Phase-Based Direct Average Strain Estimation for Elastography

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

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-


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


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).


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.


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


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.


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).


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.


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)].


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.


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


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.


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.


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.


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.


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.

Phase-Based Direct Average Strain Estimation for Elastography

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