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Ann Thorac Surg 2001;71:654-662
© 2001 The Society of Thoracic Surgeons
a Departments of Surgery and Bioengineering, University of California, San Francisco, California, USA
b Department of Surgery, Washington University, St. Louis, Missouri, USA
c Department of Biomedical Engineering, Columbia University, New York, New York, USA
Accepted for publication August 24, 2000.
Address reprint requests to Dr Guccione, Division of Surgical Services (112D), Department of Veterans Affairs Medical Center, 4150 Clement St, San Francisco, CA 94121
e-mail: Julius.Guccione{at}med.va.gov
| Abstract |
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Methods. An anteroapical transmural myocardial infarct was created by coronary arterial ligation in an adult Dorset sheep and was allowed to mature into left ventricular aneurysm for 10 weeks. The animal was imaged subsequently using magnetic resonance imaging with simultaneous recording of intraventricular pressures. A realistic mathematical model of the three-dimensional ovine left ventricle with an anteroapical aneurysm was constructed from multiple short-axis and long-axis magnetic resonance imaging slices at the beginning of diastolic filling.
Results. Three model simulations are presented: (1) normal border zone contractility and normal aneurysmal material properties; (2) greatly reduced border zone contractility (by 50%) and normal aneurysmal material properties; and (3) greatly reduced border zone contractility (by 50%) and stiffened aneurysmal material properties (by 1000%). Only the latter two simulations were able to reproduce experimentally observed stretching of border zone fibers during isovolumic systole.
Conclusions. The mechanism underlying mechanical dysfunction in the border zone region of left ventricular aneurysm is primarily the result of myocardial contractile dysfunction rather than increased wall stress in this region.
| Introduction |
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There have been no successful methods developed to measure stress in the intact heart wall, primarily because of its large deformations and the tissue injury caused by implanted transducers [6]. An alternative approach to quantifying ventricular wall stress is mathematical modeling based on the conservation laws of continuum mechanics. To solve the governing equations of equilibrium for a pressure vessel with such a complex geometry, boundary conditions, and material properties, computational techniques are required. The most versatile technique for analyzing cardiac mechanics is the finite element (FE) method [7]. For example, Bogen and colleagues [8] developed an isotropic, initially spherical membrane model based on the FE method to examine the effects of infarct size, stiffness, and age on LV performance. Even though that study was published two decades ago, it represented the state-of-the-art until now.
The present study was undertaken to develop a more realistic anisotropic, three-dimensional (3-D) FE model of the infarcted LV and use it to study the mechanism underlying mechanical dysfunction in the BZ region of LV aneurysm. Specifically, we sought to address the following questions: (1) How much stretching of BZ fibers during isovolumic systole is the result of an increase in wall stress alone? (2) How much of a reduction in the contractility of these fibers is required to have the model simulate the observed stretching? (3) How much does this reduction in BZ fiber contractility depend on the stiffness of the infarcted myocardium? To answer these questions and investigate our hypotheses, we used MR imaging of an ovine model of LV aneurysm and a 3-D FE method for large elastic deformations of ventricular myocardium [9] to create the most realistic mathematical model for the regional mechanics of an infarcted LV to date.
| Material and methods |
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Animal protocol for magnetic resonance imaging
At 10 weeks after infarct, the animal was sedated with ketamine (15 mg/kg intramuscularly) and then intubated and ventilated with a mixture of isoflurane and oxygen. Nonferromagnetic catheters (Mikro-Tip, 5F, model SPC-350 MR; Millar Instruments Inc, Houston, TX) were placed in the LV and right ventricle and in the aortic root of the animal through the left carotid artery, jugular vein, and femoral artery, respectively. The animal was positioned supine in the magnet (Magnetom Vision, Siemens Medical Systems, Iselin, NJ) with its chest centered in a Helmholtz coil. After MR imaging, the animal was sacrificed using pentobarbital (120 mg/kg intravenously) and heparin (5000 U intravenously) followed by a rapid infusion of KCl (80 mEq intravenously).
Magnetic resonance image acquisition and processing
A series of scout images was obtained to locate the heart and the true long-axis and short-axis planes. Subsequently, a set of eight short-axis imaging planes (8 mm thick) were obtained parallel to the true short-axis plane and at 8-mm intervals beginning at the level of the mitral valve and ending at a short-axis imaging plane that contained only apical myocardium and no LV or right ventricular endocardium. An additional set of four long-axis imaging planes was obtained according to the following criteria: (1) orthogonal to the true short-axis imaging plane, (2) intersecting the centroid of the LV, and (3) oriented in a radial fashion with 45-degree separation between long-axis imaging planes.
