Simulations were divided into approximately 100 segments of varying duration with a constant drug concentration. The fractional drug concentration for each segment was changed from the previous value by ±0.01 so as to model the synaptic concentration profile of the Sheiner model (Fig. 2B). In all simulations, the fraction of unliganded receptors R(t) was converted to twitch amplitude, T1(t), using:
where γ is the Hill factor describing the steepness of the twitch-concentration curve and R50 is the fraction of unliganded receptors at 50% twitch suppression. Literature values6 of γ = 6.16 and R50 = 0.09 were used, and this produced a minimal twitch amplitude of 0.1 (used in all simulations). This value of R50 when combined with the standard simulation parameter values (Table 1) gives an effective concentration at 50% twitch height, EC50, of 178 nM.
Simulations proceeded until the fractional equilibrium receptor occupancy, R∞, was achieved.
The antagonist concentration was chosen so that, at equilibrium, approximately 94% of the receptors have at least 1 antagonist molecule bound, that is, R = 0.0625. This corresponds to a steady-state twitch amplitude of 0.1. Table 2 lists the published clinical parameters, the time to 50% twitch height, and EC50 that are used for comparison with the simulations in this paper. Analysis calculations were performed using Igor Pro 6.04 (WaveMetrics Inc., Lake Oswego, OR).
The ability of the morphological model (Fig. 1A) and MCell, to describe the time course and variability of miniature endplate currents (mepc, the current produced by the release of ACh from 1 vesicle), has been demonstrated.22,25 To validate the model with respect to competitive antagonism, mepcs were simulated under conditions of different receptor densities. The results were compared with published experimental data from mouse diaphragm muscle where (+)-tubocurarine was used to reduce the effective receptor density.51 Figure 3A shows simulated mepcs using the standard receptor density of 10,000 µm−2 and a density of 2000 µm−2. This illustrates one aspect of the safety margin at the neuromuscular junction: when only 20% of the receptors remained, the mepc amplitude was still 40% of control. There was excellent agreement between simulations and experimental results both when AChE was active (Fig. 3B) and inhibited (not shown). The simulations also recapitulated the decrease in decay time constant seen with increasing concentrations of (+)-tubocurarine when AChE is inhibited51 (not shown). The high safety margin is not diminished by altering the parameters in the model. Figure 3C shows the effect of 2-fold increases or decreases in receptor gating kinetics, ACh binding, the amount of ACh contained in a vesicle, and the receptor density. The figure shows the ratio of mepc amplitude when 50% of the receptors are inhibited to the mepc amplitude without inhibition. All of these changes preserve a margin of safety because the ratio is >0.5.
Instantaneous Change of Drug Concentration Near the Synapse
Figure 4 illustrates a simulation using the standard antagonist (Table 1). This EC50 of 178 nM closely corresponds to the properties of pancuronium at human AChRs.31,35 Equilibration of the synapse with an antagonist concentration of 237 nM results in 94% occupancy of the receptors and an endplate current (epc, the current produced by release of ACh from all 30 vesicles) whose amplitude is 10% of control. Figure 4A shows the time course of state occupancy for the unliganded receptor, R, and the antagonist-bound states, RC, RC and CRC. Figure 4B shows the time course of the free antagonist concentration within the synapse. The figures illustrate 2 general features of buffered diffusion. (1) Equilibration is limited by the time required for sufficient antagonist molecules to enter the synapse. Here, 168,500 antagonist molecules are required (64,000 for CR, 2 × 49,000 for CRC, and 6,500 for RC). The initial flux of antagonist into the synapse is 45,000 s−1. If this flux persisted (it does not because the concentration gradient dissipates with time), approximately 4 seconds would be needed to satisfy the receptors. In the absence of receptors, the synapse would equilibrate with 220 antagonist molecules in approximately 5 milliseconds. Thus, the presence of the receptors slows the effective diffusion of the antagonist by a factor of approximately 1000. (2) With buffered diffusion, the time required for 50% receptor occupancy is shorter than for 50% free antagonist equilibration (3.6 seconds vs 9.3 seconds in Fig. 4, A and B respectively). This illustrates the high buffering capacity of the receptors. Figure 4C shows the time dependence of twitch amplitude calculated from Equation (1). Due to the steep dependence of twitch on receptor occupancy (γ = 6.16) and the high margin of safety (R50 = 0.09), twitch depression occurred over a narrow range of unliganded receptors (R). It required 4.2 seconds to go from 80% to 20% twitch; during this time, the number of unliganded receptors decreased from 12% to 7.6%. Twitch amplitude decreased to 50% at 12.6 seconds.
