A model-based time-reversal of left ventricular motion improves cardiac motion analysis using tagged MRI data
© Alrefae et al; licensee BioMed Central Ltd. 2008
Received: 14 June 2007
Accepted: 19 May 2008
Published: 19 May 2008
Myocardial motion is an important observable for the assessment of heart condition. Accurate estimates of ventricular (LV) wall motion are required for quantifying myocardial deformation and assessing local tissue function and viability. Harmonic Phase (HARP) analysis was developed for measuring regional LV motion using tagged magnetic resonance imaging (tMRI) data. With current computer-aided postprocessing tools including HARP analysis, large motions experienced by myocardial tissue are, however, often intractable to measure. This paper addresses this issue and provides a solution to make such measurements possible.
To improve the estimation performance of large cardiac motions while analyzing tMRI data sets, we propose a two-step solution. The first step involves constructing a model to describe average systolic motion of the LV wall within a subject group. The second step involves time-reversal of the model applied as a spatial coordinate transformation to digitally relax the contracted LV wall in the experimental data of a single subject to the beginning of systole. Cardiac tMRI scans were performed on four healthy rats and used for developing the forward LV model. Algorithms were implemented for preprocessing the tMRI data, optimizing the model parameters and performing the HARP analysis. Slices from the midventricular level were then analyzed for all systolic phases.
The time-reversal operation derived from the LV model accounted for the bulk portion of the myocardial motion, which was the average motion experienced within the overall subject population. In analyzing the individual tMRI data sets, removing this average with the time-reversal operation left small magnitude residual motion unique to the case. This remaining residual portion of the motion was estimated robustly using the HARP analysis.
Utilizing a combination of the forward LV model and its time reversal improves the performance of motion estimation in evaluating the cardiac function.
Cardiac dysfunction is associated with a variety of cardiovascular conditions that lead to heart failure. Significant cardiac events such as those in infarcted or diabetic hearts are associated with impaired relaxation and contraction of myocardium or abnormalities in left ventricular (LV) wall motion [1–5]. Tagged magnetic resonance imaging (tMRI) is an established cardiac imaging modality used to visualize regional myocardial motion within the LV wall  and references therein. Application of tMRI combined with sensitive motion estimation techniques, such as harmonic phase analysis (HARP), has proven to be feasible and diagnostically valuable in evaluating the performance of normal or diseased hearts in live subjects using conventional global and regional measures [7–11]. In the presence of large motions, however, the current analysis techniques including HARP fail to accurately describe the absolute displacement of the myocardial tissue. This paper addresses this issue and offers a solution – providing motion estimates with improved performance even when the tissue is subject to large movements. The solution requires characterizing the LV wall motion using a simple time-dependent model and performing time-reversal of tMRI data prior to analysis using HARP or other motion tracking techniques. In the following, we describe the forward LV model and give details of its implementation and algorithms involved during a priori data processing steps. Next, we explain the time-reversal operation. Using examples with real and simulated data, we demonstrate how the time-reversal approach improves the computer-aided regional myocardial motion measurements by minimizing the errors made during the digital estimation process.
Tagged MRI employs electrocardiogram (ECG) gated acquisition. The imaging sequence contains initial saturation pulses followed by a repetitive image acquisition . At a specific time point between the QRS peaks of the ECG waveform, spatial modulation of magnetization pulses is applied to saturate the spins perpendicular to the imaging plane in the body. Then, a series of images is acquired repetitively in equal time intervals covering the whole period of the heart beat. The resulting images provide snapshot views of the heart along either its short axis or long axis at different phases of the cardiac cycle. Depending on the nature and orientation of the saturation pulses, the first image in the series contains dark parallel lines, known as tags, which may be organized in horizontal, vertical or in both directions in a grid fashion. In the grid organization, the tag lines define boundaries of square-shaped cells. Thickness and separation of the tag lines are set to the desired values prior to the initiation of the data acquisition. As the heart contracts or relaxes, the tag lines follow the motion of the underlying myocardial tissue. The changes in relative distances between myocardial tissues as they move during the cardiac cycle define the tissue deformation. Measuring the regional motion and quantifying the resulting deformation experienced by each cell provides a sensitive measure to assess the viability of the myocardium within the cell.
