# Automated postoperative blood pressure control

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<ul><li><p>Journal of Control Theory and Applications 3 (2005) 207- 212 </p><p>Automated postoperative blood pressure control </p><p>Hang ZHENG, Kuanyi ZHU (School of Electrical and Electronic Engineering, Nanyang Technological University, Nanyang Avenue, 639798 Singapore) </p><p>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. </p><p>Keywords: Multiple-model adaptive control; Fuzzy control; Blood pressure control; Cardiovascular modeling </p><p>1 Introduction </p><p>After completion of open-heart surgery, some patients </p><p>develop high mean arterial pressure (MAP) . .Such </p><p>hypertension should be treated timely to prevent severe </p><p>complication. Continuous infusion of vasodilator drags, </p><p>such as sodium nitropmsside (SNP) ,wou ld quickly lower </p><p>the blood pressure in most patients. However, a wide range </p><p>of patient drag sensitivities to SNP occurs and critical </p><p>patient care o~en requires the infusion of blood, fluid, or </p><p>drags to regulate cardiovascular system variables. Therefore, </p><p>manual control by clinical personnel could be very tedious, </p><p>time consuming and inconsistent. Automation of these </p><p>therapies by application of feedback control techniques can </p><p>be effective. </p><p>To improve the quality of patient care, automatic </p><p>closed-loop control systems for SNP delivery have been </p><p>developed. A proportional-integral (PI) controller-based </p><p>multiple-model adaptive control (MMAC) system was built </p><p>in [ 1 ] , and was improved by Martin [ 2 ~ 5 ]. MMAC is a </p><p>feasible system identification method even in the presence </p><p>of large noise. However, our simulations have shown that </p><p>PI controller cannot work well because the system is </p><p>nonlinear. A fuzzy control system was built in [ 6,7 ] . Based </p><p>on the fuzzy controller, general fuzzy control method is </p><p>proposed in [8 ,9 ] , and the nonlinear controller can </p><p>overcome the shortcomings that PI controller possesses </p><p>inherently. However, no identification of the plant model is </p><p>considered. Furthermore, the wide range of patient drag </p><p>sensitivities to SNP is not considered. In this paper, we use </p><p>multiple-model adaptive control (MMAC) algorithm to </p><p>identify the patient model, and use fuzzy control algorithm </p><p>to design controller bank. Thus, the proposed control </p><p>method can provide satisfactory control performance and </p><p>can handle changes in the system dynamics. </p><p>2 Plant characteristics </p><p>A dynamic model of patient mean arterial pressure </p><p>response to SNP infusion rate was developed by Slate </p><p>[ 10]. It is obtained by using animals' as well as actual </p><p>patients' data together with physiological knowledge about </p><p>SNP effect. Improvement is made to State's model </p><p>according to our nonlinear, pulsatile-flow patient model, </p><p>which will be illustrated in Section 4, and dog experiment </p><p>results. The modified Slate's model is given by (1) and </p><p>(2) . </p><p>MAP = MAP 0 +AMAP + MAPn, (1) </p><p>where MAP is the actual mean arterial pressure; MAP0 is </p><p>the initial blood pressure; AMAP is the change in blood </p><p>pressure due to infusion of SNP; MAPn is the plant </p><p>background noise. MAP0 is usually 115 - 140 mmHg[ 1 ]. The variance of MAPn is typically 4 mmHg (for normal </p><p>noise levels) or 16 ~ 36 mmHg (for high noise levels). </p><p>Received 8 June 2005. </p></li><li><p>208 H. ZHENG et al. I Journal of Control Theory and Applications 3 (2005) 207- 212 </p><p>Many people might consider MAP0 as a constant. </p><p>However, MAP 0 in fact can change with time in many </p><p>cases. For example, after the patient is cured, his MAP 0 </p><p>gready declines, and the infusion of SNP can be stopped. </p><p>"the " Therefore, defining MAP 0 as initial blood pressure is </p><p>imprecise, we can consider that MAP 0 is just a parameter in </p><p>(1 ) , and that the parameter can change with time. </p><p>The transfer function describing the relationship between </p><p>the change in the blood pressure AMAP and the drug </p><p>infusion rate is given by </p><p>AMAP(s) Ke-r,k'(1 + ae- r~) SNP(s) - (1 + Tids)(1 + rs ) ' (2) </p><p>where AMAP(s) is the change in blood pressure, SNP( s ) </p><p>is the SNP infusion rate, K represents the sensitivity of </p><p>patients to SNP, Tik is the transport delay from the </p><p>injection site, a is the recirculation constant, Tc is the </p><p>recirculation time delay, Tid results fi'om the gradual </p><p>relaxing of the vascular muscle in response to SNP, r is the </p><p>lag time constant resulting from the uptake, distribution, </p><p>and biotransformation of the drug. Note