Abstract
The objective was to determine the dynamics of contractile processes from pressure responses to smallamplitude, sinusoidal volume changes in the left ventricle of the beating heart. Hearts were isolated from 14 anesthetized rabbits and paced at 1 beats/s. Volume was perturbed sinusoidally at four frequencies ( f ) (25, 50, 76.9, and 100 Hz) and five amplitudes (0.50, 0.75, 1.00, 1.25, and 1.50% of baseline volume). A prominent component of the pressure response occurred at thef of perturbation [infrequency response,
 crossbridge model
 crossbridge detachment
 crossbridge power stroke
 heart muscle
pressure responses to controlled volume perturbation of the left ventricle (LV) of the beating heart have long been used to characterize ventricular mechanodynamics. These pressure responses have been elicited using a variety of volume perturbation protocols including 1) sustained constantflow volume withdrawal over periods sufficient to achieve a given volume at a specified time in the cardiac cycle (2931, 35);2) rapid smallvolume withdrawal at each of several times during the cardiac cycle (1, 13, 25, 26, 38);3) smallvolume withdrawal of varying amplitude and rate at the time of peak systolic pressure (4, 5,9, 27); and 4) sinusoidal volume change over the entire time course of the cardiac cycle (32, 33). Analyses of the resultant data have related pressure to volume and flow. Conclusions from all these analyses point to an organ with complex mechanodynamics that varies with time over the course of the cardiac cycle. These analyses have emphasized global features such as chamber elastance, viscous resistance, and series elastance and have made only indirect associations between these global features and the underlying muscle properties responsible for them. Recently, there has been an effort to interpret LV pressure responses directly in terms of underlying muscle mechanisms (3, 5, 7, 27). The promise from these early studies is that by using carefully controlled smallamplitude perturbations and appropriate modelbased analysis, detailed kinetic behavior of cardiac muscle may be elucidated from observations of pressure responses to volume changes in the whole heart.
In this study, we pursued that promise by using one of the original volume perturbation protocols: a continuous highfrequency, smallamplitude, sinusoidal volume change delivered over the entire cardiac cycle (32, 33). We chose this protocol not only because of the history of its previous use but also because it provides a means for identifying underlying contractile behavior from observations in the whole heart; these observations extend over the entire cycle period and are not confined to a very brief 20ms interval at the time of peak isovolumic systole as in our previous studies (5, 27, 28). Furthermore, we employed a crossbridge model with pre and postpower stroke elastance states to describe, predict, and explain the observed pressure responses. We concluded that these experimental and analytic techniques could be used to extract information about underlying crossbridge mechanisms from observations made in the whole heart.
Glossary
 A_{1}
 Amplitude scaling factor for pressure response component that varies proportionately with P_{r}(t)
 A_{2}
 Amplitude scaling factor for pressure response component that varies proportionately with time drivative ofP˙_{r}(t)
 AIC
 Aikake Information Criterion
 A_{p}
 Amplitude of dynamic passive component
 b, d
 Rate constants governing formation and dissolution of prepower stroke state
 B_{n}
 Amplitude of the nth harmonic of ΔP_{d}(t)
 ΔP(t)
 Pressure response to volume perturbation
 ΔP_{d}(t)
 Depressive component of pressure response
 ΔP_{f}(t)
 Infrequency component of pressure response
 ΔP_{fa}(t)
 Active part of infrequency response
 ΔP_{p}(t)
 Dynamic passive component of pressure response
 ΔV
 Measured amplitude of volume perturbation
 ΔV(t)
 Timevarying volume perturbation
 ΔV_{c}
 Computercommanded amplitude of volume perturbation
 E_{e0}(t)
 Elastance of pressure generators in the prepower stroke state
 E_{ep}(t)
 Elastance of pressure generators in the postpower stroke state
 E_{SE}
 Elastance of seriescoupled noncontractile element
 c_{i}
 regression coefficient
 ε
 Elastance of a single generator
 f
 Frequency of volume perturbation
 g
 Rate constant governing crossbridge detachment
 h
 Rate constant governing power stroke
 K
 Number of parameters
 Δl_{m/2}
 Change in halfmass wall circumference
 N
 Number of sampled data points
 N_{e0}(t)
 Number of pressure generators in the prepower stroke state
 N_{ep}(t)
 Number of pressure generators in the postpower stroke state
 P(t)
 Pressure of perturbed beat
 P_{iso}(t)
 Pressure of isovolumic (unperturbed) beat
 P_{r}(t)
 Pressure around which ΔP_{f} (t) occurred
 Q_{10}
 Relative rate of change with a 10°C increase in temperature
 ϕ_{1}
 Phase of pressure response component that varies proportionately with P_{r}(t)
 ϕ_{2}
 Phase of pressure response component that varies proportionately withP˙_{r}(t)
 ϕ_{p}
 Phase of dynamic passive component
 RSS
 Residual Sum of Squares
 SC
 Schwartz Criterion
 T
 Period of a heartbeat
 θ_{n}
 Phase of the nth harmonic of ΔP_{d}(t)
 V_{BL}
 Baseline volume
 V_{W}
 Wall volume
 ς
 Wall stress
 ω
 Angular frequency
 X_{0}
 Average isovolumic distortion of postpower stroke generators
 X_{e0}(t)
 Average distortion of prepower stroke generators
 X_{ep}(t)
 Average distortion of postpower stroke generators
 Z_{ep}
 Total distortion among all ep generators
EXPERIMENTAL METHODS AND PROCEDURES
Experimental Preparation
Hearts were isolated from 14 adult male rabbits (avg wt = 3.1 kg). Procedures for isolating the heart and attaching it to a volumeservo device have been described in detail elsewhere (7, 19). Briefly, the brachiocephalic artery was cannulated, and perfusion was begun with oxygenated relaxing solution (in mM: 121.4 Na^{+}, 35.0 K^{+}, 137.4 Cl^{−}, 0.1 Ca^{2+}, 1.1 Mg^{2+}, 21.0
The heart was transferred to a perfusion support system consisting of a gasexchange chamber, a roller pump, a constantpressure chamber, and an environmental chamber. The heart was placed within an environmental chamber where the coronary arteries were perfused at 90 mmHg. Temperature was kept constant at 30°C. The heart was submerged in perfusate at all times by allowing the coronary effluent to accumulate in the environmental chamber until it reached the chamber overflow at the level of the base of the heart. The perfusate was not recirculated.
