Introduction

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Previous reports have documented experimental data regarding the myocardial kinetics of meperidine as determined by measurements of the product of its arteriovenous concentration difference across the heart and myocardial blood flow after bolus administration in sheep (Huang et al., 1994a). In the same experiments, it also was shown that meperidine [at doses that are sufficiently low to avoid overt central nervous system (CNS) stimulation] caused transient reductions in myocardial contractility (Huang et al., 1994b). This was taken to be a direct effect on the heart, as reductions in contractile performance also have been documented in isolated heart tissues without CNS innervation (Rendig et al., 1980) and appears to be related to the action of meperidine on calcium currents rather than opioid receptors (Wu et al., 1997).

As part of developing a physiological model of meperidine disposition, these previously published data were examined to determine an optimal model of the myocardial kinetics of meperidine and its dynamic effect on myocardial contractility. The general strategy used first was to examine the kinetic-dynamic relationship by using model-independent methods based on the analysis of hysteresis loops. Previously, this approach has been used for lidocaine in the heart, with mass balance principles (Upton et al., 1988) used to calculate the lidocaine concentration in the myocardium. It was found that the reduction in myocardial contractility caused by lidocaine was better related to this myocardial concentration than to its concentration in either arterial or coronary sinus (effluent from the heart) blood (Huang et al., 1993). Second, an optimal dynamic model (e.g., linear, maximum effect (E_max), or sigmoid E_max) was determined by directly fitting the observed reductions in myocardial contractility to the calculated myocardial concentrations, a method that required no assumptions about the underlying myocardial kinetics of meperidine. Third, the myocardial kinetics were studied in detail by determining the best fit to the data of a variety of one-, two-, and three-compartment models of myocardial kinetics (including a traditional single flow-limited compartment and a traditional membrane-limited model). These were also combined with the optimal dynamic model to produce a variety of kinetic-dynamic models. The optimal models were chosen based on hybrid modeling of the time courses of the coronary sinus concentrations of meperidine and changes in contractility together with deductions from the model-independent analysis.

There are some outstanding issues with respect to the modeling of myocardial kinetics and dynamics. It is not clear from the present literature whether multicompartment models of the myocardium can be resolved from data regarding the arteriovenous concentration difference across the myocardium, as collected previously for meperidine. Furthermore, it is not known whether a more realistic description of myocardial kinetics can be achieved by adding the extra compartments in series to represent dispersion (Roberts et al., 1988) or by adding the extra compartments as "peripheral" or "deep" compartments as used in traditional membrane-limited models (Gerlowski and Jain, 1983). Finally, when considering the relationship of the model to simultaneous effect measurements in the heart, it is clear that the optimal model is one in which at least one of the compartments is in pseudoequilibrium with the site of drug action responsible for the effect. Clearly, increasing the number of compartments used to describe the myocardium provides greater flexibility when constructing such a combined kinetic-dynamic model. The aims of the present study were to explore these issues by examining our previously published data on the myocardial kinetics and dynamics of meperidine. The specific aims were to:

1. determine whether the time course of the effect of meperidine on myocardial contractility was better related to the time courses of its concentration in arterial blood or coronary sinus blood or its calculated mean concentration in the myocardium;

2. determine the optimal dynamic model describing the relationship between the calculated myocardial concentration of meperidine and changes in myocardial contractility;

3. determine whether data regarding the arteriovenous concentration difference of meperidine across the myocardium have sufficient information to distinguish between a variety of compartmental models of myocardial kinetics and, if so, determine which model is optimal for describing the data; and

4. determine whether such models also can describe the effects of meperidine on myocardial contractility.

The general principles elucidated here may be applicable to other models of drug kinetics and dynamics in individual organs.

	Materials and Methods

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Data Source

The data on the effect of meperidine on myocardial contractility and other cardiovascular parameters (Huang et al., 1994b) and on its blood and myocardial pharmacokinetics (Huang et al., 1994a) after bolus i.v. injection in adult female sheep have been reported previously. In this paper, the relationship between myocardial contractility and myocardial kinetics is examined for the first time. A summary of the study design and the experimental methods relevant to this paper is presented here for completeness.

