Extended habituating model predictive control of propofol and remifentanil anesthesia

Biomedical Signal Processing and Control 55 (2020) 101656

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Biomedical Signal Processing and Control journal homepage: www.elsevier.com/locate/bspc

Extended habituating model predictive control of propofol and remifentanil anesthesia Neda Eskandari, Klaske van Heusden, Guy A. Dumont ∗ Department of Electrical and Computer Engineering, The University of British Columbia, Vancouver, BC V6T 1Z4, Canada

a r t i c l e

i n f o

Article history: Received 30 July 2018 Received in revised form 24 June 2019 Accepted 17 August 2019 Keywords: Automated drug delivery Automated anesthesia Medical control systems Model predictive control

a b s t r a c t This paper proposes a model predictive control (MPC) solution for closed-loop propofol and remifentanil infusion based on feedback from a depth-of-hypnosis monitor, a multi-input single-output system. A clinically relevant objective function is proposed, extending the habituating MPC solution. The presented MPC controller meets the following clinical objectives; it (1) gives the anesthesiologist control over the analgesic-hypnotic balance, (2) reacts fast in response to disturbances, (3) maintains a bounded response to noise, and (4) is robustly stable toward patient variability. The proposed MPC framework and controller design will facilitate practical implementation of patient safety constraints compared to previously proposed LTI solutions. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction During surgery under general anesthesia, a combination of drugs is administered to allow the surgeon to perform a procedure, while maintaining patient safety and comfort. In total intravenous anesthesia, the anesthestic drug propofol is administered to achieve hypnosis, in combination with analgesia using for example remifentanil, which is required to avoid harmful physiological responses to painful stimulation. Individual drug requirements vary between patients and depend on the procedure and clinical context. For certain procedures muscle relaxation is required. In current practice, the anesthesiologist manually adjusts intravenous drug dosing using feedback from indirect measures of the patient state, like blood pressure, heart rate and sweating, taking the type of procedure and clinical context into account. Closed-loop control of anesthesia can take over the low-level task of drug infusion adjustments, and decrease the workload. It has been shown that automated drug delivery systems can decrease the risks of under- and over-dosing which can cause patient awareness during surgery or may negatively affect patient outcomes. Closed-loop control of anesthesia has been shown to outperform manual control in terms of time in range of adequate anesthesia (e.g. [1,2]). Various closed-loop control techniques have been suggested for inducing and maintaining depth-of-hypnosis (DOH), for SISO

∗ Corresponding author. E-mail address: [email protected] (G.A. Dumont). https://doi.org/10.1016/j.bspc.2019.101656 1746-8094/© 2019 Elsevier Ltd. All rights reserved.

control of propofol infusion, or for control of both propofol and remifentanil. Feasibility of PID control of propofol anesthesia has been shown in adults ([3,4]) and children ([5]). Model predictive controllers (MPC) have been proposed to automate propofol administration [6–8]. In a clinically evaluated adaptive MPC controller, data from induction of anesthesia was used to identify a patient model [9]. This patient model was then used to control propofol infusion during maintenance of anesthesia. An extension to MIMO control of both propofol and remifentanil was proposed [10]. While clinical anesthesia is a MIMO problem, the use of MISO control has been proposed because of the lack of a reliable measure of analgesia. MISO MPC control has been proposed based on interaction models of propofol and remifentanil [11,7], focussing on the effect of remifentanil on DOH in the absence of stimulation. Focussing on the analgesic properties of remifentanil, other MISO control systems have been motivated by the observation that variability in the DOH has been associated with lack of analgesia [12,13]. Mid-ranging control has been proposed to exploit the fast dynamics of remifentanil [14]. Control solutions for MISO systems that satisfy steady state requirements are not unique. Habituating control [15] proposed a MISO model-matching and an MPC solution that exploit the additional degree of freedom, using an additional objective appropriate for valve position control. In the context of control of anesthesia, the MISO designs cited above do not explicitly use the second degree of freedom provided by remifentanil infusion. The analgesic-hypnotic balance is defined indirectly by the controller or tuning parameters. Consequently, the user does not have control over this balance, and

