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1 Department of Cardiovascular Dynamics, National Cardiovascular Center Research Institute, Suita, Osaka 565, Japan; and 2 Department of Biomedical Engineering, Vanderbilt University, Nashville, Tennessee 37235
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ABSTRACT |
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Top
Abstract
Introduction
Materials & Methods
Results
Discussion
Appendix
References
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We developed a new technique to evaluate regional myocardial elastance using minute vibration. In 13 isolated cross-circulated canine hearts, we applied small sinusoidal vibrations of displacement to the left ventricular surface at various frequencies (50-100 Hz). Using the measured displacement and force between the vibrator head and myocardium, we derived myocardial elastance on the basis of the equation of motion for a given moment of the cardiac cycle. Simultaneous solution of the equations of motion at different frequencies yielded a unique value of elastance. Time-varying myocardial elastance increased from diastole (0.028 ± 0.211 × 106 dyn/cm) to systole (0.833 ± 0.391 × 106 dyn/cm). The end-systolic elastance (ees) linearly correlated with end-systolic left ventricular elastance (r = 0.717, P < 0.001) and also with the end-systolic Young's modulus (r = 0.874, P < 0.0001). We also measured ees at both ischemic and nonischemic regions during coronary occlusion. Young's modulus, estimated by normalizing ees by the wall thickness and by the estimated mass, did not change significantly at the nonischemic regions, whereas it decreased significantly from 2.303 ± 0.556 to 1.173 ± 0.370 × 106 dyn/cm2 at the ischemic region after coronary occlusion (P < 0.005). We conclude that this technique is useful for the quantitative assessment of regional myocardial elastance.
cardiac mechanics; myocardial contractility; muscle properties; stress-strain relationship; regional ischemia
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INTRODUCTION |
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Top
Abstract
Introduction
Materials & Methods
Results
Discussion
Appendix
References
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END-SYSTOLIC LEFT VENTRICULAR ELASTANCE (Ees) has been known to be a relatively load-insensitive index of cardiac chamber contractility (25, 26). The index has several drawbacks, however. Because Ees is dependent not only on muscle contractility (myocardial mechanical properties) but also on chamber geometry, muscle fiber orientation, heterogeneity, and so on (2, 22, 27, 29), it is difficult to determine in the face of a decreased Ees whether the myocardium itself has become weak or whether chamber geometry has changed. In addition, in the case of regional myocardial dysfunction such as that resulting from regional ischemia or myocardial infarction, evaluation of regional myocardial properties becomes necessary (21).
A number of studies have been performed aiming at in vivo determination of load-insensitive myocardial contractility, or myocardial elastance. Investigators have tried to measure the stress-strain relationship by measuring strain and calculating stress using various models. Recent advances in quantifying strain have included optical techniques (17), ultrasonic methods (15, 19), and radiopaque marker methods (11, 32). However, even though precision in measuring strain has greatly improved, one cannot rely on the accuracy of calculated stress values, which depend greatly on the specific model used to derive the stresses (20, 34). Thus a method to directly evaluate regional myocardial time-varying elastance is required. Although some other studies (3, 7, 8, 30) have quantified stiffness in isolated muscle using vibration or indentation, those studies have not measured myocardial elastance or chamber elastance and, therefore, have had no "gold standard." The purpose of this investigation was to develop a technique to directly measure myocardial elastance using minute vibration and the equation of motion for the regional myocardium. We measured the stress-strain relationship in the orientation perpendicular to the myocardial layer as a representative measure of the myocardial stiffness. The results indicate that the proposed method was capable of measuring the myocardial regional time-varying elastic property and thus contractility.
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THEORETICAL CONSIDERATIONS |
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We used high-frequency pure sinusoidal vibration as the input to the region of interest, as shown in Fig. 1A, because in studying the ventricular properties we are dealing with a time-varying system. Although it is well appreciated that an impulse would be the most suitable stimulus to use in quantifying time-varying elastance (10), we relinquished the idea of using an impulse because the myocardium behaved differently when too much acceleration was applied. We chose the frequency of input to be fast enough that we could assume constant myocardial stiffness during one sinusoidal cycle length. We chose the input amplitude to be small enough not to apply too much acceleration but large enough to produce detectable changes in elastance.
