Hierarchical Motion Estimation with Bayesian Regularization in Cardiac Elastography: Simulation and in-vivo Validation

Abstract

Cardiac elastography (CE) is an ultrasound-based technique utilizing radiofrequency (RF) signals for assessing global and regional myocardial function. In this work, a complete strain estimation pipeline for incorporating a Bayesian regularization-based hierarchical block matching algorithm, with Lagrangian motion description and myocardial polar strain estimation is presented. The proposed regularization approach is validated using finite element analysis (FEA) simulations of a canine cardiac deformation model that is incorporated into an ultrasound simulation program. Inter-frame displacements are initially estimated using a hierarchical motion estimation framework. Incremental displacements are then accumulated under a Lagrangian description of cardiac motion from end-diastole (ED) to end-systole (ES). In plane Lagrangian finite strain tensors are then derived from the accumulated displacements. Cartesian to cardiac coordinate transformation is utilized to calculate radial and longitudinal strains for ease of interpretation. Benefits of regularization is demonstrated by comparing the same hierarchical block matching algorithm with and without regularization. Application of Bayesian regularization in the canine FEA model provided improved end-systole radial and longitudinal strain estimation with statistically significant (p< 0.001) error reduction of 48.88% and 50.16% respectively. Bayesian regularization also improved the quality of temporal radial and longitudinal strain curves with error reductions of 78.38% and 86.67% (p<0.001) respectively. Qualitative and quantitative improvements were also visualized for in-vivo results on a healthy murine model after Bayesian regularization. Radial strain signal to noise ratio (SNR_e) increased from 3.83 dB to 4.76 dB, while longitudinal strain SNR_e increased from 2.29 dB to 4.58 dB with regularization.

I. Introduction

Objective quantification of myocardial function non-invasively has been a key area of interest in clinical cardiology [1]. Echocardiography has been routinely used to assess myocardial function as it is cost-effective, fast, portable and provides high temporal resolution for real-time visualization of heart in a clinical setting [2, 3]. Qualitative assessment of echocardiographic image sequences over several cardiac cycles (visual wall motion scoring and wall thickening evaluation) by expert clinicians was used to quantify myocardial function [4]. However, the accuracy of these assessments is dependent on extensive training, expertise [1, 5] and suffers from inter-observer variability. Myocardial deformation imaging has therefore been utilized to obtain clinically valuable information based on an objective assessment of regional and global ventricular function [6]. Several approaches for myocardial deformation imaging have been developed and reported in the peer-reviewed literature. Cardiac strain imaging (CSI) is one such approach that utilizes two-dimensional (2-D) echo data to extract the local deformation field of the cardiac musculature [7]. Widespread application of cardiac strain imaging in both human and animal studies has been reported in literature [6]. Applications in human imaging [8] include detection of patients with coronary heart disease [9], myocardial ischemia [10] or dilated cardiomyopathies [11]. CSI has also been used in detection of myocardial infarction in murine models [12, 13] and assessment of response to cardiac therapy [14]. These wide ranges of applications were the driving force behind innovations and improvements in cardiac strain imaging.

CSI can be performed either using a sequence of ultrasound B-mode images (envelope detected and log compressed signals) or using radio-frequency (RF) signal [15, 16]. A common approach with motion estimation is to search for similar speckle patterns in a sequence of cardiac B-mode images using block matching [17, 18]. In this approach, similarity between matching blocks is quantified using similarity metrics like normalized cross-correlation coefficients (NCC) or sum of absolute difference. A second popular approach for motion tracking using B-mode ultrasound images are optical flow based motion estimation approaches [19, 20]. These methods assume brightness consistency of a pixel over a short period of time and derive motion by matching pixel intensity across frames. Optical flow-based motion estimation using RF signals was also reported in the literature [21, 22]. Finally, a third approach for motion estimation from B-mode images is the utilization of non-rigid B-mode image registration [23–26]. These methods aim to find a spatio-temporal deformation field by iteratively minimizing difference between motion compensated images and a reference [23]. This task is formulated as a global optimization problem where optimal deformation field minimizes a specific cost function. Smooth basis functions such as B-splines are typically used to parametrize the myocardial deformation field with additional regularization terms in the cost function to be minimized [23, 26]. The regularization enforces additional constraints such as smoothness on the derived motion field [23, 24]. Recently, non-rigid image registration was extended to RF domain and validated with a simulation study [27].

In this paper we focus on cardiac elastography (CE) [28–31] which is a RF echo-signal based speckle tracking technique for CSI. Elastography was originally developed as a technique to estimate local [32] tissue strain via normalized cross-correlation (NCC) of time-shifted RF signals under an external compression along the ultrasound beam propagation direction [33, 34]. CE on the other hand uses the natural contraction and relaxation of myocardium as a mechanical stimulus for strain estimation [28, 31]. Initially CE was performed using 1-D cross-correlation of time-shifted signals along the ultrasound beam propagation direction [31, 35]. As myocardium undergoes a three-dimensional (3-D) deformation during a cardiac cycle [1], several approaches of CE have been proposed to estimate cardiac motion and strain in 2-D [2, 4, 7, 13, 17, 27, 36–40] and even 3-D [3, 32, 41, 42].

One added advantage of CE is the presence of phase information with RF signals resulting in accurate deformation estimation when compared to B-mode or envelope-based methods in detecting small deformations [15, 27, 43]. However, the lack of phase information in the lateral direction (perpendicular to beam direction) makes motion estimation challenging resulting in noisier lateral strain estimates [7, 44]. Further difficulty in accurate 2-D motion estimation results from the “out-of-plane” motion artifacts due to imaging 3-D myocardial deformation using 2-D imaging planes [3, 7, 32, 42]. To improve lower quality lateral strain estimation, several innovative approaches have been implemented such as spatial angular compounding [45–47] and transverse oscillation approaches [48, 49]. Other approaches include regularization [50–58] of the deformation field where a boundary condition is imposed based on apriori information [15]. Regularization is achieved either by correcting an initial estimate of motion or by incorporating a constraining term in the motion estimation cost function itself. Boundary condition based regularization includes geometric regularizers where the estimated motion field is forced to follow a specific curve [59]. Examples of geometric regularizers include the use of polynomial curve fitting [60] or constraining the wall motion to be piece-wise linear [39]. Regularization of estimated motion field can also be achieved by median filtering [15] or Gaussian smoothening [59]. Smoothness penalty constraints used in non-rigid image registration-based approaches is an example of incorporating regularization in the motion estimation framework itself [23, 26].

Bayesian regularization has been previously implemented by our group [50] for a hierarchical block matching algorithm for carotid elastography [61]. Application of the proposed method provided clinically significant results for in-vivo plaque imaging [62–64]. Our group previously reported the use of a hierarchical block matching algorithm for CE [2, 4, 37, 43]. However, Bayesian regularization was not applied and validated for CE. Therefore, in this article, we focus on examining the feasibility of applying Bayesian regularization with the hierarchical block matching algorithm for CE.