A thin latex balloon, secured to the piston cylinder of a volumeservo system, was drawn into the LV chamber such that its tip was anchored through a puncture in the apex, which also served as a vent for any fluids between the balloon and chamber wall. A drawstring suture in the mitral annulus was tightened around the obturator of a pistoncylinder device, which secured the balloon in the LV chamber. The balloon was filled with degassed distilled water until passive chamber pressure reached 10 mmHg. Balloons were sized to fill the LV without excessive folding and without developing pressure at the volumes encountered in these ventricles. Thus balloons did not contribute to measured pressure.
The perfusing solution was changed from the relaxing solution to one that allowed spontaneous beating (in mM: 148.4 Na^{+}, 7.4 K^{+}, 139.1 Cl^{−}, 1.24 Ca^{2+}, 1.1 Mg^{2+}, 21.0
The volumeservo system consisted of a linear motor, a pistoncylinder device, and a linear variable differential transformer (LVDT, model 0294–0000, TransTek). The pistoncylinder device was a modified 5ml glass syringe (East Rutherford Syringes) with two side ports. One side port allowed calibrated infusion of fluid into the LV balloon to establish a baseline volume (V_{BL}). The second port was used to introduce a 5Fr cathetertip pressure transducer (Millar, Houston, TX) into the balloon. The piston was driven by the armature shaft of the linear motor. Motions of the piston produced LV volume changes around V_{BL} at a resolution of 0.001 ml. Both the pressure measurement system and the LVDT system had frequency responses of 1 kHz.
Motion of the motor armature, and consequently piston motion, was controlled to achieve specified changes in LV volume by feeding back the position signal from the LVDT transducer, comparing it with a reference position signal from a supervisorycontrol computer, and passing the difference through an analog proportionalintegralderivative compensator. Output from the compensator was used to drive a highcurrent amplifier, which delivered electrical current to the motor, causing piston position to match the volume command.
The supervisorycontrol computer controlled experimental protocols according to programmed instructions and also acquired data for later analysis. Pressure and volume signals were amplified to make maximal use of the 12bit range of an analogtodigital converter and were acquired at a 2kHz sampling rate.
Experimental use of animals was approved by the Animal Care and Use Committee at Washington State University. The investigation conforms with the Guide for the Care and Use of Laboratory Animals published by the National Institutes of Health (NIH publication No. 85–23, Revised 1985).
Protocols
A singlebeat FrankStarling protocol (7) was conducted to establish V_{BL} for each heart. V_{BL} was chosen as the volume equal to 80% of the volume at which maximum pressure was developed. This protocol was also used to establish the passive pressurevolume relationship. A monoexponential equation was fit to points over the range of enddiastolic pressure and volume values generated in this protocol. Thus the contribution to pressure by parallel passive structures at any volume was estimated and removed from all ensuing data records in order to allow us to focus on just active contractile properties.
After V_{BL} was established, a highfrequency volume perturbation protocol was conducted as follows. Twenty pairs of data records consisting of pressure and volume signals were taken. One record in a pair contained a single volumeperturbed beat, and the other record contained an unperturbed beat that served as a reference. Volume perturbation was administered only on a selected single beat. On the perturbed beat, the linear motor was commanded to deliver a sinusoidal volume change at one of four frequencies (100, 76.9, 50, or 25 Hz, corresponding to periods of 10, 13, 20, or 40 ms) and one of five amplitudes (0.5, 0.75, 1.0, 1.25, or 1.5% of V_{BL}). Repeated records of perturbed beats were taken until all combinations of frequencies and amplitudes (20 perturbed beats) were recorded. Pressure responses to the volume perturbation were then analyzed.
