Introduction

Physiologically-based pharmacokinetic (PBPK) models are progressively being used to relate tissue physiology, anatomy, and biochemistry in the prediction of tissue concentrations vs. time profiles. The premise is to interconnect tissues of discrete volumes by blood flow to describe the transport, elimination, and pharmacologic effects in select, target organs and tissues for metabolism, excretion, sampling and activity. Physical (binding and distribution) and biochemical (Michaelis–Menten parameters, Vmax/Km) data are fabrics for model building. Other requisite constants include enzymatic constants for metabolism (CLint,met based on Vmax/Km), passive diffusion clearance (CLd), and transport clearances for influx (CLin) or efflux (CLef) at the basolateral membrane as well as at the apical membrane for secretion (CLint,sec) of the eliminating organ. These, together with recent, improved estimates of tissue to plasma or blood partitioning coefficients (RT, Ctissue/Ctissue blood, which equals Ctissue/Cvenous blood) for weak bases and acids [2–4], have greatly improved the modeling outcomes.

The technique is extremely useful and uniquely pertinent to correlate in vitro and in vivo preclinical data from animals such as the mouse, rat, dog or monkey and extrapolate these to describe drug behavior in humans [5–9]. PBPK modeling/simulations have provided the basis for the identification and selection of candidates with desirable pharmacokinetic properties in drug discovery and drug development [10–13] and health risk assessment [14–20]. The technique is well suited in the appraisal of how alterations of physiological or biochemical conditions such as age [21, 22], disease states [23–26], or genetic variants in transporters, enzymes, and/or protein binding affect drug disposition [27, 28]. The recent PBPK study on the role of the anionic transporter, organic anion transporting polypeptide 1B1 (OATP1B1) related the sensitivity of the systemic pravastatin exposure to hepatic uptake, and further predicted the effects of OATP1B1 polymorphism in humans [28]. PBPK modeling has been applied to account for differences in metabolism due to enzyme abundance (for example, CYP2D6 or CYP2C9 variants) and ethnic differences in alprazolam, caffeine, chlorozoxazone, cyclosporine, midazolam, omeprazole, sildenafil, tolbutamide, triazolam, S-warfarin, and zolpidem metabolism [29].

Indeed, drug metabolism brings about termination of drug action in the formation of inactive metabolites. But some metabolites are active, and constitute the premise of prodrug therapy in forming active drugs. Metabolism also leads to the formation of toxic metabolites. There is a serious concern for safety considerations on the investigation of metabolites, as with metabolite-in-safety testing or MIST [30, 31], or metabolites as inhibitors of enzymes or transporters in drug–metabolite interactions [32, 33], or when metabolites are too reactive to be monitored [34]. However, useful information may not be guaranteed by the administration of the metabolite since differences between the fates of formed vs. administered or preformed metabolites are known to exist [35, 36]. The primary purpose of this review is to summarize the utility of PBPK models in the description of drug absorption, transport, metabolism and excretion and newer aspects in PBPK model development, including the net events in transport and metabolism of a drug undergoing futile cycling with its metabolite in the liver, and integration of organ models to whole body PBPK modeling.

Uniqueness of PBPK modeling for metabolite kinetics

The pros and cons of the PBPK and compartmental modeling methods are summarized, and these readily reveal that major differences exist between compartmental and physiologically-based modeling (Table 1). The summary attests to the appropriateness and usefulness of PBPK models in the examination of metabolite kinetics. The compartmental model usually combines all the metabolite formation organs and elimination organs as the central or peripheral compartment, whereas in PBPK modeling, each organ/tissue is treated as a separate entity. In this way, the PBPK model is able to account for the amount of metabolite formed and the amount of metabolite which will not reach the systemic circulation due to immediate excretion or metabolism within the metabolite formation organ [36, 37]. Hence, elimination (metabolism and excretion) within each tissue will reduce the rate of appearance of the metabolite into the systemic circulation by the fraction that is removed, or the extraction ratio of the formed metabolite, E{mi,P} [37–39]. Only the available fraction of the amount of formed primary metabolite (F{mi,P}) is able to reach the systemic circulation (Fig. 1a). This concept, the immediate, sequential first-pass removal of the formed metabolite in situ the formation organ may be viewed analogously to first-pass removal during drug absorption. In order to account for the lesser amount of metabolite appearing systemically, F{mi,P} is multiplied to kmi, formation rate constant of the primary metabolite (Mi), to account for sequential elimination in compartmental modeling. What is lost materially should yield the secondary metabolite or that amount of Mi eliminated (Fig. 1b).