This paper reports on three main contributions. First, a complete strain estimation pipeline for incorporating Bayesian regularization-based hierarchical block matching algorithm, Lagrangian description of motion and myocardial polar strain estimation is presented. Second, we present results with a canine cardiac deformation model [2] and an in-vivo healthy murine model to evaluate the performance of the hierarchical block matching algorithm with and without Bayesian regularization. Rigorous quantitative analysis demonstrates that Bayesian regularization improves the quality of strain imaging for CE. Third, we present results from an initial comparison study of the proposed strain estimation pipeline against a commercially available CSI software to demonstrate its in-vivo applicability.

II. Materials and Methods

A. Finite Element Analysis (FEA) Model for Cardiac Elastography

A 3-D FEA model of a healthy canine heart [2] was employed in this study to validate the performance of the proposed strain estimation framework. The original 3-D deformation model of canine heart was developed by the Cardiac Mechanics Research Group at the University of California San Diego (UCSD) [65]. This experimentally validated model allowed simulation of the complex left ventricular mechanics accurately providing a realistic validation setup for cardiac motion estimation algorithms [66, 67]. The original model contained movement of 1296 points located in the canine heart wall acquired at a temporal sampling rate of 250 Hz. Each time point of these movements will correspond to one frame of RF data in the simulation study. These positional deformation information were adapted for ultrasound simulation by a reconstructing a 3D continuous smooth surface of the canine heart model [2]. Finally, to ensure Rayleigh scattering, over 1 × 10⁶ scatterers were randomly positioned in the myocardium of the cardiac model. From this 3-D model, a 2-D plane was selected to simulate a parasternal long axis (PLAX) ultrasonic imaging view. To obtain independent strain estimation we utilized the stochastic nature of scatterer generation, generating five independent speckle realizations of the FEA simulation.

A frequency domain ultrasound simulation program [68] was utilized to generate the RF data that incorporated realistic 3D ultrasound propagation on FEA generated cardiac deformations seeded with a randomly generated scatterer distribution. This simulation approach is used due to its greater flexibility in modelling frequency-dependent ultrasonic imaging properties such as attenuation, dispersion and backscattering over time domain simulation models. The 1-D linear array modeled, consisted of 0.2×10 mm elements with a pitch of 0.2 mm. Conventional Delay and Sum (DAS) beamforming with 128 consecutive elements were utilized to form each A-line. The incident pulse was modeled to be Gaussian shaped with 8 MHz center frequency and 80% bandwidth. The speed of sound and attenuation coefficient were set to 1540 m/s and 0.5 dB/cm-MHz respectively. Each simulated ultrasound image had an 80 × 100 mm² field of view. The ultrasound simulation program related parameters are summarized in Table I. Three region of interests (ROIs) were placed in anterior, apical and posterior walls of myocardium to quantify the simulated sonographic signal-to-noise ratio (SNR_s). SNR_s calculation was performed using a frequency domain approach described in [69]. The calculated SNR_s for over five scatterer realizations at end-systole frame were 30.38 ± 5.026 dB, 29.34 ± 5.20 dB and 33.76 dB ± 7.64 dB respectively for anterior, apical and posterior wall. Simulation of electronic noise was not performed in this study.

TABLE I.

FEA Simulation program Parameters.

Parameter	Value
Probe specific parameters
Transducer type	1-D linear array
Number of active elements	128
Single element geometry [width × length]	[0.2 mm × 10 mm]
Pitch	0.2 mm
Aperture size	25.6 mm
Focusing mode	Single
Transmit focus location	40 mm
F-number (Dynamic receive focusing)	1
Number of A-lines	500
Parameters for RF signal reconstruction from scatter frequency response
Incident pulse	Gaussian-shaped
Center frequency	8 MHz
Pulse bandwidth	80%
Attenuation coefficient	0.5 dB/cm-MHz
Assumed speed of sound	1540.0 m/s
Beamforming method	Delay and sum
RF Sampling frequency	78.84 MHz
Lateral sampling spacing	0.2 mm
Frame rate of acquisition	250 Hz

B. In-vivo experimental protocol and image acquisition

To validate in-vivo use of the proposed framework, a 12 weeks old male BALB/CJ mouse obtained from Jackson Labs (ME, USA) was scanned using a Vevo 2100 LAZR imaging system (FUJIFILM VisualSonics, Inc., Toronto, Canada). All in-vivo procedures were approved by the Institutional Animal Care and Use Committee (IACUC) at the University of Wisconsin-Madison. During the imaging session, the mouse was anesthetized using 1.5% isoflurane with a constant flow of oxygen. Hair was removed from the chest region using depilatory cream. Mouse was placed in the supine position on a heated imaging platform with continuous monitoring of physiological parameters. High frequency ultrasound imaging was performed using a MS 550D transducer (broadband frequency range of 22– 55 MHz) operating at a center frequency of 40 MHz. 2-D RF data were collected in parasternal long axis (PLAX) view. The field of view was 11×12.08 mm² with a sampling frequency of 512 MHz resulting into acquisition of 220 A-lines. Single transmit focusing with the focal depth set at 7 mm from the face of the transducer was used. Imaging frame rate was 235 Hz. 2-D gain (25dB) and TGC were adjusted carefully to acquire RF data with optimal signal-to-noise ratio (SNR) for CE. We acquired 1000 frames per imaging plane, which was stored for off-line analysis.

C. Inter-frame Displacement Estimation

In this study, a multi-level block matching algorithm with Bayesian regularization [50] was used for displacement tracking of both simulation and in-vivo RF data. Inter-frame displacement estimation was performed over a cardiac cycle starting from end-diastolic phase. We use the term pre-deformation and post-deformation image to describe the current and the next frame used for inter-frame displacement estimation, respectively. Initially, both pre-deformation and post-deformation RF data were up-sampled in the lateral direction (perpendicular to beam propagation direction) by a factor of two using a windowed sinc interpolator to improve lateral displacement estimates [39, 70, 71]. Following upsampling, a coarse to fine pyramid with three levels were constructed for performing an iterative coarse-to-fine motion estimation [4, 37, 50, 72, 73]. Pyramid construction was performed by applying decimation factors presented in Table II to original RF data with Gaussian smoothening having a variance of ${(\frac{d_f}{2})}^2$ where d _f denotes the decimation factor.

TABLE II.

Motion Estimation Algorithm Processing Parameters.

