Methods to Predict Volume of Distribution

Abstract

Purpose of Review

Prior to human studies, knowledge of drug disposition in the body is useful to inform decisions on drug safety and efficacy, first in human dosing, and dosing regimen design. It is therefore of interest to develop predictive models for primary pharmacokinetic parameters, clearance, and volume of distribution. The volume of distribution of a drug is determined by the physiological properties of the body and physiochemical properties of the drug, and is used to determine secondary parameters, including the half-life. The purpose of this review is to provide an overview of current methods for the prediction of volume of distribution of drugs, discuss a comparison between the methods, and identify deficiencies in current predictive methods for future improvement.

Recent Findings

Several volumes of distribution prediction methods are discussed, including preclinical extrapolation, physiological methods, tissue composition-based models to predict tissue:plasma partition coefficients, and quantitative structure-activity relationships. Key factors that impact the prediction of volume of distribution, such as permeability, transport, and accuracy of experimental inputs, are discussed. A comparison of current methods indicates that in general, all methods predict drug volume of distribution with an absolute average fold error of 2-fold. Currently, the use of composition-based PBPK models is preferred to models requiring in vivo input.

Summary

Composition-based models perfusion-limited PBPK models are commonly used at present for prediction of tissue:plasma partition coefficients and volume of distribution, respectively. A better mechanistic understanding of important drug distribution processes will result in improvements in all modeling approaches.

Introduction

Prediction of drug pharmacokinetics in humans is critically important early in drug discovery and development. Noncompartmental analysis, compartmental models, and physiologically based pharmacokinetic (PBPK) models are commonly utilized to characterize the absorption, distribution, metabolism, and elimination of drugs. Primary PK parameters—drug clearance (CL), volume of distribution (V, or V_ss for steady-state volume of distribution), and bioavailability (F) upon a non-systemic dose—aid in the characterization of drug plasma concentration-time profiles. This manuscript focuses on V_ss, and methods for the prediction of V_ss in drug discovery.

Volume of distribution (V) of a drug is a proportionality constant to describe the relationship between the concentration of drug in the plasma to the amount of drug in the body [1]. It is often described as an “apparent volume of distribution,” describing a mathematical volume in which the drug distributes in the body [2]. The V_ss of a drug is determined by the physiological properties of the body and physiochemical properties of the drug, and is used to determine secondary parameters, including the half-life [2]. These factors include plasma protein binding, tissue partitioning, and drug transporters as discussed below (Fig. 1). The volume of distribution can be experimentally determined with a variety of PK modeling techniques, with clinical plasma concentration-time data. Any estimation of V_ss requires IV dosing (bolus or infusion) data. If only oral data are available, an oral volume (V/F) can be calculated, but the extent of distribution is unknown. In a drug discovery setting, while the volume of distribution is often considered in drug design, it does not necessarily need to be optimized for drugs. For example, increasing volume of distribution not only increases the elimination half-life of a compound but also decreases the unbound target-site concentration. Understanding the disposition of a drug early in drug discovery is important in order to determine efficacious human doses, to design dosing regimens, as well as to characterize the therapeutic window of a drug. Only unbound drug is assumed to cross membranes and to distribute into cells out of the plasma, but as discussed below, poor permeability across membranes and/or the presence of transporters can alter the equilibrium assumed by the free drug hypothesis [1]. Unbound drug concentrations at the target site are relevant in order to characterize the therapeutic index of a drug, and in vitro assays as well as predictive models should ideally provide a useful correlation between the unbound target-site drug concentration and the total plasma drug concentration.

Fig. 1.

Determinants of drug distribution. Drug (D) can bind to erythrocytes as well as plasma proteins in the blood. Unbound drug diffuses out of capillaries into extracellular fluid (ECF), where it can bind to plasma proteins in the ECF. Unbound drug reversibly diffuses across the plasma membrane, or can be actively transported into or out of cells. Cell partitioning in intracellular organelles such as lysosomes, as well as partitioning into phospholipids and neutral lipids in the cell, is depicted

To understand the kinetics of drug distribution, both the rate and extent of distribution must be considered. The rate of distribution for permeable drugs in many tissues is often perfusion-limited (blood-flow limited), as the only restriction to the rate of distribution is the blood perfusion rate to the tissue. In contrast, some drugs in some tissues exhibit permeability-limited distribution, when membrane permeability is a barrier that restricts partitioning into tissues. Early in drug discovery/development, predictive models often assume perfusion-limited distribution of drugs. The extent of distribution is determined by tissue:plasma partition coefficients (K_p) and tissue volumes, which together determine the V_ss of a drug.

