Journal of Control Theory and Applications 3 (2005) 207- 212

Automated postoperative blood pressure control

Hang Z H E N G , 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 ( M A P ) . . S u c h

hypertension should be treated timely to prevent severe

complication. Continuous infusion of vasodilator drags,

such as sodium nitropmsside ( S N P ) , w o u l d 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 o f feedback control techniques can

be effective.

To improve the quality o f patient care, automatic

closed-loop control systems for SNP delivery have been

developed. A proportional-integral (PI ) controller-based

multiple-model adaptive control ( M M A C ) 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 o f 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 m m H g (for normal

noise levels) or 16 ~ 36 m m H g (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 + a e - r ~ ) SNP(s ) - (1 + Tids)(1 + r s ) ' (2)

where AMAP(s) is the change in blood pressure, SNP( s )

is the SNP infusion rate, K represents the sensitivity o f

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 o f 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 ( n o n n a l = 2 0 s ) , To = 30 - 75s ( n o r m a l = 4 5

s ) , a = 0 - 0 . 4 , TAd = 10 - 40s ( n o m a a l = 2 0 s ) , v =

3 0 - 6 0 s (normal = 40 s ) . The unit o f AMAP is m m H g .

The unit o f the infusion rate of SNP is m l / h . The

concentration o f 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 o f 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 ) = M A P ( 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 ) = [ m a P ( k ) - M i P ( 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)

f o r x > 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 a l . / 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 k g ) , i m is

maximum recommended dose (the value is 10 Pg" kg -1 .

m in -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 t h

model due to infusion of SNP, SNP(s ) is the input of

these 8 models, MAPj is the output of the j t h 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 . 5 6 ~ 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 [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

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 a lgor 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) = ~ , [ W j ( k ) . A S N P j ( k ) ] , (9) j=l

where W~(k) is the weighting factor of the j th

controller, ASNPj(k) is the output of the j th 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, f o r j = 1 , " ' , 7 , Wj(k) = 1, f o r 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 j th model.

V is a parameter controlling the convergence rate of

Wj(k ) with R~(k ) . We compare [1] with [12] , and

find that 2

V 2 = tTn (MAP0 - MAPc) 2' (14)

3) Normalization

[ W i ' ( 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[- R 2 ( k ) / 2 V 2 ] W i ( 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 ( f o r j = 1 , - - - ,8 ) .

The relative residual R 2 ( k ) is defined as the normalized

squared error between plant and model, i. e.

[ M A P j ( k ) - MAP(k)]2

R2(k ) = M---~OINIT--- ~ ' j = 1, °°* , 8 ,

(13)

H. ZHENG et al . / Journal o f Control Theory and Applications 3 (2005) 2 0 7 - 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 = 4 0 , 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.

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t l a n ~ 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.

K u a n y 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.