INTRODUCTION

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EXCITATION-CONTRACTION (E-C) coupling requires Ca²⁺ on the order of 20-100 µmol/kg myocardium to be bound with various intracellular Ca²⁺ binding sites, including troponin C and calmodulin, in each cardiac contraction (1, 3, 6, 8, 10, 18, 22-24). These total Ca²⁺ handling (flux or transport) values have been obtained either biochemically or physiologically using isolated contracting myocardial preparations. However, these analytic methods cannot be used readily to quantify total Ca²⁺ handling in a beating whole heart (9, 29). Although the popular Ca²⁺ transient methods detect sarcoplasmic free Ca²⁺ concentrations on the order of 0.1-2 µmol/l in beating myocardium and whole heart preparations, they represent only a small fraction of the total Ca²⁺ handling, left unbound with Ca²⁺ binding sites (3, 5, 6, 11, 13, 16, 18, 22). Therefore, there has been no appropriate method to quantify total Ca²⁺ handling in a beating whole heart.

We recently attempted to quantify total Ca²⁺ handling in the left ventricle (LV) of the canine beating whole heart by combining LV myocardial O₂ consumption (VO₂) and the so-called intracellular Ca²⁺ recirculation fraction [RF, the fraction of total released Ca²⁺ that is sequestered by the sarcoplasmic reticulum (SR) Ca²⁺-ATPase pump] (29). RF was obtained from the exponential decay beat constant of the postextrasystolic potentiation (PESP) (2, 9, 12, 14, 20, 21, 29, 33). Assuming no futile Ca²⁺ cycling via the SR in normal hearts, we were able to estimate total Ca²⁺ handling in control and enhanced contractile states produced with Ca²⁺ and epinephrine (29). However, we could not apply this method to the failing hearts produced by infusing intracoronary ryanodine because this intervention led to energy-wasting, futile Ca²⁺ cycling from the SR rendered leaky to Ca²⁺ (9, 31). This is a serious limitation of our previous method (29). Accordingly, we have developed a new method that enables us to quantify total Ca²⁺ handling in failing hearts that includes the energy-wasting Ca²⁺ handling due to futile Ca²⁺ cycling.

This new integrative method was applied to the mechanoenergetics and RF data of hearts treated with ryanodine to produce contractile failure and, presumably, futile Ca²⁺ cycling (31) to estimate the feasibility of this method.

	METHODS

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Background. Our cardiac mechanoenergetic framework separates VO₂ for E-C coupling from the total VO₂ of a beating heart (15, 25-27, 31). E-C coupling VO₂ is primarily due to the energy requirement of Ca²⁺ handling because Na⁺ handling energy for membrane excitation is negligibly small (1, 3, 25, 29). The present method integrates this framework with the RF concept and the different molar Ca²⁺:ATP stoichiometries of the energetically major Ca²⁺ handling routes (1, 3, 8, 9, 17, 30). Figure 1 depicts these major intracellular (internal) and transsarcolemmal (external) Ca²⁺ handling routes, including the sarcolemmal and SR Ca²⁺ channels, the Ca²⁺- and Na⁺-K⁺-ATPase pumps, and the Na⁺/Ca²⁺ exchangers (1, 3, 4, 8, 9, 11, 12, 17, 19, 22, 29, 30). Although we neglected the Na⁺ handling energy for membrane excitation, we retained the Na⁺ handling energy by the Na⁺-K⁺-ATPase pump coupled with the Na⁺/Ca²⁺ exchange (see below) (1, 3, 9, 17, 25, 29).