Because the volumeservo system was underdamped, the actual volume perturbation did not exactly equal the commanded sinusoid from the supervisorycontrol computer. The frequency ( f ) of actual and commanded signals was the same, but there were differences between actual and commanded amplitudes (ΔV and ΔV_{c}, respectively), and there was 1–2 ms delay in the actual signal relative to the commanded signal. Consequently for some analyses (see below), each of the measured volume perturbation signals was fitted with the analytic function
After the highfrequency volume perturbation protocol, a second singlebeat FrankStarling protocol was conducted to generate a FrankStarling curve that could be compared with the one collected previously. This allowed detection of any deterioration of the preparation during the course of an experiment. No detectable deterioration occurred.
PRESSURE RESPONSE
Peak isovolumic pressure generated by these 14 hearts (averaged over all 280 observations) was 120.2 ± 13.6 (SD) mmHg at an average V_{BL} of 2.11 ± 0.09 ml. The average LV weight, including the septum plus LV free wall, was 5.96 ± 0.54 g.
The pressure response [ΔP(t)] to ΔV(t) was defined as the difference between active pressure of the reference isovolumic beat [P_{iso}(t)], i.e., the pressure that would have developed had no volume perturbation been administered, and active pressure of the perturbed beat, P(t)
This report concerns just
It is clear from Fig. 1 that, in accordance with results from several studies (1, 13, 25, 29, 32, 38),
MODEL DESCRIPTION
A model for describing and predicting the active part of the pressure response [
It was further assumed that during a heartbeat mechanodynamics were from two sources: 1) dynamics of activation as activator Ca^{2+} comes and goes and numbers of forcebearing crossbridges rise and fall, and2) dynamics of crossbridge cycling as myosin heads cyclically attach to and detach from the actin binding site. In accordance with an earlier hypothesis (6), we argue that the only dynamics expressed within the brief cycle period of frequencies ≥25 Hz were those associated with steps in the crossbridge cycle and that the dynamics of activation were too slow to contribute to changes within these brief time periods. Such separation of time scales in the study of muscle dynamics is in accordance with analyses conducted by Kawai and coworkers (17, 24, 39) and in accordance with our recent demonstration that cooperativity between forcebearing crossbridges and activation can cause activation to be slow relative to crossbridge dynamics (2).
We refer to crossbridges as force generators. Generators contributing to the pressure response were assumed to be in two states:1) a state that possessed elastance but did not, under isometric (isovolumic) conditions, generate pressure (state e0), and2) a state that both possessed elastance and also generated isometric (isovolumic) pressure (state ep). When we assume linear, independent, and parallel generators, the elastance associated with each state is the number of parallel generators in that state (N) times the elastance of a single generator (ε). Assuming that all generators instates e0 andep possess the same ε, we show the net elastance of all parallel generators in each of the two states as
Generators are in continual transition as they progress from one state to another in the crossbridge cycle (Fig.3). We assumed thatstate e0 precededstate ep. Furthermore, transitions into, out of, and between states e0 andep were assumed to be governed by rate constants b,d, g, and h.State 0 in Fig. 3 is without elastance and is a precursor to state e0;state 00 is also without elastance and follows state ep. These nonelastance states represent all other states needed to complete a crossbridge cycle. Given these relationships between states and assuming that there is no noncontractile series elastance (4), it is shown in the
that
Because of their elastic nature, generators within each of these states are distorted during volume perturbation. The net volumeinduced distortion is determined by the rate of volume change relative to the rates of formation and dissolution of the respective states. Differential equations for these volumeinduced distortions [Δ
Given these relationships, the model equation predicting the active part of the infrequency pressure response [ΔP̂
Equations 913
constitute a 2state, 4parameter model. In this model, ΔV(t) is the input [although distortion is driven directly by ΔV˙(t)];
MODEL PREDICTIONS
Two Dynamic Components of InFrequency Response
Infrequency response consists of two dynamic components:1) a component with an amplitude varying with P_{r}(t) and 2) a component with an amplitude varying with the time derivative of P_{r}(t). The model output equation Eq. 13
can be rearranged by substituting elastance Eqs.9
and
10
intoEq. 13
to create an alternative formulation.
The first term on the righthand side of Eq.14 is a response component with an amplitude varying proportionately with P_{r}(t), and the second term on the righthand side is a component with an amplitude varying proportionately withP˙_{r}(t). This development clearly identifies the contribution of the prepower stroke, e0 state as the sole source of the dynamic response component with an amplitude rising and falling in proportion to the derivative of the pressure around which the response is occurring.
To determine the relative roles of these two components, an approximate sinusoidal solution of the model equations Eqs.11
and
12
was developed as follows. When we ignore the influence of the timevarying part of the coefficients in Eqs. 11
and
12
, a steadystate solution of these equations for Δ
The relative role of theP˙_{r}(t) component in the observed responses was determined by an incremental approach. Equations 6
and
16
were combined and fit to the observed
To test for degradation or improvement in the representation of
When fit to
To summarize these results, the model predicted that there would be an infrequency response component with an amplitude rising and falling in proportion to the pressure around which the response occurred [P_{r}(t)] and another component with an amplitude rising and falling in proportion to the derivative of the pressure around which the response occurred [P˙_{r}(t)]. Analysis of all the response data revealed that the response was dominated by the P_{r}(t) component, although a small but significant component existed with an amplitude of which was proportional toP˙_{r}(t). Given the very small contribution by theP˙_{r}(t) component, the model was further used to test the importance of including this term in validation procedures described below.
