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British Journal of Anaesthesia 1995; 75: 109–112

COMMENTARY

Calculation of an effect compartment rate constant using recovery indices obtained with an isolated arm technique C. A. SHANKS, M. J. AVRAM, T. C. KREJCIE AND T. K. HENTHORN

Summary We have examined the implications of the theoretical single pharmacokinetic compartment associated with blocker-induced paralysis, in relation to the isolated arm technique. It is assumed that the blocker concentration–effect relationship can be characterized by a sigmoid curve, which incorporates an exponent, s. After tourniquet release, the concentration gradient between the effect compartment and plasma should be large, and elimination related to the rate constant, k eo. The major measurement of spontaneous recovery with the isolated arm is the time interval between 75 % and 25 % twitch depression, T25–T75. The general equation relating these three variables is developed: k eo : 2.2/(s
(T25–T75)). Insertion of published values for T25–T75 with isolated arm studies into this equation gave estimates for an intrinsic k eo for atracurium, vecuronium, rocuronium and pancuronium. (Br. J. Anaesth. 1995; 75: 109–112) Key words Neuromuscular block, atracurium. Neuromuscular block, vecuronium. Neuromuscular block, pancuronium. Model, pharmacokinetic.

Spontaneous recovery from neuromuscular block through the range 75 % to 25 % depression appears to be linear, and the time interval between these (T25–T75) is often termed the recovery index. Recovery indices observed after systemic i.v. administration of a neuromuscular blocking agent are longer than those obtained with an isolated arm technique. In the latter, block is produced with i.v. injection of a small dose of blocker below a tourniquet, wirh tourniquet release 3 min later [1–4]. Kinetic–dynamic modelling of neuromuscular block usually involves initial characterization of the concentration–time curve. The pharmacokinetics, often multicompartmental, become the forcing function for drug entry into an additional (effect) compartment used to model the pharmacodynamics of block. In the effect compartment, the relationship between blocker concentrations and intensities of neuromuscular block is assumed to be consistent for both onset and offset of paralysis, a relationship modelled traditionally as a sigmoid curve [5]. Disparities in the plasma concentrations observed during onset and offset of paralysis are reconciled by

optimization of the rate constant for the effect compartment, keo. Use of the isolated arm technique precludes the need for complex pharmacokinetics in modelling the dynamics, as a step change in plasma drug concentrations to zero may be assumed. In this study we have explored the use of T25–T75, obtained with the isolated arm technique, to derive a value for keo.

Methods When the concentration–effect relationship in the effect compartment is postulated to be a sigmoid curve, this can take the general form s fractional effect = ECs/(EC50s +EC)

(1)

where the fractional effect is determined by any concentration (EC) in terms of the midpoint of the curve (that concentration related 50 % effect, EC50) and the Hill exponent (s). Derivations for the effect compartment concentrations associated with 25 % and 75 % effects (EC25 and EC75, respectively) are:

25%/100% = 1/ 4 = EC25s /( EC50s + EC75s)

(2)

75% /100% = 3/ 4 = EC75s /(EC50s + EC75s)

(3)

Solving equations (2) and (3) to eliminate EC50s by forming a ratio gives: EC75s/EC25s = 9

(4)

which, with loge transform becomes: In (EC75/EC25) = In (9)/ s

(5)

Concentrations in a one-compartment model, for any time interval (t) from time zero, may be characterized by the general equation: C(t ) = C(o)e−k *t

(6)

where k : elimination rate constant and C(o) is the initial concentration at time zero.

C. A. SHANKS, MD, CHB, FRCA, M. J. AVRAM, PHD, T. C. KREJCIE, MD, T. K. HENTHORN, MD, Northwestern University Medical School, Department of Anesthesia, Ward Building, 12–189, 303 E Chicago Avenue, Chicago, IL 60611, USA. Accepted for publication: February 17, 1995.