View larger version (36K): [in this window] [in a new window]   Fig. 1.   Schematic diagram of energetically major Ca2+ handling routes adopted in our present total Ca2+ handling model in the heart. A: without futile Ca2+ cycling. B: with futile Ca2+ cycling. In A and B, sarcoplasm is surrounded by largest rectangle indicating sarcolemma (SL). Sarcoplasmic reticulum (SR) is shown by largest shaded oval. Thick solid loop passing through SR and remaining within SL is intracellular Ca2+ recirculation route via Ca2+ pump and ryanodine-sensitive Ca2+ release channel of SR. Thin solid loop running across SL is the Ca2+ extrusion route via the Na+/Ca2+ exchanger and the dihydropyridine-sensitive, L-type Ca2+ channel of the SL. Hatched loop only in B is the futile Ca2+ cycling via the SR. Na+/Ca2+ exchange is assumed to be ionically and energetically coupled with Na+-K+-ATPase pump for Ca2+ and Na+ homeostasis in each steady-state beat. Ca2+ pump: Ca2+-ATPase pump of SR. Sarcolemmal Ca2+ ATPase pump is not shown, under the assumption of its minor contribution. RF, internal recirculation fraction of Ca2+ via SR. XF, 1  RF: transsarcolemmal extrusion fraction of Ca2+. N, number of futile Ca2+ cycling via SR. N + 1 is total number of internal Ca2+ cycling, where 1 cycle is number of normal Ca2+ cycling via the SR. A corresponds to N = 0. B corresponds to N > 0 due to the increased Ca2+ permeability of the SR. Total Ca2+, total amount of Ca2+ handling (flux, or transport) in each contraction. R, reactivity of an index of ventricular contractility (Emax) to total Ca2+: proportional constant relating Emax to total Ca2+ handling.

RF decreases and 1  RF (the other fraction of total Ca²⁺ transported transsarcolemmally primarily by the Na⁺/Ca²⁺ exchanger) reciprocally increases in various types of failing hearts (9, 14, 20). A constant RF and Ca²⁺ and Na⁺ homeostasis are maintained in steady-state beats. Otherwise, gradual changes in RF and internal Ca²⁺ and Na⁺ concentrations would alter myocardial contractility over beats, disrupting the steady state (3, 12, 29). To maintain the Ca²⁺ and Na⁺ homeostasis, the Na⁺-K⁺ pump is coupled with the Na⁺/Ca²⁺ exchange to remove the exchanged Na⁺ influx (3).

Ca²⁺ handling VO₂ will increase with an internal-to-external shift of Ca²⁺ handling because the Na⁺/Ca²⁺ exchange-Na⁺-K⁺ pump system has the 1 Ca²⁺:1 ATP stoichiometry in contrast to the 2 Ca²⁺:1 ATP stoichiometry of the SR Ca²⁺ pump (see below) (1, 3, 9, 29). In fact, our studies have shown that ryanodine infused at nanomolar (not micromolar) concentrations into the coronary circulation of the excised cross-circulated canine heart preparation stably decreased both LV contractility and RF without significantly decreasing Ca²⁺ handling VO₂ (9, 31). Therefore, we suspected that not only total Ca²⁺ handling but also RF are major determinants of the Ca²⁺ handling VO₂ (9, 29).

We previously proposed an original formula to calculate total Ca²⁺ handled or transported in the E-C coupling process from experimentally obtained Ca²⁺ handling VO₂ and RF values in normal hearts with presumably no futile Ca²⁺ cycling (29). This formula was successfully applied for the first time to canine normal LVs in control and contractile states enhanced by intracoronary Ca²⁺ and epinephrine (29). However, we could not apply this formula legitimately to pathological hearts that are presumably wasting Ca²⁺ handling VO₂ due to futile Ca²⁺ cycling via the Ca²⁺-leaky SR (9, 31). Therefore, we could not obtain an estimate of total Ca²⁺ handling in ryanodine-treated failing hearts, although we successfully estimated both Ca²⁺ handling VO₂ and RF in these hearts (9). If futile Ca²⁺ cycling is occurring, total Ca²⁺ handling cannot simply be divided by RF into the internal and external fractions (9). Therefore, we developed the following new method.

