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Ann Thorac Surg 2001;71:654-662
© 2001 The Society of Thoracic Surgeons

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Original article: cardiovascular

Mechanism underlying mechanical dysfunction in the border zone of left ventricular aneurysm: a finite element model study

^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

Top Abstract Introduction Material and methods Results Comment Acknowledgments References

 
Background. The global left ventricular dysfunction characteristic of left ventricular aneurysm is associated with muscle fiber stretching in the adjacent noninfarcted (border zone) region during isovolumic systole. The mechanism of this regional dysfunction is poorly understood.

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

Top Abstract Introduction Material and methods Results Comment Acknowledgments References

 
Left ventricular (LV) aneurysm is a significant complication of myocardial infarction that may lead to global LV dysfunction, ventricular arrhythmias, or thromboembolic complications [1]. For acutely ischemic myocardium, the pathophysiology of the global LV dysfunction has been linked to dysfunction in the border zone (BZ) region of normally perfused but poorly functioning myocardium [2]; however, the exact mechanism underlying the abnormal function has not been elucidated. Using magnetic resonance (MR) tagging and two-dimensional, regional strain analysis, Moulton and colleagues [3] recently observed abnormal midwall circumferential lengthening strains in the BZ regions of an ovine model of LV aneurysm during isovolumic systole. In a similar study, Kramer and colleagues [4] demonstrated that mechanical dysfunction in the BZ of LV aneurysm persists up to 6 months after transmural infarction. These results, combined with information on the histologic [3] and cellular biochemistry [5] of BZ muscle fibers, imply that the stretching of BZ fibers during isovolumic systole may be explained by some combination of (1) an intrinsic contractile abnormality of the BZ region and (2) an increase in wall stress.

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

Top Abstract Introduction Material and methods Results Comment Acknowledgments References

 
Animal protocol
Creation of left ventricular aneurysm
The study was performed in compliance with the animal welfare regulations and the "Guide for the Care and Use of Laboratory Animals" as revised in 1996 [10]. The extremely reproducible ovine model of LV aneurysm described by Markovitz and colleagues [11] was used. An adult Dorset sheep was sedated with ketamine (15 mg/kg intramuscularly), masked, and then intubated and ventilated with a mixture of isoflurane and oxygen. The surface electrocardiogram and the arterial pressure (left femoral artery) were monitored on a continuous oscilloscope display. A left thoracotomy was performed in the fifth intercostal space; the pericardium was opened and 2-0 silk sutures were placed around the distal homonymous coronary artery at a point 40% of the distance from the apex to the base of the heart. An additional ligature was placed around the second diagonal branch of the homonymous coronary artery at its inception from the homonymous. During the left thoracotomy incision a lidocaine infusion (1.5 mg · kg^-1 · min^-1) was started. Before securing the ligature, the sheep received a bolus of lidocaine (2 mg/kg) while the infusion rate was increased (3 mg · kg^-1 · min^-1). After coronary ligation, the animal was monitored closely for 40 to 45 minutes for the occurrence of ventricular tachycardia, which was treated with additional boluses of lidocaine (1.5 mg/kg each) and bretylium (50 mg each). Subsequently, the thoracotomy incision was closed in layers and a 28F chest tube was placed. Lidocaine infusion was discontinued 10 minutes postoperatively, and the chest tube was removed before the animal was extubated. The animal was treated with Naxcel (ceftiofur sodium, 1 mg/pound per day intramuscularly) for 3 days after the operation, and was allowed to recover for 10 weeks.

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).

View larger version (89K): [in this window] [in a new window]   Fig 1. (Left) Magnetic resonance image showing a short-axis slice at the beginning of diastolic filling, transverse to the long-axis of the left ventricle at the level of the equator, with the left ventricular endocardial and epicardial surfaces denoted by white lines. (Right) Magnetic resonance image showing a corresponding long-axis slice through the lateral free walls of the left ventricle and right ventricle and the interventricular septum; notice the abnormally thin apical region (aneurysm), which is relatively axisymmetric (ie, extends longitudinally a similar amount on the septal and lateral free walls) and the tapering border zone region.

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.

View larger version (32K): [in this window] [in a new window]   Fig 2. (Left) Left ventricular endocardial and epicardial data points denoted by symbols (crosses) and a prolate spheroidal finite element surface mesh with four elements circumferentially and four elements longitudinally. (Right) Close agreement between the left ventricular endocardial surface of the mesh and the corresponding data points. The agreement was equally close between the left ventricular epicardial mesh surface and the corresponding data points, not shown for clarity.

View larger version (56K): [in this window] [in a new window]   Fig 3. Border zone and aneurysmal regions, indicated by the light and dark gray areas, respectively, as assumed in the model. Anterior border zone (ABZ), posterior border zone (PBZ), remote (REM), and aneurysmal (AN) locations are indicated. In the transmural direction, these regions extend from the endocardium to the epicardium.

