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A 2D FE model of the heart demonstrates the role of the pericardium in ventricular deformation

Departments of ¹Cardiac Sciences, ²Physiology and Biophysics, and ³Civil Engineering; and ⁴Libin Cardiovascular Institute of Alberta, University of Calgary, Calgary, Alberta, Canada

Submitted 18 January 2006 ; accepted in final form 1 June 2006

	ABSTRACT

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During pulmonary artery constriction (PAC), an experimental model of acute right ventricular (RV) pressure overload, the interventricular septum flattens and inverts. Finite element (FE) analysis has shown that the septum is subject to axial compression and bending when so deformed. This study examines the effects of acute PAC on the left ventricular (LV) free wall and the role the pericardium may play in these effects. In eight open-chest anesthetized dogs, LV, RV, aortic, and pericardial pressures were recorded under control conditions and with PAC. Model dimensions were derived from two-dimensional echocardiography minor-axis images of the heart. At control (pericardium closed), FE analysis showed that the septum was concave to the LV; stresses in the LV, RV, and septum were low; and the pericardium was subject to circumferential tension. With PAC, RV end-diastolic pressure exceeded LV pressure and the septum inverted. Compressive stresses developed circumferentially in the septum out to the RV insertion points, forming an arch-like pattern. Sharp bending occurred near the insertion points, accompanied by flattening of the LV free wall. With the pericardium open, the deformations and stresses were different. The RV became much larger, especially with PAC. With PAC, the arch-like circumferential stresses still developed in the septum, but their magnitudes were reduced, compared with the pericardium-closed case. There was no free wall inversion and flattening was less. From these FE results, the pericardium has a significant influence on the structural behavior of the septum and the LV and RV free walls. Furthermore, the deformation of the heart is dependent on whether the pericardium is open or closed.

ventricular function, right ventricular overload; finite element analysis

Using finite element (FE) analysis, our group studied the behavior of the interventricular septum during diastole with acute pulmonary artery constriction (PAC) (17, 18). Under control conditions, the septum behaved as part of the LV and was under slight circumferential tension; the dominant stress was a low radial compression over the width of the septum. Previous FE analyses estimating normal wall stress (2, 3, 10) showed similar results.

The stress pattern changed dramatically with PAC, where RV pressure was higher than LV pressure (17, 18). Two-dimensional (2D) echocardiography showed that the septum flattened and often inverted toward the LV. Under these conditions, large quasi-circumferential compressive stresses developed in the septum, forming an arch-like pattern from the LV side of one RV insertion point to the LV side of the other, rising to the RV side of the septum at midspan. In this septal model, bending moments had to be applied at the insertion points (i.e., the ends of the model) with PAC to match the deformed shape of the model with that observed experimentally. If the entire cross section of the heart had been considered, these bending moments would have had to be balanced by equal and opposite bending moments in the ventricular free walls. Because the thickness of the LV is greater than that of the RV, it would be expected that most of these bending moments would be carried into the LV free wall, which would then become flatter. The extent of septal flattening and inversion with PAC may be dependent on the stiffness of the pericardium and the constraint it exerts. The concomitant transfer of moments into the LV free wall may also be affected by the pericardium. We used a 2D model to match the geometric information obtained from echocardiography. Results from this model, as described here, led us to blood flow experiments (6) that showed that the hydrostatic stresses induced by PAC reduced blood flow in specific areas of the myocardium; those blood flow experiments may help to explain signs of reduced blood flow in patients with normal coronary arteries.

Therefore, with the use of FE analysis, the goal of this study was to describe quantitatively the deformation of the septum and LV free wall with acute PAC and to define the role of the pericardium in this response.

	METHODS

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Animal preparation. With the use of a protocol approved by the University of Calgary Animal Care Committee, experiments were performed in 14 open-chest anesthetized dogs. Animals in the weight range of 22–24 kg were chosen to minimize heart size differences. Anesthesia was induced with 25 mg/kg iv thiopental sodium and maintained with an infusion of 25 mg/ml solution (100 ml/h) of fentanyl citrate. The dogs were ventilated with a constant-volume respirator (model 607, Harvard Apparatus, Millis, MA). A single-lead ECG was used for cardiac monitoring, and body temperature was maintained in the physiological range with a heating pad.

