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2054 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 65, NO. 11, NOVEMBER 2018
Autoregressive Signal Processing Applied to
High-Frequency Acoustic Microscopy
of Soft Tissues
Daniel Rohrbach ,Member, IEEE, and Jonathan Mamou ,Senior Member, IEEE
Abstract— Quantitative acoustic microscopy (QAM) at fre-
quencies exceeding 100 MHz has become an established imaging
tool to depict acoustical and mechanical properties of soft biologi-
cal tissues at microscopic resolutions. In this study, we investigate
a novel autoregressive (AR) model to improve signal processing
and parameter estimation and to test its applicability to QAM.
The performance of the AR model for estimating acoustical
parameters of soft tissues (i.e., acoustic impedance, speed of
sound, and attenuation) was compared to the performance of
the Hozumi model using simulated ultrasonic QAM signals and
using experimentally measured signals from thin (i.e., 12 and
6µm) sections of human lymph-node and pig-cornea tissue
specimens. Results showed that the AR and Hozumi methods
performed equally well (i.e., produced an estimation error of 0)
in signals with low, linear attenuation in the tissue and high
impedance contrast between the tissue and the coupling medium.
However, the AR model outperformed the Hozumi model in
estimation accuracy and stability (i.e., parameter error variation
and number of outliers) in cases of 1) thin tissue-sample thickness
and high tissue-sample speed of sound, 2) small impedance
contrast between the tissue sample and the coupling medium,
3) high attenuation in the tissue sample, and 4) nonlinear
attenuation in the tissue sample. Furthermore, the AR model
allows estimating the exponent of nonlinear attenuation. The
results of this study suggest that the AR model approach can
improve current QAM by providing more reliable, quantitative,
tissue-property estimates and also provides additional values of
parameters related to nonlinear attenuation.
Index Terms—Autoregressive model, hozumi model, scanning
acoustic microscopy, signal processing.
I. INTRODUCTION
SCANNING acoustic microscopy (SAM) has become an
established imaging tool to characterize acoustical (e.g.,
speed of sound, acoustic impedance, and attenuation) and
mechanical (e.g., bulk modulus and mass density) proper-
ties of soft [1]–[13] and hard biological tissues [14]–[16] at
microscopic resolutions [17]–[19] using ultrasound frequen-
cies between 100 MHz and 1.5 GHz. Although the spatial
resolution of SAM is coarser than that of optical microscopy,
SAM has a high relevance for several research areas because
SAM not only has the ability to scan unstained and unfixed
tissues but it also can be used to derive unique tissue
Manuscript received June 22, 2018; accepted September 6, 2018. Date of
publication September 13, 2018; date of current version November 7, 2018.
This work was supported by NIH Grant EB016117. (Corresponding author:
Daniel Rohrbach.)
The authors are with the F. L. Lizzi Center for Biomedical Engi-
neering, Riverside Research, New York, NY 10038 USA (e-mail:
daniel.rohrbach.
@.
web.de).
Digital Object Identifier 10.1109/TUFFC.2018.2869876
properties that cannot be obtained using other microscopy
methods [20]. Although SAM is a mature technology, estimat-
ing all material properties in a single measurement remains
a challenging task, particularly at frequencies greater than
200 MHz. In a previous study, we developed a novel quan-
titative acoustic microscopy (QAM) system and used it to
obtain high-quality images of acoustic parameters of soft
tissues at 250 and 500 MHz [21]. The system scans thin
(i.e., 5–14 μm) soft-tissue sections that are attached to a
glass plate (i.e., substrate) and records radio frequency (RF)
signals that originate from ultrasound reflections from the
sample surface and sample-glass-plate interface, respectively.
This common approach [3], [11], [22], [23] allows a single
measurement to obtain an estimate of acoustic impedance
(Z), speed of sound (c), and ultrasound attenuation (α)as
well mass density (ρ) and bulk modulus (K), which are
directly related to cand Z[24]. Several signal processing
algorithms exist to estimate these parameters for each scan
location in a C-mode (raster scanning) configuration. Our
previous studies [21], [24] used the most common method,
which is similar to the one described by Hozumi et al. [3]
which we term the Hozumi-based model. For measurements on
thin sections of tissue, attached to a substrate (i.e., microscopy
glass plate), two reflected signals occur in the recorded RF
data. The first reflected signal emanates from the water–sample
interface (front reflection, s1), and the second reflected signal
emanates from the sample–substrate interface (back reflec-
tion, s2). Acoustical parameter values (i.e., Z,c,andα)
are estimated by comparing the time of flight (TOF) and
amplitude of s1and s2to the TOF and amplitude of a reference
signal, which is obtained from a glass-plate-only reflection.
Accurately decomposing the acquired RF signals into these
two reflection signals is the signal processing challenge in
soft-tissue QAM applications.
In an ideal scenario, the thickness of the investigated
soft-tissue section should be in the order of the available
axial resolution and a fraction of the −6-dB depth of field
for the following four reasons: to mitigate possible interfering
signals (e.g., scattering) from inside the tissue; to prevent reso-
lution deterioration caused by out-of-focuseffects; to minimize
attenuation effects; and to maintain a nearly planar wavefront.
However, if the sample is too thin, then the reflected signals
overlap in time and frequency domains, which make separat-
ing the signals with conventional signal processing methods
challenging. The bandwidth of a QAM system dictates the
lower limit for sample thickness. Another challenge is the low
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ROHRBACH AND MAMOU: AR SIGNAL PROCESSING APPLIED TO HIGH-FREQUENCY ACOUSTIC MICROSCOPY OF SOFT TISSUES 2055
amplitude of the first reflected signal, which is typical because
the acoustic impedance of most soft tissues is close to that of
the coupling fluid (e.g., water or saline). This problem makes
detection of the first signal and correct parameter estimation
difficult.
Although our previously developed algorithms [21] worked
satisfactorily in several applications [20], [24]–[30], we expe-
rienced suboptimal acoustic parameter estimation when ana-
lyzing signals from thin tissue samples. In our previous study,
we compared measurements obtained using QAM systems
equipped with transducers operating at 250 and 500 MHz. For
such dual-frequency experiments, signal-processing algorithms
must reliably operate over a large range of sample thicknesses.
We found that the current algorithm works best on 6- and
12-μm-thin sections when 500- and 250-MHz transducers
are employed, respectively. However, to directly compare
measurements from both transducers, ideally, the same 6-μm
sections are scanned with both transducers instead of scanning
adjacent slides with different thicknesses (i.e., a 6-μmandan
adjacent 12-μm-thin cut). Another limitation of the current
approach is its inability to estimate nonlinear attenuation.
Numerous studies described in the literature suggest that ultra-
sound attenuation in most soft tissues is best described by an
exponential model as a nonlinear function of frequency [31].
In particular, at the very high frequencies employed in QAM,
frequency-dependent attenuation in all the materials, including
the coupling medium as well as the tissue specimen, can have
a significant impact on the parameter estimation.
In this paper, we investigate a novel autoregressive (AR)
model approach coupled with a denoising algorithm to further
improve signal decomposition and parameter estimation and
to test its applicability to QAM. In the following, we will
refer to it as the AR-based model. In addition, we compared
the performance of the AR model with the Hozumi model in
simulated data and experimental data from soft tissues.
