INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

THE MECHANISM OF Ca²⁺-contraction coupling in cardiac muscle is attributed in large part to changing myoplasmic Ca²⁺ concentration ([Ca²⁺]_m) and myofilament sensitivity to Ca²⁺. Modeling the cyclic relationship between left ventricular (LV) pressure (LVP) and [Ca²⁺]_m in intact beating hearts may help to elucidate the impact of regulatory proteins and kinetics of actin and myosin cross-bridge interaction on Ca²⁺-contraction coupling. It is well known that the relationship between [Ca²⁺]_m and force is complicated by the various cellular mechanisms that alter Ca²⁺ handling. Previous model (14, 23, 27, 32, 38, 40, 53-55) either measured or used [Ca²⁺] and force recordings from isolated myocardial fibers or skinned cardiac muscle strips collected from various species. These preparations were typically studied at room temperature and stimulated to contract at slow rates. Although an isolated muscle strip is simpler to validate, the intact heart model allows a more physiological assessment of Ca²⁺-contraction coupling and cross-bridge kinetics at body temperature and spontaneous heart rate.

Our first goal was to determine whether a four-state mathematical model described previously (4, 10, 43) (Fig. 1) reproduces the dominant features of Ca²⁺-contraction coupling from the relationship between [Ca²⁺]_m and developed LVP in guinea pig isolated, spontaneously beating, and normothermic hearts. The model is based on interactions between actin and myosin for cross-bridge formation and includes binding of Ca²⁺ to troponin C (TnC) on the actin myofilament (TnCA) (32, 40, 53, 55) while assuming rapid switching of tropomyosin between permissive and nonpermissive states (4, 10, 12, 38, 43). In contrast to previous studies that used aequorin bioluminescence, we measured [Ca²⁺]_m using the fluorescent probe indo 1. These two techniques result in different Ca²⁺ transient shapes, which may impact on model predictions and interpretations.

View larger version (13K): [in this window] [in a new window]   Fig. 1.   Block diagram of biochemical model relating the input/output relationship between myoplasmic [Ca2+] ([Ca2+]m) and left ventricular pressure (LVP) adapted from Baran et al. (4) and Shimizu et al. (43). The four-state model is governed by five differential equations (see APPENDIX) along with model initial conditions. TnCA, troponin C (TnC) protein molecule on the actin (A) myofilament; M, myosin head; +, weak bonds; · , strong bonds. The sequence of events from phasic [Ca2+]m to contraction are as follows: Ca2+ binds to TnCA, tropomyosin shifts so M and A can bind forming an actinomyosin cross bridge, Ca2+ dissociates from TnCA with cross-bridge attached, and finally the cross-bridge breaks. Model parameters are rate constants (see Table 1); K1 and K3 are measures of Ca2+-binding affinity of myofilaments before cross-bridge formation, and K2 and K4 are measures of Ca2+-binding affinity of myofilaments in the presence of an attached cross bridge. Ka and Kd are measures of cross-bridge association and dissociation rates, respectively, in the presence of bound Ca2+, and Kd' is a measure of the cross-bridge dissociation rate in the absence of bound Ca2+. Model rate constants and their units are indicated by their sites of action. Note that A and M cannot form cross bridges in the absence of Ca2+; however, because this is a loose coupling model, once a cross bridge has been formed it no longer needs associated Ca2+ to remain attached.

