Distribution Clearance: Significance and Underlying Mechanisms

Purpose Evaluation of distribution kinetics is a neglected aspect of pharmacokinetics. This study examines the utility of the model-independent parameter whole body distribution clearance (CLD) in this respect. Methods Since mammillary compartmental models are widely used, CLD was calculated in terms of parameters of this model for 15 drugs. The underlying distribution processes were explored by assessment of relationships to pharmacokinetic parameters and covariates. Results The model-independence of the definition of the parameter CLD allowed a comparison of distributional properties of different drugs and provided physiological insight. Significant changes in CLD were observed as a result of drug-drug interactions, transporter polymorphisms and a diseased state. Conclusion Total distribution clearance CLD is a useful parameter to evaluate distribution kinetics of drugs. Its estimation as an adjunct to the model-independent parameters clearance and steady-state volume of distribution is advocated.


Introduction
Total distribution clearance (CL D ) is defined as measure of distribution kinetics of drugs in the body that is independent of a specific structural model.In order to segregate the distribution from the elimination process, it is based on the area under the curve in a hypothetical noneliminating system [1].Due to this clear interpretability, the parameter distribution clearance can be estimated from drug disposition data using compartmental models, physiologically-based pharmacokinetic (PBPK) models or circulatory models (based on transit time densities) [2], and therefore it allows a systematic comparison of drugs with respect to their distributional properties.Together with elimination clearance CL D quantifies the extent of deviation of drug disposition curves from monoexponential disposition.
Here the distribution clearances of 15 drugs were calculated and the underlying mechanisms were investigated.The results of this review show that additional useful information and physiological insight can be gained by estimating CL D .That the evaluation of distribution kinetics, ie of the rate of drug distribution, is a neglected feature of pharmacokinetics was already pointed out by Atkinson [3].The objective for the present analyses was to demonstrate the utility of the parameter CL D for this purpose.

Noneliminating System
The parameter distribution clearance is determined by the area under concentration-time curve (AUC D ) in a hypothetical noneliminating (closed) system (CL = 0), that means the area between C D (t) after bolus injection of dose D iv and the concentration reached at steady state.
C SS = D iv ∕V SS (Fig. 1) [1]: (1) The C D (t) curve (Fig. 1) reflects what "kinetics" means in a physical context: the tendency of a system to reach a state of equilibrium.
The measure CL D can be calculated from the parameters of different pharmacokinetics models, as mammillary compartment models, sum of exponentials, PBPK models and recirculatory models [2].Since compartmental models (Eq.6) or sums of exponentials (Eq.8) are commonly used in practice for analyzing drug disposition data, these models were applied in the present study to calculate CL D , despite their failure to describe the initial distribution (see Limitations below).Note that Eqs. 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 and 16 were moved to Appendix in order to preserve the readability of the text.
The parameter CL D was originally defined by Eq. 1 in terms of a recirculatory model.Later it became clear that a definition based on mass transfer out of the initial distribution volume V 0 (where the drug distributes instantaneously at t = 0) may be more appropriate from a physical point of view.The distribution process can be described by where A 0 (t) = V 0 C D (0) is the amount of drug in V 0 = D iv ∕C D (0) .After integrating both sides of Eq. 2, we obtain CL D as ie CL D,corr = 1 − V 0 ∕V SS CL D .Since in all previous publi- cations CL D (also called CL M ) was calculated by Eq. 1, the term CL D,corr will be used for the definition by Eq. 3. (2) To this end, based on the individual parameters of compartment models V 0 , V 1 , V 2 , CL 02 , CL 02 and sum of expo- nentials B i , i , i = 1..3 the distribution clearances were calculated using Eq.6 and Eq. 8, respectively.Apart from parameters estimated in our own studies, parameters from the literature were employed.The selection of drugs was somewhat arbitrary since only those publications could be considered where the individual parameter estimates were available.Furthermore, only results obtained with a threecompartment or a triexponential model were selected.Here we report the means and coefficients of variations of the CL D values of the drugs.Note that in contrast to the sum intercompartmental clearances CL 0i between central volume V 0 and volumes V i , ∑ n i=1 CL 0i (eg Ref. [4]), which character- izes only the initial distribution process at time t = 0 [5,6], CL D describes the overall distribution behavior in the body.
Linear regression analysis with pharmacokinetic parameters and available covariates were performed to reveal information on which factors determine the distribution process.For comparison with the conventionally used empirical measure CL D , the physically more realistic parameter CL D,corr (Eq. 3) as also reported.