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N. Eskandari, K. van Heusden and G.A. Dumont / Biomedical Signal Processing and Control 55 (2020) 101656

cannot tailor the anesthetic to the procedure and clinical context. To overcome this limitation, previous work in our group proposed an LTI model matching control framework motivated by habituating control, which includes a clinically relevant second control objective [16]. In clinical practice, control of anesthesia will require multiple safety constraints [8]. For clinical evaluation, the abovementioned MISO LTI controller was implemented with additional patient safety constraints and anti-windup [17]. MPC allows straightforward implementation of constrained control, as well as optimal control when the constraints are active. This paper therefore proposes an MPC controller that meets objectives for MISO propofol-remifentanil anesthesia as defined in [16]. The proposed solution is motivated by the model predictive formulation of habituating control [15], which is extended to include a clinically relevant objective for remifentanil infusion. Section 2 gives additional background on clinical anesthesia, describes the design objectives and the models used in this study. Section 3 summarizes habituating MPC and introduces the proposed extension. The propofol-remifentanil MISO controller design is detailed in Section 4, followed by simulation results. Finally Section 5 focuses on a brief discussion about the outcomes and a conclusion. 2. Background and problem formulation 2.1. Background Anesthetic and analgesic drug requirements in the operating room depend on the patient’s drug sensitivity and health as well as the procedure. The level of surgical stimulation following incision, for example, is much higher than during surgical preparation. To achieve an opioid-hypnotic balance that is appropriate for the patient, procedure and clinical context, the anesthesiologist adjusts anesthetic and analgesic drugs individually. Indirect measures of hypnosis and anti-nociception such as blood pressure, heart rate, sweating and movement are commonly used to guide drug infusion. Monitors such as the Bispectral Index (BIS) (Medtronic), Entropy (GE Healthcare), and NeuroSENSE (NeuroWave Systems) can be used to monitor DOH in clinical practice. Increasing dosing of analgesia with propofol anesthesia did not significantly change the BIS for hypnotic endpoints, while it did reduce the variability in the BIS following stimulation [18]. Addition of remifentanil affected BIS only following painful stimulation [13]. Interaction models quantifying the effect of propofol and remifentanil indicate some effect of remifentanil on BIS during sedation [19]. In the absence of stimulation, limited effect was reported at DOH levels corresponding to general anesthesia [20]. In contrast, strong synergy has been reported for blunting responses to noxious stimuli [21].

Fig. 1. Controller structure: Feedback WAVCNS from the NeuroSENSE monitor with dynamics GNS is used to control both propofol infusion and remifentanil infusion. The patient response is determined by the propofol dynamics GP and the remifentanil dynamics GR , and affected by noise n and disturbance d. The second output Ce is predicted using GPKPDR . Setpoints for the controller are WAVref and uRbase .

1. Can control both propofol and remifentanil, but can also function when propofol is closed-loop controlled while an analgesic is administered manually: The controller should provide adequate control of DOH for the complete study population during induction and maintenance of anesthesia, both when only propofol is closedloop controlled, and when both propofol and remifentanil are adjusted automatically. 2. Uses primarily propofol to control DOH, and remifentanil to suppress the effect of nociceptive stimulation: In response to a nociceptive stimulation, the predicted remifentanil effect site concentration Ce should increase by 2 ng/ml following a rapid increase of 10 DOH units. 3. Gives the user control over the analgesic-hypnotic balance: In the absence of stimulation, the level of analgesia should match the level defined by the user through uRbase . The remifentanil infusion should return to this setpoint also in the presence of noise and in the case of static errors and errors at low frequencies. 4. Is robust in the presence of interpatient variability: in the presence of a realistic interpatient variability and in the presence of bounded nonlinear interaction of propofol and remifentanil. These clinical objectives were translated to technical objectives for the design of LTI controllers [16]. Objective 1 was achieved through the design of the propofol loop. Objective 2 determines performance requirements on the transfer function between a disturbance d and the (predicted) Ce , while objective 3 and robustness to nonlinear behaviour of objective 4 impose limits on the achievable performance, which can be described as frequency dependent limits on the gain of this transfer function. To provide user control over the analgesic-hypnotic balance in the case of static or low frequency errors, the remifentanil controller needs to achieve zero steady-state gain. Robust stability can be evaluated using wellknown LTI robust stability results. This paper proposes an extended habituating model predictive controller design to meet these technical objectives. 2.3. Modeling the effect of propofol and remifentanil