Table 1 Differences expected of PBPK vs. compartmental models
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Fig. 1
figure 1

Schematic depiction of sequential metabolism of a precursor drug (P) in the formation of the primary (Mi) and secondary (Mii) metabolites with rate constants, kmi and km{mi}, respectively, within an elimination compartment (a), and hidden events in sequential metabolism of the primary metabolite in compartmental modeling, showing the effective formation rate constant of Mi as F{mi,P} kmi and not kmi due to immediate removal of the formed metabolite; what disappeared yields the secondary metabolite with the effective formation rate constant, E{mi,P} kmi; F{mi,P} and E{mi,P} are the hepatic availability and the extraction ratio of the formed metabolite, respectively (b)

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The PBPK model not only addresses the difference in transporters among tissues [35, 40] but also describes how the transport processes: passive diffusion and/or active transporters, facilitate entry of the parent drug (P) and/or the metabolite(s) into eliminating organs. Discrepant metabolite handling has been shown to occur when the metabolites display poor permeability across biological membranes [35, 41–43]. The membrane barrier can limit or even bar the metabolite from entering or leaving the tissue, thereby rendering differences in fate of the formed vs. preformed metabolite kinetics. Another unique feature of PBPK models is the incorporation of a deep/sequestered compartment to explain coupled metabolic reactions in order to account for atypical kinetic profiles of sequentially formed metabolites. Although cytosolic sulfation is the normal conjugation pathway of gentisamide, glucuronidation is the preferential sequential metabolic pathway for gentisamide which is nascently formed from salicylamide within the endoplasmic reticulum space [44]. Another example may be found in the preferential glucuronidation of estrone formed via desulfation of estrone sulfate within the same endoplasmic reticulum space, rather than the re-sulfaion of estrone in the cytosolic space [45]. A deep, intracellular compartment in liver, representing the mitochondria, has been evoked by Schwab et al. [46] for their appraisal of the glycine conjugation of benzoate in the formation of hippurate.

The PBPK model also addresses the interplay of competing pathways within the metabolite formation organ and sequential elimination of the metabolite within the same or in other downstream organs. This aspect will be addressed in the sections to follow. The various PBPK models, with the attendant differential equations, have provided solutions, defined by the various determinants, for the area under the curve (AUC) of the formed, primary metabolite Mi or AUC{mi,P}, pursuant to precursor (P) administration. As shown in the sections to follow, the AUC{mi,P} differs from the AUC of the administered primary metabolite, AUC{pmi}. The formed metabolite area is influenced by parent drug characteristics, whereas the preformed metabolite is not, and the difference is captured in published, theoretical solutions [36, 47–50]. The discrepancy is further caused by differences in transport characteristics of the primary metabolite in each of the organs and the enzymes involved in its formation or further metabolism [36, 40, 49]. Despite these observations, administration of the preformed metabolite is often employed in metabolite-in-safety testing, with the expectation that the strategy exposes the toxicity potential of the formed metabolite arising from drug [35, 36]. However, the answer is not always positive. But when properly executed, PBPK models that utilize preformed metabolite data to enrich model parameters prove to be extremely successful in modeling of sequential metabolism [44–46] and provide accurate predictions associated with drug metabolites and in risk assessment.

