| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
|
|
|
|||||||
|
|||||||||
1 Section of Cardiovascular Medicine, Department of Internal Medicine, 2 Department of Diagnostic Radiology, and 3 Department of Electrical Engineering, Yale University School of Medicine, New Haven, Connecticut 06520-8042
 |
ABSTRACT |
|---|
 TOP
 ABSTRACT
 INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
|
|---|
A comprehensive three-dimensional (3-D) shape-based approach for quantification of regional myocardial deformations was evaluated in a canine model (n = 8 dogs) with the use of cine magnetic resonance imaging. The shape of the endocardial and epicardial surfaces was used to track the 3-D trajectories of a dense field of points over the cardiac cycle. The shape-based surface displacements are integrated with a continuum biomechanics model incorporating myofiber architecture to estimate both cardiac- and fiber-specific endocardial and epicardial strains and shears for 24 left ventricular regions. Whereas radial and circumferential end-systolic strains were fairly uniform, there was a significant apex-to-base gradient in longitudinal strain and radial-longitudinal shear. We also observed transmural epicardial-to-endocardial gradients in both cardiac- and fiber-specific strains. The increase in endocardial strain was accompanied by increases in radial-longitudinal shear and radial-fiber shears in the endocardium, supporting previous theories of regional myocardial deformation that predict considerable sliding between myocardial fibers.
myocardial strain; finite element analysis; fiber architecture; cardiac mechanics
| |
INTRODUCTION |
|---|
|
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
|
|---|
THE HEART UNDERGOES A COMPLEX three-dimensional (3-D) deformation with each cardiac cycle. Regional myocardial deformation (strain) is determined by several factors, including local material properties, local contractile function, and global forces such as left ventricular pressure. Many approaches (3, 5, 14, 36, 42-45) have been employed to evaluate the complex cyclic regional deformations of the heart. Most studies that have attempted to analyze the detailed deformations of the heart have employed methodology that permits analysis of only a very limited area of the myocardium. These approaches are highly invasive and involve implantation of arrays of either radio-opaque beads or sonomicrometers in the myocardium (9, 14, 15, 22, 29, 42-44). In the analysis of implanted radio-opaque beads, deformations are defined by analysis of the 3-D displacement of the beads by using biplane cine angiography. This approach offers reasonable spatial and temporal resolution, although it is quite tedious and can be confounded by errors associated with localization of the beads in 3-D space (14, 29). The sonomicrometry approach offers both higher spatial and temporal resolution but is also highly invasive (10, 15, 27, 36). In both approaches, implantation of the beads or sonomicrometers may themselves alter the pattern of regional deformation (14).
The development of magnetic resonance (MR) imaging tagging has enabled a more comprehensive noninvasive analysis of the complex deformations of the heart (2, 5, 45). This approach requires specific imaging sequences, which impart signal voids in the myocardium that are tracked over time. This approach generally provides analysis of regional deformation in fixed imaging planes. A few studies (5, 6) have attempted to derive 3-D deformations by integrating orthogonal image sets. These analyses demonstrated that deformation of the left ventricle is not uniform.
We have developed a comprehensive 3-D shape-based approach for quantification of regional myocardial deformations that can be applied to any high-resolution cine 3-D image set (11-13, 26, 33, 39). Local shape properties of the endocardial and epicardial surfaces are used to derive 3-D trajectories for a dense field of points over the entire cardiac cycle. These trajectories are then used to deform a mesh that represents the left ventricular myocardial volume. This analysis is embedded in a unifying framework by using a continuum biomechanics model. The resulting equations are solved with the use of the finite-element method. This shape-based approach for tracking regional myocardial deformation has been (39) validated in both 2- and 3-D, by comparing algorithm-derived trajectories with displacements derived with the use of implanted endocardial and epicardial markers. Recently, Papademetris (30) extended this validation for determination of regional myocardial strain by using the same implanted marker system.
We now extend this shape-based analysis of MR-derived displacements to compute regional cardiac- and fiber-specific strains and shears over the cardiac cycle for all regions of the left ventricle. In this study, we use shape-based surface displacements together with a continuum biomechanics model incorporating myofiber architecture to estimate endocardial and epicardial fiber-specific strains. The normal pattern of myocardial deformation was studied by using our shape-based approach in closed-chest dogs. Whereas circumferential and radial strains were fairly uniform, there was a significant apex-to-base gradient in longitudinal strain and radial-longitudinal shear. We also observed transmural epicardial-to-endocardial gradients in both cardiac- and fiber-specific strains. The increase in endocardial strains was accompanied by increases in radial-longitudinal shear and radial-fiber shears in the endocardium. Transmural gradients in cardiac- and fiber-specific strains and shears support previous theories of regional myocardial deformation, which predict considerable sliding between myocardial fibers (9).
| |
METHODS |
|---|
|
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
|
|---|
Experiments were performed on fasting adult mongrel dogs with approval of the Yale Animal Care and Use Committee and in compliance with the guiding principles of the American Physiological Society on research animal use. All dogs were anesthetized intravenously with 10-12 mg/kg of thiopental sodium (Pentothal, Abbott; Chicago, IL), intubated, and mechanically ventilated on a respirator with a mixture of halothane (0.5%-1.5%) and N2O and O2 (3:1). Arterial pH, PCO2, and PO2 were measured serially, and the ventilator was adjusted to maintain these parameters within the physiological range. An electrocardiographic lead was monitored continuously.
Surgical Preparation/Experimental Protocol
A femoral vein and both femoral arteries were isolated and cannulated for administration of fluids and drugs, pressure monitoring, and arterial sampling. A thoracotomy was performed in the fifth intercostal space. An incision was made in the pericardium, and the proximal left anterior descending coronary artery was isolated for placement of a snare occluder. This occluder was externalized and the chest was closed in layers.After surgical preparation was completed, the dogs were positioned in the MR scanner for imaging. An electrocardiographic limb lead was monitored continuously during MR imaging and used for gating. Resting MR images were completed in 1 h. Heart rate (HR) and aortic pressure (AoP) were recorded immediately before and after each complete image acquisition. Dogs subsequently underwent repeat MR imaging after coronary occlusion as part of a separate investigation. Dogs were euthanized with a bolus of potassium chloride after completing all imaging.