Amplitude Ratio and Phase of Response
Additional model predictions resulted from considering the nature of the response just around the time of peak P_{r}(t), whenP˙_{r}(t) approximated zero. When transients are ignored and it is assumed that steady state had been achieved at this time, an argument can be made that during a short interval around the time of peak pressure, thee0 state does not contribute to the response and the model reduces to
Two predictions result from Eq. 21
:1) the amplitude ratio will increase with frequency up to some plateau, provided the frequencies examined are in the vicinity of the characteristic frequency,g. At frequencies either far below or far above g, the amplitude ratio will change only weakly with frequency.2) The phase of
To test these model predictions, the amplitude ratio,A
_{1}/ΔV, was evaluated for its dependence on f and ΔV. As noted above,A
_{1}/ΔV will change sharply withf over a frequency range around the characteristic frequency of the underlying process. Additionally,A
_{1}/ΔV is not expected to be dependent on the amplitude of the ΔV input. Any dependence ofA
_{1}/ΔV on the amplitude of the input is an indication of nonlinear processes that are not part of the current model. Nonlinearities may also show up as dependence ofA
_{1}/ΔV on product combinations of f and ΔV. Stepwise regression analysis was used for these determinations. Regression equations were formulated as
The dependence ofA
_{1}/ΔV onf at the various commanded ΔV_{c} for one heart is shown in Fig. 4. At all ΔV_{c},A
_{1}/ΔV increased with f. Furthermore, atf equal to 25, 50, and 76.9 Hz,A
_{1}/ΔV had virtually no dependence on ΔV_{c}; there was an apparent small dependence on ΔV_{c} at 100 Hz. Regression analysis of pooled data from all hearts revealed that, of all potential predictor variables, there was a significant dependence onf and ΔV^{2}; the simplest best regression equation (leaving out terms for between subjects variability) was
Predictions with regard to the phase lead were not analyzed exhaustively. Rather, single cycles, spanning the time of peak P_{r}(t) of response to 1% ΔV_{c} at each frequency, were examined. In these, the phase of
To summarize these results, model predictions with regard to frequency dependence of inputoutput amplitude ratio and phase relations were confirmed. These indirect confirmations of the model suggested a more rigorous model validation test.
MODEL VALIDATION
Ability to Fit the Data
Model validation was, in part, by evaluating how well the model fit the full time course of the pressure response over an entire cardiac cycle. Model fitting was by the following procedure. Initial values of the four model parameters (g,h, d, and X
_{0}) and the two dynamic passive pressure parameters (A
_{p} and ϕ_{p}) were assigned. Derivatives of measured ΔV(t) and P_{r}(t) were calculated using a fivepoint Lagrangian polynomial method. Measured P_{r}(t), the calculated derivatives, and the thencurrent parameter values were fed into the differential equations Eqs.11
and
12
, allowing these equations to be solved numerically by integrating with a fourthorder RungeKutta algorithm (integration step size = 0.0005 s) to obtain predictions of Δ
Unlike the sinusoidal approximation of Eq.18 , which could be fit only to individual responses to a single perturbation, model Eqs.913 are more general and could be fit to groups of responses to perturbations of multiple frequencies and amplitudes. By fitting simultaneously to responses to the 20 perturbations imposed in any one heart, we required this single model with a single set of parameters to account for a wide range of behaviors resulting from many different perturbations as in Fig. 2. Successful reproduction of this broad range of dynamic responses was taken as compelling evidence for validity of the basic model.
By all measures, the basic 2state, 4parameter model fit the response data very well. An example of this good fit in one heart is shown for a single heartbeat ( f = 50 Hz; ΔV_{c} = 1% V_{BL}) in Fig.5. In evaluating Fig. 5, it must be kept in mind that the fit that generated the predicted response was to responses obtained from the complete set of four frequencies and five amplitudes of inputs delivered to each of 20 beats and not just to the single beat shown in Fig. 5. Because rapid cycling at 50 Hz generated a dense pattern on display from which modelpredicted and measured waveforms could not be discriminated, individual cycles in the response were identified that 1) spanned the point on the ascending limb of P_{r}(t) at which P_{r}(t) = 1/2 its peak value, 2) spanned the peak value of P_{r}(t), and 3) spanned the point on the descending limb at which P_{r}(t) = 1/2 its peak value. These three cycles were then expanded inrow B of Fig. 5 such that predicted and observed waveforms could be compared. The comparatively small values of the differences between predicted and observed waveforms (residuals) are given in row C of Fig.5. In this particular example, but not true in all cases, the residuals appeared to be random at all times during the heart period and exhibited no transient systematic character. Systematic patterns in the residuals will exhibit as a periodicity at the frequency of perturbation.
