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BioMed CentralModelling

Open AccessResearch

Volume of the effect compartment in simulations of neuromuscular

block

1 2 3Vladimir Nigrovic* , Johannes H Proost , Anton Amann and

1Shashi B Bhatt

1 2Address: Department of Anesthesiology, Medical University of Ohio, Toledo, OH, USA, Research Group for Experimental Anesthesiology and

3Clinical Pharmacology, University Hospital Groningen, Groningen, The Netherlands and Department of Anesthesiology and Critical Care

Medicine, Leopold-Franzens University, Innsbruck, Austria, and Department of Environmental Sciences, The Swiss Federal Institute of Technology,

Zürich, Switzerland

Email: Vladimir Nigrovic* - vnigrovic@meduohio.edu; Johannes H Proost - j.h.proost@rug.nl; Anton Amann - Anton.Amann@uibk.ac.at;

Shashi B Bhatt - sbhatt@meduohio.edu

* Corresponding author

Published: 03 October 2005 Received: 02 September 2005

Accepted: 03 October 2005

Theoretical Biology and Medical Modelling 2005, 2:41 doi:10.1186/1742-4682-2-41

This article is available from: http://www.tbiomed.com/content/2/1/41

© 2005 Nigrovic et al; licensee BioMed Central Ltd.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0),

which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Background: The study examines the role of the volume of the effect compartment in simulations

of neuromuscular block (NMB) produced by nondepolarizing muscle relaxants.

Methods: The molar amount of the postsynaptic receptors at the motor end plates in muscle was

assumed constant; the apparent receptor concentration in the effect compartment is the ratio of

this amount and the volume arbitrarily assigned to the effect compartment. The muscle relaxants

were postulated to diffuse between the central and the effect compartment and to bind to the

postsynaptic receptors. NMB was calculated from the free concentration of the muscle relaxant in

the effect compartment.

Results: The simulations suggest that the time profiles of NMB and the derived pharmacokinetic

and pharmacodynamic variables are dependent on the apparent receptor concentration in the

effect compartment. For small, but not for large, volumes, times to peak submaximal NMB are

projected to depend on the magnitude of NMB and on the binding affinities.

Conclusion: An experimental design to estimate the volume of the effect compartment is

suggested.

compartment using the equation of Hill. Binding of mus-Background

In the majority of the pharmacokinetic-pharmacody- cle relaxants to the postsynaptic receptors at the motor

namic (PK-PD) models proposed to simulate neuromus- end plates is not considered. Because muscle relaxants

cular block (NMB) [1-3], the volume of the effect produce NMB by binding to these receptors, considera-

compartment is postulated to be negligibly small or the tion of the interaction of muscle relaxants with the recep-

coent is postulated to contain a negligibly small tors represents a more realistic approach and an

amount of the muscle relaxant. The models simulate NMB advancement in simulations [4-6]. Donati and Meistel-

based on the concentration of the muscle relaxant in this man [5] were the first to consider binding of muscle

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relaxants to the receptors. These investigators suggested

that the receptor concentration in the effect compartment −⋅λλtt − ⋅ −⋅λ tNO PDd=⋅ose()N⋅e +O⋅e +P⋅e{}

-7 plasmaM, but the volume of the effect comparis 2.8·10

was assumed to be negligibly small. Given a fixed amount Here, N, O, and P (P = 1 - N - O) are fractions of the dose

of postsynaptic receptors, a finite receptor concentration that are eliminated from plasma with the first order rate

is not compatible with a negligibly small volume of the constants λ , λ , and λ , respectively. The dose is inN O P

-1effect compartment. mol·kg . Division of the equation by V , V expressed inC C

-1L·kg , converts the amounts in plasma to molar concen-

We decided to examine the role of the volume of the effect trations. V represents the volume of the space into whichC

compartment in a pharmacokinetic-pharmacodynamic the muscle relaxant is uniformly diluted at time t = 0, i.e.,

model for NMB and were interested in answering the fol- at the moment of bolus intravenous injection.

lowing questions: (1) Is it necessary to postulate a negligi-

bly small amount of a muscle relaxant in the effect The values assigned to the parameters in the triexponen-

compartment? (2) Do the projections from simulations tial equation were based on the following postulates: For

approximates theusing a small or a large volume of the effect compartment the hypothetical muscle relaxant D, VC

differ? If so, what are the differences? (3) Can the simula- volume of plasma and V , the volume of distribution atSS

tions suggest an experimental design suitable to test steady state, approximates the volume of the extracellular

whether the volume of the effect compartment is negligi- space. The dose that produces NMB50, i.e., ED50, is

bly small or a large volume may be more appropriate? defined by the postulate that the concentration in plasma

at 4.5 min after bolus intravenous injection is [D] =plasma

IC50. The definition of IC50 is provided below. The fol-Methods

General approach lowing values satisfy these requirements:

