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Neuromuscular Blockade Nonlinear Model Identification

B. Andrade Costa, M. Silva, T. Mendonca and J. M. Lemos

Abstract This paper presents a methodology for parameterestimation of a nonlinear neuromuscular blockade dynamicmodel to be used as a predictive model for automated control,in general anesthesia. The neuromuscular blockade dynamicmodel comprises two blocks connected in series, a pharma-cokinetic model and the pharmacodynamic model. The phar-macokinetic model is a second order linear dynamic modeland describes the redistribution of the drug in the body. Thepharmacodynamic model is a nonlinear function, named asthe Hill equation, and it describes the interaction between theconcentration of the drug in the effect site and the measuredpatients muscle paralysis state. The identification methodologyuses four data points taken from the neuromuscular blockaderesponse obtained with the administration of the first bolus.The four data points are chosen to avoid the identification

difficulties caused by the presence of the nonlinear behaviorof the Hill equation. This approach enables the identificationof the pharmacokinetic dynamics, that is, the two poles of thesecond order linear dynamic model followed by the estimationof the normalized parameters of the Hill equation.

Computer simulations show that the proposed identificationmethodology is able to provide good results even when thepharmacokinetic dynamics has an order higher that two. Thissuggests that the methodology may be employed in neuromus-cular blockade automated control as a predictive model, to helpthe initial tuning of the controller parameters or in adaptivecontrol to get a first model that can be improved with onlineidentification using some recursive minimization techniques toadjust the adaptive controller or as an advising mechanism tohelp the anesthesiologist during the anesthesia.

Keywords: Biomedical engineering, Neuromuscular Blockade,Model Identification, Nonlinear Model

I. INTRODUCTION

Computer controlled systems are being considered as a

promising technology to improve the practice of anesthesia

[1] [4]. In principle, it is possible to adjust the amount of

drugs to the patients characteristics, that is, to keep the drug

concentration in the patients body at a constant, safe and

adequate level, and by that to have a quicker recovering time

from anesthesia [3]. Another important field is the utilization

of automated systems to control anesthesia in animals to

reduce the costs and work of the veterinary.

At the core of any computer controlled system for anes-

thesia automation is a model describing the transport of the

drug in the human body (pharmacokinetic model) and the

This work was developed in the project IDeA framework - Integrated De-sign for Automation of Anesthesia, contract PTDC/EEA-ACR/69288/2006

B. Andrade Costa, is with INESC-ID/DEEC/IST/TU Lisbon, R. AlvesRedol 9, 1000-029 Lisboa Portugal, bac@comp.ist.utl.pt

M. Silva, and T. Mendonca, are with Dep. Matematica Apli-cada FCUP, Rua do Campo Alegre, 687 4169-007 Porto, Portugaltmendo@fc.up.pt

J. M. Lemos, is with INESC-ID/DEEC/IST/TU Lisbon, R. Alves Redol9, 1000-029 Lisboa Portugal, jlml@inesc-id.pt

effect of the drug in the patients state (pharmacodynamic

model). These are the cases of TCI and TIVA systems

[2]. These systems use compartmental models obtained by

processing data, collected from a large sample to characterize

the properties of a population. However the TCI and

TIVA systems can be described as operating based on one

mean model, there is one model able to describe the

dynamics of all patients! But the current practice shows that

for neuromuscular blockade control there is a huge variability

between patients [5] [6]. This suggests that the administration

of drugs during anesthesia must be based on methods able

to estimate patients characteristics.

With compartmental models, each compartment is as-sumed to have homogenous properties, that is, at each time

instant the drug distribution inside a compartment is uniform.

This approach is similar as modeling a lumped parameter

system. Those models may have from 2 compartments to

12 compartments [7][8][9][10]. Other modeling techniques

use more deep knowledge of physiology to build complex

models [11], but this approach needs data and experimental

procedures that are not available in a standard operating

room.

