Determinants of Biological Half-Lives and Terminal Slopes in Physiologically Based Pharmacokinetic Systems: Assessment of Limiting Conditions

Abstract

In pharmacokinetic (PK) analyses, the biological half-life T_1/2 is usually determined in the terminal phase after drug administration, which is readily calculated from the relationship T_1/2 = ln2/λ_z where λ_z is the terminal-phase slope obtainable from non-compartmental analysis (NCA). Since kinetic understanding of λ_z has been limited to the theory of a one-compartment model, this study seeks kinetic determinants of λ_z in more complex plasma concentration-time profiles. We utilized physiologically based pharmacokinetic (PBPK) systems that are consistent with the assumptions of NCA (e.g., linear PK and elimination occurring from plasma) to interrelate λ_z and disposition kinetic parameters of PBPK models. In a mammillary form of PBPK models, the two boundary conditions of λ_z are the inverses of the mean residence time in the body (1/MRT_B = CL/V_SS) and the mean transit time through the kinetically largest tissue (1/MTT_max = Q_Tf_dR_b/V_TK_p). Importantly, the limiting conditions of λ_z between 1/MRT_B and 1/MTT_max are dependent on a simple product MRT_Bλ_z (P_det) and a simple ratio MTT_max/MRT_B (K_det), leading to introduction of the unitless product-ratio plot for determination of the limiting condition of λ_z in linear PK. We found that the MRT_Bλ_z value of 0.5 serves as a practical threshold determining whether λ_z is more closely associated with 1/MRT_B or 1/MTT_max. The current theory was applied for assessment of the terminal slope λ_z for observed PK data of various compounds in man and rat.

Introduction

The biological half-life (T_1/2) is typically defined as the time required for the concentration of therapeutic agents in the body or plasma to fall by half. The T_1/2 is a crucial parameter in clinical pharmacology and pharmacotherapeutics since this parameter is one of the major determinants of treatment regimens (e.g., dosing frequency) and the time required to reach the steady-state concentration (e.g., 4 times T_1/2 for 93.8% C_SS) with repeated drug administration [1]. During multiple dosing, T_1/2 is also practically useful for predicting the extent of drug accumulation and the washout rate after termination of dosing [2]. In pharmacokinetic (PK) analyses, T_1/2 is usually determined from the terminal phase in plasma after drug administration, which is readily calculated from the relationship T_1/2 = ln2/λ_z where λ_z is the terminal-phase slope.

Due to the physiological and anatomical complexity of the body, however, the kinetic mechanisms underlying the terminal slope λ_z emerging in plasma concentration-time relationships are not explicitly understood. One of the most ideal cases of a physiological one-compartment model (1CM) would be that (i) the drug distribution space is limited to the plasma and (ii) its elimination occurs directly from the systemic circulation (e.g., metabolized by esterases in the blood and/or excreted by glomerular filtration without renal reabsorption). In this case, the PK profile will exhibit a mono-exponential decline where λ_z is described by the ratio of the systemic clearance (CL) to the volume of distribution (V_d; the plasma volume for this example), or the inverse of the mean residence time in the body (MRT_B) [3]. If the drug disposition kinetics are consistent with a 1CM, therefore, the product of MRT_B and λ_z will be unity. However, drugs usually exhibit distribution to peripheral tissues (i.e., V_d now defined as the steady-state volume of distribution, V_SS) and λ_z becomes mathematically no longer the same as 1/MRT_B or CL/V_SS (i.e., rather CL/V_β where V_β is the terminal-phase volume of distribution) [4]. Although it was suggested that λ_z values under the limited condition (e.g., V_SS ≈ V_β) are closely associated with 1/MRT_B [5], this approach based on a 1CM assumption is not sufficient for explicit understanding of λ_z in more general drug cases.

In the application of standard moment analysis, T_1/2 and λ_z along with CL and V_SS are calculated based on the assumption of linear PK (e.g., biopharmaceutical and physiological processes mediated by the first-order kinetics) [6]. Recently, whole-body physiologically based pharmacokinetic (WB-PBPK) models were examined with respect to the expression of the system matrix of simultaneous differential equations [7, 8]. By a rational tissue lumping approach, mathematically complicated WB-PBPK models can be expressed as significantly less complex equations in a form of minimal PBPK (mPBPK) models [9]. The shape of plasma concentration-time profiles (e.g., the number of exponential phases) appears to be dependent on the ratio of the mean transit time (MTT) through the kinetically largest tissue (MTT_max) to MRT_B [7, 8]. In a multi-compartment mammillary form of WB-PBPK model structure, a theoretical upper bound of λ_z was found to be the inverse of MTT_max (i.e., Browne’s theorem) [10] while biopharmaceutical conditions are not clear when λ_z becomes limited by 1/MTT_max. Although MTT_max is a tissue kinetic parameter that may be difficult to identify solely from plasma concentration profiles, the limiting condition of λ_z if determinable only with plasma data without utilizing PBPK models would provide insight for explicit understanding of the terminal-slope T_1/2.

This study proposes a theoretical method for determination of the limiting condition of λ_z utilizing fundamental PK parameters obtainable from plasma data. To interrelate λ_z and physiologically relevant disposition kinetic parameters, mathematical derivations are provided for basic two- and three-compartment mPBPK models as well as a 10-compartment WB-PBPK model with key relationships parameterized as mean transit and residence times (e.g., MTT_max and MRT_B). In such PBPK models that are consistent with the kinetic assumptions considered for non-compartmental analysis (NCA), we found that the limiting condition of λ_z between 1/MRT_B and 1/MTT_max can be determined in general based on (i) the product of MRT_B and λ_z (P_det) that are readily obtainable from NCA (i.e., for top-down kinetic analyses), and/or (ii) the ratio of MTT_max to MRT_B (K_det) that can be calculated from WB-PBPK model parameters (i.e., for bottom-up kinetic analyses). In addition, we propose an approximation method to estimate a model-independent MTT_max from NCA of plasma data, which was found to be useful when the predicted MTT_max is larger than MRT_B. The current theory was applied for assessment of the terminal slope λ_z from the observed PK data of 1352 (in man) [11] and 113 (in rat) [8] compounds in the literature.

Methods

General assumptions considered in typical NCA include (i) linear PK and (ii) elimination occurring from plasma. In practice, NCA is carried out using observed PK data based on its SHAM (i.e., slope, height, area, and moment) properties: λ_z (as a slope) is simply obtained from data points located in the terminal phase while MRT_B is determined as the ratio of the area under the first moment curve AUMC to the area under the curve AUC (i.e., AUMC/AUC). Although NCA is a model-independent method that relies on intact observed data, PK models that comply with the two assumptions above are needed to interrelate λ_z and disposition kinetic parameters of such models. In a physiological system having a multi-compartmental kinetic behavior, typical plasma PK profiles eventually show an exponential decline in the terminal phase (with the slope λ_z) after reaching drug equilibrium between plasma and peripheral tissues. Therefore, we considered physiological model structures to obtain a mathematical expression of λ_z for such PBPK systems that would be useful for determination of the limiting condition of λ_z based on NCA parameters.

Two-Compartment Model

One of the simplest forms of a physiological two-compartment model (2CM) is a one-tissue mPBPK model [9] consisting of the blood pool and one tissue compartment (Fig. 1a). Based on the assumptions of (i) drug elimination from the blood pool and (ii) instantaneous drug distribution within blood, differential equations for the two compartments considered in a one-tissue mPBPK model are:

$$V_B{R}_b\frac{d{C}_p}{dt}=-Q_{CO}\cdot f_d\cdot R_b\cdot \left(C_p-\frac{C_T}{K_p}\right)-CL\cdot C_p$$ (1a)

$$V_T\frac{d{C}_T}{dt}=Q_{CO}\cdot f_d\cdot R_b\cdot \left(C_p-\frac{C_T}{K_p}\right)$$ (1b)

where V_B and V_T are the anatomical volumes of the blood and peripheral tissue; C_p and C_T are the plasma and tissue concentrations; Q_CO is the cardiac output; and R_b and K_p are the blood-to-plasma and tissue-to-plasma partition coefficients. Utilizing a fractional distribution parameter (f_d) in mPBPK [9] and WB-PBPK [12] models, the apparent distributional clearance to the tissue (CL_app) is expressed as the product of the blood flow (e.g., Q_CO) and f_d so that the tissue MTT for drugs with tissue permeability limitations can be simply described as the division of the apparent volume of distribution to the tissue (i.e., V_TK_p/R_b) by CL_app [7, 8]. According to the theory for a typical 2CM [13], therefore, the mean transit times through the central blood (MTT_c) and tissue (MTT_T) compartments along with the mean residence times in the body (MRT_B) and the central compartment (MRT_c) can be expressed by using the PBPK parameters as:

$$MT{T}_c=\frac{V_B}{CL/R_b+Q_{CO}{f}_d}$$ (2a)

$$MT{T}_T=\frac{V_T{K}_p}{Q_{CO}{f}_d{R}_b}$$ (2b)

$$MR{T}_B=\frac{V_B{R}_b+V_T{K}_p}{CL}=\frac{V_{SS}}{CL}$$ (2c)

$$MR{T}_c=\frac{V_B{R}_b}{CL}$$ (2d)

Fig. 1.