Plasma Protein Binding

Plasma protein binding of drugs is a key determinant of the volume of distribution. Drugs generally bind to plasma proteins (including albumin, alpha acid glycoprotein or AAG, and lipoproteins) in a relatively nonspecific manner. In general, acidic drugs bind primarily to plasma albumin, bases predominantly to AAG, and hydrophobic compounds to lipoproteins. However, there are many exceptions, e.g., some bases and many neutral molecules bind albumin. The fraction of drug unbound in plasma (f_up) is calculated as the unbound plasma concentration compared with the total concentration in the plasma at equilibrium. Drugs with high plasma protein binding tend to have smaller volumes of distribution compared to drugs that are not highly plasma protein bound [1]. Plasma protein binding in the extracellular fluid also occurs, and the smallest volume that a highly albumin-bound drug can have is about 7 L in a healthy human [1]. This corresponds to the volume of distribution of plasma proteins. In contrast to plasma, the blood additionally has red blood cells (RBCs), and drug binding to RBC membranes is typically measured with a blood/plasma ratio (BP). The hematocrit of the blood sample, f_up, and the affinity of the drug for the RBCs determines the value of BP [1].

Accurate determination of f_up is critical, since this value is used for the prediction of drug clearance as well as volume of distribution [3]. Equilibrium dialysis under controlled pH conditions (in the presence of 5–10% CO₂) provides more consistent estimates of f_up compared with assays conducted in the absence of CO₂, or with ultracentrifugation methods. The determination of BP is relatively straightforward.

Tissue Partitioning

Another key determinant of the extent of drug distribution is the partitioning of drugs into tissues. Tissue partitioning is determined by the physicochemical properties of the drug, as well as the composition of each specific tissue. Drugs with high tissue partitioning will generally have a large V_ss. Drugs typically partition into tissue membrane components, including phospholipids and neutral lipids, as well as cellular lipid components and tissue proteins. Tissue partitioning is characterized by the equilibrium tissue to plasma partition coefficient of a drug in a tissue (K_p). Currently, the most commonly used collection of experimental data for tissue K_p values is provided in Rodgers et al. [4, 5] and includes the K_p of 67 drugs in various tissues in rats. As discussed below, various methods are employed to predict tissue K_p values of drugs, for prediction of V_ss, and for use in PBPK modeling.

Overall, the observed V_ss of drugs is a result of the competition between plasma protein binding and tissue partitioning. However, extensive plasma protein binding does not necessarily result in a low V_ss, when tissue partitioning is extensive. For example, tamoxifen is > 98% plasma protein bound but has a V_ss of ~ 4000 L [6].

Blood Flow and Perfusion-Limited Distribution

In perfusion-limited distribution, the rate-limiting step to distribution is the blood flow to the organ/tissue. The rate of distribuion of highly permeable drugs is typically perfusion-limited, and uptake or efflux transporters generally do not have a significant impact. Distribution occurs rapidly in highly perfused tissues, and in tissues with fenestrated or discontinuous capillaries, e.g., liver and spleen. Tissues like muscle and skin, which have continuous capillaries and low perfusion rates, often exhibit slower rates of drug distribution [7]. Even more restrictive is the blood-brain barrier (BBB), which has tight junctions between endothelial cells, and drugs must cross membranes to reach the extracellular space. For most PBPK modeling efforts in early drug discovery, drugs are assumed to exhibit perfusion-limited distribution in all organs, and the impact of membrane permeability and transporters is often ignored. Resulting models may predict the extent of drug distribution accurately but poorly predict the rate of distribution.