We (1, 2, 11, 44, 52) have reported that hypothermia and ischemia-reperfusion injury result in dissociation between Ca²⁺ and contraction and relaxation. Our second goal was to determine whether this four-state model is also capable of describing imposed dissociations in Ca²⁺-contraction coupling induced by hypothermic perfusion and warm and cold ischemia-reperfusion injury in the isolated heart preparation. We used the model to predict LVP in response to [Ca²⁺]_m during 17°C perfusion, after short-term 37°C ischemia, and after long-term 17°C ischemia. We postulated that changes in the modeled rate constants will help to interpret altered contractile function after cold versus warm ischemia due to changes in myofilament Ca²⁺ handling and cross-bridge kinetics.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Langendorff heart preparation. The investigation conformed to the National Institutes of Health Guide for the Care and Use of Laboratory Animals (NIH Publication No. 85-23, Revised 1996). Prior approval was obtained from the Medical College of Wisconsin and Marquette University Animal Studies Committees. The preparation of guinea pig hearts at our laboratory has been described in detail (1, 15, 47, 52). Briefly, 30 mg of ketamine and 1,000 units of heparin were injected intraperitoneally into albino English short-haired guinea pigs (n = 16) 15 min before the animals were decapitated. The animals were euthanized after they became unresponsive to noxious stimulation. After thoracotomy, the inferior and superior vena cavae were cut, and the aorta was cannulated distal to the aortic valve. Each heart was immediately perfused via the aortic root with cold oxygenated modified Krebs-Ringer (KR) solution (equilibrated with 97% O₂-3% CO₂) at an aortic root perfusion pressure of 55 mmHg, and the heart was then rapidly excised. The KR perfusate (pH 7.39 ± 0.01, PO₂ 620 ± 10 mmHg) was filtered (5 µm pore size) in-line and had the following calculated composition (nonionized, in mM): 137 Na⁺, 5 K⁺, 1.2 Mg²⁺, 2.5 Ca²⁺, 134 Cl, 15.5 HCO, 1.2 H₂PO, 11.5 glucose, 2 pyruvate, 16 mannitol, 0.05 EDTA, and 0.1 probenecid and 5 U/l insulin. Perfusate and bath temperatures were maintained at 37.2 ± 0.1°C with the use of a thermostatically controlled water circulator.

Measurement of myoplasmic free Ca²+ in intact hearts. Experiments were carried out in a light-blocking Faraday cage. The heart was partially immobilized by hanging it from the aortic cannula, the pulmonary artery catheter, and the LV balloon catheter. The heart was immersed continuously in the bath at 37°C. The distal end of a trifurcated fiber silica fiber-optic cable (optical surface area 3.85 mm²) was placed gently against the LV epicardial surface through a hole in the bath. A rubber O-ring was placed over the fiber optic tip to seal the hole and netting was applied around the heart for optimal contact with the fiber-optic tip. This maneuver did not affect LVP. Background autofluorescence was determined for each heart after initial perfusion and equilibration at 37°C.

Experimental protocol and data collection. Both normothermic and hypothermic experimental protocols (Fig. 2) began with a 30-min stabilization period, during which initial background data was obtained at 37°C in each heart. Control data were obtained 30 min after the dye washout and before global ischemia. Eight hearts were subjected to 30 min of global ischemia at 37°C, followed by a 2-h, 37°C reperfusion period (group A). We reported that this resulted in an infarct size of 52 ± 3% of total ventricular weight (2). Eight other hearts were cooled to 17°C and subjected to 240 min of global ischemia at 17°C, followed by a 2-h 37°C reperfusion period (group B). This resulted in an infarct size of 36 ± 3% (11). Thus there were five experimental conditions: 1) 37°C control, 2) 17°C control, 3) 17°C perfusion, 4) 37°C ischemia-reperfusion, and 5) 17°C ischemia-reperfusion. Perfusate and bath were maintained at 37°C by a heated water circulator and at 17°C by a parallel-refrigerated water circulator. The order of experiments was randomized. At the end of each experiment, 100 µM MnCl₂ was infused for 10 min to quench the myoplasmic indo 1 Ca²⁺ signal so that the resulting signal consisted only of the nonmyoplasmic fraction. This allowed for back correction of the nonmyoplasmic component from the total cell Ca²⁺ transients, as described in the APPENDIX. All analog signals were digitized (PowerLab/8 SP, ADInstruments; Castle Hill, Australia) and recorded at 125 Hz (Chart and Scope version 3.63, ADInstruments) on Power Macintosh G4 computers (Apple; Cupertino, CA) for later analysis using MATLAB (MathWorks; Natick, MA) and Excel software (Microsoft; Redmond, WA).

View larger version (10K): [in this window] [in a new window]   Fig. 2.   Experimental protocols for normothermic (A; group A) and hypothermic (B; group B) ischemia (I)-reperfusion (R) model. Reported results are for the five conditions shown in italics. These conditions are 1) 37°C control, 2) 17°C control, 3) 17°C perfusion, 4) 37°C ischemia-reperfusion, and 5) 17°C ischemia-reperfusion. Data for conditions 4 and 5 are at 30 min into warm reperfusion.

View larger version (35K): [in this window] [in a new window]   Fig. 3.   A: group A; B: group B. Plot of simultaneously recorded [Ca2+]m and isovolumic LVP (LVPexp) under several conditions. Diastolic [Ca2+] events were detected using a threshold-blanking period algorithm and the points were used as time markers for creating the averaged [Ca2+] and LVP transient. Note the longer time scale for 17°C perfusion.

