Ann Thorac Surg 2002;74:1581-1587
© 2002 The Society of Thoracic Surgeons
Original article: cardiovascular
Biomechanical modeling of hemodynamic factors determining bulging of ventricular aneurysms
Thomas Bartel, MDa*,
Hans Vanheiden, MDb,
Johannes Schaar, MDa,
Wolfgang Mertzkirch, MDb,
Raimund Erbel, MD, FACCa
a Department of Internal Medicine, University of Essen, Essen, Germany
b Department of Fluid Mechanics, University of Essen, Essen, Germany
Accepted for publication June 13, 2002.
* Address reprint requests to Dr Bartel, Division of Cardiology, Department of Internal Medicine, Hufelandstr 55, 45122 Essen, Germany.
e-mail: thomas.bartel{at}uni-essen.de
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Abstract
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BACKGROUND: Ventricular aneurysm formation is a frequent complication of transmural myocardial infarction. The hemodynamic determinants of aneurysmal bulging remain unclear.
METHODS: A rubber heart placed in a water tank served as an in vitro model. Rhythmic injections of specific volumes into the tank simulated heart beats. The heart rate was adjustable in increments. A section of the heart model’s wall was shielded from compression to simulate an aneurysm. To quantitate the relation between hemodynamics and bulging, pressures, echocardiographic measurements of maximal expansion, and mean velocity were recorded. Bulging volume, stroke volume, aneurysmal wall stress, and systemic resistance were calculated.
RESULTS: The mean velocity was the echocardiographic factor most closely related to bulging volume (r = 0.92, p < 0.01). When bulging indices were compared with hemodynamics, bulging volume and mean velocity were found to directly depend on heart rate (r = 0.66, p < 0.01; r = 0.70, p < 0.01). Polynomial regression revealed bulging volume to reach minimal values near 80 beats/min. Maximal systolic aneurysmal wall stress was closely related to the peak positive rate of pressure change (r = 0.94, p < 0.01) and moderately to stroke volume (r = 0.75, p < 0.01). Filling pressures were unrelated to bulging. The greatest bulging volume reduction occurred below 790 dynes . s . cm-5; bulging was practically eliminated at systemic resistance values less than 395 dynes . s . cm-5.
CONCLUSIONS: Aneurysmal bulging and aneurysm formation depend mainly on heart rate, contractility, and afterload. This suggests that hemodynamic management may affect the extent of bulging in a clinical setting.
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Introduction
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After a transmural myocardial infarction, a well-delineated area of nonuniform interstitial fibrosis develops that can increasingly bulge outward during systole [1]. In addition to ventricular arrhythmias [2] and mural thrombus formation with the risk of thrombembolization [3], a true aneurysm can develop, leading to an increase in both volume and thickness of the nonaneurysmal portion of the left ventricle and eventually to congestive heart failure [1]. Echocardiography is considered the method of choice for left ventricular aneurysm detection and follow-up [4, 5]. Nevertheless, exact quantification of outward bulging remains difficult [6]. Up to now, it is unknown to what extent global hemodynamics impact the development of ventricular aneurysms. Thrombolytic therapy contributes to the reduced incidence of aneurysm formation and so does control of hypertension [3], a fact that suggests that a hemodynamic approach to minimize bulging may be useful. Aneurysm formation depends on the strength of the scar tissue and on traction forces acting on this tissue [7]. Recent results suggest that structural changes after myocardial infarction slowly increase myocardial stiffness during scar formation, therefore the aneurysm will gradually expand over a critical period of 4 to 6 weeks [1, 2].
The present study sought to identify hemodynamic factors that might prevent aneurysm formation. The aims were (1) to quantify outward bulging in ventricular aneurysm by simple standard M-mode echocardiographic measurements and (2) to elucidate the relation between bulging and various hemodynamic factors. It was hypothesized that specific hemodynamic settings would effectively limit average end-systolic wall stress and aneurysm bulging.