II. THEORY
A. Forward Model
Fig. 1 depicts the experimental approach used to collect
data. The sample is raster scanned in 2-D, and RF echo
signals are acquired at each scan location. The RF echo signal
acquired from a location devoid of tissue is termed as the
reference signal and is symbolized by s0(t−t0). This notation
indicates that the reference signal is composed of only one
echo at the glass–water interface. Table I lists all symbols
used to describe the theory.
Similarly, we refer to signals derived at all other scanned
locations as sample signals, symbolized by s(t). Our forward
model is described using the following expression:
s(t)=s1(t)+s2(t)+···+sn(t)(1)
=C1s∗
0(t−t1)+C2s∗
0(t−t2)+···+Cns∗
0(t−tn)
(2)
which means that the sample signals are the sum of nweighted
and delayed versions of the reference signal and the “*”
symbol represents frequency-dependent attenuation effects.
Among these nsignals, two signals (i.e., sk1and sk2with
Fig. 1. Diagram of the experimental QAM approach. The RF echo acquired
from a location devoid of tissue is referred to as the reference signal s0(t−t0).
Other scanned locations are referred as sample signal, with s1(t)being the
reflection from the sample–water interface and s2(t)being the reflection from
the sample–substrate interface. ddenotes the sample thickness.
TAB LE I
USED SYMBOLS
k1= k2and tk1<tk2) are the echoes from the water–tissue
and tissue–glass interfaces, respectively. These two signals are
the ones needed to obtain quantitative tissue properties. The
remaining n−2 signals (i.e., sk3,...,skn) account for potential
multiple reflections, scattering, or noise.
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2056 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 65, NO. 11, NOVEMBER 2018
By taking the Fourier transform of (2), we obtain
S(f)=S0(f)[C1exp(f(β1+j2π(t1−t0))) +···
+Cnexp(f(βn+j2π(tn−t0)))](3)
where βkis the attenuation coefficient of the kth signal in
Np/Hz and S0(f)is the Fourier transform of the reference
signal (i.e., S0(f)=FFT{s0}). By dividing S(f)by S0(f),
(3) yields the normalized spectrum
N(f)=S(f)
S0(f)
=
n
k=1
Ckexp(f[βk+j2π(tk−t0)]). (4)
B. Hozumi Inverse Model
Our recent manuscript details the Hozumi model [21] and
is compared here to the AR model. Briefly, the Hozumi
inverse model assumes n=2 in the forward model and our
implementation starts by assessing the extrema of the squared
magnitude (|N|2) of the normalized spectra. From (4), using
n=2 and assuming β1=0 (i.e., the first signal suffers no
attenuation), following expression can be derived:
|N|2=
FFT{s1(t)+s2(t)}
FFT{s0(t)}
2
(5)
=C2
1+2C1C2·efβ2·cos(2πf(t2−t1)) +C2
2e2fβ2.
(6)
Then
ϕu(N)=2πfmax 2d
cw
−p,p∈Z(7)
and
ϕu(N)=4πfmin d
cw
−π(2p−1), p∈Z(8)
where dis the tissue thickness, fmin and fmax are the frequen-
cies of the extrema of |N|2,andϕu(N)is the unwrapped phase
of N. Equations (7) and (8) are used to estimate the thickness
of the specimen. The speed of sound (c) can be estimated
using
d=ϕu(N)
4πfmax1
cw−1
c(9)
where cwis the speed of sound in the coupling medium.
C1,C2,andα=β2/2dcan be found from the amplitudes of
|N|2. The acoustic impedance of the sample (Z) was estimated
from C1using first principles (see [21, eqs. (17) and (18)]).
To estimate α, we used a dichotomy method applied to |N|2
as shown by [21, eq. (16)].
C. Autoregressive Inverse Model
Initially, the AR model assumes that the signals are com-
posed of more than one signal, i.e., nis assumed to be
≥2, which provides robustness and stability. Most of the
ndecomposed signals are related to noise and estimation
artifacts. The aim of this section is to find the two signals
that are related to the water–sample and sample–substrate
interfaces.
The inverse model consists of rewriting (4) at discrete
frequencies denoted by fi=if. The step size (f)is
related to the total duration of the sampled signal in points
(M) and the sampling frequency ( fs) and is defined by
fs/(2M). (Note that zero-padding the signal will make f
smaller as it is typically done in Fourier analysis, but in
this application, it does not provide new information and is,
therefore, unnecessary and was not done.) Discretization yields
the following equation for Ni=N(if):
Ni=
n
k=1
Ck[exp( f[βk+j2π(tk−t0)])]i(10)
=
n
k=1
Ckλi
k(11)
where λk=exp( f[βk+j2π(tk−t0)]).
Then, the AR process is introduced by assuming that Ni
can be estimated using a linear combination of the ˆn-previous
frequencies [32]
Ni=
ˆn
k=1
xkNi−k+i(12)
where iis an error term and xkare the AR coefficients and
we choose ˆn=n.
Based on (12), the AR inverse model consists of the
following four steps: 1) estimating Ckand λk; 2) Cadzow
denoising; 3) determining k1and k2; and 4) estimating all
acoustic parameters from Ck1,Ck2,λk1,andλk2.
1) Estimation of Ckand λk:Equation (12) is rewritten in
matrix notation and only is used between frequencies f1=
F1fand f2=F2f, which are determined by the −20-dB
bandwidth of the transducer:
N=−RX +(13)
where Nis the column vector of length F2−F1−n+1andis
composed of the values of Nifrom F1+nto F2,Ris the nby
F2−F1−n+1 matrix whose element (q,p)is NF1+n−q+p−1,
and is the column vector of length F2−F1−n+1 composed
of the values qfrom q=F1+nto q=F2.
Equation (13) is solved for Xusing a least-squares min-
imization, the sum of the magnitude-squared errors (i.e.,
t, where the superscript tsymbolizes the transposition
operation)
X=−(RtR)−1(RN). (14)
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ROHRBACH AND MAMOU: AR SIGNAL PROCESSING APPLIED TO HIGH-FREQUENCY ACOUSTIC MICROSCOPY OF SOFT TISSUES 2057
Equations (10) and (12) are combined to establish a rela-
tionship between Xand λk
Ni=−
n
l=1
xi
n
k=1
Ckλi−l
k(15)
⇔−
n
k=1
xkNi−k=−
n
l=1
xi
n
k=1
Ckλi−l
k(16)
which yields
n
k=1
Ckλi
kP1
λk=0 (17)
where Pis a polynomial of degree ndefined by
P(z)=1+
n
l=1
xlzl.(18)
Equation (17) must be true at all frequencies, thus,
P(1/λk)=0forallk. Therefore, λkare the reciprocal of
the nzeros of P.P(z)=0 can be easily solved, because the
coefficients xlare known through (14).
To find Ck, (10) is written for all ibetween f1and f2
yielding the following matrix equation:
=N2(19)
where is a Vandermonde-like matrix of size nby F2−F1+1
whose element q,pis λp
q,is a column vector of unknown
Cq,andN2is a column vector whose qth element is NF1+q−1.
This equation is solved using a least-squares approach to yield:
=(t)−1(N2). (20)
2) Cadzow Denoising: To improve the signal decomposition
by further taking advantage of the low expected rank of R,
Cadzow denoising was applied to preprocess Nandtoyield
a new vector ND[33]. The method makes iterative use of
singular-value decomposition (SVD) and antidiagonal averag-
ing to reduce the rank of the Hankel matrix [Hq,p=Nq+p−1,
see (12)] as described briefly in the following.