MR Image Acquisition
MR imaging was performed on a GE Signa 1.5 T scanner with version 4.8 GE software by using the head coil (26-cm diameter) for transmission and reception. Short-axis images through the left ventricle were obtained with the gradient echo cine technique by using the following parameters: echo time = 6 ms, repetition time = 40 ms, flip angle = 30°, 16 phases collected, 5-mm slices, matrix 256 × 256, two averages, and field of view = 40 cm. A total of 16 contiguous 5-mm-thick slices were collected by acquiring four sets of staggered short-axis slices (4 slices/set) with a separation gap of 20 mm and 5 mm offset. This sequence provides images with an in plane resolution of 1.64 × 1.64 mm for a 256 × 256 matrix and a 5-mm resolution perpendicular to the imaging plane. This sequence also provides excellent temporal resolution (16 frames/cardiac cycle,
40
ms/frame).
Image Analysis
All image analysis was completed off-line after the completion of each study with the use of a Silicon Graphics workstation (Octane, R10000, 195 MHz, 128 MB RAM).Visually interactive semiautomated 3-D segmentation of MR images. A semiautomated technique for left ventricular endocardial and epicardial boundary segmentation was utilized to improve reproducibility of surface contour generation. A software package was developed at Yale University specifically for this purpose on the Silicon Graphics platform (31). The package has an intuitive user interface and can simultaneously display orthogonal views of the 3-D image and multiple surface sections as well as multiple 3-D surface renderings from any angle. All components of the display can also be shown in cine mode. The colors and transparency of the surfaces can be edited to allow the user to display one surface inside another. After the contours in an initial slice are defined, the algorithm automatically propagates contours to adjacent slices. Defined contours are used as an initialization for the subsequent or preceding time frames, and therefore contours are also propagated through each time frame (7, 8). All contours were reviewed by superimposing them on the corresponding image slices, and, if necessary, corrected by using simple manual point and click operations within the platform. Contours are then assembled into complete endocardial and epicardial surfaces within this platform (31) by using Delaunay triangulation.
Shape-based displacement computation. Once the surfaces are extracted, we compute the principal curvatures for local regions on each surface at each time point. The initial frame-to-frame correspondences are then established between points on the surfaces by using a shape-tracking approach. For each point on a surface in the initial time frame, a search window is generated on the associated surface of the next time frame containing all plausible candidate corresponding points. We define the corresponding point as the point in the search window that has the closest shape properties to the original point. The shape properties are defined by using the curvatures of local surface regions (see APPENDIX). The strength and uniqueness of each local match was preserved as a confidence measure for the match (38). Regional endocardial and epicardial displacements were computed by using this previously described shape-based surface tracking approach for surfaces derived from two temporally successive images (12). We have previously validated this approach for tracking points on the surface of the heart by comparing shape-derived trajectories with displacements derived from endocardial and epicardial surface markers (39).
Deformation modeling and analysis.
We have developed a framework from which we can compute myocardial
deformation by integrating shape-tracked surface displacement data with
knowledge about the mechanical properties of the tissue. The details of
this approach were summarized in a previous publication (32) and outlined in the APPENDIX. We model
the imaged 3-D section of the myocardium as a transversely isotropic
elastic solid continuum with preferential stiffness along fiber
directions as defined by Guccione and McCulloch (19). This
approach results in a set of partial differential equations, which
provide relations between the displacements throughout the left
ventricle. These equations are solved with the use of the
finite-element method (21). The first step in this method
is to divide the continuous structure of the object into finite pieces
or elements to construct a finite element mesh. The equations are
assembled in matrix form as KU = A(Ue
U), where K is the stiffness matrix
representing the material properties of the tissue, A is
loading matrix representing the confidence in the original
shape-tracking-derived nodal displacements Ue
(where available), and U is the nodal displacement vector
field, which we are trying to estimate. This equation is solved to find
the displacement field U, which can then be differentiated
to compute strain.
Strain analysis.
In the finite-element discretization, the displacement field in each
element is described by u = [N][u], where [N] is the
interpolation matrix and [u] is the vector of
displacements at the nodes. The strain 
is computed as = [B][u], where [B] is the strain
to displacement Jacobian matrix. This produces strains in the global coordinate space. Strains were computed for each image frame relative to end diastole. The strains can be subsequently rotated to either a
fiber-specific or a cardiac-specific coordinate system by using an
appropriate rotation matrix.
Methods of Analysis
Application of our shape-based algorithm with finite element analysis and continuum mechanics provided an estimate of systolic and diastolic strains and shears for multiple small regions of the left ventricle. For purposes of describing the normal pattern of left ventricular deformation, the left ventricle was divided into three major short-axis slices (apical, midventricular, and basal), excluding the true apex and the base of the left ventricle at the membranous septum. Each slice was then divided into 8 circumferential wedges, resulting in 24 myocardial segments. Regional systolic myocardial strains were obtained for each of 24 "pie-shaped" myocardial segments. These transmural sections were then further divided into endocardial and epicardial segments. The end-systolic frame was determined for each dog by selecting the 3-D image frame with the smallest global left ventricular volume. Principal strains were computed for each image frame within the systolic period relative to the end-diastolic reference frame. Regional strains and shears were then derived with the use of two different axis systems. Strains were first generated with the use of a standard cardiac-specific axis, whereby the orthogonal strains were radial, circumferential, and longitudinal. The following cardiac-specific shears were also computed: radial-circumferential, radial-longitudinal, and circumferential-longitudinal. Strains were also rotated to fiber-specific axes previously described by Guccione and McCulloch (19) derived from histological studies by Streeter et al. (40), whereby the orthogonal strains and shears were cross-fiber strain, radial strain, fiber strain, cross-fiber-radial shear, cross-fiber-fiber shear, and radial-fiber shear. Regional myocardial strain data derived by using the two different coordinate systems were analyzed separately.Statistical Analysis
By using the univariate procedure in Statistical Analysis Systems, it was determined that utilizing a normal approximation for the distributions of strain was appropriate. Analyses of variance (ANOVA) were performed to determine if there were significant differences in strain between the apical region, the midventricular region, and the basal region of the heart. ANOVA were also performed to determine if there were significant differences in strain between the eight circumferential regions of the heart (across the entire heart) as well as for each of the three slices separately. For all of these analyses, Tukey's test (honestly significant difference test) was performed on all main-effects means to determine significant differences. In comparison of endocardial and epicardial differences, paired Student's t-tests were performed on average values for each dog. All data are presented as means ± SD. A level of significance,
= 0.05, was utilized for all of the
analyses performed.
| |
RESULTS |
|---|
|
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
|
|---|
Eight dogs completed the MR imaging protocol. Hemodynamic parameters and cardiac rhythm remained stable during the MR image acquisition. The HR and systolic AoP immediately before the MR acquisition (HR, 93 ± 22 beats/min; AoP, 108 ± 14 mmHg) were not significantly different from that obtained at the completion of the acquisition (HR, 94 ± 25 beats/min; AoP, 107 ± 20 mmHg). All image analyses were performed on all dogs.