Parameter	Value
Hierarchical Block-matching parameters
Levels in multi-resolution pyramid	3
Lateral interpolation factor	2
Axial decimation factors	[3, 2, 1]
Kernel overlaps [Axial, Lateral*]	[10%, 90%]
Lateral decimation factors	[2, 1, 1]
Axial kernel length (Wavelengths)	[8 λ, 3 λ, 1 λ]
Lateral kernel length (A-lines)	[15, 12, 10]
Axial search range (Wavelengths)	[3 λ, 2 λ, 1 λ]
Lateral search range (A-lines)	[5, 5, 3]
Strain filtering threshold [axial strain, lateral strain]	[2.5%, 2.5%]
Bayesian regularization specific parameters
Axial strain rate sigma	0.150
Lateral strain rate sigma	0.075
Number of iterations**	1/3

^*Lateral overlap of 90% corresponds to lateral window shift of 3 A-lines

^**Number of iterations was chosen empirically based on the application. For FEA simulation study, one iteration provided good results while in-vivo study required three iterations

Following pyramid construction, pre and post-deformation data were divided into a rectangular grid of 2-D kernels. 2-D NCC [73] calculation was performed to compare a kernel between the pre to post-deformation RF frame. NCC calculation was restricted within an empirically chosen search region in the post-deformation frame. This process results in a 2-D similarity metric for each estimation location of the rectangular grid. Parameters employed for 2-D NCC are shown in Table II. Progressively decreasing block sizes were used to improve spatial resolution of the estimated displacement vectors.

To improve motion estimation accuracy, each similarity metric was regularized using a recursive Bayesian regularization algorithm [50, 61]. In brief, the algorithm tries to remove noisy NCC estimates from a given similarity metric using guidance from left, right, top and bottom neighbors of the similarity matrices. This is achieved by formulating regularization as a maximum aposteriori estimation problem in a Bayesian framework. The algorithm requires a parameter referred to as strain sigma, which is related to the maximum expected strain in both axial and lateral directions. Strain rate sigma parameter values for axial and lateral directions were chosen empirically and listed in Table II. This process results in regularized similarity metrics, which were used in the next stage to generate displacement vectors.

For subsample displacement estimation, parabolic interpolation was used for level 1 to level 2 while the final level employed sinc interpolation to achieve unbiased estimation [61, 70]. Subsample estimation using 2-D windowed sinc interpolation was performed using a multilevel global peak finder scheme [74]. A central-difference gradient was used to estimate strain from corresponding displacement vectors for replacing erroneous displacement estimates due to peak-hopping errors. Inter frame displacement vectors generating strain magnitude > 2.5% were replaced using linear interpolation from neighboring displacement estimates with strain magnitudes less than 2.5%. This is done to inhibit the propagation of peak-hopping errors which present as irrationally high strains [61]. After obtaining displacement vectors and strains at the current level, signal re-correlation using matching block alignment and local temporal stretching is performed for the next level. Signal alignment and stretching improves displacement and strain accuracy by reducing signal decorrelation within the matching block [44, 75–77].

To achieve matching block alignment in our multi-level framework, estimated displacement vectors at each level are used to translate the center of post-deformation matching block in the next higher level. Next, in the align and stretch stage a 9 point least squares to estimate strains. Using the estimated strains we stretch the post-deformation block of next level using a 2-D windowed sinc interpolation for resampling and using a scale factor: S_i = 1 + e_ii where e_ii the normal strain in that direction. This estimation process is repeated until we reach the final level, i.e. level three in this study. At the final level, we perform a 2D median filtering of estimated displacement vectors with a [5 pixel × 5 pixel] window to remove any outliers. The displacement estimation procedure is summarized in the flowchart in Figure 1.

Fig. 1.

Flowchart depicting the various steps involved in the multi-level block matching displacement estimation algorithm with Bayesian regularization. The dotted line indicates that estimated displacement and strain from the current level that guides the search region initialization in the next level.

D. Lagrangian Description of Motion for Displacement Accumulation and Polar Strain Estimation

To determine cumulative displacements and strains occurring over a cardiac cycle, inter-frame displacements are integrated over time based on a Lagrangian description of motion. This accumulation process is not trivial as the myocardium changes its location over the cardiac cycle. An end-diastole (ED) frame is considered to be the reference frame. Location of each pixel in this frame is defined as the reference state. For every frame, these locations are updated by translation of axial and lateral coordinates of pixel locations using the estimated inter-frame axial and lateral displacements respectively. Displacement values derived from updated locations are registered back to reference initial location in the ED frame and accumulated. Pixel registration ensures that the cumulative displacement represents motion along the same geometry [78]. In this way, incremental inter-frame displacements are integrated over a cardiac cycle to obtain cumulative displacement over the cycle. For each individual points, baseline drift is compensated by performing a linear de-trending of temporal displacement and strain curves with the constraint that curves should return to zero after a cardiac cycle [24].

We perform cumulated strain estimation using the resulting cumulated displacement maps. First, the displacement gradient tensor, Gis calculated, defined as:

$$\text{G}=\left(\begin{array}{cc}\frac{\partial u_x}{\partial x}& \frac{\partial u_x}{\partial y}\\ \frac{\partial u_y}{\partial x}& \frac{\partial u_y}{\partial y}\end{array}\right)$$ (1)

In equation (1), u_x and u_y denotes estimated displacement in lateral and axial direction respectively. Gis obtained by differentiating lateral and axial displacement maps using least squares estimator [79] with 0.2 mm and 1 mm kernels respectively. To account for the large myocardial deformation (~30–40%) that occurs from end-diastole (ED) to end-systole (ES), an in-plane Lagrangian finite strain tensor, E[6, 66, 80] is used. Eis formulated using displacement gradient tensor, Gas follows [81]:

$$\text{E}=\frac{1}{2}\left(\text{G}+{\text{G}}^T+{\text{G}}^T\text{G}\right)$$ (2)

The diagonal components of Edenoted by E_xx and E_yy are the cumulative lateral and axial strains respectively.

The strain measure, Eis coordinate dependent as it involves spatial derivatives in the orthogonal coordinate system used for ultrasound imaging [66, 80]. This poses a challenge in interpretation of these strains in a cardiac coordinate system [16]. In the cardiac coordinate system for apical and parasternal long axis views, we are interested in strains along radial and longitudinal directions, which are defined as follows [6, 16]:

• Radial direction is perpendicular to the endocardial border and provides positive radial strain during contraction. Positive and negative radial strain illustrates thickening and thinning of myocardial walls respectively [82].

• Longitudinal direction is tangential to the endocardial border and provides negative longitudinal strain during contraction. Positive and negative longitudinal strain illustrates lengthening and shortening of the ventricle respectively [82].

To be consistent with the interpretation of strains in the cardiac coordinate system, radial and longitudinal strains are derived from E using the coordinate transformation [66]:

$${\mathbf{E}}^{rl}={\mathbf{MEM}}^T$$ (3)

where M is a rotation matrix defined as:

$$\text{M}=\left(\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right)$$ (4)

In equation (3), the superscript rldenotes strain in radial and longitudinal direction respectively in the cardiac coordinate system. The diagonal components of E^rl denoted by E^rr and E^ll are radial and longitudinal strains respectively. The angle θ used in equation (4) is calculated locally along a sampling grid encompassing the entire myocardium. The heart is segmented manually in the ED frame using the B-mode image. This process results in a binary label image. We generate a mesh of points using this binary label image containing 600 points longitudinally (tangentially) and 40 points radially to the myocardial wall resulting into 24,000 point of interest in the entire myocardium. For a sample point with coordinate value (x_n, y_n), the angle θ was calculated by considering a neighborhood of ten sample points around it along the longitudinal direction and using the following equation for angle of a normal to a line.

$$\theta ={\tan }^{-1}(\frac{x_{n+5}-x_{n-5}}{y_{n-5}-y_{n+5}})$$ (5)

This angle denotes the radial direction for the point located at (x_n, y_n).