Membrane Permeability, Permeability-Limited Distribution, and Transporters

Poorly permeable drugs are limited in their distribution into tissues by membrane barriers at the capillary or the cell. Organs such as the brain are protected by the BBB, and most drugs exhibit permeability-limited distribution in the brain. Efflux transporters can lower the unbound drug concentration in the tissue compared to the plasma, as seen with the efflux of drugs by P-glycoprotein at the BBB [8]. Drugs with poor permeability are also often substrates for uptake transporters. The impact of uptake transporters in drug distribution is exemplified by atorvastatin uptake in the liver [9]. Active uptake transport results in higher unbound drug concentration in the tissue compared to the plasma at equilibrium.

The various physiologic complexities interplay with physicochemical properties of drug molecules to eventually determine V_ss of drugs. Various methods have been developed for the prediction of V_ss, including use of in vitro data to predict V_ss directly, use of in vitro and in silico methods to predict K_p, and quantitative structure activity relationships (QSAR). Some of these methods are discussed below.

V_ss Prediction Methods

Volume of distribution prediction methods are important, as they allow an early understanding of drug behavior, before clinical data is obtained, in order to drive decisions in drug discovery and development. Several methods have been used to predict V_ss, including (a) preclinical extrapolation [10], (b) physiological equations, (c) K_p prediction models, and (d) QSAR models [11]. All these methods attempt to incorporate physicochemical (drug) and physiological (body) determinants of distribution.

(a). Preclinical Extrapolation

Prior to the clinic, human volume of distribution values are often extrapolated from a preclinical species, often through direct correlation or allometry. Preclinical parameters can be empirically extrapolated to predict human V_ss [3]. These methods are often accurate because tissue composition is generally conserved across mammalian species. Preclinical V_ss is first used to predict the fraction of drug unbound in tissue (f_ut), and human V_ss is then predicted based on the predicted f_ut and human unbound plasma fraction (f_up) from in vitro assays. This method is based on the assumptions that partitioning into tissue lipids is relatively independent of species, and plasma protein binding may show significant deviations. While V_ss is often scaled from a preclinical species, classical allometric relationships are also used as described below.

Allometric scaling relates body weight (BW) to a given physiological parameter or pharmacokinetic parameter [10]. This allows prediction of these parameters across species, based on their body weight. An allometric equation can be expressed as:

$$Y=aB{W}^b$$ (1)

where Y is the parameter, a and b are the coefficients which are parameterized from data from different species, and BW is the body weight. Boxenbaum initially showed that allometry could be used for the prediction of human volume of distribution [10]. The allometric exponents vary for different PK parameters, and the exponents used for clearance and V_ss are generally 0.75 and 1, respectively [12]. An exponent of 1 implies that weight normalized V_ss is constant across species, which is a trivial allometric relationship. Allometric predictions of PK parameters are empirical and do not account for physiological differences across species, including but not limited to, enzymes, transporters, and plasma protein binding. Plasma protein binding of drugs can differ greatly from one species to the next [13]. Sugita et al. showed that allometry could not be used for tolbutamide, which is a highly protein-bound drug [14]. Obach et al. reported that correction for protein binding led to better allometric predictions [3]. Sawada et al. compared V_ss predictions for 10 weakly basic drugs with direct extrapolation across species versus with an equation using f_up and BP (a form of the Oie-Tozer equation, discussed below) [15], and reported that V_ss predictions were erroneous upon simple extrapolation. Jones et al. evaluated 24 different methods for the prediction of V_ss and found that generally, methods using in vivo preclinical data (when available for more than two species) were more predictive than those that relied solely on in vitro data [16].

(b). Physiological Models

Various models that use in vitro data to directly predict V_ss have been proposed. The commonly used Gillette and Oie-Tozer equations, modifications thereof, and a method based on membrane partitioning published previously by our laboratory are briefly described.

