INTRODUCTION

Top Abstract Introduction Methods Conclusions Appendix References

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 constant-flow volume withdrawal over periods sufficient to achieve a given volume at a specified time in the cardiac cycle (29-31, 35); 2) rapid small-volume withdrawal at each of several times during the cardiac cycle (1, 13, 25, 26, 38); 3) small-volume 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 small-amplitude perturbations and appropriate model-based 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 high-frequency, small-amplitude, 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 20-ms interval at the time of peak isovolumic systole as in our previous studies (5, 27, 28). Furthermore, we employed a cross-bridge 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 cross-bridge mechanisms from observations made in the whole heart.

Glossary

A1	Amplitude scaling factor for pressure response component that varies proportionately with Pr(t)
A2	Amplitude scaling factor for pressure response component that varies proportionately with time drivative of r(t)
AIC	Aikake Information Criterion
Ap	Amplitude of dynamic passive component
b, d	Rate constants governing formation and dissolution of prepower stroke state
Bn	Amplitude of the nth harmonic of Pd(t)
P(t)	Pressure response to volume perturbation
Pd(t)	Depressive component of pressure response
Pf(t)	In-frequency component of pressure response
Pfa(t)	Active part of in-frequency response
Pp(t)	Dynamic passive component of pressure response
V	Measured amplitude of volume perturbation
V(t)	Time-varying volume perturbation
Vc	Computer-commanded amplitude of volume perturbation
Ee0(t)	Elastance of pressure generators in the prepower stroke state
Eep(t)	Elastance of pressure generators in the postpower stroke state
ESE	Elastance of series-coupled noncontractile element
ci	regression coefficient
	Elastance of a single generator
f	Frequency of volume perturbation
g	Rate constant governing cross-bridge detachment
h	Rate constant governing power stroke
K	Number of parameters
lm/2	Change in half-mass wall circumference
N	Number of sampled data points
Ne0(t)	Number of pressure generators in the prepower stroke state
Nep(t)	Number of pressure generators in the postpower stroke state
P(t)	Pressure of perturbed beat
Piso(t)	Pressure of isovolumic (unperturbed) beat
Pr(t)	Pressure around which Pf (t) occurred
Q10	Relative rate of change with a 10°C increase in temperature
1	Phase of pressure response component that varies proportionately with Pr(t)
2	Phase of pressure response component that varies proportionately with r(t)
p	Phase of dynamic passive component
RSS	Residual Sum of Squares
SC	Schwartz Criterion
T	Period of a heartbeat
n	Phase of the nth harmonic of Pd(t)
VBL	Baseline volume
VW	Wall volume
	Wall stress
	Angular frequency
X0	Average isovolumic distortion of postpower stroke generators
Xe0(t)	Average distortion of prepower stroke generators
Xep(t)	Average distortion of postpower stroke generators
Zep	Total distortion among all ep generators

	EXPERIMENTAL METHODS AND PROCEDURES

Top Abstract Introduction Methods Conclusions Appendix References

Experimental Preparation

The heart was transferred to a perfusion support system consisting of a gas-exchange chamber, a roller pump, a constant-pressure 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 volume-servo 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 draw-string suture in the mitral annulus was tightened around the obturator of a piston-cylinder 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 Ca²⁺, 1.1 Mg²⁺, 21.0 , 0.36 , and 11.1 glucose and 2.5 U/l insulin). The heart, suspended in the spent perfusate, was paced at 1 Hz with field stimulation using 5-ms pulses of 15 mV and 250 mA from 5 cm × 5 cm copper plates placed 4.5-cm apart on either side of the heart.

The volume-servo system consisted of a linear motor, a piston-cylinder device, and a linear variable differential transformer (LVDT, model 0294-0000, Trans-Tek). The piston-cylinder device was a modified 5-ml 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 5-Fr catheter-tip 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 supervisory-control computer, and passing the difference through an analog proportional-integral-derivative compensator. Output from the compensator was used to drive a high-current amplifier, which delivered electrical current to the motor, causing piston position to match the volume command.