ß

((1))

((2))

where x^k are deformed rectangular Cartesian coordinates and _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 fiber–cross-fiber and fiber–radial 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, b_f = 18.48, b_t = 3.58, and b_fs = 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))

where the contravariant metric tensor referred to fiber coordinates . 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))

where T_max is the isometric tension achieved at the longest sarcomere length and maximum peak intracellular calcium concentration (Ca₀)_max. The length-dependent calcium sensitivity and the internal variable are given by:

((5))

and

((6))

where B is a constant, l₀ is the sarcomere length at which no active tension develops, and

((7))

In Equation 7, the duration of relaxation, t_r, is a linear function of sarcomere length:

((8))

where m and b are constants. Previously, we found that the measured values T_max = 135.7 kPa, Ca₀ = 4.35 µmol/L, (Ca₀)_max = 4.35 µmol/L, B = 4.75 µm^-1, l₀ = 1.58 µm, m = 1.0489 seconds · µm^-1, and b = -1.429 seconds allowed an FE model of the beating dog heart [12] to predict end-systolic in-plane normal and shear strain distributions from a midventricular region of the anterior LV free wall consistent with experimental measurements [19].

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 T_max 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))

where l_R is the sarcomere length in the unloaded reference configuration. We assumed a linear transmural variation in l_R from 1.78 µm (endocardium) to 1.91 µm (epicardium), in accordance with experimental observations [23]. The Cauchy stress tensor referred to new locally orthonormal fiber ⁱ coordinates in the deformed body and the second Piola-Kirchhoff stress tensor are related by the transformation:

((10))

Although the fiber coordinates are curvilinear in general, orthonormal scaling of the base vectors means that T^ij represent physical components of stress. Figure 4 shows predicted end-diastolic, end-systolic, and intermediate fiber stresses (T¹¹) plotted as functions of sarcomere length for the case of uniform extension in the fiber direction.

View larger version (16K): [in this window] [in a new window]   Fig 4. Computed end-diastolic, end-systolic, and intermediate fiber stresses plotted as a function of sarcomere length for the case of uniform extension in the fiber direction when lR = 1.85 µm. The corresponding values of the variable Tmax in Equation 4 are indicated. In the unloaded reference model configuration, lR varies linearly from 1.78 µm (endocardium) to 1.91 µm (epicardium).

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

Top Abstract Introduction Material and methods Results Comment Acknowledgments References

 
Normal border zone contractility
Figure 5A shows the model configuration obtained for an LV chamber pressure of 1 kPa (7.5 mm Hg) and normal diastolic material properties throughout the wall (including the aneurysm). Figure 5B shows the model configuration obtained for an LV chamber pressure of 8 kPa (60 mm Hg), normal contractility in the remote and BZ regions, and no contractility in the apical aneurysm. Although the chamber volumes are the same in both figures, the corresponding myocardial strain distributions are quite different. Figure 6 shows average midwall fiber strain in four different LV regions plotted as a function of LV pressure during the isovolumic contraction phase of the cardiac cycle. Here, the BZ has the same contractility as the remote regions, and the aneurysm has no contractility and the same diastolic material properties as the rest of the LV wall. These strains are referred to the unloaded or zero transmural pressure configuration. In contrast to the experimental observations of Moulton and colleagues [3], our normal BZ contractility model predicted shortening of midwall muscle fibers in the BZ regions (especially in the anterior wall) during isovolumic systole.

View larger version (51K): [in this window] [in a new window]   Fig 5. (Left) Model configuration obtained for a left ventricular chamber pressure of 1 kPa (7.5 mm Hg) and normal diastolic material properties throughout the wall. (Right) Model configuration obtained for a left ventricular chamber pressure of 8 kPa (60 mm Hg), normal contractility in the remote and border zone regions, and no contractility in the apical aneurysm.

View larger version (18K): [in this window] [in a new window]   Fig 6. Average midwall fiber strain from the four different locations (see Fig 3 for abbreviations and definitions) plotted as a function of left ventricular (LV) pressure during the isovolumic contraction phase of the cardiac cycle when the border zone has the same contractility as the remote regions and the aneurysm has no contractility and the same diastolic material properties as the rest of the left ventricular wall. These strains are referenced to the unloaded or zero transmural pressure configuration.

View larger version (18K): [in this window] [in a new window]   Fig 7. Corresponding predictions when the contractility in the border zone is reduced to only 50% of that in the remote regions. (See Fig 3 for abbreviations and definitions.)

View larger version (18K): [in this window] [in a new window]   Fig 8. Corresponding predictions when the contractility in the border zone is reduced to 50% of that in the remote regions and the diastolic stiffness of the aneurysm is 10 times that in the rest of the left ventricular (LV) wall. (See Fig 3 for other abbreviations and definitions.)

	Comment

Top Abstract Introduction Material and methods Results Comment Acknowledgments References

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 model’s 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

Top Abstract Introduction Material and methods Results Comment Acknowledgments References

 
The authors gratefully acknowledge Glen W. Foster, RT, for his patience and expertise in performing the imaging experiments, Ruth Okamoto, DSc, for assistance with the analysis of the images, and Diane Toeniskoetter and Dennis Gordon for assistance with the animal experiments. This study was supported by National Institutes of Health grant R01-HL-58759 (J.M.G.).

	References

Top Abstract Introduction Material and methods Results Comment Acknowledgments References

 

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