The first step consists of calculating the SVD such that
H=U∗S∗V(21)
where Sis a diagonal matrix with nonnegative diagonal
elements (i.e., singular values) in decreasing order and Uand
Vare unitary matrices. In the next step, ˜
(Ni)is reconstructed
by keeping only the nlargest singular values of S
˜
Ni=U∗Sn∗V(22)
where Sn(q,p)=S(q,p)for q,p<nand Sn=0otherwise.
A denoised version (ND)ofNcan be reconstructed by taking
the average of all antidiagonals of ˜
Ni
ND
i=mean
q+p=i+1(˜
N). (23)
Equations 21–23 are repeated iteratively (i.e., five times in
our QAM applications). Although we choose n>2, the aim
is to find the two signals, sk1and sk2[k1= k2and tk1<tk2,
see (2)], that represent the water–sample and sample–substrate
reflections, respectively. Other signal components (i.e., sk3–
skn) either are related to noise, estimation artifacts or are
actual signals originating from structures inside the tissue (e.g.,
caused by scattering). To simplify the parameter estimation
further, we eliminated signals related to noise and estimation
artifacts. To do this, we tested the following two strategies:
we set nto a fixed predefined constant (e.g., n=2) or we
estimated it based on a threshold procedure. We estimated n
to be the Ksingular values that contribute more than 10%
to the overall signal. During the first iteration of the Cadzow
denoising, we calculated
n=1+
qs.t. q≥1and S(q,q)2
p≥1(S(p,p))2>0.1
(24)
where represents the cardinal symbol that expresses the
number of elements in the enclosed set ({}). This approach
can produce signals with higher orders (i.e., n>2). To find
sk1and sk2to estimate the acoustic parameters, we applied the
Cadzow denoising followed by replacing Nby NDin (10).
3) Determining k1and k2:The next step of the inverse
model consists of finding the indices (k1and k2) corresponding
to the water–tissue reflection defined by sk1and the tissue-glass
reflection defined by sk2. The AR inverse model can find
signals that are related to noise or artifacts and are not related
to the original signal. Some of these unreliable signals easily
can be identified and excluded by thresholding the signal
amplitude or phase shift (i.e., tk−t0). For QAM applications,
we implemented a simple, but fast and efficient, algorithm to
find sk1and sk2. The first step of the algorithm consists of
calculating the amplitude ( AMin millivolts) and TOF (TOFM
in microseconds) of the maximum of the Hilbert-transformed
signals. Assuming that the amplitude [Cp=1/ˆ
Rp,where ˆ
Rp
is the reflection coefficient of signal spwith p∈{1,...,n}
and is estimated using (40)] and TOF [TOFp=tp−t0,
see (26)] of at least one of the signals sk1or sk2must be
close to AMand TOFMof the measured signal, the algorithm
first finds the spwith
p=arg min
p(AM−Cp)2+(TOFM−TOFp)2.(25)
The second signal is then selected by finding Cqwith q=
arg minq= p|AM−Cq|. The final step consists of sorting sp
and sqso that sp=sk1;sq=sk2if TOFp≤TOFqand
sq=sk1;and sp=sk2otherwise. This approach assumes
that the two amplitudes from sk1to sk2are the two largest
among the nsignals of the forward model. If the amplitude
from a third signal, originating from scattering or reflections
from within the sample, is larger, then Ck1or Ck2in the
above-mentioned algorithm will misidentify the desired sig-
nals. However, we never encountered this situation in any of
our experiments in soft tissue with a glass plate as a substrate.
4) Acoustic Parameter Estimation: The final step of the
AR inverse model consists of estimating, acoustic impedance,
speed of sound, and attenuation from Ck1,Ck2,λk1,andλk2.
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2058 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 65, NO. 11, NOVEMBER 2018
The definition of λk[see (10)] directly yields
tk1−t0=imag(log(λk1))
f(26)
tk2−t0=imag(log(λk2))
f(27)
and
βk1=real(log(λk1))
f(28)
βk2=real(log(λk2))
f(29)
where imag and real symbolize the functions that take the
imaginary and real parts of the argument.
First principles also lead to the following expressions:
tk1−t0=2d
cw
(30)
tk2−t0=2d
c−2d
cw
(31)
which can be simultaneously solved to yield
d=cw
2
imag(log(λk1))
f(32)
c=cw
imag(log(λk1))
imag(log(λk1)) +imag(log(λk2)) .(33)
To estimate Z, we exploit the fact that Ck1is the ratio
between the pressure reflection coefficients at the water–tissue
and at the water–glass interfaces (i.e., Rwtand Rwg,
respectively)
Ck1=Rwt
Rwg=Z−Zw
Z+Zw
Zg+Zw
Zg−Zw
(34)
which yields
Z=Zw
1+Ck1
Rwg
1−Ck1
Rwg
.(35)
In (34) and (35), Zwand Zgstand for the known acoustic
impedance of water and glass, respectively.
To estimate the attenuation, we note that βk2is the attenua-
tion coefficient suffered by the signal through round trip travel
into a tissue that has a thickness of d, therefore,
α=βk2
2d=real(log(λk2))
2df(36)
which is then converted to dB/cm/MHz.
First principles also provide an estimate of Zbased on Ck2
using
Ck2=2Z
Zw+Z
Zg−Z
Zg+Z
2Zw
Zw+Z
Zw+Zg
Zw−Zg(37)
which results because the tissue–glass echo propagates from
water to tissue, reflects at the glass–tissue interface, and
propagates back from tissue to water. The last term in (37) also
appears in (34) from the division by S0and is the reciprocal
of the water–glass pressure reflection coefficient.
Equation (37) can be rewritten as
δZ3+[δ(2Zw+Zg)+1]Z2+δZ2
w+2ZwZg−ZgZ
+δZ2
wZg=0 (38)
where
δ=Ck2Zg−Zw
4Zw(Zg+Zw).(39)
Equation (38) is a third-degree polynomial, which is solved
using closed-form equations to provide three roots. Typically,
finding the correct root is straightforward, because the correct
root is close to the acoustic impedance value of water. One of
the other two roots is usually smaller than one MRayl and the
last one greater than three MRayl.
Initial tests with measured signals showed in some cases that
the attenuation βk1for the first signal (sk1), which originates
only from ultrasound paths in water (see Fig. 1), is not negli-
gible (i.e., βk1= 0). Hence, assuming that the attenuation in
water is negligible at frequencies of 250 MHz and higher can
lead to an inaccurate estimate of the acoustic impedance (Z).
To obtain an attenuation-corrected estimate of the first signal
amplitude (i.e., ˆ
Ck1), we used
ˆ
Ck1=Ck1|λk1|
fm
f=Ck1exp(βk1fm)(40)
where fmis the center frequency of the transducer. Equa-
tion (40) is used to correct the amplitude of the first signal for
a potentially negative value of βk1by providing a corrected
value at the center frequency; ˆ
Ck1is then used instead of Ck1to
estimate Zin (35). [Note that because ˆ
C=Ck1when βk1=0,
(40) is always used.]