The supervisory-control 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 12-bit range of an analog-to-digital converter and were acquired at a 2-kHz 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

After V_BL was established, a high-frequency 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 volume-perturbed 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 volume-servo system was underdamped, the actual volume perturbation did not exactly equal the commanded sinusoid from the supervisory-control computer. The frequency ( f ) of actual and commanded signals was the same, but there were differences between actual and commanded amplitudes (V and V_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

(1)

where  was a phase relative to the recorded time window. Equation 1 fitted all measured V(t) with a correlation coefficient (R²) > 0.99 and was thus judged to be an adequate representation of the actual perturbation signal for specific analyses. The underdamped character of the volume-servo system produced an actual V that, for a given V_c, increased with frequency over the 25- to 100-Hz frequency range with the result that the actual V at 100 Hz was 145% of that at 25 Hz. Thus V from the fit to the measured signal was used rather than the commanded V_c in all data analyses.

After the high-frequency volume perturbation protocol, a second single-beat Frank-Starling protocol was conducted to generate a Frank-Starling 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 [P(t)] to V(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)

(2)

Representative P_iso(t), P(t), and P(t) are shown in Fig. 1 (f = 50 Hz, V_c = 1% V_BL). All responses [P(t)] contained two components: a depressive response, P_d(t) [called "depressive" because it represented a sustained decrease in pressure below P_iso(t) that was not at the perturbation frequency] and an in-frequency response [(t)] (that part of the response at the perturbation frequency). Thus

(3)

The P_d(t) was extracted from P(t) by fitting a curve to P(t) that did not contain frequency content of the perturbation frequency. P_d(t) was taken as the sum of the first 10 harmonics in the Fourier series

(4)

where n is the harmonic number, B_n and _n are harmonic amplitude and phase, respectively, of n, and T is the heart period. Because the shortest heart period used in these studies was 1 s, the 10th harmonic (10 Hz) was well below 25 Hz, the lowest frequency used for volume perturbation. The amplitude and phase parameters for the ith harmonic (B_i and ) had no particular significance other than to give a P_d(t) waveshape, identifiable within P(t), that did not include components of the in-frequency response. Once P_d(t) was identified by fitting with Eq. 4, it was subtracted from P(t) to yield (t). Subtraction of P_d(t) from P_iso(t) generated a signal representing the pressure around which (t) took place [P_r(t)]. P_r(t), with corresponding P(t), P_d(t), and (t), is shown in Fig. 1.

View larger version (18K): [in this window] [in a new window]   View larger version (14K): [in this window] [in a new window]   Fig. 1.   Method of determining pressure response [volume perturbation = 50 Hz, 1% baseline volume (VBL)]. A: 2 left panels: Piso(t), pressure of an isovolumic beat in which no volume perturbation was applied; P(t), pressure of a beat that received volume perturbation. Middle panel: P(t), pressure response to volume perturbation [= P(t)  Piso(t)]. Right panel: Pd(t), depressive component of P(t) obtained by low-pass filtering P(t). Bottom panel: (t), in-frequency component of P(t) [=P(t)  Pd(t)]. B: pressure around which in-frequency response occurred [Pr(t)] was obtained by subtracting Pd(t) from Piso(t). Vertical scale is in mmHg, horizontal scale is in s.

This report concerns just (t); the P_d(t) is the subject of another report (unpublished observations). Twelve of twenty (t) responses obtained in one heart are shown in Fig. 2. (t) rose and fell during the course of the heartbeat but also contained a small contribution that was present during diastolic periods when there was no active contraction. This small component was assumed to be due to dynamic features of parallel passive properties that were not included in the static passive pressure-volume relationship, which had already been subtracted from the response. Furthermore, these passive dynamic properties were assumed to be expressed continuously throughout the period of the heartbeat and, for a given V(t) sinusoid, they were represented as

(5)

This passive dynamic response was subtracted from the response signal using data-fitting procedures described below. The great majority of (t) waxed and waned as pressure rose and fell and was considered to be an expression of active processes. Thus

(6)

where (t) was the contribution of active process to (t).