E. Myocardial Region Definition for Segmental Analysis

In this study, we perform segmental analysis of estimated displacement and strains over the entire myocardial wall using American Heart Association (AHA) recommended standard six segment model (employed for global 16-segment model) [83]. Figure 2 shows the definition of segments employed for the PLAX view. Segmental analysis is achieved easily as we have already warped the cumulative displacement and strain maps for the ED geometry during accumulation. Six segments are defined such that all segments have equal length in the ED frame. All the results reported in this work denote displacement and strain measures averaged over the entire cardiac walls on a segment basis.

Fig. 2.

Definition of cardiac segments for studying regional variation in displacement and strains. Cardiac segments defined in the PLAX view are: (1) Anterior Base, (2) Anterior Mid, (3) Anterior Apex, (4) Posterior Apex, (5) Posterior Mid and (6) Posterior Base.

F. Comparative Performance Analysis

We evaluated the performance of our hierarchical block matching algorithm using NCC with and without Bayesian regularization. Displacement and strain estimation accuracy using the FEA simulation model was compared over n=5 randomly generated independent collection of scatterers. True inter-frame displacement estimates were derived from the FEA canine heart model and integrated over time to obtain cumulative true displacement and strain as described in Section II D. True and estimated temporal displacement and strain curves for six segments were extracted and compared qualitatively for both approaches. Radial and longitudinal strains were compared in terms of two error metrics, namely - strain error (%) at ES and total temporal relative (TTR) strain error (%). Strain error (%) at ES [84] and total temporal relative strain error (%) were computed using the following equations:

$$\begin{gathered} \text{Strain}\text{Error}(\%)\text{at}\text{ES}= \\ \frac{\sum _{i=1}^P|\text{ES}-\text{TS}|}{\sum _{i=1}^P\left|\text{TS}\right|}\times 100 \end{gathered}$$ (6)

$$\begin{gathered} \text{Total}\text{Temporal}\text{Relative}\text{Strain}\text{Error}(\%)= \\ \frac{\sum _{t=1}^T|\text{ES}(t)-\text{TS}(t)|}{\sum _{t=1}^T|\text{TS}(t)|}\times 100 \end{gathered}$$ (7)

where ES and TS denote estimated and true strain respectively, ES (t) and TS (t) denote average of estimated and true strain values respectively at time t, P is the number of points in the segment of interest and T is the number of frames in the cardiac cycle of interest. Strain error (%) at ES quantified the deviation of estimated from ideal FEA strain image at ES while total temporal relative strain error (%) quantified the deviation of estimated from ideal temporal strain curve. Statistical significance was evaluated using paired t-test with p values less than 0.001.

To show in-vivo feasibility, our proposed approach was compared to a commercially available strain estimation software, VevoStrain on the Vevo 2100 LAZR imaging system (FUJIFILM VisualSonics, Inc., Toronto, Canada) for a healthy murine model. Suitable B-mode cine loop containing a cardiac cycle of deformation was loaded into VevoStrain for analysis based on clear visualization of myocardial borders and absence of respiratory artifacts. After endocardial and epicardial borders were delineated in the ED frame, the software automatically tracks myocardial wall deformations using speckle tracking echocardiography. Manual correction of wall tracings was performed to improve quality of tracking and obtain segmental strain curves. VevoStrain reports longitudinal strain curves for epicardial and endocardial wall separately and radial strain curves for the entire myocardial wall. Therefore, endo and epicardial strain curves were averaged along the longitudinal direction. Finally, global radial and longitudinal strain curves were calculated by averaging segmental strain curves for comparison to our approach on the same cine loop.

For the in-vivo study, the elastographic SNR (SNR_e) at ES was computed as:

$${\text{SNR}}_e[\text{in}\text{decibels}]=20{\log }_{10}(\frac{\mu }{\sigma })$$ (8)

where, μ and σ denote the mean and standard deviation of strain (radial/longitudinal) image respectively at ES.

III. Results

A. FEA Simulation Results

Axial displacement maps at ES from FEA model, along with those estimated using our multi-level NCC without and with Bayesian regularization are shown in Figures 3 (a) – (c) respectively. We refer to the approach without regularization as “NCC” and approach with regularization as “Bayesian” for simplicity in the rest of this paper. Positive axial displacements shown in red color shades indicate motion away from the transducer and negative axial displacements in blue color shades indicate motion towards the transducer. From Fig 3 (b) and (c), we observe good agreement between FEA and estimated axial displacements with both methods. Note that in the apical region indicated using arrows in Fig 3 (b) and (c), we observe a smoother transition from positive to negative displacement values with regularization when compared to NCC without regularization. Figures 3 (d) – (f) represent lateral displacement maps at ES for FEA, NCC and Bayesian respectively. Positive lateral displacements in red color indicate motion to the right and negative lateral displacements in blue color indicate motion to the left. In the displacement transition regions (shown using arrows in Fig 3 (e) and (f)), application of Bayesian regularization provided smoother transitions when compared to the NCC approach. Moreover, in the unregularized lateral displacement image (Fig 3 (e)), we observe heterogeneity in estimated lateral displacement vectors in the apical region (seen as white bands and indicated using an arrow) and not seen in the FEA result in Fig 3 (d). These artifacts were also absent with Bayesian regularization, which provides smooth apical lateral displacement estimation.

Fig. 3.

End-systole accumulated axial displacement maps from (a) FEA model, (b) NCC and (c) Bayesian. ES accumulated lateral displacement maps from (d) FEA model, (e) NCC and (f) Bayesian. NCC = no regularization. Bayesian = with regularization.

Figure 4 summarizes the comparison between ES radial and longitudinal strains between the FEA model and estimation results. ES radial strain images from FEA, NCC and Bayesian are shown in Figures 4 (a) – (c) respectively. Radial thickening of the myocardial wall was observed from positive radial strain at ES in the FEA model. NCC provides reliable radial strain estimates in anterior and posterior walls as shown in Fig. 4 (b). However erroneous negative radial strain values were observed in significant portions of the apical region (indicated using arrows). On the other hand, the radial strain image with Bayesian regularization (Fig 4 (d)) provided reliable strain estimation around the myocardium with significantly lower number of negative radial strain values in the apical region when compared to NCC. Figures 4 (d) – (f) represent longitudinal strain images for FEA, NCC and Bayesian respectively. Myocardial wall shortening was observed from negative longitudinal strain values in FEA model. Like radial strain, NCC provides reliable longitudinal strain estimation in the anterior and posterior walls. However, the method was prone to errors in the apical region showing positive longitudinal strain values in significant portions of the apex (indicated using arrows). In Fig 4 (f), note the significant improvement in strain estimation through incorporation of Bayesian regularization in the longitudinal direction. Regions with highest improvement after regularization are indicated using arrows in Fig 4 (d) and (f). Overall, regularized radial and longitudinal strain images showed better qualitative agreement with FEA results.

