Linear Model Predictive Controller for Closed-LoopControl of Intravenous Anesthesia with Time Delay by ides editor

Full Paper ACEEE Int. J. on Control System and Instrumentation, Vol. 4, No. 1, Feb 2013

Linear Model Predictive Controller for Closed-Loop Control of Intravenous Anesthesia with Time Delay Deepak D. Ingole1, Dayaram N. Sonawane1, Vihangkumar V. Naik1, Divyesh L. Ginoya1, Vedika V. Patki1 1

Department of Instrumentation and Control, College of Engineering, Pune - 411 005 Maharashtra, India Email: dingole21@yahoo.in, dns.instru@coep.ac.in, naikvihang@gmail.com, dlginoya007@gmail.com, patki.vedika@gmail.com loop control minimize the drug consumption, intraoperative awareness and recovery times, thereby decreasing the cost of the surgery and also the cost of the postoperative care. Overall, this is to improve the patient’s safety and rehabilitation during and after the surgery [3]. However, in order to design and implement closed-loop control schemes, mathematical models of the patient/drug delivery system are required. The standard modeling paradigm that has been commonly used to describe the relationships between anesthetic inputs and patient output indicators (or effects) is that of compartment models [4]. Closed-loop control in anesthesia emerged as a serious contender for many forms of control in the late 1970s. It was pioneered by Sheppard et al. and Asbury et al., who demonstrated through clinical experiments that this form of control is safe, effective and in many cases better than manual control [5]. After the commercial availability of BIS monitor in 1998, many researchers proposed a closed-loop Proportional Integral Derivative (PID) control system for Propofol administration using BIS as the controlled variable [6]. However, one problem with fixed-parameter PID controller in controlling anesthesia is the lesser output response and vast deviation from the output. Adaptive strategies cheerfully overcome the problem of inter-intra patient’s variability. In [7] a nonlinear adaptive controller has been used for controlling the BIS level. Some of the researchers applied non-model based control strategies such as lookup tables and fuzzy controllers [8] for controlling depth of anesthesia. With syringe pump and DoA monitors providing technological conditions and models an adequate theoretical basis for controller design, a number of control algorithms tested both in simulation and clinical trials are described in the literature. Significative but nonexhaustive examples include Generalized Predictive Control (GPC), Internal Model Control (IMC) [9], and Extended Dynamic Matrix Control (EDMC) [10]. Among the possible model based design strategies, Model Predictive Control (MPC) [11] presents a number of advantages, in particular the ability to tackle constraints, BIS reference tracking, disturbances and noise handling and is currently the subject of increasing attention in what concerns its application, in various forms, to DoA [12], [13]. The proposed LMPC uses the developed mathematical model of the BIS response to Propofol infusion. The model is a series connection of three elements: a Pharmacokinetic (PK)

Abstract—During intravenous anesthesia, anesthetic drugs must be administered at a suitable rate to prevent over dosing and under dosing in a patient. A developed PharmacokineticPharmacodynamic (PK/PD) model, which has been used to study the relationship between administered anesthetic dose and its effect on the patient in terms of hypnosis, is considered. In this paper, Linear Model Predictive Controller (LMPC) framework based on Active Set Method (ASM) with modified approach for closed loop control of intravenous anesthesia is presented for Single-Input (Propofol infusion rate) SingleOutput (Bispectral Index (BIS)) model of a patient. Effectiveness of the designed LMPC has been studied for BIS reference tracking as well as constraints, disturbances and noise handling in the measured variables. Performance of proposed approach is compared with conventional Proportional-Integral-Derivative (PID) controller, considering closed loop time delays. Simulation result shows that, proposed LMPC outperforms conventional PID controller. Index Terms—Closed-Loop Control of Anesthesia, Bispectral Index (BIS)), Propofol, Linear Model Predictive Control (LMPC), Active Set Method (ASM), Cholesky factorization.

