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Anesth Analg 2006;102:1032-1039 © 2006 International Anesthesia Research Society doi: 10.1213/01.ane.0000202537.19646.10
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Abstract |
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The logistic time constant (
L) has been proposed as a better index of the rate of left ventricular (LV) relaxation or lusitropism than the conventional monoexponential time constant (E). However, whether and how the Frank-Starling effect influences L remains to be elucidated. We compared the effect of LV volume (LVV) loading on both logistic and monoexponential fittings. The isovolumic LV relaxation pressure curves from the maximum negative time derivative of pressure (-dP/dtmax) were analyzed at 3 different end-points at 4 LVVs of 10, 12, 14, and 16 mL in 8 excised, cross-circulated canine hearts. We found that the logistic fitting was superior to the monoexponential fitting at all LVVs and end-points. LVV loading did not affect L but affected E slightly. Although the advancing end-point increased both L and E, the increases were significantly smaller for L than for E at all LVVs. Moreover, the changes in both the amplitude constants and nonzero asymptotes with the advancing end-point were significantly smaller for the logistic fitting than for the monoexponential fitting. We conclude that L served as a more reliable index of lusitropism that is independent of the change in LVV loading or the Frank-Starling effect. |
Introduction |
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Nearly half of all cases of heart failure are caused by relaxation failure. Therefore, the measurement of the rate of left ventricular (LV) relaxation, that is to say, lusitropism, will be very useful to estimate cardiac relaxation. Many physicians have investigated and attempted to derive reliable indices to evaluate LV relaxation from observed cardiac hemodynamics. Monoexponential function has been proposed as a reasonably good fitting curve for isovolumic LV pressure (LVP) in the excised canine heart (1). Thereafter, the monoexponential time constant (
E) has been widely used as a reasonable lusitropism index in both experimental (2–4) and clinical settings (5,6). However, the LV relaxation pressure curve could not be precisely characterized by monoexponential fitting (7,8). In contrast, the logistic function fit the isovolumic relaxation pressure curve more precisely than the monoexponential function in terms of correlation coefficient (r) and residual mean squares (RMS) (6), which quantify the goodness of fit in the excised canine heart (9). The logistic time constant (L) has been proposed as a better lusitropism index than the conventional E. We subsequently found that the logistic function could fit the isometric relaxation force curve more precisely than the monoexponential function in the ferret isolated right ventricular (RV) papillary muscle (10). Moreover, the logistic fitting has previously been used in rabbit RV papillary muscle and trabeculae (11), calf heart (12), and the human heart (13).
We questioned whether and how the superiority of the logistic fitting to the monoexponential fitting would hold in the setting of fluid infusion to augment blood volume, a mainstay of anesthesia practice (14). Fluid therapy is used for immediate correction of hypovolemia in emergency departments or for support of arterial blood pressure and cardiac output in operating rooms and intensive care units. However, excessive infusion can produce excessive intravascular volume, edema, and impaired hemodynamic status (15). Increasing preload may impair cardiac mechanics and functions. The impaired myocardial relaxation causes diastolic dysfunction in patients with heart failure (5,16). It is therefore valuable to investigate whether L remains more reliable than E, independent of LV preload or the Frank-Starling effect.
Various end-points of the relaxation pressure curve have been used and there has been no consensus about which end-point is best. For example, the end-point of the isovolumic relaxation period was defined as the time when LVP decreases to 5 mm Hg above LV end-diastolic pressure (LVEDP) of the subsequent beats (2,17). This end-point was chosen to ensure that it would occur before the mitral valve opens. Furthermore, there is a considerable gap between the best-fit monoexponential curve and the actual relaxation curve near the end-point of relaxation. We have found that L is independent of and E dependent on the end-point of relaxation (9,10). Therefore, it is desirable to evaluate lusitropism at the different end-points.
In the present study, we investigated whether and how LV volume (LVV) loading affects both the best-fit logistic and monoexponential parameters, including L and E in the isovolumic relaxation pressure curves at different end-points.
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Methods |
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This study protocol was approved by the animal investigation committee of Okayama University Graduate School of Medicine, Dentistry and Pharmaceutical Sciences. All procedures were conducted in conformity with the Guiding Principles for Research Involving Animals and Human Beings endorsed by both the American Physiological Society and the Physiological Society of Japan.
We used the same excised, cross-circulated canine heart preparation we have been using consistently in cardiac mechanoenergetic studies (18). The details of the surgical preparations are described elsewhere (9,19).
