Due to the physiological features of the body (Fig. 1a), the outflow blood exiting from splanchnic tissues (i.e., gut and spleen) enters the liver via the hepatic portal vein. The WB-PBPK equations for the rate of drug distribution to liver, gut, and spleen may be described as (Eqs. A1a, A1b, and A1c):

$$_\frac_}=__\cdot \left(_-\frac\cdot R}}\right)-C___$$

(A1a)

$$_\frac}=__\cdot \left(_-\frac\cdot R}}\right)$$

(A1b)

$$_\frac}=__\cdot \left(_-\frac\cdot R}}\right)$$

(A1c)

where LI, GU, and SP in the subscripts denote the liver, gut, and spleen and CLu,int is the intrinsic clearance of the unbound drug in the liver. The Cin,liver is the drug concentration in the blood entering the liver, which could be expressed as (Eq. A2):

$$_=\frac_-_-_\right)_+__+__}}$$

(A2)

where Cout,SP and Cout,GU are the outflow blood concentrations from spleen and gut (Eq. 4). When it was necessary to lump splanchnic tissues (e.g., liver, gut, and spleen) into a single compartment (Fig. 1b), each side of Eqs. A1a, A1b, and A1c may be added and the mathematical relationship, when rearranged, results in a differential equation for a lumped splanchnic compartment (Eq. A3):

$$\frac_+_+_\right)}=\frac_R}_}\cdot \left(_\cdot __/R-_\right)\kern7.5em +\frac_\left(1-_\right)R}_}\cdot \left(_\cdot __/R-_\right)\kern1.5em +\frac_\left(1-_\right)R}_}\cdot \left(_\cdot __/R-_\right)-C___$$

(A3)

When each term in front of the parentheses in Eq. A3 is regarded as the inverse of MTT for liver (1/MTTLI) and modified MTTs for spleen and gut (1/MTTSP′, and 1/MTTGU′), it was noted that the shape of Eq. A3 was analogous to the addition of each side of Eqs. 7a and 7b for tissues in parallel connection. Of note, the Kp term used in the calculation of MTT (Eq. 5) for lumping cases that involve eliminating organs (from Eq. A1a to A3) is different than Kp,ss (i.e., steady-state tissue-to-plasma concentration ratio). The Kp,ss could be theoretically converted to the PBPK-operative Kp by taking into account an organ extraction ratio (ER) and fd (30) when a peripheral elimination and a tissue permeability limitation are involved. Regarding fd,SP(1 − fd,LI) and fd,GU(1 − fd,LI), the modified fd values for spleen and gut (i.e., fd,SP′ and fd,GU′), a similar approach to deriving Eqs. 10a and 10b may be applicable where the splanchnic tissues were lumped resulting in the following equation for the rate of drug distribution (Eq. A4a) along with the related parameters calculated as (Eqs. A4b, A4c, and A4d):

$$\frac_+_+_\right)}=\frac_}=\left(_+_+_\right)\frac_}=__\cdot \left(_-\frac\cdot R}}\right)-C___$$

(A4a)

$$_=\frac_+__+__}+_+_}$$

(A4b)

$$_=\frac_\left(1-_\right)+__\left(1-_\right)+__}}$$

(A4c)

$$_=\frac}}}+\frac___}}\cdot \left(\frac}-\frac}\right)}$$

(A4d)

where Spl in the subscript of the corresponding parameters denotes the lumped splanchnic compartment. The elimination term in Eq. A4a could be obtained by the correction of fu,LICLI in Eq. A1a to fu,SplCSpl in Eq. A4a, utilizing Eq. A4d to preserve the systemic clearance CLsys (i.e., QLI ∙ ER) during the lumping (30). If the lumping of the splanchnic compartment does not affect the tissue permeability limitation (i.e., fd,Spl = fd,LI), estimation of an apparent free fraction of drug in the splanchnic compartment (fu,Spl) would be dependent on the change of Kp by the lumping (i.e., fu,Spl = fu,LIKp,LI/Kp,Spl).