New method. Figure 1A illustrates the Ca²⁺ handling model for our original method (29) and Fig. 1B illustrates that for our new method. Both have the internal and external Ca²⁺ handling routes in common. Both methods require Ca²⁺ handling VO₂ and RF to divide the Ca²⁺ handling VO₂ into the two major Ca²⁺ handling routes with the twofold different molar Ca²⁺:ATP stoichiometries (see below) (1, 3, 9, 17, 29, 30). However, the new method incorporates the futile Ca²⁺ cycling and the resultant extra VO₂ into the original method (29), as shown by the third route (hatched loop) in Fig. 1B.

As shown in both Fig. 1A and Fig. 1B, Ca²⁺ enters the cell via the sarcolemmal Ca²⁺ channel and simultaneously Ca²⁺ is released into the sarcoplasm from the SR via its Ca²⁺ release channel (3, 11). Most of the Ca²⁺ is then bound to the Ca²⁺ binding sites, including not only troponin C to elicit cross-bridge cycling but also calmodulin, mitochondria, etc. (3, 6, 10, 11, 13, 16, 18, 22-25). The SR Ca²⁺ pump sequesters a considerable fraction, which is the RF, of the total Ca²⁺ by hydrolyzing ATP at a molar stoichiometry of 2 Ca²⁺:1 ATP (1, 3, 4, 7, 8, 30). The remaining fraction (1  RF) of the total Ca²⁺ is predominantly extruded by the Na⁺/Ca²⁺ exchanger and secondarily by the sarcolemmal Ca²⁺ pump (although not shown, but see below). The Na⁺/Ca²⁺ exchange coupled with the Na⁺-K⁺ pump maintains Ca²⁺ and Na⁺ homeostasis at a molar stoichiometry of 1 Ca²⁺:1 ATP (see below) (1, 3, 4, 8, 17). Note that this stoichiometry is one-half that of the SR Ca²⁺ pump. In other words, the SR Ca²⁺ pump is economical and the Na⁺/Ca²⁺ exchange coupled with the Na⁺-K⁺ pump is one-half economical, or twice as wasteful, in myocardial Ca²⁺ handling.

We have assumed that Ca²⁺ cycling via the SR during each contraction is only once in normal hearts; i.e., the Ca²⁺ transiently released and then sequestered by the SR Ca²⁺ pump during the same beat is no more released until the next beat (3, 29, 30). However, extra Ca²⁺ cycling is assumed to occur when the SR becomes leaky to Ca²⁺ in abnormal hearts such as those treated with ryanodine at nanomolar concentrations (3, 4, 9, 19, 31). Ryanodine at nanomolar concentrations bound to the SR Ca²⁺ release channel fixes the channel one-half open and makes it permeable to Ca²⁺ (in contrast to the complete channel closure achieved at micromolar concentrations of ryanodine) (4, 19, 31). In the Ca²⁺-leaky SR, part of the Ca²⁺ sequestered by the Ca²⁺ pump may leak out and be sequestered again during the same beat. Hence, the internal Ca²⁺ handling includes extra Ca²⁺ cycling (N) above and beyond the normal Ca²⁺ cycle in each beat. N > 0 represents the existence of the futile Ca²⁺ cycling (4, 31) as modeled in Fig. 1B. The new method described in this paper takes the futile Ca²⁺ cycling into consideration. As for the reactivity (R), see below.

Formulation. We propose the following equation to calculate the amount of ATP consumed for total Ca²⁺ handling that includes the futile Ca²⁺ cycling

(1)

where the units of both Ca²⁺ handling ATP and total Ca²⁺ handling are micromoles per kilogram wet myocardium. N is dimensionless. As for the details of RF, see Recirculation fraction below.

In Eq. 1, (total Ca²⁺ handling)RF gives the amount (µmol/kg) of Ca²⁺ handling via the internal route (i.e., SR), and (total Ca²⁺ handling)(1  RF) gives the amount (µmol/kg) of Ca²⁺ handling via the external route. These two terms were used in the original method (Fig. 1A) that we previously proposed to quantify total Ca²⁺ handling in normal hearts, which were assumed to have no futile Ca²⁺ cycling (29). The new term in this equation, (total Ca²⁺ handling)N × RF, which is equal to N × (total Ca²⁺ handling)RF, estimates the amount (µmol/kg) of Ca²⁺ handled by futile cycling (N > 0) via the SR (Fig. 1B).

