave
k w
ave
num
ber
m
iner
tia
of
reso
nant
sys
tem
M
nu
mbe
r o
f ind
ivid
ual s
ubpu
lses
in c
oded
seq
uenc
e n
nois
e si
gnal
N
no
ise
para
met
er
p tr
ansm
itte
d si
gnal
p
acou
stic
pre
ssur
e Pi
in
cide
nt p
ress
ure
P
tota
l pre
ssur
e P
0 am
bien
t pre
ssur
e f
orth
ogon
al c
oord
inat
es f
or 3
-dim
ensi
onal
spa
ce
r le
ngth
off
r
rece
ived
sig
nal
s st
iffn
ess
of r
eson
ant s
yste
m
s sc
atte
red
sign
al
t ti
me
coor
dina
te
X
u ra
dial
vel
ocit
y U
bu
bble
vel
ocit
y du
e to
rad
iati
on f
orce
z
prop
agat
ion
dire
ctio
n fo
r pl
ane
wav
e Z
ac
oust
ical
im
peda
nce
Gre
ek le
tter
s
(3
nonl
inea
rity
par
amet
er
Nom
encl
atur
e
812
ph
ase
angl
e be
twee
n 1
st a
nd 2
nd
har
mon
ic c
ompo
nent
in r
adiu
s os
cill
atio
n "'
com
pres
sibi
lity
>.
wav
e le
ngth
f.lv
vi
scos
ity
1r
rati
o o
f cir
cum
fere
nce
of
a ci
rcle
to i
ts d
iam
eter
p
dens
ity
O'e
ex
tinc
tion
cro
ss s
ecti
on
~
sum
mat
ion
sym
bol
r ti
me
dela
y ¢
phas
e an
gle
<P12
ph
ase
angl
e be
twee
n p
t an
d 2n
d ha
rmon
ic c
ompo
nent
in t
rans
mit
fie
ld
'ljJ
part
icle
dis
plac
emen
t ,(p
pa
rtic
le v
eloc
ity
;j; pa
rtic
le a
ccel
erat
ion
w
angu
lar
freq
uenc
y, w
= 2
n f
n no
rmal
ized
ang
ular
fre
quen
cy
Sub
scri
pts
and
supe
rscr
ipts
* co
mpl
ex c
onju
gate
0
equi
libr
ium
or
ambi
ent s
tate
0
reso
nant
sta
te
c co
ntra
st a
gent
sig
nal
d D
oppl
er s
hift
i
inci
dent
fie
ld
I in
vers
e fi
lter
M
mat
ched
filt
er
n te
rm n
umbe
r n
in P
ower
Ser
ies
expa
nsio
n s
isen
trop
ic s
tate
t
tiss
ue s
igna
l W
W
iene
r fi
lter
Nom
encl
atur
e
Abb
revi
atio
ns
CN
R
Con
tras
t to
Noi
se R
atio
C
TR
C
ontr
ast t
o T
issu
e R
atio
PI
P
ulse
Inv
ersi
on
SN
R
Sig
nal t
o N
oise
Rat
io
Sym
bols
* co
nvol
utio
n op
erat
ion
in th
e ti
me
dom
ain
t
xi
Cha
pter
1
Intr
oduc
tion
1.1
Med
ical
Ult
raso
und
Imag
ing
Ult
raso
und
is s
ound
wav
es w
ith
freq
uenc
y ab
ove
the
audi
ble
rang
e w
hich
is
arou
nd 2
0 kH
z. T
he f
requ
ency
ran
ge u
sed
in m
edic
al u
ltra
soun
d im
agin
g is
typ
ical
ly f
rom
1 to
10
MH
z, g
ivin
g w
ave
leng
ths
from
1.5
to
0.15
mm
in
soft
tis
sue,
alt
houg
h ap
plic
atio
ns o
f hi
gher
freq
uenc
ies
are
inte
rest
ing
whe
n im
agin
g ov
er s
mal
l reg
ions
clo
se to
the
ultr
asou
nd
tran
sduc
er.
Ult
raso
und
tran
sduc
ers
are
usua
lly
mad
e as
a p
late
of
piez
oele
ctri
c m
ater
ial
wit
h m
et
alli
zed
surf
aces
act
ing
as e
lect
rode
s fo
r ap
plyi
ng e
lect
rica
l vo
ltag
e ac
ross
the
thi
ckne
ss
dire
ctio
n o
f the
pla
te.
The
pie
zoel
ectr
ic p
late
is o
pera
ted
at t
hick
ness
res
onan
ce t
o m
axi
miz
e th
e di
spla
cem
ent o
f th
e pl
ate
and
the
tran
sduc
er is
thu
s ef
fici
ent o
nly
over
a li
mit
ed
freq
uenc
y ra
nge.
The
thic
knes
s vi
brat
ions
of t
he tr
ansd
ucer
pro
duce
pre
ssur
e w
aves
whe
n pl
aced
in c
onta
ct w
ith
soft
tiss
ue.
The
se p
ress
ure
wav
es p
ropa
gate
wit
h th
e sp
eed
of s
ound
in
the
med
ium
whi
ch is
com
pres
sed
and
deco
mpr
esse
d w
ith
a sp
atia
l pe
riod
equ
al to
the
sp
atia
l wav
e le
ngth
alo
ng t
he p
ropa
gati
on d
irec
tion
of t
he w
ave.
Sof
t ti
ssue
is
in t
he p
rese
nt c
onte
xt c
onsi
dere
d a
hete
roge
neou
s co
ntin
uum
mad
e up
of
com
pone
nts
such
as
mus
cle,
fat
, an
d co
nnec
tive
tis
sue
whi
ch a
gain
, on
a s
mal
ler
scal
e,
are
hete
roge
neou
s. A
cous
tic
para
met
ers
thus
hav
e sp
atia
l va
riat
ions
, al
thou
gh r
elat
ivel
y sm
all,
and
the
vari
atio
ns in
mas
s de
nsit
y an
d co
mpr
essi
bili
ty p
rodu
ce u
ltra
soun
d sc
atte
rin
g fr
om s
oft t
issu
e [3
, Cha
pter
7].
Due
to v
aria
tion
s in
con
cent
rati
ons
of b
lood
cel
ls, b
lood
is a
lso
a he
tero
gene
ous
med
ium
. T
he h
eter
ogen
eity
is,
how
ever
, le
ss t
han
for
soft
tis
sue
and
ultr
asou
nd s
catt
erin
g fr
om
bloo
d is
muc
h w
eake
r th
an f
rom
sof
t tis
sue
[13,
Tab
le 4
.21-
4.22
].
Med
ical
ult
raso
und
imag
ing
is p
erfo
rmed
by
plac
ing
a tr
ansd
ucer
in c
onta
ct w
ith
the
skin
an
d tr
ansm
itti
ng u
ltra
soun
d w
aves
, us
uall
y in
the
form
of f
ocus
ed b
eam
s, i
nto
the
regi
on
2 In
trod
ucti
on
of
inte
rest
in
the
body
. T
he t
rans
mit
ted
wav
es a
re t
hen,
due
to
inho
mog
enei
ties
in
the
med
ium
, sc
atte
red
in v
ario
us d
irec
tion
s an
d th
e w
aves
bei
ng s
catt
ered
in t
he d
irec
tion
of
the
rece
ivin
g tr
ansd
ucer
are
pic
ked
up a
s de
laye
d ec
hoes
of
the
tran
smit
ted
wav
e.
The
sc
atte
red
wav
es a
re,
due
the
rela
tivel
y sm
all v
aria
tions
in m
ass
dens
ity
and
com
pres
sibi
lity
, low
in a
mpl
itud
e re
lativ
e to
the
tra
nsm
itte
d w
aves
.
Aco
usti
c ab
sorp
tion
als
o re
duce
s th
e am
plit
ude
of
the
rece
ived
ult
raso
und
echo
es [
2,
Cha
pter
4.5
]. F
or e
ach
wav
e le
ngth
the
wav
e pr
opag
ates
, a
smal
l am
ount
of
the
me
chan
ical
ene
rgy
in th
e w
ave
is i
rrev
ersi
bly
conv
erte
d to
hea
t, an
d th
is l
oss
of
ener
gy d
ue
to a
bsor
ptio
n is
thu
s pr
opor
tion
al to
the
num
ber
of
wav
e le
ngth
s tr
avel
ed.
Thi
s ac
oust
ic
abso
rpti
on m
echa
nism
lim
its
the
max
imum
dep
th f
or im
agin
g w
ith
ultr
asou
nd a
t a c
erta
in
freq
uenc
y du
e to
the
the
rmal
and
ele
ctro
nic
nois
e pr
esen
t in
a pu
lse
echo
imag
ing
syst
em.
The
am
plit
ude
of
the
tran
smit
pre
ssur
e pu
lses
dep
ends
on
the
freq
uenc
y ap
plie
d bu
t may
ty
pica
lly
be f
rom
0.1
to
2 M
Pa
in m
edic
al u
ltra
soun
d im
agin
g. D
epen
ding
on
the
tran
sm
it a
mpl
itude
, th
e re
ceiv
ed e
choe
s ar
e di
stor
ted
rela
tive
to t
he t
rans
mit
ted
puls
es.
Thi
s di
stor
tion
mai
nly
occu
rs d
ue t
o th
e no
nlin
ear
natu
re o
f tis
sue
elas
ticity
. T
he r
elat
ions
hip
betw
een
pres
sure
and
vol
ume
com
pres
sion
is
not
line
ar u
nles
s ve
ry l
ow a
mpl
itud
es,
as
in t
he s
catt
ered
wav
es,
are
cons
ider
ed a
nd t
he d
isto
rtio
n o
f the
wav
e he
nce
occu
rs i
n th
e tr
ansm
it fi
eld
resu
ltin
g in
a fo
rwar
d no
nlin
ear
dist
orti
on o
f the
tra
nsm
it p
ulse
. A
s w
ith
the
acou
stic
abs
orpt
ion,
the
loc
al e
ffec
t o
f no
nlin
eari
ty i
s lo
w b
ut a
ccum
ulat
es a
s th
e w
ave
prop
agat
es.
Thi
s no
nlin
ear
effe
ct h
as g
iven
ris
e to
the
sec
ond
harm
onic
im
agin
g te
ch
niqu
e [6
] [7
] [3
8] w
hich
toda
y is
wid
ely
in u
se in
med
ical
dia
gnos
tic
ultr
asou
nd im
agin
g.
In t
his
tech
niqu
e, t
he s
econ
d ha
rmon
ic c
ompo
nent
s o
f th
e di
stor
ted
echo
es a
re u
sed
to
crea
te th
e ul
tras
ound
imag
e.
The
spa
tial
var
iatio
ns o
f ac
oust
ic p
aram
eter
s ar
e, a
s in
dica
ted,
res
pons
ible
for
the
sca
tte
ring
of t
he i
ncid
ent t
rans
mit
ted
wav
es a
nd a
re h
ence
the
bas
is f
or im
age
reco
nstr
ucti
on.
Insi
de o
rgan
s, t
hese
spa
tial
var
iatio
ns a
re u
sual
ly lo
w,
and
the
scat
teri
ng c
an b
e ap
prox
im
ated
by
a fi
rst
orde
r sc
atte
ring
oft
en c
alle
d th
e B
orn
appr
oxim
atio
n [3
, C
hapt
er 7
]. I
n th
is a
ppro
xim
atio
n, t
he s
catt
ered
fie
ld i
s ca
lcul
ated
bas
ed o
n th
e un
dist
urbe
d ho
mog
ene
ous
tran
smit
ted
fiel
d an
d th
e he
tero
gene
ities
. T
he r
esul
ting
sca
tter
ed f
ield
is
low
in
ampl
itud
e re
lativ
e to
the
inc
iden
t fi
eld
and
prop
agat
es a
s if
in a
hom
ogen
eous
med
ium
, i.e
. w
itho
ut b
eing
sca
ttere
d.
Sca
tter
ing
that
can
be
adeq
uate
ly d
escr
ibed
by
this
Bor
n ap
prox
imat
ion
give
s th
e be
st u
ltra
soun
d im
ages
.
In t
he t
wo
or th
ree
firs
t ce
ntim
eter
s be
low
the
ski
n, t
he b
ody
wal
l co
nsis
ts o
f co
mpo
site
m
uscu
lar
tissu
e an
d fa
t. M
uscl
e an
d fa
t ar
e th
e tw
o co
nsti
tuen
ts i
n so
ft t
issu
e w
ith
the
larg
est
diff
eren
ces
in a
cous
tic
para
met
ers.
The
Bor
n ap
prox
imat
ion
is t
hus
not
valid
in
the
body
wal
l an
d m
ulti
ple
scat
teri
ng,
or r
ever
bera
tions
, ar
e pr
oduc
ed.
Dif
fere
nt p
arts
of
the
tran
smit
ted
beam
wil
l ty
pica
lly
trav
erse
une
qual
dis
tanc
es o
f fa
t an
d m
uscl
e an
d sm
ooth
pha
se f
ront
s o
f th
e tr
ansm
itte
d w
ave
are
pote
ntia
lly
dest
roye
d, g
ivin
g di
stor
ted
phas
e fr
onts
of
vary
ing
ampl
itude
. T
his
phas
e fr
ont
aber
ratio
n de
stro
ys t
he f
ocus
of
the
tran
smit
ted
beam
and
hen
ce r
educ
es t
he r
esol
utio
n in
the
im
age
whi
le t
he r
ever
bera
tion
s ap
pear
as
adde
d no
ise
in t
he i
mag
e [3
, C
hapt
er 1
1].
Try
ing
to c
ompe
nsat
e fo
r th
ese
1.2
Ult
raso
un
d C
ontr
ast A
gen
ts
3
phas
e fr
ont
aber
rati
ons
is t
oday
a m
ajor
res
earc
h ar
ea i
n th
e fi
eld
of m
edic
al u
ltra
soun
d im
agin
g [1
7] [
25]
[24]
.
If th
e tr
ansm
itte
d w
ave
is s
catt
ered
fro
m m
ovin
g ti
ssue
or
bloo
d, t
he D
oppl
er e
ffec
t ca
n be
use
d to
mea
sure
the
vel
ocit
y o
f th
e m
ovin
g sc
atte
rer
[14]
. A
cha
nge
in f
requ
ency
of
the
rece
ived
sca
tter
ed s
igna
l can
be
dete
cted
if t
he s
catt
erer
has
a v
eloc
ity
in th
e di
rect
ion
of
the
ultr
asou
nd b
eam
. S
ince
the
vel
ocit
y o
f th
e sc
atte
rer
is m
uch
less
tha
n th
e sp
eed
of
soun
d, t
his
freq
uenc
y sh
ift
wil
l be
sm
all
rela
tive
to
the
tran
smit
ted
freq
uenc
y an
d is
ty
pica
lly
a fe
w k
Hz,
i.e
. in
the
aud
ible
fre
quen
cy r
ange
. D
oppl
er t
echn
ique
s ar
e to
day
wid
ely
in u
se in
dia
gnos
tic
med
ical
ult
raso
und.
1.2
Ult
raso
und
Con
tras
t Age
nts
Ult
raso
und
scat
teri
ng f
rom
blo
od is
, as
men
tion
ed,
muc
h w
eake
r th
an u
ltra
soun
d sc
atte
rin
g fr
om s
oft
tissu
e, a
nd t
he s
catt
ered
blo
od s
igna
l ca
n th
eref
ore
not
easi
ly b
e se
para
ted
and
visu
aliz
ed in
a m
edic
al u
ltra
soun
d im
age.
Obt
aini
ng in
form
atio
n ab
out b
lood
flow
in
vess
els
and
and
bloo
d flo
w t
hrou
gh v
ario
us o
rgan
s is
fro
m a
med
ical
dia
gnos
tic
poin
t of
view
ver
y he
lpfu
l.
Blo
od f
low
in
larg
er v
esse
ls m
ay b
e de
tect
ed u
sing
Dop
pler
tec
hniq
ues
wit
h hi
ghpa
ss
filte
ring
to
sepa
rate
the
blo
od s
igna
l fr
om t
he t
issu
e si
gnal
. In
sm
all
vess
els,
the
blo
od
velo
city
is
typi
call
y to
o lo
w f
or D
oppl
er t
echn
ique
s to
be
appl
icab
le a
nd c
ontr
ast
agen
ts
are
nece
ssar
y. A
s an
exa
mpl
e, t
he b
lood
vel
ocit
y in
the
cap
illa
ries
is
typi
call
y le
ss t
han
3 m
m/s
[39
]. A
lso,
boa
rder
det
ecti
on o
f th
e he
art
cavi
ties
may
be
impr
oved
app
lyin
g co
ntra
st a
gent
s.
The
sca
tter
ing
from
blo
od c
an b
e si
gnif
ican
tly
incr
ease
d by
add
ing
ultr
asou
nd c
ontr
ast
agen
ts, u
sual
ly m
ade
as s
olut
ions
of s
mal
l gas
bub
bles
in a
liqu
id,
to t
he b
lood
. F
rom
un
derw
ater
aco
usti
cs, i
t is
know
n th
at g
as b
ubbl
es a
re b
oth
pow
erfu
l and
non
line
ar s
catt
erer
s o
f ul
tras
ound
wav
es.
The
gas
bub
bles
hav
e hi
gh c
ompl
ianc
e re
lati
ve t
o th
e su
rrou
ndin
g w
ater
or b
lood
and
in m
edic
al u
ltra
soun
d, t
he g
as b
ubbl
es a
re m
uch
smal
ler t
han
the
wav
e le
ngth
s of
the
tran
smit
pul
ses.
Con
trar
y to
the
wea
k lo
cal t
issu
e no
nlin
eari
ty, t
he c
ontr
ast
bubb
les
show
str
ong
nonl
inea
r lo
cal
resp
onse
s du
e to
lar
ge r
adiu
s ex
curs
ions
wit
h re
sult
in
g sh
ear
defo
rmat
ion
of t
he s
urro
undi
ng f
luid
. T
he c
ontr
ast b
ubbl
es m
ainl
y be
have
as
mon
opol
e sc
atte
rers
and
hen
ce s
catt
er e
nerg
y in
al
l di
rect
ions
, in
clud
ing
the
forw
ard
prop
agat
ion
dire
ctio
n. T
he c
ontr
ast
sign
al s
catt
ered
in
the
for
war
d di
rect
ion
adds
in
phas
e w
ith
the
prop
agat
ing
tran
smit
pul
se a
nd i
ntro
duce
s an
add
itio
nal d
isto
rtio
n o
f th
e tr
ansm
it p
ulse
in r
egio
ns t
hat
in r
ange
dir
ecti
on a
re b
eyon
d a
cont
rast
fill
ed a
rea.
Thi
s ad
diti
onal
dis
tort
ion
may
the
n be
lin
earl
y sc
atte
red
from
sof
t ti
ssue
and
int
erpr
eted
as
cont
rast
age
nt.
In c
ontr
ast
imag
ing
of th
e he
art
or la
rge
vess
els,
th
is i
s ty
pica
lly
a pr
oble
m w
hen
the
tran
smit
pul
se h
as tr
avel
ed th
roug
h th
e la
rge
cont
rast
fi
lled
regi
ons.
A
pply
ing
a li
near
con
tras
t ag
ent
dete
ctio
n te
chni
que
is t
he o
nly
way
to
4 In
trod
ucti
on
avoi
d or
red
uce
the
prob
lem
.
Lor
d R
ayle
igh
[33]
stu
died
the
beh
avio
r of
the
liqu
id s
urro
undi
ng a
col
laps
ing
sphe
rica
l ca
vity
in
1917
and
lat
er i
n 19
33,
Min
naer
t [2
7] p
ubli
shed
a m
odel
whe
re t
he b
ubbl
e w
as v
iew
ed a
s a
harm
onic
osc
illa
tor.
In
194
9, P
less
et [
31]
incl
uded
a d
rivi
ng a
cous
tic
pres
sure
, by
lett
ing
the
back
grou
nd p
ress
ure
vary
wit
h tim
e, t
o th
e eq
uati
on b
ased
on
the
wor
k by
Lor
d R
ayle
igh.
The
res
ulti
ng R
ayle
igh-
Ple
sset
equ
atio
n is
the
fou
ndat
ion
for
the
num
eric
al b
ubbl
e si
mul
atio
ns o
n w
hich
the
pres
ent t
hesi
s is
par
tial
ly b
ased
.
The
fir
st c
ontr
ast
agen
ts f
or m
edic
al u
ltra
soun
d w
ere
appr
oved
by
heal
th c
are
auth
orit
ies
in 1
991
and
the
sear
ch f
or g
ood
cont
rast
age
nts
and
cont
rast
age
nt d
etec
tion
tec
hniq
ues
has
been
rel
ativ
ely
inte
nse
duri
ng t
he l
ast
15 t
o 20
yea
rs.
Tw
o si
gnal
pow
er ra
tios
hav
e vi
tal
impo
rtan
ce f
or th
e qu
alit
y o
f per
form
ance
of t
he m
ed
ical
ult
raso
und
cont
rast
age
nt i
mag
ing
syst
em.
Fir
st,
the
Con
tras
t si
gnal
to T
issu
e si
gnal
R
atio
(C
TR
) w
hich
giv
es t
he r
atio
of t
he s
igna
l pow
er f
rom
the
con
tras
t age
nt in
a r
egio
n to
the
sig
nal
pow
er f
rom
the
tis
sue
in t
hat
regi
on.
Sec
ond,
the
Con
tras
t si
gnal
to
Noi
se
Rat
io (
CN
R)
whi
ch g
ives
the
rati
o o
f the
sig
nal p
ower
from
the
con
tras
t age
nt in
a r
egio
n to
the
noi
se p
ower
in t
hat r
egio
n. T
he C
TR
des
crib
es t
he a
bili
ty t
o di
ffer
enti
ate
cont
rast
si
gnal
and
tis
sue
sign
al i
n an
ult
raso
und
imag
e w
here
as C
NR
des
crib
es t
he e
nhan
cem
ent
of t
he c
ontr
ast s
igna
l abo
ve th
e no
ise
sign
al a
nd d
eter
min
es th
e m
axim
um d
epth
for
imag
in
g th
e co
ntra
st a
gent
. In
add
itio
n, t
he r
esol
utio
n in
the
imag
e is
of
grea
t im
port
ance
as
in m
ost i
mag
ing
sys
tem
s. B
ette
r res
olut
ion
typi
call
y de
man
ds a
pply
ing
high
er tr
ansm
it fr
eque
ncie
s, a
nd t
here
is
a tr
ade-
off b
etw
een
imag
e re
solu
tion
and
max
imum
dep
th o
f im
agin
g.
1.2.
1 C
ontr
ast
Age
nt D
etec
tion
Tec
hniq
ues
Spe
cial
tec
hniq
ues
for
dete
ctin
g th
e co
ntra
st a
gent
in
the
bloo
d is
nec
essa
ry b
ecau
se t
he
stro
ng l
inea
rly
scat
tere
d ti
ssue
sig
nal
typi
call
y is
lar
ger
than
or
of th
e sa
me
orde
r as
the
sc
atte
red
cont
rast
age
nt s
igna
l. In
sm
all v
esse
ls, o
nly
a fe
w c
ontr
ast b
ubbl
es w
ill b
e in
side
a
sam
ple
volu
me
and
the
resu
ltin
g ba
ck-s
catt
ered
con
tras
t age
nt s
igna
l is
wea
k co
mpa
red
to t
he s
urro
undi
ng t
issu
e si
gnal
whe
reas
in
the
larg
e bl
ood
fille
d ca
viti
es o
f th
e he
art,
the
num
ber
of c
ontr
ast
bubb
les
wil
l be
lar
ge g
ivin
g a
stro
ng b
ack-
scat
tere
d co
ntra
st a
gent
si
gnal
from
the
cav
ity.
A s
uper
ior
cont
rast
age
nt d
etec
tion
tec
hniq
ue p
rodu
ces
a ba
ck-s
catt
ered
con
tras
t ag
ent
sign
al f
or i
mag
e re
cons
truc
tion
whi
ch i
s ea
sily
and
ade
quat
ely
diff
eren
tiat
ed f
rom
the
sc
atte
red
tiss
ue s
igna
l an
d th
e no
ise
sign
al o
f th
e im
agin
g sy
stem
. In
add
itio
n, t
o ob
tain
hi
gh i
mag
e re
solu
tion
, th
e sc
atte
red
cont
rast
sig
nal
used
for
im
age
reco
nstr
ucti
on s
houl
d ha
ve h
igh
band
wid
th.
Sev
eral
con
tras
t age
nt d
etec
tion
tech
niqu
es h
ave
been
pro
pose
d an
d I
will
her
e gi
ve a
bri
ef
desc
ript
ion
of s
ome
of th
e m
ost
impo
rtan
t tec
hniq
ues.
Com
mon
for
all
thes
e m
etho
ds is
1.2
Ult
raso
un
d C
ontr
ast A
gent
s 5
that
they
are
bas
ed o
n th
e no
nlin
ear a
cous
tic
prop
erti
es o
f the
con
tras
t age
nt.
As
indi
cate
d,
a pr
oble
m w
ith
all
cont
rast
har
mon
ic d
etec
tion
tec
hniq
ues
is a
spr
ead
of c
ontr
ast
sign
al
beyo
nd th
e ac
tual
con
tras
t fill
ed r
egio
n. T
he n
onli
near
par
t of t
he c
ontr
ast s
igna
l sca
tter
ed
in th
e fo
rwar
d pr
opag
atio
n di
rect
ion
wil
l add
in p
hase
wit
h th
e tr
ansm
it fi
eld
and
may
the
n be
line
arly
bac
k-sc
atte
red
from
the
tis
sue.
Non
e o
f th
e pr
opos
ed te
chni
ques
do
fulf
ill t
he
men
tion
ed c
rite
ria
for
the
supe
rior
det
ecti
on t
echn
ique
and
thi
s su
peri
or te
chni
que
mig
ht
turn
out
to b
e im
poss
ible
to d
eriv
e.
The
fir
st g
roup
of
cont
rast
im
agin
g te
chni
ques
con
sist
s o
f th
e ha
rmon
ic i
mag
ing
met
hod
s. T
hese
met
hods
are
bas
ed o
n tr
ansm
issi
on o
f on
e pu
lse
dow
n ea
ch l
ine
of
sigh
t an
d th
en a
ppli
cati
on o
f va
riou
s ha
rmon
ic f
ilter
s on
the
rec
eive
d ec
hoes
to
obta
in t
he d
esir
ed
harm
onic
com
pone
nts
used
for
im
age
reco
nstr
ucti
on.
Sec
ond
Har
mon
ic I
mag
ing
A p
ulse
cen
tere
d ar
ound
a f
unda
men
tal
freq
uenc
y co
mpo
nent
is t
rans
mit
ted
and
the
re
sult
ing
rece
ived
ech
os a
re b
andp
ass
filte
red
arou
nd t
wic
e th
is t
rans
mit
fre
quen
cy.
The
re
ceiv
ed e
nerg
y at
this
sec
ond
harm
onic
ban
d is
then
use
d fo
r im
age
reco
nstr
ucti
on.
Thi
s is
the
sim
ples
t and
pos
sibl
y m
ost r
obus
t of
the
nonl
inea
r de
tect
ion
met
hods
. A
lim
itat
ion
is t
hat i
n or
der
to p
reve
nt le
akag
e fr
om th
e fu
ndam
enta
l fr
eque
ncy
band
into
the
pass
band
o
f th
e se
cond
har
mon
ic f
ilter
app
lied,
the
tra
nsm
it p
ulse
mus
t be
suf
fici
entl
y na
rrow
ba
nded
res
ulti
ng i
n li
mit
ed r
ange
res
olut
ion.
A
lso,
the
rec
eive
d ti
ssue
sig
nal
typi
call
y co
ntai
ns a
sig
nifi
cant
am
ount
of
ener
gy a
t th
e se
cond
har
mon
ic c
ompo
nent
lim
itin
g th
e C
TR
. The
CN
R m
ay a
lso
be in
adeq
uate
, esp
ecia
lly
whe
n im
agin
g at
larg
e de
pths
. S
econ
d ha
rmon
ics
from
con
tras
t age
nts
are
stud
ied
and
repo
rted
in t
he l
iter
atur
e [2
6] [
11]
[12]
.
Hig
her
Har
mon
ic I
mag
ing
The
se te
chni
ques
are
a g
ener
aliz
atio
n o
f the
sec
ond
harm
onic
imag
ing
tech
niqu
e ap
plyi
ng
a di
ffer
ent f
requ
ency
ban
d, f
or e
xam
ple
the
thir
d ha
rmon
ic b
and,
for i
mag
e re
cons
truc
tion
. T
he r
ecei
ved
tiss
ue s
igna
l ty
pica
lly
cont
ains
ver
y li
ttle
ene
rgy
at t
hese
hig
her
harm
onic
co
mpo
nent
s an
d th
e C
TR
is b
ette
r tha
n w
ith
the
seco
nd h
arm
onic
tech
niqu
e. T
he re
ceiv
ed
cont
rast
sig
nal
typi
call
y al
so c
onta
ins
less
ene
rgy
at t
hese
hig
her
harm
onic
com
pone
nts
and
the
CN
R i
s a
bigg
er p
robl
em th
an w
ith
the
seco
nd h
arm
onic
tec
hniq
ue.
Des
ign
and
man
ufac
ture
of
broa
dban
d tr
ansd
ucer
s th
at a
re e
ffic
ient
ove
r se
vera
l fr
eque
ncy
band
s is
to
day
very
cha
llen
ging
lim
itin
g th
e ex
peri
men
tal w
ork
carr
ied
out u
sing
thes
e hi
gher
har
m
onic
com
pone
nts.
6 In
trod
uct
ion
Su
b H
arm
onic
Im
agin
g
Con
tras
t bu
bble
s ha
ve t
he p
oten
tial
to
scat
ter
ener
gy a
t fr
eque
ncie
s be
low
the
inc
iden
t dr
ivin
g fr
eque
ncy.
Mos
t im
port
ant i
s he
re th
e sc
atte
red
ener
gy a
t hal
f the
dri
ve f
requ
ency
. S
ubha
rmon
ics
typi
call
y re
quir
e lo
ng d
rive
pul
ses
to d
evel
op r
esul
ting
in
degr
aded
ran
ge
reso
luti
on.
Res
ults
fro
m i
mpl
emen
tati
on o
f sub
harm
onic
im
agin
g ar
e re
port
ed [
34]
[15]
.
Non
lin
ear
Fre
quen
cy M
ixin
g
If tw
o se
para
ted
freq
uenc
y ba
nds
are
sim
ulta
neou
sly
tran
smitt
ed,
the
nonl
inea
r re
spon
se
wil
l con
tain
ene
rgy
at th
e su
m a
nd d
iffe
renc
e fr
eque
ncie
s o
f the
two
tran
smit
ted
freq
uenc
y ba
nds.
A
str
ong
nonl
inea
r bu
bble
res
pons
e m
ay,
in t
he s
ame
man
ner
as w
ith
the
othe
r ha
rmon
ic i
mag
ing
tech
niqu
es,
be d
etec
ted
in t
he p
rese
nce
of a
wea
ker
nonl
inea
r ti
ssue
re
spon
se.
The
sec
ond
grou
p o
f con
tras
t im
agin
g te
chni
ques
can
be
grou
ped
into
wha
t may
be
call
ed
mul
tipl
e pu
lse
met
hods
whe
re a
t lea
st tw
o pu
lses
are
tra
nsm
itte
d do
wn
each
line
of s
ight
. T
he i
mag
e re
cons
truc
tion
is
then
bas
ed o
n co
mbi
nati
ons
of
the
resu
ltin
g ec
hoes
alo
ng
each
line
of s
ight
.
Pu
lse
Inve
rsio
n T
echn
ique
s
In i
ts s
impl
est
embo
dim
ent,
the
puls
e in
vers
ion
tech
niqu
e co
nsis
ts o
f tr
ansm
itti
ng t
wo
puls
es,
whe
re t
he s
econ
d pu
lse
is a
rep
lica
of
the
firs
t pu
lse
but
wit
h in
vert
ed p
olar
ity
, w
ith
a re
lativ
e tim
e de
lay
so t
hat
the
resu
ltin
g tw
o ec
hoes
are
sep
arat
ed.
The
tw
o ec
hoes
are
the
n ad
ded
toge
ther
and
the
im
age
is b
ased
on
this
sum
mat
ion
sign
al.
In t
he
idea
l cas
e, o
dd h
arm
onic
com
pone
nts
in th
e re
sult
ing
sign
al, i
n pa
rtic
ular
the
fund
amen
tal
com
pone
nt,
are
canc
eled
whi
le e
ven
harm
onic
com
pone
nts
pers
ist.
The
pul
se i
nver
sion
te
chni
que
henc
e tu
rns
out
as a
n al
tern
ativ
e w
ay o
f do
ing
seco
nd h
arm
onic
im
agin
g. T
he
mai
n ad
vant
age
rela
tive
to t
he s
impl
e se
cond
har
mon
ic i
mag
ing
tech
niqu
e is
red
ucti
on
of le
akag
e fr
om t
he f
unda
men
tal
band
into
the
sec
ond
harm
onic
ban
d, t
hus
allo
win
g fo
r m
ore
broa
dban
d tr
ansm
it p
ulse
s. T
he m
ain
disa
dvan
tage
is a
rtif
acts
res
ulti
ng f
rom
tis
sue
mot
ion
betw
een
the
two
tran
smit
ted
puls
es.
A c
ombi
ned
puls
e in
vers
ion
and
Dop
pler
te
chni
que
has
been
stu
died
by
Sim
pson
et a
l [3
6].
1.3
Ove
rvie
w o
f the
Th
esis
7
Pow
er M
odu
lati
on T
echn
ique
s
If tw
o tr
ansm
it p
ulse
s w
ith
diff
eren
t am
plit
udes
are
tra
nsm
itte
d, t
he l
inea
r co
mbi
nati
on
of
the
two
resu
ltin
g ec
hoes
can
be
used
for
bub
ble
dete
ctio
n. If
the
tiss
ue r
espo
nse
is
clos
e to
lin
ear,
it c
an b
e st
rong
ly s
uppr
esse
d in
the
lin
ear
com
bina
tion
of t
he t
wo
echo
es
and
mai
nly
the
nonl
inea
r co
ntra
st e
cho
rem
ains
. A
lso,
the
fun
dam
enta
l co
mpo
nent
of
the
cont
rast
ech
o in
this
line
ar c
ombi
nati
on is
, alt
houg
h si
gnif
ican
tly
redu
ced,
usu
ally
not
ca
ncel
ed.
Bu
bb
le D
estr
uct
ion
Met
hods
If su
bjec
t to
high
inte
nsit
y dr
ive
puls
es th
e co
ntra
st b
ubbl
es, u
sual
ly e
ncap
sula
ted
in a
thin
st
abil
izin
g sh
ell,
tend
to
get d
estr
ucte
d du
e to
a r
uptu
re o
f th
e sh
ell.
Thi
s ru
ptur
e re
sult
s in
fra
gmen
tati
on o
f the
bub
ble
into
sm
alle
r bub
bles
and
/or
diff
usio
n o
f the
enc
apsu
lati
ng
gas.
Mec
hani
sms
of c
ontr
ast a
gent
des
truc
tion
are
stu
died
by
Cho
mas
et a
l [9]
. S
uch
bub
ble
dest
ruct
ion
wil
l al
ter
the
acou
stic
sca
tter
ing
prop
erti
es o
f th
e co
ntra
st a
gent
. P
ower
D
oppl
er t
echn
ique
s us
e pu
lse-
to-p
ulse
dec
orre
lati
on i
n co
ntra
st a
gent
ech
oes,
cau
sed
by
bubb
le d
isru
ptio
n, t
o di
stin
guis
h be
twee
n co
ntra
st a
gent
and
tis
sue
usin
g D
oppl
er p
ro
cess
ing
tech
niqu
es.
Kir
khom
et a
l [2
2] s
ugge
sted
app
lyin
g a
rele
ase
burs
t to
rupt
ure
the
cont
rast
bub
bles
and
then
usi
ng d
ecor
rela
tion
met
hods
on
cont
rast
sig
nals
bef
ore
and
afte
r th
e re
leas
e bu
rst t
o de
tect
the
cont
rast
age
nt.
1.3
Ove
rvie
w o
f the
The
sis
Thi
s th
esis
is
mad
e up
of
four
sep
arat
e pa
pers
. In
the
fir
st p
aper
, th
e ef
fect
of
the
seco
nd
harm
onic
com
pone
nt,
intr
oduc
ed d
ue to
the
non
line
arit
y o
f ult
raso
und
wav
e pr
opag
atio
n in
sof
t tis
sue,
in
the
wav
e fi
eld
inci
dent
to
the
cont
rast
age
nt,
is i
nves
tigat
ed.
The
sec
on
d pa
per
inve
stig
ates
a n
ew t
hird
har
mon
ic c
ontr
ast a
gent
det
ecti
on te
chni
que,
app
lyin
g a
puls
e co
mpr
essi
on m
etho
d fa
mil
iar
in r
adar
sys
tem
s an
d co
mm
unic
atio
n th
eory
. T
he
thir
d pa
per
prop
oses
a n
ew t
hird
or
four
th h
arm
onic
con
tras
t ag
ent
dete
ctio
n te
chni
que,
us
ing
dual
fre
quen
cy b
and
tran
smit
pul
ses
and
a ge
nera
l fo
rm o
f pu
lse
inve
rsio
n.
And
fi
nally
, th
e fo
urth
pap
er p
ropo
ses
a ne
w d
etec
tion
tec
hniq
ue u
tili
zing
the
tot
al s
catt
ered
co
ntra
st s
igna
l fo
r im
age
reco
nstr
ucti
on,
henc
e ov
erco
min
g pr
oble
ms
in r
elat
ion
to h
ar
mon
ic i
mag
ing
met
hods
. T
his
last
met
hod
is m
ainl
y a
line
ar c
ontr
ast
agen
t de
tect
ion
tech
niqu
e.
The
con
tent
of t
he f
our
pape
rs is
sum
mar
ized
bel
ow.
8 In
trod
ucti
on
Pap
er A
Red
ucti
on o
f N
onli
near
Con
tras
t Age
nt
Scat
teri
ng d
ue t
o N
onli
near
Inc
iden
t W
ave
Pro
paga
tion
Ult
raso
und
wav
e pr
opag
atio
n in
tis
sue
and
scat
teri
ng f
rom
ult
raso
und
cont
rast
age
nts
are
both
kno
wn
to b
e no
nlin
ear
proc
esse
s at
typ
ical
fre
quen
cies
and
am
plit
udes
use
d in
med
ic
al u
ltra
soun
d im
agin
g. T
he n
onli
near
ity
of w
ave
prop
agat
ion
man
ifes
ts i
tsel
f mai
nly
as
a se
cond
har
mon
ic c
ompo
nent
whi
ch,
due
to d
iffr
acti
on,
wil
l hav
e a
vary
ing
phas
e an
gle
rela
tive
to t
he l
inea
r fu
ndam
enta
l com
pone
nt in
a f
ocus
ed b
eam
com
mon
ly u
sed
in m
edi
cal u
ltra
soun
d im
agin
g. B
ased
on
num
eric
al s
imul
atio
ns, t
his
pape
r sho
ws
that
, dep
endi
ng
on t
he r
elat
ive
phas
e an
gle
betw
een
the
fund
amen
tal
and
seco
nd h
arm
onic
com
pone
nt o
f th
e w
ave
fiel
d in
cide
nt to
the
con
tras
t ag
ent,
nonl
inea
r co
ntra
st a
gent
sca
tter
ing
may
be
sign
ific
antl
y di
min
ishe
d du
e to
the
inci
dent
pul
se d
isto
rtio
n ca
used
by
the
nonl
inea
rity
of
wav
e pr
opag
atio
n.
Pap
erB
Usi
ng
Bar
ker
Cod
es in
Con
tras
t Har
mon
ic I
mag
ing
Ult
raso
und
wav
e pr
opag
atio
n is
gen
eral
ly a
wea
k no
nlin
ear p
roce
ss r
elat
ive
to t
he n
onli
nea
rity
of u
ltra
soun
d sc
atte
ring
from
con
tras
t age
nts.
Thi
s di
ffer
ence
in d
egre
e o
f non
line
ar
resp
onse
mak
es h
ighe
r ha
rmon
ic i
mag
ing
tech
niqu
es i
nter
esti
ng.
A n
onli
near
gen
erat
ed
harm
onic
ban
d o
f the
fun
dam
enta
l tra
nsm
itte
d ba
nd is
the
n us
ed f
or d
etec
ting
the
cont
rast
ag
ent s
igna
l an
d di
ffer
enti
atin
g it
fro
m t
he t
issu
e si
gnal
. R
ecei
ved
harm
onic
com
pone
nts
typi
call
y co
ntai
n le
ss e
nerg
y th
an t
he l
inea
rly
rece
ived
fun
dam
enta
l co
mpo
nent
and
, in
co
ntra
st h
arm
onic
imag
ing
tech
niqu
es, t
he li
mit
ing
fact
or i
s of
ten
the
nois
e si
gnal
alw
ays
pres
ent
in a
pul
se e
cho
imag
ing
syst
em a
nd n
ot th
e m
aski
ng o
f the
con
tras
t si
gnal
by
the
tiss
ue s
igna
l. P
ulse
com
pres
sion
tec
hniq
ues,
fam
ilia
r in
rad
ar s
yste
ms
and
com
mun
ica
tion
theo
ry,
are
tech
niqu
es f
or i
ncre
asin
g th
e si
gnal
leve
l re
lati
ve to
the
noi
se le
vel o
f th
e pu
lse
echo
im
agin
g sy
stem
whi
ch is
ass
umed
to b
e ev
enly
dis
trib
uted
ove
r al
l fre
quen
cies
o
f in
tere
st.
The
sig
nal
leve
l is
inc
reas
ed b
y tr
ansm
itti
ng a
n el
onga
ted
puls
e an
d no
t by
in
crea
sing
the
tra
nsm
it a
mpl
itud
e. T
he r
esul
ting
ech
oes
mus
t th
en b
e co
mpr
esse
d to
re
stor
e ra
nge
reso
luti
on.
Thi
s pa
per
inve
stig
ates
the
use
of
Bar
ker
code
s, w
hich
are
a t
ype
of p
ulse
com
pres
sion
cod
es,
and
thei
r pot
enti
al to
inc
reas
e th
e si
gnal
leve
l of t
he r
ecei
ved
thir
d ha
rmon
ic c
ompo
nent
from
con
tras
t bub
bles
rel
ativ
e to
the
con
stan
t noi
se le
vel.
1.3
Ove
rvie
w o
f the
The
sis
9
Pap
er C
A N
ew D
ual F
requ
ency
Ban
d C
on
tras
t Age
nt D
etec
tion
Tec
hniq
ue
The
fac
t th
at th
e co
ntra
st a
gent
s re
spon
d m
uch
mor
e no
nlin
earl
y th
an s
oft
tiss
ue to
ultr
aso
und
puls
es h
as g
iven
rise
to t
he c
ontr
ast h
arm
onic
imag
ing
tech
niqu
es w
here
a h
arm
onic
co
mpo
nent
of t
he to
tal s
catt
ered
sig
nal,
typi
call
y th
e se
cond
har
mon
ic c
ompo
nent
, is
used
fo
r im
age
reco
nstr
ucti
on.
In a
med
ical
ult
raso
und
imag
ing
situ
atio
n, b
oth
the
harm
onic
sc
atte
red
tissu
e si
gnal
acc
umul
atin
g in
the
for
war
d pr
opag
atio
n di
rect
ion
and
the
unco
rre
late
d th
erm
al a
nd e
lect
roni
c no
ise
sign
al w
ill p
oten
tial
ly m
ask
the
scat
tere
d co
ntra
st
harm
onic
sig
nal.
The
pre
sent
pap
er d
eals
wit
h a
new
thi
rd a
nd f
ourt
h ha
rmon
ic c
ontr
ast
agen
t im
agin
g te
chni
que,
des
igne
d to
inc
reas
e th
e co
ntra
st h
arm
onic
sig
nal
rela
tive
to
both
the
noi
se s
igna
l as
wel
l as
the
har
mon
ic t
issu
e si
gnal
. In
ord
er to
ach
ieve
thi
s, t
he
new
met
hod
mak
es u
se o
f du
al f
requ
ency
ban
d tr
ansm
it p
ulse
s, t
oget
her
wit
h a
gene
ral
puls
e in
vers
ion
tech
niqu
e.
Pap
erD
Lin
ear
Co
ntr
ast
Age
nt D
etec
tion
thr
ough
Low
Fre
quen
cy M
anip
ulat
ion
of
Hig
h F
requ
ency
Sca
tter
ing
Pro
pert
ies
In m
edic
al u
ltra
soun
d co
ntra
st h
arm
onic
det
ecti
on t
echn
ique
s, o
nly
a co
mpo
nent
of
the
tota
l sc
atte
red
cont
rast
sig
nal,
typi
call
y th
e se
cond
har
mon
ic c
ompo
nent
, is
uti
lize
d fo
r im
age
reco
nstr
ucti
on.
The
se h
arm
onic
det
ecti
on t
echn
ique
s m
ake
it p
ossi
ble
to d
iffe
ren
tiate
con
tras
t si
gnal
and
tis
sue
sign
al s
catt
ered
fro
m t
he p
art
of
the
body
bei
ng i
mag
ed,
and
as h
ighe
r ha
rmon
ic c
ompo
nent
s ar
e ut
ilize
d, t
his
diff
eren
tiat
ion
typi
call
y be
com
es
bette
r. A
ll pu
lse
echo
im
agin
g sy
stem
s ar
e, h
owev
er,
infe
sted
by
unw
ante
d th
erm
al a
nd
elec
tron
ic n
oise
whi
ch u
sual
ly c
an b
e co
nsid
ered
uni
form
ly d
istr
ibut
ed o
ver t
he f
requ
ency
ra
nge
of
inte
rest
. R
ecei
ved
harm
onic
com
pone
nts
are
typi
call
y re
duce
d in
am
plit
ude
as
high
er c
ompo
nent
s ar
e co
nsid
ered
, an
d al
thou
gh th
e di
ffer
enti
atio
n o
f con
tras
t sig
nal a
nd
tiss
ue s
igna
l m
ight
be
exce
llen
t app
lyin
g th
ese
high
er h
arm
onic
s, t
he c
ontr
ast s
igna
l w
ill
be m
aske
d by
the
noi
se s
igna
l. H
arm
onic
im
agin
g te
chni
ques
als
o re
quir
e ap
plic
atio
n of
re
lativ
ely
narr
ow b
ande
d tr
ansm
it p
ulse
s th
us d
egra
ding
ran
ge r
esol
utio
n in
the
ult
raso
und
imag
e.
Ano
ther
pro
blem
wit
h al
l co
ntra
st h
arm
onic
det
ecti
on t
echn
ique
s is
a s
prea
d of
co
ntra
st s
igna
l be
yond
the
act
ual
cont
rast
fill
ed r
egio
n.
The
non
line
ar p
art
of
the
con
tras
t sig
nal s
catt
ered
in th
e fo
rwar
d pr
opag
atio
n di
rect
ion
adds
in
phas
e w
ith
the
tran
smit
pu
lse
and
may
the
n be
lin
earl
y ba
ck-s
catt
ered
fro
m t
he t
issu
e.
The
pre
sent
pap
er p
ro
pose
s a
met
hod
appl
ying
the
tota
l sca
tter
ed c
ontr
ast s
igna
l for
im
age
reco
nstr
ucti
on,
thus
la
rgel
y ov
erco
min
g th
e pr
oble
ms
enco
unte
red
in h
arm
onic
im
agin
g te
chni
ques
. In
the
ne
w m
etho
d, t
he c
ontr
ast
sign
al a
nd t
issu
e si
gnal
are
dif
fere
ntia
ted
appl
ying
a s
impl
e pu
lse
subt
ract
ion
tech
niqu
e w
hich
can
cels
or
sign
ific
antly
red
uces
the
sca
tter
ed t
issu
e si
gnal
. T
he s
catt
ered
con
tras
t ag
ent
sign
al i
s, h
owev
er,
pres
erve
d in
thi
s pr
oces
s du
e to
10
Intr
odu
ctio
n
tran
smit
ted
low
fre
quen
cy p
ulse
s al
teri
ng th
e ac
oust
ic s
catt
erin
g pr
oper
ties
of t
he c
ontr
ast
agen
t in
a h
igh
freq
uenc
y ra
nge
used
for
im
age
reco
nstr
ucti
on.
The
mai
n m
echa
nism
th
roug
h w
hich
thi
s im
agin
g te
chni
que
sele
cts
the
cont
rast
age
nt s
igna
l is
the
lin
ear
reso
na
nt p
rope
rtie
s of
the
cont
rast
bub
ble.
Ch
apte
r 2
Red
uct
ion
of N
onli
nea
r C
ontr
ast
Age
nt
Sca
tter
ing
du
e to
Non
lin
ear
Inci
den
t W
ave
Pro
pag
atio
n
Abs
trac
t
Ult
raso
und
wav
e pr
opag
atio
n in
tis
sue
and
scat
teri
ng f
rom
ult
raso
und
cont
rast
age
nts
are
both
kno
wn
to b
e no
nlin
ear
proc
esse
s at
typ
ical
fre
quen
cies
and
am
plit
udes
use
d in
med
ic
al u
ltra
soun
d im
agin
g. T
he n
onli
near
ity
of
wav
e pr
opag
atio
n m
anif
ests
its
elf m
ainl
y as
a
seco
nd h
arm
onic
com
pone
nt w
hich
, du
e to
dif
frac
tion
, w
ill h
ave
a va
ryin
g ph
ase
an
gle
rela
tive
to
the
line
ar f
unda
men
tal
com
pone
nt i
n a
focu
sed
tran
smit
bea
m c
omm
only
us
ed i
n m
edic
al u
ltra
soun
d im
agin
g. B
ased
on
num
eric
al s
imul
atio
ns,
this
pap
er s
how
s th
at, d
epen
ding
on
the
rela
tive
pha
se a
ngle
bet
wee
n th
e fu
ndam
enta
l and
sec
ond
harm
onic
co
mpo
nent
of t
he w
ave
fiel
d in
cide
nt to
the
con
tras
t age
nt,
nonl
inea
r co
ntra
st a
gent
sca
tte
ring
may
be
sign
ific
antl
y di
min
ishe
d du
e to
the
inc
iden
t pu
lse
dist
orti
on c
ause
d by
the
no
nlin
eari
ty o
f wav
e pr
opag
atio
n.