LV, RV, and aortic pressures were measured with 8-Fr micromanometer-tipped catheters (Millar Instruments, Houston, TX), which were inserted via peripheral vessels. Pericardial pressures were recorded with flat, fluid-filled balloon catheters loosely attached to the epicardium (12). The heart was repositioned in the pericardium, and the edges of the pericardium were reapproximated with loose interrupted sutures (21).

A pneumatic constrictor (14–16 mm, In-Vivo Metrics, Healdsburg, CA) was placed on the pulmonary artery to increase RV pressure transiently and change the end-diastolic transseptal pressure gradient [LV end-diastolic pressure (LVEDP) – RV end-diastolic pressure (RVEDP)]. Pressures were recorded simultaneously with 2D echocardiography (Sonos 1000, Hewlett-Packard, Palo Alto, CA); an LV minor-axis view at the level of the papillary muscles was recorded by using a 2.5-MHz transesophageal probe held on the surface of the RV free wall. The shapes and dimensions used for the FE analysis (i.e., LV and RV diameters and wall thicknesses) were derived from analysis of the echocardiographic images (Echo Analysis version 1.05, Advanced Measurements; Calgary, AB, Canada). Examples of the echo data used are shown in Fig. 1. End-diastolic echocardiographic images were digitized by manually tracing the endocardium and epicardium of the LV and the RV endocardial surface of the septum and its insertion points. The curvatures (i.e., 1/R, where R is the radius of curvature measured from the echo images) of the septum and the LV free wall were calculated. An LV area index was also calculated from the echo images as the product of the anterior-posterior and septum-to-free wall epicardial diameters.

View larger version (41K): [in this window] [in a new window]   Fig. 1. Typical examples of echocardiograms obtained under control conditions (A) and with pulmonary artery constriction (PAC; B).

FE model. A 2D FE model of an equatorial plane of the heart was developed by using PATRAN (version 5.4; MSC Software, Santa Ana, CA) and is shown in Fig. 2. The thicknesses of the LV and RV walls were typical of values at end diastole. The pericardium was modeled as congruent with the LV and RV free walls, separated from them tangentially near the RV insertion points. Slide-line elements were used to separate the free walls from the pericardium and to allow free sliding movement within the pericardium. All solid elements were eight-node quadrilaterals. Over 1,600 elements were used in the model. One node was fixed in space to allow the model to converge. Hyperelasticity was assumed for all materials, and geometric nonlinearity was utilized, because large displacements were expected.

View larger version (44K): [in this window] [in a new window]   Fig. 2. Two-dimensinal finite element mesh used to model left and right ventricles and pericardium.

The 2D model was chosen for its simplicity and convenience. As illustrated by our several previous experimental studies of ventricular interaction (1, 9, 11, 14, 23, 24), minor-axis dimensions can be used to analyze the phenomena of ventricular interaction in ways that are both useful and consistent.

The model was solved using Abaqus (version 6.27, Abaqus; Pawtucket, RI). As shown in the previous study (18), both linear and nonlinear stress-strain relationships produced similar results after analysis. Linear analysis was used because it required much less computer time.

Data recorded under control conditions and during PAC, with the pericardium closed and with the pericardium opened, were analyzed and modeled.

	RESULTS

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Only statistically significant differences are cited; P values can be found in Table 1.

Results of the FE model for control conditions for one dog at end diastole are shown in Fig. 3A and Fig. 4, A and C. Under control conditions (pericardium closed), the displaced mesh is very similar to the initial unstressed mesh (Fig. 3A); the LV (including the septum) was quasi-circular.

View larger version (64K): [in this window] [in a new window]   Fig. 3. Mesh plots showing displacement from initial model under control conditions with pericardium intact (A), PAC with pericardium intact (B), control conditions with pericardium removed (C), and PAC with pericardium removed (D).

View larger version (41K): [in this window] [in a new window]   Fig. 4. A and B: maximal principal stresses for control and PAC (pericardium intact), respectively, plotted on a common scale. C and D: minimal principal stresses for control and PAC, respectively, plotted on a similar scale.

As seen in the previous septum-only model (18), the dominant stress was radial compression (Fig. 4, A and C), with the inner endocardial layer showing greater compression than the outer epicardial layer. The pattern was uniform throughout the wall. The maximal compression seen was –10.5 mmHg (or –1.4 kPa; negative values indicate compression and positive values indicate tension), whereas mean pixel-average radial compression ranged between –3.2 ± 0.7 and –4.3 ± 0.7 mmHg. To obtain a solution for the model, we used a pericardial stiffness (0.06–0.090 MPa, or 452–677 mmHg) that was an order of magnitude higher than the myocardial (muscle) stiffness (22–46 mmHg). The pericardium was subject to circumferential tension of >150 mmHg (Fig. 4A). Tensions at the RV insertion points were low. The stress distribution was similar in all hearts analyzed.