View larger version (33K): [in this window] [in a new window]   Fig. 2.   (t) over single heartbeat for 12 of 20 responses in one heart, showing extremes and midrange of responses to various amplitudes and frequencies of V(t). Note growth in amplitude of response with both amplitude and frequency of V(t). A: 0.5%; B: 1.0%; C: 1.5%.

It is clear from Fig. 1 that, in accordance with results from several studies (1, 13, 25, 29, 32, 38), (t) waxed and waned during the heart period as P_r(t) rose and fell. One objective of the current work was to predict from model considerations whether other time-varying components were contained within (t) and, then, to test these predictions.

	MODEL DESCRIPTION

A model for describing and predicting the active part of the pressure response [(t)] was constructed on the assumption that elements responsible for force generation in cardiac muscle (i.e., cross-bridges between thick and thin myofilaments) were also responsible for pressure generation in the LV chamber. Furthermore, it was assumed that there was a straightforward linear transformation between force-length relationships in the wall of the heart and pressure-volume relationships in the LV chamber. Acceptability of the linear transformation assumption requires small-amplitude perturbations and homogenous myocardium. Criteria for satisfying the small-amplitude requirement are detailed in the APPENDIX. Evidence that the homogenous myocardium requirement is satisfied is given in the findings relative to the unimportance of noncontractile series elasticity in these hearts (see MODEL VALIDATION). However, a strong reason for employing the linear transformation assumption is the success that has been achieved with its application in earlier studies (3, 5, 6, 27).

It was further assumed that during a heartbeat mechanodynamics were from two sources: 1) dynamics of activation as activator Ca²⁺ comes and goes and numbers of force-bearing cross-bridges rise and fall, and 2) dynamics of cross-bridge 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 cross-bridge 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 co-workers (17, 24, 39) and in accordance with our recent demonstration that cooperativity between force-bearing cross-bridges and activation can cause activation to be slow relative to cross-bridge dynamics (2).

We refer to cross-bridges 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), and 2) 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 (). Assuming that all generators in states e0 and ep possess the same , we show the net elastance of all parallel generators in each of the two states as

(7)

(8)

where, because of our linear transformation assumption, E_e0(t) and E_ep(t) may be taken as volumetric elastances (with units of mmHg/ml).

Generators are in continual transition as they progress from one state to another in the cross-bridge cycle (Fig. 3). We assumed that state e0 preceded state ep. Furthermore, transitions into, out of, and between states e0 and ep 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 cross-bridge cycle. Given these relationships between states and assuming that there is no noncontractile series elastance (4), it is shown in the APPENDIX that E_e0(t) and E_ep(t) may be calculated from P_r(t) and its first time derivative [_r(t)] according to

(9)

and

(10)

where X₀ is a parameter representing average volumetric distortion among generators in state ep during isovolumic conditions. The transitional step between states e0 and ep, which is governed by h, is the cross-bridge power stroke, and this step is responsible for inducing X₀ distortion in generators as they go through the power stroke to enter the ep state. The dissolution of the postpower stroke (ep) state, which is governed by g, is the cross-bridge detachment step.

View larger version (17K): [in this window] [in a new window]   Fig. 3.   Schematic drawing of states in cross-bridge cycle. Ni is the number of generators in ith state. Generators in states e0 and ep possess elastance, whereas generators in states 0 and 00 do not. Generators possessing elastance may be distorted during a volume perturbation such that both contribute to pressure response. Under isovolumic conditions, generators enter state e0 without distortion and do not generate pressure. Transitions between states are governed by rate constants b, d, h, and g. Transition between state e0 and state ep is the power stroke and induces a baseline distortion in postpower stroke (ep) generators, which, as a result of their elastance, causes development of isovolumic pressure. Isovolumic pressure is modified during a volume perturbation by induced distortion in postpower stroke and prepower stroke states, ep and e0, respectively, and by whatever influence volume perturbation has on recruitment of generators into or out of cross-bridge cycle.

