Automated postoperative blood pressure control
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Journal of Control Theory and Applications 3 (2005) 207- 212 Automated postoperative blood pressure control Hang ZHENG, Kuanyi ZHU (School of Electrical and Electronic Engineering, Nanyang Technological University, Nanyang Avenue, 639798 Singapore) Abstract: It is very important to maintain the level of mean arterial pressure (MAP). The MAP control is appfied in many clinical situations,including limiting bleeding during cardiac surgery and promoting healing for patient's post-surgery. This paper presents a fuzzy controller-based multiple-model adaptive control system for postoperative blood pressure management. Multiple-model adaptive control (MMAC) algorithm is used to identify the patient model, and it is a feasible system identification. method even in the presence of large noise. Fuzzy control (FC) method is used to design controller bank. Each fuzzy controller in the controller bank is in fact a nonlinear proportional-integral (PI) controller, whose proportional gain and integral gain are adjusted continuously according to error and rate of change of error of the plant output, resulting in better dynamic and stable control performance than the regular PI controller, especially when a nonlinear process is involved. For demonstration, a nonlinear, pulsatile-flow patient model is used for simulation,and the results show that the adaptive control system can effectively handle the changes in patient' s dynamics and provide satisfactory performance in regulation of blood pressure of hypertension patients. Keywords: Multiple-model adaptive control; Fuzzy control; Blood pressure control; Cardiovascular modeling 1 Introduction After completion of open-heart surgery, some patients develop high mean arterial pressure (MAP) . .Such hypertension should be treated timely to prevent severe complication. Continuous infusion of vasodilator drags, such as sodium nitropmsside (SNP) ,wou ld quickly lower the blood pressure in most patients. However, a wide range of patient drag sensitivities to SNP occurs and critical patient care o~en requires the infusion of blood, fluid, or drags to regulate cardiovascular system variables. Therefore, manual control by clinical personnel could be very tedious, time consuming and inconsistent. Automation of these therapies by application of feedback control techniques can be effective. To improve the quality of patient care, automatic closed-loop control systems for SNP delivery have been developed. A proportional-integral (PI) controller-based multiple-model adaptive control (MMAC) system was built in [ 1 ] , and was improved by Martin [ 2 ~ 5 ]. MMAC is a feasible system identification method even in the presence of large noise. However, our simulations have shown that PI controller cannot work well because the system is nonlinear. A fuzzy control system was built in [ 6,7 ] . Based on the fuzzy controller, general fuzzy control method is proposed in [8 ,9 ] , and the nonlinear controller can overcome the shortcomings that PI controller possesses inherently. However, no identification of the plant model is considered. Furthermore, the wide range of patient drag sensitivities to SNP is not considered. In this paper, we use multiple-model adaptive control (MMAC) algorithm to identify the patient model, and use fuzzy control algorithm to design controller bank. Thus, the proposed control method can provide satisfactory control performance and can handle changes in the system dynamics. 