Structures of a one-tissue mPBPK, b two-tissue mPBPK, and c 10-compartment WB-PBPK models. Intravenous bolus input and drug elimination from the blood compartment were considered. All symbols are defined with description of the equations in the text and in Supplementary Table S1

One-tissue mPBPK models have an analytical solution for C_p consisting of two exponential functions for an impulse input (e.g., $C_p(t)=C_1{e}^{-{\lambda }_{1}t}+C_2{e}^{-{\lambda }_{2}t}$ where λ₁ > λ₂). Utilizing the system matrix representing Eq. 1a and b among the possible methods to obtain the terminal slope λ₂ (e.g., Laplace transformation of Eq. 1a and b as an alternative), a quadratic equation (Eq. 3a; 2 solutions represented as λ) can be solved to express the terminal slope λ₂ as (Eq. 3b):

$$\frac{\frac{Q_{CO}{f}_d}{MT{T}_T}}{\lambda -\frac{1}{MT{T}_T}}=V_B\left(\lambda -\frac{1}{MT{T}_c}\right)$$ (3a)

$$\begin{gathered} {\lambda }_2=\frac{1}{2}\left(\frac{1}{MT{T}_c}+\frac{1}{MT{T}_T}-\sqrt{{\left(\frac{1}{MT{T}_c}+\frac{1}{MT{T}_T}\right)}^2-4\frac{1}{MT{T}_T}\cdot \frac{1}{MR{T}_c}}\right) \\ =\frac{1}{2MR{T}_c}\left(1+\frac{MR{T}_B}{MT{T}_T}-\sqrt{{\left(1+\frac{MR{T}_B}{MT{T}_T}\right)}^2-4\frac{MR{T}_c}{MT{T}_T}}\right) \end{gathered}$$ (3b)

When rearranged with respect to the product of MRT_B and λ₂ (i.e., λ_z for one-tissue model):

$$MR{T}_B{\lambda }_2=\frac{MR{T}_B}{2MR{T}_c}\left(1+\frac{MR{T}_B}{MT{T}_T}-\sqrt{{\left(1+\frac{MR{T}_B}{MT{T}_T}\right)}^2-4\frac{MR{T}_B}{MT{T}_T}\frac{MR{T}_c}{MR{T}_B}}\right)$$ (4)

Based on Eq. 4, the product MRT_Bλ₂ is now expressed as a function of two determinants, namely, MRT_B/MRT_c (i.e., V_SS/V_BR_b, which is larger than 1) and MTT_T/MRT_B.

Limiting Conditions of λ_z for Two-Compartment Model

The next objective is to identify the biopharmaceutical meaning of MRT_Bλ_z (e.g., MRT_Bλ₂ for 2CM) in the interpretation of the terminal slope or half-life. Therefore, when Eq. 4 is further rearranged:

$$MR{T}_B{\lambda }_2=\frac{MR{T}_B}{2MR{T}_c}\cdot \left(1+\frac{MR{T}_B}{MT{T}_T}\right)\cdot \left(1-\sqrt{1-4\frac{MT{T}_T\cdot MR{T}_c}{{\left(MR{T}_B+MT{T}_T\right)}^2}}\right)$$ (5)

For the case of MTT_T ≪ MRT_B, a Taylor series expansion of $\sqrt{1-x}(\text{i}.\text{e}.,1-x/2-x^2/8-\dots \text{; when }\left|x\right|<1)$ can be applied for Eq. 5, which converges as:

$$MR{T}_B{\lambda }_2=\frac{MR{T}_B}{2MR{T}_c}\cdot \left(1+\frac{MR{T}_B}{MT{T}_T}\right)\cdot \left(2\frac{MT{T}_T\cdot MR{T}_c}{{\left(MR{T}_B+MT{T}_T\right)}^2}+\dots \right)\approx \frac{MR{T}_B}{MR{T}_B+MT{T}_T}\to 1$$ (6)

where this 2CM structure (Fig. 1a) becomes kinetically close to a 1CM (i.e., 1/MRT_B as a left-hand limit of λ₂). On the other end, under the condition of MRT_c ≤ MRT_B ≪ MTT_T, the same expansion principle can also apply, leading from Eq. 5 to:

$$MR{T}_B{\lambda }_2\approx \frac{MR{T}_B}{MR{T}_B+MT{T}_T}\to \frac{MR{T}_B}{MT{T}_T}$$ (7)

where the terminal slope λ₂ now converges to 1/MTT_T (i.e., also as a left-hand limit of λ₂). In this case, λ₂ appears distinct from 1/MRT_B in the plasma concentration profiles where a consideration of an additional peripheral compartment is necessary to explain this distinct exponential phase.

In line with Eqs. 6 and 7, it is noteworthy that the boundary conditions of λ₂ as a right-hand limit are also found to be 1/MRT_B and 1/MTT_T when the same mathematical principle considered for a three-compartment model (3CM; see below) is applied for the 2CM. Based on Eq. 4, the limiting conditions of λ₂ between 1/MRT_B and 1/MTT_T can be numerically/graphically presented in an MRT_Bλ₂ versus MTT_T/MRT_B plot depending on the condition of MRT_B/MRT_c (see “Results”).

Three-Compartment Model

As shown in Fig. 1b, a physiological 3CM was considered based on a two-tissue mPBPK model consisting of the blood pool and two peripheral tissue compartments. Analogously, differential equations for the blood pool and two tissue compartments (denoted as subscripts 1 and 2) can be written as:

$$V_B{R}_b\frac{d{C}_p}{dt}=-Q_1\cdot f_{d1}\cdot R_b\cdot \left(C_p-\frac{C_1}{K_{p1}}\right)-Q_2\cdot f_{d2}\cdot R_b\cdot \left(C_p-\frac{C_2}{K_{p2}}\right)-CL\cdot C_p$$ (8a)

$$V_1\frac{d{C}_1}{dt}=Q_1\cdot f_{d1}\cdot R_b\cdot \left(C_p-\frac{C_1}{K_{p1}}\right)$$ (8b)

$$V_2\frac{d{C}_2}{dt}=Q_2\cdot f_{d2}\cdot R_b\cdot \left(C_p-\frac{C_2}{K_{p2}}\right)$$ (8c)

where the symbols are consistent with Eq. 1a and b. The mean transit times through the blood pool (MTT_c) and Tissues 1 and 2 (MTT₁ and MTT₂) along with MRT_B are expressed as:

$$MT{T}_c=\frac{V_B}{CL/R_b+Q_1{f}_{d1}+Q_2{f}_{d2}}$$ (9a)

$$MT{T}_1=\frac{V_1{K}_{p1}}{Q_1{f}_{d1}{R}_b}$$ (9b)

$$MT{T}_2=\frac{V_2{K}_{p2}}{Q_2{f}_{d2}{R}_b}$$ (9c)

$$MR{T}_B=\frac{V_B{R}_b+V_1{K}_{p1}+V_2{K}_{p2}}{CL}=\frac{V_{SS}}{CL}$$ (9d)

For an analytical solution of a two-tissue mPBPK model after a bolus input (i.e., $C_p(t)=C_1{e}^{-{\lambda }_{1}t}+C_2{e}^{-{\lambda }_{2}t}+C_3{e}^{-{\lambda }_{3}t}$ where λ₁ > λ₂ > λ₃), the 3 slopes (represented as λ) are determined by [7, 8]:

$$\frac{\frac{Q_1{f}_{d1}}{MT{T}_1}}{\lambda -\frac{1}{MT{T}_1}}+\frac{\frac{Q_2{f}_{d2}}{MT{T}_2}}{\lambda -\frac{1}{MT{T}_2}}=V_B\left(\lambda -\frac{1}{MT{T}_c}\right)$$ (10)

which can be obtained from the system matrix representing Eq. 8a to c. It is noteworthy that the number of fraction terms in the left-hand side of Eq. 10 is the same as the number of peripheral compartments. Based on Browne’s theorem [10], which was further assessed for mammillary models by Sheppard and Householder [14] and Vaughan and Dennis [15], it is noted that the 3 roots of Eq. 10 (i.e., the eigen-values of the system matrix for mammillary models) are separated by the reciprocal form of two tissue MTTs (i.e., λ₃ < 1/MTT₂ < λ₂ < 1/MTT₁ < λ₁): Thus, λ₃ is determined in the range lower than 1/MTT₂ (i.e., the theoretical upper bound of λ₃). For the range of λ < 1/MTT₂, Eq. 10 can be expanded by a Taylor series as:

$$-Q_1{f}_{d1}\cdot \left(1+MT{T}_1\lambda +\dots \right)-Q_2{f}_{d2}\cdot \left(1+MT{T}_2\lambda +\dots \right)=V_B\lambda -Q_1{f}_{d1}-Q_2{f}_{d2}-CL/R_b$$ (11)

When λ is sufficiently lower than 1/MTT₂ (i.e., MTT₂λ → 0 and MTT₁λ → 0), the expression of λ₃ can be rearranged to:

$$\underset{MT{T}_2\lambda \to 0}{\text{lim}}{\lambda }_3=\underset{MT{T}_2\lambda \to 0}{\text{lim}}\frac{CL/R_b-{\sum }_{i=1,2}{Q}_i{f}_{d,i}\cdot \left[{\left(MT{T}_i\lambda \right)}^2+{\left(MT{T}_i\lambda \right)}^3+\dots \right]}{V_{SS}/R_b}=\frac{1}{MR{T}_B}$$ (12)

indicating that the theoretical bound of λ₃ is now found to be 1/MRT_B (i.e., as a right-hand limit of λ₃). It is noteworthy that the boundary conditions of λ₃ as a left-hand limit can also be readily found as 1/MRT_B and 1/MTT₂ based on the same mathematical principle applied for multi-compartment models (see below). Collectively, the boundary conditions of the terminal slope are consistent between the 2CM (1/MRT_B and 1/MTT_T) and 3CM (1/MRT_B or 1/MTT₂).