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Material and methods
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Mechanical model of aneurysmatic left ventricle
A recent biomechanical model of the left ventricle (Fig 1)
was used. The ventricular model is suspended in an elaborate pressure wave chamber to simulate ventricular and aortic pressures as well as dyskinetic ventricular aneurysms [8]. The chamber is filled with water (Fig 2).
With respect to size and geometry, the rubber model, cast from an actual left ventricle, resembles an average normal left ventricle. A computer-controlled mechanical pump makes the ventricle eject by pushing predefined volumes into the tank. The ventricle is driven by the pressure changes in the surrounding water and by the elastic forces of the rubber walls. The model ventricle is attached to a tube system with inlet and outlet valves to simulate the circulation. The fluid pumped through the system has the same rheologic properties as blood. Transducers in the ascending part of the circulatory system, which simulates the aorta, in the model ventricle, and in the descending part of the circulatory system, which simulates the left atrium, provide instantaneous pressure recordings. Transmission data throughout the system are computed from these pressures. The system was shown to accurately reproduce the ventricular and aortic pressure curves of a healthy individual [8].
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Fig 1. Experimental set-up. 1 = pressure sensor; 2 = pulse rate sensor; 3 = drive wheel; 4 = circulation; 5 = mechanical model of an aneurysmatic ventricle (see also Fig 2); 6 = piston pump.
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Fig 2. Dynamic model of an aneurysmatic ventricle. 1 = atrial inflow; 2 = model atrium; 3 = inlet and outlet valve; 4 = model ventricle; 5 = water tank generating ventricular pressure; 6 = aneurysm chamber; 7 = model aorta; 8 = systemic outflow tract; 9 = echocardiographic transducer.
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A small water-filled compartment (aneurysm chamber) completely shields a section of the model ventricle from the pressure generated in the water tank. This isolated section of the ventricular wall represents aneurysmal scar tissue. The volume in the aneurysm chamber does not contribute to ventricular ejection. To simulate a dyskinetic aneurysm, the water in the aneurysm compartment is gradually removed and replaced by a corresponding amount of air until approximately one-third of the aneurysm chamber is filled with air. A throttle valve connects the air-filled part of the chamber with the ambient air, permitting the aneurysm to drive air out of the chamber, while the fluid (used as acoustic interface for echocardiographic examinations) remains in the chamber when the aneurysm bulges outward (Fig 2). This technique maintains a mean pressure of about 15 mm Hg in the aneurysm chamber to oppose outward bulging. The slight counterpressure simulates the effect of the pericardium and the thoracic wall on the aneurysm. The elastic properties of the model aneurysmal sac allow the wall to bulge into the aneurysmal chamber during systole. The effective forward output of the ventricle depends on the volume increase in the water tank (compression volume, CV), the pressures, and the distribution between bulging volume (BV) of the aneurysm sac and stroke volume (SV):
 | ((1)) |
Th system allows CV settings of 33, 46, 76, 96, and 110 mL, and heart rates (HR) of 60, 70, 80, or 90 beats/min. Mean (m) atrial pressure (ATP) and arterial pressure (AP) are continuously variable. Consequently, the model can simulate even those hemodynamic conditions that cannot be studied in vivo.
Hemodynamic settings and analysis
Four hundred fourteen consecutive experiments with increasing HR, ATP, AP, and CV settings were performed. Starting with the HR at 60 beats/min, ATPm of 12 mm Hg, APm of 26 mm Hg, and CV of 33 mL/beat, APm was incrementally increased to 73 mm Hg. The ATPm was increased in steps of 2 mm Hg until reaching 18 mm Hg. Subsequently, the same experimental set-up was repeated while raising CV to 110 mL. In a second, third, and fourth run, these hemodynamic conditions were again simulated, but the HR was now set to 70, 80, and 90 beats/min, respectively. Ventricular pressure, peak positive rate of pressure change (dP/dtpeak), ATP, and AP were monitored and pressure curves and data stored.
