In this context, system identification is an important component

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1 211 th IEEE Conference on Decision and Contro and European Contro Conference (CDC-ECC) Orando, FL, USA, December 12-1, 211 An improved strategy for NeuroMuscuar Bockade contro with parameter uncertainty J. Ameida, T. Mendonça and P. Rocha Abstract This paper presents a contro strategy for exact reference tracking in the presence of parameter uncertainty for the NeuroMuscuar Bockade (NMB) eve of patients undergoing genera anesthesia. For this contro appication, a compartmenta reaization of a minimay parameterized Wiener mode was used in [1] in which the parameters were estimated from a bank of coected rea data using the Extended Kaman Fiter (EKF). Due to the parameter uncertainty this procedure did not achieve the desired tracking goa. Here a modified contro strategy is presented to overcome this drawback. I. INTRODUCTION Contro and modeing of dynamica systems is an important step in appied mathematics, in particuar, when appying mathematica methods to address biomedica probems. Among the commony used modes are the nonnegative compartmenta modes. These modes are characterized by conservation aws and composed by a finite number of interconnected homogeneous, we-mixed subsystems caed compartments. The exchange of nonnegative quantities of materia among the compartments of the system and with the environment is described by aws that take into account the conservation, dissipation and transfer of materia (mass) among the compartments and to the environment. Compartmenta modes are widey used in pharmacoogy and in particuar in the contro of drug dosing in genera anesthesia. One exampe of this appication is the automatic contro of the NeuroMuscuar Bockade (NMB) eve, that is usuay monitored during genera anesthesia. This condition is obtained by the administration of a musce reaxant to provide adequate surgica conditions. Mathematicay, the NMB eve is modeed by PharmacoKinetic/PharmacoDynamic (PK/PD) modes that reate the administration drug dose with the NMB measure. In this context, system identification is an important component in controer design since it is used to get adequate modes for the conception of a prediction agorithm or simuation, see e.g. [9]. In this paper a contro strategy for the NMB eve of patients undergoing genera anesthesia is deveoped based on a new minimay parameterized mode. This contro strategy consists of a positive contro aw for J. Ameida and T. Mendonça are from Departamento de Matemática, Facudade de Ciências da Universidade do Porto, Rua do Campo Aegre, Porto (juameida@fc.up.pt and tmendo@fc.up.pt) P. Rocha is form Facudade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, Porto (mprocha@fe.up.pt) T. Mendonça and P. Rocha are aso from Center for Research and Deveopment in Mathematics and Appications, Departamento de Matemática, Campus Universitário de Santiago, Aveiro, Portuga Fig. 1. Bock diagram that reates the atracurium dose, u(t), with the NMB response,. The signa y(t, α) corresponds to the effect concentration feedback stabiization of compartmenta systems proposed by []. The Extended Kaman Fiter (EKF), see e.g. [9], is used to estimate the individua patient parameters to be used in that contro aw. As expected, due to the parameter uncertainty, this procedure does not yied the desired tracking goa. However, the anaysis of the system response aows to take advantage of the mode structure and estimate the steady-state parameter error. In a second stage, this knowedge is used to correct the origina contro aw and achieve the desired contro objective. II. CONTROL LAW FOR NMB DRUG In this section a reationship between the drug dose of atracurium and its measured effect (NMB) is described using a minimay parameterized mode. Furthermore, a positive contro aw for feedback stabiization of compartmenta systems, [], is used and its accommodation to the specific dynamica system structure under study is anayzed. A. NeuroMuscuar Bockade minimay mode This section describes a mode for the reationship between the administrated drug dose of the musce reaxant atracurium and the measured effect, which in this case is the NMB eve. This reationship can be modeed by a SISO Wiener mode, as iustrated in Figure 1.The new minimay parameterized mode for NMB eve used in this paper was proposed by [4] and resuts from an approximation of the PharmacoKinetic/PharmacoDynamic (PK/PD) mode presented in the iterature, [7].This PK/PD mode invoves a tota of eight parameters and the poor excitation of the input and output signas is not enough to enabe the identification of a such a high number of parameters. In order to overcome this difficuty, [4] proposed this new mode with ony two patient-dependent parameters to be identified: one in the inear dynamics (parameter α in first bock in Figure 1) and the other in the static noninearity of the Wiener structure (parameter γ in second bock in Figure 1). Contrary to what happens with the PK/PD mode, this new minimay parameterized mode does not have a direct physioogica meaning, /11/$ IEEE 867