2.2. Design objectives for MISO propofol-remifentanil control Previous work in our group proposed a MISO propofolremifentanil closed-loop system, based on feedback from the WAVCNS measure of depth-of-hypnosis (DOH) provided by the NeuroSENSE monitor (NeuroWave Systems Inc.) [16]. The design proposed an extension to the direct habituating control framework including two user defined setpoints; a setpoint for the desired DOH WAVref and a second setpoint for the baseline remifentanil infusion uRbase . A second, not measured output that reflects the predicted remifentanil effect site concentration Ce was added to formulate a clinically relevant objective for the remifentanil controller. The controller structure is shown in Fig. 1. This framework allowed for the design of a control system that:

A set of thirty patient models was available for this study [16]. This model set includes (1) individual propofol effect models identified from clinical data, (2) corresponding remifentanil pharmacokinetic (PK)-pharmacodynamic (PD) models defined according to the Minto model [22] using patient demographics, (3) a worst-case effect model of remifentanil on DOH based on interaction models, with the worst-case effect occurring at sedation levels of anesthesia, therefore overestimating the effect at DOH levels corresponding to general anesthesia. The propofol models include population average propofol pharmacokinetic (PK) models [23], and an identified first-order plus time-delay pharmacodynamic (PD) model with nonlinear Hill function. For controller design, the drug infusion (model input) was

N. Eskandari, K. van Heusden and G.A. Dumont / Biomedical Signal Processing and Control 55 (2020) 101656

scaled by lean body mass (LBM), and the Hill equation in the PD model was linearized. A nominal model and corresponding unstructured uncertainty were defined for this linearized model set. The nominal model is given in Eq. (1). The sampling time of this model is 5 s, the input units are mg/LBM/h, the output is the DoH on a scale from 0–100. Remifentanil models were defined using allometric scaling (ALO = (bwt/70)0.75 ) and nominal remifentanil models were defined both for the worst-case effect model GR and the PKPD model GPKPDR used to predict Ce . GPKPDR is given in Eq. (2). The sampling time is 5 s, the input units are mcg/ALO/h, the output units are nl/ml. The worst-case effect model is given by GR = 3.1GPKPDR , see [16] for details. The sensor dynamics of the NeuroSENSE monitor are given by GNS (s) =

1 (8s + 1)2

3.2. Extended habituating model predictive control

2.4173(z + 5.9568)(z − 0.99983)(z − 0.99261)(z + 0.15637) ,(1) (z − 0.93542)(z − 0.95468)(z − 0.99913)(z − 0.99993) 7.055e−06 (z + 0.9645)(z − 0.9854)(z − 0.9992) . (z − 0.9992)(z − 0.9917)(z − 0.9579)(z − 0.9305)

(2)

3. Extended habituating model predictive control 3.1. Habituating MPC Habituating control offers a systematic design approach or systems with two inputs and one measured output, introducing a control objective for valve position control [15]. Habituating MPC solutions have been proposed for control of blood glucose in the ICU [24,25]. The MPC formulation uses a modified SISO MPC objective function with a penalty on the difference between the second (fast) input and its setpoint. For the control of propofol and remifentanil, using propofol as the first input and remifentanil as the second (fast) input, the habituating MPC objective function can be written as [15]: J(t) =

HP  

2

WAV ref (t + i) − yˆ (t + i|t)



2

uR (t + l − 1|t) − uRbase

2

(3)

l=1

lim {WAV CNS − WAV ref } = 0

(4a)

lim {uR − uRbase } = 0.