Quantification of the actual deformation in the LV wall involves accurately measuring the motion and relative displacements experienced by the myocardial tissue. For measuring the cardiac motion, HARP makes use of the information embedded in Fourier spectrum of the tMRI . The tag patterns give rise to spectral peaks, also known as harmonic peaks, in the spectrum. The peak corresponding to the fundamental frequency defined by the separation of the tagged lines is extracted by means of a band-pass filter. Two filters were designed to select the vertical or horizontal tag lines. Essentially, the inverse Hilbert transform of the filtered spectrum has two signal parts, namely magnitude H and phase ϕ. The resulting complex signal H can be written as H = He jϕ . While the magnitude is not of importance in motion quantification, the phase is considered to be the basis of the HARP analysis. The HARP is considered a material property of the moving cardiac tissue. Therefore, it can be used for tracking and quantifying the Lagrangian motion of myocardium.
where ∠ denotes phase angle and * denotes comlex conjugate.
An outstanding problem in cardiac motion estimation
In addition, statistical errors made during the estimation of large motions increase with the magnitude of the motion. Although, sophisticated signal processing approaches may improve the estimation performance by minimizing such errors, these methods also demand significantly increased computation time, as in the case of motion tracking algorithms employing window-based search techniques. In the presence of large motions, signal decorrelation between two image frames becomes a prominent factor in determining the errors. That is, a severe decorrelation between two windowed signals subsequently contributes to a motion estimation outcome with poor performance [13–15].
A potential solution
It is possible to compensate for the degrading effects of signal decorrelation when the tissue motion is large. In our previous work in elastography imaging, we introduced global and adaptive regional signal stretching methods with the ultimate goal of producing better performance in displacement estimation . Signal stretching procedures employed in our earlier works were, however, directionally linear and uniform in space. In ultrasonic elastography imaging, we demonstrated that the estimation performance is improved after globally decompressing the post-compression ultrasonic wave field composed of a backscattered signal received by each element of an array transducer by the same stretch value equivalent to the applied strain on the sample. In the adaptive stretching approach, the post-compression signals localized to a region within the wave field were expanded or compressed to increase the signal similarity with the pre-compression signals. Here, we adapt a similar strategy and introduce a time-reversal operation to improve the performance of tracking the LV wall motion by processing the tMRI data. The adaptation first requires modeling of the forward motion of the LV wall during systole, which is described in Appendix B. The model mimics the motion of the myocardial tissue by utilizing a spatiotemporal nonlinear mapping operator, T (Eq. B1). The operator involves a time-dependent parameter α accounting for the temporal changes in the LV size in the short-axis view at the mid-ventricle level. The function of the operator T can be thought of as radially contracting the image canvas I 1 by an amount defined by α.
The true displacement d of myocardial tissue at a spatial location in the LV wall can be decomposed into two components: d = Δ + δ.
Here, Δ denotes the bulk motion predicted by the time-reversal model. The term δ is the residual displacement that is not accounted for by the time-reversal operation and remains to be estimated between the image I 1 and the time-reversed (TR) image. However, because the residual displacement is smaller in magnitude than the true displacement, its estimation results in small estimates with smaller errors when standard motion estimation algorithms are used.
Obtain time-reversed image by applying T-1 to the desired image frame I i with the corresponding α i .
Calculate the amount of bulk motion Δ analytically from T-1.
Estimate the amount of residual motion δ between the time-reversed image TR = T-1 [I i ] and the first systolic frame I 1 using a method of preference for the motion estimation, e.g., HARP.
Add δ to Δ to get an estimate for the true total displacement d.
Methods and procedures
Cardiac MRI data from the mid-ventricle level were collected from four male Sprague-Dawley rats. The rats were anesthetized using 1.5% isofluorane in a mixture of air and oxygen (60% and 40% respectively) and scanned using a 9.4 T horizontal bore scanner (Varian Inc., Palo Alto, CA) and 60 mm radio frequency volume coil. ECG gated gradient echo based tagged images were captured from the short-axis view of the heart. The cardiac cycle was temporally resolved into ten equally incremented phases. The first five were the systolic frames. The following settings were used for the image acquisition: repetition time = 25 ms, echo time = 2.44 ms, number of averages = 1, field of view = 60 × 60 mm, image matrix = 256 × 256, slice thickness = 2.0 mm. The square grid tags had dimensions of width = 0.3 mm and separation = 0.8 mm. All experimental procedures were approved by the University of Kansas Medical Center Institutional Animal Care and Use Committee.
