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IEEE TransacTIons on UlTrasonIcs, FErroElEcTrIcs, and FrEqUEncy
conTrol, vol. 60, no. 11, novEmbEr 20132266
08853010/$25.00 2013 IEEE
Phase-Based Direct Average Strain Estimation for
Elastography
sharmin r. ara, Faisal mohsin, Farzana alam, sharmin akhtar
rupa, rayhana awwal, soo yeol lee, and md. Kamrul Hasan
AbstractIn this paper, a phase-based direct average strain
estimation method is developed. A mathematical model is pre-sented
to calculate axial strain directly from the phase of the zero-lag
cross-correlation function between the windowed pre-compression and
stretched post-compression analytic signals. Unlike phase-based
conventional strain estimators, for which strain is computed from
the displacement field, strain in this paper is computed in one
step using the secant algorithm by exploiting the direct
phasestrain relationship. To maintain strain continuity, instead of
using the instantaneous phase of the interrogative window alone, an
average phase function is defined using the phases of the
neighboring windows with the assumption that the strain is
essentially similar in a close physical proximity to the
interrogative window. This method accounts for the effect of
lateral shift but without requiring a prior estimate of the applied
strain. Moreover, the strain can be computed both in the
compression and relaxation phases of the applied pressure. The
performance of the proposed strain estimator is analyzed in terms
of the quality metrics elasto-graphic signal-to-noise ratio (SNRe),
elastographic contrast-to-noise ratio (CNRe), and mean structural
similarity (MSSIM), using a finite element modeling simulation
phantom. The re-sults reveal that the proposed method performs
satisfactorily in terms of all the three indices for up to 2.5%
applied strain. Comparative results using simulation and
experimental phan-tom data, and in vivo breast data of benign and
malignant masses also demonstrate that the strain image quality of
our method is better than the other reported techniques.
I. Introduction
Elastography is an imaging modality for noninvasive assessment
of tissue elasticity by measuring its degree of deformation under
the application of an external force. Tissue elasticity, a
mechanical characteristic which may
change under the influence of pathophysiologic processes, has
clinical benefit in the diagnostic evaluation of differ-ent
diseased organs [1][5]. Elastography provides a quan-titative
evaluation of the elastic parameter of tissue and promises
detection of pathological changes at the primary stage for
diagnosing breast cancer [6][8], prostate cancer [9], liver
cirrhosis [10], vascular plaques [11], and lymph node and thyroid
cancer [12], [13].
In quasi-static elastography, the time-domain post-compression
rF signal is modeled as a compressed and delayed version of the
pre-compression rF signal. To ascertain the displacement between
these two signals, which is eventually used to calculate strain,
the correla-tion of these pre- and post-compression signals are
ana-lyzed. algorithms based on this principle are categorized as
displacement-based strain estimation techniques. some of the
notable displacement-based strain estimators are time delay
estimation (TdE) [6], [8], [14], time delay es-timation with prior
estimate (TdPE) [15], and analytic minimization (am) [16]. TdE
fails at high strain because of decorrelation noise and is
unsuitable for real-time ap-plication because of its high
computational cost. TdPE is an improved version of TdE, in which
prior estimates from the neighboring windows are used for reducing
the search region of the correlation peak. at high strain, when the
correlation falls below a predefined threshold, TdPE switches to
TdE. TdPE, however, suffers from the noise introduced by the
gradient operator in calculating strain from the displacement
field, which is recognized as a ma-jor drawback of the
gradient-based algorithms. all of these windowing approaches must
trade-off between the good spatial resolution of small windows and
the accuracy of large windows which help to reduce jitter errors
[16]. ana-lytic minimization (am) [16] uses an individual sample of
rF data, omitting the window-based analysis. The effect of
decorrelation noise is minimized in this real-time 2-d elastography
technique through the use of regularization terms. However, 2-d
analytic minimization (am2d) is highly sensitive to the optimal
setting of eight tuning pa-rameters and attenuation effects.
another group of algo-rithms known as direct strain estimators
[17][20] employ global or local adaptive scaling of the
post-compression signal to estimate strain directly from the
stretching fac-tor. strain estimation using the neighborhood in
[20] was based on a Fourier spectrum equalization technique where
the mean strain at the interrogative window was com-puted by
minimizing, with respect to stretching factor, a cost function
derived from the exponentially weighted windowed segments in both
the axial and lateral direc-
manuscript received February 26, 2013; accepted august 7, 2013.
This work was supported by the Higher Education quality Enhancement
Program (HEqEP), University Grants commission (UGc)
(cP#96/bUET/Win-2/sT(EEE)/2010), bangladesh, and in part by
national research Foundation of Korea grant funded by the Korean
government (2009-0078310).
s. r. ara, F. mohsin, and m. K. Hasan are with the department of
Electrical and Electronic Engineering, bangladesh University of
Engi-neering and Technology (bUET), dhaka, bangladesh (e-mail:
[email protected]).
F. alam is with the department of radiology and Imaging,
bang-abandhu sheikh mujib medical University (bsmmU), dhaka,
bangla-desh.
s. a. rupa is with the department of radiology and Imaging, Enam
medical college and Hospital, savar, dhaka, bangladesh.
r. awwal is with the department of Plastic surgery, dhaka
medical college, dhaka, bangladesh.
s. y. lee and m. K. Hasan are with the department of biomedi-cal
Engineering, Kyung Hee University, Kyungki, south Korea (e-mail:
[email protected]).
doI http://dx.doi.org/10.1109/TUFFc.2013.2825
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ara et al.: phase-based direct average strain estimation for
elastography 2267
tions. one of the limitations of this algorithm is the
ex-haustive search region for the stretching factor. moreover, this
method requires a prior knowledge of the strain, the deciding
factor for the range of the search region, and/or the global
stretching. because these methods do not suffer from the effect of
gradient noise, image quality improves at the cost of computational
expense.
Phase-zero methods are found to be suitable for real-time
elastography because they offer high computational efficiency [21].
Phase root seeking (Prs) [22] was the first published phase-based
method for displacement mapping. The simplicity of this algorithm
lies in the fact that it seeks the root of phase from a
phase-versus-displacement function employing the newtonraphson
algorithm. ma-jor sources of error in the Prs algorithm are
incorrect dis-placement estimation using an erroneous estimate of
the phase-zero that results from poor correlation between the
concerned windows, and propagation of this error through the
remainder of a scan-line. combined autocorrelation method (cam)
[23] is another real-time two-step algo-rithm in which the phase of
the maximum of envelope autocorrelation is used to calculate the
axial and lateral displacements. a recent phase-based algorithm
known as weighted phase separation (WPs) [24] determines the
dis-placement location along with the displacement estimate to
eliminate the need of signal scaling. strain is finally estimated
from the displacement field using least-squares [25] or
gradient-based algorithms. These algorithms, therefore, suffer from
the noise amplification associated with the gradient operation and
are usually not robust to decorrelation noise that results from
non-axial motion. Until now, to the best of our knowledge, the
phase-based strain estimation techniques [22], [23], [26], [27],
determine the phase of zero-lag cross-correlation for a single pair
of pre- and post-compression windows. Using these methods,
displacement or strain continuity cannot be ensured. a smoothing
filter is generally used as a post-processor for restoring
continuity or removing the noise from the dis-placement or strain
field. direct estimation of strain in the cross-correlation phase
domain is an open problem, because all of these techniques relied
on the variants of displacement estimate.
