Control of Neuromuscular Blockade: A Method for the Autocalibration of Pid Controllers

Copyright © IFAC Modelling and Control in Biomedical Systems, Warwick, UK, 1997

COlVIPARATIVE STUDY OF DESIGN lVIETHODS FOR DRUG DOSAGE REGIMENS P. J .A. Lago, M. A. Gomes

Departamento de Matematica .4plicada, Universidade do Porto Rua das Taipas 135. 4050 Porto, Portugal

Abstract: A number of different methods can be used for the design of drug dosage regimens. In particular, stochastic control methods provide a proper way to deal with the interindividual variability of the patients responses since stochastic methods incorporate in the design a measure of the intervariability of the patient dynamics. This paper presents and evaluates alternative techniques for the design of open-loop . optimal drug dosage regimens. Results are illustrated with the dosage of lidocaine. Keywords : Open-loop, Stochastic control. Pharmacokinetic data.

1.

I~TRODUCTION

perposition of a loading dose coA -1 to achieve the target concentration in a short time and a maintenance dose coAUC- l to induce the desired concentration at steady st ate (Norman, 1983) .

One of the applications of pharmacokinetic models is the design of drug dosage regimens . The experimental data for pharmacokinetic analysis is often the plasma concentration time curve induced by a fast injection (bolus) . Assuming linear timeinvariant dynamics the system impulse response is et sum of exponentials

It is well known that the patients concentration responses, c(t) , to the same drug dosage regimen are fairly different. When the patient dynamics are unknown and measurements of the indu ced concentration are not available, the design of a dosage regimen is non ind ividualized . tinder these circumstances the design of a dosage regimen is often based on the average pharmacokinetic parameters deduced from a set of individual pharmacokinetic models . The average do not convey information on the spread of the individual values . Therefore it is reasonable to assume that the use of additional information on the interindividual variability of the pharmacokinetic parameters may lead · to better techniques with improved results .

p

g(t)

=

L

aie-,l.·1

(1)

;=1

which can be deduced from input-output experimental data. With this representation the pharmacokinetic model of a patient can be described by the parameter vector e = [al ... ap Al ... Ap ]. All drubs have a therapeutic window: a too low concentration is ineffective whereas a too high concentration may lead to undesirable effects and toxicity. In it.s simplest form the design of a drug dosage regimen is based on a few characteristic values of the system impulse response , namely the .4 and the .4.UC values .4.

=

g(O); AUC

=

100 g(t)dt

This study evaluates different techniques for the design of optimal drug dosage regimens. The techniques are illustrated with the dosage of lidocaine .

(2)

2.

For a constant target concentration c(t) = Co, an acceptable drug dosage regimen is often the su-

PROBLE~I

STATEMENT

The design of an optimal drug dosage regimen can be based on a therapeutic cost function J

169

~ ~

-C,(t)

which weights the differences between the target concentration and the concentration induced by the dosage regimen on a set of previously specified times. The therapeutic cost J is a function of the dosage regimen and of the patient pharmacokinetic parameter vector. Like in previous studies (O·Argenio. et al.. 1988: Lago, 1992: Bayard. er al .. 1994) a quadratic therapeutic cost is assumed

Q

-C:!(t) ••

~~(t)••

~~JC'l -·c".,(t) -----...--;I::. - cn(t)

u(t)

..-

----~.~~~---~ e(t) = E(e(t)]

u(t)

m

,](0. u)

=

L u:(k)(c(t .. ; u , 0) -

cdt .. »):!

(3)

Fig. 1. The individual responses to the same dosage regimen are different. Model PAll gives the expected value of the concentration time curve induced by u(t) in the population.

.. =1

In equation (3) u is the dosage regimen (a time function). tl; a set of m target times, cr(t .. ) the target concentration at time te, c(t .. ;u , 0) the concentration at time t .. induced by u(t) on a patient with parameter vector 0 and w(k) a positive weighting factor . The dosage regimen is usually a piecewise constant function with specified discontinuity times. or a time sequence of 5 functions (the model for a bolus) or a combination of both.

pected values in the population A(l)

= E[A]; AUC(l) = E[AUCj

(5)

In equation (4) ~ is the the covariance matrix of 0. For additional details and a discussion of the merits of model P~\l1 see Lago (1989).

The patient parameter vector 0 is assumed to be unknown . Therefore, rather than calculating the optimal dosing regimen for a particular model. 0 is regarded as a multidimensional random variable and the optimal dosage is deduced taking into consideration the probabilistic distribution of the patient model parameters in the population . This can be done either using in J (0, u) a representative model for the distribution of 0 , or by considering the expected value of J(0 , u). In both cases the optimal dosage regimen mimimizes the therapeutic cost over the class of allowable regimens.