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We lumped myocardial mechanical properties into three elements connected in parallel (23). These included elastance, viscosity, and inertia (mass). Because force produced with these elements is proportional to displacement, velocity, and acceleration, respectively, the relationship between contact force (f) and displacement (x) at a given instant of the cardiac cycle may be formulated as
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(1) |
t, the complex contact force divided by the amplitude of input
(D) would be expressed as
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(2) |
) is the complex
force-displacement relationship, i
represents the imaginary unit, and is angular frequency. The real
part of the complex force-displacement relationship represents the
amplitude of the in-phase sinusoidal wave (sine for sine), and the
imaginary part represents that of the orthogonal sinusoid wave (cosine
for sine). Equation 2 implies that, at
a given instant of the cardiac cycle, the real part of the complex
force-displacement relationship is linearly and negatively related to
the square of the angular frequency. The intercept of the linear
regression line at the angular frequency of zero is a direct measure of
myocardial elastance. Thus, if we measure the real part of the complex
force-displacement relationship at various frequencies and determine
the intercept, we can estimate myocardial elastance,
e.
Figure 1B depicts a hypothetical relationship between the square of the angular frequency and the real part of the complex force-displacement relationship. Isochronal lines during one cardiac cycle are shown. If mass remains unaltered throughout one cardiac cycle, the isochronal lines shift in a parallel fashion with contraction. Because the intercepts represent myocardial elastance at particular instants, an upward shift of parallel lines occurs with systole and a downward shift with diastole.
The mechanical properties of the myocardium are known to be frequency dependent (4). Because the presence of frequency dependence would lead to a deviation from linearity of the isochronal lines, we investigated the frequency range over which the above hypothesis could be considered valid.
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MATERIALS AND METHODS |
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Top
Abstract
Introduction
Materials & Methods
Results
Discussion
Appendix
References
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Preparation
The study was performed in 13 excised, blood-perfused, cross-circulated canine ventricles as previously described in detail (28, 29). Briefly, two mongrel dogs (body wt 12-25 kg) were anesthetized with pentobarbital sodium (30 mg/kg iv) after premedication with ketamine hydrochloride (5 mg/kg im). Both dogs were heparinized (1,000 U/kg). A heart isolated from the "donor" dog was metabolically supported by arterial blood from the second "support" dog. A thin water-filled latex balloon, connected to a computer-controlled ventricular volume servo-pump system, was placed in the left ventricle. Left ventricular pressure was measured using a micromanometer-tipped catheter (SPC-350, Millar Instruments, Houston, TX) placed inside the balloon. Left ventricular volume was measured with a linear variable differential transformer. The mean level of the systemic arterial pressure of the support dog was reasonably stable and >80 mmHg throughout each experiment. The support dog was ventilated with room air. The arterial blood was repeatedly sampled for measurements of pH, PO2, and PCO2. Supplemental oxygen and intravenous sodium bicarbonate were given as necessary to maintain these parameters within their physiological ranges. In six hearts, the proximal left anterior descending coronary artery (LAD) was dissected from surrounding tissues and encircled by a snare for the purpose of creating regional ischemia.Application of Minute Vibration
We used a vibrator (DPS-270, Dia Medical System, Tokyo, Japan) to impose sinusoidal displacement changes. This device was a servo-controlled oscillator that was able to precisely reproduce the command signal provided for displacement. The displacement command signal was generated and provided by a personal computer (PC-9821Ap, NEC, Tokyo, Japan) that was also used for data acquisition. The vibrator was equipped with both displacement and force transducers. The orientation of the vibration was perpendicular to the myocardium (Fig. 1A, top inset). We fastened the vibrator to the frame of the ventricular volume servo-pump so as to maintain stable contact with the myocardial surface throughout the cardiac cycle and at all phases of sinusoidal vibration. We used a contact force (averaged over a cardiac cycle) ranging between 2 × 105 and 3 × 105 dyn, because our pilot study indicated that force variations in this range had little effect on the estimated mass term and, thus, on the region of the imposed vibration (2.2 ± 1.5 g at 2 × 105 dyn and 2.3 ± 1.3 g at 3 × 105 dyn). We selected the area of myocardium perfused by the first diagonal branch of the LAD as the area to which the vibration was to be imposed. In a regional ischemic model, we also applied the vibration to the area perfused by the posterolateral branch of the left circumflex coronary artery (LCX). The contact surface between the vibrator head and the heart was 0.2 cm2. The myocardium was subjected to continuous oscillation by the vibrator throughout the entire cardiac cycle. The peak amplitude of displacement was set at 0.18-0.25 mm so as to fulfill the conditions previously stated. We repeated measurements at frequencies ranging from 50 to 100 Hz in stepwise increments of 5 Hz.Experimental Protocol
We kept left ventricular volume constant at a level that resulted in a ventricular systolic pressure of ~85 mmHg during control conditions (9.2-25.5 ml). To maintain adequate contact between the head of the vibrator and the myocardium, we chose to limit our study to the isovolumic contraction mode. We also fixed heart rate by left atrial pacing (140 ± 8 beats/min). Only when the heart beat regularly without arrhythmias and with a stable contractility was it subjected to the minute oscillation protocol.In seven dogs, we ran the following protocols after the control measurements.
