Logistic time constant of isometric relaxation force curve of ferret ventricular papillary muscle: reliable index of lusitropism

Analysis of Right Ventricular Myocardial Stiffness and Relaxation Components in Children and Adolescents With Pulmonary Arterial Hypertension

Background

The rate of left ventricular pressure decrease during isovolumic relaxation is traditionally assessed algebraically via 2 empirical indices: the monoexponential and logistic time constants (τ_E and τ_L). Since the pattern of right ventricular (RV) pressure decrease is quite different from that of the left ventricular, we hypothesized that novel kinematic model parameters are more appropriate and useful to evaluate RV diastolic dysfunction.

Methods and Results

Eight patients with pulmonary arterial hypertension (age 12.5±4.8 years) and 20 normal subjects (control group; age 12.3±4.4 years) were enrolled. The kinematic model was parametrized by stiffness/restoring Ek and damping/relaxation μ. The model predicts isovolumic relaxation pressure as a function of time as the solution of d²P/dt²+(1/μ)dP/dt+EkP=0, based on the theory that the pressure decay is determined by the interplay of inertial, stiffness/restoring, and damping/relaxation forces. In the assessment of RV diastolic function, τ_E and τ_L did not show significant differences between the pulmonary arterial hypertension and control groups (46.8±15.5 ms versus 32.5±14.6 ms, and 19.6±5.9 ms versus 14.5±7.2 ms, respectively). The pulmonary arterial hypertension group had a significantly higher Ek than the control group (915.9±84.2 s⁻² versus 487.0±99.6 s⁻², P<0.0001) and a significantly lower μ than the control group (16.5±4.3 ms versus 41.1±10.4 ms, P<0.0001). These results show that the RV has higher stiffness/elastic recoil and lower cross‐bridge relaxation in pulmonary arterial hypertension.

Conclusions

The present findings indicate the feasibility and utility of kinematic model parameters for assessing RV diastolic function.

Temperature-dependent inotropic and lusitropic indices based on half-logistic time constants for four segmental phases in isovolumic left ventricular pressure–time curve in excised, cross-circulated canine heart

Varying temperature affects cardiac systolic and diastolic function and the left ventricular (LV) pressure–time curve (PTC) waveform that includes information about LV inotropism and lusitropism. Our proposed half-logistic (h-L) time constants obtained by fitting using h-L functions for four segmental phases (Phases I–IV) in the isovolumic LV PTC are more useful indices for estimating LV inotropism and lusitropism during contraction and relaxation periods than the mono-exponential (m-E) time constants at normal temperature. In this study, we investigated whether the superiority of the goodness of h-L fits remained even at hypothermia and hyperthermia. Phases I–IV in the isovolumic LV PTCs in eight excised, cross-circulated canine hearts at 33, 36, and 38 °C were analyzed using h-L and m-E functions and the least-squares method. The h-L and m-E time constants for Phases I–IV significantly shortened with increasing temperature. Curve fitting using h-L functions was significantly better than that using m-E functions for Phases I–IV at all temperatures. Therefore, the superiority of the goodness of h-L fit vs. m-E fit remained at all temperatures. As LV inotropic and lusitropic indices, temperature-dependent h-L time constants could be more useful than m-E time constants for Phases I–IV.

The isovolumic relaxation to early rapid filling relation: kinematic model based prediction with in vivo validation

Although catheterization is the gold standard, Doppler echocardiography is the preferred diastolic function (DF) characterization method. The physiology of diastole requires continuity of left ventricular pressure (LVP)‐generating forces before and after mitral valve opening (MVO). Correlations between isovolumic relaxation (IVR) indexes such as tau (time‐constant of IVR) and noninvasive, Doppler E‐wave‐derived metrics, such as peak A‐V gradient or deceleration time (DT), have been established. However, what has been missing is the model‐predicted causal link that connects isovolumic relaxation (IVR) to suction‐initiated filling (E‐wave). The physiology requires that model‐predicted terminal force of IVR (F_tIVR) and model‐predicted initial force of early rapid filling (F_{i E‐wave}) after MVO be correlated. For validation, simultaneous (conductance catheter) P‐V and E‐wave data from 20 subjects (mean age 57 years, 13 men) having normal LV ejection fraction (LVEF>50%) and a physiologic range of LV end‐diastolic pressure (LVEDP) were analyzed. For each cardiac cycle, the previously validated kinematic (Chung) model for isovolumic pressure decay and the Parametrized Diastolic Filling (PDF) kinematic model for the subsequent E‐wave provided F_tIVR and F_{i E‐wave} respectively. For all 20 subjects (15 beats/subject, 308 beats), linear regression yielded F_tIVR = α F_{i E‐wave} + b (R = 0.80), where α = 1.62 and b = 1.32. We conclude that model‐based analysis of IVR and of the E‐wave elucidates DF mechanisms common to both. The observed in vivo relationship provides novel insight into diastole itself and the model‐based causal mechanistic relationship that couples IVR to early rapid filling.