I. INTRODUCTION The activities of anesthesiologists include numerous repeated and isolated tasks. Anesthesiologist has to maintain certain patient state variables within an acceptable operating range. Propofol, an intravenous anesthetic drug, is frequently used for hypnosis control during surgical operations because of its good solubility, short onset time and quick recovery [1]. The anesthetist use different variables for estimating the Depth of Anesthesia (DoA), some of them are not measurable. However, the automation of the control of DoA needs measurable outputs. The BIS is a numerical processed, clinically-validated EEG (electroencephalogram), parameter, used as an indicator of depth of hypnosis, measuring the degree of depression in the central nervous system [2]. The ease of BIS monitoring and its ready availability in the operating theaters opens the possibility of closed-loop control of anesthetic drug administration, using BIS as the performance and measurement variable. In anesthesia, automatic regulation, in a closed-loop control of infusion of drugs has been shown to provide more advantages as compared to manual administration. A well designed automatic control system can avoid both overdosage and under-dosage of the drugs to the patient. Closed© 2013 ACEEE DOI: 01.IJCSI.4.1.1063

Full Paper ACEEE Int. J. on Control System and Instrumentation, Vol. 4, No. 1, Feb 2013 model, a time delay, and a Pharmacodynamic model (PD) [4]. Because of large inter-patient variability in Propofol pharmacokinetics (PK) and pharmacodynamics (PD), a number of simulations are conducted to verify the robustness of the LMPC controller. The proposed LMPC strategy has also been tested for disturbance and measurement noise. Performance obtained with the LMPC controller is compared with the conventional PID controller. The main contribution of the paper is in solving Quadratic Programming (QP) problem using Active Set Method (ASM) algorithm with modified approach which is the core of LMPC algorithm. Present study envisages to investigate the viability of LMPC for closed loop control of intravenous anesthesia. The paper is organized as follows: In Section II, the mathematical model of patient is presented. In Section III, the formulation of anesthesia control problem as a LMPC problem and proposed ASM QP method is presented. The simulation results and comparison of LMPC controller with PID is presented in Section IV. The paper is ended by concluding remarks.

Figure 1. PK/PD model of the patient

Central Compartment (c1): The central compartment is the volume in which initial mixing of the drug occurs, and thus can be thought to include the vascular system (blood volume) and for some drugs the interstitial fluid. The concentration of Propofol within the central compartment x1(t) is given by:

II. HUMAN BODY MODEL FOR DEPTH OF ANESTHESIA CONTROL

The most commonly used models in the literature to describe the Pharmacokinetics-Pharmacodynamics (PK/PD) of the Propofol (hypnotic) effects in the patient are described in this section.

V alue

1 .7 2 BW 0 .71 AG E  0 .39 L

V2 V3

3 . 32 BW 0 . 61 L 2 66 L

V1 L 100

0 .0595 BW 0.75 L/min for (A GE = 60)

dx3 t   k3 x1  k 3 x3 . dt

(3)

dx2 t   k 4 x1  k 4 x4 . dt

(4)

The PK model can be described in state space form as

L/min for (AG E > 60 )

(2)

Effect-Site Compartment (c4): The PD nonlinearity relates the drug concentration with the effect, DoA, and has a first linear dynamic part approximated by a first order differential equation [4]. The concentration of Propofol within the central compartment x4(t) is modelled as:

0 . 0595 BW 0 .75  0 . 045 AG E  2 . 7

dx2 t   k 2 x1  k 2 x2 . dt

Deep Peripheral Compartment (c3): The deep or slow peripheral compartment is used to mathematically represent a compartment into which re-distribution occurs more slowly, and thus can be thought of as including tissues with a poor blood supply (such as adipose tissue). The concentration of Propofol within the deep peripheral compartment x3(t) is modelled as:

TABLE I. PHARMACOKINETIC PARAMETER VALUES PUBLISHED BY SCHÜTTLER AND I HMSEN [4]

(1)

Shallow Peripheral Compartment (c2): The shallow or fast peripheral compartment represents a compartment of the body that absorbs drug rapidly from the central compartment, and thus can be thought of as comprising tissues of the body that are well-perfused (such as muscles and vital organs). The concentration of Propofol within the shallow peripheral compartment x2(t) is given by:

A. Pharmacokinetic (PK) Model of Patient Pharmacokinetics refers to the study of Absorption, Distribution, Metabolism and Excretion (ADME) of bioactive compounds in a higher organism. In this paper, the Schuttler and Ihmsen PK model [4] was used. It is a four-compartment model, outlined in Fig. 1, Each compartment represents a class of tissues. Here, xi is the concentration of Propofol in compartment i; compartments 1, 2, 3 and 4 correspond, to the central, shallow peripheral, deep peripheral compartment and effect-site compartment, respectively. In addition, u is the infusion rate of Propofol, and ki and Vi are the clearance and volume of compartment, respectively, given as functions of the patient’s age and weight, as given in Table I.