To demonstrate the goodness of the fit over the full set of 20 perturbed beats collected in the single heart featured in Fig. 5, predicted and measured values were plotted against one another to generate Fig. 6. In keeping with anR
^{2} value of 0.98 in this heart, all the points cluster tightly around a line that is not clearly distinguishable from the line of identity. It can be seen that deviations of predicted from observed
An additional demonstration of model goodness of fit comes from comparing the actual responses displayed in Fig. 2 with the corresponding modelpredicted responses displayed in the same format in Fig. 7. Visual comparison of these two figures reveals no important differences. Yet another demonstration of correspondence between model prediction and measured responses is seen in Fig. 8, where it is shown that there was good agreement betweenA _{1}/ΔV for all 20 responses from the sinusoidal analysis and the 20 modelderived equivalents. This agreement was with respect to both absolute magnitudes and to systematic variations withf and ΔV, strong dependence onf, and little or no dependence on ΔV.
That good fits were obtained in all 14 hearts is supported by a medianR
^{2} value of 0.978 with a narrow range between 0.968 and 0.987. TheseR
^{2} values were slightly smaller than theR
^{2} value obtained when the sinusoidal approximation (Eq.18
) was fit to a single response; theseR
^{2} values often exceeded 0.98. However, it must be kept in mind that the sinusoidal equation was fit individually to each of the 20 responses in each heart, and the resultantR
^{2} values reflected the ability of the sinusoidal equation to reproduce only the variability seen in a single response with no consideration of the causal relation between ΔV(t) and the response. This contrasts with theR
^{2} obtained with the model in which causal relationships between ΔV(t)and
This good fit by the basic model left only 2.2% of the total variation in the data unaccounted. Of this 2.2% it was very difficult to ascertain how much was simply random noise that could not be reduced with model improvement and how much was systematic and subject to reduction with model improvement. Systematic error would appear as systematic variation in the residuals at the frequency of perturbation, whereas random error would appear as residual variation unrelated to the frequency of perturbation. Unfortunately, the residuals did not lend themselves to analysis because systematic error, when it could be observed, did not persist throughout the period of the cardiac cycle. Rather, detectable systematic error would appear briefly during, say, just early systole or just during late systole and then disappear into apparently random error for the rest of the cycle. Such comings and goings of systematic error made it impossible to quantify the degree to which the residuals were random. Qualitatively, our impression was that systematic variation in the residuals was only a small part of the total residual variation. Because there was only a very small systematic error, our impression was that there was very little room for improvement with model modifications. However, results of this analysis do not eliminate the possibility that other features are present or even that a simpler model may be as good.
Basic Model vs. Competing Alternatives
Further model validation was confirmed by comparing the basic model against both simpler and more complex alternatives.
Simpler model, without the prepower stroke state, does not work as well. Because of the very small contribution of theP˙_{r}(t) component, it was important to test whether the 2state, 4parameter model, which predicted that there would be aP˙_{r}(t) response component, was actually needed, or whether a simpler model would suffice. A model that discounts the contribution by prepower stroke state e0 and considers only contributions from the postpower stroke state ep is more parsimonious. Such a model predicts a response that is composed of a single component proportional only to P_{r}(t) and possesses just 1 state and 2 parameters as given by
The basic 2state, 4parameter model was superior to the simple 1state, 2parameter model in fitting the data. The improvement in median R ^{2} for data from the 14 hearts was on the order of 3% (from 0.951 to 0.978). Smaller AIC and SC were associated with the 2state, 4parameter model in all 14 hearts with a median reduction of 8.5% (range = −1.22 to −13.6%) and 8.5% (range = −1.22 to −13.6%), respectively. Furthermore, the incrementalFtest generated anFstatistic that was significant atP < 0.01 in all 14 hearts. These are strong results for accepting the 2state, 4parameter model over the simpler 1state, 2parameter model. Another way of stating the outcome of the incremental Ftest is that the precision of the estimates of parameters remain within an acceptable range in the 2state, 4parameter model. Precision of parameter estimates were quantified by using the standard error of parameter estimate (11) using the RSS and covariance information returned from the optimization algorithm. Over all 14 hearts, the median and range of the standard error of the estimates ofX _{0},g, h, and d were (expressed as percentage of the respective parameter estimate) 0.19 (0.08–0.44), 0.38 (0.00–0.65), 1.88 (0.78–31.32), and 0.60 (0.20–26.93). The heart that generated the upper extremes in the ranges forh andd was an outlier among these 14 hearts because the heart with the next highest standard error of the estimate for these two parameters exhibited values of 8.48 and 8.60%, respectively. Except for estimates ofh andd in one heart, the respective parameters were generally estimated with more than adequate precision.