2 but, as sha be seen in the seque, produces good resuts when used for contro purposes. The reationship between the drug dose administration and the drug concentration in the effect compartment can be described by the foowing third-order inear dynamic mode, Y(s) = k1 k2 k α U(s), (1) (s+k1α)(s+k2α)(s+kα) where Y(s) is the Lapace transform of the output of the inear of dynamic mode that corresponds to the effect concentration, and the U(s) is the Lapace transform of the input signa, i.e., the atracurium dose. The vaues of k1 < k2 < k are positive constants and fixed according to [4]. Furthermore, note that the inear part is stabe if α >. The mode (1) can be written through the foowing statespace mode, { ẋ(t) = A(α) x(t) + B(α) u(t) (2) = C x(t) with, and α are positive, it is easiy seen that the matrix A(α) in (2) is indeed compartmenta and that the components of the vectors B(α) and C are nonnegative. The contro aw proposed by [] is shown to stabiize the tota mass M(x(t)) of a compartmenta system at a given positive set point. For the sake of simpicity M(x(t)) is here after denoted by. This contro aw is obtained from the foowing equation, ( ) 1 u(t)= bi(α) [(1 1 1) A(α)x(t)+λ( )], i=1 where i=1 b i(α) is equa to (1 1 1)B(α), by imposing a positivity constraint, which yieds: u(t)=max(,ũ(t)) (4) ( ) 1 ũ(t)= bi(α) [(1 1 1)A(α)x(t) + λ( )] i=1 A(α)= k α k2 α k2 α, B(α)= k α, k1 α k1 α C= ( 1 ). where x(t) is the state vector. The state x corresponds to the effect concentration,, and the states x1 and x2 are auxiiary variabes. The second bock in Figure 1 represents the static noninearity of the Wiener structure given by the Hi equation, [11]: = 1Cγ C γ + () yγ (t), where C, i.e. the effect concentration at haf of the maxima effect, and γ are patient-dependent parameters. The output of the mode corresponds to the NMB eve and varies between 1% (for norma muscuar activity) and % (for totay paraysis). B. Mass contro For the purpose of appying a positive contro aw for feedback stabiization of compartmenta systems proposed in [] it is necessary to check that the state-space mode (2) has a compartmenta structure. For this sake the components of the vectors B(α) and C must be nonnegative and A(α) must satisfy the foowing conditions, []. A(α) is a Metzer matrix, i.e., it has nonnegative offdiagona entries, ai j(α), i, j and i j A(α) has negative diagona entries, aii(α) i A(α) is diagonay dominant, aii(α) j j a ji(α) i If the matrix A(α) satisfies these conditions it is caed as a compartmenta matrix. Since, the vaues for k1 < k2 < k where λ is a positive design parameter, x(t) is the state vector and the tota mass of the system is given by the amount of mass in each compartment, xi(t), = xi(t). () i=1 The foowing theorem guarantees the convergence of the state trajectories of the cosed-oop system to the set Ω = { x R : =}, known as the iso-mass corresponding to the vaue. Theorem 2.1: For the cosed-oop system (2)-(4) with the arbitrary initia conditions x() the iso-mass Ω is forward invariant; the state vector x(t) is bounded for a t > and converges to the iso-mass Ω. The proof of this theorem is given in []. The idea now is to use the mass contro in order to achieve the contro of the NMB eve. More concretey, it turns out that the contro aw (4) not ony drives the system mass to the vaue, but it aso drives the system to a steady-state in the iso-mass Ω, and consequenty to a certain vaue of the effect concentration. Therefore, by suitaby choosing the vaue of, one can obtain the desired steady-state vaue for the effect concentration y re f, and consequenty drive the NMB response to a desired reference vaue. Here this reference vaue is taken to be 1%, as usuay required by cinica practice. Note that, when =, the cosed-oop system (2)- (4) can be written as { ẋ(t) = Ã(α)x(t) (6) = Cx(t) where Ã(α)=A(α)+ B(α) i=1 b i(α) (1 1 1) A(α). 868