2.1
Intr
oduc
tion
Wav
e pr
opag
atio
n in
tis
sue
is u
sual
ly a
wea
k no
nlin
ear
proc
ess
at f
requ
enci
es a
nd a
m
plit
udes
typ
ical
ly u
sed
in m
edic
al u
ltra
soun
d im
agin
g, a
nd t
he t
rans
mit
pul
se i
s sl
ight
ly
dist
orte
d as
it
prop
agat
es t
hrou
gh t
he m
ediu
m.
Alt
houg
h th
e lo
cal
effe
ct o
f no
nlin
eari
ty
is s
mal
l, th
e cu
mul
ativ
e ef
fect
whe
n th
e w
ave
has
prop
agat
ed s
ever
al w
avel
engt
hs i
s no
t ne
glig
ible
. If
the
wav
e tr
ansm
itte
d fr
om a
n ul
tras
ound
tra
nsdu
cer
has
its e
nerg
y co
ncen
tr
ated
in
som
e fu
ndam
enta
l fr
eque
ncy
band
, th
e no
nlin
eari
ty o
f w
ave
prop
agat
ion
give
s ri
se to
har
mon
ics
of th
is f
unda
men
tal b
and
[3, C
hapt
er 1
2] [
30, C
hapt
er 1
1].
Lev
els
of r
e-
12
Pap
er A
ceiv
ed s
econ
d ha
rmon
ic f
rom
tis
sue
is t
ypic
ally
fou
nd to
be
arou
nd 2
0 dB
bel
ow th
e le
vel
of th
e re
ceiv
ed f
unda
men
tal
com
pone
nt b
ut th
is l
evel
dep
ends
on
acou
stic
par
amet
ers
in
addi
tion
to i
mag
ing
para
met
ers
such
as
freq
uenc
y, a
mpl
itud
e, a
nd d
epth
of i
mag
ing.
The
se
cond
har
mon
ic c
ompo
nent
acc
umul
ates
gra
dual
ly a
s th
e w
ave
prop
agat
es a
nd w
ill,
due
to d
iffr
acti
on,
have
a v
aryi
ng p
hase
ang
le r
elat
ive
to t
he f
unda
men
tal
com
pone
nt in
a f
ocu
sed
tran
smit
ted
ultr
asou
nd b
eam
. T
his
phas
e an
gle
wil
l be
a re
lativ
ely
com
plex
func
tion
o
f bo
th a
xial
and
lat
eral
pos
itio
n re
lati
ve t
o th
e ul
tras
ound
bea
m a
xis
[3,
Cha
pter
12.
6].
Alt
houg
h m
uch
wea
ker t
han
the
seco
nd h
arm
onic
com
pone
nt, h
ighe
r ha
rmon
ics
may
als
o ex
ist i
n th
e tr
ansm
it fi
eld.
Med
ical
ult
raso
und
cont
rast
age
nts
are
typi
call
y m
ade
as s
olut
ions
of
smal
l ga
s bu
bble
s (d
iam
rv 3
/-lm
) in
a f
luid
and
sca
tter
ing
from
suc
h ga
s bu
bble
s is
typ
ical
ly a
str
ong
non
line
ar p
roce
ss [
23]
[11]
[12
]. A
s in
dica
ted,
the
wav
e fi
eld
inci
dent
to t
he c
ontr
ast
agen
t co
ntai
ns e
nerg
y in
a s
econ
d ha
rmon
ic b
and
of
the
fund
amen
tal
band
tra
nsm
itte
d fr
om
the
ultr
asou
nd tr
ansd
ucer
due
to t
he n
onli
near
tiss
ue e
last
icity
. D
epen
ding
on
the
rela
tive
ph
ase
angl
e be
twee
n th
e in
cide
nt f
unda
men
tal
and
seco
nd h
arm
onic
ban
d, t
he p
rese
nce
of
this
sec
ond
harm
onic
ban
d is
in
the
pres
ent
pape
r sh
own
to p
oten
tial
ly h
ave
a m
ajor
di
min
ishi
ng e
ffec
t on
the
nonl
inea
r res
pons
e fr
om a
con
tras
t bub
ble.
2.2
The
ory
2.2.
1 Si
ngle
Bub
ble
Osc
illat
ion
The
con
tras
t age
nt is
ass
umed
to b
e sp
heri
cal g
as b
ubbl
es e
ncap
sula
ted
in a
thin
she
ll.
The
di
amet
er o
f the
bub
ble
is m
uch
less
than
the
wav
elen
gth
of th
e in
com
ing
wav
e fi
eld
and
the
bubb
le th
us e
xper
ienc
es a
n ap
prox
imat
ely
unif
orm
spa
tial
fie
ld a
nd t
he b
ubbl
e os
cill
atio
n is
ass
umed
to b
e pu
rely
sph
eric
al.
Sim
ulat
ions
for
bubb
le r
adiu
s os
cill
atio
ns a
nd a
cous
tic
scat
teri
ng a
re d
one
usin
g th
e nu
mer
ical
mod
el d
evel
oped
by
Ang
else
n et
al
[4].
Thi
s m
odel
inc
lude
s an
equ
atio
n fo
r th
e re
lati
on b
etw
een
pres
sure
and
rad
ial
stra
in i
n a
thin
sh
ell
enca
psul
atin
g a
gas
bubb
le.
The
mod
el a
llow
s fo
r a
fini
te s
peed
of
soun
d in
the
m
ediu
m s
urro
undi
ng th
e bu
bble
, thu
s ta
king
radi
atio
n lo
sses
from
the
bubb
le in
to a
ccou
nt.
Oth
erw
ise
it is
com
para
ble
to t
he w
ell k
now
n R
ayle
igh-
Ple
sset
equ
atio
n [3
3] [
31]
and
the
two
mod
els
give
sim
ilar
resu
lts
for
inci
dent
pre
ssur
e pu
lses
and
bub
ble
para
met
ers
stud
ied
in th
is p
aper
.
The
Ray
leig
h-P
less
et e
quat
ion
is a
sec
ond
orde
r non
line
ar d
iffe
rent
ial e
quat
ion.
For
sm
all
ampl
itud
es o
f th
e in
cide
nt d
rive
pre
ssur
e, t
he b
ubbl
e os
cill
atio
n ca
n be
ass
umed
to
be
appr
oxim
atel
y li
near
and
we
have
the
fol
low
ing
seco
nd o
rder
line
ar d
iffe
rent
ial
equa
tion
fo
r th
e ra
dial
dis
plac
emen
t, '1/J
, aro
und
an e
quil
ibri
um ra
dius
a
(2.1
)
Her
e, m
is
the
iner
tia
of
the
syst
em,
b is
the
dam
ping
fac
tor
of th
e sy
stem
, an
d s
is t
he
2.2
The
ory
13
20
I
\
10
i:Q ~
0
-10
0 0.
2 0.
4 0.
6 0.
8 1
1.2
1.4
1.6
1.8
2 [Q
]
4 3
12
\
\ .!
::, 0
----
--0
0.2
0.4
0.6
0.8
1 1.
2 1.
4 1.
6 1.
8 2
[Q]
Fig
ure
2.1:
Tra
nsfe
r fu
nctio
n fr
om d
rive
pre
ssur
e to
rad
ial
disp
lace
men
t in
Eq.
2.5
. T
he
para
met
er d
in E
q. 2
.3 i
s se
t to
0.5
and
0.1
giv
ing
the
soli
d lin
e an
d da
shed
line
, re
spec
tiv
ely.
U
pper
pan
el:
Abs
olut
e va
lue
of
tran
sfer
fun
ctio
n.
Low
er p
anel
: P
hase
ang
le o
f tr
ansf
er fu
nctio
n.
stif
fnes
s o
f the
gas
and
enc
apsu
lati
ng b
ubbl
e sh
ell.
Eq.
2.1
typ
ical
ly d
escr
ibes
the
for
ced
line
ar o
scil
lati
on o
f a
syst
em c
onsi
stin
g of
a m
ass
m a
ttac
hed
on a
spr
ing
wit
h st
iffn
ess
s w
here
as b
acc
ount
s fo
r th
e da
mpi
ng i
n th
e sy
stem
. B
y ta
king
the
Fou
rier
Tra
nsfo
rm o
f E
q. 2
.1 w
e ob
tain
whe
re
d=
-b
-2
s W
o =
-,
m
Wom
Rea
rran
ging
Eq.
2.2
we
obta
in
47ra
2
1/J(
w) =
-H
(rl)
p;(
w)
s
whe
re t
he tr
ansf
er f
unct
ion
from
dri
ve p
ress
ure
to r
adia
l dis
plac
emen
t is
1 H
(rl)
=
rl2
-1
-ir
ld
(2.2
)
(2.3
)
(2.4
)
(2.5
)
and
whe
re t
he a
bsol
ute
valu
e an
d ph
ase
angl
e o
f H
(rl)
are
sho
wn
in F
ig.
2.1.
In
the
lo
wer
pan
el o
f th
is f
igur
e, w
e se
e th
at f
or d
rive
fre
quen
cies
wel
l be
low
res
onan
ce t
he
14
Pap
er A
disp
lace
men
t is
1r
out
of p
hase
with
the
dri
ving
pre
ssur
e. T
his
mea
ns t
hat
the
bubb
le i
s ex
pand
ed a
nd i
ncre
ased
in
size
dur
ing
the
nega
tive
pre
ssur
e cy
cle
whi
le c
ompr
esse
d an
d re
duce
d in
siz
e du
ring
the
pos
itiv
e pr
essu
re c
ycle
. F
or f
requ
enci
es w
ell
abov
e re
sona
nce
the
bubb
le r
espo
nds
diff
eren
tly to
the
driv
e pr
essu
re.
The
dis
plac
emen
t and
dri
ve p
ress
ure
are
now
in
phas
e so
tha
t the
bub
ble
is i
ncre
ased
in s
ize
duri
ng th
e po
sitiv
e pr
essu
re c
ycle
an
d vi
ce v
ersa
. A
roun
d re
sona
nce
the
disp
lace
men
t is
appr
oxim
atel
y ~
out o
f pha
se w
ith
the
driv
e pr
essu
re.
The
abs
olut
e va
lue
of t
he a
mpl
itud
e o
f the
tra
nsfe
r fu
ncti
on i
s se
en i
n th
e up
per p
anel
of t
he fi
gure
. G
oing
fro
m f
requ
enci
es b
elow
res
onan
ce to
war
ds r
eson
ance
th
e am
plit
ude
incr
ease
s gr
adua
lly c
ulm
inat
ing
wit
h a
prom
inen
t pe
ak a
roun
d re
sona
nce
for
the
situ
atio
n w
ith
low
dam
ping
(d
=
0.1)
and
a c
onsi
dera
ble
smal
ler
peak
for
the
si
tuat
ion
wit
h hi
gher
dam
ping
(d
=
0.5)
. In
bot
h ca
ses,
the
am
plitu
de is
see
n to
dec
reas
e ra
pidl
y ab
ove
reso
nanc
e.
2.2.
2 Se
cond
Har
mon
ic C
ompo
nent
in T
rans
mit
Fie
ld
Wav
e pr
opag
atio
n fr
om a
foc
used
ult
raso
und
tran
sduc
er to
the
con
tras
t-fi
lled
reg
ion
be
ing
imag
ed i
s no
nlin
ear
beca
use
the
tiss
ue e
last
icit
y re
spon
ds s
ligh
tly
nonl
inea
rly
whe
n su
bjec
t to
an
osci
llat
ing
tran
smit
pul
se.
The
wav
e in
cide
nt t
o th
e co
ntra
st a
gent
wil
l th
eref
ore
cont
ain
seco
nd a
nd p
ossi
bly
high
er h
arm
onic
com
pone
nts.
In
this
con
text
the
re
lativ
e ph
ase
angl
e be
twee
n th
e in
cide
nt f
unda
men
tal
and
seco
nd h
arm
onic
com
pone
nt
is o
f sp
ecia
l im
port
ance
. T
his
phas
e an
gle
is i
n th
e pr
esen
t pa
per
give
n as
a f
ract
ion
of
the
tem
pora
l pe
riod
of
the
seco
nd h
arm
onic
com
pone
nt.
Whe
n tr
ying
to
dete
rmin
e th
e re
lativ
e ph
ase
angl
e be
twee
n th
e fu
ndam
enta
l an
d se
cond
har
mon
ic c
ompo
nent
, w
e w
ill
here
mai
nly
be c
once
rned
with
pul
ses
whe
re t
he l
evel
of
the
seco
nd h
arm
onic
is
arou
nd
20 d
B b
elow
the
fund
amen
tal c
ompo
nent
and
whe
re le
vels
of h
ighe
r har
mon
ics
are
so lo
w
that
the
se c
an b
e ne
glec
ted.
The
men
tion
ed p
hase
ang
le i
s th
en f
ound
app
roxi
mat
ely
by
visu
al i
nspe
ctio
n of
the
puls
es i
n th
e ti
me
dom
ain.
We
defi
ne t
he i
ndic
ated
pha
se a
ngle
to
zero
whe
n th
e ze
ro-c
ross
ings
of
the
fund
amen
tal
pres
sure
com
pone
nt c
oinc
ide
wit
h ev
ery
seco
nd z
ero-
cros
sing
of
the
seco
nd h
arm
onic
pr
essu
re c
ompo
nent
. W
ith t
his
defi
nitio
n, z
ero
phas
e an
gle
give
s a
saw
-too
th s
hape
d pu
lse
as s
how
n in
the
upp
er p
anel
of F
ig.
2.2
whe
re t
he c
ompr
essi
on p
erio
d, g
oing
fro
m
posi
tive
to n
egat
ive
valu
es,
is l
ess
stee
p th
an t
he e
xpan
sion
per
iod,
goi
ng f
rom
neg
ativ
e to
pos
itiv
e va
lues
. A
pha
se a
ngle
of-~, s
how
n in
the
low
er p
anel
of
Fig.
2.2
, gi
ves
a pu
lse
wit
h sh
arpe
ned
cres
ts a
nd r
ound
ed t
roug
hs w
here
mag
nitu
des
of c
rest
s ar
e la
rger
th
an m
agni
tude
s o
f tr
ough
s. T
he a
ctua
l ph
ase
angl
e in
the
tra
nsm
it fi
eld
from
a f
ocus
ed
med
ical
ult
raso
und
tran
sduc
er is
a r
esul
t of w
ave
diff
ract
ion.
Dep
endi
ng o
n th
e ax
ial
and
late
ral
posi
tion
rel
ativ
e to
the
ult
raso
und
beam
axi
s, t
his
phas
e an
gle
is u
sual
ly f
ound
to
be s
omew
here
bet
wee
n th
e tw
o in
dica
ted
case
s in
Fig
. 2.
2 [3
, C
hapt
er 1
2.6]
. W
e w
ill
in
the
pres
ent p
aper
fro
m n
ow o
n re
fer
to t
his
phas
e an
gle
as <
P 12
•
2.2
Th
eory
0.5 0
1-----
-0.5
1.5
0.5 0
1-------
-0.5
1.5
2 2.
5 3
2
3.5
[J.ts
] 4
4.5
5 5
15
5.5
6
5.5
6
Fig
ure
2.2:
D
efin
ition
of
phas
e an
gle,
<P 1
2,
betw
een
inci
dent
fun
dam
enta
l an
d se
cond
ha
rmon
ic c
ompo
nent
. T
he s
um o
f a
fund
amen
tal
and
a se
cond
har
mon
ic c
ompo
nent
is
depi
cted
. U
pper
pan
el:
Phas
e an
gle,
<P 1
2,
betw
een
fund
amen
tal
and
seco
nd h
arm
onic
co
mpo
nent
is
zero
. L
ower
pan
el:
Phas
e an
gle,
<P 1
2,
betw
een
fund
amen
tal
and
seco
nd
harm
onic
com
pone
nt is
-i.
16
Pap
er A
2.3
Res
ults
2.3.
1 Si
mul
atio
n of
Tra
nsm
itte
d W
ave
Fie
ld
A n
onli
near
sim
ulat
ion
prog
ram
for
wav
e pr
opag
atio
n de
velo
ped
in o
ur g
roup
[41
], i
s us
ed t
o ca
lcul
ate
the
acou
stic
tra
nsm
it f
ield
. T
his
prog
ram
is
capa
ble
of
mak
ing
a 3-
dim
ensi
onal
sim
ulat
ion
of
the
acou
stic
tra
nsm
it f
ield
fro
m a
n an
nula
r tr
ansd
ucer
tak
ing
nonl
inea
r el
astic
ity, f
requ
ency
dep
ende
nt a
bsor
ptio
n, a
nd d
iffr
acti
on in
to a
ccou
nt.
The
sim
ulat
ed f
unda
men
tal
and
seco
nd h
arm
onic
tra
nsm
it f
ield
fro
m a
n an
nula
r tr
ans
duce
r is
dis
play
ed in
dec
ibel
sca
le in
the
left
and
rig
ht p
anel
of F
ig. 2
.3, r
espe
ctiv
ely.
The
ve
rtic
al a
xis
in t
he f
igur
e is
ran
ge d
irec
tion
whi
le t
he h
oriz
onta
l ax
is i
s th
e la
tera
l di
rec
tion.
T
he r
adiu
s of
the
annu
lar
tran
sduc
er is
1 e
m a
nd t
he g
eom
etri
cal
focu
s w
as s
et a
t 7
em w
hile
the
fun
dam
enta
l tr
ansm
it f
requ
ency
was
set
to 1
MH
z. A
cous
tic
para
met
ers
for
mus
cle
foun
d in
the
lite
ratu
re [
13]
wer
e us
ed i
n th
e si
mul
atio
n. F
rom
the
fund
amen
tal
fiel
d, s
how
n in
the
lef
t pan
el in
the
fig
ure,
it i
s se
en t
hat m
ost o
f th
e en
ergy
em
itte
d fr
om
the
tran
sduc
er f
ollo
ws
a ge
omet
rica
l con
e in
wha
t is
usu
ally
ref
erre
d to
as
the
near
-fie
ld.
The
aco
usti
c fi
eld
in th
is n
ear-
fiel
d is
, due
to i
nter
fere
nce
from
dif
fere
nt p
arts
of t
he tr
ans
duce
r, s
omew
hat i
rreg
ular
, as
can
be s
een
from
the
inte
nsit
y va
riat
ions
. In
the
regi
on f
rom
ab
out 4
em
to 8
em
, th
e en
ergy
is
conc
entr
ated
in a
nar
row
er r
egio
n an
d th
en s
low
ly d
ive
rges
in
the
far-
fiel
d. A
cous
tic
abso
rpti
on m
akes
the
am
plit
ude
in t
he f
ar-f
ield
fal
l m
ore
rapi
dly
than
the
rela
tive
ly w
eak
dive
rgin
g ef
fect
of t
he f
ield
wou
ld s
ugge
st.
Fro
m th
e ri
ght
pane
l in
the
figu
re,
the
seco
nd h
arm
onic
com
pone
nt is
see
n to
be
negl
igib
le d
own
to a
bout
2
em.
The
gen
erat
ed s
econ
d ha
rmon
ic c
ompo
nent
is s
omew
hat b
ette
r foc
used
in
the
foca
l re
gion
and
als
o m
ore
coll
imat
ed i
n th
e fa
r-fi
eld
rela
tive
to
the
fund
amen
tal
com
pone
nt.
The
se r
esul
ts a
re i
n go
od a
gree
men
t wit
h an
alyt
ical
con
side
rati
ons
[3,
Cha
pter
12.
6].
2.3.
2 Si
mul
atio
ns o
f Bub
ble
Osc
illat
ion
Num
eric
al s
imul
atio
ns o
f bub
ble
osci
llat
ions
are
don
e us
ing
a si
ngle
bub
ble
wit
h ac
oust
ic
prop
erti
es c
ompa
rabl
e to
the
con
tras
t ag
ent
Sona
zoid
[19
]. T
his
agen
t co
nsis
ts o
f bu
bbl
es c
onta
inin
g pe
rflu
orca
rbon
gas
enc
apsu
late
d in
a th
in s
urfa
ctan
t mem
bran
e. T
he b
ulk
mod
ulus
of
a ty
pica
l bu
bble
is a
roun
d 60
0 kP
a w
hich
is a
bout
6 t
imes
the
sti
ffne
ss o
f a
free
gas
bub
ble
[19]
. T
he r
eson
ance
fre
quen
cy o
f th
e bu
bble
use
d in
num
eric
al s
imul
ati
ons
is a
roun
d 4
MH
z. S
imul
atio
ns f
or b
ubbl
e ra
dius
osc
illa
tion
s an
d ac
oust
ic s
catt
erin
g ar
e do
ne u
sing
the
num
eric
al m
odel
dev
elop
ed b
y A
ngel
sen
et a
l [4
]. A
s m
enti
oned
, th
is
mod
el a
nd t
he w
ell
know
n R
ayle
igh-
Ple
sset
equ
atio
n [3
3] [
31]
give
sim
ilar
res
ults
for
dr
ive
pres
sure
s an
d bu
bble
par
amet
ers
stud
ied
in t
he p
rese
nt p
aper
.
Firs
t, th
e bu
bble
is d
rive
n in
to s
mal
l ra
dius
osc
illa
tion
s so
tha
t a
clos
e to
lin
ear
resp
onse
ca
n be
ass
umed
. Fi
g. 2
.4(a
) de
pict
s th
e ra
dius
res
pons
e fr
om t
he b
ubbl
e (l
ower
pan
el)
whe
n dr
iven
wel
l be
low
res
onan
ce b
y a
0.5
MH
z pr
essu
re p
ulse
(up
per
pane
l).
We
see
2.3
Res
ults
17
-20
-5
-25
-10
-30
I -1
5 !
-35
-20
-40
-25
-45
-30
-50
-1
0 -1
0
(em
] (e
m]
Figu
re 2
.3:
Sim
ulat
ed t
rans
mit
fie
ld i
n de
cibe
l sc
ale
from
an
annu
lar
tran
sduc
er w
ith
radi
us e
qual
to
1 em
and
geo
met
ric
focu
s at
7 e
m.
Aco
ustic
pro
pert
ies
foun
d in
the
li
tera
ture
for
mus
cle
is u
sed
[13]
. L
eft
pane
l: F
unda
men
tal
tran
smit
fie
ld.
Rig
ht p
anel
: S
econ
d ha
rmon
ic t
rans
mit
fie
ld.
18
Pap
er A
that
the
bubb
le r
adiu
s is
1r ou
t of p
hase
wit
h th
e dr
ive
pres
sure
whi
ch is
in
agre
emen
t wit
h ou
r li
near
con
side
rati
ons
that
led
to F
ig.
2.1.
By
chan
ging
the
fre
quen
cy o
f ou
r dr
ive
pres
sure
to
4 M
Hz
we
driv
e th
e bu
bble
at
res
onan
ce a
nd t
he r
adiu
s re
spon
se a
nd d
rive
pre
ssur
e ar
e di
spla
yed
in t
he l
ower
and
upp
er
pane
l of
Fig
. 2.
4(b)
, re
spec
tivel
y.
The
pha
se o
f th
e ra
dius
res
pons
e re
lativ
e to
the
dri
ve
pres
sure
has
now
cha
nged
to ~
as
expe
cted
fro
m F
ig.
2.1.
We
also
not
ice
that
the
bubb
le
"rin
gs"
for
a sh
ort
peri
od o
f ti
me
afte
r th
e dr
ive
pres
sure
pul
se h
as e
nded
whe
n dr
iven
at
res
onan
ce.
Fina
lly,
in F
ig.
2.4(
c),
the
radi
us r
espo
nse
whe
n dr
ivin
g th
e bu
bble
wel
l ab
ove
reso
nanc
e by
a 1
0 M
Hz
pres
sure
pul
se c
an b
e se
en.
The
rad
ius
osci
llat
ion
is n
ow
alm
ost
in p
hase
with
the
dri
ve p
ress
ure
mea
ning
tha
t th
e bu
bble
is
expa
nded
dur
ing
the
posi
tive
pre
ssur
e cy
cle
and
com
pres
sed
duri
ng t
he n
egat
ive
pres
sure
cyc
le.
Aga
in,
this
is
in a
gree
men
t w
ith
the
line
ar r
esul
ts f
rom
Fig
. 2.
1.
Sim
ilar
sim
ulat
ion
are
then
per
form
ed w
ith
high
er a
mpl
itud
es o
f th
e in
cide
nt d
rive
pre
ssu
re s
o th
at th
e bu
bble
in e
ach
case
is
driv
en in
to n
onli
near
osc
illa
tion
s. I
n al
l ca
ses,
the
se
cond
har
mon
ic c
ompo
nent
in th
e ra
dius
res
pons
e is
abo
ut 2
0 dB
bel
ow th
e fu
ndam
enta
l co
mpo
nent
whe
reas
hig
her
harm
onic
com
pone
nts
are
negl
igib
le.
The
pha
se a
ngle
be
twee
n th
e fu
ndam
enta
l an
d se
cond
har
mon
ic c
ompo
nent
in th
e ra
dius
osc
illa
tion
is f
rom
no
w o
n re
fen·
ed to
as
812
and
the
sam
e de
fini
tions
as
for
<P 12
in
Fig.
2.2
are
use
d.
Fig.
2.5
(a)
show
s th
e si
tuat
ion
whe
n th
e bu
bble
is d
rive
n in
to n
onli
near
osc
illa
tion
s w
ell
belo
w r
eson
ance
. T
he f
unda
men
tal
com
pone
nt i
n th
e ra
dius
res
pons
e is
sti
ll a
ppro
xi
mat
ely
7r o
ut o
f pha
se w
ith
the
driv
e pr
essu
re.
By
com
pari
ng th
e lo
wer
pan
el o
f Fig
. 2.5
(a)
wit
h Fi
g. 2
.2,
the
rela
tive
phas
e an
gle
betw
een
the
fund
amen
tal
and
seco
nd h
arm
onic
co
mpo
nent
in t
he r
adiu
s re
spon
se,
812
, is
fou
nd t
o be
aro
und
-~.
The
bub
ble
is t
hen
driv
en i
nto
nonl
inea
r os
cill
atio
ns a
t re
sona
nce
and
the
radi
us r
espo
nse
and
driv
e pr
essu
re a
re s
how
n in
Fig
. 2.
5(b)
. A
s in
the
lin
ear
situ
atio
n, t
he f
unda
men
tal
com
pone
nt o
f th
e ra
dius
osc
illa
tion
is
appr
oxim
atel
y ~
out
of p
hase
with
the
dri
ve p
res
sure
. W
e se
e th
at,
due
to t
he s
econ
d ha
rmon
ic c
ompo
nent
in
the
radi
us o
scil
lati
on,
the
mag
nitu
des
of c
rest
s ar
e no
w l
ower
tha
n m
agni
tude
s of
trou
ghs
and
the
cres
ts h
ave
be
com
e ro
unde
d w
hile
the
tro
ughs
are
sha
rpen
ed.
Thi
s is
the
opp
osit
e of
wha
t fou
nd i
n th
e lo
wer
pan
el o
f Fi
g. 2
.5(a
).
The
rea
son
is t
hat
812
ha
s ch
ange
d fr
om -
~ to
-3 ;.
The
to
tal
phas
e sh
ift
on 8
12 is
thu
s -1
r w
hen
chan
ging
the
dri
ve f
requ
ency
fro
m w
ell
belo
w
reso
nanc
e to
res
onan
ce.
The
res
ult w
hen
the
bubb
le is
dri
ven
wel
l ab
ove
reso
nanc
e is
dep
icte
d in
Fig
. 2.
5(c)
. W
e se
e fr
om t
he u
pper
pan
el t
hat
a ve
ry h
igh
pres
sure
am
plit
ude
(rv3
MPa
) m
ust b
e us
ed i
n or
der
to d
rive
the
bub
ble
into
non
line
ar o
scil
lati
ons
whi
ch a
gree
s w
ith
the
uppe
r pa
nel
in F
ig.
2.1
indi
cati
ng t
hat
the
ampl
itud
e of
the
tran
sfer
fun
ctio
n fr
om d
rive
pre
ssur
e to
ra
dius
osc
illa
tion
falls
rap
idly
abo
ve r
eson
ance
. T
he fu
ndam
enta
l com
pone
nt o
f the
rad
ius
osci
llat
ion
is,
as i
n th
e li
near
sit
uati
on, i
n ph
ase
wit
h th
e dr
ive
pres
sure
. T
he p
hase
ang
le
812
is,
how
ever
, aga
in a
ppro
achi
ng a
val
ue c
lose
to -
3 ;.
2.3
Res
ults
(t!\
/\[j
~ 1!\/
\f ; ;
I 1
2 3
4 5
6 7
8 9
2 2.
2 2.
4 2.
6 2.
8 3.
2 3.
4 ~
~
·J?\/\1
\d ·~
61\/E
12
3
4 5
6
7 8
9 2
2.2
2.4
2.6
2.8
3
3.2
3.4
~
~
(a)
Bub
ble
driv
en w
ell
belo
w r
eson
ance
. (b
) B
ubbl
e dr
iven
at r
eson
ance
.
(c)
Bub
ble
driv
en w
ell
abov
e re
sona
nce.
Figu
re 2
.4:
Bub
ble
driv
en i
n cl
ose
to l
inea
r os
cilla
tions
. U
pper
pan
el in
sub
fi
gure
s:
Dri
ve p
ress
ure
puls
es.
Low
er p
anel
in
sub-
figu
res:
B
ubbl
e ra
dius
re
spon
ses.
19
20
Pap
er A
<M
/0 i~F
1/\f:
. : I
l 2
3 4
5 6
7 8
9 2
2.2
2.4
2.6
2.8
3.2
3.4
~
~
,l~\1\d ·JE
J\/6
1
2
3 4
56
7
8 9
2 2
.2
2.4
2.6
2.8
3
3.2
3.4
~
~
(a)
Bub
ble
driv
en w
ell
belo
w r
eson
ance
.
(c)
Bub
ble
driv
en w
ell
abov
e re
sona
nce.
(b)
Bub
ble
driv
en a
t res
onan
ce.
(d)
Bub
ble
driv
en a
t a
norm
aliz
ed f
requ
ency
, n,
equa
l to
0.5
.
Fig
ure
2.5:
Bub
ble
driv
en in
to n
onli
near
osc
illa
tion
s. S
econ
d ha
rmon
ic in
ra
dius
osc
illa
tion
is a
roun
d 20
dB
bel
ow f
unda
men
tal
com
pone
nt.
Upp
er p
anel
in
sub
-fig
ures
: D
rive
pre
ssur
e pu
lses
. L
ower
pan
el i
n su
b-fi
gure
s:
Bub
ble
radi
us r
espo
nses
.
2.3
Res
ults
,:LV\
/!\1 ~P
\/Sfl
2 2.
5 3
3.5
4 4.
5 5
5.5
6 1.
5 2
2.5
3 3.
5 4
4.5
5 5.
5 6
~
~
·jtlh
: .• j ·J
/~
0 0.
5 1
1.5
2 2.
5 3
3.5
4.5
5 0
0.5
1 1.
5 2
2.5
3 3.
5 4
4.5
5 [1
1Hz]
[M
Hz]
(a)
Rad
ius
osci
llat
ion
and
ampl
itud
e o
f its
fre
qu
ency
spe
ctru
m.
(b)
Sca
tter
ed p
ress
ure
puls
e an
d am
plit
ude
of
its
freq
uenc
y sp
ectr
um.
Fig
ure
2.6:
Bub
ble
driv
en b
y no
n-di
stor
ted
pres
sure
pul
se a
t n =
0.2
5.
21
Fig.
2.5
(d)
show
s th
e bu
bble
res
pons
e w
hen
the
bubb
le i
s dr
iven
at
a no
rmal
ized
fre
qu
ency
, n,
equ
al t
o 0.
5.
In t
his
case
812
is
clo
se t
o -Jr a
nd m
agni
tude
s of
cre
sts
are
appr
oxim
atel
y eq
ual
to m
agni
tude
s o
f the
tro
ughs
.
Whe
n va
ryin
g th
e dr
ive
freq
uenc
y fr
om w
ell
belo
w r
eson
ance
to
wel
l ab
ove
reso
nanc
e th
e ph
ase
on th
e fu
ndam
enta
l co
mpo
nent
of t
he r
adiu
s re
spon
se r
elat
ive
to t
he d
rive
pre
ssu
re c
hang
es f
rom
bei
ng 1
r ou
t o
f ph
ase
to b
eing
in
phas
e, r
espe
ctiv
ely.
Thi
s re
sult
was
ob
tain
ed b
oth
whe
n dr
ivin
g th
e bu
bble
in
clos
e to
lin
ear
osci
llat
ions
and
whe
n dr
ivin
g it
int
o no
nlin
ear
osci
llat
ions
and
Fig
. 2.
1 is
the
refo
re a
lso
a go
od a
ppro
xim
atio
n o
f th
e ph
ase
of t
he f
unda
men
tal
radi
us o
scil
lati
on r
elat
ive
to t
he d
rive
pre
ssur
e fo
r th
e no
nlin
ear
bubb
le.
The
rad
ius
phas
e an
gle
812
is
, ho
wev
er,
chan
ging
mor
e ra
pidl
y w
hen
vary
ing
the
driv
e fr
eque
ncy
from
-~
wel
l be
low
res
onan
ce t
o ap
prox
imat
ely
-3 ;
at r
eson
ance
. A
bove
res
onan
ce,
812
fi
rst
decr
ease
s to
war
ds -
Jr a
nd t
hen
incr
ease
s ag
ain
appr
oach
ing
-3 ;
wel
l ab
ove
reso
nanc
e.
The
upp
er p
anel
of F
ig.
2.6(
a) s
how
s th
e ra
dius
osc
illa
tion
in th
e tim
e do
mai
n o
f the
sam
e bu
bble
whe
n dr
iven
by
a 1
MH
z in
cide
nt p
ress
ure
puls
e. T
he a
mpl
itude
of
the
inci
dent
pu
lse
is a
ppro
xim
atel
y 25
0 kP
a.
In t
he l
ower
pan
el o
f Fi
g. 2
.6(a
) th
e ab
solu
te v
alue
of
the
Fou
rier
Tra
nsfo
rm o
f the
rad
ius
osci
llat
ion
is s
how
n an
d w
e cl
earl
y se
e th
at th
e ra
dius
os
cill
atio
n is
non
line
ar a
nd t
hat
it m
ainl
y ha
s a
seco
nd h
arm
onic
com
pone
nt in
add
itio
n to
the
line
ar fu
ndam
enta
l com
pone
nt.
The
bub
ble
is d
rive
n be
low
res
onan
ce,
at n
= 0.
25,
and
we
see
that
the
pha
se a
ngle
812
in
the
rad
ius
osci
llat
ion
is c
lose
to
-~.
A
lso,
the
fu
ndam
enta
l co
mpo
nent
of t
he r
adiu
s os
cill
atio
n is
clo
se to
1r ou
t of p
hase
wit
h th
e dr
ive
pres
sure
.
The
far
-fie
ld c
ompo
nent
of
the
scat
tere
d pr
essu
re f
rom
the
sam
e bu
bble
is
show
n in
22
Pap
er A
,F37\J
&j ~ESE
1.5
2 2.
5 3
3.5
4 4.
5 5
5.5
6 1.
5 2
2.5
3 3.
5 4
4.5
5 5.
5 6
~
~
+zv;
;:: 1-~1 1~
0 0.
5 I
1.5
2 2.
5 3
3.5
4 4.
5 5
0 0.
5 I
l.5
2 2.
5 3
3.5
4 4.
5 5
[Ml:l
z]
[MH
z]
(a)
Rad
ius
osci
llat
ion
and
ampl
itud
e o
f its
fre
qu
ency
spe
ctru
m.
(b)
Sca
tter
ed p
ress
ure
puls
e an
d am
plit
ude
of
its
freq
uenc
y sp
ectr
um.
Fig
ure
2.7:
Bub
ble
driv
en b
y di
stor
ted
pres
sure
pul
se a
t n
= 0
.25.
S
econ
d ha
rmon
ic c
ompo
nent
in d
rive
pre
ssur
e is
23
dB b
elow
fun
dam
enta
l co
mpo
ne
nt,
whi
le <
I>12
= 0.
Fig.
2.6
(b).
T
his
is t
he p
ress
ure
com
pone
nt s
ever
al w
avel
engt
hs a
way
fro
m t
he b
ubbl
e w
hich
we
may
pic
k up
wit
h an
ult
raso
und
tran
sduc
er.
The
pre
ssur
e pu
lse
show
n ha
s no
t be
en m
odif
ied
by a
cous
tic
abso
rpti
on d
ue t
o w
ave
prop
agat
ion
and
we
see
that
it
has
resp
onde
d no
nlin
earl
y w
ith
sign
ific
ant
amou
nts
of
ener
gy a
t ha
rmon
ic c
ompo
nent
s no
t pr
esen
t in
the
driv
e pr
essu
re p
ulse
.
We
now
dri
ve th
e sa
me
bubb
le w
ith
a di
stor
ted
inci
dent
pre
ssur
e pu
lse
whi
ch h
as th
e sa
me
fund
amen
tal c
ompo
nent
as
the
pres
sure
dri
ve p
ulse
use
d in
Fig
. 2.6
but
now
als
o co
ntai
ns
a se
cond
har
mon
ic c
ompo
nent
whi
ch i
s 23
dB
bel
ow t
he f
unda
men
tal
com
pone
nt.
The
se
cond
har
mon
ic c
ompo
nent
in
the
driv
e pr
essu
re h
as a
pha
se a
ngle
<I>
12,
as d
efin
ed i
n F
ig. 2
.2, e
qual
to z
ero.
The
res
ulti
ng r
adiu
s re
spon
se a
nd s
catt
ered
pre
ssur
e ar
e di
spla
yed
in F
ig.
2.7(
a) a
nd 2
.7(b
), r
espe
ctiv
ely.
A
ddin
g th
e se
cond
har
mon
ic c
ompo
nent
in
the
driv
e pr
essu
re h
as s
ligh
tly
incr
ease
d th
e se
cond
and
thi
rd h
arm
onic
com
pone
nts
in t
he
radi
us r
espo
nse
of th
e bu
bble
as
can
be s
een
by c
ompa
ring
Fig
. 2.7
(a)
and
2.6(
a).
A s
mal
l fo
urth
har
mon
ic c
ompo
nent
is n
ow a
lso
visi
ble.
The
eff
ect o
n th
e sc
atte
red
pres
sure
pul
se
from
the
bub
ble
is s
imila
r, a
sm
all i
ncre
ase
in a
ll of
the
scat
tere
d ha
rmon
ic c
ompo
nent
s is
fo
und
by c
ompa
ring
Fig
. 2.7
(b)
and
2.6(
b).
The
bub
ble
is t
hen
exci
ted
wit
h a
dist
orte
d in
cide
nt p
ress
ure
puls
e si
mil
ar to
the
one
ap
plie
d in
Fig
. 2. 7
, the
onl
y di
ffer
ence
bei
ng th
at th
e ph
ase
angl
e <I>
12
in th
e in
cide
nt p
ress
ure
puls
e ha
s ch
ange
d fr
om z
ero to-~. T
he u
pper
pan
el o
f Fig
. 2.8
(a)
show
s th
e ra
dius
osc
il
lati
on o
f the
bub
ble
in th
e ti
me
dom
ain
whe
n dr
iven
by
this
new
dis
tort
ed p
ress
ure
puls
e.
In t
he l
ower
pan
el,
the
abso
lute
val
ue o
f th
e F
ouri
er T
rans
form
of
the
radi
us o
scil
lati
on
is d
epic
ted
and
we
see
that
the
sec
ond
harm
onic
com
pone
nt o
f th
e ra
dius
osc
illa
tion
is
2.3
Res
ults
£;Fi/\/
SCJ ~
E!\1\t?
1.5
2 2.
5 3
3.5
4 4.
5 5
5.5
6 1.
5 2
2.5
3 3.
5 4
4.5
5 5.
5 6
~
~
·ll\~
: hJ lE
d
0 0
5
l 1
5
2 2
5
3 3
5
4 4
5
5 0
OS
l 1
5
2 2
5
3 3
5
4 4
5
5 [M
Hz]
[M
Hz]
(a)
Rad
ius
osci
llat
ion
and
ampl
itud
e o
f its
fre
qu
ency
spe
ctru
m.
(b)
Sca
tter
ed p
ress
ure
puls
e an
d am
plit
ude
of
its f
requ
ency
spe
ctru
m.
Fig
ure
2.8:
Bub
ble
driv
en b
y di
stor
ted
pres
sure
pul
se a
t n
=
0.25
. S
econ
d ha
rmon
ic c
ompo
nent
in d
rive
pre
ssur
e is
23
dB b
elow
fun
dam
enta
l co
mpo
ne
nt,
whi
le <
P12 =
-~.
redu
ced
com
pare
d to
the
low
er p
anel
of F
ig.
2.6(
a).
23
The
fun
dam
enta
l co
mpo
nent
of
the
radi
us o
scil
lati
on is
app
roxi
mat
ely
1r o
ut o
f pha
se
wit
h th
e fu
ndam
enta
l co
mpo
nent
of
the
driv
ing
pres
sure
. F
rom
our
line
ar c
onsi
dera
tion
s re
sult
ing
in F
ig.
2.1,
we
may
con
clud
e th
at t
he l
inea
r ra
dius
res
pons
e to
the
sec
ond
har
mon
ic c
ompo
nent
in t
he d
rive
pre
ssur
e, a
t n
=
0.5
in t
his
case
, is
als
o cl
ose
to 1
r ou
t of
ph
ase.
It
was
see
n in
the
upp
er p
anel
of
Fig.
2.6
(a)
that
812
in
the
radi
us o
scil
lati
on w
as
clos
e to
-~
whe
n dr
iven
by
a no
n-di
stor
ted
driv
e pr
essu
re.
Whe
n th
e bu
bble
then
is d
rive
n by
a d
isto
rted
pre
ssur
e pu
lse
cont
aini
ng a
sec
ond
harm
onic
com
pone
nt w
ith
phas
e an
gle
clos
e to
-~
rela
tive
to t
he f
unda
men
tal
com
pone
nt, t
his
dist
orti
on w
ill h
ave
the
pote
ntia
l to
lin
earl
y ca
ncel
out
or
redu
ce t
he s
econ
d ha
rmon
ic c
ompo
nent
in t
he r
adiu
s os
cill
atio
n ob
tain
ed w
hen
driv
ing
the
bubb
le w
ith
the
non-
dist
orte
d pr
essu
re p
ulse
as
show
n in
the
lo
wer
pan
el o
f Fig
. 2.
6(a)
. A
lso
impo
rtan
t, is
tha
t the
thir
d ha
rmon
ic c
ompo
nent
inhe
rent
in th
e ra
dius
osc
illa
tion
in
the
low
er p
anel
of
Fig.
2.6
(a),
obt
aine
d by
usi
ng t
he n
on-d
isto
rted
dri
ve p
ress
ure,
is
canc
eled
or
sign
ific
antl
y re
duce
d w
hen
usin
g th
is n
ew d
isto
rted
dri
ve p
ress
ure
as s
een
from
the
low
er p
anel
in F
ig. 2
.8(a
). T
his
is a
non
line
ar e
ffec
t res
ulti
ng f
rom
the
mix
ing
of
the
fund
amen
tal
and
seco
nd h
arm
onic
com
pone
nt in
the
dis
tort
ed d
rive
pre
ssur
e pu
lse.
Fi
g. 2
.8(b
) di
spla
ys th
e sc
atte
red
pres
sure
fro
m t
he b
ubbl
e dr
iven
by
the
new
dis
tort
ed
pres
sure
pul
se.
We
obse
rve
that
the
leve
ls o
f bot
h th
e se
cond
, th
ird,
and
fou
rth
harm
onic
co
mpo
nent
s ar
e si
gnif
ican
tly
redu
ced
com
pare
d to
wha
t w
e fo
und
in t
he l
ower
pan
el o
f Fi
g. 2
.6(b
) us
ing
a no
n-di
stor
ted
driv
ing
pres
sure
pul
se.
The
dim
inis
hing
eff
ects
on
the
thir
d an
d fo
urth
har
mon
ic c
ompo
nent
s in
the
sca
tter
ed p
ress
ure
puls
e ar
e no
nlin
ear.
24
Pap
er A
;f!\T
\f3 S
b!\)
8 u
2 u
3 ~
4 u
5 ~
6 ~.
2 u
3 ~
4 u
5 ~
6 ~
~
-~r r~1 -~r
fkC
;J
0 0.
5 l
1.5
2 2.
5 3
3.5
4 4.
5 5
0 0.
5 l
1.5
2 2.
5 3
3.5
4 4.
5 5
[MH
z]
[MH
z]
(a)
Sca
tter
ed p
ress
ure
puls
e an
d am
plit
ude
of
its
freq
uenc
y sp
ectr
um.
(b)
Sca
tter
ed p
ress
ure
puls
e an
d am
plit
ude
of
its
freq
uenc
y sp
ectr
um.
Fig
ure
2.9:
Bub
ble
driv
en b
y di
stor
ted
pres
sure
pul
se a
t n
=
0.25
. S
econ
d ha
rmon
ic c
ompo
nent
in d
rive
pre
ssur
e is
25
dB a
nd 2
1 dB
bel
ow f
unda
men
tal
com
pone
nt, l
eft a
nd r
ight
pan
el, r
espe
ctiv
ely,
whi
le <
!> 12
= -
~.
The
res
ulti
ng s
catt
ered
pre
ssur
e pu
lse
from
the
bub
ble
whe
n re
duci
ng t
he le
vel o
f sec
ond
harm
onic
in
the
dist
orte
d dr
ive
pres
sure
fro
m 2
3 dB
to
25 d
B b
elow
the
fun
dam
enta
l co
mpo
nent
, w
hile
kee
ping
<!>
12 e
qual
to -~,is s
een
in F
ig.
2.9(
a).
Com
pari
ng t
he l
ower
pa
nel o
f thi
s fi
gure
wit
h th
e lo
wer
pan
el o
f Fig
. 2.
8(b)
we
see
that
ther
e is
now
a s
tron
ger
redu
ctio
n of
the
scat
tere
d th
ird
and
four
th h
arm
onic
com
pone
nts
whe
reas
the
sca
tter
ed
seco
nd h
arm
onic
com
pone
nt is
som
ewha
t les
s re
duce
d.
Incr
easi
ng t
he l
evel
of
seco
nd h
arm
onic
in
the
dist
orte
d dr
ive
pres
sure
to
21 d
B b
elow
th
e fu
ndam
enta
l co
mpo
nent
, an
d st
ill
not
chan
ging
<!> 1
2, w
e ob
tain
the
resu
lts
depi
cted
in
Fig.
2.9
(b)
for
the
scat
tere
d pr
essu
re p
ulse
fro
m t
he b
ubbl
e. I
n th
is c
ase,
by
com
pari
son
wit
h th
e lo
wer
pan
el i
n Fi
g. 2
.8(b
), th
e re
duct
ion
of
the
scat
tere
d se
cond
har
mon
ic c
om
pone
nt is
str
onge
r w
hile
the
thir
d an
d fo
urth
har
mon
ic c
ompo
nent
s ar
e le
ss d
imin
ishe
d.
We
now
inc
reas
e ou
r dr
ive
freq
uenc
y to
2 M
Hz
so t
hat n
= 0
.5 f
or t
he f
unda
men
tal
com
pone
nt i
n th
e dr
ive
pres
sure
. T
he a
mpl
itud
e o
f th
e dr
ive
pres
sure
is
now
180
kPa
. E
xpos
ing
the
sam
e bu
bble
to t
his
new
dri
ve p
ress
ure
cont
aini
ng o
nly
a fu
ndam
enta
l co
m
pone
nt w
e ob
tain
the
res
ults
sho
wn
in t
he u
pper
and
low
er p
art
of F
ig.
2.10
(a)
for
the
radi
us o
scil
lati
on i
n th
e tim
e do
mai
n an
d th
e ab
solu
te v
alue
of
the
Fou
rier
Tra
nsfo
rm
of
the
radi
us o
scil
lati
on,
resp
ectiv
ely.
B
y co
mpa
ring
the
upp
er p
anel
of
this
fig
ure
wit
h Fi
g. 2
.2 w
e ca
n co
nclu
de th
at 8
12 i
n th
e ra
dius
osc
illa
tion
is s
imil
ar to
the
sit
uati
on in
the
uppe
r pa
nel
of F
ig.
2.2
whe
re <
!>12
is
equa
l to
zer
o. T
he d
iffe
renc
e is
, ho
wev
er,
that
wit
h ou
r de
fini
tion,
zer
o ph
ase
angl
e re
sult
s in
a s
aw-t
ooth
sha
ped
puls
e w
here
the
com
pres
si
on p
erio
d fr
om h
igh
to l
ow v
alue
is
less
ste
ep t
han
the
expa
nsio
n pe
riod
fro
m l
ow t
o
2.3
Res
ults
-~~h
r:K~
: l·~[ :c
vs;; I
0
12
3
4 5
6
7 8
01
2
3 4
56
7
8 [M
Hz]
[M
Hz]
(a)
Rad
ius
osci
llat
ion
and
ampl
itud
e o
f its
fre
qu
ency
spe
ctru
m.