The denominator 2 of two RF/2 terms in Eq. 1 is the coefficient 2 in the molar stoichiometry of 2 Ca²⁺:1 ATP of the SR Ca²⁺ pump ATPase (1, 8, 30). The value of 2 for both (total Ca²⁺ handling)RF and (total Ca²⁺ handling)N × RF terms in Eq. 1 converts the respective quantities of Ca²⁺ handling via the SR into the respective amounts of ATP hydrolyzed for the normal and futile Ca²⁺ handling by the SR Ca²⁺ pump. The externally handled term, (total Ca²⁺ handling)(1  RF), has 1, but not 2, in the denominator because its net stoichiometry is simply 1 Ca²⁺:1 ATP as the result of the 3 Na⁺/Ca²⁺ exchange and the 3 Na⁺-2 K⁺-ATP pump (1, 8, 17). The sarcolemmal Ca²⁺ pump, which secondarily removes Ca²⁺ from the cytoplasm, also has a 1 Ca²⁺:1 ATP stoichiometry (3). As a result, RF/2 + (1  RF) + N × RF/2 is an overall stoichiometric factor to convert total Ca²⁺ handling into Ca²⁺ handling ATP.

Mitochondrial oxidative phosphorylation starting with such metabolic substrates as lactate and pyruvate has a nominal stoichiometry of atomic ratio of 3 P:1 O or molecular ratio of 6 P:1 O₂, where P is the high-energy phosphate of ATP. The P-to-O ratio increases by ~5% when the metabolic substrate is glucose and decreases by a similar percentage when the substrates are free fatty acids. Therefore, the P-to-O ratio falls between 2.83 and 3.17 in practice (25). On the average, 3 is a reasonable P-to-O ratio in blood-perfused hearts under aerobic conditions where the metabolic substrates are physiological mixtures of lactate, glucose, and free fatty acids (25). Table 1 is a numerical listing of the assumptions and parameters of the present Ca²⁺ handling model.

Equation 1 is modified to obtain Ca²⁺ handling VO₂

(2)

where the unit of Ca²⁺ handling VO₂ is micromoles per kilogram, and converts ATP in micromoles per kilogram to VO₂ in micromoles per kilogram.

Solving Eq. 2 for total Ca²⁺ handling yields

(3)

where both total Ca²⁺ handling and Ca²⁺ handling VO₂ have the same dimensions of micromoles per kilogram.

To use a unit of milliliters per 100 grams for Ca²⁺ handling VO₂, we modified Eq. 3 to

(4)

where the unit of total Ca²⁺ handling is micromoles per kilogram and that of Ca²⁺ handling VO₂ is milliliters per 100 grams myocardium (STPD, namely, 0°C, 1 atm, and dry).

Dividing both terms by Ca²⁺ handling VO₂, Eq. 4 became

(5)

where 2,680 is a rounded-off value of 6 × 10⁷/22,400. The left-hand term (total Ca²⁺ handling)/(Ca²⁺ handling VO₂) is the ratio of total Ca²⁺ handling to the VO₂ needed for its handling. We designated this ratio as the "Ca²⁺ handling economy."

Figure 2A shows a family of curves derived from Eq. 5 with N as a parameter. This graph shows that total Ca²⁺ handling cannot be estimated even when both Ca²⁺ handling VO₂ and RF are known unless we know N, which is not directly measurable.

View larger version (31K): [in this window] [in a new window]   Fig. 2.   Theoretical relations among total Ca2+ handling, Ca2+ handling VO2, RF, Emax, R, and N. A: family of curves relating Ca2+ handling economy [(total Ca2+ handling)/(Ca2+ handling VO2)] to RF with N as a parameter, derived from Eq. 5. B: family of curves relating Emax to total Ca2+ handling with R as a parameter, derived from Eq. 6. Arrows 1-8 indicate steps to obtain estimates of unknown variables from known variable values.