Eliminating System
To separate distribution and elimination process, CL D was defined above by setting the elimination clearance, CL = 0 .In the real eliminating (open) system CL D determines the departure from the one-compartment behavior (monoexponential decay of the disposition curve).A measure of this deviation is given by [7] where RD 2 D denotes the relative dispersion (normalized variance) of the disposition residence time distribution (Eq.11).For an instantaneous distribution in the body, ie CL D → ∞ , one gets RD 2 D − 1 → 0 .The role of the measure (Eq.4) char- acterizing the effect of the distribution kinetics can be best demonstrated by defining a time-varying volume of distribution according to where A(t) and C(t) are the drug amount in the body and plasma concentration after bolus injection.V(t) was proposed by Niazi [8], who calculated its time course for a multiexponential drug disposition curve.As an example Fig. 2 shows the time courses of V(t) calculated from the parameters published for chlormethiazole in healthy volunteers and patients with cirrhosis of the liver [9], that are characterized by 2CL/CL D values of 1.0 and 4.6, respectively.The time Fig. 1 The area AUC D between the concentration-time curve and the steady state concentration C ss in a noneliminating system The curve was simulated using the population mean parameter estimates for rocuronium [2] setting CL = 0 courses of V(t) clearly shows the effect of slow and rapid distribution (CL D = 0.8 and 3.08).(The origin of the difference in CL D will be discussed below.)The increase in V(t) reflects the evolution towards a thermodynamic equilibrium state in the distribution process where the equilibrium volume V z is reached asymptotically.However, in contrast to the distribution process in the closed system (Fig. 1) where the static equilibrium is characterized by V ss (C ss = const.),V Z denotes a dynamic equilibrium (A(t)/C(t) = const.).The ratio V Z /V ss increases in parallel to RD 2 D and generally the following relationship holds [7]: with equality in the limiting case of instantaneous distribution at t = 0.
Since a one-compartment disposition model (Bateman function) is often used for oral data, one can expect that in this case a better fit is obtained for drugs with a low 2CL/ CL D value (especially if the absorption rate is relatively high).

Results and Discussion
The results are summarized in Table I where the means and coefficients of variation of CL D as well as of CL D,corr are reported together with the measure 2CL/C D for the eliminating system.Furthermore, the results of linear regression analysis are presented indicating the relationship to pharmacokinetic parameters and contribution of covariates to Note that these correlations are of empirical nature.Particularly striking is the difference between the CL D values of 0.19 L/min for gadoxetate and of 6.64 L/min for propranolol, which is connected with the highest and lowest deviation from a monoexponential disposition curve, respectively (with 2CL/CL D values of 15.9 and 0.5).Significant changes in CL D were observed under certain conditions for gadox- etate [13], talinolol [14], thiopental [16] and chlormethiazole [9].
In order to understand the differences between the estimated CL D values of the 15 drugs in Table I, we have to identify potential influencing factors of the distribution process.

Capillary Permeability
Generally mass transfer out of the vascular space is limited by a permeability barrier.Thus apart from the unbound fraction (f u ) discussed below, CL D is determined by the permeability-surface area product (PS) and blood flow.The prediction obtained from the recirculation model (organs of the systemic circulation lumped in a single subsystem, see Eqs. 13 and 14) was visualized for alfentanil by a response surface plot in order to illustrate the effect of tissue permeability (f u PS) and cardiac output (Q) on CL D (Fig. 3).Note that in this simplest model of the systemic circulation the parameters, V B ,V T and f u PS are the apparent parameters for the systemic circulation averaged over all organs.Figure 3 shows that CL D is mainly determined by f u PS, and it becomes nearly independent of Q for low values of f u PS.Although the increase of CL D with Q observed for alfentanil (Table I), is in principal accordance with the prediction of the simplified recirculation model (assuming RD 2 B = 3 [23]), the present compartmental approach does not properly describe the flow dependency, since the effect of initial distribution is neglected (see discussion below).Note that based on estimating the relative dispersion of the circulatory transit time distribution RD 2  C with a circulatory model, inulin and antipyrine were characterized by barrier-limited and perfusion-limited distribution, respectively [23].Thus the distribution of alfentanil presents an intermediate situation between these two extremes.
Since plasma protein binding controls the free drug concentration in plasma, and only unbound drug molecules can cross the membrane barrier, it is a main determinant of distribution clearance.This is demonstrated by the significant increase of distribution clearance of chlormethiazole in patients with cirrhosis (from 0.82 to 3.05 L/min) (Table I).This may be mainly attributed to the increase in the free fraction f u as result of the decrease in serum albumin.First, f u was statistically significantly increased in patients with cirrhosis by 33% [9]; and second, CL D was negatively correlated with the serum albumin level (R = 0.66, p < 0.05),  [22] which is important in the light of the significant correlation of serum level of albumin and percentage chlormethiazole bound to plasma proteins [9].The effect of the higher free fraction of drug in cirrhotic patients on the time course of V(t), ie the more rapid distribution, is shown in Fig. 2.
One may speculate that the significant positive correlation between CL D and V ss or CL found for some of the drugs (trospium, propiverine, R-ketamine and vancomycin) could be explained by inter-individual differences in the serum albumin levels, since both CL D and V ss increase with the fraction unbound.An effect of free fraction on inter-subject variability in distribution was also suggested by Upton et al. [24] (due to variability in albumin concentration and/or binding affinity).For alfentanil, for example, an inter-individiual variability in f u of 42% was observed [25].More obvious is the role of total body weight (TBW) in this respect.For vancomycin, a significant correlation was found between between CL D and TBW [21].In this case the positive correlations between CL D and V ss as well as between CL D and CL, can be explained by significant correlations between V ss and TBW (R = 0.89, p < 0.01) and between CL and TBW (R = 0.94, p < 0.01).That also for alfentanil [4] CL D correlates with TBW indicates that part of the inter-individual variability in CL D is dues to the variability in TBW.
Interestingly, no correlation between CL D and lipophilicity (log P) was observed.This may be due to the role of plasma protein binding as a cofounding factor.If the degree of binding is similar as for thiopental and fentanyl (about 70%) [26], the higher lipophilicty of fentanyl (Log P: 3.89) compared to thiopental (Log P: 1.85) [27] explains higher CL D (3.95 vs. 0.90 l/min).That the CL D of alfentanil (1.21 l/ min) is lower than that of fentanyl, on the other hand, may be caused by both a higher protein binding (about 90% [26]) and a lower lipophilicity (log P: 2.16) [27].Note also that the ratio of CL D of alfentanil to that of fentanyl (0.31) is quite similar to the ratio of uptake clearances of alfentanil and fentanyl measured in rat muscle (0.44) [28].

Cardiac Output
For rocuronium and alfentanil where an independent estimate of cardiac output was available, a significant correlation between CL D and Q was found (Table I).However, in these cases CL D increases only moderately with Q (with slopes between 0.11 for rocuronium and 0.26 for alfentanil).However, even for highly permeable drugs, the distribution out of the vascular space occurs not instantaneously and flow-limited distribution RD 2 C → RD 2 B for f u PS → ∞ , Eq. 14) is a theoretical limiting case for which CL D cannot be defined.Interestingly, the CL D /Q ratio of 0.25 predicted by Eqns.13 and 14 for alfentanil (Q = 6 l/min, f u PS = 1.4 l/ min) is quite similar to the ratio of 0.2 calculated for thiopental [29].That the CL D of thiopental of 0.902 L/min was significantly decreased by 40% in patients treated with the α 2 -adrenoceptor agonist dexmedetomidine may be attributed to a decrease and redistribution of cardiac output [14].This is similar to the result calculated form the parameters reported in a porcine model of hemorrhagic shock [30], where the reduction in CL D of fentanyl from 2.99 ± 0.97 L/ min to 1.91 ± 0.82 L/min in the shock group was likely due to a redistribution of cardiac output accompanied by the decrease in cardiac index by 43%.
In discussion the hemodynamic influences on CL D using Eq. 13, we have to consider the neglected the effect of regional distribution of cardiac output (organ blood flows) and the heterogeneity of distributional properties of organs, which affect the relative dispersion of circulation times,RD 2 C (Eq. 14) and therefore CL D (Eq.13).This means in applying Eq. 14 a compromise was made between necessary simplifications and the aim to reveal the main influencing factors of CL D .At this point the relationship to PBPK models (eg Refs.[28,31,32]) should be discussed.Assuming a multi-organ model where the i organs/tissues of the systemic circulation (with blood flows Q i ) are described by well-mixed vascular and tissue compartments( distribution volumes V Bi and V Ti ), separated by a permeability barrier (f u PS i ), CL D can be easily defined in terms of these parameters (in analogy to the case of well-mixed organs (Eq.15)) [33].But although the distribution clearance could be more accurately predicted using a PBPK model, this approach is not practicable in the present context due to the large number of unknown parameters.Note also that the RD 2  C predicted for an intravascular marker (Eq.15) shows that not Q per se but a redistribution of blood flow affects RD 2 C .A feasible approach, however, could be the reduction of the systemic circulation into two heterogeneous subsystems (Eq.17 with n = 2).For the lipophilic drug thiopental, for example, a splitting into fat and non-fat tissues improved the fit and allowed a prediction of CL D as a function of percentage body fat in obese subjects [29].The predicted CL D /Q ratio of 0.2 in lean subjects is in accordance with the estimate reported in Table I.

Active Transport
Of particular note is the fact that the distribution clearance of gadoxetate is about one magnitude lower than those of the other compounds (Table I).It takes almost 40 h until distributional equilibrium is achieved (Fig. 3 in Ref. [13]) compared to about 1 h for rocuronium (Fig. 1).The reason is that gadoxetate leaves the vascular space primarily by uptake into hepatocytes whereas extravascular permeation is relatively low in comparison to the other compounds [13].Gadoxetate is transported into hepatocytes by the organic anion transporting polypeptide (OATP) 1B1, OATP1B3, and backflux into the sinusoids occurs via the transport protein MRP3 and/or bidirectional-acting OATPs.Since the efflux from hepatocytes is slower than the uptake, gadoxetate accumulates in the liver.The finding that CL D of gadoxetate in carriers of the variant OATP1B1*15/*15 was significantly smaller than in carriers of the wildtype protein, OATP1B1*1a/*1a could be explained by an increased sinusoidal efflux rate in subjects with the variant *15/*15 protein compared with the wild type *1a/*1a.This corresponds to the linear increase in CL D with the ratio of hepatic uptake to efflux constant as well as with the tissue-to-plasma transport constant estimated with the PBPK model.The fuPS value of gadoxetate was ten-fold lower than that of rocuronium [10].
Another example where active transport may play a role in distribution kinetics is talinolol, where CL D increased significantly under rifampicin treatment (Table I).Since talinolol is a substrate of the efflux transporter P-glycoprotein (P-gp), this suggests that rifampicin-mediated P-gp induction leads to an increase in CL D .Furthermore, this increase correlated with the increase in elimination clearance (R = 0.79, p < 0.05), which has been attributed to an increase of Pgp mediated intestinal secretion [34,35].Although no clear explanation is available for the increase in CL D as a consequence of rifampicin-mediated P-gp induction, it could be due to an intestinal reabsorption of talinolol.

Slow Tissue Binding
An interesting special case is the distribution kinetics of digoxin.The high distribution volume of digoxin of about 600 l is mainly determined by binding to skeletal muscular Na + /K + -ATPase (sodium pumps) at the extracellular side of the plasma membrane [36].Digoxin permeates into the interstitial space (with negligible cellular uptake), but although the distribution is not diffusion limited with a fuPS value that is about 20-fold higher than that of rocuronium [10], their CL D values are quite similar.This unexpected low distribution clearance of digoxin is primarily determined by slow tissue binding (mainly to skeletal muscle), ie binding of digoxin molecules to receptors (sodium pumps, Na + / K + -ATPase) at the extracellular side of the plasma membrane.Pharmacokinetic modeling predicts a time constant of 34 min for binding equilibration and suggests that a ~ 1.5 fold increase in digoxin binding leads to a ~ 20% increase in CL D [22].This corresponds to the finding that stimulation of the Na + /K + -pump in skeletal muscle by the β 2 -agonist salbutamol affects distribution kinetics of digoxin in a similar fashion [37].

Limitations
In this study, as mostly in pharmacokinetics, drug disposition curves are regarded as decreasing (ie RD 2 D > 1 in Eq. 11 and RD 2 C > 1 in Eq. 13).The implicit assumption of an instantaneous distribution into an initial distribution volume V 0 (volume of the central compartment, see also Eq. 10) is a limitation in the estimation and interpretation of CL D values.Assuming that initial distribution is already complete (C(0) = D iv /V 0 ), rapid distribution within the first minutes after bolus injection is not covered by the present approach.Thus it cannot be applied to highly permeable drugs like antipyrine with flow-limited distribution kinetics where CL D is mainly determined by the initial distribution phase.One approach to deal with initial distribution is recirculatory modeling based on frequent early arterial blood sampling (within the first 3 min) and the simultaneous injection of an intravascular marker distribution (like indocyanine green, ICG).That the estimate of RD 2 C for antipyrine is then only slightly higher than that for ICG (characterizing intravascular mixing) [23], indicates that antipyrine distributes in the whole body similarly fast like ICG in the vascular space.Note that the circulatory mixing time of an intravascular indicator increases with RD 2  C and decreases with Q [38].The assessment of early drug distribution is of special importance for intravenously administered anesthetic drugs [39] and circulatory models have a high heuristic value in explaining the effect of drug interactions and hemodynamic changes (eg [5,23,40]).
It should be further noted that also in the present case of decreasing C(t) curves (compartmental modeling), the estimated V 0 may depend on the early sampling schedule.If sampling starts relatively late or if a 2-compartment model is applied, unrealistically high V 0 values may be obtained.In this case (when V 0 is not much smaller than V ss ), CL D values higher than cardiac output can be obtained, while theoretically CL D ≤ Q (Eq.13).This was not the case in the present examples where a 3-compartment model or a sum of three exponentials was applied, and it would not appear if distribution kinetics is assessed by CL D,corr .Generally, however, the CL D,corr values were not much different from those of CL D ; with the exception of drugs like propiverine with a ratio of V 0 /V ss of about 0.5.
Thus, although the definition of CL D is independent of a specific structural model, the limitations inherent in its estimation and explanation (eg in terms of a mechanistic model) should be always taken into account.Both are dependent on the underlying assumptions.Finally note also that our model is based on system linearity (as assumed in the underlying studies).

Summary
The results demonstrate that the distribution clearance CL D is a useful measure of distribution kinetics of drugs.It enables a comparison of the distributional properties of different drugs and reflects the effect of drug-drug interactions, transporter polymorphisms and diseased states.The estimation of CL D offers physiological insights due to its interpretability by mechanistic models.CL D could serve as a sensitive parameter to characterize intersubject variability of distribution kinetics and analysis of covariates may reveal influencing factors.The physiological interpretability of CL D makes this parameter useful in drug development.It is hoped that this study will stimulate the estimation of CL D as an adjunct to the other modelindependent pharmacokinetic parameters CL and V ss .For a comprehensive review of mechanisms of drug distribution and the relationship between the drug distribution and pharmacologic response, see eg Ref. [41].

Mammillary Compartment Model
The distribution clearance for a mammillary compartment model with peripheral compartment volumes V i and intercompartmental clearances CL 0i (between central volume V 0 and volumes V i ) is given by [6]:

Sum of Exponentials
If the disposition data were fitted by a sum of exponentials we have [1] where and the initial distribution volume is given by

Relative Dispersion of Disposition Residence Time Distribution
As proved in Ref. [7], the relative dispersion (normalized variance) of disposition residence time distribution, RD 2 D , determines the deviation from the one-compartment behavior or monoexponential disposition curve (ie instantaneous distribution in the body) where RD 2 D = 1 and the deviation is given by RD 2 D − 1 , ie for decreasing drug dis- position curves RD 2 D > 1 holds.Furthermore one can show that [1] Thus in terms of CL D and CL, a measure for deviation from the one-compartment behavior is given by.
The whole body distribution clearance CL D should not confused with the organ distribution clearance defined in physiological base pharmacokinetic (PBPK) models [28,42].
2CL CL D

Recirculatory model
For a recirculatory model with cardiac output Q and relative dispersion of the circulatory transit time distribution RD 2 C one gets [1] Applying an equation that has been derived for the relative dispersion of transit times across a single organ [43] for the systemic circulation, the systemic transit time dispersion is determined by intravascular mixing RD 2 B , ie the relative dispersion of circulatory transit time distribution of the intravascular marker and a second term describing tissue distribution kinetics with relative dispersion of tissue residence times RD 2 T [43].Since all systemic organs were lumped into one subsystem, V B ,V T and f u PS are the apparent parameters for the systemic circulation.Although intratissue distribution occurs in reality not instantaneously ( RD 2 T > 1), a well mixed extravascular space ( RD 2 T = 1) can be assumed for simplicity.Note that RD 2  C > 1 follows the condition RD 2 D > 1 (Eqs.11 and 13).Although Eq. 14 represents a gross oversimplification of the real multi-organ structure, it may be helpful in explaining the main factors determining CL D , intravascular mixing and blood-tissue exchange.Note that using circulatory models, RD 2  C can be estimated directly and Eq. 14 is only used to interpret the result [23].
To understand the role of RD 2 B in defining RD 2 C , it can be derived using a minimal PBPK model; with n organs of the systemic circulation arranged in parallel and assuming the organs/tissues as well-mixed subsystems (where RD 2 C = RD 2 B ) one obtains [33] Equation 15 shows that RD 2 B is dependent on flow heterogeneity in the systemic circulation.Since Eq. 15 gives RD 2 B =1 for a well-mixed whole body system (n = 1), CL D → ∞ is predicted by Eq. 13 for RD 2  C → 1 (f u PS → ∞) in accordance with the limiting case RD 2 D → 1 (monoexponential disposition, Eq. 11).That a two-compartment recirculatory pharmacokinetic model [2,44] is not capable of predicting the dependence of CL D on Q becomes clear from Eqs. 13 and 14 for RD 2 B = 1 Which, as expected is structural identical to CL D obtained for the two-compartment model (Eq.6) for f u PS = CL 01 , V B = V 0 and V T = V 1 .
Although Eq. 14 can be easily extended to n organs in parallel [38] where RD 2  Ci is given by Eq. 14 with RD 2 Bi V B,i , V T,i and f u PS i , the application of Eq. 17 is not feasible in the present context as discussed above for the PBPK model.

Fig. 3
Fig. 3 Permeability-surface area product (f u PS) and cardiac output (Q) as determinants of distribution clearance (CL D ).The plot was calculated by Eqs. 13 and 14 assuming RD 2 B = 3, V B = 5 l and V T = 25 l.The volume parameters are those of alfentanil[4]