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The isolated arm technique assumes negligible plasma blocker concentrations enabling the dynamic model to be characterized as a single effect compartment. The monoexponential equation, made specific for the effect compartment rate constant, keo, and for concentrations over the time interval, T25–T75, becomes: EC25 = EC75e

− keo *(T25-T75)

(7)

which, with rearrangement and inversion, becomes: EC75 / EC25 = e k

(T25- T75)

eo *

(8)

Log transform and combination with equation (5) gives: (9) In (9) /s = keo * (T25 - T75) and thus keo = 2.2 /( s * (T25 − T75)) (10) giving a general equation for the relationship between T25–T75, the Hill exponent and the intrinsic keo.

Results As a special case, when the concentration EC75 is double that of EC25, then: EC75 / EC25 = 2 (11) It follows from equations (4) and (11) that s

s

(EC75 / EC25 ) = 2 = 9

(12)

giving a value of s = 3.17 (13) Combining equations (8) and (11), with log transforms the special case gives: In(2) = keo *(T25 − T75) (14) and

T25−T75 = 0.693/keo = T½ keo

(15)

Thus, if the EC75/EC25 ratio : 2, the recovery index obtained with an isolated arm technique gives a value for an intrinsic half-life for the effect compartment, with quantification of the T½ keo . A recent article comparing atracurium and vecuronium in the isolated arms of the same subjects reported that the mean recovery index for vecuronium was significantly shorter (7.9 min) than that for atracurium (10.6 min) [2]. A similar study gave recovery indices for rocuronium (9.32 min) and pancuronium (12.5 min) [3]. Table 1 shows the relationships between selected EC75/EC25 ratios and

the Hill exponent. It also gives values for the intrinsic keo , estimated from equation (10) and the recovery indices for these four blockers [2, 3].

Discussion Kinetic–dynamic modelling with a sigmoid curve to define the concentration–effect relationship (equation (1)) assumes that concentrations in the effect compartment are identical during both onset and offset of neuromuscular block. As the concentration–effect relationship is independent of time, curves inherent in equations (l)–(5) resemble those which might be realized with laboratory organ-bath studies. In patient-based studies, the non-linear relationship between plasma concentrations and effect is obvious, particularly during onset of paralysis. The time domain is added by a link model, here using an effect compartment rate constant, keo , reported in reciprocal minutes (table 1). Classically, its derivation minimizes the hysteresis in the effect compartment, a curve depicted when onset and offset plasma concentrations are plotted against the concomitantly observed paralysis. The intrinsic keo , calculated in table 1, uses T25–T75 values obtained in isolated arm studies [2, 3]. These assume that the small dose of blocker administered in an isolated arm study does not result in adequate plasma concentrations to affect the gradient between the neuromuscular junction and plasma. Separate simulations with kinetic–dynamic models of systemic administration of blockers confirmed that concentration gradients in the effect compartment would not be altered materially by the entry of the blocker into the body with release of the tourniquet. In the isolated arm study comparing atracurium and vecuronium, it was proposed that the different recovery times (T25–T75) reflect different affinities with biophase binding sites [2]. Our calculations are consistent with this concept, and indicate that the value of T25–T75 with the isolated arm technique is proportional to the elimination half-life of the effect compartment, T½keo (equation (10)), enabling quantification of removal of blocker from the biophase. As a general rule, the rate-limiting step in spontaneous recovery from blocker-induced paralysis must be the rate of decrease in plasma concentrations, the concurrent “context-sensitive” plasma half-life [6]. From systemic studies it seems extraordinarily rare for the plasma half-life to be less than the intrinsic T½keo . This exception might be true for the trans-

Table 1 Relationships between the EC75/EC25 ratio and the Hill exponent (s), and related estimates of the intrinsic keo. Estimates of keo for four neuromuscular blocking agents based on recovery indices reported in isolated arm studies and calculated from equation (10) [2, 3]. * Calculated from equation (5) keo (min91) EC75/EC25 ratio