Construction of the LV motion model
The time-reversal operation T-1 was applied on an image frame I i with the corresponding optimal α i value. The resulting image TR was compared with the end-diastole image I 1 and the HARP analysis was performed in between.
Results and discussion
Forward LV motion model
This study was initiated to seek a simple, yet comprehensive model with a sufficient number of elements to accurately represent the LV wall motion at the mid-ventricle level. With a set of basic assumptions and simplifying approximations, we proposed the transformation in Eq. (B1) to model the myocardial motion during the systole. We specifically targeted the mid-ventricle for modeling the LV motion because this level experiences minimal amount of torsion and twisting during the heart beat, allowing the model to only account for the radial motion of the wall. We empirically constructed the model parameters using real tMRI data gathered from the hearts of four rats. This implementation involved data processing steps, which were described algorithmically in Appendix A. The initial step involved windowing and scaling the images to consistently depict the LV size and shape with the same dimensions at the end-diastole. This approach made the model versatile enough to analyze the LV motion in different species or minimize the inter-variation of the measurements when the same species is considered. This feature of the model increases its capacity by simulating the LV motion in both humans and animals.
The values (mean ± std) for the parameter α estimated using the data acquired from the control rats (n = 4).
12.0 ± 0.8
16.0 ± 0.6
19.0 ± 0.4
20.0 ± 0.8
Our model is also open to other clinical relevance. For example, strain calculation and its relation to the parameter α is a good area for future exploration in cardiac imaging research. Such a relation could, in principle, indicate a certain systolic dysfunction associated with certain diseases or abnormalities, such as diabetes. Similarly, another useful parameter well accepted by clinicians is the myocardial strain rate. By calculating the differences in strains exhibited by the LV tissues from one systolic phase to another, more insights may be gained regarding the myocardial behavior and its time dependence. These and other potential merits of the proposed motion model of the LV are left for future work.
The correlation coefficients computed to quantitatively assess goodness of fit between the real and simulated pseudo images for each rat.
Mean ± Std
0.83 ± 0.14
0.83 ± 0.17
0.85 ± 0.17
0.84 ± 0.17
A literature search reveals numerous approaches that have been employed to model the LV wall motion [16, 17]. Examples include techniques based on finite elements, finite differences, B-spline methods, and prolate spheroidal basis functions [18–20]. Such techniques have attained good results in describing the cardiac wall motion and computing useful clinical data like displacement and strain profiles. Nevertheless, to accurately describe the LV wall motion, these models require large numbers of parameters and intense numerical computations. The need for a simple, yet reliable model has therefore been met by the current implementation.
Limitations of the forward LV motion model
Despite its ability to describe the myocardial forward motion, our model suffers from two main limitations. The first limitation is that the model treats the entire myocardium as if it were contracting homogeneously. While such treatment greatly simplifies the problem, it ignores the heterogeneous nature of myocardial contractility. Evidently, contractility varies with the position of the myocardium such that tissues in the wall exhibit deformation profiles that differ from those of the septum. This variation in deformation is due to the fact that, depending on the location, some tissues have constraints that limit their motion while others do not. Accommodating this behavior with regional dependencies requires more complicated models to better describe the myocardial motion.
Another limitation of the model is its inability to handle twisting (torsion) motion that takes place in the basal and apical levels of the LV wall. Thus, the model is limited to work at the mid-ventricle level where the twist is minimal. Nonetheless, adding a twisting ability is plausible and computationally possible by having position-dependent angles to account for rotation in the transform operator in Eq. (B1). Such integrations are likely to increase the capability of the model to represent the myocardial motion at various levels of the LV in short axis views.
Although a typical cardiac cycle is composed of both systolic and diastolic phases, we chose to focus the model on systolic motion only. This approach was chosen to simplify the problem, knowing that we could easily extrapolate the analysis to include the diastolic motion if desired.