2 Plant characteristics A dynamic model of patient mean arterial pressure response to SNP infusion rate was developed by Slate [ 10]. It is obtained by using animals' as well as actual patients' data together with physiological knowledge about SNP effect. Improvement is made to State's model according to our nonlinear, pulsatile-flow patient model, which will be illustrated in Section 4, and dog experiment results. The modified Slate's model is given by (1) and (2) . MAP = MAP 0 +AMAP + MAPn, (1) where MAP is the actual mean arterial pressure; MAP0 is the initial blood pressure; AMAP is the change in blood pressure due to infusion of SNP; MAPn is the plant background noise. MAP0 is usually 115 - 140 mmHg[ 1 ]. The variance of MAPn is typically 4 mmHg (for normal noise levels) or 16 ~ 36 mmHg (for high noise levels). Received 8 June 2005. 208 H. ZHENG et al. I Journal of Control Theory and Applications 3 (2005) 207- 212 Many people might consider MAP0 as a constant. However, MAP 0 in fact can change with time in many cases. For example, after the patient is cured, his MAP 0 gready declines, and the infusion of SNP can be stopped. "the " Therefore, defining MAP 0 as initial blood pressure is imprecise, we can consider that MAP 0 is just a parameter in (1 ) , and that the parameter can change with time. The transfer function describing the relationship between the change in the blood pressure AMAP and the drug infusion rate is given by AMAP(s) Ke-r,k'(1 + ae- r~) SNP(s) - (1 + Tids)(1 + rs ) ' (2) where AMAP(s) is the change in blood pressure, SNP( s ) is the SNP infusion rate, K represents the sensitivity of patients to SNP, Tik is the transport delay from the injection site, a is the recirculation constant, Tc is the recirculation time delay, Tid results fi'om the gradual relaxing of the vascular muscle in response to SNP, r is the lag time constant resulting from the uptake, distribution, and biotransformation of the drug. Note that steady-state gain of dose response is K(1 + a) .The parameters in (2) are chosen as: K = - 0.25 to - 9 (normal = - 1 ) , Tik = 10 ~ 20s (nonna l=20s) , To = 30 - 75s (normal=45 s), a =0-0 .4 , TAd = 10 - 40s (nomaal=20s) , v = 30-60s (normal = 40 s). The unit of AMAP is mmHg. The unit of the infusion rate of SNP is ml /h . The concentration of SNP is 200 rng/1. 3 Control system design The MMAC procedure is based upon the assumption that the plant can be represented by one of a finite number of models and that for each such model a controller can be a priori designed. All of these controllers constitute a controller bank. An adaptive mechanism is then needed for deciding which controller should be dominant for a given plant. One procedure for solving this problem is to form a weighted sum of all the controller outputs, where the weighting factors are determined by the relative residuals between the plant response and the model responses. Fig. 1 shows the structure of the MMAC system. Setpoint + ~1 Controller 1 Controller Bank) ~NP2] I Npi; sN : / Integrate _ _ MAP2 MA ?s Ws (Model Bank [~ RI [ Operation L of .- R 8 i Residuals w1 Operation w2. of [ Weights t~ Fig. 1 Since the plant gain is negauve, the system error is expressed as e(k) = MAP(k) - MAPo(k) , where k is the sampling time and MAPo(k) is the commanded or set-point pressure level. The rate change of error is defined by r (k ) = [maP(k) - M iP (k - 1 ) ] / r , where T is the sampling interval, and T = l0 s. MMAC system structure. For patient safety, two nonlinear units are built into the system. The nonlinear unit limiting SNP infusion rate change is given as f l (x ) = for - 40 ~< x ~< 7, (3) fo rx >7. The reason for this constraint is that SNP is metabolized by the body into cyanide, and hence, too much SNP can be H. ZHENG a al . / Jounuzl of Control Theory and Applications 3 (2005) 207- 212 209 toxic to the patient. The nonlinear unit limiting SNP infusion rate is given as 0, for x < 0, f i (x ) = x, for 0 ~ X ~ U m, (4) Um, for x > Urn, where Um is the maximum infusion rate. This is to prevent rapid decreases in pressure which can cause diminished blood flows or circulatory collapse. We use the equation in [ 1 ] to compute Urn, U m --- 60 x Wp x i m X C; I, (5) where Wp is patient weight (the unit is kg) , i m is maximum recommended dose (the value is 10 Pg" kg-1. min-1) , C, is drag concentration (the unit is pg/ml) . For example,if Wp = 70kg, then (60min/h) x 70kg x (lOtzg kg -1 min -1) U m = 200/~g/ml = 210ml/h. With respect to the MMAC development, the design of the model bank is described in Section 3.1 ; in Section 3. 