However, when we assumed N = 0 in the normal hearts, we could estimate (total Ca²⁺ handling)/(Ca²⁺ handling VO₂) from RF (listed in Table 2), as indicated by the direction and order of arrows 1 and 2 in Fig. 2A. This is nothing but solving Eq. 5 for (total Ca²⁺ handling)/(Ca²⁺ handling VO₂) by substituting the RF value and N = 0 into Eq. 5. We actually obtained (total Ca²⁺ handling)/(Ca²⁺ handling VO₂) by solving Eq. 5 and multiplied it by Ca²⁺ handling VO₂ to obtain the total Ca²⁺ handling listed in Table 2.

The next question was how to estimate N in the ryanodine-treated, failing hearts, in which N was reasonably expected to be nonzero after ryanodine treatment at nanomolar concentrations (31). We introduced a new index of Ca²⁺ sensitivity or responsiveness of the contractile machinery to relate total Ca²⁺ handling to LV contractility (E_max) (28). We could not use the conventional Ca²⁺ sensitivity or responsiveness of the contractile machinery (3), because they already have their own specific definitions and are not applicable to our present study without confusion. These conventional ones refer to mechanical responses relative to sarcoplasmic free Ca²⁺ concentration (3), which we did not measure in the canine hearts subjected to the ryanodine studies (31). In contrast, our new index refers to the mechanical response relative to total Ca²⁺ handling (flux or transport) that we wanted to obtain. We designated the new index as the "reactivity of E_max to total Ca²⁺ handling" and abbreviated it to R. Figure 1, A and B, shows the R as a proportional constant between total Ca²⁺ handling and E_max.

Here, E_max is a mechanical load-independent index of ventricular contractility that Suga et al. proposed in 1973 (28; see also Ref. 26). E_max represents the maximum elastance of the LV chamber at the end of systole or the slope of the LV end-systolic pressure-volume relation line (26, 28). E_max has been considered to quantify the incremental number or amount of attached cross-bridges per unit increment in ventricular volume (26). E_max has been widely used as a practically useful index of contractility to characterize ventricular performance and mechanical energy (15, 25, 26).

By definition, R was given as

(6)

where the unit of total Ca²⁺ handling is micromoles per kilogram and that of E_max is millimeters mercury per milliliter per 100 grams. Therefore, the unit of R is millimeters mercury per milliliter per 100 grams per micromole per kilogram.

Figure 2B shows a family of curves derived from Eq. 6 with R as a parameter. This graph shows that the response of E_max to total Ca²⁺ handling increases as an increasing function of R. This indicates that total Ca²⁺ handling can be uniquely determined from E_max if R is given and vice versa. Inversely, the R value could be determined if both E_max and total Ca²⁺ handling are given.

In the normal hearts, we obtained R values from their E_max and total Ca²⁺ handling (listed in Table 2), the latter having been estimated in Fig. 2A, as the intersection of arrows 3 and 4 in Fig. 2B.

We then assumed that R remained unchanged by ryanodine at nanomolar concentrations, taking advantage of its pharmacological specificity. It selectively fixes the ryanodine-sensitive Ca²⁺ channel of the SR one-half open without affecting the Ca²⁺ sensitivity of the troponin C and Ca²⁺ responsiveness of the contractile proteins (4, 19, 31). Therefore, we obtained total Ca²⁺ handling from the known E_max (as listed in Table 2) using the same R line as indicated by arrows 5 and 6 in Fig. 2B.

Then, this total Ca²⁺ handling was divided by its Ca²⁺ handling VO₂ to obtain Ca²⁺ handling economy on the y-axis of Fig. 2A. Arrows 7 and 8 start from the Ca²⁺ handling economy and RF to obtain N as the intersection of these arrows. In reality, we solved Eqs. 5 and 6 for N by first substituting the same R as the normal R into Eq. 6 and then substituting RF and the obtained total Ca²⁺ handling into Eq. 5.