For the sake of simplicity, however, we introduced an elimination process that occurred only in the systemic compartment, employing the systemic clearance CLsys instead of CLu,int (i.e., CLu,int = 0). This mathematical transformation is on the basis that, only with the plasma data, all elimination kinetics may have to be assumed to occur from the systemic compartment even if there were a peripheral elimination: Therefore, the experimental Kp,ss should be considered as the PBPK-operative Kp if a central elimination is assumed alternatively [i.e., liver regarded as a non-eliminating organ where Kp = Kp,ss (30)]. Collectively, the splanchnic compartment lumped from liver, gut, and spleen could be considered as one of the peripheral tissues connected in parallel to the arterial/venous blood compartment (Fig. 1b).

According to the literature (9), a consideration of the systemic circulation was attempted as an approach to lump venous blood, lung, and arterial blood, which were serially connected in WB-PBPK (Figs. 1a and 1b). The rate of drug distribution in the vein, lung, and artery may be expressed as (Eqs. A5a, A5b, and A5c):

$$_\frac_}= Dose\ rate+\sum _i_-__= Dose\ rate+_\cdot \left(_-_\right)$$

(A5a)

$$_\frac}=__\cdot \left(_-\frac\cdot R}}\right)$$

(A5b)

$$_\frac_}=_\cdot \left(_-_\right)-C_\cdot _$$

(A5c)

where ven,blood, LU, art,blood, and art,plasma in the subscript denote the venous blood, lung, arterial blood, and arterial plasma; Dose rate is the dosing rate of the drug where the duration of infusion could be applied in accordance with the experimental procedure; and Qi and Cout,i denote the blood perfusion rate and outflow blood concentration from the adipose, bone, brain, heart, kidney, liver, muscle, and skin, in addition to the residual blood flow [11.1% of the cardiac output (QCO)] and Cart,blood. The summation of these products of Qi and Cout,i may be simply expressed as the multiplication of QCO and the inflow blood concentration into the venous blood (Cin,ven). In this study, we assumed that elimination follows linear kinetics and CLsys obtained from the non-compartmental analysis of plasma data was used as the total drug clearance as described in Eq. A5c.

Since the blood pool (i.e., composed of vein and artery) would be used in a typical mPBPK model structure, we first considered the blood compartment, by the addition of Eqs. A5a and A5c, in which the lung was mathematically regarded as one of the peripheral tissues (Fig. 1c). Considering the splanchnic compartment as one of the peripheral tissues (see above), the equation for the blood pool was expressed as (Eq. A6):

$$_\frac_}+_\frac_}=\left(_+_\right)\frac_B}= Dose\ rate-\sum _i_\cdot \left(_B-\fracR}}\right)-C_\cdot _B/R$$

(A6)

where CB and Cp were the blood and plasma concentrations of the lumped blood pool. Of note, the residual blood flow directly delivered from artery to vein (11.1% of QCO) was structurally no longer of significance in Eq. A6.

When necessary for the systemic circulation composed of the venous/arterial blood and the lung, a similar approach may also be applied by the addition of each side of Eqs. A5a, A5b, and A5c (Eq. A7):

$$\frac_+_+_\right)}=\frac_}=_\frac_B}= Dose\ rate-\sum__i_\cdot \left(_B-\fracR}}\right)-C_\cdot _B/R$$

(A7)

where Vsys is the summation of the apparent volumes of distribution of drug in the vein, lung, and artery (i.e., Vven + Vart + VLUKp,LU/R). In cases of the pharmacokinetic analysis with regard to the plasma concentration (e.g., Cp), the blood concentration term (e.g., CB) may be divided by R resulting in the corresponding plasma concentration, assuming instantaneous partitioning of drugs within the blood.