PBPK models of the intestine

PBPK modeling offers a new perspective of how drug and metabolite parameters, transporters, and enzymes of the intestine modulate drug absorption and metabolite kinetics. PBPK intestinal models exist to describe absorption and intestinal elimination, based on the perfused intestine preparation [51, 52]. Various PBPK models have been used to relate data for intestinal absorption, secretion, and metabolism. The traditional model (TM) is the PBPK model that describes the intestinal tissue as a whole tissue compartment that receives blood from the superior mesenteric artery. This model describes the complement of enzymes, absorptive transporters, and ATP transporters at the apical and basolateral membrane that mediate efflux back to the lumen or circulation, respectively (Fig. 2a). The drug equilibrates across the basolateral membrane with influx (CL Id1 ) and efflux (CL Id2 ) clearances, summative terms for passive diffusion and transporter-mediated processes. However, the TM has been found to be inadequate in describing the lower extent or absence of metabolite formation following intravenous drug dosing when compared to oral dosing. The observation has prompted the development of the segregated flow model (SFM) [51], a model that presupposes that the intestinal flow to various regions to the intestine is segregated; a minor flow (5–30%, assumed as 10% of the superior mesenteric artery, QSMA, designated to equal the intestinal blood flow QI for simplicity) [53] perfuses the active enterocyte area which mediates absorption, metabolism and efflux, and a larger flow (90%) perfuses the remaining, nonactive serosal region, the submucosa and mucosa regions (Fig. 2b). The influx (CL Id3 ) and efflux (CL Id4 ) clearances allow equilibration of the drug between serosal blood and this serosal tissue region. The SFM is able to explain that a greater extent of metabolism occurs after oral dosing over intravenous dosing than the TM [51, 53], an observation that is distinguished otherwise as pre- and postabsorptive intestinal elimination. The concept is being adopted by others; for instance, the strategy of reduced flow is adopted by the simulation program, SimCYP® to describe intestinal metabolism of oral vs. intravenous dosing [54].

Fig. 2
figure 2

Physiologically-based pharmacokinetic intestine models, the TM and the SFM that depict the intestine as the only elimination tissue as in the perfused intestine preparation. For the TM, the intestinal blood (QI) perfuses the entire intestinal tissue for metabolism, secretion, and absorption from the lumen. For the SFM, segregated intestinal blood flows perfuse the nonmetabolizing (90% QI) and enterocyte-mucosal (10% QI) regions. The precursor drug, P, equilibrates with those in the corresponding tissue layers with intrinsic transfer clearances CL Id1 and CL Id2 for TM, and as CL Id1 and CL Id2 between enterocyte (en) and enterocyte blood (enB), and CL Id3 and CL Id4 between the serosal region (s) and serosal blood (sB) for the SFM. The absorptive, metabolic, and efflux activities within the villus tips of the mucosal layer are represented by the rate constant, ka, and metabolic (CLint,met1,I for Mi formation and CLint,met2,I for other metabolites) and secretory (CLint,sec,I) intrinsic clearances; kg is the rate constant that represents the loss in lumen either due to degradation or ineffective absorption. Similar processes exist for the metabolite, with the parameters classified as “{mi}” (modified from reference [40], with permission)

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With the building of rate equations for the various PBPK models, AUC relationships have been developed under linear conditions upon matrix inversion [40]. Our laboratory had described these PBPK intestinal models that consider the presence of competing metabolic pathways for the drug and the metabolite within the intestine extensively (Fig. 3). In the model, the drug forms the metabolite MiI with CLint,met1,I, forms other metabolites with CLint,met2,I, and is secreted (intrinsic clearance, CLint,sec,I), and MiI undergoes metabolism/secretion in the intestine with intrinsic clearances, CLint,met,I{mi} and CLint,sec,I{mi}, respectively. The secretory intrinsic clearance represents the sum of both passive and carrier-mediated processes. The drug and metabolite may be absorbed from the lumen with the respective rate constants, ka and ka{mi}, and removed luminally by the intestine with rate constants, kg and kg{mi}; the fraction absorbed, Fabs is ka/(ka + kg).