In this paper, a new method for direct average strain estimation
from the phase of zero-lag cross-correlation function of the
pre-compression and the correspond-ing stretched post-compression
rF echoes is proposed. The novelty of this phase-based direct
strain estimation (PdsE) method lies in two facts. First, it
searches for the root of the phase using a phase-versus-strain
function employing the secant algorithm. Unlike other phase-based
methods, phase here is considered as a function of strain instead
of the displacement, which opens up the option for adaptive
stretching in the axial direction. lateral dis-placement is also
adjusted adaptively using this scaling factor in conjunction with
the Poissons ratio. second, the strain calculation of the
interrogative window includes the effect of phase of the
neighboring windows without requir-ing any prior estimate of it.
This inherent use of neighbor-
hood results in a continuity of strain in the axial direction.
The use of the secant-type phase-root seeking algorithm usually
makes the proposed PdsE converge in 3 to 5 iter-ations, unlike the
time- and spectral-domain direct strain estimation techniques
[17][20], for which the number of iterations depends on the
resolution of the adopted total search technique. The PdsE can
estimate strain both in the compression and relaxation phase
without prior in-formation regarding the strain. all of these
features make PdsE an attractive candidate for freehand
quasi-static elastography. The performance of the algorithm is
evalu-ated using a finite element modeling (FEm) simulation
phantom, an experimental phantom, and in vivo patient data.
II. Proposed method
A. The Signal Model
The distribution of axial strain in a block of tissue re-sulting
from an external force applied vertically on its top can be
calculated from the backscattered rF echo sig-nals received before
and after compression. because of the axial compression, in
general, the tissue will experience a three-dimensional
displacement in the axial, lateral, and out-of-plane directions. If
the out-of-plane displacement is neglected, then the pre- and
post-compression signal can be modeled as
r t s t p t1( ) = ( ) ( ) (1)
r t st
a x y t p t2 0( ) = ( , ) ( )
, (2)
where r1(t) and r2(t) are the pre- and post-compression rF echo
signals acquired by the transducer, respectively; s(t) is the 1-d
ultrasound scattering function; p(t) denotes the point spread
function (PsF) or ultrasound system re-sponse; t0 denotes the time
shift; x and y are the axial and lateral coordinates; and a(x, y)
is the 2-d local stretching factor. If it is assumed that p(t/a(x,
y) t0) p(t) [17], [20], which holds reasonably up to a certain
value of strain and t0 0, then the post-compression signal can be
re-lated to the pre-compression signal as
r t rt
a x y t2 1 0( ) = ( , )
. (3)
If the lateral expansion is compensated by utilizing the
Poissons ratio, a(x, y) can be replaced by a(x). Then, a(x) can be
written as a function of the axial strain (x) of the tissue segment
as [28]
a x x( ) = 1 ( ) . (4)
If a(x) 1 and t0 0, then the post-compression signal can be
considered as a replica of the pre-compression sig-
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IEEE TransacTIons on UlTrasonIcs, FErroElEcTrIcs, and FrEqUEncy
conTrol, vol. 60, no. 11, novEmbEr 20132268
nal. In such a case, the cross-correlation function of the pre-
and post-compression analytic signals behaves almost like an
autocorrelation function, and tracking the phase or correlation
coefficient at zero-lag will suffice for checking the similarity
between these two signals. In the following, a(x) will be
abbreviated as a for readability.
The symbols and acronyms used in this paper are pre-sented in
Table I.
B. Phase-Based Direct Average Strain Estimation
Unlike the conventional phase-root seeking algorithms for
elastography which are basically displacement esti-mators, our goal
here is to introduce a phase-based di-rect strain estimation (PdsE)
technique with a built-in smoothing feature to reduce the
decorrelation noise, and also to get rid of the gradient noise
through direct esti-mation of strain utilizing the phase of the
zero-lag cross-correlation between the stretched version of the
post-com-pression and the corresponding pre-compression analytic
signals. The continuity of strain is ensured through the inclusion
of neighborhood rF echo segments in the aver-age phase calculation,
in contrast to conventional phase-root seeking algorithms in which
only the phase associated with the interrogative window is
used.
1) Basic Concept: The pre- and post-compression rF echo signals
can be converted to analytic signals as
r t r t jr t R t e j t1 1 1 1 ( )( ) = ( ) ( ) = ( ) 1+ +
(5)
r t rta t jr
ta t R t e
j t2 1 0 1 0 2
( )( ) = = ( ) 2+
+
, (6)
where r t1( ) is the Hilbert transform of r1(t); 1(t) and 2(t)
are the phases of the analytic pre- and post-compression signals,
respectively; and R1(t) and R2(t) are the enve-lopes of the pre-
and post-compression rF signals, respec-tively. The
post-compression analytic signal in (6) stretched by a factor ,
denoted by r t2 ( )+ , can be ex-pressed as
r t rta t jr
ta t R t e
j2 1 0 1 0 2( ) = = ( ) 2+
+
( )t , (7)
where 2( )t and R t2 ( ) are the phase and envelope of the
stretched post-compression analytic signal, respectively. The
cross-correlation function of r t1 ( )+ and r t2 ( )+ can be
written as
( ) = ( ), ( )
= ( ) ( )
1 2
1( )
2( )1 2
+
+
+ +
+r t r t
R t e R t ej t j t dtt, (8)
TablE I. symbols and acronyms.
symbols description
r1(t) Pre-compression rF echor2(t) Post-compression rF echo
stretching factorr1+(t) Pre-compression rF analytic signalr t2 ( )+
stretched post-compression rF analytic signalR1(t) Envelope of the
pre-compression rF analytic signalR t2 ( ) Envelope of the
stretched post-compression rF analytic signal() cross-correlation
function of the pre- and stretched post-compression rF analytic
signal() Zero-lag phase of the cross-correlation functionWl length
of rF echo segmentsWs Interwindow shift between two consecutive rF
echo segmentsNc number of scan lines in a single ultrasound image
Poissons ratio{La, Ll} neighborhood factors in the axial and
lateral directions, respectivelya Weighting factors of the
exponential weight function in both the axial and lateral
directions Wavelength of the rF signal at the center frequencyj
scan line indexk Window indexn Iteration indexavg(k) Estimated
average strain of the kth interrogative window using neighborhood
windowsavg( )j arithmetic mean strain of the jth scan lineam
analytic minimizationassE adaptive spectral strain estimationcnre
Elastographic contrast-to-noise ratioFEm Finite element
modelingssIm structural similaritymssIm mean structural
similaritynF neighborhood factorPrs Phase root seekingPsF Point
spread functionsnre Elastographic signal-to-noise ratioTGc
Time-gain-control
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elastography 2269
where is the cross-correlation lag. The phase ( ) of the
cross-correlation function ( ) with = 0, called the zero-lag phase,
can be obtained as
( ) = ( ) ( )1( )
2( )1 2arg . ( )R t e R t e tj t j t d (9)
If the effect of t0 is neglected in (7) and an appropriate
stretching factor can be estimated such that = a, then (0)|=a in
(8) represents the autocorrelation function of the analytic
pre-compression signal itself. because the autocorrelation function
is positive-real at zero lag, the phase, ()|=a will be zero. This
means that the un-known compression factor, and hence the strain,
can be estimated directly from the cross-correlation function phase
() by finding an appropriate stretching factor that makes () zero.
We assume here that there is no is-sue with phase wrapping, i.e.,
strain window length , where is the wavelength of the rF signal at
the center frequency of the transducer [21], [24].