Dependence on Antagonist Affinity, Selectivity, and Concentration
Because the rate of buffered diffusion is determined by the flux of antagonist molecules into the synapse, a higher affinity antagonist is expected to exhibit a longer equilibration time. Such an antagonist would be administered at a lower concentration, giving rise to a smaller concentration gradient and a lower flux. Figure 4D shows the results of simulations with different values of antagonist dissociation rate,
, (1–300 s−1). The antagonist concentration, [C], was changed in proportional to
so that that the equilibrium occupancy of R and equilibrium twitch were the same for each of the simulations. The corresponding EC50 values are 59 to 17800 nM; encompassing the range of values for clinically used muscle relaxants (Table 2). The logarithmic time scale of Figure 4D shows that a 3-fold decrease in dissociation rate led to a 3-fold increase in time to 50% twitch.
So far, the simulations considered situations for which the antagonist was selective for one binding site over the other by a factor of 10. The site selectivity of muscle relaxants ranges from nonselective to >100-fold selective.31 For a nonselective drug, L1eq = L2eq = 79 nM and [C] = 3 × L1eq = 237 nM produces the same equilibrium level of receptor occupancy as the “standard” antagonist and concentration (Table 1). Buffered diffusion of a nonselective drug is slower than for a highly selective drug. Time to 50% receptor occupancy was 4.2 seconds (compared with 3.6 seconds), and the time to 50% twitch was 15 seconds (compared with 12.6 seconds). The slower equilibration time is due to 1.4-fold more bound drug molecules being required, mostly due to a larger number of doubly bound receptors (CRC), and these must be supplied by the same external concentration. Increasing the selectivity factor from 10- to 100-fold did not produce a significant change in twitch onset times.
For a given affinity antagonist, equilibration time depended on the concentration of the antagonist used; higher concentrations led to faster equilibration. For a simple rectangular box model of a synapse, it was shown with high concentrations of antagonist (>L1eq) that the reciprocal of the time to 50% equilibration is directly proportional to antagonist concentration.21 This also applies to the model used in this paper (not shown).
Time-Dependent Change in Plasma Concentration of Drug
After a bolus injection of a drug, the plasma concentration increases to a peak and fluctuates as the drug circulates through the body.52 A 2-compartment model has been used to describe the onset of paralysis by vecuronium.11,38,53 In this model, the plasma concentration of the drug varies as shown in Figure 2A. The drug concentration at the synapse is determined by the movement of drug from plasma into the effect compartment (Fig. 2B). I assumed that this relationship describes the drug concentration immediately outside the synapse and used this to simulate the diffusion of drug into the neuromuscular junction per se.
Figure 5 shows the results of a simulation using a concentration change profile corresponding to a slowly perfused muscle, the adductor pollicis, (keo = 0.15 min−1)11 and the “standard” antagonist (EC50 = 178 nM). Figure 5B shows how the drug concentration in the interior of the synapse (dark line) did not increase as quickly as the external drug concentration (thin line) for the first 70 seconds. A similar lag is seen (Fig. 5A) in the simulation of unliganded receptors (R, dark line) compared with what would be expected if buffered diffusion did not occur (dashed line). However, by the time the number of unliganded receptors decreased to levels that produce twitch inhibition (Fig. 5C), the lag was nearly gone. Buffered diffusion caused an increase of only 12 seconds in 50% twitch onset time. A larger shift was observed for more potent drugs, 18 seconds for EC50 = 59 nM and 80 seconds for EC50 = 18 nM (Fig. 5D).
Figure 6 summarizes the results of simulations with 4 different values of keo (keo = ∞ corresponds to an instantaneous concentration change at the synapse, Fig. 4) and compares them with clinical onset times. The contribution of buffered diffusion can be seen as the increase in onset time from the high concentration (low potency) asymptote of each curve. Although an instantaneous concentration change at the synapse produced the steepest concentration dependence of onset times (keo = ∞), these times were much faster than clinical onset. For the slowest concentration versus time profile (keo = 0.15 min−1), onset times were essentially determined by the rate of drug perfusion of the muscle; buffered diffusion played a very small role except for an antagonist with EC50 = 28 nM. Higher equilibration rates (keo = 0.3 and 0.6 min−1) resulted in a greater fractional increase in onset time due to buffered diffusion for the most potent drugs. However, no single value of keo can account for all of the clinical data.