Computing averaged [Ca²+]_m and LVP transients. Times of occurrence of peak diastolic [Ca²⁺]_m were obtained over the 2.5-s recordings with the use of a simple threshold and blanking period event-detection algorithm triggered on the filtered [Ca²⁺]_m signal. The amplitude threshold was set at 65% of the minimum signal amplitude, and local minima were detected in a 144-ms window after threshold crossing. These local minima corresponded to the diastolic Ca²⁺ for each cardiac cycle. The Ca²⁺ and simultaneously obtained LVP signals between each consecutively detected Ca²⁺ diastolic point were aligned and averaged on a point-by-point basis to form the averaged Ca²⁺ and LVP transient signals. Cycle length, heart rate, and time-dependent indexes of LVP and [Ca²⁺]_m were recorded for use in data analysis and modeling.

Description and assumptions of mathematical model. The mathematical model applied to our data is based on the interaction between TnCA and myosin, in the presence of Ca²⁺, for cross-bridge formation and force development (4, 10, 43). The model (Fig. 1) consists of four stages and is governed by five differential equations, as described in the APPENDIX. The rate constants are described in Table 1. LVP, as predicted by the model (LVP_mod), is proportional to the number of cross-bridges formed: [Ca · TnCA · M] + [TnCA · M], where A · M is the actin and myosin cross-bridge (4). Chemomechanical coupling in cardiac muscle includes a positive feedback mechanism, termed cooperativity (32), which is responsible for the rise in force. There are various hypotheses for the mechanism of cooperativity, including the effect of Ca²⁺ binding to TnCA on myofilament Ca²⁺ sensitivity, effect of cross-bridge formation on the rate of formation of neighboring cross-bridges, and effect of Ca²⁺ binding to TnCA on neighboring tropomyosin units (32, 38, 40). In accordance with Baran et al. (4) and Shimizu et al. (43), we accounted for cooperativity by allowing K₁ and K_a to vary according to the following functions, as described in Table 1

where  and  are adjustable second-order rate constants and t refers to time. The -parameter represents the slopes of the K₁ and K_a curves and is a measure of the sensitivity of the cooperative mechanism; an increase in  represents a stronger positive biochemical feedback. The -parameter represents the static value of K₁ and K_a at 0 LVP; an increase in  indicates an increase in the basal value of either parameter.

Model implementation. The model was executed with the use of customized software developed in MATLAB and evaluated solely from data we obtained in guinea pig isolated hearts, where [Ca²⁺]_m was measured with the use of the fluorescent dye indo 1 and isovolumetric LVP (LVP_exp) was measured using a transducer connected to a saline-filled latex balloon placed in the LV. The governing differential equations were solved numerically using fourth-order Runge-Kutta with a 0.4-ms step size and the appropriate initial conditions (10, 43), as shown in the APPENDIX. Experimentally measured and averaged Ca²⁺ transients were the inputs to our model. Model rate constants were optimized using commercially available algorithms based on constrained quasi-Newton methods that guarantee linear convergence in MATLAB and were estimated to minimize the root-mean-square error between LVP_mod and LVP_exp at the sampled time points
Several constraints were imposed on the model rate constants during optimization: 1) to ensure a positive feedback, ₁ and _a must be >0, 2) K'_d must be greater than K_d because cross bridges dissociate more readily in the absence of attached Ca²⁺; K'_d accounts for the physiological difference between contraction and relaxation kinetics (4), and 3) the maximum rate of Ca²⁺ binding to TnCA with attached cross bridges must be greater than the maximum rate of Ca²⁺ binding to TnCA with no attached cross bridges (K₂ > K₁); this concept incorporates the idea of a positive feedback mechanism to explain the delay in rise in LVP during contraction (25).

Statistical analysis. All experimental measurements and model rate constants are expressed as means ± SE. All experimental observations and model rate constants computed during 17°C perfusion and after 37°C and 17°C ischemia-reperfusion injury were compared with the 37°C baseline control by one-way analysis of variance for repeated measures, followed by Tukey's comparison of means post hoc test (MINITAB statistical software version 13.3, Minitab; State College, PA). Differences among means were considered statistically significant at P < 0.05 (two-tailed).

