Observer-Based Strategies for Anesthesia Drug Concentration Estimation

Knowledge of the anesthesia drug concentration in a patient during a surgical procedure has potential to improve patient monitoring and control of drug administration. High-fidelity estimates of anesthesia drug concentration would allow anesthesiologists to make relevant low-level dose adjustment as well as high-level clinical decisions. The estimated drug concentrations may also be useful for computerized anesthesia delivery, including, but not limited to, targeting/limiting the drug concentration itself (Van Poucke et al., 2004; Gentilini et al., 2002) as well as monitoring and maintaining hemodynamic stability (Rao et al., 2000).

describe the clinical effect of interest for feedback correction. The observers were evaluated against the traditional open-loop prediction using a Monte-Carlo simulation of surgical procedures using a wide range of patient models. H ∞ O boasted the best overall performance with its capacity to deliver statistically significant reduction in plasma and effect site propofol concentration errors over open-loop prediction, regardless of the presence of a surgical stimulation disturbance. RNO was more effective than open-loop prediction in suppressing the effect site propofol concentration error, but its efficacy for plasma propofol concentration was susceptible to surgical stimulation disturbance. RLO was good at reducing the effect site propofol concentration error in the absence of surgical stimulation disturbance, but perfomed poorly in the presence of a surgical stimulation disturbance. This chapter is organized as follows. Section II describes the patient model. Section III details the design of observers for anesthesia drug concentration estimation. Section IV details the Monte-Carlo simulation and statistical analysis to compare the performance of observer-based estimation and open-loop prediction strategies. The results are presented and discussed in Section V, followed by conclusions in Section VI.

Patient model
An anesthetized patient can be described by a series connection of a PK model, a PD model, and a monitor model, as shown in Fig. 2. The input to this process is the drug infusion rate (I H ), while the output is the clinical effect measurement (WAV CNS ). The surgical stimulation is modeled as an output disturbance, because it counteracts the effect of anesthesia drug in an unpredictable manner and thereby distorts the WAV CNS measurement. Fig. 2. PK/PD and effect monitor models of anesthesia drug delivery process.

PK and PD models
A 3-compartment model of propofol developed by Schüttler and Ihmsen (2000) was used to describe the PK (Fig. 3). Denoting x 1 , x 2 , and x 3 as drug concentrations in the plasma, fast peripheral, and slow peripheral compartments, the state-space representation of the PK is given by (1) where x PK =[x 1 x 2 x 3 ] T , u=I H , k ij are the rate constants, and V 1 is the volume of the plasma compartment. To account for the PD lag associated with the distribution of drug into the effect site (i.e. the brain), the following delay-plus-first-order model was used (Zikov et al., 2006): where T d and k d are respectively the transport delay and the rate of drug distribution from the plasma to the brain. Finally, the saturating Hill equation was used to describe the relation between C e and the anesthesia effect: where 50 e e C C EC  is the effect site drug concentration normalized by the 50% effect concentration (EC 50 ), and  is the cooperativity coefficient.

WAV CNS monitor model
The WAV CNS index reflects the combined effect of the anesthesia drug and the unknown surgical stimulation. It takes values ranging from 100 to 0, where WAV CNS =100 corresponds to fully awake state, WAV CNS =50 corresponds to adequate anesthesia, and WAV CNS =0 corresponds to an isoelectric electroencephalogram (EEG). The WAV CNS monitor model derived by Zikov et al. (Zikov et al., 2006) was used: where d is the unknown surgical stimulation. Note that maximum (=1) and minimum (=0) clinical effects correspond to WAV CNS =0 and WAV CNS =100, respectively.

Observer design model
The observer design model is obtained by locally linearizing the PD model in the neighborhood of the operating regime of E=0.5 during the maintenance phase of anesthesia, which yields Combining (1), (2), (4), and (5)

Observer design
A patient undergoing anesthesia is subject to a significant amount of uncertainty arising from the inherent inter-patient physiologic variability and unknown surgical stimulation. The population-based PK and PD models have thus limited predictive accuracy (Schüttler & Ihmsen, 2000). Besides, the PK and PD model parameters may fluctuate in response to the individual patient's physiologic condition. Furthermore, the clinical effect measurement is persistently corrupted by unpredictable surgical stimulation. The observer must be designed to be robust against these multiple confounding factors.