(1) The amount of the postsynaptic receptors at the motor

end plates in muscle, in terms of mol per kg body weight, N = 0.71; O = 0.192; P = 0.098

was assumed constant and the receptors uniformly

-1 -1 -1diluted in the effect compartment. (2) The plasma con- λ = 1.3 min ; λ = 0.31 min ; λ = 0.0231 minN O P

centrations of a hypothetical muscle relaxant after admin-

-1 -1istration of an intravenous bolus dose, defined by an V = 0.044 L·kg V = 0.28 L·kgC SS

arbitrary multiexponential equation, are labeled target

concentrations. In the simulations, the target plasma con- Compartmental interpretation of the triexponential decay

centrations fulfill the role of the experimentally deter- of the plasma concentrations yields the following param-

mined plasma concentrations. (3) A PK-PD model was eters for the standard 3-compartment pharmacokinetic

designed a priori to include an effect compartment of an model assuming a mammillary arrangement of the com-

assigned volume. The pharmacokinetic parameters in the partments and elimination only from compartment [7]:1

model were defined by the postulate that the concentra-

-1 -1tions in the central compartment (compartment1) fit the V = V = 0.044 L·kg k = 0.1848 min1 C 10

target plasma concentrations. (4) The muscle relaxant dif-

-1 -1fuses from the central to the effect compartment. (5) Phar- k = 0.3771 min k = 0.5581 min12 21

macodynamic parameters were obtained from the

-1 -1postulate that peak neuromuscular block from a bolus k = 0.4229 min k = 0.0902 min13 31

ED50 dose occurs at 4.5 minutes after injection. The peak

concentration of the muscle relaxant in the effect com- Estimation of the receptor amount

partment at this moment corresponds to the IC50 concen- The molar amount of receptors per kg body weight was

tration. (6) The relationship between NMB and the free estimated based on the following assumptions: One hun-

concentrations of the muscle relaxant in the effect com- dred g of muscle is represented as a cube with side length

partment is defined by the Hill equation. of 4.64 cm, i.e., specific density of muscle ~ 1. There is 430

g muscle per kg body weight. The muscle fibers are

The target plasma concentrations densely packed cylinders with the diameter of 50 µm and

Muscle relaxant D was postulated to display linear phar- the length of 4.64 cm (928 rows × 928 columns of fibers

macokinetics. The triexponential equation that defines in a cross section perpendicular to the length of the fib-

the time course of the molar amounts of the muscle relax- ers). Each muscle fiber has one motor end plate with

7 ant in plasma is given by (braces indicate molar 2.1·10 receptors at each end plate [8,9].

amounts):

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The PK-PD Model For an assigned volume of the effect compartment (V ),e

The pharmacokinetic model consists of four compart- the pharmacokinetic parameters in the PK-PD model were

), two peripheralments: the central (compartment estimated in a two-step procedure. In the first step, the1

(compartment and compartment ), and the effect com- parameter k was obtained using the following con-2 3 e1

partment in mammillary arrangement with elimination straints: dose = ED50, the amounts in plasma as defined

from the central compartment. The model is defined in by the triexponential equation, and the maximal NMB =

terms of the amounts of the muscle relaxant present in 50% attained at 4.5 min after administration of the mus-

each compartment and the amount eliminated from the cle relaxant. In the second step, the parameters V , k , k ,1 10 12

body. Transport between the central and the effect com- k , k , and k were estimated using the following con-21 13 31

partment is defined as diffusion according to the concen- straints: dose = ED50 and k fixed to the value obtainede1

tration gradient of the free muscle relaxant in both in the first step. The parameters were fitted by minimizing

compartments. As a result, at the moment when the free the sum of squared differences between the logarithms for

muscle relaxant attains the peak concentration in the the calculated concentrations in compartment and the1

effect compartment and there is no net transport between target concentrations in plasma. The evaluations were car-

the compartments (steady state), the concentrations in ried out at 250 time points from t = 0 to t = 25 min and at

the two compartments are equal. In the model, this con- 50 points for t = 25 to t = 50 min after administration.

straint necessitates that the transport rate constant into the Goodness-of fit was expressed as the coefficient of varia-

1effect compartment, k , be defined in terms of the trans- tion (CV in % ) of the differences between the two time1e

port rate constant from the effect to the central compart- profiles.

ment, k . Hence, k = (V /V )·k , where V and Ve1 1e e 1 e1 e 1

represent the volumes of the effect and the central com- Interaction between the muscle relaxant and the postsyn-

partments, respectively. The volume of the central com- aptic receptors was defined in terms of the association, kas-

pa is known (V ≈ V in the triexponential , and dissociation, k , rate constants. We assumed that1 C soc dis

function). The volume of the effect compartment was each receptor possesses only a single binding site for the

assigned different values. Hence, the amounts of D in the muscle relaxant. The ratio k /k defines the equilib-dis assoc

central and the effect compartments may be converted to rium dissociation constant, K . The inverse of K definesD D

and compartment are the affinity of the receptors for the muscle relaxant.concentrations. Compartment2 3

defined only in terms of the amounts present in them.

The values of all the mentioned parameters are listed in

The amount of receptors in the effect compartment is con- Table 1. The set of five ordinary differential equations

stant and independent of the volume assigned to the defining the amounts of the muscle relaxant in the four

effect compartment. A small assigned volume results in a compartments and the amount of the complex with the

high receptor concentration, while the concentration is receptors in the effect compartment is presented in the

low in the large effect compartment. Appendix.

Table 1: Pharmacokinetic and pharmacodynamic parameters for the PK-PD model. The volume of the effect compartment (V ) was e

postulated to be either small (SMALL) or large (LARGE).