Adult Merino sheep were prepared under anesthesia with chronic intravascular catheters in the aorta (for sampling afferent blood to the heart), the coronary sinus (for sampling pure efferent blood from the heart once the hemiazygous vein is ligated), and the right atrium (for drug administration). An ultrasonic Doppler flow probe was placed on the left main-stem coronary artery to measure myocardial blood flow and was calibrated ex vivo. A pressure transducer-tipped catheter was acutely placed in the left ventricle (LV) on experimental days via a chronic introducer catheter, and the pressure trace was integrated digitally to give the maximum positive rate of change of LV pressure (LV dP/dt_max), an index of myocardial contractility.

On an experimental day, conscious sheep were administered 100-, 200-, or 300-mg i.v. doses of meperidine over 1 s. Myocardial blood flow and LV dP/dt_max were recorded continuously for the next 10 min and at 15 min, while paired arterial and coronary sinus blood samples were taken at intervals of as short as 5 s for 15 min after the dose. Meperidine blood concentrations were determined by using a gas chromatographic method. The myocardial concentrations of meperidine were calculated by using mass balance principles (Huang et al., 1994a; Upton, 1994). By this method, the mass of meperidine in the myocardium at a given time is the integral of the difference in the flux of meperidine entering the myocardium [i.e., myocardial blood flow (Qh) times the arterial meperidine concentration, C_art] and the flux leaving the myocardium (i.e., Qh times the coronary sinus meperidine concentration, C_cs). The myocardial concentration (C_myo) is this mass divided by the mass of the myocardium perfused by the left main coronary artery (M_myo), estimated previously (Huang et al., 1994a):
Only the 100-mg data were analyzed in the present paper because the higher doses produced CNS agitation and sympathetic stimulation, which masked the underlying myocardial depressant effect of meperidine at clinically relevant doses (Huang et al., 1994b). Four sheep were studied.

Data Analysis

Hysteresis Analysis. For the model-independent analysis, the time course of changes in LV dP/dt_max (expressed as percent reduction from baseline) were plotted against the time courses of the arterial, coronary sinus, and calculated myocardial concentrations of meperidine. To quantify the hysteresis, each plot was divided into two sections at the point of the maximum meperidine concentration, with one section representing concentration-effect relationships when meperidine concentrations were increasing and the other representing the relationships when the meperidine concentrations were decreasing. The area under the curve (AUC) of each section of the concentration-effect loop were calculated by using the trapezoid rule. Concentration-effect hysteresis was considered to be present when the differences between the AUC of these two sections of the hysteresis loop were statistically different. The times of the peak values of the concentrations and percent reduction in contractility also were compared.

Optimal Dynamic Model. To determine the optimal dynamic model, the relationship between the calculated concentration of meperidine in the myocardium and the contractility data was analyzed directly by using Scientist for Windows (Micromath Scientific Software, Salt Lake City, UT). The fit of linear, linear with a threshold, E_max, and sigmoid E_max dynamic relationships to these data were examined by using a least-squares method. The equations for all but the linear with a threshold models are in common usage and have been reported previously by our laboratory (Huang et al., 1998). The equations of the threshold model (in a form common to many programming languages) were as follows, where C_myo is the calculated myocardial concentration, "dpdt" is the percent reduction in LV dP/dt_max from baseline, and "dpdtemp" is a temporary variable:

For the least-squares curve-fitting, the best fit was determined as that with the highest model selection criteria (MSC) and without nonidentifiable parameters. The MSC is essentially the inverse of the Akaike Information Criterion scaled to compensate for data sets of different magnitudes (Scientist for Windows manual; Micromath Scientific Software) and is calculated from the following formula:
where w_i is a weighting term and p is the number of parameters. The higher the value of the MSC, the better the fit. Note that this term adjusts for the number of parameters of a model, so that models with more parameters do not necessarily produce a higher MSC. No weighting was considered necessary because there was no evidence that the data were heteroscedastic.