Fig. 4.

End-systole radial strain images from (a) FEA model, (b) NCC and (c) Bayesian. End-systole longitudinal strain images from (d) FEA model, (e) NCC and (f) Bayesian. NCC = no regularization. Bayesian = with regularization.

Figure 5 (a) presents the segmental and global radial strain error (%) and Fig. 5 (b) summarizes the results for longitudinal strain error (%). A logarithmic scale was used for the y-axes in both figures. Application of Bayesian regularization showed statistically significant error reduction of 48.88% (p<0.001) globally with highest improvements in anterior and apical regions (see segments 1–4 in Fig. 5 (a)). Benefit of Bayesian regularization was also evident in longitudinal strain error (%) results (Fig. 5 (b)) with statistically significant global error reduction of 50.16% (p<0.001). For longitudinal strain, highest reductions in error percentages were observed in anterior mid and apical regions (see segments 2–4 in Fig. 5 (b)). Table III summarizes quantitative comparison results between NCC and Bayesian at ES.

Fig. 5.

Segmental and global strain errors (%) at end-systole. (a) ES radial strain error (%), (b) ES longitudinal strain error (%).

TABLE III.

Comparison Of End-Systole Strain Error (%) Between NCC And Bayesian.

	Radial (Err) Strain Error (%) *		Longitudinal (Ell) Strain Error (%) **
	NCC	Bayesian	NCC	Bayesian
Segment 1	54.63 ± 3.33	38.51 ± 0.86	24.20 ± 2.09	26.66 ± 1.76
Segment 2	65.36 ± 8.57	28.87 ± 1.88	55.19 ± 15.86	31.22 ± 0.98
Segment 3	217.90 ± 39.97	67.35 ± 5.39	182.53 ± 38.48	56.18 ± 3.39
Segment 4	367.03 ± 28.97	197.45 ± 6.63	193.51 ± 32.16	64.29 ± 6.10
Segment 5	58.40 ± 5.93	48.90 ± 8.79	37.96 ± 5.88	43.60 ± 4.67
Segment 6	39.28 ± 4.97	36.16 ± 2.88	24.25 ± 3.53	23.90 ± 3.42
Global	98.57 ± 8.50	50.39 ± 1.55	77.92 ± 8.77	38.83 ± 2.57

^*Segments 5 and 6 did not show statistically significant difference with p<0.001.

^**Segments 1, 5 and 6 did not show statistically significant difference with p<0.001.

Figures 6 and 7 summarize results from comparative segmental analysis between NCC and Bayesian for radial and longitudinal strain estimations respectively. Figures 6 (a)–(f) represent FEA and estimated radial strain curves while Fig. 7 (a)–(f) represent longitudinal strain curves for the six segments. In Fig 6 (a)–(f), FEA radial strain curves (shown in black) demonstrated positive peak systolic strains in all six segments indicating cumulative radial thickening over a cardiac cycle. Strain curves estimated using Bayesian regularization (shown in blue) exhibit good agreement with FEA results in all six segments with positive peak systolic strains and very small standard deviations over realizations. On the contrary, unregularized NCC strain curves (shown in red) showed good agreement with FEA results in the posterior and anterior basal regions (segments 1, 5 and 6) as shown in Fig 6 (a), (e) and (f). But NCC strain curves significantly deviated from FEA results for the anterior mid and apical regions (segments 2, 3 and 4) with negative peak systolic strain at segment 4. A similar trend in the estimation performance was observed for the longitudinal strain curves. Good qualitative agreements between FEA and Bayesian strain curves were seen in all segments. NCC produced good strain curves in posterior and anterior basal regions (segments 1, 5 and 6) as shown in Fig 7 (a), (e) and (f) but significantly deviated from FEA results for the anterior mid and apical regions (segments 2, 3 and 4). NCC radial and longitudinal strain curves also exhibited higher standard deviations over scatterer realizations when compared to regularized curves. Overall, Bayesian regularization provided better quality strain curves in all six segments and showed very good qualitative agreement with FEA results when compared to the unregularized strain curves utilizing only NCC processing.

Fig. 6.

Regional radial strain curves from (a) Anterior Base, (b) Anterior Mid, (c) Anterior Apex, (d) Posterior Apex, (e) Posterior Mid and (f) Posterior Base segments respectively. These segments are referred as segments 1–6 respectively in the discussion.

Fig. 7.

Regional longitudinal strain curves from (a) Anterior Base, (b) Anterior Mid, (c) Anterior Apex, (d) Posterior Apex, (e) Posterior Mid and (f) Posterior Base segments respectively.

Figure 8 (a) presents the segmental and global TTR radial strain error (%) results and Fig. 8 (b) summarizes the TTR longitudinal strain error (%). Statistically significant TTR radial strain error reduction of 78.38 % (p<0.001) globally was observed with highest improvements in anterior mid and apical regions (see segments 2–4 in Fig. 8 (a)) after incorporating Bayesian regularization. Benefits of Bayesian regularization was also clearly evident in TTR longitudinal strain error (%) results (Fig. 8 (b)) with statistically significant global error reduction of 86.67 % (p<0.001). For longitudinal strain, highest reductions in error percentages were observed in the apical region (see segments 3 and 4 in Fig. 8 (b)). Although, we observe a 2.17 × increased error for the anterior basal segment (segment 1) with Bayesian regularization, this reduction of performance was negligible when compared to improvements in the apical region where NCC has 15.44 and 12.36 × TTR longitudinal strain error (%) in the anterior apical and posterior apical segments respectively. Table IV summarizes the quantitative comparison results between NCC and Bayesian estimated strain curves.

Fig. 8.

Segmental and global TTR strain error (%) results. (a) Temporal radial strain error (%), (b) Temporal longitudinal strain error (%).

TABLE IV.

Comparison Of Temporal Total Relative Strain Error (%) Between NCC And Bayesian.

	TTR Radial (Err) Strain Error (%) *		TTR Longitudinal (Ell) Strain Error (%) **
	NCC	Bayesian	NCC	Bayesian
Segment 1	17.40 ± 4.97	11.81 ± 1.63	5.79 ± 1.22	12.62 ± 1.87
Segment 2	19.48 ± 5.69	6.09 ± 2.85	13.72 ± 8.02	19.28 ± 3.69
Segment 3	199.06 ± 63.33	37.23 ± 7.07	325.05 ± 153.76	21.05 ± 5.17
Segment 4	376.90 ± 156.96	55.68 ± 20.35	545.35 ± 144.52	44.11 ± 15.80
Segment 5	10.16 ± 9.02	11.74 ± 3.32	9.81 ± 2.71	10.45 ± 1.05
Segment 6	21.33 ± 14.03	29.62 ± 3.41	16.47 ± 0.72	14.58 ± 0.88
Global	107.39 ± 19.08	25.36 ± 5.93	152.70 ± 29.69	20.35 ± 3.77

^*Segments 5 and 6 did not show statistically significant difference with p<0.001.