2) Weighted Average Phase Using Neighborhood: In general, strain
throughout any arbitrary scan-line is not uniform because of tissue
inhomogeneity and/or nonuni-form distribution of the applied stress
field inside the tis-sue medium; however, it can be considered
invariant in a small interrogative window of the post-compression
rF data segment. The received discrete-time rF signals must
therefore be windowed to obtain an estimate of the local strain
from a pair of pre- and post-compression rF data segments that
represent the same part of the tissue. let U1(i, j) and U2(i, j)
denote, respectively, the pre- and post-compression rF data frames
of the same imaging plane in discrete format acquired by an
ultrasonic transducer. Here, i and j are the axial and lateral
indices, respectively. The corresponding pair of windowed rF
segments of the jth scan-line are denoted by r1(k, j) and r2(k, j),
where k is the widow index. The pre-compression windows are
se-lected at a regular interval in a given scan-line; however,
because of compression, the corresponding positions of the
post-compression windows become a function of the previ-ous strain
estimates [26]. The aligned and laterally- and axially-interpolated
corresponding pair of pre- and post-compression 1-d windowed rF
signals for the jth scan line that are to be used for local strain
estimation are given by
r k j U i j k W i k W W1 1( , ) = ( , ), 1 ( 1) ( 1)+ +s s l
(10)
r k j U i j2 2( , ) = ( , ) (11)
where
i i ki
=1
,, ,
={ otherwise (12)with
1 (1 ( )) (1 ( ))1 1=1
1
1=1
1
1+
k
k
k
k
k W i k W avg s avg s
+ l avgW k n(1 ( , 1)) (13)
jj j
j jN
j=
, 1
2 ( 1),
=
+ ( )
cavg otherwise.
(14)
Here, Wl denotes the window length for the pre-compres-sion
signal, Ws denotes the window shift, Nc is the total number of
columns in the scanned image, avg(k, n) is the weighted average
estimate of strain [which will be defined in (19), in the next
section] for the kth window of an arbi-trary scan-line j, avg( )j
is the arithmetic mean strain of the jth column, and [0.2, 0.5] is
the Poissons ratio. It has been assumed that the compression is
applied at the middle of the transducer appropriately so that the
center line corresponds to Nc/2. To consider the effect of lateral
expansion (see Fig. 1), j is adjusted adaptively according to (14).
This simple adaptive approach for compensating the motion in the
lateral direction eliminates the need for a 2-d search or lateral
strain estimation [29] and contrib-utes to the improvement in the
quality of the strain image. because the values of i and j in (13)
and (14) are usually fractional, a linear interpolation technique
is used to re-solve this problem. although the length of the kth
window of the post-compression signal is different from that of the
pre-compression signal, the number of samples within each of the
windows will be maintained the same with the help of a suitable
interpolation technique [19]. The signal represented by (11) is by
now adaptively aligned and lat-erally compensated and henceforth
will be used for calcu-lating the stretching factor [(k) = 1
avg(k)] by an adaptive phase-root seeking algorithm.
In this paper, the phase defined in (9) is called the
in-stantaneous phase, which is evaluated by comparing a sin-gle
pair of windows, without taking into effect the phase of the
neighboring windows. Unlike the conventional Prs approach and its
variants, the proposed PdsE algorithm inherently includes the
effect of neighborhood without any prior knowledge of the phase of
the neighboring windows
Fig. 1. Illustration of lateral shift resulting from applied
axial pressure in non-slip boundary conditions. In the selection of
a post-compression window, this shift has been adjusted according
to (14).
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IEEE TransacTIons on UlTrasonIcs, FErroElEcTrIcs, and FrEqUEncy
conTrol, vol. 60, no. 11, novEmbEr 20132270
to ensure strain continuity. because the youngs modu-lus (i.e.,
stiffness) of a biological tissue is almost similar to that of the
neighboring ones because of their physical proximity, phase-roots
of the zero-lag cross-correlation of the analytic pre-compression
and stretched post-compres-sion signals in the neighboring windows
should be similar too. This means that an average value of the
phase-root can be used for strain calculation instead of the
instanta-neous one, which is very much susceptible to noise.
There-fore, to ensure strain continuity and to obtain a direct
av-erage value of strain, the phase in this paper is calculated as
the weighted average of phases of the neighborhood, and is given
by
avgl
l
a
a
( , ) =( , ) ( , )
( , )
1 1 1 1==
1 1
11k jk j w k j
w k jj j Lj L
k k Lk L
+
+
jj j Lj L
k k Lk L
11 == +
+
l
l
a
a (15)
w k j e a k k j j( , ) =1 1 (| | | |)1 1 + , (16)
where (k, j) is the instantaneous phase of the (k, j)th win-dow
calculated as in (9), La and Ll are the neighborhood factors (nF)
in the axial and lateral directions, respec-tively, a is the
weighting factor in both directions, and w(k1, j1) is the
exponential weight of the neighboring win-dow (k1, j1). The greater
the difference between the index of the interrogative window and
the neighboring window, the lesser will be the weight. an
exponential weighting is desired because the correlation between
the interroga-tive and the neighborhood windows decreases
exponen-tially with the increase of distance of the neighborhood
[20]. The role of this weight is to ensure strain continuity within
the neighboring windows that have similar strain. note that the
interrogative window will have the high-est weight, equal to 1.
Fig. 2 illustrates the method of weighting in the neighborhood,
where two windows from each side of the interrogative window are
considered as the neighborhood, i.e., La = Ll = 2 (i.e., nF =
2).
The PdsE algorithm will be implemented without re-quiring any
prior estimate of strain, and unlike the total search technique
adopted in [20], an adaptive root finding algorithm can be
formulated for direct estimation of strain using the average phase
defined in (15).
3) Adaptive Phase-Root Seeking Algorithm: The prob-lem of strain
estimation in the PdsE has been reduced to the problem of finding
the root of avg(k, j) in (15) as a function of (k), the stretching
factor for the (k, j)th window of the post-compression signal. In
this work, we employ the secant method based on the same
generalized assumption of all numerical root finding algorithms,
which states that around its root, (k, j) can be approximated as
[26]
( , ) = ( , ) ( , ) ( , )k j m k j k j c k j+ , (17)
where (k, j) is the strain of the (k, j)th window. Unlike the
Prs technique [22], where the phase was assumed to be a linear
function of the displacement and t = d/dt was
approximated by 0 (the transducer center frequency), = d/d in
(17) cannot be approximated by a simple pa-rameter. as a result,
the secant method instead of the newtonraphson iteration has been
adopted, where a fi-nite difference is used to approximate the
derivative of phase with respect to strain. In (17), m(k, j)
represents the slope of the approximated line around the root of
the in-stantaneous phase of the interrogative window and c(k, j) is
the corresponding intercept. The neighboring windows will have a
similar relationship between phase and strain but with different
slope and intercept. To include the ef-fect of phase of the
neighboring windows in the calculation of strain, an average
gradient instead of the instantaneous gradient of the interrogative
window will be used. For the average phase function defined in
(15), (17) can be ex-pressed as
avg avg avg avg( , ) = ( , ) ( , ) ( , )k j m k j k j c k j+ ,
(18)
where
m k jm k j w k j
w k jj j Lj L
k k Lk L
avgl
l
a
a
( , ) =( , ) ( , )
( , )
1 1 1 1==
1 1
11 +
+
jj j Lj L
k k Lk L
11 == +
+
l
l
a
a
c k jc k j w k j
w k jj j Lj L
k k Lk L
avgl
l
a
a
( , ) =( , ) ( , )
( , )
1 1 1 1==
1 1
11 +
+
jj j Lj L
k k Lk L
11 == +
+
l
l
a
a.