With PAC, RVEDP exceeded LVEDP: RVEDP was 8.9 ± 2.8 mmHg, whereas LVEDP was 5.5 ± 2.5 mmHg, making the transseptal pressure gradient negative (–3.4 ± 1.7 mmHg; Table 1). These values are similar to those observed by other groups in previous studies (5, 22). To reach a solution with FE analysis for PAC, the stiffness of the myocardium was increased slightly compared with control (i.e., it ranged from 0.006 to 0.009 MPa). End-diastolic pericardial pressure increased from 6.9 ± 0.8 to 8.5 ± 0.8 mmHg with PAC (P < 0.01). Pericardial model stiffness was also increased (ranging from 0.09 to 0.12 MPa).

Results of the FE model during PAC (pericardium closed) for one dog are shown in Fig. 3B and Fig. 4, B and D. The displaced mesh (Fig. 3B) shows the enlarged RV and the flattened and/or inverted septum that occurs with PAC. The LV free wall also shows some flattening when compared with control, a result of bending moments induced by the flattened septum.

Figure 4B shows a contour plot of the maximal principal stresses (exhibiting the greatest tensions), and Fig. 4D shows a contour plot of minimal principal stresses (exhibiting the greatest compressions). The circumferential tension seen in the pericardium with PAC was significantly higher than seen with control (Fig. 4, comparing A and B). Within the constraint of the pericardium, the septum flattened and, except for two dogs, inverted, with compressive circumferential stresses developing out to the insertion points (Fig. 4D). Again, this is similar to the results from the septum-only model (18). Rather than seeing a constant curvature (as with control conditions), there was a sharp bend near the RV insertions and flattening and inversion of the septum. The end-diastolic curvature of the septum changed from 0.276 ± 0.015 cm^–1 (concave to the LV) to –0.125 ± 0.011 cm^–1 with PAC (inverted into the LV, concave to RV; P < 0.01). The inverted septum and the larger RV size (Fig. 3B) caused greater pull and increased tension at the RV insertion points (Fig. 4, B and D). Not including the extreme tensions at the RV insertion that may be model dependent, the average stress over the region of the epicardial RV insertion points was seen to change from –4.1 ± 0.6 at control to +22.2 ± 1.4 mmHg with PAC. Maximal tensions in this region were as great as 75.8 ± 5.9 mmHg. The inverted septum also caused a zone of tension (to develop) on the endocardial side of the midseptum (with PAC) (regional average pixel stress changed from –5.4 ± 0.6 at control to +19.7 ± 2.0 mmHg; P < 0.01). The maximum individual-pixel tensile stress in this region was 36.8 ± 3.2 mmHg.

The maximum individual-pixel compressive stress (–98.3 mmHg or –13.0 kPa) occurred on the LV side of the insertion points (Fig. 4D; minimal principal stresses). This maximal compression is almost 10 times that seen in the control model (Fig. 4, C vs. D), and this compression was circumferentially oriented rather than radially. The maximum individual-pixel compressive stress in the endocardial insertion points increased significantly from –5.3 ± 0.5 at control to –75.0 ± 7.7 mmHg during PAC.

Another zone of significant compression developed on the RV side of the inverted septum, although these stresses were not as great as those at the insertion points (Fig. 4D). The maximum individual-pixel compression on the RV side of the septum increased from –4.1 ± 0.7 to –46.6 ± 5.0 mmHg. The circumferential stresses extended beyond the insertions, into the LV free wall. The LV free wall flattened significantly (P < 0.01) but never inverted, with the curvature at midspan changing from 0.349 ± 0.023 (control) to 0.274 ± 0.015 cm^–1 (PAC). The average circumferential compression at the midpoint of the LV free wall epicardium, where flattening was most obvious, changed from –3.7 ± 0.7 to –28.1 ± 9.8 mmHg (P < 0.01). This stress distribution was similar for all hearts analyzed.

The hydrostatic component of stress is the mean of the principal stresses and tends to cause changes in tissue volume, as opposed to changes in shape, which are caused by changes in the deviatoric components of stress. Plots of the hydrostatic component of stress were shown in our previous study (6). Consistent with the results shown in the present paper, the hydrostatic stress plots showed regions of compression at the endocardial RV insertion points as well as the RV side of the midseptum.