Because of their elastic nature, generators within each of these states are distorted during volume perturbation. The net volume-induced 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 volume-induced distortions [X_e0(t) and X_ep(t) in states e0 and ep, respectively] are derived in the APPENDIX as

(11)

(12)

A dot over a variable indicates its first time derivative. In words, Eqs. 11 and 12 state that the time rate of change of generator distortion is negatively related to distortion itself and is positively related to the first time derivative of volume. Thus volume-induced distortion is driven directly by flow (velocity) and dies away at a rate proportional to the distortion.

Given these relationships, the model equation predicting the active part of the in-frequency pressure response [_fa(t)] is

(13)

where the "hat" in _fa(t) indicates that the quantity is model predicted.

Equations 9-13 constitute a 2-state, 4-parameter model. In this model, V(t) is the input [although distortion is driven directly by (t)]; E_e0(t), E_ep(t), and their time derivatives combine to form time-varying parameters; X_e0(t) and X_ep(t) are state variables; and _fa(t)is the output variable. Equations 11 and 12 are a set of linear, uncoupled, first-order, time-varying differential equations.

	MODEL PREDICTIONS

Two Dynamic Components of In-Frequency Response

(14)

The first term on the right-hand side of Eq. 14 is a response component with an amplitude varying proportionately with P_r(t), and the second term on the right-hand side is a component with an amplitude varying proportionately with _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 time-varying part of the coefficients in Eqs. 11 and 12, a steady-state solution of these equations for X_e0(t) and X_ep(t) when V(t) is a volume sinusoid of frequency f results in an expression for the first and second terms on the right-hand side of Eq. 14 as

(15)

Such that for a volume sinusoid of frequency f

(16)

The relative role of the _r(t) component in the observed responses was determined by an incremental approach. Equations 6 and 16 were combined and fit to the observed (t) by first excluding the _r(t) component

(17)

and then including the _r(t) component

(18)

Fitting of Eqs. 17 and 18 to (t) was by an heuristic search algorithm (Levenberg-Marquardt algorithm, Argonne National Laboratory) to minimize the residual sum of squares (RSS).

To test for degradation or improvement in the representation of (t) signal with the addition of the two additional parameters in Eq. 18 that relate to the _r(t) component, the Aikake Information Criterion (AIC) and the Schwartz Criterion (SC) were calculated from fits with both Eqs. 17 and 18 according to Landaw and DiStefano (20) as

(19)

where N is the number of sampled data points, and K is the number of parameters. Note that the first term on the right-hand side of Eq. 19 is a measure of how well the model fit the data, whereas the second term is a penalty function based on the number of model parameters. Therefore, increasing the number of parameters increases AIC and SC unless there is a more than compensating reduction in RSS. In considering two competing equations such as Eqs. 17 and 18, the better representation is the one with the smallest AIC and SC. To further determine whether significant reduction in the RSS occurred with Eq. 18 compared with Eq. 17, an incremental F-test was used (11).

When fit to (t) of each of the 280 data records obtained in these hearts (14 hearts times 20 records/heart), both Eq. 17, which did not include the _r(t) component, and Eq. 18, which did include this component, fit (t) very well with median R² of 0.980 and 0.981, respectively. The contribution of the _r(t) component was quite small as judged by the fact that A₂ was always two-orders of magnitude less than A₁. However, despite this small contribution, the inclusion of the _r(t) component in Eq. 18 consistently reduced the AIC (only one exception in 280 instances; median reduction: 0.78%; range: 0.02 to 10.1%) and SC (14 exceptions in 280 instances; median reduction: 0.63%; range: 0.09 to 9.93%), suggesting that the addition of the two parameters associated with the _r(t) component improved the representation of the information in the (t) signals. Furthermore, of the 280-response records analyzed, the incremental F-test generated an F-statistic that was significant at the P < 0.01 level in all 280 instances. Thus, although making only a small contribution in accounting for the total variability in (t), the _r(t) component contributed significantly to representing its information content and in reducing the RSS.