P arameter

dx1 t   k1  k 2  k3  k 4 x1  k 2 x2 dt  k 3 x3  k 4 x4  u .

0 .62

L/min 0 .0 96 9 BW 0 .0 88 9 BW 0 .55 L/min 0 . 12 L/min

xt  1  Axt   Bu t  y t   Cx t  . 9

(5)

Full Paper ACEEE Int. J. on Control System and Instrumentation, Vol. 4, No. 1, Feb 2013 where, x(t) is the vector of Propofol concentrations in the compartments, u(t) is the infusion rate of Propofol, y(t) is the effect site concentration of Propofol, and A, B, C are PK parameters and given as

 k1  k 2  k3  k 4  V1  k 2   V1 A k3  V3   k4  V4  1 B V1

k2 V1 k  2 V1

k3 V1 0 

0 0

k3 V3 0

The response of the BIS(t) to drug administration involves considerable closed-loop time delays caused by movement of Propofol from a syringe pump to the patient’s body in an intravenous fluid line, distribution of Propofol in blood vessels, and BIS monitor also takes time to calculate the BIS (approximately 15–60 s [4]). The closed-loop system considered is a Single-Input (Propofol infusion rate) Single-Output (Bispectral index (BIS)). Therefore, sum of the delays mentioned above are added to find the single-output delay Lc, so that, the current BIS(t) value is determined by the past value y(t) of the effect-site concentration [4],

k4  V1   0    0 ,  k4    V4 

y t   Cx t  Lc  . where



BIS t   E0  Emax

B. Pharmacodynamic (PD) Model of Patient The Pharmacodynamic model is the static relation between the effect site concentration of Propofol and the BIS value. A PD model relates y(t) to the clinical effect E, which in this paper is the DoA. E= E0 0 indicates the value of BIS in fully awake state and E=1 indicates the value of BIS in total absence of cortical activity. The relationship between y(t) and E is given by the following Hill’s sigmoid Emax model (6),

A. Basics of LMPC Model predictive control (MPC), also known as Moving Horizon Control (MHC) or Receding Horizon Control (RHC) is an optimal control algorithm that predicts the behavior of a plant through the use of an explicit process model and solving an on-line optimization problem at each sampling interval. Linear MPC has shown great success in applications, especially in the process industry and is spreading to other application areas [11]. MPC have many advantages, in particularly it can easily apply to Multi-Input Multi-Output (MIMO) systems, can compensate the effect of closed-loop delay by the prediction, inducing the anticipate effect in closed loop, handles constraints on inputs, outputs and internal states [11]. Hence, MPC seems to be a promising approach to be applied to safety critical applications like anesthesia control and glucose control. The basic elements present in all model-based predictive controllers are: prediction model, objective function, calculation of the control action. The prediction model is the central part of the MPC, because it is important to predict the future outputs of the system. In LMPC, the state space model is used as prediction model. The objective function defines the criteria to be optimized in order to force the generation of a control sequence that drives the system as desired.

(6) where c 50 is the effect-site concentration y(t) value corresponding to 50% clinical effect and γ>1 is a parameter determining the shape of the sigmoid known as Hill’s coefficient. During surgery the BIS signal is obtained directly from the EEG and not (6). D. Time Delay (TD) In this section, we present impact of the time-delay Lc of the patient and instrumentations such as BIS monitor, sensors and actuators on closed-loop control of anesthesia during surgery. Because plasma Propofol concentration measurement is unavailable, it is estimated through the nominal PK model as given in (5). The Propofol infusion is applied to the patient and the real BIS signal is recorded by the BIS monitor. The controller has to maintain BIS between 40 and 60 during the surgery. Firstly, it is assumed that the patient is in a fully awake state (BISH 100) and then the controller is turned on the set-point is changed from 100 to 50. At 50, the BIS value is calculated by (7).