Volume run. To test whether our measured myocardial elastance was independent of loading conditions, we increased end-diastolic volume so as to elevate ventricular systolic pressure by ~30% above the control level and repeated measurements.
Contractility run. To examine whether enhanced and depressed myocardial contractile states were detectable via myocardial elastance measurements in our study, we altered contractility of the ventricle as follows. First, we infused dobutamine via the coronary perfusion tubing at the constant rate of 2-5 µg/min to enhance the contractile state. When contractility increased and reached a new steady state, we repeated the measurements. Subsequently, dobutamine infusion was stopped. After contractility was allowed to return to the first control level, intracoronary propranolol was initiated with an initial bolus dose of 0.3 mg followed by constant infusion of 15 µg/min. When the contractile state was thereby depressed to a new steady-state level, measurements were repeated.
To evaluate whether myocardial elastance as measured by our technique could detect regional differences in myocardial properties, we studied a series of six other dogs wherein, in addition to control measurements, we occluded the LAD for 20 min and repeated measurements both in myocardial regions in the distribution of the LAD (ischemic) and in regions supplied by the LCX (nonischemic). Measurements were done both before and during the LAD occlusion.Data Analysis
All data were recorded simultaneously on a multichannel thermal array recorder (Omnicorder 8M24, NEC San-Ei, Tokyo, Japan) as well as digitized at 1-ms intervals using a 12-bit analog-to-digital converter and stored on a personal computer hard drive (PC-9821Ap, NEC). Off-line data analysis was performed using an IBM PC-compatible personal computer system (586-60 Vix, Proside, Tokyo, Japan).Myocardial elastance was calculated using the following procedures. We
first detected the force response to vibration by removing the
low-frequency signal below 10 Hz by means of Fourier series expansion
(Fig. 2, B
and C); such low-frequency signals
reflected the motion of the heart as a whole and were thus regarded for our purposes as noise. We then obtained the pure sinusoidal force response at each sampling instant within a cardiac cycle by reapplying the Fourier series expansion to the force response to the vibration. We
matched the period of the Fourier series expansion identically to that
of one wavelength of vibration spanning one-half of the wavelength
before and after the specified instant (see the time window between the
two vertical lines in Fig. 2) to avoid spectral leakage. This procedure
was required to remove the residual nonsinusoidal components introduced
by the time-varying and nonlinear nature of the myocardium. Finally,
division of the complex sinusoidal force by displacement yielded the
complex force-displacement relationship. We derived the real part
(e 
m2)
of the complex force-displacement relationship, which was then corrected for the mass effect of the vibrator head (Fig.
2D). We repeated these procedures at
each 1 ms over the entire cardiac cycle at various angular frequencies.
We calculated myocardial elastance, e,
from the intercepts of the linear regressions of the real part of
complex force-displacement relationship on
2.
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We examined the relationship between the estimated raw myocardial elastance and the chamber elastance values. Moreover, because the myocardial elastance as determined by minute vibration, when properly normalized (see Normalization of Myocardial Elastance), should represent myocardial material properties, we also examined the relationship between normalized myocardial elastance and the stress-strain relationship derived from the ventricular pressure-volume relationship.
Calculation of Conventional Contractility Indexes Against Which to Compare Myocardial Elastance by Vibration
Under each experimental condition, we measured left ventricular peak pressures of isovolumic beats at different ventricular volumes. In each case, from these data we determined the end-systolic pressure-volume relation (ESPVR) via linear regression. Using the volume-axis intercept of ESPVR, we then estimated ventricular time-varying elastance E(t) as
![<IT>E </IT>( <IT>t</IT> ) = P( <IT>t</IT> )/ [V( <IT>t</IT> ) − V<SUB>0</SUB>]](/content/vol274/issue4/fulltext/H1404/img003.gif)
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(3) |
For comparison, we performed a stress-strain analysis using a thick-walled spherical ventricular model based on the same data used to obtain ESPVR and V0 (see APPENDIX). Because the end-systolic stress-strain relationship was nonlinear, we calculated end-systolic incremental elastic modulus, i.e., incremental Young's modulus (Yinc, the tangential slope) around the strain used for measuring myocardial elastance. Incremental elastic modulus was determined from the expression
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(4) |
is the change in stress induced by a small change in strain, d
,
around the operating strain value that corresponded to the ventricular
volume at which myocardial elastance was measured. To obtain
incremental Young's modulus, the end-systolic stress-strain relationship was fit to a linear function at the reference strain ±0.01. When d was ±0.01, the linear function was found to be reasonable.