(4b)

t→∞

 ,

uR (t)

yˆ (k) = C xˆ (k)



Ap

(5)

T

⎤

0

0

0

Ar

0

0⎥

⎣ BNS Cp BNS Cr ANS Bp

rr (l),

uP (t)

ˆ where xˆ (k) = xp (k) xr (k) xNS (k) d(k) , with xp corresponding to the states of GP , xr to the states of GR , xNS to the states of the sensor, dˆ the estimated disturbance, and

⎡0

where t denotes the time step yˆ (t + i|t)) is the predicted WAVCNS output assuming a step disturbance model, q, rp and rr are weights, uP (t + k − 1|t) = uP (t + k − 1|t) − uP (t + k − 2|t) denotes the difference in propofol infusion, uR denotes the remifentanil input, HP is the prediction horizon, HCp = HCr are the control horizon for propofol and remifentanil respectively. The predicted output yˆ (t + i|t)) is calculated assuming uR (t + i|t) = uR (t + HCr − 1|t), ∀ i > HCr and uP (t + i|t) = uP (t + HCp − 1|t), ∀ i > HCp . The use of uP introduces integral action. It can be shown that, if the closed-loop controlled system is stable with no input saturation, this controller is offset free and provides asymptotic remifentanil baseline tracking [15], i.e.: t→∞

xˆ (k + 1) = Aˆx(k) + B

⎢ ⎢0

k=1

+



A=⎢ ⎢

(uP (t + k − 1|t)) rp (k)

HCr  

To meet the design objectives, the proposed extension to habituating model predictive control explicitly uses a second, not measured output Ce to define a clinically relevant control objective function JE (t). Define the discrete state-space description of GP and GR as (Ap , Bp , Cp , Dp ) and (Ar , Br , Cr , Dr ) respectively, with sampling time Ts = 5 seconds. Note that the individual propofol models as well as the nominal model contain a time delay. Define a discrete time statespace description of the monitor dynamics GNS , with sampling time Ts = 5 seconds, as (ANS , BNS , CNS , DNS ). Let yˆ (t) denote the predicted WAVCNS , as predicted by the following model,

⎡ q(i)

i=1 HCp

+

This control solution does provide a unique solution, however, penalizing the difference between the remifentanil infusion and its setpoint does not provide a clinically relevant objective. Furthermore, if the propofol input becomes saturated, this controller will provide a compromise between the output tracking and tracking of the remifentanil baseline, meaning that objective 3, user control over the analgesic-hypnotic balance, cannot be guaranteed. Since this approach considers the effect of both propofol and remifentanil on the WAVCNS , there is no guarantee that the controller designed for propofol and remifentanil will function in SISO mode. This MISO MPC solution can therefore not meet the design objectives for the propofol-remifentanil control system.

.

GP = z −16

GPKPDR =

3

0 0

⎤0

⎥ ⎥, ⎥ 0⎦ 1

⎢ ⎥ Br ⎢0 ⎥ ⎥ B=⎢ ⎢B D B D ⎥, ⎣ NS p NS r ⎦ 0 C= 0 0

0 CNS

1

(6)



The states are estimated using a standard Luenberger observer with L = [0 0 0 1], corresponding to a step disturbance model for the ˆ disturbance state d(t). Define Md (q−1 ), with q−1 the backward shift operator, as the desired transfer function between d(t) and Ce (t), which meets objectives 2, 3, and 4. The error function

ˆ ed (t) = Md (q−1 )d(t) − GPKPDR (q−1 )uR (t)

4

N. Eskandari, K. van Heusden and G.A. Dumont / Biomedical Signal Processing and Control 55 (2020) 101656 Table 1 Controllers’ tuning parameters.

is introduced to extend the objective function to

JE (t)

=

HP  

2

WAV ref (t + i) − yˆ (t + i|t)

qp (i)

i=1 HP



+

2

(ed (t + j|t)) qr (j)

j=1 HCp 

+

(7) 2

(uP (t + k − 1|t)) rp (k)

k=1 HCr



+

uR (t + l − 1|t) − uRbase

2

rr (l),

Parameter

Value

HP HCp HCr

150 3 25

Indices

qp (i) qp (i) qp (i) qr (j) qr (j)