Briefly, in each of 8 experiments, we anesthetized a metabolic support dog (mean ± sd, 21.8 ± 5.2 kg) and a heart donor dog (13.2 ± 1.7 kg), both adult mongrels, with pentobarbital sodium (25 mg/kg IV) and fentanyl citrate (0.1–0.2 mg/h IV) after premedication with ketamine hydrochloride (25 mg/kg IM). They were tracheally intubated, ventilated with room air, and heparinized (15,000 U IV per support dog and 10,000 U IV per donor dog).
We cannulated the bilateral common carotid arteries and unilateral external jugular vein of the support dog and connected them to the arterial and venous cross-circulation tubes, respectively. The chest of the donor dog was opened midsternally. The arterial and venous cross-circulation tubes from the support dog were cannulated into the left subclavian artery and the RV via the right atrial appendage of the donor dog, respectively. All systemic and pulmonary vascular connections to the heart, i.e., the descending aorta, inferior vena cava, brachiocephalic artery, superior vena cava, azygos vein, and bilateral pulmonary hili, were ligated. The metabolically supported beating heart was excised from the chest of the donor dog under continuous cross-circulation from the support dog. The coronary perfusion of the excised donor heart was never interrupted during the surgical preparation.
We opened the left atrium of the donor heart and cut all the LV chordae tendineae. Complete atrioventricular block was created chemically (0.2–0.5 mL injection of 36% formaldehyde solution) or electrically (direct current of 20–30 J) with ablation of the His bundle. A bipolar pacing electrode was placed on the upper portion of the ventricular septal endocardium via the left atrium for para-Hisian pacing. The LV was electrically paced at 500-ms intervals (120 bpm) throughout the experiment.
We placed a thin latex balloon with an unstretched volume of about 50 mL mounted on a rigid connector into the LV. The connector was secured at the mitral annulus. The balloon was connected to our custom-made volume servo pump (AR-Brown, Tokyo, Japan). Both the balloon and the water housing of the servo pump were primed with water. The servo pump enabled us to measure LVV accurately and to control it precisely. LVP was measured with a miniature pressure gauge (model P-7; Konigsberg Instruments, Pasadena, CA) placed inside the apical end of the balloon.
We measured LV temperature of the excised donor heart with a thermistor placed between the endocardium and the LV balloon. It was maintained at 36°C by regulating the temperature of the arterial cross-circulation tube with a thermostatic bath.
We recorded an LV epicardial electrocardiogram with a pair of screw-in electrodes to trigger data acquisition and to identify end diastole. All LVP, LVV, and electrocardiogram signals were digitized at 2-ms intervals and processed on a LabVIEW (National Instruments, Austin, TX)-installed computer.
The systemic arterial blood pressure of the support dog provided the coronary perfusion pressure for the excised donor heart. Systemic arterial blood pressure was maintained at stable values by infusing whole blood saved from the donor dog, 6% hydroxyethylated starch solution, or methoxamine (5–30 mg/h) as needed. Arterial pH, Pao2, and Paco2 of the support dog were repeatedly measured with a blood gas analyzer and maintained within physiologic range with supplemental oxygen, IV sodium bicarbonate, or ventilator setting changes.
The LV, including the septum, weighed 92.8 ± 13.5 g and the RV 35.8 ± 6.2 g after each experiment.
Steady-state isovolumic contractions were attained by fixing LVV with the volume servo pump. Four different isovolumic LVVs were set at 10, 12, 14, and 16 mL in every heart. At each LVV, 3 beats separated by more than 15 beats were sampled for analysis. A total of 12 separate beats in each heart and 96 separate beats in 8 hearts were subjected to analyses.
LVP signals were sampled at 2-ms intervals and stored in a computer. To suppress small noises in the digitized pressure data, the sampled data were digitally smoothened by a five-point, nonweighted moving average.
We limited the isovolumic relaxation time to the period from the maximum negative time derivative of pressure (-dP/dtmax) to the end-point of relaxation. -dP/dt was obtained by differentiating digitized LVP signals, and the time at -dP/dtmax was identified as the onset of relaxation. We identified the full end-point of relaxation as the time when the decrease in isovolumic pressure reached LVEDP of the preceding diastole. This end-point corresponded to the time when LV-developed pressure decreased to zero.
We used the same logistic function that Matsubara et al. (9) proposed to fit the isovolumic relaxation pressure curve and the least squares method with DeltaGraph 4.0 (DeltaPoint, Monterey, CA):
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where PA is the amplitude constant, t is the time from -dP/dtmax, L is the time constant of the exponential term in the denominator of the logistic function, and PB is the non-zero asymptote, as shown in Figure 1A. We designated L as the logistic time constant to distinguish it from the conventional monoexponential time constant that Weiss et al. proposed (1).