D. Nonlinear Autoregressive Inverse Model
This describes a refinement of the AR inverse model in the
case of acoustic attenuation that is not linear with frequency.
1) Affine Attenuation: We start from the following approx-
imation for attenuation:
α( f)=α0+α1f(41)
where α0is not equal to zero as assumed earlier. In this case,
simple algebraic manipulations can be used to estimate α0and
to establish that α1can be estimated using (36) because
Ck2Sk2(f)
S0(f)
=Ck2exp(2πf(βk2+j(tk2−t0)))
=Ck2exp(2π[2d(α0+α1f)]+2πfj(tk2−t0))
=Ck2exp(2π[2dα0])exp(2πf(2dα1+2πfj(tk2−t0)))
=C∗
k2exp 2πfβ∗
k2+j(tk2−t0) (42)
where
C∗
i2=Ci2exp(2π[2dα0])(43)
β∗
i2=2dα1.(44)
Therefore, (44) confirms that α1can be directly obtained
from (36). To obtain α0, we use (37) with the value of Zfound
from (35) to obtain Ci2. Because of the identity of (42), (20)
yields C∗
i2. Equation (43) then can estimate α0.
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ROHRBACH AND MAMOU: AR SIGNAL PROCESSING APPLIED TO HIGH-FREQUENCY ACOUSTIC MICROSCOPY OF SOFT TISSUES 2059
2) Power-Law Attenuation: Finally, although the affine law
of (41) is often used to obtain an attenuation approximation
when attenuation values are known over a finite bandwidth,
we consider the power-law model of attenuation as expressed
by
α( f)=σfη.(45)
To estimate σand η, we use the affine approach
over mdistinct frequency bandwidths (i.e., BWl∈
{BW1,BW2,...,BWm}) that are composed of frequencies
BW1=f1
1,..., f1
N1,BW2=f2
1,..., f2
N2,...,BWm=
fm
1,..., fm
Nm. This procedure yields αl
0and αl
1where the
superscript lspecifies which bandwidth was used to obtain
these estimates. Then, we derive the following equations for
all flincluded in each BW:
log(σ ) +ηlog(fl)=log αl
0+αl
1fl(46)
which can be written in the following matrix form:
M[log(σ ), η]=T(47)
where
M=
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⎜
⎜
⎜
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log(α1
0+α1
1f1
1)
.
.
.
log(α1
0+α1
1f1
N1)
.
.
.
log(αm
0+αm
1fm
1)
.
.
.
log(αm
0+αm
1fm
Nm)
⎞
⎟
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(48)
which is solved using a least-squares matrix formulation to
obtain σ[from log(σ )]andη.
In the current implementation, we used three bandwidths
(i.e., m=3): BW1was the full −20-dB bandwidth of the
transducer extending from f1to f2(i.e., the same as used
in the AR approach), BW2extended from f1to f2−1/11
(f2-f1), and BW3extended from f1to f2−2/11( f2-f1).
We limit our presentation below of results of the non-
linear AR (NLAR) approach only to results obtained using
the power-law attenuation model. Therefore, the remainder
of this paper the NLAR approach always is discussed in
terms of the power-law attenuation method described in this
section.
III. MATERIAL AND METHODS
Our group has designed, fabricated, tested, and applied a
working QAM system operating at frequencies ranging from
100 to 500 MHz [21], [24], and although all the methods
described in Section II are applicable over a wide range of
frequency, the remaining discussion pertains to the QAM
system described in [21] and equipped with a broadband
transducer operating a center frequency of 250 MHz.
TAB LE I I
SIMULATED PARAMETER VARIATIONS
A. Simulations
We conducted simulations to evaluate performance of the
inverse models. To mimic our experiments closely, we used
a measured reference signal obtained using the apparatus
described in Section III-B. Simulations consisted of setting
values for d,c,Z,η,andσand reconstructing the simulated
signal sSim(t)accordingly
sSim(t)=C1sSim
1(t)+C2sSim
2(t)+Cperts∗
0(ttpert )+N(t)
(49)
where the first two terms are the simulated reflections from
the water–tissue and tissue–glass interfaces. These two signals
are defined by the simulation parameters. The third term is a
“perturbation” term having a shape similar to the first two
terms, which can be turned off by selecting Cpert =0. The
last term is a white Gaussian noise of power 2with
=10(20 log10((ζ (s0)−SNR)/20)) (50)
where SNR is the signal-to-noise ratio expressing the dif-
ference in decibel between the amplitudes of the reference
signal and noise. The symbol ζexpresses the maximum
of the Hilbert transform of s0. Equation (49) depends on
a very large number of parameters; therefore, we limited
the range of parameter variations to experimentally relevant
values. We tested the effects of decreasing the signal-to-noise
ratio (SNR), decreasing the signal separation (i.e., sample
thickness, d), decreasing the amplitude of the first signal (i.e.,
acoustic impedance, Z), increasing the attenuation coefficient
[i.e., σ, see (45)], and increasing the frequency exponent (η).
Table II gives a summary of all parameter value ranges
used in the simulations. These ranges were selected to be
representative of realistic scenarios for QAM applications and
were based on preliminary tests to find the optimal range
between easily separable cases (i.e., large SNR, d,Z,smallσ,
and η=1) and difficult cases (i.e., small SNR, d,Z,largeσ,
and η>1). In each simulation scenario, the value of the
parameter under investigation was varied and all the remaining
parameter values were kept constant with SNR =60 dB,
d=8μm, Z=1.63 MRayl, c=1600 m/s, σ=10 dB/cm
at 250 MHz, Cpert =0, and η=1. To assess statistical
variations, each case was performed for 200 realizations of
N(t). This procedure required (7+6+10+6+6)×200 =7000
simulations.
To investigate the impact of a third signal, we varied Cpert
from 0 ·Cs1to 0.9·Cs1(increment 0.1·Cs1) and randomly
placed the third signal between s1and s2(i.e., t1≤tpert ≤t2).
This procedure ensured that Cpert wasalwayssmallerthen
Cs1and Cs2. The other acoustic parameter values were kept
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2060 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 65, NO. 11, NOVEMBER 2018
constant at SNR =60 dB, d=8μm, Z=1.63 MRayl,
c=1600 m/s, and σ=10 dB/cm.
To assess the performance of the inverse models, we cal-
culated the mean error and standard deviation of estimated
parameters (i.e., c,Z,α,σ,andη) with the simulated
parameter. In addition, we used Grubb’s [34] test to detect
and report outliers in terms of percentages to provide another
metric to compare the performance of the AR and Hozumi
model approaches in the simulation experiments.
B. Experiments
The inverse models were tested using experimental data,
which were collected with our previously described 250-MHz
QAM system [21], [24]. The device was equipped with a
250-MHz transducer and signals were digitized at 2.5 GHz
with 12-bit accuracy. The data were acquired from a 12-μm-
thin human lymph node and a 6-μm-thin section of a human
cornea. The thickness of the samples was chosen to be thin
enough for scattering effects to be strongly mitigated. The
specimens were raster scanned in 2-D with a 2-μm step size
in both directions. RF was data acquired at each location
and processed individually using the inverse models. 2-D
maps of d,c,Z, and attenuation were generated. Adjacent
3-μm-thin section was stained with Haemotoxylin and Eosin
(H&E) and digitally imaged at 20×to provide a reference
for tissue microstructure. Outliers in experimental data were
detected by using absolute thresholds for values of cand
Zparameters, because Grubb’s test could not be applied.