Design strategies
The following alternatives were considered as observers for drug concentration estimation: 1) RLO: a linear observer with a robust feedback gain, 2) RNO: a nonlinear observer with a robust feedback and a nonlinear disturbance rejection gain, and 3) H ∞ O: a robust dynamic observer designed based on H ∞ control theory. The RLO was intended to explore the limit of performance of a simple linear observer. The RNO was motivated by the effort to separate the tasks of robustness and disturbance rejection: the linear feedback was designed to achieve robustness against PK/PD model uncertainty, whereas the nonlinear feedback was designed to compensate for the unknown surgical stimulation. The requirements of disturbance rejection and the observer matching condition necessitates the use of WAV CNS and its derivative as measurements in the design of RNO. The purpose of the H ∞ O was to examine if its added complexity is worth exploiting in order to improve the fidelity of drug concentration estimation.

Robust linear observer (RLO) design
A standard linear observer assumes the following form: where the upper bar on a matrix denotes its nominal value. L is designed to satisfy the inequality (8) for a positive definite matrix P in order to place the closed-loop poles in the left-half plane: Considering the patient model (6) and the observer (7), the propofol concentration estimates provided by the observer are susceptible to the model perturbations included in Ed that stem from the PK/PD uncertainty and unknown surgical stimulation. The observer gain matrix L must be designed so as to achieve robustness against these confounding factors. It is possible to determine L such that the state estimation error is bounded within a prescribed ellipsoid in the presence of model perturbations and/or disturbances by exploiting Theorem 1 below: Theorem 1 (Set-Theoretic Robustness Condition) (Blanchini & Miani, 2008): Consider the system of the form   xA xF w  which is subject to a persistent external disturbance w. Assume that the disturbance is bounded by w T w≤1. Then the ellipsoid {x: Rewrite (6) as follows where the matrices A and B are decomposed into their nominal and uncertain parts: Subtracting (7) from (9) yields the following error dynamics: where F specifies the uncertainty bound and is chosen such that w T w≤1. Then according to Theorem 1 the state estimation error e is bounded within the ellipsoid {e: e T Pe≤1} if Q=P -1 satisfies the following inequality for some >0: Pre-and post-multiplying (11) by P results in (12): Noting that P and PFF T P in (12) are positive definite and positive semi-definite, respectively, the inequality (12) is stronger than (8) and the pair of matrices L and P satisfying (12) automatically satisfies (8). Also, the "size" of P is directly related to the size of the error ellipsoid within which the state estimation error is bounded. Therefore, the optimal observer gain L for (7) can be determined by finding the pair of matrices L and P which maximizes the size of P while satisfying the set-theoretic robustness condition (12).
The optimal RLO gain design problem can be cast into a linear matrix inequality (LMI) problem by manipulating (12) (14) the inequality (13) can be converted to the following LMI: Summarizing, the optimal RLO gain design is equivalent to solving the following LMI: where the size of P is measured in terms of its trace norm, which is equal to the sum of its eigenvalues.

Robust nonlinear observer (RNO) design
The RNO to be designed in this chapter assumes the following form: where L is the linear feedback designed to suppress the adverse effect of the PK/PD uncertainty, and  is the nonlinear feedback to compensate for the unknown surgical stimulation. The error dynamics is derived by subtracting (17) from (9): where G specifies the uncertainty bound and is chosen such that w T w≤1. Note that which suggests that within the RNO framework the PK/PD uncertainty Gw is taken care of by L, whereas the surgical stimulation is accounted for by , in contrast to the RLO where both of these confounding factors Fw=Gw+Ed must be dealt with by L.
To exploit  to cancel out the effect of d, it is required that the following observer matching condition be satisfied for some function h(x,u): where P satisfies (8). Note that (20) cannot be satisfied by the output matrix C in (6); it violates the positive definite requirement on P by imposing p 66 =0. In order to resolve this problem, we assume that WAV CNS and its time derivative are available as sensor measurements: With this choice of the output, the following constraints are imposed on P: Then the linear feedback gain L of the RNO can be designed by solving the LMI (16) with F substituted by G, subject to (22). Once a feasible L and P are obtained from the LMI, the function h(x,u) is directly obtained from (22).
Assuming that the linear feedback gain L and the positive matrix P are designed to satisfy (16) with F substituted by G and the observer matching condition (20), the nonlinear feedback  with  satisfying h(x,u) guarantees the boundedness of the observer error e: Note that 1) the term (a) is strictly negative whenever e is outside of the ellipsoid defined by {e: e T Pe≤1}, and 2) the term (b) is always strictly negative. Thus, the time derivative of V(e) is strictly negative if e T Pe≤1 is violated. Therefore, the observer error is guaranteed to converge to and be bounded within the ellipsoid {e: e T Pe≤1}.