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We chose the following equation as the monoexponential function (7,8):
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where P0 is the amplitude constant, t is time from -dP/dtmax, E is the monoexponential time constant, and P
is the non-zero asymptote, as shown in Figure 1B. The non-zero asymptote of Equation 2 is a better monoexponential function than Weiss et al.’s original monoexponential function with zero asymptote (1).
The logistic function (Equation 1) has the same number of parameters as the monoexponential function (Equation 2). PA, L, and PB in Equation 1 have similar theoretical meaning with P0, E, and P in Equation 2, respectively.
We studied the relationships with 3 different end-points of curve fitting. The end-point was advanced from the full end-point on the isovolumic relaxation pressure curve at zero developed pressure (0%EP), which is the same pressure level as LVEDP, to 5% and 10% higher-developed pressure levels (+5%EP and +10%EP). Here, 100% means the developed pressure at -dP/dtmax but not LV end-systolic pressure (LVESP).
We averaged the L, E, PA, P0, PB, and P, r, and RMS values of both the logistic and monoexponential fittings for the relaxation pressure curves at 0%EP, +5%EP, and +10%EP of the 3 separate beats at each LVV.
We evaluated the goodness of fit by comparing r between the logistic and monoexponential fittings. The paired Student’s t-test was applied to r after Fisher’s Z transformation: Z = 1/2[ln(1+ r) – ln(1 – r)] (20).
We analyzed the residuals that were the differences between either the best-fit logistic or monoexponential curves and the corresponding relaxation curve at all the sampling points. The RMS that also quantifies the goodness of fit was calculated as the residual sum of squares divided by the residual degrees of freedom (the number of data points analyzed minus the number of parameters in the function) (6). We compared the RMS value of the logistic fitting with that of monoexponential fitting by the paired Student’s t-test.
We compared the L, E, PA, P0, PB, and P values among the 4 LVVs and the 3 end-points by one-way repeated-measures analysis of variance. When the analysis of variance was significant (F-test, P < 0.05), we performed multiple comparisons by the post hoc Scheffé’s test.
Further, we analyzed nL and nE for +5%EP and +10%EP after normalizing them relative to their 100% values at 0%EP.
The logistic curve described by Equation 1 decayed monotonically from P (0) (= [PA/2 + PB]) toward PB. Therefore, the maximum amplitude of the logistic curve was PA/2. The monoexponential curve described by Equation 2 decayed monotonically from P (0) (= [P0 + P]) toward P. Therefore, the maximum amplitude of the monoexponential curve was P0. We analyzed the difference in PA/2 between +5%EP and 0%EP (dPA/2 at +5-0%EP) and that in P0 between +5%EP and 0%EP (dP0 at +5-0%EP). We also analyzed the difference in PA/2 between +10%EP and 0%EP (dPA/2 at +10-0%EP) and that in P0 between +10%EP and 0%EP (dP0 at +10-0%EP).
Similarly, we analyzed the difference in PB between +5%EP and 0%EP (dPB at +5-0%EP) and that in P between +5%EP and 0%EP (dP at +5-0%EP). We also analyzed the difference in PB between +10%EP and 0%EP (dPB at +10-0%EP) and that in P between +10%EP and 0%EP (dP at +10-0%EP).
We compared the nL at +5%EP with nE at +5%EP as well as nL at +10%EP with nE at +10%EP, dPA/2 at +5-0%EP with dP0 at +5-0%EP, as well as dPA/2 at +10-0%EP with dP0 at +10-0%EP, and dPB at +5-0%EP with dP at +5-0%EP, as well as dPB at +10-0%EP with dP at +10-0%EP by the paired Student’s t-test.
We analyzed all data using the Microsoft Excel 98 (Microsoft-Japan, Tokyo, Japan), Statcel (OMS, Saitama, Japan) and StatView 5.0 (SAS Institute Inc, Cary, NC) software. The measured and calculated values were presented as mean ± sd. P value < 0.05 indicated statistical significance.
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Results |
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Table 1 summarizes hemodynamic data of the isovolumic LV relaxation pressure curves at the 4 LVVs and the 3 end-points in the 8 hearts, showing that isovolumic LVESP became larger with increasing LVV. Simultaneously, -dP/dtmax became smaller with increasing LVV. LVP at -dP/dtmax became larger with increasing LVV. LVEDP at 0%EP, +5%EP, and +10%EP became larger with increasing LVV.