Normalization of Myocardial Elastance
To directly compare incremental elastic modulus with myocardial elastance determined by minute vibration, we normalized myocardial elastance according to the following two methods. Details concerning the derivation of normalized myocardial elastance are provided in the APPENDIX.First, myocardial elastance was normalized using the calculated stress and strain imposed by vibration. We calculated stress by dividing contact force by the area of the vibrator head (0.2 cm2). We calculated strain by assuming the reference thickness to be the thickness at reference volume (V0) in the spherical model. The ratio of stress to strain gives the normalized myocardial elastance. It has the same dimensions as the incremental elastic modulus (i.e., dyn/cm2).
Second, instead of using the actual contact surface area of the vibrator head, we calculated the effective contact surface area by dividing the mass being vibrated by the ventricular wall thickness. The mass was estimated from the slope of the relationship between the real part of the complex force-displacement relationship and the square of angular frequency at end systole. We derived stress by dividing force by the effective area. As a result of this normalization, the normalized myocardial elastance has the same dimensions as the incremental elastic modulus.
Statistical Analysis
Data are expressed as means ± SD. The two-tailed paired t-test was used to compare variables before and after volume loading and before and during coronary occlusion. Correlation analysis was performed using a standard least-square method. Goodness of fit was expressed as Pearson's r value, and multiple comparisons were performed using analysis of variance (6). Values of P < 0.05 were considered statistically significant.| |
RESULTS |
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Top
Abstract
Introduction
Materials & Methods
Results
Discussion
Appendix
References
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Figure 2 shows representative data obtained during minute vibration at
70 Hz. The amplitude of vibration was held constant at 0.2 mm
throughout the cardiac cycle (Fig.
2A). There was no noticeable
oscillation in the left ventricular pressure curve (Fig.
2E). Raw contact force shows the
fast response to the vibration on top of the slow baseline change with
cardiac contraction (Fig. 2B). We
removed the slow baseline change through use of Fourier series
expansion and obtained the response to the vibration (Fig. 2C). We then reapplied Fourier
series expansion to the force response over the period of one
wavelength of vibration and obtained the pure sinusoidal force
response. We derived the real part of the complex force-displacement
relationship, i.e., e m2 (Fig.
2D), which was then corrected for
the mass of the vibrator head. The real part of the complex
force-displacement relationship corrected for the mass of vibrator head
will hereafter be referred to simply as the real part of complex
force-displacement relationship.
Validity of the Mechanical Model in Representing Myocardial Properties
To test the validity of the mechanical model, we correlated the real part of the complex force-displacement relationship with the square of the input angular frequency. If the model properly describes the mechanical properties of the heart, at each time instant in the cardiac cycle, the real part should correlate linearly with the square of input frequency (2). Actual
findings in a representative case are shown in Fig. 1C. Over the entire range of
displacement frequencies, the real part became smaller as the frequency
of displacement increased. The relationship was reasonably linear when
we confined our analysis to the frequency range above 70 Hz. Thus we
determined isochronal lines using linear regression applied only to
data obtained above 70 Hz. The isochronal lines showed a parallel
upward shift from end diastole to end systole. These findings suggested
that the slope of the regression line, corresponding to mass in the
model, was constant throughout the entire cardiac cycle. The intercept values taken from the isochronal lines correspond to time-varying elastance.
The pooled data in Fig. 3 from all animals show the intercept (myocardial elastance), slope (mass), and square of the correlation coefficient as a function of time. We determined these values by applying linear regression in the frequency range above 70 Hz. Normalized time as used in Fig. 3 results from normalization by the time from the onset of contraction to end systole. As represented in Fig. 3A, myocardial elastance, i.e., the intercept, increased from end diastole to end systole (0.028 ± 0.211 × 106 dyn/cm at end diastole and 0.833 ± 0.391 × 106 dyn/cm at end systole) in a manner just like that of left ventricular chamber elastance. In contrast, mass, i.e., the slope, remained remarkably constant throughout the cardiac cycle (Fig. 3B). The linear correlation coefficient was also high throughout the cardiac cycle (Fig. 3C). This would imply that instantaneous myocardial elastance can be derived so long as we measure contact force at a minimum of two different frequencies above 70 Hz.