2, the design of the controller bank is given; and finally, the actual control computation is described in Section 3.3. 3.1 Model bank design The model bank consists of 8 models with constant parameters characterizing the individual plant subspace. are described by the following transfer These models functions: AMAPi ( s ) SNP( s ) Kie-2*(1 + 0.2e -5') = (1 + 40s)(1 + 45s) ' j = 1 , . . . ,8 . (6) MAPj = MAPoINIT + AMAPj, j = 1 , " ' ,8 , (7) where AMAPi(s) is the blood pressure change of the j th model due to infusion of SNP, SNP(s) is the input of these 8 models, MAPj is the output of the j th model, MAPoINrr is the estimated initial value of patient's MAP0. MAPoINrr is calculated in the first 30 seconds, as the average of MAP(0), MAP( 1 ) and MAP(2). Tc has trivial effect on the control system because the value of a is small, so it can be frozen in the model bank. Tid and r are lag time constants, and they can affect the dynamic behavior of the system. Smaller Tid and v restflt in smaller settling time. However, the range of Tid and r is small, so their effects are reduced. Therefore, Tid and r can also be frozen. In the model bank, the three parameters are selected as Tcj = 50, Tidj = 25, and rj = 45 (for j = 1, "" ,8). It is noted that while K reflects patient response sensitivity, the effective steady state gain is K( 1 + a ). That is, the recirculation enhances the effect of infusion [ 11 ]. However, the effect of a can be compensated by different Kj (for j = 1 , ' " , 8). Therefore, a can also be frozen. In the model bank, aj = 0.2 (for j = 1 , ' " , 8). The plant parameter Tik significantly affects undershoot. Controlling a high-infusion-dehy patient with a low-infusion-delay model will cause large undershoot. For a blood pressure control system,large undershoot is not allowed, so that Tik j (for j = 1 , ' " ,8 ) should be chosen as the ~ u m expected plant delay time. Although controlling a low-infusion-delay patient with a high-infusion-delay model causes long settling time, it can be partly compensated by different Kj (for j = 1 , - " , 8). The plant parameter K has the greatest effect on controller performance. If the plant has high gain and is being controlled by a low-gain model, large overshoots occur; if the plant has low gain and is being controlled by a high-gain model, long settling time Hill occur. Table 1 shows the eight models and the range of plants they cover. The plant gain partition factor is about 1.56, i .e . , Kjn~,,/Kjmi,~ ~ 1.56. Notice that ~/1 .56~ 1.25, therefore Kj,,ax/K j ~ Kj/Kimi,, ~ 1.25. Kjmin is the mini- mum plant gain that Kj covers, and Kjm~x is the maximum plat gain that Kj covers. Table 1 Gains of the eight models (column 2) , along with the range of plants they cover (column 3). Model Model Gain/ Plant Gain/ Number (mmHg/(ml" h- ' )) (mmHg/(ml-h- ' ) ) 1 -0.3125(K1) ffom-O.25(K~a,)to-O.39(K~,~) 2 - 0.4872 (K2) from- 0.39 (K2~a.) to - 0.61 (K2n~x) 3 -0.7625(/(3) from- 0.61 (K3,~,) to - 0.95 (K3,~) 4 - 1. 