^**Segments 2,5 and 6 did not show statistically significant difference with p<0.001.

B. In-vivo Murine Model Results

Figure 9 summarizes the displacement estimation results over a cardiac cycle. Figures 9 (a) – (c) show axial displacement maps of the entire myocardium at ES estimated using NCC, and for one iteration and three iterations of Bayesian regularization respectively. A visual analysis of the results shows that the estimated axial displacement vectors were consistent with the expected inward motion of myocardium during contraction. No qualitative difference in estimated axial displacement maps was observed for NCC and Bayesian regularized images. Figures 9 (d) – (f) illustrate lateral displacement maps at ES estimated using NCC, one iteration and three iterations of Bayesian respectively. All approaches provided displacement estimations consistent with the inward myocardial deformation during systole. However, application of Bayesian regularization (both one and three iterations) resulted in higher lateral motion estimation at the posterior wall (Fig. 9 (e) and (f)) when compared to NCC in Fig 9 (d).

Fig. 9.

ES in-vivo axial displacement images (a) without regularization, with (b) one iteration and, (c) three iterations of Bayesian regularization respectively. Lateral displacement images (d) without regularization, with (e) one iteration and, (f) three iterations of Bayesian regularization respectively.

In Figure 10, we present strain estimation results for the same mouse over a cardiac cycle. Figures 10 (a) – (c) show radial strain images of the entire myocardium at ES estimated using NCC, and one and three iterations of Bayesian regularization respectively. Radial wall thickening was observed in the estimated radial strain results with all the methods. But, the NCC strain image exhibited some erroneous negative strain values in the basal segment of the posterior wall. Both one and three iterations of Bayesian regularization were able to correct these erroneous radial strain estimates shown using arrows in Fig. 10 (b) and (c). In general, better quality radial strain images were obtained using both one and three iterations of regularization (SNR_e = 6.89 and 4.76 dB respectively) compared to the NCC only strain image (SNR_e = 3.83 dB).

Fig. 10.

ES in-vivo radial strain images (a) without regularization, with (b) one iteration and, (c) three iterations of Bayesian regularization respectively. Longitudinal strain images (d) without regularization, with (e) one iteration and, (f) three iterations of Bayesian regularization respectively.

Figures 10 (d) – (f) show longitudinal strain images of the entire myocardium at ES estimated using NCC without regularization, along with one iteration and three iterations of Bayesian regularization respectively. Longitudinal strain images exhibit ventricular shortening at ES based on negative strain values. NCC provided incorrect positive longitudinal strain values in significant portions of anterior wall and basal segment of posterior wall indicated with yellow arrows in Fig 10 (c). With one iteration of Bayesian regularization, improvements in the posterior and apical walls were observed but the anterior wall still suffered from erroneous positive strain values (shown with yellow arrows) as seen in Fig 10 (e). Significant improvement of longitudinal strain estimation was achieved with three iterations of Bayesian regularization (Fig 10 (f)). Positive strain estimates in the anterior wall observed with only NCC and one iteration of regularization were corrected using three iterations of regularization. Notable improvement was also seen in the basal segment of the posterior wall with more uniform negative strain values. Highest ES SNR_e was achieved with three iterations (SNR_e = 4.58 dB) compared to one iteration and no regularization (SNR_e = 1.62 dB and 2.29 dB respectively). A small portion of apical wall shown in red arrows in Fig 10 (e) and (f) indicated positive strain values in the regularized strain images when compared to the NCC image shown in Fig 10 (c). However, this effect was negligible when compared to the improvement achieved by utilizing Bayesian regularization.

Figure 11 presents the results for temporal segmental radial and longitudinal strain curves estimated with and without regularization. Radial strain curves are shown in Fig. 11 (a) and (c). Both approaches were able to resolve radial myocardial wall thickening by exhibiting peak positive radial strains at ES. But regularized radial strain curves in the posterior apical and basal segments (segments 4 and 6 respectively) were smoother compared to unregularized curves indicated using arrows in Fig. 11 (c). Figures 11 (b) and (d) show estimated longitudinal strain curves. The regularized strain curves showed negative peak systolic longitudinal strain indicating ventricular shortening during systole. We observed smooth temporal variation of strain in all six segments over the cardiac cycle as expected from a healthy murine model. But we observed deterioration of performance without regularization as shown in Fig. 11 (d). All six segments resulted in noisier strain curves when compared to the regularized cases. Erroneous positive longitudinal strain values in segments 2 and 4 as high as 10% was observed towards the end of the cardiac cycle as indicated using arrows Fig 11 (d). Higher variations in peak systolic strain values were also observed in Fig 11 (d) compared to Bayesian regularized strain curves in Fig 11 (b). Overall, the benefit of Bayesian regularization for estimating regional longitudinal strain curves is clearly visualized from Fig. 11 (b) and (d).

Fig. 11.

In-vivo segmental radial and longitudinal strain curves. (a) Radial and (b) Longitudinal strain curves with Bayesian regularization, (c) Radial and (d) Longitudinal strain curves with no regularization.

In Figure 12, we present comparison results between global strain estimation using regular NCC, Bayesian regularized NCC and speckle tracking echocardiography using VevoStrain. Fig. 12 (a) shows radial strain results while Fig. 12 (b) shows longitudinal strain results. In Fig 12 (a), all three methods provided positive peak systole strain magnitudes with close resemblance in the overall shape indicating radial wall thickening. For longitudinal strain results in Fig 12 (b), negative peaks systolic strain values were observed in estimation from all three methods indicating ventricular shortening at end systole. However, variations of the strain magnitude among CSI and CE approaches were observed.

Fig. 12.

Comparison between cardiac strain estimation between cardiac elastography and speckle tracking echocardiography using VevoStrain (FUJIFILM VisualSonics). (a) Radial strain results and (b) Longitudinal strain results.

C. Computational Complexity

The algorithm was implemented in MATLAB (Mathworks Inc., MA) using a standard gateway interface (MEX) in conjunction with C++ and CUDA for cross-platform acceleration. GPU acceleration of computationally intense sections such as Bayesian Regularization and Sinc subsample estimation was achieved by writing a mex wrapper for the original CUDA implementation reported in [74]. All tests were performed on an Intel(R) Xeon(R) CPU E5–2640 v4 at 2.40 GHz, while the CUDA C++ code was run on a Tesla K40c GPU belonging to the Kepler architecture with compute capability 3.5. In the simulation study with RF data dimension of 8192×500 samples, the proposed algorithm with and without regularization takes 128.63 and 66.67 secs respectively (mean value) to calculate the displacement map between two consecutive frames with parameters presented in Table II. In the in-vivo study with final RF data dimension of 6016×440 samples, inter-frame displacement estimation execution time with and without regularization was 91.83 and 55.10 secs respectively.