3-D Maps of Cardiac- and Fiber-Specific Strain and Shear
Regional differences in myocardial systolic strains and shears were observed in normal hearts and are summarized below. An example of typical cardiac- and fiber-specific strain and shear patterns are shown for short-axis cutaway cross sections (see Fig. 1) at several different times points during the systolic interval. All six strain-tensor components were computed for both cardiac- and fiber-specific axes. The radial strain is common to both axes systems. The volume of the left ventricle can be cut and viewed along any axis to visualize the entire transmural strain pattern.
|
Analysis of Transmural Strain and Shear
Cardiac-specific strains and shears.
Figure 2 illustrates the average regional
transmural end-systolic cardiac-specific strains (Fig. 2A)
and shears (Fig. 2B), which were computed for each of the 24 myocardial segments (3 slices and 8 radial sectors per slice). Radial
(0.21 ± 0.03) and circumferential (0.10 ± 0.01) strains
were fairly uniform over the left ventricle. A significant
(P = 0.0001) apex-to-base gradient in transmural
longitudinal strain was observed (apical, 0.02 ± 0.02;
midventricle, 0.01 ± 0.01; and basal, 0.03 ± 0.01).
Apical segments tended to lengthen along the long axis of the heart, whereas basal segments demonstrated longitudinal shortening.
|
0.21 ± 0.05; midventricle,
0.11 ± 0.03; and basal, 0.001 ± 0.06; P = 0.0001). There appeared to be a hinge point in the basal septum in the
region of the membranous septum.
Fiber-specific strains and shears.
Figure 3 illustrates the average
variation in regional transmural end-systolic fiber-specific strains
(Fig. 3A) and shears (Fig. 3B). Fiber
(0.05 ± 0.01) and cross-fiber (0.06 ± 0.02) strains
were similar in magnitude and also fairly uniform around the left
ventricle. There were small but significant apex-to-base differences in
both fiber (apical, 0.04 ± 0.01; midventricle, 0.05 ± 0.01; and basal, 0.06 ± 0.01; P = 0.002) and
cross-fiber strain (apical, 0.04 ± 0.01; midventricle,
0.06 ± 0.01; and basal, 0.08 ± 0.01; P = 0.0001).
|
0.17 ± 0.02;
midventricle, 0.07 ± 0.03; and basal, 0.02 ± 0.05;
P = 0.0001) and fiber-radial (apical, 0.07 ± 0.03; midventricle, 0.06 ± 0.02; and basal, 0.03 ± 0.04; P = 0.0005) shears were also observed.
Analysis of Endocardial and Epicardial Strain and Shear
Average endocardial and epicardial cardiac-specific and fiber-specific strains and shears were computed for each dog for all of the predefined 24 left ventricular segments. Overall average endocardial and epicardial strain tensors were computed from the dog averages.Cardiac-specific endocardial and epicardial strains and shears.
Figure 4 provides a summary of the
overall endocardial and epicardial cardiac-specific strains and shears.
There were significant endocardial-epicardial gradients in both radial
(endocardial, 0.26 ± 0.09; epicardial, 0.17 ± 0.07;
P = 0.0009) and circumferential (endocardial,
0.14 ± 0.04; epicardial, 0.07 ± 0.02; P = 0.0002) strains. As previously described with the use of a variety
of techniques, greater radial and circumferential strain occurred in
the subendocardial region. These differences in strains between the endocardial and epicardial segments were associated with small but significant differences in all three shears.
|
Fiber-specific endocardial and epicardial strains and shears.
Figure 5 summarizes of the
endocardial and epicardial fiber-specific strains and shears. There was
slightly greater strain along the fiber direction in the epicardium.
However, a prominent endocardial-epicardial gradient in cross-fiber
strain was observed (endocardial, 0.11 ± 0.03; epicardial,
0.02 ± 0.01; P = 0.0001). This increase in
endocardial cross-fiber strain was associated with large
fiber-cross-fiber (endocardial, 0.09 ± 0.02; epicardial, 0.01 ± 0.01; P = 0.00002) and radial-fiber
(endocardial, 0.10 ± 0.05; epicardial, 0.03 ± 0.02;
P = 0.0002) shears, suggesting that much of the
deformation in the endocardium can be attributed to cross-fiber sliding
or changes in fiber separation. The increase in endocardial deformation
with respect to the epicardium does not appear to be attributable to
increases in endocardial fiber shortening.
|
Temporal Changes in Systolic Strains
The proposed approach for analysis of regional myocardial deformation can also provide temporal changes in endocardial and epicardial systolic strain and shear. An example of the temporal changes in systolic cardiac- and fiber-specific strains is shown in Fig. 6. The observed endocardial-epicardial gradients in radial, circumferential, and cross-fiber strains were generally preserved for all 24 myocardial regions throughout the systolic interval.
|
Temporal Changes in Average Systolic and Diastolic Strains
Our approach can also provide information regarding diastolic strain. An example of the temporal changes in average cardiac- and fiber-specific strain over the entire cardiac cycle is shown for a representative dog (Fig. 7). Diastolic strain could not be derived for all of the dogs in this series of experiments because of image flow artifacts at the endocardial surface of the heart during the diastolic period complicating surface segmentation. The strains and shears generally returned to baseline at end diastole with an appropriate leveling off in late diastole. The failure of average longitudinal strains to return to baseline during diastole reflects regional variation in longitudinal strain, particularly in basal segments.
