## Visual Overview

## Abstract

Drug distribution is a necessary component of models to predict human pharmacokinetics. A new membrane-based tissue-plasma partition coefficient (*K*_{p}) method (*K*_{p,mem}) to predict unbound tissue to plasma partition coefficients (*K*_{pu}) was developed using in vitro membrane partitioning [fraction unbound in microsomes (*f*_{um})], plasma protein binding, and log *P*. The resulting *K*_{p} values were used in a physiologically based pharmacokinetic (PBPK) model to predict the steady-state volume of distribution (*V*_{ss}) and concentration-time (C-t) profiles for 19 drugs. These results were compared with *K*_{p} predictions using a standard method [the differential phospholipid *K*_{p} prediction method (*K*_{p,dPL})], which differentiates between acidic and neutral phospholipids. The *K*_{p,mem} method was parameterized using published rat *K*_{pu} data and tissue lipid composition. The *K*_{pu} values were well predicted with *R*^{2} = 0.8. When used in a PBPK model, the *V*_{ss} predictions were within 2-fold error for 12 of 19 drugs for *K*_{p,mem} versus 11 of 19 for** **K_{p,dPL}. With one outlier removed for *K*_{p,mem} and two for *K*_{p,dPL}, the *V*_{ss} predictions for *R*^{2} were 0.80 and 0.79 for the *K*_{p,mem} and *K*_{p,dPL} methods, respectively. The C-t profiles were also predicted and compared. Overall, the *K*_{p,mem} method predicted the *V*_{ss} and C-t profiles equally or better than the *K*_{p,dPL} method. An advantage of using *f*_{um} to parameterize membrane partitioning is that *f*_{um} data are used for clearance prediction and are, therefore, generated early in the discovery/development process. Also, the method provides a mechanistically sound basis for membrane partitioning and permeability for further improving PBPK models.

**SIGNIFICANCE STATEMENT** A new method to predict tissue-plasma partition coefficients was developed. The method provides a more mechanistic basis to model membrane partitioning.

## Introduction

The volume of distribution and clearance equally determine the half-life of a drug. The steady-state volume of distribution (*V*_{ss}) can be predicted using empirical methods (Obach et al., 1997), computational approaches (Ghafourian et al., 2004; Lombardo et al., 2006; Zhivkova and Doytchinova, 2012), physiologic equations (Oie and Tozer, 1979; Lombardo et al., 2004; Korzekwa and Nagar, 2017a), and tissue:plasma partition coefficients (*K*_{p}). The *K*_{p} prediction methods are widely used since they describe the distribution in physiologically based pharmacokinetic (PBPK) models. While some methods require an in vivo component (Arundel, 1997; Björkman, 2002; Jansson et al., 2008; Poulin and Theil, 2009), others use more readily available in vitro inputs.

Several factors influence drug distribution, including partitioning into membranes and other lipids, binding to proteins (primarily plasma proteins), pH partitioning (e.g., lysosomes), transporters, and membrane permeability. Most models represent tissue interactions with in vitro surrogates. The Poulin and Krishnan (1995) model originally described phospholipid partitioning with the octanol:water partition coefficient (*P*) and assumed phospholipid composition to be represented by 30% octanol and 70% water. They developed *K*_{p} prediction equations that included an additional surrogate for neutral lipid partitioning in adipose tissue (Poulin and Theil, 2000; Poulin et al., 2001), which was modified by Berezhkovskiy (2004). Rodgers and Rowland (2006) developed two equations for prediction of unbound *K*_{p} (*K*_{pu}): one for acids, neutrals, and weak bases, and another for moderate-to-strong bases (Rodgers et al., 2005). Drug partitioning into erythrocytes was used to parameterize the interaction of bases with acidic phospholipids (APs). It was assumed that ionized bases interact only with APs, while uncharged molecules interact only with neutral phospholipids (Rodgers et al., 2005; Rodgers and Rowland, 2006).