Despite these results, certain caveats in representing the prepower stroke e0 state in the model must be made clear. In contrast to the good correspondence betweenA
_{1}/ΔV and its modelderived equivalent (Fig. 8), which is a response primarily attributable to the postpower stroke state, only a rough correspondence was obtained betweenA
_{2}/ΔV and its modelderived equivalent (Fig. 9), which is a response attributable to the prepower stroke state. The order of magnitude of values ofA
_{2}/ΔV and its modelderived equivalent were the same, but dependencies of these quantities on f and ΔV were not the same: there was no dependence on f and strong dependence on ΔV forA
_{2}/ΔV, whereas there was weak dependence on f and no dependence on ΔV for the modelderived equivalent. Employing the procedures used in the derivation of Eq.25
, we show that an approximate modelderived equivalent ofA
_{2}/ΔV is
The reason for the relatively small dynamic contribution of the prepower stroke state in the 25 to 100Hz range is the very high value of the characteristic frequency of responsible underlying processesh + d= 2,555 s^{−1}. As a consequence, the prepower stroke state is very labile and, once a generator enters this state, it does not last long enough to undergo appreciable distortion when the volume is perturbed at frequencies in the 25 to 100Hz frequency range. It would take frequencies in the 400Hz (≅2,555/2π) range to generate appreciable distortion in thee0 state and make a sizable contribution to the response. Consequently, thee0 state makes only a small contribution to the response over the frequencies used in these studies, and the associated parameters are estimated with less precision than those associated with the postpower stroke state.
There were additional consequences to incorporating the prepower stroke state in the model. Including this state doubled the estimatedg from a median of 69.0 s^{−1} in the 1state, 2parameter model to 115.6 s^{−1} in the 2state, 4parameter model. Also, including this state increased the robustness of the parameter estimates, making their values less sensitive to the manner in which the data were prepared for analysis, i.e., values of filter settings, complete removal of trends, etc. (results not shown). We concluded that although it was important to include contributions from the prepower stroke state to account for variability in the data and for achieving robustness in the estimates of the parameters, we could identify only the rough magnitude of the underlying dynamic processes, because it made only small contributions to the overall response at even the highest frequency used in these studies (100 Hz).
Addition of noncontractile series elastance did not improve model. An important assumption in the model was that only crossbridges were contributing to active chamber properties, i.e., there was no contribution by noncontractile structures in parallel or in series with the crossbridge force generators. Contributions by noncontractile parallel structures were identified from passive behavior and removed by previously described methods. Our previous findings (4) indicated that there was no important noncontractile series elasticity participating in LV mechanical behavior to rapid ramp changes in volume. However, there was a need to confirm that this was so for sinusoidal volume changes.
To test for the contributions by noncontractile series elastic elements, the basic 2state, 4parameter model was compared with an elaborated model containing series elastance and consisting of 3 states and 5 parameters. The thirdstate variable was required to account for internal shortening during P_{r}(t), and the fifth parameter was a noncontractile series elastance. Equations for the model with series elastance are given in the
. In addition to the extra state and parameter, these equations differed from the basic modelEqs. 913
in that the two distortional variables Δ
The improvement in fit when series elastance was added to the basic model was scarcely observable, the medianR
^{2} value was 0.978 both with and without the series elastance. Furthermore, the AIC was reduced in only 9 of 14 hearts, and the SC was reduced in only 8 of 14 hearts. The incremental Fstatistic was significant in 8 of 14 hearts (P< 0.05). In the six hearts where the series elastic element was found not to be significant, the estimated value was >10^{10} mmHg/ml. In these six hearts, the model was equally well fit with any large value of series elastance, leading to considerable imprecision in parameter estimates and correspondingly insignificant reduction in RSS. In the eight hearts where series elastance was found to be significant, the average estimated value was 17,892 mmHg/ml. This compares with an average peak generator elastance [i.e., the maximal value of
Incorporation of distortiondependent crossbridge detachment did not improve model. Another fundamental model assumption was that the kinetic rate constants were truly constants and did not depend on behavior such as distortion of the generators. There are several reasons why this assumption required testing. First, distortiondependent, crossbridge rate constants have been a central tenet in much of the theoretical work in muscle mechanics (14) and have been a major feature of several models to explain aspects of mechanical behavior (10, 15, 22, 23, 34). Second, enhanced crossbridge detachment (i.e., elevated detachment rate constant) secondary to vibrationinduced, crossbridge distortion in cardiac muscle has been hypothesized to be the mechanism by which force was depressed (12, 18, 21, 36, 37) and relaxation period shortened (16) in cardiac muscle, and this depressive response was a prominent part of the response observed in these experiments (as in Fig. 1). Third, our results indicated a small but significant effect of ΔV onA _{2}/δV through the ΔV^{2} term inEq. 23 , which suggested amplituderelated nonlinearity such as distortiondependent detachment.
To test whether distortiondependent kinetic factors accounted for any of the observed behavior, we incorporated these effects into an expanded form of the basic model and compared the results with those from the basic 2state, 4parameter model. To introduce distortional effects, we considered that both distortional variables, Δ
A fundamental difference between equations of the model with distortiondependent g compared with those of the basic model (Eqs.913
) was that, with distortiondependentg, it could no longer be assumed that there were no changes in elastance at the perturbation frequency. Accordingly, it was necessary to add a thirdstate variable to account for infrequency variation in
Comparison of competing model formulations was by fitting the model with each g formulation, then comparing results from Eq. 29 with those from the basic model using AIC, SC, and incrementalFtest, and then comparing results from Eq. 30 with those from the basic model using the same criteria.