To summarize these results, the model predicted that there would be an in-frequency 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 [_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 to _r(t). Given the very small contribution by the _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

(20)

(21)

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 (t) will lead the phase of V(t) by as much as 90° at low frequencies, but this phase lead will decline and approach zero as frequencies increase above g.

To test these model predictions, the amplitude ratio, A₁/V, was evaluated for its dependence on f and V. As noted above, A₁/V will change sharply with f over a frequency range around the characteristic frequency of the underlying process. Additionally, A₁/V is not expected to be dependent on the amplitude of the V input. Any dependence of A₁/V 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 of A₁/V on product combinations of f and V. Stepwise regression analysis was used for these determinations. Regression equations were formulated as

(22)

where the c_i values are regression coefficients and (f,V) represents one or more of four candidate interaction terms: f · (V), root-mean-squared (rms) flow; f  ² · (V), rms acceleration; V², squared rms volume amplitude; and (f · V)², squared rms flow amplitude. The regression procedure used dummy variables and effects coding to account for between-subjects differences, and the subject dummy variables were forced into the stepwise regression (11). A candidate predictor variable was considered significant only when the P value for its inclusion was <0.05.

The dependence of A₁/V on f at the various commanded V_c for one heart is shown in Fig. 4. At all V_c, A₁/V increased with f. Furthermore, at f equal to 25, 50, and 76.9 Hz, A₁/V had virtually no dependence on V_c; there was an apparent small dependence on V_c at 100 Hz. Regression analysis of pooled data from all hearts revealed that, of all potential predictor variables, there was a significant dependence on f and V²; the simplest best regression equation (leaving out terms for between subjects variability) was

(23)

No interaction terms were found to be significant in this regression. Of the two significant predictor variables, f was the more important variable. For example, at f = 50 Hz and V = 1%, the term in Eq. 23 due to f adds 0.55 units to A₁/V, whereas the term due to V² subtracts only 0.03 units, an 18-fold difference in sensitivity of A₁/V on these variables. The three important outcomes of this regression analysis are 1) A₁/V increased with f in accordance with (t) acting as the effective driving function; 2) because of the strong dependence on f, the characteristic frequency of the dynamic processes responsible for the P_r(t) response component lies either within or not far distant from the 25- to 100-Hz frequency range; and 3) the very small dependence of the P_r(t) response component on V indicated that nonlinearities were not a large part of the underlying processes.

View larger version (7K): [in this window] [in a new window]   Fig. 4.   Amplitude ratio (A1/V) of (t) component proportional to Pr(t) as a function of frequency (f) and commanded amplitude (Vc) of perturbation. , Vc = 0.50% VBL; +, Vc = 0.75% VBL; ×, Vc = 1.00% VBL; * Vc = 1.25% VBL; , Vc = 1.50% VBL. A1/V increases with frequency and has very little dependence on Vc, as predicted by model.

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% V_c at each frequency, were examined. In these, the phase of (t) led that of V(t) at 25 Hz by ~30-40°, and this phase lead decreased progressively to approach zero at 76.9 and 100 Hz.

To summarize these results, model predictions with regard to frequency dependence of input-output 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

Unlike the sinusoidal approximation of Eq. 18, which could be fit only to individual responses to a single perturbation, model Eqs. 9-13 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 2-state, 4-parameter 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; V_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 model-predicted 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 in row 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.