Optimal Kinetic and Combined Kinetic-Dynamic Models. Models of myocardial kinetics with a maximum of three compartments were considered. To illustrate the possible configurations, consider methods for improving the fit of a single compartment model by adding a second compartmenttwo compartments either may be arranged "in series" to improve the description of the dispersion process (Roberts et al., 1988) or as a more traditional "membrane-limited" model to account for deep distribution (Gerlowski and Jain, 1983). When three compartments are considered, it is clear that a three-"tank in series" model, a "double" membrane-limited model, and other arrangements that combine the features of both of these extremes are possible. The resultant nine kinetic models are shown in Figs. 1, 2, and 3; the equations describing the models are shown in Appendix 1.

View larger version (12K): [in this window] [in a new window]   Fig. 1.   Pictorial representations of models 1 through 3. The arterial concentration and myocardial blood flow were designated Cart and Qh, respectively. The volume of the compartment in equilibrium with coronary sinus blood (Ccs) was always designated Vh. The additional compartments were designated either V1 or V2 and had drug concentrations of C1 and C2, respectively. The permeability constant of these compartments (units of flow) were PS1 and PS2, respectively.

View larger version (13K): [in this window] [in a new window]   Fig. 2.   Pictorial representations of models 4 through 6. See the legend to Fig. 1 for details.

View larger version (8K): [in this window] [in a new window]   Fig. 3.   Pictorial representations of models 7 through 9. See the legend to Fig. 1 for details.

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Statistical Analysis

Data in the text are presented as mean (lower to upper 95% confidence intervals), assuming a t distribution (Gardner and Altman, 1989). Two means were considered significantly different at the 95% level if each of the means lay outside of the confidence intervals of the other or if the 95% confidence intervals of the difference did not include zero.

	Results

Top Abstract Introduction Materials and Methods Results Discussion Appendix 1 References

Hysteresis. The hysteresis loops of effect versus the various concentrations are shown in Fig. 4. There was a pronounced counter-clockwise hysteresis for the arterial concentrations, a smaller clockwise hysteresis for the coronary sinus concentrations, and an again smaller clockwise hysteresis for the calculated mean myocardial concentrations. The mean size of the hysteresis loops (and 95% confidence intervals) as given by the difference between AUC under the sections of the plots when the concentrations were increasing and decreasing was 449 mg s l¹ (242-655) for the arterial concentrations. This statistically significant hysteresis is clearly evident in Fig. 4A. Figure 4, B and C, shows the AUC differences of 37 mg s l¹ (39 to 113) for the coronary sinus concentrations and 69 mg s l¹ (12 to 150) for the calculated myocardial concentrations; these later two sites were not statistically different from zero, suggesting no significant hysteresis.

View larger version (19K): [in this window] [in a new window]   Fig. 4.   Examples of the relationships between mean concentrations of meperidine in arterial blood (A), coronary sinus blood (B), myocardium (C), and the mean percent reduction of LV dP/dtmax after the bolus injection of 100 mg of meperidine over 1 s. Significant hysteresis was found for arterial concentrations. Error bars are S.E.M.

Optimal Dynamic Model. The MSC of fits of the dynamic models to the data are shown in Table 1. Both the E_max and sigmoid E_max models were underdetermined for the value of E_max leading to nonidentifiability for this parameter. Although it could be argued that using higher meperidine doses would allow the measurement of E_max, this is physically impossible for myocardial contractility in vivo. The maximum reduction in contractility is the balance between direct depressant effects and indirect CNS effects, as well as the risk of fatality. The linear model was a reasonable fit, but returned values for the slope of 3.12, S.D. 0.28, and an intercept of 7.53, S.D. 1.92. This negative intercept suggests a "threshold" effect for measurable contractility changes, which is broadly consistent with Fig. 4C. The final model chosen, therefore, was the linear model with a threshold such that at concentrations below the threshold the reductions in contractility were zero. This form of the dynamic relationship (but not the parameter values) therefore was used for the subsequent combined kinetic-dynamic models, which enabled a considerable reduction in the number of combined kinetic-dynamic that needed to be examined in the following section.

