Journal of Pharmacokinetics and Biopharmaceuiics, Vol. 26, No. 3, 1998

Rapid Attainment of Steady State Plasma DrugConcentrations Within Precise Limits

Ben Korman,1,3,4 Les S. Jennings,2 and John R. A. Rigg3

Received October 21, 1997—Final June 9, 1998

INTRODUCTION

In several common clinical situations in anesthesia and critical care, itis necessary to rapidly establish and maintain therapeutic drug concen-trations in the circulation. Examples are general anesthesia, myocardialinfarction with life-threatening arrhythmias, severe systemic hypertension,severe asthma, and massive pulmonary embolism. Many drugs used in thesecircumstances have a narrow therapeutic range or a low therapeutic index.Hence, in clinical practice there is considerable potential for inadequate orexcessively high drug concentrations in the blood.

Supported by a grant from the Royal Perth Hospital Medical Research Foundation.1Department of Anaesthesia, Royal Perth Hospital, Perth, W.A. 6000, Australia.2Center for Applied Dynamics and Optimization, Department of Mathematics, University ofWestern Australia, Stirling Highway, Nedlands, W.A. 6907, Australia.

3322 Rokeby Road, Subiaco, W.A. 6008, Australia.4To whom correspondence should be addressed.

We describe a method of rapidly obtaining a specified steady state plasma concentration of anintravenous drug within precise limits. The technique requires an initial bolus to raise the plasmaconcentration to the upper limit followed by a series of constant-rate infusions each of which isassociated with a minimum plasma concentration equal to the tower limit. The infusion rate isstepped down when the plasma concentration returns to the upper limit. Computer simulation,based on the method, is used to generate plasma concentration-time curves with fluctuations ofup to 10% about selected steady state concentrations of amrinone, esmolol, lidocaine, midazolam,propofol, and theophylline. The utility of this general approach to intravenous dosing and potentiallimitations of the method are discussed.

KEY WORDS: anesthetic techniques; continuous infusion; pharmacokinetics.

3190090-466X/98/0600-0319$15.00/0 © 1998 Plenum Publishing Corporation

320 Kornun, Jennings, and Rigg

Fig. 1. Schematic depiction of plasma concentration-time profilefor multiple infusion procedure. Initial bolus dose followed byinfusion I1, for total time=0 to T1, I2 from total time= T1, to(T1 + T2), and I„ from (T1, + T2) onwards. Infusion rates havebeen chosen to maintain the plasma concentration between(1 - P)Css and (1 + P)Css with a minimum for I1, and I2 occurringat a plasma concentration of (1 - P)Css.

Several publications indicate how a loading dose accompanied by aseries of different consecutive constant infusion rates can be used to rapidlyobtain a steady state plasma drug concentration (1-5). Most of these tech-niques have been summarized by Bailey (4). These methods are applicableto any drug, with a disposition determined by first-order kinetics, but theextent of fluctuation of plasma drug concentration about the steady stateconcentration has not previously been the subject of investigation. In thispaper, the theory of computing an infusion regimen, with precisely deter-mined limits of fluctuation is described using as an example drugs whosedisposition may be characterized by an open two-compartment system.

METHODS

Suppose that a steady state plasma concentration Css, with a fractionaldeviation (C-Css)/Css, limited to ±P is specified (Fig. 1). A loading dose isgiven to provide an initial plasma concentration of (1 + P)Css. This is fol-lowed by a constant-rate infusion I1, which results in a decrease of C to(1 - P)Css followed by an increase to (1 + P)Css at time T1, when I1, ends. Asuitably smaller constant infusion T2 begins at T1 and a similar patternoccurs, with (1+P)C s s being reached at time (T1 + T2) (Fig. 1). Furtherinfusion rates I3, I4, etc. may be needed to continue to maintain C within thepreselected limits of (1 ±P)Css,. The number of different infusions requireddepends on the pharmacokinetic behavior of the drug and the magnitudeof the specified fluctuation limits. For the purposes of this analysis, eachinfusion may be regarded as a separate event, so that t is rezeroed at thestart of every new infusion. During the first infusion, t varies from 0 to T1;

Rapid Attainment of Steady State Drug Concentrations 321

during the second infusion, t varies from 0 to T2, etc. The total time thathas elapsed from the initial bolus to the beginning of the nth infusion isT1 + T2+... + Tn-1.

