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Abstract
This work addresses the problem of computing the time evolution of the probability density function (pdf) of the state in a nonlinear neuromuscular blockade (NMB) model, assuming that the source of uncertainty is the knowledge about one parameter. The NMB state is enlarged with the parameter, that verifies an equation given by its derivative being zero and has an initial condition described by a known pdf. By treating the resulting enlarged state-space model as a stochastic differential equation, the pdf of the state verifies a special case of the Fokker-Planck equation in which the second derivative terms vanish. This partial differential equation is solved with a numerical method based on Trotter’s formula for semigroup decomposition. The method is illustrated with results for a reduced complexity NMB model. A comparison of the predicted state pdf with clinical data for real patients is provided. © Springer International Publishing Switzerland 2014.

Abstract
This paper addresses the problem of joint estimation of the state and parameters for a deterministic continuous time system, with discrete time observations, in which the parameter vector is constant but its value is not known, being a random variable with a known distribution. Along time, the uncertainty in the parameter induces uncertainty in the plant state. The joint probability density function (pdf) satisfies the Liouville partial differential equation that is a limit case of the Fokker-Planck equation for vanishing diffusion. The continuous-discrete filter proposed operates as follows: Between two consecutive output sampling time instants, the pdf is propagated by solving the Liouville equation for an augmented state and is then corrected by using the last observation and Bayes law. An application to state estimation of the neuromuscular blockade of patients subject to general anesthesia, where parameter uncertainty is due to inter-patient variability, is described.