View larger version (28K): [in this window] [in a new window]   Fig. 5.   A: response to a single perturbation f = 50 Hz; Vc = 1% VBL. Model was fit to complete set of 4 frequencies and 5 amplitudes of inputs delivered to 20 beats, not just to single beat shown. B: model-predicted vs. measured waveforms. These overlain waveforms cannot be discriminated in left panel. Waveforms of single cycles (identified by period between pairs of vertical lines in left panel) are displayed in 3 panels to right. These cycles spanned: 1) the point on ascending limb of Pr(t) at which Pr(t) = 1/2 its peak value, 2) the peak value of Pr(t), and 3) the point on descending limb at which Pr(t) = 1/2 its peak value. C: residuals between model-predicted and observed waveforms. Residuals are small valued and do not have periodicity of perturbation. Thus, in this example, they are apparently random.

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 an R² 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 (t) were never large, and an excellent fit was achieved at all frequencies and amplitudes of perturbation.

View larger version (17K): [in this window] [in a new window]   Fig. 6.   Model predicted vs. measured (t) for all 20 perturbed beats in one heart; 40,000 points shown (20 beats, each of 1-s duration, sampled at 2 kHz). Points in densest part of cluster around origin represent data generated at smallest Vc, at lowest f, at lowest values of Pr(t), and during zero crossings at all perturbations. Points in less dense vertices of cluster represent data generated at highest Vc, at highest f, and at peak of Pr(t). All points cluster tightly around line of identity. Deviations of predicted from observed (t) were never large, and an excellent fit was achieved at all frequencies and amplitudes of perturbation.

An additional demonstration of model goodness of fit comes from comparing the actual responses displayed in Fig. 2 with the corresponding model-predicted 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 between A₁/V for all 20 responses from the sinusoidal analysis and the 20 model-derived equivalents. This agreement was with respect to both absolute magnitudes and to systematic variations with f and V, strong dependence on f, and little or no dependence on V.

View larger version (30K): [in this window] [in a new window]   Fig. 7.   Alternative view of some of data plotted in Fig. 6. Twelve of 20 model-predicted responses are shown, corresponding to 12 measured responses shown in Fig. 2. A: 0.5%; B: 1.0%; C: 1.5%.

View larger version (7K): [in this window] [in a new window]   Fig. 8.   , Model-predicted equivalent of A1/V for each of 20 volume perturbations. There is virtually no dependence of model-predicted A1/V on perturbation amplitude but a strong dependence on frequency. , A1/V from sinusoidal analysis, i.e., same data shown in Fig. 4.

That good fits were obtained in all 14 hearts is supported by a median R² value of 0.978 with a narrow range between 0.968 and 0.987. These R² values were slightly smaller than the R² value obtained when the sinusoidal approximation (Eq. 18) was fit to a single response; these R² 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 resultant R² 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 V(t) and the response. This contrasts with the R² obtained with the model in which causal relationships between V(t)and (t) were stipulated and in which the model was asked to reproduce not just the variability in a single response but rather the variability due to changes in V and f in the input V(t) over the whole family of 20 responses in each heart. Given the resulting range of response variation to these broad ranges in input V and f (Fig. 2), the median R² value of 0.978 obtained with the basic model is remarkable.

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

Simpler model, without the prepower stroke state, does not work as well. Because of the very small contribution of the _r(t) component, it was important to test whether the 2-state, 4-parameter model, which predicted that there would be a _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

(24)

(25)

The simple 1-state, 2-parameter (Eqs. 10 and 24-25) and the basic 2-state, 4-parameter (Eqs. 9-13) models were each fit to the data and compared to determine which one was the best based on reduction in the AIC and SC and on the incremental F-test as described above.