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Relationship Between Myocardial and Left Ventricular Chamber Time-Varying Elastance
Figure 4A shows the instantaneous relationship between myocardial elastance [e(t)] and left ventricular chamber elastance [E(t)] under control conditions and at two different levels of contractility. For all given contractile states, instantaneous myocardial elastance tightly and linearly correlated with ventricular elastance. Moreover, the slopes of the regression lines remained in very good agreement despite differences in contractile states.
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We summarized the regression analysis of the E(t)-e(t) relationship in Table 1. In all animals, e(t) linearly correlated with E(t) regardless of contractility. The intercepts were relatively small. The correlation coefficients varied between 0.891 and 0.997 (median 0.985) for all animals for all contractile states. Although the magnitude of the regression slope varied among animals (P < 0.005), the strength of correlation between E(t) and e(t) remained despite the various contractile states (P = 0.54). The differences in slope among animals suggested the presence of possible geometric effects that are unique for individual hearts.
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Influence of Changes in Left Ventricular Contractility on Myocardial Elastance
As mentioned earlier in reference to Fig. 4A, changes in ventricular contractility as measured by chamber elastance correlated well with myocardial elastance.Figure 4B shows the relationship between end-systolic myocardial elastance (ees) and end-systolic left ventricular elastance (Ees) plotted for all animals for all contractile states. The relationship between the mean value of end-diastolic left ventricular elastance (Eed) and end-diastolic myocardial elastance (eed) at control conditions was also plotted in Fig. 4B. Averaged values of end-diastolic data also stay on the same end-systolic myocardial elastance-ventricular elastance line. Although the relationship between ees and Ees was reasonably linear, the correlation coefficient (r = 0.717, P < 0.001) was not as large as that for the E(t)-e(t) relationship obtained for individual animals (Table 1). Again, this supports the idea that differences in chamber size among individual hearts, i.e., geometric effects, might be present, thus contributing to the lowering of the correlation for pooled observations.
Influence of Volume Loading on End-Systolic and End-Diastolic Myocardial Elastance
We increased ventricular volume from 14.3 ± 5.4 to 19.9 ± 6.3 ml. Neither Ees (from 9.0 ± 3.2 to 9.1 ± 3.4 mmHg/ml) nor Eed (from 1.5 ± 1.1 to 1.7 ± 1.1 mmHg/ml) changed significantly. Figure 5 shows the effects of volume loading on ees. Both ees and eed also remained unchanged with volume loading (ees: from 0.833 ± 0.391 to 0.841 ± 0.409 × 106 dyn/cm; eed: from 0.028 ± 0.211 to 0.029 ± 0.214 × 106 dyn/cm). Thus estimated myocardial elastance was insensitive to changes in ventricular volume in our measurement range.
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Relationship Between Normalized ees and End-Systolic Incremental Elastic Modulus
Figure 6 shows the relationships between normalized ees and end-systolic incremental elastic modulus, Yinc, derived from the end-systolic pressure-volume relationship using a spherical ventricular model (see APPENDIX). As we mentioned in Data Analysis, we normalized ees according to two different methods. As shown in Fig. 6A, normalized ees as determined by the actual contact surface area of the vibrator head and wall thickness correlated well with Yinc (r = 0.769, P < 0.0001). The correlation showed significant improvement over that determined earlier between Ees and ees (Fig. 4B). This improvement in correlation is expected because both normalized ees and Yinc are, in principle, geometry independent, whereas Ees and ees are geometry dependent. In other words, the lower correlation between Ees and ees might result from intersubject variations in geometry. The absolute value of normalized ees was ~10 times larger than that of Yinc, however.
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Normalized ees as determined by the effective surface area estimated from the vibrated ventricular mass and wall thickness also correlated very closely with Yinc (r = 0.874, P < 0.0001). This is illustrated in Fig. 6B. The slope of the relationship became closer to unity than that achieved with the other normalization method, which was based on the actual vibrator head contact area (Fig. 6A).