|
| |
DISCUSSION |
|---|
|
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
|
|---|
We employed a novel 3-D shape-based analysis of standard cine MR cardiac images for the comprehensive assessment of regional left ventricular deformation. The normal regional patterns of transmural left ventricular systolic cardiac- and fiber-specific deformations were defined in an experimental canine model. Regional endocardial and epicardial strains were also computed noninvasively and are consistent with measures previously derived by using highly invasive methodology. These object-centered parameters of deformation have a clear advantage over frame-centered parameters like displacements and velocities, because regional myocardial strain is independent of translational and rotational motion of the heart. Transmural myocardial radial and circumferential strains were fairly uniform throughout the left ventricle. Significant apex-to-base gradients were observed for longitudinal strain and several of the shear strains, in particular radial-longitudinal (transverse) shear. An epicardial-to-endocardial gradient in radial and circumferential strain was observed throughout the left ventricle. There was also an increase in cross-fiber strain in the endocardium. An increase in endocardial strains was accompanied by increases in radial-longitudinal and radial-fiber shears in the endocardium. Thus the transmural gradients in cardiac- and fiber-specific strains and shears support previous theories of regional myocardial deformation, which predict considerable sliding between myocardial fibers.
Quantitative Analysis of Left Ventricular Deformation
The left ventricle is a 3-D object that moves linearly and nonlinearly in space over the cardiac cycle. The myocardium is also a complex 3-D structure consisting of myocytes interconnected by a dense collagen weave that course in different directions. Under normal conditions, the myocardium undergoes a complex cyclic deformation. We can expect that regional ischemia and infarction will produce even more complex changes in regional and global deformation. Ideally, one would like to track these changes in regional deformation in a 3-D space.Most standard approaches for regional analysis of function have focused on the measurement of endocardial motion or left ventricular thickening from two-dimensional (2-D) image sequences. Some of these approaches have relied on volumetric information, such as regional ejection fractions, derived from radial segments extending from the center of the ventricle. This approach is dependent on the reproducible location of references from which to make regional measurements. To alleviate the problems associated with reliably marking reference points on the left ventricle, the reference-free centerline method was developed (37). However, the centerline approach can only compute changes in deformation along the radial direction and does not account for potential deformations out of the imaging plane (20).
Invasive Measurements of Regional 3-D Deformation and Strain
Much of the information gained regarding 3-D deformation of the left ventricle comes from analysis of implanted radio-opaque markers that use biplane cine radiography. Whereas this approach has certain merits, there are also significant limitations. One obvious concern is the effect of bead implantation. Fenton et al. (14) reported an average flow decrease of 25% within 5 mm of marker implantation. Additionally, there are errors associated with localization of the markers in a 3-D space (± 0.15-0.2 mm) (14, 29). This error is due to the pincushion effect, cone effect, and identification of the marker centroids, which is in part dependent on the marker size.Other investigators have computed regional strains with implanted sonomicrometers. This approach offers greater temporal (sampling rate 120-150 Hz) and spatial resolution (27). The higher spatial resolution of the crystals may permit an equally accurate determination of transmural strain with smaller sized and fewer numbers of implanted crystals. However, analysis with this approach is also limited to a finite region of the left ventricle. Again, implantation of the crystals, like implantation of the beads, may alter regional deformation and flow.
Analysis of 3-D Deformation
Many studies evaluating 3-D deformation in the heart assume that myocardial deformation is homogeneous. An important finding of some earlier studies is that systolic myocardial deformation is indeed not homogeneous. Significant transmural gradients of strain have been observed. Continuous transmural strain distributions can be obtained from the nonhomogeneous model, as proposed by McCulloch and Omens (24). This model computes continuous transmural variations in strain and rotation by using 3-D nonlinear finite element interpolating functions for nonhomogeneous deformation analysis (24). To date, only a few recent studies (29) have incorporated this more sophisticated analysis of myocardial deformation because of the time intensive computations required. We employ a 3-D shape-based approach for analysis of cine MR images, which integrates image-derived shape-tracked surface displacement data and fiber orientation data with a transversely isotropic elastic solid continuum model with preferential stiffness along fiber directions.Calculation of Cardiac-Specific Strains and Shears
The cardiac-specific strains and shears derived using our 3-D image-based approach are similar to those previously derived in the dog by using more invasive approaches. Villarreal et al. (42) evaluated regional myocardial deformation in dogs with the use of biplane radiography of implanted transmural arrays of radio-opaque beads (as described above). Cardiac-specific strains were computed for subepicardial, midwall, and subendocardial regions. These investigators demonstrated systolic radial strain (rr) was positive under control conditions.
In contrast, circumferential (cc) and
longitudinal (ll) systolic strains were
negative. They also observed small positive systolic transverse shear
deformations [circumferential-radial (cr),
0.05; and longitudinal-radial (lr), 0.08]
under control conditions. During control conditions, these
investigators identified the usual epicardial to endocardial gradient
in both circumferential and radial strains, with the greatest
deformations occurring in the endocardial region. An earlier
publication by Fenton et al. (14), using a similar
implanted bead methodology, also demonstrated different patterns
of deformation in endocardial regions relative to epicardial and
midwall regions. The endocardial deformation was complex and also had
large transverse shear components under control conditions.
The cardiac-specific strains and shears determined by using our comprehensive noninvasive shape-based approach are generally concordant with the results reported by the use of invasive measures. However, we were able to define spatial heterogeneity in these cardiac-specific indices of deformation. We observed a large apex-to-base gradient in the longitudinal strain and radial-longitudinal (transverse) shear. A similar gradient in longitudinal strain has been recently identified in human subjects with the use of color-coded pulsed-wave tissue Doppler (16). Regional differences in longitudinal tissue velocities in the endocardium have also been reported in dogs with the use of tissue Doppler (18). The observed regional variability in strains and shears may explain some of the discordant observations in the literature.
One potential limitation of this type of regional analysis with implanted markers relates to their application under conditions of regional ischemia. Observed changes in the central ischemic region may be counterbalanced by unmeasured complex changes in the peri-ischemic border or remote regions. Thus important changes in the spatial patterns of deformation associated with regional ischemia cannot be adequately evaluated by using the implanted bead approach. The potential effects of the altered shape of the ischemic region also cannot be assessed with a transmural array of beads. These studies highlight the necessity for methods, which can provide a comprehensive 3-D measurement of myocardial deformation.