Fig. 3
figure 3

A whole body PBPK model depicting the intestine as the only tissue for metabolite formation and sequential metabolism. Metabolism of drug to other metabolites also occurs in the intestine, and both drug and metabolite are secreted back to the intestinal lumen. Although the drug and metabolite distribute into the liver with CL Hin , CL Hef , CL Hin {mi} and CL Hef {mi}, respectively, there is no elimination within the liver. Both the drug and metabolite is excreted by the kidneys with CLr and CLr{mi}, respectively. The symbols have the same meaning as in Fig. 2, and the parameters associated with the metabolite are classified as “{mi}” (modified from reference [51], with permission)

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With the intestine as the only metabolizing tissue in a whole body PBPK model (Fig. 3), solutions for the AUCs clearly show the influence of the binding parameters, flow, and transport and elimination intrinsic clearances of drug and metabolite on the AUC{mi,P}, and stress the importance of competing pathways (alternate pathway of intestinal metabolism or secretion) within the formation organ and in other competing organs (renal clearance) [40]. The AUC{mi,P} resulting from intestinal metabolism only is dependent on metabolite binding, transport and eliminatory clearances, and parameters for the drug (Table 2). From the equations, the secretory intrinsic clearance is effectively reduced when there is high reabsorption of the drug (Fabs ≈ 1) and metabolite (Fabs{mi} ≈ 1), suggesting that rapid reabsorption negates secretion. The solutions for the AUC{mi,P} of the intestinally formed metabolite, MiI, after intravenous and oral drug administration are virtually identical when renal clearance is absent (Table 2); the only missing term is Fabs. Furthermore, there are clearly recognizable differences of AUC{mi,P) compared to the area after preformed metabolite administration, AUC{pmi} (solutions for AUC{pmi} is the same as those for the drug, except now the solution describes the preformed metabolite). The greatest difference occurs when there is low permeability of the metabolite [36, 40–42, 49]. These discrepancies question the legitimacy of the approach of metabolite administration to ascertain MIST [35, 36].

Table 2 Area under the curves for parent drug and formed metabolite under conditions of intestinal metabolism only (modified from reference [40], with permission)
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The relations derived from metabolite areas can enhance bioequivalence/bioavailability estimates and risk assessments. Solutions for the parent drug (AUC) and its intestinally formed primary metabolite (AUC{mi,P}) for the PBPK model approach [40] clearly show that the AUCpo/AUCiv of the drug yields the systemic bioavailability (Fsys or FabsFI), whereas the ratio of the metabolite areas or AUC{mi,P}po/AUC{mi,P}iv after po and iv drug dosing yields the fraction of dose absorbed (Fabs) when renal clearance of drug is absent (CLr = 0) (Table 2) [40]. The quotient of these area ratios (drug/metabolite) yields the intestinal availability of the drug, FI, when the intestine is the only eliminating organ [40]. In both instances, regardless of whether the drug is renally excreted or not (CLr > 0 or = 0), the metabolite/drug area ratio after po and iv dosing of drug—[AUCpo{mi,P}/AUCpo]/[AUCiv{mi,P}/AUCiv]—equals \( {\frac{{\left[ { 1+ {\frac{{{\text{CL}}_{\text{r}} ( {\text{CL}}_{\text{d1}}^{\text{I}} + {\text{Q}}_{\text{PV}} )}}{{{\text{Q}}_{\text{PV}} {\text{CL}}_{\text{d1}}^{\text{I}} }}}} \right]}}{{{\text{F}}_{\text{I}} }}} \) and \( {\frac{1}{{{\text{F}}_{\text{I}} }}} \), respectively, and exceed unity [40], indicative of first-pass intestinal metabolism.

Extensions of the TM and SFM have been developed, with the recognition that there are segmental differences in distribution of enzymes and transporters [53, 55]. Expansion of the tissue compartment into three segments (duodenal, jejunal, and ileal) and their corresponding flows for the enterocyte and serosal regions allows an examination of the effects of different enzymes and transporters which are heterogeneously distributed [55]. The impact of proximal distribution of CYP3A and not of P-gp (distal) as the strategic factor that affects drug bioavailability has been described [55].

PBPK models of the liver

The PBPK model of the liver and the body/reservoir based on the liver organ only (Fig. 4) has greatly enriched the development of conceptual frameworks of hepatic drug clearances [1, 35, 37, 40, 56]. The reservoir (blood compartment) and liver tissue are interconnected by the hepatic blood flow rate, QH. Rate equations have been developed for the compartments: reservoir (R), liver blood (HB), liver (H) and bile compartments. The unbound fraction, normally described as the unbound fraction in blood, fu, corrects for the binding to plasma and/or red blood cells. QH and Qbile denote the total hepatic blood flow and bile flow rates, respectively. When the liver is the only metabolite formation organ, the split flow (QHA and QPV) can be presented summatively as QH. The model allows for both passive and transporter-mediated processes at the sinusoidal (basolateral) membrane to be expressed collectively as the influx (CL Hin ) and efflux (CL Hef ) clearances to denote the entry and exit transfer clearances of the precursor drug (P) or of the hepatically formed primary metabolite or MiH. The formation intrinsic clearance of the metabolite (CLint,met1,H) as well as for other metabolites (CLint,met2,H) and the secretory (CLint,sec,H) intrinsic clearance constitute the total hepatic drug intrinsic clearance (CLint,H), and MiH may be further metabolized in the liver (with intrinsic clearance, CLint,met,H{mi}) or excreted into bile (with intrinsic clearance, CLint,sec,H{mi}). In absence of other drug eliminating organs, AUC{mi,P} solutions for the liver as an eliminating organ in isolation (Fig. 4) or as the only metabolic organ in a whole body PBPK (Fig. 5) are identical (Table 3). Within this whole body PBPK, the liver is the only metabolizing organ; there is no contribution by the intestine to drug metabolism or excretion (Fig. 5).

Fig. 4
figure 4

Physiologically-based pharmacokinetic model of the liver as the only elimination organ. The model is divided into four compartments: the reservoir (R), liver blood (HB), liver tissue (H) and bile compartment (bile). The influx (CL Hin ) and efflux (CL Hef ) clearances for drug at the basolateral membrane control the entry and exit of the parent drug (P) or metabolite (Mi) (classified as {mi}) between blood and liver tissue. Only the unbound drug or metabolite undergoes transport and metabolism and is considered in the rate equations, though these are not shown graphically for simplification. Metabolism occurs with the metabolite formation intrinsic clearance, CLint,met1,H for the assigned metabolite MiH, or CLint,met2,H for alternate metabolites; the drug also undergoes biliary excretion, with the secretory intrinsic clearance, CLint,sec,H. Hepatic blood flow rate and bile flow rate are denoted by QH and Qbile, respectively. Note the panels of transporters for influx, enzymes for metabolism, and transporters for biliary excretion

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Fig. 5
figure 5

A whole body PBPK model depicting the liver as the only tissue for metabolite formation and sequential metabolism. The drug is metabolized to other metabolites; both drug and metabolite are secreted by the liver. The drug and metabolite distribute into the intestinal tissue, though no elimination occurs within this tissue, and both the drug and metabolite are excreted by the kidneys. The symbols have been defined in Figs. 2, 3, 4 (modified from reference [51], with permission)

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Table 3 Area under the curves for parent drug and formed metabolite under conditions of hepatic metabolism only, with and without renal drug excretion (modified from reference [40], with permission)
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These solutions for AUC{mi,P} show that they are defined by precursor parameters such as CLint,met,H and CLint,sec,H, regardless of hepatic transfer clearances (Table 3). Differences are immediately recognizable from the solutions for AUC{mi,P} and AUC{pmi} when the liver is the only metabolite formation organ. The AUC{pmi} is only dependent on metabolite parameters (as in solutions for drug, now for pmi; Table 3). When the liver is the only elimination organ (CLr = 0), the dose corrected AUCpo{mi,P}/AUCiv{mi,P} ratio yields Fabs, the fraction of dose absorbed to enter the intestinal tissue (Table 3), but this ratio is influenced by CLr when the drug is also renally cleared. When CLr = 0, division of the AUCpo{mi,P}/AUCiv{mi,P} ratio into AUCpo/AUCiv yields the hepatic availability, FH (Table 3). In either instances, regardless of whether the drug is renally excreted or not (CLr > 0 or = 0), the metabolite/drug area ratio after po and iv dosing of drug—[AUCpo{mi,P}/AUCpo]/[AUCiv{mi,P}/AUCiv]—equals \( {\frac{{1 + {\text{CL}}_{\text{r}}/{\text{Q}}_{\text{H}} }}{{{\text{F}}_{\text{H}} }}} \) and \( {\frac{1}{{{\text{F}}_{\text{H}} }}} \), respectively, and exceeds unity [40], indicative of first-pass liver metabolism. Acinar heterogeneity of enzymes and transporters further modifies the kinetics of drugs and metabolites [50, 57–59].