The basic 2-state, 4-parameter model was superior to the simple 1-state, 2-parameter model in fitting the data. The improvement in median R² 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 2-state, 4-parameter 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 incremental F-test generated an F-statistic that was significant at P < 0.01 in all 14 hearts. These are strong results for accepting the 2-state, 4-parameter model over the simpler 1-state, 2-parameter model. Another way of stating the outcome of the incremental F-test is that the precision of the estimates of parameters remain within an acceptable range in the 2-state, 4-parameter 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 of X₀, 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 for h and d 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 of h and d 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 between A₁/V and its model-derived equivalent (Fig. 8), which is a response primarily attributable to the postpower stroke state, only a rough correspondence was obtained between A₂/V and its model-derived equivalent (Fig. 9), which is a response attributable to the prepower stroke state. The order of magnitude of values of A₂/V and its model-derived equivalent were the same, but dependencies of these quantities on f and V were not the same: there was no dependence on f and strong dependence on V for A₂/V, whereas there was weak dependence on f and no dependence on V for the model-derived equivalent. Employing the procedures used in the derivation of Eq. 25, we show that an approximate model-derived equivalent of A₂/V is

(26)

where

(27)

Because median h + d was 2,555 s¹ (see Table 1), at 100 Hz

(28)

This model approximate value, which is on the order of values found for A₂/V, is more than two orders of magnitude less than that for A₁/V, implicating virtually no role for the prepower stroke state. However, the contribution of the prepower stroke state is not only determined by the relative values of A₁/V and A₂/V, but also because the amplitudes of components associated with pre- and postpower stroke states depend on the relative values of P_r(t) and _r(t), which change with time over the cycle. A more valid assessment of contributions by the prepower stroke state is in the ratio of A_2r(t)/A₁P_r(t). This ratio varied with frequency, but representative values ranged between 1 and 5%. Even with these considerations, the contribution by the prepower stroke state is relatively small. Thus despite the statistical evidence suggesting good precision of the estimates in most cases and significant reduction in RSS, AIC, and SC, the prepower stroke state is contributing very little to the response and caution must be exercised in placing unqualified confidence in the estimates of associated parameters h and d.

View larger version (10K): [in this window] [in a new window]   Fig. 9.   , Model-predicted equivalent of A2/V for each of 20 volume perturbations. There is virtually no dependence of model-predicted A2/V on perturbation amplitude and only a weak dependence on frequency. , A2/V from sinusoidal analysis. There is only a rough correspondence between model-predicted and sinusoidal A2/V.

The reason for the relatively small dynamic contribution of the prepower stroke state in the 25- to 100-Hz range is the very high value of the characteristic frequency of responsible underlying processes h + d = 2,555 s¹. 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 100-Hz frequency range. It would take frequencies in the 400-Hz (2,555/2) range to generate appreciable distortion in the e0 state and make a sizable contribution to the response. Consequently, the e0 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 estimated g from a median of 69.0 s¹ in the 1-state, 2-parameter model to 115.6 s¹ in the 2-state, 4-parameter 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 cross-bridges were contributing to active chamber properties, i.e., there was no contribution by noncontractile structures in parallel or in series with the cross-bridge 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 2-state, 4-parameter model was compared with an elaborated model containing series elastance and consisting of 3 states and 5 parameters. The third-state 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 APPENDIX. In addition to the extra state and parameter, these equations differed from the basic model Eqs. 9-13 in that the two distortional variables X_e0(t) and X_ep(t) were no longer uncoupled, i.e., the differential equations describing the derivatives of each of these variables depended on both variables. The model with series elastance was fit to the data and then compared with the basic model without series elastance using the AIC, SC, and the incremental F-test.

The improvement in fit when series elastance was added to the basic model was scarcely observable, the median R² 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 F-statistic 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 >10¹⁰ 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 E_e0(t) + E_ep(t)] of 411 mmHg/ml. Thus when it was found to be significant (in roughly half of all hearts), series elastance was 43 times greater than maximal generator elastance and, consequently, played only a minor role in chamber mechanical properties. Furthermore, the addition of the series elastance element did not appreciably change the estimated value of the other model parameters (Table 1). In accordance with previous findings (4), we concluded that there was no important role for noncontractile series elastance in these hearts and no compelling reason for complicating the basic model by incorporating noncontractile series elastance.