(b)
Sca
tter
ed p
ress
ure
puls
e an
d am
plit
ude
of
its
freq
uenc
y sp
ectr
um.
Fig
ure
2.10
: B
ubbl
e dr
iven
by
non-
dist
orte
d pr
essu
re p
ulse
at n
= 0
.5.
25
high
val
ue.
The
rad
ius
resp
onse
in F
ig. 2
.10(
a) is
a k
ind
of s
aw-t
ooth
sha
ped
puls
e w
here
th
e co
mpr
essi
on p
art i
s st
eepe
r th
an th
e ex
pans
ion
part
whi
ch re
sult
s in
a p
hase
ang
le 8
12
in t
he r
adiu
s re
spon
se t
hat i
s cl
ose
to -
1r
wit
h ou
r de
fini
tion
s.
Fig
. 2.
10(b
) di
spla
ys t
he s
catt
ered
pre
ssur
e pu
lse
from
the
bub
ble
whe
n dr
iven
by
the
non-
dist
orte
d dr
ive
pres
sure
whi
ch is
see
n to
be
nonl
inea
r.
Fro
m F
ig.
2.1
the
line
ar r
adiu
s re
spon
se w
hen
driv
ing
the
bubb
le a
t n
= 0
.5 i
s cl
ose
to 1
r
out o
f pha
se w
ith
the
driv
e pr
essu
re.
A s
econ
d ha
rmon
ic c
ompo
nent
in t
he d
rive
pre
ssur
e is
now
at
reso
nanc
e an
d fo
r th
is c
ase
the
line
ar r
adiu
s re
spon
se i
s ~
out
of
phas
e w
ith
the
driv
e pr
essu
re a
ccor
ding
to
Fig.
2.1
. G
oing
fro
m d
rive
pre
ssur
e to
rad
ius
osci
llat
ion,
th
e fu
ndam
enta
l co
mpo
nent
in t
he d
rive
pre
ssur
e, a
t n =
0.5
, is
shi
fted
app
roxi
mat
ely
7r,
the
seco
nd h
arm
onic
com
pone
nt in
the
dri
ve p
ress
ure,
at n
=
1, i
s sh
ifte
d ~.
whi
le t
he
phas
e re
lati
on b
etw
een
the
fund
amen
tal
and
the
seco
nd h
arm
onic
com
pone
nt o
f the
radi
us
osci
llat
ion,
81
2.
is c
lose
to -
1r
acco
rdin
g to
the
upp
er p
anel
of F
ig.
2.10
(a).
Set
ting
<!> 1
2
in t
he d
isto
rted
dri
ve p
ress
ure
equa
l to
-~
will
thu
s po
tent
iall
y ca
ncel
out
or
redu
ce t
he
inhe
rent
sec
ond
harm
onic
com
pone
nt in
the
low
er p
anel
of F
ig.
2.10
(a).
The
bub
ble
resp
onse
is n
ow c
alcu
late
d us
ing
the
sam
e fu
ndam
enta
l com
pone
nt in
the
driv
e pr
essu
re a
s in
Fig
. 2.
10 b
ut a
ddin
g a
seco
nd h
arm
onic
com
pone
nt w
hich
is 2
3 dB
bel
ow
the
fund
amen
tal
com
pone
nt a
nd w
here
<I> 1
2 is
equ
al to
-~.
R
adiu
s os
cill
atio
ns r
esul
ting
fr
om d
rivi
ng th
e bu
bble
wit
h su
ch a
dis
tort
ed p
ress
ure
puls
e is
sho
wn
in F
ig.
2.11
(a).
By
com
pari
ng w
ith
Fig.
2.1
0(a)
we
noti
ce t
hat
the
nonl
inea
rity
of
the
radi
us r
espo
nse
has
been
sig
nifi
cant
ly r
educ
ed w
hen
driv
en b
y th
e di
stor
ted
pres
sure
pul
se r
elat
ive
to w
hen
driv
en b
y th
e no
n-di
stor
ted
pres
sure
pul
se.
26
Pap
er A
'::bl\&
:J ;5
6/\/\
3 1.
8 2
2.2
2.4
2.6
2.8
3 3.
2 3.
4 3.
6 3.
8 1.
8 2
2.2
2.4
2.6
2.8
3 3.
2 3.
4 3.
6 3.
8 ~
~
·~tl
\~ ••
1 ·J .T
b :•
I 0
12
3
4 5
8 0
I 2
3 4
56
7
8 [W
lz)
[MI-
Iz]
(a)
Rad
ius
osci
llat
ion
and
ampl
itud
e o
f its
fre
quen
cy s
pect
rum
. (b
) S
catt
ered
pre
ssur
e pu
lse
and
ampl
itud
e o
f its
fre
quen
cy s
pect
rum
.
Fig
ure
2.11
: B
ubbl
e dr
iven
by
dist
orte
d pr
essu
re p
ulse
at
D =
0.5
. S
econ
d ha
rmon
ic c
ompo
nent
in
driv
e pr
essu
re i
s 23
dB
bel
ow f
unda
men
tal
com
po
nent
, w
hile
<!>
12 =
-~.
The
sca
tter
ed p
ress
ure
from
the
bubb
le is
dis
play
ed in
Fig
. 2.
11(b
) an
d by
com
pari
ng w
ith
Fig
. 2.
1 O
(b)
we
can
conc
lude
that
the
bubb
le re
spon
se is
muc
h le
ss n
onli
near
whe
n dr
iven
by
the
dis
tort
ed p
ress
ure
puls
e.
2.3.
3 D
rivi
ng t
he B
ubbl
e w
ith
the
Sim
ulat
ed T
rans
mit
Fie
ld
The
pre
viou
s se
ctio
n sh
owed
that
the
pres
ence
of
a sm
all s
econ
d ha
rmon
ic c
ompo
nent
in
the
pres
sure
pul
se d
rivi
ng th
e bu
bble
into
osc
illa
tion
s po
tent
iall
y ha
s a
maj
or d
imin
ishi
ng
effe
ct o
n th
e no
nlin
ear
scat
teri
ng p
rope
rtie
s o
f th
e bu
bble
. It
was
als
o se
en t
hat t
he p
hase
re
lati
on b
etw
een
the
fund
amen
tal
and
seco
nd h
arm
onic
com
pone
nt in
the
dri
ve p
ress
ure
was
of c
ruci
al i
mpo
rtan
ce.
The
obv
ious
que
stio
n is
the
refo
re i
f th
e ph
ase
rela
tion
s fo
und
in a
ctua
l tr
ansm
it f
ield
s ar
e su
ch th
at a
red
ucti
on o
f no
nlin
ear
scat
teri
ng f
rom
the
bub
ble
wil
l oc
cur
rela
tive
to
wha
t w
ould
be
obta
ined
if
the
seco
nd h
arm
onic
com
pone
nt i
n th
e tr
ansm
it f
ield
was
rem
oved
. In
our
inv
esti
gati
ons
here
we
are
goin
g to
use
the
aco
usti
c tr
ansm
it f
ield
obt
aine
d fr
om n
umer
ical
sim
ulat
ions
wit
h ou
r si
mul
atio
n pr
ogra
m f
or n
on
line
ar w
ave
prop
agat
ion
[41]
, sh
own
in F
ig.
2.3,
to d
rive
the
con
tras
t bub
ble.
The
leve
l of
the
seco
nd h
arm
onic
com
pone
nt, s
how
n in
the
righ
t pan
el o
f the
figu
re,
is i
n th
e ne
ar-f
ield
to
o lo
w t
o ha
ve a
ny e
ffec
t on
the
sca
tter
ing
from
the
con
tras
t bub
ble
and
the
near
-fie
ld i
s he
nce
not i
nter
esti
ng in
the
pre
sent
con
text
.
The
cal
cula
ted
pres
sure
pul
se a
t 10
em
on
the
sym
met
ry a
xis
of t
he tr
ansm
it fi
eld
is s
how
n in
Fig
. 2.
12(a
). T
he s
olid
line
is t
he d
isto
rted
pre
ssur
e pu
lse
obta
ined
fro
m t
he s
imul
atio
n
2.3
Res
ults
<~:I
1 1.
5 2
2.5
3 3.
5 4
4.5
5 5
5
6 [J.
ts]
·1ll~
I 0
0.5
I 1.
5 2
2.5
3 3.
5 4
[Mllz
]
(a)
Dri
ve p
ress
ure
puls
es a
nd a
mpl
itud
e o
f th
eir
freq
uenc
y sp
ectr
a.
0.2
0.1
-0.1
-0.2
I
20
10
~ 0
-10
-200
1.5
2.5
3.5
[f.ls]
4.
5 5.
5
(b)
Sca
tter
ed p
ress
ure
puls
es a
nd a
mpl
itud
e o
f th
eir
freq
uenc
y sp
ectr
a.
Fig
ure
2.12
: S
olid
lin
es:
Cal
cula
ted
tran
smit
pre
ssur
e pu
lse
at 1
0 em
on
the
sym
met
ry a
xis
in F
ig.
2.3
(lef
t pa
nel)
and
res
ulti
ng s
catt
ered
pre
ssur
e pu
lse
(rig
ht p
anel
). D
ashe
d lin
es:
Fun
dam
enta
l com
pone
nt o
nly
of c
alcu
late
d tr
ans
mit
pre
ssur
e pu
lse
at 1
0 em
on
sym
met
ry a
xis
in F
ig.
2.3
(lef
t pa
nel)
and
re
sult
ing
scat
tere
d pr
essu
re p
ulse
(ri
ght p
anel
).
27
prog
ram
whe
reas
the
das
hed
line
rep
rese
nts
the
fund
amen
tal
com
pone
nt o
nly
of
the
ob
tain
ed d
isto
rted
pre
ssur
e pu
lse.
In
the
tim
e do
mai
n, i
n th
e up
per
pane
l o
f th
e fi
gure
, w
e se
e th
ere
is a
rat
her
smal
l di
ffer
ence
bet
wee
n th
e or
igin
al a
nd f
ilter
ed p
ulse
, w
here
as i
n th
e fr
eque
ncy
dom
ain,
in
the
low
er p
anel
, th
e gr
aphs
are
plo
tted
in
deci
bel
scal
e an
d th
e di
ffer
ence
is m
ore
clea
rly
seen
. In
Fig
. 2.
12(b
), t
he s
catt
ered
pre
ssur
e pu
lse
from
the
bub
ble
whe
n dr
iven
by
the
dis
tort
ed a
nd th
e fi
ltere
d pu
lse
in F
ig. 2
.12(
a) is
dis
play
ed, s
olid
and
das
hed
line
, res
pect
ivel
y.
The
leve
l of
the
scat
tere
d fu
ndam
enta
l co
mpo
nent
is s
imil
ar f
or t
he t
wo
scat
tere
d pu
lses
w
hich
is n
atur
al s
ince
the
fun
dam
enta
l co
mpo
nent
in t
he t
wo
driv
e pu
lses
are
ide
ntic
al.
The
sec
ond
harm
onic
com
pone
nt s
catt
ered
fro
m t
he b
ubbl
e is
, ho
wev
er,
redu
ced
by a
ppr
oxim
atel
y 3
dB w
hen
driv
en b
y th
e di
stor
ted
puls
e co
mpa
red
to w
hen
driv
en b
y th
e pu
lse
cont
aini
ng t
he f
unda
men
tal
com
pone
nt o
nly.
Loo
king
at
the
thir
d ha
rmon
ic c
om
pone
nt s
catt
ered
fro
m t
he b
ubbl
e, w
e se
e th
at t
he r
educ
tion
in
scat
teri
ng i
s so
mew
hat
stro
nger
, th
e re
duct
ion
now
bei
ng a
roun
d 7
dB w
hen
driv
en b
y th
e to
tal d
isto
rted
pre
ssur
e pu
lse
rela
tive
to
whe
n dr
iven
by
the
fund
amen
tal
com
pone
nt o
nly.
The
sca
tter
ed f
ourt
h ha
rmon
ic c
ompo
nent
is
even
sli
ghtl
y m
ore
redu
ced
due
to t
he d
isto
rtio
n o
f th
e in
cide
nt
driv
e pu
lse
wit
h a
redu
ctio
n o
f abo
ut 9
dB
.
We
then
take
out
the
sim
ulat
ed p
ress
ure
puls
e at
12
em b
ut 8
mm
to th
e si
de o
f the
sym
me
try
axis
of t
he tr
ansm
it fi
eld
and
this
pre
ssur
e pu
lse
is d
ispl
ayed
in F
ig. 2
.13(
a).
Aga
in, t
he
soli
d li
ne is
the
tota
l dis
tort
ed p
ress
ure
puls
e ob
tain
ed w
hile
the
dash
ed li
ne r
epre
sent
s th
e
28
Pap
er A
0.1
0.01
~ 0 0.2~
-0.1
--
0.01
--0.\~-:"1.5:---:-2 --
--;2:
'::-.5
-----:
Jo---
---:",
,:---c
-, -,:'::-
,,-,=--
-::,,:--
, _j6
[j.
!S]
·Jib
::: I
0 Q
5 1
1.5
2 25
3
5
4.5
5 [M
Hz]
(a)
Dri
ve p
ress
ure
puls
es a
nd a
mpl
itud
e o
f th
eir
freq
uenc
y sp
ectr
a.
~ -to
-20
-30
4.5
(b) S
catt
ered
pre
ssur
e pu
lses
and
am
plit
ude
of
thei
r fr
eque
ncy
spec
tra.
Fig
ure
2.13
: S
olid
lin
es:
Cal
cula
ted
tran
smit
pre
ssur
e pu
lse
at 1
2 em
but
8
mm
off
the
sym
met
ry a
xis
in F
ig.
2.3
(lef
t pa
nel)
and
res
ulti
ng s
catt
ered
pr
essu
re p
ulse
(ri
ght
pane
l).
Das
hed
lines
: F
unda
men
tal
com
pone
nt o
nly
of
calc
ulat
ed t
rans
mit
pre
ssur
e pu
lse
at 1
2 em
and
8 m
m o
ff s
ymm
etry
axi
s in
Fi
g. 2
.3 (
left
pan
el)
and
resu
ltin
g sc
atte
red
pres
sure
pul
se (
righ
t pan
el).
fund
amen
tal
com
pone
nt o
nly
of
the
calc
ulat
ed p
ress
ure.
In
the
tim
e do
mai
n, i
n th
e up
per
pane
l of t
he f
igur
e, t
he o
rigi
nal a
nd f
ilter
ed p
ulse
are
now
alm
ost i
ndis
ting
uish
able
due
to
the
low
lev
el o
f se
cond
har
mon
ic i
n th
e di
stor
ted
puls
e at
thi
s lo
catio
n. T
he d
iffe
renc
e in
sc
atte
red
pres
sure
fro
m t
he b
ubbl
e w
hen
driv
ing
it w
ith
the
dist
orte
d ve
rsus
the
filt
ered
pr
essu
re p
ulse
is, h
owev
er,
sign
ific
ant
as s
how
n in
Fig
. 2.
13(b
). A
s se
en f
rom
the
fig
ure,
th
e se
cond
har
mon
ic c
ompo
nent
is
now
hea
vily
red
uced
by
appr
oxim
atel
y 10
dB
whe
n ap
plyi
ng th
e di
stor
ted
driv
e pr
essu
re r
elat
ive
to a
pply
ing
the
fund
amen
tal c
ompo
nent
onl
y w
here
as t
he e
ffec
t on
the
scat
tere
d th
ird
harm
onic
com
pone
nt is
les
s si
gnif
ican
t.
In T
able
2.1
the
res
ults
obt
aine
d by
tak
ing
out
the
sim
ulat
ed t
rans
mit
pre
ssur
e pu
lse
at
vari
ous
loca
tion
s in
the
tran
smit
fiel
d in
Fig
. 2.
3 an
d ap
plyi
ng it
on
the
sam
e co
ntra
st b
ub
ble
are
sum
mar
ized
. U
nles
s ot
herw
ise
note
d, p
ress
ure
driv
e pu
lses
fro
m t
he s
ymm
etry
ax
is a
re u
sed.
Thi
s ta
ble
show
s th
e re
duct
ion
of
nonl
inea
rly
scat
tere
d ha
rmon
ic c
ompo
ne
nts
from
the
bub
ble
whe
n dr
iven
by
the
dist
orte
d tr
ansm
it p
ress
ure
puls
e re
lati
ve to
the
fu
ndam
enta
l com
pone
nt o
nly
of t
he tr
ansm
it p
ress
ure
puls
e. F
or th
e co
mbi
nati
on o
f tra
ns
mit
fie
ld a
nd c
ontr
ast b
ubbl
e us
ed i
n th
e pr
esen
t pap
er,
redu
ctio
n o
f no
nlin
ear
scat
teri
ng
from
the
bub
ble
usin
g th
e di
stor
ted
driv
e pu
lse
vers
us f
unda
men
tal
com
pone
nt o
nly,
was
fo
und
to b
e in
sign
ific
ant u
sing
tra
nsm
it d
rive
pre
ssur
e pu
lses
in t
he l
ocat
ion
betw
een
the
tran
sduc
er a
nd d
own
to a
bout
6 e
m in
Fig
. 2.
3. D
own
to a
bout
2 o
r 3
em,
the
leve
l of
the
seco
nd h
arm
onic
com
pone
nt in
the
tra
nsm
it f
ield
is
too
low
to h
ave
any
sign
ific
ant e
ffec
t on
the
sca
tter
ing
from
the
con
tras
t bub
ble.
Bet
wee
n 3
and
6 em
, a
sign
ific
ant a
mou
nt o
f
2.4
Con
clus
ions
29
seco
nd h
arm
onic
bui
lds
up i
n th
e tr
ansm
it fi
eld,
as
seen
in th
e ri
ght p
anel
of
Fig.
2.3
, bu
t th
e ph
ase
angl
e <1>
12 be
twee
n th
e fu
ndam
enta
l an
d se
cond
har
mon
ic c
ompo
nent
is
such
th
at a
red
ucti
on o
f non
line
ar s
catt
erin
g fr
om t
he c
ontr
ast b
ubbl
e do
es n
ot o
ccur
. B
elow
6
em t
he e
ffec
t of
the
puls
e di
stor
tion
in
the
tran
smit
fie
ld i
s, h
owev
er,
sign
ific
ant
as s
een
from
Tab
le 2
.1.
At t
he s
ymm
etry
axi
s, t
he d
imin
ishi
ng e
ffec
t due
to t
he d
isto
rted
inc
iden
t dr
ive
puls
e on
the
non
line
ar s
catt
erin
g fr
om t
he b
ubbl
e is
mos
t se
vere
on
the
scat
tere
d th
ird
and
four
th h
arm
onic
com
pone
nts
whe
reas
the
sca
tter
ed s
econ
d ha
rmon
ic c
ompo
ne
nt i
s re
duce
d by
2 o
r 3
dB.
Usi
ng d
rive
pul
ses
off
the
sym
met
ry a
xis,
we
see
that
the
di
min
ishi
ng e
ffec
t on
the
scat
tere
d se
cond
har
mon
ic c
ompo
nent
is s
tron
ger.
Tab
le 2
.1:
Red
uctio
n o
f sc
atte
red
cont
rast
har
mon
ic
com
pone
nts
due
to i
ncid
ent p
ulse
dis
tort
ion
2na
harm
onic
6
cm
2d
B
Scm
3
dB
lOcm
3
dB
1
2cm
2
dB
1
4cm
2
dB
8
em a
3 dB
1
0cm
a 4
dB
1
2cm
a 3
dB
14 e
m a
2d
B
lOcm
b
8d
B
12
cm b
10dB
14
em
b
6d
B
a 4
mm
off
sym
met
ry a
xis
b 8
mm
off
sym
met
ry a
xis
2.4
Con
clus
ions
3ra
harm
onic
4t
h ha
rmon
ic
3 dB
3
dB
5 dB
5
dB
7 dB
9
dB
8
dB
15
dB
4
dB
6
dB
4
dB
4
dB
8
dB
8
dB
12dB
7
dB
2d
B
3d
B
2d
B
Thi
s pa
per
has,
bas
ed o
n nu
mer
ical
sim
ulat
ions
, in
vest
igat
ed b
ubbl
e ra
dius
osc
illa
tion
s an
d sc
atte
red
pres
sure
pul
ses
from
a c
ontr
ast b
ubbl
e dr
iven
by
vari
ous
inci
dent
pre
ssur
e pu
lses
. R
esul
ts f
rom
usi
ng s
ever
al d
isto
rted
inc
iden
t pr
essu
re p
ulse
s ar
e co
mpa
red
wit
h th
e re
sult
s fr
om u
sing
the
fun
dam
enta
l co
mpo
nent
onl
y o
f th
e di
stor
ted
inci
dent
dri
ve
pres
sure
pul
ses.
The
dis
tort
ion
of th
e pr
essu
re p
ulse
inc
iden
t to
the
con
tras
t bu
bble
oc
curs
due
the
the
nonl
inea
r na
ture
of
tissu
e el
asti
city
and
the
rel
ativ
e ph
ase
angl
e be
twee
n th
e fu
ndam
enta
l an
d se
cond
har
mon
ic c
ompo
nent
in t
he t
rans
mit
fie
ld i
s a
resu
lt o
f w
ave
diff
ract
ion.
A
cal
cula
ted
tran
smit
fie
ld f
rom
an
annu
lar
ultr
asou
nd t
rans
duce
r w
as o
bta
ined
usi
ng a
non
line
ar n
umer
ical
sim
ulat
ion
tool
dev
elop
ed i
n ou
r gr
oup.
D
rivi
ng a
30
Pap
er A
cont
rast
bub
ble
wit
h th
e di
stor
ted
tran
smit
fie
ld o
btai
ned
from
thi
s si
mul
atio
n pr
ogra
m
and
com
pari
ng t
he r
esul
ting
sca
tter
ing
wit
h th
e sc
atte
ring
obt
aine
d by
dri
ving
the
sam
e bu
bble
wit
h th
e fu
ndam
enta
l co
mpo
nent
onl
y o
f th
e ca
lcul
ated
tra
nsm
it f
ield
cle
arly
in
dica
tes
that
the
rela
tive
ly s
mal
l di
stor
tion
int
rodu
ced
due
to n
onli
near
wav
e pr
opag
atio
n ha
s a
pote
ntia
lly
stro
ng d
imin
ishi
ng e
ffec
t on
the
non
line
ar s
catt
erin
g fr
om t
he c
ontr
ast
bubb
le.
In t
he t
rans
mit
nea
r-fi
eld,
the
pul
se d
isto
rtio
n is
typ
ical
ly v
ery
low
and
its
eff
ect
on t
he n
onli
near
sca
tter
ing
from
the
bub
ble
is f
ound
to
be n
egli
gibl
e.
How
ever
, in
the
fo
cal
regi
on a
nd e
spec
iall
y in
the
far
-fie
ld o
f th
e tr
ansm
it f
ield
, th
e di
stor
tion
of
the
pres
su
re p
ulse
inc
iden
t to
the
bub
ble
is f
ound
to
have
a s
igni
fica
nt d
imin
ishi
ng e
ffec
t on
its
no
nlin
ear
scat
teri
ng p
rope
rtie
s.
The
dim
inis
hing
eff
ect,
due
to p
ulse
dis
tort
ion
caus
ed b
y no
nlin
ear
wav
e pr
opag
atio
n, o
n th
e sc
atte
red
seco
nd h
arm
onic
com
pone
nt f
rom
the
con
tras
t bu
bble
can
to
a la
rge
exte
nt
be e
xpla
ined
as
a li
near
pro
cess
, i.e
. se
para
tely
dri
ving
the
bub
ble
wit
h th
e fu
ndam
enta
l an
d se
cond
har
mon
ic c
ompo
nent
s o
f th
e di
stor
ted
inci
dent
dri
ve p
ulse
and
then
sum
min
g th
e tw
o sc
atte
red
cont
ribu
tion
s to
geth
er.
The
dim
inis
hing
eff
ects
on
the
scat
tere
d th
ird
and
four
th h
arm
onic
com
pone
nts
from
the
con
tras
t bub
ble
are,
how
ever
, no
nlin
ear.
2.5
Akn
owle
dgem
ets
Thi
s w
ork
was
sup
port
ed b
y th
e R
esea
rch
Cou
ncil
of N
orw
ay.
Ch
apte
r 3
Usi
ng
Bar
ker
Cod
es in
Con
tras
t H
arm
onic
Im
agin
g
Abs
trac
t
Rec
eive
d sc
atte
red
harm
onic
com
pone
nts
typi
call
y co
ntai
n le
ss e
nerg
y th
an t
he l
inea
rly
rece
ived
fun
dam
enta
l co
mpo
nent
and
, in
con
tras
t har
mon
ic i
mag
ing
tech
niqu
es,
the
lim
it
ing
fact
or i
s of
ten
the
nois
e si
gnal
alw
ays
pres
ent
in a
pul
se e
cho
imag
ing
syst
em a
nd
not t
he m
aski
ng o
f the
con
tras
t sig
nal b
y th
e tis
sue
sign
al.
Puls
e co
mpr
essi
on te
chni
ques
, fa
mil
iar
from
rad
ar s
yste
ms
and
com
mun
icat
ion
theo
ry,
are
tech
niqu
es f
or i
ncre
asin
g th
e si
gnal
leve
l rel
ativ
e to
the
noi
se le
vel o
f the
pul
se e
cho
imag
ing
syst
em w
hich
is a
ssum
ed
to b
e co
nsta
nt a
nd e
venl
y di
stri
bute
d ov
er a
ll fr
eque
ncie
s o
f in
tere
st.
The
tra
nsm
itte
d si
gnal
lev
el i
s in
crea
sed
by t
rans
mit
ting
an
elon
gate
d pu
lse
and
not
by i
ncre
asin
g th
e tr
ansm
it a
mpl
itud
e. T
o re
stor
e ra
nge
reso
lutio
n, t
he r
esul
ting
rec
eive
d si
gnal
s m
ust
then
be
com
pres
sed.
Thi
s pa
per
inve
stig
ates
the
use
of B
arke
r cod
es, w
hich
are
a ty
pe o
f pul
se
com
pres
sion
cod
es,
and
thei
r po
tent
ial
to i
ncre
ase
the
sign
al l
evel
of
the
scat
tere
d th
ird
harm
onic
com
pone
nt fr
om c
ontr
ast b
ubbl
es r
elat
ive
to t
he c
onst
ant n
oise
leve
l.
3.1
Intr
oduc
tion
Med
ical
ult
raso
und
cont
rast
age
nts
are
used
to
enha
nce
the
ultr
asou
nd s
igna
l sc
atte
red
from
blo
od s
ince
ult
raso
und
scat
teri
ng f
rom
blo
od is
muc
h w
eake
r th
an s
catt
erin
g fr
om
soft
tis
sue.
T
he c
ontr
ast
agen
ts a
re t
ypic
ally
mad
e as
sol
utio
ns o
f sm
all
gas
bubb
les
in a
flu
id a
nd s
catt
erin
g fr
om s
uch
gas
bubb
les
is p
oten
tial
ly a
hig
hly
nonl
inea
r pr
oce
ss [
23]
[11]
[12
].
Ult
raso
und
wav
e pr
opag
atio
n is
als
o kn
own
to b
e a
nonl
inea
r pr
oces
s at
fre
quen
cies
and
32
Pap
erB
ampl
itud
es c
omm
only
use
d in
med
ical
ult
raso
und
imag
ing.
The
non
line
arit
y o
f th
e w
ave
prop
agat
ion
is,
how
ever
, si
gnif
ican
tly
wea
ker
than
for
the
sca
tter
ing
proc
ess
from
the
co
ntra
st b
ubbl
es,
and
cont
rast
har
mon
ic i
mag
ing
is t
hus
an i
nter
esti
ng i
mag
ing
tech
niqu
e si
nce
scat
teri
ng f
rom
tis
sue
is a
ssum
ed to
be
a li
near
pro
cess
.
In c
ontr
ast
harm
onic
im
agin
g, a
har
mon
ic b
and
of
the
fund
amen
tal
tran
smit
ted
band
is
used
for
bub
ble
dete
ctio
n. T
he s
catt
ered
con
tras
t si
gnal
is
mor
e di
stor
ted
than
the
sca
tte
red
tiss
ue s
igna
l an
d by
usi
ng h
ighe
r ha
rmon
ic c
ompo
nent
s o
f th
e re
ceiv
ed s
igna
l fo
r im
age
reco
nstr
ucti
on,
one
wou
ld p
resu
mab
ly a
chie
ve b
ette
r di
ffer
enti
atio
n be
twee
n tis
su
e si
gnal
and
con
tras
t si
gnal
. T
he a
mpl
itud
e o
f th
e re
ceiv
ed t
issu
e ha
rmon
ics
are
sig
nifi
cant
ly l
ower
than
for
the
line
arly
rec
eive
d fu
ndam
enta
l co
mpo
nent
and
as
high
er h
ar
mon
ics
are
cons
ider
ed,
this
am
plit
ude
drop
s st
eepl
y. T
he a
mpl
itud
e of
the
rece
ived
con
tr
ast h
arm
onic
s ar
e al
so t
ypic
ally
low
er th
an f
or th
e re
ceiv
ed f
unda
men
tal c
ompo
nent
. A
s hi
gher
con
tras
t har
mon
ics
are
cons
ider
ed, t
he le
vel o
f the
am
plit
ude
usua
lly
falls
alt
houg
h no
t as
fast
as
for
the
tiss
ue h
arm
onic
s.
The
ult
raso
und
med
ical
im
agin
g sy
stem
is
a pu
lse
echo
im
agin
g sy
stem
suf
feri
ng f
rom
th
erm
al a
nd e
lect
roni
c no
ise
and,
as
high
er s
igna
l ha
rmon
ics
are
used
for
im
age
reco
nst
ruct
ion,
the
leve
l of t
hese
har
mon
ics
wil
l be
com
para
ble
to t
he n
oise
leve
l of t
he s
yste
m
resu
ltin
g in
im
age
degr
adat
ion.
App
lica
tion
of
high
er h
arm
onic
com
pone
nts
impl
ies
re
duce
d pe
netr
atio
n de
pth
due
to t
he f
requ
ency
dep
ende
nt a
bsor
ptio
n in
tis
sue
and
a re
duc
tion
of t
he r
ecei
ved
harm
onic
sig
nal
ampl
itud
e fr
om t
he c
ontr
ast a
gent
.
If th
e no
ise
leve
l in
a pu
lse
echo
im
agin
g sy
stem
is a
ssum
ed to
be
cons
tant
, th
e on
ly w
ay
to i
ncre
ase
the
Sig
nal
to N
oise
Rat
io (
SNR
) is
by
incr
easi
ng t
he s
igna
l le
vel.
In m
edic
al
ultr
asou
nd i
mag
ing
the
tran
smit
ted
acou
stic
am
plit
ude
cann
ot b
e ar
bitr
aril
y hi
gh d
ue t
o pa
tien
t sa
fety
[8]
[5]
. A
lso,
the
bub
bles
ten
d to
get
des
troy
ed e
ven
duri
ng i
mag
ing
at
rela
tive
ly lo
w i
nten
siti
es s
o th
at th
e in
tens
ity
in th
e tr
ansm
it f
ield
mus
t be
kept
rel
ativ
ely
low
if b
ubbl
e de
stru
ctio
n is
unw
ante
d.
Pul
se c
ompr
essi
on te
chni
ques
, kno
wn
from
rad
ar s
yste
ms
and
com
mun
icat
ion
theo
ry,
have
the
abi
lity
to
impr
ove
the
SN
R w
itho
ut i
ncre
asin
g th
e pe
ak a
mpl
itud
e le
vels
tra
ns
mitt
ed.
A p
ulse
of l
ong
dura
tion
com
pare
d to
a c
onve
ntio
nal t
rans
mit
pul
se is
tra
nsm
itte
d an
d th
e re
ceiv
ed e
cho
sign
al i
s pr
oces
sed
wit
h a
mat
ched
filt
er [
21,
Cha
pter
4.8
] [ 4
0,
Cha
pter
5.4
] to
ach
ieve
suf
fici
ent
rang
e re
solu
tion
. T
hese
tec
hniq
ues
intr
oduc
e ra
nge
side
lobe
art
ifac
ts w
hich
typ
ical
ly a
re i
nade
quat
e fo
r m
edic
al u
ltra
soun
d im
agin
g.
The
ra
nge
side
lobe
s ca
n, h
owev
er, b
e fu
rthe
r re
duce
d w
ith
deco
nvol
utio
n ty
pe f
ilter
s.
Bar
ker c
odes
[21
, C
hapt
er 6
.9]
are
a ty
pe o
f pul
se c
ompr
essi
on c
odes
whe
re a
long
pul
se
is m
ade
up o
f M
sub
puls
es w
hich
are
pha
se c
oded
and
whe
re t
he p
hase
of
the
subp
ulse
s is
eit
her
0 or
1r.
The
se c
odes
are
seq
uenc
es f
or w
hich
the
sid
elob
es o
f the
aut
ocor
rela
tion
fu
ncti
on d
o no
t exc
eed
1/ M
. O
nly
nine
suc
h co
des
exis
t, th
e lo
nges
t one
bei
ng o
f len
gth
M=
13.
Con
tras
t ha
rmon
ic i
mag
ing
usin
g th
e th
ird
harm
onic
com
pone
nt w
ill
typi
call
y gi
ve s
upe
rior
Con
tras
t to
Tis
sue
Rat
io (
CT
R)
but
infe
rior
Con
tras
t to
Noi
se R
atio
(C
NR
) w
hen
3.2
The
ory
33
com
pare
d to
sec
ond
harm
onic
con
tras
t im
agin
g. T
he C
TR
is b
ette
r du
e to
the
hig
h no
nli
near
ity
of t
he s
catt
erin
g fr
om t
he c
ontr
ast b
ubbl
es c
ompa
red
to t
he w
eake
r no
nlin
eari
ty
of t
he w
ave
prop
agat
ion
in s
oft
tissu
e. C
NR
wil
l ge
nera
lly
be lo
wer
due
to
the
rela
tivel
y lo
wer
lev
el o
f re
ceiv
ed t
hird
har
mon
ics
from
the
bub
ble
assu
min
g th
e no
ise
leve
l to
be
cons
tant
ove
r th
e fr
eque
ncy
rang
e o
f int
eres
t. B
y us
ing
Bar
ker
code
s to
inc
reas
e th
e C
NR
at
the
rec
eive
d th
ird
harm
onic
com
pone
nt,
it m
ay b
e po
ssib
le t
o ac
hiev
e ad
equa
te C
NR
an
d C
TR
wit
h th
ird
harm
onic
con
tras
t age
nt i
mag
ing.
The
con
tras
t bub
bles
add
ed t
o th
e bl
ood
flow
will
, co
ntra
ry t
o th
e si
tuat
ion
in r
adar
sys
te
ms
and
com
mun
icat
ion
theo
ry,
repr
esen
t a
nonl
inea
r, r
eson
ant,
and
unst
able
med
ium
fo
r sc
atte
ring
of
the
inci
dent
wav
es.
The
non
line
ar r
espo
nse
from
the
con
tras
t bu
bble
s,
util
ized
in c
ontr
ast h
arm
onic
imag
ing,
dep
ends
str
ongl
y on
the
ampl
itude
, fre
quen
cy,
and
enve
lope
of
the
inci
dent
pul
se.
Con
tras
t bub
bles
are
typ
ical
ly e
ncap
sula
ted
in a
thin
sta
bi
lizi
ng s
hell
wit
h ac
oust
ic p
rope
rtie
s th
at m
ay b
e al
tere
d du
ring
ins
onif
icat
ion
by u
ltr
asou
nd.
Due
to
the
pum
ping
of
the
hear
t m
uscl
e, t
he b
ubbl
es w
ill
have
tim
e va
ryin
g ve
loci
ties.
In
add
ition
, th
ere
may
be
inte
ract
ions
bet
wee
n th
e bu
bble
s an
d th
e in
cide
nt
wav
e fi
eld
wil
l im
pose
a r
adia
tion
for
ce o
n th
e bu
bble
s w
ith a
res
ulti
ng t
ime
vary
ing
ve
loci
ty i
n th
e pr
opag
atio
n di
rect
ion
of
the
field
. T
he p
rese
nt p
aper
wil
l tr
y to
inv
estig
ate
som
e of
thes
e in
dica
ted
effe
cts
in r
elat
ion
to t
he a
ppli
cati
on o
f B
arke
r co
des
in c
ontr
ast
harm
onic
imag
ing.
3.2
The
ory
3.2.
1 O
verv
iew
The
mot
ivat
ion
for
intr
oduc
ing
puls
e co
mpr
essi
on te
chni
ques
in
cont
rast
har
mon
ic im
ag
ing
is a
nee
d to
inc
reas
e th
e C
NR
, esp
ecia
lly
at th
e th
ird
or h
ighe
r ha
rmon
ic c
ompo
nent
s.
Dec
onvo
luti
on o
f th
e re
ceiv
ed s
igna
l ca
n be
don
e us
ing
a m
odif
ied
inve
rse
filte
r w
here
th
e de
conv
olut
ion
kern
el f
or e
xam
ple
is b
ased
sol
ely
on a
bin
ary
Bar
ker
sequ
ence
. T
his
deco
nvol
utio
n op
erat
ion
on th
e re
ceiv
ed s
igna
l is
nece
ssar
y to
res
tore
the
rang
e re
solu
tion
w
hich
is
lost
whe
n tr
ansm
itti
ng a
n el
onga
ted
puls
e. T
he i
dea
behi
nd p
ulse
com
pres
sion
te
chni
ques
is t
o in
crea
se th
e S
NR
by
incr
easi
ng th
e si
gnal
leve
l tra
nsm
itte
d an
d as
sum
ing
the
nois
e le
vel
to b
e co
nsta
nt.
The
tra
nsm
itte
d si
gnal
lev
el i
s in
crea
sed
by t
rans
mit
ting
an
elo
ngat
ed p
ulse
and
not
by
incr
easi
ng t
he a
mpl
itud
e le
vel
tran
smit
ted.
Wit
h no
ise
we
here
mea
n no
ise
unco
rrel
ated
wit
h th
e tr
ansm
itte
d si
gnal
suc
h as
the
rmal
and
ele
ctro
nic
nois
e.
34
~t I :
:1
1· j
-1,'---~, --~10---cl'o-5 --
'---=
-,,--
--"c
,,---
---_
__J,,
[).
1.5]
"FBB
EE
0.1
~ 0
-11.
1
..-{),2 o
5 10
15
20
25
30
(J.
I.SJ
Pap
erB
Fig
ure
3.1:
Upp
er p
anel
: F
our b
it b
inar
y B
arke
r seq
uenc
e. L
ower
pan
el:
Tra
nsm
itte
d fo
ur
bit B
arke
r se
quen
ce.
3.2.
2 B
arke
r C
odes
The
Bar
ker
code
s ar
e pu
lse
com
pres
sion
cod
es,
mad
e up
of
M c
onti
guou
s su
bpul
ses,
w
hich
are
mod
ulat
ed i
n ph
ase.
T
he p
hase
mod
ulat
ion
is e
ithe
r 0
or 1
r an
d th
e co
de i
s a
bina
ry s
eque
nce
taki
ng v
alue
s 1
and
-1.
The
upp
er p
anel
of F
ig.
3.1
show
s an
exa
mpl
e of
a
four
bit
Bar
ker c
ode
whe
re th
e th
ird
bit h
as a
pha
se s
hift
of 1
r re
lati
ve to
the
othe
rs.
In t
he
low
er p
anel
, an
exam
ple
of a
tran
smit
ted
four
bit
Bar
ker c
ode
is s
how
n, a
nd th
e bi
nary
fou
r bi
t cod
e ha
s th
en b
een
conv
olve
d w
ith
a co
nven
tion
al tr
ansm
it p
ulse
. A
s ca
n be
see
n, t
he
indi
vidu
al s
ubpu
lses
in
the
Bar
ker
sequ
ence
hav
e be
en s
omew
hat s
epar
ated
in t
ime.
Thi
s is
don
e to
ade
quat
ely
sepa
rate
the
con
tras
t ech
os f
rom
eac
h su
bpul
se s
ince
the
sca
tter
ing
from
con
tras
t bu
bble
s is
a h
ighl
y no
nlin
ear
proc
ess
and
the
prin
cipl
e o
f su
perp
osit
ion
is
ther
efor
e no
t val
id. If
the
rece
ived
sig
nal,
obta
ined
usi
ng th
e co
nven
tion
al tr
ansm
it p
ulse
, is
def
ined
as
r 1(t
) =
s(t
) +
n(t)
(3
.1)
whe
re s
(t)
is t
he s
um o
f th
e ti
ssue
sig
nal
and
cont
rast
sig
nal
and
n(t
) is
the
noi
se s
igna
l du
e to
the
rmal
and
ele
ctro
nic
nois
e, t
he s
igna
l re
ceiv
ed w
hen
tran
smit
ting
the
elo
ngat
ed
Bar
ker c
ode,
is
r 2(t
) =
b(t)
* s(
t) +
n(t)
(3
.2)
whe
re b
( t) i
s th
e bi
nary
Bar
ker c
ode.
Her
e, t
he a
ster
isk
deno
tes
the
conv
olut
ion
oper
atio
n.
The
rec
eive
d si
gnal
r2(t
) m
ust
be p
roce
ssed
wit
h a
deco
nvol
utio
n fi
lter
to r
esto
re r
ange
re
solu
tion
. In
the
nois
e fr
ee c
ase,
app
lyin
g th
e in
vers
e fi
lter
1 H
r(f)
= B
(f)
(3.3
)
whe
re B
(f)
is t
he F
ouri
er T
rans
form
of
b(t),
wou
ld p
rodu
ce th
e sa
me
sign
al a
s ob
tain
ed
usin
g th
e co
nven
tion
al t
rans
mit
pul
se.
In a
rea
l si
tuat
ion
whe
re n
oise
is
a pa
rt o
f th
e
3.3
Num
eric
al R
esul
ts
35
rece
ived
sig
nal,
this
inve
rse
filte
r w
ould
, ho
wev
er,
caus
e a
larg
e am
plif
icat
ion
of t
he n
oise
si
gnal
at
freq
uenc
ies
whe
re j
B(f
)j i
s lo
w.
To c
ompe
nsat
e fo
r th
is u
nwan
ted
effe
ct w
e m
ay u
se a
sta
bili
zed
inve
rse
filte
r in
stea
d. T
his
deco
nvol
utio
n fi
lter
take
s th
e fo
rm
H
-B
*(f)
w
(f)
-jB
(f)j
2 +
maxJ~
(J)J2
(3.4
)
whe
re B
*(f)
is
the
com
plex
con
juga
te o
f B
(f),
and
N i
s a
nois
e pa
ram
eter
, w
hile
the
te
rm m
axJ~
(J)J
2
can
be t
houg
ht o
f as
the
inv
erse
of
the
SN
R o
f th
e sy
stem
. T
his
filte
r is
si
mil
ar to
the
opti
mal
Wie
ner
filte
r [1
, Cha
pter
8.1
] [3
2, C
hapt
er 8
.5]
obta
ined
per
form
ing
a le
ast-
squa
res
deri
vatio
n. W
e se
e th
at i
f N i
s la
rge
com
pare
d to
jB
(f) I
the
filte
r is
sim
ilar
to
the
inv
erse
filt
er i
n E
q. 3
.3.
On
the
othe
r ha
nd,
as N
is
redu
ced
the
filte
r w
ill
appr
oach
B*(
f)
HM
(f)
= N
ma
xjB
(f)j
2
(3.5
)
whi
ch is
the
mat
ched
filt
er c
omm
only
use
d in
rad
ar im
agin
g [2
1, C
hapt
er 4
.8]
[40,
Cha
pte
r5.4
].
3.2.
3 B
ubbl
e O
scill
atio
n
The
con
tras
t bu
bble
is
assu
med
to
be a
sph
eric
al b
ubbl
e of
gas
enc
apsu
late
d in
a t
hin
shel
l. T
he d
iam
eter
of
the
bubb
le i
s m
uch
less
tha
n th
e w
avel
engt
h o
f th
e in
com
ing
wav
e fi
eld
and
the
bubb
le t
hus
expe
rien
ces
an a
ppro
xim
atel
y sp
atia
l un
ifor
m f
ield
and
th
e bu
bble
osc
illa
tion
is
assu
med
to
be p
urel
y sp
heri
cal.
Sim
ulat
ions
for
bub
ble
radi
us
osci
llat
ions
and
aco
usti
c sc
atte
ring
are
don
e us
ing
the
num
eric
al m
odel
dev
elop
ed b
y A
ngel
sen
et a
l [4
]. T
his
mod
el i
nclu
des
an e
quat
ion
for
the
rela
tion
bet
wee
n pr
essu
re
and
radi
al s
trai
n in
a th
in s
hell
enc
apsu
lati
ng a
gas
bub
ble.
The
mod
el a
llow
s fo
r a
fini
te
spee
d o
f so
und
in t
he m
ediu
m s
urro
undi
ng t
he b
ubbl
e, t
hus
taki
ng r
adia
tion
los
ses
from
th
e bu
bble
int
o ac
coun
t. O
ther
wis
e it
is
com
para
ble
to t
he w
ell
know
n R
ayle
igh-
Ple
sset
eq
uati
on [3
3] [
31]
and
the
two
mod
els
give
sim
ilar
resu
lts
for
inci
dent
pre
ssur
e am
plit
udes
an
d bu
bble
par
amet
ers
stud
ied
in t
his
pape
r.
3.3
Num
eric
al R
esul
ts
3.3.
1 N
oise
Fre
e Se
quen
ce w
itho
ut H
arm
onic
Com
pone
nts
If a
nois
e fr
ee B
arke
r se
quen
ce,
as s
how
n in
the
low
er p
anel
of F
ig. 3
.1, i
s pr
oces
sed
wit
h th
e st
abil
ized
inve
rse
filte
r in
Eq.
3.4
, the
rang
e si
delo
be le
vel o
f the
com
pres
sed
sequ
ence
ca
n be
set
arb
itra
rily
by
choo
sing
the
para
met
er N
. A
low
ran
ge s
idel
obe
leve
l is
esse
ntia
l in
med
ical
ult
raso
und
imag
ing
whe
re t
he d
ispl
ayed
dyn
amic
ran
ge i
s la
rge
to e
nsur
e th
e
36 (a
) Si
delo
bes
afte
r pu
lse
com
pres
sion
.
Pap
erB
(b)
Wid
th o
f m
ainl
obe
afte
r pu
lse
com
pres
si
on.
Fig
ure
3.2:
Lef
t pa
nel:
Exa
mpl
e o
f a
nois
e fr
ee c
ompr
esse
d se
quen
ce w
here
B
(f)
is n
orm
aliz
ed to
1 a
nd N
in
Eq.
3.4
is s
et to
103
. R
ight
pan
el:
Abs
olut
e va
lue
of o
ne o
f the
sub
puls
es in
the
Bar
ker
sequ
ence
sho
wn
in th
e lo
wer
pan
el
in F
ig.
3.1
(sol
id l
ine)
. R
ange
mai
nlob
e af
ter
deco
nvol
utio
n o
f th
e B
arke
r se
quen
ce i
n th
e lo
wer
pan
el i
n F
ig.
3.1
wit
h th
e st
abil
ized
inv
erse
filt
er i
n E
q. 3
.4 (
dash
ed li
ne).
abil
ity
to s
epar
ate
stro
ng a
nd w
eak
scat
tere
rs.
Fig.
3.2
(a)
show
s an
exa
mpl
e w
here
B(f
) is
nor
mal
ized
to
1 an
d th
e pa
ram
eter
N h
as b
een
set
to 1
03 an
d th
e ra
nge
side
lobe
lev
el
in t
he c
ompr
esse
d no
ise
free
seq
uenc
e is
thu
s do
wn
at 6
0 dB
.
The
wid
th o
f th
e ra
nge
mai
nlob
e in
the
com
pres
sed
sequ
ence
is a
lso
an i
mpo
rtan
t par
am
eter
and
sho
uld
be a
s sm
all
as p
ossi
ble
to e
nsur
e go
od r
ange
res
olut
ion
in t
he u
ltra
soun
d im
age.
T
he w
idth
of
the
mai
nlob
e is
giv
en b
y th
e le
ngth
of
the
indi
vidu
al s
ubpu
lses
in
the
tran
smit
ted
Bar
ker
sequ
ence
as
show
n in
Fig
. 3.
2(b)
. H
ere,
the
sol
id l
ine
is t
he a
bso
lute
val
ue o
f one
of
the
subp
ulse
s in
the
low
er p
anel
of F
ig.
3.1,
whi
le t
he d
ashe
d li
ne
depi
cts
the
rang
e m
ainl
obe
obta
ined
aft
er d
econ
volu
tion
wit
h th
e st
abil
ized
inve
rse
filte
r in
Eq.
3.4
. W
e ob
serv
e th
at t
he m
ainl
obe
of
the
com
pres
sed
sequ
ence
is
equa
l to
the
en
velo
pe o
f the
sub
puls
es p
roce
ssed
by
the
stab
iliz
ed in
vers
e fil
ter.
In a
typi
cal
med
ical
ult
raso
und
cont
rast
imag
ing
situ
atio
n th
ere
are
seve
ral
effe
cts
whi
ch
pote
ntia
lly
wil
l inc
reas
e th
e ra
nge
side
lobe
leve
l sho
wn
in F
ig.
3.2(
a).
We
wil
l now
try
to
inve
stig
ate
som
e of
the
pres
umab
ly m
ost i
mpo
rtan
t eff
ects
.
3.3
Nu
mer
ical
Res
ults
37
3.3.
2 P
rese
nce
of S
ever
al H
arm
onic
s in
Sig
nal f
or P
roce
ssin
g
Ult
raso
und
scat
teri
ng f
rom
the
con
tras
t bub
bles
is h
ighl
y no
nlin
ear
and
the
rece
ived
con
tr
ast
agen
t si
gnal
wil
l co
ntai
n en
ergy
in
seve
ral
harm
onic
ban
ds.