0 1 100 105 100

i  16 i ∈ [17, 60] i ∈ [61, 150] j  50 j ∈ [51, 150]

rp (k) rp (k) rr (l)

10 500 0.001

k=1 k = 2, k = 3 ∀l

l=1

Cp magnitude (dB)

60

Frequency response LTI equivalent controllers

40 20 0 -20 10 -6

10 -5

10 -4

10 -3

10 -2

10 -1

10 -5

10 -4

10 -3

10 -2

10 -1

50

Cr magnitude (dB)

where qp and qr denote the weights for the propofol dominated prediction error and the remifentanil objective error respectively. uP (t + i|t) = 0, ∀ i > HCp and uR (t + i|t) = uRbase ∀i > HCr . Note that this optimization objective function remains quadratic in the decision variables, and an analytic solution to the unconstrained problem is given by the least squares solution. Verification of design objectives The clinically relevant control objective is formulated using the desired Md . However, the control solution depends on the weights and the prediction and control horizon, and a posteriori verification of the design is required. Verification whether the design meets the specifications can be done assuming no constraints are active, by calculating the equivalent LTI control system to verify if the constraints on the transfer function between d and Ce are met, and to verify if the robust stability condition is met. Note that closed-loop stability and robustness are not guaranteed in the design phase, and need to be verified a posteriori as indicated above.

0

-50 10 -6

Frequency (rad/sec) Fig. 2. Bode magnitude diagram of LTI equivalnt feedback propofol and remifentanil controllers. The top plot shows the Bode magnitude diagram for the LTI equivalent propofol controller Cp , the bottom plot for the LTI equivalent remifentanil controller Cr .

4. Extended HMPC for controlling depth of anesthesia 4.1. Controller design and tuning The models described in Section 2 were used in this design. In addition to the requirements for the remifentanil control loop, the propofol controller needs to provide zero steady-state error, robust stability and complete induction of anesthesia for the entire study population within ≈5 min with limited overshoot. Following the approach in [16], the WAVCNS is predicted (equation (6)) assuming GR = 0 as remifentanil has limited effect on the DOH during general anesthesia (see Section 2). The worst-case effect model GR is used to verify robustness. The prediction horizon HP = 150 was chosen to include the expected settling time for the majority of the patients in the model set, including the nominal model. The propofol and remifentanil control horizons HCp and HCr as well as the weights qp , qr , rp and rr were tuned to achieve the desired performance. qp (i) = 0 for values of i within the delay of the nominal model, and the weight increases for i > 60 to penalize the lower frequency tracking of DOH. qr (j) was selected higher for j  50 than for j ∈ [51, 150] to achieve fast disturbance rejecting. The tuning is summarized in Table 1. In addition to the MPC controller, a feedforward filter FRbase (s) is used for the reference uRbase (t) to ensure a steady state Ce corresponding to this baseline is rapidly achieved following changes in the baseline infusion. FRbase (s) =

250s + 0.25 . (5s + 0.25)(s + 1)

This filter is also discretized with Ts = 5 s.

(8)

4.2. Simulation results The LTI equivalent controllers, assuming no constraints are active, were calculated and their Bode magnitude diagrams are presented in Fig. 2. As designed, the propofol controller includes integral action, while the remifentanil controller has large highfrequency gain and small low-frequency gain. 4.2.1. Verification of performance The closed-loop response of the nominal model controlled by the proposed extended HMPC controller is shown in Fig. 3. This simulation shows induction of anesthesia, followed by nociceptive stimulation after 20 min, resulting in a step increase in the measured WAVCNS of 10 units. This disturbance leads to a fast increase in predicted remifentanil Ce , and meets objective 2 as described in Section 2. Fig. 4 visualizes the response of uR , remifentanil infusion, and Ce , the predicted remifentanil effect-site concentration, of the nominal model to a step increase in the remifentanil baseline infusion uRbase (t). Using FRbase (s), the remifentanil infusion rapidly reaches the baseline infusion, while Ce reaches steady state after roughly two minutes and continues to track this new baseline after a small overshoot. Fig. 5 visualizes the effect of stimulation d(t) on the predicted remifentanil Ce on the nominal model. Following a step disturbance of 10 WAVCNS units, remifentanil plasma concentration increases ≈2 ng/ml, and meets the clinical requirements for disturbance

N. Eskandari, K. van Heusden and G.A. Dumont / Biomedical Signal Processing and Control 55 (2020) 101656

5

rejection. Note that this figure indicates the change in remifentanil infusion.