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The logistic r values were always significantly greater than the monoexponential r values, as shown in Table 2. Pooling all the 96 fittings, the mean logistic r value was 0.9996 ± 0.0002 and the mean monoexponential r value was 0.9976 ± 0.0006. The former was notably significantly larger than the latter (P < 0.000005).
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The RMS values for the logistic fitting were always significantly smaller than the RMS values for the monoexponential fitting, as shown in Table 2. Pooling all the 96 fittings, the mean RMS value for the logistic fitting was 0.090 ± 0.104 (mm Hg)2 and the mean RMS value for the monoexponential fitting was 0.526 ± 0.288 (mm Hg)2. The former, on average, was 1/6 of the latter.
Thus, the goodness of the logistic fitting was consistently superior to that of the monoexponential fitting, in terms of r and RMS, at the 4 LVVs and the 3 end-points.
Multiple comparisons among the 4 LVVs showed no significance of L at the 3 end-points but showed significance of E between LVV of 10 and 16 mL at 0%EP, as shown in Figure 2A. With the advancing end-point, L remained almost constant or decreased significantly only slightly, whereas E significantly became markedly larger at the 4 LVVs. The nL values at +5%EP and +10%EP were significantly smaller than the nE values at the 4 LVVs, as shown in Figure 2B.
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PA and P0 became significantly larger with increasing LVV from 10 to 16 mL at the 3 end-points as shown in Figure 3A. PA remained almost constant, or became slightly larger with significance with the advancing end-point, whereas P0 significantly became markedly larger with the advancing end-point at the 4 LVVs. dPA/2 at +5-0%EP and dPA/2 at +10-0%EP were significantly smaller than dP0 at the 4 LVVs, as shown in Figure 3B.
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PB and P became significantly larger with increasing LVV from 10 to 16 mL at the 3 end-points, as shown in Figure 4A. PB remained almost constant but decreased only slightly with significance with the advancing end-point, whereas P significantly became markedly smaller with the advancing end-point at the 4 LVVs. dPB at +5-0%EP and +10-0%EP were significantly smaller than dP at the 4 LVVs, as shown in Figure 4B.
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Discussion |
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The above results demonstrate that LVV loading does not affect
L but almost keeps or decreases E slightly, although LVV loading increases LVESP and LVEDP. Therefore, our finding indicates that the rate of isovolumic pressure relaxation or lusitropism is independent of the Frank-Starling effect.
There are reports that the change in LVV resulted in no significant change in E in the isolated canine (1), ejecting right-heart bypass preparation (8), conscious canine (21), and anesthetized closed-chest canine (22). Similarly, a decreased preload by inferior vena cava occlusion does not influence E in humans (23). Furthermore, L and E are unaffected by preload reduction induced by transient obstruction of inferior vena caval inflow in normal and failing human hearts (13). Moreover, the muscle length does not affect L and E (10). These reports support the present results.
In contrast, there are reports that E becomes larger with intravascular fluid administration in the closed-chest (2) and open-chest (4,24,25) anesthetized canines. Moreover, both L and E become larger with increasing LVEDP (9). These results are contrary to what would have been predicted. The different result is attributable to different range in LVV between the study of Matsubaras et al. (9) and our study, although we used the same heart model. In short, they changed LVV over a wide range from small volumes (mean, 15.2 mL) to large volumes (mean, 20.6 mL), whereas we examined lusitropism in a narrow LVV range from 10 to 16 mL in 2-mL increments. In small volume infusions, there are no changes in E whereas small increases in LVEDP occur, but in large volume infusions E is prolonged and large increases in LVEDP occur in conscious (26) and anesthetized (24) canines. Our investigations cannot exclude the possibility that a larger increase in LVV loading may prolong the rate of isovolumic relaxation pressure. If we had increased LVV more vigorously above a certain threshold level, a greater contraction might have resulted and a prolongation of isovolumic relaxation detectable by L and E might have been produced.
There is, however, a report that E becomes smaller with LVV loading (27). In our results, too, there is a significant decrease in E from 10 to 16 mL at 0%EP.
Concerns about the influence of preload on the rate of isovolumic and isometric relaxation are conflicting. It is possible that the magnitude and methodology of the change in LVV loading affected previous reports on the preload dependency of L and E. There may also be different changes in L and E in preloading condition as a result of different animal species and specific conditions. Further, we suspect that an increased preload may increase afterload and influence heart rate via reflex change in the sympathetic tone and the neurohumoral regulation in those previous models.
E is prolonged when afterload is increased by aortic cross-clamping (21,26), phenylephrine (26), and methoxamine (24) infusions. We suspect that increased afterload might affect an increase in preload. Therefore, afterload in the specific preparation may influence the result of preload dependency of L and E. However, in our model, we cut and ligate the aorta and the subclavian and brachiocephalic arteries and excise the heart to eliminate the effect of afterload.