Effects of Regional Ischemia on Myocardial Elastance
To evaluate the effect of regional ischemia on myocardial elastance, we used the normalized ees as determined using the effective surface area estimated from the vibrated ventricular mass and wall thickness, because normalization by the effective surface area (Fig. 6B) was better than that by the actual contact surface (Fig. 6A). Figure 7 shows the effects of coronary occlusion on E(t) and e(t) measured at nonischemic (broken line) and ischemic regions (solid line) for a representative case. After LAD occlusion, the peak value of E(t) decreased (Fig. 7A), but that of e(t) at the nonischemic region did not sizably change (Fig. 7B). However, the peak value of e(t) at the ischemic region significantly decreased after coronary occlusion (Fig. 7C).
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Figure 8 shows the changes in the left ventricular chamber elastance and in the myocardial elastance after coronary occlusion plotted for all six animals. The variance of ees among myocardial regions in the distribution of either the LAD or the LCX before coronary occlusion was 15 ± 18% (P was not significant between LAD and LCX). Ees decreased after coronary occlusion from a value of 10.4 ± 4.5 to 7.7 ± 4.3 mmHg/ml (P < 0.005; Fig. 8A). Although ees normalized by the effective area at the nonischemic region did not change significantly (2.066 ± 0.781 and 1.975 ± 0.749 × 106 dyn/cm2 for before and after the coronary occlusion, respectively, Fig. 8B), normalized ees at the ischemic region decreased significantly from 2.303 ± 0.556 to 1.173 ± 0.370 × 106 dyn/cm2 (P < 0.005; Fig. 8C).
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DISCUSSION |
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Top
Abstract
Introduction
Materials & Methods
Results
Discussion
Appendix
References
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Validity of Fundamental Assumptions
We have shown that regional myocardial elastance estimated by the minute vibration method linearly correlated with left ventricular chamber elastance under various conditions of contractility and preload. Moreover, the measured myocardial elastance was sensitive to regional differences in myocardial contractility induced by regional myocardial ischemia. Our findings overall indicated that we were able to estimate regional myocardial elastance reasonably well.We piecewise analyzed short-duration data and obtained the three lumped mechanical components (mass, viscosity, and elasticity) all as functions of the time within the cardiac cycle. We found from the results that mass on which vibration was imposed was relatively constant, but elastance dramatically changed within a cardiac cycle. Because these mechanical properties might change with frequency of vibration (4), we further made sure that our measurement was independent of the selection of the two vibration frequencies that were necessary for unique determination of elastance. This was true when we selected vibration frequencies above 70 Hz (Fig. 1C). We speculate that potential sources of the deviation from linear regression in the low-frequency range are the result of nontransmural penetration of vibration, frequency dependence of elastance, translation of the heart as a whole, and nonstationarity of elastance within one cycle of vibration.
Templeton et al. (30) have already demonstrated that axial stiffness (that oriented along the principal myocardial axis) measured by imposing sinusoidal length changes correlated well with mean tension determined at the same instant. Because the stress-strain relationships in other directions (cross fiber and axial fiber) might mutually affect each other, Halperin and co-workers (3, 7, 8) compared transverse stiffness obtained using 20-, 50-, or 67-Hz indentation stimuli (analogous to our myocardial elastance) with the biaxial stress-strain relationship. They showed that in the arterially perfused canine ventricular septa, transverse (radial) stiffness correlated strongly with both in-plane stresses and the in-plane strains (e.g., circumferential and meridional). Thus the transverse stiffness was a good index for estimation of the in-plane mean wall stresses of the regional myocardium, although the individual in-plane stress components were not distinguished. We applied a similar technique to the epicardial surface using a higher frequency stimulus, because we thought that in the lower frequency range the heart would move away from the vibrator head or that the myocardial elastance would change within one vibration cycle length. Indeed, the term of mass in our model obtained from lower frequency data became negative when the myocardial elastance from higher frequency data was assumed to be correct. Thus we only used high-frequency indentations for determinations of regional elastance.