3 869 γ = γ + γ by y re f = Cx e. Nevertheess, this parameter infuences how quick is the converge of the mass of the system,, to. Thus, the ony parameter that affects the convergence of the system mass is the parameter γ present in noninear static equation. It is assumed that the reference vaue of the effect concentration y re f, given In this section simuated cases of the appication of the NMB compartmenta contro aw are presented, based on a bank P of hundred noninear dynamic modes Pj { j = 1,...,1} used [6]. As mentioned before, the nomina vaues of the parameters α(t ) and γ(t ) given by the EKF agorithm at time instant t were used for the appication of the compartmenta contro aw. Moreover, desired NMB eve is taken to be 1%, as aso mentioned before. The identified parameters, α and γ may be affected by uncertainties and the main goa of this paper is to anayze and derive a contro aw for reference tracking of the NMB eve which is not sensitive to these uncertainties. The performance of this new approach is aso iustrated by simuation studies. Note that the vaue of the desired mass given by (8), to be used in the contro aw, is independent of the parameter α, so the identification of this parameter does not affect III. APPLICATION TO THE NMB CASE STUDY D. Parameter uncertainties where sd i is the standard deviation of the estimated parameter cacuated between the time t and the end of the surgery, and x i is the corresponding mean vaue for each parameter and each patient. Thus, an adequate choice for the parameter uncertainties is given by the mean vaue of c i v, {i=1,...,6}. The identification of the dynamic system by the Extended Kaman Fiter (EKF) is described in this section. Simiary to what happens in [4] the unknown parameters are identified using the EKF by a couped identification mode. In order to impement the proposed mode structure in the identification agorithm, the system is discretized using the zero-order hod method [2], considering the time samping frequency h=1/ imposed by the samping in the surgery room. The contro aw previousy described is appicabe to modes with fixed parameters [], which does not agree with the identification procedure given by the EKF agorithm. Therefore and meeting the cinica procedures, the idea is to give the patients an initia bous, run the EKF unti a certain time t, and simuate the system (2) using the ast obtained estimate for each parameter, α(t ) and γ(t ), respectivey. The time t is given by the OLARD (OnLine tuned Agorithm for Recovery Detection) agorithm [1], that produces an estimate for the beginning of the patients recovery after the initia atracurium bous administration, according to the cinicians point of view. c i v = sdi x i {i=1,...,6} (1) Figure 2 iustrates the variation of the parameters α and γ identification given by the EKF. The EKF was appied to each rea case and the onine estimates of α and γ were obtained. The OLARD was aso appied to detect the recovery time for each rea case ( in Figure 2) in order to evauate γ, nomina vaue for the parameter γ. Hence, for each patient i in this database and for each parameter a coefficient of variation c i v, was cacuated, given by Fig. 2. The estimates of parameters given by EKF agorithm in rea database R. The dashed ines represent the minimum, mean and maximum vaues for the nomina vaue of the parameters α and γ at recovery time t γ 2 2. C. Identification of system parameters and y re f corresponds to the NMB response reference of 1%, forces the system to foow this desired reference y re f (γ)= ( 1 y re f 1 ) 1/γ C, (9) α.6 where y re f (γ) can be cacuated by the inversion of the Hi equation.8.1 Moreover, it can be shown that this equiibrium point is asymptoticay stabe. Therefore, setting as desired system mass = x e = y re f (γ), (8) x e =( / / /) T. (7) This system has an equiibrium point x e which is the soution of the equation Ã(α)x e = that satisfies M(x e ) =, which is given by where γ = γ(t ) is considered as the nomina vaue of the parameter γ, and γ is the parameter uncertainty. In order to obtain reaistic vaues for simuations, the uncertainties α and γ present in parameters α and γ, respectivey, are obtained by anayzing the set of parameter estimates obtained in previousy coected rea data R. This database presents 6 rea patients submitted to genera anesthesia for abdomina surgeries where the cosed-oop contro software Hipocrates, [8], was appied.