In this section, we aimed to mathematically derive a relationship between FCT [(λter′ − λter)/λter] and UETSEG. Eqs. 14 and 15b in the main text are repeated here as Eqs. B1 and B2:

$$\frac_}_1}}_1}}+\frac_}_2}}_2}}+\dots +\frac_}_9}}_9}}=_B\left(\lambda -\frac_c}\right)$$

(B1)

$$-\sum_^9_i_\cdot \left(1+ MT_i_+_i_\right)}^2+\dots \right)=_B_-\left(\sum_^9_i_+C_\right)$$

(B2)

Considering up to the 2nd-order expansion terms in Eq. B2, the following approximation could be obtained (Eq. B3):

$$-__-p}^2\approx -_$$

(B3)

where \(p=_1_ MT_1^2+\dots +_9_ MT_9^2\). From the quadratic formula, λter (λter > 0) could be expressed as (Eq. B4):

$$_\approx \frac\left(-1+\sqrt_}_\right)}^2}}\right)}$$

(B4)

For the case of lumping the longest (9th) and second-longest (8th) MTT tissues into the SEG, the equation for determining the eigenvalues (λ′) could be expressed as (Eq. B5):

$$\frac_}_1}}-\frac_1}}+\dots +\frac_}_7}}-\frac_7}}+\frac_}_}}-\frac_}}=_B \left(^-\frac_c}\right)$$

(B5)

It was noteworthy that the size of matrix A was reduced to a 9×9 structure (i.e., matrix A′). For the case of λ′ = λter′, Eq. B5 could be expanded by a Taylor series as (Eq. B6):

$$-\sum_^7_i_\cdot \left(1+ MT_i}^+_i}^\right)}^2+\dots \right)\kern14.75em -__\cdot \left(1+ MT_}^+_}^\right)}^2+\dots \right)=_B}^-\left(\sum_^9_i_+C_\right)$$

(B6)

Considering up to the 2nd-order expansion terms in Eq. B6, the following approximation could be obtained (Eq. B7):

$$-_}^-q_}^ \approx -_$$

(B7)

where \(q=_1_ MT_1^2+\dots +_7_ MT_7^2+__ MT_^2\). From the quadratic formula, λter′ (λter′ > 0) could be expressed as (Eq. B8):

$$_}^\approx \frac\left(-1+\sqrt_}_\right)}^2}}\right)}$$

(B8)

It should be noted that Eqs. B3 and B8 were not directly applicable for the prediction of the terminal phase slope (data not shown). Considering up to the 2nd-order term in Taylor series expansion of \(0.75<1-\frac_}_\right)}^2}<1\) for Eq. B3 and \(0.75<1-\frac_}_\right)}^2}<1\) for Eq. B8), the fractional change of the terminal phase slope could be approximated as (Eq. B9):

$$\frac}^-_}}\approx \frac_}_\right)}^2}\cdot \left(p-q\right)$$

(B9)

In this section, we aimed to mathematically predict the deviation of λβ from λmajor as a ratio of λβ/λmajor. For the case of the 9-tissue WB-PBPK model where λ = λmajor, each term in the left side of Eq. 14 could be expanded by a Taylor’s series, depending on the mathematical condition for convergence, as Eqs. C1a and C1b:

For terms satisfying \(_<\frac\),

$$\frac_}}}=-_i_\cdot \left(1+ MT_i\lambda +_i\lambda \right)}^2+\dots \right)$$

(C1a)

For terms satisfying \(_>\frac\),

$$\frac_}_i}}_i}}=_i_\cdot \left(\frac_i\lambda }+_i\lambda}\right)}^2+\dots \right)$$

(C1b)

For the case of Model D, since it was assumed that peripheral tissues were kinetically separated into two groups, the eigenvalues (in terms of λ′; the roots corresponded to λα, λβ, and λγ) could be determined by Eq. C2:

$$\frac_}_}}-\frac_}}+\frac_}_}}-\frac_}}=_B\left(^-\frac_c}\right)$$

(C2)

where REG in the subscript denoted the rapidly-equilibrating tissue group. Here, Browne’s theorem II (20) could also be applied so as to obtain the inequalities of \(\frac}<_<\frac}\). Similar to the expansions in Eqs. C1a and C1b, the two terms in the left side of Eq. C2 could be expanded for the case of λ′ = λβ (Eqs. C3a and C3b):