Comprehensive Image-Based Analysis of 3-D Deformation
Several imaging approaches for the temporal analysis of true 3-D cardiac deformation have also been pursued. The MR imaging techniques include radial tagging, specific spatial modulation of magnetization, and phase-contrast imaging. MR tagging of the myocardium is one of the most popular approaches to track motion (3, 45). The advent of MR myocardial tagging techniques enabled the visualization of myocardial deformations associated with contraction at rest and after an ischemic insult. MR tagging involves creation of signal voids in the tomographic data, which deform with the myocardium, permitting the nontraumatic measurement of regional deformation. Whereas initial approaches for MR tagging provided 2-D information, recent approaches have permitted analysis of true 3-D deformations (4-6). Azhari et al. (4) evaluated several MR image-derived 3-D strain parameters for discrimination of ischemic and nonischemic regions. Three-dimensional myocardial deformations were computed for 24 myocardial segments and related to regional microsphere flow and postmortem histochemical staining. However, analysis of regional function did not correlate well with the level of ischemia, highlighting a potential limitation of this well-accepted MR tagging approach.Bogaert and Rademakers (6) recently employed the MR tagging approach to define the normal patterns of 3-D deformation in the adult human left ventricle. In this large clinical study, both cardiac- and fiber-specific strains were generated. These investigators demonstrated a regional nonuniformity of left ventricular strains, which was similar to our findings that used a comprehensive nontagging approach. However, their analysis was limited by lower image resolution. In addition, their strain calculations were restricted to analysis of a sparse set of nodal points defined by the intersections of the tagging stripes with the endocardial and epicardial surface of the heart.
The tagging approach has the following limitations: 1) MR tags are difficult to detect throughout the cardiac cycle as tags tend to decay due to T1 relaxation, 2) tags are not easily extended to tracking points from 3-D image sequences, and 3) for regional wall motion analysis, additional methods are required for identifying the precise tag locations and defining myocardial boundaries.
As an alternative to MR tagging, several investigators have analyzed changes in phase due to motion of tissue within a fixed volume of interest to estimate instantaneous, localized velocities, and ultimately cardiac deformation (1, 28, 34, 41). This technique identifies the change in MR signal phase (which is proportional to velocity) associated with uniform motion of tissue in the presence of a magnetic field gradient.
These widely employed MR approaches for assessment of myocardial deformation assume that the MR images provide absolute uncorrupted information regarding myocardial strains, ignoring potential errors associated with image acquisition, image processing, or through-plane motion. In our comprehensive 3-D shape-based approach for quantification of regional myocardial deformation, we impart a biomechanical model in the analysis of the image data to help constrain the image-derived displacements, by reducing the errors associated with imaging. Our biomechanical model is based on previously derived information of myocardial mechanics obtained with the use of highly invasive measures. The use of prior knowledge in combination with a comprehensive 3-D image set strengthens our more physiological approach. In the future, we may be able to improve our biomechanical model as additional information becomes available.
In the current study, the shape of the endocardial and epicardial surfaces was used to track the 3-D trajectories of a dense field of points approach follows the complex regional deformation of the myocardium, guided by shape-matched displacements found at the bounding surfaces of the myocardium (13, 25, 38, 39). The shape-based surface displacements are integrated with a continuum biomechanics model incorporating myofiber architecture to estimate fiber-specific endocardial and epicardial strains and shears (32). This shape-based approach has the advantage of being able to track deformation over the entire cardiac cycle, and would be applicable to any high-resolution 3-D imaging modality, which defines the myocardial surfaces over the entire cardiac cycle.
Calculation of Fiber-Specific Strains and Shears
The fiber-specific strains and shears derived in this study correlate with findings derived with the use of biplane cine radiography of implanted beads or implanted sonomicrometers. Rodriguez et al. (36), employing implanted sonomicrometers in combination with careful histological analysis, demonstrated that fiber-specific strains were fairly uniform across the wall (epicardial,0.14; midwall, 0.14; and endocardial, 0.12) and similar in
magnitude to our estimated fiber strains. Waldman et al.
(44), employing biplane cine radiography of implanted
beads, demonstrated epicardial (0.09 ± 0.04) and endocardial
(0.06 ± 0.06) fiber strains nearly identical to those observed
by using our shape-based analysis. LeGrice et al. (22)
observed regional differences in longitudinal-radial (transverse) shear
using biplane cine radiography of implanted beads placed in the septum
and anterior free wall. These investigators reported positive
transverse shear in the endocardium of the anterior left ventricular
wall and negative shears in the septum. The differences in shear were
attributed to regional differences in myocardial cleavage planes in
these two regions. The approach employed by these investigators limits
the analysis to very small regions of the left ventricle. Our findings
suggest that there are significant regional differences in
longitudinal-radial shear from apex to base, as well as around the
circumference of the left ventricle. This would indicate that the apex
to base location of the implanted beads could tremendously alter the
observations. Thus there appears to be significant regional variation
between regional wall thickening (radial strain) and
longitudinal-radial shear. In our analysis, there was a weak direct
linear relationship between regional transmural radial strain and
longitudinal-radial shear (r = 0.43, P = 0.03).
Limitations
Our analysis of regional myocardial deformation was limited to normal resting conditions. In our analysis of fiber-specific strains, we assume uniform fiber architecture as previously established in the literature. This standard fiber architecture may not be applicable following myocardial infarction or in the presence of diseased states like asymmetric septal hypertrophy, which would distort the fiber architecture. In the future, we may be able to noninvasively derive information regarding the fiber architecture by using MR diffusion imaging (17, 35). These additional MR imaging sequences may need to be incorporated in future applications of our strain computations.Our model may also force endocardial-to-epicardial gradients in strain, based on assumptions regarding conservation of volume. This bias could be minimized in the future by incorporating midwall information, derived by using either MR tagging or phase-velocity mapping. We have primarily focused our analysis to the systolic portion of the cardiac cycle. Analysis of regional myocardial strains in the diastolic phase of the cardiac cycle was complicated by image flow artifacts in some of the dogs. Analysis of diastole was also limited by the time required for segmentation of the entire cardiac cycle. However, our approach could be extended to analysis of the entire cardiac cycle, particularly if the segmentation algorithms could be further automated. An illustration of the typical changes in regional myocardial deformation over the entire cardiac cycle is provided in Fig. 7, A-D.