However, an explicit expression of λ₁, λ₂, or λ₃ solved from Eq. 10 (i.e., a cubic equation) appears mathematically too complex to identify which root represents the terminal slope λ_z because of the involvement of the imaginary number i [16]. Alternatively, therefore, seven PBPK parameters (i.e., Q₁, f_d1, f_d2, V₁, K_p1, K_p2, and CL, while V₂ is back calculated by subtraction of V_B and V₁ from body weight; and Q₂ is calculated as Q_CO − Q₁) were assigned a given set of numbers to generate a variety of PK datasets (i.e., 9 values evenly spaced on a log scale assigned for each parameter; 9⁷ = 4782969 cases; see Supplementary Table S1). Utilizing these diverse sets of PK parameters, Eq. 10 was numerically solved to obtain λ₁, λ₂, or λ₃ so that the product of the resulting λ₃ and MRT_B (Eq. 9d) can be plotted against MTT₂/MRT_B.

Multi-Compartment Model

Due to the physiological and anatomical complexity of the body, WB-PBPK models consisting of physiological tissue compartments (i.e., multi-compartment model) are widely utilized in PK analyses. Since we were interested in assessing the limiting condition of λ_z based on NCA of observed plasma concentration-time data, we assumed that a poly-exponential function of WB-PBPK models would sufficiently represent the experimental PK profiles.

To assess the generalizability of the current approach of MRT_Bλ_z from the mPBPK models to standard moment analyses, therefore, we considered a WB-PBPK structure in a form of a mammillary compartment model (e.g., consisting of the blood pool and 9 peripheral tissues, i.e., adipose, bone, brain, heart, kidney, lung, muscle, skin, and a splanchnic compartment; Fig. 1c). Based on the assumption of drug elimination occurring from the blood (i.e., consistent with NCA of the plasma PK), the 10 slopes (represented as λ) of the analytical solution of the model (i.e., $C_p(t)={\sum }_{i=1}^{10}{C}_i{e}^{-{\lambda }_{i}t}$) are determined by [7, 8]:

$$\frac{\frac{Q_1{f}_{d1}}{MT{T}_1}}{\lambda -\frac{1}{MT{T}_1}}+\frac{\frac{Q_2{f}_{d2}}{MT{T}_2}}{\lambda -\frac{1}{MT{T}_2}}+\cdots +\frac{\frac{Q_9{f}_{d9}}{MT{T}_9}}{\lambda -\frac{1}{MT{T}_9}}=V_B\left(\lambda -\frac{1}{MT{T}_c}\right)$$ (13)

where MTT_i is the mean transit time through the ith tissue (e.g., MTT₁ < … < MTT₉). Analogously, the number of fraction terms in the left-hand side of Eq. 13 is the same as the number of peripheral compartments. Consistent with Browne’s theorem, the 10 solutions of Eq. 13 are separated by the inverse of MTT_i (i.e., λ₁₀ < 1/MTT₉ < λ₉ < … < λ₂ < 1/MTT₁ < λ₁). Based on the same principle applied for the 3CM (i.e., Eq. 12 and Browne’s theorem for right-hand limits) and Appendix A (for left-hand limits), it is readily provable from Eq. 13 that the two theoretical bounds of λ₁₀ are 1/MRT_B and 1/MTT₉.

However, an explicit solution for λ₁₀ appears to be difficult to obtain from Eq. 13. Different from the case of the 3CM, creation of diverse sets of PBPK parameters (e.g., 19 parameters such as CL, K_p,i, and f_d,i for i = 1, …, 9; 9¹⁹ cases) requires extremely high computational costs. Instead, we considered an approximation approach for an explicit expression of λ₁₀ in the multi-compartment model, as described next.

Mathematical Approximation for the Terminal Slope of a Multi-Compartment Model

When the left-hand side of Eq. 13 is defined as f(λ), f(λ) satisfies the mathematical properties in the range of 0 < λ < 1/MTT₉, such as:

$$\underset{MT{T}_9\lambda \to 0}{\text{lim}}f(\lambda )=-\sum _{i=1}^9{Q}_i{f}_{d,i}$$ (14a)

$$\underset{MT{T}_9\lambda \to 0}{\text{lim}}{f}^{\prime }(\lambda )=-\sum _{i=1}^9{V}_i{K}_{p,i}/R_b$$ (14b)

$$\underset{MT{T}_9\lambda \to 1}{\text{lim}}f(\lambda )=-\infty$$ (14c)

Analogously, we introduced a function g(λ) in a mathematically simplified form (e.g., a + b/(λ − c)) that satisfies the three conditions above, expressed as:

$$g(\lambda )=\frac{\frac{\sum V_i{K}_{p,i}}{MT{T}_9^2\cdot R_b}}{\lambda -\frac{1}{MT{T}_9}}+\frac{\sum V_i{K}_{p,i}}{MT{T}_9\cdot R_b}-\sum Q_i{f}_{d,i}$$ (15)

When g(λ) is equated with the right-hand side of Eq. 13, an approximated λ₁₀ (i.e., λ_10,app, which falls within the range lower than 1/MTT₉) can be calculated using:

$${\lambda }^2-\frac{1}{MR{T}_c}\left(1+\frac{MR{T}_B}{MT{T}_9}\right)\lambda +\frac{1}{MR{T}_c}\frac{1}{MT{T}_9}=0$$ (16a)

$${\lambda }_{10,app}=\frac{1}{2MR{T}_c}\left(1+\frac{MR{T}_B}{MT{T}_9}-\sqrt{{\left(1+\frac{MR{T}_B}{MT{T}_9}\right)}^2-4\frac{MR{T}_B}{MT{T}_9}\frac{MR{T}_c}{MR{T}_B}}\right)$$ (16b)

where MRT_c is consistent with Eq. 2d, and MRT_B is V_SS/CL (i.e., V_SS = V_BR_b + ∑ V_iK_p,i). It is noted that Eq. 16b is structurally equivalent to Eq. 3b where MTT₉ (or more generally, MTT_max) in the multi-compartment model takes a role of MTT_T in the 2CM. Therefore, if the numerical error arising from this approximation between f(λ) and g(λ) in terms of the terminal slope λ₁₀ (or λ_z) is negligible, the limiting condition of λ_z between 1/MRT_B and 1/MTT_max would be equally determined from Eq. 16b based on the product MRT_Bλ_z and the ratio MTT_max/MRT_B, analogous to Eqs. 6 and 7 shown for the 2CM.

Estimation of MTT_max from the Plasma Concentration-Time Data

When plasma concentration-time data are available, the PK parameters (i.e., CL, V_SS, and λ_z) can be determined from NCA. As an extension, the parameter of interest to be estimated from the plasma data is now MTT₉ or MTT_max. Assuming that the error of the current approximation is negligible in the 10-compartment model (i.e., λ₁₀ ≅ λ_10,app), a predicted MTT₉ (i.e., MTT_9,pred) can be calculated by rearrangement of Eq. 16a as:

$${MTT}_{9,\mathit{\text{pred}}}=\frac{1}{{\lambda }_{10}}\cdot \frac{1-MR{T}_B{\lambda }_{10}}{1-MR{T}_c{\lambda }_{10}}=\frac{1}{{\lambda }_{10}}\cdot \frac{CL-V_{SS}\cdot {\lambda }_{10}}{CL-V_B{R}_b\cdot {\lambda }_{10}}$$ (17)

or more generally,

$$MT{T}_{max,\mathit{\text{pred}}}=\frac{1}{{\lambda }_z}\cdot \frac{CL-V_{SS}\cdot {\lambda }_z}{CL-V_B{R}_b\cdot {\lambda }_z}$$ (18)

Thus, if the plasma concentration-time relationship that allows the calculation of CL, V_SS, and λ_z is available along with the volume of the central compartment (e.g., V_BR_b or Dose/C_p(0) extrapolated from the initial data points), the tissue kinetic parameter MTT_max can be estimated by Eq. 18 without considering PK models.

Assessment of the Validity of the Mathematical Approximations

The current approximation approaches considered for the terminal slope of the 3CM (Eq. 29b; see Appendix B) and multi-compartment model (Eq. 16b) result in a mathematical relationship equivalent to Eq. 3b for a 2CM. Based on these relationships, a practically applicable method was developed for estimation of MTT_max from only plasma data (Eqs. 30 and 18). To first assess the validity of this approach for a 3CM, comparisons were made between λ₃ (numerically solved from Eq. 10) and λ_3,app (Eq. 29b), and between MTT₂ (Eq. 9c) and MTT_2,pred (Eq. 30), utilizing the seven PBPK parameters generated for the two-tissue mPBPK models for rat and man (i.e., 9⁷ cases each; Supplementary Table S1).

To validate the current approach for the case of a multi-compartment model circumventing the computational costs, a physiologically plausible set of WB-PBPK models were produced from a series of in silico predictions for PBPK parameters. Since the lumping of (i) vein and artery into the blood pool and (ii) gut, liver, and spleen into a splanchnic compartment appears to have insignificant effects on the shape of the plasma concentration-time relationship [7, 8], we regarded the current 10-compartment mammillary model (Fig. 1c) as representative of a WB-PBPK model structure. Physiological input parameters (i.e., tissue volume and blood flow) used for the model calculations [12, 17] are summarized in Supplementary Table S2. For the case of rat, CL values observed and physicochemical properties for 113 model compounds that were dosed intravenously were obtained from Jeong et al. [8], and K_p values for the 9 tissue compartments were calculated based on the Rodgers and Rowland method [18, 19]. For the case of man, CL values and/or physicochemical properties of 1352 compounds that were dosed intravenously are available in Lombardo et al. [11]. Due to the absence of information on their ionization status in the literature (i.e., pKa), however, we generated a diverse set of K_p values for the 9 tissue compartments in man based on the Poulin and Theil method [17, 20], assuming the same value between logP_vo:w and logD_vo:w (i.e., vegetable oil-to-water partition (non-ionized species) and distribution (non-ionized/ionized species) coefficients) for the calculation of the adipose K_p.

