INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

ALTHOUGH MANY EARLY STUDIES of isolated cardiac muscle showed a complex dependence of myocardial contractile force on length, rate, and extent of shortening, a simpler concept of load-independent behavior of the intact heart initially evolved from studies in isolated, intact blood-perfused hearts. These studies led to widespread acceptance of the end-systolic pressure-volume relationship (ESPVR) as a load-independent index of ventricular contractile state (29). Although the ESPVR approach has proven invaluable as a tool to quantify and track changes in ventricular contractile state under a wide range of conditions and has enabled new understanding of ventricular-vascular coupling, it is a phenomenological description of ventricular properties with no link to basic mechanisms of myofilament contraction. Additionally, it has become increasingly clear that loading conditions can influence the ESPVR (6, 7, 15).

Attempting to establish a link between the growing understanding of the biochemical interactions involved in muscle contraction and whole organ properties, we demonstrated the feasibility of a four-state biochemical scheme of calcium, actin, and myosin interactions (Fig. 1) to explain the complex contractile behavior of the intact heart under isovolumic conditions at different volumes (3, 5). Initial modeling studies led to experiments focused on characterizing length dependence of myocardial calcium sensitivity in intact hearts (28). We identified important quantitative differences in calcium sensitivity and load dependence of calcium sensitivity between intact hearts and isolated, superfused cardiac muscle (28). Our prior investigations, however, were limited to experiments performed under isovolumic conditions.

View larger version (12K): [in this window] [in a new window]   Fig. 1.   Schematic representation of biochemical models proposed to account for interactions between calcium and myofilament and force generation. Tn, calcium-binding subunit of troponin; A, actin; M, myosin; , strain; K1-K4, rate constants for respective chemical reaction; Ka, association constant (actin-myosin affinity); Kd, dissociation constant for state 3; Kd', dissociation constant for state 4.

In the present study, calcium and ventricular pressure transients were measured on isovolumic and ejecting contractions in isolated, blood-perfused, physiologically afterloaded canine hearts. Using these data, we tested the hypothesis that the four-state model (Fig. 1) with length-dependent rate constants could explain contractile behavior of the intact, ejecting heart. Length dependence of rate constant values was determined from the isovolumic beats, and these were then used to successfully predict contractile performance on ejecting beats. The characteristics of intracellular calcium transients ([Ca²⁺]_i) measured during abrupt changes in loading conditions challenged the notion that myofilament calcium binding is length dependent. Model analysis further showed that length-dependent calcium binding was not required for the model to accurately predict ventricular behavior over a wide range of loading conditions.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Surgical Preparation

Measurements of Calcium Transients

(1)

1

Experimental Protocol

View larger version (31K): [in this window] [in a new window]   Fig. 2.   Representative left ventricular (LV) pressure (LVP; A), volume (LVV; B), and calcium transients ([Ca2+]i; C) measured during steady-state ejecting contractions and on the first isovolumic beat of variously timed volume clamps. D: LVP-LVV loops corresponding to A and B. There is an approximately linear relationship between peak pressure and volume on the isovolumic contractions, and the pressure-volume loop of the ejecting beat "breaks through" this line. E: superimposed averaged [Ca2+]i during steady-state ejecting contractions (dotted line) and during isovolumic contractions at the 3 different volumes; there is no detectable difference between these tracings.

Calculation of Strain and Stress from Observed LV Volume and Pressure

(2)

(3)

Determination of Load-Dependent Rate Constants

(4)

Because it is controversial as to whether or not troponin C calcium binding affinity is length dependent, two different model analyses were performed. In analysis 1, all parameters of the model were allowed to vary with ventricular volume under the assumption that both calcium binding affinity of troponin and actin-myosin binding affinity vary with strain. In analysis 2, however, calcium-binding affinity of troponin was assumed to be independent of strain. In analysis 2, therefore, length dependence of myofilament performance lies solely in the cross-bridge interaction. For each analysis, the value of each rate constant was plotted as a function of strain. Results showed that there was always a reasonably linear relationship between parameter values and strain, so these were each summarized by linear regression.

Application of Four-State Model to Ejecting Contractions

(5)

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1

1

1

1

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Intracellular Calcium Transients Are Not Significantly Different Between Ejecting and Isovolumic Contractions

Four-State Model Predicts Stress on Isovolumic Contractions at Different Strains

View larger version (18K): [in this window] [in a new window]   Fig. 3.   Representative measured (solid lines) and model-predicted pressure curves of the first isovolumic contractions at different volumes after the same preceding steady-state ejecting conditions. Dashed lines are stress curves predicted by analysis 1, and dotted lines are stress curves predicted by analysis 2.