Exponent s*

Atracurium

Vecuronium

Rocuronium

Pancuronium

1.4 1.6 1.8 2.0 2.2 2.4

6.5 4.7 3.7 3.2 2.8 2.5

0.03 0.04 0.06 0.07 0.07 0.08

0.04 0.06 0.07 0.09 0.10 0.11

0.04 0.05 0.06 0.07 0.09 0.09

0.03 0.04 0.05 0.06 0.06 0.17

Effect compartment with the isolated arm

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Table 2 Sample means for keo and the Hill exponent (equation (1)).* The pharmacokinetic models reported here all assume complete mixing at time zero

Pharmacokinetic model* Atracurium Non-compartmental [11] Two-compartment[12] Two-compartment [13] Non-compartmental [14] Vecuronium Three-compartment [15] Three-compartment [16] Two- and three-compartment [17] Non-compartmental [17] Three-compartment [18] Rocuronium Three-compartment [19] Pancuronium Two- and three-compartment [18] Three-compartment [19] Three-compartment [20]

keo (min91)

Exponent s

EC75/EC25 (from equation (5))

0.07 0.12 0.10 0.12

6.1 4.6 4.3 3.4

1.4 1.6 1.7 1.8

0.10 0.17 0.06 0.09 0.11

5.5 5.8 5.3 3.9 —

1.5 1.5 1.5 1.8 —

0.12

5.0

1.5

0.48 0.11 0.15

5.5 4.8 —

1.5 1.6 —

trans and cis-trans isomers of mivacurium, assuming the cis-cis isomer to have insignificant action. In studies with i.v. administration of mivacurium, the trans-trans and cis-trans isomers were reported to have a plasma half-life of approximately 2 min [7], yet the T25–T75 for the parent mixture was between 6 and 10 min [7–9]. Equation (15) suggests that the T½keo , for the active isomers would be greater than 6 min, more than three times the plasma half-life. If the blocker is administered systemically, the rate-limiting step in the offset of block may be the rate of decrease in its plasma concentrations, and the resultant concentration gradient between the plasma and the effect compartment. Alternatively, intrinsic elimination from the effect compartment may govern the time-course of spontaneous recovery from block. One study examined the midpoint of the dose– response relationship (ED50), when a dose–response curve was obtained subsequent to recovery from initial block established either with an isolated technique or with systemic administration of vecuronium [4]. The authors suggested that the reduction in ED50, which occurred only after prior systemic blocker, was consistent with drug persisting in the plasma, and that there was no effect from residual drug remaining at the effect site following an isolated arm study [4]. Equation (11) is a special case assumed for the isolated arm calculations; the EC75/EC25 ratio from systemic studies is usually between 1.3 and 2.3 [10]. Table 1 shows the values for the exponent for ratios between 1.4 and 2.4, and for the related exponents between 2.5 and 6.5. Table 2 shows published values for studies in which blocker was administered systemically, where the sample means for the exponent were between 3.4 and 6.1. The EC75/EC25 ratios associated with these are calculated to be in the range 1.4 to 1.8. The values in table 1 calculated for the intrinsic keo for the four blockers are in the range 0.03–0.11 min91, which are comparable with the low range of those reported in table 2. The values for vecuronium keo have been shown to be model-dependent [17]. The mean value for keo

(0.09 min⫺1) shown in table 2 is for a traditional back-extrapolated non-compartment model, but inclusion of data with an initial time delay reduced this (to 0.06 min⫺1). Table 1 gives calculated values for keo that are in the lower range of those in the literature (table 2). The largest difference between the tables is for the long-acting blocker pancuronium. This difference emphasizes the predominant influence of the context-sensitive half-life over the T½ of the effect compartment, where the value in table 1 reflects the lack of pharmacokinetic influence, the relative purity of the effect compartment with the isolated arm technique, in the derivation of the “intrinsic” value for the rate constant.

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