Explaining anaesthetic hysteresis with effect-site equilibration

Abstract

Background: Anaesthetic induction occurs at higher plasma drug concentrations than emergence in animal studies. Some studies find evidence for such anaesthetic hysteresis in humans, whereas others do not. Traditional thinking attributes hysteresis to drug equilibration between plasma and the effect site. Indeed, a key difference between human studies showing anaesthetic hysteresis and those that do not is in how effect-site equilibration was modelled. However, the effect-site is a theoretical compartment in which drug concentration cannot be measured experimentally. Thus, it is not clear whether drug equilibration models with experimentally intractable compartments are sufficiently constrained to unequivocally establish evidence for the presence or absence of anaesthetic hysteresis.

Methods: We constructed several models. One lacked hysteresis beyond effect-site equilibration. In another, neuronal dynamics contributed to hysteresis. We attempted to distinguish between these two systems using drug equilibration models.

Results: Our modelling studies showed that one can always construct an effect-site equilibration model such that hysteresis collapses. So long as the concentration in the effect-site cannot be measured directly, the correct effect-site equilibration model and the one that erroneously collapses hysteresis are experimentally indistinguishable. We also found that hysteresis can naturally arise even in a simple network of neurones independently of drug equilibration.

Conclusions: Effect-site equilibration models can readily collapse hysteresis. However, this does not imply that hysteresis is solely attributable to the kinetics of drug equilibration.

Keywords: effect-site concentration; hysteresis; mechanisms of anaesthesia; modelling; pharmacokinetics.

Fig 1

Pharmacokinetic–pharmacodynamic (PK–PD) model. (a) The general structure of the PK–PD model used in Fig 1, Fig 2, Fig 3, Fig 4, Fig 5. Infusion I(t) is administered into the plasma compartment. The drug distributes to other compartments labelled ‘effect-site’ or ‘rest of body’ or be eliminated. The distribution and elimination are modelled using standard first order chemical kinetics with the rate constants k_xy denoting the rate constant governing distribution from compartment x to compartment y. This constitutes the PK module of the PK–PD model. The PD model is a sigmoid concentration–response curve that relates the concentration in the effect-site compartment to ‘depth of anaesthesia’. (b) Time course of concentration changes after a brief infusion (10 time steps) of the drug (bottom panel). All drug concentrations in this and other figures are normalised to the maximum concentration in the compartment. (c) After a prolonged infusion, the normalised concentration in the effect-site and plasma compartments converge. au, arbitrary unit.

Fig 2

Pharmacokinetic–pharmacodynamic (PK–PD) mechanism of anaesthetic hysteresis. (a) Combined PK–PD model is simulated for a fixed rate infusion (as in Fig. 1c) of drug followed by discontinuation. Note that effect-site concentration lags behind plasma concentration. Depth of anaesthesia is computed as a sigmoid function of the effect-site concentration. (b) When depth of anaesthesia (from a) is plotted as a function of the plasma concentration, hysteresis is readily observed (left). However, when the same data are plotted as a function of the concentration in the effect-site, hysteresis collapses (right). Thus, in this case anaesthetic hysteresis is solely attributable to drug equilibration. au, arbitrary unit.

Fig 3

Neuronal dynamics mechanism of anaesthetic hysteresis. (a) ‘Energy’ landscape plotted as a function of anaesthetic concentration in the effect-site (colour bar). When there is no anaesthetic only the awake state is stable. At high anaesthetic concentrations, only the anaesthetised state is stable. At intermediate anaesthetic concentrations both the awake and anaesthetised states are equally stable. This corresponds to EC₅₀ as such a system will spend 50% of the time being awake or anaesthetised. Rather than using the sigmoid concentration–response curve (Fig. 2) we simulated the system where effect-site concentration (driven by the same pharmacokinetic–pharmacodynamic [PK–PD] model as in Fig 1, Fig 2) changed the shape of the energy landscape. The fraction of time spent anaesthetised is plotted on the ordinate in (b) and (c). Because relaxation of the system to the steady-state in this case is given by neuronal dynamics, here hysteresis is observed when anaesthetic depth is plotted as a function of both plasma (b) and the effect-site (c) concentrations. Thus, in this case, hysteresis is not attributable solely to drug distribution. anes., anaesthetised.

Fig 4

Pharmacokinetic–pharmacodynamic (PK–PD)models can collapse hysteresis caused by neuronal dynamics. (a) Time evolution of drug concentration and depth of anaesthesia. Data are identical to that in Figure 4. Here, a new PK–PD model is created such that plasma concentration and depth of anaesthesia are identical. However, the effect-site concentration is now different. That is effect was produced as a function of correct effect-site concentration (dashed red line), but the new model predicts the incorrect effect-site concentration (red solid line). Because this new model produces identical plasma concentration and effect, it is experimentally indistinguishable from the correct model used to generate the data. (b) Hysteresis exhibited with respect to plasma concentration. (c) Hysteresis collapses with respect to the incorrect effect-site concentration. In this way, PK–PD models can be used to minimise the effect of neuronal dynamics. anes., anaesthetised. au, arbitrary units.

Fig 5

A simple neuronal network can produce stochastic switching between different states at a fixed anaesthetic concentration. (a) The network consists of two neurones, A and B (excitatory and inhibitory). At the low effect-site concentration, self-inhibition is minimal. Strength of self-inhibition is related to the effect-site concentration according to the PK–PD model in Figure 1. (b) Examples of activity produced by the network at three anaesthetic concentrations. At an intermediate anaesthetic concentration the network flips between two distinct oscillatory regimes. (c) Wavelet spectra of neuronal activity in a network simulated at the high (left) and low (right) levels of noise. Power in log units is shown by colour (blue low power, yellow high power). In the high noise regime, the system fluctuates frequently between two oscillatory patterns. In the low noise regime, the system tends to persist in one state. This resistance to state transitions manifests as anaesthetic hysteresis.

Fig 6

Simulation of target-controlled infusion. (a) Simulation of target-controlled infusion (TCI). Infusion rate is adjusted such that effect-site concentration is constant at every step. The TCI algorithm assumes that the effect-site equilibrates rapidly such that during each concentration step effect-site concentration is at steady-state. (b) Dynamics of effect-site concentration under the infusion scheme in (a). Different effect-site concentration curves (that assume different equilibration kinetics) are shown. Colour coding is the same for panels (b)–(e) (colour bar). TCI was computed assuming that equilibration kinetics are rapid relative to step duration (yellow). Slower kinetics of effect-site equilibration (orange and red) are shown for comparison purposes. Because effect-site concentration is assumed rather than measured, it is shown by dashed lines. (c) Mean of 500 simulations of the noisy two well potential system driven by effect-site concentrations shown in (b) colour coded by the equilibration kinetics. Observed plasma concentration is replotted from (a) to facilitate evaluation of the relationship between plasma concentration and the effect. (d) Concentration–response curves computed from data in (c) colour coded by effect-site equilibration kinetics. If the effect-site equilibration is fast, little to no hysteresis is observed because the two well potential is simulated in the noisy regime (Supplementary Fig. S1 demonstrates behaviour as a function of noise). When slower effect-site equilibration is assumed hysteresis is clearly present (induction curves solid lines, emergence curves dotted lines). (e) Magnitude of hysteresis computed as area between induction and emergence concentration curves in (d) over a range of equilibration kinetics (shown by colour). Error bars show standard deviation computed across multiple simulations. One can conclude that hysteresis is either present or absent depending on the assumed kinetics of effect-site equilibration. au, arbitrary units.

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