In most currently used composition-based models, log *P* is used to model the phospholipid partitioning (0.3*P*). A shortcoming of using log *P* to represent phospholipid partitioning is the lack of orientation-specific interactions with phospholipid membranes (Balaz, 2009; Nagar and Korzekwa, 2012, 2017). Additionally, both neutral and ionized bases are known to interact with all phospholipids, and not just net-neutral and net-acidic phospholipids, respectively. Therefore, current methods to calculate *K*_{p} appear to be based on mechanistically unsound assumptions. Previously, we used microsomal partitioning [fraction unbound in microsomes (*f*_{um})] instead of log *P* to parameterize phospholipid partitioning in a *V*_{ss} model (Korzekwa and Nagar, 2017a). Partitioning into microsomes (unsorted phospholipid vesicles) is used extensively in clearance predictions and can be determined experimentally or predicted (Austin et al., 2002; Hallifax and Houston, 2006; Poulin and Haddad, 2011; Nagar and Korzekwa, 2017). A benefit of using *f*_{um} to represent phospholipid partitioning is that it measures interactions with all phospholipids for both charged and uncharged species.

Previous studies have compared different *K*_{p} prediction methods and their ability to predict both tissue *K*_{p} and/or *V*_{ss} (De Buck et al., 2007; Poulin and Theil, 2009; Jones et al., 2011; Graham et al., 2012; Zou et al., 2012; Chan et al., 2018). These studies came to different conclusions on the most accurate *K*_{p} model, which was primarily dependent on the drug data set used (De Buck et al., 2007; Graham et al., 2012). Graham et al. (2012) showed that the Rodgers et al. (2005) method was able to better predict *K*_{p} and *V*_{ss} for different classes of drugs than other composition-based models. The Poulin and Theil (2009) method led to good *V*_{ss} predictions, but required in vivo data (*K*_{pu} muscle) (Graham et al., 2012). More recently, Chan et al. (2018) compared the ability of composition-based *K*_{p} models and preclinical extrapolation to predict *V*_{ss}. Composition-based models predicted *V*_{ss} with accuracy similar to preclinical extrapolation. They noted that the Rodgers method was able to predict *V*_{ss} well for drugs with log *P* values less than 3, and that many drugs with large errors in *V*_{ss} for composition-based models also had errors in preclinical extrapolation (Chan et al., 2018).

This report evaluates a model to predict *K*_{p} using *f*_{um} to represent membrane partitioning. Plasma protein binding and microsomal partitioning values were determined experimentally for 19 drugs. Tissue *K*_{p} values were calculated for each compound using a differential phospholipid *K*_{p} prediction method (*K*_{p,dPL)} (Rodgers et al., 2005; Rodgers and Rowland, 2006), as well as a method that uses *f*_{um} to parameterize membrane partitioning [the membrane-based *K*_{p} prediction method (*K*_{p,mem})]. Simulations were run and their ability to predict *V*_{ss} and concentration-time (C-t) profiles was determined. More mechanistically sound assumptions for *K*_{pu} will be required when expanding current perfusion-limited PBPK models to include explicit membrane partitioning and permeability. Models to predict *K*_{pu} bases on experimental partitioning into membranes may allow a facile transition to models that limit permeability with explicit membrane compartments (Nagar et al., 2014).

## Materials and Methods

#### Materials.

A Harvard Apparatus (Holliston, MA) 96-well equilibrium dialyzer and single-plate Harvard Apparatus plate rotator were used for equilibrium dialysis experiments. Human plasma was obtained from US Biologic (Salem, MA) and Innovative Research Inc. (Novi, MI). Rat liver microsomes were obtained from BD Biosciences (San Jose, CA) and Corning Life Sciences (Tewksbury, MA). Warfarin, fluconazole, glyburide (glybenclamide), ketoprofen, fenofibrate, +/*cis*-diltiazem hydrochloride, +/− verapamil hydrochloride, caffeine, betaxolol hydrochloride, DMSO, nicardipine hydrochloride, metoprolol tartrate, felodipine, and nafcillin sodium were obtained from Sigma Aldrich (St. Louis, MO). Quinidine gluconate, formic acid, acetonitrile, and diphenhydramine hydrochloride were obtained from Fisher Scientific (Norristown, PA). Mibefradil hydrochloride was obtained from Cayman Chemical Company (Ann Arbor, MI). Diclofenac sodium was obtained from Calbiochem (Burlington, MA). Fenofibric acid was obtained from Kano Laboratories (Nashville, TN). One milligram per milliliter solutions of phenytoin, diazepam, and midazolam in methanol were obtained from Cerilliant (a Sigma Aldrich company). The 100 mM PBS and 0.3 mM MgCl_{2} dialysis buffer was composed of magnesium hydrochloride hexahydrate (Fisher Scientific), potassium phosphate monobasic (Sigma Aldrich), and potassium phosphate dibasic (Fisher Scientific). An Agilent 1100 HPLC and API 4000 mass spectrometer and Agilent 1100 HPLC and API 4000 Q-Trip mass spectrometer were used to determine the concentrations for equilibrium dialysis. Mathematica version 11.0 (Wolfram, Champaign, IL) was used for all compartmental modeling and simulations. Literature data from plots were digitized using Engauge Plot Digitizer version 10.4 (GitHub, San Francisco, CA).

