An Adaptive Pulse Compression Filter for Ultrasound Contrast Harmonic Imaging

doi:10.4236/eng.2013.510B024

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Engineering, 2013, 5, 118-122

http://dx.doi.org/10.4236/eng.2013.510B024 Published Online October 2013 (http://www.scirp.org/journal/eng)

An Adaptive Pulse Compression Filter for Ultrasound

Contrast Harmonic Imaging

Jenho Tsao, Ming-Huang Chen

Graduate Institute of Biomedical Electronics and Bioinformatics, National Taiwan University

Email: [email protected]

Received December 2012

ABSTRACT

Coded excitation is useful for ultrasound contrast imaging to increase penetration and SNR, and improve the contrast to

tissue ratio (CTR). The waveform of bubble response depends greatly on bubble size, the frequency and bandwidth of

the excitation chirp signal. This makes the pulse compression filter based on square-law be wrong for bubbles with

changing sizes. In this paper, an adaptive pulse compression (APC) filter for the second harmonic of microbubble with

varying size distribution is proposed. The APC filter is designed based on the estimated power spectrum of the received

bubble harmonic echoes. Theoretical analysis and simulation studies are presented for evaluating performance of the

APC filter. For monodisperse bubble, the power improvement factor of the APC filter can be more than 20 dB.

Keywords: Ultrasound Contrast Imaging; Square Law Pulse Compression; Adaptive Pulse Compression

1. Introduction

Coded excitation is useful for ultrasound imaging to in-

crease penetration and SNR. For improving the CTR

(contrast to tissue ratio) in ultrasound contrast imaging,

coded excitation is preferred also. In a previous research,

Borsboom et al. suggest a nonlinear compression f ilter to

do pulse compression for bubble second harmonic imag-

ing [1]. They made use of a square law (SQL) model to

design the pulse compression filter, which had the twice

instant a neous fre quency of the transmitted chirp.

The performance of microbubble agents depends on

their resonant character to differ fro m tissue; th e resonant

character of bubble depends greatly on th e size of bubble ,

and the frequency, bandwidth and waveform of the exci-

tation signal. Novell et al. indicated that for up-sweep

chirp, bubbles below resonance size provided a longer

response; whereas bubbles above resonance size produced

much more damped responses with large amplitude ex-

cursions [2]. Macdonald et al. showed that larger bubbles

had more shift in resonance frequency than smaller bub-

bles. Also, the resonance frequency shifted to lower fre-

quency when driving pressure was increased [3].

In this study, it is shown by simulation that chirp re-

sponse of bubble depends greatly on the size of bubble

and the frequency, bandwidth the excitation chirp signal.

The pulse compression filter designed based on the SQL

model (SPC) is a fixed pulse compression filter; equiva-

lently it assumes that the chirp response of bubble is in-

variant. In reality, microbub bles have a time-varying size

distribution in the p erfusion region of interest, bubbles of

different sizes respond to chirp excitation differently.

This will lead to restrictions on the performance of the

SPC filter. Consequently adaptive pulse compression

(APC) is required to maximize the backscatter power of

microbubbles for contrasting imaging to increase SNR.

In this paper, we propose an adaptive pulse compres-

sion filter to compress the second harmonic of bubble

while microbubbles have a time var ying size distribu tion.

It is found that bandwidth of the second harmonic will be

reduced while bubbles are excite to resonate. The APC

filter is designed based on the estimated power spectrum

of the received bubble harmonic echoes. Thus, it ch anges

the pulse compression filter for bubble of different sizes

adaptively to optimize the output power.

In the follows, theoretical analysis of the pulse com-

pression problem is given in Section 2. The proposed

adaptive pulse compression filter is given in Section 3.

The performance of the APC filter is studied using s imu -

lation signals generated by BubbleS im [4]

2. Theories

Since the bubble response to chirp excitation is compli-

cated, the theoretical foundations for the pulse compres-

sion problems are given below, step by step.

2.1. Bubble Response

The ultrasound contrast agents are encapsulated micro -

This work is supported by the Natio nal Science Co uncil, Taiw an, ROC

(NSC101-2221-E-002-081).