Thresholds were selected based on results of our previ-
ous studies [21], [24], [25], [30], [35]. Specifically, estimates
were rejected if c<1500, c>2200, Z<1.48, or Z>2.2.
IV. RESULTS
A. Simulations
1) General Simulations: Fig. 2 shows the results of parame-
ter variation and resulting error in estimating acoustic property
values, which are c(i.e., column 1), Z(i.e., column 2), and
α(i.e., column 3). Column 4 shows the percentage of outliers
for each model.
The inverse AR and Hozumi models performed well (i.e.,
average errors were approximately zero for all three properties
and small number of outliers) in the “easy” cases (i.e., large
SNR, d,andZand small σand η=1). Both methods were
stable in estimating c,Z,andαdown to 50-dB SNR, which
is lower than values found in our measurements (i.e., an SNR
∼60 dB was obtained in an experimental water–glass reflection
signal).
However, in the harder cases, standard deviations, bias, and
the number of outliers, of the Hozumi model were significantly
greater than those of the AR model. This difference in the
performance of the two inverse models is particularly, apparent
for less separated signals (i.e., variation in d, second row
Fig. 2) and for small amplitudes of the first signal (i.e.,
variation of Z, third row Fig. 2). The Hozumi model is unable
to estimate cand Zwhen dis smaller than 6 μmandZ
is smaller than 1.56 MRayl. (In Figs. 2–4, the absence of a
data point and error bars indicate 100% outlier as shown in
column 4, i.e., the two parameters could not be estimated and
the two signals could not be detected in all 200 cases.) The AR
model remains stable (i.e., chas an average error of ∼0m/s
and Zhas an average error of ∼0 MRayl and the standard
deviation in cis <10 m/s and in Zis <0.05 MRayl) down to
adof 4 μmandaZof 1.52 MRayl.
Simulated variations in αhad a limited impact on estimates
of cand Z(fourth row of Fig. 2), although the Hozumi model
showed a trend toward increasing negative bias in cand Z
when αincreases. In addition, results clearly indicate that the
NLAR and AR models significantly outperform the Hozumi
model in estimating α. The simulation results also indicate
that the NLAR model yielded smaller standard deviation (even
though the simulated attenuation was linear). Furthermore,
the AR and NLAR models did not produce any outliers,
whereas the Hozumi model produced an average outlier per-
centage of approximately 7%.
2) Nonlinear Attenuation Results: Simulating nonlinear
attenuation (i.e., η>1) had no impact on estimates of
cand Z(last row of Fig. 2) obtained using the AR and
Hozumi models. However, an important impact on estimating
αwas visible with the Hozumi and AR models showing a
strongly increasing positive bias when ηincreased. However,
when αwas estimated using a power-law model for attenuation
estimation [see (45)] the NLAR model produced satisfactory
estimates of αup to η=1.4.
To investigate the NLAR model performance further,
Fig. 3(a)–(d) shows errors in ηestimates as a function of
simulated SNR, d,Z,andα, respectively, η=1 (black) and
η=1.2 (red). The results obtained for η=1 indicate that the
NLAR model was capable of performing satisfactorily of esti-
mating ηup to an SNR of 50 dB, d=5μm, Z=1.52 MRayl,
and α=20 dB/MHz/cm. Similarly, results obtained for
η=1.2 were nearly identical in terms of standard deviation,
but a small positive bias of 0.05 was observed in most
cases.
Finally, Fig. 3(e) shows errors in estimates of ηas a function
of simulated ηfrom 1 to 1.5. Results indicate very small
standard deviations (i.e., <0.02); however, a slowly increasing
positive bias occurs as the simulated value of ηincreases up
to a maximum bias of 0.13 when the simulated ηwas 1.5.
3) Perturbation Signal Results: Estimation results based
on simulations that included a perturbation signal showed
that, as the amplitude of the perturbation signal increased,
estimation errors increased for all four parameters and for
all models (Fig. 4). However, the AR model generally out-
performed the Hozumi model. Nevertheless, the impact of
a perturbation signal on estimates of cand Zwas small
[Fig. 4(a) and (b)] with nearly no bias and with standard
deviations smaller than 5 m/s and 0.05 MRayl for all models.
The error in Zwas nearly zero for all tested amplitudes of
Cpert when the AR model was used, whereas it increased with
Cpert when the Hozumi model is used.
The existence of a perturbation signal had an important
effect on estimates of αand η[Fig. 4(c) and (d)]. For α,theAR
models (i.e., NLAR and AR) significantly outperformed the
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ROHRBACH AND MAMOU: AR SIGNAL PROCESSING APPLIED TO HIGH-FREQUENCY ACOUSTIC MICROSCOPY OF SOFT TISSUES 2061
Fig. 2. Results of simulations. Columns 1–3: error of the estimated acoustic property [i.e., speed of sound (c), acoustic impedance (Z), and attenuation
coefficient (α)]. Rows: different parameter-set variations for SNR, separation (i.e., d, thickness of the specimen), amplitude of the first signal (i.e., Z),
attenuation coefficient (i.e., α), and attenuation exponent (i.e., η). Blue error bars: error and standard deviation of the Hozumi-based estimation. Black: error
and standard deviation of AR-based estimation. Gray bars: results if αis estimated using the nonlinear estimation procedure [see (45)]. Third column: outliers
(in %) that were removed from the parameter estimation. Mean and standard deviations were obtained after outliers were removed. Blue and gray bars show
results for the Hozumi and AR model, respectively.
Hozumi model. In fact, the error bars shown in Fig. 4(c) in the
case of the Hozumi model were greater than 0.5 dB/MHz/cm if
Cpert was greater than 0.2, essentially making αestimates unre-
liable. Finally, ηestimates obtained using the NLAR model
showed an overall trend toward increasing standard deviation
as Cpert increased, but the standard deviation remained below
0.1 up to Cpert =0.7 except for the outlier result obtained for
Cpert =0.6.
In summary, the existence of a perturbation signal has
limited effects on estimates of cand Z, but strong effects on
attenuation parameters. Overall, the AR and NLAR models
perform much better than the Hozumi model by producing
much smaller standard deviation and more reliable αestimates
for nearly all values of Cpert.
4) Illustrative Simulation Fits: Fig. 5 shows representative
simulated and fit signals [Fig. 5(a)–(f)] using the Hozumi and
AR inverse models, respectively. The depicted signals are for
representative cases of the parameter variation experiments
with SNR =50 dB [Fig. 5(a) and (b)], d=6μm [Fig. 5(c)
and (d)], and Z=1.56 MRayl [Fig. 5(e) and (f)]. These
cases were selected at thresholds where the Hozumi or AR
models start to fail more frequently (as illustrated in Fig. 2
rows 1–3). Both models show moderately good fits even in
the low SNR case [Fig. 5(a) and (b)]. However, the AR
model shows a smaller estimation error, in particular for
the more challenging case with a small sample thickness
[i.e., d=6μm, Fig. 5(c) and (d)]. Although the fit looks
reasonable, the errors in estimating Zand care 25% and
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2062 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 65, NO. 11, NOVEMBER 2018
Fig. 3. Simulation results showing estimation error and standard deviation of exponent η[see (45)] for varying (a) SNR, (b) d,(c)Z,(d)α,and(e)ηusing
the NLAR model.