Fig. 2. Illustration of weighting of the neighboring windows
with re-spect to the interrogative window at (k, j) to calculate
the average strain, where La = Ll = 2.
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elastography 2271
Thus, we see that the use of the weighted average phase in (18)
results in a straight line around the root with weighted average
slope of the neighboring windows. This averaging approach is
expected to minimize the disconti-nuity in strain calculation.
omitting the scan-line index j for notational simplicity, the
finite difference technique adopted from the secant algorithm in
terms of strain and phase for the jth scan-line can be written
as
avg avgavg
avg( , ) = ( , 1)
( , 1)( , 1)
,
> 2, = 1,2,
k n k nk nk n
k n
(19)
with avg avg ( ,0) = ( 1, )k k N (20)
avg avg ( ,0) = ( 1, )k k N , (21)
where k and n are the window and iteration indices,
re-spectively; avg(k, n) and avg(k, n) are the average strain and
zero-lag phase of the cross-correlation function de-fined in (15)
for the kth window at the nth iteration; N is the index of
iteration at which the solution is obtained; avg(k, 0) is the
initial strain for the kth window; and avg(k 1, N) is the
calculated final strain for the (k 1)th win-dow. based on (18), the
derivative of the phase, avg ( , )k n , can be expressed as
avg avg
avg avg
avg avg
( , ) = ( , )
=( , ) ( , 1)( , ) (
k n m k n
k n k nk n k,, 1) , > 2, = 1,2,n k n
(22)
note that (22) holds only if the phase versus strain
rela-tionship can be assumed linear, as in (17) or (18), in the
vicinity of the root. although the first two windows (i.e., k = 1,
2) dont employ the secant algorithm, to execute the secant
algorithm for k 3, we need to know avg (2, )N to initialize avg
(3,0) according to (21). now, using (18), we can write
avgavg avg
avg avg
(2,0) =
(1, ) (2,0)(1, ) (2,0)
NN , (23)
where it is assumed that cavg(2, 0) cavg(1, N), and avg (2,0) =
avg (1, )N . The strain of the first two windows [i.e., avg(1, N)
and avg(2, N)] for each scan-line are com-puted using (25). Then,
following (23), avg (2, )N can be computed as
avgavg avg
avg avg
(2, ) =
(1, ) (2, )(1, ) (2, )N
N NN N . (24)
note that avg(k, n) will be calculated using avg(k, n). The
final estimate of the strain from the secant algorithm will be
denoted as avg(k) avg(k, n).
now, to estimate strain for the first two windows (i.e., for k =
1, 2) of each of the scan-line, a total search tech-nique will be
used to find the minimum of |avg(k, n)|. The strain can be
estimated as
avg avg
avgavg max
( ) ( , )
= ( , ) , = 1,2, 0 { =1 }
k k N
k n k
argmin nn = 1,2,,
(25)
where max is the maximum possible value of the strain. This
total search technique will also be accessed when there is an
unusual jump between two consecutive strain values on the same
scan-line, which may result because of the divergent nature of the
secant algorithm. The jump is defined by a threshold strain value
given by C avg(k, n 1), where C 2 is a constant. If the difference
in strain of the present and previous window [i.e., avg(k, n)
avg(k, n 1)] exceeds [C avg(k, n 1)], then the total search is
accessed.
a list of the steps involved in the proposed PdsE method is
presented in Table II.
4) Convergence of the Algorithm: The convergence of the proposed
PdsE algorithm with respect to iterations for an arbitrary
simulation data set is depicted in Fig. 3. The presented values of
phase and strain for five iterations in Fig. 3 reveal that the
zero-phase is achieved at the third iteration for the corresponding
strain value and almost no changes in the strain and phase values
occur for the sub-sequent iterations, implying that the secant
algorithm has converged. To ensure the convergence of the
algorithm, up to five iterations are executed.
III. simulation and Experimental results
In this section, we present some performance test results of our
proposed PdsE method using an FEm simulation phantom, a cIrs Inc.
(norfolk, va) experimental phan-tom, and in vivo patient data.
These results are compared with those of the phase-root seeking
(Prs) [22], am2d [16], and adaptive spectral strain estimation
(assE) [19] methods. The qualitative performance is determined
visu-ally but the quantitative performance is evaluated with the
help of three numerical metrics: elastographic signal-to-noise
ratio (snre) [30], elastographic contrast-to-noise ratio (cnre)
[31], and mean structural similarity (ms-sIm) [32].
A. Simulation Phantom Results
ansys (ansys Inc., canonsburg, Pa) and Field II [33] are used
for finite element simulation and ultrasound sim-ulation,
respectively, of a rectangular 40 40 mm FEm phantom. a 2-d FEm
model was used for which the total number of nodes was 26 749. In
ultrasound simulation us-ing Field II, the sampling frequency was
50 mHz. This
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phantom had a homogeneous background with youngs modulus (i.e.,
stiffness) of 10 kPa, which is close to the average stiffness of
normal glandular tissue in a female breast [34][36]. There are
three 7.5-mm-diameter circular inclusions to model different types
of breast lesions. The top, bottom left, and the bottom right
inclusions have youngs moduli of 3, 30, and 80 kPa, respectively
[see Fig. 4(a)]. The phantom was compressed from the top by a
compressor that was wider than the phantom. The bot-tom of the
phantom was rested on a planar surface and the phantom was allowed
to freely expand both at the top and bottom (full-slip condition).
It was scanned from the top with a 10 mHz center frequency
transducer of 1.5 mm beam width. The number of scan-lines was 128
and the number of samples used in a scan-line was 2597. The ideal
elastogram for 1.5% applied strain is shown in Fig. 4(b). The
corresponding ideal strain profile of the marked line in Fig. 4(b)
is plotted in Fig. 4(c). The two strain wells in
Fig. 4(c) represent the regions occupied by the 30-kPa and
80-kPa lesions. although the phantom has a background of
homogeneous stiffness, the varied strain profile in the rest of the
region reflects the interaction between the le-sions.
before making a comparative analysis, the definitions of the
three quality metrics snre [30], cnre [31], and mssIm [32] are
provided in the following:
SNRe bb
= , (26)
where b and b denote the statistical mean and standard deviation
of the strain computed in a homogeneous area, respectively.
CNRe l bl b
=2( )2
2 2 +
, (27)
where is the mean strain and is the standard devia-tion of the
strain in a homogeneous background area. The subscripts l and b
refer to the lesion and background, respectively.
mssIm is an excellent predictor of the perceived image quality.