Echocardiographic analysis
Ultrasound studies were performed with a phased array echocardiographic system HDI 5000 (Advanced Technology Laboratories, Mountain View, CA). An echocardiographic 2- to 4-MHz transducer was partially inserted into the aneurysm chamber surrounded by water. The transducer was aligned with the paradoxical motion of the ventricular wall and the insertion site sealed. The transducer was used to record two-dimensional and M-mode scans. The ultrasonic probe was aimed at the central aneurysmal portion of the ventricular wall bulging into the aneurysm chamber, maintaining a distance between 0.5 and 2.5 cm during the simulated cardiac cycle, as dictated by the hemodynamic conditions. Conventional M-mode (M-mode velocity 50 mm/s) recordings were used to measure the maximal expansion (EXP) of the aneurysm (vertical distance between diastole and systole), and the mean wall velocity (WVm) (slope during outward bulging).
Hemodynamic calculations
As proposed by Buck and colleagues [6], BV was calculated as a half-ellipsoid from EXP (minor hemi-axis in the x and y directions) and the radius (r1 = hemi-long axis = 2.5 cm; r2 = hemi-short axis = 2 cm) of the elliptical aneurysm camber opening:
 | ((2)) |
Subsequently, effective SV was calculated by substituting BV (in equation 1) by equation 2. Cardiac output (CO) was obtained by multiplying SV with HR:
 | ((3)) |
With respect to the experimental system, systemic resistance (SR) is defined [9] as:
 | ((4)) |
Using equations 3 and 4, SR is calculated as follows:
 | ((5)) |
SR is primarily determine d by HR, CV, AP, and ATP, but also depends on the extent of bulging.
Relative bulging volume
To demonstrate how severely an aneurysm impairs ventricular function and how this impairment is related to hemodynamics, the relative bulging volume (RBV) was calculated:
 | ((6)) |
Aneurysmal wall stress
Analogously to the average circumferential wall stress, the average systolic increase of the aneurysm’s wall stress (
) is directly related to the product of systolic ventricular pressure and internal radius (r1) of the aneurysm, and inversely related to wall thickness (h), which was assumed to remain constant at 0.3 mm. Using Laplace’s law for an ellipsoidal ventricle [10], aneurysmal wall stress was calculated as follows:
 | ((7)) |
The dependence of , a measure closely related to aneurysmal development, on hemodynamic variables was analyzed to define which hemodynamic factors should be modified to minimize growth of the aneurysm.
Statistical and other mathematical analysis
Numerical data are expressed as mean ± standard deviation. Independent measures (HR, SV, ATP, AP, SR, and dP/dtpeak) were compared with the dependent variables (WVm, BV, RBV, and ) using simple linear, stepwise multiple, and nonlinear correlation analysis. The SV was not compared with RBV, as there is a relation by definition. To correct for multiple testing, particularly when calculating 21 correlation coefficients, an appropriate Bonferroni 
-adjustment was applied multiplying p values by 21. Then, values of p less than 0.05 were considered significant. For polynomial regression equations, a local minimum of the corresponding function for the independent variable was calculated (xo represents a minimum for y = f(x) if y' = 0) to define the conditions associated with minimal bulging. In cases of logarithmic relation, values of the independent variable (xn) were calculated for specific slopes of the corresponding curve (12.25°, 22.50°, and 45.0°) to determine how bulging can be most effectively controlled.
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Results
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General echocardiographic and hemodynamic findings
Descriptive statistics of hemodynamic and echocardiographic data are summarized in Table 1.
Except for dP/dtpeak and AP, hemodynamics were found to be consistent with the range observed during clinical aneurysm formation. The APm and dP/dtpeak could not be increased to more than 85 mm Hg and 672 mm Hg/s to avoid overload of the aneurysmal portion of the left ventricle.
Influence of heart rate on bulging
Polynomial correlation analysis reveals a specific relation between HR and BV and between HR and WVm (Fig 3a,b).
Regression curves (second order polynomial functions) show explicit local minimums at 82 and 79 beats/min. This is the HR range where outward bulging is minimal. Thus, a HR of about 80 beats/min can be considered optimal for minimizing BV. Conversely, any higher and lower HR increases bulging. In contrast, is unrelated to heart rate (p > 0.05). was unrelated and RBV poorly related to HR (Tables 2 and 3).