Parameter Unit SMALL LARGE

-1 -7ED50 mol·kg 2.2325·10

-1V L·kg 0.0440 0.0434

1

-1k min 0.1847 0.179510

-1k min 0.3769 0.357412

-1k min 0.5587 0.766321

-1min 0.4226 0.1981k13

-1k min 0.0909 0.058131

-1 -1 10k M ·min 2.4·10assoc

-1 -10{R} mol·kg 1.2921·10total

-1 -5 -2V L·kg 4.4·10 9.23·10e

-1k min 0.6159 0.1477e1

-6 -9[R] M2.9367·10 1.4·10total

Onset time min 4.50

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Calculation of NMB in the small and the large effect compartments are graph-

The intravenous bolus dose of the muscle relaxant ically presented in the upper panel of Figure 1. The three

curves for the amounts in plasma overlap. The good fit ofrequired to produce a half-maximal NMB, NMB50, is

labeled ED50. We postulated that NMB50 is attained at the amounts in compartment to the target amounts in1

4.5 min after the bolus injection. The peak concentration plasma is evident from the small values of the coefficient

of the free muscle relaxant in the effect compartment of variation, 0.0007% for the small and 0.7% for the large

established by ED50 is IC50. At 4.5 min after injection, volume of the effect compartment. The peak free amount

[D] = peak [D] = IC50. The fractional receptor occu- of D in the small effect compartment constitutes a smallplasma e

-4pancy by the muscle relaxant (Occ) at NMB50 is labeled fraction of ED50, 1.38·10 . On the other hand, the peak

Occ and assigned a value of 0.875 [10]. Because K free amount of D in the large effect compartment accountsNMB50 D

= [D] ·(1 - Occ)/Occ, and at NMB50 [D] = IC50 and Occ for a sizable fraction of ED50, 0.289 (upper panel in Fig-e e

= Occ = 0.875, it follows that IC50 = 7·K . ure 1). The PK-PD model that includes a large effect com-NMB50 D

partment requires intercompartmental transport rate

Neuromuscular block (NMB) was calculated using the constants different from those for the small volume of the

Hill equation, the free concentrations of the muscle relax- effect compartment (Table 1). The peak receptor occu-

ant in the effect compartment, [D] , and two parameters: pancy, Occ = Occ = 0.875, and the peak [D] = IC50e NMB50 e

γ and IC50 (γ = 4 and IC50 = 7·K , Eq 1 in Appendix). = 7·K , were attained at 4.50 min for either volume of theD D

effect compartment. Hence, for both volumes the simu-

To describe quantitatively the simulated NMB as a func- lated peak NMB = NMB50 and occurs at 4.5 min after

tion of doses used to establish the peak concentrations in injection, but the time course of NMB is different between

the effect compartment, the values for NMB calculated the small and large volumes of the effect compartment

from peak [D] were plotted as a function of doses of the (lower panel in Figure 1). To reach the respective peaks ate

muscle relaxant. A modified equation of Hill (Eq. 2, 4.50 min after the injection required k that was approx-e1

Appendix) was fitted to these points using the program imately four times higher for the small than for the large

TableCurve2D from SPSS, Chicago, IL, and the fitted esti- effect compartment (Table 1).

mates of the exponent γ and ED50 are reported.f f

The calculations were verified by calculating the sum of

All calculations were performed independently using the the amounts in the four compartments plus the amount

programs MATHEMATICA (version 5.1) from Wolfram eliminated from the body. For all times between 0 and 50

Research, Inc., Champaign, IL, MULTIFIT and PKPDFIT min after injection, the sum was equal to ED50. Expressed

written by J.H. Proost, and MATLAB (version as fractions of the administered dose (= ED50), the peak

6.1.0.450(R12.1)) from The Mathworks Inc., Natick, MA. amounts in compartment and compartment and the2 3

times after injection when the peaks were attained are for

Results the small volume of the effect compartment 0.199 at 1.6

The estimated total molar amount of receptors at the min and 0.483 at 7.3 min, respectively. For the large vol-

-10motor end plates in muscles is {R} = 1.2921·10 ume, the corresponding values are 0.158 at 1.3 min andtotal

-1mol·kg . Receptor concentration in the effect compart- 0.268 at 11.9 min.

ment is the ratio of this amount and the volume assigned

to the effect compartment. Two additional observations were made during these sim-

ulations. First, exclusion of the small effect compartment

Simulations with a small or a large volume assigned to the from the PK-PD model only minimally influences the fit

effect compartment of the amounts in compartment to the target plasma1

For the initial simulations, V was assigned the value of amounts. The result is not unexpected, because the inter-e

-5 -1 0.001·V , i.e., V = 4.4·10 L·kg [11], for the small and compartmental transport rate constants (microconstants)C e

-1 0.0923 L·kg for the large effect compartment. The latter in the model with a small volume of the effect compart-

approximates the volume of the interstitial space in mus- ment (Table 1) are close to those in the standard 3-com-

cle. Receptor concentrations in the effect compartment partment model. Second, when the effect compartment in

-6 -9 were: [R] = 2.94·10 M and 1.4·10 M for the small the PK-PD model was postulated not to contain the recep-total

and large volume, respectively. The hypothetical muscle tors, i.e., {R} = 0, identical values of k establish thetotal e1