For the calculation of a tissue permeability coefficient P, an empirical correlation between the physicochemical properties and a parallel artificial membrane permeability assay P (i.e., P_app,PAMPA, with a unit of cm/s; 2% phosphate-dylcholine in dodecane) value was used [8]:

$$\text{log\hspace{0.17em}}{P}_{app,\mathit{\text{PAMPA}}}=-10.1039+0.1663\cdot \text{logP}-0.2158\cdot \left|\text{logP}-\text{logD}\right|+1.8332\cdot \text{log\hspace{0.17em}}MW+0.3797HA+0.0801H{D}^2-0.0453\cdot \mathit{\text{TPSA}}$$ (19)

where logP and logD are the octanol-to-water partition and distribution coefficients, MW is molecular weight, HD and HA are the number of hydrogen bond donors and acceptors, and TPSA is the topological polar surface area (Ų). Utilizing P_app,PAMPA values, two distribution models (i.e., Models 1 and 2 referred to as Tube and Jar models) were considered in the calculation of f_d [12], which are expressed as:

$$f_{d,\mathit{\text{Model}}1}=1-e^{-\frac{f_{up}P{S}_{eff}}{Q_T{R}_b}}$$ (20a)

$$f_{d,\mathit{\text{Model}}2}=\frac{f_{up}P{S}_{eff}}{Q_T{R}_b+f_{up}P{S}_{eff}}$$ (20b)

where the effective surface area values (S_eff) for each tissue normalized by tissue weight (in a unit of cm²/g tissue) [12] are assumed to be consistent across species (Supplementary Table S2). The free fraction in plasma (f_up) was calculated based on empirical correlations depending on the charge states of compounds [21], which was subsequently used for estimation of R_b utilizing empirical relationships [22]:

$$\text{log\hspace{0.17em}}{K}_b=0.617\cdot \text{log\hspace{0.17em}}\left[\left(1-f_{up}\right)/f_{up}\right]+0.208$$ (21a)

$$R_b=\left(K_b\cdot f_{up}-1\right)\cdot Hct+1$$ (21b)

where K_b is the ratio of the drug concentration in red blood cells to that in plasma water, and Hct is the hematocrit (0.45). Assuming that a given set of in silico data (i.e., 226 cases for rat; and 2674 cases for man, except for 15 compounds whose physicochemical properties needed were not fully available) is sufficient for assessment of the approximation, comparisons were conducted between λ₁₀ (numerically computed from Eq. 13) and λ_10,app (Eq. 16b), and between MTT₉ (the longest MTT) and MTT_9,pred (Eq. 17). All numerical calculations of PBPK parameters were conducted by Python^™ 3.8.3 (www.python.org) using Numpy and Scipy libraries.

Application of the Product MRT_Bλ_z for PK Analysis of Various Compounds in Man and Rat

Based on the simple product of MRT_B and λ_z, we also applied our method of determining the limiting condition of λ_z for observed PK data of various compounds. From NCA, the CL, V_SS, and/or T_1/2 were available for 1352 compounds in man [11] and 113 compounds in rat [8]. Therefore, MRT_B (i.e., V_SS/CL) and λ_z (i.e., ln2/T_1/2) were used to calculate the MRT_Bλ_z product for those compounds in man and rat. For the estimation of MTT_max, however, additional information (V_B and R_b) is required for the application of Eq. 18. Since compound-specific R_b values were not completely available, we used the plasma volume V_p as the volume of the central compartment (i.e., 3 L for 70 kg man and 7.8 mL for 0.25 kg rat) [23] considering red blood cells as one of the potential distribution spaces, instead of using V_BR_b in Eq. 18. Utilizing the experimental datasets in man and rat, MRT_Bλ_z was plotted against MTT_max,pred/MRT_B for an intuitive understanding of λ_z.

Results

MRT_Bλ₂ for Two-Compartment Model

For the case of the 2CM in the form of a one-tissue mPBPK, the relationship between MRT_Bλ₂ and MTT_T/MRT_B (Eq. 4) can be graphically presented, depending on MRT_B/MRT_c (Fig. 2a, named as “the unitless product-ratio plot”). Under the condition of MRT_B/MRT_c = 1, the MRT_Bλ₂ appears to be consistently (i) the value of 1 when MTT_T/MRT_B is smaller than 1 and (ii) the ratio MRT_B/MTT_T when MTT_T/MRT_B is larger than 1, although the MTT_T of this imaginary tissue compartment is not rationally definable under this condition (i.e., 1CM). At the other end, an extremely high extent of distribution (i.e., MRT_B/MRT_c = ∞) leads to the relationship MRT_Bλ₂ = 1/(1 + MTT_T/MRT_B) (see Eq. 6 for derivation). Based on these boundary relationships dependent on the two extreme conditions of MRT_B/MRT_c, it was found that MRT_Bλ₂ goes to 1 when MTT_T ≪ MRT_B, whereas MRT_Bλ₂ converges to MRT_B/MTT_T when MRT_B ≪ MTT_T.

Fig. 2.

The unitless product-ratio plot for a two- (2CM) and b three-compartment models (3CM), with the Y-axis plotted on rectangular (upper panels) and logarithmic (lower panels) coordinates. a For a 2CM, a theoretical curve for the product MRT_Bλ₂ and the ratio MTT_T/MRT_B can be defined (Eq. 4) depending on MRT_B/MRT_c values from 1 (blacked dashed) to infinity (red dashed). b For a 3CM, a variety of PK datasets (9⁷ = 4782969 cases) were generated to plot the theoretical points of the product MRT_Bλ₃ and the ratio MTT₂/MRT_B depending on the range of MRT_B/MRT_c

In terms of the plasma data analysis, Fig. 2a demonstrates that if MRT_Bλ₂ is observed to be 1, the plasma concentration-time profile would essentially show a mono-exponential decline with the slope λ₂ of 1/MRT_B (i.e., a kinetically negligible MTT_T). Whereas, λ₂ converges to the inverse of MTT_T when MRT_Bλ₂ is sufficiently close to 0 (i.e., λ₂ being distinct from 1/MRT_B). Thus, in line with the theoretical convergences of λ₂ depending on MTT_T/MRT_B (e.g., Eqs. 6 and 7), Fig. 2a also graphically indicates that the two boundary conditions of λ₂ (1/MRT_B and 1/MTT_T) can be determined depending on MRT_Bλ₂ in the 2CM. All kinetic factors (MRT_Bλ₂ and MTT_T/MRT_B) used for the unitless product-ratio plot originate from time-related constants, indicating that the current approach is applicable across species.

MRT_Bλ_z for Three- and Multi-Compartment Models

Analogously, the theoretical bounds of λ₃ in the 3CM as a right-hand limit were found to be 1/MRT_B (Eq. 12) and 1/MTT₂ [10]. In addition, the boundary conditions of λ₃ as a left-hand limit were also found to be 1/MRT_B and 1/MTT₂ (Appendix A where the longest MTT is herein MTT₂). For numerical assessment of the 3CM, a diverse set of two-tissue mPBPK models (i.e., 9⁷ cases; Supplementary Table S1) was produced to calculate MRT_Bλ₃ and MTT₂/MRT_B (Fig. 2b). When the points plotted in Fig. 2b are compared with theoretical curves of Fig. 2a for the 2CM, the cases with the low MRT_B/MRT_c values (e.g., < 1.5) in Fig. 2b produced the points around the curves plotted in Fig. 2a at MRT_B/MRT_c values of 1.1 and 1.5. Whereas, the higher MRT_B/MRT_c values (e.g., > 3) produced the points that fall within a more variable range in Fig. 2b by the kinetics of an additional compartment (Tissue 1). Importantly, it should be noted that all the points of Fig. 2b fall within the range bordered by the upper black (MRT_B/MRT_c = 1) and lower red (MRT_B/MRT_c = ∞) dashed lines/curves of Fig. 2a for the 2CM, indicating that an addition of one more peripheral compartment (that has a shorter MTT) to the 2CM structures does not produce any points that fall outside the specified range of the unitless product-ratio plot.

When the unitless product-ratio plot was generated for multi-compartment models using the theoretical PK datasets for rat and man (i.e., 226 and 2674 cases), all the points of MRT_Bλ₁₀ versus MTT₉/MRT_B were also found to fall within the specified range of the plot (Supplementary Fig. S1), consistent with the cases of the 2CM and 3CM (Fig. 2a and b). Collectively, therefore, we inductively reasoned that all the points of MRT_Bλ_z versus MTT_max/MRT_B with linear PK in general would fall within the range bordered by the lower (i.e., MRT_Bλ_z = 1/(1 + MTT_max/MRT_B)) and upper (i.e., MRT_Bλ_z = 1 when MTT_max < MRT_B and MRT_Bλ_z = MRT_B/MTT_max when MRT_B < MTT_max) boundary lines/curves.