Since \(_<\frac}\),

$$\frac_}}}-\frac}}=-__\cdot \left(1+ MT_^+_^\right)}^2+\dots \right)$$

(C3a)

whereas \(_>\frac}\) led to:

$$\frac_}_}}-\frac_}}=__\cdot \left(\frac_^}+_^}\right)}^2+\dots \right)$$

(C3b)

When all the fraction terms in the left-hand side of Eq. 14 expanded by Eq. C1a (i.e., MTTiλmajor < 1) were summed with respect to tissues included in the REG, the resulting coefficients of the first two expansion terms would be mathematically comparable to those found in the right side of Eq. C3a. Since the higher-order errors included from the second-order expansion terms may be assumed to be mathematically insignificant, our lumping procedure of preservation (Eqs. 10a and 10b) may be applied.

For the case of λ = λmajor in Model C having 9 peripheral tissues along with the blood pool, Eq. B1 could be expanded by a Taylor series as (Eq. C4):

$$-\sum_<\frac_i}}_i_\cdot \left(1+ MT_i_+_i_\right)}^2+\dots \right)+\sum__i}<_}_i_\cdot \left(\frac_i_}+\frac_i_\right)}^2}+\dots \right)=_B_-\left(\sum _i_+C_\right)$$

(C4)

Considering the terms up to the first-order of MTTiλmajor or 1/MTTiλmajor in Eq. C4, the following approximation could be obtained (Eq. C5):

$$\left(V_-\left(V_K_+V_K_\right)/R\right)\cdot }^2-\left(CL_+Q_f_+Q_f_\right)\cdot\lambda_\approx\sum\limits_\frac}$$

(C5)

where all tissues in Model C such that \(\frac_i}<_\) are included in either SEG or AMB. Based on the quadratic formula, the following approximation could be obtained (Eq. C6):

$$_\approx \frac_+__+__}_-\left(__+__\right)/R\right)}\cdot \left\_-\left(__+__\right)/R\right)_\frac_}_i}}_+__+__\right)}^2}}\right\}$$

(C6)

When \(\frac_-\left(__+__\right)/R\right)_\frac_}_i}}_+__+__\right)}^2}<1\), Eq. C6 could be approximated considering a Taylor expansion of \(\sqrt\), as (Eq. C7):

$$_\approx \frac_+__+__}-\left(__+__\right)/R}+\frac\frac_}_i}}_+__+__}+\cdots$$

(C7)

Similarly for the λ = λβ in Model D, Eq. C2 could be expanded by a Taylor series, resulting in (Eq. C8):

$$-__\cdot \left(1+ MT__+__\right)}^2+\dots \right)+__\cdot \left(\frac__}+\frac__\right)}^2}+\dots \right)=_B_-\left(\sum __+C_\right)$$

(C8)

Considering the terms up to the first-order of MTTREGλβ or 1/MTTSEGλβ in Eq. C8, the following approximation could be obtained (Eq. C9):

$$\left(_-__/R\right)\cdot }^2-\left(C_+__\right)\cdot _\approx \frac_}_}$$

(C9)

Based on the quadratic formula, the following approximation could be obtained (Eq. C10):

$$_\approx \frac_+__}_-__/R\right)}\left\_-__/R\right)\cdot \frac_}_}}_+__\right)}^2}}\right\}$$

(C10)

When \(\frac_-__/R\right)\cdot \frac_}_}}_+__\right)}^2}<1\), Eq. C10 could be approximated using a Taylor expansion of \(\sqrt\), as (Eq. C11):

$$_\approx \frac_+__}-__/R}+\frac_}_}}_+__}+\dots$$

(C11)

Considering the first term in Eqs. C7 and C11 to be most dominant, the following approximation could be derived for the fold difference between λβ and λmajor, using UETREG (Eq. C12):

$$\frac}}\approx UE_=\frac_+__}_+__+__}\cdot \frac-\left(__+__\right)/R}-__/R}$$

(C12)