-7 relaxant D was assigned K = 1·10 M. The assignment peak free amount of D in the respective effect compart-D

-7 -1defined ED50 as ED50 = 2.23·10 mol·kg . Optimal ment at identical times (data not presented).

estimates of the pharmacokinetic parameters, including

k , were obtained as described in the Methods section. Based on the derived pharmacokinetic rate constants,e1

The target amounts of the muscle relaxant in plasma and NMB was simulated with different doses of D. One thou-

those estimated in compartment as well as the amounts sand points were selected for a 10-fold increase in doses.1

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-2 -1 -5 -1 -7 Uppeeffect complarge (M) in pFigure 1 r panel: Vlasma and compartment= 9.23·10artment asThe amounts of muscle relaxant L·kgsign) voed alu smmeand the free amall (V = 4.4·10D (K ounts inL·kg= 1·10) or the a e 1 e D

-7 UppeD (KD

M) in plounts in the 1

-5 -1effect compartment assigned a small (V = 4.4·10 L·kg ) or a e

-2 -1large (V = 9.23·10 L·kg ) volume. All the amounts are nor-e

-7 -1malized to the injected dose (= ED50 = 2.23·10 mol·kg ).

Solid and dashed lines indicate the amounts contained in the

small and the large effect compartment, respectively. Filled

circles denote the target amounts in plasma defined by the -7 the effect fudix (Upper panFigure 2nction of the peak concentraγ = 4.ecompartment using Eq 1 presented in the Ap0 al: Neuromuscnd IC50 = 7· u10lar block (NMB) calculaM)tions of muscle rela ted as a xant D in pen-

triexponential function. The three curves for the amounts in el: ulart

plasma overlap. The estimates were obtained at 0.1 min

futions of xant D in

intervals. Lower panel: Time course of the neuromuscular

the effect pen-

block (NMB) by ED50 of the muscle relaxant D. NMB was -7 dix (γ = 4.0 and IC50 = 7·10 M). The doses presented along

calculated using Eq 1 (Appendix), [D] for the small and large e the abscissa refer to the doses that established the peak con-

volume of the effect compartment presented in the upper

centrations. The range of NMB is from NMB05 to NMB95. -7 panel, and by setting γ = 4.0 and IC50 = 7·10 M. The lines

One thousand logarithmically equidistant values were used

are identical to those in the upper panel for the small and

for a 10-fold increase in doses. The volumes of the effect

large volume of the effect compartment.

compartment and the lines are identical to those presented

in Figure 1. Lower panel: Onset times as a function of the mag-

nitude of NMB. Onset times are defined as the times after

the bolus intravenous injection of muscle relaxant D needed

to establish peak NMB, from NMB05 to NMB95. Other

NMB was calculated with the peak free concentrations of details are identical to those in the upper panel.

D in the effect compartment using Eq 1 in the Appendix

-7 (γ = 4 and IC50 = 7·10 M). The relationship between

NMB and the doses that produced the peak concentra-

tions differed between the models (upper panel in Figure

2 2 for NMB = 0.05 to NMB = 0.95, i.e., NMB05 to NMB95). doses. The fit was excellent for both sets (r > 0.9999, the

To obtain a quantitative estimate for the difference, equa- number of points, n, = 381 for the small and n = 641 for

tion of Hill (Eq 2) was fitted to both sets of points to the large volume of the effect compartment). The 95%

describe the relationship between NMB and the injected confidence interval (95%CI) for the fitted γ was 6.819 tof

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onset times and the magnitudes of NMB. For NMB <

NMB50, the onset times were longer, and for NMB >

NMB50 the onset times were shorter than those projected

by the model with a large effect compartment.

Simulations with different volumes assigned to the effect

compartment using ED50

Next, the influence of the volume assigned to the effect

compartment was examined systematically. The volumes

-6 -1 -1 varied from 1·10 to 1·10 L·kg (11 logarithmically

equidistant values). The pharmacokinetic parameters,

including k , were estimated as previously stipulated, i.e.,e1

ED50 dose establishes peak receptor occupancy = Occ875

and peak [D] = IC50 at 4.5 min after injection. The coef-e

ficient of variation for the fit of the concentrations of D in

compartment to the target plasma concentrations was CV1

= 0.84% for the largest and = 0.0006% for the smallest

volume. The values for k as a function of the assignede1 s, estimated with ED50 and using the same muscle

-7 relaxant (K = 1·10 M), are presented in the upper panelD

of Figure 3. The results demonstrate that k increasese1

markedly for the smaller values of V. The relativee

amounts of D bound to the receptors, the amounts free in

the effect compartment, and the ratio of the bound to the

total amount in the effect compartment with ED50 show

(lower panel in Figure 3) that for all volumes the amounts

of D bound to the receptors are constant. For smaller vol-

UppefunctionVFigure 3betweenr panel: of the volume assign the effect and the cenValues estimated for th ed to tral comthe e transport rate constaneffect compartment, partment, k , as a t umes the bound amounts make up nearly all of D presente e1

Uppee t in the effect compartment, while for larger volumes the

betweentral compartment, k , as a e1 total amount of D is nearly completely accounted for by

functioned to the

the free amount.