According to [7][8][9][12] a second order linear dynamic

model is used to describe the pharmacokinetics of drugs

belonging to the atracuriums family. This motivated the use

of the second order linear dynamic model in the identificationmethodology presented in this paper, the model has a defined

structure but the parameters must be estimated. An important

issue comes from the constraint imposed by ethical and

practical reasons that constrains the design of identification

signals to be used in model identification, this makes low or-

der dynamic models more suitable to describe local dynamics

than higher order dynamic models where overparametrization

cause identification problems. The problem is that low order

models do not fully describe the all dynamics and may

impose additional constrains in the control design. To tackle

the problem online identification may be used to adjust the

low order models.

The aim of this work is to model the neuromuscu-lar blockade dynamics using the information that a anes-

thetist/anesthesiologist has when performing her/his job in

the operating room, by using the index evolution provided

by the neuromuscular blockade monitor and using the infor-

mation obtained from evolution of the infusion or from the

sequence of bolus.

This paper is structured as follows: Section II describes the

neuromuscular blockade model based on a 3 compartmental

model and gives a possible cue for the assumption of a phar-

macokinetic second order linear model. Section III presents

17th Mediterranean Conference on Control & Automation

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June 24 - 26, 2009

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Fig. 1. The neuromuscular blockade model with a three compartmentalmodel for the pharmacokinetics, where C1 represents the central compart-ment and C2 represents the effect compartment. The r(t) signal representsthe neuromuscular blockade response.

the identification methodology. Section IV shows the results

obtained with computer simulation using a database of 100neuromuscular blockade models that have a pharmacokinet-

ics of order 4 with a zero. Conclusions are presented in

the last section.

II. NEUROMUSCULAR BLOCKADE MODEL

In this section a neuromuscular blockade model based on

a three compartmental model is presented [13]. Fig. 1 shows

the pharmacokinetics diagram with three compartments. The

central compartment (C1) includes organs such as the cardio-vascular and pulmonary system, brain and other organs that

are highly irrigated. The second compartment (C2) represents

the effect site in this case the muscle. The third compartment(C3) represents organs/tissues that have a low rate of drugabsorption/diffusion. Considering a compartment i, Vi repre-sents the total volume ([l]) of the fluid (plasma/blood) in thecompartment. The drugs mass is represented by mi ([mg]),and the volumetric flow rate of fluid leaving the compartment

i towards compartment j is denoted by qji ([ml.min1]),this quantity in directly correlated with the hearts volumetric

flow rate, and ki represents the drugs elimination flow rateor the rate at which the drug loses its power.

By performing a mass balance, the drugs concentrations

in the three compartments are described by the following set

of differential equations, where ci(t)= mi(t)/Vi.

c.1(t) = 11c1(t) + 12c2(t) + 13c3(t) + u(t)

c.2(t) = 22c2(t) + 21c1(t) (1)

c.3(t) = 33c3(t) + 31c1(t)

y(t) = c2(t)

The ji parameters are shown in table I, q represens the totalflow rate leaving the central compartment and f representsthe fraction of q entering/leaving compartment C2, ki isassumed to be identical in all compartments, this is justified

by [12], where the neuromuscular blockade drug such as

TABLE I

PARAMETERS OF THE THREE COMPARTMENT MODEL 1.

11 =q+kV1

12 =fq

V213 =

(1f)qV3

22 =fq+kV2

21 =fq

V1

33 =(1f)q+k

V331 =

(1f)q+kV1

atracurium is transformed by chemical reactions that depend

on the bodys temperature and on the bloods acidity which is

assumed identical in all compartments. Note that the model

(1) may be considered a linear varying parameter model

because the ji parameters depend on the hearts volumetricflow rate and on the bodys temperature and bloods acidity

that may change over time.

The pharmacodynamic model representing the effect of

the drug is described by the Hill function, eq. (2),

r(t) =C50

C50 + (y(t))

(2)

where C50 represents the drugs concentration to obtain halfof the full effect, and is a non-dimensional parameter thatcharacterizes the slope of the Hill function.