Choice of Constant-Rate InfusionAssume we are dealing with an open two-compartment mammillary

model with central compartment of volume V and elimination constant k.The expressions for C1 and C2, the concentrations in the central and periph-eral compartments respectively at time t during the nth infusion are

where

Co= In/Vk with In the infusion rate during the nth infusion. C1(0), C2(0) arethe concentrations of the agent in the central and peripheral compartmentsat the beginning of the nth infusion. At the beginning of the first infusion,the initial bolus of (1 + P)CssV raises C1(0) to (1 + P)Css while C2(0) is zero.At the beginning of the nth infusion, C1(0) and C2(0) are equal to C1 ( T n - 1 )and C2(Tn-1), the concentrations in the central and peripheral compart-ments at the end of the (n-l)th infusion. Differentiating Eq. (1) withrespect to t, we find that the minimum value of C\ occurs when

We denote this time as tmin.We first investigate the possibility of using Iss= VkCss, the infusion rate

which, on its own, eventually produces Css. We do this by substituting Iss

for In in the expression for C0, then determining A and B in Eqs. (3) and(4). The value of tmin with this infusion is then determined from Eq. (5) andthe corresponding value of C1 from Eq. (1). If C1 >(1 -P)Css, then Iss maybe used and no further changes of infusion rate are necessary. If not, alarger infusion rate is required, and can be estimated as follows: At thevalue of t determined in Eq. (5), the minimum plasma concentration is to

322 Korman, Jennings, and Rigg

The only unknown in Eq. (6) is In which appears in the expressions for A,B, and C0. [The value of t is given explicitly by Eq. (5) in terms of In.] Thisnonlinear equation may be solved in each phase by iteration or inter-polation. Since the solution is easily bracketed, iteration using the bisectiontechnique is the method of choice (6). This technique relies on using tworeference points; infusion rates each associated with a particular minimumplasma concentration, one greater than, the other less than (1 -P)Css, thedesired minimum concentration. ISS serves as the infusion rate which givesa minimum below (1-P)CSS. For the other reference point we use theinfusion rate which is so great that C\ does not fall below (1 + P)CSS, i.e.,the associated minimum occurs at time t=0 (Fig. 2). An equation fordetermining this infusion rate is obtained by setting t=0 in Eq. (5). Thisgives

Fig. 2. Schematic depiction of the technique used to bracketthe value of the nth unknown constant-rate infusion In. Iu

denotes an upper bound for In; IL denotes a lower boundfor In. For the first iteration Iu is calculated using Eq. (7)and IL is equal to Iss. These values of Iu and IL are averagedto produce a new infusion rate Iav that is associated with aconcentration profile lying between those of Iu and IL.Since Iav produces a minimum concentration which isgreater than the desired minimum, (1 -P)C s s , Iav becomesIu for the next iteration and IL. remains unchanged. Thedifference between Iv and IL is systematically bisected untilthe curve associated with In, the infusion rate giving a mini-mum plasma concentration of (1 -P)Css, is obtained.

Rapid Attainment of Steady State Drug Concentrations 323

The infusion rates associated with the two reference points are averagedto give a new infusion rate whose minimum is determined from Eqs. (5) and(1). If the minimum is greater that (1 -P)Css, the upper reference point isreplaced by the new reference point; otherwise the lower reference point isreplaced. The bisection process is continued until In is determined.

Once a solution has been obtained for In, and provided In#Iss, thisinfusion rate is continued for Tn minutes, at which time the plasma concen-tration reaches the upper limit, (1 + P)Css. To find Tn, we evaluate Eq. (1)at Tn and replace C 1 (T n ) by (1 + P)Css. Since A1 is considerably larger thanA2, the term e~^1 diminishes more rapidly than e~^21', and the second sum-mation term in Eq. (1) may be omitted, which gives the following expression

RESULTS

Computer-predicted plasma concentration-time profiles were generatedto test the method of rapidly obtaining steady state plasma drug concen-trations, using published pharmacokinetic data for amrinone (7), esmolol,(8), lidocaine (9), midazolam (10), propofol (11), and theophylline (12). Foreach drug, a precise limit of deviation from Css of 10% was selected. Therelevant data and the results are summarized in Table I.

DISCUSSION

Parameters of the general biexponential expression based on an opentwo-compartment mammillary model that describes the plasma concen-tration-time profile of a drug following an intravenous injection were util-ized to develop a dosing regimen which rapidly provides a specified steadystate plasma drug concentration and maintains this concentration preciselywithin specified limits. Computer simulations using data for man from theliterature for amrinone, esmolol, lidocaine, midazolam, propofol, and theo-phylline indicated a more rapid attainment and greater precision of the tar-get concentration would be obtained than with previously recommendedcombinations of loading dose and single infusion rate (9,12,13).

Unlike the so-called BET infusion scheme (14,15), the technique doesnot require a computerized infusion pump. These are expensive and are notyet widely available. On the other hand, if necessary, our technique can beused to program such a pump, in which case Css is approximated by a bolusfollowed by a series of constant-rate infusions.