Myocardial Elastance as a Predictor of Chamber Elastance
We demonstrated that instantaneous myocardial elastance changed in a fashion similar to that of instantaneous ventricular chamber elastance during a cardiac cycle. In addition, we observed a relatively similar relationship in the same heart under conditions of enhanced and depressed contractility. Furthermore, when we plotted Ees versus ees for all animals, we again found a linear relationship, albeit with a lower correlation coefficient. In the absence of a gold standard for myocardial elastance, we examined whether we could measure myocardial elastance with vibration by comparing it with chamber elastance. This seems reasonable without regional heterogeneity of myocardial elastance within the ventricle. The lower correlation coefficient between Ees and ees for pooled data seems to be indicative of an intersubject variability that might arise from the different geometry. Thus, although there is undoubtedly some intersubject variability, we may conclude that ees is a good predictor of Ees under conditions in which homogeneity of myocardial elastance can be reasonably assumed.Insensitivity of End-Systolic Regional Elastance to Preload
End-systolic myocardial elastance did not change significantly when left ventricular end-diastolic volume was increased, although the myocardial tension-length relationship has been shown to be nonlinear in a number of studies (5, 24). This apparent discordance can be explained as follows. First, although most studies of the tension-length relationship examined uniaxial data, we have included multiaxial properties. Second, the operating points in this study were within relatively linear ranges of the exponential tension-length relationship curve. Third, the apparent insensitivity could be due to the small number (two levels) of preload settings we investigated. However, even with consideration of all of these points, ees would still seem to be a powerful index of myocardial properties, because it was scarcely influenced by loading conditions within a physiological range of preload.Normalization of Myocardial Elastance
Theoretically, normalization of myocardial elastance would result in an expression of material properties (Young's modulus) of the myocardium that would be independent of geometry. We found that ees normalized either by the area of the vibrator head or by the effective contact surface area correlated better with the incremental elastic modulus at end systole, Yinc, than with the end-systolic chamber elastance, Ees. As stated earlier, the lower correlation between Ees and ees might have resulted from intersubject variations in chamber geometry. The fact that normalized ees correlated better with Yinc suggests that normalized ees is less affected by geometry than is Ees.Rather than using the area of the vibrator head, we calculated effective surface area from the estimated mass from the slope in Fig. 1C (2.0-3.4 g, median 2.5 g), assuming transmural penetration of vibration (see Limitations regarding this assumption). The effective surface area was calculated as ~2.5 cm2, an area 13 times as large as the physical contact area. The relationship between ees normalized by the effective area and Yinc became reasonably tighter, and the slope of the relation became closer to unity than that obtained using the vibrator head area for normalization (Fig. 6, A and B). The absolute value of the normalized ees was in close agreement with the transverse stiffness measured in canine ventricular septa (7). From these results, we found that our regional elastance certainly reflected myocardial mechanical properties.
Advantages of Directly Measuring Regional Myocardial Elastance
In experimental studies, some investigators have tried to estimate regional elastance as a muscle property based on the relationship between stress and strain (1, 9, 15). However, it is difficult to measure regional stress reasonably accurately without depending on a particular model. The fact that our method does not require assumptions about cardiac geometry makes it a unique tool for estimating regional elastance.Although Ees has been reported as a load-insensitive index of ventricular contractility (13, 28, 29), Sunagawa and co-workers (28, 29) reported that Ees barely changed with regional ischemia. We succeeded in this study in detecting regional myocardial dysfunction in regional ischemia with the normalized ees (Fig. 8). This might also be important for other cardiac disease states, such as dilated cardiomyopathy, wherein there may exist a significant heterogeneity of myocardial contractility.
Another advantage of our method is in its potential to separately evaluate myocardial viscous properties and elastic properties. Some investigators have reported (14, 16, 18) that viscous properties could be important in characterizing the left ventricular diastolic properties. Obviously, further investigations are needed to explore such a possibility.
Limitations
There are some limitations in this study. The first limitation concerns the question of whether the mass term might change between measurements. Our pilot study indicated that the average contact force that ranged from 2 × 105 to 3 × 105 dyn between the vibrator and the heart affected the estimated mass term a little (see MATERIALS AND METHODS). The fact that the real part of the regional force linearly correlated with the square of frequency indicated that variations in the contact force were relatively minimal under our experimental conditions.Second, there is no guarantee that there was complete transmural penetration of the imposed vibration. However, because for the estimated mass (on which the vibration was imposed) a lesser degree of penetration would imply a surface area significantly larger than what was encountered, we reasoned that the vibration penetrated most of the myocardial layer.
Third, as Halperin demonstrated (7, 8), the stress-strain relationships in transverse and in-plane directions (cross fiber and axial fiber) might mutually affect each other. Although we cannot quantify the relative contributions of transverse and in-plane stiffness to our measurement, the parallel changes in stiffness when muscle was activated or contractility was enhanced enabled us to estimate regional myocardial elastance with transverse vibration.
Fourth, left ventricular geometry might be too simple for the estimation of incremental Young's modulus to compare with the normalized elastance. Although normalized ees correlated well with incremental elastic modulus, the accuracy was still not extremely high. We conjecture that the residual variance results from the oversimplified model rather than from a basic defect in our method.