In 1971, Gillette described a relationship between V_ss and physiologic plasma and tissue volumes, drug protein binding, and drug tissue partitioning [17]:

$$V_{ss}=V_P+V_T\frac{f_{up}}{f_{ut}}$$ (2)

where V_p is the volume of the plasma and V_T is the volume of the tissue. This relationship was further explored mechanistically in several seminal PK discussions [18, 19]. Equation 2 is the origin of essentially all methods to predict V_ss. For example, if various tissues are considered, the sum of all tissue volumes would equal V_T. Also, the term f_up/f_ut equals K_p, since unbound concentration in the plasma is assumed to equal unbound concentration in tissue. The Oie-Tozer equation [20] was based on the Gillette equation and made the distinction between drug bound to plasma proteins in the plasma versus drug bound to plasma proteins in the extracellular fluid:

$$V_{ss}=V_p\left(1+R_{E/I}\right)+V_p{f}_{up}\left(\frac{V_E}{V_P}\right)-R_{E/I})+V_R\frac{f_{up}}{f_{ut}}$$ (3)

where R_E/I is the ratio of total binding sites in the extracellular fluid outside plasma to the total binding sites in plasma, V_E is the extracellular fluid volume, and V_R is the aqueous volume outside extracellular fluid into which drug distributes. The Oie-Tozer equation has been used in an attempt to better predict the V_ss, as well as be more mechanistic [21, 22]. Lombardo used the Oie-Tozer equation to predict f_ut (n = 64 drugs) and V_ss (n = 14 drugs) of neutral and basic drugs [21]. Of the 14 drugs in the validation set, V_ss for 8 drugs was predicted within 2-fold of the experimental value. These results were expanded and corroborated in a subsequent study by the same group [22].

Membrane partitioning is an important mechanism for drug distribution and the subsequent parameterization of V_ss. Using two experimental inputs—plasma protein binding (f_up) and microsomal partitioning (f_um) to represent membrane partitioning—V_ss predictions were attempted for 63 drugs (acids, bases, and neutrals) [23••]. The resulting model had an R² of 0.83 and 1.6-fold error for 60 drugs, with three outliers. This base model was expanded to include other factors including neutral lipid interactions as well as interactions with lysosomes (for bases). These different additions did not significantly improve V_ss predictions.

(c). K_p Models

V_ss can also be predicted by using K_p values [15]. V_ss is determined by taking the sum of the tissue volumes multiplied by the respective drug K_p values, and adding this sum to the plasma volume. This method requires knowledge of drug K_p for different tissues in the body. K_p values can be determined in preclinical species, but this is costly and time-intensive. Early in drug discovery, it is therefore helpful to predict K_p values in order to predict V_ss. These K_p values are also useful in PBPK modeling to describe tissue distribution and are used to simulate drug concentration-time profiles. K_p values are generally predicted from physicochemical properties (e.g., LogP, pKa), in vitro studies, in vivo preclinical studies, and physiological parameters including tissue composition and protein ratios. Equation 4, derived from the Gillette equation, is used:

$$Vss=Vp+{\sum }_{i=1}^{n}V{t}_i\cdot K{p}_i$$ (4)

where V_ti is the volume of the ith tissue and K_pi is the corresponding K_p. Table 1 provides a brief comparison of various K_p models in the literature. Bjorkman used the relationship in Eq. 4 to determine which tissue K_p values were necessary in order to predict an initial V_ss [26]. Distribution to adipose and muscle accounted for an average 84% of the total estimated V_ss for the bases (n = 17) evaluated. For acids (n = 18), distribution to adipose and muscle accounted for only 65% of V_ss. Many published K_p prediction models require an in vivo component such as a K_p,muscle term or experimental V_ss [30,33–35]. Clausen and Bickel compared drug binding in blood to drug binding in tissue homogenate for in vitro determination of K_p. These predicted values generally under-predicted the experimental K_p values in rat tissue homogenates for 10 drugs [34]. Berry et al. [33] used a similar technique by using ultracentrifugation to determine the fraction unbound in tissues as well as in plasma, to determine V_ss for different compounds. These values were compared to V_ss determined by composition-based K_p prediction methods, as well as to experimental V_ss values. Using tissue binding data, V_ss was predicted within 2-fold of experimental values for 61% of all drugs studied, compared to 42 or 53% with composition-based methods (Berezhokivsiy and Rodgers and Rowland methods, respectively) within Symcyp Rat version 8.0. These in vitro binding methods had some limitations. Homogenization likely disrupted elements of the tissues which contribute to distribution, including lysosomal partitioning and transporters. Also, choice of drugs used, experimental errors, and literature inputs probably influenced the results. While these methods provide reasonable predictions, they are more labor intensive and less high-throughput than computational methods. Jansson et al. used preclinical V_ss values and drug lipophilicity to predict tissue K_p for 22 drugs. The K_p values were generally well predicted for non-eliminating tissues (R² = 0.81), with 72% of predicted values within 2-fold of experimental values. Various additional algorithms for predicting K_p have been previously reported [30, 35, 36] and have been compared in the literature [37].