J. TSAO, M.-H. CHEN

119

bubbles, which provide strong scattering echo through

the resonance behavior of bubbles. In general, bubble

resonance is a complicated process, especially to drive

the bubbles to generate high harmonics [4-6]. For a given

driving pressure

()xt

, the scattered pressure

()

can

be found as

( )

()

()2()() ()

y tR tRtRt

= +

 

(1)

where R is the bubble radius, dynamics radius

()

Rt , r is

the distance from bubble and

is the liquid density

[7]. The dynamics radius

()

Rt can be found by solving

the nonlinear Rayleigh-Plesset like differential equation:

4 ()

RR R

pp xt

RRR R

σσ







 



=+−+ −+

 





 

 



(2)

where

and

are the initial and instantaneous ra-

dius of the bubble,

and

are the density and vis-

cosity of the liquid,

is the surface tension,

is the

polytropic exponent,

is the ambient pressure, and

()xt

is the incident pressure.

When driving pressure is a simple sinusoidal, the scat-

tering cross section (SCS) can be found analytically as a

function of driving frequency and bubble radius to be

()

(,)

[4-7], where n represents n-th harmonic. For

complicated driving waveform, such as chirp, the only

way to find the scattered pressure is by solving the Ray-

leigh-Plesset like equation. In this study, the chirp res-

ponses of bubbles are found by using BubbleSim [4] to

solve the Rayleigh-Plesset like equation.

2.2. The SCS of Microbubbles

The scattering cross section of microbubbles provides an

easy way to investigate the scattering properties of mi-

crobubbles. Based on Church’s formulation, Equation

(26b) in [6], for a microbubble coated with an elastic

solid, its second-harmonic SCS can found to be a func-

tion of driving frequency and bubble radius. The bubble

parameters used are: shell thickness = 4 nm, shell shear

modulus = 50 MPa, shell viscosity = 0.8 Pas, ambient

pressure = 100 KPa, polytropic exponent = 1, viscosity

of the liquid = 0.001 Pas, density of the surrounding liq-

uid = 1000 kg/m3 and density of the shell material = 1100

kg/m3 [4].

The SCS’s of the second harmonic of three bubbles

with radius 1.5, 2.5 and 3.0 μm are shown in Figure 1.

Since bubbles are resonant scatters, and resonance is in

general a narrow-band phenomenon, the SCS’s have dif-

ferent resonant bandwidths. It can be found that smaller

bubble has larger resonant bandwidth than that of larger

bubble. The resonant bandwidth of the 3.0 μm is quite

-10

Driving Frequency [MHz]

(dB μm2)

3.0 μm

2.5 μm

1.5 μm

Figure 1. The scattering cross sections of three microbub-

bles.

small, which may not be good for imaging system to

have high range resolution. Although SCS can not show

the effect of resonant bandwidth on the scattered wave-

form, it does show the problem of insufficient bandwidth

for large microbubbles. The apparent differences among

the bubbles with different sizes also predict that their

chirp responses will be very different. More detail about

the chirp responses of microbubbles are given next.

2.3. The Chirp Response

The transmitted chirp signal is defined as:

() ()2π2c

xtw tcostft



=⋅+





t− ≤≤

(3)

where

()

wt is a window function, B is excitation band-

width,

is the signal duration defined by

()

wt and

is center (driving) frequency [8]. In latter uses, the

window function is a Hann ing window discretized at 100

MHz and signal duration is 10 μs. The center frequency

and bandwidth are set when they are used in different

cases.

Based on square law, the second harmonic of micro-

bubble is modeled to have a waveform as

()xt

[1,9,

10]. Thus the pulse compression correlator has an im-

pulse res ponse as

()()4 2

SQ c

htw tcostft



=⋅π+





(4)

To exploit the properties of the scattered signal of

bubbles excited by the chirp signal, BubbleSim is em-

ployed to solve a modified Raleigh-Plesset equation in [4]

with parameters set same as i n computing the SCS given

in last section. The chirp signal is set to have bandwidth

= 2 MHz and transmission acoustic pressure = 50 KPa.

In F igure 2, the top panel shows the impulse response

of the pulse compression correlator based on square-law,

the others are the simulated waveforms of bubbles with

radius = 1, 2, 3 and 4 μm. Their waveforms are different

significantly different from each other. Based on matched

filter theory, the impulse response of a pulse compression

filter should have same waveform as the scattered pres-

sure to be compressed. However, the scattered pressures

of bubbles with different radiuses are so different, it’s

J. TSAO, M.-H. CHEN

120

0.1

-0.1

-1

1 μm

2 μm

-1

μs

3 μm

4 μm

-1

SQL

Figure 2. The pre dicted and sim ulated wave forms. The SQL

predicted waveform by square-law (Top panel) and the si-

mulated waveforms of bubbles with radius = 1, 2, 3 and 4

μm.

almost impossible to have an ideal pulse compression

filter. By comparing the waveforms in Figure 2, it can be

found that the waveform predicted by the square-law (top

panel) has high similarity to the scattered pressures of 1

and 2-μm bubbles, but it is very bad for the 3 and 4-μm

bubbles. This shows that the pulse compression filter

should be designed differently for bubbles with different

sizes and predicts that there will be compression loss for

the square-law pulse compression filter.