Alt
houg
h it
is p
ossi
ble
to u
se t
he i
ndic
ated
pul
se c
ompr
essi
on t
echn
ique
wit
hout
ban
dpas
s fi
lteri
ng,
i.e.
on t
he
tota
l re
ceiv
ed s
igna
l, th
is i
s no
t at
trac
tive
in
cont
rast
age
nt i
mag
ing
sinc
e th
e st
rong
fun
da
men
tal
com
pone
nt s
catt
ered
fro
m t
he t
issu
e w
ould
mas
k th
e si
gnal
sca
tter
ed f
rom
the
co
ntra
st a
gent
. U
sing
thi
s fo
rm o
f ph
ase
codi
ng o
n th
e re
ceiv
ed s
econ
d ha
rmon
ic c
ompo
nent
wou
ld
not y
ield
goo
d re
sult
s as
can
be
seen
fro
m t
he f
ollo
win
g eq
uati
on
00
s(t)
= L
::>
n(t)
pn(t
-T)
(3
.6)
n=
l
whe
re a
n(t)
are
posi
tive
am
plit
ude
func
tion
s an
d w
here
the
rece
ived
sig
nal,
s(t)
, is
mod
el
ed b
y a
sim
ple
pow
er e
xpan
sion
of
the
tran
smit
ted
puls
e, p
(t),
wit
h a
tim
e de
lay
T.
We
obse
rve
that
the
sec
ond
harm
onic
com
pone
nt is
a q
uadr
atic
eff
ect o
f th
e tr
ansm
itte
d si
gnal
and
pha
se c
odin
g w
ith
Bar
ker
sequ
ence
s, w
here
the
tra
nsm
itte
d se
quen
ce is
pha
se
code
d w
ith
1r, ut
iliz
ing
the
seco
nd o
r an
y ot
her
even
har
mon
ic c
ompo
nent
of t
he r
ecei
ved
sign
al i
s no
t fr
uitf
ul.
Usi
ng t
his
puls
e co
mpr
essi
on t
echn
ique
on
the
rece
ived
thi
rd h
ar
mon
ic c
ompo
nent
is,
how
ever
, in
tere
stin
g. T
he th
ird
harm
onic
com
pone
nt f
rom
tis
sue
is,
as p
revi
ousl
y in
dica
ted,
low
whe
n im
agin
g w
ith
rela
tive
ly l
ow i
nten
sity
, w
hile
the
thi
rd
harm
onic
com
pone
nt fr
om t
he c
ontr
ast a
gent
is c
onsi
dera
ble
due
to t
he h
igh
nonl
inea
rity
o
f th
e ul
tras
ound
sca
tter
ing
from
gas
bub
bles
.
Usi
ng t
he s
catt
ered
thir
d ha
rmon
ic c
ompo
nent
mea
ns t
hat
a th
ird
harm
onic
ban
dpas
s fil
te
r m
ust
be a
ppli
ed o
n th
e re
ceiv
ed s
igna
l in
add
itio
n to
the
sta
bili
zed
inve
rse
filte
r in
E
q. 3
.4.
The
upp
er p
anel
of F
ig.
3.3
show
s an
exa
mpl
e of
a f
our
bit B
arke
r co
de c
onsi
st
ing
of
thre
e ha
rmon
ic c
ompo
nent
s ac
cord
ing
to E
q. 3
.6,
and
in t
he l
ower
pan
el w
e se
e th
e ab
solu
te v
alue
of
the
Fou
rier
tran
sfor
m o
f th
e ti
me
sequ
ence
in
the
uppe
r pa
nel.
The
sp
ecia
l ap
pear
ance
of
the
thre
e ha
rmon
ic c
ompo
nent
s in
fre
quen
cy d
omai
n is
a r
esul
t o
f th
e co
mbi
ng e
ffec
t int
rodu
ced
by t
he B
arke
r se
quen
ce.
The
thir
d ha
rmon
ic c
ompo
nent
of t
he s
eque
nce
is o
btai
ned
wit
h a
thir
d ha
rmon
ic b
andp
ass
filte
r an
d th
e st
abil
ized
inv
erse
filt
er i
s th
en a
ppli
ed t
o co
mpr
ess
the
code
. R
esul
ts u
sing
tw
o di
ffer
ent b
andw
idth
s in
the
thi
rd h
arm
onic
ban
dpas
s fi
lter
are
show
n in
Fig
. 3.
4. T
he
soli
d li
ne i
s th
e id
eal
com
pres
sed
code
obt
aine
d fo
r an
im
agin
ed c
ase
whe
n th
ere
is n
o fu
ndam
enta
l or
sec
ond
harm
onic
com
pone
nt p
rese
nt in
the
rece
ived
sig
nal,
i.e.
only
a3(t
) is
dif
fere
nt f
rom
zer
o in
Eq.
3.6
. T
he d
otte
d li
ne i
s ob
tain
ed u
sing
a t
hird
har
mon
ic
Gau
ssia
n fi
lter
wit
h a
-6 d
B b
andw
idth
equ
al t
o 0.
4 M
Hz
(nar
row
band
) on
the
sig
nal
in
Fig
. 3.
3, w
hile
the
das
hed
line
is
obta
ined
usi
ng a
sim
ilar
filt
er w
ith
a -6
dB
ban
dwid
th
equa
l to
0.85
MH
z (b
road
band
) on
the
sam
e si
gnal
. F
or th
e da
shed
line
, the
ran
ge s
ide l
obe
leve
l has
incr
ease
d si
gnif
ican
tly
com
pare
d to
the
idea
l sit
uati
on w
hen
ther
e is
onl
y a
thir
d ha
rmon
ic c
ompo
nent
pre
sent
in th
e to
tal
sign
al.
The
reas
on is
an
over
lap
from
the
sec
ond
38
"'H
llj
0.2
0.1 0
-0.1
-o.2 o
5 10
15
20
25
30
[J
.ls]
·!.l :U~
M~~.~:
I 0
0.5
1 1.
5 2
2.5
3 3.
5 4
{MH
z]
Pap
erB
Fig
ure
3.3:
U
pper
pan
el:
Fou
r bi
t B
arke
r se
quen
ce c
onsi
stin
g o
f th
ree
harm
onic
com
po
nent
s ac
cord
ing
to E
q. 3
.6.
Low
er p
anel
: A
bsol
ute
valu
e of
Fou
rier
tra
nsfo
rm o
f th
e se
quen
ce in
the
uppe
r pa
nel.
harm
onic
com
pone
nt i
nto
the
thir
d ha
rmon
ic b
and
proc
esse
d by
the
sta
bili
zed
inve
rse
filte
r. Thi
s ov
erla
p in
fre
quen
cy i
s se
en i
n Fi
g. 3
.5 w
hich
dis
play
s th
e ab
solu
te v
alue
of
the
Fou
rier
Tra
nsfo
rm o
f on
e o
f th
e su
bpul
ses
in t
he u
pper
pan
el o
f Fi
g. 3
.3.
Whe
n us
ing
the
broa
dban
d th
ird
harm
onic
ban
dpas
s fi
lter
(dot
ted
line
wit
h ci
rcle
s) t
he l
evel
of
seco
nd h
arm
onic
(da
shed
line
) in
the
pass
band
of t
he f
ilter
is m
uch
high
er th
an w
hen
the
narr
owba
nd t
hird
har
mon
ic b
andp
ass
filte
r (d
otte
d lin
e w
ith
diam
onds
) is
app
lied.
T
he
pres
ence
of
a se
cond
har
mon
ic c
ompo
nent
in t
he s
igna
l fed
to
the
stab
iliz
ed in
vers
e fi
lter
easi
ly in
crea
ses
the
rang
e si
delo
bes,
as
seen
fro
m F
ig.
3.4,
due
to th
e fa
ct t
hat t
his
seco
nd
harm
onic
com
pone
nt is
not
pha
se c
oded
as
prev
ious
ly e
xpla
ined
. W
hen
the
narr
owba
nd
filte
r is
use
d w
e se
e fr
om F
ig.
3.4
that
the
inc
reas
e in
ran
ge s
idel
obes
is
not
so d
rast
ic.
App
lyin
g th
e na
rrow
band
filt
er t
here
is,
how
ever
, a
sign
ific
ant
incr
ease
in
the
wid
th o
f th
e m
ainl
obe
whi
ch w
ill d
egra
de r
ange
res
olut
ion
and
henc
e th
e ab
ility
to s
epar
ate
targ
ets
whi
ch a
re c
lose
.
3.3.
3 E
ffec
t of A
cous
tic
Pow
er A
bsor
ptio
n
Bot
h th
e fo
rwar
d pr
opag
atin
g tr
ansm
it p
ulse
and
the
sca
tter
ed p
ulse
s fr
om t
he c
ontr
ast
bubb
les
prop
agat
ing
back
to
the
ultr
asou
nd t
rans
duce
r w
ill b
e m
odif
ied
by a
fre
quen
cy
depe
nden
t ab
sorp
tion
in t
he t
issu
e. T
his
freq
uenc
y de
pend
ent
abso
rptio
n w
ill,
depe
ndin
g o
f nu
mbe
r of
wav
e le
ngth
s pr
opag
ated
, sh
ift
the
scat
tere
d pu
lses
som
ewha
t dow
n in
fre
qu
ency
. In
the
low
er p
anel
of
Fig.
3.6
, it
is
show
n ho
w t
he n
orm
aliz
ed t
hird
har
mon
ic
com
pone
nt o
f a
subp
ulse
aff
ecte
d by
an
abso
rpti
on o
f 10
dB
/MH
z (d
ashe
d li
ne)
may
ap
pear
com
pare
d to
one
not
aff
ecte
d by
abs
orpt
ion
(sol
id li
ne).
The
upp
er p
anel
of F
ig.
3.6
3.3
Nu
mer
ical
Res
ult
s
-10
-20
-30
~._-4(
1 ' I I
\ I
I \
I I
-50
I I I
-<0
I' \
I I I
I I
I I I
{I"]
('\
/\
I I
: I
I I I I I I
~··.,
!'"J
I'
I'
I I I I I ,!
39
Fig
ure
3.4:
Eff
ect
of b
andw
idth
in t
he t
hird
har
mon
ic b
andp
ass
filte
r on
ran
ge s
idel
obe
leve
ls a
nd w
idth
of
mai
nlob
e.
Sol
id l
ine
is t
he r
esul
ts a
fter
pul
se c
ompr
essi
on f
or t
he
idea
l ca
se w
itho
ut a
fun
dam
enta
l or
sec
ond
harm
onic
com
pone
nt p
rese
nt i
n th
e or
igin
al
sequ
ence
. T
he d
ashe
d li
ne is
obt
aine
d us
ing
the
broa
dban
d th
ird
harm
onic
Gau
ssia
n fi
lter
on t
he s
eque
nce
in F
ig.
3.3
whi
le t
he d
otte
d li
ne i
s ob
tain
ed u
sing
the
nar
row
band
thi
rd
harm
onic
Gau
ssia
n fi
lter
on th
e sa
me
sequ
ence
.
w,--.---r~-o---.---.--~~.o=·bro=,d~~~.,~~~~
35
25
20
~ 15
10
-5
' '
' \
I \
I \
I I
I \ I
·~·narrowband
filter
Fig
ure
3.5:
Pul
se c
onsi
stin
g of
thre
e ha
rmon
ic b
ands
, se
cond
har
mon
ic b
and
disp
laye
d as
das
hed
line.
Thi
rd h
arm
onic
Gau
ssia
n fi
lters
, br
oadb
and
and
narr
owba
nd,
dott
ed l
ine
wit
h ci
rcle
s an
d do
tted
line
wit
h di
amon
ds, r
espe
ctiv
ely.
40
Pap
erB
~:~
-I U
7 ~
I U
9
~
W
=
""'
Fig
ure
3.6:
Upp
er p
anel
: N
orm
aliz
ed a
bsol
ute
valu
e o
f F
ouri
er t
rans
form
of
Bar
ker
se
quen
ce n
ot a
ffec
ted
by a
bsor
ptio
n (s
olid
line
) an
d af
fect
ed b
y an
abs
orpt
ion
of
10 d
B/M
Hz
(das
hed
line)
. L
ower
pan
el:
Nor
mal
ized
thi
rd h
arm
onic
com
pone
nt o
f on
e o
f th
e su
bpu
lses
in a
Bar
ker
sequ
ence
not
aff
ecte
d by
abs
orpt
ion
(sol
id li
ne)
and
one
affe
cted
by
an
abso
rpti
on o
f 10
dB
/MH
z (d
ashe
d li
ne).
depi
cts
the
abso
lute
val
ue o
f the
Fou
rier
tran
sfor
m o
f the
Bar
ker s
eque
nces
bef
ore
the
thir
d ha
rmon
ic f
ilter
is
appl
ied
whe
re t
he s
olid
line
is t
he o
rigi
nal s
eque
nce
and
the
dash
ed li
ne
is t
he s
eque
nce
affe
cted
by
abso
rpti
on.
The
tw
o se
quen
ces
have
bee
n no
rmal
ized
to t
he
sam
e m
axim
um v
alue
.
In F
ig.
3.7
the
resu
lts
afte
r ap
plyi
ng t
he s
tabi
lize
d in
vers
e fi
lter
on t
he o
rigi
nal
sequ
ence
no
t af
fect
ed b
y ab
sorp
tion
and
on
the
one
affe
cted
by
an a
bsor
ptio
n o
f 10
dB
/MH
z ar
e sh
own.
The
upp
er p
anel
is f
or th
e ca
se w
hen
the
broa
dban
d th
ird
harm
onic
Gau
ssia
n fi
lter
is u
sed
whi
le th
e lo
wer
pan
el s
how
s re
sult
s ob
tain
ed u
sing
the
narr
owba
nd th
ird
harm
onic
ba
ndpa
ss f
ilter
. W
e se
e th
at w
hen
appl
ying
the
narr
owba
nd a
nd b
road
band
thir
d ha
rmon
ic
band
pass
filt
er,
the
freq
uenc
y de
pend
ent
abso
rpti
on i
ncre
ases
the
ran
ge s
idel
obe
leve
l by
ar
ound
3 a
nd 8
dB
, re
spec
tivel
y.
3.3.
4 B
ubbl
e Si
gnal
with
Inf
init
e C
NR
The
upp
er p
anel
of
Fig.
3.8
sho
ws
the
scat
tere
d pr
essu
re p
ulse
fro
m a
bub
ble
wit
h re
so
nanc
e fr
eque
ncy
arou
nd 4
MH
z w
hen
driv
en b
y th
e ac
oust
ic p
ress
ure
puls
e in
the
low
er
pane
l of F
ig.
3.1
whi
ch h
as a
cen
ter
freq
uenc
y o
f 1
MH
z.
In t
he l
ower
pan
el o
f Fig
. 3.
8,
the
abso
lute
val
ue o
f th
e F
ouri
er T
rans
form
of
the
sequ
ence
in
uppe
r pa
nel
is d
ispl
ayed
an
d th
e sc
atte
red
pres
sure
pul
se i
s cl
earl
y se
en t
o co
ntai
n en
ergy
at
seve
ral
harm
onic
ba
nds.
The
thi
rd h
arm
onic
com
pone
nt o
f th
e sc
atte
red
pres
sure
pul
se i
s ob
tain
ed u
sing
th
e tw
o ha
rmon
ic G
auss
ian
filte
rs f
rom
Fig
. 3.
5 an
d th
e fi
ltere
d se
quen
ces
are
proc
esse
d
3.3
Num
eric
al R
esul
ts
41
Fig
ure
3.7:
E
ffec
t o
f ab
sorp
tion
on
rang
e si
delo
be l
evel
. S
olid
lin
e is
obt
aine
d fr
om
the
orig
inal
seq
uenc
e w
hile
the
das
hed
line
is o
btai
ned
from
the
seq
uenc
e af
fect
ed b
y an
abs
orpt
ion
of 1
0 dB
/MH
z.
Upp
er p
anel
: T
he b
road
band
Gau
ssia
n th
ird
harm
onic
ba
ndpa
ss f
ilter
has
bee
n ap
plie
d. L
ower
pan
el:
The
nar
row
band
thir
d ha
rmon
ic G
auss
ian
filte
r ha
s be
en a
pplie
d. O.
Q2tt±
ffi
O.o
! 0
--0.
01
-0.0
2
-0.0
3
-0.0
40,._
__._
----
-'c-, --~10'-------':-15 -
'----":,,---'-:,~-----:',.
'"''
·J ~~~
M~.~J.
1 0
0.5
I 1.
5 2
2.5
3 3.
5 4
4.5
5 IM
HzJ
Fig
ure
3.8:
U
pper
pan
el:
Sca
tter
ed p
ress
ure
puls
e fr
om a
4 J
-lm b
ubbl
e w
ith
reso
nanc
e fr
eque
ncy
arou
nd 4
MH
z w
hen
driv
en b
y th
e ac
oust
ic p
ulse
in th
e lo
wer
pan
el o
f Fig
. 3 .
1.
Low
er p
anel
: A
bsol
ute
valu
e o
f th
e F
ouri
er T
rans
form
of
the
tim
e se
quen
ce i
n th
e up
per
pane
l.
42
Pap
erB
Fig
ure
3.9:
Res
ults
aft
er a
pply
ing
the
stab
iliz
ed in
vers
e fi
lter
on th
e th
ird
harm
onic
com
po
nent
of t
he s
catt
ered
pre
ssur
e pu
lse
show
n in
Fig
. 3.
8. S
olid
lin
e is
obt
aine
d us
ing
the
narr
owba
nd t
hird
har
mon
ic f
ilter
whi
le t
he d
ashe
d lin
e is
obt
aine
d us
ing
the
broa
dban
d th
ird
harm
onic
ban
dpas
s fil
ter.
wit
h th
e st
abil
ized
inve
rse
filte
r fr
om E
q. 3
.4.
Res
ults
aft
er d
econ
volu
tion
are
dep
icte
d in
Fig
. 3.
9 w
here
the
sol
id a
nd d
ashe
d lin
es a
re
obta
ined
usi
ng t
he n
arro
wba
nd a
nd b
road
band
thi
rd h
arm
onic
ban
dpas
s fil
ter,
resp
ec
tivel
y.
Aft
er d
econ
volu
tion,
the
sid
elob
e le
vel
is a
ppro
xim
atel
y do
wn
at 4
0 an
d 20
dB
us
ing
the
narr
owba
nd a
nd b
road
band
ban
dpas
s fil
ter,
resp
ectiv
ely.
The
se r
esul
ts a
re o
bta
ined
on
sequ
ence
s fo
r w
hich
unc
orre
late
d no
ise
stil
l has
not
bee
n ad
ded
and
the
side
lobe
le
vel
in t
he c
ompr
esse
d pu
lse
is m
ainl
y du
e to
the
pre
senc
e of
sec
ond
(and
fou
rth)
har
m
onic
sig
nal c
ompo
nent
s in
the
filt
ered
sig
nals
pro
cess
ed b
y th
e st
abil
ized
inve
rse
filte
r. Fr
om F
ig. 3
.4 it
was
see
n th
at th
e pr
esen
ce o
f a s
econ
d ha
rmon
ic c
ompo
nent
in t
he s
igna
l fe
d to
the
sta
bili
zed
inve
rse
filte
r si
gnif
ican
tly i
ncre
ased
the
ran
ge s
idel
obe
leve
l in
the
co
mpr
esse
d si
gnal
rel
ativ
e to
the
im
agin
ed s
itua
tion
hav
ing
only
the
thi
rd h
arm
onic
sig
na
l co
mpo
nent
. A
lso,
as
seen
fro
m t
he l
ower
pan
el o
f Fi
g. 3
.8,
the
harm
onic
com
pone
nts
scat
tere
d be
low
res
onan
ce a
re s
omew
hat s
hift
ed u
p in
fre
quen
cy r
elat
ive
to h
arm
onic
s o
f th
e dr
ive
freq
uenc
y w
hich
is c
ente
red
arou
nd 1
MH
z. T
he u
pwar
d fr
eque
ncy
shif
t of t
he s
catt
ered
th
ird
harm
onic
com
pone
nt is
opp
osit
e to
the
dow
nwar
d fr
eque
ncy
shif
t dis
play
ed i
n th
e up
per
pane
l of
Fig
. 3.
6 ob
tain
ed d
ue t
o ab
sorp
tion.
T
his
freq
uenc
y sh
ift
wil
l th
us a
lso
cont
ribu
te s
omew
hat
to t
he s
idel
obe
leve
l obt
aine
d in
Fig
. 3.
9 in
a s
imil
ar m
anne
r as
the
ab
sorp
tion
inc
reas
ed th
e si
delo
bes
in F
ig.
3.7.
In
add
ition
, as
dis
cuss
ed i
n th
e ne
xt s
ectio
n, t
he r
espo
nse
from
the
non
linea
r, r
es
onan
t bu
bble
wil
l de
pend
on
the
band
wid
th o
f th
e in
cide
nt d
rive
pul
se a
nd t
he b
ubbl
e m
ay r
espo
nd d
iffe
rent
ly o
n th
e th
ird
subp
ulse
whe
re th
e po
lari
ty is
inv
erte
d re
lativ
e to
the
thre
e ot
her
subp
ulse
s. T
he in
cide
nt d
rive
pul
se in
the
low
er p
anel
of F
ig.
3.1
is,
how
ever
,
3.3
Num
eric
al R
esul
ts ~TEB
3 1
-O.l
o
5 lO
15
20
25
{l
.l.s]
lEff
il
-O.l
o
5 10
15
2
0
25
[I.J.s
]
43
Fig
ure
3.10
: U
pper
pan
el: B
road
band
inc
iden
t Bar
ker
sequ
ence
. L
ower
pan
el:
Res
ulti
ng
scat
tere
d pr
essu
re p
ulse
fro
m a
4 J
.Lm b
ubbl
e.
suff
icie
ntly
nar
row
ban
ded
for
this
eff
ect t
o be
mar
gina
l.
3.3.
5 T
rans
mit
Pul
se B
andw
idth
The
con
tras
t bub
ble
repr
esen
ts a
hig
hly
nonl
inea
r an
d re
sona
nt m
ediu
m f
or s
catt
erin
g o
f th
e in
cide
nt tr
ansm
it p
ulse
. F
or a
wid
e ba
ndw
idth
inci
dent
pul
se c
onsi
stin
g o
f on
ly a
few
ha
lf p
erio
ds, t
he b
ubbl
e m
ay r
espo
nd d
iffe
rent
ly w
hen
the
inci
dent
osc
illa
tion
beg
ins
wit
h a
rare
fact
ion
peri
od ra
ther
than
a c
ompr
essi
on p
erio
d [2
9] [
28].
In
addi
tion
, pro
blem
s w
ith
the
pres
ence
of
seco
nd (
and
four
th)
harm
onic
sig
nal
com
pone
nts
in t
he p
assb
and
of
the
thir
d ha
rmon
ic b
andp
ass
filte
r di
scus
sed
in S
ec.
3.3.
2 w
ill
be b
igge
r ap
plyi
ng b
road
band
tr
ansm
it p
ulse
s.
Fig
. 3.1
0 di
spla
ys a
n ex
ampl
e o
f a b
road
band
tran
smit
Bar
ker
sequ
ence
, up
per
pane
l, an
d th
e re
sult
ing
scat
tere
d pr
essu
re p
ulse
fro
m t
he b
ubbl
e, l
ower
pan
el.
In t
he u
pper
pan
el o
f Fig
. 3.
11 w
e se
e th
e th
ird
harm
onic
env
elop
e, o
btai
ned
wit
h th
e na
rrow
band
ban
dpas
s fi
lter,
of
the
scat
tere
d pu
lse
from
the
low
er p
anel
of F
ig.
3.10
as
the
soli
d lin
e. T
he d
ashe
d li
ne s
how
s th
e re
sult
ing
thir
d ha
rmon
ic e
nvel
ope
obta
ined
by
usin
g a
driv
e pu
lse
sim
ilar
to t
he o
ne in
the
upp
er p
anel
of F
ig.
3.10
but
wit
hout
the
phas
e sh
ift
on th
e th
ird
subp
ulse
, i.e
. co
nsis
ting
of
four
ide
ntic
al s
ubpu
lses
. T
he n
onli
near
res
pons
e fr
om t
he b
ubbl
e w
hen
driv
en b
y th
e br
oadb
and
subp
ulse
wit
h in
vert
ed p
olar
ity
is c
lear
ly
seen
to b
e di
ffer
ent
from
res
ults
obt
aine
d w
ith
the
thre
e ot
her
subp
ulse
s. W
e no
tice
tha
t th
e th
ird
harm
onic
env
elop
e fr
om t
he t
hird
sub
puls
e in
the
sca
tter
ed B
arke
r se
quen
ce i
s so
mew
hat t
ime
dela
yed
and
has
low
er a
mpl
itud
e re
lati
ve to
the
oth
er s
catt
ered
sub
puls
es.
In t
he l
ower
pan
el o
f Fig
. 3.
11,
the
resu
lt a
fter
pul
se c
ompr
essi
on is
dep
icte
d an
d w
e se
e th
at t
he s
idel
obe
leve
l is
ver
y hi
gh c
ompa
red
to F
ig.
3.9
obta
ined
by
usin
g th
e m
ore
44
·li~AhJ
--4~
00
310
320
330
340
350
360
370
380
390
400
(IJ.S
]
Pap
erB
Fig
ure
3.11
: U
pper
pan
el:
Thi
rd h
arm
onic
env
elop
e o
f sc
atte
red
sign
al f
rom
low
er p
anel
o
f Fig
. 3.
1 0,
sol
id li
ne.
Das
hed
line
rep
rese
nts
thir
d ha
rmon
ic e
nvel
ope
obta
ined
by
driv
in
g th
e bu
bble
wit
h fo
ur i
dent
ical
sub
puls
es.
Low
er p
anel
: R
esul
t aft
er p
ulse
com
pres
sion
of
scat
tere
d br
oadb
and
Bar
ker
sequ
ence
.
narr
owba
nded
inc
iden
t dri
ve p
ulse
fro
m t
he l
ower
pan
el o
f Fig
. 3.
1.
A n
ew b
road
band
Bar
ker
sequ
ence
, si
mil
ar to
the
one
in t
he u
pper
pan
el o
f Fig
. 3.
10 b
ut
wit
h hi
gher
am
plit
ude,
is t
hen
used
to d
rive
the
bubb
le.
The
new
inc
iden
t Bar
ker
sequ
ence
an
d th
e re
sult
ing
scat
tere
d pr
essu
re p
ulse
fro
m t
he b
ubbl
e ca
n be
see
n in
the
upp
er a
nd
low
er p
anel
of F
ig.
3 .12
, res
pect
ivel
y.
The
thi
rd h
arm
onic
env
elop
e, o
btai
ned
wit
h th
e na
rrow
band
ban
dpas
s fi
lter,
is d
is
play
ed a
s th
e so
lid
line
in
the
uppe
r pa
nel
of F
ig.
3.13
. A
gain
, th
e da
shed
lin
e re
pre
sent
s th
e re
sult
obt
aine
d by
usi
ng a
dri
ve p
ulse
sim
ilar
to t
he o
ne i
n th
e up
per
pane
l of
Fi
g. 3
.12
but
wit
h no
pha
se in
vers
ion
on th
e th
ird
subp
ulse
. C
ompa
ring
the
upp
er p
anel
s o
f Fig
. 3.1
1 an
d Fi
g. 3
.13,
we
noti
ce th
at th
e th
ird
harm
onic
env
elop
e o
f the
thir
d su
bpul
se
in t
he s
catt
ered
Bar
ker
sequ
ence
s ha
ve s
imil
ar t
ime
shif
ts w
hen
the
broa
dban
d in
cide
nt
Bar
ker
sequ
ence
s of
dif
fere
nt a
mpl
itud
es a
re a
pplie
d. B
y in
spec
ting
the
sca
tter
ed p
ulse
s in
the
low
er p
anel
of
Fig.
3.1
0 an
d 3.
12,
it i
s cl
ear
that
the
thi
rd s
ubpu
lse
star
ts w
ith
mai
nly
low
fre
quen
cy s
igna
l co
mpo
nent
s w
hile
the
mai
n co
ntri
buti
ons
to t
he h
igh
fre
quen
cy c
ompo
nent
s oc
cur
late
r in
the
subp
ulse
. T
he a
mpl
itud
e of
the
thir
d ha
rmon
ic e
nvel
ope
was
in
the
uppe
r pa
nel
of
Fig
. 3.
11
seen
to b
e lo
wer
for
the
thir
d su
bpul
se w
hen
usin
g th
e lo
w a
mpl
itud
e br
oadb
and
inci
dent
B
arke
r se
quen
ce.
Usi
ng th
e hi
gh a
mpl
itud
e br
oadb
and
inci
dent
Bar
ker
sequ
ence
, the
am
pl
itud
e of
the
thir
d ha
rmon
ic e
nvel
ope
is,
how
ever
, hi
gher
for
the
sub
puls
e w
ith
inve
rted
po
lari
ty e
ven
if th
e am
plit
ude
of
the
tota
l sc
atte
red
sign
al i
n th
e lo
wer
pan
el o
f Fig
. 3.
12
is l
ower
for
this
sub
puls
e.
3.3
Nu
mer
ical
Res
ults
45
~Tf±
B 1
-0.40~----L-7, ----
-L~l
O;--
----
'--c
l:';
-5 --
'--:
;';:
20
----
-:!,
. [IJ
.s]
l++
B i
0 5
to
15
20
25
().1S
]
Fig
ure
3.12
: U
pper
pan
el:
Bro
adba
nd i
ncid
ent B
arke
r se
quen
ce.
Low
er p
anel
: R
esul
ting
sc
atte
red
pres
sure
pul
se f
rom
a 4
J-Lm
bub
ble.
{!IS
]
Fig
ure
3.13
: U
pper
pan
el:
Thi
rd h
arm
onic
env
elop
e o
f sca
tter
ed s
igna
l fro
m l
ower
pan
el
of F
ig.
3.12
, so
lid
line.
Das
hed
line
rep
rese
nts
thir
d ha
rmon
ic e
nvel
ope
obta
ined
by
dri
vin
g th
e bu
bble
wit
h fo
ur id
enti
cal s
ubpu
lses
. L
ower
pan
el:
Res
ult a
fter
pul
se c
ompr
essi
on
of
scat
tere
d br
oadb
and
Bar
ker
sequ
ence
.
46
-fff
fi
~ -Q.0
4
0 5
10
15
20
25
30
("
'I
Pap
erB
Fig
ure
3.14
: U
pper
pan
el:
Sig
nal
scat
tere
d fr
om b
ubbl
e dr
iven
by
the
acou
stic
pul
se i
n th
e lo
wer
pan
el o
f Fi
g. 3
.1 w
here
the
equ
ilib
rium
bub
ble
radi
us i
s in
crea
sed
by 5
% f
or
the
last
sub
puls
e. L
ower
pan
el:
Res
ults
aft
er th
ird
harm
onic
ban
dpas
s fi
lteri
ng o
n th
e th
e pu
lse
in t
he u
pper
pan
el a
nd c
ompr
essi
ng u
sing
the
sta
bili
zed
inve
rse
filte
r. S
olid
line
is
obta
ined
usi
ng t
he n
arro
wba
nd b
andp
ass
filte
r an
d th
e da
shed
lin
e is
obt
aine
d us
ing
the
broa
dban
d ba
ndpa
ss f
ilter
.
3.3.
6 V
aria
ble
Aco
usti
c B
ubbl
e P
aram
eter
s
Con
tras
t ag
ent
gas
bubb
les
are
not
enti
rely
sta
ble
and
as t
he b
ubbl
es a
re s
ubje
cted
to
the
inci
dent
pre
ssur
e pu
lses
the
y m
ay,
to s
ome
degr
ee,
chan
ge t
heir
aco
usti
c pr
oper
ties
. T
he t
hin
shel
l enc
apsu
lati
ng t
he g
as b
ubbl
e dr
amat
ical
ly c
hang
es t
he a
cous
tic p
rope
rtie
s an
d st
abil
ity
of th
e co
ntra
st b
ubbl
e co
mpa
red
to a
fre
e ga
s bu
bble
not
enc
apsu
late
d by
th
is s
hell
[19
]. T
he p
rope
rtie
s o
f the
she
ll h
ence
, to
a l
arge
ext
ent,
dete
rmin
e th
e ac
oust
ic
prop
erti
es o
f the
bub
ble.
Whe
n th
e bu
bble
is s
ubje
cted
to a
Bar
ker s
eque
nce
it is
the
refo
re,
cont
rary
to t
he s
itua
tion
in P
ower
Dop
pler
tech
niqu
es, i
mpo
rtan
t for
the
she
ll to
mai
ntai
n its
pro
pert
ies
for
the
enti
re d
urat
ion
of th
e se
quen
ce.
The
upp
er p
anel
in
Fig.
3.1
4 sh
ows
an e
xam
ple
of th
e sc
atte
red
pres
sure
pul
se f
rom
a
bubb
le w
hen
the
bubb
le h
as i
ncre
ased
its
equi
libr
ium
rad
ius
by 5
% f
or t
he l
ast s
ubpu
lse
com
pare
d to
the
thr
ee f
irst
sub
puls
es.
Thi
s in
crea
se i
n ra
dius
cou
ld f
or e
xam
ple
be d
ue
to a
cha
nge
in t
he p
rope
rtie
s of
the
enc
apsu
lati
ng s
hell
or a
hea
ting
of
the
gas
insi
de
the
shel
l. T
he r
adiu
s in
crea
se r
esul
ts i
n a
smal
l re
duct
ion
of th
e re
sona
nce
freq
uenc
y o
f th
e bu
bble
. In
the
low
er p
anel
of
Fig.
3.1
4 w
e se
e th
e re
sults
aft
er t
he t
hird
har
mon
ic
com
pone
nt o
f the
seq
uenc
e in
the
upp
er p
anel
has
bee
n fi
ltere
d ou
t and
pro
cess
ed b
y th
e st
abil
ized
inv
erse
filt
er.
Com
pare
d to
res
ults
obt
aine
d in
Fig
. 3.
9, r
ange
sid
elob
e le
vels
ar
e si
gnif
ican
tly in
crea
sed
both
whe
n ap
plyi
ng th
e br
oadb
and
band
pass
filte
r (d
ashe
d lin
e)
and
the
narr
owba
nd b
andp
ass
filte
r (s
olid
line
).
Fig.
3.1
5 de
pict
s th
e no
rmal
ized
env
elop
e o
f the
thir
d ha
rmon
ic c
ompo
nent
in th
e la
st s
ub-
3.3
Nu
mer
ical
Res
ults
47
0.9
0.3
Fig
ure
3.15
: N
orm
aliz
ed e
nvel
ope
of th
e th
ird
harm
onic
com
pone
nt in
the
las
t su
bpul
se
in t
he u
pper
pan
el o
f Fig
. 3.
8 an
d 3.
14, s
olid
and
das
hed
line,
res
pect
ivel
y.
puls
e in
the
upp
er p
anel
of F
ig.
3.8
and
3.14
, sol
id a
nd d
ashe
d li
ne r
espe
ctiv
ely.
A s
mal
l ch
ange
in
the
equi
libr
ium
rad
ius
of th
e bu
bble
sli
ghtl
y ch
ange
s th
e ac
oust
ic p
rope
rtie
s,
and
henc
e th
e re
sona
nce
freq
uenc
y of
the
bubb
le.
Thi
s ha
s an
eff
ect o
n th
e sc
atte
red
thir
d ha
rmon
ic c
ompo
nent
env
elop
e.
The
env
elop
e o
f th
e th
ird
harm
onic
com
pone
nt i
n th
e su
bpul
se s
catt
ered
fro
m t
he b
ubbl
e w
ith
the
incr
ease
d eq
uili
briu
m ra
dius
(da
shed
line
) is
so
mew
hat d
elay
ed a
nd s
tret
ched
rel
ativ
e to
the
env
elop
e ob
tain
ed fr
om t
he s
ubpu
lse
scat
te
red
from
the
bub
ble
wit
h th
e un
chan
ged
orig
inal
equ
ilib
rium
rad
ius
(sol
id l
ine)
. T
he
ampl
itud
e o
f the
thir
d ha
rmon
ic c
ompo
nent
sca
tter
ed f
rom
the
incr
ease
d bu
bble
is a
roun
d 3
dB l
arge
r th
an f
rom
the
ori
gina
l bu
bble
. In
add
itio
n to
thi
s am
plit
ude
vari
atio
n, t
he
smal
l de
lay
and
stre
tch
of
the
thir
d ha
rmon
ic e
nvel
ope
sign
ific
antly
inc
reas
es t
he r
ange
si
delo
be le
vel i
n th
e co
mpr
esse
d pu
lse
as s
how
n in
the
low
er p
anel
of F
ig.
3.14
.
If in
stea
d th
e bu
bble
rad
ius
is i
ncre
ased
by
5 %
for
the
fir
st s
catt
ered
sub
puls
e, w
hile
th
e th
ree
last
sub
puls
es a
re s
catt
ered
fro
m t
he o
rigi
nal
4 p,
m b
ubbl
e, w
e ob
tain
the
re
sults
dis
play
ed i
n Fi
g. 3
.16.
Aga
in,
the
soli
d an
d da
shed
lin
es i
n th
e lo
wer
pan
el o
f th
e fi
gure
is
obta
ined
usi
ng t
he n
arro
wba
nd a
nd b
road
band
thi
rd h
arm
onic
ban
dpas
s fi
lter,
resp
ectiv
ely.
In
the
low
er p
anel
of
Fig.
3.1
4 th
e si
delo
bes
to t
he l
eft
of t
he m
ainl
obe
are
sign
ific
antly
low
er t
han
the
side
lobe
s to
the
rig
ht a
nd o
ne m
ight
ini
tial
ly e
xpec
t th
e si
delo
bes
in t
he l
ower
pan
el o
f Fi
g. 3
.16
to a
ppea
r in
an
exac
tly
oppo
site
man
ner.
T
he
side
lobe
s to
the
left
of t
he m
ainl
obe
in th
e lo
wer
pan
el o
f Fig
. 3.
16 a
re,
inde
ed,
som
ewha
t hi
gher
tha
n th
e si
delo
bes
to t
he r
ight
but
the
sid
elob
es a
t th
e le
ft a
nd r
ight
of
the
mai
nlo
be a
re s
igni
fica
ntly
mor
e ba
lanc
ed i
n le
vel
than
wha
t w
as f
ound
in
the
low
er p
anel
of
Fig.
3 .1
4. A
lso,
the
hig
hest
sid
elob
e le
vel i
n Fi
g. 3
.16
is s
omew
hat l
ower
than
wha
t fou
nd
in F
ig.
3.14
.
48
··~
<: -<>.04
-to
-20
0 5
10
15
20
25
30
""'
~-30
/
-40
I I
-50
I
4~,0~~~~~~~~~~~~~~~~
Pap
erB
Fig
ure
3.16
: U
pper
pan
el:
Sig
nal
scat
tere
d fr
om b
ubbl
e dr
iven
by
the
acou
stic
pul
se i
n th
e lo
wer
pan
el o
f Fig
. 3.
1 w
here
the
equ
ilib
rium
bub
ble
radi
us i
s in
crea
sed
by 5
% f
or
the
firs
t su
bpul
se.
Low
er p
anel
: R
esul
ts a
fter
thir
d ha
rmon
ic b
andp
ass
filte
ring
on
the
the
puls
e in
the
upp
er p
anel
and
com
pres
sing
usi
ng t
he s
tabi
lize
d in
vers
e fil
ter.
Sol
id l
ine
is
obta
ined
usi
ng t
he n
arro
wba
nd b
andp
ass
filte
r an
d th
e da
shed
lin
e is
obt
aine
d us
ing
the
broa
dban
d ba
ndpa
ss f
ilter
.
3.3.
7 B
ubbl
e M
ovem
ent
The
con
tras
t ag
ent
in a
med
ical
ult
raso
und
imag
ing
situ
atio
n is
usu
ally
mov
ing
due
to
bloo
d flo
w.
Eve
n th
ough
the
bloo
d flo
w v
eloc
ity
is l
ow c
ompa
red
to t
he s
peed
of s
ound
it
may
hav
e a
degr
adin
g ef
fect
on
the
rang
e si
delo
be le
vel o
btai
ned
afte
r pu
lse
com
pres
sion
o
f a
Bar
ker
sequ
ence
. T
he m
ovem
ent
of th
e co
ntra
st b
ubbl
e w
ill
caus
e a
Dop
pler
shi
ft
acco
rdin
g to
the
fol
low
ing
equa
tion
(3.7
)
whe
re f
is
the
tran
smit
fre
quen
cy,
v is
the
vel
ocit
y o
f th
e co
ntra
st a
gent
in
the
beam
di
rect
ion,
and
c is
the
spe
ed o
f sou
nd in
the
med
ium
. S
etti
ng v
=
1 rn
!s g
ives
a D
oppl
er
freq
uenc
y ar
ound
5 k
Hz
for
the
thir
d ha
rmon
ic c
ompo
nent
wit
h a
tran
smit
freq
uenc
y eq
ual
to 1
MH
z w
hich
is r
athe
r lo
w c
ompa
red
to t
he f
requ
ency
shi
fts
obta
ined
fro
m a
bsor
ptio
n (u
pper
pan
el in
Fig
. 3.
6) a
nd s
catt
erin
g (l
ower
pan
el in
Fig
. 3.
8).
The
fre
quen
cy s
hift
due
to
the
Dop
pler
eff
ect i
s th
us n
ot b
elie
ved
to h
ave
a si
gnif
ican
t eff
ect o
n th
e ra
nge
side
lobe
le
vel o
btai
ned
in t
he c
ompr
esse
d pu
lse.
Whe
n th
e co
ntra
st b
ubbl
e is
sub
ject
ed to
an
inci
dent
pre
ssur
e w
ave
it is
als
o af
fect
ed b
y a
radi
atio
n pr
essu
re d
ue to
the
fact
that
the
bubb
le a
bsor
bs a
nd s
catt
ers
part
s o
f the
ene
rgy
in
the
inco
min
g w
ave.
Thi
s ra
diat
ion
pres
sure
wil
l hav
e th
e po
tent
ial t
o in
trod
uce
a ve
loci
ty
to t
he b
ubbl
e in
the
dire
ctio
n o
f th
e in
com
ing
wav
e. T
he in
tens
ity
of
a pl
ane
or s
pher
ical
3.3
Nu
mer
ical
Res
ult
s 49
inci
dent
wav
e ca
n be
cal
cula
ted
as
(3.8
)
whe
re p
is
the
acou
stic
pre
ssur
e am
plit
ude
and
Z i
s th
e ac
oust
ic i
mpe
danc
e. T
he e
nerg
y ab
sorb
ed a
nd s
catt
ered
by
the
bubb
le d
urin
g a
tim
e in
terv
al b
..t i
s I (
J eb.
.t w
here
(J e
is t
he
exti
ncti
on c
ross
sec
tion
of t
he b
ubbl
e. E
xtin
ctio
n cr
oss
sect
ion
is t
he s
um o
f the
sca
tter
ing
cros
s se
ctio
n an
d th
e ab
sorp
tion
cro
ss s
ecti
on a
nd is
thu
s th
e to
tal l
oss
of e
nerg
y fr
om t
he
inci
dent
pre
ssur
e w
ave.
T
he w
ork
done
on
the
bubb
le d
urin
g b.
.t is
Frc
b..t
whe
re F
r is
th
e ra
diat
ion
forc
e on
the
bubb
le a
nd c
is th
e sp
eed
of s
ound
in t
he m
ediu
m.
Equ
atin
g th
e en
ergy
abs
orbe
d/sc
atte
red
by th
e bu
bble
and
the
wor
k do
ne o
n th
e bu
bble
we
obta
in
"' _
I (Je
rr-
c (3
.9)
whe
re v
alue
s fo
r (Je
for
the
agen
t Son
azoi
d ca
n be
est
imat
ed [
19, p
p. 1
52-1
57].
If t
he fl
ow
arou
nd t
he b
ubbl
e is
ass
umed
to b
e la
min
ar a
nd t
he s
hape
of
the
bubb
le is
ass
umed
to b
e pu
rely
sph
eric
al,
the
bubb
le v
eloc
ity
due
to t
he r
adia
tion
for
ce c
an b
e ca
lcul
ated
as
[ 42,
pp
. 18
3-18
4]
(3.1
0)
Her
e, a
is t
he b
ubbl
e ra
dius
, an
d f.iv
is
the
visc
osit
y o
f th
e am
bien
t fl
uid.
Ins
erti
ng ty
pica
l nu
mer
ical
val
ues
give
s a
velo
city
cau
sed
by r
adia
tion
for
ces
whi
ch is
of t
he s
ame
orde
r as
co
mm
on v
eloc
itie
s du
e to
reg
ular
blo
od fl
ow a
nd t
his
effe
ct is
the
refo
re n
ot n
egli
gibl
e.
Day
ton
et a
l [1
0] e
xam
ined
the
mag
nitu
de o
f ra
diat
ion
forc
es o
n ul
tras
ound
con
tras
t ag
ents
tak
ing
a m
ore
rigo
rous
app
roac
h th
an d
one
here
. A
vel
ocit
y o
f up
to
0.5
mls
due
to
rad
iati
on f
orce
s on
a 4
f.im
bub
ble
inso
nifi
ed w
ith
sim
ilar
pre
ssur
e am
plit
udes
as
in th
e pr
esen
t pap
er ( r
v 0
.1 M
Pa)
and
a tr
ansm
it fr
eque
ncy
arou
nd 2
.25
MH
z w
as t
hen
obta
ined
. T
he v
eloc
ity
intr
oduc
ed d
ue to
rad
iati
on f
orce
s w
ill b
e ti
me-
vary
ing.
If th
e ve
loci
ty o
f the
bub
ble
is a
ssum
ed to
be
1 m
/s,
the
bubb
le w
ill h
ave
mov
ed a
dis
tanc
e o
f 25
f.im
in
a ti
me
peri
od o
f 25
f.L
S (w
hich
is
appr
oxim
atel
y th
e du
rati
on o
f th
e fo
ur b
it
Bar
ker
sequ
ence
use
d).
In a
pul
se e
cho
imag
ing
syst
em t
his
wil
l in
trod
uce
a ti
me
shif
t eq
ual
to 3
3 ns
if
the
spee
d o
f so
und
is s
et e
qual
to 1
500
m/s
. D
ue to
the
pum
ping
of
the
hear
t m
uscl
e an
d th
e pr
esen
ce o
f ra
diat
ion
forc
es,
the
bloo
d flo
w w
ill
not
be s
tati
onar
y an
d th
e up
per p
anel
in F
ig.
3.17
sho
ws
an e
xam
ple
whe
re th
e la
st s
ubpu
lse
in th
e sc
atte
red
Bar
ker
sequ
ence
has
bee
n ti
me
shif
ted
by a
n am
ount
equ
al to
30
ns r
elat
ive
to t
he o
rigi
nal
sequ
ence
in
the
uppe
r pa
nel
of
Fig.
3.8
. H
ere,
the
sol
id l
ine
is t
he n
orm
aliz
ed t
hird
ha
rmon
ic e
nvel
ope
of
the
orig
inal
uns
hift
ed l
ast
subp
ulse
, w
hile
the
das
hed
line
is
the
norm
aliz
ed t
hird
har
mon
ic e
nvel
ope
of
the
resu
ltin
g ti
me-
shif
ted
subp
ulse
. In
the
low
er
pane
l it
is s
how
n ho
w t
his
tim
e sh
ift
effe
cts
the
rang
e si
delo
be l
evel
obt
aine
d af
ter
the
stab
iliz
ed i
nver
se f
ilter
is
appl
ied.
W
e se
e th
at t
he B
arke
r se
quen
ce i
s ve
ry s
ensi
tive
to
even
a m
inor
tim
e sh
ift b
etw
een
its s
ubpu
lses
. T
he c
ompr
esse
d pu
lse
in t
he l
ower
pan
el o
f Fig
. 3 .
14,
obta
ined
by
slig
htly
cha
ngin
g th
e ac
oust
ic p
rope
rtie
s o
f th
e co
ntra
st b
ubbl
e fo
r th
e la
st s
ubpu
lse,
and
the
com
pres
sed
50
Pap
erB
Fig
ure
3.17
: U
pper
pan
el:
Nor
mal
ized
env
elop
e o
f th
ird
harm
onic
com
pone
nt f
rom
tw
o su
bpul
ses
whe
re o
ne o
f th
em h
as b
een
tim
e sh
ifte
d by
an
amou
nt e
qual
to 3
0 ns
rel
ativ
e to
the
oth
er.
Low
er p
anel
: D
ashe
d li
ne s
how
eff
ect o
n ra
nge
side
lobe
leve
l in
trod
uced
by
tim
e sh
ifti
ng t
he l
ast s
ubpu
lse
of
a fo
ur b
it B
arke
r se
quen
ce b
y an
am
ount
equ
al to
30
ns
rela
tive
to t
he s
idel
obe
leve
l obt
aine
d fr
om t
he o
rigi
nal u
nshi
fted
seq
uenc
e (s
olid
line
).
puls
e in
the
low
er p
anel
of F
ig.
3.17
, ob
tain
ed b
y sl
ight
ly t
ime
shif
ting
the
last
sub
puls
e,
are
seen
to b
e ve
ry s
imila
r. A
s se
en in
Fig
. 3.1
5, a
n im
port
ant e
ffec
t int
rodu
ced
by s
ligh
tly
chan
ging
the
aco
usti
c pr
oper
ties
of
the
bubb
le w
as a
tim
e sh
ift
of
the
scat
tere
d th
ird
harm
onic
env
elop
e w
hich
exp
lain
s th
e si
mil
arit
ies
in t
he l
ower
pan
els
of
Fig.