1875 (K4) t~om- 0.95 (K4~) to - 1.48 (K4~) 5 -1.8500(/(5) ~om- 1.48 (Ksmm) to - 2.30 (Ks,,~) 6 -2.8750(/(6) from- 2.30 (Kt~) to - 3_60 (Kt,~) 7 -4.5000(/(7) from- 3.60 (K7~an) to - 5.60 (K7,~) 8 - 7.2000 (Ks) fi'om- 5.60 (Ks~) to - 9.00 (gs~) 3.2 Controller bank design The details about the fuzzy controller are described in [6 ,7 ] . It converted the expert-system-shell-based fuzzy controller to nonfuzzy control algorithms, and a total of ten different nonfuzzy control algorithms for 20 different input combinations were obtained. We compare the nonfuzzy control algorithms, and find that they can be substituted in fact only by one algorithm ASNP( k ) f ( GE x e(k)) + f ( GR x r(k)) =-G I ' L 'T" 3L - max[f( GE x e (k ) ) , f ( gR x r (k ) ) ] ' (8) 210 H. ZHENG et al . /Journal of Control Theory and Applications 3 (2005) 207- 212 where e(k) is the error, r (k ) is the rate of change of error, G1 is the scalar for incremental SNP infusion rate, GE is the scalar for the error, GR is the scalar for the rate of change of error, L is a turning point of the fuzzy sets in the fuzzy controller. T is the sampling interval. Ftmctionf(x) = sgn(x)" min(L, Ix 1). After simplifying the fuzzy control algorithms as (8) , it is much easier to design the controller bank. There are eight controllers in the controller bank. All of the parameters except GI are the same in the eight controllers,and they are chosen as L = 16, GE = 0.25, GR = 15. In  , GR = 13.5;in this paper, we increase GR to 15, because simulations show that the controller works better when GR = 15. Table 2 shows GI values of the eight models. Note that Ks" GIj ..~ O. 0460 (for j = 1, ...,8). Table 2 GI values of the eight controllers (column 2),and the gains of the eight models (column 3). Model Model Gain/ GI Number (mmHg/(rrfl" h- 1 ) ) 1 - 0. 1843 (G I l ) -0.3125 (Kl) 2 -0.1183 (GI2) - 0.4875 (K2) 3 -0.0755 (GI3) -0.7625 (K3) 4 -0.0485 (G/a) - 1.1875 (K4) 5 - 0.0311 (CI5) - 1.8500 (Ks) 6 - 0.0200 (g16) - 2.8750 (K6) 7 - 0.0128 (g17) - 4.5000 (K7) 8 - 0.0080 (GIs) -7.2000(/(8) 3.3 Identification algor i thm To achieve desirable system performance and guarantee patient safety, the identification algorithm should converge quickly to the desired values and should react to time-varying plant characteristics, as well as ensure a reasonable rate of blood pressure change. Thus, the control output is computed as a weighted sum of controller bank signals, i. e . , 8 ASNP(k) = ~, [Wj (k ) .ASNP j (k ) ] , (9) j=l where W~(k) is the weighting factor of the jth controller, ASNPj(k) is the output of the jth controller. In the first 30 s, MAPoINr r is calculated. In the meantime, the SNP infusion rate should be equal to zero. Therefore, in the first 30s, Wj(k) = 0, j = 1, . . . ,8 . The convergence algorithm of Wj will be described later. Since the control variable SNP(k) will be in error before the convergence of Wj, large overshoots could occur for plants with high gain. Therefore, from 0.5 min mark to 3 rain mark, the infusion rate is based on the assumption that the most matched model is Model 8, i. e. from 0.5 rain mark to 3 min mark, 0, for j = 1 , " ' ,7 , Wj(k) = 1, for j 8. The convergence algorithm is run at 30 s mark, and the weighting factors begin to converge. However, their values won't be used until 3 rain mark, because their values haven't converged in the first 3 minutes. The convergence algorithm in [ 1 - 4] is improved, so that the computation precision of the weighting factors can be enhanced. In the algorithm, the weights are selected as follows: 1) Recursive update exp[-R~( k )/2V2]Wj( k -1 ) Wj'(k) = 8 ' J = 1,-..,8; ~] exp[-R2(k)/2V2]Vc~i(k-l) i=1 2) Bounding from zero a, for E . ' (k ) ~< a, Wj(k) = Wj'(k), for Wj'(k) > c~, j = 1, (10) . . . ,8; (11) where MAPj (k) is the output of the jth model. V is a parameter controlling the convergence rate of Wj(k) with R~(k). We compare  with , and find that 2 V 2 = tTn (MAP0 - MAPc) 2' (14) 3) Normalization [Wi ' (k ) ] 2 Wj(k) = 8 , J = 1,--- ,8, (12) 2 i=1 where R2(k) is the relative residual which will be defined in (13), V is a parameter controlling the convergence rate of (k) with Eq. (11) is