Half-logistic time constants as inotropic and lusitropic indices for four sequential phases of isometric tension curves in isolated rabbit and mouse papillary muscles

The waveforms of myocardial tension and left ventricular (LV) pressure curves are useful for evaluating myocardial and LV performance, and especially for inotropism and lusitropism. Recently, we found that half-logistic (h-L) functions provide better fits for the two partial rising and two partial falling phases of the isovolumic LV pressure curve compared to mono-exponential (m-E) functions, and that the h-L time constants for the four sequential phases are superior inotropic and lusitropic indices compared to the m-E time constants. In the present study, we tested the hypothesis that the four sequential phases of the isometric tension curves in mammalian cardiac muscles could be curve-fitted accurately using h-L functions. The h-L and m-E curve-fits were compared for the four phases of the isometric twitch tension curves in 7 isolated rabbit right ventricular and 15 isolated mouse LV papillary muscles. The isometric tension curves were evaluated in the four temporal phases: from the beginning of twitch stimulation to the maximum of the first order time derivative of tension (dF/dt(max)) (Phase I), from dF/dt(max) to the peak tension (Phase II), from the peak tension to the minimum of the first order time derivative of tension (dF/dt(min)) (Phase III), and from dF/dt(min) to the resting tension (Phase IV). The mean h-L correlation coefficients (r) of 0.9958, 0.9996, 0.9995, and 0.9999 in rabbit and 0.9950, 0.9996, 0.9994, and 0.9997 in mouse for Phases I, II, III, and IV, respectively, were higher than the respective m-E r-values (P < 0.001). The h-L function quantifies the amplitudes and time courses of the two partial rising and two partial falling phases of the isometric tension curve, and the h-L time constants for the four partial phases serve as accurate and useful indices for estimation of inotropic and lusitropic effects.

Stiffness and relaxation components of the exponential and logistic time constants may be used to derive a load-independent index of isovolumic pressure decay

In current practice, empirical parameters such as the monoexponential time constant tau or the logistic model time constant tauL are used to quantitate isovolumic relaxation. Previous work indicates that tau and tauL are load dependent. A load-independent index of isovolumic pressure decline (LIIIVPD) does not exist. In this study, we derive and validate a LIIIVPD. Recently, we have derived and validated a kinematic model of isovolumic pressure decay (IVPD), where IVPD is accurately predicted by the solution to an equation of motion parameterized by stiffness (Ek), relaxation (tauc), and pressure asymptote (Pinfinity) parameters. In this study, we use this kinematic model to predict, derive, and validate the load-independent index MLIIIVPD. We predict that the plot of lumped recoil effects [Ek.(P*max-Pinfinity)] versus resistance effects [tauc.(dP/dtmin)], defined by a set of load-varying IVPD contours, where P*max is maximum pressure and dP/dtmin is the minimum first derivative of pressure, yields a linear relation with a constant (i.e., load independent) slope MLIIIVPD. To validate the load independence, we analyzed an average of 107 IVPD contours in 25 subjects (2,669 beats total) undergoing diagnostic catheterization. For the group as a whole, we found the Ek.(P*max-Pinfinity) versus tauc.(dP/dtmin) relation to be highly linear, with the average slope MLIIIVPD=1.107+/-0.044 and the average r2=0.993+/-0.006. For all subjects, MLIIIVPD was found to be linearly correlated to the subject averaged tau (r2=0.65), tauL(r2=0.50), and dP/dtmin (r2=0.63), as well as to ejection fraction (r2=0.52). We conclude that MLIIIVPD is a LIIIVPD because it is load independent and correlates with conventional IVPD parameters. Further validation of MLIIIVPD in selected pathophysiological settings is warranted.