#### Probe Drug Selection and Data Collection.

A diverse set of drugs made up of acids, bases, and neutrals was selected to compare the prediction methods (Table 1). Drugs were considered neutral when primarily uncharged at physiologic pH (7.4). Unless noted otherwise, the p*K*_{a} values of acids (p*K*_{a,a}) and bases (p*K*_{a,b}) for neutrals were set at 14 and 1, respectively. Any significant ionized and neutral fraction was considered by both methods. The probe drugs were selected based on the availability of literature intravenous pharmacokinetics data, as well as drug-specific parameters. Average experimental intravenous bolus and/or infusion C-t profiles for 19 drug studies were collected from the literature. If the data were represented as graphical C-t profiles, the plots were digitized. When average subject weight was available, simulations were conducted to reproduce the observed *V*_{ss} for that average weight. The observed clearance and the steady-state volume of distribution were determined by compartmental analysis using standard equations with Mathematica. Specifically, one- to three-compartment models were evaluated to generate the C-t profiles from the experimental data. Experimental clearance and *V*_{ss} values were determined from these compartmental models. The best model for the experimental data was determined by the corrected Akaike information criterion values (Akaike, 1974) and residual plots. For all drugs, microrate constants were well defined and use of noncompartmental analysis was not required. Experimental clearance values were assumed for all further modeling efforts and experimental *V*_{ss} values were compared with predicted values.

Literature physiologic data were used for *K*_{p} predictions and PBPK modeling (Brown et al., 1997; Poulin and Theil, 2002; Fenneteau et al., 2010; Ye et al., 2016). For drug-specific parameters [log *P*, p*K*_{a}, and blood to plasma (BP)] (Table 2), experimental values from the literature were preferred over calculated/predicted values, and if more than one experimental value was found, then the experimental values were averaged. Human BP values could not be found for betaxolol and nafcillin. For betaxolol, the values of the fraction unbound in plasma (*f*_{up}) were similar for rat and human; therefore, the rat BP value of 2.0 was used. For nafcillin, a value of 0.55 (1-hematocrit) was used, which is the BP ratio of similar compounds in humans (Greene et al., 1978). Also, this will not affect *V*_{ss} predictions since BP is not included in the *K*_{p} equations for acids. Protein binding was experimentally determined for all compounds (Table 2), with the exception of caffeine due to caffeine contamination in all plasma samples. A caffeine *f*_{up} value of 0.72 was determined by averaging values found in the literature.

The log *P*_{vo} (log of the vegetable oil:buffer partition coefficient) was calculated from log *P* using eq. 1 (Leo et al., 1971). This term is used to represent neutral lipid partitioning in adipose tissue in the *K*_{p,dPL} method; however, it is not used in the *K*_{p,mem} prediction method:(1)

#### Microsomal Partitioning and Plasma Protein Binding.

Equilibrium dialysis was used to determine the *f*_{up} and *f*_{um} for the probe drugs using a protocol modified from prior studies (Kochansky et al., 2008; Curran et al., 2011; Di et al., 2017). Human plasma was adjusted to pH 7.4 by adding 1 M HCl. For *f*_{um} determination, a 0.5 mg/ml rat liver microsomal solution was prepared from a 20 mg/ml pooled rat liver microsome stock solution. For highly bound compounds, a dilution method was used. Plasma was diluted using a 100 mM phosphate buffer and 3 mM MgCl_{2} solution to either 50% or 10% plasma. A 50% dilution of plasma was used for warfarin, while a 10% dilution was used for ketoprofen, nicardipine, glyburide, diclofenac, felodipine, and mibefradil. Drug solutions (2 *μ*M) in either plasma or microsomes were added to wells on one side of the dialyzer, and blank 100 mM phosphate buffer with 3 mM of MgCl_{2} was added to the other side. The dialyzer plate was placed in the plate rotator, set to a speed of approximately 22 rotations per minute, and incubated for 22 hours at 37°C and 5% CO_{2}. Liquid chromatography–tandem mass spectrometry was used to determine the concentration of drug in the buffer and the matrix.

The fraction unbound in a given matrix was determined by dividing the concentration of drug on the buffer side by the concentration of drug on the matrix side. For protein binding experiments using the dilution method, the *f*_{up} value is calculated by eq. 2:(2)where *D* is the dilution factor; *f*_{u,d} is the fraction unbound in plasma measured in the diluted matrix; and *f*_{up} is the fraction unbound in plasma.

Experimental *f*_{um} values were measured at microsomal concentrations between 0.5 and 2 mg/ml and converted to values for 1 mg/ml (eq. 3) (Austin et al., 2002):(3)where *f*_{u2} is the corrected unbound fraction; *C*_{2} is the 1 mg/ml microsomal protein concentration; *C*_{1} is the microsomal protein concentration used in assay; and *f*_{u1} is the fraction unbound in matrix measured during the assay. The average fraction unbound, S.D., and CV values were determined for each assay.

#### Simulations.

A generic PBPK model was used, with different tissues represented by 10 compartments (adipose, bone, brain, gut, kidney, liver, lungs, muscle, skin, and spleen), representing the major tissues in the body (Fig. 1). These compartments are linked via arterial and venous blood flows. The *K*_{p} values were predicted for each of the tissues using both methods (*K*_{p,dPL} and *K*_{p,mem}). The original *K*_{p,dPL} method uses two separate equations, one for acids, weak bases, and neutrals, and another for moderate-to-strong bases (eqs. 4–7) (Rodgers et al., 2005; Rodgers and Rowland, 2006).

The *K*_{p,dPL} equation used for prediction of *K*_{pu} for strong-to-moderate bases is(4)where *f*_{ew} is the fractional volume of extracellular water; *f*_{iw} is the fractional volume of intracellular water; [AP] is the concentration of acidic phospholipids in the tissue; *K*_{AP} is the association constant for acidic phospholipids in the tissue; *f*_{nl} is the fractional volume of neutral lipids; *f*_{npl} is the fractional volume of neutral phospholipids; p*K*_{a,b} is the basic ionization constant; and p*K*_{a,a} is the acidic ionization constant.

The Rodgers equation used for prediction of *K*_{pu} for acids, neutrals, and weak bases is(5)where [PR]_{T}/[PR]_{P} is the plasma protein tissue (extracellular fluid) to plasma ratio, and NP denotes neutral phospholipid.

For eq. 4, the association constant for blood cells (BCs) is defined in eq. 6, and the tissue:plasma water partitioning coefficient for the BCs is defined in eq. 7:(6)(7)The tissue:plasma partition coefficient (*K*_{p}) can be determined from *K*_{pu} by eq. 8:(8)For the *K*_{p,mem} method we use the previously reported equation (Korzekwa and Nagar, 2017a) that considers both phospholipid membrane partitioning with *f*_{um} and neutral lipid partitioning with *P* (eq. 9):(9)where *r* is the protein ratio between the tissue and plasma; *f*_{pl} is the fractional volume of phospholipid; LKL is the lipid binding constant; and a and b are parameterized coefficients. Plasma and tissue pH values were assumed to be 7.4 and 7.0, respectively. As described previously for *V*_{ss} predictions (Korzekwa and Nagar, 2017a), the tissue-specific *r* values for bases were decreased by 2.23-fold due to the lower amount of *α*-acid glycoprotein in the extracellular fluid relative to albumin (Rowland and Tozer, 2011).

Neutral lipid partitioning in adipose tissue is described by the vegetable oil:water partition coefficient in the *K*_{p,dPL} method, which is generally calculated from log *P*. In the *K*_{p,mem} prediction method log *P* is used directly. The a and b terms in eq. 9 were parameterized using the tissue *K*_{p} values and tissue composition data from Rodgers et al. (2005) and Rodgers and Rowland (2006). We excluded zwitterions and combined neutral and acidic phospholipids to obtain a fraction of the total phospholipids. Most *f*_{um} values were calculated with our previously reported model (Nagar and Korzekwa, 2017) since experimental values are not available for this data set. The log *K*_{pu} values and the log of eq. 9 were used to fit a and b, with no additional weighting (log transformation results in 1/*Y* weighting). Outliers were identified using the BoxWhisker function (Frigge et al., 1989) in Mathematica with outliers defined as >1.5 times the interquartile range. Another model, which included an additional parameter for partitioning into adipose, was evaluated but did not improve predictions. The lipid concentrations multiplied by the lipid binding constant, L times K_{L}, (LKL), was calculated from *f*_{um} with eq. 10, using *f*_{um} values normalized to 1 mg/ml microsomal protein:(10)Exposure overlap coefficients (EOCs) were used to quantify the ability to predict the shape of the C-t profile (eq. 11) (Nagar et al., 2017). They are calculated by determining the overlapping portion of the experimental and predicted C-t profile curves and dividing that area by the experimental area under the curve (AUC). Since the experimental clearance values were used for all predictions, both experimental and predicted C-t profiles will have the same AUC. This allows the EOC to be used as a direct comparison of curve shapes. Differences in the average EOCs were determined using the *t* test:(11)

*V*_{ss} Predictions.

The *V*_{ss} value was determined from the predicted *K*_{p} values and physiologic volumes (eq. 12):(12)where *V*_{p} is the plasma volume, and *V*_{t} represents the tissue volume.

Predicted *V*_{ss} values were compared with observed values determined by compartmental modeling. To evaluate the predictive precision of the two methods, the absolute fold-error and absolute average fold error (AAFE) values were determined for all compounds and different subsets. Significance of differences in absolute fold errors were determined with a one-tailed *t* test. The AAFE determines the geometric mean of the absolute fold error (eq. 13) and is a measure of how precisely the two methods predict *V*_{ss}:(13)where *n* is the number of drugs, and FE is the fold error.

## Results

The parameters a and b in eq. 9 were fit to the experimental tissue reported *K*_{pu} values for the rat (Rodgers et al., 2005; Rodgers and Rowland, 2006). After sequential removal of 20 outliers (out of 401), the optimized parameters were a = 1383 ± 85 and b = 0.096 ± 0.029. The predicted versus observed *K*_{pu} values for 381 drugs are shown in Fig. 2. The *R*^{2} value for the fit was 0.80. There was no consistent characteristic for the removed outliers with the exception of the overprediction of phencyclidine in four tissues and the underprediction of basic compounds in the lung (five drugs).

Figure 3 and Table 3 show the observed versus predicted *V*_{ss} values using either the *K*_{p,dPL} or *K*_{p,mem} prediction method. The accuracy of the *V*_{ss} predictions was analyzed by determining the percentage of predictions within a range of absolute fold errors (Table 4) and AAFEs (Table 5). For the total data set of 19 drugs, the *K*_{p,mem} and *K*_{p,dPL} methods had comparable AAFE values (2.12 and 2.27, respectively). The *K*_{p,mem} and *K*_{p,dPL} methods had one (mibefradil) and two (diphenhydramine, felodipine) outliers, respectively. When these were excluded, the AAFEs were again comparable.

Ten different categories are compared in Tables 4 and 5. For predictions that were less than 1.5-fold error, *K*_{p,mem} scored higher in five categories, lower in two categories, and the same in three categories compared with *K*_{p,dPL} (Table 4); for predictions that were less than 3-fold error, *K*_{p,mem} scored lower in five categories and the same in five categories compared with *K*_{p,dPL}. When comparing all 19 drugs, the AAFEs with *K*_{p,mem} versus *K*_{p,dPL} were 2.12 versus 2.27, respectively (Table 5). Across the 10 categories, the AAFE was lower for *K*_{p,mem} than for *K*_{p,dPL} in eight categories. When outliers were excluded in each method, the AAFE was lower for *K*_{p,mem} than for *K*_{p,dPL} in five categories. For all categories listed in Table 5, there were no statistically significant differences in average fold error between *K*_{p,mem} and *K*_{p,dPL}.

#### Simulations.

An example of EOC calculation is shown in Fig. 4 for verapamil. Concentration profiles were simulated for 19 drugs with both *K*_{p} prediction methods (Figs. 5–7). EOCs were determined for all 19 drugs (Table 3). Overall, there was no significant difference in the average EOC values for the *K*_{p,mem} or *K*_{p,dPL} method. However, there were some interesting deviations between the methods. The C-t profiles were poorly predicted by both methods for five drugs: mibefradil, diazepam, felodipine, diclofenac, and nafcillin. In addition, the *K*_{p,dPL} method poorly predicted the profiles for betaxolol and diphenhydramine. Some possible explanations for these discrepancies are discussed subsequently.

As discussed previously (Korzekwa and Nagar, 2017b), eqs. 5–7 indicate that the *K*_{pu,tissue} value, and ultimately the unbound *V*_{ss} value, should be proportional to *K*_{pu,BC} (eq. 7), which was first experimentally observed by Hinderling (1997). Also, *f*_{up} in the denominator of eq. 7 is ultimately multiplied by *f*_{up} in the PBPK framework (eq. 8). Therefore, the predicted *V*_{ss} values for bases should be relatively insensitive to *f*_{up} when using eqs. 4–7. The impact on the predicted *V*_{ss} after a 2-fold decrease in *f*_{up} is shown in Table 6. As expected, changing *f*_{up} for acids has little effect since plasma protein binding is high and *V*_{ss} is low. For neutrals, a 2-fold decrease in *f*_{up} results in an average 1.7-fold increase in *V*_{ss} for both methods. For bases, a 2-fold decrease in *f*_{up} results in an average 2-fold decrease in *V*_{ss} for *K*_{p,mem}, but an average 1.05-fold increase with *K*_{p,dPL}.

## Discussion

Tissue partition coefficients, and ultimately *V*_{ss}, are determined primarily by competition between plasma protein binding and lipid partitioning. The *K*_{p,dPL} method considers binding to neutral lipids such as triglycerides and neutral phospholipids in membranes, with acidic phospholipids considered separately (Rodgers et al., 2005; Rodgers and Rowland, 2006). The method assumes that neutral molecules only interact with neutral phospholipids, and only ionized bases interact with APs, an assumption questioned previously (Korzekwa and Nagar, 2017b). The major acidic phospholipid is phosphatidylserine and the major neutral phospholipid is phosphatidylcholine. Although phosphatidylcholine is net neutral and phosphatidylserine is net acidic, both molecules are zwitterions. Interactions between charged species in the polar head group region are dynamic processes with conformational changes occurring in a picosecond time frame (Tieleman et al., 1997). Balaz (2009) compiled experimental data evaluating the orientation of exogenous molecules in membranes. Hydrophilic molecules accumulate in the polar head group region, amphiphilic molecules accumulate at the interface, and hydrophobic molecules accumulate in the hydrophobic core. We have used this concept to develop quantitative models for membrane partitioning (Nagar and Korzekwa, 2017).

The *K*_{p,mem} model is based on previously reported *V*_{ss} models (Korzekwa and Nagar, 2017a). The *K*_{p} model described herein uses the phospholipid component of tissues and *f*_{um} to model membrane partitioning and log *P* for neutral lipid interactions. Equation 9 was parameterized using reported tissue composition data and tissue *K*_{p} values (Rodgers et al., 2005; Rodgers and Rowland, 2006). Only two constants were parameterized: the scaling factor for 1) membranes and 2) neutral lipids. Although the two methods use different mechanistic assumptions, the resulting fit for *K*_{p,mem} (Fig. 2) is similar to that reported by Rodgers et al. (2005) for *K*_{pu} parameterization.

The volume of distribution is generally low for acids due to high plasma protein binding and low partitioning into membranes and neutral lipids. At physiologic pH, most acids are negatively charged, and membranes have few hydrogen bond donors. Therefore, microsomal partitioning is low for acids, with *f*_{um} values ranging from 0.72 to 0.98 (Table 2). Both the *K*_{p,mem} and *K*_{p,dPL} methods assume that only the neutral acids partitions into tissues, and both methods predict *V*_{ss} with similar accuracy (AAFE = 2.12 and 2.27 for *K*_{p,mem} and *K*_{p,dPL}, respectively) (Table 5). This is expected since any model that restricts a compound with low *f*_{up} to the plasma and extracellular space will predict a *V*_{ss} value approximately equal to that of plasma proteins (∼7.5 l) (Rowland and Tozer, 2011). Underprediction of acids with *V*_{ss} values >9, e.g., nafcillin, is frequently observed (Chan et al., 2018). Transporter activity (e.g., organic anion transporting polypeptides) could be one reason for the underprediction.

When two outliers for the *K*_{p,dPL} analysis (diphenhydramine and felodipine) are excluded, *V*_{ss} predictions for neutrals using *K*_{p,dPL} improved from an AAFE value of 2.82 to 1.91 (Table 5). For diphenhydramine, the *K*_{p,dPL} method resulted in a 6.9-fold underprediction and *K*_{p,mem} gave a 3-fold underprediction (not an outlier). For felodipine, *K*_{p,dPL} resulted in a 7.4-fold underprediction and *K*_{p,mem} gave a 4.3-fold underprediction (not an outlier). The reason for these poor predictions is unknown but it may be difficult to predict *V*_{ss} of a highly protein bound and partitioned neutral compound (felodipine).

The *K*_{p,mem} method uses a single equation for bases, neutrals, and acids and predicts *V*_{ss} for bases with similar accuracy to *K*_{p,dPL}, which uses different equations for bases (eqs. 5–7). The *K*_{p,dPL} method assumes that ionized bases only interact with acidic phospholipids. This interaction is parameterized with BP, using the erythrocyte partition coefficient to parameterize binding of the ionized base to acidic phospholipids. Mechanistically, the assumption that bases bind only to acidic phospholipids is questionable. Hydrophobic bases bind to neutral phospholipids as well, with key interactions between the cation and negatively charged phosphate, and the hydrophobic region with the hydrophobic membrane core. From eqs. 5–7, it is clear that for moderate-to-strong bases with *V*_{ss} values greater than the total body water, the *K*_{p} values are dominated by the acidic phospholipid binding terms. Therefore, although binding to only acidic phospholipids was assumed, a similar relationship is possible, assuming that ionized bases bind to all phospholipids. The ratio of neutral to acidic phospholipids is relatively constant across tissues (CV = 15%) (Rodgers et al., 2005), and total phospholipids can be substituted for neutral phospholipids. The *K*_{BC} term in eq. 6 would be smaller, but the relevant phospholipid term in eq. 5 would be larger.

Another implication of using BP to predict *K*_{pu} for bases is the insensitivity of *V*_{ss} to measured *f*_{up}. For acids, *V*_{ss} is insensitive to *f*_{up} since tissue partitioning is minimal. The *V*_{ss} values for bases are expected to be proportional to *f*_{up} since they partition heavily into tissues from the unbound concentration in cytosol (assumed to be equal to the unbound concentration in plasma). This is observed for *K*_{p,mem} but not for *K*_{p,dPL} (Table 6). This is a consequence of using BP (which includes *f*_{up}) to calculate *K*_{pu}. The *V*_{ss} predictions can be relatively accurate when BP is used to predict *K*_{p} for bases, since errors in *f*_{up} are not manifest and unbound *V*_{ss} for bases is proportional to erythrocyte partitioning (Hinderling, 1997). However, errors in *f*_{up} can still result in many other inaccuracies, including in predictions of clearance and target activity.

Overall, the *K*_{p,mem} and *K*_{p,dPL} models give similarly accurate predictions, explaining 68% and 63% of the variance in *V*_{ss} (80% and 79% without outliers), respectively. Several factors may explain the remaining variance. First, there can be significant variability in the *V*_{ss} values measured across clinical studies. Not all pharmacokinetic data sets provide body weights. Also, experimental data from multiple sources are used, e.g., BP values; for example, Graham et al. (2012) observed a 7% decrease in accuracy when predicted instead of experimental log *P* values were used. For *f*_{up}, differences between laboratories can be very large. Several recent publications discuss assay conditions for protein binding, including dilution and use of CO_{2} (Kochansky et al., 2008; Curran et al., 2011; Di et al., 2017). In this study, we measured a value of *f*_{up} of 0.03 for mibefradil, whereas a value of <0.005 was reported previously (Clozel et al., 1991). Use of a smaller *f*_{up} value would result in a better prediction with *K*_{p,mem}, but exclusion of our data is not justified.

Lysosomal partitioning affects tissue distribution for bases. For strong bases, partitioning into lysosomes due to pH differences results in lysosomal concentrations >200 times cytosolic concentrations. Assuming 5% lysosomes and 60% intracellular water in cells, partitioning of a strong base into lysosomes can increase the *K*_{pu} value by 6-fold. As discussed previously, while lysosomal partitioning certainly occurs it is likely to be highly correlated with phospholipid partitioning of bases (Korzekwa and Nagar, 2017a). Finally, although the vegetable oil:water partition constant would be a good surrogate for adiposomes if measured, this value is typically modeled using log *P*, which is not necessarily accurate (Korzekwa and Nagar, 2017b).

