# Identification of Wiener Box-Jenkins Model for Anesthesia Using Particle Swarm Optimization
## 1 Introduction  
Anesthesia is the process of preventing the pain and mitigating the stress of the patient's body to injury during medical operations [1]. The process is conducted by injecting gases and/or drugs by a specialist before and during surgical operations. Anesthesia can be performed by inhalationof smelled gases or intravenous injections using needles [2]. As a result of the advanced devices available in the market for inhalation, the current study is focused on intravenous anesthesia. In both cases, the process can be divided into three stages: (1) induction, which starts from administration of drug to the patient until he/she reaches unconscious, (2) maintenance during which the surgery is performed, and (3) recovery, in which the patient regains consciousness [3]. The use of advanced control and automation in anesthesia focuses on the maintenance stage in order to inject the correct dosage, which keeps the level of patient's unconsciousness within safe limits during operations. This is of vital importance, especially if there is a shortage of anesthesiologists in, e.g., war times, natural disasters, or remote areas [4]. The level of Anesthesia is usually expressed in terms of the so-called Depth of Anesthesia (DoA), which measures the degree of suppression of the central nervous system [1]. Various techniques have been used for the control of the DoA including Proportional Integral Derivative (PID) [5, 6, 7], fractional PI [8], model predictive controller (MPC) [9], and neuro-fuzzy control [10]. Attempts to model DoA began around the middle of the twentieth century through monitoring brain activity using Electroencephalogram (EEG) signal. Later, in the eighties, other attempts are made predominantly on inhaled anesthesia [11]. In the past two decades,modeling of intravenous anesthesia received more attention from researchers [3; 5; 9; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23]. Although there are attempts to use linear models such as [24], most of the proposed models are nonlinear, employing a linear transfer function subsystem followed by the so-called hill static nonlinearity function. This model structure is termed Wiener model in the system identification literature and is widely used in the modeling of anesthesia [2; 3; 14; 16; 23; 25]. One common model approximating the DoA process and giving deep insight is the so-called Pharmacokine-Pharmacodynamic nonlinear model (PK/PD), where the infusion rates of the propofol and remifentanil are the input signals and the DoA obtained by the spectral Bispectral index (BIS), which monitors the depth of anesthesia, is the output. The approximated model is represented by Wiener model, where the PK part comprises the linear dynamic subsystem in cascade with PD static nonlinearity [21; 22]. There are few iterative approaches to model drug dynamics during anesthesia such as artificial neural network [13] where the prediction results are very promising, however, the method did not perform well due to the presence of delays in this application. Additionally, the multi-input multi-output (MISO) output error model using closed loop instrumental variables (IV) method is employed [18]. Although this approach is favorable to the black-box neural network model, it is not very accurate either. As the PK/PD model of drug dynamics during anesthesia consists of linear block followed by nonlinearity, it is proposed, in this paper, to use the Wiener Box-Jenkins model to the relationship between the infusion rate of muscle relaxant, e.g., atracurium, \(u(t)\) in [ugkg\({}^{-1}\)min\({}^{-1}\)] and the level of neuromuscular blockade NMB or DoA in [%]. By simulating the process in closed-loop configuration, a set of input-output data, sampled every 0.1 min, is generated to be used in offline identification of model parameters. The model parameters are estimated by minimizing the mean square one-step ahead prediction error (MSE) using Particle Swarm Method (PSO), a very successful meta-heuristic optimization technique [26; 27; 28].