Effects of Synapse Geometry
The model of Figure 1 was constructed from measurements of rat neuromuscular junction. Equally detailed models for human neuromuscular junction are not available, so it is essential to determine the extent to which differences in anatomy may alter the results of buffered diffusion simulations. Simulations were performed with the width of the nerve-muscle contact (nominally about 2 µm, y-dimension of Fig. 1) being doubled or tripled while the receptor density was held constant. A value of keo = 0.6 min−1 was chosen for these simulations because this approximates the onset speed for low potency drugs (Fig. 6). Increasing the nerve-muscle contact width slowed the rate of antagonist binding to receptors (Fig. 7A), the rate at which the free antagonist concentration equilibrated (Fig. 7B) and the rate of twitch suppression (Fig. 7C). Figure 7D summarizes the results of simulations using a range of EC50 values and compares these with the clinical data. Onset times for nerve-muscle contact widths of 4 or 6 µm (2× or 3×) encompass most of the clinical onset times. It would be expected that a similar result would be seen if the distance between the nerve and muscle (z-dimension in Fig. 1) were decreased by a factor of 2 or 3. Conversely, a decrease in nerve-muscle contact area or an increase in the width of the synaptic cleft would be expected to decrease the importance of buffered diffusion.
The goal of this investigation was to determine whether buffered diffusion might be expected to affect the onset of paralysis by muscle relaxants. I considered a range of muscle relaxant potencies. I considered 4 scenarios for the rate at which drug arrives immediately outside the synapse (neuromuscular junction): instantaneously (Fig. 4), and with the rate of transfer variable (keo) being 0.15, 0.3, and 0.6 min−1 (Figs. 5 and 6). I considered a standard model of the rat neuromuscular junction (Fig. 1A) and some modifications of this morphology (Fig. 7). I conclude that for keo = 0.6 min−1 and a 2- to 3-fold wider (than the standard rat model) nerve-muscle contact width, buffered diffusion could account for the differences in paralysis onset times among clinically used muscle relaxants (Fig 7D). Other morphological differences, particularly a smaller gap between nerve and muscle, may act in conjunction with the contact width to increase the role of buffered diffusion.
Critique of the Model
A complex computer model was used in this study. Modeling is crucial for understanding phenomena that occur under nonequilibrium conditions. One must, however, assess the sensitivity of the results to the assumptions used. The kinetic model of muscle nAChR activation, desensitization, and competitive inhibition (Fig. 1B) is well supported by experiments. The kinetic parameters are appropriate for room temperature rather than 37°C; very little kinetic data are available for 37°C. Increasing temperature decreases antagonist affinity20 and increases diffusion coefficients. Thus, buffered diffusion would be faster at higher temperatures.
The morphological model of the neuromuscular junction (Fig. 1A) is based on rat diaphragm muscle. Vertebrate species differ in the nerve-muscle contact area, degree of muscle membrane folding, quantal content, distribution of contact areas, and transverse width of individual nerve-muscle contacts.54,55 These studies indicate that the human neuromuscular junction consists of a 20 to 30 µm wide cluster of strands. However, each strand within the cluster may be only several microns in width. The extent to which Schwann cells extend over the cluster is unknown but even if they formed a narrow barrier around the strands, this would not significantly impede the movement of antagonist molecules because the path would not be lined with receptors. I performed simulations assuming a range of widths for the nerve-muscle contact region (Fig. 7). The human neuromuscular junction has 7 to 8 times more muscle area than nerve area.54 Thus, the model, with an area ratio of 5.7:1, is reasonable. Species differences in nerve-muscle contact area and in quantal content may compensate for each other.54 The length of the contact region (i.e., the extent along the x-direction of Fig. 1A) does differ among species, but this does not affect the time required for antagonists to diffuse into the interior as they enter only through the narrow upper regions of the front and rear x-z surfaces (green surface and arrows in Fig. 1A).
The simulations show that the onset time for receptor equilibration was inversely proportional to L1eq when the concentration change was instantaneous (Fig. 4D). This indicates that antagonist diffusion into the synapse is strongly buffered by the receptors. Deviations from this rapid binding scenario occur when (a) the diffusion coefficient is 100-fold larger, (b) the antagonist association constant is 100-fold smaller, (c) the receptor density is 100-fold smaller, or (d) the synaptic gap is 10-fold larger.21 None of these alternate scenarios appears very likely.
The models used in some pharmacokinetic/pharmacodynamic studies appropriately consider buffering effect of receptors on the concentration of drug in the neuromuscular junction. However, they did not consider either the way in which the narrow junction limits the flux of drug molecules or the kinetics of drug association and dissociation. In these models, the receptor concentration is a fitting parameter and the resulting values grossly underestimate the actual receptor concentration (280 nM7,12 or 1.5 µM10 compared with approximately 320 µM, calculated from 104 receptors/µm2/0.05 µm). This paper demonstrated that Monte Carlo simulation of diffusion and pharmacodynamics is a robust method for determining the time course of receptor binding in a realistic synapse.
Buffered Diffusion with an Instantaneous Concentration Change Near the Synapse
The first set of simulations was performed using the assumption that the volume surrounding the synapse was clamped instantaneously to a given concentration of antagonist. The slope of the log–log plot of onset time against L1eq is −1.0 (not shown). Experimentally, concentration clamp conditions were approximated with iontophoretic application of antagonists and ACh to isolated frog neuromuscular junctions.14,15 (+)-Tubocurarine applied at 4 × Leq (Leq = 500 nM for frog muscle AChR) inhibited endplate potentials with a time constant of 0.5 seconds.14 This is within a factor of 2 of our simulated value. Similar experiments using 7 antagonists with a 50-fold range in potency found the slope of a log(onset time) versus log (Leq) plot between −0.7 and −0.8.15 This is close to the theoretical relationship especially considering the experimental difficulties and the potential confounding effect of different binding site selectivities. In contrast, the dependence of onset time for human paralysis on L1eq is weaker, giving a slope of −0.22 on a log–log plot (not shown).
Buffered Diffusion with a Time-Dependent Change in Plasma Concentration
The antagonist concentration immediately outside the synapse was calculated from published values of plasma vecuronium concentrations11 and a 2-compartment pharmacokinetic model38 (Fig. 2). The simulations indicate that buffered diffusion slows the initial rate of equilibration of antagonist molecules with the receptor (Fig. 5B). Due to the high safety margin, twitch depression does not occur until t >80 seconds. By this time, however, buffered diffusion had only a small effect on receptor occupancy, antagonist concentration and twitch (Fig. 5, A–C). The effect was significantly increased for simulations using a 3- to 10-fold higher potency drug (Fig. 5D). Figure 6 shows that there is little agreement between simulated and experimentally observed twitch onset times. In particular, simulations of buffered diffusion using any single value of keo cannot account for the differences among the drugs.
The morphological model of Figure 1, based on rat neuromuscular junction, may not faithfully represent the geometry of the human junction. Modification of the model to allow for a 2- or 3-fold wider extent of nerve-muscle contact provided a better congruence between simulated and observed onset times (Fig. 7D). A similar result would be obtained if it were assumed that the width of the junction was unchanged, but the width of the synaptic cleft was decreased by a factor of 2 or 3. Results from simulations assuming a 2- to 3-fold wider synapse (4–6 µm), and keo = 0.6 min−1 encompass all of the clinical data (Fig. 7D). Until more specific anatomical information about the human neuromuscular junction is available, it will not be possible to know for certain whether buffered diffusion can account for the dependence of onset time on drug potency.
Name: James P. Dilger, PhD.
Contribution: This author designed and conducted the study, analyzed the data, and wrote the manuscript.
Attestation: James Dilger has seen the original study data, reviewed the analysis of the data, approved the final manuscript, and is the author responsible for archiving the study files.
This manuscript was handled by: Tony Gin, FANZCA, FRCA, MD.
The author thanks Dr. Aaron Hartman, graduate of the Medical School Program, Stony Brook University, for assistance with writing and testing MCell3 model files; Dr. Jeevendra Martyn, Department of Anesthesia, Critical Care and Pain Medicine, Massachusetts General Hospital for his helpful advice on an early version of the manuscript; and Dr. Mihai Sadean, Department of Anesthesiology, Stony Brook University for use of his spreadsheet to make pharmacokinetic/pharmacodynamics calculations.
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