Physical determinants of left ventricular isovolumic pressure decline: model prediction with in vivo validation

The rapid decline in pressure during isovolumic relaxation (IVR) is traditionally fit algebraically via two empiric indexes: τ, the time constant of IVR, or τL, a logistic time constant. Although these indexes are used for in vivo diastolic function characterization of the same physiological process, their characterization of IVR in the pressure phase plane is strikingly different, and no smooth and continuous transformation between them exists. To avoid the parametric discontinuity between τ and τL and more fully characterize isovolumic relaxation in mechanistic terms, we modeled ventricular IVR kinematically, employing a traditional, lumped relaxation (resistive) and a novel elastic parameter. The model predicts IVR pressure as a function of time as the solution of d2P/d t2 + (1/μ)dP/d t + EkP = 0, where μ (ms) is a relaxation rate (resistance) similar to τ or τL and Ek (1/s2) is an elastic (stiffness) parameter (per unit mass). Validation involved analysis of 310 beats (10 consecutive beats for 31 subjects). This model fit the IVR data as well as or better than τ or τL in all cases (average root mean squared error for dP/d t vs. t: 29 mmHg/s for model and 35 and 65 mmHg/s for τ and τL, respectively). The solution naturally encompasses τ and τL as parametric limits, and good correlation between τ and 1/μ Ek (τ = 1.15/μ Ek − 11.85; r2 = 0.96) indicates that isovolumic pressure decline is determined jointly by elastic ( Ek) and resistive (1/μ) parameters. We conclude that pressure decline during IVR is incompletely characterized by resistance (i.e., τ and τL) alone but is determined jointly by elastic ( Ek) and resistive (1/μ) mechanisms.

HYPOVOLEMIA DOES NOT AFFECT SPEED OF ISOVOLUMIC LEFT VENTRICULAR CONTRACTION AND RELAXATION IN EXCISED CANINE HEART

Hypovolemia results in hypotension due to a decrease in left ventricular (LV) stroke volume. We have showed a logistic relaxation time constant (tauL) that is a superior lusitropic index during the LV pressure (LVP) falling phase independent of LV preload compared with the conventional monoexponential relaxation time constant (tauE). In the present study, we investigated the effect of decreasing LV preload on tauL and tauE during the LV contraction and other relaxation phases. The isovolumic LVP curve was analyzed at LV Volumes (LVVs) of 18, 14, and 10 mL during 2-Hz pacing in seven excised, cross-circulated canine hearts. TauL and tauE were evaluated using logistic and monoexponential analyses of the four phases of the cardiac cycle: the period from the onset to the maximum time derivative of LVP (LV dP/dtmax), from LV dP/dtmax to peak LVP, from peak LVP to the minimum time derivative of LVP (LV dP/dtmin), and from LV dP/dtmin to LV end-diastolic pressure. TauL and tauE during the four phases did not change significantly with the decrease in LVV. During the change in LVV, the logistic function always fit significantly better compared with the monoexponential function. In conclusion, hypovolemia does not affect the speed of isovolumic LV contraction and relaxation. Each phase of the LVP curve is of a logistic nature. TauL is as a useful index for estimation of the speed of alteration during each phase of cardiac systole and diastole.

Half-logistic time constant: a more reliable lusitropic index than monoexponential time constant regardless of temperature in canine left ventricle

Temperature changes influence cardiac diastolic function. The monoexponential time constant (tauE), which is a conventional lusitropic index of the rate of left ventricular (LV) pressure fall, increases with cooling and decreases with warming. We have proposed that a half-logistic time constant (tauL) is a better lusitropic index than tauE at normothermia. In the present study, we investigated whether tauL can remain a superior measure as temperature varies. The isovolumic relaxation LV pressure curves from the minimum of the first time derivative of LV pressure (dP/dtmin) to the LV end-diastolic pressure were analyzed at 30, 33, 36, 38, and 40 °C in excised, cross-circulated canine hearts. tauL and tauE were evaluated by curve-fitting using the least squares method and applying the half-logistic equation, P(t) = PA/[1 + exp(t/tauL)] + PB, and the monoexponential equation, P(t) = P0exp(–t/tauE) + P∞. Both tauL and tauE increased significantly with decreasing temperature and decreased with increasing temperature. The half-logistic correlation coefficient (r) values were significantly higher than the monoexponential r values at the 5 above-mentioned temperatures. This implies that the superiority of the goodness of the half-logistic fit is not temperature dependent. The half-logistic model characterizes the amplitude and time course of LV pressure fall more reliably than the monoexponential model. Hence, we concluded that tauL is a more useful lusitropic index regardless of temperature.