Table 1.

Summary of key K_p prediction methods

Reference	Summary of study	Comments
Arundel [24]	A minimal PBPK model with 6 tissues was proposed, with perfusion-limited distribution Literature VT and blood flow values were used Known Vss values were used to determine tissue distribution rate constants and subsequently to calculate tissue Kp values Assumed that the ratio of tissue distribution rate constants for any pair of tissues is constant for different drugs	Requires an in vivo component, i.e., Vss Would not necessarily be used for prediction of Vss n = 10 drugs was used, of which 4 were barbiturates
Jansson et al. [25]	Empirical method for Kp prediction in preclinical species, using experimental Kpu values and lipophilicity descriptors (e.g., LogP, LogD) Based on previous predictions for Kp, by Bjorkman [26], Arundel [24], and Poulin [27] Training set contained 49 compounds. For n = 22 validation set compounds, 72% of predicted Kp values were within 2-fold error of experimental Kp Perfusion-limited distribution was assumed Kpu,muscle was correlated with Kp of all other tissues, except adipose	Requires in vivo component, i.e., Kp,muscle from preclinical species Improved predictions when preclinical data are available
Poulin et al. [27–29]	In vitro data on drug lipophilicity and plasma protein binding was used as input, and 3 equations were initially developed Rat and human Vss predictions attempted (n = 123 compounds), and 80% of predicted values were within 2-fold of experimental values [27] Perfusion-limited distribution was assumed	Relies on assumptions about tissue interactions and phospholipid composition Vegetable oil: water partition coefficient used for lipid partitioning in adipose tissue [28]
Poulin and Theil [30]	Developed a linear regression model, evaluating the relationship between Kpu,muscle and Kpu of other tissues for moderate to strong bases Results were compared with other composition-based methods in the literature For basic drugs (n = 61), 77% of predicted Kpu fell within less than 2-fold error of observed values Perfusion-limited distribution was assumed Correlation was observed between partitioning in RBCs and Kpu,muscle For adipose tissues, an adjusted skin Kpu value was used	Able to accurately predict Vss in rats [31] Requires in vivo component, i.e., Kpu,muscle
Berezkovskiy [32]	Theoretical equation was developed, and tissue partitioning was modeled from extracellular fluids	Requires only fup, and lipid partitioning data
Rodgers and Rowland [4, 5]	Used 28 moderate-strong bases in test set, to compare predictions with experimental Kpu values as well as with Poulin and Theil method 89% of predictions, compared to 45% with Poulin method, fell within 3-fold error of experimental Kpu values for most tissues Based on free drug hypothesis, and assuming perfusion-limited distribution Assumed that ionized bases interact only with acidic phospho lipids	Requires BP Insensitive to changes in fup
Rodgers and Rowland [5]	For 39 neutral, weak acid, and weak base drugs (in 13 tissues, for a total of 370 Kpu values), the predicted Kpu values were compared with experimental Kpu 84% of the predictions fell within a factor of 3, compared with 61% with the Berezkovskiy method Perfusion-limited distribution was assumed	Relies on assumptions about phospholipid composition Uses LogP to represent lipid interactions
Korzekwa and Nagar [23••]	Modeled membrane partitioning with fum for 19 test compounds Free drug hypothesis and perfusion-limited distribution were assumed Vss predicted directly	No assumptions are made about the type of phospholipid and interactions with charged/uncharged drugs