16% of an assumed natural variations of 1.5–1.9 MRayl and
1500–1800 m/s in soft biological tissues, respectively. Similar
results were observed for the variations in signal amplitude
[Fig. 5(e) and (f)] with an underestimation of Z.
B. Experiments
1) Lymph-Node Results: Figs. 6–8 depict parameter maps of
a 12-μm lymph-node section. Parameter maps obtained with
both models show similar features and allow distinguishing
between different tissue types [i.e., fibrous tissue of the
capsule (Ca) and lymph-node parenchyma tissue (LT)]. For
example, Z,c,andαvalues were significantly higher in Ca
compared to LT (T-test, p<0.01). Table III summarizes
average values for Ca and LT estimated using the Hozumi
and AR models. Average values of cand Zestimated with
the Hozumi model are slightly lower than those estimated
using the AR model. On average, αvalues are higher when
estimated with the AR model. However, all differences are
within the standard deviations of the parameter variations. Dif-
ferences are apparent in the structural features. In particular,
the variation of c[Figs. 6(e) and (f) and 7(c) and (d)] is
more noticeable in images based on the AR model than those
based on the Hozumi model, and the images based on the AR
model more clearly depict anatomic structures matching those
visible in the histology images [Fig. 6 (b)]. Note that QAM
parameter maps and H&E stained images are obtained from
adjacent but different slides, which can result in dissimilarities
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ROHRBACH AND MAMOU: AR SIGNAL PROCESSING APPLIED TO HIGH-FREQUENCY ACOUSTIC MICROSCOPY OF SOFT TISSUES 2063
Fig. 4. Simulations results showing estimation error and standard deviation of (a) c,(b)Z,(c)α, and (d) exponent ηfor varying amplitude of the perturbation
signal (Cpert).
TABLE III
ACOUSTIC PARAMETERS OF LYMPHOCYTE TISSUE AND Ca OF LYMPH
NODES AND Ep AND St TISSUE OF CORNEA SAMPLES ESTIMATED
USING THE HOZUMI AND THE AR MODEL,RESPECTIVELY
between H&E and QAM images. The color maps in Fig. 6
are optimized to highlight features in the fibrous tissue of
the Ca, and Fig. 7 optimizes the color maps to compare the
Hozumi and AR models in LT. No difference in the number
of outliers was observed between the Hozumi and AR models.
Parameter maps of σand ηgenerated using the NLAR model
are shown in Fig. 8. The contrast between Ca and LT is weak
for σ. Values for ηshow no significant difference between
Ca and LT.
2) Cornea Results: The parameter maps of the 6-μm cornea
sample are shown in Fig. 9 and the average values are
summarized in Table III. Two tissue types [i.e., epithelium (Ep)
and stroma (St)] were identified with both models. Differences
in acoustic properties and the structural contrast between the
Hozumi and AR models are evident. They are particularly
apparent in St. The AR model shows higher c,Z,andαvalues.
Furthermore, the features follow the collagen fiber orientation
as seen in the H&E stained histology image [Fig. 9(b)] and
the Hozumi model failed in additional locations (i.e., white
areas in the parameter map) compared to the AR model. The
values of c,Z,andαin the Ep are only slightly lower in the
Hozumi-based estimates.
3) Illustrative Experimental Fits: The findings mentioned
earlier are confirmed by comparing the fit models with
the measured signals as shown in Fig. 5(g)–(j). The fit
amplitudes of the first signals (i.e., reflections from the
tissue–water interface) are underestimated when the Hozumi
model is used, which leads to lower estimated Zvalues.
Lower Zvalues were also observed in experimental results
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2064 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 65, NO. 11, NOVEMBER 2018
Fig. 5. (a)–(f) Simulated and (g)–(h) measured signals (i.e., black) together with the fit signals (i.e., red dotted) using the Hozumi model (i.e., a, c,e,g,
and i) and AR model (i.e., b, d, f, h, and j), respectively. (a) and (b) Signal with a low SNR (i.e., simulated SNR =40 dB, c=1600 m/s, Z=1.63 MRayl,
and d=8μm). (c) and (d) Signal with small d(i.e., simulated d=6μm, c=1600 m/s, and Z=1.63 MRayl). (e) and (f) Signal with small Z(i.e.,
Z=1.56 MRayl, c=1600 m/s, and d=8μm). (g) and (h) Measured signal of the Lymph node sample. (i) and (j) Measured signal from the cornea sample.
Estimated parameters of the signals are given in the figures upper right cornea.
[Figs. 6(e) and (f) and 7(c) and (d)], and the number of
outliers was significantly higher in the thin cornea samples
[white pixels in Fig. 9(c), (e), and (g)]. Furthermore, simu-
lations suggest that nonlinear attenuation (i.e., η>1) can
cause an overestimation of αif the Hozumi model is used,
as shown in Fig. 2 column 3 and row 5. The experimental
data show ηvalues larger than 1 for Ca and LT (see Table III).
The Hozumi model shows higher attenuation values compared
to the average αvalues obtained using the AR model. How-
ever, the St tissue showed an ηvalue of approximately 1, and
in this case, attenuation estimated with the AR model was
almost twice as high when compared to the Hozumi model.
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ROHRBACH AND MAMOU: AR SIGNAL PROCESSING APPLIED TO HIGH-FREQUENCY ACOUSTIC MICROSCOPY OF SOFT TISSUES 2065
Fig. 6. QAM-amplitude map derived from the maximum amplitude of (a) envelope signals and (b) H&E stained histology map of a human lymph node.
(c), (e), and (g) Parameter maps generated using the Hozumi model. (d), (f), and (h) Parameter maps generated using the AR model. The second row
(i.e., c and d) plots Z, the third (i.e., e and f) plots c, and the last row (i.e., g and h) shows α. Two tissue types are identified in the histology section, which
are Ca and lymphocyte tissue. (c)–(h) are from ROI1.
C. Results Summary
Overall experiment and simulation results are consistent.
For challenging cases (e.g., small d,smallZ,orlargeα),
the Hozumi model underestimates Z(Fig.2,column2and
rows 2 and 3) and the percentage of outliers is larger than
that of the AR model (Fig. 2 column 4, rows 2 and 3).
Simulations and experimental results show that both methods
perform equally well in well-separated signals with low, linear
attenuation and high Zcontrast between the tissue and the
coupling medium. However, the AR model outperforms the
Hozumi model in terms of parameter estimation accuracy
and stability (i.e., parameter error variation and number of
outliers) if 1) the signals overlap (e.g., because of small sample
thickness or high c), 2) the contrast in Zbetween the tissue
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2066 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 65, NO. 11, NOVEMBER 2018
Fig. 7. (a), (c), and (e) Parameter maps from ROI2 of Fig. 6(a) generated using the Hozumi model. (b), (d), and (f) Parameter maps generated using the AR
model. The first row (i.e., a and b) plots Z, the second (i.e., c and d) plots c, and the last row (i.e., e and f) shows α. The depicted tissue type is lymphocyte
tissue.
and the coupling medium decreases (i.e., smaller amplitude of
the first signal), 3) the attenuation increases, and 4) in case of
nonlinear attenuation (i.e., η>1).