-7 V . The dose of the muscle relaxant D = ED50 = 2.23·10e

-1 -7 mol·kg and K = 1·10 M. Lower panel: Amounts of the mus-D Simulations using different volumes and different doses

cle relaxant D (left Y-axis) bound to the receptors (filled

We used the set of pharmacokinetic parameters obtainedupright triangles) and the amounts free (filled diamonds). The

for each assigned volume, but now varied the dose usingamounts are normalized to the injected dose presented in

1000 values for a ten-fold increase. The results are pre-the upper panel. The ratio of the bound to the total amounts

sented in Figure 4. Increasing doses increase the peak free(empty circles; total = bound + free; right Y-axis) in the effect

compartment is presented as a function of the volume concentrations of D for each volume of the effect com-

assigned to the effect compartment. partment (upper panel in Figure 4). The increase is

steepest for the smallest volume and the slopes decrease

for the larger assigned volumes. The estimated peak free

concentrations of D in the effect compartment were used

-7 to calculate NMB (IC50 = 7·10 M and γ = 4, Eq 1 in

6.838 for the small and 4.0040 to 4.0041 for the large Appendix). The values of NMB from NMB05 to NMB95 as

effect compartment. The 95%CI for the fitted ED50 were calculated using [D] were plotted against the injectedf e

-7 -1 (2.235 to 2.236)·10 mol·kg and (2.23256 to doses separately for each assigned volume, similarly to the

-7 -12.23258)·10 mol·kg , respectively. results presented in the upper panel of Figure 2. The mod-

ified equation of Hill (Eq 2, Appendix) was fitted to each

The onset times for NMB05 to NMB95 differed between of these 11 sets of points to define NMB as a function of

2 the models assigned different volumes of the effect com- the injected doses. The fit was excellent (r > 0.9996 for n

partment (lower panel in Figure 2). The model with the between 314 to 641 points). The fitted values of γ are pre-f

large effect compartment projected that the onset times sented in the lower panel in Figure 4. The values increase

were nearly independent of the magnitude of NMB. The markedly for smaller volumes. The 95%CI for the eleven

model incorporating a small volume of the effect com- fitted estimates of ED50 varied between (2.225 tof

-7 -1 partment projected an inverse relationship between the 2.226)·10 mol·kg for the smallest and (2.25720 to

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NMB95 (onset times). The results are presented in the

lower panel of Figure 4. Onset times for NMB05 and

NMB95 differ widely for the small volumes, but the differ-

ences progressively decrease for larger volumes of the

effect compartment. The onset times for NMB05 and

NMB95 are nearly identical for the largest assigned

volume.

Simulations with different binding affinities assigned to

muscle relaxants

The PK-PD model was also tested with two additional

muscle relaxants using the previously defined small and

large volumes of the effect compartment. One muscle

relaxant, D , was assigned a 10 times lower affinity for the2

-6 = 1·10 M. The other,binding sites at the receptors, KD2

-8D , was assigned a 10 times higher affinity, K = 1·103 D3

M. The assignments changed the respective k , but notdiss

k . Two series of simulations were performed. In theassoc

first series, all the pharmacokinetic constants, including

k , were those defined previously for either the small ore1

the large volume of the effect compartment and for the

-7 muscle relaxant with K = 1·10 M (Table 1). A 100-foldD

increase in affinities was projected to require 16.3 times

-6lower ED50 for the small volume (ED50 = 1.144·10

-1 -8 -1 mol·kg for D and ED50 = 7.018·10 mol·kg for D ),2 3

-6 -but a 98.9 times lower ED50 (ED50 = 2.230·10 mol·kg

1 -8 -1 for D and ED50 = 2.255·10 mol·kg for D ) for the2 3

model with a large volume of the effect compartment. For

the small effect compartment, the times to reach NMB50

compartmentFigure 4D as a function of the voluD in the effect compartUpper panel: The peak free concentrationment mes assigned to the effect calculated wi sth vari of muscle able dorelaxases nt of

were 1.82 min for D and 34.74 min for D . In the model2 3Uppel: snt

with a large effect compartment, the times to NMB50 dif-ment calculated with variable doses of

fered only minimally, from 4.50 min for D to 4.54 min2 D as a function of the volumes assigned to the effect com-

-6 -6 - for D .3partment. The assigned volumes were: 1·10 , 3.16·10 , 1·10

5 -5 -4 -4 -3 -3 -2 -, 3.16·10 , 1·10 , 3.16·10 , 1·10 , 3.16·10 , 1·10 , 3.16·10

2 -1 -1, and 1·10 L·kg . The bold solid and the dotted lines indi- In the second series of simulations, we postulated that

cate the lowest and the highest assigned volumes, respec- ED50 of either D or D produces NMB50 at 4.5 min after2 3

tively. Concentrations for the intermediate volumes are injection using either the small or the large volume of the

indicated in sequence by thin solid lines. The three dashed effect compartment. The doses producing NMB50 were

lines parallel with the X-axis represent the free concentra- -values, i.e., ED50 = 2.2325·10related to the assigned KD

tion of D for NMB95 (IC95, upper line), for NMB50 (IC50, 6 -1 -8 -1 mol·kg for D and ED50 = 2.2325·10 mol·kg for2 middle line) and for NMB05 (IC05, lower line). The concen-