H ∞ observer (H ∞ O) design
The H ∞ O was designed as described in Hahn et al. (2011) and the detailed discussion on its design is therefore omitted here. Briefly, the H ∞ O assumes the same form as the RLO (6). However, in contrast to the RLO whose observer gain L is a constant matrix, the observer gain L of the H ∞ O is a transfer matrix: L=L(s), where s is the Laplace variable. Therefore, the state estimation is given by: where the feedback  is the output of a dynamic system with the clinical effect estimation error as its excitation signal: The transfer matrix between  and ŷy The objective is to design a feedback gain L(s) of the H ∞ O that is robust against Fw. In the frequency domain, (6) and (25) where W e (s) is a weighting function specifying the desired error bound. The linear fractional transformation (LFT) (Skogestad & Postlethwaite, 1996) can be used to convert (30) to the standard H ∞ optimization set-up (Skogestad & Postlethwaite, 1996) shown in Fig. 4 where k specifies the error bound at steady state,  1 specifies the observer bandwidth, and  2 > 1 . Then the state-space representation (33)  Based on (33), the optimal H ∞ O can be designed using the D-K iteration (Skogestad & Postlethwaite, 1996). Note that although (33) does not satisfy all the regularity assumptions required for H ∞ optimization (Skogestad & Postlethwaite, 1996), the problem can be resolved by any standard regularization procedure (e.g. Pertew et al., 2006). www.intechopen.com

Design of observers
The demographic and PD data of 15 patients aged 18-30 (22+/-3.2) listed in Dumont et al. (2009) were used to obtain a nominal PK and PD model (1)-(4). Then 100 random Monte-Carlo models were created for the nominal model by applying up to +/-15% perturbations to all the PK and PD parameters. The PK/PD variability was specified as follows: where  i , i=1,...,6 were chosen to encompass the resultant effect of up to +/-15% random PK/PD parametric perturbations as well as surgical stimulation of up to +/-20 WAV CNS units.
Before designing the observers, the observability of the system was analyzed by evaluating the observability grammian matrix (35) with the nominal patient model: using which the relationship between the dominant direction of each eigenvalue and the propofol concentration at each compartment was examined. The condition number of W o was also calculated to assess the well-posedness of the observer design problem.
Using the nominal patient model, the observer gain matrix L for RLO and RNO were designed by solving the LMI (8) and (16) with =1 and =10 -5 , respectively. In the RNO design, the constraint (22) was imposed on P. The H ∞ O was designed by applying the D-K iteration procedure to the standard set-up (33). The weighting function (32) was specified as k=5.0,  1 -1 =2.0×10 -5 Hz, and  2 -1 =1.0×10 9 Hz, which requires the error to be bounded by 20% up to 2.0×10 -5 Hz, beyond which an increase in size of the error is permitted.

Monte-Carlo simulation
The 100 randomly created patient models were used for the Monte-Carlo simulation. TCI administration of propofol to target the effect site propofol concentration of 3mcg/ml was chosen as the simulation scenario (Fig. 5). The duration of the simulated procedure was 2000s. The surgical stimulation shown in Dumont et al. (2009) was adopted and applied to the patient models from 1200s to 1700s (Fig. 5). The open-loop-predicted propofol concentrations were calculated by solving the PK and PD models (1) and (2).

Statistical analysis
From the simulation of each of the 100 random patient models, the following plasma and effect site percentage errors (PE) were calculated for both observer-based estimation ( (Varvel et al., 1992). The errors were calculated for the induction phase of anesthesia (t<600s) and the entire procedure (t<2000s)

Observability analysis and observer design
The effect site propofol concentration x 4 (which is the primary concentration of interest) was associated with the second smallest eigenvalue of W o , whereas the peripheral compartmental concentrations x 2 and x 3 corresponded to its largest and second largest eigenvalues. This implies that the influence of x 4 on the output (x 5 ) is relatively weak compared to x 2 and x 3 , because x 2 and x 3 can dominate x 4 in terms of output energy through their slowly decaying dynamics (i.e. x 2 and x 3 decay much slower than x 1 and x 4 ). The condition number of W o was very large (approximately 821), indicating that the degree of observability for propofol concentrations in different compartments can be significantly different. Essentially, the observability analysis clearly illustrates that estimating x 4 is inherently a challenging task.
RLO design resulted in L with optimal tr[P]=6.7×10 8 . The real part of the closed-loop eigenvalues ranged from -4.5 to -0.4. RNO design resulted in L with optimal tr[P]=8.2×10 8 . The real parts of the closed-loop eigenvalues range from -2.1×10 2 to -1.2×10 -4 . H ∞ O design resulted in L(s) that has robust stability margin of ∞ and robust performance margin of 1.1, indicating that the closed-loop system is always stable, and is robust against up to 1.1 times the uncertainty incorporated in the observer design (Balas et al., 2009).