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Fig 3. Regression plots show relation between bulging measures and hemodynamics. (a) Relation between bulging volume and heart rate; (b) between mean wall velocity of the aneurysm and heart rate; (c) between bulging volume and peak-positive pressure change; (d) between mean end-systolic wall stress and peak-positive pressure change; (e) between mean end-systolic wall stress and stroke volume; (f) between bulging volume and systemic resistance. (BV = bulging volume; dP/dtpeak = peak-positive pressure change; HR= heart rate; = mean end-systolic wall stress; SR = systemic resistance; SV = stroke volume; WVm = mean wall velocity.)
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Relation between contractility, output, and bulging
Compared to a healthy ventricle, dP/dtpeak was low when an aneurysm was simulated (Table 1). Correlation analysis revealed a polynomial relation between BV and dP/dtpeak (Fig 3c), and RBV was similarly related to dP/dtpeak (Table 3). The corresponding regression curve shows a local minimum at 635 mm Hg/s. The trend seems to indicate a second increase at dP/dtpeak levels higher than 635 mm Hg/s. In contrast, linear correlation analysis demonstrated to be closely related to dP/dtpeak (Fig 3d) and to a certain extent also to SV (Fig 3e and Table 2). The relation between dP/dtpeak and WVm was found to be comparatively poor (r = 0.38, p < 0.01). The BV was related to SV; the fact that the regression equation is a fourth order polynomial function (r = 0.76, p < 0.01) indicates, however, that BV is mainly determined by other hemodynamic variables. A comparable function, but rather poor correlation, resulted with respect to WVm (WVm versus SV: r = 0.46, p < 0.05).
Relation between preload, afterload, and bulging
Simple linear regression analysis revealed no relation between ATPm, considered a measure of preload, and BV (r = 0.13, p > 0.05). Relative bulging volume, WVm, and were found to be also unrelated to ATPm (p > 0.05 even before Bonferroni -adjustment; Table 2). Among various procedures, logarithmic correlation analysis revealed a function that described the relation between BV and SR quite precisely (Fig 3f), that is, increasing SR causes more bulging. The logarithmic relation expresses mathematically that the BV slope becomes much flatter with increasing SR. Conversely, BV decreases with a down-slope of more than 22.5 degrees if SR decreases below 790 dynes . s . cm-5. Below 395 dynes . s . cm-5, BV slopes steeply downward at an angle of more than 45° (Fig 4).
The RBV was even closer related to SR (Table 3). In comparison to BV, logarithmic correlation demonstrated WVm and to be less closely related to SR (r = 0.45, p < 0.01; Table 2).
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Fig 4. Calculated systemic resistance for local slope values of 45, 22.5, and 11.25 degrees at the logarithmic function between bulging volume and systemic resistance. (BV = bulging volume; SR = systemic resistance.)
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Comment
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A left ventricular aneurysm produces mechanical impairment not only because of the lost contractile tissue but also because of significant paradoxical systolic expansion [5]. Using a mechanical model, we simulated severe regional dysfunction and exposed the resulting aneurysm to various hemodynamic conditions. The effect of different hemodynamic factors on the aneurysm wall—BV, RBV, WVm, and —were assessed. The results add to the understanding of hemodynamic effects on ventricular aneurysm formation and have diagnostic and therapeutic implications. M-mode echocardiographic estimation of BV and WVm is proposed as a simple tool, which can be used in addition to the well-established cross-sectional analysis of left ventricular aneurysms [5, 11]. Relative bulging volume demonstrates how severely an aneurysm impairs cardiac output, independently of the ventricle’s actual functional state.
Hemodynamic determinants of aneurysmal bulging
Clinical outcome with and without surgical repair clearly relates to aneurysm size [12], but size alone is not a good predictor for further enlargement. However, specific guidelines for the hemodynamic management would be desirable. For the first time, our results suggest that it may be advantageous to avoid bradycardia as well as tachycardia while an aneurysm is forming. From a purely mechanistic view (neglecting myocardial ischemia and tissue characteristics, as well as elastic properties of both the aneurysmal and the nonaneurysmal portion of the ventricle), a HR of approximately 80 beats/min might minimize bulging and the underlying forces acting on the aneurysmal ventricular wall.