that steady-state </p><p>gain of dose response is K(1 + a) .The parameters in (2) </p><p>are chosen as: K = - 0.25 to - 9 (normal = - 1 ) , Tik = </p><p>10 ~ 20s (nonna l=20s) , To = 30 - 75s (normal=45 </p><p>s), a =0-0 .4 , TAd = 10 - 40s (nomaal=20s) , v = </p><p>30-60s (normal = 40 s). The unit of AMAP is mmHg. </p><p>The unit of the infusion rate of SNP is ml /h . The </p><p>concentration of SNP is 200 rng/1. </p><p>3 Control system design </p><p>The MMAC procedure is based upon the assumption </p><p>that the plant can be represented by one of a finite number </p><p>of models and that for each such model a controller can be </p><p>a priori designed. All of these controllers constitute a </p><p>controller bank. An adaptive mechanism is then needed for </p><p>deciding which controller should be dominant for a given </p><p>plant. One procedure for solving this problem is to form a </p><p>weighted sum of all the controller outputs, where the </p><p>weighting factors are determined by the relative residuals </p><p>between the plant response and the model responses. Fig. 1 </p><p>shows the structure of the MMAC system. </p><p>Setpoint + </p><p>~1 Controller 1 </p><p>Controller Bank) </p><p>~NP2] I Npi; sN : / Integrate _ _ MAP2 </p><p>MA ?s </p><p>Ws </p><p>(Model Bank </p><p>[~ RI [ Operation L </p><p>of </p><p>.- R 8 i Residuals </p><p>w1 Operation w2. of [ Weights </p><p>t~ </p><p>Fig. 1 </p><p>Since the plant gain is negauve, the system error is </p><p>expressed as </p><p>e(k) = MAP(k) - MAPo(k) , </p><p>where k is the sampling time and MAPo(k) is the </p><p>commanded or set-point pressure level. The rate change of </p><p>error is defined by </p><p>r (k ) = [maP(k) - M iP (k - 1 ) ] / r , </p><p>where T is the sampling interval, and T = l0 s. </p><p>MMAC system structure. </p><p>For patient safety, two nonlinear units are built into the </p><p>system. The nonlinear unit limiting SNP infusion rate </p><p>change is given as </p><p>f l (x ) = for - 40 ~< x ~< 7, (3) </p><p>fo rx >7. </p><p>The reason for this constraint is that SNP is metabolized by </p><p>the body into cyanide, and hence, too much SNP can be </p></li><li><p>H. ZHENG a al . / Jounuzl of Control Theory and Applications 3 (2005) 207- 212 209 </p><p>toxic to the patient. The nonlinear unit limiting SNP </p><p>infusion rate is given as </p><p>0, for x < 0, </p><p>f i (x ) = x, for 0 ~ X ~ U m, (4) </p><p>Um, for x > Urn, </p><p>where Um is the maximum infusion rate. This is to prevent </p><p>rapid decreases in pressure which can cause diminished </p><p>blood flows or circulatory collapse. We use the equation in </p><p>[ 1 ] to compute Urn, </p><p>U m --- 60 x Wp x i m X C; I, (5) </p><p>where Wp is patient weight (the unit is kg) , i m is </p><p>maximum recommended dose (the value is 10 Pg" kg-1. </p><p>min-1) , C, is drag concentration (the unit is pg/ml) . For </p><p>example,if Wp = 70kg, then </p><p>(60min/h) x 70kg x (lOtzg kg -1 min -1) U m = </p><p>200/~g/ml </p><p>= 210ml/h. </p><p>With respect to the MMAC development, the design of </p><p>the model bank is described in Section 3.1 ; in Section 3. </p><p>2, the design of the controller bank is given; and finally, </p><p>the actual control computation is described in Section 3.3. </p><p>3.1 Model bank design The model bank consists of 8 models with constant </p><p>parameters characterizing the individual plant subspace. </p><p>are described by the following transfer These models </p><p>functions: </p><p>AMAPi ( s ) SNP( s ) </p><p>Kie-2*(1 + 0.2e -5') = (1 + 40s)(1 + 45s) ' j = 1 , . . . ,8 . </p><p>(6) </p><p>MAPj = MAPoINIT + AMAPj, j = 1 , " ' ,8 , (7) </p><p>where AMAPi(s) is the blood pressure change of the j th </p><p>model due to infusion of SNP, SNP(s) is the input of </p><p>these 8 models, MAPj is the output of the j th model, </p><p>MAPoINrr is the estimated initial value of patient's MAP0. </p><p>MAPoINrr is calculated in the first 30 seconds, as the average </p><p>of MAP(0), MAP( 1 ) and MAP(2). </p><p>Tc has trivial effect on the control system because the </p><p>value of a is small, so it can be frozen in the model </p><p>bank. Tid and r are lag time constants, and they can affect </p><p>the dynamic behavior of the system. Smaller Tid and v restflt </p><p>in smaller settling time. However, the range of Tid and r is </p><p>small, so their effects are reduced. Therefore, Tid and r can </p><p>also be frozen. In the model bank, the three parameters are </p><p>selected as Tcj = 50, Tidj = 25, and rj = 45 (for j = 1, </p><p> "" ,8). It is noted that while K reflects patient response </p><p>sensitivity, the effective steady state gain is K( 1 + a ). That </p><p>is, the recirculation enhances the effect of infusion [ 11 ]. </p><p>However, the effect of a can be compensated by different </p><p>Kj (for j = 1 , ' " , 8). Therefore, a