3.1
4 an
d Fi
g. 3
.17.
3.3.
8 E
ffec
t of H
avin
g a
Fin
ite
CN
R
In a
n ac
tual
ult
raso
und
imag
ing
situ
atio
n, t
here
wil
l al
way
s be
a f
inite
CN
R a
nd t
he m
oti
vati
on f
or i
ntro
duci
ng B
arke
r se
quen
ces
was
to
impr
ove
the
CN
R a
t th
e re
ceiv
ed t
hird
ha
rmon
ic c
ompo
nent
fro
m t
he c
ontr
ast
agen
t. T
wo
diff
eren
t lev
els
of
unco
rrel
ated
noi
se
are
now
add
ed t
o th
e or
igin
al s
catt
ered
fou
r bi
t B
arke
r se
quen
ce d
ispl
ayed
in
Fig.
3.8
. In
bot
h ca
ses
the
nois
e le
vel
is s
o hi
gh t
hat
the
resu
ltin
g si
delo
bes
in t
he c
ompr
esse
d se
quen
ce is
lim
ited
by
the
nois
e si
gnal
and
not
the
effe
cts
disc
usse
d in
rela
tion
to F
ig.
3.9.
In th
e up
per p
anel
of F
ig. 3
.18,
the
enve
lope
of t
he s
catt
ered
thir
d ha
rmon
ic c
ompo
nent
of
the
Bar
ker
sequ
ence
, w
hen
the
narr
owba
nd b
andp
ass
filte
r ha
s be
en a
pplie
d, i
s di
spla
yed
as t
he s
olid
line
. T
he d
ashe
d lin
e re
pres
ents
the
enve
lope
of t
he th
ird
harm
onic
com
pone
nt
of a
n un
corr
elat
ed n
oise
sig
nal
and
the
CN
R b
efor
e ap
plyi
ng t
he s
tabi
lize
d in
vers
e fi
lter
is a
roun
d 4
dB.
In t
he l
ower
pan
el o
f th
e sa
me
figu
re,
the
resu
lts
afte
r pu
lse
com
pres
sion
ar
e sh
own.
T
he s
olid
lin
e is
the
res
ult
obta
ined
fee
ding
the
sum
of
the
two
sign
als
in
the
uppe
r pa
nel
to t
he s
tabi
lize
d in
vers
e fi
lter
whi
le t
he d
ashe
d li
ne is
the
res
ult o
btai
ned
3.4
Exp
erim
enta
l R
esul
ts
51
-50
Fig
ure
3.18
: U
pper
pan
el:
Env
elop
e o
f sca
tter
ed th
ird
harm
onic
com
pone
nt, o
btai
ned
wit
h na
rrow
band
ban
dpas
s fil
ter,
from
a 4
bit
Bar
ker s
eque
nce
(sol
id li
ne)
and
enve
lope
of t
hird
ha
rmon
ic u
ncor
rela
ted
nois
e si
gnal
(da
shed
line
). T
hird
har
mon
ic C
NR
is a
ppro
xim
atel
y 4
dB b
efor
e pu
lse
com
pres
sion
. L
ower
pan
el:
Res
ult a
fter
app
lyin
g th
e st
abil
ized
inve
rse
filte
r on
the
sum
of t
he tw
o se
quen
ces
in th
e up
per p
anel
(so
lid
line
) an
d on
the
nois
e si
gnal
on
ly (
dash
ed li
ne).
Thi
rd h
arm
onic
CN
R is
app
roxi
mat
ely
13 d
B a
fter
pul
se c
ompr
essi
on.
proc
essi
ng o
nly
the
nois
e si
gnal
in
the
uppe
r pa
nel.
The
CN
R a
fter
pul
se c
ompr
essi
on is
ap
prox
imat
ely
13 d
B a
nd t
he in
crea
se in
CN
R a
chie
ved
usin
g th
e B
arke
r seq
uenc
e is
thu
s ar
ound
9 d
B.
In F
ig.
3.19
the
add
ed n
oise
sig
nal
has
been
som
ewha
t re
duce
d.
The
sol
id l
ine
in t
he
uppe
r pa
nel
show
s th
e en
velo
pe o
f th
e sc
atte
red
thir
d ha
rmon
ic B
arke
r se
quen
ce w
hile
th
e da
shed
lin
e re
pres
ents
the
env
elop
e o
f th
e th
ird
harm
onic
com
pone
nt o
f th
e no
ise
sign
al w
hen
the
narr
owba
nd f
ilter
is
used
. T
he th
ird
harm
onic
CN
R b
efor
e th
e st
abil
ized
in
vers
e fi
lter
is a
ppli
ed is
now
app
roxi
mat
ely
13 d
B.
In t
he l
ower
pan
el w
e se
e th
e re
sult
s af
ter
puls
e co
mpr
essi
on.
Aga
in,
the
soli
d li
ne i
s ob
tain
ed p
roce
ssin
g th
e su
m o
f th
e tw
o si
gnal
s in
the
uppe
r pan
el a
nd th
e da
shed
line
is o
btai
ned
from
the
noi
se s
igna
l onl
y in
the
up
per
pane
l. A
n in
crea
se o
f ab
out 7
dB
in
the
thir
d ha
rmon
ic C
NR
is n
ow a
chie
ved
afte
r pu
lse
com
pres
sion
is p
erfo
rmed
.
3.4
Exp
erim
enta
l Res
ults
In v
itro
mea
sure
men
ts a
re d
one
on a
tis
sue
mim
icki
ng p
hant
om u
sing
an
annu
lar
tran
sdu
cer
cons
isti
ng o
f 5
ring
s. T
he o
uter
rin
g is
a lo
w f
requ
ency
rin
g us
ed f
or t
rans
mit
ting
th
e ul
tras
ound
sig
nal
whi
le t
he f
our
inne
r ri
ngs
are
high
fre
quen
cy r
ings
use
d fo
r re
ceiv
in
g th
e sc
atte
red
thir
d ha
rmon
ic c
ompo
nent
s o
f th
e tr
ansm
itte
d si
gnal
. A
sig
nal s
imil
ar to
th
e on
e in
the
low
er p
anel
of F
ig.
3.1,
wit
h a
cent
er fr
eque
ncy
equa
l to
1.3
MH
z, i
s st
ored
52
Pap
erB
-50
-55
~-60
I
-65
:,
•',
..
• :'t
, r,
: -7
0 ':!
~ .. J:
: '"•: :·
\:,/
:: :
_ 75
!~(
I 1 1
1 I
I 11
11
I
0 20
Fig
ure
3.19
: U
pper
pan
el:
Env
elop
e o
f sca
tter
ed th
ird
harm
onic
com
pone
nt, o
btai
ned
wit
h na
rrow
band
ban
dpas
s fil
ter,
from
a 4
bit
Bar
ker s
eque
nce
(sol
id li
ne)
and
enve
lope
of t
hird
ha
rmon
ic u
ncor
rela
ted
nois
e si
gnal
(da
shed
lin
e).
Upp
er p
anel
: T
hird
har
mon
ic C
NR
is
appr
oxim
atel
y 13
dB
bef
ore
puls
e co
mpr
essi
on.
Low
er p
anel
: R
esul
t af
ter
appl
ying
the
st
abil
ized
inve
rse
filte
r on
the
sum
of t
he tw
o se
quen
ces
in t
he u
pper
pan
el (
soli
d li
ne)
and
on th
e no
ise
sign
al o
nly
(das
hed
line
). T
hird
har
mon
ic C
NR
is a
ppro
xim
atel
y 20
dB
aft
er
puls
e co
mpr
essi
on.
in a
sig
nal
gene
rato
r an
d us
ed a
s an
exc
itat
ion
puls
e on
the
out
er r
ing
of
the
ultr
asou
nd
tran
sduc
er.
The
tiss
ue m
imic
king
pha
ntom
con
tain
s a
smal
l pol
yeth
ylen
e tu
be w
ith
inne
r di
amet
er e
qual
to
0.28
mm
thr
ough
whi
ch a
sol
utio
n o
f w
ater
and
con
tras
t ag
ent
may
flo
w.
The
con
tras
t ag
ent S
onaz
oid
[19]
whi
ch h
as a
res
onan
ce f
requ
ency
aro
und
4 M
Hz,
w
as u
sed
in t
he p
rese
nt m
easu
rem
ents
. M
easu
rem
ents
wer
e co
nduc
ted
on a
Sys
tem
FiV
e ul
tras
ound
sca
nner
mad
e by
GE
Vin
gmed
Ult
raso
und
and
the
cont
rast
bub
bles
wer
e su
bje
cted
to r
elat
ivel
y lo
w a
mpl
itud
e ul
tras
ound
pul
ses
wit
h an
am
plit
ude
arou
nd 0
.1 M
Pa.
The
nar
row
band
thir
d ha
rmon
ic b
andp
ass
filte
r sh
own
in F
ig.
3.5
wit
h a
cent
er fr
eque
ncy
equa
l to
3.9
MH
z w
as u
sed
on t
he r
ecei
ved
sign
als
to o
btai
n th
e sc
atte
red
thir
d ha
rmon
ic
com
pone
nt.
The
upp
er p
anel
of F
ig.
3.20
dep
icts
the
thi
rd h
arm
onic
env
elop
e fr
om a
rec
eive
d B
arke
r se
quen
ce w
here
the
CN
R is
aro
und
4 dB
. W
e no
tice
tha
t for
the
thir
d su
bpul
se, o
ccur
ring
at
aro
und
29 J
l,S o
n th
e ti
me
axis
, th
e si
gnal
lev
el i
s pa
rtic
ular
ly r
educ
ed.
Thi
s re
duct
ion
is m
ost
like
ly d
ue t
o de
stru
ctiv
e in
terf
eren
ce w
ith
the
unco
rrel
ated
noi
se s
igna
l an
d no
t a
chan
ge i
n th
e ac
oust
ic p
rope
rtie
s of
the
bubb
le s
ince
for
the
las
t su
bpul
se,
occu
rrin
g at
aro
und
35 !
JS,
the
sign
al l
evel
is
agai
n in
crea
sed.
In
the
low
er p
anel
of
Fig
. 3.
20,
the
resu
lt a
fter
the
sta
bili
zed
inve
rse
filte
r ha
s be
en a
ppli
ed i
s sh
own
and
the
CN
R i
s no
w
appr
oxim
atel
y 13
dB
.
Fig.
3.2
1 sh
ows
anot
her e
xam
ple
of a
rece
ived
sca
tter
ed B
arke
r seq
uenc
e fr
om th
e co
ntra
st
bubb
les
flow
ing
thro
ugh
the
tiss
ue m
imic
king
pha
ntom
. In
the
upp
er p
anel
of
the
figu
re,
3.4
Exp
erim
enta
l R
esul
ts
53
·~~j
0 10
2
0
30
40
50
6
0
70
80
90
'"'l
Fig
ure
3.20
: U
pper
pan
el:
Env
elop
e o
f m
easu
red
thir
d ha
rmon
ic c
ompo
nent
, ob
tain
ed
wit
h th
e na
rrow
band
ban
dpas
s fi
lter,
from
a 4
bit
Bar
ker
sequ
ence
. T
hird
har
mon
ic C
NR
is
app
roxi
mat
ely
4 dB
bef
ore
puls
e co
mpr
essi
on.
Low
er p
anel
: R
esul
t af
ter
appl
ying
th
e st
abil
ized
inv
erse
filt
er o
n th
e si
gnal
in
the
uppe
r pa
nel.
Thi
rd h
arm
onic
CN
R i
s ap
prox
imat
ely
13 d
B a
fter
pul
se c
ompr
essi
on.
the
thir
d ha
rmon
ic e
nvel
ope
is d
ispl
ayed
and
the
CN
R is
for
thi
s ca
se a
roun
d 10
dB
. T
he
low
er p
anel
of
the
figu
re s
how
s th
e re
sult
aft
er p
ulse
com
pres
sion
and
the
CN
R i
s no
w
seen
to
be a
ppro
xim
atel
y 17
dB
. T
here
is
a gr
adua
l in
crea
se i
n th
e re
ceiv
ed s
igna
l le
vel
from
the
fir
st t
o th
e th
ird
subp
ulse
occ
urri
ng b
etw
een
27 a
nd 3
8 JlS
on
the
tim
e ax
is.
Thi
s in
crea
se i
s al
so m
ost
like
ly d
ue t
o in
terf
eren
ce b
etw
een
the
bubb
le s
igna
l an
d th
e no
ise
sign
al s
ince
for
the
las
t su
bpul
se,
occu
rrin
g at
aro
und
43 J
lS,
the
sign
al l
evel
is
slig
htly
re
duce
d.
Fig.
3.2
2 de
pict
s ye
t an
othe
r ex
ampl
e o
f a
rece
ived
sca
tter
ed B
arke
r se
quen
ce.
In t
he
uppe
r pa
nel,
befo
re p
ulse
com
pres
sion
, th
e C
NR
is s
een
to b
e ar
ound
16
dB.
In t
he l
ower
pa
nel,
afte
r pul
se c
ompr
essi
on, t
he C
NR
has
inc
reas
ed to
aro
und
23 d
B.
The
pre
viou
s th
ree
figu
res
show
exa
mpl
es o
f re
ceiv
ed s
catt
ered
Bar
ker
sequ
ence
s w
here
th
e pr
esen
ce o
f un
corr
elat
ed n
oise
is b
elie
ved
to b
e th
e li
mit
ing
fact
or r
egar
ding
the
side
lo
be le
vel o
btai
ned
afte
r pul
se c
ompr
essi
on is
per
form
ed.
The
bub
bles
are
hen
ce a
ssum
ed
to m
aint
ain
rath
er c
onst
ant
valu
es w
ith
resp
ect
to a
cous
tic
prop
erti
es a
nd b
ubbl
e m
ove
men
t is
beli
eved
not
to c
ause
a li
mit
atio
n on
the
side
lobe
leve
l obt
aine
d.
In th
e up
per p
anel
of F
ig. 3
.23,
we
see
an e
xam
ple
of a
rece
ived
seq
uenc
e w
here
the
sign
al
leve
l fro
m th
e co
ntra
st a
gent
is s
een
to g
radu
ally
incr
ease
from
the
firs
t sub
puls
e oc
cmT
ing
at a
roun
d 27
JlS
, to
the
las
t su
bpul
se o
ccur
ring
at
arou
nd 4
2 Jl
S.
Thi
s st
eady
inc
reas
e in
si
gnal
leve
l may
ind
icat
e a
grad
ual c
hang
e of
the
acou
stic
pro
pert
ies
of th
e co
ntra
st a
gent
in
side
the
sam
ple
volu
me.
One
pla
usib
le e
xpla
nati
on m
ight
be
that
the
thin
enc
apsu
lati
ng
shel
l ge
ts s
ligh
tly
mor
e fl
exib
le d
urin
g th
e os
cill
atio
ns o
f ea
ch s
ubpu
lse
and
thus
res
ults
54
Pap
erB
Fig
ure
3.21
: U
pper
pan
el:
Env
elop
e o
f m
easu
red
thir
d ha
rmon
ic c
ompo
nent
, ob
tain
ed
wit
h th
e na
rrow
band
ban
dpas
s fil
ter,
from
a 4
bit
Bar
ker
sequ
ence
. T
hird
har
mon
ic C
NR
is
app
roxi
mat
ely
10 d
B b
efor
e pu
lse
com
pres
sion
. L
ower
pan
el:
Res
ult
afte
r ap
plyi
ng
the
stab
iliz
ed i
nver
se f
ilter
on
the
sign
al i
n th
e up
per
pane
l. T
hird
har
mon
ic C
NR
is
appr
oxim
atel
y 17
dB
aft
er p
ulse
com
pres
sion
.
Fig
ure
3.22
: U
pper
pan
el:
Env
elop
e o
f m
easu
red
thir
d ha
rmon
ic c
ompo
nent
, ob
tain
ed
wit
h th
e na
rrow
band
ban
dpas
s fil
ter,
from
a 4
bit
Bar
ker
sequ
ence
. T
hird
har
mon
ic C
NR
is
app
roxi
mat
ely
16 d
B b
efor
e pu
lse
com
pres
sion
. L
ower
pan
el:
Res
ult
afte
r ap
plyi
ng
the
stab
iliz
ed i
nver
se f
ilter
on
the
sign
al i
n th
e up
per
pane
l. T
hird
har
mon
ic C
NR
is
appr
oxim
atel
y 23
dB
aft
er p
ulse
com
pres
sion
.
3.5
Con
clus
ions
55
Fig
ure
3.23
: U
pper
pan
el:
Env
elop
e o
f m
easu
red
thir
d ha
rmon
ic c
ompo
nent
, ob
tain
ed
wit
h th
e na
rrow
band
ban
dpas
s fi
lter,
from
a 4
bit
Bar
ker
sequ
ence
. L
ower
pan
el:
Res
ult
afte
r ap
plyi
ng th
e st
abil
ized
inve
rse
filte
r on
the
sign
al in
the
upp
er p
anel
.
in a
sli
ght i
ncre
ase
in th
e sc
atte
ring
cro
ss s
ecti
on f
rom
the
con
tras
t age
nt.
The
low
er p
anel
o
f F
ig.
3.23
sho
ws
the
resu
lt a
fter
pul
se c
ompr
essi
on a
nd b
y co
mpa
ring
wit
h th
e up
per
pane
l in
the
fig
ure,
we
see
that
the
CN
R is
app
roxi
mat
ely
the
sam
e be
fore
and
aft
er p
ulse
co
mpr
essi
on is
per
form
ed.
The
upp
er p
anel
of F
ig. 3
.24
show
s a
fina
l ex
ampl
e o
f the
thir
d ha
rmon
ic e
nvel
ope
from
a
rece
ived
sca
tter
ed B
arke
r se
quen
ce.
We
noti
ce th
at th
e tw
o la
st re
ceiv
ed s
ubpu
lses
, occ
ur
ring
bet
wee
n 29
and
37
J-tS
on t
he t
ime
axis
, ar
e si
gnif
ican
tly
low
er in
am
plit
ude
than
the
tw
o fi
rst
subp
ulse
s in
the
seq
uenc
e. T
his
effe
ct m
ight
als
o be
a r
esul
t o
f a
chan
ge i
n th
e ac
oust
ic p
rope
rtie
s o
f th
e co
ntra
st a
gent
. T
he C
NR
in
the
uppe
r pa
nel
is a
ppro
xim
atel
y 19
dB
for
the
two
firs
t sub
puls
es a
nd h
ence
too
high
to e
xpla
in th
e si
gnif
ican
t red
ucti
on in
am
plit
ude
of
the
two
last
sub
puls
es in
the
seq
uenc
e as
res
ulti
ng f
rom
des
truc
tive
inte
rfer
en
ce b
etw
een
the
cont
rast
sig
nal a
nd t
he n
oise
sig
nal.
In t
he l
ower
pan
el o
f Fig
. 3.
24, w
e se
e th
e re
sult
aft
er a
pply
ing
the
stab
iliz
ed in
vers
e fi
lter.
We
noti
ce th
at,
agai
n, t
here
is li
ttle
or
no
impr
ovem
ent i
n th
e C
NR
in th
e co
mpr
esse
d se
quen
ce r
elat
ive
to t
he u
ncom
pres
sed
sequ
ence
.
3.5
Con
clus
ions
The
app
lica
tion
of a
fou
r bi
t Bar
ker
code
to i
ncre
ase
the
CN
R fo
r th
ird
harm
onic
con
tras
t ag
ent
imag
ing
has
been
stu
died
num
eric
ally
and
exp
erim
enta
lly.
Res
ults
fro
m n
umer
ical
si
mul
atio
ns, a
ddin
g va
riou
s le
vels
ofu
ncor
rela
ted
nois
e to
a fo
ur b
it B
arke
r seq
uenc
e sc
at
tere
d fr
om a
con
tras
t bub
ble
and
appl
ying
a s
tabi
lize
d in
vers
e fi
lter
indi
cate
d an
inc
reas
e in
the
thir
d ha
rmon
ic C
NR
bet
wee
n 6
and
9 dB
aft
er p
ulse
com
pres
sion
. In
vit
ro m
easu
re-
56
Pap
erB
Fig
ure
3.24
: U
pper
pan
el:
Env
elop
e o
f m
easu
red
thir
d ha
rmon
ic c
ompo
nent
, ob
tain
ed
wit
h th
e na
rrow
band
ban
dpas
s fil
ter,
from
a 4
bit
Bar
ker
sequ
ence
. L
ower
pan
el:
Res
ult
afte
r ap
plyi
ng t
he s
tabi
lize
d in
vers
e fi
lter
on t
he s
igna
l in
the
upp
er p
anel
.
men
ts p
erfo
rmed
on
a ti
ssue
mim
icki
ng p
hant
om w
ith
the
cont
rast
age
nt S
onaz
oid,
car
ried
ou
t usi
ng t
he s
ame
four
bit
Bar
ker
sequ
ence
, ty
pica
lly
gave
sim
ilar
impr
ovem
ents
in
the
thir
d ha
rmon
ic C
NR
whe
n th
e ac
oust
ic p
rope
rtie
s o
f th
e co
ntra
st a
gent
wer
e as
sum
ed to
be
clo
se to
con
stan
t for
the
who
le B
arke
r se
quen
ce,
and
resu
lts
from
mea
sure
men
ts w
ere
henc
e in
goo
d ag
reem
ent
wit
h th
e nu
mer
ical
res
ults
. R
ecei
ved
sequ
ence
s fo
r w
hich
the
ac
oust
ic p
rope
rtie
s o
f th
e co
ntra
st b
ubbl
es w
ere
beli
eved
to c
hang
e du
ring
ins
onif
icat
ion
by t
he B
arke
r se
quen
ce s
how
ed l
ittl
e or
no
impr
ovem
ent
in C
NR
whe
n co
mpa
ring
the
co
mpr
esse
d an
d un
com
pres
sed
sign
als.
A l
onge
r B
arke
r se
quen
ce w
ould
hav
e th
e po
ten
tial
to
furt
her
incr
ease
the
CN
R.
Usi
ng a
sig
nifi
cant
ly l
onge
r se
quen
ce o
ne w
ould
, ho
w
ever
, al
so p
resu
mab
ly e
ncou
nter
pro
blem
s re
gard
ing
bubb
le m
otio
n du
ring
ins
onif
icat
ion
by t
he s
eque
nce.
The
pos
sibi
lity
that
the
cont
rast
bub
bles
wou
ld,
to s
ome
exte
nt, c
hang
e th
eir
acou
stic
pro
pert
ies
duri
ng in
soni
fica
tion
by
such
a lo
ng s
eque
nce
is a
lso
grea
ter
than
fo
r th
e sh
orte
r cod
e. B
oth
thes
e ef
fect
s ar
e in
the
pres
ent p
aper
sho
wn
to p
oten
tial
ly h
ave
a m
ajor
deg
radi
ng e
ffec
t on
the
rang
e si
delo
be le
vel o
btai
ned
in th
e fi
nal
com
pres
sed
puls
e.
For
seq
uenc
es w
ith
a lo
w C
NR
bef
ore
puls
e co
mpr
essi
on, t
he a
djus
tabl
e st
abil
ized
inve
rse
filte
r is
set
so
that
the
filt
er i
s sh
ifte
d to
war
ds a
mat
ched
filt
er,
whi
le f
or s
eque
nces
wit
h hi
gher
CN
R b
efor
e pu
lse
com
pres
sion
, th
e fi
lter
mor
e cl
osel
y re
sem
bles
the
inv
erse
filt
er.
3.6
Ack
now
ledg
men
ts
Thi
s w
ork
was
sup
port
ed b
y th
e R
esea
rch
Cou
ncil
of N
orw
ay.
Ch
apte
r 4
A N
ew D
ual
Fre
qu
ency
Ban
d C
ontr
ast
Age
nt
Det
ecti
on T
echn
ique
Abs
trac
t
Ult
raso
und
wav
e pr
opag
atio
n in
sof
t tis
sue
and
scat
teri
ng f
rom
ult
raso
und
cont
rast
age
nts
are
both
kno
wn
to b
e no
nlin
ear
proc
esse
s at
fre
quen
cies
and
am
plit
udes
com
mon
ly a
ppl
ied
in m
edic
al u
ltra
soun
d im
agin
g.
The
fac
t th
at t
he c
ontr
ast
agen
ts u
sual
ly r
espo
nd
muc
h m
ore
nonl
inea
rly
than
sof
t tis
sue
has
give
n ri
se t
o th
e co
ntra
st h
arm
onic
im
agin
g te
chni
ques
whe
re o
nly
a ha
rmon
ic c
ompo
nent
of
the
tota
l sc
atte
red
sign
al,
typi
call
y th
e se
cond
har
mon
ic c
ompo
nent
, is
use
d fo
r im
age
reco
nstr
ucti
on.
In a
med
ical
ult
raso
und
imag
ing
situ
atio
n, b
oth
the
harm
onic
sca
tter
ed ti
ssue
sig
nal a
nd t
he u
ncor
rela
ted
ther
mal
an
d el
ectr
onic
noi
se s
igna
l w
ill
pote
ntia
lly
mas
k th
e sc
atte
red
cont
rast
har
mon
ic s
igna
l. T
he p
rese
nt p
aper
dea
ls w
ith
a ne
w c
ontr
ast
hatm
onic
im
agin
g te
chni
que,
des
igne
d to
in
crea
se t
he c
ontr
ast
harm
onic
sig
nal
rela
tive
to b
oth
the
nois
e si
gnal
as
wel
l as
the
har
m
onic
tiss
ue s
igna
l. In
ord
er to
ach
ieve
this
, the
new
met
hod
mak
es u
se o
f dua
l fre
quen
cy
band
tran
smit
pul
ses,
tog
ethe
r w
ith
a ge
nera
l pu
lse
inve
rsio
n te
chni
que.
4.1
Intr
oduc
tion
Wav
e pr
opag
atio
n in
sof
t ti
ssue
is
usua
lly
a w
eak
nonl
inea
r pr
oces
s fo
r fr
eque
ncie
s an
d am
plitu
des
typi
call
y us
ed i
n m
edic
al u
ltra
soun
d im
agin
g, a
nd t
he t
rans
mit
ted
puls
e is
sl
ight
ly d
isto
rted
as
it p
ropa
gate
s th
roug
h th
e m
ediu
m.
Alt
houg
h th
e lo
cal e
ffec
t of
non
linea
rity
is s
mal
l, th
e cu
mul
ativ
e ef
fect
whe
n th
e w
ave
has
prop
agat
ed s
ever
al w
avel
engt
hs
is n
ot n
eglig
ible
. If
the
wav
e tr
ansm
itte
d fr
om a
n ul
tras
ound
tra
nsdu
cer
has
its e
nerg
y co
ncen
trat
ed in
som
e fu
ndam
enta
l fr
eque
ncy
band
, th
e no
nlin
eari
ty o
f wav
e pr
opag
atio
n gi
ves
rise
to
harm
onic
s o
f th
is f
unda
men
tal
band
[3,
Cha
pter
12]
[30
, C
hapt
er 1
1].
The
58
Pap
er C
leve
l of r
ecei
ved
seco
nd h
arm
onic
com
pone
nt fr
om t
issu
e is
typ
ical
ly f
ound
to b
e ar
ound
20
dB
bel
ow t
he l
evel
of
the
fund
amen
tal
com
pone
nt b
ut t
his
leve
l de
pend
s on
aco
usti
c pa
ram
eter
s in
add
itio
n to
im
agin
g pa
ram
eter
s su
ch a
s fr
eque
ncy,
am
plit
ude,
and
dep
th o
f im
agin
g.
Ult
raso
und
scat
teri
ng f
rom
sof
t tis
sue
is a
ssum
ed to
be
a li
near
pro
cess
so
that
the
ener
gy
rece
ived
fro
m t
issu
e ou
tsid
e th
e fu
ndam
enta
l tr
ansm
itte
d ba
nd is
gen
erat
ed in
the
forw
ard
nonl
inea
r pr
opag
atio
n o
f th
e w
ave
unle
ss a
spr
eadi
ng o
f sca
tter
ed c
ontr
ast
sign
al b
eyon
d th
e ac
tual
con
tras
t-fi
lled
reg
ion
occu
rs.
Alt
houg
h m
uch
wea
ker
than
the
sec
ond
harm
onic
co
mpo
nent
, hig
her
harm
onic
s m
ay a
lso
exis
t in
the
scat
tere
d ti
ssue
sig
nal.
Med
ical
ult
raso
und
cont
rast
age
nts
are
typi
call
y m
ade
as s
olut
ions
of
smal
l ga
s bu
bble
s (d
iam
rv 3
JLm
) in
a f
luid
and
sca
tter
ing
from
suc
h ga
s bu
bble
s is
pot
enti
ally
a s
tron
g no
nlin
ear
proc
ess
[23]
[11
] [1
2].
The
hig
h co
mpl
ianc
e o
f th
e ga
s bu
bble
s re
lati
ve to
sof
t ti
ssue
mak
e th
em h
ighl
y ef
fect
ive
scat
tere
rs o
f ac
oust
ic e
nerg
y.
A s
pher
ical
gas
bub
ble
in a
liq
uid
unde
rgoi
ng s
impl
e ha
rmon
ic m
otio
n ha
s a
natu
ral
or
reso
nanc
e fr
eque
ncy
firs
t ca
lcul
ated
by
Min
naer
t [2
7].
Whe
n dr
iven
by
acou
stic
pul
ses
used
in
med
ical
ult
raso
und
imag
ing,
and
esp
ecia
lly
whe
n su
bjec
ted
to p
ulse
s w
ith
fre
quen
cy c
ompo
nent
s cl
ose
to o
r be
low
its
res
onan
ce f
requ
ency
, th
e ga
s bu
bble
rep
rese
nts
a hi
ghly
non
line
ar s
catt
erer
of
ultr
asou
nd g
ivin
g ri
se t
o th
e co
ntra
st h
arm
onic
im
agin
g te
chni
ques
[35
].
The
Con
tras
t to
Tis
sue
Rat
io (
CT
R),
or
spec
ific
ity, i
s de
fine
d as
the
rat
io o
f si
gnal
pow
er
from
the
con
tras
t ag
ent
in a
reg
ion
to t
he s
igna
l po
wer
fro
m t
he t
issu
e in
tha
t re
gion
. In
or
der
to a
dequ
atel
y di
ffer
enti
ate
cont
rast
sig
nal
and
tiss
ue s
igna
l it
is t
here
fore
nec
essa
ry
wit
h a
high
CT
R.
Con
tras
t bu
bble
s ge
nera
lly
resp
ond
muc
h m
ore
nonl
inea
rly
than
sof
t ti
ssue
whe
n su
bjec
ted
to u
ltra
soun
d pu
lses
and
the
CT
R t
ypic
ally
inc
reas
es a
s hi
gher
ha
rmon
ic c
ompo
nent
s ar
e co
nsid
ered
.
Ult
raso
und
puls
e ec
ho i
mag
ing
syst
ems
are
infe
sted
by
ther
mal
and
ele
ctro
nic
nois
e an
d th
e C
ontr
ast
to N
oise
Rat
io (
CN
R),
or
sens
itivi
ty,
is d
efin
ed a
s th
e ra
tio o
f si
gnal
pow
er
from
the
con
tras
t ag
ent
in a
reg
ion
to t
he n
oise
pow
er i
n th
at r
egio
n.
The
the
rmal
and
el
ectr
onic
noi
se s
igna
l (n
oise
not
cor
rela
ted
wit
h th
e tr
ansm
itte
d si
gnal
) ca
n be
con
sid
ered
con
stan
t ove
r th
e fr
eque
ncy
rang
e o
f in
tere
st i
n m
edic
al u
ltra
soun
d im
agin
g. I
n th
e re
ceiv
ed s
igna
l fr
om a
con
tras
t bub
ble,
the
am
plit
ude
of
harm
onic
com
pone
nts
typi
call
y de
crea
ses
as h
ighe
r har
mon
ic c
ompo
nent
s ar
e co
nsid
ered
and
the
CN
R th
us d
ecre
ases
for
hi
gher
har
mon
ic c
ompo
nent
s.
In a
n im
agin
ed i
mag
ing
situ
atio
n w
itho
ut ti
ssue
, th
e be
st C
NR
wou
ld b
e ac
hiev
ed t
rans
m
itti
ng a
hig
h am
plit
ude
puls
e an
d th
en u
sing
the
tot
al r
ecei
ved
sign
al f
or i
mag
e re
con
stru
ctio
n o
f th
e co
ntra
st a
gent
. A
hig
h am
plit
ude
tran
smit
pul
se i
s, h
owev
er,
lim
ited
by
pati
ent
safe
ty [
5] [
8].
Als
o, t
he b
ubbl
es t
end
to g
et d
estr
oyed
eve
n du
ring
ins
onif
icat
ion
by r
elat
ivel
y lo
w a
mpl
itud
e ul
tras
ound
pul
ses.
O
n th
e ot
her h
and,
in a
n im
agin
ed n
oise
free
sit
uati
on, t
he b
est C
TR
wou
ld b
e ac
hiev
ed
4.2M
etho
d 59
by t
rans
mit
ting
a l
ow a
mpl
itud
e pu
lse,
mak
ing
the
tiss
ue b
ehav
e ap
prox
imat
ely
line
arly
w
hile
sti
ll d
rivi
ng th
e bu
bble
s in
to n
onli
near
osc
illa
tion
s an
d us
ing
the
thir
d or
hig
her h
ar
mon
ic c
ompo
nent
of
the
rece
ived
sig
nal
for
imag
e re
cons
truc
tion
. In
an
actu
al i
mag
ing
situ
atio
n, ti
ssue
sig
nals
and
noi
se s
igna
ls a
re p
rese
nt,
and
the
CT
R a
nd C
NR
bot
h ne
ed to
be
as
high
as
poss
ible
in o
rder
to c
onst
ruct
a r
elia
ble
and
adeq
uate
im
age
of
the
cont
rast
ag
ent
adde
d to
the
blo
od.
The
pre
sent
pap
er d
escr
ibes
a n
ew c
ontr
ast h
arm
onic
im
agin
g m
etho
d de
sign
ed to
sig
nif
ican
tly
incr
ease
the
CN
R a
t th
e th
ird
or f
ourt
h ha
rmon
ic c
ompo
nent
whi
le m
aint
aini
ng,
or e
ven
incr
easi
ng,
the
pote
ntia
lly
good
CT
R f
ound
for
the
se h
arm
onic
com
pone
nts.
4.2
Met
hod
4.2.
1 W
ave
Pro
paga
tion
and
Sca
tter
ing
from
Sof
t T
issu
e
Ult
raso
und
wav
e pr
opag
atio
n in
sof
t tis
sue
can,
as
indi
cate
d, b
e co
nsid
ered
a w
eak
nonl
in
ear
proc
ess
for
ampl
itud
es a
nd f
requ
enci
es c
omm
on in
med
ical
ult
raso
und
imag
ing.
The
lo
cal n
onli
near
ity
is lo
w b
ut th
e di
stor
tion
acc
umul
ates
gra
dual
ly in
the
forw
ard
prop
agat
in
g w
ave.
The
tiss
ue e
last
icit
y be
have
s sl
ight
ly n
onli
near
ly a
nd b
y do
ing
a T
aylo
r se
ries
ex
pans
ion
of th
e eq
uati
on o
f st
ate
P =
P(p
, s)
alon
g an
ise
ntro
pe s
= s
0 [1
6, C
hapt
er
2],
whe
re P
is
the
sum
of
the
ambi
ent
and
acou
stic
pre
ssur
e an
d p
is t
he d
ensi
ty o
f th
e m
ediu
m,
we
can
incl
ude
this
non
line
arit
y. B
y di
scar
ding
ter
ms
of
orde
r hi
gher
tha
n th
e se
cond
in t
he T
aylo
r se
ries
exp
ansi
on,
we
obta
in th
e fo
llow
ing
nonl
inea
r ti
ssue
ela
stic
ity
equa
tion
[3,
Cha
pter
12.
3]
-\7
·-!i(
r, t)
= ~p(
r, t)-
f3{~p(r, t)
}2 +
h * ~
p(r,
t)
t (4
.1)
whe
re 1$
is p
arti
cle
disp
lace
men
t, ~
is t
he c
ompr
essi
bili
ty o
f th
e m
ediu
m, p
is
the
acou
stic
pre
ssur
e, (
3 is
a n
onli
near
ity
para
met
er,
h is
a c
ausa
llow
pass
filt
er,
and
f an
d t
are
the
spac
e an
d ti
me
coor
dina
tes,
res
pect
ivel
y. T
he f
irst
and
sec
ond
term
on
the
righ
t-ha
nd
side
of E
q. 4
.1 r
epre
sent
the
line
ar a
nd n
onli
near
tis
sue
elas
ticity
, res
pect
ivel
y, w
hile
the
la
st c
onvo
luti
on t
erm
is
taki
ng c
are
of
acou
stic
abs
orpt
ion.
B
y ap
plyi
ng t
his
nonl
inea
r ti
ssue
ela
stic
ity
equa
tion
we
may
der
ive
the
foll
owin
g no
nlin
ear
equa
tion
for
wav
e pr
op
agat
ion
[3, C
hapt
er 1
2.3]
(4.2
)
whe
re c
is th
e sp
eed
of s
ound
in th
e pr
opag
atio
n m
ediu
m.
We
obse
rve
that
the
nonl
inea
rit
y de
scri
bed
by t
his
wav
e eq
uati
on i
s pr
oduc
ed b
y a
quad
rati
c ef
fect
and
tha
t th
is e
ffec
t en
ters
as
a so
urce
ter
m i
n th
e ho
mog
eneo
us w
ave
equa
tion
obt
aine
d by
set
ting
the
ter
m
on r
ight
-han
d si
de in
Eq.
4.2
equ
al to
zer
o.
60
Pap
er C
Fir
st c
alcu
lati
ng th
e tr
ansm
itte
d fu
ndam
enta
l fi
eld
from
the
hom
ogen
eous
wav
e eq
ua
tion
and
usin
g th
is f
ield
as
a so
urce
ter
m i
n E
q. 4
.2 t
o ca
lcul
ate
the
tran
smit
ted
seco
nd
harm
onic
fie
ld i
s of
ten
refe
rred
to a
s th
e qu
asi-
line
ar a
ppro
xim
atio
n o
f the
non
line
ar w
ave
equa
tion.
Thi
s qu
asi-
line
ar a
ppro
xim
atio
n, w
hich
is
a fi
rst
appr
oxim
atio
n to
the
non
lin
ear
wav
e eq
uati
on,
is t
ypic
ally
goo
d w
hen
the
nonl
inea
r di
stor
tion
is r
elat
ivel
y lo
w a
s in
ab
sorb
ing
soft
tiss
ue.
The
sig
nal
scat
tere
d fr
om s
oft
tiss
ue i
s a
resu
lt o
f th
e in
here
nt i
nhom
ogen
eous
nat
ure
of
the
prop
agat
ion
med
ium
. T
here
is
a sp
atia
l va
riat
ion
of th
e m
ass
dens
ity
and
com
pr
essi
bili
ty o
n se
vera
l sca
les
resu
ltin
g in
sca
tter
ing
of t
he t
rans
mit
ted
acou
stic
wav
e. T
he
acou
stic
sca
tter
ing
proc
ess
from
sof
t ti
ssue
is
assu
med
to
be l
inea
r an
d th
e pr
esen
ce o
f hi
gher
har
mon
ic c
ompo
nent
s in
the
sca
tter
ed t
issu
e si
gnal
is,
unl
ess
spre
adin
g o
f sc
at
tere
d co
ntra
st a
gent
sig
nal b
eyon
d a
cont
rast
-fil
led
regi
on o
ccur
s, t
here
fore
a r
esul
t of
the
nonl
inea
rity
in th
e fo
rwar
d pr
opag
atin
g w
ave
[3, C
hapt
er 7
].
4.2.
2 Sc
atte
ring
from
Con
tras
t Age
nts
Ult
raso
und
cont
rast
age
nts
gene
rall
y re
spon
d m
uch
mor
e no
nlin
earl
y th
an s
oft
tiss
ue
whe
n su
bjec
t to
an
imag
ing
ultr
asou
nd p
ulse
. H
ighe
r or
der
term
s w
ill
not
be n
egli
gibl
e in
the
sca
tter
ed s
igna
l an
d a
quas
i-li
near
app
roxi
mat
ion
of t
he n
onlin
eari
ty,
as e
xpla
ined
in
Sec
. 4.2
.1, i
s th
eref
ore
usua
lly
not a
goo
d ap
prox
imat
ion
for
the
cont
rast
age
nt.
A w
ell
know
n no
nlin
ear
equa
tion
for
the
flu
id s
urro
undi
ng a
sph
eric
al p
ulsa
ting
gas
bub
ble
is
the
Ray
leig
h-P
less
et e
quat
ion
[33]
[31
]. T
he r
adiu
s os
cill
atio
n of
the
bubb
le i
s in
thi
s eq
uati
on e
xpre
ssed
as
3 p(
aii +
2a?) =
p(
a, t)-
Po
-p;
(t)
(4.3
)
whe
re a
is
the
radi
us, a
and
ii i
s th
e ve
loci
ty a
nd a
ccel
erat
ion
of
the
bubb
le r
adiu
s, p
is
the
den
sity
of
the
surr
ound
ing
flui
d, P
0 is
the
am
bien
t eq
uili
briu
m p
ress
ure,
p;(
t) i
s th
e in
cide
nt d
rivi
ng p
ress
ure,
and
p( a
, t)
is t
he p
ress
ure
at t
he b
ubbl
e su
rfac
e. N
onli
near
el
asti
city
of
the
gas
and
the
enca
psul
atin
g th
in s
hell,
sur
face
ten
sion
, an
d vi
scos
ity
may
be
inco
rpor
ated
in t
he f
irst
ter
m o
n th
e ri
ght-
hand
sid
e in
Eq.
4.3
.
A n
umer
ical
mod
el d
eriv
ed b
y A
ngel
sen
et a
l [ 4
] ha
s be
en u
sed
for
calc
ulat
ing
bubb
le
radi
us o
scil
lati
ons
and
scat
tere
d pr
essu
re i
n th
e pr
esen
t pa
per.
Thi
s m
odel
inc
lude
s an
eq
uati
on f
or t
he r
elat
ion
betw
een
pres
sure
and
rad
ial s
trai
n in
a th
in s
hell
enc
apsu
lati
ng a
ga
s bu
bble
. T
he m
odel
als
o al
low
s fo
r a
fini
te s
peed
of s
ound
in t
he m
ediu
m s
urro
undi
ng
the
bubb
le,
thus
tak
ing
radi
atio
n lo
sses
fro
m t
he b
ubbl
e in
to a
ccou
nt.
Oth
erw
ise
this
m
odel
is
com
para
ble
to t
he w
ell
know
n R
ayle
igh-
Ple
sset
equ
atio
n an
d th
e tw
o m
odel
s gi
ve s
imil
ar re
sults
for
sm
all a
mpl
itud
e ra
dius
osc
illa
tion
s of
the
cont
rast
bub
ble.
4.2
Met
hod
61
4.2.
3 A
New
Pul
se I
nver
sion
Tec
hniq
ue
In t
he c
onve
ntio
nal
puls
e in
vers
ion
tech
niqu
e [2
0],
two
sing
le f
requ
ency
ban
d pu
lses
are
tr
ansm
itte
d, s
catte
red,
and
rec
eive
d se
para
tely
whe
re t
he p
olar
ity
of t
he s
econ
d tr
ansm
it
ted
puls
e is
inv
erte
d re
lativ
e to
the
firs
t. T
he tw
o re
ceiv
ed p
ulse
s ar
e th
en a
dded
tog
ethe
r an
d, i
n th
e id
eal
situ
atio
n, a
ll od
d ha
rmon
ic c
ompo
nent
s in
the
sum
med
sig
nal,
incl
ud
ing
the
fund
amen
tal
com
pone
nt,
are
canc
eled
whi
le t
he e
ven
harm
onic
com
pone
nts
are
pres
erve
d. I
n th
is w
ay t
he c
onve
ntio
nal
puls
e in
vers
ion
tech
niqu
e ty
pica
lly
turn
s ou
t as
a
seco
nd h
arm
onic
tec
hniq
ue,
usin
g th
e se
cond
har
mon
ic c
ompo
nent
for
im
age
reco
nst
ruct
ion.
The
sec
ond
harm
onic
com
pone
nt is
, in
the
sum
mat
ion,
ide
ally
inc
reas
ed b
y 6
dB f
or t
he c
ontr
ast
sign
al a
s w
ell
as t
he t
issu
e si
gnal
, he
nce
incr
easi
ng t
he C
ontr
ast
to
Noi
se R
atio
(C
NR
) by
3 d
B a
ssum
ing
a w
hite
noi
se s
igna
l, bu
t giv
ing
the
sam
e C
ontr
ast
to T
issu
e R
atio
(C
TR
) re
lativ
e to
the
seco
nd h
arm
onic
imag
ing
tech
niqu
e w
here
onl
y on
e tr
ansm
it p
ulse
is a
pplie
d an
d th
e re
ceiv
ed s
econ
d ha
rmon
ic s
igna
l co
mpo
nent
is
filte
red
out a
nd u
sed
for
imag
e re
cons
truc
tion.
The
pre
sent
pap
er d
eals
with
a n
ew a
nd m
ore
cont
rast
spe
cifi
c pu
lse
inve
rsio
n te
chni
que
whe
re d
ual
freq
uenc
y ba
nd p
ulse
s, w
ith
freq
uenc
y co
mpo
nent
s ov
erla
ppin
g in
tim
e, a
re
tran
smitt
ed.
Fir
st,
a pu
lse
cont
aini
ng t
wo
freq
uenc
y ba
nds,
in
part
icul
ar a
fun
dam
enta
l ba
nd a
nd i
ts s
econ
d ha
rmon
ic b
and,
is
tran
smit
ted,
and
the
sca
tter
ed s
igna
l is
rec
eive
d an
d st
ored
. T
hen
a se
cond
pul
se i
s tr
ansm
itte
d, w
here
the
pol
arit
y on
eit
her
the
firs
t or
se
cond
fre
quen
cy b
and
is i
nver
ted
rela
tive
to t
he f
irst
tra
nsm
itte
d pu
lse,
and
a g
ener
al
form
of p
ulse
inve
rsio
n is
per
form
ed o
n th
e re
ceiv
ed s
catt
ered
sig
nals
.
By
perf
orm
ing
this
new
pul
se in
vers
ion
tech
niqu
e, n
onli
near
itie
s de
scri
bed
by t
he q
uasi
li
near
app
roxi
mat
ion
are
canc
eled
. T
his
grea
tly
favo
rs i
mag
ing,
and
in
part
icul
ar th
ird
or
four
th h
arm
onic
im
agin
g, o
f a
regi
on c
onta
inin
g co
ntra
st a
gent
s em
bedd
ed i
n so
ft ti
ssue
si
nce
the
cont
rast
age
nts
are
assu
med
to
resp
ond
muc
h m
ore
nonl
inea
rly
than
the
sof
t ti
ssue
whe
n su
bjec
t to
the
tra
nsm
it p
ulse
s. T
he tw
o tr
ansm
itte
d pu
lses
may
for
exa
mpl
e be
exp
ress
ed a
s
P1(t)
= a
1(t)
sin(
wt)
+ az
(t)s
in(2
wt +
¢)
pz(t
) =
a3(
t)si
n(w
t)-
a4(t
)sin
(2w
t + ¢)
(4.4
)
(4.5
)
whe
re a
n(t)
are
pos
itive
am
plit
ude
func
tions
, w
is t
he a
ngul
ar f
requ
ency
, an
d ¢
is a
n ar
bitr
ary
phas
e an
gle.
The
fac
t th
at a
3(t)
may
be
diff
eren
t fr
om a
1(t
) an
d a 4
(t) m
ay b
e di
ffer
ent f
rom
a2(t)
giv
es t
he f
lexi
bilit
y to
fur
ther
uti
lize
the
assu
mpt
ion
of s
tron
g ve
rsus
w
eak
nonl
inea
rity
of
the
prop
agat
ion
med
ium
. If
the
rece
ived
sig
nals
are
mod
eled
as
pow
er s
erie
s ex
pans
ions
of t
he tr
ansm
itte
d si
gnal
, the
y m
ay b
e w
ritte
n
2
St(t)
= L
)n(t
)pn
(t-
T)
(4.6
) n
=l
(X)
Sc(t)
= L
Cn(
t)pn
(t-
T)
(4.7
) n
=l
62
Pap
er C
for
the
tiss
ue s
igna
l an
d co
ntra
st s
igna
l, re
spec
tivel
y.
Her
e p
is t
he t
rans
mit
ted
sign
al,
bn(t)
and
cn(
t) ar
e am
plit
ude
func
tions
, an
d T
is
a t
ime
dela
y.
If a
3(t)
=
a1(t
) an
d a 4
(t) =
a2(t)
in
Eq.
4.4
and
4.5
, su
mm
ing
the
two
rece
ived
sig
nals
, B
tl a
nd s
t2,
from
a
regi
on o
f so
ft t
issu
e no
t co
ntai
ning
con
tras
t ag
ents
, an
d w
here
non
line
arit
ies
henc
e ar
e as
sum
ed to
be
desc
ribe
d by
the
qua
si-l
inea
r ap
prox
imat
ion,
giv
es
sn(t
) +
st2(
t) =
2b
1(t
)al(
t)si
n(w
t)
+ 2
b2(t
)ai(
t)si
n2 (wt)
+ 2b
2(t)
a~(t
)sin
2(2
wt +
¢).
(4
.8)
Her
e, t
he f
irst
ter
m o
n th
e ri
ght-
hand
sid
e is
the
lin
ear
com
pone
nt i
n th
e su
mm
ed s
ig
nal
whi
le t
he s
econ
d an
d th
ird
term
s ar
e th
e fi
rst
orde
r no
nlin
ear
harm
onic
com
pone
nts,
i.e
. th
e re
sult
of
the
quad
rati
c ef
fect
in
Eq.
