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

Abstract

Given a function that describes the uptake of a substance into the body with time, an analytical technique is described which transforms that function into a model of parallel first-order compartments that converges to the same uptake profile as the number of compartments is increased. The fitting of the compartmental model to the given uptake function is optimized to minimize the squared error. A necessary condition of the analytical method is that the uptake function be capable of being successively integrated at least as many times as the number of desired compartments. The uptake function should also be monotonically decreasing as all parallel first-order compartment models predict monotonically decreasing uptake. We applied this technique and ascertained the compartmental structure of the Severinghaus relationship, a longstanding observation in the field of clinical anesthesia that the uptake of nitrous oxide follows an inverse-square-root of time profile. The Severinghaus relationship is numerically poorly behaved at a time of zero elapsed minutes, predicting an instantaneously infinite uptake. Nevertheless, modeling of the first minute of anesthesia is necessary for characterizing the initial induction of anesthesia and methods of maintaining closed-circuit anesthesia such as the unit dose method. Using solely analytical methods, solutions for the compartmental properties of a mammillary model that matches the Severinghaus relationship for any expressed time interval are produced. These properties are compared to currently accepted values for the uptake of nitrous oxide. When matched to the Severinghaus relationship in the range of 0–100 min with a three-compartment model, we identified time constants of 0.28, 4.69 and 33.49 min with associated apparent volumes of 1.44, 2.14 and 7.97 l, respectively. The time constants in particular contrast to our earlier findings for the range of 1–100 min (1.46, 7.41 and 42.0 min). Our earlier findings were well matched to published time constants for tissues in classical pharmacokinetic models for volatile uptake. Consequently, we conclude that rigid adherence to the Severinghaus relationship from a time of zero minutes may lead to the over-administration of anesthetic agent due to an implicit mischaracterization of the relevant compartmental properties.