Fifth, the elastance measured by our method might be different from that measured with slower volume changes. Although we demonstrated that these correlated well, these elastances might change discrepantly under other conditions. Better understanding of similarities and potential differences in sensitivity between these elastances requires further study.
Finally, it is well known that mechanical vibration lowers contractility of the myocardium (12, 31). This negative inotropic effect has been known to depend on the acceleration of the vibration. In this experiment, the maximal acceleration we used was 47.0 ± 15.4 m/s2. Although this value is much smaller than that present in experiments manifesting negative inotropic effects, we should be aware of the possibility that our technique might underestimate regional contractility through this negative inotropism, especially in low contractile states.
In summary, we quantitatively evaluated regional myocardial elastance using regional minute vibration and the equation of motion for the regional myocardium. The regional myocardial elastance showed a linear correlation with left ventricular elastance under various contractile states. Within a physiological range of loading conditions, ees was insensitive to changes in preload. Furthermore, ees was sensitive to regional differences in myocardial contractility induced by regional myocardial ischemia. Our data suggest that it may be possible, using minute vibration, to estimate Young's modulus for the myocardium. We conclude that this technique is useful for quantitative assessment of regional myocardial time-varying elastic properties and thus of contractility.
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APPENDIX |
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|
Top
Abstract
Introduction
Materials & Methods
Results
Discussion
Appendix
References
|
|---|
Model
We used a thick-walled spherical model for the left ventricle in which the geometric parameters may be estimated using the formulas Vc = (4
/3)r3i and
Vc + Vw = (4/3)r3o, where
Vc and
Vw are cavity and wall volume
(wall mass/1.05), respectively, and
ri and
ro are the inner
and outer radii, respectively.
Strain
Instantaneous natural strain () for the midwall layer was determined
from instantaneous left ventricular chamber volume and wall mass
|
(A1) |
|
(A2) |
Stress
According to Wolff et al. (33), on the basis of the balanced-force equation for a thick-walled spherical model, we calculated instantaneous circumferential stress () as
|
(A3) |
2 · mmHg1
is the constant for unit conversion. We defined
A as
|
(A4) |
Normalization of Myocardial Elastance by Physical Contact Surface Area
For an incompressible Hookean substance, there is a constant that relates stress and strain. This constant is characterized as incremental elastic modulus, i.e., incremental Young's modulus, the ratio of unit force per unit area to relative change in dimension. Our myocardial elastance e(t) does not have the same dimensions as incremental elastic modulus. Accordingly, it may be difficult to compare elastances obtained under conditions with various ventricular geometries. To compare our myocardial elastance with the stress-strain relationship, we normalized e with the wall thickness at the reference volume and with the contact surface area. From Eq. 4
|
(A5) |
Normalization of Myocardial Elastance by Effective Surface Area
We calculated the wall thickness at the operating volume under the assumption of spherical geometry for the left ventricle in the same manner as described in the above section. Assuming that the vibration had transmural and cylindrical effects, we calculated the effective contact surface area from the effective mass and the wall thickness. Effective mass subjected to vibration was first calculated from the slope of the relationship between the real part of contact force and the square of frequency. We then calculated the normalized myocardial elastance based on this effective area instead of the area of the vibrator head. This is expressed as
|
(A6) |
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ACKNOWLEDGEMENTS |
|---|
This study was supported by Grants-in-Aid for General Scientific Research (Nos. 05454281, 05770505, 06770530, 06770531, 06770533, and 07770553) from the Ministry of Education, Science and Culture of Japan; by Research Grants for Cardiovascular Diseases (5A-3, 6A-4, 7C-2, and 7A-1) from the Ministry of Health and Welfare of Japan; by a grant from the Science and Technology Agency, Encourage System of the Center of Excellence, and by a grant from the Sankyo Foundation of Life Science.
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FOOTNOTES |
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Address for reprint requests: T. Shishido, Dept. of Cardiovascular Dynamics, National Cardiovascular Center Research Institute, 5-7-1 Fujishirodai, Suita, Osaka 565, Japan.
Received 6 February 1997; accepted in final form 3 December 1997.
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Appendix
References
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). A pair of dashed curves indicates 95%
confidence limits of regression. In addition, the mean value of
end-diastolic myocardial elastance and left ventricular elastance under
control conditions was plotted (
). Error bars indicate SD.