Our methods remain sensitive to variations in image quality and resolution and are primarily limited by accurate and efficient segmentation of the left ventricular surface. In our current 3-D algorithms, we have not incorporated a potentially important constraint in the analysis, which would require periodicity (i.e., a return of the final coordinate position back to the point of origin). We have found this constraint useful in our previous 2-D algorithms for improving image segmentation and strain analysis. In the current analysis, resolution was also somewhat limited along the longitudinal axis due to the 5-mm MR short-axis slice thickness. This would inherently increase the error associated with calculation of longitudinal strain. The acquisition of overlapping short-axis images would improve through plane resolution and minimize this limitation. Preliminary studies using this overlapping approach confirm the current observations regarding the normal gradients in longitudinal strain.
We have also ignored respiratory motion in our acquisition of the MR short-axis MR images. Respiratory motions are averaged over the course of acquiring the full 3-D data set. With the advancement of MR imaging hardware and software these motions could also be eliminated. Correction for respiratory motion will likely improve the analysis by using our shape-tracked approach.
Clinical Implications
To date, quantitative noninvasive assessment of regional myocardial strain from 3-D images has been limited to specific MR acquisitions, in particular, MR tagging or MR phase-contrast velocity analysis. Our shape-based approach, which employs finite element analysis and a biomechanical model-based strategy, provides a comprehensive analysis of regional 3-D myocardial deformation. This surface-based approach would be applicable to standard cine-gradient MR images, 3-D cine computed tomographic images, and 3-D echocardiography. However, widespread application of our shape-based approach will require improved image segmentation. The application of our approach to 3-D echocardiography may be facilitated by the use of echocardiographic contrast agents and would potentially bring accurate quantitative analysis of regional myocardial deformation to the patient's bedside.| |
APPENDIX |
|---|
|
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
|
|---|
Description of Strain Estimation Algorithm
Shape-tracking displacement estimates. In this study, the original displacements on the inner and outer surfaces of the left ventricular myocardium were obtained by using the shape-tracking algorithm previously described in detail (39). This approach is outlined below.
An illustration of this process is provided by Fig. 8. Consider point p1 on a surface at time t1, which is to be mapped to a point p2 on the deformed surface at time t2. First, a search is performed on a physically plausible region W on the deformed surface and the point p2, which has the local shape properties closest to those p1 is selected. The shape properties here are captured in terms of the principal curvatures
1
and 2. The distance measure used is the bending energy
required to bend a curved plate or surface patch to a newly deformed
state. This is labeled as dbe and is defined as
![d<SUB>be</SUB>(<IT>p</IT><SUB>1</SUB><IT>, p</IT><SUB>2</SUB>)<IT>= </IT><FENCE><FR><NU>[<IT>&kgr;</IT><SUB>1</SUB>(<IT>p</IT><SUB>1</SUB>)<IT>−&kgr;</IT><SUB>1</SUB>(<IT>p</IT><SUB>2</SUB>)]<SUP>2</SUP><IT>+</IT>[<IT>&kgr;</IT><SUB>2</SUB>(<IT>p</IT><SUB>1</SUB>)<IT>−&kgr;</IT><SUB>2</SUB>(<IT>p</IT><SUB>2</SUB>)]<SUP>2</SUP></NU><DE>2</DE></FR></FENCE>](/content/vol281/issue2/fulltext/H698/img001.gif)
|
(A1) |
![&mgr;<SUP>m</SUP><SUB>1</SUB>=<A><AC>p</AC><AC>˜</AC></A><SUB>2</SUB>−p<SUB>1</SUB>, <A><AC>p</AC><AC>˜</AC></A><SUB>2</SUB>=<LIM><OP>arg min</OP><LL><IT>p</IT><SUB>2</SUB><IT>∈W</IT><SUB>2</SUB></LL></LIM>[<IT>d</IT><SUB>be</SUB>(<IT>p</IT><SUB>1</SUB><IT>, p</IT><SUB>2</SUB>)]](/content/vol281/issue2/fulltext/H698/img003.gif)
|
(A2) |
|
Confidence measures in the match. Bending energy measures for all the points inside the search region W are recorded as the basis to measure the "goodness and uniqueness" of the matching choices. The confidence is high if the best match is both good and unique and falls off otherwise.
Modeling the Myocardium
The passive properties of the left ventricular myocardium are captured using a biomechanical model. We use a transversely isotropic linear elastic model, which allows us to incorporate information about the preferential stiffness of the tissue along fiber directions as previously defined (19). These fiber directions are shown in Fig. 9. The model described in terms of an internal or strain energy function of the form
|
(A3) |
is the strain and C is the 6 × 6 matrix containing the elastic constants, which define the material
properties.
|
(A4) |
f and p are the
corresponding Poisson's ratios, and Gf is the
shear modulus across fibers [Gf 
Ef/2(1 + fp)]. If Ef = Ep and fp =
p, this model reduces to the more common
isotropic linear elastic model. The fiber stiffness was set to be 3.5 times greater than the cross-fiber stiffness (19). The
Poisson's ratios were both set to 0.4 to model approximate
incompressibility. More details can be found in a continuum mechanics
textbook (23).
|
To evaluate the potential significance of these assumptions regarding
the fiber architecture, differential fiber stiffness, and index of
incompressibility, we computed end-systolic cardiac- and fiber-specific
strains and shears in one of the dogs by using different model
pararmeters. These differences are illustrated in Fig.