4.2
. T
he l
inea
r co
mpo
nent
of t
he t
rans
mit
ted
seco
nd h
arm
onic
ban
d is
can
cele
d in
the
sum
mat
ion
proc
ess,
due
to
the
phas
e in
vers
ion
in E
q. 4
.4 a
nd 4
.5,
whe
reas
the
lin
ear
com
pone
nt o
f th
e tr
ansm
itte
d fu
ndam
enta
l ba
nd
is d
oubl
ed.
The
tw
o no
nlin
ear
term
s co
nsis
t o
f a
seco
nd h
arm
onic
com
pone
nt f
rom
the
tr
ansm
itte
d fu
ndam
enta
l ba
nd a
nd a
sec
ond
harm
onic
com
pone
nt f
rom
the
tra
nsm
itte
d se
cond
har
mon
ic b
and
(i.e.
a
four
th h
arm
onic
com
pone
nt).
The
non
line
ar s
um a
nd d
if
fere
nce
freq
uenc
ies,
pro
duci
ng th
ird
and
fund
amen
tal
harm
onic
com
pone
nts,
hav
e he
nce
been
can
cele
d in
the
pul
se i
nver
sion
pro
cess
for
a p
ropa
gati
on m
ediu
m o
f w
eak
nonl
in
eari
ty,
such
as
soft
tiss
ue.
Str
ong
nonl
inea
r sc
atte
rers
, su
ch a
s co
ntra
st a
gent
s, w
ill
not b
e ad
equa
tely
des
crib
ed b
y th
e qu
asi-
line
ar a
ppro
xim
atio
n o
f th
e no
nlin
eari
ty a
nd s
catt
ered
thi
rd h
arm
onic
com
po
nent
s fr
om t
he b
ubbl
es w
ill n
ot b
e ca
ncel
ed in
the
resu
ltin
g pu
lse
inve
rsio
n pr
oces
s.
In a
dditi
on,
adju
stin
g th
e ar
bitr
ary
phas
e an
gle
in E
q. 4
.4 a
nd 4
.5 o
ne i
s, w
hen
in
vert
ing
the
pola
rity
of
the
seco
nd h
arm
onic
com
pone
nt,
able
to p
rodu
ce tw
o as
ymm
etri
c tr
ansm
it p
ulse
s w
ith
resp
ect
to p
osit
ive
and
nega
tive
pres
sure
am
plitu
des,
hen
ce a
lter
ing
the
acou
stic
sca
tteri
ng p
rope
rtie
s of
the
cont
rast
age
nt f
or t
he t
wo
tran
smit
ted
puls
es a
s sh
own
in S
ec. 4
.3.3
.
If w
e in
stea
d su
btra
ct th
e tw
o re
ceiv
ed s
igna
ls, S
t 1 a
nd s
t2,
from
a re
gion
of s
oft t
issu
e no
t co
ntai
ning
con
tras
t ag
ents
, an
d w
here
non
line
arit
ies
henc
e ar
e as
sum
ed t
o be
des
crib
ed
by t
he q
uasi
-lin
ear
appr
oxim
atio
n, w
e ob
tain
sn(t
)-St
2(t)
= 2
b 1(t
)a2(
t)si
n(2w
t +
¢)
+ 4b
2(t)
a1(t
)a2(
t)si
n(w
t)si
n(2w
t +
¢).
(4
.9)
Now
, th
e li
near
tra
nsm
itte
d fu
ndam
enta
l ba
nd is
can
cele
d in
the
pul
se i
nver
sion
pro
cess
w
hile
the
lin
ear
tran
smit
ted
seco
nd h
arm
onic
ban
d is
dou
bled
. T
he f
irst
ord
er n
onli
near
te
rm,
seco
nd t
erm
on
the
righ
t-ha
nd s
ide
in E
q. 4
.9,
now
con
sist
s of
a n
onli
near
mix
in
g te
rm b
etw
een
the
line
ar t
rans
mit
ted
fund
amen
tal
and
seco
nd h
arm
onic
com
pone
nts
prod
ucin
g no
nlin
ear
fund
amen
tal
and
thir
d ha
rmon
ic c
ompo
nent
s. P
erfo
rmin
g a
subt
rac
tion
inst
ead
of
a su
mm
atio
n of
the
rece
ived
sca
tter
ed s
igna
ls h
ence
can
cels
the
fir
st o
rder
no
nlin
ear
four
th h
arm
onic
com
pone
nts
from
the
sof
t tis
sue.
4.3
Num
eric
al S
imul
atio
ns
63
Bot
h th
e fi
rst
orde
r no
nlin
ear
thir
d an
d fo
urth
har
mon
ic ti
ssue
com
pone
nts
can
pote
ntia
lly
be c
ance
led
by s
ubtr
acti
ng t
he s
catt
ered
sig
nals
fro
m t
wo
iden
tica
l du
al f
requ
ency
ban
d tr
ansm
it p
ulse
s w
here
ther
e is
no
phas
e in
vers
ion
on e
ithe
r the
fun
dam
enta
l no
r th
e se
cond
ha
rmon
ic b
and.
Sub
trac
ting
the
sca
tter
ed s
igna
ls f
rom
tw
o su
ch i
dent
ical
tra
nsm
it p
ulse
s w
ould
, how
ever
, pro
babl
y al
so s
igni
fica
ntly
red
uce
the
scat
tere
d th
ird
and
four
th h
arm
onic
co
mpo
nent
s fr
om t
he c
ontr
ast a
gent
.
4.3
Num
eric
al S
imul
atio
ns
Num
eric
al s
imul
atio
n o
f no
nlin
ear
wav
e pr
opag
atio
n ha
s be
en d
one
usin
g a
sim
ulat
ion
tool
dev
elop
ed i
n ou
r gr
oup
[41]
. T
he s
imul
atio
n pr
ogra
m i
s ca
pabl
e of
mak
ing
a 3-
dim
ensi
onal
sim
ulat
ion
of t
he a
cous
tic
tran
smit
fiel
d fr
om a
n an
nula
r tr
ansd
ucer
incl
udin
g ef
fect
s o
f no
nlin
ear
wav
e pr
opag
atio
n, f
requ
ency
dep
ende
nt a
bsor
ptio
n, a
nd d
iffr
acti
on.
The
non
line
ar e
last
icit
y ef
fect
of
the
prop
agat
ion
med
ium
is
in t
he s
imul
atio
n pr
ogra
m
not l
imit
ed b
y th
e qu
asi-
line
ar a
ppro
xim
atio
n di
scus
sed
in S
ec.
4.2.
1.
An
annu
lar
tran
sduc
er w
ith
radi
us e
qual
to
1 em
and
a g
eom
etri
c fo
cus
at 8
em
was
us
ed t
o ca
lcul
ate
the
tran
smit
fie
ld w
hen
usin
g ac
oust
ic p
rope
t1ie
s o
f m
uscl
e fo
und
in
the
lite
ratu
re [
13].
T
he t
rans
mit
fie
ld o
n th
e sy
mm
etry
axi
s o
f th
e tr
ansd
ucer
is
then
in
Sec
. 4.
3.2
and
4.3.
3 us
ed t
o ca
lcul
ate
the
scat
tere
d si
gnal
fro
m a
n in
sert
ed "
tiss
ue
scat
tere
r" a
nd a
con
tras
t bub
ble,
res
pect
ivel
y.
Bub
ble
radi
us o
scil
lati
ons
and
scat
tere
d fa
r-fi
eld
pres
sure
pul
ses
wer
e ca
lcul
ated
usi
ng
the
num
eric
al m
odel
dev
elop
ed b
y A
ngel
sen
et a
t [4
].
Aco
usti
c pr
oper
ties
fou
nd i
n th
e li
tera
ture
[37
] [ 1
8] f
or t
he c
ontr
ast a
gent
Son
azoi
d w
ere
used
for
the
cont
rast
bub
ble.
Th
e eq
uili
briu
m r
adiu
s w
as s
et t
o 1.
5 {t
m a
nd t
he r
eson
ance
fre
quen
cy o
f th
e bu
bble
is
then
ar
ound
4 M
Hz.
4.3.
1 T
he T
rans
mit
Fie
ld
Fir
st,
a si
ngle
fre
quen
cy b
and
1 M
Hz
tran
smit
pul
se,
desc
ribe
d by
Eq.
4.4
whe
re a
2(t
) =
0, i
s us
ed a
s a
sour
ce o
n th
e an
nula
r tr
ansd
ucer
and
the
axi
sym
met
ric
tran
smit
fie
ld i
s ca
lcu
late
d. F
ig.
4.1
depi
cts
the
obta
ined
fun
dam
enta
l an
d se
cond
har
mon
ic t
rans
mit
fie
ld i
n de
cibe
l sc
ale,
lef
t an
d ri
ght
pane
l, re
spec
tive
ly.
The
hor
izon
tal
axis
is
the
late
ral
dire
ctio
n w
hile
the
ver
tica
l ax
is i
s th
e ra
nge
dire
ctio
n. T
he d
ynam
ic r
ange
is
30 d
B i
n bo
th p
anel
s bu
t th
e se
cond
har
mon
ic f
ield
is
plot
ted
in a
dyn
amic
sca
le w
hich
is
20 d
B b
elow
the
fu
ndam
enta
l fi
eld.
We
noti
ce t
hat
the
seco
nd h
arm
onic
fie
ld b
uild
s up
gra
dual
ly i
n ra
nge
dire
ctio
n an
d th
at i
ts i
nten
sity
is
arou
nd 2
5 dB
bel
ow t
he i
nten
sity
of
the
fund
amen
tal
fiel
d.
A d
ual
freq
uenc
y ba
nd p
ulse
, de
scri
bed
by E
q. 4
.4 w
here
now
a2(t
) f=
0 an
d w
here
a1 (t
)
64
Pap
er C
-2
0
_, -2
l
-10
-
30
a -I
S
I -
35
"' -2
0 -4
0
-2
l -4
l
-30
-
SO
-I
0 [e
m)
(em
)
Fig
ure
4.1:
Tra
nsm
it f
ield
fro
m s
ingl
e fr
eque
ncy
band
sou
rce
puls
e. L
eft
pane
l: F
unda
m
enta
l fi
eld.
Rig
ht p
anel
: S
econ
d ha
rmon
ic f
ield
.
is u
ncha
nged
rel
ativ
e to
the
pre
viou
s pu
lse,
is t
hen
used
as
a so
urce
on
the
tran
sduc
er. T
his
dual
fre
quen
cy b
and
puls
e co
nsis
ts o
f a 1
MH
z an
d a
2 M
Hz
puls
e w
hich
are
ove
rlap
ping
in
tim
e. T
he o
btai
ned
tran
smit
fie
lds
for
the
fund
amen
tal
(1 M
Hz)
and
sec
ond
harm
onic
(2
MH
z) c
ompo
nent
s ar
e di
spla
yed
in F
ig.
4.2,
lef
t an
d ri
ght
pane
l, re
spec
tive
ly.
The
dy
nam
ic r
ange
in
the
plot
s ar
e st
ill
30 d
B b
ut t
he t
wo
fiel
ds a
re n
ow p
lott
ed i
n th
e sa
me
dyna
mic
sca
le.
We
obse
rve
that
the
int
ensi
ty o
f th
e fu
ndam
enta
l an
d se
cond
har
mon
ic
fiel
d is
of
the
sam
e or
der.
The
sec
ond
harm
onic
fie
ld i
s no
w m
ainl
y a
resu
lt o
f th
e li
near
pr
opag
atio
n of
the
seco
nd h
arm
onic
sou
rce
at th
e tr
ansd
ucer
and
not
a fi
rst o
rder
non
line
ar
effe
ct a
s in
the
rig
ht p
anel
of F
ig.
4.1.
By
com
pari
ng th
e le
ft p
anel
in
Fig
. 4.1
and
4.2
we
obse
rve
that
the
tra
nsm
itte
d fu
ndam
enta
l fi
eld
is v
ery
sim
ilar
in t
he t
wo
situ
atio
ns.
In F
ig.
4.3
, th
e ob
tain
ed t
hird
and
fou
rth
harm
onic
fie
lds
usin
g th
e du
al f
requ
ency
ba
nd s
ourc
e pu
lse
are
show
n, l
eft a
nd r
ight
pan
el, r
espe
ctiv
ely.
The
se tw
o ha
rmon
ic fi
elds
ar
e di
spla
yed
in t
he s
ame
dyna
mic
sca
le a
s th
e se
cond
har
mon
ic f
ield
in t
he r
ight
pan
el o
f F
ig. 4
.1.
The
thir
d an
d fo
urth
har
mon
ic f
ield
s fr
om t
he d
ual
freq
uenc
y ba
nd s
ourc
e pu
lse
are
of th
e sa
me
orde
r as
the
sec
ond
harm
onic
fie
ld f
rom
the
sin
gle
freq
uenc
y ba
nd s
ourc
e pu
lse
whi
ch i
s na
tura
l si
nce
they
all
mai
nly
are
resu
lts
of th
e fi
rst
orde
r no
nlin
ear
effe
ct
disc
usse
d in
Sec
. 4.2
.1.
A s
econ
d du
al f
requ
ency
ban
d pu
lse,
des
crib
ed b
y E
q.
4.5,
whe
re t
he p
olar
ity
of t
he
tran
smit
ted
seco
nd h
arm
onic
com
pone
nt i
s in
vert
ed r
elat
ive
to t
he f
irst
dua
l fr
eque
ncy
band
pul
se,
and
whe
re a
3 ( t)
=
a
1 ( t)
and
a4
( t)
=
a 2 ( t)
, is
the
n us
ed a
s a
sour
ce o
n th
e tr
ansd
ucer
and
the
tra
nsm
it f
ield
is
calc
ulat
ed.
The
tw
o tr
ansm
it f
ield
s, o
btai
ned
by
usin
g th
e tw
o di
ffer
ent
dual
fre
quen
cy b
and
sour
ces
on t
he t
rans
duce
r, a
re t
hen
sum
med
to
geth
er i
n a
puls
e in
vers
ion
proc
ess.
S
ince
sca
tter
ing
from
tis
sue
is a
ssum
ed t
o be
a
line
ar p
roce
ss,
the
sign
al o
btai
ned
by a
ddin
g th
ese
two
tran
smit
fie
lds
give
s an
im
pres
sion
of
how
the
pul
se s
umm
atio
n pr
oces
s w
ill a
ffec
t th
e re
sult
ing
rece
ived
tis
sue
sign
als.
4.3
Num
eric
al S
imul
atio
ns
65
-S
-S
-10
-
10
-IS
8
-IS
-"
-20
-2
0
_, _,
-30
-
30
-I
0 -I
[em
]
Fig
ure
4.2
: Tra
nsm
it fi
eld
from
dua
l fre
quen
cy b
and
sour
ce p
ulse
. L
eft p
anel
: F
unda
men
ta
l fi
eld.
Rig
ht p
anel
: S
econ
d ha
rmon
ic f
ield
.
-20
-20
_, _,
-30
-3
0
! -3
5
-35
-40
-40
-4S
-4
S
-SO
-
SO
-I
0 -I
0 (e
m]
(em
]
Fig
ure
4.3
: T
rans
mit
fie
ld f
rom
dua
l fr
eque
ncy
band
sou
rce
puls
e.
Lef
t pa
nel:
Thi
rd
harm
onic
fie
ld.
Rig
ht p
anel
: F
ourt
h ha
rmon
ic f
ield
.
66
Pap
er C
-15
-20
-5
-25
5 "'--1
0
-30
-15
-
35
-20
-4
0
-I
0 -I
[ern
]
Fig
ure
4.4:
Wav
e fi
eld
obta
ined
as
sum
of
tran
smit
fie
lds
from
tw
o du
al f
requ
ency
ban
d so
urce
s w
ith
iden
tica
l fun
dam
enta
l co
mpo
nent
s an
d in
vert
ed p
olar
ity
on s
econ
d ha
rmon
ic
com
pone
nts.
Lef
t pa
nel:
Fun
dam
enta
l fie
ld.
Rig
ht p
anel
: S
econ
d ha
rmon
ic f
ield
.
In F
ig.
4.4,
the
fun
dam
enta
l an
d se
cond
har
mon
ic f
ield
obt
aine
d af
ter
puls
e su
mm
atio
n is
sho
wn,
lef
t an
d ri
ght
pane
l, re
spec
tivel
y. T
he d
ynam
ic r
ange
is
stil
l 30
dB
but
the
dy
nam
ic le
vel
has
been
incr
ease
d by
6 d
B i
n bo
th p
anel
s re
lati
ve to
Fig
. 4.
1. C
ompa
ring
the
le
ft p
anel
s in
Fig
. 4.
4 an
d 4.
1 w
e se
e th
at t
he i
nten
sity
of
the
fund
amen
tal
com
pone
nt is
in
crea
sed
by a
roun
d 6
dB i
n th
e pu
lse
sum
mat
ion
proc
ess
rela
tive
to t
he s
itua
tion
app
lyin
g a
sing
le f
requ
ency
ban
d so
urce
. T
his
is d
ue to
the
fac
t tha
t the
pha
se a
nd a
mpl
itud
e o
f th
e fu
ndam
enta
l co
mpo
nent
s in
the
tw
o tr
ansm
itte
d du
al f
requ
ency
ban
d pu
lses
are
ide
ntic
al,
and
the
inte
nsit
y o
f the
fun
dam
enta
l com
pone
nt in
the
sum
med
sig
nal i
s he
nce
doub
led
as
show
n in
Eq.
4.8
. C
ompa
ring
the
righ
t pan
els
in F
ig.
4.4
and
4.1
we
obse
rve
that
the
sec
on
d ha
rmon
ic c
ompo
nent
in t
he s
igna
l obt
aine
d af
ter
puls
e su
mm
atio
n al
so h
as i
ncre
ased
its
int
ensi
ty b
y ap
prox
imat
ely
6 dB
rel
ativ
e to
the
sit
uati
on w
hen
a si
ngle
fre
quen
cy b
and
sour
ce is
use
d. A
gain
, th
is is
in
agre
emen
t w
ith
Eq.
4.8
. In
Fig
. 4.
5 th
e th
ird
and
four
th h
arm
onic
fie
ld o
btai
ned
from
the
pul
se s
umm
atio
n pr
o
cess
are
dis
play
ed,
left
and
rig
ht p
anel
, re
spec
tive
ly.
The
dyn
amic
sca
le i
n th
e le
ft p
anel
ha
s be
en r
educ
ed b
y 20
dB
rel
ativ
e to
the
lef
t pa
nel
in F
ig.
4.3
whi
le t
he d
ynam
ic s
cale
in
the
rig
ht p
anel
has
bee
n in
crea
sed
by 6
dB
rel
ativ
e to
the
rig
ht p
anel
in
Fig.
4.3
. B
y co
mpa
ring
the
left
pan
els
in F
ig. 4
.3 a
nd 4
.5 w
e ca
n co
nclu
de th
at th
e th
ird
harm
onic
com
po
nent
is s
igni
fica
ntly
red
uced
in
the
puls
e su
mm
atio
n pr
oces
s as
exp
ecte
d fr
om E
q. 4
.8.
Fina
lly,
com
pari
ng t
he r
ight
pan
els
in F
ig.
4.3
and
4.5
we
obse
rve
that
the
fou
rth
har
mon
ic f
ield
has
inc
reas
ed b
y an
exp
ecte
d 6
dB i
n th
e pu
lse
sum
mat
ion
proc
ess
rela
tive
to
the
field
obt
aine
d fr
om t
he d
ual
freq
uenc
y ba
nd s
ourc
e pu
lse.
4.3
Num
eric
al S
imul
atio
ns
67
-15
_,
-20
-2
0
-25
-2
5
-30
1
-30
-35
-3
5
-40
-4
0
-I
0 -
I 0
(em
} [e
m]
Figu
re 4
.5:
Wav
e fi
eld
obta
ined
as
sum
of
tran
smit
fie
lds
from
tw
o du
al f
requ
ency
ban
d so
urce
s w
ith
iden
tica
l fun
dam
enta
l co
mpo
nent
s an
d in
vert
ed p
olar
ity
on s
econ
d ha
rmon
ic
com
pone
nts.
Lef
t pa
nel:
Thi
rd h
arm
onic
fie
ld.
Rig
ht p
anel
: F
ourt
h ha
rmon
ic f
ield
.
4.3.
2 Sc
atte
ring
from
Tis
sue
The
cal
cula
ted
tran
smit
fie
ld a
t th
e sy
mm
etry
axi
s is
now
use
d to
cal
cula
te t
he s
catt
ered
si
gnal
fro
m a
"ti
ssue
sca
tter
er"
inse
rted
at v
ario
us l
ocat
ions
alo
ng t
he s
ymm
etry
axi
s. T
he
scat
teri
ng f
rom
sof
t tis
sue
is a
ssum
ed t
o be
a l
inea
r pr
oces
s pr
opor
tion
al w
ith
the
fre
quen
cy.
Onl
y th
e re
lativ
e le
vels
of s
catt
ered
har
mon
ic c
ompo
nent
s ar
e co
nsid
ered
in th
e pr
esen
t pa
per
and
abso
lute
leve
ls o
f sca
tter
ed t
issu
e si
gnal
s an
d co
ntra
st b
ubbl
e si
gnal
s ar
e he
nce
not
give
n.
Firs
t, th
e ob
tain
ed t
rans
mit
fie
ld f
rom
the
sin
gle
freq
uenc
y ba
nd s
ourc
e pu
lse,
des
crib
ed
by s
etti
ng a
2(t)
= 0
in E
q. 4
.4,
is u
sed
to c
alcu
late
the
sca
tter
ed s
igna
l fr
om t
issu
e. T
he
inte
nsit
y of
the
scat
tere
d ti
ssue
sig
nal
alon
g th
e sy
mm
etry
axi
s is
dis
play
ed a
s th
e da
shed
li
nes
in F
ig. 4
.6.
The
pan
els
in t
his
figu
re i
ndic
ate
scat
tere
d ha
rmon
ic c
ompo
nent
s, g
oing
fr
om t
he f
unda
men
tal
com
pone
nt in
the
lef
t pa
nel,
to t
he f
ourt
h ha
rmon
ic c
ompo
nent
in
the
righ
t pa
nel.
The
dyn
amic
ran
ge i
n al
l pa
nels
is
set
to 3
0 dB
whe
reas
the
dyn
amic
le
vel
vari
es b
etw
een
the
pane
ls.
As
seen
in
the
firs
t pa
nel,
the
scat
tere
d ti
ssue
sig
nal
is
norm
aliz
ed s
o th
at th
e fu
ndam
enta
l co
mpo
nent
at 8
em
is s
et to
0 d
B.
In t
he s
econ
d pa
nel
we
see
that
the
sca
tter
ed s
econ
d ha
rmon
ic t
issu
e co
mpo
nent
is a
roun
d 20
dB
bel
ow t
he
scat
tere
d ti
ssue
fun
dam
enta
l co
mpo
nent
, w
hile
the
scat
tere
d th
ird
harm
onic
com
pone
nt is
in
the
thir
d pa
nel f
ound
to b
e ar
ound
35
dB b
elow
the
sca
tter
ed f
unda
men
tal
com
pone
nt.
The
sol
id l
ines
in
Fig.
4.6
are
har
mon
ic t
issu
e co
mpo
nent
s of
the
puls
e su
mm
atio
n si
gna
l. T
his
puls
e su
mm
atio
n si
gnal
is a
cqui
red
by s
umm
ing
the
two
scat
tere
d ti
ssue
sig
nals
ob
tain
ed b
y ap
plyi
ng th
e tw
o du
al f
requ
ency
ban
d so
urce
s, d
escr
ibed
by
Eq.
4.4
and
4.5
, w
here
a3(t)
= a
1(t)
and
a4(t
) =
a2(t
).
In t
he l
eft
pane
l, w
e ob
serv
e th
at t
he i
nten
sity
68
Pap
er C
Fig
ure
4.6:
S
catt
ered
har
mon
ic t
issu
e si
gnal
alo
ng s
ymm
etry
axi
s ba
sed
on c
alcu
late
d tr
ansm
it f
ield
s, g
oing
fro
m f
unda
men
tal c
ompo
nent
in le
ft p
anel
to f
ourt
h ha
rmon
ic c
om
pone
nt i
n ri
ght
pane
l. D
ashe
d lin
es:
Sin
gle
freq
uenc
y ba
nd s
ourc
e pu
lse
used
. S
olid
lin
es:
Tw
o du
al f
requ
ency
ban
d so
urce
pul
ses
used
, ph
ase
inve
rsio
n on
sec
ond
harm
onic
co
mpo
nent
, w
ith
puls
e su
mm
atio
n of
scat
tere
d si
gnal
s.
of th
e fu
ndam
enta
l ti
ssue
com
pone
nt i
s in
crea
sed
by a
ppro
xim
atel
y 6
dB i
n th
e pu
lse
sum
mat
ion
proc
ess
of
the
two
scat
tere
d du
al f
requ
ency
ban
d so
urce
sig
nals
rel
ativ
e to
the
sc
atte
red
sing
le f
requ
ency
ban
d so
urce
sig
nal.
In t
he s
econ
d pa
nel,
the
seco
nd h
arm
onic
co
mpo
nent
of
the
puls
e su
mm
atio
n si
gnal
is a
lso
seen
to b
e ar
ound
6 d
B a
bove
the
sign
al
obta
ined
usi
ng t
he s
ingl
e fr
eque
ncy
band
sou
rce
puls
e. T
he t
hird
har
mon
ic c
ompo
nent
, di
spla
yed
in t
he t
hird
pan
el, i
s si
gnif
ican
tly
high
er th
an t
he s
catt
ered
thir
d ha
rmon
ic c
om
pone
nt o
btai
ned
usin
g th
e si
ngle
fre
quen
cy b
and
sour
ce p
ulse
. H
owev
er,
by c
ompa
ring
w
ith
the
scat
tere
d fo
urth
har
mon
ic c
ompo
nent
of t
he p
ulse
sum
mat
ion
sign
al in
the
four
th
pane
l, w
e se
e th
at t
he s
catt
ered
thi
rd h
arm
onic
pul
se s
umm
atio
n si
gnal
is
sign
ific
antl
y re
duce
d in
the
pul
se s
umm
atio
n pr
oces
s.
The
se a
re a
ll ex
pect
ed r
esul
ts i
n re
ason
able
ag
reem
ent w
ith
Eq.
4.8
.
If th
e tw
o sc
atte
red
sign
als,
obt
aine
d us
ing
the
two
dual
fre
quen
cy b
and
sour
ces,
are
tim
e sh
ifte
d re
lativ
e to
eac
h ot
her
by a
sm
all
amou
nt b
efor
e su
mm
atio
n, t
he r
esul
ting
thi
rd
harm
onic
com
pone
nt in
the
puls
e su
mm
atio
n si
gnal
can
be
furt
her
redu
ced.
In
Fig
. 4.7
the
two
scat
tere
d si
gnal
s ha
ve b
een
tim
e sh
ifte
d re
lati
ve to
eac
h ot
her b
y up
to
10
ns b
efor
e pu
lse
sum
mat
ion
is p
erfo
rmed
. Fi
g. 4
.7 r
epre
sent
s th
e sa
me
as F
ig. 4
.6 a
nd
the
disp
laye
d dy
nam
ic r
ange
and
sca
le a
re id
enti
cal i
n th
e tw
o fi
gure
s. B
y co
mpa
ring
thes
e tw
o fi
gure
s w
e ob
serv
e th
at th
e sm
all t
ime
shif
ts i
ntro
duce
d be
fore
pul
se s
umm
atio
n ha
ve
litt
le o
r no
eff
ect o
n th
e fu
ndam
enta
l an
d fo
urth
har
mon
ic p
ulse
sum
mat
ion
com
pone
nts.
T
his
is d
ue to
the
fact
that
thes
e ha
rmon
ic c
ompo
nent
s ar
e in
pha
se w
hen
sum
med
toge
ther
an
d a
smal
l ti
me
shif
t, re
lati
ve t
o th
e w
ave
leng
th,
will
not
hav
e a
maj
or e
ffec
t on
the
re
sults
of t
he s
umm
atio
n. T
he s
econ
d an
d th
ird
harm
onic
com
pone
nts
are,
how
ever
, clo
se
4.3
Nu
mer
ical
Sim
ula
tion
s 69
Fig
ure
4.7:
Sam
e as
Fig
. 4.
6 bu
t w
ith
smal
l ti
me
shif
ts b
etw
een
the
two
scat
tere
d du
al
freq
uenc
y ba
nd p
ulse
s be
fore
pul
se s
umm
atio
n.
to 1
r ou
t of p
hase
whe
n su
mm
ed to
geth
er a
nd th
e sm
all t
ime
shif
ts i
ntro
duce
d ar
e se
en to
ha
ve m
ajor
eff
ects
on
thes
e ha
rmon
ic c
ompo
nent
s. R
elat
ive
to t
he s
itua
tion
wit
hout
tim
e sh
ifts
bet
wee
n th
e tw
o sc
atte
red
puls
es a
s in
Fig
. 4.6
, bo
th th
ese
com
pone
nts
are
redu
ced
whe
n in
trod
ucin
g th
e sm
all
tim
e sh
ifts
wit
h a
3 dB
red
ucti
on f
or t
he s
econ
d ha
rmon
ic
com
pone
nt a
nd a
maj
or re
duct
ion
for
the
thir
d ha
rmon
ic c
ompo
nent
. W
ith th
e in
trod
uced
ti
me
shif
ts,
the
thir
d ha
rmon
ic c
ompo
nent
aft
er p
ulse
sum
mat
ion
is o
f th
e sa
me
orde
r as
ob
tain
ed f
rom
the
sin
gle
freq
uenc
y ba
nd s
ourc
e pu
lse
(das
hed
line)
.
If in
stea
d th
e tw
o sc
atte
red
sign
als
from
the
dua
l fr
eque
ncy
band
sou
rces
are
sub
trac
ted,
th
e sc
atte
red
four
th h
arm
onic
com
pone
nt f
rom
tis
sue
may
be
redu
ced
as p
revi
ousl
y in
di
cate
d in
rel
atio
n to
Eq.
4.9
. T
he d
ashe
d lin
es i
n Fi
g. 4
.8 s
till
ind
icat
e th
e sc
atte
red
leve
l ob
tain
ed b
y us
ing
a si
ngl
e fr
eque
ncy
band
sou
rce
and
the
scat
tere
d fu
ndam
enta
l co
mpo
nent
in t
he l
eft
pane
l is
no
rmal
ized
to 0
dB
at
8 em
as
in t
he t
wo
prev
ious
fig
ures
. T
he s
olid
lin
es i
n th
e pa
nels
in
dica
te h
arm
onic
lev
els
obta
ined
aft
er p
ulse
sub
trac
tion
of
the
scat
tere
d ti
ssue
sig
nals
ob
tain
ed f
rom
the
two
dual
fre
quen
cy b
and
sour
ces,
des
crib
ed b
y E
qs. 4
.4 a
nd 4
.5,
whe
re
a 3(t)
=
a1(t)
and
a4(t)
=
a2(t)
. In
the
lef
t pa
nel
we
clea
rly
see
that
the
fun
dam
enta
l co
mpo
nent
is h
eavi
ly r
educ
ed i
n th
e pu
lse
subt
ract
ion
proc
ess
rela
tive
to t
he c
ase
usin
g a
sing
le f
requ
ency
ban
d so
urce
. T
he l
inea
r tr
ansm
itte
d fu
ndam
enta
l co
mpo
nent
is c
an
cele
d in
the
pro
cess
and
onl
y a
nonl
inea
r fu
ndam
enta
l co
mpo
nent
, se
cond
ter
m o
n th
e ri
ght-
hand
sid
e in
Eq.
4.9
, re
mai
ns.
A s
tron
g li
near
tra
nsm
itte
d se
cond
har
mon
ic c
om
pone
nt,
firs
t te
rm o
n th
e ri
ght-
hand
sid
e in
Eq.
4.9
, is
evi
dent
in
the
seco
nd p
anel
. In
th
e th
ird
pane
l, a
sign
ific
ant f
irst
ord
er n
onli
near
thir
d ha
rmon
ic c
ompo
nent
, sec
ond
term
on
the
rig
ht-h
and
side
in
Eq.
4.9
, al
so r
emai
ns a
fter
the
pul
se s
ubtr
acti
on p
roce
ss.
The
fo
urth
har
mon
ic c
ompo
nent
in t
he l
ast p
anel
is,
alth
ough
sig
nifi
cant
ly r
educ
ed c
ompa
red
to F
ig. 4
.6,
stil
l rel
ativ
ely
stro
ng.
70
Pap
er C
Fig
ure
4.8:
S
catt
ered
har
mon
ic t
issu
e si
gnal
alo
ng s
ymm
etry
axi
s ba
sed
on c
alcu
late
d tr
ansm
it fi
elds
, go
ing
from
fun
dam
enta
l co
mpo
nent
in l
eft p
anel
to f
ourt
h ha
rmon
ic c
om
pone
nt i
n ri
ght
pane
l, D
ashe
d lin
es:
Sin
gle
freq
uenc
y ba
nd s
ourc
e pu
lse
used
. S
olid
lin
es:
Tw
o du
al f
requ
ency
ban
d so
urce
pul
ses
used
, ph
ase
inve
rsio
n on
sec
ond
harm
onic
co
mpo
nent
, wit
h pu
lse
subt
ract
ion
of s
catt
ered
sig
nals
.
To f
urth
er r
educ
e th
e fo
urth
har
mon
ic c
ompo
nent
in F
ig. 4
.8,
smal
l tim
e sh
ifts
and
am
pli
tude
cor
rect
ions
mus
t be
intr
oduc
ed in
the
tw
o sc
atte
red
puls
es b
efor
e th
ey a
re c
ombi
ned
in th
e pu
lse
inve
rsio
n pr
oces
s. I
n Fi
g. 4
.9, a
tim
e sh
ift o
f up
to 2
2 ns
and
am
plit
ude
corr
ec
tion
s up
to 4
dB
are
int
rodu
ced
befo
re p
ulse
sub
trac
tion
. W
e no
tice
tha
t the
eff
ects
on
the
resu
ltin
g fu
ndam
enta
l an
d fo
urth
har
mon
ic c
ompo
nent
s, a
re o
f sp
ecia
l im
port
ance
. T
he
four
th h
arm
onic
com
pone
nt in
the
rig
ht p
anel
is h
eavi
ly r
educ
ed b
y ap
prox
imat
ely
20 d
B
com
pare
d to
wha
t was
obt
aine
d in
Fig
. 4.8
wit
hout
the
tim
e an
d am
plit
ude
com
pens
atio
ns
whe
reas
the
res
ulti
ng f
unda
men
tal
com
pone
nt, i
n th
e le
ft p
anel
, is
sign
ific
antl
y in
crea
sed
by a
roun
d 20
dB
.
In F
ig. 4
.6 a
nd 4
.8,
we
saw
tha
t the
sca
tter
ed th
ird
or fo
urth
har
mon
ic t
issu
e co
mpo
nent
s,
mai
nly
intr
oduc
ed d
ue t
o ap
plic
atio
n o
f du
al f
requ
ency
ban
d so
urce
s, c
ould
be
sign
ifi
cant
ly r
educ
ed b
y pe
rfor
min
g th
e si
mpl
e pu
lse
sum
mat
ion
or p
ulse
sub
trac
tion
pro
cess
ac
cord
ing
to E
q. 4
.8 a
nd 4
.9.
Intr
oduc
ing
smal
l ti
me
shif
ts a
nd a
mpl
itud
e co
rrec
tion
s in
th
e tw
o sc
atte
red
puls
es b
efor
e su
mm
atio
n or
sub
trac
tion
, th
e th
ird
or f
ourt
h ha
rmon
ic
com
pone
nts
coul
d, h
owev
er,
be f
urth
er r
educ
ed.
The
nee
d to
int
rodu
ce t
hese
com
pen
sati
ons
in t
he s
catt
ered
sig
nals
bef
ore
perf
orm
ing
the
puls
e in
vers
ion
proc
ess
is a
con
se
quen
ce o
f th
e fa
ct t
hat t
he r
esul
ting
wav
e pr
opag
atio
n fr
om t
he a
ppli
ed s
ourc
es d
evia
tes
som
ewha
t fro
m t
he q
uasi
-lin
ear
appr
oxim
atio
n di
scus
sed
in S
ec. 4
.2.1
.
In T
able
4.1
, th
e re
sult
ing
thir
d an
d fo
urth
har
mon
ic s
igna
l co
mpo
nent
s al
ong
the
sym
m
etry
axi
s fo
r th
e tw
o du
al f
requ
ency
ban
d tr
ansm
it f
ield
s ar
e co
mpa
red.
T
wo
sour
ces
desc
ribe
d by
Eq.
4.4
and
4.5
, w
here
a3(t)
= a
1 (t
) an
d a 4
(t) =
a2(t)
an
d¢
= 0
, ar
e us
ed
as s
ourc
es o
n th
e in
dica
ted
tran
sduc
er.
4.3
Nu
mer
ical
Sim
ulat
ions
71
Fig
ure
4.9:
Sam
e as
Fig
. 4.
8 bu
t w
ith
tim
e sh
ifts
and
am
plit
ude
corr
ecti
ons
betw
een
the
two
scat
tere
d du
al f
requ
ency
ban
d pu
lses
bef
ore
subt
ract
ion.
Insp
ecti
ng th
e th
ird
harm
onic
com
pone
nts
we
obse
rve
that
the
rela
tive
tim
e sh
ifts
be
twee
n th
e tw
o tr
ansm
it fi
elds
are
alm
ost c
onst
ant f
rom
aro
und
6 em
in r
ange
dir
ecti
on a
nd
that
dif
fere
nces
in
ampl
itud
es g
ener
ally
are
sm
all.
Cha
ngin
g to
the
fou
rth
harm
onic
com
pone
nts,
we
see
that
dif
fere
nces
in
ampl
itud
es
are
muc
h la
rger
and
in
Fig.
4.9
, am
plit
ude
com
pens
atio
ns h
ad to
be
incl
uded
in o
rder
to
furt
her
supp
ress
the
four
th h
arm
onic
com
pone
nt in
the
pul
se s
ubtr
acti
on s
igna
l.
The
dev
iati
ons
betw
een
the
full
nonl
inea
r w
ave
prop
agat
ion
and
the
quas
i-li
near
app
rox
imat
ion
can
be f
urth
er e
xam
ined
by
usin
g ou
r si
mul
atio
n pr
ogra
m in
a q
uasi
-lin
ear
mod
e to
cal
cula
te t
he t
wo
quas
i-li
near
wav
e fi
elds
res
ulti
ng f
rom
the
tw
o du
al f
requ
ency
ban
d so
urce
s.
The
se t
wo
tran
smit
wav
e fi
elds
are
the
n ca
lcul
ated
by
firs
t us
ing
the
dual
fre
qu
ency
ban
d so
urce
pul
se d
escr
ibed
by
Eq.
4.4
on
the
tran
sduc
er a
nd t
hen
appl
ying
the
so
urce
des
crib
ed b
y E
q. 4
.5,
whe
re a
3(t)
= a
1(t)
and
a4(t)
= a
2(t)
an
d¢
= 0
. K
eepi
ng
in m
ind
that
the
scat
teri
ng f
rom
tis
sue
is a
ssum
ed to
be
a li
near
pro
cess
, th
e tw
o tr
ansm
it
fiel
ds a
re t
hen
com
bine
d in
the
pul
se i
nver
sion
pro
cess
and
com
pare
d w
ith t
he p
revi
ous
resu
lts o
btai
ned
by p
erfo
rmin
g a
full
nonl
inea
r so
luti
on o
f Eq.
4.2
.
Fig.
4.1
0 di
spla
ys a
com
pari
son
of
the
thir
d ha
rmon
ic p
ulse
sum
mat
ion
tran
smit
fie
lds
obta
ined
fro
m t
he t
he f
ull
nonl
inea
r so
luti
on o
f th
e no
nlin
ear
wav
e eq
uati
on a
nd t
he
quas
i-li
near
sol
utio
n of
the
nonl
inea
r w
ave
equa
tion
, le
ft a
nd r
ight
pan
el,
resp
ectiv
ely.
W
e ob
serv
e th
at t
he t
hird
har
mon
ic l
evel
fro
m t
he q
uasi
-lin
ear
solu
tion
is
sign
ific
antly
lo
wer
tha
n fr
om t
he f
ull
nonl
inea
r so
lutio
n. C
ompa
riso
ns o
f th
e tw
o tr
ansm
it f
ield
s ob
ta
ined
fro
m t
he f
ull
nonl
inea
r so
luti
on o
f th
e w
ave
equa
tion
res
ulte
d in
tim
e de
lays
and
am
plit
ude
vari
atio
ns s
how
n in
Tab
le 4
.1, w
hich
are
not
pre
sent
in th
e qu
asi-
line
ar s
olut
ion
of t
he w
ave
equa
tion.
F
rom
Eq.
4.8
, th
e th
ird
harm
onic
com
pone
nt o
f th
e pu
lse
sum
mat
ion
sign
al s
houl
d,
in t
he q
uasi
-lin
ear
situ
atio
n, i
deal
ly b
e to
tally
can
cele
d.
In t
he r
ight
pan
el o
f Fi
g. 4
.10,
72
Pap
er C
Tab
le 4
.1:
Rel
ativ
e ti
me
dela
ys a
nd a
mpl
itud
e va
riat
ions
alo
ng s
ymm
etry
axi
s fo
r th
e tw
o du
al f
requ
ency
ban
d tr
ansm
it f
ield
s.
3rct
har
mon
ic
3rct
har
mon
ic
4th
har
mon
ic
4th
har
mon
ic
1cm
0
.2d
B
0.7
dB
2cm
0.
4 dB
1.
3 dB
3
cm
-2 n
s 0
.6d
B
-2 n
s 2
.6d
B
4cm
5
ns
0.1
dB
24
ns
4d
B
Scm
6
ns
0.5
dB
15 n
s 2.
9 dB
6
cm
8 ns
0.
5 dB
15
ns
3 dB
7
cm
9
ns
0.5
dB
15 n
s 3
.2d
B
Scm
9
ns
0.5
dB
16 n
s 3.
4 dB
9
cm
9 ns
0.
5 dB
18
ns
3.6
dB
1
0cm
9
ns
0.5
dB
19 n
s 3.
8 dB
11
em
9
ns
0.4
dB
20
ns
3.9
dB
1
2cm
10
ns
0.3
dB
22
ns
4d
B
the
thir
d ha
rmon
ic c
ompo
nent
is,
how
ever
, no
t fu
lly
canc
eled
. T
he r
esul
ting
sec
ond
and
four
th h
arm
onic
com
pone
nts
whi
ch a
re n
ot th
at s
tron
gly
supp
ress
ed in
the
pul
se s
umm
ati
on s
igna
l w
ill,
depe
ndin
g on
the
ban
dwid
th o
f th
e tr
ansm
itte
d fu
ndam
enta
l an
d se
cond
ha
rmon
ic c
ompo
nent
s an
d th
e ba
ndw
idth
of
the
thir
d ha
rmon
ic f
ilter
app
lied
, in
trod
uce
wea
k si
gnal
com
pone
nts
in t
he p
assb
and
of t
he t
hird
har
mon
ic f
ilter
.
In F
ig.
4.11
we
see
a co
mpa
riso
n be
twee
n th
e fo
urth
har
mon
ic p
ulse
sub
trac
tion
tran
smit
fi
elds
obt
aine
d fr
om t
he t
he f
ull
nonl
inea
r so
luti
on o
f th
e no
nlin
ear
wav
e eq
uati
on a
nd
the
quas
i-li
near
sol
utio
n o
f the
non
line
ar w
ave
equa
tion
, lef
t an
d ri
ght p
anel
, res
pect
ivel
y.
We
obse
rve
that
the
inte
nsit
y o
f th
e fo
urth
har
mon
ic f
ield
obt
aine
d fr
om t
he q
uasi
-lin
ear
appr
oxim
atio
n o
f th
e no
nlin
ear
wav
e eq
uati
on is
sig
nifi
cant
ly lo
wer
in t
he r
esul
ting
pul
se
subt
ract
ion
sign
al.
4.3.
3 Sc
atte
ring
from
a C
ontr
ast B
ubbl
e
We
now
pro
ceed
to i
nves
tiga
te th
e sc
atte
ring
from
a c
ontr
ast b
ubbl
e. T
he m
ain
goal
now
is
twof
old.
Fir
st,
to c
ompa
re th
e si
gnal
sca
tter
ed f
rom
the
bub
ble
whe
n dr
iven
by
a co
nven
ti
onal
sin
gle
freq
uenc
y ba
nd p
ulse
and
the
sign
al o
btai
ned
afte
r pul
se in
vers
ion
usin
g du
al
freq
uenc
y ba
nd d
rive
pul
ses.
And
sec
ond,
to
com
pare
the
rela
tive
leve
l of h
arm
onic
com
po
nent
s in
the
dua
l fr
eque
ncy
band
pul
se i
nver
sion
bub
ble
sign
al a
nd t
he d
ual
freq
uenc
y ba
nd p
ulse
inve
rsio
n so
ft ti
ssue
sig
nal f
rom
the
pre
viou
s se
ctio
n. U
ltra
soun
d co
ntra
st b
ub
bles
hav
e m
uch
high
er c
ompl
ianc
e th
an s
oft t
issu
e an
d th
e bu
bble
wil
l res
pond
dif
fere
ntly
re
lati
ve to
sof
t tis
sue
whe
n dr
iven
by
the
dual
fre
quen
cy b
and
puls
es.
4.3
Num
eric
al S
imul
atio
ns
73
-35
-3
5
-40
-4
0
-45
-4
5
~ -5
0
1 -5
0
-55
-5
5
-60
-o
o
-I
0 -I
0 [e
m)
(em
]
Fig
ure
4.10
: W
ave
fiel
d ob
tain
ed a
s su
mm
atio
n o
f tra
nsm
it fi
elds
fro
m tw
o du
al f
requ
ency
ba
nd s
ourc
es w
ith
iden
tica
l fun
dam
enta
l co
mpo
nent
s an
d in
vert
ed p
olar
ity
on s
econ
d ha
rm
onic
com
pone
nts.
Lef
t pa
nel:
Thi
rd h
arm
onic
fie
ld o
btai
ned
from
ful
l no
nlin
ear
solu
ti
on o
f E
q. 4
.2.
Rig
ht p
anel
: T
hird
har
mon
ic f
ield
obt
aine
d fr
om q
uasi
-lin
ear
solu
tion
of
Eq
. 4.2
.
-35
-3
5
-40
-4
0
-45
-45
I -5
0
I -5
0
-55
_,
-60
-o
o
-I
0 -I
0 [e
m]
(em
]
Fig
ure
4.1
1: W
ave
fiel
d ob
tain
ed a
s di
ffer
ence
of
tran
smit
fie
lds
from
tw
o du
al f
requ
ency
ba
nd s
ourc
es w
ith
iden
tica
l fun
dam
enta
l co
mpo
nent
s an
d in
vert
ed p
olar
ity
on s
econ
d ha
rm
onic
com
pone
nts.
Lef
t pa
nel:
Fou
rth
harm
onic
fie
ld o
btai
ned
from
ful
l no
nlin
ear
solu
ti
on o
f E
q. 4
.2.
Rig
ht p
anel
: F
ourt
h ha
rmon
ic f
ield
obt
aine
d fr
om q
uasi
-lin
ear
solu
tion
of
Eq.
4.2
.
74
7.::~
~3·
~
0
-0.1
-0.2
-O.]
l 1.
5 2
2.5~·-=-,_~, ~-4.5
5 5.
5 6
"")
0.2
~ o
f-----..
..
-0.2
-0.4
1'-
--~~-,~-~~,_,:--~-"---c,~.,-~~._,,-------L...__,,~_, _
__
_!
[).tS
]
Pap
er C
Figu
re 4
.12:
U
pper
pan
el:
Sing
le f
requ
ency
ban
d pu
lse.
L
ower
pan
el:
Dua
l fr
eque
ncy
band
pul
ses
wit
h id
enti
cal
fund
amen
tal
com
pone
nts
and
phas
e in
vers
ion
on s
econ
d ha
rm
onic
com
pone
nts.
The
upp
er p
anel
in F
ig.
4.12
sho
ws
an e
xam
ple
of a
1 M
Hz
sing
le f
requ
ency
ban
d so
urce
pu
lse
whe
re a
2(t
) in
Eq.
4.4
is
set
to z
ero.
If
a1(t)
in
Eq.
4.4
is
unch
ange
d an
d a 2
(t)
now
is
diff
eren
t fr
om z
ero,
so
that
a 2
MH
z pu
lse
is a
dded
, w
e ca
n ge
t th
e so
lid
line
in t
he l
ower
pan
el o
f Fi
g. 4
.12
repr
esen
ting
a d
ual
freq
uenc
y ba
nd s
ourc
e pu
lse.
T
hen,
si
mpl
y by
inve
rtin
g th
e po
lari
ty o
n th
e ad
ded
2 M
Hz
puls
e w
hile
kee
ping
the
1 M
Hz
puls
e un
chan
ged,
we
can
cons
truc
t the
pul
se d
ispl
ayed
as
the
dash
ed l
ine
in t
he l
ower
pan
el o
f th
e fi
gure
. T
hese
tw
o pu
lses
, in
the
low
er p
anel
, re
pres
ent
the
two
dual
fre
quen
cy b
and
puls
es u
sed
in t
he e
xpla
ined
pul
se i
nver
sion
tec
hniq
ue.
We
noti
ce t
hat
thes
e pu
lses
are
as
ymm
etri
c w
ith
resp
ect
to p
ositi
ve a
nd n
egat
ive
pres
sure
am
plit
udes
and
the
con
tras
t bu
bble
will
pre
sum
ably
res
pond
dif
fere
ntly
whe
n dr
iven
by
one
of th
em r
elat
ive
to t
he
othe
r.