View larger version (29K): [in this window] [in a new window]   Fig. 4.   Results from 1 representative heart showing how rate constant values vary as a function of strain. Dashed lines are for analysis 1 and dotted lines are for analysis 2. Note that rate constants K1-K4 are constrained to be independent of strain in analysis 2.

Ejecting Contractions

1

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View larger version (24K): [in this window] [in a new window]   Fig. 5.   A: representative calcium transient during steady-state ejecting contraction. B: midwall strain () and change in  with time (d/dt). C: measured (solid line) and predicted midwall stress curves. D: measured (solid line) and predicted stress-strain loops corresponding to data in B and C. E: calculated instantaneous force per unit cross bridge (CB) during the ejecting contractions. F: predicted force-velocity relationship from Eq. 3. In each panel, the dashed line shows results from analysis 1 and the dotted line shows results of analysis 2.

View larger version (27K): [in this window] [in a new window]   Fig. 6.   Additional examples of measured (thin solid line) and prospectively predicted (by analysis 2; thick solid line) stress curves during ejecting contractions from model rate constant values determined under isovolumic contractions. The stress curves of isovolumic beats clamped at end-diastolic volume (dashed line) and at end-systolic volume (dotted line) are also shown. A and B: high ejection fractions (50%). C and D: low ejection fraction (~15%). Results obtained with analysis 1 were indistinguishable.

Physiological Behavior of Four-State Model

Myofilament calcium sensitivity. Myofilament calcium sensitivity is classically indexed by measuring the relationship between force () and calcium concentration under steady state, equilibrium (nontwitch) conditions (force-pCa curves). This was simulated by imposing constant calcium concentrations on the four-state model. The resulting data provided typical sigmoidal curves (Fig. 7A, data from analysis 2), which were fit to Hill equations

(6)

where _H is the Hill coefficient and K_1/2 is the calcium concentration for 50% maximal activation. K_1/2 decreased as strain increased, a consequence of the experimentally determined strain dependence of the rate constants. For strains varying between 1.00 and 0.92, K_1/2 values ranged between 0.061 and 0.123 µM. Hill coefficients averaged ~4, did not vary significantly with strain, and did not vary significantly between analysis 1 and analysis 2 (Table 3).

View larger version (23K): [in this window] [in a new window]   Fig. 7.   A: representative force-calcium curves based on analysis 2 at various strains ( = 1.00, ; 0.96, ; 0.92, ). These curves shifted leftward and upward with increasing strain. B: predicted time course of bound calcium (i.e., sum of amount of state 2 and state 3 of Fig. 1) for analysis 1, in which calcium binding rate constants (K1-K4) are allowed to vary with strain; calcium binding increases with length. C: for analysis 2, with length-independent calcium binding rate constants, bound calcium is similar for isovolumic contractions at different strains.

Time course of calcium binding. It has been suggested that the amount of calcium bound to the myofilament varies as a function of length and that this contributes importantly to the Frank-Starling relationship (1). The amount of bound calcium as a function of time predicted by each analysis at three different strains is shown in Fig. 7, B and C. For analysis 1 (Fig. 7B), where calcium binding rate constants K₁-K₄ are allowed to vary with strain, calcium binding increases with length. For analysis 2 (Fig. 7C), with invariant calcium binding constants, calcium binding is similar for contractions at different lengths. This aspect of predicted muscle physiology fundamentally differentiates analyses 1 and 2.

Pressure-flow relationship. To determine how the model predicts that ejection velocity would impact on contractile performance we explored the relationship between ventricular flow and pressure at specific times during the contraction. Representative results are shown in Fig. 8. For each condition tested, strain patterns were set to achieve a common value (_s) at a common time (t_s). For example, in Fig. 8A, a family of strain curves is shown all of which start ejection 115 ms after the onset of contraction and achieve a strain of 0.990 at a time of 125 ms after the onset of contraction but start from different initial strains and therefore attain different ejection velocities at t_s. Model-predicted stress curves are shown in the graph. Similar curves are shown for _s values of 0.985 and 0.980 in Fig. 8, B and C, respectively. The stresses and strain rates were converted into LVP and flow rates, respectively, as described in METHODS. Pressure was then plotted as a function of flow, as shown in Fig. 8D for the data of Fig. 8, A-C. As shown, there is a negative, linear relationship between flow and pressure. Neither the slope of the relationship nor the flow-axis intercept (Q_max) varied with _s (indistinguishable results obtained with analyses 1 and 2). These simulated experiments were repeated for a common value of _s but for various values of t_s; a representative example is shown in Fig. 8E for an experiment with an _s value of 0.985. As shown again, the pressure-flow relation is linear with a negative slope that varies with t_s but with a relatively invariant Q_max value. For the parameter values obtained in the present study, Q_max values of ~800 ml/s were obtained, which is only slightly greater than the ~700 ml/s reported in the original studies of this phenomenon (27). All of the fundamental features of the pressure-flow relationships are thus reproduced by the four-state model.

