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A Branch and Bound method for the exact parameter identification of the PK/PD model for anesthetic drugs

Giulia Di Credico, Luca Consolini, Mattia Laurini, Marco Locatelli, Marco Milanesi, Michele Schiavo, Antonio Visioli

TL;DR

The paper tackles exact parameter identification for a PK/PD model of anesthetic drugs by formulating it as a nonlinear, non-convex regression problem within a Wiener model framework. A global Branch and Bound method is developed, using a computable lower bound to guarantee convergence to the global optimum of the one-step-ahead prediction-error objective. The authors apply the method to a Wiener model of propofol-induced BIS response, validating on a dataset of 12 patients and showing exact recovery of the nonlinear Hill parameters $(\gamma,E_{max})$ under appropriate ARX modeling choices. The work provides a rigorous optimization-based approach that can improve patient-specific dosing by delivering globally optimal parameter estimates, with potential for broader clinical adoption and extension to other PK/PD scenarios.

Abstract

We address the problem of parameter identification for the standard pharmacokinetic/pharmacodynamic (PK/PD) model for anesthetic drugs. Our main contribution is the development of a global optimization method that guarantees finding the parameters that minimize the one-step ahead prediction error. The method is based on a branch-and-bound algorithm, that can be applied to solve a more general class of nonlinear regression problems. We present some simulation results, based on a dataset of twelve patients. In these simulations, we are always able to identify the exact parameters, despite the non-convexity of the overall identification problem.

A Branch and Bound method for the exact parameter identification of the PK/PD model for anesthetic drugs

TL;DR

The paper tackles exact parameter identification for a PK/PD model of anesthetic drugs by formulating it as a nonlinear, non-convex regression problem within a Wiener model framework. A global Branch and Bound method is developed, using a computable lower bound to guarantee convergence to the global optimum of the one-step-ahead prediction-error objective. The authors apply the method to a Wiener model of propofol-induced BIS response, validating on a dataset of 12 patients and showing exact recovery of the nonlinear Hill parameters under appropriate ARX modeling choices. The work provides a rigorous optimization-based approach that can improve patient-specific dosing by delivering globally optimal parameter estimates, with potential for broader clinical adoption and extension to other PK/PD scenarios.

Abstract

We address the problem of parameter identification for the standard pharmacokinetic/pharmacodynamic (PK/PD) model for anesthetic drugs. Our main contribution is the development of a global optimization method that guarantees finding the parameters that minimize the one-step ahead prediction error. The method is based on a branch-and-bound algorithm, that can be applied to solve a more general class of nonlinear regression problems. We present some simulation results, based on a dataset of twelve patients. In these simulations, we are always able to identify the exact parameters, despite the non-convexity of the overall identification problem.
Paper Structure (14 sections, 3 theorems, 39 equations, 3 figures, 2 tables, 1 algorithm)

This paper contains 14 sections, 3 theorems, 39 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related literature
  3. PK/PD model
  4. Problem formulation
  5. Reinterpretation as a Wiener model
  6. Formulation of the identification problem
  7. Reduction to a nonlinear regression problem
  8. Statement of contribution
  9. A BnB method for solving Problem \ref{['prob_nonl_regr']}
  10. Application to the identification of the Wiener model
  11. Experimental results
  12. Patients database
  13. Numerical tests
  14. Conclusions and future works

Key Result

Proposition 1

Let $M,N \in \mathbb{R}^{m \times n}$, let $k \in \mathbb{R}$, with $k>0$, then

Figures (3)

  • Figure 1: Hill function plot for different identification parameters. Fixed constants are set as $E_0=100$ and $C_{e50}=40$.
  • Figure 2: Plot of $BIS$ for fixed $E_0$, $C_{e50}$ and $C_e$, as a function of $\gamma$ and $E_{\max}$.
  • Figure 3: Plot of function $h$ defined in \ref{['eqn_for_plot']}, for $M=N=2$ and Partient $Id=1$.

Theorems & Definitions (6)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof