Mechanistic modeling of digoxin distribution kinetics incorporating slow tissue binding

e u r o p e a n j o u r n a l o f p h a r m a c e u t i c a l s c i e n c e s 3 0 ( 2 0 0 7 ) 256–263

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Mechanistic modeling of digoxin distribution kinetics incorporating slow tissue binding Michael Weiss ∗ Section of Pharmacokinetics, Department of Pharmacology, Martin Luther University Halle-Wittenberg, 06097 Halle, Germany

a r t i c l e

i n f o

a b s t r a c t

Article history:

This study aims to develop a mechanistic pharmacokinetic model that accounts for the

Received 21 July 2006

kinetics of tissue binding in order to evaluate the effect of slow binding of digoxin to skele-

Accepted 15 November 2006

tal muscular Na+ /K+ -ATPase in humans. The approach is based on a minimal circulatory

Published on line 23 November 2006

model with a systemic transit time density function that accounts for vascular mixing, transcapillary permeation and extravascular binding of the drug. The model parameters were

Keywords:

estimated using previously published disposition data of digoxin in healthy volunteers and

Digoxin

physiological distribution volumes taken from the literature. A time constant of the binding

Pharmacokinetics

process of 34 min was estimated indicating that receptor binding and not permeation clear-

Model

ance is the rate-limiting step of the distribution process. Model simulations suggest that up-

Binding kinetics

or downregulation of sodium pumps, typically observed under physiological or pathophys-

Skeletal muscular Na+ /K+ -ATPase

iological conditions, could be detected with this method. The model allows a quantitative prediction of the effect of changes in skeletal muscular sodium pump activity on plasma levels of digoxin. © 2006 Elsevier B.V. All rights reserved.

1.

Introduction

Although cardiac glycosides have been used for 200 years in the treatment of congestive heart failure, digoxin still remains one of the most widely prescribed drugs (Gheorghiade et al., 2004). Because of their narrow therapeutic range, knowledge of pharmacokinetics is essential in optimizing dosing and minimizing side effects. The disposition kinetics of digoxin and the basic parameters clearance (CL) and steady-state volume of distribution (Vss ) have been well characterized in humans for more than 40 years (Kramer et al., 1974). However, surprisingly little is known about the determinants of distribution kinetics of digoxin in the body and particularly about the role of binding to skeletal muscular Na+ /K+ -ATPase (sodium pump) that acts as receptor for digoxin. The facts that (1) the skeletal muscle pool of sodium pumps constitutes the main determinant of the Vss of digoxin (Schmidt et al., 1993), (2) receptor

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binding is relatively slow (Crambert et al., 2004; Weiss and Kang, 2004), and (3) transcapillary permeation of digoxin is rapid relative to tissue binding (Kang and Weiss, 2002) suggest that tissue binding is the rate-limiting step in digoxin distribution kinetics. This information about binding kinetics is important to understand whether the activity of skeletal muscular sodium pumps could affect plasma levels of digoxin (McDonough et al., 1995), given the fact that disease states or exercise change the sodium pump activity/capacity (e.g., Joreteg and Jogestrand, 1984; Schmidt et al., 1993). In order to explain the determinants digoxin of distribution kinetics, we need a mechanistic model of whole body distribution kinetics; traditional compartmental models of drug distribution lack physiological reality and their parameters of distribution kinetics cannot be interpreted in terms of underlying transport and binding kinetics (e.g., Weiss, 1998; Weiss et al., 2007). Although the importance of slow binding of ouabain

e u r o p e a n j o u r n a l o f p h a r m a c e u t i c a l s c i e n c e s 3 0 ( 2 0 0 7 ) 256–263

to cardiac and skeletal muscle tissue has been pointed out previously using destructive sampling in guinea pigs (Harashima et al., 1992), this methodology cannot be applied to humans. Therefore, the purpose of this paper is to describe the development and application of a model based on the mechanisms of drug distribution kinetics in the body, namely convective transport (vascular mixing), transcapillary transport (permeation), and tissue binding. This is accomplished through a minimal circulatory model in which all organs of the systemic circulation are lumped into one heterogeneous subsystem. Thus, two types of models have been combined in our recent work developing a mechanistic model for drug distribution kinetics (Weiss et al., 2007), a recirculatory model (Weiss et al., 1996) and a model of a heterogeneous tissue system assuming capillary permeation and noninstantaneous binding in the extravascular phase (Weiss and Roberts, 1996). Our model allows for the analysis of permeation and tissue binding kinetics of digoxin in humans and has been applied to literature data (Kramer et al., 1974).

2.

Methods

2.1.

Data

The present study uses digoxin disposition data of five healthy male volunteers reported by Kramer et al. (1974). Briefly, 21 blood samples were taken between 2 min and 72 h times after rapid intravenous injection of a 1 mg dose of digoxin. The individual blood (VB ) and interstitial (VIS ) volumes were calculated using individual weights (mean 76.6 ± 5.3 kg) and VB = 0.071 l/kg (Niemann et al., 2000) or VIS = 0.277 l/kg (Prescott et al., 1991). A cardiac output Q of 5 l/min was assumed.

2.2.

Model

2.2.1.

Transit time density functions

257

To develop a mechanistic pharmacokinetic model that can be identified solely on the basis of plasma concentration–time data, its structural complexity must be reduced to a minimum. The most rigorous structural simplification is given in terms of only two subsystems, the pulmonary and systemic circulation. A concept to characterize these subsystems without assuming well-mixed compartments is by their transit time density (TTD) functions of molecules (Fig. 1). The TTD of drugs through the subsystems are based upon a model, which accounts for the physiological heterogeneity assuming blood and tissue separated by a permeability barrier with noninstantaneous mixing/distribution in the blood and tissue phases (Fig. 1). This lumped organ model of the systemic circulation has been previously described in detail and applied to solutes with negligible tissue binding (Weiss et al., 2007). Briefly, the information on the intravascular mixing process (an important determinant of the kinetics of blood tissue exchange) is incorporated into the model using the measured TTD of an intravascular indicator, e.g., indocyanine green (ICG), which is described by an empirical density function (inverse Gaussian density). The Laplace transform of drug TTD through the systemic circulation is given by (Weiss and Roberts, 1996; Weiss et al., 2007): fˆs (s) = fˆB,s [s + kin (1 − fˆy (s))]

(1)

where fˆB,s (s) denotes the Laplace transform of the vascular marker. The uptake rate constant kin is determined by permeation clearance CLperm and vascular or blood volume of the

Fig. 1 – Schematic of the recirculatory model (cardiac output, Q) developed to study distribution kinetics of digoxin. The basic model consists of two heterogeneous subsystems, the pulmonary and systemic circulation, characterized by transit time density functions fˆp (s) and fˆs (s), respectively, in the Laplace domain. Elimination of digoxin in the systemic circulation is characterized by extraction ratio, E. The model for transcapillary transport from vascular (B) to interstitial space (IS) with rate constants kin and kout , as well as specific binding to extracellular receptors (kon and koff ) and rapid nonspecific binding (Krapid ), is derived in a virtual microelement.

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systemic circulation, VB,s :

(Hung et al., 2001):

CLperm = kin VB,s

(2)

Note that CLperm = fub PS, where PS denotes the permeability-surface product and fub is the free fraction in blood. The rate constant kout is obtained from Eq. (2), when VB,s is replaced by VIS,s , i.e., kout = (VB,s /VIS,s ) kin . All information on distribution kinetics within tissue (after leaving the vascular space by passive permeation) is provided by the density of extravascular sojourn times of drug molecules, fˆy (s).

2.2.2.

Intravascular mixing

The Laplace transform of the density function of the inverse Gaussian distribution that is used as empirical TTD function for the vascular marker across subsystem i is given by

 fˆB,i (s) = exp

1 RD2i

 −

Vi /Q



RD2i /2

s+

1 2(Vi /Q)RD2i

1/2  (3)

where i = s or p denotes the systemic or pulmonary circulation, respectively, VB,i and RD2i are the distribution volume and the relative dispersion of the TTD of the vascular marker across subsystem i. This function has been found most appropriate in fitting the disposition data of the vascular marker in dogs (Weiss et al., 2006). Note that the mean transit time MTTi = VBi /Q, is determined by the vascular volume VBi and cardiac output Q, while the relative dispersion RD2i provides information about the mixing process (Weiss and Pang, 1992).

2.2.3.

The interaction of digoxin molecules in interstitial fluid DIS with unoccupied receptors (sodium pumps) at the extracellular side of the plasma membrane, R, to form a complex, DR, with association and dissociation kinetic rate k∗on and koff , can be described by the following differential equations (Kenakin, 1993): (4)

where Rtot is the total number of receptors. For relatively small occupancies (DR  Rtot ), Eq. (4) simplifies to that of linear, nonsaturable binding: dDR(t) = kon Dis (t) − koff DR(t) dt

where v = VIS,s /VB,s is the ratio of interstitial to blood volume, kon and koff represent the tissue binding and unbinding rate constant, respectively, of the slow binding process with equilibrium amount ratio Kslow = kon /koff . Given the TTD of the vascular marker [Eq. (3)], the TTD of digoxin through the systemic circulation is obtained by substituting Eq. (7) into Eq. (1). Neglecting the contribution of the lung (Seale et al., 2002), the interstitial volume of the systemic circulation VIS,s and the equilibrium binding ratios Kslow and Krapid determine the tissue distribution volume VT,s : VT,s = VIS,s (1 + Kslow + Krapid )

(8)

and the whole body steady-state distribution volume is given by Vss = VB,p + VB,s + VT,s

(9)

Note that the binding process (step response) is characterized by a time constant: bind =

1 kon + koff

(10)

2.2.4.

Minimal circulatory model

The recirculatory model (Fig. 1) is based on the TTD of drug molecules through the systemic and pulmonary circulation, fp (t) and fs (t), and their extraction during one transit through the systemic circulation E described as extraction ratio E = CL/Q. Working in the Laplace domain simplifies model building; applying elementary network theory, the Laplace ˆ transform C(s) of the concentration–time curve C(t) observed as response to an intravenous bolus dose Div is obtained as follows (Weiss et al., 1996, 2006; Weiss et al., 2007): fˆp (s) D ˆ C(s) = iv Q 1 − (1 − E)fˆs (s)fˆp (s)

(11)

(5)

where kon = k∗on Rtot

(s+koff )kin s2 v(1+Krapid )+s(v(koff +Krapid koff +kon )+kin )+vkoff +kin koff (7)

(e.g., Weiss and Kang, 2004).

Tissue binding kinetics

dDR(t) = k∗on [Rtot − DR(t)]Dis (t) − koff DR(t) dt

fˆy (s)=

(6)

is a pseudo-first order rate constant. The distribution and binding of digoxin in the extravascular space of the systemic circulation is determined by the sojourn time density fˆy (s). Assuming that the distribution kinetics in the interstitial space (VIS,s ) is primarily determined by slow tissue binding as described by Eq. (5), fˆy (s) was derived for slow drug binding in the liver (Weiss, 1999; Weiss et al., 2000) and later extended to include quasi-instantaneous unspecific binding to tissue constituents characterized by an equilibrium amount ratio Krapid

Since digoxin is mainly bound in the skeletal muscle, tissue distribution in the lungs can be neglected and fˆp (s) is simply determined by VB,p and RD2B,p of vascular marker [Eq. (3)]. The TTD of digoxin through the systemic circulation [Eq. (1)], in contrast, depends on mixing in the vascular space (VB,s , RD2B,s ) and additionally on the transfer across a permeability barrier (kin = CLperm /VB,s ) as well as on the distribution kinetics in the extravascular space (VIS,s /VB,s , kon , koff , Krapid ) [Eq. (7)]. It is important to note that in deriving the model equation, no well-mixed assumption has been made for the vascular or interstitial space, respectively. We use blood flow (cardiac output) in the circulatory model since digoxin is bound to red blood cells and the total blood to plasma ratio does not differ more than 3% from 1.0.

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(Hinderling, 1984). Due to the relatively fast equilibration and the small contribution to the total amount of binding sites, the kinetics of erythrocyte binding has been neglected (Hinderling and Hartmann, 1991).

tivity function

2.3.

determines the relative change in C(t) caused by a small relative change in the model parameter p. Since Sp is nondimensional, it allows a comparison of results obtained for different parameters. The sensitivity functions [Eq. (14) using Eq. (11)] were calculated using MAPLE 8 after implementing a numerical method of inverse Laplace transformation (Schalla and Weiss, 1999). All simulations were based on average parameter estimates.

Parameter estimation

Since the model of the digoxin disposition curve [Eq. (11)] has been derived in the Laplace domain, one obtains C(t) ˆ by numerical inverse Laplace transformation, C(t) = L−1 [C(s)]. This procedure is implemented in nonlinear regression software SCIENTIST (MicroMath Scientific Software, Salt Lake City, UT) which has been used in fitting the C(t)-data of digoxin with weighting according to 1/C2 . Simulations were also performed using SCIENTIST. We fitted the available digoxin concentration–time data of five subjects (Kramer et al., 1974) in order to estimate five parameters, namely kin , kon , koff , Krapid and E. The calculated or assumed values of the other model parameters Q, VB , VIS , RD2B,p and RD2B,s were held constant in fitting the individual data sets. Since in contrast to Q, VB and VIS no estimates of the vascular mixing parameters RD2B,p and RD2B,s are available in the literature, the ICG disposition data measured in a healthy human volunteer during propranolol infusion by Niemann et al. (2000) were fitted by Eq. (11) [after substituting Eq. (3)] with the result that RD2B,p = 0.32 and RD2B,s = 0.74. For comparison of fitting quality between novel and traditional model, the digoxin data were also fitted by a 3-exponential function. All data are presented as mean ± S.D. Fitting quality was assessed using the model selection criterion (MSC), a modified (inverse) Akaike information criterion (normalized to the number and magnitude of data points) where higher MSC values indicate a better fit. The coefficients of variation of individual parameter estimates (CV) represent the uncertainty in parameter estimates (imprecision). As criteria for evaluating the numerical identifiability of estimates, we used CV < 0.5 and a correlation coefficient threshold of 0.8.

2.4.

p ∂C(t) p −1 L = Sp (t) = C(t) ∂p C(t)

3.



ˆ ∂C(s) ∂p

 (14)

Results

The observed and predicted pharmacokinetic profiles are depicted in Fig. 2 for one of the five subjects as an illustrative example. It is apparent that the novel model perfectly fitted the data. The inset to Fig. 2 shows the shape of the model curve predicted for the first two minutes after injection (within 10 s), i.e., before the first sampling point (at 2.4 min). This time course is concordant with data by Powell et al. (1990) if the longer input time due to injection into an i.v. line is taken into account. Parameter estimates and derived parameters are reported in Table 1 together with the imprecision (CV) of the estimates. Distribution kinetics of digoxin in the systemic circulation was well characterized by transport across the capillary barrier with a permeation clearance CLperm of 9.8 ± 1.3 l/min followed by binding to membrane receptors with association and dissociation rate constants of kon = 0.028 ± 0.003 min−1 and koff = 0.0012 ± 0.0003 min−1 , respectively. These estimates correspond to an equilibration time constant  bind ∼ 34 min. The CVs of all individual parameter estimates were less than 20% and the elements of the correlation matrix less than 0.4. There was no significant difference in “goodness of fit”

Simulations

As a measure of rate of whole body distribution kinetics, the mean equilibration time, MEQT, can be calculated from the moments mn of the systemic TTD [Eq. (1)]: n

∂n fˆs (s) s→0 ∂sn

mn = (−1) lim

(12)

according to Weiss and Pang (1992): MEQT =

1 m

3

3

m2

−

1 m

2

2

m1

(13)

Using Eqs. (12) and (13), the dependency of MEQT on the binding rate constant kon has been simulated with the help of MAPLE 8 (Waterloo Maple Inc., Waterloo, ON). Sensitivity analysis shows on how a model parameter affects the concentration–time profile C(t). Information about a model parameter p may be most accurately gained at time points with a high sensitivity to the parameter p. The sensi-

Fig. 2 – Representative fit of the slow binding model to digoxin disposition data by Kramer et al. (1974) (subject TF, Div = 1 mg). The inset shows the shape of the model curve within the first 2 min after injection.

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Table 1 – Pharmacokinetic parameters of digoxin estimated from the disposition data in healthy volunteers by Kramer et al. (1974) (n = 5) Parameter (units) kin (min−1 ) kon (min−1 ) koff (min−1 ) Krapid E (%)  bind (min) CLperm (l/min) CL (l/min) Vss (l) MEQT (min) a

Mean

S.D.

2.74 0.2 0.0288 0.003 0.0012 0.0004 1.44 0.25 4.33 1.02 33.6 3.3 9.82 1.28 0.216 0.051 605.8 159.0 69.8 7.8

CV (%)a

S.D.

18 9 15 11 0.4 Eq. (10) Eq. (2) = QE Eq. (9) Eqs. (12) and (13)

3 2 5 3 0.2

Asymptotic coefficients of variation of parameter estimates obtained from individual fits (mean and S.D.).

obtained with our slow-binding model compared with the classical 3-compartment model both according to the MSC values (4.2 ± 0.8 versus 4.7 ± 0.6, p > 0.05) and visual inspection. The estimates of parameters that are independent of a specific model of distribution kinetics, Vss and CL, were also not significantly different. In order to demonstrate the effect of an increase in digoxin binding, two disposition curves were simulated, one using the average parameter estimates (Table 1) and another with a 1.5-fold higher kon value (Fig. 3). The increase in kon led to a decrease in C(t) ∼2–4 h after injection. Since the kinetics of drug distribution in the body can be best visualized in a hypothetical noneliminating system, we simulated an average C(t)-curve for E = 0 (Fig. 4). After a very rapid decay within the first 10 min, the concentration in serum declined apparently exponentially with a time constant of ∼50 min and then more slowly towards an equilibrium concentration of C∞ = 1.75 ng/ml. This distributional equilibration process can be quantified by mean equilibration time MEQT, the first

Fig. 4 – Simulated time course of digoxin plasma concentration in a hypothetical system without elimination (E = 0). The equilibrium concentration C∞ = 1.75 ng/ml is indicated as dashed line (Div = 1 mg).

Fig. 5 – Dependency of the mean equilibration time of the distribution process, MEQT, on the binding rate constant kon . The estimated parameters are indicated (䊉). Note that the range of change in kon corresponds to a 0.7–3.5-fold change in receptor capacity.

Fig. 3 – Effect of a 1.5-fold increase in binding capacity on digoxin disposition curve (dashed line) after a bolus dose of 1 mg compared with the control simulated on the basis of average parameter estimates.

normalized moment AUMC*/AUC* of the curve C*(t) = C(t) − C∞ (Weiss and Pang, 1992). Using Eqs. (12) and (13), the value of MEQT = 69.8 ± 7.8 min was calculated for the distribution in the systemic circulation. The dependency of MEQT on the binding rate constant kon is shown in Fig. 5. The respective influence of model parameters RD2B,p , RD2B,s , kin , kon and koff on the digoxin disposition curve is illustrated by their sensitivity functions which give at each time point a measure of the relative change in the curve at a relative change of the model parameters (Fig. 6).

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Fig. 6 – Sensitivity functions Sp (t) describing the predicted relative change in digoxin disposition curve in response to a change in the value of the parameter p.

4.

Discussion

Although modeling of digoxin pharmacokinetics began 40 years ago, a comprehensive model that explains the roles of transcapillary permeation and tissue binding in distribution kinetics is still lacking. Therefore, the goal of this study was to develop a mechanistic pharmacokinetic model that allows the evaluation of slow binding kinetics of digoxin on the basis of drug disposition data in humans. Furthermore, this model provides for the first time a basis for subsequent quantitative prediction of the effect of changes in capacity and/or affinity of digoxin binding to sodium pumps in skeletal muscle on serum concentration during digitalis therapy. The model is an extension of the approach developed in Weiss et al. (2007) to analyse disposition data of inulin and antipyrine, marker compounds that do not bind to tissues. While the goodness of fit and the reliability of parameter estimates (indicated by the reasonably low CVs and correlation coefficients) supports the validity of the model, this is only a necessary but not sufficient step in the validation process. Thus, it is important to point out that our results are supported by those obtained in independent experiments. Given the fact that all ␣-subunit isoforms of the human Na+ /K+ ATPase have similar affinity for cardiac glycosides (Wang et al., 2001), a time constant of the in-vitro binding process of  bind ∼ 30 min is obtained from the results of the binding kinetics of ouabain binding to the dominant ␣1 isoform (Crambert et al., 2004). This value is well in accordance with our estimate. Furthermore, based on a pharmacokinetic/pharmacodynamic analysis it has been suggested that slow binding of digoxin to myocardial Na+ /K+ -ATPase determines the delay of the inotropic effect with respect to plasma concentration (Weiss and Kang, 2004). Interestingly, this delay of 1.3 h is in the same order of magnitude as our estimate of the mean distributional equilibration time, MEQT. Note that this relatively long delay also explains why the high concentration peak immediately after bolus injection (Fig. 2) does not translate into a pharmacological response. That receptor binding dominates

261

the distribution process is obvious from the fact that only 5.8 ± 1.5% of total binding are due to rapid unspecific tissue binding. The simulated digoxin disposition curve (Fig. 4) shows that a 1.5-fold increase in kon [due to an increase in receptor capacity Rtot and/or k∗on , cf. Eq. (6)], leads to a maximal decrease in concentration of ∼40% at ∼3 h after injection. A similar change in digoxin disposition curves was observed during salbutamol infusion in healthy volunteers and was discussed in terms of increased digoxin binding (Edner et al., 1992). Note that an increase in skeletal muscular sodium pump capacity of ∼20–40% has been reported, for example, after glucocorticoid treatment (Nordsborg et al., 2005) or physical training (Evertsen et al., 1997). This suggests that changes in kon due to upregulation or downregulation of the total content of sodium pumps in skeletal muscle under physiological or pathophysiological conditions could be detected with our method. The maximum sensitivity of the digoxin disposition curve with respect to kon after about 3 h (Fig. 6) is in accordance with the simulation results (Fig. 3) and reported data (Edner et al., 1992) discussed above. Furthermore, sensitive analysis indicates that both the intravascular mixing process (RD2B,p and RD2B,s ) and transcapillary transport (kin ) have practically no influence on the disposition curve for sampling times >2 h (Fig. 6). Importantly, these simulations (Figs. 3 and 6) can serve as useful predictive tools for optimization of sampling designs. How slow binding governs whole body distribution kinetics is reflected in Fig. 4 where the C(t)-curve simulated without elimination approaches an equilibrium concentration C∞ = 1.75 ng/ml, that is in accordance with the expected value of C∞ = Div /Vss = 1.65 ng/ml. An analogous distribution curve simulated for the interstitial marker sorbitol shows that distributional equilibration is nearly reached within 20 min (Weiss et al., 1996), in contrast to about 6 h for digoxin (Fig. 4). That the mean distributional equilibration time MEQT is mainly determined by the binding rate constant kon is illustrated in Fig. 5. Since kon is proportional to Rtot [Eq. (6)] this curve also predicts the effect of upregulation or downregulation of Na+ /K+ -ATPase on the rate of digoxin distribution. Note that the rapid decline of the predicted distribution curve within the first 10 min is due to permeation clearance, i.e., mixing of digoxin into the interstitial space. As expected from the time constant 1/(kin + kout ) = 0.31 ± 0.03 min, permeation is very rapid relative to binding and its influence on MEQT can be neglected. For comparison, a rough prediction for CLperm can be obtained by allometric upscaling (e.g., Nestorov, 2003) of the value of 2.4 ml/min g estimated in rat heart (Kang and Weiss, 2002). The predicted value of ∼40 l/min is in the same order of magnitude as our estimate of ∼10 l/min. That tissue binding and not permeation is the rate-limiting process follows also from a recent study of digoxin distribution kinetics in perfused rat hearts (Kang and Weiss, 2002). However, since the dominating Na+ /K+ -ATPase isoform in the rat has a much lower affinity than that in humans, binding occurs much faster. That pharmacokinetics is determined by specific tissue binding with a time constant of about 30 min is a unique feature of cardiac glycosides. For unspecific tissue binding of drugs equilibration times less than one minute are expected according to the results obtained in rat liver (Weiss et al., 2000; Hung et al., 2001) and muscle (Weiss and Roberts, 1996).

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Based on the relatively low occupancy of skeletal muscular sodium pumps of about 13% observed during digoxin therapy (Schmidt et al., 1993), linear binding is regarded as a reasonably valid approximation. Furthermore, dose-independent pharmacokinetics of digoxin was reported in humans after i.v. dosing (Hinderling and Hartmann, 1991) and a proportional increase in skeletal muscle digoxin concentration was observed after doubling of the dose (Jogestrand et al., 1981). As theoretically expected, there was no significant difference between our estimates of Vss and CL and those obtained with the 3-compartmental model (Kramer et al., 1974) or a fit of a 3-exponential function to the data (not shown) since in all these models it is assumed that the elimination rate is proportional to plasma concentration. Under this condition, the parameters Vss and CL are independent of the underlying pharmacokinetic model (e.g., Weiss, 1992). To avoid misunderstandings, it should be also pointed out, that in general monotonically decreasing disposition curves of drugs can be well fitted by a sum of exponentials (Weiss, 1986); but this fact says nothing about the underlying tissue binding kinetics. Thus, it would be fundamentally wrong to argue that a 3-exponential disposition function would imply slow binding. For example, if we fit our model to the disposition data of sorbitol (Weiss et al., 1996), that are also characterized by a 3-exponential disposition function, we obtain a good fit for kon = 0 and Krapid = 0 since sorbitol permeates into interstitial space but does not bind to tissue constituents. Although our approach is physiologically more realistic than classical pharmacokinetic models, it is necessarily also based on simplifying assumptions. First, in order to guarantee identifiability, all organs of the systemic circulation were lumped into one heterogenous subsystem. Concerning parameter estimates of distribution kinetics, this implies that they represent average values across all organs. For digoxin, however, one could assume that skeletal muscle where digoxin primarily binds to sodium pumps, is the main determinant of distribution kinetics. Second, since blood sampling started later than 2 min, practically no information on the initial distribution process, i.e., intravascular mixing (RD2B,s and RD2B,s ) and permeation into the interstitial space (kin ), is available from the data as indicated by the sensitivity analysis (Fig. 5). In compartment modeling this creates no problems since instantaneous mixing in an initial distribution volume of ∼50 l is assumed. As a consequence, however, the central volume and also the other compartmental volumes have no physiological meaning. We think that assuming reasonable values of physiological distribution volumes (VB , VIS ) and rate of vascular mixing (RD2B,s and RD2B,s ) in a more complex but physiologically more realistic model is a better way of dealing with this identifiability problem. Note that the incorporation of preassigned physiological/anatomical parameters into the physiologically based kinetic model to ensure that the number of adjustable parameters can be kept to a minimum has been termed “forward modeling” (Mukkamala and Cohen, 2001). In this sense, the present approach mainly aims to address the heuristic value of the slow binding model. Future prospective studies, however, could be designed as multiple injection studies, where the disposition of indicators and drug are determined in the same individual.