Numerical Assessment of the Current Approximation Approaches for the Terminal-Phase Slope in Multi-Compartment Models

Due to the mathematical difficulties present in obtaining an analytical solution for the terminal-phase slope of the 3CM and multi-compartment model, we considered an alternative approach for acquisition of an (approximately) explicit expression of the terminal slope. When utilizing the seven PBPK parameters for rat and man (i.e., 9⁷ datasets each) in the 3CM, λ₃ numerically solved from Eq. 10 appears to be consistent with λ_3,app calculated from Eq. 29b (Fig. 3a). When the fold-difference (λ_3,app/λ₃) is plotted against MTT₂/MRT_B, the approximation error for both rat and man appears to be the highest at the MTT₂/MRT_B value of 1 while all the λ_3,app/λ₃ values fall within a factor of 2. However, the approximation error (λ_3,app/λ₃ ranges in parentheses) becomes insignificant when MTT₂/MRT_B < 0.1 (from 0.924 to 1.0 for rat and from 0.919 to 1.0 for man) and MTT₂/MRT_B > 10 (from 0.920 to 1.0 for rat and 0.916 to 1.0 for man).

Fig. 3.

a Comparisons between the approximated terminal slope λ_3,app and the numerically solved λ₃ (left) and their fold-difference (λ_3,app/λ₃) depending on MTT₂/MRT_B (right), utilizing 9⁷ datasets for the three-compartment model (Supplementary Table S1). b Comparisons between the approximated terminal slope λ_10,app and the numerically solved λ₁₀ (left) and their fold-difference (λ_10,app/λ₁₀) depending on MTT₉/MRT_B (right), utilizing a physiologically plausible set of PBPK parameters for the 10-compartment model (Supplementary Table S2)

Based on the assumption that a poly-exponential function of multi-compartment WB-PBPK models would sufficiently represent the experimental PK profiles, we also assessed the appropriateness of the current approximation for the terminal slope, comparing λ₁₀ (i.e., numerically solved from Eq. 13) with the approximated λ_10,app (Eq. 16b). As a result, λ_10,app values are fairly consistent with λ₁₀ values based on their fold-differences (λ_10,app/λ₁₀) between 0.748 and 1.0 (for rat) and between 0.777 and 1.0 (for man) (Fig. 3b). When λ_10,app/λ₁₀ is plotted against MTT₉/MRT_B, the approximation error (λ_10,app/λ₁₀ ranges in parentheses) appears the highest at MTT₉/MRT_B = 1 while being insignificant when MTT₉/MRT_B < 0.1 (from 0.959 to 0.993 for rat and from 0.951 to 1.0 for man) and MTT₉/MRT_B > 10 (from 0.947 to 0.996 for rat and 0.937 to 1.0 for man). Assuming that the approximation error of the current approach is negligible (i.e., within a factor of 1.34 at most), we found that λ_10,app (Eq. 16b) is useful for calculation of MRT_Bλ₁₀ and thereby for assessment of the limiting condition of λ₁₀ between 1/MRT_B and 1/MTT₉ depending on MTT₉/MRT_B, based on the same principle applied for Eq. 4 in the 2CM.

Estimation of MTT_max from Plasma Concentration-Time Data

When only a plasma concentration profile is available, the tissue kinetic parameter MTT_max may be difficult to accurately determine. Since Eqs. 29b and 16b appear to be useful for estimation of the terminal slope in the 3CM and multi-compartment models (Fig. 3a and b), these relationships can be rearranged for the estimation of MTT_max in NCA of the plasma data. Due to the usual absence of the MTT_max information in the actual PK analysis, the ratio MTT_max,pred/MRT_B could be regarded as a useful alternative of MTT_max/MRT_B for assessment of plasma concentration-time profiles.

To evaluate the predictability of Eqs. 30 (MTT_2,pred) and 17 (MTT_9,pred), the fold-differences MTT_2,pred/MTT₂ and MTT_9,pred/MTT₉ were calculated for the cases of rat and man (Fig. 4a): As a result, Eqs. 30 (MTT_2,pred) and 17 (MTT_9,pred) lead to the underestimation of MTT_max and MTT₂ for all 4 cases (i.e., rat and man; and 3- and 10-compartment models). For the 3CM, the MTT_2,pred/MTT₂ values for both rat and man fall within a factor of 2 at MTT_2,pred/MRT_B higher than 1. The estimation becomes more accurate at MTT_2,pred/MRT_B higher than 10 with the MTT_2,pred/MTT₂ values between 0.920 and 1.0 (for rat) and between 0.916 and 1.0 (for man). Analogously, when utilizing a plausible set of WB-PBPK models, the estimation errors appear less significant with the MTT_9,pred/MTT₉ values between 0.752 and 0.996 (for rat) and between 0.777 and 1.0 (for man) at MTT_9,pred/MRT_B higher than 1. At MTT_9,pred/MRT_B higher than 10, the MTT_9,pred/MTT₉ values fall between 0.947 and 0.996 (for rat) and between 0.937 and 1.0 (for man). For all 4 cases shown in Fig. 4a, however, the longest MTT from the plasma data appears not always accurately identifiable when MTT_max,pred is estimated to be lower than MRT_B.

Fig. 4.

a Predictability of the longest MTT presented as the fold-difference MTT_2,pred/MTT₂ depending on MTT_2,pred/MRT_B in a 3CM (upper panels) and the fold-difference MTT_9,pred/MTT₂ depending on MTT_9,pred/MRT_B in a 10-compartment model (lower panels). b Practical usefulness of MTT_2,pred/MRT_B and MTT_9,pred/MRT_B for assessment of the limiting condition of λ₃ in a 3CM (upper panels) and λ₁₀ in a 10-compartment model (lower panels). The calculations were conducted for the cases of rat (left) and man (right). Kinetic parameters used for the calculations are listed in Supplementary Tables S1 (3CM) and S2 (10-compartment model)

Nevertheless, the ratio MTT_max,pred/MRT_B is useful for assessment of the limiting condition of λ_z. When the product MRT_Bλ₃ is plotted against the ratio MTT_2,pred/MRT_B in the 3CM (Fig. 4b), this ratio utilizing the estimated MTT_2,pred can be used to determine the limiting condition of λ₃ (i.e., MRT_Bλ₃ → 1 with MTT_2,pred/MRT_B decreased, and MRT_Bλ₃ → MRT_B/MTT_2,pred with MTT_2,pred/MRT_B increased). These characteristics are also consistently found in Fig. 4b in terms of MRT_Bλ₁₀ and MTT_9,pred/MRT_B for the 10-compartment model (i.e., MRT_Bλ₁₀ → 1 with MTT_9,pred/MRT_B decreased, and MRT_Bλ₁₀ → MRT_B/MTT_9,pred with MTT_9,pred/MRT_B increased).

The MRT_Bλ_z Value of 0.5 Serves as a Practical Threshold Determining Whether λ_z Is More Closely Associated with 1/MRT_B or 1/MTT_max

In this study, we found that the limiting condition of λ_z can be determined depending on MTT_max/MRT_B (i.e., x-axis of the unitless product-ratio plots; Fig. 2a and b) when the information on MTT_max and MRT_B is available (i.e., a bottom-up kinetic analysis). Taken together with the inductive reasoning that all the points of MRT_Bλ_z versus MTT_max/MRT_B with linear PK would fall within the range bordered by the lines/curves at MRT_B/MRT_c of 1 to infinity in Fig. 2a, λ_z can theoretically exist within the range of:

$$\frac{1}{MR{T}_B+MT{T}_{max}}<{\lambda }_z<\text{Min}\left(\frac{1}{MT{T}_{max}},\frac{1}{MR{T}_B}\right)$$ (22)

which is mathematically consistent with the derivations in Appendix A for the multi-compartment model (see also Supplementary Fig. S1). However, for a typical PK analysis where only plasma data is available (e.g., the absence of the MTT_max information, especially in clinical situations), a top-down approach for determination of the limiting condition of λ_z is also necessary. When the inequality relationship 22 is rearranged with respect to MTT_max/MRT_B:

$$\frac{1-MR{T}_B{\lambda }_z}{MR{T}_B{\lambda }_z}<\frac{MT{T}_{max}}{MR{T}_B}<\frac{1}{MR{T}_B{\lambda }_z}$$ (23)

Based on these relationships, MRT_Bλ_z obtainable from the plasma data is found to abide by:

1. When the MRT_Bλ_z value of 0.5 is determined for a certain compound, MTT_max can have a value within the range between MRT_B and 2MRT_B, depending on the distribution kinetics of the other peripheral tissue(s) (or simply depending on the V_SS for the 2CM).

2. When MRT_Bλ_z is less than 0.5, MTT_max/MRT_B is greater than 1 where λ_z becomes kinetically associated with 1/MTT_max within a factor of 2 (i.e., In Eq. 22). For the case of MRT_Bλ_z = 0.1, the terminal slope λ_z falls within a more narrow range from 0.9/MTT_max to 1/MTT_max. When the MRT_Bλ_z value sufficiently goes to 0, MRT_B/MTT_max also convergently goes to 0, leading to λ_z limited by 1/MTT_max.

3. The MRT_Bλ_z value greater than 0.5 literally means that the fold-difference between 1/MRT_B and λ_z is less than 2. For an example of MRT_Bλ_z value of 0.8, however, MTT_max can fall within a relatively variable range from 0.25MRT_B to 1.25MRT_B, depending on the distribution kinetics of the other peripheral compartment(s) (or simply depending on the V_SS for the case of a 2CM). When MRT_Bλ_z approaches 1, λ_z becomes limited by 1/MRT_B.

Application of the Product MRT_Bλ_z for PK Analysis of Various Compounds in Man and Rat

Based on the theoretical examinations of linear PK, we applied our approaches for the observed plasma concentration-time data for various compounds that were dosed intravenously in man [11] and rat [8]. The reported V_SS, CL, and T_1/2 (or λ_z) values were used for the calculation of MRT_Bλ_z and MTT_max,pred/MRT_B. Although many of the MRT_B values are directly available in the literature, a representative MRT_B was recalculated as the ratio of the reported V_SS to the reported CL in order to rule out a propagated error from the inter-individual variability (e.g., the reported MRT_B regarded as an averaged V_SS/CL). However, due to such propagated errors still present in T_1/2 values, some compounds (8 for rat (7.08%) and 104 for man (7.69%)) showed MRT_Bλ_z values greater than 1 (e.g., up to 1.95 for caffeine in rat and 2.65 for deoxycytidine-5F-2’ in man) and thus negative MTT_max,pred values, which are in theory not obtainable. Of note, indocyanine green is observed to have a negative MTT_max,pred value because its V_SS (0.035 L/kg) is smaller than the V_p (0.0429 L/kg) assigned in the current analysis. If V_SS, CL, and T_1/2 as well as a suitable volume of the central compartment (e.g., Dose/C_p(0)) are available for each individual, therefore, an individually calculated MRT_Bλ_z would fall between 0 and 1, leading to a non-negative MTT_max,pred value. Three additional exceptions (i.e., porcine secretin, cangrelor, and suprofen) in man were found to have MTT_max,predλ_z values larger than 1, which are also in theory not obtainable. Numerical calculation results are provided in Supplementary Table S3.

Bearing in mind some data noise that may exist in the parameters involved, MRT_Bλ_z for 1191 (man) and 105 (rat) compounds except for these atypical cases is plotted against MTT_max,pred/MRT_B (Fig. 5). As a result, MTT_max,pred/MRT_B ranges from 0.000276 (lanicemine) to 50.2 (fosaprepitant) in man and from 0.00375 (butyrate) to 33.4 (DA6034) in rat. The number of compounds that have MRT_Bλ_z less than 0.5 (i.e., the range where λ_z is kinetically associated with MTT_max and thus an acceptable estimation of MTT_max is allowed) is 272 (22.8% out of 1191) in man and 47 (44.8% out of 105) in rat. For these compound cases, λ_z is kinetically distinct from 1/MRT_B and a consideration of additional peripheral compartment(s) would have been necessary to explain this distinct exponential phase.

Fig. 5.

Application of the current theory for determining the limiting condition of λ_z of 1191 (man) and 105 (rat) compounds, based on the product MRT_Bλ_z versus an alternative ratio MTT_max,pred/MRT_B. Red and black dashed curves denote the theoretical curves indicating MRT_B/MRT_c values of 1 and infinity as shown in Fig. 2a. Numerical values used are listed in Supplementary Table S3

Among the 1191 compounds that were administered to man, 5 calcium channel blockers having a variable range of MRT_Bλ_z values (e.g., model-dependent values of 0.884 for amlodipine, 0.768 for nifedipine, 0.591 for felodipine, 0.317 for isradipine, and 0.0882 for clevidipine) [24–28] were selected to exemplify how MRT_Bλ_z is related to the shape (e.g., λ_z) of the plasma (blood for clevidipine) concentration-time profiles. Utilizing the maximum likelihood method of ADAPT 5 [29], a two-tissue mPBPK model (Eqs. 8a to 8c) was applied for analyzing the plasma/blood data. The primary PBPK parameters (i.e., V₁K_p1, V₂K_p2, Q₁f_d1, Q₂f_d2, and CL where R_b was assumed to be 1) along with the secondary estimated parameters (i.e., MTT₁, MTT₂, MRT_B, MRT_Bλ_z, and MTT₂/MRT_B) are summarized in Table I. According to the threshold MRT_Bλ_z value of 0.5 proposed in this study (see above), amlodipine, nifedipine, and felodipine are expected to have their λ_z closely associated with 1/MRT_B, whereas λ_z values of isradipine and clevidipine would be closely associated with 1/MTT₂.

Table I.

Summary of the Primary and Secondary Estimated Parameters (CV%) by the mPBPK Model Fitting for the Pharmacokinetics of the Indicated 5 Calcium Channel Blockers in Man as Shown in Fig. 6

Parameter	Definition (unit)	Amlodipine	Nifedipine	Felodipine	Isradipine	Clevidipine
Primary estimated parameters
V 1 K p1	Apparent volume of distribution to Tissue 1 (L)	34.4 (19.9)	21.4 (17.5)	84.8 (8.94)	10.1 (9.70)	2.98 (16.5)
V 2 K p2	Apparent volume of distribution to Tissue 2 (L)	1140 (4.32)	42.0 (12.3)	194 (8.32)	35.9 (8.21)	3.95 (10.4)
Q 1 f d1	Apparent distributional clearance to Tissue 1 (L/hr)	150 (21.9)	297 (27.0)	305 (12.9)	35.1 (14.4)	138 (33.5)
Q 2 f d2	Apparent distributional clearance to Tissue 2 (L/hr)	188 (8.56)	43.8 (26.4)	36.2 (13.0)	8.87 (9.72)	11.4 (12.6)
CL	Systemic clearance (L/hr)	26.4 (3.00)	28.5 (4.02)	44.4 (3.00)	28.5 (2.77)	312 (4.30)
Secondary parameters
MTT 1	Mean transit time through Tissue 1 (hr)	0.230 (15.1)	0.0720 (26.6)	0.278 (14.0)	0.289 (12.7)	0.0216 (24.3)
MTT 2	Mean transit time through Tissue 2 (hr)	6.04 (8.68)	0.960 (22.9)	5.37 (11.7)	4.04 (9.53)	0.348 (8.85)
MRT B	Mean residence time in the body (hr)	44.5 (3.49)	2.33 (8.44)	6.35 (5.40)	1.72 (5.90)	0.0319 (4.74)
Model-dependent MRTBλz and MTT2/MRTB
MRT B λ z	The product of MRTB and λz (Pdet)	0.884	0.768	0.591	0.317	0.0882
MTT 2 /MRT B	The ratio of MTT2 to MRTB (Kdet)	0.136	0.412	0.846	2.35	10.9

Figure 6a shows the results of mPBPK modeling for the plasma/blood concentration-time data of these 5 drugs and the sensitivity of the model depending on the different CL values (e.g., 0.3- and 3-fold changes). Consistent with expectations, the alterations of CL (i.e., the alterations in MRT_B) in the mPBPK model appear to directly affect the terminal slope of the plasma PK of amlodipine and nifedipine. However, for the case of isradipine and clevidipine having the terminal slopes kinetically independent of CL (i.e., see Table I for MTT₂λ_z values calculated to be 0.745 for isradipine and 0.961 for clevidipine), different CL values mostly altered the overall exposure of the drugs, while λ_z remains not significantly changed. Interestingly, felodipine also shows the CL-dependent λ_z values (i.e., 2.34-fold decrease at 0.3CL and 1.54-fold increase at 3CL). This result indicates that felodipine, with an MRT_Bλ_z value (0.591) around the threshold, has a terminal slope which becomes more sensitive to the change of CL towards the condition of MRT_Bλ_z > 0.5, and less sensitive to the change of CL towards the condition of MRT_Bλ_z < 0.5. Figure 6b shows the 5 example drug cases located in a variable range of the unitless product-ratio plot. It is noteworthy that the transfer of the point by the change of CL across the threshold of MRT_Bλ_z = 0.5 may be involved as described for felodipine, while amlodipine and clevidipine remain in each of their MRT_Bλ_z ranges even with altered CL values (Fig. 6b).

Fig. 6.

a Pharmacokinetic profiles for 5 calcium channel blockers having divergent MRT_Bλ_z values. Closed circles denote the observed data points, and two-tissue mPBPK models (solid curves) were fitted to the data for sensitivity analysis of the model in relation to the CL values (i.e., 0.3- and 3-fold; dashed curves). b The unitless product-ratio plot for the 5 example drugs. Red and black dashed curves denote the theoretical curves indicating MRT_B/MRT_c values of 1 and infinity shown in Fig. 2a. Arrows exemplify the transfer of points by alterations of CL of amlodipine and clevidipine

Discussion

In PK analysis of plasma data (e.g., intravenous administration), the biological half-life T_1/2 is an important parameter that characterizes the drug disposition kinetics in the body. Due to the complexity of the body, however, kinetic understanding for T_1/2 has been somewhat limited on a qualitative basis to the theory of a 1CM (e.g., the lower CL or the higher V_SS, the longer T_1/2, and from the expectation of T_1/2 ∝V_SS/CL) [5]. Since T_1/2 can typically be defined as a stationary parameter (i.e., an exponential decline with the terminal slope λ_z), many types of PK models (e.g., compartment and PBPK models) have been developed for describing/predicting the plasma concentration-time data, based on the assumption of linear (i.e., first-order) kinetic processes in the body. To simulate a manifestation of the complicated anatomy and physiology of the body, WB-PBPK models incorporate the system-specific (e.g., V_T and Q_T) and drug-specific input parameters that can well address the distribution (e.g., K_p) and elimination (e.g., CL) kinetics in the plasma/tissues. Despite the practical utility of WB-PBPK models, however, numerical interpolations are needed to estimate T_1/2 from the model simulations because of the lack of explicit understanding and mathematical solution of T_1/2 in WB-PBPK models.

Recently, reconciliation was achieved between WB-PBPK (mathematically too complicated) and mPBPK (physiological but much simpler) models, by lumping tissues of WB-PBPK models into the slowly- or rapidly-equilibrating tissue group (SEG or REG) of mPBPK models [7, 8]. In the literature, a parameter K_det viz., MTT_max/MRT_B, was found as a determining coefficient for the number of tissue groups (the number of exponential phases) of distinct transit times, which governs the fractional change of λ_z during the progressive lumping from the tissue having MTT_max. Interestingly, it is noteworthy that all the ratios MTT_T/MRT_B (Fig. 2a), MTT₂/MRT_B (Figs. 2b and 3a), and MTT_max/MRT_B (Fig. 3b and Supplementary Fig. S1) are other expressions of K_det for two-, three-, and multi-compartment models. Based on the unitless product-ratio plot proposed in this study (Fig. 2), the K_det was found to be also crucial for the determination of the limiting condition of λ_z between 1/MRT_B and 1/MTT_max.

When WB-PBPK parameters are available to determine MTT_max and MRT_B, a drug with a low K_det value (e.g., less than 0.3 for rat) was suggested to have a robust λ_z that is not significantly affected even by lumping all the peripheral tissues into one tissue group (i.e., WB-PBPK model kinetically approximated to a one-tissue mPBPK). From a point of view of the current study, the reason for the robust λ_z under this condition of K_det < 0.3 for rat is because λ_z is sufficiently limited by 1/MRT_B (i.e., MRT_Bλ_z > 0.769; Fig. 2) rather than affected by the tissue transit kinetics. At a higher K_det value, the lumping from the tissue of MTT_max is expected to cause a marked change in λ_z. Under this latter condition, only few tissue(s) may be capable of being lumped into the SEG (i.e., two-tissue mPBPK model), since the change of the longest MTT by the lumping of such kinetically large tissues would readily affect λ_z that is closely associated with 1/MTT_max at a higher K_det. As shown for the 5 example drugs having divergent K_det values (Table I), the contribution of the initial distribution phases (i.e., λ₁ and λ₂ phases) to the overall shape of the PK profiles appears to become less significant when assessed from a higher K_det drug clevidipine (10.9) to a lower K_det drug amlodipine (0.136) (Fig. 6a). However, since a two-tissue mPBPK model was still needed for an adequate description of the amlodipine PK, further theoretical investigations are warranted to seek a more conservative K_det threshold that may be necessary for the use of a one-tissue mPBPK model in man.

To adequately predict an overall shape of the plasma PK, WB-PBPK parameters such as the steady-state tissue-to-plasma concentration ratio (K_p,ss) [30] and the intrinsic elimination clearance (CL_int) [31] (i.e., bottom-up) should be able to explain model-independent parameters V_SS and CL (i.e., top-down). If λ_z is limited by 1/MRT_B, model predictability will be adequate only with V_SS and CL values that are consistent between the bottom-up and top-down approaches. If λ_z is limited by 1/MTT_max, however, even appropriate K_p,ss and CL_int values may not be sufficient to capture the terminal slope of the plasma concentration-time data. In this case, MTT_max needs to be assigned an adequate set of V_T, Q_T, K_p, R_b, and f_d values (e.g., see Eq. 9c for MTT₂ of 3CM). When tissue permeability PS is not available, the perfusion-limited distribution (CL_app = Q_T) is widely assumed in the WB-PBPK model building. In order to capture the terminal-phase PK based on the perfusion-limited model, sometimes K_p (e.g., K_p,ss for a non-eliminating organ) [32] appears further adjusted (typically enlarged) using an unknown scaling factor (e.g., K_p,scalar), even with an adequate K_p,ss value that can be measured/supported by experiment. However, this K_p-adjustment approach could be misleading since MTT_max is determined not only by the extent (V_TK_p/R_b) but also by the rate (Q_Tf_d) of tissue distribution where a consideration of f_d would have been necessary instead of K_p,scalar. Although an inclusion of PS in WB-PBPK models usually requires differential equations for vascular/interstitial fluids that are mathematically inconvenient, a consideration of f_d in WB-PBPK [12] and mPBPK [9] models enables a simple expression of the tissue distribution kinetics (e.g., Eq. 1b) and MTTs (e.g., Eq. 2b) for the drugs having tissue permeability limitations. In addition, an f_d term is necessary along with an organ extraction ratio (ER) for a valid conversion of K_p,ss to a PBPK-operative K_p (i.e., K_p = K_p,ss/(1 − ER/f_d)) [32] when a peripheral elimination and a tissue permeability limitation are involved. Considering the importance of f_d in PBPK models, therefore, the rate of drug distribution to tissues that seems currently undervalued in the field should be regarded as one of the crucial factors for PBPK model building along with K_p,ss and CL_int parameters, especially for drug cases having a λ_z limited by 1/MTT_max.

In this study, we propose an intuitively understandable plot that relates the product MRT_Bλ_z and the ratio MTT_max/MRT_B, named as the unitless product-ratio plot. This graphical presentation for the 2CM (Fig. 2a) was expanded to the case of the 3CM using a variety of PK datasets (i.e., 9⁷ cases; Fig. 2b). As a result, all the in silico generated points for the 3CM fall within the range bordered by the upper black (MRT_Bλ₂ = 1 when MTT_T < MRT_B, and MRT_Bλ₂ = MRT_B/MTT_T when MRT_B < MTT_T) and lower red (MRT_Bλ₂ = MRT_B/(MRT_B + MTT_T)) dashed lines/curves of Fig. 2a, which leads us to inductive reasoning for the inequality relationship 22 generalized for the multi-compartment model. Indeed, a left-hand limit of λ_z is also theoretically found to be 1/(MRT_B + MTT_max) in the multi-compartment model (Appendix A), as also shown in Supplementary Fig. S1. Taken together with the upper limits of the terminal slope (i.e., 1/MRT_B when MTT_max < MRT_B (Eq. 12), and 1/MTT_max for MRT_B < MTT_max (Browne’s theorem)), the 3 statements above appear to be generally applicable for linear PK.

However, we encountered many drug cases where the limiting condition of λ_z needs to be determined only based on a top-down analysis of the plasma data. Since K_det is not accurately determinable without the MTT_max information, we herein propose the product MRT_Bλ_z as a theoretical indicator (i.e., named as P_det) and a useful alternative of K_det for determining the limiting condition of λ_z between 1/MRT_B and 1/MTT_max. We found that the MRT_Bλ_z value of 0.5 is a useful threshold to determine whether λ_z is kinetically associated with 1/MRT_B or 1/MTT_max. As shown for amlodipine, nifedipine, and felodipine (MRT_Bλ_z > 0.5), λ_z appears to be dependent on CL, whereas isradipine and clevidipine (MRT_Bλ_z < 0.5) show CL-independent λ_z values (Fig. 6). Therefore, it is indicated that an improvement in the biological stability of a drug would increase mostly its overall exposure to the body, but may not be sufficient for the extension of the terminal-phase T_1/2 when the drug has the λ_z limited by 1/MTT_max. In this case, one may need to consider a formulation strategy that can lead to a prolongation of MTT_max (e.g., increase of K_p and/or decrease of f_d for the kinetically largest tissue).

Due to the mathematical difficulties recognized in multi-compartment models, we considered an approximation method for an explicit relationship between λ_z and PBPK parameters (Eq. 16b). Therefore, if the parameters involved (MTT_max, MRT_B, and MRT_c) are available, an approximate estimate of λ_z can be obtained without performing any numerical integration of differential equations, based on the appropriateness of Eq. 16b (Fig. 3). Our approach of approximation involving f(λ) and g(λ) is consistent with the assumption that there would be only one peripheral compartment that is identifiable from λ_z, considering that the terminal-slope data alone may not be sufficient to figure out how many exponential phase(s) disappeared before reaching this pseudo-equilibrium phase (e.g., MTT₁ set to be 0 for the two-tissue mPBPK model). Even with this kinetic assumption, Eq. 16b can address the characteristics of λ_z between 1/MRT_B (when MTT_max ≪ MRT_B) and 1/MTT_max (when MRT_B ≪ MTT_max). It is noteworthy that the maximum approximation error from Eqs. 16b and B3b with respect to λ_z cannot exceed a factor of 2 regardless of the number of compartments (Fig. 3), since the range of MRT_Bλ_z that can vary at a certain value of K_det in the unitless product-ratio plot for linear PK (Fig. 2) falls within a factor of 2 (i.e., a maximal 2-fold error at the K_det value of 1).

In addition, the rearrangement of Eq. 16b enables the estimation of MTT_max from the NCA of plasma data. When MRT_Bλ_z is observed to be less than 0.5, MTT_max,pred/MRT_B is expected to be greater than 1 (Fig. 4b) and thus the predictability for MTT_max becomes more acceptable within a factor of 2 (Fig. 4a). This is consistent with the Statement 2 above that the theoretical boundaries of MTT_max are narrowed down to be less than a factor of 2 under this condition. When MRT_Bλ_z > 0.5, however, MTT_max,pred/MRT_B can fall within a variable range (Fig. 4b) and the predictability for MTT_max could not be secured only with the MRT_Bλ_z information (i.e., Statement 3).

Previously [33], an attempt was made to obtain a mathematical expression of λ₂ for a two-compartment physiological model based on the assumption of perfusion-limited distribution (Eq. 3b where f_d = 1). In the publication, the MTT_T (i.e., denoted as V_t/k₂ in the publication) was “predicted” using the physiological (i.e., Q_CO and V_B) and drug-specific parameters (i.e., R_b and V_SS), i.e., MTT_T,pred = (V_SS/R_b − V_B)/Q_CO, which is equivalent to Eq. 18 when rearranged for the case of the 2CM. Then, this expression of MTT_T,pred was used in the place of MTT_T in Eq. 3b, in order to calculate an “approximate” estimate of λ₂ (or λ_2,app) for many compounds (the number of cases in parenthesis) dosed intravenously in rat (34), monkey (6), dog (8), and man (27). Despite a considerable number of compounds assessed (a total of 75), however, all of their MRT_Bλ_z values are larger than or equal to 0.5: Since their λ_z values appear to be kinetically associated with 1/MRT_B, a reasonable correlation between observed and calculated T_1/2 values was a partial presentation with the 75 points that would be located only in the range of MRT_Bλ_z ≥ 0.5 in Fig. 2a. Considering that (i) perfusion-limited distribution may not always be applicable (i.e., a valid expression of MTT_T,pred as (V_SS/R_b − V_B)/(Q_COf_d) where f_d ≤ 1), (ii) a one-tissue mPBPK model may not always be applicable (e.g., only when K_det < 0.3 in rat) [8] (i.e., a valid expression of MTT_2,pred as (V_SS/R_b − V_B)/(Q₂f_d2) for the 3CM), and (iii) the Q₂f_d2 information may not always be available with plasma data (i.e., Eq. 30 for MTT_2,pred expressed only with NCA parameters), Eq. 18 that complies with NCA assumptions can be considered useful for prediction of a model-independent MTT_max especially when the MTT_max,pred is larger than MRT_B. The practical applicability of Eq. 18 is accentuated by the proportion of compounds having 1/MTT_max-associated λ_z values (i.e., 22.8% in man and 44.8% in rat; Fig. 5).