Incorporation of distortion-dependent cross-bridge 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, distortion-dependent, cross-bridge 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 cross-bridge detachment (i.e., elevated detachment rate constant) secondary to vibration-induced, cross-bridge 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 V on A₂/V through the V² term in Eq. 23, which suggested amplitude-related nonlinearity such as distortion-dependent detachment.

To test whether distortion-dependent 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 2-state, 4-parameter model. To introduce distortional effects, we considered that both distortional variables, X_e0(t) and X_ep(t), vary with V(t) according to Eqs. 11 and 12. Because the rate constant governing detachment of the prepower stroke state (h + d) was very large compared with both the frequency of perturbation and the detachment rate constant for the postpower stroke state (g), a V(t) would induce a much greater X_ep(t) than X_e0(t) and, because of this, it is reasonable to consider only the effects of V-induced changes in X_ep(t) on g, the rate constant of cross-bridge detachment. We used two functional forms of distortion-dependent g, where g varied around its isometric (isovolumic) value (g₀) according to either

(29)

or

(30)

These two formulations differ importantly as follows. In Eq. 29, when X_ep(t) is negative, g takes on values less than g₀, i.e., reduced generator distortion below baseline would slow rate of detachment, whereas increased distortion would increase rate of detachment. This contrasts with the formulation of Eq. 30, where g is greater than g₀ for both positive and negative values of X_ep(t), i.e., both reduced and increased distortion increase g and rate of detachment.

A fundamental difference between equations of the model with distortion-dependent g compared with those of the basic model (Eqs. 9-13) was that, with distortion-dependent g, it could no longer be assumed that there were no changes in elastance at the perturbation frequency. Accordingly, it was necessary to add a third-state variable to account for in-frequency variation in E_ep(t). Equations for this formulation are given in the APPENDIX.

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 incremental F-test, 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 (R²) in reducing the AIC and SC or in yielding a significant incremental F-statistic when either form of distortion-dependent 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 F-test, 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 distortion-dependent 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 distortion-dependent g into the basic model, in either of the forms tested, was not worth the negligible increase in ability to explain the in-frequency component of the response to high-frequency volume perturbations.

From the ability of the basic 2-state, 4-parameter 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 in-frequency 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

Top Abstract Introduction Methods Conclusions Appendix References

Dynamic Processes Between 25 and 100 Hz Are Driven by Derivative of Volume Rather Than Volume Itself

Nonlinearities Are Not an Important Part of Response to Small-Amplitude Volume Sinusoids

Parameters From Current Sinusoidal Technique Agree With Those From Previous Studies and With Those Obtained From Isolated Muscle

1

1

1

1

1

1

In earlier studies with the ramp-withdrawal technique (28), a Q₁₀ for g 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 Q₁₀ for g 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 Q₁₀ of 4.6 for maximal velocity of shortening (which is proportional to g) in membrane-intact rat trabecular muscle (9). At this time we do not know whether the 2-state, 4-parameter model detects features that respond more strongly to temperature than do features of the 1-state, 2-parameter model. Resolution of these contrasting results and the effects of temperature will have to await the conclusion of on-going studies.

Results obtained here in isolated hearts may also be compared with results from isolated muscle studies by Kawai and co-workers (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 and h correspond with their rate constants for the detachment step, 2c, and the energy transduction step, 2b, respectively. Because 2c and 2b 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 fiber-bathing solution concentrations were comparable to a membrane-intact condition. Reported values of comparable 2c and 2b were approximately 30 and 8.6 s¹ (24) or 37 and 15 s¹ (17), respectively, in ferret heart muscle; and 23 and 15 s¹, 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 (2b/2c = 1/3 to 2/3) as it was in our study (h/g = 2/3). Second, given the potential high Q₁₀ 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.

Cross-Bridge Mechanisms Can be Observed in Whole Heart Behavior

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 small-amplitude, high-frequency volume perturbations in the isolated heart provide dynamic information relative to cross-bridge kinetics. The use of a model to analyze these responses gives estimates of specific kinetic parameters relative to the power stroke and cross-bridge 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.