Image acquisition was synchronized to the R wave of the electrocardiogram signal. During the actual image data acquisition, the ventilator (Hallowell 2000, Hallowell EMC, Pittsfield, MA) was stopped for 20 to 30 seconds at maximum inspiration (to minimize respiratory motion and the associated motion artifacts in our MR images). During this period, a series of images was acquired at 29-ms intervals until the approximate completion of the entire cardiac cycle. Data acquisition time was about 45 minutes. Imaging variables were a repetition time equal to the cardiac cycle (RR duration), an echo time of 29 ms, an excitation angle of 15 degrees, and an acquisition matrix of 256 x 256. The field of view was set to 350 x 350 mm2 and 400 x 400 mm2 for the short-axis and long-axis images, respectively. Raw 256 x 256-pixel images were transferred to a Silicon Graphics workstation (Silicon Graphics Inc, Mountain View, CA) and were converted from Siemens format to SGI format using custom software. Images were cropped and scaled by 300% (Fig 1).
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Hemodynamic data acquisition and analysis
During the scanning interval, the LV pressure, right ventricular pressure, thoracic aortic pressure, and the trigger signal were recorded continuously using customized data acquisition and manipulation software (LabView 5, National Instruments, Inc, Austin, TX). The MR images resulting from the experiment represented an average over many heartbeats. Therefore, a signal-averaging algorithm was performed on the hemodynamic data set to yield a 20-beat average for each hemodynamic variable. The algorithm computed the mean signal and standard deviation for each hemodynamic variable during the isovolumic interval. In particular, the average LV pressures at the beginning and end of isovolumic systole were approximately 1 kPa (7.5 mm Hg) and 8 kPa (60 mm Hg), respectively.
Mathematical model
The 3-D FE method of Costa and colleagues [9] for large elastic deformations of ventricular myocardium was used, together with the mathematical descriptions for diastolic and systolic myocardial material properties (stress-strain relations) of Guccione and colleagues [12]. A nonsymmetric high-order FE model of the 3-D ovine LV with LV aneurysm was constructed in prolate spheroidal coordinates (focal length, 41.0 mm) using the method of Nielsen and colleagues [13] to fit smooth bicubic Hermite FE surfaces to epicardial (304 points) and endocardial (247 points) boundary data from eight short-axis and four long-axis MR imaging slices at the beginning of diastolic filling (Fig 2). With 16 elements, the root-mean-squared errors in the surface fits were 0.8 and 1.2 mm, respectively. Wall thickness of the model ranged from 3 mm near the apex to 12 to 15 mm near the base and around the truncated papillary muscles, and wall and cavity volumes were 104.8 and 198.1 mL, respectively. To more accurately describe the non-axisymmetric geometries of the BZ and aneurysm, the original 16-element mesh in Figure 2 was subdivided into 12 elements circumferentially and 14 elements longitudinally (Fig 3). The normal, BZ, and aneurysmal portions of the model comprised approximately 76%, 14%, and 10%, respectively, of the total wall volume. All three deformed geometric (prolate spheroidal) nodal coordinate directions (transmural, longitudinal, circumferential) were interpolated using trilinear basis functions, and the hydrostatic pressure variable was constant in each element. A physiologic transmural fiber angle distribution from 83 degrees (endocardium) to -37 degrees (epicardium) with respect to the circumferential direction [14] was homogeneously used.
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ß referred to fiber 
coordinates:
![]() | ((1)) |
![]() | ((2)) |

ß is the Kronecker delta. E11 is the fiber strain, E22 is cross-fiber in-plane strain, E33 is the radial strain, E23 is the shear strain in the transverse plane, and E12 and E13 are shear strain in fibercross-fiber and fiberradial coordinate planes, respectively. Inasmuch as the fiber coordinates are locally orthonormal, these covariant strain components are also physical components. The form of Q describes a material that is transversely isotropic with respect to the muscle fiber axis [15]. Previously, we found that the material constants C = 0.88 kPa, bf = 18.48, bt = 3.58, and bfs = 1.63 allowed a cylindrical model of the left ventricle to match epicardial strains measured in an intact canine heart preparation [16] during passive LV filling. That analysis and biaxial testing of thin sheets from the LV free wall [17] both indicate that the normal passive tissue is significantly stiffer in the fiber direction than in the transverse plane. Moreover, Equation 1 allowed an FE model of the beating dog heart [12] to predict end-diastolic finite strain distributions from a midventricular region of the anterior LV free wall consistent with 3-D strain measurements for passive inflation [14].