3. The Adaptive Pulse Compression Filter

The impulse response of the adaptive pulse compression

filter (correlator) is defined as the chirp signal with pa-

rameters to be set adaptively:

()()2 2

htw tcostft



=⋅π +





t− ≤≤

(5)

The center frequency and bandwidth of APC,

are the estimated center frequency and bandwidth of the

received scattering signal of bubbles. The duration

is set as

TTBB=

, which affects the length of

()

wt .

Performances of the proposed APC filter are studied

using simulation signals of BubbleSim. Since the scat-

tered pressure of bubble depend simultaneously on the

driving frequency, excitation bandwidth and bubble size,

the simulations are done for two driving frequencies,

2.45 and 3.5 MHz, and two excitation bandwidths, 2 and

4 MHz. The pulse compressions are done for bubbles

with size ranged from 0.5 to 5 μm by the square-law

(SPC) and adaptive PC (APC) filters. Results are pre-

sented in Figures 3 to 5. The top panels are the output

powers of pulse compression done by SPC and by APC.

The output powers of APC’s are all larger than that of

SPC’s. The dB difference between APC and SPC are

-40

-20

(b) Radius, μm

SPC and APC: 2.45, B=2MHz

SPC

APC

-5

(a)

IF of APC

Figure 3. (a) Output pow er of pulse compressi on by square-

law (SPC) and by adapti ve P C (APC), for fc = 2.45 M Hz and

B = 2 MHz; (b) The im p ro v em ent factor of APC o v er SPC.

-40

-20

(a)

SPC and APC: 2.45, B=4MHz

SPC

APC

-10

(b) Radius, μm

IF of APC

Figure 4. (a) Output pow er of pulse compressi on by square-

law (SPC) and by a daptiv e P C (APC), for fc = 2.45 M Hz and

B = 4 MHz; (b) Th e im p ro v em ent factor of APC o v er SPC.

J. TSAO, M.-H. CHEN

121

-40

-20

SPC

APC

-5

IF of APC

(b) Radius, μm

(a)

SPC and APC: 3.5, B=4MHz

Figure 5. (a) Output pow er of pulse compressi on by square-

law (SPC) and by adaptive PC (APC), f or fc = 3.5 MHz and

B = 4 MHz; (b) The im p ro v em ent factor of APC o v er SPC.

defined as the improvement factor for the APC and

shown in the lower panels. Two points found by com-

paring the three cases are given below.

1) All three cases show a same property that APC fil-

ter has power improvement over SPC only for bubbles

within a range of sizes. That is the performance of APC

is bubble-size dependent. This result has a close relation

with the resonance property of bubble; for a given exci-

tation signal, only bubbles with certain sizes can be ex-

cited to resonate. Too small or too large bu bbles will not

be excited to resonate, then there is no waveform distor-

tion; in such case, SPC can predict the waveform of

scattered pressure correctly and has a same result as

APC.

2) The APC filter provides more improvement for

larger excitation bandwidth. This property can be ob-

tained by comparison that both Figures 4 and 5 has im-

provement factor (IF) larger than 20 dB, however Figure

3 has IF ar ound 8 dB only .

Based on the bubble-size dependent property of APC,

it can be concluded that, for bubbles with a given size

distribution, to have optimal power by PC, not only the

PC filter must be designed adaptively, the driving fre-

quency must be selected to match the size distribution

also. In addition, the bubble-size dependent property of

APC confirms that SPC is not proper for chirp harmonic

imagi ng when bubbl e s are excit e d t o re s o na te.

4. Conclusions

In this study, the square-law based pulse compression

filter is shown to be improper for contrast harmonic im-

aging. Based on simulation studies, the performance of

APC is found to be bubble-size dependent due to bubble

resonance. To have optimal power by APC, the driving

frequency must be adapted to match the size distribution

too. The APC filter is shown to be able to have a power

imp ro ve men t factor to be more than 20 dB over the SPC

filter.

Since the resonant character of micro bubble depends

complicatedly on the size of bubble and the frequency,

bandwidth and waveform of the excitation signal, more

studies including experiments are necessary to prove the

validity and performance of the APC filter.

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