A. Analysis of the three compartment model

The three compartment model given by (1) is represented

in the state-space form and it is necessary to estimate seven

parameters, that may be difficult. In order to simplify the

identification procedure, an input-output representation using

a transfer functiony(s)u(s) = P(s) will be obtained from (1).

The transfer function P(s) has three poles and one zero and

it can be written as shown in (3), with Q(s)= s+22

s+33.

P(s) =21

(s + 11)(s + 22) 1221 1331Q(s)(3)

It is easy to conclude that P(s) can be simplified to asecond order transfer function if Q(s) 1, that is, 22 33. Now the issue is to demonstrate that there is somephysical/physiological evidence to support that 22 33.Taking the definition of both parameters from table I and

assuming that (1f)q >> k and fq >> k, this is acceptableotherwise the drug is quickly removed and anesthesia is not

possible, then(1f)q

V3 fq

V2this implies that

f V2V2 + V3(4)

Defining now T2 = V2/(f q) and T3 = V3/((1 f)q) asthe time needed to renew the fluids inside compartment 2and 3, and using eq. (4) then T2 = (V2 + V3)/f q andT3 = (V2 + V3)/q. That is, the renewal times for bothcompartments are identical. This can be justified if one

considers that blood must be renewed in such a way that

each cell must be supplied with a constant rate of oxygen,

that is, the flow rate must be proportional to the number of

cells (volume of the compartment). Assuming that the above

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Fig. 2. Description of the time instantes t1 < t2 < t3 < t4 to be usedby the identification methodology.

argument is true then Q(s) 1 and the transfer function (3)is written as

P(s) =21

(s + a)(s + b)(5)

witha

andb

being the transfer function poles computed from

the parameters 11, 22, 12, 21, 13 and 31.

III. IDENTIFICATION METHODOLOGY

In order to present the identification methodology, the

second order linear model is normalized such that its static

gain is set to one, and the Hill function parameters are

modified in such a way that the input-output properties of

{r(t), u(t)} are maintained. The new model is represented byz(s)u(s) = M(s), with

M(s) =ab

(s + a)(s + b)(6)

and

r(t) = (C50)

(C50) + (z(t))

(7)

where C50= (a.b.C50)/21. This enables the separation of

the parameters in two classes, {a, b} are estimated first, and{, C50} are estimated later using a, and b.

The development of the identification methodology is

based on the fact that the first neuromuscular blockade bolus

usually forces the patient to a state of total paralysis, the

index monitor r(.) goes from the value 1.0 to 0.0 or to avalue near 0.0, as in fig. (2). After some time and dependingon the pharmacokinetics, r(.) starts to increase. As a standardpractice the anesthesiologist manipulates the drug infusion

rate or the bolus sequence to maintain r(.) near the value of0.1 (10%). This means that there is a time window definedby {t0 < t < tf : r(t) < 0.1} where it is possible to select4 data points with time instantes such that

r(t1) = r(t4)r(t2) = r(t3)

(8)

with t1 < t2 < t3 < t4, fig.(2).The constraints imposed by (8) imply that

z(t1) = z(t4)z(t2) = z(t3)

(9)

Because the pharmacokinetic model response to the first

bolus is an approximation of the impulsive response of a

dynamic system, the impulse response equation of a second

order model hm(t), (10), is used

hm(t) =ab

b a(expat expbt) (10)

to estimate the parameters a and b from the measured values

t1 < t2 < t3 < t4. These time instants must be selected insuch a way to avoid numeric problems.

A. Estimating the dynamics ofM(s)

Using the constraint z(t1) = z(t4) and eq. (10) then a isgiven by

a =1

t4 t1ln[

1 exp0t4

1 exp0t1] (11)

with 0 = b a. A similar equation can now be written withparameters t2 and t3

a =1

t3 t2ln[

1 exp0t3

1 exp0t2] (12)

The eq.s (12) and (11) are now used to define a function,

f(), such that

f() =1

t3 t2ln[

1 expt3

1 expt2]

1

t4 t1ln[

1 expt4

1 expt1] (13)