The bending moments around the circumference of the LV were calculated from data recorded under normal conditions and during PAC; the FE model results are illustrated in Fig. 5 for an individual pericardium-closed case. Under control conditions (Fig. 5A), the bending moments around the LV circumference were small. Slight tension at the RV insertion points was consistent with the inflections in the LV endocardial circumference. Axial forces were uniform. With PAC, the patterns were magnified. Large bending moments developed in the septum, and tensions at the RV insertion points increased because of larger RV circumferential forces. As a result, the midpoint of the septum was displaced toward the LV, and, in this case, the septum inverted. The LV free wall flattened (Fig. 3, A vs. B) as a result of the induced bending.

View larger version (20K): [in this window] [in a new window]   Fig. 5. Bending moment diagrams for normal loading (A) and abnormal loading (B) in a single animal.

View larger version (30K): [in this window] [in a new window]   Fig. 6. A and B: maximal principal stresses for control and PAC (pericardium removed), respectively, plotted on a common scale. C and D: minimal principal stresses for control and PAC (pericardium removed), respectively, plotted on a similar scale. Gray regions in D indicate tensions >20 mmHg. By convention, positive values indicate tension and negative values indicate compression.

With the pericardium open, PAC still produced a negative transseptal gradient (–4.2 ± 1.1 mmHg), although LVEDP and RVEDP were significantly lower than during PAC with the pericardium closed. LVEDP decreased from 5.2 ± 0.9 (control, open pericardium) to 3.0 ± 0.6 mmHg during PAC while RVEDP increased from 4.0 ± 1.1 (control, open pericardium) to 7.1 ± 1.2 mmHg (see Table 1). During PAC with the pericardium open, the RV showed a large increase in size. In all cases, the septum flattened and in eight dogs it inverted, but there was no change in the LV free wall curvature associated with this flattening and/or inversion (Fig. 3D).

FE results of a PAC for one dog with the pericardium opened are shown in Fig. 6, B and D. When the pericardium was opened, the septum inverted significantly less with PAC than when it was closed. In five of the 14 dogs, the septum only flattened, whereas in the others, it inverted. Mean septal curvature during PAC was –0.007 ± 0.059 cm^–1, compared with –0.125 ± 0.011 cm^–1 during PAC with the pericardium closed.

Arch-like circumferential stresses still developed in the septum but were smaller than when the pericardium was closed (compare Fig. 6D with Fig. 4D). The maximal compressive stresses were again seen on the LV endocardial side of the insertion points but were smaller than with the pericardium-closed model. For this individual case (Fig. 6), a maximum compression of –48.6 mmHg occurred again at the insertion points, compared with –98.3 mmHg when the pericardium was closed. The maximum individual-pixel compression with PAC (–22.7 ± 1.4) was also reduced with pericardium open (compared with –75.0 ± 7.7 for the pericardium-closed case). The compression within the septum (midpoint, RV side) was also reduced in magnitude over the pericardium-closed model. The overall size of the compression zones at both the insertion points and septum appeared to be reduced (Fig. 6D compared with the pericardium-closed model in Fig. 4D).

When the pericardium was open, PAC did not flatten the LV free wall significantly (0.326 ± 0.019 cm^–1 compared with 0.318 ± 0.020 cm^–1 for control, pericardium opened; not significantly different) nor did stresses change, in spite of substantial septal deformation.

	DISCUSSION

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In the present study, we used a 2D FE model to demonstrate that the pericardium affects the deformation of the heart and the magnitudes of the strains and stresses that develop. Although this simple model does not account for the third dimension, fiber orientation, or anisotropy and although our results cannot be used to predict reliably the behavior of every other 2D plane of the heart, the model has proved capable of predicting significant deformation-induced compression, compression that has been shown to correlate with substantial reductions in local blood flow (6).

Although our data were obtained during acute RV pressure loading, our findings may also apply to chronic RV pressure loading that is associated with similar septal flattening and inversion (16, 19). Substantial pericardial remodeling occurs with experimental intracardiac shunting (18) and in patients with chronic RV pressure overload, such as in chronic embolic thromboembolic pulmonary hypertension (19). However, those patients had enlarged RVs, their septums flattened (i.e., moved leftward during diastole), and, after embolectomy, septal motion returned to normal. This would suggest that stress patterns in those patients were not substantially different from those we observed in this study. Stress patterns with the pericardium open also support this: PAC still caused septal flattening, albeit to a lesser degree. Thus pericardial remodeling may attenuate the effects; however, it is likely that decreasing the transseptal pressure gradient still deforms the heart and causes similar stresses.