10. We first
calculated the strains with and without the fiber architecture in our
mechanical model. The mechanical model became isotropic without the
differential fiber stiffness. We then computed strains with a 10°
rotation in the assumed fiber axes. Finally, we computed strains by
using a lower Poisson's ratio of 0.3, modeling the tissue as more
compressible. The effects of these changes were relatively small for
our calculated transmural strains. Changing the fiber angle produced
predictable changes in the fiber and cross-fiber strains. Slightly
greater effects were seen for the calculated shears. On the basis of
these additional analyses, our mechanical model does not appear overly
sensitive to small changes in the model parameters or assumed fiber
architecture. However, these differences might be greater for the
calculated strains and shears in each of the smaller midwall elements
in our finite element analysis because calculation of the strains within the myocardium would be more dependent on the mechanical model
and fiber architecture.
|
Integrating the Data and Model Terms
Having defined both the data term model and the model term, we discretize the equations by using the finite element method to generate the final equation, which takes the form
|
(A5) |
| |
ACKNOWLEDGEMENTS |
|---|
We gratefully acknowledge the technical assistance of Dr. Cindy Shi, Dr. Jennifer Hu, Dr. Eliot N. Heller, Dr. Edouard Daher, Dr. Michael Shen, Michael Singer, Peter Borek, Jason Soares, and Donna Dione. We also thank Lawrence Staib for invaluable suggestions and guidance.
| |
FOOTNOTES |
|---|
Address for reprint requests and other correspondence: A. J. Sinusas, Yale Univ. School of Medicine, P.O. Box 208017, 3FMP, New Haven, CT 06520-8017 (E-mail: albert.sinusas{at}yale.edu).
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Received 24 January 2000; accepted in final form 11 March 2001.
| |
REFERENCES |
|---|
|
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
|
|---|
1.
Arai, A,
Gaither C,
Epstein F,
Balaban R,
and
Wolff S.
Myocardial velocity gradient imaging by phase contrast MRI with application to regional function in myocardial ischemia.
Magn Reson Med
42:
98-109,
1999[Web of Science][Medline].
2.
Axel, L,
and
Dougherty L.
MR imaging of motion with spatial modulation of magnetization.
Radiology
171:
841-845,
1989
3.
Axel, L,
Goncalves R,
and
Bloomgarden D.
Regional heart wall motion: two-dimensional analysis and functional imaging with MR imaging.
Radiology
183:
745-750,
1992
4.
Azhari, H,
Weiss J,
Rogers W,
Siu C,
and
Shapiro E.
A noninvasive comparative study of myocardial strains in ischemic canine hearts using tagged MRI in 3-D.
Am J Physiol Heart Circ Physiol
268:
H1918-H1926,
1995
5.
Azhari, H,
Weiss J,
Rogers W,
Siu C,
Zerhhouni E,
and
Shapiro E.
Noninvasive quantification of principal strains in normal canine hearts using tagged MRI images in 3-D.
Am J Physiol Heart Circ Physiol
264:
H205-H216,
1993
6.
Bogaert, J,
and
Rademakers F.
Regional nonuniformity of normal adult human left ventricle.
Am J Physiol Heart Circ Physiol
280:
H610-H620,
2001
7.
Chakraborty, A,
Staib L,
and
Duncan J.
An integrated approach to boundary finding.
In: IEEE Workshop on Biomedical Image Analysis, Seattle, WA, 1994, p. 13-22.
8.
Chakraborty, A,
Worring M,
and
Duncan J.
On multi-feature integration for deformable boundary finding.
In: International Conference on Computer Vision, 1995, p. 846-851.
9.
Costa, KD,
Takayama Y,
McCulloch AD,
and
Covell JW.
Laminar fiber architecture and three-dimensional systolic mechanics in canine ventricular myocardium.
Am J Physiol Heart Circ Physiol
276:
H595-H607,
1999
10.
Dione, D,
Shi P,
Smith W,
DeMan P,
Soares J,
Duncan J,
and
Sinusas A.
Three-dimensional regional left ventricular deformation from digital sonomicrometry.
In: Proc 19th Annu Int Conf IEEE Eng Med and Biol Soc, 1997, p. 848-851.
11.
Duncan, J,
Lee F,
Smeulders A,
and
Zaret B.
A bending energy model for measurement of cardiac shape deformity.
IEEE Trans Med Imaging
10:
307-320,
1991.
12.
Duncan, J,
Shi P,
Constable R,
and
Sinusas A.
Physical and geometrical modeling of image-based recovery of left ventricular deformation.
Prog Biophys Mol Biol
69:
333-351,
1998[Web of Science][Medline].
13.
Duncan, J,
Shi PC,
Amini A,
Constable R,
Staib L,
Dione D,
Shi QX,
Heller E,
Singer M,
Chakraborty A,
Robinson G,
Gore J,
and
Sinusas A.
Towards reliable noninvasive measurement of myocardial function from 4D images.
In: SPIE Proc Medical Imaging. Newport Beach, CA:, 1994.
14.
Fenton, T,
Cherry J,
and
Klassen G.
Transmural myocardial deformation in the canine left ventricular wall.
Am J Physiol Heart Circ Physiol
235:
H523-H530,
1978.
15.
Freeman, GL,
LeWinter MM,
Engler RL,
and
Covell JW.
Relationship between myocardial fiber direction and segment shortening in the midwall of the canine left ventricle.
Circ Res
56:
31-39,
1985
16.
Galiuto, L,
Ignone G,
and
DeMaria A.
Contraction and relaxation velocities of the normal left ventricle using pulsed-wave tissue Doppler echocardiography.
Am J Cardiol
81:
609-614,
1998[Web of Science][Medline].
17.
Garido, L,
Wedeen V,
Kwong K,
Spencer U,
and
Kantor H.
Anisotropy of water diffusion in the myocardium of the rat.
Circ Res
74:
789-793,
1994
18.
Gorscan, J,
Strum D,
Mandarino W,
Gulati V,
and
Pinsky M.
Quantitative assessment of alterations in regional left ventricular contractility with color-coded tissue Doppler echocardiography: comparison with sonomicrometry and pressure-volume relations.
Circulation
95:
2423-2433,
1997
19.
Guccione, J,
and
McCulloch A.
Finite element modeling of ventricular mechanics.
In: Theory of Heart, edited by Hunter P,
McCulloch A,
and Nielsen P.. Berlin: Springer-Verlag, 1991, p. 122-144.
20.
Holman, E,
Buller V,
deRoos A,
van der Geest R,
Baur L,
van der Laarse A,
Bruschke A,
Reiber J,
and
van der Wall E.
Detection and quantification of dysfunctional myocardium by magnetic resonance imaging: a new three-dimensional method for quantitative wall-thickening analysis.
Circulation
95:
924-931,
1997
21.
Huebner, K,
Thornton E,
and
Byrom T.
The Finite Element Method For Engineers. New York: Wiley, 1995.
22.