IV. Discussion

A. FEA Simulation Study

In this study, we investigated the feasibility of using a multilevel motion estimation framework using NCC coupled with Bayesian regularization for CE. The primary findings of the FEA simulation study can be summarized as follows.

1. Cumulative axial and lateral displacements at ES estimated using both NCC and Bayesian showed good qualitative agreement with FEA results. But Bayesian regularization provided smoother displacement estimates (Figure 3).

2. Bayesian regularization improved radial and longitudinal strain images estimated at ES when compared to NCC alone (Figure 4). Highest improvements were observed in apical segments (segments 3 and 4). Regularized images had fewer negative radial and positive longitudinal strain values respectively.

3. Quantitative analysis of ES strain images revealed that ES radial strain error (%) decreases from 98.57 ± 8.50% without regularization to 50.39 ± 1.55 % with regularization (Figure 5). Similarly, the ES longitudinal error reduces from 77.92 ± 8.77 % without regularization to 38.83 ± 2.57 % with regularization. In both cases, Bayesian regularization resulted in statistically significant error reduction (p<0.001) globally (Table III).

4. Bayesian regularization improved the quality of radial and longitudinal temporal strain curves when compared to NCC (Figure 6 and 7). In anterior and posterior walls, both approaches provided strain curves of comparable quality (segments 1,2,5 and 6). NCC alone fails to estimate strain for apical segments (segments 4 and 5) with higher deviation from FEA results. Use of Bayesian regularization significantly improved NCC results (Fig. 6 (c) and (d), Fig 7 (c) and (d)). Bayesian regularization also provided consistent estimation with lower standard deviation (see error bars in Fig. 6 and 7).

5. Temporal radial strain error (%) decreased from 107.39 ± 19.08 % without regularization to 25.36 ± 5.93 % with regularization (Figure 8). Similarly, the temporal longitudinal error reduces from 152.70 ± 29.69 % without regularization to 20.35 ± 3.77 % with regularization. In both cases, Bayesian regularization provided statistically significant error reduction (p<0.001) globally (Table IV).

Bayesian regularization resulted in smoother displacement vectors when compared to utilizing only NCC (arrows in Figure 3). The Bayesian inference process utilizing a regularized similarity metric (from NCC map) incorporates information from neighboring location as a likelihood function significantly reduces errors by not allowing for any abrupt changes resulting in a smooth deformation field. NCC alone can result in some incorrect displacement discontinuities amplified into noisier strain images by the gradient operation. In addition, the lower spatial resolution and lack of phase information in lateral direction pose significant difficulty in lateral motion tracking with NCC [44]. However, within this limitation Bayesian regularization provided reasonable lateral motion estimation.

We observed the highest improvement in strain estimation from Bayesian regularization for the apical segments (segments 3 and 4). In these segments radial and longitudinal strains had significant contributions from shearing components of Eand we hypothesize that the smoothly varying deformation field with regularization contributed to better estimation of these components. Although we obtained significant improvements some of these errors were not fully corrected using the approach. Several factors might contribute to this. First, the FEA model used in this study contains all deformation information (compression, translation, and torsion) derived experimentally from a canine heart resulting in a realistic complex 3-D deformation model [2, 65, 67]. Imaging this 3-D deformation using 2-D image planes result in significant “out-of-plane” motion [7, 35]. This issue can be resolved by extending the proposed approach to 3-D image planes using 2-D matrix transducers. Secondly, there exists spatial variations in the elasticity of the myocardial wall (see Figure 4 (a) and (d)). Our proposed algorithm attempts to remove noisy NCC estimates from a given similarity metric using guidance from neighboring values. However, in some regions most of the initial similarity metric estimates may be noisy resulting in a noisy final estimate even after regularization. Finally, as mentioned in the discussion on lateral displacement estimation, lower spatial resolution and lack of phase information also introduces errors in lateral displacement estimation. Strain estimation is performed on the cumulated displacement. Any small error in inter-frame displacement estimates are propagated through the accumulation process. Strain estimation also has a tendency for amplification of displacement estimation noise [79]. Thus, any small error in lateral displacement estimates will cause significantly noisier lateral strain estimates. These issues indicate that motion estimation in lateral direction requires additional improvement.

In the proposed framework, regularization is performed in an iterative manner and the performance of the algorithm is dependent on correctly chosen number of iterations. For the FEA study, we found that a single iteration was sufficient to improve image quality. The number of iterations should be increased with caution as over-regularization might adversely affect the image quality resulting into “banding” artifacts due to over smoothening. Figure 13 illustrates the effect of over-regularization on a longitudinal strain image at ES with three iterations of Bayesian regularization.

Fig. 13.

Effect of overregularization in strain estimation. End-systole longitudinal images with (a) one iteration and (b) three iterations of Bayesian regularization. Overregularization resulted into “banding” artifacts in the estimated strain image.

B. In-vivo Healthy Murine Model

Myocardial contraction in-vivo during systole was clearly visible in axial displacement maps (Fig 9 (a) – (c)), with anterior wall moving away from the transducer (red shades) and posterior wall moving towards the transducer (blue shades). Estimated lateral displacement maps from all three methods were consistent with the healthy myocardial contraction during systole (Fig 9 (d) – (f)). Both one and three iterations of Bayesian regularization provided smoother lateral motion estimations in the apical region. This contributed to improved radial and longitudinal strains. Physiologically inaccurate radial and longitudinal strain values incurred using NCC alone were corrected using Bayesian regularization. Optimal estimation performance required three iterations for the in-vivo murine model. This emphasizes the importance of correctly choosing the number of iterations for Bayesian regularization. In clinical practice, quantitative estimates of the SNRe could be utilized to determine the optimal number of iterations. In future work, we will look into maximization of the conditional expected value of the SNRe [43, 85] to determine the optimal number of iterations.

In addition, we were able to resolve to clinically relevant details [6] from longitudinal strain curves such as peak positive strain, ES strain and post-systole strain (see segment 1 and 6 in Figure 11 (b)) with Bayesian regularization. These details were suppressed by noise in the NCC only longitudinal strain curves (Figure 11 (d)). In Fig. 11 (d), we observed that strain curves from segments 2 and 3 were noisier. One potential reason for this finding in the apical region of this mouse is that an acoustic shadowing artifact (most likely from a rib or the sternum) is present in the image. As reported in literature [86], acoustic shadowing may result in underestimation of strain and/or the appearance of a regional wall motion abnormality. This made tracking more challenging in those segments and consequently lower quality strain curves.