It considers contrast, luminance, and structural similarity between
the estimated and actual strain images to compute the value of the
index. For calculating the ms-sIm index, at first, the actual and
estimated strain images are locally windowed. Each of the windowed
actual and estimated signals, x (x = [x1, x2 xL]) and y (y = [y1,
y2 yL]), respectively, are of length L. These two sig-nal vectors
are then Gaussian function (w = [w1, w2 wL]) weighted, with a
standard deviation of 1.5 samples, where
iL
iw=0 = 1. Then, the estimates of local statistics of x and y are
calculated as
TablE II. summary of the PdsE algorithm for the jth scan
line.
step 1:a. Get the pre- and post-compression rF frames,U1(i, j)
and U2(i, j), respectively.b. select the window length (Wl), window
shift (Ws), Poissons ratio ( ), maximum number of iterations Nmax
for the secant algorithm. define the termination condition and set
a = 0.25.step 2:a. set window index, k = 1, 2, , K.b. compute r1(k,
j) and r1+(k, j) using (10) and (5), respectively.step 3:Estimate
strain using the total search technique given in (25) for k 2. In
each iteration, calculate the post-compression signal r2(k, j)
using (11)(14) and r k j2 ( , )+ as defined in (7), where (k) = 1
avg(k, n 1).step 4:Estimate strain using the secant algorithm for k
> 2 as in the following: i) Initialize avg(k, 0) = avg(k 1, N)
as in (20), and avg ( , 0)k = avg ( 1, )k N according to (21).
ii) set n = 1, 2, , Nmax. iii) calculate the post-compression
signal r2(k, j) using (11)(14) and r k j2 ( , )+ as defined in (7),
where (k) = 1 avg(k, n 1). iv) compute phase avg(k, n 1) using
(15). v) compute strain avg(k, n) using (19). vi) check if the
termination condition is reached. If not go to step 4(ii).step
5:store the estimated strain as avg(k) = avg(k, N ), where n = N
denotes the index of iteration at which the solution is
obtained.step 6:compute the arithmetic mean strain, avg( )j = k
K k K=1 ( ) avg / , if k = K is reached. otherwise go to step
2(a).
Fig. 3. convergence of the proposed PdsE algorithm. results are
plot-ted for an arbitrary window of the FEm simulation phantom
data.
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xi
L
i iL w x=1
,=1 (28)
yi
L
i iL w y=1
,=1 (29)
xi
L
i i xL w x=1
1 ( ) ,=1
2
(30)
yi
L
i i yL w y=1
1 ( ) ,=1
2
(31)
xyi
L
i i x i yL w x y=1
1 ( )( ).=1
(32)
The ssIm index between the signals x (x = [x1, x2 xL]) and y (y
= [y1, y2 yL]) is calculated as [32]
SSIM =(2 )(2 )
( )( )1 2
2 21
2 22
x y xy
x y x y
C CC C+ +
+ + + +, (33)
where C1 = (K1MI)2, C2 = (K2MI)2, K1 = 0.01, and K2 = 0.03.
Here, MI is the dynamic range of the pixel values. Finally, the
mean ssIm (mssIm) is calculated as
MSSIM =1
( , )=1
M x yk
M
k kSSIM , (34)
where xk and yk are the image contents at the kth local window
and M is the number of local windows in the im-age.
The quantitative performance of the Prs, assE, am2d, and the
proposed PdsE for a range of applied strain varying from 1% to 4%
is evaluated and presented in Fig. 5. The window length and inter
window-shift for both the PdsE and Prs were 0.74 mm (= 4.8) and
0.148 mm (= 0.96), respectively, and those for the assE were 2.29
mm (= 14.8) and 0.286 mm (= 1.858), re-spectively. The weighting
factor a in (16) was selected to be 0.25. The am2d computes the
individual pixel strain
with a pixel length of 0.0148 mm (= 0.096). For strain 2.5%, nF
= 2 (i.e., La = 2, Ll = 2), and La = 0, Ll = 2 were used otherwise
for the proposed PdsE. The snre (averaged over four homogeneous
background regions) and mssIm values are plotted in Figs. 5(a) and
5(b), re-spectively, and the cnre (background strain is averaged
over four homogeneous regions) for the three different le-sions are
plotted in Figs. 5(c)5(e). The proposed PdsE performs significantly
better than the assE and phase-based Prs up to 2.5% applied strain
in terms of snre, mssIm, and cnre. Except for the mssIm and cnre
for lesion-1 (i.e., 3-kPa lesion), the proposed PdsE out-performs
the am2d up to the same range of strain. The am2d performs slightly
better than the PdsE in terms of mssIm at 2.5% strain. The
exclusion of a few boundary rows and columns helps to reduce border
error and con-tributes to higher mssIm for the am2d. It can,
however, be concluded from Figs. 5(a)5(e) that the performance of
PdsE is satisfactory up to 2.5% to 3.0% applied strain for the FEm
simulation phantom.
The generated elastograms of the FEm simulation phantom for Prs,
assE, PdsE, and am2d are pre-sented in Fig. 6 for various applied
strains (1%, 1.5%, 2%, 2.5%, 3%, and 4%) for qualitative
evaluation. The Prs produces noisier image compared with the other
three methods as is observed from Figs. 6(a)6(f). Figs. 6(m)6(r)
demonstrate the effect of increasing the applied strain in strain
imaging by am2d. There are eight tun-ing parameters that must be
adjusted to produce a good quality strain image when using am2d,
and unlike the other three methods, am2d is highly sensitive to
attenu-ation effect. The am2d results are relatively noisy at low
strain. comparatively, the performance of assE is ob-served to be
robust for a wide range of strain, as shown in Figs. 6(g)6(l).
However, noise increases in the two sides of the strain image as
the applied strain increases. The proposed PdsEs efficacy in
generating elastogram for the specified strain range is illustrated
in Figs. 6(s)6(x). The background imaged by the proposed PdsE is
smoother in comparison with that of Prs and assE. The boundaries of
the lesions are comparable in all methods except the
Fig. 4. FEm simulation phantom. (a) stiff inclusions in a
homogeneous background of 10 kPa, (b) corresponding ideal
elastogram at 1.5% applied strain, (c) strain profile of the marked
line in (b).
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Prs. The best visual quality strain image shown in Fig. 6(t) is
observed at 1.5% strain, which is consistent with the snre and
mssIm performance of the proposed PdsE presented in Figs. 5(a) and
5(b), respectively.
To observe the distortion in the strain curves, 1-d lat-eral
strain profiles are drawn in Figs. 7(a)7(d) for the four methods
(Prs, assE, am2d and the proposed PdsE). These profiles are taken
from the strain image in Fig. 6, including the 30- and 80-kPa
lesions for the 1.5% applied strain. It can be observed from these
1-d strain profiles that although the strain profiles estimated by
all four methods closely follow the reference curve, the assE and
PdsE generated wells are more accurate. The strain curve generated
by the PdsE method, however, is the smoothest.
B. Experimental Phantom Results
We performed elastography experiments with an 18 12 9.5 cm cIrs
tissue-mimicking phantom (TmP). The phantom consists of a
13.6-mm-diameter spherical inclu-sion in a homogeneous background;
it is made of zerdine with a sound speed of 1540 m/s. The youngs
moduli of the background and lesion are 17 and 75 kPa (as per the
order specification), respectively. The attenuation coeffi-cients
of the background and lesion are 0.68 and 0.73 db/cm/mHz,
respectively. a sonixToUcH research (Ultra-sonix medical corp.,
richmond bc, canada) scanner in-tegrated with a l14-5/38 probe
operating at 10 mHz and at a sampling frequency of 40 mHz was used
to acquire
rF echo-signals from this cIrs phantom. The experiment was
performed at the bangladesh University of Engineer-ing and
Technology (bUET) medical center, dhaka, ban-gladesh.
The strain images are generated employing the Prs, assE, am2d,
and the proposed PdsE at approximately 0.5% and 1.5% applied strain
and shown in Figs. 8(a) and 8(b), Figs. 8(c) and 8(d), Figs. 8(e)
and 8(f), and Figs. 8(g) and 8(h), respectively. assE, am2d, and
PdsE produce the strain images of the lesion with good con-trast
and boundaries, although for the am2d method, the strain image
appears to be little over-smoothed at 1.5% strain [see Fig. 8(f)].
