The Severinghaus square root of time relationship for anesthetic uptake and its implications for the stability of compartmental pharmacokinetics

Abstract

For nitrous oxide, the first anesthetic for which uptake was measured in humans, Severinghaus noted empirically that a plot of the log of uptake against the log of elapsed time produced a straight line with slope -0.5, suggesting that uptake is proportional to the inverse square root of time. This result is something of a black box model, based on empirical curve fitting without regard to physiology. Some authors (e.g., Lowe) repeatedly returned to this inverse square root of time model as a benchmark while others (e.g., Hendrickx) questioned its validity and demanded the relationship be expressed with a physiologic model whose structure matches the known physiology being modeled. Nevertheless, the fact that authors have repeatedly come back to this inverse square root of time model as a benchmark suggests that it might have some underlying validity which has not previously been recognized. We re-explored this mathematically in an attempt to reveal hitherto undiscovered insights or limitations. In this study, we examined the square root of time model (viewed as a power function) and compared it with multi-compartment models. Further, we explored the stability of this relationship to systematic variation in the power value and also to the superimposition of noise-like perturbations, seeking conditions under which it might not work. Based upon this theoretical analysis, we also speculate on the existence of a physiological compartment with a time constant between that of the vessel-rich group (VRG) and muscle, and what the identity of such a compartment might be.

Closed-form solutions for the optimum equivalence of first-order compartmental models and their implications for classical models of closed-circuit anesthesia

“…Nevertheless, the inverse-square-root of time relationship became the empirical basis for the practice of closedcircuit anesthesia Ernst 1981, Lowe 1979). Previously (Connor and Philip 2008), we clarified the behavior of this relationship, demonstrating that the Severinghaus relationship corresponds to a narrow range of appropriate time constants for equivalent mammillary models, and that these values are stable in the presence of measurement noise and plausible inter-patient physiological variation over 1-100 min of exposure. A criticism of the Severinghaus relationship is that it predicts that the uptake should be infinite at the instant of initiation of constant alveolar tension ventilation.…”

“…. t N (Connor andPhilip 2008, Kaufmann 2003). This linear combination is introduced by the linear weighting coefficients r 1 .…”

“…This result has had remarkable persistence in the field of anesthesia, including extension to inhaled vapors (Lowe and Ernst 1981) and finding use as a reference for other more physiologic models that have followed (Lowe and Ernst 1981, Lowe 1972, Lerou and Booij 2001, da Silva et al 1997. These models are mammillary, linear, first-order, multi-compartment models (Cowles et al 1973, Philip 1986, Bouillon and Shafer 2000 of the form illustrated in figure 2 of our earlier paper (Connor and Philip 2008). In contrast to the empirical Severinghaus relationship, models of this type can be adjusted to take account of known physiological variations, such as hyperventilation (Munson and Bowers 1967), anesthesia-depressed ventilation (Munson and Bowers 1967), changes in cardiac output and distribution (Munson 1968) or circulation time (Zwart et al 1972).…”

“…In this study, we therefore elaborate a novel and wholly analytical approach to producing a closed-form model of parallel first-order compartments and then apply this technique to the Severinghaus relationship. In doing so, we deduce the significance of the first minute of uptake on the validity of the Severinghaus relationship-a matter that had previously been unresolved (Connor and Philip 2008).…”

“…this is a property of all nonlinear numerical methods, and no program or mathematical procedure can be guaranteed to keep one out of local minima.' In a previous work (Connor and Philip 2008), we applied a numerical transformation (Kaufmann 2003) to convert this problem into a simpler, linear fitting task to determine the underlying equivalent compartmental properties of the Severinghaus relationship (Severinghaus 1954)-a longstanding empirical relationship in the field of clinical anesthesia.…”