Application of the time-reversal of the LV model to aid the analysis of the cardiac motion using HARP
The simple prior time-reversal operation applied to a systolic image with the forward model of the LV wall motion accounts for the large displacements experienced by the myocardial tissue in sequentially-acquired tMRI data. The remaining smaller magnitude regional myocardial motions in the time-reversed image can be computed by an estimation technique selected by the user. Using this two-step approach, we showed that HARP analysis became more immune to phase wrapping and consequently the motion estimation performance is enhanced. Robust measurements obtained with our approach therefore allow better interpretation of the cardiac motion. This should enable computation of accurate strain fields to characterize the regional deformation aimed at evaluating the function, viability and pathological state of the underlying myocardial tissue.
Appendix A. Preprocessing of tMRI data
Appendix B. A simple model of the LV wall motion in short-axis view of heart
Current research efforts are focused on spatio-temporal modeling of the LV shape and motion. To aid these efforts and provide a model for the time-reversal operation in the main body of the paper, here, we introduce a mathematical construct as a means of quantitatively describing the LV wall motion in a beating heart.
Anatomical and structural characterization of the LV
For clinical analysis or evaluation purposes, the short axis view of the heart is divided into three sections, namely base, mid-ventricle and apex. This division is in accordance with the standardized myocardial segmentation published by the American Heart Association . To perform its pumping task, the LV and its myocardial fibers are organized in a complex architecture both anatomically and structurally within these three levels . As a result, the LV wall twists with respect to its long axis at the base and apex in opposing directions, and consequently the deformations at these levels are governed by torsion together with a combination of radial and circumferential strains. The motion at the mid-ventricular level, on the other hand, is relatively twist-free and hence the deformation in this region is described mainly by the radial and circumferential strains. Moreover, the patterns and degrees of the LV wall motion are also known to depend on the size of the ventricle at the level observed. At the level of the mid-ventricle, the LV cavity diameter experiences its greatest radial shortening (elongation) during the systolic (diastolic) phase. If all aspects of the LV motion are considered, prior knowledge and experimental observations suggest that its complete characterization with a model would be a challenging task to undertake. Nevertheless, if the mid-ventricle is chosen as the specific level of interest, a simple yet powerful model can be constructed. Due to the minimal torsion effects at the mid-ventricle, we propose a model that is capable of describing the LV wall motion comprehensively at this almost twist-free level.
The myocardial tissue deforms the same manner whether it is in the septum or outer LV wall.
Although a full cardiac cycle includes both systolic and diastolic phases, to keep the analysis simple, the model is built to mimic the systolic LV motion only. Nonetheless, it is straightforward to extrapolate the analysis to include the diastolic phase if desired. The digital cardiac images acquired sequentially in equally spaced time intervals in the systolic phases are identified by the frames F i for i = 1,2 ... 5, where i = 1 is the end diastole frame and i = 5 is the end systole frame.
The cardiac images are constructed on a virtual canvas with rubber sheet properties that can be digitally stretched according to a spatially varying deformation pattern. With this elastic feature, the canvas serves as a basis upon which the model acts through the application of a spatial perturbation in the coordinates of the myocardial tissue. Thus, such a spatial transformation describes the local myocardial motion. As shown below, the canvas' elastic behavior is exploited through the use of time-forward operations applied on the images to simulate the real myocardial motion.
Motion model of the LV
To accurately describe the myocardial motion, it is necessary to satisfy two main criteria observed in the MR experimental data, as discussed in the literature. First, the transformation function should exhibit radial dependence, so that image coordinates closer to the LV cavity's center are deformed more than those further away from the origin. This criterion is clearly satisfied in Figs. 13-b and 13-d. The bow-shaped behavior exhibited by the deformed lines indicates that the resulting deformation has radial dependence. Second, the transformation function causes an increase in the thickness of the LV walls, and a decrease in the size of the LV cavity as observed in the real systolic cardiac motion. This requirement is also met by the proposed model. The resulting donut in Fig. 13-d clearly exhibits increase in the wall thickness, and decrease in the cavity size, which are both desired features to accurately model the LV motion. In these regards, the parameter α alone can be seen as capable of mimicking the changes in the LV wall's diameter and thickness.
TA was supported by Kuwait Education Ministry scholarship and IVS was supported by American Heart Association Scientist Development award.
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