Optimal Kinetic and Combined Kinetic-Dynamic Models. The MSC of the various models are shown in Table 2. The parameters of the kinetic models and their S.D. returned by the fitting program are given below. Model 1, the single flow-limited compartment, was the worst description of both the kinetic and combined kinetic and dynamic data sets. The general problem was a poor account of the time of the peak and "washout" coronary sinus concentrations (Fig. 5). The volume of the heart (V_h) was 0.73 S.D. 0.02 liter, which equated to an equilibrium half-life between arterial and coronary sinus blood of 4.13 min.

View this table: [in this window] [in a new window]   TABLE 2 The best fits of the various compartmental models of myocardial kinetics The models can be related to the pictorial representations in Figs. 1, 2, and 3 and the equations in Appendix 1. The MSC is the model selection criteriathe higher the number, the better the fit. "Non-ident" refers to nonidentifiability and is defined as the coefficient of variation of the parameter exceeding 100%. The class leader for each has been highlighted in bold. "Best Conc" refers to the concentration in the kinetic model that gave the best fit of the combined kinetic-dynamic data when linked to the "threshold" dynamic model.

View larger version (18K): [in this window] [in a new window]   Fig. 5.   The best fit (solid line) of the observed coronary sinus concentration data (symbols) for the single flow-limited compartment model (model 1), a two tank in series model (model 2), a traditional membrane-limited model (model 3), and compilation model A, the optimal kinetic model (model 5). The latter model clearly has the least systematic deviation from the observed values. The dotted lines are the 95% confidence intervals of the observed data assuming a t distribution.

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Summary of Model Performance. Note that across the range of models a good fit to the kinetic data did not necessarily mean a good fit to the kinetic-dynamic data, and vice versa. This is indicative of the fact that a good fit to the dynamic data generally arose when a compartment of the model linked to the dynamic model was "upstream" of the coronary sinus compartment. This is consistent with the model-independent analysis, which also showed that the effect preceded the coronary sinus concentrations.

	Discussion

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An initial observation from this analysis is the general utility of the mass-balance method for describing this type of regional kinetic-dynamic data. In this case, it was used to determine whether the mean myocardial concentration of a drug was related to its proposed effect on the myocardium and was able to do so even if there was incomplete equilibrium between the effect and the venous concentrations emerging from the organ. This itself is good evidence that a single-compartment description of the organ may not be appropriate, as this model assumes instantaneous equilibration between tissue and effluent venous drug concentrations.

A general picture also is emerging for a variety of drugs that changes in myocardial contractility caused by the drug are better related to its myocardial rather than arterial concentrations; this has been demonstrated for thiopental (Upton et al., 1996), lidocaine (Huang et al., 1993), verapamil (Huang et al., 1998), and, now, meperidine in the present study. The mass-balance method also was useful for determining the optimal dynamic model independent of any assumptions about the underlying myocardial (or effect-compartment) kinetics. It is not clear whether the observed threshold effect of meperidine on myocardial contractility was a real effect or was due to lack of sensitivity in the method. Although the error bars in Fig. 4 are relatively small, such a threshold effect has not been observed in vitro (Wu et al., 1997).