D . These doses establish plasma concentrations at 4.53trations for IC05 and IC95 were calculated based on γ = 4.0

min [D] = IC50 = 7·K = peak [D] . In the modelplasma D e(Eq 1, Appendix). Lower panel: Times to NMB05 (open cir-

containing a small volume of the effect compartment, thecles) and NMB95 (filled circles, left Y-axis) as a function of

-1 postulate was satisfied by k = 0.196 min for D and kthe volumes assigned to the effect compartment, V . Neu- e1 2 e1e

-1 -7 = 4.710 min for D . For the large volume, the estimatesromuscular block was calculated using Eq 1 (IC50 = 7·10 M, 3

-1 -1 γ = 4) and the peak free concentrations of D presented in of k were 0.1475 min for D and 0.1498 min for D .e1 2 3

the upper panel. The values of the exponent γ (filled dia-f

monds, right Y-axis) were obtained by fitting Eq 2 to the cal- Discussion

culated NMB. The simulations suggest that the volume of the effect com-

partment per se is not the critical parameter in a PK-PD

model for nondepolarizing muscle relaxants. If the effect

compartment is postulated to be void of the postsynaptic

receptors, then the peak concentration of free muscle

-7 -1 2.25722)·10 mol·kg for the largest volume. These sim- relaxant in this compartment is attained at identical times

ulations permitted us to estimate the times to NMB05 and using identical transport rate constant k for any volumee1

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of the effect compartment. These conclusions agree with fuse through the pores in the capillary wall into the sur-

those obtained from PK-PD models assuming a negligibly rounding interstitial spaces. Diffusion across the cellular

small volume of the effect compartment and not taking membranes is very unlikely due to the high hydrophilicity

into account binding of a muscle relaxant to the postsyn- of the molecules. Therefore and as a first approximation,

aptic receptors [1-3]. However, NMB is produced not by muscle relaxants remain diluted in a space limited to

the free molecules of muscle relaxants in the effect com- plasma and the interstitial space. The pharmacokinetic

partment, but by the molecules bound to the postsynaptic compartments for muscle relaxants likely represent the

receptors at the motor end plates. Therefore, considera- amounts of muscle relaxants in plasma and the interstitial

tion of binding of muscle relaxants to the postsynaptic spaces of different tissues.

receptors in the effect compartment is advantageous in

PK-PD modeling. The present simulations confirm the In the muscle, muscle relaxants diffuse throughout the

conclusion from the reports [4-6] that the receptor con- interstitial space, including the synaptic clefts at the motor

centration in the effect compartment is a critical end plates. There are no anatomical barriers between the

parameter in PK-PD modeling. The predictions from our interstitial space in muscle and the synaptic clefts to pre-

simulations assuming a low or a high receptor concentra- vent diffusion of muscle relaxants into the synaptic clefts

tion differ with respect to (1) the onset times to the peak [13]. These considerations qualify the interstitial space in

but submaximal neuromuscular block for a single muscle muscle, including the synaptic spaces, as a single

relaxant (lower panel in Figure 2), (2) the time course of pharmacokinetic compartment. The volume of the inter-

NMB using ED50 (lower panel in Figure 1), (3) the shape stitial space in muscle defines the volume of this

of the NMB-versus-dose curves (upper panel in Figure 2), compartment.

(4) the estimates of k (Table 1 and upper panel in Figuree1

3), and (5) the estimates of ED50 and the onset times as a Due to the presence of the postsynaptic receptors in the

function of affinities assigned to the muscle relaxants for synaptic clefts, the

binding to the receptors (D and D , Results).2 3

compartment represents the effect compartment for mus-

In the aforementioned models [4-6], receptor concentra- cle relaxants. The functional receptors are immobile and

tion was an explicit model parameter. In contrast, the are located exclusively within the synaptic clefts. Hence,

present model defines the receptor concentration as the interaction between the receptors and the free molecules

ratio between the constant amount of postsynaptic recep- of a muscle relaxant occurs due to diffusion of the free

tors and the variable volume assigned to the effect com- molecules of the muscle relaxant to the receptors. In

partment. This approach allows PK-PD modeling without effect, interaction between the two partners may be repre-

the constraint of a negligibly small effect compartment. sented as proceeding in a space common to both, i.e., the

The earlier models taking into account receptor concentra- interstitial space in muscle. Volume of this space defines

tion [4-6] assumed a negligibly small volume of the effect the volume of the effect compartment, V . We suggest thate

compartment. Given a fixed amount of postsynaptic the apparent or the effective concentration of the postsyn-

receptors, a finite receptor concentration is not compati- aptic receptors for the interaction with muscle relaxants is

ble with the negligibly small volume of the effect com- the ratio of the amount of receptors and the volume of

partment. This is an inherent weakness of such models. interaction, [R] = {R} /V .total total e