End-systolic wall stress of the aneurysm is markedly influenced by contractility and, to a lesser extent, by ventricular ejection. The relationship between afterload and is not as strong as between afterload and BV or afterload and RBV. This result suggests that mainly depends on ventricular function, whereas BV and RBV mainly depend on afterload. This finding is in line with the observation that there was a weak relation between contractility and extent of bulging and only limited correlation between contractility and aneurysm wall velocity. This finding may be one explanation for the increased mortality observed in association with digitalis in a high-risk subset of patients within the first several months after myocardial infarction [13, 14]. Preload appears to be unrelated to bulging, or to be of only secondary importance. In contrast, afterload is closely related to BV and even more to RBV and therefore, to bulging and to impairment of ventricular output due to formation of an aneurysm. The logarithmic relation clearly shows that afterload reduction would dramatically reduce bulging. In contrast, bulging seems difficult to avoid if resistance exceeds a certain level. This finding is consistent with clinical results of angiotensin converting enzyme inhibitors decreasing the 4- to 6-week mortality and to improve ventricular function after acute myocardial infarction [15, 16]. These effects might be secondary to effective prevention of aneurysm formation under significant afterload reduction. The importance of afterload for aneurysm development was previously demonstrated, but only in isolated strips of tissue that were exposed to stress [17]. In contrast, the present study attempts to shed light on the interplay between global hemodynamics and the aneurysm as an integral part of the complete circulatory system.
Limitations
It is recognized that the present system is only capable of partially simulating the performance of aneurysmal ventricles, the behavior of aneurysms in particular, as the material properties of the simulated aneurysm are identical to the rest of the model ventricle, and neither one is matched to actual tissue. The biomechanical model ignores adaptive processes such as increasing scar tissue stiffness and ventricular remodeling [18] or possible epicardial adhesions after infarction. Nonuniform scar tissue thinning over time, different ratios of scar tissue thickness/contractile tissue thickness, and anisotropy of both aneurysmal tissue and remote noninfarcted myocardium [18] cannot be simulated. In addition, the underlying disease—in most cases coronary heart disease [19]—is not taken into account. Thus, loss of contractility in the border zone of the infarcted area (the result of reduced coronary perfusion) and regional variations are imperfectly simulated, because a homogeneous contraction pressure is generated by external compression and not by the ventricle itself. In addition to size, curvature of the aneurysm wall also determines geometric and mechanical changes [20]. All calculations were done under the condition that the size of the aneurysm base remains fixed during compression and filling of the model ventricle. Approximating BV as a half-ellipsoid must be also considered an oversimplification. If ATPm exceeds the pressure in the aneurysm chamber (15 mm Hg), the aneurysmal wall may start with a bit of a bulge before the contraction begins. Wall thickness and elasticity of the model ventricle are different from those of the aneurysmal heart wall. Especially elastic properties of the rubber cast are obviously more uniform than infarcted myocardium [21]. Neither the force–velocity relation nor the velocity–afterload relation defining contractility were determined, as the model did not allow accurate measurement of force and velocities other than aneurysmal wall velocity. Alternatively, dP/dtpeak was considered a measure of contractility. Comparatively low dP/dtpeak values are caused by the bulging aneurysm itself; compensatory mechanisms of the remaining ventricle are lacking. The effect of high dP/dtpeak on bulging could not be tested and is only based on assumptions. The model represents an abstraction to selectively clarify interrelations between contracting ventricle, bulging, and circulation. There is clearly a need to confirm the results in a clinical setting.
In conclusion, even if the tissue characteristics of the human heart are very different from our model, the underlying hemodynamic principles still apply. The present results suggest that adequate hemodynamic management focusing on optimal HR, contractility, and afterload, may affect the extent of bulging. Clinical studies evaluating the usefulness of hemodynamic optimization will be of critical importance to assess the value of the present findings in the overall management of patients with extended myocardial infarction.
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T. Bartel
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Abstract
Introduction