used to prevent 8 ___a~exp[- R2(k) /2V2]Wi(k - 1), the denominator on i=1 the fight in (10), from being zero. In the algorithm, the initia ! weighting factors Wj(O)(j = 1, ' " , 8) must be determined a priori. We choose them as Wj(0) = 1/8 ( for j = 1,-- - ,8). The relative residual R 2 ( k ) is defined as the normalized squared error between plant and model, i. e. [ MAP j (k ) - MAP(k)]2 R2(k) = M---~OINIT--- ~ ' j = 1, * ,8 , (13) H. ZHENG et al . / Journal of Control Theory and Applications 3 (2005) 207- 212 211 2 where a , is the variance of noise associated with MAP. For rapid convergence of Wj(k) , a smaller value of V is desired. However, if we choose a small value of V when high noise level is present, the system ~ be unstable. In this paper, we choose V as V = 0.1. 4 Simulation study Fig. 2 shows the structure of a nonlinear, pulsafile-flow patient model, and we develop it based on [5,13 - 15]. It is a much more complex patient model than modified Slate' s model, and will be used in computer simulation. In SNP pharmacokinetic model and pharmacod3mmnic model, the effect of different values of the infusion delay, sensitivity to SNP, amotmt of recirculation, and speed of action of SNP, can all be studied. In cardiovascular model, it is possible to change, independently or in any combination, the resistances of the arteries and veins, their comphances, the strength of the heat's contraction, heart rate and many other parameters. We only introduce some most important parameters in the patient model. ;t and T d are two parameters in SNP pharmacokinetic model. The gradual relaxing of the vascular muscle in response to SNP is determined by Td, and 2 represent SNP recirculation effect. /31 and f12 are two parameters in SNP pharmacodynamic model, and they represent the sensitivity of patients to SNP. Concentration of Arterial resistance SNP SNP in the arteries SNP SNP (t) Pharmacokinetic Phamacodynamic model Concentration of model Venous unstressed SNP in the veins volume change Cardiovascular I model I Arterial pressure Fig. 2 A nonlinear, pulsatile-flow patient model. The set-point pressure level is initially chosen as MAP~ = 100. Some important parameters of the patient model are initially chosen as ;t = 0.4, T d = 40 , fll = 90 and/32 = 120. Then, Td is changed to 25 at the 4 rain mark, /31 and /32 are increased to 135 and 180 respectively at the 20 rain mark, MAPc is decreased to 80 at the 35 min mark, 2 is changed to 0.6 at the 50 rain mark. White Gaussian noise (mean 0, variance 9) is included. Figs. 3 and 4 show the simulation results. 140 130 120 ZZ E 110 ~- 100 ~ 9o 80 70 L, i0 80 ' I~tl/r ~ ' - '~ , l l . Z:0.4~0.6 Td:40--25 fi1:90~135 ~ . ~ , /~2,:120~!80 , ,~h~ ' '" 10 20 30 40 50 60 70 t / min Fig. 3 Mean arterial pressure. e- .o 35[ " fl1:90]~135 " ! " ~ " 30 / t2:120~180 i ,'" i~. :04_06 - ls[ f MAp0:100-80 l0 } , 0 10 20 30 40 50 60 70 t / rain Fig. 4 Mean arterial pressure. 5 Conclusions A fiazzy controller-based MMAC algorithm is proposed to control postoperative hypertension. Modification has been made to Slate's model and a nonlinear, pulsatile-flow patient model is developed for computer simulation. The results of simulations indicate that the control system can automatically control blood pressure over a fairly wide plant parameter envelope, even in the presence of large background noise. Further animal experiments and a flail clinical test should be conducted. References [ 1 ] W.G. He, H. Kaufinan, K. Roy. Multiple-model adaptive control proce- dure for blood pressure control [J ]. IEEE Trans. on Biomedical Engineering, 1986,33( 1 ) : 10 - 19. [ 2 ] J .F . Martin, A. M. Schneider, N. T. Smith. Multiple-model adaptive control of blood pressure using sodium ultroprusside [J]. IEEE Trans. on Biomedical Engineering, 1987,34(8):603- 611.  