Although calculations are simpler using the technique described byBailey (5), in that they may be performed on a hand-held calculator, once

324 Korman, Jennings, and Rigg

Rapid Attainment of Steady State Drug Concentrations 325

Fig. 3. Plasma concentration-time profile predicted for amrinone with P= 0.1. Aninitial bolus of 370.7jg . kg-1 is followed by infusions of 46,8 and15.6 fig . kg-1 . min-1 for 19.5 and 43.5 min, respectively, followed by an infusion of8.2 ftg . kg-1 . min-1 (=Iss). Note the absence of a minimum during the third infusionas the concentration decreases monotonically from the upper concentration limitof (1 + P)Css(=2.75 Mg . ml-1) to Css(=2.5 /ig . ml-1).

solutions have been obtained with our method for a particular drug usingspecific values for Css and relevant pharmacokinetic parameters, the calcu-lations do not need to be repeated. In the Bailey technique, C1 is said toalways underestimate Css whenever it does not equal it but there is no cer-tainty that the plasma concentration will remain above a specified lowerlimit. Our technique offers greater certainty regarding the plasma concen-tration at any time.

The technique may be applied to maintain the concentration of drugin a particular compartment of a multicompartment model (e.g., the com-partment representing the site of drug effect) but the mathematics are morecomplicated because there may be no simple explicit relationships equivalentto Eqs. (5), (7), and (8).

Occasionally, Eq. (5) produces the logarithm of a negative number withIss. This means that there is no time at which a minimum occurs in C1 andthat this variable is continuously decreasing from its initial value to Css (thisis shown for amrinone in Fig. 3).

Although it did not occur in our calculations, one difficulty that mightarise is a protracted return towards (1 + P)Css. One way of dealing with this

326 Korman, Jennings, and Rigg

Fig. 4. Plasma concentration-time profile predicted for amrinone with P= 0.05. Aninitial bolus of 353.9 jig-kg - 1 is followed by infusions of 53.9, 30.8, and14.8/Jg . kg-1 . min-1 for 11.3, 17, and 38 min respectively followed by an infusion of8.2/ig . kg-1 . min - 1 (=Iss). In this case Iss becomes asymptotic to Css in the standardfashion.

is to apply a test for prolongation of Tn, for example is Tn> 10 x tmin? If so,we suggest that in Eq. (8), C1(rn) be set to Css. It is then almost alwayspossible to switch to ISS and remain within the specified plasma concen-tration limits.

The effect of reducing the tolerance P is shown for amrinone in Figs.4 and 5 in which P has been dropped to 0.05 and 0.01, respectively. It canbe seen that the number of infusions required increases. As P approacheszero, the algorithm becomes exact but the number of infusions neededincreases dramatically. Its application would then require a computer-con-trolled infusion pump. We have chosen the value of P=0.1 because it isassociated with a manageable number of infusions and should be more thanadequate for most clinical situations.

The method outlined can facilitate the induction of clinical anesthesia,smoothly and safely, with intravenous agents such as propofol. The resultsgiven in Table I correspond to values of 9.5mg/kg per hr for I1 (T1 =66min), 6mg/kg per hr for I2 (T2=254min) and 5mg/kg per hr for I3,following a loading dose of 1.5mg/kg. These predictions are based on atwo-compartment model (11) and differ from the currently used 12/9/6

Fig. 5. Plasma concentration-time profile predicted for amrinone with P= 0.01. Aninitial bolus of 340.4 jjg . kg-1 is followed by infusions of 62.8, 50.9, 40.6, 31.6, 24.1,17.9, 13.1, and 9.8Mg-kg - 1 ' min-1 for 4.4, 5.5, 5.7, 6.8, 8.2, 11, 16, and 33min,respectively, followed by an infusion of 8.2 fig . kg-1 min-1 (=Iss). Note the change inscale on the y-axis compared with Figs. 3 and 4.

(16,17) and 10/8/6 (18-20) regimens. More recent data suggest a three-compartment model is more appropriate for propofol (21,22). In addition,Css, varies depending on the type of surgery and the nature of other agentsused concomitantly (e.g., nitrous oxide and opioids) (23).

The potential and demonstrated pitfalls in using pharmacokinetic par-ameters to predict dose regimens must be taken into consideration if thisapproach to intravenous dosing is applied in practice; the precision andutility of the technique for different drugs requires testing on an individualbasis. Although the method has been described for a two-compartmentmodel, it can be extended to other multicompartment models.

ACKNOWLEDGMENTS

We acknowledge the contribution of Tze Wong, MSc, formerly of theDepartment of Mathematics, McMaster University, Hamilton, Ontario,Canada.

Rapid Attainment of Steady State Drug Concentrations 327

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