During PAC with the pericardium closed, the LV free wall flattened in a similar manner as the septum, although not to the same degree. This appears to be the result of the bending moments induced by the septal inversion. When the pericardium was opened, however, LV free wall flattening did not occur, probably because the bending moments were smaller than when the pericardium was closed and the angle of pull of the RV was increased. The effect of the changes caused by the shift in RV pull can be seen by comparing Figs. 4 and 7. With PAC at end diastole when the pericardium is taut, pericardial constraint causes a greater negative transseptal pressure gradient, greater septal inversion, and thus greater bending moments to be transferred to the LV free wall, compared with when the pericardium is open. Therefore, it is clear that data from studies performed with an open pericardium and FE or other analytic models may not accurately reflect structural and mechanical responses of the organ and/or individual tissues when the pericardium is normally closed. The potential role of the pericardium should be considered in any experimental study or analytic model of the heart.

To reach a solution under PAC conditions, the stiffnesses of the muscle and pericardium had to be increased, compared with the values needed under control conditions. Such increases had also been required with analysis of the septum-only model (18). We therefore hypothesized that the compressive stresses that developed with PAC might reduce coronary blood flow, perhaps by impeding the drainage of venous blood. Congestion of blood within the wall has been shown by others (19, 25) to increase tissue stiffness. If compression reduced blood flow, it might help to explain the apparent perfusion defects in patients with normal coronary arteries who have pulmonary hypertension, left bundle branch block, or other conditions that elevate RV pressure (13, 16, 20).

A coronary perfusion study using colored microspheres was performed to test this hypothesis . Under control conditions with the pericardium closed, there was similar flow in all regions of the LV, and these flow values compared well with previously published flow data (4, 8). With PAC, however, the flow pattern changed significantly. The areas showing the greatest decrease in flow occurred at the endocardial side of the RV insertion points as well as on the RV side of the septum. These zones of high flow reduction compared well with the zones of high circumferential compressive stress shown in Fig. 4D. The data show a strong correspondence between the circumferential compressive stresses and blood flow. Compressive stresses are greater on the endocardial side than the epicardial side, and this was also where there was reduced flow (except in zones opposite a flattening/inversion zone). This transmural difference in flow has also been demonstrated by other research groups (8, 22) but never explained in terms of compression changes. Thus our results provide a possible mechanism for septal hypoperfusion with RV pressure overload.

The following points summarize our conclusions.

1) With the pericardium closed, bending moments induced in the septum during acute PAC are transmitted into the LV free wall, flattening that structure. The decrease in constraint due to opening the pericardium allows the RV to enlarge considerably, changes the "pull" angle of the RV on the LV wall, and reduces the moments induced in the flattened (inverted) septum and in the LV free wall. With the pericardium open, the LV free wall does not flatten like it does during PAC with the pericardium closed.

2) The pericardium is a major modulator of the behavior of the structural elements of the heart during diastole. Its presence substantially affects the deformation of the heart during diastole and, consequentially, strains and stresses. Our results suggest that data from experiments and models in which the pericardium is open should be reconsidered in this light.

	GRANTS

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This study was supported by a grant (MT-12886) from the Canadian Institutes for Health Research (Ottawa, Ontario) (to J. V. Tyberg and N. G. Shrive).

	ACKNOWLEDGMENTS

 
We appreciate the technical assistance of Cheryl Meek and the helpful criticisms of Dr. I. Belenkie.

C. Gibbons Kroeker held a postdoctoral research fellowship from the Natural Science and Engineering Research Council (Ottawa), N. G. Shrive is a Killam Memorial Professor, and J. V. Tyberg was a Scientist of the Alberta Heritage Foundation for Medical Research (Edmonton).

	FOOTNOTES

 

Address for reprint requests and other correspondence: C. A. Gibbons Kroeker, Dept. of Cardiac Sciences, Faculty of Medicine, Univ. of Calgary, Health Sciences Centre, 3330 Hospital Dr. NW, Calgary, AB, Canada T2N 4N1 (e-mail: kcagibbo{at}ucalgary.ca)

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.

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