Fig. 8(i) of the 1-d strain profile drawn from the calculated
strains at 0.5% strain is used to compare the lateral diameter
measured from differ-ent methods. The measured diameters from PdsE,
Prs, assE, and am2d are 14.55, 15, 14.35, and 14.2 mm,
re-spectively.
C. In Vivo Breast Experiment
The in vivo breast data in this paper are taken from an existing
database of 175 patients (age range: 1568 years, average age: 34
years) who appeared for free-hand elastog-raphy. The data were
acquired at the bUET medical cen-ter, dhaka, bangladesh, using a
sonixToUcH research (Ultrasonix medical corp.) scanner integrated
with a l14-5/38 probe operating at 10 mHz and at a sampling rate of
40 mHz. The study was approved by the institutional review board
(Irb) and conducted with informed consent
Fig. 5. Performance comparison of different methods using
numerical performance metrics for the FEm simulation phantom data.
(a) snre versus applied strain, (b) mssIm versus applied strain,
and cnre versus applied strain for (c) 3 kPa, (d) 30 kPa, and (e)
80 kPa lesions. For the PdsE method, La = 2, Ll = 2 for strain 2.5%
and La = 0, Ll = 2 otherwise.
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Fig. 6. strain images of the FEm simulation phantom generated by
different methods. (af) Prs method, (gl) assE method, (mr) am2d
method, (sx) proposed PdsE method [La = 2, Ll = 2 for (sv) and La =
0, Ll = 2 for (wx)].
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from the patients. out of 175 patients, four were selected to
represent four different diagnosis, i.e., fibroadenoma, lactating
adenoma, abscess, and carcinoma. details of these four patients are
presented in Table III. The results obtained using the four
different algorithms are presented in Fig. 9. The b-mode images of
patients IIv are shown in Figs. 9(a)9(d), respectively.
The elastograms generated by Prs, assE, am2d, and the proposed
PdsE for the in vivo patient data are shown in Figs. 9(e)9(h),
Figs. 9(i)9(l), Figs. 9(m)9(p), and Figs. 9(q)9(t), respectively.
Four different cases (i.e., fibroadenoma, lactating adenoma,
abscess, and carcinoma) are selected to check the effectiveness of
the proposed algorithm in the detection of a lesion, along with its
size. It is well reported that the size of a malig-nant lesion in
an elastogram is often larger than that on the b-mode ultrasonogram
[7], but they are usually simi-lar for a benign tumor, with
exception for some inflam-matory cases. For Patient I, the proposed
PdsE, assE and am2d can clearly differentiate the lesion from the
background. The size of this benign lesion imaged by the PdsE and
assE is closer to that of the b-mode image in comparison with that
from the am2d, as is evident from Figs. 9(q), 9(i), and 9(m),
respectively. For Patient
II, the horizontal orientation of the lactating adenoma, one of
its characteristic features, is more prominent in the elastograms
in Figs. 9(f), 9(j), 9(n), and 9(r) than in the b-mode image in
Fig. 9(b). although the lesion size on the elastogram is larger
than that on the b-mode image, it was diagnosed benign based on
fine needle as-piration cytology (Fnac). among these four methods,
the size of the lesion imaged by the PdsE and assE in Figs. 9(r)
and 9(j), respectively, closely matches that on the b-mode in Fig.
9(b). For Patient III, the generated strain images of the abscess
in Figs. 9(o) and 9(s) by the am2d and PdsE, respectively, clearly
differentiate the softer and harder regions, which are not very
evident in Figs. 9(g) and 9(k) from the Prs and assE,
respec-tively. The elastograms of Patient Ivs lesion, shown in
Figs. 9(h), 9(l), 9(p), and 9(t), reveal that the size of the
lesion (diagnosed as carcinoma) on elastogram is larger than that
on b-mode. In case of the Prs algorithm, the generated elastograms
are relatively noisy. The strain images obtained from the am2d are
usually free of large artifacts. as is evident from Figs. 9(q)9(t),
the PdsE with enforced strain continuity (resulting from the
inclu-sion of neighboring windows) also produces strain images with
good contrast.
Fig. 7. strain curves obtained for FEm simulation phantom by
using different methods at 1.5% strain. (ad) lateral strain profile
of the marked line (including the 30-kPa and 80-kPa inclusions) in
Fig. 4(b) for the Prs, assE, am2d, and proposed PdsE methods,
respectively.
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The 1-d strain profile of lateral and axial diameter lines for
Patient II and Patient Iv, respectively, are shown in Figs. 10(a)
and 10(b). For Patient II, the proposed PdsE produces a clear
strain well with no oscillation in strain inside the lesion. This
means that the lesion boundary can be identified clearly by the
PdsE. With some variation inside the strain well, the Prs can also
detect the bound-
ary of the lesion. The assE shows no definite boundary, i.e.,
the calculated strains vary significantly within the well. The am2d
fails to draw a sharp boundary com-pared with the PdsE, resulting
in the largest boundary among these four methods. The approximate
lateral diam-eters of the lesion measured from the PdsE, am2d, Prs,
and machine b-mode are 19.05, 25.4, 21.1, and 15.4 mm,
respectively. significant increase in the size as measured from
elastograms may be considered as being imparted by the stiffness of
the surrounding tissue because of the non-capsulation of the
lactating adenoma. as in Patient II, the proposed PdsE outperforms
the other three meth-ods in differentiating the lesion from its
background for Patient Iv. The approximate axial diameters of the
lesion measured from the PdsE, am2d, and assE are 20, 19.6, and
18.5 mm, respectively. The edges of the strain well
Fig. 8. strain images of the experimental (cIrs) phantom
generated by different methods. (a and b) Prs method, (c and d)
assE method, (e and f) am2d method, (g and h) proposed PdsE method
with nF = 2, (i) lateral strain well generated by the different
methods at 0.5% strain.
TablE III. Patient details.
Patient number age
method of diagnosis diagnosis
I 15 Fnac FibroadenomaII 20 Fnac lactating adenomaIII 55 Fnac
abscessIv 38 core biopsy carcinoma
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produced by the assE and am2d are not as sharp as by the PdsE.
The axial diameters measured from the Prs and machine b-mode are
20.25 and 20 mm, respectively. although the axial diameter measured
from the b-mode UsG and elastogram remains more or less the same,
the carcinogenic feature of the lesion resulted in an increased
size/area on elastograms in comparison with that on the b-mode.
note that the size of the lesion as determined by an expert
radiologist in b-mode is taken as the reference value.
To observe the effect of including the neighborhood, the
quantitative and qualitative results obtained by the proposed PdsE
for different neighborhood factors (nF = 0, 1, 2, 3) are depicted
in Figs. 11 and 12, respectively. The performance metrics, snre,
mssIm, and cnre [see Figs. 11(a)11(e)] obtained for nF = 2 excels
that for nF = 0 and nF = 1 up to the maximum range of applied
strain (i.e., 3%). Insignificant improvement in snre up to 3%
applied strain and in mssIm up to 2.5% applied strain is observed
for nF = 3 compared with that for nF
= 2. The calculated cnres for nF = 3 are either compa-rable or
better than those for nF = 2, except for lesion 1 (i.e., the 3-kPa
lesion). overall, there is an improvement in quantitative
performance metrics with increase in nF at the expense of
computational cost and edge blurring. To illustrate the qualitative
results, strain images of the FEm simulation phantom for nF = 0, 1,
2, and 3 at 1.5% applied strain are presented in Figs. 12(a)12(d).
smooth-ness in the background improves at the expense of con-trast
as nF increases, which is evident from Figs. 12(a)12(d). The
boundary of the lesions also becomes blurred with the increase of
the neighborhood factor. To examine the effect of neighborhood in
strain imaging with the ex-perimental data, elastograms were
generated using the in vivo patient data (Patient II) for nF = 0,
1, 2, and 3, as shown in Figs. 12(e)12(h). The results are
consistent with the elastograms of Figs. 12(a)12(d) for the FEm
simulation phantom data. considering both the subjective and
objective quality evaluation results for different nF factors, we
suggest nF = 2 for general use.