B. Mathematical Formulation of the LMPC The LMPC can be formulated either as an LP (linear problem) or as a QP (quadratic problem). With both formulations, the objective is to minimize a cost function [11]. In this paper we have formulated LMPC problem as a QP. For



(9)

In this section a short introduction to linear model predictive control (LMPC) and an outline of the problem formulation of LMPC is given.



y t  .   y t   c50

III. LINEAR MODEL PREDICTIVE CONTEOLLER (LMPC )

y t  .   y t   c50

BIS t   E0  Emax

C  0 0 0 0 1

Taking into account the time delay Lc the real time value of BIS(t) can be expressed by the following relation:

 0 0 0 ,C  0 0 0 1 

E t   E0  Emax

(8)

(7) 10

Full Paper ACEEE Int. J. on Control System and Instrumentation, Vol. 4, No. 1, Feb 2013 simplicity, given a single input single output discrete linear time-invariant plant in the state space form as described in (5),

xk  1  Axk   Bu k ,

0  CB  CAB CB      N p 1 N p 2 B CA B CA

(10)

y k   Cx k   Du k .

(11) where x is the state vector, u is the input vector and y is the vector measured outputs which are to be controlled A, B and C are constant matrices with compatible dimensions.





1 p 2 Jy   Y  R Q. 2 k 1

Jy 

Subject to (14) (15)

k 1

Y  x0  U . where, Y,  , U, and  are given below,,

(26)

Regularization: Regularization can be done by introducing a new term JΔu, in the objective function, where Δu(k)=u(k)u(k-1). Control problem, with regularization

(18)

1 p 2 1 N u 1 2 J   y k   r k  Q   u k  S . 2 k 1 2 k 0

(19)

(27)

Subject to (14) and (15). This new term minimizes the difference between two consecutive steps in u, which gives more “smooth” input. Again this should be formulated as a QP problem.

 y k  1  u k     CA   y k  2    CA 2   u k  1  ,     , U   Y                Np   y k  N p  CA  u k  N u  1

1 Nu 2 u k   u k  1 S .  2 k 0

(28)

1 N u 1 u k   u k  1T S u k   u k  1.  2 k 0

(29)

J u  

(25)

(17)

This can be written as;

y 0   Cx 0 .

1 J y  U T H yU  FyT U   . 2

1 J y  U T H yU  FyT U . 2

(16)

k 1 j 0

(24)

where, Hy, Fy and ρ are given by; Hy=ΓTQΓ, T T T Fy=Γ QΦx(0)-Γ QR, and ρ=1/2b Qb. Since, ρ doesn’t influence the solution to the problem, it can be discarded. Equation (25) is equivalent to standard QP problem; QP formulation of (12) resulted into;

from [14] it’s know that;

y k   CA k x0   C  A k 1 j u 0  .

(23)

1 U  b T QU  b . 2

Jy 

(13)

j 0

(22)

To make this problem easier to solve, it is convenient to express it as a QP problem;

discarded. Here Np and Nu are prediction and control horizons respectively. This gives the first control problem;

y k   CA k x 0  C  A k 1 j u  j  .

1 2 x0  U  R Q . 2

J y  U  b Q , b  R  x 0.

(21)

Putting (19) into (21) gives;

since y(0) can’t be influenced, the term y 0   r 0 Q is

y k   Cx k , k  0,1 N p .

(20)

The objective function (13) can be written as,

(12)

xk  1  Axk   Bu k , k  0,1 N u  1.

      N p  Nu   CA B

R  r k  1 r k  2   r k  N p  .

1 p 2 J y   y k   r k  Q . 2 k 1

0 0 0

By introducing R as a vector containing the set points

Unconstraint MPC: The goal of the controller is, to make the difference between the output y(k), and the reference r(k), as small as possible. This can be done by using a least squares problem. The weighted 2-norm is used [14, 15]:

1 p 2 y k   r k  Q .  2 k 1



Full Paper ACEEE Int. J. on Control System and Instrumentation, Vol. 4, No. 1, Feb 2013 constraints. Algorithm to find optimal solution of (35) is described in Algorithm 1; Algorithm 1. Step 1: Initialization Start with a feasible x0, Let the initial working set of W0 be empty Set k=0 to kmax Step 2: KKT System The solution of the sub problem (34) is of course given by the solution of the Karush-Kuhn-Tucker (KKT) system

1 T  U T H SU  N u  1u  1 U  u 1Su1 . 2 (30)





where HS, N(u(-1)) and FΔu are given as;

S   2S 0   S   N u  1   , H u     0    0   0

S 2S  0

0   S

0  0   S  2S 

H M T 

Fu  N u  1u  1 1 J u  U T H uU  FTuU . 2

Like with ρ, the term 1/2u-1Su-1 is discarded, because of the lack of influence on the solution to the problem. Combining (26) and (31) we obtain new QP problem as; (32)



q1 0  s1 0 0  0      Q      , S      . 0 0 q   0 0 s N 1  Np  u   

 d  M i xk   min  i  M i x and xk+1 (a)

  . 