A compartment is defined by Jacquez [12] as "an amount Transport of a drug between two compartments is repre-

of a material that acts kinetically like a distinct, sented in a standard pharmacokinetic model by two first-

homogenous amount". This is the reason that the five order rate constants. A modification of this approach is

equations defining the transport and the distribution of a needed, if the transport is assumed to proceed via diffu-

muscle relaxant in the body (Appendix) were formulated sion. Occurrence of a peak amount in a non-central com-

in terms of amounts rather than concentrations. The total partment suggests that at that moment there is no net

amount of a drug in the body is represented by two, three, transport. The postulate of transport via diffusion implies

or more such compartments. The necessity to invoke that the concentration of a muscle relaxant in the central

more than a single compartment arises from the physico- and the peak concentration in the effect compartment are

chemical properties of the drug in relation to those struc- identical at that moment. In the simulations, the transport

tures in the body that prevent drug's uniform dilution. rate constant out of the effect compartment into

Anatomical structures and/or physiologic processes repre- compartment is represented by the symbol k . The rate1 e1

sent these barriers. For muscle relaxants, small constant in the opposite direction, k , is expressed as a1e

hydrophilic cations with MW < 1000 da, the principal function of k , viz., k = k ·(V /V ). The expressione1 1e e1 e 1

barriers are the capillary wall and the cellular membranes. results from the postulate that the transport occurs via dif-

It seems plausible to postulate that muscle relaxants dif- fusion. We suggest that k may be interpreted as the ratioe1

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of the plasma flow to the muscle and the volume of the pharmacodynamic models, NMB in the proposed model

interstitial space in muscle. For the adductor pollicis mus- was calculated using the peak free concentration of a mus-

cle and assuming plasma flow to the forearm or the hand cle relaxant in the effect compartment and two constants:

-1 -1 of 0.9 to 4.7 mL·min ·(100 g muscle) [14] and the vol- γ and IC50 (Eq 1, Appendix). When the NMB, calculated

ume of the interstitial space in muscle of 15 to 22 using peak [D] , was plotted as a function of the doses thate

-1mL·(100 g muscle) , the value of the transport rate produced these peak concentrations in the effect compart-

constant k may be estimated to between 0.041 and ment, the fitted value of γ (Eq 2, Appendix) was largere1 f

-10.313 min . than γ used in the calculation of NMB from [D] (uppere

panel in Figure 2 and lower panel in Figure 4) and the fit-

The postulate seems plausible that the molar amount of ted values of γ increase progressively for smaller volumesf

the postsynaptic receptors is a physiologic constant. The assigned to the effect compartment (lower panel in Figure

-3 -1value of the constant may be 10 times lower or 10 times 4). For volumes > 10 L·kg , the fitted values of γf

higher than the assigned value (Table 1) without mark- approach the value of γ used in the calculations of NMB

edly altering the results of the simulations. The postulate from [D] (lower panel, Figure 4). The difference is due toe

of a constant amount of postsynaptic receptors permits the relationship between the peak concentrations of the

the definition of the apparent receptor concentration in free muscle relaxant in the effect compartment and the

the effect compartment via the relationship [R] = injected doses (upper panel in Figure 4). For volumes <total

-3 -1{R} /V . 10 L·kg , the peak concentrations increase rapidly withtotal e

increasing doses. The steeper slope implies that the differ-

The results of the simulations demonstrate that a PK-PD ence in doses producing IC05 and IC95, corresponding to

model may be constructed for a wide range of volumes NMB05 and NMB95, respectively, is smaller the smaller

assigned to the effect compartment. We examined V from the volume assigned to the effect compartment. The nar-e

-6 -1 -11·10 to 1·10 L·kg and the corresponding apparent rower spread of these doses leads, in turn, to higher fitted

concentrations of the receptors. In general, smaller values of γ when NMB is represented as a function of thef

-3 -1volumes require higher values of k (upper panel in Fig- injected dose. To summarize, if V < 10 L·kg , then a cor-e1 e

ure 3), are associated with smaller total amounts of the relation of NMB to the doses needed to establish the peak

γ higher than themuscle relaxant in the effect compartment and larger frac- concentrations requires fitted values for f

-tions of the muscle relaxant in the bound form (lower value of γ used in calculating NMB from [D] . For V > 10e e

3 -1panel in Figure 3). The smaller volumes are compatible L·kg , the estimates of the fitted γ approach the value off

with the intercompartmental transport rate constants γ used in calculating NMB as a function of [D] . Therefore,e

close to those in the standard 3-compartment pharmacok- a comparison of γ, estimated in a PK-PD model and based

-3 -1inetic model. For volumes < 1·10 L·kg , the onset times on [D] , with γ , obtained experimentally in a NMB-versus-e f

of submaximal NMB are negatively correlated with the dose study, provides information about the volume of the

magnitude of NMB (lower panels in Figures 2 and 4). The effect compartment and the receptor concentration in it.

onset times are also markedly dependent on the values The fitted values of ED50 are rather independent of thef

assigned to the equilibrium dissociation constants for volumes assigned to the effect compartment and the esti-

binding of the muscle relaxants to the receptors (muscle mates are close to the a priori defined ED50 used in calcu-

relaxants D and D ), higher affinities associated with pro- lating the target plasma concentrations.2 3

longed onset times. All these findings change for V >e

-3 -1 -7 1·10 L·kg and the receptors concentrations < 1·10 M The PK-PD models are based on two sets of experimental