Performance evaluation and statistical analysis
The distributions of MDAPE, MNAPE, and MXAPE of the propofol concentration errors at the plasma and the effect site are shown in Fig. 6 and Fig. 7 for the induction phase of anesthesia and the entire procedure, respectively. Tables 1-3 summarize mean (SD) of these errors. The RNO and the H ∞ O outperformed the RLO. Except for its benefit in reducing MXAPE of C e during the induction phase of anesthesia (by 21%, p<10 -5 ; Table 3), the RLO was no better than the open-loop prediction (Table 1 and Table 2). Compared with the open-loop prediction, the RNO was capable of reducing MXAPE of C e (by 59%, p<10 -5 ) and C p (by 38%, p<10 -5 ) during the induction phase of anesthesia (Table 3), and MXAPE of C e (by 43%, p<10 -5 ) during the entire procedure (Table 3). Its MDAPE and MNAPE were as good as those of the open-loop prediction during the induction phase of anesthesia, but they deteriorated more than the open-loop prediction when evaluated for the entire procedure (Table 1 and  Table 2). The degraded performance of the observers in the presence of surgical stimulation is expected because it distorts the clinical effect measurement, resulting in an inappropriate feedback correction, whereas the open-loop prediction is not affected by surgical stimulation (see Fig. 1). In contrast to RLO and RNO, the H ∞ O was able to deliver statistically significant reduction in all the PE metrics over the open-loop prediction, regardless of the presence of surgical stimulation (see Table 4). Moreover, the SD of all the metrics for the H ∞ O were significantly smaller than those of the open-loop prediction, indicating that the propofol concentrations provided by the H ∞ O are more reliable. This suggests that, in contrast to the open-loop prediction in which the PK/PD variability directly deteriorates the propofol concentration estimates, the observer's feedback correction based on the clinical effect measurement is viable in suppressing the propagation of the adverse influences of the PK/PD variability into the propofol concentration errors. C e 13% (p<10 -10 ) 21% (p<10 -10 ) 73% (p<10 -10 ) 24% (p<10 -10 ) 25% (p<10 -10 ) 72% (p<10 -10 ) C p 11% (p<10 -10 ) 15% (p<10 -10 ) 59% (p<10 -10 ) 22% (p<10 -10 ) 21% (p<10 -10 ) 56% (p<10 -10 ) Several important observations can be made from the overall results. First, the observers were effective in reducing MXAPE (except for the RLO in the presence of surgical stimulation, i.e. during the maintenance phase). This is attributed to the philosophy behind the design of robust observers -the observers considered in this chapter were designed to improve the worst-case performance (Skogestad & Postlethwaite, 1996). Due to the design objective of guaranteeing performance and robustness even in extreme situations, the behaviour of these observers on patients whose dynamics are rather close to the nominal design model may not be optimal (this may partly explain why MDAPE and MNAPE metrics of RLO and RNO are not as good as the open-loop prediction).
Second, in spite of its robustified feedback gain, the standard linear observer was not an attractive strategy for anesthesia drug concentration estimation, due to its limited performance in the presence of surgical stimulation. In our attempt to overcome this limitation, we presented an RNO and an H ∞ O as alternatives to the RLO. The RNO could provide a marginal improvement that was not sufficient to justify the cost of additional measurement (i.e. the time derivative of the clinical effect). The H ∞ O exhibited significantly superior performance to RLO and RNO without any extra measurement requirements, by virtue of the additional flexibility in its structure: the feedback gain L(s) is a dynamic system rather than a constant matrix as in RLO and RNO. Despite their limited performance in the presence of surgical stimulation, RLO and RNO can still be considered as valid options for applications in the induction phase of anesthesia and the operations involving minor surgical procedures, by virtue of their structural simplicity in implementation in comparison with H ∞ O.
From the design perspective, one advantage of the observer-based strategies investigated in this chapter is the explicit specification of the amount of PK/PD uncertainty and surgical stimulation it must tolerate (see the matrix F in (16) for RLO and RNO and in (33) for H ∞ O) and its systematic use for optimizing the observer performance so that the observer is tuned to appropriately exploit the clinical effect measurement distorted by surgical stimulation, in order to better estimate the drug concentrations over the open-loop prediction in spite of the PK/PD uncertainty.

Conclusion
This chapter investigated alternative observer-based strategies for estimation of anesthesia drug concentrations. The design objective was dedicated to robustness of observers against the PK/PD model uncertainty and the unknown surgical stimulation. In this chapter, we considered linear (RLO), nonlinear (RNO), and dynamic (H ∞ O) observers. The performance of the observers in comparison with traditional open-loop prediction was evaluated using a www.intechopen.com