It should be mentioned that in the pharmacokinetics literature and related areas the use of a compartmental model description is much more favoured than the description by the system impulse response. vVith a compartmental model representation the population dynamics is usually described by a compartmental model with parameters equal to the expected values in the population.

4. DESIGN OF DOSAGE REGnIE\'S

U.

The J\l AP Bayesian dosage regimen (D'Argenio, et al., 1988) is found by minimizing J(0. u) over the class of allowable dosage regimens U, using in equation (3) as patient model description the expected value 0 0 of the compartmental model parameters

3. \lODELS FOR THE POPULATIO\, Given the probability distribution of the individual model parameters . 0, the population dynamics can be described by the expected value 00 E[0]. This is the simplest global description of a population with dynamics described by a vector of parameters of random nature. Although easily justified. the expected value does not provide the best global discription of the dynamics of a population . This issue is discussed elsewere in some detail (Lago, 1989), where an alternative population model (the model P:\11) has been introduced . The meaning of model P.U1 is illustrated in figure 1. The impulse response parameters of model P.\fl are the solution of the optimization problem

=

(6)

The multiple model linear quadratic control ( J-f M LQ) (Bayard. et al., 1994) is based on the assumption that the probability distribution of the parameters 0 of the compartmental model is either discrete or has been previously approximated by a discrete distribution. Under this restrictive assumption. the expected value J 1 (u) = E[J(0, u)] is given by .]1 (U) =

N

m

;=1

.. =1

L p(0;) L w(k)(c(t .. ; u, 0;) -

cc(t .. )f

(7) where p( 0 i ) is the probability of occurrence of model 0; and N the number of distinct models. The M M LQ dosage regimen is the solution of the

with constraints imposed on e(l) to ensure that model PJtfl has an impulse response with values for A and AU C equal to the corresponding ex-

170

Table 1 Models for lidocaine parameters al(ml-') ~2(ml-1 ) ·\dmin-I) .\2(min- l )

.\[AP

.~(ml-l)

AUC(min ml- I \/(1) Ct(ml/min)

fig . 2. Tw
= minJdu )

(8)

uE U

The two other design methods that are compared in this study are based on the model P ~'vfl . For the design of dosage regimens with model P ;\;f 1 two alternatives have been considered . The first alternative is identical to the MAP method since only a single model is used in the design . The difference lies on the choice of the model; instead of t he expected value 0 0 of the compartmental model parameters. the method uses the parameter vector 0( I 1 of model PM 1. Therefore the optimal dosage is the solution of

= minJ(e l1 ), u)

(9)

The second method has similarities with AI ,\! LQ si nce it is based on the expected value of the cost function ( 3) . Ye t . instead of imposing a discrete dis tribution for the model parameters . the evaluation of the expected value is based on a fi rst order approximation for the variance of the induced concent-ration . l;sing this approximation (Lago, 1992). h ( u) = E[J(u, e)J can be obtained from m

L

u'( k)( (c( t!: ; u , e (I») -cr(t.c))2+IIS (t k)U I1 2 )

k= l

(10 )

and the optimal dosage is the solut ion of P HI

u .

= min h ( u)

( 11 )

uEU

In equation (10 ) C(tk; u, 0(1 l) is the concentration at time tk induced by u(t) on the model P.\fl . S(t!: ) is the concentration sens itivity matrix Sri) = oc(t ; u . 0 )

(12)

00

=

computed at t tl; and e gular mat.r ix such that VL

= e(1) and L a trian-

=L

The lidocaine pharmacokinetics can be modelled by the tW
e( t)

[

=

-(kIO

[l/V

+ kd

k12

0] X(t)

]+[

~

]

AIMLQ

[0.5] [5,20] [20.50] (50,110] [110.170] (170.230]