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Figure 3 shows representative tracings of simultaneously obtained LVP_exp and [Ca²⁺]_m under control, 17°C perfusion, and during 37°C reperfusion after warm and cold ischemia. Note the different time scales for cold perfusion data compared with warm perfusion and reperfusion data after warm and cold ischemia. Detected diastolic Ca²⁺ points are shown for control data. There was beat-to-beat variability in Ca²⁺ transient morphology after warm or cold ischemia-reperfusion injury compared with before ischemia. From Tables 2 and 3, there were no significant differences in experimental observations or model rate constants at 37°C between the two temperature control groups, i.e., before a change in temperature and ischemia-reperfusion.

Hypothermic perfusion. Figure 4 displays a typical plot of the coupling between [Ca²⁺]_m and LVP_exp during 37°C perfusion and during 17°C perfusion for one heart. LVP_mod is plotted as a function of [Ca²⁺]_m and is represented by the solid line for both data. The model described the Ca²⁺-contraction coupling with an error of 3.1 ± 0.4% between LVP_exp and LVP_mod at 37°C and an error of 3.4 ± 0.6% at 17°C perfusion. As seen in Figs. 3 and 4, and listed in Table 2, hypothermia markedly reduced heart rate by eightfold while markedly increasing diastolic LVP and systolic and diastolic [Ca²⁺]_m. Cold perfusion resulted in a marked slowing in contraction and relaxation rates as noted from the reduced values of the maximal and minimal rates of pressure increase and decrease over time, respectively, compared with control.

View larger version (19K): [in this window] [in a new window]   Fig. 4.   Plot of sample tracings of LVP vs. [Ca2+]m averaged over all beats in the 2.5-s recordings during normothermic (37°C) and hypothermic (17°C) perfusion. Experimental data are represented by open symbols and the model fit is represented by a smooth line.

View larger version (20K): [in this window] [in a new window]   Fig. 5.   Plot of the cooperative rate constants (see Table 1) K1 (A) and Ka (B) as a function of the total number of estimated cross bridges ([Ca · TnCA · M] + [TnCA · M]) during controls (groups A and B), 17°C perfusion (group B), normothermic reperfusion after warm ischemia (group A), and normothermic reperfusion after cold ischemia (group B). Note the similarity in estimated values of K1 and Ka for control in groups A and B and reperfusion in group B. In contrast, note the flat curves of K1 and Ka for the 17°C perfusion in group B and after warm short-term ischemia in group A, which indicate a loss of sensitivity in the two cooperative mechanisms.

View larger version (37K): [in this window] [in a new window]   Fig. 6.   Plot of model-predicted changes in the various contractile elements over one cardiac cycle. Note the eightfold longer time scale for cold perfusion (right) than for warm perfusion (left).

Reperfusion after normothermic ischemia and hypothermic ischemia. Figure 7 shows the cyclic relationship between [Ca²⁺]_m and LVP_exp during 1) 37°C control, 2) 30 min after 37°C global ischemia, and 3) 30 min after 4-h 17°C global ischemia. LVP_mod as a function of [Ca²⁺]_m is shown in solid lines for all data sets. The model successfully described the cyclic Ca²⁺-contraction relationship with errors of 3.1 ± 0.4%, 2.4 ± 0.3%, and 3.2 ± 0.3% between LVP_exp and LVP_mod for all three conditions, respectively. Normothermic perfusion for 30 min after 37°C ischemia-reperfusion injury did not change heart rate or phasic [Ca²⁺]_m, but it depressed systolic LVP and elevated diastolic LVP markers of contractile dysfunction compared with 37°C and 17°C baseline values. In addition, rates of contraction and relaxation were slowed after short-term warm ischemia-reperfusion injury.

View larger version (18K): [in this window] [in a new window]   Fig. 7.   Plot of sample tracings of LVP vs. [Ca2+]m averaged over all beats in the 2.5-s recordings during normothermic (37°C) and hypothermic (17°C) perfusion. Experimental data are represented by open symbols and the model fit is represented by a smooth line.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

The main findings of this study were 1) the four-state model with cooperativity is capable of interpreting actinomyosin cross-bridge kinetics and myofilament Ca²⁺ handling in guinea pig isolated hearts from the instantaneous relationship between indo 1-measured [Ca²⁺]_m and LVP under normothermic, nonischemic conditions, 2) the model predicts a marked decrease in association and dissociation rates between actin and myosin heads and reduced myofilament Ca²⁺ binding during hypothermic perfusion with a markedly slowed heart rate, and 3) the model predicts much better preservation of cooperativity in both cross-bridge kinetics and myofilament Ca²⁺ handling after long-term cold ischemia-reperfusion injury than after short-term warm ischemia-reperfusion injury despite much higher [Ca²⁺]_m on reperfusion after cold ischemia.