A new linearly-combined bi-exponential model for kinetic analysis of the isometric relaxation process of Bufo gastrocnemius under electric stimulation in vitro

There was a slow-relaxing tail of skeletal muscles in vitro upon the inhibition of Ca(2+)-pump by cyclopiazonic acid (CPA). Herein, a new linearly-combined bi-exponential model to resolve this slow-relaxing tail from the fast-relaxing phase was investigated for kinetic analysis of the isometric relaxation process of Bufo gastrocnemius in vitro, in comparison to the single exponential model and the classical bi-exponential model. During repetitive stimulations at a 2-s interval by square pulses of a 2-ms duration at 12 V direct currency (DC), the isometric tension of Bufo gastrocnemius was recorded at 100 Hz. The relaxation curve with tensions falling from 90% of the peak to the 15th datum before next stimulation was analyzed by three exponential models using a program in MATLAB 6.5. Both the goodness of fit and the distribution of the residuals for the best fitting supported the comparable validity of this new bi-exponential model for kinetic analysis of the relaxation process of the control muscles. After CPA treatment, however, this new bi-exponential model showed an obvious statistical superiority for kinetic analysis of the muscle relaxation process, and it gave the estimated rest tension consistent to that by experimentation, whereas both the classical bi-exponential model and the single exponential model gave biased rest tensions. Moreover, after the treatment of muscles by CPA, both the single exponential model and the classical bi-exponential model yielded lowered relaxation rates, nevertheless, this new bi-exponential model had relaxation rates of negligible changes except much higher rest tensions. These results suggest that this novel linearly-combined bi-exponential model is desirable for kinetic analysis of the relaxation process of muscles with altered Ca(2+)-pumping activity.

Superior logistic model for decay of Ca2+ transient and isometric relaxation force curve in rabbit and mouse papillary muscles

A decrease in myocardial intracellular calcium concentration ([Ca(2+)](i)) precedes relaxation, and a monoexponential function is typically used for fitting the decay of the Ca(2+) transient. However, a logistic function has been shown to be a better fit for the relaxation force curve, compared to the conventional monoexponential function. In the present study, we compared the logistic and monoexponential functions for fitting the [Ca(2+)](i) declines, which were measured using the aequorin method, and isometric relaxation force curves at 4 different onsets: the minimum time-derivative of [Ca(2+)](i) (d[Ca(2+)](i)/dt (min)) and force (dF/dt(min)), and the 10%, 20% and 30% lower [Ca(2+)](i) levels and forces over the data-sampling period in 7 isolated rabbit right ventricular and 15 isolated mouse left ventricular papillary muscles. Logistic functions were significantly superior for fitting the [Ca(2+)] (i) declines and relaxation force curves, compared to monoexponential functions. Changes in the normalized logistic [Ca(2+)] (i) decline and relaxation force time constants at the delayed onsets relative to their 100% values at d[Ca(2+)] (i)/dt(min) and dF/dt(min) were significantly smaller than the changes in the normalized monoexponential time constants. The ratio of the logistic relaxation force time constant relative to the logistic [Ca(2+)](i) decline time constant was significantly smaller in mouse than in rabbit. We conclude that the logistic function more reliably characterizes the [Ca(2+)](i) decline and relaxation force curve at any onset, irrespective of animal species. Simultaneous analyses using the logistic model for decay of the Ca(2+) transient and myocardial lusitropism might be a useful strategy for analysis of species-specific myocardial calcium handling.

Starling-Effect-Independent Lusitropism Index in Canine Left Ventricle: Logistic Time Constant

The logistic time constant (tau(L)) has been proposed as a better index of the rate of left ventricular (LV) relaxation or lusitropism than the conventional monoexponential time constant (tau(E)). However, whether and how the Frank-Starling effect influences tau(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/dt(max)) 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 tau(L) but affected tau(E) slightly. Although the advancing end-point increased both tau(L) and tau(E), the increases were significantly smaller for tau(L) than for tau(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 tau(L) served as a more reliable index of lusitropism that is independent of the change in LVV loading or the Frank-Starling effect.