LeGrice, IJ,
Takayama Y,
and
Covell JW.
Transverse shear along myocardial cleavage planes provides a mechanism for normal systolic wall thickening.
Circ Res
77:
182-193,
1995
23.
Malvern, L.
Introduction to the Mechanics of a Continuous Medium. Englewood Cliffs, NJ: Prentice Hall, 1969.
24.
McCulloch, AD,
and
Omens JH.
Non-homogeneous analysis of three-dimensional transmural finite deformation in canine ventricular myocardium.
J Biomech
24:
539-548,
1991[Web of Science][Medline].
25.
McEachen, J,
and
Duncan J.
Shape-based tracking of naturally occurring annuli in image sequences.
In: IEEE Conf Computer Vision and Pattern Recognition, 1993.
26.
McEachen, J,
and
Duncan J.
Shaped-based tracking of left ventricular wall motion.
IEEE Trans Med Imaging
16:
270-283,
1997[Web of Science][Medline].
27.
Meoli, D,
Mazhari R,
Dione D,
Omens J,
McCulloch A,
and
Sinusas A.
Three dimensional digital sonomicrometry: comparison with biplane radiography.
In: IEEE 24th Annual NE Bioeng Conf, 1998, p. 64-67.
28.
Nayler, G,
Firmin D,
and
Longmore D.
Blood flow imaging by CINE magnetic resonance.
J Comput Assist Tomogr
10:
715-722,
1986[Web of Science][Medline].
29.
Ono, S,
Waldman L,
Yamashita H,
Covell J,
and
Ross J.
Effect of coronary reperfusion on transmural myocardial remodeling in dogs.
Circulation
91:
1143-1153,
1995
30.
Papademetris, X.
Estimation of 3D left ventricular deformation from medical images using biomechanical models. (PhD Thesis). New Haven, CT: Yale University, 2000.
31.
Papademetris, X,
Rambo J,
Dione D,
Sinusas A,
and
Duncan J.
Visually interactive semi-automated 3D segmentation of cine cardiac magnetic resonance images (Abstract).
J Am Coll Cardiol
31:
7A,
1998.
32.
Papademetris, X,
Shi P,
Dione D,
Sinusas A,
Constable R,
and
Duncan J.
Recovery of soft tissue object deformation from 3D image sequences using biomechanical models.
In: Information Processing in Medical Imaging. Visegrad, Hungary: Springer Lecture Notes in Computer Science, 1999, p. 352-356.
33.
Papademetris, X,
Sinusas A,
Dione D,
and
Duncan J.
Estimation of 3D left ventricular deformation from echocardiography.
Med Image Anal
5:
17-28,
2001[Web of Science][Medline].
34.
Pelc, N,
Bernstein M,
Shimakawa A,
and
Glover G.
Encoding strategies for three direction phase contrast MR imaging of flow.
J Magn Reson Imaging
1:
405-413,
1991[Web of Science][Medline].
35.
Reese, TG,
Weisskoff RM,
Smith RN,
Rosen BR,
Dinsmore RE,
and
Wedeen VJ.
Imaging myocardial fiber architecture in vivo with magnetic resonance.
Magn Reson Med
34:
786-791,
1995[Web of Science][Medline].
36.
Rodriguez, E,
Hunter W,
Royce M,
Leppo M,
Douglas A,
and
Weisman H.
A method to reconstruct myocardial sarcomere lengths and orientation at transmural sites in beating canine hearts.
Am J Physiol Heart Circ Physiol
263:
H293-H306,
1992
37.
Sheehan, FH,
Bolson EL,
Dodge HT,
Mathey DG,
Schofer J,
and
Woo HW.
Advantages and applications of the centerline method for characterizing regional ventricular function.
Circulation
74:
293-305,
1986
38.
Shi, P,
Sinusas A,
Constable R,
and
Duncan J.
Volumetric deformation analysis using mechanics-based data fusion: applications in cardiac motion recovery.
Int J Computer Vision
35:
87-107,
1999.
39.
Shi, P,
Sinusas AJ,
Constable RT,
Ritman E,
and
Duncan JS.
Point-tracked quantitative analysis of left ventricular surface motion from 3-D image sequences.
IEEE Trans Med Imaging
19:
36-50,
2000[Web of Science][Medline].
40.
Streeter, DJ,
Spotnitz H,
Patel D,
Ross JJ,
and
Sonnenblick E.
Fiber orientation in the canine left ventricle during diastole and systole.
Circ Res
24:
339-347,
1969
41.
Van Dijk, P.
Direct cardiac NMR imaging of heart wall and blood flow velocity.
J Comput Assist Tomogr
8:
429-436,
1984[Web of Science][Medline].
42.
Villarreal, F,
Lew W,
Waldman L,
and
Covell J.
Transmural myocardial deformation in the ischemic canine left ventricle.
Circ Res
68:
368-381,
1991
43.
Waldman, L,
Fung Y,
and
Covell J.
Transmural myocardial deformation in the canine left ventricle: normal in vivo three-dimensional finite strains.
Circ Res
57:
152-163,
1985
44.
Waldman, LK,
Nosan D,
Villarreal F,
and
Covell JW.
Relation between transmural deformation and local myofiber direction in canine left ventricle.
Circ Res
63:
550-562,
1988
45.
Zerhouni, E,
Parish D,
Rogers W,
Yang A,
and
Shapiro E.
Human heart: tagging with MR imaging-a method for noninvasive assessment of myocardial motion.
Radiology
169:
59-63,
1988
This article has been cited by other articles:
|
|
 |
|
  A. Cheng, F. Langer, F. Rodriguez, J. C. Criscione, G. T. Daughters, D. C. Miller, and N. B. Ingels Jr. Transmural cardiac strains in the lateral wall of the ovine left ventricle Am J Physiol Heart Circ Physiol, April 1, 2005; 288(4): H1546 - H1556. [Abstract] [Full Text] [PDF]
|
|
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
| Visit Other APS Journals Online |
) and epicardial (
) segments. Shown
are R (A), C (B), L (C), F
(D), and C-F (E) systolic strains for 8 radial
segments for three short-axis sections: apical, mid (midventricular),
and basal. The observed endocardial-epicardial gradients in R, C, and
CF strains were generally preserved for the 24 myocardial regions
throughout the systolic interval.
