# Fractional-Order Closed-Loop Model Reference Adaptive Control for Anesthesia
## 3 Anesthesia
### Pharmacokine/Pharmacodynamic Model  
The PK/PD models most frequently used for propofol (hypnotic drug) are the fourth-order compartmental models (Figure 2) [58; 59]. These models have been developed, tested, and validated on a wide range of real patient data and have often used in the literature for the control of anesthesia [60]. In this paper, we use the model presented in [58] given by  
Figure 1: Input–output variables in anesthesia. Figure 2: PK/PD (pharmacokinetic/pharmacodynamic) model. \[\dot{x}(t)= \begin{bmatrix}-(a_{11}+a_{21}+a_{31})&a_{12}&a_{13}\\ a_{21}&-a_{12}&0\\ a_{31}&0&-a_{13}\end{bmatrix} x(t)+ \begin{bmatrix}1\\ 0\\ 0\end{bmatrix} u(t), \tag{10}\]  
where \(x_{1}\) represents the concentration of the drug in the central compartment (intravascular blood); \(x_{2}\) y \(x_{3}\) represent the concentration of the drug in the peripheral compartments; \(a_{12}\), \(a_{13}\), \(a_{21}\), \(a_{31}\) are positive constants representing the flow between compartments; \(a_{11}\) is the elimination rate of the drug through the metabolism; and \(u(t)[mg/min]\) is the rate of infusion of anesthetic (propofol) in the central compartment. \(a_{ij}\) are in \([min^{-1}]\) and \(x_{i}\) are in \([mg]\). An additional compartment, namely, the effect compartment, is introduced to represent the time delay between the observed effect and the plasma concentration. The effect compartment model links the plasma concentration (concentration in the central compartment) to the effect concentration with a first-order differential equation:  
\[\dot{y}(t)=a_{eff}(x_{1}(t)-y(t)),\;\;y(0)=x_{1}(0),\;\;t\geq 0, \tag{11}\]  
where \(a_{eff}\) is the time constant; \(x_{1}(t)\) is the concentration in the central compartment defined in (10); and \(y(t)\) is the concentration of the effect compartment. The pharmacokinetic parameters can be obtained through the following equations [58]:  
\[V_{1} = 4.27[l]\] \[V_{2} = 18.9-0.391(age-53)[l]\] \[V_{3} = 2.38[l]\] \[C_{11} = 1.89+0-0456(weight-77)-0.06681(ibm-59)\] \[+0.0264(height-177)[l/min]\] \[C_{12} = 1.29-0.024(age-53)[l/min]\] \[C_{13} = 0.836[l/min]\] \[a_{11} = \frac{C_{11}}{V_{1}}[min^{-1}];\;a_{12}=\frac{C_{12}}{V_{1}}[min^{ -1}];\;a_{13}=\frac{C_{13}}{V_{1}}[min^{-1}]\] \[a_{21} = \frac{C_{21}}{V_{2}}[min^{-1}];\;a_{31}=\frac{C_{13}}{V_{3}}[min^{ -1}]\] \[V_{1} = 4.27[l]\]0 \[V_{1} = 4.27[l]\]1 \[V_{1} = 4.27[l]\]2  
As we can observe in the previous equations, the PK parameters depend on the biometrical characteristics of the patient. The bispectral index (BIS) is a signal derived from the electroencephalogram (EEG) used to assess the level of consciousness in anesthesia. A BIS value of 0 equals a flat line in the EEG, while a BIS value of 100 is the expected value of a fully conscious adult patient. The 60-70 and 40-60 ranges represent light and moderate hypnotic conditions, respectively. The target value during surgery is 50, giving us a gap between 40 and 60 to guarantee adequate sedation [6]. The BIS can be related to the concentration of the effect compartment by the nonlinear static function, termed the Hill equation [6]:\[V_{1} = 4.27[l]\]3  
where \(E_{0}\) denotes the base value (awaken state) and by convention, typically is given a value of 100; \(E_{max}\) is the maximum effect achieved by drug infusion; \(EC_{50}\) is the drug concentration to half maximal effect and represents the patient's sensibility to the drug; and \(\gamma\) determines the degree of nonlinearity of the function. One of the significant challenges in the control of anesthesia is the variability among patients. This variability can occur as a result of patient physiology (age, gender, disease), variations in the PK processes (rates of absorption, distribution, metabolism, and elimination), and differences in PD (sensitivity of receptor, etc.) [61]. Also, in the medical practice, no state of this model is available for measurement, only the output (BIS) is measurable for feedback.