Transporter-mediated distribution can result in inaccurate predictions of *K*_{p} and *V*_{ss}, particularly for some acids. Organic anion transporting polypeptide transporters can alter hepatic intracellular concentrations by two orders of magnitude (Kulkarni et al., 2016). Therefore, uptake into this organ alone can result in a 2-fold increase in *V*_{ss}. Efflux transporters (e.g., P-glycoprotein and breast cancer resistance protein) will have a smaller impact. The decrease in *V*_{ss} due to P-glycoprotein and breast cancer resistance protein at the blood-brain barrier would result in a 2% decrease in *V*_{ss}. The impact from the liver would be even smaller since efflux transporters in the apical membrane would only decrease liver concentrations by 50% (Korzekwa and Nagar, 2014).

Since the AUC is determined by experimental clearance and dose in both methods, the AUC values for the simulations are normalized, and the EOC captures differences in the shapes of the C-t profile (Fig. 4). Several C-t profiles in Figs. 5–7 are not well predicted by either method. Although there are differences in the EOC for some drugs, the average EOCs for the *K*_{p,mem} and *K*_{p,dPL} prediction methods were not significantly different. Since clearance is constant, when the *V*_{ss} value is overpredicted (e.g., betaxolol using *K*_{p,dPL} and mibefradil using *K*_{p,mem}) (Fig. 5), the terminal half-life is overpredicted. When the *V*_{ss} value is underpredicted (e.g., diphenhydramine using *K*_{p,dPL}) (Fig. 5), the terminal half-life is underpredicted. Perhaps the most significant deficiency of the reported modeling approaches is the assumption of perfusion-limited distribution. As seen with diazepam, felodipine, diclofenac, and nafcillin, accurate C-t profiles are not predicted even when the *V*_{ss} value is well predicted. For verapamil (Figs. 4 and 5), the distribution phase is not well predicted, presumably due to a combination of using a perfusion-limited model and experimental clearance. Clearly, multicompartmental distribution is not accurately modeled with perfusion-limited distribution.

In conclusion, *K*_{p,mem} can be used to predict *K*_{pu} with accuracy similar to *K*_{p,dPL}. An advantage of using *f*_{um} to parameterize membrane partitioning is that *f*_{um} is used for clearance prediction and is generated early in the discovery/development process. Also, differentiating between acidic and neutral phospholipids for bases and using 0.3*P* for neutral compounds is not mechanistically justified. Finally, since both the extent and rate of membrane partitioning and permeability are important, a mechanistically sound basis for membrane interactions is necessary for improved PBPK models.

## Authorship Contributions

*Participated in research design:* Holt, Nagar, Korzekwa.

*Conducted experiments:* Holt, Ye.

*Performed data analysis:* Holt, Nagar, Korzekwa.

*Wrote or contributed to the writing of the manuscript:* Holt, Nagar, Korzekwa.

## Footnotes

- Received May 13, 2019.
- Accepted July 9, 2019.
This research was supported by the National Institutes of Health [Grants R01GM104178 and R01GM114369].

## Abbreviations

- AAFE
- absolute average fold error
- AP
- acidic phospholipid
- AUC
- area under the curve
- BC
- blood cell
- BP
- blood to plasma
- C-t
- concentration-time
- EOC
- exposure overlap coefficient
*f*_{ew}- fractional volume of extracellular water
*f*_{iw}- fractional volume of intracellular water
*f*_{nl}- fractional volume of neutral lipids
*f*_{um}- fraction unbound in microsomes
*f*_{up}- fraction unbound in plasma
*K*_{p}- tissue:plasma partition coefficient
*K*_{p,dPL}- differential phospholipid tissue-plasma partition coefficient prediction method
*K*_{p,mem}- membrane-based tissue-plasma partition coefficient prediction method
*K*_{pu}- unbound tissue:plasma partition coefficient
*P*- octanol:water partition coefficient
- PBPK
- physiologically based pharmacokinetic
- p
*K*_{a,a} - ionization constant for acids
- p
*K*_{a,b} - ionization constant for bases
*V*_{ss}- steady-state volume of distribution

- Copyright © 2019 by The American Society for Pharmacology and Experimental Therapeutics