The method and steps we used to obtain the Ca²⁺ handling-related variables listed in Table 2 are described above. However, if we were not allowed to assume the unchanged R in the failing hearts, Eqs. 5 and 6 could not have been solved for R and N. For such a case, refer to APPENDIX.

Recirculation fraction. We obtained the RF in each contractile state by two different methods. The first method was the conventional one based on using the monotonically decaying PESP (12, 14, 20, 33), and the second one was our recently developed method to use the transient alternans PESP (2, 9, 21, 29). We had shown that RFs obtained by the two different methods were essentially the same in normal hearts (21). We had also shown that the same held either before or after ryanodine (9). However, RFs obtained by the two different methods were significantly decreased equally by ryanodine treatment (9). We utilized these previously obtained RF data (9) in the present study.

(7)

(8)

Ryanodine experiments. We used the standard excised cross-circulated (blood-perfused) canine heart preparation (15, 25). The details of our ryanodine experiments were documented in the original paper (31). Briefly, ryanodine was continuously infused at a constant rate of 1.4 ± 0.5 nmol/min into the cross-circulated heart via the coronary arterial circulation for ~1 h (31). Because coronary flow was 51 ± 21 ml · min¹ · 100 g LV¹, we calculated the intracoronary concentration of ryanodine to be 29 ± 13 nmol/l coronary blood. Note that ryanodine at nanomolar (not micromolar) concentrations did not abolish the Ca²⁺ release function of the SR (19, 31). E_max gradually fell to nearly one-half of control over 1 h. We obtained mechanoenergetics data, including LV pressure and volume, E_max, total VO₂, and unloaded VO₂, in steady-state beats with constant atrial pacing. We found that spontaneous PESP occurred sporadically. Basal metabolic VO₂ was measured under KCl arrest at the end of each experiment. Ca²⁺ handling VO₂ was obtained by subtracting basal metabolic VO₂ from unloaded VO₂ at zero pressure-volume area.

	RESULTS AND DISCUSSION

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To test the feasibility of the present method in cases with N > 0, we applied it to our previous mechanoenergetics data obtained in the ryanodine-treated failing canine LVs (9, 31). Table 2 lists the experimentally obtained mechanoenergetics data (31). The E_max and VO₂ values are the mean ± SD values during control conditions before the ryanodine treatment and in the failing condition after the ryanodine treatment (9, 31).

The RF values in Table 2 are the data that we had calculated from the beat constants (_e) of the exponential decay of the monotonically decaying PESP and the exponential decay component of the transient alternans PESP (9).

We assumed N = 0 in the control contractile state before ryanodine and N > 0 in the depressed contractile state after ryanodine treatment (9, 31). We also assumed that R remained virtually unchanged by the ryanodine treatment at nanomolar concentrations (4, 9, 31). The bases of these assumptions are described in METHODS.

Table 2 lists all the resultant Ca²⁺-related values calculated from the mechanoenergetics and RF data by the present method. Although the RF values obtained from the monotonic and alternans PESPs were not significantly different (9), we combined these RF values separately with the same representative set of mechanoenergetic data as shown by the two PESP-type columns (patterns A and C) in Table 2. The resultant total Ca²⁺ handling values for both RF values are comparable either before or after ryanodine. The resultant total Ca²⁺ handling values were considerably smaller after ryanodine than before. Moreover, these total Ca²⁺ handling values in both control and ryanodine-treated hearts fell within the physiological range (20-100 µmol/kg) documented in the literature (1, 3, 6, 8, 10, 18, 22-24).

We obtained total Ca²⁺ handling of 56 (from pattern A) and 60 (from pattern C) µmol/kg at a baseline E_max of 4.2 mmHg · ml¹ · 100 g. This E_max value is approximately one-third to one-fourth of the maximum E_max with high doses of intracoronary Ca²⁺ or catecholamines (15, 28). Considering the maximum Ca²⁺ binding capacity of 20-100 µmol/kg of intramyocardial sites including troponin C and calmodulin (3, 6, 10, 16, 22, 23), the ~60 µmol/kg of Ca²⁺ could be interpreted as a rather high value for the baseline E_max.

However, our present Ca²⁺ handling values are dynamic values calculated on a per-beat basis, whereas the apparently maximum Ca²⁺ binding capacity values were biochemically determined under steady-state conditions. The rate constants of Ca²⁺ binding to troponin C and the other Ca²⁺ binding sites (18, 22) indicate that Ca²⁺ binding to them is not immediate after the Ca²⁺ release. This suggests that part of the released Ca²⁺ may be removed before effective binding to the Ca²⁺ binding sites. Our simulation study of Ca²⁺ kinetics, similar to that of Robertson et al. (18), has shown that a significant fraction (20-30%) of the total released Ca²⁺ has been sequestered by the SR by the time of one-half peak force development (13). Therefore, our total Ca²⁺ handling values could be considered reasonable if part of the total released Ca²⁺ is removed by the SR Ca²⁺ pump immediately after its release and is not used effectively for E-C coupling (3, 7).

The high-affinity Ca²⁺ binding sites (50-70 µmol/kg, 2 Ca²⁺ per 1 troponin C molecule) of troponin C (25-35 µmol/kg) seem to be almost fully saturated even at a low diastolic free Ca²⁺ level (3, 6, 10, 23). Therefore, the total Ca²⁺ handling we obtained appears to be related to the beat-to-beat changes in the amount of Ca²⁺, which are proportional to contractile force development in excess of the stably bound amount of Ca²⁺ independent of the twitch force development. The beat-to-beat change in Ca²⁺ includes that bound to the low-affinity Ca²⁺-specific sites (25-35 µmol/kg, 1 Ca²⁺ per 1 troponin C molecule) of troponin C as well as other Ca²⁺ bound to calmodulin and all other Ca²⁺ binding sites on a per-beat basis (3, 7).

Because we assumed the same R after ryanodine (see Table 2) as that before ryanodine, the resultant total Ca²⁺ handling was proportional to E_max before and after ryanodine (Fig. 2B). Despite the considerably decreased total Ca²⁺ handling from 56-60 to 31-33 µmol/kg after ryanodine, Ca²⁺ handling VO₂ was only little decreased after ryanodine, as seen in Table 2. This disproportionately high value for Ca²⁺ handling VO₂ in the ryanodine-treated heart is now accounted for by the combination of the significantly decreased RF and markedly increased N despite the nearly halved total Ca²⁺ handling associated directly with E-C coupling.

The present method has several limitations. First of all, the present model may appear too simple to calculate accurately total Ca²⁺ handling because of the numerous assumptions incorporated into the model. However, we have included the major ATP-consuming processes of Ca²⁺ handling and neglected minor ones on the basis of the contemporary knowledge (1, 3, 4, 8, 9, 11, 12, 17, 19, 22, 29, 30). Although the new knowledge of other minor Ca²⁺ handling processes is available (3, 11, 22), it cannot readily be incorporated into our model, because it was obtained by reductionistic methodology in isolated, but not in situ, myocytes of animals other than dogs. For example, we neglected the contribution of the reverse mode of the Na⁺/Ca²⁺ exchanger to the transsarcolemmal Ca²⁺ entry according to its relatively small contribution in normal rabbit myocytes (3). However, this contribution is yet unknown in canine hearts. Therefore, even if we adopt reductionistic knowledge obtained in rabbit myocytes into the present model, this may not guarantee improvement of the model. Nevertheless, the present study warrants further efforts to improve the integrative approach toward better understanding of myocardial Ca²⁺ handling by integrative methods.

Despite this limitation, we would consider that our estimates of total Ca²⁺ handling in Table 2 appear reasonably realistic in the light of the literature data (1, 3, 6, 8, 10, 18, 22-24). Any of Eqs. 1-5 indicates mathematically that a relatively small error in RF, N, and Ca²⁺ handling VO₂ due to our neglect of minor Ca²⁺ handling processes would produce a comparably small error in total Ca²⁺ handling.