V. DISCUSSION AND CONCLUSION
In this study, we investigated the application of an AR
inverse model for estimating acoustical properties of thin,
soft-tissue sections at microscopic resolutions. Although AR
methods are well known in signal processing for appli-
cations ranging from filtering and denoising to parameter
estimation [36]–[38], the suitability of such approaches for
QAM applications was not investigated at the level of details
described here. In addition, our extension to better deal with
power-law attenuation and the elegant use of the Cadzow
denoising have never been investigated and provide significant
improvements in QAM imaging. The proposed AR model is
similar to Prony’s method, which was successfully applied
in ultrasound research to separate fast and slow waves as
observed in ultrasound through-transmission experiments of
bone specimens [39]. Furthermore, the inversion method used
to obtain the modes of the AR process based on the zeros
of a polynomial [see. (18)], also exists in signal processing
literature and is often termed annihilating filter or polynomial
and has applications in sparsity-based methods [40].
The AR and Hozumi model are ultrasound-frequency-
spectrum approaches. However, time-domain methods were
also suggested in SAM and QAM applications [41], [42] and
work well when the two signals are completely separated in
time. However, the spectral methods (e.g., Hozumi, AR, and
cepstrum [41]) are the only methods available to separate the
two signals when they overlap in time. In fact, our results
(Figs. 2 and 5) demonstrate that the AR approach is far
superior when the signals overlap significantly.
The most widely used inverse methods for QAM-based
thin-section assessment remain those based on the approach
suggested by Hozumi et al. [3], [5], [6]. However, the Hozumi
approach has limitations. For example, it only models order
two signals, strongly depends on the transducer bandwidth,
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ROHRBACH AND MAMOU: AR SIGNAL PROCESSING APPLIED TO HIGH-FREQUENCY ACOUSTIC MICROSCOPY OF SOFT TISSUES 2067
Fig. 8. Parameter maps of σ[(a), (c)] and η[(b), (d)] of a human lymph node of (a) and (b) ROI1 and (c) and (d) ROI2 of Fig. 6(a) derived using the
NLAR-model.
relies on peak detection in the frequency domain, and does not
allow estimating nonlinear attenuation. Therefore, the moti-
vation of this study was to find an algorithm that improves
performance for tissues with 1) acoustic impedance close
to water (i.e., signals with small amplitudes and low SNR),
2) strongly overlapping signals, 3) high attenuation, and
4) nonlinear attenuation. Our simulation results show better
performance (e.g., a smaller parameter estimation error) and
better stability (e.g., a smaller variation of estimation error
and fewer outliers) for our AR inverse model approach in all
four investigated cases (Fig. 2) when compared to the Hozumi
approach. Particularly relevant to soft-tissue measurements is
the better performance of the AR model when Zis close to
the impedance of water (i.e., when the amplitude of the first
signal is small and the SNR is low). Furthermore, in very
thin sections that produce overlapping signals, the AR model
is more stable. This feature is a very important attribute
for soft-tissue measurements because it allows performing
acoustic microscopy and optical microscopy on the same
sample without sacrificing histology image quality caused by
thick samples.
Interestingly, our results suggest that nonlinear attenuation
in ranges typical for soft tissues has only a small impact
on cand Zestimates for the Hozumi and AR models. The
Hozumi model shows only a small bias for Zestimates
with increasing attenuation. Nevertheless, the Hozumi inverse
model has difficulties estimating αwith increasing attenuation
and η(Fig. 2). This occurs because the Hozumi model esti-
mates Zand αvalues only from the resonance peak amplitudes
in the frequency domain [21], rather than from the normalized
spectrum over the entire bandwidth as it is done for the AR
model. The peak locations and amplitudes also are affected by
the attenuation which can lead to estimation errors (Fig. 10).
The results from ex vivo tissue experiments appear to be
consistent with those obtained from simulations and demon-
strate a better performance of the AR model as follows.
1) Analysis of the parameter maps of the very thin sections
(e.g., the cornea sample, Fig. 9) that were derived using
the AR model revealed anatomical structures that match
histology more closely than parameter maps generated
by the Hozumi model.
2) The Hozumi model failed to separate the two signals in
more cases than the AR model as indicated by the white
areas (i.e., outliers) in Fig 9.
3) In the thicker, 12-μm lymph-node sections, the parame-
ter maps of the AR model showed enhanced structural
features in the Zand cparameter maps, which is a result
of greater robustness and sensitivity of the AR model to
small parameter variations, as shown in the simulation
results. The simulations indicate that, in low-Zand
low-dconditions, the AR model produces lower vari-
ation in parameter estimation errors and still provides
reliable results when the Hozumi model completely fails
(see Fig. 2).
4) If a third signal is introduced (Fig. 4), then the AR model
performs better than the Hozumi model, which may be
one reason for the better contrast of the parameter maps
in the 12-μm sections (Figs. 6–8) obtained from the
experimental data.
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2068 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 65, NO. 11, NOVEMBER 2018
Fig. 9. (a) Amplitude map and (b) histology map of a pig cornea. (c), (e), and (g) Parameter maps generated using the Hozumi model. (d), (f), and (h)
Parameter maps generated using the AR model. The second row (i.e., c and d) plot Z, the third (i.e., e and f) plot c, and the last row (i.e., g and h) shows α.
Two tissue types are identified on the histology section, which are Ep and St tissue.
Perturbations can arise from scattering or strong reflections
inside the tissue. The impact of scattering is assumed to be
small because echoes from the scattering signals would be
about one order of magnitude weaker than the sample–tissue
and tissue–substrate interface signals, which essentially orig-
inate from planar reflectors. Scattering is also mitigated by
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ROHRBACH AND MAMOU: AR SIGNAL PROCESSING APPLIED TO HIGH-FREQUENCY ACOUSTIC MICROSCOPY OF SOFT TISSUES 2069
Fig. 10. Normalized spectra (N(f)) of the standard signal (i.e., “Easy,” black
spectra with SNR =60 dB, d=8μm, Z=1.63 MRayl, c=1600 m/s,
σ=10 dB/cm at 250 MHz, Cpert =0andη=1) and spectra with (a) varied
d=6μ(blue curve) and Z=1.53 MRayl (red curve), (b) varied α=
30 dB/cm (blue curve) and η=1.5 (red curve), and (c) varied SNR =40 dB
(blue curve) and the Cadzow denoised spectra (red curve).
the use of thin samples (i.e., with thicknesses of one to two
wavelength). In fact, the extensive experimental data from
our previous studies [20], [24]–[26] did not show any large
signals between the signals from the two planar surfaces,
which led to the assumption that the reflections from the
water–tissue and tissue–substrate reflections have the largest
amplitudes. In different scenarios, the AR approach could be
tailored to pick the correct two signals out of several signals by
using a criterion other than signal amplitude (e.g., TOF and
attenuation), but the existence of a prominent middle signal
also would mean that our forward model (i.e., assuming a
single layer) is most likely violated, and estimated parameters
would almost certainly need to be excluded. Multilayer media
could also be taken into account using a different version of
AR model.