4 87 The strategy for the NMB contro with parameter uncertainties used here is schematicay represented in Figure and can be summarized in the foowing steps: 1 o The rea patient dynamics is simuated from the modes of the database P, after the administration of an initia bous of musce reaxant of µg/kg; 2 o Unti time t, the mode parameters α and γ and the state vector are identified by the EKF agorithm; o The OLARD agorithm determines the time instant t and from this instant on the contro aw (tuned for the nomina parameter vaues α and γ ) is appied to the simuated patient mode with α = α(t )+ α and γ = γ(t )+ γ considering two distinct simuation scenarios. In a first stage it is considered that α = γ = and in a second stage it is considered that α and γ. With the purpose of testing the goba approach comprising the dedicated identification and contro agorithms previousy described, simuation studies have been carried out using the bank mode P. The behavior of the system mass, the desired system mass, the drug dose profie u(t) obtained by the compartmenta contro aw and the NMB response controed by this dose profie are potted in order to iustrate the performance of the contro strategy. From a previous study [1], it turns out that an adequate choice of the design parameter λ is.2. Figure 4 shows the contro agorithm performance when the exact parametrization of the simuated patient is given by (2) and () and the α = γ =. In particuar for patient P6 the parameters are α =.68 and γ = and the contro objective of achieving a NMB eve of 1% amounts to stabiize the system mass on the vaue = 22.8µg/kg. As shown in Figure 4, the NMB eve was driven to the target vaue (1%) and the system mass converges to the desired mass. Figure iustrates a simuation for patient mode P6 assuming non zero vaues for the uncertainties α and γ,.14 and.42, respectivey, obtained using, for each parameter, the mean vaue of the coefficient of variation c i v {i=1,...,6} (1). As shown in Figure, the system mass converges to y re f (γ + γ)=y re f (γ )+ y (1) In particuar, the term y re f (γ + ˆ γ) in the ast equation can be written as = y re f (γ + γ). (12) This is due to the fact that depends on γ and is hence affected by the uncertainty in this parameter. More concretey, the vaue of is obtained from (8) with γ = γ, i.e.: = y re f (γ ), (11) whereas the correct vaue for the desired mass shoud be Fig.. Simuation for the NMB contro eve of the patient P6 with α and γ, = 22.8µg/kg, α =.68+ α and γ = γ. Upper pot: system mass (soid ine) and the desired mass (dashed ine). Controer NMB eve (eft bottom pot) using the contro signa represented in the right bottom pot. The star in xx-axis represents t = 27.min detected by the OLARD (%) Fig. 4. Simuation for the NMB contro eve of the patient P6 with α = γ =, = 22.8µg/kg, α =.68 and γ = Upper pot: system mass (soid ine) and the desired mass (dashed ine). Controer NMB eve (eft bottom pot) using the contro signa represented in the right bottom pot. The star in xx-axis represents t = 27.min detected by the OLARD. Fig.. Feedback contro system bock diagram composed by the rea patient, the patient mode, the identification and the controer bocks (%) the desired = 22.8µg/kg and the NMB eve converges to a vaue which is different from the desired target.

5 871 Figure 6 shows what happens when the initia guess for the is rectified and repaced in the contro aw by the This work was supported by FEDER founds through COMPETE Operationa Programme Factors of Compet- Fig. 6. Simuation for the NMB contro eve of the patient P6 with α =.14 and rectification of, assuming α =.68 and γ = Upper pot: system mass (soid ine) and the desired mass (dashed ine). Controer NMB eve (eft bottom pot) using the contro signa represented in the right bottom pot. The star in xx-axis represents t = 27.min detected by the OLARD. V. ACKNOWLEDGMENTS (%) This paper presents a new strategy for the contro of the NeuroMuscuar Bockade (NMB) of patients undergoing genera anesthesia in the presence of parameter uncertainties. This consists in using a minimay parametrized mode previousy presented in the iterature, which is here shown to have a compartmenta structure. This specia structure enabes the design of a controer for tracking a desired NMB eve, by driving the system to an adequate tota mass. The presence of uncertainties in the mode parameters does not affect the mass contro agorithm, but affects the computation of the adequate set-point for the tota mass that corresponds to the desired NMB eve. However an anaysis of the system response aows to estimate the uncertainties in the parameters and rectify the contro aw in order to obtain the desired NMB eve set-point. 