In F
ig.
4.13
, th
e so
lid
line
s di
spla
y th
e ab
solu
te v
alue
of
the
Fou
rier
Tra
nsfo
rm o
f th
e sc
atte
red
bubb
le s
igna
l fr
om a
3 p
m b
ubbl
e w
ith
reso
nanc
e fr
eque
ncy
arou
nd 4
MH
z w
hen
driv
en b
y th
e si
ngle
fre
quen
cy b
and
puls
e in
the
upp
er p
anel
of
Fig.
4.1
2.
We
clea
rly
see
that
the
bub
ble
has
resp
onde
d no
nlin
earl
y to
the
dri
ve p
ulse
with
sig
nifi
cant
am
ount
s of
scat
tere
d en
ergy
at h
arm
onic
com
pone
nts.
The
sca
tter
ed b
ubbl
e si
gnal
is
then
cal
cula
ted
usin
g th
e tw
o du
al f
requ
ency
ban
d pu
lses
in
the
low
er p
anel
of F
ig.
4.12
as
driv
e pu
lses
. A
ddin
g th
e re
sult
ing
two
scat
tere
d pu
lses
, w
e ob
tain
the
das
hed
line
in t
he u
pper
pa
nel
of F
ig.
4.13
. A
ll sc
atte
red
harm
onic
com
pone
nts
are
incr
ease
d in
the
pul
se s
um
mat
ion
proc
ess
rela
tive
to w
hen
usin
g th
e si
ngle
fre
quen
cy b
and
driv
e pu
lse
(sol
id l
ine)
. W
e pa
rtic
ular
ly n
otic
e th
e in
crea
se o
n th
e fu
ndam
enta
l an
d th
ird
harm
onic
com
pone
nt,
appr
oxim
atel
y 6
and
15 d
B,
resp
ectiv
ely.
If
we
inst
ead
subt
ract
the
two
scat
tere
d bu
bble
sig
nals
fro
m t
he d
ual
freq
uenc
y ba
nd
4.3
Num
eric
al S
imu
lati
ons
75
20 ·] ~---]
0 0.
5 1
1.5
2 2.
5 3
3.5
4 4.
5 5
{MH
z]
Fig
ure
4.13
: A
bsol
ute
valu
e o
f F
ouri
er T
rans
form
of
scat
tere
d bu
bble
sig
nal.
Sol
id li
ne:
Sin
gle
freq
uenc
y ba
nd d
rive
pul
se f
rom
upp
er p
anel
in F
ig. 4
.12
used
. D
ashe
d li
ne u
pper
pa
nel:
Sum
of s
catt
ered
pul
ses
whe
n dr
iven
by
the
two
puls
es in
low
er p
anel
of F
ig. 4
.12.
D
ashe
d lin
e lo
wer
pan
el:
Sub
trac
tion
of
the
scat
tere
d pu
lses
whe
n dr
iven
by
the
two
puls
es in
low
er p
anel
of F
ig. 4
.12.
driv
e pu
lses
, w
e ob
tain
the
das
hed
line
in t
he l
ower
pan
el o
f Fi
g. 4
.13.
A
ll s
catt
ered
ha
rmon
ic c
ompo
nent
s ex
cept
fro
m t
he f
unda
men
tal,
are
now
inc
reas
ed r
elat
ive
to w
hen
usin
g th
e si
ngle
fre
quen
cy b
and
driv
e pu
lse
(sol
id li
ne).
Her
e, w
e pa
rtic
ular
ly n
otic
e th
e m
ajor
incr
ease
on
the
scat
tere
d fo
urth
har
mon
ic c
ompo
nent
of
abou
t 30
dB
.
In F
ig.
4.14
, th
e da
shed
lin
es d
ispl
ay h
arm
onic
com
pone
nts
scat
tere
d fr
om a
con
tras
t bu
bble
whe
n dr
iven
by
the
calc
ulat
ed tr
ansm
it fi
eld
from
Sec
. 4.3
.1 o
btai
ned
wit
h a
sing
le
freq
uenc
y ba
nd s
ourc
e pu
lse
desc
ribe
d by
Eq.
4.4
whe
re a
2(t)
=
0.
As
prev
ious
ly,
the
scat
tere
d fu
ndam
enta
l co
mpo
nent
in t
he l
eft
pane
l is
nor
mal
ized
to
0 dB
at
8 em
. B
oth
the
dyna
mic
ran
ge a
nd s
cale
var
ies
betw
een
the
pane
ls d
ue t
o th
e fa
ct t
hat
the
leve
l o
f sc
atte
red
harm
onic
bub
ble
sign
al d
epen
ds s
tron
gly
on t
he a
mpl
itud
e o
f the
inc
iden
t dri
ve
puls
e.
Sim
ilar
to F
ig.
4.7,
the
sol
id l
ines
in
Fig.
4.1
4 sh
ow t
he s
catt
ered
har
mon
ic c
ompo
nent
s ob
tain
ed u
sing
the
tra
nsm
it f
ield
s fr
om t
he d
ual
freq
uenc
y ba
nd s
ourc
e pu
lses
in
Eqs
. 4.
4 an
d 4.
5 to
dri
ve t
he b
ubbl
e, a
nd t
hen
sum
min
g th
e tw
o sc
atte
red
sign
als
in a
pul
se s
um
mat
ion
proc
ess.
The
sm
all t
ime
shif
ts in
trod
uced
bet
wee
n th
e tw
o sc
atte
red
sign
als
befo
re
the
puls
e su
mm
atio
n pr
oces
s in
Fig
. 4.7
hav
e al
so b
een
appl
ied
in F
ig. 4
.14
and
it is
ther
efo
re o
f spe
cial
inte
rest
to c
ompa
re th
ese
two
figu
res.
Int
rodu
cing
the
indi
cate
d ti
me
shif
ts
to t
he s
catt
ered
bub
ble
sign
als
befo
re s
umm
atio
n do
, ho
wev
er, o
nly
have
mar
gina
l eff
ects
on
the
obt
aine
d pu
lse
sum
mat
ion
sign
al.
Sta
rtin
g w
ith
the
scat
tere
d fu
ndam
enta
l co
mpo
nent
s in
the
lef
t pa
nel
we
see
that
the
le
vel
is i
ncre
ased
by
arou
nd 6
dB
in
the
puls
e su
mm
atio
n pr
oces
s re
lati
ve t
o th
e le
vel
obta
ined
usi
ng a
sin
gle
freq
uenc
y ba
nd s
ourc
e fo
r th
e co
ntra
st b
ubbl
e (F
ig.
4.14
) as
wel
l
76
Pap
erC
as f
or t
he s
oft t
issu
e (F
ig.
4.7)
. T
he s
econ
d ha
rmon
ic b
ubbl
e si
gnal
obt
aine
d af
ter p
ulse
sum
mat
ion
is s
een
to li
e fr
om
arou
nd 1
0 to
2 d
B a
bove
the
bubb
le s
igna
l ach
ieve
d us
ing
a si
ngle
fre
quen
cy b
and
sour
ce
puls
e w
here
as f
or s
oft t
issu
e, t
here
is a
n al
mos
t con
stan
t 3 d
B i
ncre
ase
in s
catt
ered
sig
nal
leve
l app
lyin
g th
e pu
lse
sum
mat
ion
proc
ess.
T
he t
hird
har
mon
ic p
ulse
sum
mat
ion
bubb
le s
igna
l is
inc
reas
ed s
igni
fica
ntly
by
15
to 2
0 dB
(at
dep
ths
from
4 t
o 12
em
) re
lati
ve t
o w
hen
appl
ying
a c
onve
ntio
nal
sing
le
freq
uenc
y ba
nd s
ourc
e pu
lse.
The
thi
rd h
arm
onic
pul
se s
umm
atio
n ti
ssue
sig
nal,
on t
he
othe
r ha
nd,
is a
roun
d th
e sa
me
leve
l as
for
the
sing
le fr
eque
ncy
band
sig
nal a
s pr
evio
usly
se
en.
Thi
s im
plic
ates
that
the
new
met
hod,
con
sist
ing
of
tran
smit
ting
two
dual
fre
quen
cy
band
sou
rce
puls
es a
nd p
erfo
rmin
g a
puls
e su
mm
atio
n pr
oces
s, i
s ca
pabl
e o
f sig
nifi
cant
ly
incr
easi
ng b
oth
the
CN
R a
nd C
TR
at
the
thir
d ha
rmon
ic c
ompo
nent
rel
ativ
e to
whe
n ap
pl
ying
the
conv
enti
onal
sin
gle
freq
uenc
y ba
nd s
ourc
e pu
lse.
L
ooki
ng a
t the
fou
rth
harm
onic
com
pone
nts
in th
e ri
ght p
anel
, we
noti
ce th
at th
ere
is a
m
ajor
incr
ease
in s
igna
l lev
el f
or b
oth
the
cont
rast
bub
ble
and
the
soft
tiss
ue a
pply
ing
the
dual
fre
quen
cy s
ourc
e pu
lses
wit
h th
e pu
lse
sum
mat
ion
tech
niqu
e. U
sing
thi
s ne
w p
ulse
su
mm
atio
n m
etho
d on
the
fou
rth
harm
onic
com
pone
nt w
ould
the
refo
re r
esul
t in
a m
ajor
in
crea
se i
n C
NR
but
als
o a
stro
ng r
educ
tion
in
CT
R r
elat
ive
to u
sing
a s
ingl
e fr
eque
ncy
band
sou
rce
puls
e.
It is
als
o in
tere
stin
g to
com
pare
the
thi
rd h
arm
onic
CN
R o
btai
ned
wit
h th
e ne
w p
ulse
su
mm
atio
n te
chni
que
and
the
seco
nd h
arm
onic
CN
R a
chie
ved
usin
g co
nven
tion
al s
ec
ond
harm
onic
im
agin
g w
ith
a si
ngle
fre
quen
cy b
and
sour
ce p
ulse
. T
he s
olid
line
in
the
thir
d pa
nel
and
the
dash
ed l
ine
in t
he s
econ
d pa
nel o
f Fig
. 4.1
4 ar
e th
en t
o be
com
pare
d.
Whe
n co
mpa
ring
the
CN
R a
t tw
o di
ffer
ent h
arm
onic
com
pone
nts,
the
effe
ct o
f abs
orpt
ion
due
to w
ave
prop
agat
ion
from
the
sca
tter
er t
o th
e tr
ansd
ucer
sho
uld
be i
nclu
ded,
and
in
the
pres
ent
com
pari
son,
an
abso
rpti
on o
f 0.
5 dB
/cm
/MH
z w
as u
sed.
A
t 4
em,
the
thir
d ha
rmon
ic C
NR
wit
h th
e ne
w m
etho
d is
aro
und
13 d
B a
bove
the
sec
ond
harm
onic
CN
R
obta
ined
wit
h a
sing
le f
requ
ency
ban
d so
urce
pul
se.
At
6 an
d 8
em,
the
incr
ease
in C
NR
is
app
roxi
mat
ely
10 a
nd 5
dB
, res
pect
ivel
y, w
hile
at
10 e
m, t
he th
ird
and
seco
nd h
arm
onic
C
NR
are
abo
ut th
e sa
me.
The
new
pul
se s
umm
atio
n te
chni
que
is s
een
to p
erfo
rm b
est
in t
he n
ear-
fiel
d an
d fo
cal
regi
on w
here
the
tran
smit
am
plit
ude
is h
ighe
r th
an i
n th
e fa
r-fi
eld.
Loo
king
at F
ig. 4
.2 th
e tr
ansm
it a
mpl
itud
e is
see
n to
fal
l ra
ther
ste
eply
in th
e fa
r-fi
eld.
The
new
pul
se s
umm
atio
n te
chni
que
wou
ld p
roba
bly
have
per
form
ed b
ette
r in
the
far-
fiel
d ap
plyi
ng a
tran
smit
fie
ld
mor
e co
llim
ated
in th
e fa
r-fi
eld.
Thi
s ca
n fo
r ex
ampl
e be
ach
ieve
d by
inc
reas
ing
the
radi
us
of t
he a
nnul
ar tr
ansd
ucer
to 1
.5 e
m a
nd th
e ge
omet
ric
focu
s to
10
em.
Als
o, a
cous
tic
abso
rpti
on i
ncre
ases
wit
h fr
eque
ncy
whe
n no
t m
easu
red
as n
umbe
r o
f w
ave
leng
ths
trav
eled
hen
ce w
orki
ng a
gain
st h
ighe
r har
mon
ic im
agin
g te
chni
ques
at l
arge
de
pths
.
The
new
met
hod,
tra
nsm
itti
ng d
ual
freq
uenc
y ba
nd s
ourc
e pu
lses
and
per
form
ing
a ge
ner
al f
orm
of
puls
e in
vers
ion,
may
als
o be
use
d ap
plyi
ng t
he s
catt
ered
fou
rth
harm
onic
4.3
Num
eric
al S
imul
atio
ns
77
Fig
ure
4.14
: S
catt
ered
har
mon
ic b
ubbl
e si
gnal
alo
ng s
ymm
etry
axi
s ba
sed
on c
alcu
late
d tr
ansm
it fi
elds
, go
ing
from
fun
dam
enta
l co
mpo
nent
in l
eft p
anel
to f
ourt
h ha
rmon
ic c
om
pone
nt i
n ri
ght
pane
l. D
ashe
d lin
es:
Sin
gle
freq
uenc
y ba
nd s
ourc
e pu
lse
used
. S
olid
lin
es:
Tw
o du
al f
requ
ency
ban
d so
urce
pul
ses
used
, ph
ase
inve
rsio
n on
sec
ond
harm
onic
co
mpo
nent
. P
ulse
sum
mat
ion
of s
catt
ered
sig
nals
wit
h sa
me
tim
e sh
ift a
s in
Fig
. 4.7
.
com
pone
nt f
or i
mag
e re
cons
truc
tion
. T
hen,
in
orde
r to
red
uce
the
resu
ltin
g fo
urth
har
m
onic
com
pone
nts
from
the
sof
t ti
ssue
as
seen
in
Sec.
4.3
.2,
the
two
scat
tere
d pu
lses
sh
ould
be
subt
ract
ed.
Fig
. 4.1
5 sh
ows
the
resu
lts
afte
r the
res
ulti
ng s
catt
ered
bub
ble
sign
als
have
bee
n su
btra
cted
in
a p
ulse
sub
trac
tion
pro
cess
. T
he tw
o sc
atte
red
bubb
le s
igna
ls h
ave
been
tim
e sh
ifte
d an
d am
plit
ude
corr
ecte
d re
lati
ve to
eac
h ot
her b
y th
e sa
me
amou
nt a
s w
as u
sed
in F
ig. 4
.9 a
nd
it is
the
refo
re o
f spe
cial
inte
rest
to c
ompa
re F
ig. 4
.9 a
nd 4
.15.
As
befo
re, t
he d
ashe
d li
nes
indi
cate
the
sca
tter
ed h
arm
onic
lev
els
obta
ined
fro
m t
he s
ingl
e fr
eque
ncy
band
sou
rce
puls
e fr
om s
oft
tissu
e an
d a
cont
rast
bub
ble,
res
pect
ivel
y.
In F
ig.
4.9,
we
see
that
all
harm
onic
com
pone
nts
from
sof
t ti
ssue
, ex
cept
the
fou
rth
harm
onic
, ar
e ve
ry s
tron
g an
d th
us o
nly
four
th h
arm
onic
imag
ing
is i
nter
esti
ng w
ith
this
pul
se s
ubtr
acti
on te
chni
que
due
to t
he r
esul
ting
lim
itat
ions
in C
TR
usi
ng t
he o
ther
har
mon
ic c
ompo
nent
s.
Inve
stig
atin
g th
e fo
urth
har
mon
ic c
ompo
nent
in th
e ri
ght p
anel
in F
ig. 4
.15,
we
noti
ce
that
the
incr
ease
in s
igna
l lev
el a
pply
ing
dual
fre
quen
cy b
and
sour
ce p
ulse
s an
d th
e pu
lse
subt
ract
ion
proc
ess
is s
igni
fica
nt.
We
thus
con
clud
e th
at u
sing
tw
o du
al f
requ
ency
ban
d so
urce
pul
ses
in c
ombi
nati
on w
ith
the
puls
e su
btra
ctio
n pr
oces
s si
gnif
ican
tly
incr
ease
s th
e C
NR
at
the
four
th h
arm
onic
com
pone
nt r
elat
ive
to a
pply
ing
a si
ngle
fre
quen
cy b
and
sour
ce p
ulse
. A
lso,
fro
m F
ig.
4.9
we
see
that
the
fou
rth
harm
onic
tis
sue
sign
al r
esul
ting
fr
om t
he p
ulse
sub
trac
tion
pro
cess
is v
ery
low
giv
ing
a hi
gh C
TR
.
Her
e, i
t is
als
o in
tere
stin
g to
com
pare
the
fou
rth
harm
onic
CN
R o
btai
ned
wit
h th
e ne
w
puls
e su
btra
ctio
n te
chni
que
and
the
seco
nd h
arm
onic
CN
R a
chie
ved
usin
g co
nven
tion
al
seco
nd h
arm
onic
imag
ing
wit
h a
sing
le f
requ
ency
ban
d so
urce
pul
se.
The
sol
id li
ne in
the
78
Pap
er C
Fig
ure
4.15
: S
catt
ered
har
mon
ic b
ubbl
e si
gnal
alo
ng s
ymm
etry
axi
s ba
sed
on c
alcu
late
d tr
ansm
it f
ield
, go
ing
from
fun
dam
enta
l co
mpo
nent
in l
eft
pane
l to
fou
rth
harm
onic
com
po
nent
in r
ight
pan
el.
Das
hed
line
s: S
ingl
e fr
eque
ncy
band
sou
rce
puls
e us
ed.
Sol
id li
nes:
T
wo
dual
freq
uenc
y ba
nd s
ourc
e pu
lses
use
d, p
hase
inve
rsio
n on
sec
ond
harm
onic
com
po
nent
. P
ulse
sub
trac
tion
of
scat
tere
d si
gnal
s w
ith
sam
e re
lati
ve t
ime
shif
ts a
nd a
mpl
itud
es
as i
n Fi
g. 4
.9.
four
th p
anel
and
the
das
hed
line
in t
he s
econ
d pa
nel
of F
ig. 4
.15
are
then
to b
e co
mpa
red.
A
n ab
sorp
tion
equ
al t
o 0.
5 dB
/cm
/MH
z is
inc
lude
d in
the
pre
sent
com
pari
son.
At
4 em
, th
e fo
urth
har
mon
ic C
NR
is a
bout
18
dB a
bove
the
seco
nd h
arm
onic
CN
R o
btai
ned
wit
h th
e si
ngle
fre
quen
cy b
and
sour
ce.
The
inc
reas
e in
CN
R u
sing
the
new
pul
se s
ubtr
acti
on
met
hod
is r
educ
ed to
aro
und
15, 8
, an
d 3
dB a
t 6,
8, a
nd 1
0 em
, re
spec
tive
ly.
Aga
in,
as i
ndic
ated
pre
viou
sly,
the
new
pul
se s
ubtr
acti
on t
echn
ique
wou
ld p
roba
bly
have
pe
rfor
med
bet
ter
in t
he f
ar-f
ield
app
lyin
g a
tran
smit
fie
ld m
ore
coll
imat
ed i
n th
e fa
r-fi
eld.
T
he e
ffec
ts o
f the
aco
usti
c ab
sorp
tion
mec
hani
sms
wil
l be
stro
nger
for
the
four
th h
arm
onic
si
gnal
com
pone
nts
from
the
pul
se s
ubtr
acti
on m
etho
d th
an f
or t
he t
hird
har
mon
ic s
igna
l co
mpo
nent
s fr
om t
he p
ulse
sum
mat
ion
met
hod.
4.4
Con
clus
ions
The
pre
sent
pap
er h
as d
escr
ibed
and
dis
cuss
ed a
new
con
tras
t har
mon
ic i
mag
ing
met
hod
desi
gned
to
incr
ease
the
sca
tter
ed h
arm
onic
con
tras
t si
gnal
rel
ativ
e to
bot
h th
e sc
atte
red
harm
onic
tis
sue
sign
al a
nd t
he n
oise
sig
nal
alw
ays
pres
ent
in a
pul
se e
cho
imag
ing
sys
tem
. T
o ac
hiev
e th
is, t
he n
ew m
etho
d co
nsis
ts o
f tr
ansm
itti
ng d
ual f
requ
ency
ban
d pu
lses
w
here
in p
arti
cula
r, a
fun
dam
enta
l ban
d an
d its
sec
ond
harm
onic
com
pone
nt a
re t
rans
mit
te
d ov
erla
ppin
g in
tim
e. T
his
dual
fre
quen
cy b
and
tran
smit
pul
se b
oost
s up
the
sca
tter
ed
thir
d an
d fo
urth
har
mon
ic c
ontr
ast
sign
al a
nd t
issu
e si
gnal
com
pone
nts.
The
inc
reas
e in
4.5
Fu
rth
er W
ork
79
scat
tere
d th
ird
and
four
th h
arm
onic
tis
sue
sign
al is
the
n re
mov
ed o
r re
duce
d tr
ansm
itti
ng
a se
cond
dua
l fr
eque
ncy
band
pul
se a
nd p
erfo
rmin
g a
gene
ral
puls
e in
vers
ion
proc
ess.
T
rans
mit
ting
dua
l fr
eque
ncy
band
pul
ses,
one
is c
apab
le o
f con
stru
ctin
g tw
o as
ymm
etri
c tr
ansm
it p
ulse
s w
ith
resp
ect t
o po
siti
ve a
nd n
egat
ive
pres
sure
am
plit
ude
and
thus
pre
vent
in
g th
e sc
atte
red
thir
d or
four
th h
arm
onic
con
tras
t sig
nal f
rom
bei
ng s
igni
fica
ntly
red
uced
in
the
puls
e in
vers
ion
proc
ess.
4.5
Fur
ther
Wor
k
Pre
sent
ed r
esul
ts b
ased
on
num
eric
al s
imul
atio
ns a
re i
nter
esti
ng a
nd s
ugge
st t
hat e
xper
im
enta
l stu
dies
sho
uld
be c
arri
ed o
ut to
fur
ther
val
idat
e th
e ne
w c
ontr
ast h
arm
onic
imag
ing
met
hod.
The
mai
n ob
stac
le i
n co
nduc
ting
the
se e
xper
imen
ts i
s th
e de
sign
and
man
ufac
tu
re o
f a
new
mul
ti ba
nd u
ltra
soun
d tr
ansd
ucer
cap
able
of t
rans
mit
ting
a f
unda
men
tal
and
seco
nd h
arm
onic
im
agin
g pu
lse
whi
le r
ecei
ving
the
sca
tter
ed t
hird
and
pos
sibl
y fo
urth
ha
rmon
ic c
ompo
nent
s.
One
int
eres
ting
tra
nsdu
cer
desi
gn i
s an
ann
ular
tra
nsdu
cer
con
sist
ing
of s
ever
al in
depe
nden
t rin
gs.
The
n, t
he o
uter
ring
s m
ight
be
used
in t
rans
mit
mod
e w
hile
the
inn
er r
ings
, ha
ving
a d
iffe
rent
fre
quen
cy r
espo
nse,
cou
ld b
e us
ed i
n re
ceiv
e m
ode.
4.6
Ack
now
ledg
men
ts
Thi
s w
ork
was
sup
port
ed b
y th
e R
esea
rch
Cou
ncil
of N
orw
ay.
Cha
pter
5
Lin
ear
Con
tras
t A
gen
t D
etec
tion
th
roug
h L
ow F
req
uen
cy M
anip
ulat
ion
of H
igh
Fre
qu
ency
Sca
tter
ing
Pro
pert
ies
Abs
trac
t
The
pre
sent
pap
er p
ropo
ses
a ne
w m
etho
d ap
plyi
ng t
he t
otal
sca
tter
ed c
ontr
ast
sign
al
for
imag
e re
cons
truc
tion
, thu
s la
rgel
y ov
erco
min
g th
e pr
oble
ms
enco
unte
red
in h
arm
onic
im
agin
g te
chni
ques
. In
the
new
met
hod,
the
con
tras
t si
gnal
s an
d th
e ti
ssue
sig
nals
are
di
ffer
enti
ated
app
lyin
g a
sim
ple
puls
e su
btra
ctio
n te
chni
que
whi
ch c
ance
ls o
r sig
nifi
cant
ly
redu
ces
the
scat
tere
d tis
sue
sign
al,
whe
reas
the
sca
tter
ed c
ontr
ast
sign
al,
due
to a
ssis
ting
tr
ansm
itte
d lo
w f
requ
ency
pul
ses
alte
ring
the
acou
stic
sca
tter
ing
prop
erti
es o
f the
con
tras
t ag
ent,
is p
rese
rved
in
this
pro
cess
. T
he m
ain
mec
hani
sm t
hrou
gh w
hich
thi
s im
agin
g te
chni
que
sele
cts
the
cont
rast
age
nt s
igna
l is
the
line
ar r
eson
ant p
rope
rtie
s o
f the
con
tras
t bu
bble
and
the
new
met
hod
is t
hus
mai
nly
a li
near
con
tras
t age
nt d
etec
tion
tech
niqu
e.
5.1
Intr
oduc
tion
Med
ical
ult
raso
und
wav
e pr
opag
atio
n is
typ
ical
ly a
wea
k no
nlin
ear
proc
ess
[3,
Cha
pter
12
]. U
ltra
soun
d sc
atte
ring
fro
m s
oft
tissu
e is
ass
umed
to
be a
lin
ear
proc
ess
and
ener
gy
rece
ived
fro
m s
oft
tiss
ue o
utsi
de t
he f
unda
men
tal
tran
smit
ted
freq
uenc
y ba
nd i
s he
nce
gene
rate
d in
the
forw
ard
prop
agat
ion
of th
e ul
tras
ound
wav
e un
less
a s
prea
ding
of c
ontr
ast
agen
t si
gnal
bey
ond
the
cont
rast
-fil
led
regi
on,
as d
iscu
ssed
bel
ow,
occu
rs.
Ult
raso
und
cont
rast
age
nts
are
intr
oduc
ed to
inc
reas
e th
e ul
tras
ound
sca
tter
ing
from
blo
od w
hich
is
82
Pap
erD
wea
k co
mpa
red
to t
he s
catt
ered
tis
sue
sign
al [
13,
Tab
le 4
.21-
4.22
]. T
he c
ontr
ast
agen
ts
are
usua
lly
mad
e as
sol
utio
ns o
f ga
s bu
bble
s in
a f
luid
whi
ch i
s in
ject
ed i
nto
the
bloo
d flo
w.
Ult
raso
und
scat
teri
ng f
rom
the
add
ed g
as b
ubbl
es in
the
blo
od is
usu
ally
str
ong
and
high
ly n
onlin
ear,
hen
ce in
crea
sing
the
scat
tere
d bl
ood
sign
al.
The
str
ong
nonl
inea
r re
spon
se f
rom
the
con
tras
t bu
bble
s w
hen
driv
en b
y an
ult
raso
und
puls
e ha
s gi
ven
rise
to
the
cont
rast
har
mon
ic i
mag
ing
tech
niqu
es [
35]
[11]
[12
], w
here
on
ly a
har
mon
ic c
ompo
nent
of
the
tota
l sc
atte
red
bubb
le s
igna
l is
use
d fo
r im
age
reco
nst
ruct
ion.
As
high
er s
catt
ered
har
mon
ic c
ompo
nent
s ar
e co
nsid
ered
, one
is a
ble
to o
btai
n a
bett
er d
iffe
rent
iati
on o
f th
e re
ceiv
ed c
ontr
ast
sign
al a
nd th
e re
ceiv
ed t
issu
e si
gnal
due
to
the
wea
ker
nonl
inea
r di
stor
tion
of
the
rece
ived
tis
sue
sign
al.
In a
ddit
ion
to t
he f
act
that
onl
y a
frac
tion
(a
harm
onic
com
pone
nt)
of
the
tota
l sc
atte
red
cont
rast
sig
nal i
s us
ed
for
imag
e re
cons
truc
tion
, th
e co
ntra
st h
arm
onic
im
agin
g te
chni
ques
do
have
im
port
ant
unw
ante
d as
pect
s, s
ome
of w
hich
we
are
brie
fly
goin
g to
dis
cuss
her
e.
Firs
t, as
hig
her
harm
onic
sig
nal
com
pone
nts
are
cons
ider
ed,
the
rece
ived
con
tras
t ha
rm
onic
sig
nal
is,
alth
ough
inc
reas
ed r
elat
ive
to t
he h
arm
onic
tis
sue
sign
al,
typi
cally
sig
ni
fica
ntly
red
uced
in a
mpl
itude
. T
he r
educ
tion
in
ampl
itud
e of
the
rece
ived
con
tras
t har
m
onic
com
pone
nts
has
impo
rtan
t co
nseq
uenc
es i
n a
puls
e ec
ho i
mag
ing
syst
em w
here
el
ectr
onic
and
the
rmal
noi
se a
lway
s is
pre
sent
, an
d as
hig
her
cont
rast
har
mon
ic c
ompo
ne
nts
are
cons
ider
ed,
the
ampl
itud
e o
f the
se w
ill b
e o
f th
e sa
me
orde
r as
the
noi
se i
n th
e im
agin
g sy
stem
. Se
cond
, fo
r th
e co
ntra
st b
ubbl
es t
o re
spon
d w
ith
stro
ng s
catt
ered
har
mon
ic c
ompo
ne
nts,
the
fre
quen
cy o
f th
e tr
ansm
itte
d dr
ive
puls
es s
houl
d be
wel
l be
low
the
res
onan
ce
freq
uenc
y o
f th
e co
ntra
st a
gent
. A
lso,
for
the
bub
ble
to r
espo
nd w
ith
dist
inct
har
mon
ic
com
pone
nts,
the
tra
nsm
it p
ulse
has
to
be r
elat
ivel
y na
rrow
band
ed.
The
se c
onst
rain
ts o
n tr
ansm
it f
requ
ency
and
pul
se l
engt
h im
pose
res
tric
tion
s in
the
rang
e re
solu
tion
of t
he u
ltr
asou
nd i
mag
e w
hich
is i
nver
sely
pro
port
iona
l to
the
leng
th o
f the
tra
nsm
it p
ulse
. T
hird
, th
e pa
rt o
f th
e co
ntra
st s
igna
l sc
atte
red
in t
he f
orw
ard
prop
agat
ion
dire
ctio
n ad
ds i
n ph
ase
wit
h th
e tr
ansm
it p
ulse
and
int
rodu
ces
an e
xtra
dis
tort
ion
of
the
tran
smit
fi
eld
due
to t
he s
tron
g no
nlin
eari
ty o
f sc
atte
ring
fro
m t
he b
ubbl
es.
Thi
s ex
tra
dist
orti
on
of
the
tran
smit
fie
ld m
ay t
hen
be l
inea
rly
back
-sca
tter
ed f
rom
tis
sue
regi
ons
and
fals
ely
inte
rpre
ted
as c
ontr
ast s
igna
l. T
his
unw
ante
d sp
read
of c
ontr
ast a
gent
sig
nal i
s es
peci
ally
tr
oubl
esom
e fo
r tis
sue
regi
ons
whi
ch in
ran
ge d
irec
tion
lies
bey
ond
larg
e bl
ood
vess
els
or
the
larg
e bl
ood
fille
d he
art c
aviti
es.
And
fin
ally
, th
e de
sign
and
man
ufac
ture
of b
road
band
ult
raso
und
tran
sduc
ers
capa
ble
of
effi
cien
tly r
ecei
ving
hig
her
than
the
sca
tter
ed s
econ
d ha
rmon
ic c
ompo
nent
, is
tod
ay
very
cha
llen
ging
.
The
pre
sent
pap
er d
eals
wit
h a
new
con
tras
t ag
ent
dete
ctio
n te
chni
que,
usi
ng t
he t
otal
sc
atte
red
cont
rast
sig
nal,
and
in p
arti
cula
r th
e li
near
com
pone
nt o
f th
is,
for
imag
e re
co
nstr
ucti
on.
By
not
bein
g de
pend
ent
on t
he s
catt
ered
con
tras
t ha
rmon
ic c
ompo
nent
s,
the
new
det
ecti
on te
chni
que
larg
ely
over
com
es th
e m
enti
oned
unw
ante
d as
pect
s re
sulti
ng
from
the
con
tras
t ha
rmon
ic i
mag
ing
tech
niqu
es.
The
new
met
hod
acco
mpl
ishe
s th
is b
y
5.2T
heo
ry
83
mak
ing
use
of t
rans
mit
ted
assi
stin
g lo
w f
requ
ency
pul
ses
whi
ch p
urpo
se a
re to
man
ipul
ate
the
acou
stic
sca
tter
ing
prop
erti
es o
f th
e re
sona
nt c
ontr
ast
bubb
les.
In
addi
tion
, a
sim
ple
puls
e su
btra
ctio
n m
etho
d is
app
lied
to
canc
el o
r si
gnif
ican
tly
redu
ce t
he s
catt
ered
tis
sue
sign
al h
ence
mak
ing
it p
ossi
ble
to e
ffic
ient
ly d
iffe
rent
iate
the
scat
tere
d co
ntra
st s
igna
l and
th
e sc
atte
red
tiss
ue s
igna
l.
5.2
The
ory
5.2.
1 W
ave
Pro
paga
tion
and
Sca
tter
ing
from
Sof
t T
issu
e
Wav
e pr
opag
atio
n in
sof
t tis
sue
can
be c
onsi
dere
d a
wea
k no
nlin
ear p
roce
ss f
or a
mpl
itud
es
and
freq
uenc
ies
com
mon
in
med
ical
ult
raso
und
imag
ing.
The
tis
sue
elas
tici
ty r
espo
nds
slig
htly
non
line
arly
giv
ing
rise
to
the
nonl
inea
rity
of
ultr
asou
nd w
ave
prop
agat
ion.
The
lo
cal n
onli
near
ity
is lo
w b
ut th
e di
stor
tion
acc
umul
ates
gra
dual
ly in
the
forw
ard
prop
agat
in
g w
ave,
man
ifes
ting
its
elf m
ainl
y as
a s
econ
d ha
rmon
ic c
ompo
nent
. B
y do
ing
a T
aylo
r se
ries
exp
ansi
on o
f th
e eq
uati
on o
f st
ate
P =
P(p
, s)
alon
g an
ise
ntro
pe s
= s
0, w
here
p
is t
he d
ensi
ty o
f th
e m
ediu
m,
we
can
acco
unt
for
this
non
line
arit
y. B
y d
isca
rdin
g te
rms
of
orde
r hi
gher
than
the
sec
ond
in t
he T
aylo
r se
ries
exp
ansi
on,
we
obta
in [
16,
Cha
pter
2]
p =
AP
1 +
B P~
.
Po
2 Po
(5
.1)
Her
e, p
0 is
the
den
sity
in t
he u
nstr
aine
d m
ater
ial w
hile
p1
= p
-Po
· The
aco
usti
c pr
essu
re
is d
efin
ed a
s p
= P
-P
0 w
here
Po
is t
he a
mbi
ent p
ress
ure,
and
the
par
amet
ers
A a
nd B
ar
e de
fine
d as
aPI
A=
po
-0
, p
O,s
azpl
B =
P6 8
p2
o,s
whe
re t
he s
ubsc
ript
s in
the
par
tial
der
ivat
ives
ind
icat
e th
at t
hey
are
eval
uate
d at
the
un
stra
ined
sta
te f
or a
n is
entr
opic
pro
cess
. W
e m
ay th
en d
eriv
e th
e fo
llow
ing
tiss
ue e
last
icit
y eq
uati
on [
3, C
hapt
er 1
2.3]
p(f
, t)
= -
A\1
· ~(r
, t) +
A{3
( \1
· ~(f,
t) r
(5.2
)
whe
re ~is
par
ticl
e di
spla
cem
ent a
nd {3
is
a no
nlin
eari
ty p
aram
eter
def
ined
as
B
{3 =
1 +
2A
.
Eq.
5.2
is a
non
line
ar m
ater
ial e
last
icit
y eq
uati
on w
here
the
nonl
inea
rity
par
amet
er c
an b
e fo
und
expe
rim
enta
lly
for
vari
ous
mat
eria
ls.
In th
e on
e-di
men
sion
al c
ase,
the
equ
atio
n fo
r co
nser
vati
on o
f m
omen
tum
giv
es
(5.3
)
84
Pap
er D
Fro
m E
q. 5
.2,
we
can
for
the
one-
dim
ensi
onal
cas
e, o
mit
ting
the
coor
dina
te d
epen
denc
y,
wri
te th
e m
ater
ial
equa
tion
as
87/;
(87/
J) 2
p=
-A
-+
A/3
-8z
8z
(5
.4)
or a
lter
nati
vely
(5.5
)
whe
re t
he c
ompr
essi
bili
ty "
' is
defi
ned
as t
he i
nver
se o
f th
e pa
ram
eter
A.
We
may
now
de
rive
the
one
-dim
ensi
onal
non
line
ar w
ave
equa
tion
as
(5.6
)
whe
re t
he li
near
pro
paga
tion
vel
ocit
y is
def
ined
as
Co=~
=~
We
noti
ce t
hat t
he n
onli
near
pro
paga
tion
vel
ocit
y va
ries
wit
h th
e vo
lum
e co
mpr
essi
on o
f th
e m
ater
ial
acco
rdin
g to
c =
co /1
-2(
38 7/J v
8z
=
cov
l +
2(3K
-p-
2(32
(K-p
)2 (5
.7)
::::;, c
ov
l +
2(3K
-p::::
;, c 0
(1 +
(3K-p
)
whe
re t
he l
ast a
ppro
xim
atio
ns a
re v
alid
for
the
cas
e w
hen
"-P <
< 1
. E
q. 5
.7 i
s la
ter
used
in
Sec
. 5.
5 to
ana
lyze
the
pro
paga
tion
of t
wo
tran
smit
ted
dual
fre
quen
cy b
and
puls
es.
The
sig
nal
scat
tere
d fr
om s
oft
tiss
ue is
a r
esul
t o
f th
e in
here
nt i
nhom
ogen
eous
nat
ure
of
the
med
ium
. T
here
is a
spa
tial
var
iati
on o
f the
mas
s de
nsit
y an
d co
mpr
essi
bili
ty o
n va
ri
ous
scal
es r
esul
ting
in s
catt
erin
g o
f the
tran
smit
ted
acou
stic
wav
e. T
he a
cous
tic
scat
teri
ng
proc
ess
from
sof
t tis
sue
is a
ssum
ed to
be
line
ar a
nd th
e pr
esen
ce o
f har
mon
ic c
ompo
nent
s in
the
sca
tter
ed t
issu
e si
gnal
is,
if
spre
adin
g o
f co
ntra
st a
gent
sig
nal
beyo
nd t
he a
ctua
l co
ntra
st-f
ille
d re
gion
s do
es n
ot o
ccur
, th
eref
ore
a re
sult
of t
he n
onli
near
ity
in th
e fo
rwar
d pr
opag
atin
g w
ave
[3,
Cha
pter
7].
In m
edic
al u
ltra
soun
d im
agin
g, s
oft
tiss
ue is
a n
on-r
eson
ant
med
ium
, th
e w
ave
prop
aga
tion
is a
wea
k no
nlin
ear
proc
ess
whi
le t
he r
esul
ting
sca
tter
ing
is a
line
ar p
roce
ss.
5.2.
2 C
ontr
ast A
gent
Sca
tter
ing
The
con
tras
t ag
ent
is a
ssum
ed t
o be
gas
bub
bles
enc
apsu
late
d in
a t
hin
shel
l. T
he d
iam
et
er o
f th
e bu
bble
is
muc
h le
ss t
han
the
wav
elen
gth
of
the
inco
min
g w
ave
fiel
d an
d th
e
5.2T
heo
ry
85
bubb
le th
us e
xper
ienc
es a
n ap
prox
imat
ely
unif
orm
spa
tial
fie
ld a
nd th
e bu
bble
osc
illa
tion
is
ass
umed
to b
e pu
rely
sph
eric
al.
Sim
ulat
ions
for
bub
ble
radi
us o
scil
lati
ons
and
acou
stic
sca
tter
ing
are
done
usi
ng t
he n
um
eric
al m
odel
dev
elop
ed b
y A
ngel
sen
et a
l [ 4
]. T
his
mod
el i
nclu
des
an e
quat
ion
for
the
rela
tion
bet
wee
n pr
essu
re a
nd r
adia
l st
rain
in a
thin
she
ll e
ncap
sula
ting
a g
as b
ubbl
e. T
he
mod
el a
llow
s fo
r a
fini
te s
peed
of s
ound
in th
e m
ediu
m s
urro
undi
ng th
e bu
bble
, th
us t
ak
ing
radi
atio
n lo
sses
fro
m t
he b
ubbl
e in
to a
ccou
nt.
Oth
erw
ise
it is
com
para
ble
to t
he w
ell
know
n R
ayle
igh-
Ple
sset
equ
atio
n [3
3] [
31]
and
the
two
mod
els
give
sim
ilar
res
ults
for
in
cide
nt p
ress
ure
puls
es a
nd b
ubbl
e pa
ram
eter
s st
udie
d in
this
pap
er.
The
Ray
leig
h-P
less
et e
quat
ion
is a
sec
ond
orde
r no
nlin
ear
diff
eren
tial
equ
atio
n fo
r th
e fl
uid
surr
ound
ing
a sp
heri
cal
puls
atin
g ga
s bu
bble
. T
his
equa
tion
des
crib
es t
he r
adiu
s os
cill
atio
n o
f the
bub
ble
as [
33]
[31]
3 Po
(aa
+ 2a?
) =
p(a
, t)-
Po
-P
i(t)
(5
.8)
whe
re a
is
the
radi
us, a
and
a is
the
vel
ocit
y an
d ac
cele
rati
on o
f th
e bu
bble
rad
ius,
p0
is t
he d
ensi
ty o
f th
e su
rrou
ndin
g fl
uid,
P0
is t
he a
mbi
ent
pres
sure
, P
i(t)
is
the
inci
dent
dr
ivin
g pr
essu
re,
and
p( a,
t) i
s th
e pr
essu
re a
t th
e bu
bble
sur
face
. N
onli
near
ela
stic
ity
of
the
gas
and
enca
psul
atin
g th
in s
hell
, su
rfac
e te
nsio
n, a
nd v
isco
sity
may
be
inco
rpor
ated
in
the
fir
st t
erm
on
the
righ
t-ha
nd s
ide
in E
q. 5
.8.
For
sm
all a
mpl
itud
es o
f the
inci
dent
dri
ve p
ress
ure,
the
bub
ble
osci
llat
ion
can
be
assu
med
to
be
appr
oxim
atel
y li
near
and
we
have
the
fol
low
ing
seco
nd o
rder
lin
ear
diff
eren
tial
eq
uati
on f
or t
he r
adia
l di
spla
cem
ent,
1/J, a
roun
d an
equ
ilib
rium
rad
ius
a
(5.9
)
Her
e, m
is
the
iner
tia
of
the
surr
ound
ing
flui
d,
b is
a d
ampi
ng c
onst
ant,
and
s is
the
st
iffn
ess
of
the
gas
and
enca
psul
atin
g bu
bble
she
ll. E
q. 5
.9 t
ypic
ally
des
crib
es t
he f
orce
d li
near
osc
illa
tion
of
a sy
stem
con
sist
ing
of a
mas
s m
att
ache
d on
a s
prin
g w
ith
stif
fnes
s s
whe
reas
b a
ccou
nts
for
the
dam
ping
in
the
syst
em.
By
taki
ng t
he F
ouri
er T
rans
form
of
Eq.
5.9
we
obta
in
(5.1
0)
whe
re w
e ha
ve d
efin
ed t
he t
rans
fer
func
tion
fro
m d
rive
pre
ssur
e to
rad
ial d
ispl
acem
ent
as
and
whe
re
1 H
l(D.)
=
D,2
-1
-iD
.d
(5.1
1)
d =
_b
_
2 s
n =
_~!___
• (5
.12)
w
om '
W
o =
m
'
Wo
Her
e, w
is t
he a
ngul
ar f
requ
ency
of
the
Fou
rier
Tra
nsfo
rm,
and
w0
is t
he r
eson
ance
fre
qu
ency
. T
he a
bsol
ute
valu
e an
d ph
ase
angl
e o
f H
1(D
.) ar
e sh
own
in F
ig. 5
.1.
We
see
that
86
·Jr:~
0 0.
2 0.
4 0
.6
0.8
1
1.2
1.4
1.6
1.8
2 [U
]
4 ~;~ ___ ___
j 0
0.2
0.4
0.6
0
.8
1 1.
2 1.
4 1.
6 1.
8 2
[U)
Pap
erD
Fig
ure
5.1:
T
rans
fer
func
tion
from
dri
ve p
ress
ure
to r
adia
l di
spla
cem
ent
in E
q. 5
.11.
T
he p
aram
eter
din
Eq.
5.1
2 is
set
to
0.5
and
0.1
givi
ng t
he s
olid
lin
e an
d da
shed
lin
e,
resp
ectiv
ely.
Upp
er p
anel
: A
bsol
ute
valu
e. L
ower
pan
el:
Pha
se a
ngle
.
for
driv
e fr
eque
ncie
s w
ell
belo
w r
eson
ance
the
dis
plac
emen
t is
1r
out
of
phas
e w
ith
the
driv
ing
pres
sure
. F
or f
requ
enci
es w
ell
abov
e re
sona
nce,
the
bub
ble
resp
onds
dif
fere
ntly
an
d th
e di
spla
cem
ent
and
driv
e pr
essu
re a
re n
ow i
n ph
ase.
A
roun
d re
sona
nce
the
dis
plac
emen
t is
appr
oxim
atel
y ~
out o
f pha
se w
ith
the
driv
e pr
essu
re.
The
abs
olut
e va
lue
of th
e am
plit
ude
of t
he t
rans
fer
func
tion
is s
een
in t
he u
pper
pan
el
of F
ig.
5.1.
G
oing
fro
m f
requ
enci
es b
elow
res
onan
ce t
owar
ds r
eson
ance
the
am
plit
ude
incr
ease
s gr
adua
lly c
ulm
inat
ing
wit
h a
prom
inen
t pe
ak a
roun
d re
sona
nce
for
the
situ
ati
on w
ith
low
dam
ping
(d
= 0
.1)
and
a co
nsid
erab
le s
mal
ler
peak
for
the
sit
uati
on w
ith
high
er d
ampi
ng (
d =
0.5
). I
n bo
th c
ases
, the
am
plit
ude
is s
een
to d
ecre
ases
rap
idly
abo
ve
reso
nanc
e.
The
far
-fie
ld p
ress
ure
at a
dis
tanc
e r
from
the
sou
rce,
rad
iate
d fr
om a
tim
e ha
rmon
ic
osci
llat
ing
bubb
le b
ehav
ing
as a
mon
opol
e so
urce
is a
div
ergi
ng s
pher
ical
wav
e th
at m
ay
be w
ritte
n p(
r) =
p(
a)a
e-ik
(r-a
) r
(5.1
3)
whe
re a
is
the
bubb
le r
adiu
s an
d k
is t
he w
ave
num
ber.
By
assu
min
g pu
re r
adia
l in
co
mpr
essi
ble
osci
llat
ions
and
neg
lect
ing
the
nonl
inea
r co
nvec
tive
acce
lera
tion
ter
m,
the
Nav
ier-
Stok
es E
quat
ions
red
uce
to [
42] 8u
r 8p
P
o-=
--
8t
8r
(5.1
4)
whe
re u
,. is
the
radi
al v
eloc
ity.
We
part
icul
arly
not
ice
that
the
visc
ous
term
s ha
ve v
anis
hed
for
the
situ
atio
n w
ith
pure
rad
ial
inco
mpr
essi
ble
flui
d m
otio
n.
We
may
now
wri
te t
he
pres
sure
gra
dien
t as
-8p
(r)
=
p(a)
(9: +
ika)
8r
r
r (5
.15)
5.3
Met
hod
87
At t
he b
ubbl
e su
rfac
e, U
r =
~ a
nd r
= a
, an
d w
e th
us g
et
Poa
2 p(
a,w
) =
-1
+ ik
a w
1/J(
w)
(5.1
6)
Ass
umin
g th
at t
he s
ourc
e is
muc
h le
ss t
han
the
wav
e le
ngth
whi
ch i
s ty
pica
lly
the
case
fo
r ul
tras
ound
con
tras
t bub
bles
, ka
< <
1, w
e ge
t
(5.1
7)
App
lyin
g th
e re
sult
fro
m E
q. 5
.10
we
fina
lly o
btai
n th
e fo
llow
ing
rela
tion
bet
wee
n th
e dr
ive
pres
sure
and
the
sca
tter
ed p
ress
ure
(5.1
8)
whe
re t
he t
rans
fer
func
tion
fro
m d
rive
pre
ssur
e to
sca
tter
ed p
ress
ure
is g
iven
by
(5.1
9)
The
upp
er a
nd l
ower
pan
el o
f Fi
g. 5
.2 d
ispl
ay t
he a
bsol
ute
valu
e an
d ph
ase
angl
e o
f H
2(f
l),
resp
ectiv
ely.
As
prev
ious
ly,
the
dash
ed l
ines
are
res
ults
obt
aine
d se
ttin
g th
e pa
ra
met
er d
equ
al to
0.1
whi
le th
e so
lid
line
s ar
e ob
tain
ed f
ord
equ
al to
0.5
. T
he a
mpl
itud
e o
f the
sca
tter
ed p
ress
ure,
as
seen
fro
m t
he u
pper
pan
el in
Fig
. 5.
2, s
igni
fica
ntly
incr
ease
s w
hen
goin
g fr
om d
rive
fre
quen
cies
bel
ow r
eson
ance
tow
ards
res
onan
ce.