A striking advantage of the proposed AR approach is its
ability to estimate nonlinear attenuation based on a power law
model, which has not been reported to the best of the authors’
knowledge. If the power law model (i.e., the NLAR model)
is used, estimating αwas significantly improved compared to
the Hozumi model, and the AR model shows stable results
(i.e., average error in estimating α∼0 dB/MHz/cm and
error variation <0.1 dB/MHz/cm) over the entire range
of simulated values and also can estimate the exponent η
correctly (i.e., error variation of η≤0.4) over the simu-
lated parameter range (Fig. 3). Biological tissues typically
exhibit nonlinear attenuation [31], but assessing ultrasound
attenuation at microscopic resolutions has been challenging.
However, our experimental results demonstrate the high value
of the nonlinear attenuation parameter of the lymph-node
sample (Fig. 8). These results demonstrate the usefulness of
the AR method, but they also indicate that further studies
on larger sample cohorts or phantoms to investigate and
improve the estimation of ηand σare warranted. Accu-
rate measurements of the nonlinear attenuation would also
allow to assess the dispersion effects that typically yield
an increase in speed of sound with increasing ultrasound
frequency. Using Kramers–Kronig causality relationship, dis-
persion can be exactly quantified using the derivative of the
frequency-dependent attenuation [43]. In the future studies,
we will investigate the frequency dependence of the speed
of sound using QAM.
In addition, higher quality information provided by the
AR model approach is important for modeling needs
and for ultrasound applications at lower frequencies. Cur-
rently, QAM is the only method that can provide multiple
acoustical and mechanical properties at fine resolutions and
over large-scale areas as is required for numerical mod-
eling of sound propagation [20], [44]–[46] or finite-element
modeling [47]–[51]. Such properties cannot be assessed using
conventional optical microscopy. Use of advanced computer
simulations is rising, which allows investigating complex
phenomena that are difficult or cannot be examined exper-
imentally. However, the results of these simulations will
only be as accurate as the underlying models whose accu-
racy, in turn, depends on realistic input data. Furthermore,
in many quantitative ultrasound (QUS) applications, assump-
tions are made about the acoustic attenuation of tissue to
correct QUS-parameter estimation [52], [53]. We hope that
QAM will provide better estimates of common tissues to
improve novel QUS methods using acoustic-impedance-map
methods [54], [55], for example. The methods described in
this paper to separate ultrasound signals and to estimate
acoustical parameter values also are suitable for applications
at lower frequencies.
This paper compares a new AR model with the current
standard method (i.e., the Hozumi model). Although some of
the improved performance of the AR model may be directly
related to implementation details, the AR model has some
general advantages that are illustrated in Fig. 10, which shows
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2070 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 65, NO. 11, NOVEMBER 2018
representative normalized spectra for different simulated sig-
nals. The Hozumi model relies on the detection of resonance
peaks in the frequency domain. In well-separated signals
[Fig. 10(a), easy] with strong amplitudes, the normalized
spectra exhibit several peaks that are easily detected. In theory,
one maximum–minimum pair is sufficient to estimate dand c
values, but additional peaks increase robustness (i.e., decrease
bias and standard deviation) to noise by averaging estimates
obtained from all possible extremum pairs [21]. Therefore,
the Hozumi model is sensitive to the bandwidth of the QAM
system [Fig. 10(a), d], which determines the number of peaks
that can be analyzed. In contrast, the AR model uses the entire
normalized spectra and not only the peaks. Thus, the AR
model is more robust at the bandwidth limits (i.e., small d),
as indicated by the simulation results. Similarly, the Hozumi
model depends strongly on the resonance peak amplitudes.
If the amplitudes are too small, peak detection methods may
fail to detect the resonance peaks accurately. This is the
case if contrast in Zbetween tissue and coupling medium
is very small [Fig. 10(a), Z]. Strong attenuation (linear and
nonlinear) produces similar effects on the peak detection
[Fig. 10(b)] and explains the better performance of the AR
model as observed in the simulation results. Furthermore,
although not discussed in great detail here in the interest of
space, the performance of the AR model is strongly enhanced
by the Cadzow denoising. Fig. 10(c) shows an example of
a noisy signal before [Fig. 10(c), SNR] and after denoising
[Fig. 10(c), Cadzow] compared to a noise-free signal.
To summarize, our AR model approach for QAM-based
parameter estimation showed better performance for three
highly relevant scenarios: 1) tissues with acoustic impedance
close to water; 2) tissue samples yielding overlapping signals;
and 3) tissues with nonlinear attenuation. We demonstrated
in experiments and simulations the improved robustness and
precision of acoustic parameter estimation of the AR model
(e.g., smaller variation and bias of errors for samples with
Z≤1.56 MRayl, d≤6μm, α≥5 dB/MHz/cm, and
η≥1). Furthermore, the AR method is easily implemented
and allows direct estimation of all acoustic properties includ-
ing those related to nonlinear attenuation. Another advantage
is the unique ability of the AR model to remove spurious
signals, such as perturbation signals (Fig. 4) and still provide
accurate estimates of acoustic properties. Our ex vivo exper-
iments demonstrate that average acoustic parameter values
are similar for both methods in 12-μm sections. However,
the AR model seems to show more contrast for thinner tissue
sections. This would allow to compare QAM measurements
at different frequencies (e.g., 250 and 500 MHz) on the same
tissue samples. Furthermore, we believe that our new AR
model approach, in general, can markedly improve current
QAM technology. Nevertheless, more research is necessary
to exploit the AR method fully and to assess its suitability
for use in a broader frequency range and in applications
involving nonbiological materials. Future studies will test the
AR method using tissue-mimicking phantoms to validate these
QAM approaches. Currently, we are testing the approach with
our 500-MHz transducer [21].
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2072 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 65, NO. 11, NOVEMBER 2018
Daniel Rohrbach (M’14) was born in Großröhrs-
dorf, Germany, in 1984. He received the German
Diplom degree in bioinformatics from the Martin
Luther University of Halle-Wittenberg, Halle,
Germany, in 2010, and the Dr.rer.nat. degree
in experimental biophysics from the Berlin-
Brandenburg School for Regenerative Therapies,
Humboldt University, Berlin, Germany, in 2013.
He is currently a member of the Research Staff
with the Lizzi Center for Biomedical Engineer-
ing, Riverside Research, New York, NY, USA.
His current research interests include tissue characterization using acoustic
microscopy and high-frequency quantitative ultrasound.
Jonathan Mamou (S’03–M’05–SM’11) graduated
from the Ecole Nationale Supérieure des
Télécommunications, Paris, France, in 2000. He
received the M.S. and Ph.D. degrees in electrical
engineering from the University of Illinois at
Urbana-Champaign, Urbana, IL, USA, in 2002 and
2005, respectively.
He is currently a Research Manager with the
F. L. Lizzi Center for Biomedical Engineering,
Riverside Research, New York, NY, USA.
He co-edited the book Quantitative Ultrasound
in Soft Tissues (Springer, 2013). His fields of interests include theoretical
aspects of ultrasonic scattering, acoustic microscopy, ultrasonic medical
imaging, ultrasound contrast agents, and biomedical image processing.
Dr. Mamou is a Fellow of the American Institute of Ultrasound in
Medicine (AIUM) and a member of the Acoustical Society of America.
He served as the Chair of the AIUM High-Frequency Clinical and Preclinical
Imaging Community of Practice. He is as an Associate Editor for Ultrasonic
Imaging and IEEE TRANSACTIONS ON ULTRASONICS,FERROELECTRICS,
AND FREQUENCY CONTROL. He is a reviewer for numerous journals.
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