There was no systematic improvement in accounting for variation in the data (R ^{2}) in reducing the AIC and SC or in yielding a significant incrementalFstatistic when either form of distortiondependent detachment (Eqs.29 and 30 ) was incorporated into the model (Table 1). Furthermore, the standard error of the estimate of the parameter associated with distortion dependence, in those cases where the parameter was found to be significant by the incremental Ftest, was, on the average 5–20 times larger than the standard error of the other parameter estimates. Such lack of precision in the parameter estimate indicated that the parameter played a lesser role in accounting for variability in the data. Finally, the values of the other four parameters were virtually unaffected by whether distortiondependent detachment was part of the model or not (Table 1). Thus the effect of including distortion dependence in the model was negligible even when statistically significant.
We concluded that the added complexity as a result of incorporating distortiondependent g into the basic model, in either of the forms tested, was not worth the negligible increase in ability to explain the infrequency component of the response to highfrequency volume perturbations.
From the ability of the basic 2state, 4parameter model to account for a wide variety of observed features and the inability of variants to bring about any significant improvement, we adopted it as the best of several options and as a proper representation of mechanisms responsible for the active part of the infrequency response. This representation not only satisfies statistical requirements, but it is consistent with all of the major phenomenology seen in this analysis.
OVERALL SUMMARY AND CONCLUSIONS
Dynamic Processes Between 25 and 100 Hz Are Driven by Derivative of Volume Rather Than Volume Itself
Evidence indicating that LV dynamic processes in the 25 to 100Hz range are responding to the derivative of volume change comes from the modelindependent observations that the phase of
Nonlinearities Are Not an Important Part of Response to SmallAmplitude Volume Sinusoids
We found no convincing evidence for important nonlinearity in these responses. First, when we plotted theA _{1}/ΔV amplitude ratio in Fig. 4 and conducted the regression analysis, there was only a slight indication that any nonlinearities were operative. Furthermore, when we fit the data with the model and examined the correspondence between modelfitted and measured pressure as in Fig. 6, there was no indication for unrepresented nonlinearity; unrepresented nonlinearity would have exhibited itself by imposing some crescent shape to the oblong cluster of points in Fig. 6. Absolutely no evidence of crescent shape to this cluster was detectable. Finally, when we introduced the most likely candidate for nonlinearity into the model in the form of distortiondependent, crossbridge detachment, we were unable to detect any improvement.
Parameters From Current Sinusoidal Technique Agree With Those From Previous Studies and With Those Obtained From Isolated Muscle
It is instructive to compare results forg andX _{0} obtained in this sinusoidal analysis with estimates of these parameters from earlier studies done in this laboratory. Using ferret hearts at 30°C, rampvolume withdrawals at the time of peak isovolumic pressure, and a model equivalent to the 1state, 2parameter model of the current study, we (5, 30) found values ofg of ∼70–80 s^{−1} and ofX _{0} of 0.31 ml. These compare with values of 115 s^{−1} forg and 0.29 ml forX _{0} in the current study. Thus g, as estimated here, was 1.5 times greater than our previous estimates with other techniques, whereas X _{0} was not different. [Interestingly, the value ofg estimated from the 1state, 2parameter model in the current sinusoidal study (69 s^{−1}) is the same as the value we previously obtained using essentially this model in the rampwithdrawal studies.] In results reported elsewhere (unpublished observations) that were obtained using the sinusoidal technique in beating rabbit hearts at a lower temperature of 25°C, we found average g from the 2state, 4parameter model to be 49 s^{−1}, which is the same as we previously obtained at 25°C from ferret hearts using rampvolume withdrawals and the 1state, 2parameter model (28). Furthermore, the 49 s^{−1} value at 25°C in beating rabbit hearts from the current sinusoidal method is the same as a value of 52 s^{−1} estimated in constantly activated (Ba^{2+}), nonbeating rabbit hearts at 25°C, using responses over a dense spectrum of sinusoidal frequencies between 0.1 and 30 Hz (6). In summary, values of g estimated in the current study at 30°C are higher than values obtained previously with the rampwithdrawal technique at 30°C, but results with the current sinusoidal technique at 25°C in beating hearts are identical with previous results with the rampwithdrawal technique in beating hearts and sinusoidal techniques in constantly activated hearts.
In earlier studies with the rampwithdrawal technique (28), a Q_{10} forg of 2.15 was obtained. When the 25°C value of g reported elsewhere (unpublished observations) is used with the 30°C value reported here to calculate a Q_{10} forg from the sinusoidal technique, a value close to 5 is obtained. Although this value is far above our previous estimate, it compares favorably with a Q_{10} of 4.6 for maximal velocity of shortening (which is proportional tog) in membraneintact rat trabecular muscle (9). At this time we do not know whether the 2state, 4parameter model detects features that respond more strongly to temperature than do features of the 1state, 2parameter model. Resolution of these contrasting results and the effects of temperature will have to await the conclusion of ongoing studies.