(36)

xk +αΔx Create Wk by adding one element of Dk to Wk

The active set method approaches the solution of KKT system by successive decent steps. Each decent step is Newton’s like decent step and the solution is obtained by solving system of linear equations (35) using appropriate numerical methods in order to determine search direction. For hardware acceleration of different computational units involved in ASM, it is necessary to know the timing contributions of each and every step involved in optimization algorithm. For this purpose, we conducted a profile study of proposed ASM QP solver described in Algorithm 1, for a randomly generated QP test problems, from that it is observed that, in ASM algorithm the major computational load is on LP solver to solve KKT system. We have implemented different decomposition methods such as Gauss elimination, QR, LU, and Cholesky factorization using LLT and LDLT [18] to solve the linear equation. From Fig. 2, it is observed that LDLT Cholesky factorization takes less time as compared to LLT and QR factorization. The coefficient matrix in a linear system involved in ASM is symmetric and positive definite so conventional and more

u min  u k   u max , ymin  y k   y max . C. Quadratic Programming Various methods have been proposed to solve the QP problem, out of which a gradient projection method, an interior point method, and an active set method [16] are wellknown methods. Compare to other algorithms, active set method requires less computational effort for small scale problems [17]. Hence, we choose active set method to solve (32). Proposed Active Set Method (ASM): Consider the general convex quadratic program (34)

Subject to MTx-d=0. where, H is (lNu× lNu) positive definite matrix, f is a (lNu× 1) vector. Moreover M and d have the size (mc× lNu) and (mc× 1), respectively, where mc is the number of inequality © 2013 ACEEE DOI: 01.IJCSI.4.1.1063

  1 

Step 3: The step size α must be chosen to maintain feasibility

(33) here, s and q are the weights of input and output respectively. Constraints: The MPC problem has to take the limits of the physical system into consideration. Constraints are imposed on the input quantity, the input rate of movement and on the output. MPC with input and output constraint;

1 T x Hx  F T x . 2

d  M xk

i i (d) Dk  i  Wk : M i xk  0, M i xk  (e) If Dk is empty then; (f) xk+1 xΔx and Wk+1 xk (g) Else go to Step 3

here, H=Hy+H”u and F=Fy+F”u. The weight matrices Q and S are given by;

f 

(35)

(a) If the solution of (35) is Δx=0 then; (b) If all entries of λ are not negative then; Terminate the solution with x*=xk (c) Else remove from Wk an index that corresponds to smallest entry of ë and then xk+1 xk else

(31)

1 J  U T HU  F T U . 2

M   x  F      for x,  .    0    d 

Full Paper ACEEE Int. J. on Control System and Instrumentation, Vol. 4, No. 1, Feb 2013 A. Comparision result of proposed QP with quadprog

Figure 2. Timing analysis of ASM algorithm using LDL T, LLT and QR factorization methods

Figure 3. Performance comparison of proposed QP with ‘quadprog’.

numerically stable Cholesky factorization (LDLT) is preferred over other numerical methods [28]. Steps to solve KKT system of (35) are described in Algorithm 2; Algorithm 2. Step 1: Cholesky factorize H=LLT Step 2: Compute K by solving LK =M Step 3: Compute w by solving Lw=F Step 4: Compute H=KTK Step 5: Compute z=KTw+b Step 6: Cholesky factorize H=NNT Step 7: Compute q by solving Nq=z Step 8: Compute λ by solving NTλ=q Step 9: Compute r=KTλ-w Step 10: Compute x by solving LTx=r The solution of Step 2, 7 and 10 are obtained by Forward Substitution and Step 3 and 8 by Backward Substitution.