12.5 3.03 1.99 1.39 1.24 1.20

P ;\fl 12.4 3.11 1.92 1.25 1.13 1.14

MA.P 19.2 5.11 3.21 2.09 1.84 1.79

(P ,Hl ) 15.1 3.93 2.44 1.60 1.41 1.38

The discrete dis t ribution of the companmental model parameters can be converted into the corresponding distribution of the system impulse response parameters 0 . From the distribution of 0 , the expected value 0 0 • the covariance matrLx L , and the expected values E[AJ and E[,·H/ CJ have been calculated . ~lodel P.Hl has been computed as described in Section 3. with the contraints imposed by the expected values E[AJ and E[A.UC], A(I ) = 0.482 ml- 1 and A.UC(I) = 2.20 min ml- l , respectively. The impulse response parameters of the two population models for lidocaine are given in table 1. It should be noticed that the models indicate quite different values for both A and .4.UC and consequently. quite different values for the volume of the first compartment (V' = A-I) and for the clearance (Cl = AU C- 1 ) • For the comparison of the design techniques the example given in Bayard et al. (1994) has been used once more. It assumes a continuous infusion u(t) with discontinuity times T = {5, 20 , 50 , 110,170 , 230 minutes}, a target concentration of 3 J.L9 ml- 1 at the target times T and equal we ights . w. Table 2 gives the optimal infusion regimens obtained with the AI ",vI LQ and the P Jfl methods. Figure 3 shows the responses of the 81 models to the two dosage regimens. which can be considered equivalent with respect to the spread of the induced concentration curves in the popula.tion.

5. CASE STUDY

X (t)

minutes

:\ discrete probability distribution for the compartmental model parameters [V, klO , k12 , k2d has been assumed. It consists of a set of 81 models with different probabilities of ocurrence. see Bayard et al. (1994) for details. The first population model that has been considered is the MA.P with compartmental parameters equal to the expected values .

uEU

h( u ) =

PMl 1.3710- 2 3.4510- 2 7.21 10- 3 1.1510- 1 4.8210- 2 2.20 20.8 4.5510- 1

Table 2 Dosage for lidocaine

u ,\/MLQ

u ( PMll

)

1.0310 • 2.7710- 2 7.1210-3 1.15 10- 1 3 .8010- 2 1.69 26.3 5.9210- 1

u(t)

(13)

171

50

100

150

time (minutes)

200

250

5D

100

150

200

250

100

150

200

250

time (minutes)

8

°0-~--~5~0----~0-0--~----~---2~50

1

150

time (minutes)

200

50

Fig. 3. Responses of the 81 models to the optimal regimens obtained by the AI lv!LQ (up) and the PM1 (down) methods .

time (minutes)

Fig. 4. Responses of the 81 models to the optimal infusion regimens obtained by the single model design technique with model lvI AP (up) and PMl (down) .

The optimal infusions obtnained with the single model technique, AIAP and (P.VI1) are also shown in table 2. Figure 4 shows the corresponding responses in the population . The induced concentration curves of figure 4 display a much larger variability than those of figure 3. The infusion obtained with AI AP (the compartmental model with average parameters) gives the worst ensemble properties for the induced concentration time curves. This is expected since the volume for the first companment, V, and the clearance, Cl, of AI AP are larger by a factor of 26% and 30% than the corresponding values for P ,vIl (see table 1). Figure 4 also illustrates that for the design of an optimal dosage regimen with a single population model (a simpler alternative to the computation of the solution of the minimum therapeutic cost), the use of model PI'vfl gives results much better t.han those obtained with the frequently used compart mental model with average parameters. This is a direct and important consequence of the constraints on the impulse response which constitute a fundamental characteristic of model P .\Il.

PM1 is described in this work.The method gives results identical to ,VI.Vf LQ and, furthermore, it can be used with equal simplicity for both discrete and continuous distributions. For the single model technique the compartmental model with parameters equal to the expected va.lues, AI AP, is currently used . It is shown that the use of lVI AP gives results much worse than those obtained by the single model technique using PM1 .

REFERENCES D'Argenio , D.Z. and D. Katz (1988) . Application of stochastic control methods to the problem of individualising intravenous theophylline therapy. Biomed. Jleas. Inf Control, 2, 115-122. Bayard , D.s. , ~1. H. ~Iilman and A. Schumitzky (1994). Design of dosage regimens: a mul tipie model stochastic control approach . Int . Journal of Bio-,. . .IedicaI Computing, 36, 103115. )Iorman , J . (1983) . The IV administration of drugs. British J. of Anaesthesia, 55, 10491052. Lago, P.J. (1989). Approximate maximum likelihood population pharmacokinetic models for the design of dosage regimens. Comp. and Biom. Researcb, 22 . 282-295. Lago, P.J . (1992). Open-loop stochastic control of pharmacokinetic systems: a new method for design of dosing regimens. Comp. and Biom. Research, 25, 85-100.

6. CONCLUSIONS The design of non individualized drug dosage regimens with some a priori information on the probability distribution of the patient dynamics. can be formulated either as a multiple/single model linear quadratic control. The multiple model technique (AI AI LQ) is rather restrictive since it requires the probability distribution of the parameters to be approximated by a discrete distribution. An a.lternative method to M M LQ based on the model

172