Design of Cycles by Impulsive Feedback: Application to Discrete Dosing

TL;DR

This paper formulates and analyzes a pulse-modulated impulsive controller for discrete dosing in a Wiener PK/PD setting, recasting closed-loop dosing into a discrete return map and targeting a stable 1-cycle with prescribed period and dose. It provides analytic fixed-point expressions for the desired periodic solution and derives necessary and sufficient orbital-stability conditions in terms of the slopes of amplitude and phase modulation functions. The authors implement a design algorithm using piecewise-affine modulation, validate stability and convergence properties, and demonstrate improved control of neuromuscular blockade with atracurium across a population of patient models. The results show meaningful reductions in underdosing events compared with open-loop regimens, along with insights on transient overshoot and the role of Hopf bifurcations in achieving fast convergence. Overall, the framework offers a biomimetic, analytically grounded approach to robust, closed-loop discrete dosing in medical and industrial contexts.

Abstract

The task of maintaining a predefined level of effect in a dynamical plant by applying periodic control actions often arises in e.g. process control and medicine. When the state variables of the plant represent the concentrations of chemical substances and the control action constitutes an instantaneous introduction of a certain quantity of a chemical or drug, this control setup is referred to as a (discrete) dosing problem. The present paper examines an amplitude- and frequency-modulated impulsive controller that, under stationary conditions, generates a desired sequence of uniform and equidistant control impulses based on continuous measurements of the output of a smooth positive nonlinear time-invariant single-input single-output plant with Wiener structure. The controller design method is based on constructing and stabilizing the fixed point of a discrete map that describes the evolution of the state vector of the continuous plant between successive impulsive control action instants. Stability of the fixed point ensures the existence of a basin of attraction along the stationary trajectory, where the solution of a perturbed closed-loop system converges to the stationary solution. The convergence rate is determined by the slopes of the amplitude and frequency modulation functions of the impulsive controller. The proposed controller is applied to the dosing of the drug \emph{atracurium} in closed-loop neuromuscular blockade, and its performance is evaluated on a database of patient-specific pharmacokinetic-pharmacodynamic models estimated from clinical data. It is demonstrated that an implementation of the standard regimen as a pulse-modulated feedback controller significantly minimizes the incidence of underdosing events.

Figures (21)

• Figure 1: Histograms (blue) and estimated lognormal distributions (red) for the model parameters in the data set.
• Figure 2: The model parameter pairs in the data set. $\alpha_{\min}= 0.0270\le \alpha \le 0.0524=\alpha_{\max}$, $\gamma_{\min}=1.4030\le\gamma\le 5.5619=\gamma_{\max}$. The extreme parameter values are indicated by the Patient Identification Number.
• Figure 3: Fixed point elements calculated as function of $\alpha\in \lbrack \alpha_{\min}, \alpha_{\max} \rbrack$. All the elements are decreasing functions of $\alpha$.
• Figure 4: Spectral radius of $Q^\prime_F(X)$ as function of $F^\prime(\bar{y}_0)$ and $\Phi^\prime(\bar{y}_0)$. The stability border, according to condition \ref{['eq.necess-stab2']}, is depicted by the red line. The mean population value of $\alpha$ is used.
• Figure 5: Absolute values of the eigenvalues of $Q^\prime_F(X)$ as function of $F^\prime(\bar{y}_0)$. The minimal spectral radius is depicted by dashed line. The absolute values of two eigenvalues (a complex pair) coincide after the bifurcation point. The third eigenvalue (yellow line) is small in comparison with the other two and slowly decreases with the decreasing $F^\prime(\bar{y}_0)$.

Theorems & Definitions (17)

• Theorem 1
• proof
• Lemma 1
• proof
• Lemma 2
• Theorem 2
• proof
• Corollary 1
• Proposition 1
• proof