View larger version (38K): [in this window] [in a new window]   Fig. 8.   Simulated stress responses to various shortening patterns designed to achieve specific strains (s) and specific times (ts) during a contraction to measure the ventricular pressure-flow relationship. A-C: representative simulations at ts = 125 ms and s = 0.990 (A), 0.985 (B), and 0.980 (C). Stress predicted by the model and d/dt (s1) were transformed into LV pressure (LVP) and change in volume with time (dV/dt; ml/s), respectively. D: averaged results (n = 13) that represent effect of strain on Qmax (extrapolated x-axis intercept) at a constant ts (125 ms). Qmax was approximately independent of s (large symbols, s = 0.990; middle-sized symbols, s = 0.985; small symbols, s = 0.980) at ts = 125 ms (P < 0.001 by ANCOVA), and slopes of the relations are not significantly different. Analysis 1 (open symbols and dashed lines) and analysis 2 (closed symbols and dotted lines) have similar results in this simulation. E: averaged results that represent effect of ts on Qmax at constant s (0.985).

Quick releases. Studies of isolated cardiac muscle have revealed that important insights into cross-bridge kinetics and calcium handling are obtained by studying how force responds to quick release-restretch loading sequences during steady-state contractures (4, 9). Such loading sequences were simulated at different calcium concentrations, and the time course of force redevelopment was analyzed. Figure 9, A and B, shows simulated force tracings during steady-state contractions at the specified calcium concentrations (0.1, 0.16, 0.25, and 1.0 µmol/l) and at a strain ( = 1.0) for analysis 2. The rate constant (K_tr) of force recovery was determined and plotted as a function of calcium for each strain (Fig. 8C). As evidenced by the tracings, K_tr increased substantially with increased calcium concentration. There was little difference in predictions between analyses 1 and 2.

View larger version (24K): [in this window] [in a new window]   Fig. 9.   Example of simulated force redevelopment after quick release and restretch during steady-state contracture at calcium concentrations of 0.10, 0.16, 0.25, and 1.00 µmol/l at a strain  = 1.00. A: predicted stress curves. B: same force traces as in A with each normalized to its maximum to show more clearly changes in the time course of force redevelopment with calcium concentration. C: relationship between calcium concentration and rate constants of force redevelopment (Ktr). Ktr in analysis 1 (filled symbols) and analysis 2 (open symbols) show clear dependence on calcium concentration at each strain ( = 1.00, squares; 0.96, circles; 0.92, triangles).

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Hemodynamic loading conditions did not influence the macroinjected aequorin luminescence transient measured in blood-perfused intact hearts at 35°C over a physiological range of loads. We conclude from this, as discussed further below, that calcium binding is not strongly influenced by length under these conditions. It was further demonstrated that contractile behavior of the intact, ejecting heart can be predicted by a model of myofilament-calcium interactions in which calcium binding is not influenced by length but other rate constants of myofilament interactions and their length dependence are determined under isovolumic conditions.

We showed in a prior study (32) with the same experimental methods that relatively small differences in calcium transients that paralleled changes in pressure development observed over longer time periods after a sustained change in loading conditions could be detected in isolated canine hearts with macroinjected aequorin luminescence measured from the epicardium. This is an important similarity to observations made originally by Allen and Kurihara (2) in isolated muscle. Aequorin luminescence is bright with this technique, requiring signal averaging of as few as two beats to obtain high-fidelity signals with low signal-to-noise ratios. Changes in strain during typical ejecting contractions in the epicardial layers from where the calcium transients are recorded are similar to those estimated for the midmyocardial layers. These factors render the methods used in the present study appropriate for studying changes in calcium kinetics under physiological conditions and would have detected these changes had they occurred.

Comparable results have been obtained in some studies of isolated, superfused papillary muscles in which peak intracellular calcium did not change significantly immediately after length changes (2, 14, 19, 21). Because it is estimated that ~95% of total released calcium is bound to the myofilament and calcium indicators (like aequorin) measure only free calcium, even small changes in myofilament affinity would be detected as substantial changes in free calcium.