Tissue composition-based models attempt to predict drug distribution based on tissue composition, drug physicochemical properties, and plasma protein binding. These methods usually require simple input parameters such as LogP, f_up, and BP. These methods do not rely on in vivo components but instead use in vitro surrogates to represent different distribution processes [4, 5, 27, 38]. Poulin et al. calculated K_p as a function of a lipophilicity measure, tissue-specific concentration of lipids and fraction unbound in plasma [27, 28, 39]. However, as discussed previously [40••], the lipophilicity measures used in these composition-based models rely on mechanistically unsound calculations of lipid partitioning. For example, a vegetable oil:water partition coefficient calculated with equations inaccurate for this purpose is used to describe neutral lipid partitioning. Also, neutral phospholipids are modeled as 70% water and 30% octanol. Berezhkovskiy revised the Poulin method by correcting for the ratio of unbound fraction in tissue to that in plasma [32]. Next, the Rodgers and Rowland models assumed that moderate to strong bases partition only in acidic phospholipids [4] and further developed two separate equations to predict the K_p of moderate-to-strong bases [4], and neutrals, weak bases, and acidic drugs [5]. A major assumption of the Rodgers and Rowland method for bases is that ionized bases only bind to acidic phospholipids, and neutral bases bind to neutral phospholipids. The validity of this assumption has been questioned previously [40••]. Also, the method uses BP to parameterize acidic phospholipid binding. It has been shown that erythrocyte partitioning is highly correlated with unbound volumes of distribution for bases [41]. Therefore, use of BP has two important consequences. First, when used to parameterize acidic phospholipid binding, the subsequent V_ss prediction becomes insenitive to errors in f_up [42]. Second, the acidic phospholipid term dominates K_p predictions, and V_ss is ultimately determined by the BP value [40••].

Additionally, several QSAR models have been used to predict K_p and/or V_ss in humans. Many groups have used descriptor-based models, to determine important interactions in tissues and physicochemical properties that dictate distribution [43–45]. Lombardo et al. used a mixture discriminant analysis-random forest method to predict V_ss and found a 1.8-fold geometric mean fold error for an external validation set [44]. This was compared to a 2.1-fold error for animal-based predictions suggesting that this method was comparable to the much more expensive in vivo method. In another study, a mixture of experimental and computed structural descriptors was used with step-wise regression to predict volume of distribution (both V_ss and V_area were used) [43]. It was found that models for unbound volumes had higher errors than total volumes (2.3-fold versus 2.0-fold, respectively). A model using genetic algorithm and stepwise regression with molecular descriptors to predict the V_ss of acidic drugs resulted in an AAFE of 2.1 for an external validation set [45]. All three methods result in an approximate fold error of two for an external validation set. It should be noted that if the external validation set contains drugs from the same class as the training set, the predictability of a compound in a new chemical class is unknown. This is a general concern when descriptor-based QSAR models are used to predict non-specific ADME properties.

Factors Influencing V_ss Predictions

Key factors that impact V_ss prediction accuracy include permeability, transporters, and accuracy of experimental inputs. BCS Class 3 and Class 4 drugs have low permeability across membranes. Polar drugs exhibit permeability-limited distribution due to low permeability across the hydrophobic lipid bilayer. In drug discovery, compound permeability is usually measured with standard Caco-2 cell assays. However, compound permeability is often ignored in V_ss predictions. For example, perfusion-limited PBPK models ignore drug permeability. Poorly permeable compounds may not reach equilibrium with unbound plasma concentrations within the exposure time of the drug. While outside the scope of this discussion, permeability is also critical to consider in the absorption of drugs via the gut. When permeability impacts the rate of drug distribution, perfusion-limited PBPK models will inaccurately predict the shape of the concentration-time profile, even though V_ss may be predicted accurately [42]. Incorporation of in vitro permeability data into simple compartmental models with explicit membranes has been previously reported [46]. Models with explicit membranes can be used to improve predictions of V_ss as well as concentration-time profiles (i.e., extent and rate of drug distribution). Although currently unavailable, full perfusion- and permeability-limited models may ultimately predict the rate and extent of distribution.