Fig. 9. strain images generated by different methods using in
vivo breast ultrasound data. (ad) b-mode images of Patients IIv.
(eh) Prs method, (il) assE method, (mp) am2d method, and (qt)
proposed PdsE method. The yellow line is the lesion border as
outlined in machine b-mode by a radiologist.
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The effect of window length and window shift on snre for the two
phase-based methods, PdsE and Prs, is shown in Figs. 13(a) and
13(b). The results are plotted for the FEm simulation phantom data
at 0.5%, 1.5%, and 2.5% applied strain. The window length ranges
from 1.9 to 11.54 in Fig. 13(a) and the window shift is selected to
be 20% of the window length in all cases. In case of the proposed
PdsE, no noticeable change is observed in snre from 7.7 to 11.54
for 0.5% and 1.5% strain; for 2.5% strain, snre falls sharply for
the window length above 3.85, as can be observed from Fig. 13(a).
For 0.5% and 1.5% strain, the snre in the Prs method increases
grad-ually with the window length, but for 2.5% strain, it starts
decreasing for window length 7.7. Figs. 13(c) and 13(d) present the
effect of window length (for 3.85 and 9.6) qualitatively for the in
vivo patient data. The same image for the window length of 4.8 is
shown in Fig. 9(q). as can be seen, the image quality is the best
for 4.8.
In Fig. 13(b), the window shift ranges from 10% to 50% of the
window length (= 4.8) with strain varying from 0.5% to 2.5%. For
the proposed PdsE, although the snre increases up to 40% window
shift for 0.5% and 1.5% strain, and up to 30% window shift for 2.5%
strain, the visual image quality deteriorates for 30% window shift,
as can be seen from Fig. 13(f). However, for 20% and 10% window
shift, the elastograms shown in Fig. 9(q) and Fig. 13(e),
respectively, are marginally different. The snre calculated at
different window shifts for 1.5% and 2.5% applied strain is very
similar up to 30% window shift for the PdsE but deteriorates
sharply for 2.5% strain beyond that shift. For different window
shifts, a steady increase in the snre is observed for the specified
range of strain for the Prs. considering the performance metrics
and visual quality, the window length and window shift have been
chosen to be 4.8 and 20% of 4.8, respectively, in all of our
results.
Iv. discussions
The performance of the proposed PdsE has been dem-onstrated both
quantitative and qualitatively in the re-sult section. The two
distinct features of the proposed PdsE that differentiate it from
the other phase-based algorithms, such as [22], [23], are the
direct strain esti-mation and inclusion of the neighborhood in
calculation of the phase of the interrogative window. although the
global-stretching-based methods are computationally ef-ficient, the
local adaptive stretching and windowing by the PdsE resulted in a
better quality strain image, with the cost paid in computational
complexity. The compu-tational cost is further increased because of
using the neighborhood, which demands adaptive stretching of the
neighboring windows along with the interrogative one in each
iteration of the algorithm, but with the benefits of robustness to
decorrelation noise and improvement in strain image quality.
Therefore, there is a trade-off be-tween the computational cost and
accurate estimation of strain when using the proposed method with
built-in smoothing feature. The computation times for the in vivo
Patient II data [cPU: 3.4-GHz core-i7 (Intel corp., santa clara,
ca), with 4 Gb of ram; software: matlab (The mathworks, natick,
ma)] for our proposed PdsE were measured to be 34.65, 108.41, and
271.09 s for nF = 0, 1, and 2, respectively. The data size, window
size, and window shift were 1040 128, 12.45, and 2.49,
respectively. For the Prs, assE, and am2d, it was 8.79, 107.97, and
0.153 s, respectively. The computation time for the cIrs phantom
data for the PdsE was 49.6, 428.7, and 1160 s, for nF = 0, 1, and
2, respectively, for the data size = 2080 128, window size = 8.73
and window shift = 0.873. For the assE, Prs, and am2d, it was
300.673, 23.3, and 0.303 s, respectively. note, however, that the
am2d method uses mex files but the others do
Fig. 10. strain profile for in vivo breast data using different
methods. (a) strain profile of the lateral diameter line of the
lesion obtained for Patient II. The estimated lateral diameter of
the lesion from Prs, am2d, PdsE, and b-mode are 21.1, 25.4, 19.05,
and 15.4 mm, respectively. (b) strain profile of axial diameter
line of the lesion obtained for Patient Iv. The estimated lateral
diameters of the lesion from Prs, assE, am2d, PdsE, and b-mode are
20.25, 18.5, 19.6, 20, and 20 mm, respectively.
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not. by using mex files, the computational time of the proposed
method can also be reduced.
as is evident from Fig. 13, the window length and over-lap
should be optimized to get the best result. according to [24], the
length beyond which performance degrades,
defined as the drop-length, becomes significant at high strain.
The sharp fall in snre at 2.5% strain and window length = 3.85
reflects this phenomenon. Improvement in snre has been observed at
higher strain with lower window length. visual quality along with
snre should be
Fig. 11. Performance evaluation of the proposed PdsE for
different neighborhood using numerical performance metrics for the
FEm simulation phantom data. (a) snre versus applied strain, (b)
mssIm versus applied strain, and (ce) cnre versus applied strain
for 3, 10, and 30 kPa lesions, respectively.
Fig. 12. Effect of using the neighborhood in the proposed PdsE
method. (ad) FEm simulation phantom data and (eh) in vivo breast
data of Patient II.
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considered to choose the optimum length. although the window
shift equal or greater than 10% of the window length improves the
resolution of the in vivo patient image [as shown in Fig. 13(e)],
this increases the computational cost. These effects are taken into
consideration to choose an appropriate window length and shift. To
account for the effect of lateral shifting in the proposed 1-d
axial strain estimation technique without using a 2-d search
region, the Poissons ratio, has been used in (14) to save some
extra computation. To select an appropriate value, has been varied
from 0 to 1 at an interval of 0.25 and the optimum image quality
for the FEm simulation phantom is observed at = 0.5 in [20]. This
value agrees with that reported in [26].
The signal model presented in (2) does not consider the effect
of additive noise, and assumes that p(t/a(x,y) t0) p(t), which is
reasonably accurate at low strain accord-ing to [17], [19].
However, the decorrelation and additive noise can be considered
partially compensated through the use of neighborhood, which is
evident from the smooth-ness of the background in Figs. 12(e)12(h).
an estima-tion technique for p(t), the PsF, is necessary to
compen-sate for this effect.