J.F. Martin, A.M. Schneider, M.L. Quinn, et aI. Improved safety and eflacacy in adaptive control of arterial blood pressure through the use of a supervisor [ J ] . IEEE Trans. on Biomedical Engineering, 1992, 39 (4) :381 - 388.  J.F. Martin, N. T. Smith, M. L. Quinn, et al. Supervisory adaptive con- trol of arterial pressure during cardiac surgery [ J ]. IEEE Trans. on Biomedical Engineering, 1992,39(4) :389 - 393.  "J. F. Martin. Multiple-Model Adaptive Control of Blood Pressure Using Sodium Nitroprnsside [ D]. San Diego: Department of Applied Mechan- ics and Engineering Science, University of California, 1987.  H.Ying,M.McEachem,D.W.Eddleman, et al. Fuzzy control of mean 212 H. ZHENG a a l . / Joumal of Control Theory and Applications 3 (2005) 207- 212 arterial pressure in postsurgical patients with sodium nitropmsside infusion [ J ]. IEEE Trans. on Biomedical Engineering, 1992,39 (10) : 1060 - 1070. [7 ] H. Ying, L. C. Sheppard. Kegnlation mean arterial pressure in postsurgical cardiac patients [J ]. IEEE Engineering in Medicine and Biology, 1994,13(5) :671 - 677. [ 8 ] H. Ying. Explicit smacmre of the typical two-input fuzzy controllers [ C] / / Proc. of the Fifth IEEE Int. Conf. on Fuzzy Systems. New Or- leans, LA, USA, 1996,1:498 - 503.  F. Lin, H. Ying. Modeling and control of fuzzy discrete event systems [J]. IEEE Trans. on Systems, Man and Cybernetics, Part B,2002,39 (4) :408 - 415.  J" B ' Slate' L" C" Sheppard' V ' C" Rideut' et al" A mdd fr design f a blood pressure controller for hypertensive patients [ C ] / / The 5 th IFAC Syrup. on Identification and System Parameter Estimation. Darmstadt: Federal Republic of Germany, 1979. [ 11 ] J .S . Delapasse, K. Behbehani, K. Tsni, et al. Accorranodation of time delay variations in automatac infusion of sodium nitroprusside [ J ] . IEEE Trans. on Biomedical Engineering, 1994, 41 ( 11 ): 1083 - 1091.  C .Yu , tL.J.R.oy, H. Kanfrnan, et al. Multiple-modd adaptive predic- tive control of mean arterial pressure and cardiac output [J ]. IEEE Trans. on Biomedical Engineering, 1992,39(8 ) : 765 - 778.  J .F.Mart in,A.M.Schneider, J .E.Mandel, et al.A new cardiovascular model for real-time applications [ J ]. Trans. Soc. Comput. Simulation, 1986,3 ( 1 ) : 31 - 66. [ 14] Eileen A. Woodruff, James F. Martin, Madonna Omens. A model for the design and evaluation of algorithms for dosed-loop cardiovascular therapy [J ]. IEEE Trans. on Biomedical Engineering, 1997,44(8) : 694 - 705. [ 15 ] Vincent C. Rideout. Mathematical and Computer Modeling of Physiological Systems [ M] . Englewood Cli~, NJ: Prentice Hall, 1991:3 -4 . [ 16] K. Behbehani, R. P,.. Cross. A controller for regulation of mean arterial blood pressure using optimum nitroprussde infusion rate [ J ]. IEEE Trans. on Biomedical Engineering, 1991,38(6) :513- 520. [ 17 ] G.I. Voss, H. J . Chizeck, P. G. Katona. Self-mmng controller for drug delivery systems [J]. Int. J. Control, 1998,47(5):1507- 1520. t lan~ ZI-IF_,NG graduated from the University of Science and Technology of China, in 2001. Currendy, he is a master student in School of Elec- trical and Electronic Engineering of Nanyang Technological University, Singapore. His research interests include nonlinear time-varying system control,multiple model adaptive control and fuzzy control. Email: zhen0008 @ ntu. edu. sg. Kuany i Z I /U graduated from No~heastem University,China,in 1982.He received his M.E . and Ph. D from University of Louvain-La-Neuve, m Belgium in 1986 and 1989, respectively. Cur- rently, he is associate professor with School of Hectrical and Electronic En~neering of Nanyang ~I Technological University, Singapore. His research interests include model-based predictive control, biomedical system control and process control. Ernail: ekyzhu @ ntu. edu. sg.
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