The opportunity was taken in this study to address a pertinent issue regarding the choice of kinetic models for individual organs or regions of the body such as the heart. In many physiological models of lipophilic drugs such as meperidine (Gabrielsson et al., 1986; Davis and Mapleson, 1993), the heart and many other organs or regions are represented as single flow-limited compartments. In agreement with the present data, it was found that the myocardial kinetics of fentanyl and alfentanil after a 1-min infusion to rats were better described by a single flow-limited compartment (equivalent to the present model 1) than either two- or three-compartment models equivalent to models 3 and 7 of the present paper, respectively (Bjorkman et al., 1994). However, although single flow-limited compartment models are in broad agreement with the available data, there is some evidence that this single-compartment representation of an organ or region is too simple, particularly for data arising from bolus or impulse administration studies. For example, when drug kinetics in organs or regions are examined individually after impulse administration, it generally is noted that the concentration profile emerging from the organ is lagged, unimodal, asymmetric, and skewed to the right (Roberts et al., 1988; Krejcie et al., 1996). This shape arises from dispersion of the idealized impulse input in the organ due to laminar flow in the vasculature, turbulence at branch points, the distribution of vascular path lengths, and heterogeneity in the perfusion of the organ (Krejcie et al., 1996). More complex, stochastic "dispersion" models of the organ are required to account for these processes, often with models that need to be solved by computationally intensive inversion of the Laplace solution to the model (Roberts et al., 1988).

It would be advantageous to find the middle ground between the computational simplicity but lower fidelity of a single-compartment model, and the opposite is true for stochastic dispersion-based models. The data suggest that a class of compartmental models known as "tank in series" models (Beek and Muttzall, 1975; Roberts et al., 1988) may provide a means of doing this, as they can represent increased dispersion by increasing the number of compartments in series. However, these models can be written as differential equations in the time domain and, therefore, solved in the same manner as the traditional single-compartment organ models used in many physiological models.

The effect of adding extra compartments in these "in series" and "peripheral" configurations is not immediately obvious. However, the simulations shown in Fig. 6 illustrate the general principles. Note that each configuration has a characteristic effect on the venous concentrations emerging from the model. First, the single-compartment model is characterized by an exponential rise and fall of the venous concentrations when exposed to a "square wave" arterial concentration input. Importantly, the peak venous concentration always occurs at the point at which the venous concentrations cross the arterial concentrationsthis is an embodiment of Zilversmit's rule originally developed for the analysis of product-precursor relationships (Rescigno and Segre, 1966; Jones and Nicholas, 1984). This behavior is characteristic of systems in which there is only one compartment directly between arterial and venous blood. Second, it is clear that the principal advantage of the tank in series configuration is that it is possible to introduce a delay in the time of the peak venous concentration. The peak is also skewed to the right in a manner consistent with real data (Krejcie et al., 1996). Finally, note that the peripheral configuration of membrane-limited models also must obey Zilversmit's rule by nature of the arrangement of their compartments, and the peak venous concentration will always occur when the venous concentrations cross the arterial, regardless of the number of additional peripheral compartments. Indeed, the principal effect of adding additional peripheral compartments is to prolong the elution process with only minor changes in the rate of uptake. The advantage of the compilation models is that both the delayed peak characteristic of dispersion and the prolonged elution characteristic of deep distribution can be accounted for to different extents as determined by the data.

View larger version (21K): [in this window] [in a new window]   Fig. 6.   A hypothetical simulation of the effect of adding extra compartments to a single flow-limited compartment in either a series configuration or as peripheral compartments. The input to the system (arterial concentration) is a pulse of 0.5-min duration and a height of 1 concentration unit. The mean transit time for each compartment in the system is 1 min. The curves shown are the venous concentrations emerging from the organ. Adding compartments in series increases dispersion, as shown by delayed peak concentration and a right-skewed curve. Adding extra peripheral compartments does not change the time of the peak concentration, but prolongs the washout from the organ.

An overall conclusion from this analysis is that arteriovenous concentration difference data across an organ can be used to differentiate between a variety of kinetic models. The redundancy of compartmental models when fitted to systemic blood concentration data, as pointed out by Wagner (1975), is not an issue in this quite different experimental paradigm. Overall, compilation model A was considered the optimal model (from those examined) for these data because it was the best fit to the kinetic data and one of the best for the kinetic-dynamic data without nonidentifiable parameters. This model is of a novel form because it incorporates both dispersion (two compartments in series) and deep distribution elements (a peripheral compartment). We currently are investigating the hypothesis that this relatively simple type of model may be of a sufficiently general form that it is suitable for describing regional kinetics for a wide range of organ and drug combinations.