Comparison of estimated strain curves using CE (NCC and Bayesian) and VevoStrain showed an overall shape agreement but variation in strain magnitudes. Strain estimation in VevoStrain is based on speckle tracking echocardiography, which calculates strain by motion tracking from ultrasound B-mode imaging sequences. In contrast, our proposed method uses ultrasound radio-frequency (RF) signals, which contains additional phase information when compared to B-mode images. A previous study from our group reported that RF signals results into more accurate strain estimates when compared to envelope/B-mode signals [43]. This could explain the magnitude variation between the two methods. Overall, comparable performance of the proposed method against a commercial system shows its potential for in-vivo CSI.

Some of the previously reported regularization approaches for elastography used assumption of continuous and smooth displacement fields, enforcing an explicit smoothness constraint as a regularizer [51, 52, 54–58]. This assumption limits the application of these approaches to CE where discontinuous deformation is expected (opposing movement of anterior and posterior wall). Incorporation of Bayesian regularization in our multi-level framework provides a balance between discontinuous motion estimation and error correction. This enables successful application of our framework for CE as shown in FEA simulation and in-vivo study.

In-vivo imaging for murine models was performed with a frame rate of 235 Hz which is comparatively lower than the literature reported values for CE in murine models with 1D tracking or plane wave imaging approaches [13, 36, 87]. Our group has previously demonstrated that a frame rate ten times the heart rate provides high SNR_e and reliable strain estimation using RF signals in a phantom study [88]. The murine model in this study had a heart rate of 5 beats per second and was imaged with a frame rate of 235 Hz, leading to 47 frames in a cardiac cycle. Our 2D hierarchical multi-level NCC approach provides deformation tracking for reliably estimating maximum strains up-to 5% axial and 2.5% lateral strain between consecutive RF frames [50, 61], whereas 1-D NCC kernels with 2-D search approaches fail due to increased signal decorrelation in this applied strain range. Using our multi-level approach with 2D kernels, we are able to reduce kernel dimensions to accurately track these high strains. Reliable strain estimation in human RF data sets with comparatively lower frame rates was previously reported by our group [4, 37, 38, 43] using this multi-level approach without regularization. Our approach with Bayesian regularization in this paper provides reliable polar strain estimation for the in-vivo murine model. However, with higher frame rates, we anticipate additional improvement in strain estimation using the proposed approach.

One limitation of our study is the discrepancy between the transducer center frequency of simulation and in-vivo experiments. The simulation study was performed based on the 3-D deformation model of a canine heart developed by the Cardiac Mechanics Research Group at the University of California San Diego (UCSD) [65]. The imaging field of view was 80 mm × 100 mm. In the ultrasound imaging simulation, we also modelled an attenuation coefficient of 0.5 dB/cm-MHz. If simulation was performed with 40 MHz center frequency, we will not be able to image the posterior part of heart due to depth-dependent attenuation co-efficient. Thus, the imaging simulation was performed using 8 MHz rather than the 40 MHz center frequency in the in-vivo study. If we are able to obtain 3-D deformation models for a mouse heart, we would be able to extend the simulation to utilize a 40 MHz center frequency.

V. Conclusion

A complete cardiac elastography framework based on hierarchical motion estimation with a Bayesian regularization scheme was presented in this paper. Feasibility of the proposed method was explored both with FEA simulation study and an in-vivo healthy murine model. Benefit of Bayesian regularization was demonstrated by performing both a qualitative and quantitative comparative study with and without regularization. Both FEA simulation and in-vivo case study showed that statistically significant improvement in myocardial polar strain estimation can be achieved with incorporation of Bayesian regularization. In-vivo estimation results were comparable with commercially available strain estimation software. In conclusion, proposed Bayesian regularization based cardiac elastography approach can resolve cardiac deformation and strain with improved and consistent performance.

Biography

Rashid A1 Mukaddim (S’ 14) received the B.S. degree in electrical and electronic engineering from the Islamic University of Technology (IUT), Board Bazar, Gazipur, Bangladesh, in 2014, and the M.S. degree in electrical and computer engineering from the University of Wisconsin-Madison, Madison, WI, USA, in 2018, where he is currently pursuing the Ph.D. degree in electrical and computer engineering.

His current research interests include cardiac elastography, photoacoustic imaging and their applications in pre-clinical and clinical imaging.

Nirvedh H. Meshram (S’16) received the B.Tech. degree in electronics engineering from the Veermata Jijabai Technological Institute, Mumbai, India, in 2014, and the M.S. degree in electrical engineering from the University of Wisconsin–Madison, Madison, WI, USA, in 2016, where he is currently pursuing the Ph.D. degree in electrical engineering.

His current research interests include elastography, ultrasound imaging, graphics processing unit computing, and medical image analysis.

Carol Mitchell received the bachelor’s degree from the University of Iowa, Iowa City, Iowa in 1993 and her graduate degrees from the University of Kansas City, Missouri-Kansas City, Missouri (M.A. 1996, Ph.D. 2002). From 1987–1990 she worked as a diagnostic medical sonographer at the University of Iowa Hospital and Clinics, Iowa City, Iowa, USA. From 1990–1999 she worked as the technical director and perinatal outreach coordinator at Saint Luke’s Hospital, Kansas City, Missouri, USA. From 1999–2010 she was the program director at the University of Wisconsin Hospital and Clinics School of Diagnostic Medical Sonography. She served as the director of the diagnostic medical imaging programs at the University of Wisconsin-Milwaukee, Milwaukee, Wisconsin, USA from 2010–2014. She currently is an assistant professor in the Department of Medicine at the University of Wisconsin – Madison, Madison, Wisconsin, USA and has an affiliate appointment with the Department of Medical Physics. Her current research interests include carotid plaque strain imaging, plaque characterization (grayscale analysis), plaque volume measurement techniques, cerebrovascular hemodynamics, protocol development and implementation of new technologies into clinical practice.

Dr. Mitchell is a fellow in the American Society of Echocardiography and the Society of Diagnostic Medical Sonography.

Tomy Varghese (S’92–M’95–SM’00) received the B.E. degree in instrumentation technology from the University of Mysore, Mysore, India, in 1988, and the M.S. and Ph.D. degrees in electrical engineering from the University of Kentucky, Lexington, KY, USA, in 1992 and 1995, respectively. From 1988 to 1990, he was an Engineer with Wipro Information Technology Ltd., Mysore. From 1995 to 2000, he was a Post-Doctoral Research Associate with the Ultrasonics Laboratory, Department of Radiology, University of Texas Medical School at Houston, Houston, TX, USA. He is currently a Professor with the Department of Medical Physics, University of Wisconsin-Madison, Madison, WI, USA. His current research interests include elastography, ultrasound imaging, quantitative ultrasound, detection and estimation theory, statistical pattern recognition, and signal and image processing applications in medical imaging.

Dr. Varghese is a fellow of the American Institute of Ultrasound in Medicine, and a member of the American Association of Physicists in Medicine and Eta Kappa Nu.