can also be frozen. In </p><p>the model bank, aj = 0.2 (for j = 1 , ' " , 8). The plant parameter Tik significantly affects undershoot. Controlling a </p><p>high-infusion-dehy patient with a low-infusion-delay model </p><p>will cause large undershoot. For a blood pressure control </p><p>system,large undershoot is not allowed, so that Tik j (for j </p><p>= 1 , ' " ,8 ) should be chosen as the ~ u m expected </p><p>plant delay time. Although controlling a low-infusion-delay </p><p>patient with a high-infusion-delay model causes long settling </p><p>time, it can be partly compensated by different Kj (for j = </p><p>1 , - " , 8). The plant parameter K has the greatest effect on </p><p>controller performance. If the plant has high gain and is </p><p>being controlled by a low-gain model, large overshoots </p><p>occur; if the plant has low gain and is being controlled by a </p><p>high-gain model, long settling time Hill occur. Table 1 </p><p>shows the eight models and the range of plants they cover. </p><p>The plant gain partition factor is about 1.56, </p><p>i .e . , Kjn~,,/Kjmi,~ ~ 1.56. Notice that ~/1 .56~ 1.25, </p><p>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 </p><p>plat gain that Kj covers. Table 1 Gains of the eight models (column 2) , along </p><p>with the range of plants they cover (column 3). </p><p>Model Model Gain/ Plant Gain/ Number (mmHg/(ml" h- ' )) (mmHg/(ml-h- ' ) ) </p><p>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~) </p><p>3.2 Controller bank design The details about the fuzzy controller are described in </p><p>[6 ,7 ] . It converted the expert-system-shell-based fuzzy </p><p>controller to nonfuzzy control algorithms, and a total of ten </p><p>different nonfuzzy control algorithms for 20 different input </p><p>combinations were obtained. We compare the nonfuzzy </p><p>control algorithms, and find that they can be substituted in </p><p>fact only by one algorithm </p><p>ASNP( k ) </p><p>f ( GE x e(k)) + f ( GR x r(k)) =-G I ' L 'T" </p><p>3L - max[f( GE x e (k ) ) , f ( gR x r (k ) ) ] ' </p><p>(8) </p></li><li><p>210 H. ZHENG et al . /Journal of Control Theory and Applications 3 (2005) 207- 212 </p><p>where e(k) is the error, r (k ) is the rate of change of </p><p>error, G1 is the scalar for incremental SNP infusion </p><p>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 </p><p>sets in the fuzzy controller. T is the sampling interval. </p><p>Ftmctionf(x) = sgn(x)" min(L, Ix 1). After simplifying </p><p>the fuzzy control algorithms as (8) , it is much easier to </p><p>design the controller bank. </p><p>There are eight controllers in the controller bank. All of </p><p>the parameters except GI are the same in the eight controllers,and they are chosen as L = 16, GE = 0.25, GR = 15. In [6] , 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 </p><p>the eight models. Note that Ks" GIj ..~ O. 0460 (for j = 1, </p><p>...,8). Table 2 GI values of the eight controllers (column 2),and </p><p>the gains of the eight models (column 3). </p><p>Model Model Gain/ GI </p><p>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) </p><p>3.3 Identification algor i thm To achieve desirable system performance and guarantee </p><p>patient safety, the identification algorithm should converge </p><p>quickly to the desired values and should react to </p><p>time-varying plant characteristics, as well as ensure a </p><p>reasonable rate of blood pressure change. Thus, the control </p><p>output is computed as a weighted sum of controller bank </p><p>signals, i. e . , 8 </p><p>ASNP(k) = ~, [Wj (k ) .ASNP j (k ) ] , (9) j=l </p><p>where W~(k) is the weighting factor of the jth </p><p>controller, ASNPj(k) is the output of the jth controller. </p><p>In the first 30 s, MAPoINr r is calculated. In the meantime, </p><p>the SNP infusion rate should be equal to zero. Therefore, in </p><p>the first 30s, </p><p>Wj(k) = 0, j = 1, . . . ,8 . </p><p>The convergence algorithm of Wj will be described later. </p><p>Since the control variable SNP(k) will be in error before </p><p>the convergence of Wj, large overshoots could occur for </p><p>plants with high gain. Therefore, from 0.5 min mark to 3 </p><p>rain mark, the infusion rate is based on the assumption that </p><p>the most matched model is Model 8, i. e. from 0.5 rain </p><p>mark to 3 min mark, </p><p>0, for j = 1 , " ' ,7 , Wj(k) = 1, for j 8. </p><p>The convergence algorithm is run at 30 s mark, and the </p><p>weighting factors begin to converge. However, their values </p><p>won't be used until 3 rain mark, because their values </p><p>haven't converged in the first 3 minutes. The convergence </p><p>algorithm in [ 1 - 4] is improved, so that the computation </p><p>precision of the weighting factors can be enhanced. In the </p><p>algorithm, the weights are selected as follows: </p><p>1) Recursi...</p></li></ul>