E decreases with increasing heart rate (1,2,7). Moreover, L and E become smaller with increasing heart rate (9). However, E is unaffected by alteration in preload status when heart rate is maintained constant by right atrial pacing (17). Therefore, heart rate may have affected these reports on the preload dependency of L and E. To eliminate the heart rate effect, we remove the sympathetic nerve and create complete atrioventricular block to maintain a fixed heart rate in our model.
We therefore propose that our model is reliable for the estimation of LVP and that L is independent of the preloading.
Both L and E increase with the advancing end-point at a fixed LVV. Both L and E become larger with the advancing end-point at arbitrary LVV (9) and fixed muscle length (10). These results show that the end-point of relaxation affects L and E.
Moreover, nL is significantly smaller than nE for the advancing end-point at a fixed LVV. nL is superior to nE at arbitrary LVV (9) and fixed muscle length (10). Furthermore, E is quite sensitive to the change in the end-point in dilated cardiomyopathy of humans and prolongs; in contrast, L shortens relatively little (13). The considerable dependence of E on the choice of the end-point is a serious concern in comparing it with L. This dependence easily changes E inadvertently even when the lusitropism remains unchanged. Therefore, L is much more independent of the choice of the end-point.
Both the logistic and monoexponential functions have poor fittings near LVEDP at the 4 LVVs. However, the residuals for the logistic fitting near LVEDP are measurably smaller than those for the monoexponential fitting. The residuals for the logistic fitting near the end of relaxation force are measurably smaller than those for the monoexponential fitting (10). Therefore, the logistic fitting near the end of relaxation is superior to the monoexponential fitting. We suspect that L reduces data noise from other determinants of relaxation.
PA/2 and P0, the pressure at the onset, extrapolate from the isovolumic relaxation pressure and increase load-dependently at any end-point. This supports the Frank-Starling effect. Furthermore, PA/2 and P0 are dependent on the choice of the end-point at any LVV. However, the differences in PA/2 between the end-points are smaller than those in P0. This indicates the superiority of PA over P0.
PB and P, which represent baselines to which pressure would decrease if these decays continued infinitely, increase load-dependently at any end-point. However, there are differing opinions on whether preloading affects the non-zero asymptote in earlier studies. The increase in P represents the important influence of slow isovolumic relaxation on increased pressures late in diastole (28). In contrast, alteration in P is not demonstrated during prolongation of isovolumic relaxation and LVEDP (29). Furthermore, PB and P are dependent on the choice of end-point of relaxation at any LVV. However, the differences in PB between the end-points are significantly smaller than those in P. This indicates the superiority of PB over P.
Moreover, the logistic function always fits the isovolumic relaxation pressure curve much more precisely than the monoexponential function at any LVV and end-point, in terms of r and RMS. Therefore, the logistic function appears to minimize systemic bias and provide a more robust assessment of relaxation.
The measurement of lusitropism is useful for not only investigators but also physicians and is important for not only science but also medicine. L and E have been used in the clinical studies of patients with cardiovascular diseases such as dilated cardiomyopathy and hypertrophic cardiomyopathy, and pharmacological stimulation with dobutamine and toborinone indicate L is superior to E (13). Our study may have important implications in the analysis and evaluation of ventricular relaxation in the clinical settings of patients with hypovolemia using L instead of E.
The logistic fitting is still an empirical model like other previous ones, including the monoexponential fitting. It remains unknown whether relaxation per se has a logistic mechanism (9). Our contention is that both ventricular and myocardial relaxation have a logistic characterization but not a logistic mechanism (9,10). The existence of a logistic mechanism remains to be determined.
We have elucidated that the logistic curve always fits the isovolumic LV relaxation pressure curve much more precisely than the monoexponential curve at any LVV, regardless of the end-point of relaxation in the canine heart. L proposed as a superior index of lusitropism proved to be independent of change in LVV loading or the Frank-Starling effect.
We would like to greatly thank Drs. Kunihisa Kohno and Byron Aoki for their excellent advice and help. We also thank Mr. Kimikazu Hosokawa for animal supply and care.
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Footnotes |
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Supported, in part, by Scientific Research Grants (13558113, 13770350, 13878185, 13878192, 14380405, 16659057) from the Ministry of Education, Culture, Sports, Science and Technology, and Cardiovascular Diseases Research Grants (11C-1, 14A-1) from the Ministry of Health, Labour and Welfare of Japan.
Accepted for publication October 20, 2005.
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