In conclusion, model analysis suggests that slow tissue binding is the rate-limiting step in digoxin distribution kinetics. The novel model may provide a tool to evaluate the effect of disease states, physical exercise, hormones and drugs like ␤2 -adrenoceptor agonists on specific binding of digoxin to skeletal muscle based on clinical pharmacokinetic data. It is hoped that this approach will stimulate others to design studies with concomitant injection of drug and vascular marker when understanding of the distribution process is a primary goal.

Acknowledgement A preliminary account of this work was presented at the 11th Workshop on Advanced Methods of Pharmacokinetic and Pharmacodynamic Systems Analysis, Marina del Rey, CA, June 2005.

references

Crambert, G., Schaer, D., Roy, S., Geering, K., 2004. New molecular determinants controlling the accessibility of ouabain to its binding site in human Na, K-ATPase alpha isoforms. Mol. Pharmacol. 65, 335–341. Edner, M., Jogestrand, T., Dahlqvist, R., 1992. Effect of salbutamol on digoxin pharmacokinetics. Eur. J. Clin. Pharmacol. 42, 197–201. Evertsen, F., Medbo, J.I., Jebens, E., Nicolaysen, K., 1997. Hard training for 5 mo increases Na(+)-K+ pump concentration in skeletal muscle of cross-country skiers. Am. J. Physiol. Regulatory Integr. Comp. Physiol. 272, R1417–R1424. Gheorghiade, M., Adams, K.F., Colucci, W.S., 2004. Digoxin in the management of cardiovascular disorders. Circulation 109, 2959–2964. Harashima, H., Mamiya, M., Yamazaki, M., Sawada, Y., Iga, T., Hanano, M., Sugiyama, Y., 1992. Kinetic modeling of ouabain tissue distribution based on slow and saturable binding to Na, K-ATPase. Pharm. Res. 9, 1607–1611. Hinderling, P.H., 1984. Kinetics of partitioning and binding of digoxin and its analogues in the subcompartments of blood. J. Pharm. Sci. 73, 1042–1053. Hinderling, P.H., Hartmann, D., 1991. Pharmacokinetics of digoxin and main metabolites/derivatives in healthy humans. Ther. Drug Monit. 13, 381–401. Hung, D.Y., Chang, P., Weiss, M., Roberts, M.S., 2001. Structure-hepatic disposition relationships for cationic drugs in isolated perfused rat livers: transmembrane exchange and cytoplasmic binding process. J. Pharmacol. Exp. Ther. 297, 780–789. Jogestrand, T., Ericsson, F., Sundqvist, K., 1981. Skeletal muscle digoxin concentration during digitalization and during withdrawal of digoxin treatment. Eur. J. Clin. Pharmacol. 19, 97–105. Joreteg, T., Jogestrand, T., 1984. Physical exercise and binding of digoxin to skeletal muscle-effect of muscle activation frequency. Eur. J. Clin. Pharmacol. 27, 567–570. Kang, W., Weiss, M., 2002. Digoxin uptake, receptor heterogeneity, and inotropic response in the isolated rat heart: a comprehensive kinetic model. J. Pharmacol. Exp. Ther. 302, 577–583. Kenakin, T., 1993. Pharmacologic Analysis of Drug–Receptor Interaction. Raven Press, New York.

e u r o p e a n j o u r n a l o f p h a r m a c e u t i c a l s c i e n c e s 3 0 ( 2 0 0 7 ) 256–263