Results obtained here in isolated hearts may also be compared with results from isolated muscle studies by Kawai and coworkers (17, 24,39). These workers used skinned, maximally activated, papillary and trabecular muscles from ferret (17, 24) and porcine (39) hearts and conducted sinusoidal perturbation experiments at 20°C. Our g andh correspond with their rate constants for the detachment step, 2πc, and the energy transduction step, 2πb, respectively. Because 2πc and 2πb change dramatically with ATP, ADP, and P_{i}, which were varied in their experiments, the proper comparison is with values of these constants obtained when fiberbathing solution concentrations were comparable to a membraneintact condition. Reported values of comparable 2πc and 2πb were approximately 30 and 8.6 s^{−1} (24) or 37 and 15 s^{−1} (17), respectively, in ferret heart muscle; and 23 and 15 s^{−1}, respectively, in porcine heart muscle (39). Despite differences in preparations, species, interpretive models, and analysis, there are remarkable similarities between our findings in the beating, isolated heart and those of Kawai’s group in skinned, constantly activated, isolated muscle. First, the rate constant of the energy transduction step was on the same order of that of the detachment step in Kawai’s studies (2πb/2πc= 1/3 to 2/3) as it was in our study (h/g= 2/3). Second, given the potential high Q_{10} for these processes, it can be argued that the isolated muscle values found at 20°C are in good agreement with those we found in the isolated heart at 30°C. Again, future studies at equivalent temperatures and with the same species are needed to establish how closely the estimates from the isolated heart agree with those from isolated muscle.
CrossBridge Mechanisms Can be Observed in Whole Heart Behavior
The most important outcome of this study is that aspects of crossbridge kinetic behavior may be observed from pressure responses to volume perturbations in the isolated heart. This is based on evidence for the validity of the proposed crossbridge model. Model validity was demonstrated from concordance between the sinusoidal analysis and model predictions, from how well the model fit the data, and from inability of elaborated models to improve representation of the data.
No model is perfect, including this one. Particularly, it is troubling that one of the model components, the component associated with the prepower stroke state, contributes so very little to the magnitude of the response. Thus a fair amount of caution must be exercised in relying on the accuracy of estimated parameter values associated with this state when estimation is made from responses to sinusoidal frequencies no higher than 100 Hz. We predict that frequencies of 400 Hz or more would be required to precisely estimate parameters associated with the prepower stroke state. More work is needed to secure greater confidence in this parameter estimate. However, given the overall good performance of the model and agreement of our estimated parameters with equivalent parameters found by others in isolated muscle, there is every reason to accept its general form and basic premises and to proceed with additional work for its verification and application.
In summary, pressure responses to smallamplitude, highfrequency volume perturbations in the isolated heart provide dynamic information relative to crossbridge kinetics. The use of a model to analyze these responses gives estimates of specific kinetic parameters relative to the power stroke and crossbridge detachment. The application of these methods to isolated heart studies promises to be a powerful experimental tool for elucidating mechanisms involved in changes in LV contractile function in response to a variety of interventions, including inotropic agents, ischemia, and chronic pathophysiological states such as cardiomyopathy.
Acknowledgments
This work was support by National Heart, Lung, and Blood Institute Grant HL21462.
Footnotes

Address for reprint requests: K. Campbell, Depts. of VCAPP & Biological Systems Engineering, Washington State Univ., Pullman, WA 99164.
 Copyright © 1997 the American Physiological Society
Appendix
Linear Transformation Between Incremental Wall Stress and Strain and Incremental Chamber Pressure and Volume
To justify the assumption that incremental linear pressurevolume elastance of the LV chamber can coexist with linear stressstrain muscle elasticity in the LV wall, we need to consider the relationship between representative lineal dimensions in the wall and chamber volume and between representative wall stress and chamber pressure. If changes in chamber volume linearly transform into changes in wall lineal dimensions and if changes in chamber pressure linearly transform into changes in wall stress, then incremental volumetric elastance can be taken as linearly related to incremental wall elasticity.
When it is assumed that the LV is a thickwalled sphere with a reference volume of V_{BL}, then it can be shown that a change in the halfmass wall circumference (Δl
_{m/2}) is related to a change in chamber volume (ΔV) according to
Relationship Between Elastance and Measured Pressure
By definition, chamber pressure equals elastance times volumetric distortion. Invoking the parallel generator assumption and the model of Fig. 3
Consider that elastances of the perturbed beat consist of isovolumic elastance plus changes in elastance induced by the perturbation. Furthermore, distortion consists of isovolumic distortion plus perturbationalinduced in distortion. Thus
Differential Equations for Distortional Variables
The differential equations for Δ
At some t + Δt,Z
_{ep}(t+ Δt) can be written asZ
_{ep}(t+ Δt) =Z
_{ep}(t) + (added distortion due to change in volume over Δt) + (added distortion due to formation of new units with baseline distortion over Δt) − (lost distortion due to detachment of distorted units over Δt), where (added distortion due to change in length over Δt) = (externally imposed ΔV) ⋅ (no. ofep generators existent att) = ΔV
Equations When There is Noncontractile Series Elastance
When there is noncontractile series elastance, there is internal shortening during isovolumic or reference beats. Under assumptions made in the text
Equations When There is DistortionDependent “g”
In addition to incorporating the functional form ofg into the differential equations of the basic model, it was necessary to expand these equations because, asg changed its value during the period of a volume sinusoid according to the distortions that were imposed,