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Volume of the effect compartment in simulations of neuromuscular block

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11 pages
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.
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Theoretical Biology and Medical
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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