The 0 = b a is obtained by solving the nonlinearequation f() = 0. Knowing the estimate 0 then a andb can be computed from

a =1

t4 t1ln[

1 exp0t4

1 exp0t1]

b = 0 + a (14)

B. Estimating the parameters of the Hill function

Knowing the estimates a and b and using eq.(10) it is pos-sible to compute z(t1) and z(t2) corresponding respectivelyto the measured values r(t1) and r(t2). Rearranging the Hillfunction it can be written as z = (C50)

(1/r 1). Thisyields (z(t1)/z(t2)) = (1/r(t1) 1)/(1/r(t2) 1). Byfurther algebraic manipulation then and C50 are obtainedfrom

= [ln(z(t1)

z(t2))]1[ln(

1

r(t1) 1) ln(

1

r(t2) 1)] (15)

C50 = z(t1)r(t1)

r(t1) 1(16)

C. Evaluating the parameter estimation error

In practice the measurements of the time instants t1 0.2 and a smallerror for r(t) 0.2, as shown in fig.(5). This estimationresult is the worst case obtained with the 100 model database.

C. Simulation results from the model n.69

The model number 69 is characterized by the following

parameters: Static gain Sg = 0.2108, z1 = 0.1230, p1 =0.0360 , p2 = 0.0723, p3 = 0.0996, p4 = 0.2999,C50 = 0.6163, = 4.2189. Note that in this model z1 isnear p3 and it can be considered a zero-pole cancelation. Thenormalized C50 is C

50 = C50/Sg = 2.9236. The measuredtime instants (in [min]) obtained by linear interpolation the

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0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

r(t)

t [min]

+ Model of order 2

o Output of the true model

Fig. 5. Neuromuscular blockade response for the first atracurium bolusof 500g/kg, using model n.18 (symbol - o). Response of the estimatedmodel, symbol +.

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

r(t)

t [min]

+ Model of order 2

o Output of the true model

Fig. 6. Neuromuscular blockade response for the first atracurium bolusof 500g/kg, using model n.69 (symbol - o). Response of the estimatedmodel, symbol +.

bolus sampled data response are t1 = 9.9364, t2 = 11.3413,

t3 = 49.7039, t4 = 54.4795.Using the estimation methodology, the estimated non-

linear second order model has p1 = 0.0185, p2 = 0.0725,C50 = 3.1387, = 7.3714.

In this case, the C50 and C

50 are similar, and are verydifferent. But if one compares the neuromuscular blockade

responses of both models they are very similar as shown in

fig.(6).

The main conclusion from the above results is that, to

obtain a good modeling of the bolus response it is necessary

to obtain a good estimation of the pharmacokinetic model.

The estimate C50 is near the true value, but has a hugeerror, however this huge error causes a degradation in the

bolus response estimation.

V. CONCLUSIONS

This paper presents an identification methodology to

obtain a second order non-linear model to describe the

patient/drug interaction for the neuromuscular blockade in

general anesthesia. The model is identified using four data

points taken from the first bolus response. The method

estimates in first place the pharmacokinetics followed by the

pharmacodynamics which is described by the Hill equation

with normalized parameters. The method was evaluated by

computer simulation using a neuromuscular blockade model

database with 100 models. The computer simulation results

show that the proposed identification methodology has a

good performance when tested with the models of the

database. The worst case was obtained with models that

cannot be approximated by a second order system, that is,

they do not have a near zero-pole cancelation. Despite

this fact, the estimated models were able to provide a good

approximation of the neuromuscular blockade response for

r(.) < 0.2 (20%).

The main conclusion from the above results is that, to

obtain a good modeling of the bolus response it is necessary

to obtain a good estimation of the pharmacokinetics model,

and this raises the problem of selecting the best structure to

model the real cases.

At the present stage there are several unanswered ques-

tions, that is:

What is the noise effect in the estimation?

There is a significant number of real cases with a flat

response at r(t) = 0.05. How to handle it? What is the closed-loop performance of a controller that

is designed using above identification methodology?

As a future work the identification methodology will be

evaluated with realistic data taken from real cases and if

possible the methodology will be extended to higher order

models.

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