Fig. 3, shows the performance comparison of MATLAB’s QP solver quadprog with proposed Cholesky factorization LDLT based ASM algorithm for randomly generated QP test problem. B. Response of proposed LMPC without considering time delay In Fig. 4, shows the simulation results, which, after the initial transient due to non-zero initial states, tracks a reference signal which changes at time 200 samples and again at time 600 samples. The lower trace on each plot is the input signal as infusion rate delivered to the patient.

IV. SIMULATION RESLUTS The PK-PD model was used for designing the LMPC. We used the PK-PD model discussed in [4] and the parameter values are given in Table I. The simulation was programmed in M-file of MATLAB. The proposed algorithm is simulated for patients with different age and weight as given in Table II. For the simulation purpose we have kept values of V3, V4, k4, c50 and γ constant as given in Table I. Here the results just for a patient of 25 year old and 63 kg male are shown. Consider that the desired target BIS is set at 50. The prediction horizon Np is 10 and control horizon Nu is 3. The weight matrices Q and S are taken as 0.0024 and 0.4 respectively.

Figure 4. Output and input signal Vs time when L c = 0 sec

C. Comparision results of LMPC and PID controller with time delay In this section we have considered time delay Lc of BIS monitor and sensor. The total time delay considered is 25 Seconds.

TABLE II. VALUES OF THE PARAMETERS FOR THE 10 PATIENTS SETS

1 2 3 4 5

1 20 50 70 110

5 55 80 100 110

5.39 9.19 8.39 8.62 7.74

08.86 38.26 48.08 55.09 58.39

0.19 1.20 1.59 2.33 4.27

0.26 1.16 1.46 1.68 1.78

0.21 0.80 0.98 1.11 1.17

6 7

80 100

50 70

5.00 5.82

36.10 44.32

2.01 3.23

1.09 1.34

0.76 0.91

Figure 5. Output and input signal Vs time when L c =25 sec

Full Paper ACEEE Int. J. on Control System and Instrumentation, Vol. 4, No. 1, Feb 2013 The Fig. 5, shows the set point tracking performance of the PID controller and LMPC. Performance of the LMPC is superior to that of the PID.

Figure 9. Effect of prediction horizon N p on BIS regulation

Figure 6. The simulation BIS when applying PID and LMPC controllers when L c = 25 sec

In Fig. 6, the response of LMPC and PID can be compared, as it is shown that LMPC response is faster with lower settling time, and also the amount of drug for LMPC is less than PID. D. Robustness test of proposed LMPC and PID Controller The controller output is constrained between 0.1 and 20. A disturbance, d, is added to the controller output from t=300 to 400 samples and t=700 to 800 samples, to create the final input to the patient. Fig.7, shows responses to disturbance, as it is clear LMPC response is smoother than PID response in presence of disturbance. Also from lower trace of plot it is observed that, the LMPC gives bounded output whereas PID shows unstable output. In Fig. 8, the performance of the controllers in presence of the noise with 0.1% magnitude is considered. The LMPC more robust and successfully works in presence of noise.

Figure 10. Effect of control horizon N c on BIS regulation

It is observed from Fig. 10, the response improves with Nu. But large Nu takes more computational time.

Figure 7. BIS value when using PID & MPC controller when disturbance is added to the output

CONCLUSIONS In this paper, Active Set Method (ASM) algorithm with proposed approach has been tested on randomly generated QP test problems. A Linear Model Predictive Control (LMPC) based on designed QP solver is presented and evaluated thoroughly for regulation of intravenous anesthesia using BIS as the controlled variable. Performance of LMPC is evaluated and compared with conventional PID controller considering the time-delays introduced by the

Figure 8. The results of PID and LMPC controllers in presence of a noise with 0.1% magnitude

D. Effect of prediction and control horizons The Fig. 9, Shows the effect of prediction horizon with time. The Fig. 10, Shows the effect of control horizon with time. ÂŠ 2013 ACEEE DOI: 01.IJCSI.4.1.1063

Full Paper ACEEE Int. J. on Control System and Instrumentation, Vol. 4, No. 1, Feb 2013 instrumentation (BIS monitor, sensors and actuators) during anesthesia control. The LMPC is found to be robust to intra and inter-patient dynamics, and better at handling constraints, disturbances as well as measurement noise.

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