The ultrasound rF signal strength attenuates as it propagates
along depth and, therefore, the signal-to-noise ratio deteriorates
deep inside the media. The effect of at-tenuation is observed to be
less in the correlation-based techniques (i.e., Prs, assE, and the
proposed PdsE)
because these methods look for the waveform similarity between
the blocks of data identified by the pre- and post-compression
windows. on the contrary, because am2d employs individual pixel
difference between the pre- and post-compression signals to
estimate strain rather than using the correlation between a pair of
windows, it is high-ly sensitive to the attenuation effect.
v. conclusions
This paper has dealt with a novel direct average strain
estimation method for ultrasound elastography utilizing the phase
of the zero-lag cross-correlation function of the pre-compression
and stretched post-compression analytic signals. a phase-strain
relationship has been exploited to calculate strain directly, in
contrast to conventional use of the phase-displacement function for
calculation of the strain via the gradient of the displacement
field. strain images obtained by the proposed method are therefore
free from the gradient noise. Instead of the instantaneous phase of
the cross-correlation function of the pre-compres-sion and
stretched post-compression analytic signals of the interrogative
window, a weighted-average phase com-puted from the neighboring
windows has been used in the estimation of strain to assure strain
continuityenforc-ing a built-in smoothing feature in the proposed
method. In every phase-root seeking iteration, the strain
obtained
Fig. 13. Effect of the window length and window shift on the
performance of the proposed PdsE and Prs method. (a) snre versus
window length, (b) snre versus window shift for simulated phantom
at different strains. (c and d) strain image of the in vivo Patient
I data for window length = 3.85, window length = 9.6. (e and f)
strain image of the in vivo Patient I data for window shift = 10%
of window length and window shift = 30% of window length,
respectively. Window length was selected to be 4.8.
-
IEEE TransacTIons on UlTrasonIcs, FErroElEcTrIcs, and FrEqUEncy
conTrol, vol. 60, no. 11, novEmbEr 20132282
is used to adjust the window displacement dynamically, stretch
the post-compression signal locally and also to cal-culate the
Poissons ratio to adjust the lateral shift. The requirement of
prior knowledge of the applied strain as in other methods is thus
eliminated in this approach.
The proposed PdsE performs satisfactorily for a prac-tical range
of strain values [8], [37] in terms of the three numerical quality
metrics, i.e., snre, cnre, and mssIm. The strain image quality of
the FEm simulation phantom has been found to be noticeably better
than the Prs and assE and comparable to the am2d for low strain
values such as those experienced in the in vivo patient
elastog-raphy [8], [37]. The comparative analysis of the PdsE with
Prs, assE, and am2d for the four different cases of the in vivo
patient data, i.e., fibroadenoma, lactating adenoma, abscess, and
carcinoma demonstrates that the visual strain image quality (e.g.,
contrast) of the proposed method is better. The regular shape of
the strain well as obtained from our PdsE for the in vivo patient
data clearly indicates the algorithms efficacy in estimating the
dimension of the lesion along with the stiffness relative to the
surrounding tissue. The proposed PdsE, however, is computationally
more expensive than the Prs, assE, and am2d methodsan issue to be
addressed in future research to make the algorithm suitable for
real-time elas-tography.
acknowledgments
The authors gratefully acknowledge the helpful com-ments from
the anonymous reviewers.
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Sharmin R. Ara was born in bangladesh in 1974. she received the
b.sc. degree in electrical and electronic engineering from the
bangladesh University of Engineering and Technology (bUET),
bangladesh, and the m.sc. degree in electrical and computer
engineering from south-ern Illinois University carbondale in 2000
and 2007, respectively. she is currently working to-ward her Ph.d.
degree in the department of Elec-trical and Electronic Engineering
at bUET. she is currently a senior lecturer in the department
of
Electrical and Electronic Engineering at East West University,
bangla-desh. Her research interests include ultrasonic imaging and
tissue char-acterization.
Faisal Mohsin received the b.sc. degree in elec-trical and
electronic engineering from the bangla-desh University of
Engineering and Technology (bUET), dhaka, bangladesh, in 2006. In
2006, he joined the radio network optimization department of
banglalink digital communications ltd., ban-gladesh. His research
interests are in digital signal processing, medical imaging, radio
network plan-ning, and optimization.
Farzana Alam received the m.b.b.s. degree from the dhaka medical
college (dmc), dhaka, bangladesh, in January, 2002. she received
the Ph.d. degree in radiology from the biomedical school of
Hiroshima University, Japan, in 2008. she is presently working as a
medical officer in the radiology and Imaging department of
bang-abandhu sheikh mujib medical University (bsmmU),
bangladesh.
Her research interests are in ultrasound breast elastography and
tissue imaging of brain.
Sharmin Akhtar Rupa received the m.b.b.s. degree from the dhaka
medical college (dmc), dhaka, bangladesh, in January, 2000. she
re-ceived her m.Phil. and Fellow of college of Physi-cians and
surgeons (FcPs) degrees in radiology and imaging from the
bangabandhu sheikh mujib medical University (bsmmU) and bangladesh
college of Physicians and surgeons (bcPs) in 2004 and 2006,
respectively. In 2008, she joined the department of radiology and
Imaging, Enam
medical college and Hospital, bangladesh, as an assistant
Professor. she is currently working as an associate Professor in
the same depart-ment.
Her research interests are in ultrasound breast elastography and
mrI.
Rayhana Awwal received the m.b.b.s. degree from dhaka medical
college (dmc), dhaka ban-gladesh in 1990. she received the
fellowship in general surgery (FcPs) from the bangladesh col-lege
of Physicians and surgeons (bcPs) in 1996 and a fellowship in
general surgery from the royal college of surgeons Edinburgh in
1999. she com-pleted a masters degree in plastic surgery in 2006
from dhaka University.
she started her career as a general surgeon in 1996 and switched
to plastic surgery in 2001.
she joined Government service of bangladesh in health care in
1993. currently, she is working as associate Professor, Plastic
surgery in the dhaka medical college and Hospital.
Her areas of interest are breast surgery, cleft surgery,
post-burn re-construction, and microsurgery. she is currently
running a breast care center in a private setting. she has authored
or co-authored more than 20 scientific publications.
Soo Yeol Lee received the m.s. and Ph.d. de-grees in electronic
engineering from the Korea ad-vanced Institute of science and
Technology (KaIsT), seoul, south Korea, in 1985 and 1989,
respectively. He was with department of biomedi-cal Engineering in
Konkuk University, south Ko-rea, from 1992 to 1999. In 1999, he
joined the department of biomedical Engineering in Kyung Hee
University, south Korea, where he is the di-rector of the
functional and metabolic imaging research center (FmIc). His
research interests are mrI, cT, elastography, and medical image
pro-cessing.
Md. Kamrul Hasan received the b.sc. and m.sc. degrees in
electrical and electronic engineering from the bangladesh
University of Engineering and Technology (bUET), dhaka, bangladesh,
in 1989 and 1991, respectively. He received his m.Eng. and Ph.d.
degrees in information and computer sciences from chiba University,
Japan, in 1995 and 1997, respectively.
In 1989, he joined bUET as a lecturer in the department of
Electrical and Electronic Engineer-ing. He is currently working as
a Professor in the
same department. He was a postdoctoral fellow at chiba
University, Japan, and a research associate at Imperial college,
london. He worked as a short-term invited research fellow at the
University of Tokyo, Japan, and as Professor of International
scholars of Kyung Hee University, Ko-rea. His current research
interests are in digital signal processing, adap-tive filtering,
speech and image processing, and medical imaging. He has authored
or coauthored more than 100 scientific publications.
dr. Hasan is currently serving as an associate editor for the
IEEE Access.