Fig. 3. Simulation results for nominal model controlled the proposed MPC controller: Induction of anesthesia is followed by nociceptive stimulation after 20 min, resulting in a step increase in the measured WAVCNS of 10 units. The top plot shows the closed-loop response. The middle plot shows the propofol input, and the bottom plot shows remifentanil input and effect site concentration of remifentanil.

4.2.2. Verification of objectives Noise in the DOH measurement can lead to input saturation of the remifentanil controller. Since drug cannot be taken out of the body, input saturation will result in an increase in average remifentanil infusion and deviation from remifentanil baseline. The worst-case increase in average remifentanil effect-site concentration assuming sinusoidal noise was quantified in [16]. This increase depends on the baseline infusion. In the following, a simplification of this result will be used assuming the worst-case increase resulting from zero baseline infusion. Using the equivalent (unconstrained) LTI controller Cr , it follows from the results of [16] that to guarantee a worst-case increase of maximally |Ce,mean |, the infinity norm of the feedback remifentanil controller has to be bounded by: Cr ∞ ≤

 |Ce,mean | , 7 |GPKPDr |s=0

(9)

where |GPKPDr |s=0 is the steady-state gain of the remifentanil PKPD model. To achieve objective 3, and give the user control over the baseline in the presence of noise, the allowed change was defined as |Ce,mean | ≤ 4.5. It follows from Eq. (9) that Cr  ∞ ≤ 234. The proposed controller satisfies this condition, with Cr  ∞ = 203. Robust stability can be verified using results from [26]; satisfaction of (10) guarantees robust stability of the MISO system given a nominal model and corresponding multiplicative uncertainty description.

 n 

2 |S(jω)Gi (jω)Ci (jω)Wi (ω)|

≤ 1,

(10)

i=1

Fig. 4. Response of uR (t) and the predicted remifentanil Ce (t) of the nominal models to a unity change in remifentanil infusion baseline uRbase (t).

where S represents the nominal sensitivity function, Gi describe the system transfer functions, Ci are controllers in the feedback loop, and Wi describes the frequency dependent model uncertainty for each input i in the multi-input single-output system. Using the equivalent LTI controllers, C1 = Cp , C2 = Cr , and G1 = GP , G2 = GR with GR the remifentanil model with worst-case gain as described in Section 2.3, the left hand side of (10) is ≈0.90 and the proposed controller achieves robust stability. 5. Conclusion and discussion

Fig. 5. Change in uR (t) and the predicted remifentanil Ce (t) of the nominal model to 10 WAVCNS units of nociceptive stimulation.

In clinical practice, closed-loop anesthesia will be subject to constraints. Such constraints can be implemented in a straighforward manner in the MPC framework. This paper proposes a model predictive control of propofol and remifentanil guided by a DoH measure, a MISO system. We propose an objective function specifically designed to meet clinically relevant control objectives. It is shown that the presented MPC controller satisfies the clinical design objectives, and this MPC framework can be used instead of the previously proposed LTI framework to facilitate practical implementation of patient safety constraints. The remifentanil controller designed in this paper reacts to the disturbances that are estimated with a standard Luenberger observer. While this observer makes precise estimations of the disturbance for the nominal model, due to interpatient variability, false disturbance detection may occur due to interpatient variability. This may lead to unnecessary remifentanil administration. While the proposed system is robustly stable and can provide safe anesthesia and analgesia for the considered patient population, improving disturbance detection may improve control of analgesia.

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N. Eskandari, K. van Heusden and G.A. Dumont / Biomedical Signal Processing and Control 55 (2020) 101656

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