In contrast, load-dependent changes in the rate of fall of free calcium have been noted in prior studies of isolated superfused muscles (2, 19). Such changes were not observed in the present study. Changes in the time course of the calcium transient were also not identified in a recent study of rat trabeculae (18). There are several possible reasons for such discrepancies. Differences in experimental conditions include blood perfusion vs. crystalloid superfusion (the latter potentially increasing cellular and interstitial water content), higher (more physiological) temperature in the intact heart, and influences of possible damage during the muscle isolation process. The findings in isolated muscles could thus reflect muscle behavior outside the boundaries of physiological conditions. Although the conditions under which isolated muscles are studied are more controllable and the measurements are less subject to artifacts and less devoid of confounding aspects of complex ventricular geometry and activation sequence than measurements in the intact heart, results obtained from intact hearts should not be dismissed on this basis.

In a study of crystalloid-perfused ferret hearts, our laboratory previously showed (3) that rate constants could be determined such that the four-state model with cooperativity (i.e., force-dependent actin-myosin binding affinity) was able to accurately fit isovolumic pressure curves from the measured calcium transient. With regard to isovolumic contractions, the present study extends these previous findings by showing how rate constants vary with volume and providing their values under the more physiological conditions of blood perfusion. On the basis of these values, model-predicted steady-state force-pCa curves were sigmoidal with _H of ~4, similar to other studies of isolated muscle preparations and isolated heart preparations (1, 28). In prior studies, but not in the present study, length-dependent increases in _H have been observed. This includes one study from our own laboratory (28) in which _H varied from 4.91 to 3.87 with strain values from 1.00 to 0.75. Differences in experimental factors noted above could have contributed to this relatively minor difference.

There is precedence for the notion that length dependence of myofilament activation can occur without length dependence of troponin C-calcium binding affinity, such as occurs in slow skeletal muscle (24). The overall force-calcium relationship is determined by both troponin C-calcium affinity and by actin-myosin interactions. Thus by influencing myofilament interaction kinetics (e.g., actin-myosin lattice spacing and binding affinities, the magnitude of cooperativity), length exerts its influence on force development. For example, the rate of force redevelopment (K_tr) reflects such intrinsic myofilament properties and the present results suggest a strong dependency of K_tr both on the level of myofilament calcium activation and on muscle (or sarcomere) length, as was shown experimentally in prior studies (23).

The application of the four-state model to ejecting contractions as implemented in the present study represents, to the best of our knowledge, the first attempt to model contractile behavior of ejecting contractions based on measured calcium transients with model parameter values obtained under isovolumic conditions. Prior theoretical studies have shown that one or another model could in general explain contractile behavior during shortening (17, 22), but there has been no prior attempt to explicitly test those models against measured data. The breadth of phenomena explained by the present model, however, should be acknowledged.

It could be argued that studies such as this should be performed in superfused isolated muscles. The use of epicardial calcium measurements from a single site and relating them to global pressure development presents several potential limitations. However, for reasons discussed above and specifically because certain key aspects of contractile behavior observed in intact hearts are not generally observed in isolated muscle, we considered it to be appropriate and important to investigate the phenomena under the more physiological conditions encountered in intact hearts. Results of several studies support the notion that, despite a seemingly complex ventricular geometry, myocardial activation sequence, and potential heterogeneity of muscle properties, conclusions pertaining to cellular and myofilament properties can be based on assessments of average midwall stresses and strains determined from intact hearts (8, 10, 11, 16, 26).

Additionally, there are a few limitations in converting LVV and LVP into strain and stress. First, strain calculated with this model does not necessarily equate with strain measured in a single sarcomere. Second, defining the reference point for strain (i.e., the point at which  = 1) at an end-diastolic pressure of 20 mmHg does not necessarily equate to L_max, the preload that provides maximal sarcomere length of ~2.3 µm. Despite the potential lack of a strict 1:1 correspondence between calculated strain and true sarcomere length, use of midwall strain has proven over the years to be a useful means of quantifying changes in muscle stretch for purposes of comparing results from different hearts and for comparing results from intact hearts and isolated muscles and has been used in countless studies of ventricular mechanics. Furthermore, no claim of 1:1 correspondence between calculated strain and sarcomere length is required for any of the interpretations of the present study.

In conclusion, two important concepts have been revealed in the present study. First, measurements with aequorin show that the intracellular free calcium transients are not influenced by abrupt changes in loading conditions, suggesting that myofilament calcium binding is not likely to be length dependent in vivo. Second, the four-state model of calcium-myofilament interactions with length and force-dependent actin-myosin binding kinetics defined during isovolumic contractions can predict the complex myocardial contractile behavior on ejecting contractions. It is not necessary to introduce length-dependent myofilament calcium affinity for the four-state model to explain contractile behavior on ejecting beats. Prior investigators advanced the concept of explaining whole heart behavior on the basis of the fundamental principles of muscle contraction (10, 11, 17, 26). The present study represents another step toward this goal by offering a comprehensive explanation for load dependence of ventricular performance, thus advancing understanding of the physiology of the intact heart under physiological conditions.