1 IV. CONCLUSIONS 2 2 once y is known. The vaue of y is obtained by making the difference between the desired effect concentration and the observed effect concentration when the system begins to stabiize, i.e. after a certain time tss. Fig. 7. Noisy simuation for the NMB contro eve of the patient P6 with α =.14 and rectification of, assuming α =.6 and γ = 2.4. Upper pot: system mass (soid ine) and the desired mass (dashed ine). Controer NMB eve (eft bottom pot) using the contro signa represented in the right bottom pot. The star in xx-axis represents t = 2.min detected by the OLARD. γ = y ( ) y re f, (17) γ=γ (%) So, it is possibe to see that the difference obtained in the NMB eve in the ast simuation case is due to the uncertainty in the desired mass that corresponds to the second term in the right-hand of equation (16). Note that, for rea patients, the uncertainty γ is unknow. However, from (14) it is possibe to approximatey determine this vaue as ( ) y re f (γ )+ y re f γ=γ γ, (16) 2 2 Therefore, the vaue for the system desired mass, is obtained by equation (12) as: ( ) y re f (γ + γ) y re f (γ )+ y re f γ=γ γ (1) So, substituting y given by the equation (14) in equation (1), the vaue of the reference for the output of the inear part is given by where ( ) y re f γ=γ = dyre f dγ (γ ). y = ( ) y re f γ=γ γ (14) where the vaue of y can be approximated as foows, vaue obtained from (16) with γ given by (17), which happens after minute 7. This correction is an onine adaptive process that changes the desired mass for the system and consequenty changes the drug administration profie and the NMB eve. The simuation resuts in Figure 7 were performed in the presence of noise taken from a typica NMB case from rea database R. The fiter agorithm presented in [8] was appied to the rea signa and the obtained residuas were used as the noise vector to be added to the output signa in the simuation. In this case the parameters are α =.6, γ = 2.4, the desired mass = 2.81µg/kg and the t = 2.min. As it is possibe to see this simuation presents a good resuts for the controer even in the presence of noise.

6 itiveness ( Programa Operaciona Factores de Competitividade ) and by Portuguese founds through the Center for Research and Deveopment in Mathematics and Appications (University of Aveiro) and the Portuguese Foundation for Science and Technoogy ( FCT Fundação para a Ciência e a Tecnoogia ), within project PEst- C/MAT/UI416/211 with COMPETE number FCOMP FEDER-2269 and the project GALENO (reference PTDC/SAU-BEB/1667/28). REFERENCES [1] J. Ameida, T. Mendonça, and P. Rocha. A compartmenta mode based strategy for neuromuscuar bockade eve. In Proc. 18th IFAC Word Congress 211, pages 99 64, Miano, Itay, August 211. [2] K. J. Åström and B. Wittenmark. Computer-Controed Systems. Prentice Ha, Engewood Ciffs, NJ, [] G. Bastin and A. Provost. Feedback stabiisation with positive contro of dissipative compartmenta systems. In Proceeding of the 1th Internation Symposium on Mathematica Theory of Networks ans Systems., Notre Dame, Indiana, USA, 22. [4] M. M. da Siva, T. Wigren, and T. Mendonca. Noninear identification of a minima neuromuscuar bockade mode in anesthesia. Contro Systems Technoogy, IEEE Transactions on, PP(99):1, 211. [] K. Godfrey. Compartmenta modes and their appication. Academic Press, 198. [6] P. Lago, T. Mendonça, and L. Gonçaves. On-ine autocaibration of a PID controer of neuromuscuar bockade. In Proc. of the 1998 IEEE Internationa Conference on Contro Appications, pages 6 67, Trieste, Itay, [7] T. Mendonça and P. Lago. PID contro strategies for the automatic contro of neuromuscuar bockade. Contro Engineering Practice, 6(1): , [8] T. Mendonça, H. Magahães, P. Lago, and S. Esteves. Hipocrates: A robust system for the contro of neuromuscuar bockade. Journa of Cinica Monitoring and Computing, 18:26 27, 24. [9] T. Söderström. Discrete-time Stochastic Systems. Springer-Verag, London, UK, 22. [1] M. M. Siva, C. Sousa, R. Sebastiao, J. Gama, T. Mendonça, P. Rocha, and S. Esteves. Tota mass TCI driven by parametric estimation. In Proc. Mediterranean Conference on Automation and Contro (MED 9), pages , Thessaoniki, Greece, June 29. [11] B. Weatherey, S. Wiiams, and E. Nei. Pharmacokinetics, pharmacodynamics and dose-response reationships of atracurium administered i. v. Br. J. Anaesth., :9s 4s,