(Figures 3 and 4). Specifically, the values of the rate con- data: the time course of the plasma concentration of a

stant k become smaller and relatively independent of the muscle relaxant and the time course of NMB. The modelse1

assigned volumes (upper panel in Figure 3), the differ- simulate, and are applicable only to, the concentrations in

ences between onset times for NMB05 and NMB95 pro- plasma and in the postulated effect compartment. The

gressively disappear (lower panel in Figure 4), the amounts or concentrations in the other compartments

affinities do not influence the onset to NMB50, and ED50 and the amount eliminated from the body are not verifia-

doses are proportional to K (muscle relaxants D and D , ble from the available experimental data. These compart-D 2 3

-3Results). It appears as if the value of V of about 1·10 ments are included in the current PK-PD model solely toe

-1 -7 L·kg and the receptor concentration ~ 1·10 M repre- preserve mass balance and to fit the amounts or concen-

sent the critical threshold for the difference between a trations of D in compartment to the target plasma1

"small" and a "large" volume of the effect compartment. amounts or concentrations. A posteriori addition of a large

effect compartment to the standard 3-compartment PK

The results of the simulations reveal a difference in the model alters the simulated amounts or concentrations in

slopes of the NMB curves when evaluated as a function of compartment and the fit of the standard 3-compartment1

the injected doses of a muscle relaxant. As in the available model to the target plasma concentrations is lost.

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Realization of this fact was the primary reason for the pos-

tulate of a negligibly small effect compartment in the pre- d

D =−kkk+ + +k ⋅D{}(){}10 12 13 1eviously introduced PK-PD model [2]. However, as 1 1dt

demonstrated in the current simulations, a PK-PD model

+⋅kD +⋅kDk+⋅D{} {} {}21 31 e12 3 emay include an effect compartment of any volume and

dcontain a sizable fraction of the injected dose, if the model {}Dk=⋅{}D −⋅k {}D12 212 1 2is designed a priori and the pharmacokinetic rate con- dt

stants, including k , adjusted so that the amounts ine1 d

Dk=⋅D −⋅k D{} {} {}compartment represent as closely as possible the 13 311 3 1 3dt

observed amounts in plasma. The fitting process is similar

d d

to that for fitting a standard pharmacokinetic model to D =⋅=kD −⋅kD − DR{} {} {} {}1ee1e 1 edt dtthe observed plasma concentrations. Alternatively and

d kwithout prejudging mass transport from plasma to any assocDR=⋅DR⋅ − DRD −⋅k DR{} {} {} {} {}() dissetotalcompartment, the amounts in plasma may be described dt Ve

using a multiexponential equation without detriment to

the pharmacodynamic part of the model. DR represents the 1 : 1 complex of D with the receptors

within the effect compartment. The differential equation

Conclusion for {DR} was derived from the differential equation for

The simulations do not indicate whether a PK-PD model [DR] written in terms of the molar concentrations [D] ,e

containing a small or a large effect compartment is more [R] , and [DR]. Multiplication of this equation by Vtotal e

appropriate. The selection should be based on the results converts the concentrations into amounts. The definition

of prospective clinical experiments. The simulations sug- of k in terms of , viz., k = k ·(V /V ), results from the1e ke1 1e e1 e 1

gest an optimal experimental design. The study needs to postulate of diffusion as the transport mechanism and

be conducted with several muscle relaxants. Several doses implies that the peak concentration of D in the effect com-

of each are selected to produce less than complete NMB, partment equals the concentration in compartment at the1

e.g., NMB10 to NMB90. The experiment needs to answer same moment. The initial conditions at t = 0 are: {D} =1

dose, and {D} = {D} = {D} = {DR} = 0.the following question: Is the onset time of submaximal 2 3 e

NMB produced by a single muscle relaxant a function of

the level of NMB? If the results with a single muscle relax- The Hill equation for the calculation of NMB from the free

ant show an inverse relationship between the level of molar concentrations of the muscle relaxant D in the

NMB and the onset times, then the model containing a effect compartment, [D] , is given by:e

small volume of the effect compartment and a high recep-

tor concentration is more appropriate. If the onset times γ

D[]eare independent of the magnitude of the submaximal NMB = Eq1

γ γNMB, then the PK-PD model containing a large volume of DI +C50[]e

the effect compartment and a low receptor concentration

is more appropriate. where [D] = peak {D} /V and IC50 = 7·K = peak [D]e e e D e

when Occ = 0.875. The exponent γ was arbitrarily assigned

a value of 4.0.Appendix

The pharmacokinetic part of the model was formulated

with the volume of the effect compartment, V , explicitly A different form of the Hill equation was used to fit thee

incorporated in the model. The following symbols are calculated NMB (Eq.1) as a function of the doses produc-

used: D for the muscle relaxant and R for the receptors. ing the peak [D] . The modified equation relates NMB toe

The braces denote molar amounts per kg body weight. The the injected doses:

first and second subscript appended to the rate constants

denote the number of the source and the target compart- γ fdose

ments, respectively. Subscript e denotes the effect com- NMB = Eq2

γγffpartment, e.g., k denotes the rate constant for the dose + ED501e f

transport from compartment to the effect compartment1

and k the transport in the reverse direction. The symbol The values for the exponent γ and ED50 were derived ine1 f f

{D} denotes the free amount of D in the effect the fitting process.e

compartment.

Competing interests

The author(s) declare that they have no competing

interests.

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