In relation to this limitation, we assumed N = 0 in the normal hearts before the ryanodine treatment and the same R value before and after ryanodine to estimate total Ca²⁺ handling. As described in Formulation and APPENDIX, if we had not used these assumptions, we could not have obtained estimates of N, R, and total Ca²⁺ handling values after ryanodine treatment. However, if one were satisfied with mechanoenergetic differentiation between normal and failing hearts, the assumptions for N and R are not required. Then, comparison of the R-N relation between them would be a very helpful new method (Fig. 3C).

View larger version (21K): [in this window] [in a new window]   Fig. 3.   Theoretical relations among RF, R, and N. A: family of curves relating N to R with RF as a parameter at a constant O2 cost of Emax (Ca2+ handling VO2/Emax) set to a representative value of 0.003 ml O2 · beat1 · mmHg1 · ml · 100 g1. B: family of curves relating N to R with O2 cost of Emax as a parameter at a given RF set to a representative value of 0.6. C: 2 pairs of R-N relations for normal hearts in control contractile state before ryanodine treatment at nanomolar concentrations and for ryanodine-treated, failing hearts. A and C stand for patterns A and C of decay pattern of postextrasystolic potentiation. All these relations were derived from Eq. 11. Arrows 9 and 10 indicate steps to obtain estimate of unknown N from assumed R in ryanodine-treated hearts.

Second, we used mechanoenergetics and PESP data documented only in our previous studies (9, 31) to test the feasibility of our new method. We had already applied the part with N = 0 to a baseline E_max in a different group of normal hearts (29). There, the representative data were E_max of 4.7 mmHg/ml, Ca²⁺ handling VO₂ of 0.011 ml O₂/beat, both per 100 grams, and RF of 0.55. All these values are close to the present control data listed in Table 2. N was assumed to be 0 in the normal hearts in control and enhanced contractile states with Ca²⁺ and epinephrine (29) as in the normal hearts in control contractile state before ryanodine in the present study. Then, the calculated total Ca²⁺ handling was 40 µmol/kg, which is of the same order as the present control value for total Ca²⁺ handling. This similarity among different groups of normal hearts supports the feasibility of the present method for the hearts with N = 0. However, the feasibility of the formula with N > 0 was tested for the first time in this study. We must admit that the present method remains to be tested using various types of failing hearts whose Ca²⁺ handling processes are abnormal, although some limitation remains, as explained in APPENDIX.

A third limitation would be our assumption that residual cross-bridge cycling does not contribute to the Ca²⁺ handling VO₂. Although residual cross-bridge cycling appeared to contribute considerably to unloaded VO₂ in rabbit hearts (34), we have obtained evidence supporting the view that residual cross-bridge cycling energy, if any, is negligibly small in the Ca²⁺ handling VO₂ of canine blood-perfused hearts (32).

Other limitations are that we only used average RF and mechanoenergetics data to calculate total Ca²⁺ handling and that the intracoronary ryanodine concentrations were, at most, 40 nmol/l. Because we only used PESPs following spontaneous extrasystoles, which occurred sporadically, we could not deal with a larger number of matched sets of mechanoenergetics and RF data. Therefore, we had to use the average data as listed in Table 2 as representative values. As for the ryanodine concentration, E_max gradually decreased to one-half of the control over 1 h during continuous intracoronary infusion of ryanodine at 1.4 nmol/min (31). E_max was already low enough to judge the heart to be in a failing state (31). Moreover, end-diastolic pressure started to rise in isovolumic contractions at a fixed intermediate volume (31). We could not obtain any stable data thereafter. Because of these limitations, future controlled studies are warranted.

In conclusion, our present heart-level method may contribute to a better understanding of total Ca²⁺ handling in ryanodine-treated, failing hearts characterized by futile Ca²⁺ cycling. We consider this to be so because no better methods are yet available to achieve the same goal. When using this method, one must realize the limitations remaining to be overcome. The utility of this integrative analysis method may be increased by conquering these limitations.