A factor closely related to drug permeability is the impact of transporters. Poorly permeable drugs are often substrates for uptake transporters. The family of organic anion transporter proteins (OATPs) has emerged as a key uptake transporter protein family expressed in the liver, with clinical implications for drug pharmacokinetics and drug-drug interactions [9, 47•]. Numerous reports have been published on active uptake of drugs such as statins by OATPs into the liver, and the impact of this uptake on drug clearance as well as drug-drug interactions [48]. Our laboratory evaluated the active hepatic uptake of atorvastatin in a rat model, and it was observed that active uptake significantly increased the total atorvastatin concentration in the rat liver [49•]. OATP-mediated transport was predicted to alter hepatic intracellular concentrations by two orders of magnitude [49•]. Therefore, uptake into the liver alone can result in a 2-fold increase in V_ss. Efflux transporters such as P-glycoprotein significantly impact distribution of drugs in the brain [8]. Incorporation of transporter kinetics into V_ss predictions will be important in improving predictions. With current V_ss prediction models, the V_ss of acids is usually under-predicted. One reason for this underprediction could be the lack of inclusion of transporters in V_ss models.

As is true for all modeling efforts, accuracy of experimental inputs is critical for accurate predictive outputs. In the context of V_ss prediction models, experimental data such as plasma protein binding measurements are one example where inaccuracies have been observed. The pH of plasma samples can greatly impact the plasma protein binding assay and the subsequent calculation of f_up [50••]. Kochansky et al. reported that for 40% of the 55 drugs tested, the ratio of f_up at pH 7.4 (assay conducted under 10% CO₂) vs. pH 8.7 (assay conducted in air) was ≥ 2.0 [51]. Another experimental input, the blood/plasma ratio, is less prone to experimental errors. Similarly, equilibrium dialysis assays are standard procedure for determination of f_um. In the authors’ experience, experimental values of LogP are preferred over values calculated with in silico methods. A bigger concern with the use of LogP is that octanol:water partitioning, although a general measure of hydrophobicity, does not correspond to the physiological process of drug distribution [40••].

Finally, errors in in vivo pharmacokinetic data can result from experimental design or data analysis. For animal studies, if data collection or bioanalytical assay sensitivity is insufficient, the resulting calculated V_ss (used as an “experimenta” V_ss value) may be under-predicted. Also, use of V_area instead of V_ss overpredicts V_ss and may not translate to the human pharmacokinetic parameters. Use of preclinical pharmacokinetics to predict human drug disposition may be subject to several species differences. With respect to distribution, plasma protein binding and transporter protein homology, substrate specificity, binding kinetics, and genetic polymorphisms are all known to differ across species. These factors may result in significant errors in V_ss prediction.

Comparison of Different Prediction Methods

Many groups have reported comparative studies on various V_ss prediction methods [3, 16, 30, 31, 52••, 53–55]. Graham et al. compared six different K_p prediction methods for their prediction of unbound K_p (K_pu) as well as 4 V_ss prediction models, for 81 compounds in 11 rat tissues [31]. The Rodgers and Rowland method [4, 5] provided the best prediction of K_pu (77% within 3-fold of experimental values). The Poulin and Theil method [27, 28] was able to predict the V_ss best (87% within 3-fold error). Jones et al. [16] compared 24 methods for V_ss prediction, and the maximum success rate (prediction within 2-fold) was 78% across methods. A recent comparison by Chan et al. compared two V_ss prediction methods for a large data set (over 150 marketed drugs) [52••]. The methods included preclinical extrapolation and in silico prediction based on the Rodgers and Rowland equations. Use of composition-based K_p prediction methods for determination of the V_ss was more accurate than preclinical extrapolation, with V_ss of 65.8% (out of 152 drugs) predicted within 2-fold.

Conclusions

Extrapolation directly from preclinical species, physiological models, composition-based K_p prediction models, and QSAR methods can all be used to predict the V_ss of a drug. Overall, the accuracy of all current methods in general is a 2-fold absolute average fold error. As with all modeling exercises, there is a trade-off between the quality of inputs and the accuracy of the predictions. For example, any model with experimental inputs (e.g., f_up, f_um, LogP) will result in better predictions than a model using calculated values. Currently, the use of composition-based PBPK models is preferred to models requiring in vivo input. Perfusion-limited PBPK models are commonly used, but we anticipate that membrane-based models that include permeability-limited distribution will be used more frequently. QSAR methods are an active area of research, but careful selection of training and validation sets is required to prevent over-fitting. A better mechanistic understanding of important drug distribution processes will result in improvements in all modeling approaches.