For
dri
ve f
re
quen
cies
abo
ve r
eson
ance
, th
e sc
atte
red
ampl
itud
e ap
proa
ches
a c
onst
ant l
evel
. In
the
low
er p
anel
of
the
figu
re,
we
see
that
for
dri
ve f
requ
enci
es w
ell
belo
w r
eso
nanc
e, t
he s
catt
ered
pre
ssur
e is
in
phas
e w
ith
the
driv
ing
pres
sure
. T
his
mea
ns t
hat
the
bubb
le i
s do
min
ated
by
s, t
he s
tiffn
ess
of
the
gas
and
shel
l. F
or f
requ
enci
es w
ell
abov
e re
sona
nce,
the
bub
ble
resp
onds
dif
fere
ntly
and
is n
ow d
omin
ated
by
m,
the
iner
tia
of t
he
co-o
scil
lati
ng m
ass.
T
he s
catt
ered
pre
ssur
e an
d dr
ive
pres
sure
are
now
1r
out
of
phas
e.
Aro
und
reso
nanc
e th
e sc
atte
red
pres
sure
is
appr
oxim
atel
y %
out
of
phas
e w
ith
the
driv
e pr
essu
re.
In m
edic
al u
ltra
soun
d im
agin
g, t
he c
ontr
ast
bubb
le i
s ty
pica
lly
a re
sona
nt s
catt
erer
and
co
ntra
st a
gent
sca
tter
ing
is a
hig
hly
nonl
inea
r pr
oces
s.
5.3
Met
hod
The
new
con
tras
t age
nt d
etec
tion
met
hod
is p
erfo
rmed
by
cons
ecut
ivel
y tr
ansm
itti
ng tw
o ul
tras
ound
pul
ses,
bot
h co
ntai
ning
tw
o fr
eque
ncy
band
s w
hich
are
ove
rlap
ping
in
the
time
dom
ain.
The
se tw
o tr
ansm
itte
d pu
lses
con
tain
a lo
w f
requ
ency
ban
d an
d a
high
fre
qu
ency
ban
d, w
here
the
pur
pose
of t
he t
rans
mit
ted
low
fre
quen
cy b
and
is t
o m
anip
ulat
e,
88
·~l Z
< __
_ ;____
: I
0 0.
2 0.
4 0.
6 0
.8
I 1.
2 1.
4 1.
6 1.
8 2
[ill
~j • =s
;mnu
l 0
0 2
0.4
0.6
0.8
1 1.
2 1.
4 1.
6 1.
8 2
[il]
Pap
erD
Fig
ure
5.2:
Tra
nsfe
r fu
nctio
n fr
om d
rive
pre
ssur
e to
sca
tter
ed p
ress
ure
in E
q. 5
.19.
The
pa
ram
eter
din
Eq.
5.1
2 is
set
to 0
.5 a
nd 0
.1 g
ivin
g th
e so
lid
line
and
dash
ed li
ne, r
espe
ctiv
ely.
Upp
er p
anel
: A
bsol
ute
valu
e. L
ower
pan
el:
Pha
se a
ngle
.
by e
xpan
ding
and
com
pres
sing
the
bub
ble,
the
aco
ustic
sca
tter
ing
prop
erti
es o
f th
e co
ntr
ast
agen
t at
the
tra
nsm
itte
d hi
gh f
requ
ency
ban
d.
The
tw
o tr
ansm
itte
d pu
lses
may
be
expr
esse
d as
P1(t)
= a
1(t)
sin(
w1t
) +
a2(t
)sin
(w2t
+ (P
t) P2
(t) =
-a3
(t)s
in(w
1t) +
a4(t
)sin
(w2t
+(h
) (5
.20)
(5.2
1)
whe
re a
n a
re p
osit
ive
ampl
itude
func
tions
, Wn
is th
e an
gula
r fre
quen
cy,
and
¢n
are
sel
ecte
d ph
ase
angl
es.
In t
he s
impl
est e
mbo
dim
ent o
f th
e ne
w i
mag
ing
tech
niqu
e, a
3 =
a1,
a4
=
a2,
and
¢2 =
qyt.
Firs
t, th
e pu
lse
desc
ribe
d by
Eq.
5.2
0 is
tran
smit
ted
and
the
scat
tere
d pu
lse
is r
ecei
ved
and
stor
ed.
The
n th
e pu
lse
in E
q. 5
.21
is t
rans
mitt
ed,
whe
re t
he p
hase
of
the
low
fre
quen
cy
com
pone
nts
not
used
for
im
age
reco
nstr
ucti
on a
re i
nver
ted,
and
the
rec
eive
d sc
atte
red
sign
al i
s su
btra
cted
fro
m t
he f
irst
rec
eive
d pu
lse.
T
he n
on-r
eson
ant
soft
tis
sue
and
the
reso
nant
con
tras
t bu
bble
will
res
pond
dif
fere
ntly
on
the
two
tran
smit
ted
dual
fre
quen
cy
band
pul
ses.
We
assu
me
that
the
time
dura
tion
of t
he t
rans
mit
ted
high
fre
quen
cy p
ulse
is l
ess
than
the
ti
me
dura
tion
of a
hal
f per
iod
of t
he tr
ansm
itte
d lo
w fr
eque
ncy
puls
e. I
f the
hig
h fr
eque
ncy
com
pone
nts
w2
in t
he f
irst
tra
nsm
itte
d pu
lse
are
plac
ed i
n th
e ne
gativ
e ha
lf p
erio
d o
f th
e lo
w f
requ
ency
com
pone
nts
w1
, th
en i
n th
e se
cond
tra
nsm
itte
d pu
lse,
the
y w
ill
due
to t
he
phas
e in
vers
ion
be p
lace
d in
the
pos
itiv
e ha
lf p
erio
d of
the
low
fre
quen
cy c
ompo
nent
s.
The
tra
nsm
itte
d lo
w f
requ
ency
com
pone
nts
are
wel
l be
low
the
res
onan
ce f
requ
ency
of
the
cont
rast
age
nt.
In t
he f
irst
sca
tter
ed b
ubbl
e si
gnal
, th
e hi
gh f
requ
ency
com
pone
nts
wil
l hen
ce b
e sc
atte
red
from
an
expa
nded
bub
ble
rela
tive
to e
quil
ibri
um s
ize,
whe
reas
in
5.4
Bub
ble
Osc
illa
tion
s 89
the
seco
nd s
catt
ered
bub
ble
sign
al,
the
high
fre
quen
cy c
ompo
nent
s w
ill b
e sc
atte
red
from
a
com
pres
sed
bubb
le (
see
low
er p
anel
of F
ig.
5.1)
.
We
saw
in
the
low
er p
anel
of
Fig
. 5.
2 th
at t
he p
hase
of
the
tran
sfer
fun
ctio
n fr
om d
rive
pr
essu
re to
sca
tter
ed p
ress
ure
chan
ges
rapi
dly
for
driv
e fr
eque
ncie
s ar
ound
the
res
onan
ce
freq
uenc
y o
f th
e bu
bble
. If
the
tran
smit
ted
high
fre
quen
cy c
ompo
nent
s th
en a
re p
lace
d ar
ound
the
equi
libr
ium
reso
nanc
e fr
eque
ncy
of th
e co
ntra
st b
ubbl
e, t
he r
esul
ting
two
scat
te
red
high
fre
quen
cy p
ulse
s fr
om t
he c
ontr
ast b
ubbl
e w
ill b
e so
mew
hat t
ime
shif
ted
rela
ti
ve to
eac
h ot
her.
Als
o, f
rom
the
uppe
r pan
el o
f Fig
. 5.2
the
ampl
itud
e o
f the
two
scat
tere
d bu
bble
sig
nals
wil
l be
som
ewha
t dif
fere
nt.
The
two
rece
ived
sig
nals
fro
m s
oft t
issu
e w
ill
not h
ave
this
rel
ativ
e ti
me
shif
t and
am
plit
ude
diff
eren
ce a
nd t
he s
oft t
issu
e si
gnal
is t
hus
canc
eled
in
the
puls
e su
btra
ctio
n pr
oces
s o
f th
e tw
o re
ceiv
ed s
igna
ls w
hile
the
rec
eive
d co
ntra
st a
gent
sig
nal i
s pr
eser
ved.
5.4
Bub
ble
Osc
illat
ions
Sim
ulat
ions
of
bubb
le r
adiu
s os
cill
atio
ns a
nd a
cous
tic
scat
teri
ng a
re d
one
usin
g th
e nu
m
eric
al m
odel
dev
elop
ed b
y A
ngel
sen
et a
l [4
]. R
esul
ts f
rom
this
mod
el a
re,
as i
ndic
ated
, si
mil
ar to
res
ults
obt
aine
d us
ing
the
wel
l-kn
own
Ray
leig
h-P
less
et e
quat
ion
[33]
[31
] fo
r pr
essu
re a
mpl
itud
es a
nd b
ubbl
e pa
ram
eter
s us
ed i
n th
e pr
esen
t pa
per.
S
imul
atio
ns a
re
done
usi
ng a
sin
gle
bubb
le w
ith
acou
stic
par
amet
ers
sim
ilar
to
the
cont
rast
age
nt S
on
azoi
d. T
he e
quil
ibri
um r
eson
ance
fre
quen
cy o
f the
con
tras
t bub
ble
is a
roun
d 5
MH
z.
Fig
. 5.
3 sh
ows
an e
xam
ple
of
two
tran
smit
pul
ses
desc
ribe
d by
Eq.
5.2
0 an
d 5.
21 w
here
a
low
fre
quen
cy 0
.5 M
Hz
and
a hi
gh f
requ
ency
5 M
Hz
puls
e ar
e tr
ansm
itte
d si
mul
ta
neou
sly.
T
he h
igh
freq
uenc
y pu
lse
in t
he u
pper
pan
el o
f th
e fi
gure
app
ears
dur
ing
the
nega
tive
hal
f per
iod
of
the
low
fre
quen
cy p
ulse
. In
the
low
er p
anel
, th
e ph
ase
of th
e lo
w
freq
uenc
y pu
lse
is i
nver
ted
rela
tive
to t
he u
pper
pan
el,
whe
reas
the
hig
h fr
eque
ncy
puls
e is
unc
hang
ed,
and
the
high
fre
quen
cy p
ulse
now
app
ears
dur
ing
the
posi
tive
hal
f pe
riod
o
f the
low
fre
quen
cy p
ulse
. T
hese
two
tran
smit
pul
ses
are
now
use
d to
dri
ve t
he c
ontr
ast
bubb
le w
hich
equ
ilib
rium
res
onan
ce f
requ
ency
is
arou
nd 5
MH
z.
In F
ig.
5.4
the
bubb
le r
adiu
s re
spon
ses
from
the
tw
o tr
ansm
itte
d dr
ive
puls
es a
re s
how
n in
the
upp
er a
nd l
ower
pan
el,
resp
ectiv
ely.
The
low
fre
quen
cy d
rive
pul
se is
muc
h be
low
th
e eq
uili
briu
m r
eson
ance
fre
quen
cy o
f th
e bu
bble
and
the
rad
ius
resp
onse
is
thus
1r
out
of p
hase
wit
h th
e lo
w f
requ
ency
dri
ve p
ulse
as
expe
cted
fro
m t
he l
ower
pan
el o
f Fig
. 5 .
1.
In t
he f
irst
tra
nsm
itte
d pu
lse,
the
hig
h fr
eque
ncy
driv
e pu
lse
occu
rs f
or a
n ex
pand
ed b
ub
ble
whi
le i
n th
e se
cond
tra
nsm
itte
d pu
lse,
the
hig
h fr
eque
ncy
driv
e pu
lse
occu
rs w
hen
the
bubb
le is
com
pres
sed.
Rel
ativ
e to
equ
ilib
rium
con
diti
ons,
the
res
onan
ce f
requ
ency
is
som
ewha
t re
duce
d an
d in
crea
sed
for
the
expa
nded
and
com
pres
sed
bubb
le,
resp
ectiv
ely.
W
e no
tice
that
the
radi
us o
scil
lati
ons
due
to t
he h
igh
freq
uenc
y pu
lse
are
larg
er in
mag
ni
tude
for
the
less
stif
f, e
xpan
ded
bubb
le th
an f
or t
he s
tiffe
r, c
ompr
esse
d bu
bble
.
90
05
LL .. /
'\ ... /\
. J
U
~,[~~LL~Jl
-O.So~~~-~2~~, -~.c
----
':-,
---'
.c--
---'
:-7
---',
---'-
. ---
-:'1
0
I!" I
Pap
erD
Fig
ure
5.3:
Dua
l fre
quen
cy b
and
driv
e pu
lses
. U
pper
pan
el:
Hig
h fr
eque
ncy
com
pone
nts
occu
r du
ring
neg
ativ
e ha
lf p
erio
d o
f lo
w f
requ
ency
com
pone
nts.
L
ower
pan
el:
Hig
h fr
eque
ncy
com
pone
nts
occu
r du
ring
pos
itiv
e ha
lf p
erio
d o
f low
fre
quen
cy c
ompo
nent
s.
·:l·:·
···~··
···:1
1.20
1
2 3
4 5
6 7
8 9
10
I!" I
2 ~::~··················
···.·.···.············
···.····.·······•.····
···· .. ·····
· 1.
4 ..
..
:·
···
; ..
....
'
....
..
. .
..
. .
.:..
...
....
..
;
. .
. .
. .
Llo'--~I--.,2'--~, --
.'---
-''--
.L, -
-.'
----':
-7
-~,'--~9 _
_JIO
I!" I
Fig
ure
5.4:
B
ubbl
e ra
dius
osc
illa
tion
s du
e to
dri
ve p
ulse
s in
upp
er a
nd l
ower
pan
el o
f Fi
g. 5
.3, r
espe
ctiv
ely.
5.4
Bu
bb
le O
scil
lati
ons ~r~ :t~: J
0 5
10
I!" I
~l••••t••·: !••
+••••
l•••i•
•·•l···
······
· 0
3 4
5 6
7 lO
I!
" I
91
Fig
ure
5.5:
S
catt
ered
pre
ssur
e pu
lses
due
to
driv
e pu
lses
in
uppe
r an
d lo
wer
pan
el o
f Fi
g. 5
.3, r
espe
ctiv
ely.
Sim
ulat
ions
of
scat
tere
d pr
essu
re p
ulse
s du
e to
the
tw
o dr
ive
puls
es a
re d
ispl
ayed
in
Fig.
5.5
, up
per
and
low
er p
anel
, re
spec
tivel
y.
Firs
t, w
e no
tice
tha
t th
e sc
atte
red
low
fr
eque
ncy
com
pone
nts,
alm
ost
invi
sibl
e in
thi
s fi
gure
, ar
e m
uch
low
er in
am
plit
ude
than
th
e sc
atte
red
high
freq
uenc
y co
mpo
nent
s. T
his
is i
n ag
reem
ent w
ith o
ur li
near
theo
reti
cal
cons
ider
atio
ns c
ulm
inat
ing
in F
ig.
5.2,
whe
re t
he a
mpl
itud
e in
the
upp
er p
anel
was
see
n to
inc
reas
e si
gnif
ican
tly w
hen
goin
g fr
om d
rive
fre
quen
cies
bel
ow r
eson
ance
tow
ards
res
on
ance
. W
e al
so s
ee t
hat
the
scat
teri
ng o
f th
e hi
gh f
requ
ency
com
pone
nts
are
som
ewha
t la
rger
in a
mpl
itude
for
the
exp
ande
d bu
bble
tha
n fo
r th
e co
mpr
esse
d on
e. T
his
is d
ue t
o th
e la
rger
hig
h fr
eque
ncy
radi
us o
scil
lati
ons
of t
he e
xpan
ded
bubb
le s
een
in t
he p
revi
ous
figu
re.
The
tw
o sc
atte
red
pres
sure
pul
ses
are
depi
cted
tog
ethe
r in
the
upp
er p
anel
of
Fig.
5.6
. H
ere,
th
e so
lid
line
is
the
puls
e sc
atte
red
from
the
exp
ande
d bu
bble
, up
per
pane
l of
Fi
g. 5
.5,
whi
le t
he d
ashe
d lin
e is
the
sca
tter
ed p
ulse
fro
m t
he c
ompr
esse
d bu
bble
, lo
wer
pa
nel
of
Fig.
5.5
, an
d w
e cl
earl
y se
e th
e di
ffer
ence
in
ampl
itud
e in
dica
ted.
Als
o im
por
tant
, we
see
that
eve
n if
the
two
tran
smit
ted
high
freq
uenc
y pu
lses
occ
ur a
t the
exa
ct s
ame
rela
tive
time,
the
tw
o sc
atte
red
high
fre
quen
cy p
ulse
s ar
e ti
me
shif
ted
rela
tive
to
each
ot
her.
Thi
s ef
fect
is d
ue t
o th
e ra
pid
chan
ge o
f the
pha
se a
roun
d re
sona
nce
of
the
tran
sfe
r fu
nctio
ns f
or r
adia
l di
spla
cem
ent
and
scat
tere
d pr
essu
re s
een
in t
he l
ower
pan
els
of
Fig.
5.1
and
Fig
. 5.
2.
The
low
er p
anel
of F
ig. 5
.6 d
ispl
ays
the
resu
lt o
btai
ned
whe
n su
btra
ctin
g th
e tw
o sc
at
tere
d hi
gh f
requ
ency
pul
ses
in t
he u
pper
pan
el.
Due
to t
he r
elat
ive
tim
e sh
ift b
etw
een
the
two
scat
tere
d pu
lses
, th
e si
gnal
obt
aine
d af
ter
subt
ract
ion
is n
ot c
ance
led
or s
igni
fica
ntly
re
duce
d.
The
sam
e bu
bble
is
now
dri
ven
by t
he h
igh
freq
uenc
y co
mpo
nent
s on
ly i
n Fi
g. 5
.3 a
nd
the
resu
ltin
g ra
dius
osc
illa
tion
and
sca
tter
ed p
ress
ure
puls
e ar
e sh
own
in t
he u
pper
and
92
::r··•····•·
~······.·.··
········.···
········
.. ··········•
.··········.
·.········.·
······ .. · .. ·.·.
·. 0
-.
-···
··:·
··
..
···-
...
···:
..
..
: .
····
·:··
· ........ .
-0.2
.
:·
..
: ~-
-..
....
...
··< .. ;
.···
-~
....
....
. : -0
.4
.........
· ....... ,.
.. ..
. .
. .
. ..
....
.. --
~-..
....
.. ;
.... ·
. .
. .
. .
. .
' 4
4.1
4.2
4.
3 4
.4
4.5
4.6
4
.7
4.8
4
.9
5 I>
" I
Pap
erD
Fig
ure
5.6:
Upp
er p
anel
: S
catt
ered
pre
ssur
e pu
lses
due
to d
rive
pul
se in
upp
er (s
olid
line
) an
d lo
wer
(da
shed
lin
e) p
anel
of
Fig.
5.3
. L
ower
pan
el:
Pul
se s
ubtr
acti
on o
f th
e tw
o pu
lses
in t
he u
pper
pan
el.
low
er p
anel
of F
ig.
5. 7
, re
spec
tivel
y. T
he s
catt
ered
pre
ssur
e pu
lse
in t
he l
ower
pan
el is
in
mag
nitu
de s
een
to l
ie b
etw
een
the
two
scat
tere
d pr
essu
re p
ulse
s in
Fig
. 5.5
obt
aine
d fr
om
the
expa
nded
and
com
pres
sed
bubb
le.
The
upp
er p
anel
of
Fig
. 5.
8 sh
ows
the
abso
lute
val
ue o
f th
e F
ouri
er T
rans
form
of
the
puls
es i
n th
e up
per
pane
l in
Fig
. 5.
6. A
s pr
evio
usly
ind
icat
ed,
we
see
that
the
sca
tter
ed
high
freq
uenc
y fu
ndam
enta
l com
pone
nt is
som
ewha
t lar
ger f
or th
e ex
pand
ed b
ubbl
e (s
olid
lin
e) th
an f
or th
e co
mpr
esse
d bu
bble
(da
shed
line
). T
he n
onli
near
ity
of t
he h
igh
freq
uenc
y sc
atte
ring
pro
cess
is,
how
ever
, st
rong
er f
or t
he c
ompr
esse
d bu
bble
. A
lso,
the
har
mon
ic
com
pone
nts
scat
tere
d fr
om t
he e
xpan
ded
bubb
le a
re s
hift
ed s
ligh
tly
dow
n in
fre
quen
cy
due
to t
he l
ower
ing
of
the
reso
nanc
e fr
eque
ncy
rela
tive
to e
quil
ibri
um c
ondi
tion
s.
In t
he l
ower
pan
el o
f Fig
. 5 .
8, t
he s
olid
line
dis
play
s th
e ab
solu
te v
alue
of t
he F
ouri
er
Tra
nsfo
rm o
f the
pul
se in
the
low
er p
anel
in F
ig. 5
.6.
As
indi
cate
d, th
ere
are
only
mar
gina
l ch
ange
s in
the
hig
h fr
eque
ncy
com
pone
nts
of
the
puls
e su
btra
ctio
n si
gnal
rel
ativ
e to
the
in
divi
dual
sca
tter
ed p
ulse
s, a
nd t
he t
otal
pul
se s
ubtr
acti
on s
igna
l m
ay t
hus
be u
sed
for
imag
e re
cons
truc
tion
. T
he d
ashe
d li
ne in
the
low
er p
anel
of F
ig.
5.8
depi
cts
the
abso
lute
val
ue o
f the
Fou
rier
T
rans
form
of
the
puls
e in
the
low
er p
anel
in
Fig.
5.7
, i.e
. w
hen
the
low
fre
quen
cy c
om
pone
nts
have
bee
n re
mov
ed f
rom
the
driv
e pr
essu
re.
As
seen
, th
e pu
lse
subt
ract
ion
sign
al
obta
ined
fro
m t
he t
wo
dual
fre
quen
cy b
and
driv
e pu
lses
is
very
sim
ilar
to t
he r
esul
t ob
ta
ined
app
lyin
g on
ly o
ne s
ingl
e hi
gh f
requ
ency
ban
d dr
ive
puls
e. A
pply
ing
only
one
suc
h si
ngle
hig
h fr
eque
ncy
band
dri
ve p
ulse
, th
e sc
atte
red
tiss
ue s
igna
l ca
n, h
owev
er,
not
be
effi
cien
tly r
emov
ed a
nd w
ill h
ence
mas
k th
e sc
atte
red
cont
rast
sig
nal.
Sim
ulat
ions
for
rad
ius
osci
llat
ion
and
scat
tere
d pr
essu
re u
sing
the
sam
e lo
w f
requ
ency
pu
lse
as i
n F
ig. 5
.3 b
ut v
aryi
ng th
e am
plit
ude
of t
he h
igh
freq
uenc
y pu
lse
wer
e al
so c
arri
ed
5.4
Bu
bb
le O
scil
lati
ons ~J •
• : 1
: : : i
j 0
2 3
5 6
7 8
9 lO
""I
93
Figu
re 5
.7:
Upp
er p
anel
: B
ubbl
e ra
dius
osc
illa
tion
whe
n dr
iven
by
high
fre
quen
cy c
om
pone
nt o
nly
in F
ig.
5.3.
L
ower
pan
el:
Sca
tter
ed p
ress
ure
puls
e re
sulti
ng f
rom
rad
ius
osci
llat
ion
in u
pper
pan
el.
Figu
re 5
.8:
Upp
er p
anel
: A
bsol
ute
valu
e o
f th
e F
ouri
er T
rans
form
of
the
puls
es i
n th
e up
per
pane
l of
Fig
. 5.
6.
Low
er p
anel
: A
bsol
ute
valu
e of
the
Fou
rier
Tra
nsfo
rm o
f th
e pu
lse
in th
e lo
wer
pan
el o
f Fig
. 5.6
, sol
id li
ne, a
nd th
e pu
lse
in th
e lo
wer
pan
el o
f Fig
. 5. 7
, da
shed
line
.
94
Pap
erD
out.
Whe
n in
crea
sing
and
dec
reas
ing
the
ampl
itud
e o
f th
e hi
gh f
requ
ency
dri
ve p
ulse
by
6 dB
, th
e re
sult
ing
ampl
itud
e o
f th
e fu
ndam
enta
l co
mpo
nent
of
the
high
fre
quen
cy p
ulse
su
btra
ctio
n si
gnal
inc
reas
ed a
nd d
ecre
ased
by
appr
oxim
atel
y 6
dB,
resp
ecti
vely
. It
is
mai
nly
the
line
ar a
cous
tic
prop
erti
es o
f the
osc
illa
ting
res
onan
t bub
ble
whi
ch a
re u
tili
zed
in t
his
new
con
tras
t ag
ent
dete
ctio
n te
chni
que,
and
the
per
form
ance
of
the
tech
niqu
e is
he
nce
not s
ensi
tive
to t
he a
mpl
itud
e o
f the
hig
h fr
eque
ncy
puls
e. A
s di
scus
sed
in th
e ne
xt
sect
ion,
the
am
plit
ude
of
the
tran
smit
ted
high
fre
quen
cy p
ulse
may
var
y co
nsid
erab
ly i
n th
e ra
nge
dire
ctio
n du
e to
aco
usti
c ab
sorp
tion
.
The
tra
nsm
itte
d lo
w f
requ
ency
com
pone
nts
alte
r th
e ac
oust
ic s
catt
erin
g pr
oper
ties
of
the
cont
rast
bub
ble
at t
he t
rans
mit
ted
high
fre
quen
cy c
ompo
nent
s an
d th
e de
scri
bed
tech
ni
que
will
be
sens
itiv
e to
the
am
plit
ude
of
thes
e lo
w f
requ
ency
pul
ses.
If
the
sam
e hi
gh
freq
uenc
y pu
lse
as i
n Fi
g. 5
.3 i
s us
ed t
o dr
ive
the
bubb
le w
here
as t
he a
mpl
itud
e o
f th
e lo
w f
requ
ency
pul
se is
inc
reas
ed a
nd d
ecre
ased
by
6 dB
, re
spec
tivel
y, w
e ge
t th
e re
sult
s di
spla
yed
in t
he u
pper
and
low
er p
anel
of
Fig.
5.9
, re
spec
tivel
y. T
his
figu
re d
epic
ts t
he
abso
lute
val
ue o
f th
e F
ouri
er T
rans
form
of
the
obta
ined
pul
se s
ubtr
acti
on s
igna
l jus
t as
th
e so
lid
line
in t
he l
ower
pan
el o
f Fig
. 5.8
and
it is
inte
rest
ing
to c
ompa
re w
ith
this
gra
ph.
In th
e up
per p
anel
of F
ig. 5
.9 t
he a
mpl
itud
e o
f the
low
fre
quen
cy p
ulse
is i
ncre
ased
by
6 dB
to
400
k:Pa
and
we
see
that
the
high
fre
quen
cy f
unda
men
tal
com
pone
nt o
f th
e pu
lse
subt
ract
ion
sign
al h
as in
crea
sed
by a
roun
d 3
dB r
elat
ive
to t
he s
olid
line
in th
e lo
wer
pan
el
of F
ig.
5.8.
The
leve
l of t
he h
igh
freq
uenc
y se
cond
har
mon
ic c
ompo
nent
in t
he p
ulse
sub
tr
acti
on s
igna
l is
clos
e to
unc
hang
ed w
hen
incr
easi
ng th
e lo
w f
requ
ency
am
plit
ude.
T
he l
ow f
requ
ency
am
plit
ude
is t
hen
redu
ced
by 6
dB
to
100
kPa
and
the
resu
ltin
g hi
gh f
requ
ency
fun
dam
enta
l co
mpo
nent
of t
he p
ulse
sub
trac
tion
sig
nal i
n th
e lo
wer
pan
el
of F
ig.
5.9
is s
een
to s
uffe
r a
redu
ctio
n o
f ar
ound
6 d
B r
elat
ive
to t
he s
olid
lin
e in
the
lo
wer
pan
el o
f Fi
g. 5
.8.
The
hig
h fr
eque
ncy
seco
nd h
arm
onic
com
pone
nt i
n th
e pu
lse
subt
ract
ion
sign
al is
now
red
uced
by
appr
oxim
atel
y 4
dB.
Red
ucin
g th
e lo
w f
requ
ency
am
plit
ude
by a
noth
er 6
dB
to
50 k
:Pa
resu
lts
in a
fur
ther
re
duct
ion
in t
he h
igh
freq
uenc
y fu
ndam
enta
l co
mpo
nent
of t
he p
ulse
sub
trac
tion
sig
nal o
f ar
ound
6 d
B.
As
the
ampl
itud
e o
f th
e m
anip
ulat
ing
low
fre
quen
cy d
rive
pul
se is
fur
ther
re
duce
d th
e tw
o sc
atte
red
high
fre
quen
cy p
ulse
s w
ill b
ecom
e m
ore
and
mor
e si
mil
ar.
5.5
Pro
paga
tion
of T
rans
mit
ted
Pul
ses
We
will
now
bri
efly
inv
esti
gate
the
forw
ard
wav
e pr
opag
atio
n o
f th
e tw
o du
al f
requ
ency
ba
nd t
rans
mit
pul
ses.
D
ue t
o th
e la
rge
diff
eren
ce i
n tr
ansm
itte
d fr
eque
ncy
(a f
acto
r 10
w
as u
sed
in t
he p
revi
ous
sect
ion)
, th
e ef
fect
s o
f di
ffra
ctio
n, a
bsor
ptio
n, a
nd n
onli
near
ity
wil
l al
l ap
pear
ver
y di
ffer
ent
for
the
two
tran
smit
ted
freq
uenc
y ba
nds.
It
may
tod
ay b
e ch
alle
ngin
g to
des
ign
and
man
ufac
ture
an
ultr
asou
nd t
rans
duce
r ca
pabl
e o
f ef
fici
entl
y tr
ansm
itti
ng f
or e
xam
ple
a lo
w f
requ
ency
0.5
MH
z pu
lse
and
a hi
gh f
requ
ency
5 M
Hz
puls
e. O
ne s
olut
ion
is t
o us
e an
ann
ular
tran
sduc
er m
ade
up o
f sev
eral
inde
pend
ent r
ings
w
hich
may
hav
e di
ffer
ent t
hick
ness
es a
nd th
us d
iffe
rent
res
onan
ce f
requ
enci
es.
5.5
Pro
paga
tion
of T
rans
mit
ted
Pul
ses
['~~{l
3·····
· ····~j
~20 o
:'-'
-'-:
-, --
-:4
--6
;---
---:
:-, -
-;';1
0,--
----
--;1
':-2 -
'---
:Ll4
--:'
:c--
--:l
':-8
------
:,0.
[MH
z]
·]~~::
1 0
2 4
6 8
10
12
14
16
18
20
[M
Hz]
95
Fig
ure
5.9:
Abs
olut
e va
lue
of th
e F
ouri
er T
rans
form
of
the
puls
e su
btra
ctio
n si
gnal
ob
tain
ed b
y in
crea
sing
and
dec
reas
ing
the
ampl
itud
e o
f th
e lo
w f
requ
ency
dri
ve p
ulse
by
6 dB
, up
per
and
low
er p
anel
, res
pect
ivel
y.
Firs
t, w
e ex
amin
e th
e ef
fect
s o
f w
ave
diff
ract
ion
from
suc
h an
axi
sym
met
ric
annu
lar
tran
sduc
er c
onsi
stin
g o
f a lo
w f
requ
ency
and
a h
igh
freq
uenc
y ri
ng.
Fig.
5.1
0 di
spla
ys th
e ca
lcul
ated
low
fre
quen
cy (
0.5
MH
z) a
nd h
igh
freq
uenc
y (5
MH
z) f
ield
s in
dec
ibel
sca
le,
left
and
rig
ht p
anel
, res
pect
ivel
y. A
Fin
ite
Dif
fere
nce
Met
hod
was
use
d in
the
cal
cula
tion
o
f th
e w
ave
diff
ract
ion.
The
low
fre
quen
cy p
ulse
is t
rans
mit
ted
from
an
unfo
cuse
d ou
ter
ring
wit
h in
ner
and
oute
r ra
dius
equ
al to
0.5
em
and
1 e
m,
resp
ectiv
ely,
whe
reas
the
hig
h fr
eque
ncy
puls
e is
tra
nsm
itte
d fr
om a
foc
used
inn
er e
lem
ent
wit
h ra
dius
equ
al to
0.5
em
an
d ge
omet
ric
focu
s at
8 e
m.
The
ver
tical
axi
s in
the
fig
ure
repr
esen
ts r
ange
dir
ecti
on
whi
le th
e ho
rizo
ntal
axi
s re
pres
ents
the
lat
eral
dir
ecti
on in
the
field
. T
he s
harp
edg
es o
ccur
ring
dow
n to
1 e
m i
n ra
nge
in t
he l
eft
pane
l o
f Fi
g. 5
.10
are
unph
ysic
al a
nd o
nly
a re
sult
of t
he w
ay t
he c
alcu
late
d fi
eld
is s
tore
d an
d vi
sual
ized
. T
he
calc
ulat
ed l
ow f
requ
ency
wav
e fi
eld
is o
nly
stor
ed a
t ea
ch c
enti
met
er in
ran
ge d
irec
tion
an
d he
nce
has
to b
e in
terp
olat
ed in
ran
ge w
hen
visu
aliz
ed.
We
see,
fro
m t
he l
eft
pane
l in
the
fig
ure,
tha
t th
e lo
w f
requ
ency
fie
ld i
s re
lati
vely
con
stan
t in
am
plitu
de a
roun
d th
e sy
mm
etry
axi
s fr
om 1
em
and
dow
n to
10
em.
Our
inte
ntio
n is
tha
t th
e hi
gh f
requ
ency
fie
ld p
ropa
gate
s in
eit
her
a po
siti
ve o
r a
neg
ativ
e ha
lf p
erio
d o
f the
ass
isti
ng lo
w f
requ
ency
fie
ld.
For
this
to
happ
en,
the
phas
e of
the
low
fre
quen
cy f
ield
has
to
chan
ge li
ttle
in r
ange
dir
ecti
on a
long
the
sym
met
ry a
xis.
The
ph
ase
shif
ts o
n th
e hi
gh f
requ
ency
com
pone
nts
are
mar
gina
l in
com
pari
son
due
to t
heir
m
uch
high
er fr
eque
ncy
and
resu
ltin
g sh
orte
r te
mpo
ral p
erio
d.
Fig.
5.1
1 di
spla
ys t
he n
orm
aliz
ed t
rans
mit
low
fre
quen
cy p
ulse
in t
he r
etar
ded
tim
e do
m
ain
at t
he s
ymm
etry
axi
s at
var
ious
loc
atio
ns in
ran
ge d
irec
tion.
The
upp
er p
anel
of t
his
figu
re s
how
s th
e pu
lse
at t
he t
rans
duce
r (0
em
). I
n th
e m
iddl
e pa
nel,
the
resu
ltin
g pu
lse
at 2
em
and
5 e
m,
soli
d an
d da
shed
lin
e, r
espe
ctiv
ely,
are
sho
wn
whi
le t
he l
ower
pan
el
96
Pap
erD
I I
-5
-5
-10
-1
0
-15
-15
Fig
ure
5.10
: C
alcu
late
d lo
w f
requ
ency
, le
ft p
anel
, an
d hi
gh f
requ
ency
, ri
ght p
anel
, tr
ans
mit
fie
lds
resu
ltin
g fr
om w
ave
diff
ract
ion
only
. L
ow f
requ
ency
sou
rce
is a
n un
focu
sed
ring
wit
h ra
dius
fro
m 0
.5 e
m to
1 e
m.
Hig
h fr
eque
ncy
sour
ce is
a fo
cuse
d di
sc w
ith
radi
us
equa
l to
0.5
em
and
foc
us a
t 8 e
m.
Fie
lds
disp
laye
d in
dec
ibel
sca
le.
depi
cts
the
puls
e at
7 e
m a
nd 1
0 em
, sol
id a
nd d
ashe
d li
ne, r
espe
ctiv
ely.
A
t 2
em t
he p
hase
shi
ft o
f th
e lo
w f
requ
ency
pul
se i
s ar
ound
~ r
elat
ive
to t
he p
ulse
at
the
tra
nsdu
cer
whi
ch i
s si
gnif
ican
t. T
his
mea
ns t
hat
if a
t th
e tr
ansd
ucer
, th
e hi
gh f
re
quen
cy p
ulse
was
cen
tere
d ar
ound
a n
orm
aliz
ed l
ow f
requ
ency
am
plit
ude
equa
l to
1 i
n th
e po
siti
ve h
alf
peri
od o
f th
e lo
w f
requ
ency
pul
se,
indi
cate
d by
the
ver
tica
l li
ne i
n th
e up
per
pane
l of t
he f
igur
e, a
t 2 e
m it
wou
ld b
e ce
nter
ed in
the
sam
e po
siti
ve h
alf p
erio
d o
f th
e lo
w f
requ
ency
pul
se b
ut a
roun
d a
norm
aliz
ed a
mpl
itud
e eq
ual
to 0
.5.
Fro
m 5
em
dow
n to
10
em,
how
ever
, th
e ph
ase
shif
t is
less
tha
n ~which c
an b
e co
nsi
dere
d li
ttle
in t
he p
rese
nt c
onte
xt.
A p
ossi
ble
solu
tion
for
this
spe
cifi
c ca
se is
to d
ivid
e th
e ra
nge
dire
ctio
n in
to tw
o se
pa
rate
reg
ions
, on
e fr
om t
he t
rans
duce
r an
d do
wn
to 4
or
5 em
, an
d on
e fr
om 4
or
5 em
and
do
wn
to 1
0 em
. T
he h
igh
freq
uenc
y co
mpo
nent
s de
dica
ted
to t
he f
irst
reg
ion
shou
ld th
en
be t
ime
shif
ted
by a
ppro
xim
atel
y ~
rela
tive
to
the
peak
of
the
low
fre
quen
cy a
mpl
itud
e at
the
tran
sduc
er to
occ
ur a
roun
d pe
ak p
osit
ive
or p
eak
nega
tive
low
fre
quen
cy a
mpl
itud
e in
the
reg
ion
of
inte
rest
. T
his
mea
ns t
hat
a to
tal
of f
our
puls
es m
ust b
e tr
ansm
itte
d do
wn
each
line
of
sigh
t ins
tead
of j
ust t
wo.
Aco
usti
c ab
sorp
tion
wil
l al
so a
ffec
t th
e tw
o ca
lcul
ated
wav
e fi
elds
in
Fig.
5.1
0 v
ery
dif
fere
ntly
. A
bsor
ptio
n in
crea
ses
wit
h fr
eque
ncy
whe
n m
easu
red
as a
fun
ctio
n o
f pr
opag
ati
on d
ista
nce
and
a ty
pica
l ex
peri
men
tal
valu
e fo
r so
ft t
issu
e is
a p
ower
red
ucti
on o
f 0.
5 dB
/cm
/MH
z.
Usi
ng t
his
valu
e, t
he i
nten
sity
of
the
low
fre
quen
cy 0
.5 M
Hz
puls
e is
re
duce
d by
onl
y 2.
5 dB
at
10 e
m,
whi
le t
he i
nten
sity
of
the
high
fre
quen
cy 5
MH
z pu
lse
is
redu
ced
by a
mas
sive
25
dB.
As
indi
cate
d ea
rlie
r, u
ltra
soun
d w
ave
prop
agat
ion
in t
issu
e is
usu
ally
a w
eak
nonl
inea
r
5.5
Pro
paga
tion
of T
rans
mit
ted
Pul
ses
97
~b;A
/1\/
fH
-lo~
o -4:
-::20
:----7
440:
:-o ---
-.,46:
;;:0,-
--:-!4
8~0 "--
--:sc!
:::oo-
'-:::0
520;;-
--~54~
0-5::0
:60:--
-:css:
:;-o ~Goo
Fig
ure
5.11
: N
orm
aliz
ed l
ow f
requ
ency
pul
se t
aken
out
at
vari
ous
loca
tion
s al
ong
the
sym
met
ry a
xis
in t
he l
eft
pane
l o
f Fi
g. 5
.10
and
plot
ted
in t
he r
etar
ded
tim
e do
mai
n as
fu
nctio
ns o
f sa
mpl
e nu
mbe
r. U
pper
pan
el:
Puls
e at
the
tra
nsdu
cer.
Mid
dle
pane
l: P
ulse
at
2 e
m a
nd 5
em
, so
lid
and
dash
ed li
ne,
resp
ectiv
ely.
Low
er p
anel
: Pu
lse
at 7
em
and
10
em,
soli
d an
d da
shed
line
, re
spec
tivel
y.
proc
ess.
The
tiss
ue e
last
icit
y be
have
s sl
ight
ly n
onlin
earl
y, g
etti
ng s
tiff
er d
urin
g co
mpr
es
sion
and
les
s st
iff d
urin
g ex
pans
ion
rela
tive
to a
lin
ear
beha
vior
. T
his
resu
lts i
n th
e fa
ct
that
the
ult
raso
und
wav
e pr
opag
atio
n ve
loci
ty w
ill b
e a
func
tion
of
the
pres
sure
and
wil
l he
nce
not b
e co
nsta
nt.
In t
he n
ew i
mag
ing
tech
niqu
e de
scri
bed
in t
he p
rese
nt p
aper
, th
is
effe
ct r
esul
ts in
the
poss
ibil
ity
that
the
high
fre
quen
cy p
ulse
pro
paga
ting
in th
e ex
pans
ion
cycl
e of
the
low
fre
quen
cy p
ulse
(up
per
pane
l of F
ig. 5
.3)
wil
l pro
paga
te w
ith
a so
mew
hat
low
er v
eloc
ity t
han
the
high
fre
quen
cy p
ulse
pro
paga
ting
in t
he c
ompr
essi
on c
ycle
of t
he
low
fre
quen
cy p
ulse
(lo
wer
pan
el o
f Fig
. 5.
3).
In E
q. 5
.7 w
e de
rive
d an
exp
ress
ion
for
the
vari
atio
n o
f the
pro
paga
tion
vel
ocity
wit
h re
spec
t to
the
pre
ssur
e fo
r a
plan
e w
ave
appr
oxim
atio
n. U
sing
typ
ical
val
ues
for
the
pa
ram
eter
s in
thi
s eq
uati
on,
j3 =
5 a
nd r
;, =
400
· I
Q-12
m
2 jN
, w
e ar
e ab
le t
o ca
lcul
ate
the
diff
eren
ce i
n pr
opag
atio
n ve
loci
ty 6
c f
or t
he t
wo
tran
smit
hig
h fr
eque
ncy
puls
es i
n Fi
g. 5
.3 a
ssum
ing
the
tran
smit
ted
low
fre
quen
cy p
ulse
s to
pro
paga
te a
s pl
ane
wav
es.
The
di
ffer
ence
in
pres
sure
, du
e to
the
low
fre
quen
cy p
ulse
s, f
or t
hese
tw
o hi
gh f
requ
ency
pu
lses
is
arou
nd 0
.5 M
Pa
and
we
may
her
e us
e th
e la
st a
ppro
xim
atio
n in
Eq.
5.7
, va
lid
whe
n r;
,p <
< 1
. W
ith
c 0
=
1500
m/s
we
get
a 6
c e
qual
to
1.5
m/s
usi
ng t
he i
ndic
ated
nu
mer
ical
val
ues.
At
a de
pth
equa
l to
10 e
m,
the
high
fre
quen
cy p
ulse
in th
e up
per
pane
l o
f Fi
g. 5
.3 w
ould
the
n ar
rive
aro
und
67 n
s af
ter
the
high
fre
quen
cy p
ulse
in
the
low
er
pane
l o
f th
is f
igur
e w
hich
is
abou
t on
e th
ird
of
the
peri
od o
f th
e 5
MH
z hi
gh f
requ
ency
pu
lse.
Whe
n su
btra
ctin
g th
e tw
o re
ceiv
ed s
catt
ered
pul
ses
in o
rder
to c
ance
l the
sca
tter
ed
high
fre
quen
cy t
issu
e co
mpo
nent
s, a
tim
e sh
ift
arou
nd o
ne t
hird
of
the
peri
od i
s en
ough
to
pre
vent
the
canc
ella
tion
of t
he t
issu
e si
gnal
. T
he t
ime
shif
ts b
etw
een
the
two
rece
ived
pu
lses
can
, du
e to
the
fac
t th
at t
he t
rans
mit
ted
low
fre
quen
cy p
ulse
s pr
opag
ate
mai
nly
98
Pap
er D
as p
lane
wav
es,
rela
tive
ly e
asil
y be
com
pens
ated
for
by
appl
ying
the
app
roxi
mat
ion
in
Eq.
5.7
to
calc
ulat
e th
e pr
oper
tim
e sh
ifts
and
int
rodu
cing
the
m i
n th
e re
ceiv
ed s
igna
ls
befo
re th
e su
btra
ctio
n pr
oces
s is
per
form
ed.
5.6
Con
clus
ions
The
pre
sent
pap
er h
as d
escr
ibed
a n
ew m
etho
d fo
r de
tect
ion
of
cont
rast
age
nts
util
izin
g th
e to
tal
scat
tere
d si
gnal
for
im
age
reco
nstr
ucti
on,
henc
e ov
erco
min
g th
e pr
oble
ms
in r
ela
tion
to
harm
onic
det
ecti
on t
echn
ique
s. I
n it
s si
mpl
est e
mbo
dim
ent,
the
new
met
hod
is
perf
orm
ed b
y tr
ansm
itti
ng t
wo
dual
fre
quen
cy b
and
puls
es,
both
con
tain
ing
a lo
w f
re
quen
cy b
and
and
a hi
gh f
requ
ency
ban
d w
hich
are
ove
rlap
ping
in t
he t
ime
dom
ain.
The
pu
rpos
e o
f th
e tr
ansm
itte
d lo
w f
requ
ency
ban
d is
to
man
ipul
ate
the
acou
stic
sca
tter
ing
prop
erti
es o
f th
e re
sona
nt c
ontr
ast b
ubbl
e at
the
tra
nsm
itte
d hi
gh f
requ
ency
ban
d, w
hich
is
use
d fo
r im
age
reco
nstr
uctio
n.
The
res
onan
t co
ntra
st b
ubbl
e w
ill
resp
ond
diff
eren
tly
on th
e tw
o tr
ansm
itte
d du
al f
requ
ency
ban
d pu
lses
rel
ativ
e to
the
non
-res
onan
t sof
t tis
sue.
T
his
is u
tili
zed
to e
ffic
ient
ly d
iffe
rent
iate
sca
tter
ed c
ontr
ast
sign
al a
nd s
catt
ered
tis
sue
sign
al in
a s
impl
e pu
lse
subt
ract
ion
proc
ess.
Mai
nly
the
line
ar a
cous
tic p
rope
rtie
s o
f th
e os
cill
atin
g re
sona
nt b
ubbl
e ar
e th
en u
tiliz
ed.
Pre
sent
ed re
sult
s sh
ow th
at th
e to
tal s
catt
ered
hi
gh f
requ
ency
sig
nal
from
the
con
tras
t bu
bble
is
pres
erve
d in
the
met
hod.
If
the
tiss
ue
elas
tici
ty i
s as
sum
ed to
res
pond
lin
earl
y to
the
tra
nsm
itte
d pr
essu
re p
ulse
s, t
he s
catt
ered
hi
gh f
requ
ency
tis
sue
sign
al c
an b
e ca
ncel
ed b
y th
e si
mpl
e pu
lse
subt
ract
ion
proc
ess.
Due
to
the
wea
k no
nlin
ear n
atur
e of
the
tissu
e el
astic
ity, s
mal
l tim
e co
mpe
nsat
ions
mus
t be
in
trod
uced
to t
he r
ecei
ved
sign
als
befo
re p
erfo
rmin
g th
e pu
lse
subt
ract
ion
proc
ess
to e
nsur
e th
at th
e sc
atte
red
high
fre
quen
cy t
issu
e co
mpo
nent
s ar
e co
mpl
etel
y re
mov
ed.
5.7
Fur
ther
Wor
k
Bas
ed o
n nu
mer
ical
sim
ulat
ions
and
the
oret
ical
con
side
rati
ons,
the
pro
pose
d ne
w c
on
tras
t det
ecti
on m
etho
d ap
pear
s in
tere
stin
g an
d sh
ould
be
furt
her
valid
ated
by
expe
rim
en
tal m
easu
rem
ents
. A
pos
sibl
e an
nula
r tr
ansd
ucer
des
ign,
con
sist
ing
of d
isti
nct r
ings
wit
h va
riab
le t
hick
ness
es,
capa
ble
of
tran
smit
ting
the
dua
l fr
eque
ncy
band
pul
ses
used
is
in
dica
ted
in r
elat
ion
to F
ig.
5.10
. It
is a
lso
poss
ible
to
use
two
sepa
rate
tra
nsdu
cers
wit
h ov
erla
ppin
g tr
ansm
it f
ield
s fo
r th
e tr
ansm
itte
d lo
w a
nd h
igh
freq
uenc
y pu
lses
. T
he r
esu
ltin
g ph
ase
rela
tion
s be
twee
n th
e tr
ansm
itte
d lo
w f
requ
ency
and
hig
h fr
eque
ncy
puls
es
wou
ld t
hen
be s
omew
hat
mor
e co
mpl
ex t
han
for
the
tran
sduc
er i
ndic
ated
in
rela
tion
to
Fig.
5.1
0.
5.8
Ack
now
led
gmen
ts
99
5.8
Ack
now
ledg
men
ts
Thi
s w
ork
was
sup
port
ed b
y th
e R
esea
rch
Cou
ncil
of N
orw
ay.
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liog
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e R
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d Im
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ww
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ltra
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raso
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n, T
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ohan
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ble
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e m
ach-
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EE
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raso
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g te
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rope
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EE
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ason
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ann
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n, J
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g, S
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anta
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a D
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h.
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aso
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