Systolic contraction was modeled by defining the second Piola-Kirchhoff stress tensor referred to fiber 
coordinates in the undeformed body t
ß as the sum of the passive 3-D stress derived from the strain energy function (Equation 1) and an active fiber-directed component T0, which was a function of time t, peak intracellular calcium concentration Ca0, and sarcomere length l [18]:
![]() | ((3)) |
. We introduced the hydrostatic pressure p as the Lagrange multiplier needed to enforce the kinematic constraint that the third principal strain invariant (I3) equals 1 for material incompressibility. The FE stress analysis of Guccione and colleagues [12] suggested that at end-systole:
![]() | ((4)) |
![]() | ((5)) |
![]() | ((6)) |
![]() | ((7)) |
![]() | ((8)) |
The variables of the active contraction model were based on experimental measurements of sarcomere length and tension in isolated rat cardiac trabeculae by ter Keurs and colleagues [20]. They found a similar relationship between peak tension and sarcomere length when isosarcomeric contractions were compared with uncontrolled twitches in which central sarcomeres shortened against more compliant ends. More recently, Guccione and colleagues [21] observed a similar lack of influence of sarcomere shortening on the sarcomere length-tension relationship at an earlier phase of the twitch (corresponding to times near the beginning of ejection). Thus, in the present study of regional mechanics during the isovolumic systole phase of the cardiac cycle, we also used a time-varying elastance approach to model systolic contraction. Analogous to the time-varying elastance model of the LV pressure-volume relationship [22], the sarcomere length-tension relationships at times between end-diastole and end-systole were obtained by varying the value of the variable Tmax between 0 and 135.7 kPa, respectively.
Sarcomere length is, in general, a function of 3-D position in the LV wall that is related to fiber strain by the following relation:
![]() | ((9)) |
i coordinates in the deformed body and the second Piola-Kirchhoff stress tensor are related by the transformation:
![]() | ((10)) |
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Figure 5 in the article by Moulton and colleagues [3] shows that cumulative circumferential strains with early diastole as a reference state between both anterior and posterior BZ regions increase above the end-diastolic value by approximately 20% to 40% (depending on the instant of time) during isovolumic systole. We varied the contractility index of our model (in increments of 5%) until it predicted similar stretching of midwall muscle fibers in both the anterior and posterior BZ regions; first with normal aneurysmal stiffness and then with increased aneurysmal stiffness.
| Results |
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0) and remote noninfarcted (0.19 µm or 9.2%) regions indicate the effect of increased wall stress in the BZ regions (Table 1). However, the effect of increased wall stress in the BZ regions alone is not sufficient to account for the BZ fiber stretching observed experimentally.
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| Comment |
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Qualitative evaluation by Moulton and colleagues [3] of histologic sections revealed preserved fiber architecture and integrity in the BZ and remote regions, but infiltration of connective tissue and disruption of normal fiber architecture in the aneurysm. Their computer-assisted planimetry demonstrated a collagen content of 90.5% ± 8% in the aneurysm, 18.5% ± 12% in the BZs, and 1.0% ± 0.1% in the remote regions. Thus, we expected our model to require some reduction in BZ contractility to predict the previously observed stretching of BZ fibers during isovolumic systole, but not a huge 50% reduction. This discrepancy suggests that a persistent ischemic injury may exist in the BZ of LV aneurysm. Stress testing of the sheep apical aneurysm model with dobutamine and MR tagging (currently being undertaken in our laboratory) should provide a more direct test of this hypothesis.
The primary limitation of the present study is our lack of knowledge concerning myocardial fibrous architecture and material properties in the ovine model of LV aneurysm. In contrast, these important determinants of regional myocardial deformation (and stress) have been studied extensively in the normal canine LV. Our preliminary data on passive material properties of noninfarcted myocardium from a sheep heart with LV aneurysm, obtained using epicardial suction [25], suggest that these properties are quite similar to those determined by Guccione and coworkers [15]. We thus implemented these canine data, together with our ovine data on 3-D geometry and LV cavity pressure, in our FE model. Myocardial deformation quantified in multiple short-axis and long-axis planes throughout diastolic filling and isovolumic systole using MR tagging (ie, a 3-D strain analysis) should provide a rigorous test of this models predictive capabilities. In any event, however, the present model study of an infarcted LV is a significant improvement over the previous state-of-the-art methods [8].
Reduction in LV size (ventriculoplasty), by either LV aneurysm repair [26] or partial ventriculectomy [27], has been proposed as surgical treatment for congestive heart failure. Although results with both aneurysm repair and partial ventriculectomy have been mixed [28, 29], a quantitative mechanical analysis of ventriculoplasty should allow the design of new surgical procedures that improve ventricular function. Changes in ventricular wall stress are believed to be stimuli for growth and remodeling [30]. Thus, it is likely that surgical aneurysm repair is successful when it results in a reduction in wall stress, a subsequent improvement in BZ contractility, and improvement in ventricular function. The present study provides a foundation for such an analysis.
| Acknowledgments |
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| References |
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