Kramer, W.G., Lewis, R.P., Cobb, T.C., Forester Jr., W.F., Visconti, J.A., Wanke, L.A., Boxenbaum, H.G., Reuning, R.H., 1974. Pharmacokinetics of digoxin: comparison of a two- and a three-compartment model in man. J. Pharmacokinet. Biopharm. 2, 299–312. McDonough, A., Wang, J., Farley, R.A., 1995. Significance of sodium pump isoforms in digitalis therapy. J. Mol. Cell. Cardiol. 27, 1001–1009. Mukkamala, R., Cohen, R.J., 2001. A forward model-based validation of cardiovascular system identification. Am. J. Physiol. Heart Circ. Physiol. 281, H2714–H2730. Nestorov, I., 2003. Whole Body Pharmacokinetic Models. Clin. Pharmacokin. 42, 883–908. Niemann, C.U., Henthorn, T.K., Krejcie, T.C., Shanks, C.A., Enders-Klein, C., Avram, M.J., 2000. Indocyanine green kinetics characterize blood volume and flow distribution and their alteration by propranolol. Clin. Pharmacol. Ther. 67, 342–350. Nordsborg, N., Goodmann, C., McKenna, M.J., Bangsbo, J., 2005. Dexamethasone up-regulates skeletal muscle maximal Na+ , K+ pump activity by muscle group specific mechanisms in humans. J. Physiol. 567, 583–589. Powell, C., Horowitz, J.D., Hasin, Y., Syrjanen, M.L., Horomidis, S., Louis, W.J., 1990. Acute myocardial uptake of digoxin in humans: correlation with hemodynamic and electrocardiographic effects. J. Am. Coll. Cardiol. 15, 1238–1247. Prescott, L.F., McAuslane, J.A., Freestone, S., 1991. The concentration-dependent disposition and kinetics of inulin. Eur. J. Clin. Pharmacol. 40, 619–624. Schalla, M., Weiss, M., 1999. Pharmacokinetic curve fitting using numerical inverse Laplace transformation. Eur. J. Pharm. Sci. 7, 305–309. Schmidt, T.A., Holm-Nielsen, P., Kjeldsen, K., 1993. Human skeletal muscle digitalis glycoside receptors (Na, K-ATPase)-importance during digitalization. Cardiovasc. Drugs Ther. 7, 175–181. Seale, K.T., Pou, N.A., Krivitski, N., Harris, T.R., 2002. Quantification of lung microvascular injury with ultrasound. Ann. Biomed. Eng. 30, 671–682.

263

Wang, J., Velotta, J.B., McDonough, A.A., Farley, R.A., 2001. All human Na(+)-K(+)-ATPase alpha-subunit isoforms have a similar affinity for cardiac glycosides. Am. J. Physiol. Cell. Physiol. 281, C1336–C1343. Weiss, M., 1986. Generalizations in linear pharmacokinetics using properties of certain classes of residence time distributions. I. Log-convex drug disposition curves. J. Pharmacokinet. Biopharm. 14, 635–657. Weiss, M., 1992. The relevance of residence time theory to pharmacokinetics. Eur. J. Clin. Pharmacol. 43, 571–579. Weiss, M., 1998. Pharmacokinetics in organs and the intact body: model validation and reduction. Eur. J. Pharm. Sci. 7, 119–127. Weiss, M., 1999. Cellular pharmacokinetics: effects of cytoplasmic diffusion and binding on organ transit time distribution. J. Pharmacokin. Biopharm. 27, 233–255. ¨ ¨ Weiss, M., Hubner, G.H., Hubner, G.I., Teichmann, W., 1996. Effects of cardiac output on disposition kinetics of sorbitol: recirculatory modelling. Br. J. Clin. Pharmacol. 41, 261–268. Weiss, M., Kang, W., 2004. Inotropic effect of digoxin in humans: mechanistic pharmacokinetic/pharmacodynamic model based on slow receptor binding. Pharm. Res. 21, 231–236. Weiss, M., Krejcie, T.C., Avram, M.J., 2006. Transit time dispersion in the pulmonary and systemic circulation: effects of cardiac output and solute diffusivity. Am. J. Physiol. Heart Circ. Physiol. 291, H861–H870. Weiss, M., Krejcie, T.C., Avram, M.J., 2007. Circulatory transport and capillary-tissue exchange as determinants of the distribution kinetics of inulin and antipyrine in dog. J. Pharm. Sci, in press. Weiss, M., Kuhlmann, O., Hung, D.Y., Roberts, M.S., 2000. Cytoplasmic binding and disposition kinetics of diclofenac in the isolated perfused rat liver. Br. J. Pharmacol. 130, 1331–1338. Weiss, M., Pang, K.S., 1992. Dynamics of drug distribution. I. Role of the second and third curve moments. J. Pharmacokinet. Biopharm. 20, 253–278. Weiss, M., Roberts, M.S., 1996. Tissue distribution kinetics as determinant of transit time dispersion of drugs in organs: application of a stochastic model to the rat hindlimb. J. Pharmacokin. Biopharm. 24, 173–196.