Mathematical modeling of cross-bridge kinetics from experimental data. Cardiac muscle cells contract and relax rhythmically and synchronously due to the cyclic influx and efflux of Ca²⁺ ions into the intracellular space (4). Changes in myocardial contractile force depend on changes in [Ca²⁺]_m, changes in the responsiveness of the myofibrils for a given [Ca²⁺]_m, or a combination of both (19, 39, 51). The contractile performance of cardiac muscle is dependent on the amplitude of the Ca²⁺ transient (upstream mechanism), the affinity of TnCA for Ca²⁺ (central mechanism), and the response of the actin and myosin myofilaments to occupancy of the Ca²⁺-binding sites on TnCA (downstream mechanism) (5, 39, 51). The upstream mechanism is limited by a plateau in the maximal contractile effort for increasing [Ca²⁺]_m (44). It is difficult to distinguish among combinations of the three mechanisms that together are responsible for altering the relationship between Ca²⁺ and the contractile effort.

Effects of hypothermic perfusion on model kinetics. It is well documented that mild hypothermia (22°-30°C) has a positive inotropic effect. We have shown that this rise in contractile force is accompanied by a rise in [Ca²⁺]_m in the isolated heart (11). The mechanism of this increased contractility is not clearly understood (30, 42). The hypotheses for this effect are prolonged duration of excitation and contraction, slowed cross-bridge cycling rates, and tissue alkalosis. Hypothermia may also cause increased myofilament Ca²⁺ loading via inhibited sarcolemmal Ca²⁺ pump, inhibited Na⁺/Ca²⁺ exchange, and slowed Ca²⁺-induced Ca²⁺ release and Ca²⁺ reuptake by the sarcoplasmic reticulum (17, 31, 35, 42, 48). Previous studies (20, 24) present contradictory results on the effects of hypothermia on myofilament Ca²⁺ sensitivity in isolated myofibrils. Our results from paced isolated hearts indicate that, whereas there is an increase in myofilament Ca²⁺ sensitivity at 27°C, at 17°C contractility and relaxation are both impaired despite a marked increase in phasic [Ca²⁺]_m (44).

Effects of normothermic ischemia-reperfusion injury on model kinetics. Reperfusion after ischemia can result in reversible injury, such as stunning, or irreversible myocardial damage, such as infarction. Ischemia-reperfusion injury is well known to cause a complex biochemical dissociation in the coupling of Ca²⁺ cycling to the generation of pressure and work (14, 52). Ischemia-reperfusion injury may occur in patients with coronary artery disease or after surgical procedures like coronary angioplasty, angiography, coronary artery bypass grafting, and cardiac valve replacement. Ischemia-reperfusion injury may result in dilated cardiomyopathy, myocardial infarction, lethal arrhythmias, valvular dysfunction, necrosis, and apoptosis (26). During cardiac ischemia, there is a gradual increase in diastolic pressure due to decreased compliance associated with the decline in ATP (49).

Effects of hypothermic ischemia-reperfusion injury on model kinetics. Hypothermia is the most cardioprotective mechanism against ischemia. Moderate hypothermia prolongs the time to stunning or permanent damage. Mild and moderate hypothermia is widely used to protect hearts during open heart surgery. As temperature is lowered, ischemic time before damage occurs is lengthened; however, hypothermia of increasing duration also has deleterious effects on Ca²⁺ handling and contractility (13, 15, 16, 44-47, 50). We have previously explored the altered association between cyclic Ca²⁺ fluctuations and contraction and relaxation in intact hearts during mild (27°C) and moderate (17°C) hypothermia (47) and after ischemia-reperfusion injury (1, 2, 15, 52). Protection afforded by hypothermia during ischemia includes better tissue perfusion, improved metabolic function, fewer reperfusion dysrhythmias, and reduced infarct size on reperfusion (15, 16).

Justification of model and techniques. Our results indicated that this four-state model adapted from the work of Baran et al. (4) and Shimizu et al. (43) was able to link observed changes in Ca²⁺-contraction coupling in the intact heart to predictable changes in myofilament Ca²⁺ association and cross-bridge formation under physiological and pathological conditions. However, there are differences between our experimental techniques, data, and predicted model parameters and those of previous investigations that must be considered.