Recently, we reviewed the PK of metformin in 9 different species based on the joint fitting of two-tissue mPBPK models [34]. Their MRT_Bλ_z values were less than 0.5 (ranging from 0.334 for cat to 0.0730 for horse) except for rabbit (0.574), and model-dependent K_det values (i.e., MTT₂/MRT_B) were mostly found to be larger than 2 (i.e., ranging from 2.15 for cat to 12.9 for horse, except for rabbit (0.844)). Collectively, these calculations indicate that λ_z is closely associated with 1/MTT₂ in those species (Statement 2). Interestingly, metformin PK in dogs has a non-stationary T_1/2 that could be described by a power function tail (e.g., C_p(t) = A · t^b) based on a gamma-Pareto convolution (GPC) model [35]. Despite its mathematical usefulness, however, a biopharmaceutical interpretation of the model was difficult for mechanistic understanding of this atypical phenomenon. Based on its MRT_Bλ_z value of 0.0925 and K_det value of 10 for dog, one possible explanation for the power function tail of metformin PK is that the terminal slope that appears continuously being prolonged (i.e., C_p(t)^′/C_p(t) = b/t) in the plasma profile would represent an array of multiple tissue components that have divergent transit times (e.g., sequestered in lysosomal/mitochondrial sub-compartments as a strong base) [36, 37], rather than CL-related mechanisms. Thus, determination of the MTTs through the peripheral tissue components may be important for mechanistic understanding of this peculiar terminal slope emerging in the plasma PK of metformin in dogs.

The present study has some limitations. First, the current theory is only applicable for linear PK where the terminal slope has a stationary value. However, since PK studies usually involve a wide range of doses that can include a linear range even for drugs having non-linear kinetics, the calculation of MRT_Bλ_z values in analyzing plasma data in the linear range would serve as a reasonable starting point of determining the limiting condition of λ_z. In addition, despite its theoretical validity, the MRT_Bλ_z value alone was not sufficient for estimation of MTT_max especially at MRT_Bλ_z > 0.5 (Statement 3). This is because MTT_max is kinetically not associated with λ_z, rather the slope of the initial distributional phases (e.g., λ₁ for a 2CM) under the condition of MRT_Bλ_z > 0.5. Further theoretical examinations are needed for explicit understanding of WB-PBPK and mPBPK models in this regard.

Conclusions

In conclusion, we found that the boundary conditions of λ_z in linear PK systems are theoretically 1/MRT_B and 1/MTT_max. The λ_z is limited by 1/MRT_B when MTT_max ≪ MRT_B, and limited by 1/MTT_max when MRT_B ≪ MTT_max. Due to the absence of MTT_max information in NCA, we propose the MRT_Bλ_z as a theoretical indicator for determining the limiting condition of λ_z in top-down PK analyses. As a result, the MRT_Bλ_z value of 0.5 was found to serve as a practical threshold of whether λ_z is more closely associated with 1/MRT_B or 1/MTT_max. For the case of MRT_Bλ_z < 0.5, MTT_max could be adequately estimated by Eq. 18 (within a factor of 2), based on the parameters readily obtainable from NCA of plasma data. Application of the MRT_Bλ_z-based approach was demonstrated for assessment of the terminal slope λ_z from the observed PK data for various compounds in man and rat. The current study should be able to provide greater insight for the biopharmaceutical interpretation of the terminal-phase λ_z or T_1/2 in SHAM analyses, utilizing the simple product MRT_Bλ_z (i.e., P_det).

Appendix A

Similar to the expansion principle shown in Eq. 11, the rearrangement of Eq. 13 for a multi-compartment model leads to (with respect to the terminal slope):

$$-\sum _{i=1}^9\frac{Q_i{f}_{d,i}\cdot {\left(MT{T}_i\lambda \right)}^2}{1-MT{T}_i\lambda }=V_{SS}/R_b\cdot \lambda -CL/R_b$$ (24)

$$1+MT{T}_9\lambda -\frac{1}{CL/R_b}\sum _{i=1}^9\frac{Q_i{f}_{d,i}\cdot \frac{MT{T}_i}{MT{T}_9}\cdot {\left(MT{T}_9\lambda \right)}^2}{\frac{MT{T}_9}{MT{T}_i}-MT{T}_9\lambda }=MR{T}_B\lambda +MT{T}_9\lambda$$ (25)

It is noted that (i) each term of the summation notation in the left-hand side of Eq. 25 and (ii) their first derivatives with regard to MTT₉λ are both 0 at MTT₉λ = 0. When MTT₉λ → 0, therefore, Eq. 25 can be rearranged in terms of the terminal slope λ₁₀ to:

$$\underset{MT{T}_9\lambda \to 0}{\text{lim}}{\lambda }_{10}=\underset{MT{T}_9\lambda \to 0}{\text{lim}}\frac{1}{MR{T}_B+MT{T}_9}\cdot \left(1+MT{T}_9\lambda -\frac{1}{CL/R_b}\sum _{i=1}^9\frac{Q_i{f}_{d,i}\cdot \frac{MT{T}_i}{MT{T}_9}\cdot {\left(MT{T}_9\lambda \right)}^2}{\frac{MT{T}_9}{MT{T}_i}-MT{T}_9\lambda }\right)=\frac{1}{MR{T}_B+MT{T}_9}$$ (26)

where 1/(MRT_B + MTT₉) is a left-hand limit of λ₁₀. Among the possible additions of the MTT_iλ terms to both sides of Eq. 25, the addition of MTT₉λ was found to lead to the identification of the lowest limit of λ₁₀ in Eq. 26. As a result, the boundary conditions of λ₁₀ are found to be 1/MRT_B when MTT₉ ≪ MRT_B, and 1/MTT₉ when MRT_B ≪ MTT₉.

Appendix B

When the left-hand side of Eq. 10 is defined as p(λ), p(λ) satisfies the mathematical properties within the range of 0 < λ < 1/MTT₂, such as:

$$\underset{MT{T}_2\lambda \to 0}{\text{lim}}p(\lambda )=-\left(Q_1{f}_{d1}+Q_2{f}_{d2}\right)$$ (27a)

$$\underset{MT{T}_2\lambda \to 0}{\text{lim}}{p}^{\prime }(\lambda )=-\left(V_1{K}_{p1}+V_2{K}_{p2}\right)/R_b$$ (27b)

$$\underset{MT{T}_2\lambda \to 1}{\text{lim}}p(\lambda )=-\infty$$ (27c)

For mathematical simplicity, we introduced a function q(λ) that satisfies the three conditions above, expressed as:

$$q(\lambda )=\frac{\frac{V_1{K}_{p1}+V_2{K}_{p2}}{MT{T}_2^2\cdot R_b}}{\lambda -\frac{1}{MT{T}_2}}+\frac{V_1{K}_{p1}+V_2{K}_{p2}}{MT{T}_2\cdot R_b}-\left(Q_1{f}_{d1}+Q_2{f}_{d2}\right)$$ (28)

When q(λ) is equated with the right-hand side of Eq. 10, an approximated λ₃ (i.e., λ_3,app, which falls within the range lower than 1/MTT₂) can be calculated using:

$${\lambda }^2-\frac{1}{MR{T}_c}\left(1+\frac{MR{T}_B}{MT{T}_2}\right)\lambda +\frac{1}{MR{T}_c}\frac{1}{MT{T}_2}=0$$ (29a)

$${\lambda }_{3,app}=\frac{1}{2MR{T}_c}\left(1+\frac{MR{T}_B}{MT{T}_2}-\sqrt{{\left(1+\frac{MR{T}_B}{MT{T}_2}\right)}^2-4\frac{MR{T}_B}{MT{T}_2}\frac{MR{T}_c}{MR{T}_B}}\right)$$ (29b)

It is noted that Eq. 29b is structurally equivalent to Eq. 3b where MTT₂ (i.e., the longest tissue MTT) in the two-tissue model takes a role of MTT_T of the one-tissue model. If the numerical error arising from this approximation for the terminal slope λ₃ in two-tissue models is negligible (i.e., λ₃ ≅ λ_3,app), Eq. 29a can be rearranged for estimation of MTT₂ (i.e., MTT_2,pred) only from the plasma data, expressed as (Eq. 30):

$$MT{T}_{2,\mathit{\text{pred}}}=\frac{1}{{\lambda }_3}\cdot \frac{1-MR{T}_B{\lambda }_3}{1-MR{T}_c{\lambda }_3}=\frac{1}{{\lambda }_3}\cdot \frac{CL-V_{SS}\cdot {\lambda }_3}{CL-V_B{R}_b\cdot {\lambda }_3}$$ (30)