Liver PBPK model for futile cycling

Although metabolism is normally considered as an irreversible process, the metabolite often re-forms the parent drug and undergoes “reversible metabolism” or “futile cycling”. The interconversion between the parent drug and its metabolite may exist for phase I [60, 61] metabolites between oxidative and reductive reactions or phase II metabolites for deconjugation and reconjugation [62–65], and acinar heterogeneity of the enzymes and transporters further modifies the kinetics of futile cycling [45, 66].

There has been some development in PBPK modeling to consider futile cycling kinetics. For modeling purposes, we consider the scenario that the precursor drug is metabolized to its interconversion metabolite by the metabolite formation intrinsic clearance, \( {\text{CL}}_{\text{int,met,H}}^{{{\text{P}} \to {\text{Mi}}}} \), and to other metabolites by the metabolic intrinsic clearance, \( {\text{CL}}_{\text{int,met,H}}^{\text{other}}. \) Similarly, the interconversion metabolite re-forms the precursor drug with the metabolic intrinsic clearance for the metabolite, \( {\text{CL}}_{\text{int,met,H}}^{{{\text{Mi}} \to {\text{P}}}} {\text{\{mi\}}}, \) or forms other secondary metabolites with the intrinsic clearance, \( \text{CL}_{\rm int,met,H}^{\rm other} \{\text{mi}\} \). Both the precursor and metabolite in the hepatocyte may be effluxed back to the sinusoid with CL Hef and \( {\text{CL}}_{\text{ef}}^{\text{H}} {\text{\{mi\},}}\) respectively, or are excreted biliarily with CLint,sec,H and CLint,sec,H{mi}, respectively (Fig. 6). The solutions for the AUC based on a simple liver model for futile cycling have been solved [67]. It becomes clear that the areas under the curve for drug (AUCR) and for the metabolite (AUCR{mi,P}) undergoing futile cycling are exceedingly similar to those in absence of futile cycling, with the exception of two new terms: \( {\text{ef}}_{\text{m}}^{\prime \prime } \) and \( {\text{ef}}_{\text{m}}^{\prime } \) that effectively modify the metabolic intrinsic clearances of the forward (\( {\text{CL}}_{\text{int,met,H}}^{{{\text{P}} \to {\text{Mi}}}} \)) and backward (\( {\text{CL}}_{\text{int,met,H}}^{{{\text{Mi}} \to {\text{P}}}} \{ {\text{mi}}\} \)) processes in futile cycling (Table 4). The fraction, \( {\text{ef}}_{\text{m}}^{\prime \prime } \) or \( {\frac{{{\text{CL}}_{\text{int,sec,H}} {\text{\{mi\} }} +{\text{CL}}_{\text{int,met,H}}^{\text{other}}{\text{\{mi\}}}}}{{{\text{CL}}_{\text{int,sec,H}} {\text{\{mi\} }}+ {\text{CL}}_{\text{int,met,H}}^{{{\rm Mi} \to {\rm P}}}}{\text{\{mi\} }}+{\text{CL}}_{\text{int,met,H}}^{\text{other}}{\text{\{mi\} }}}}\), is the effective coefficient for metabolite formation or the fraction that reduces the intrinsic clearance for metabolite formation, \( {\text{CL}}_{\text{int,met,H}}^{{{\text{P}} \to {\text{Mi}}}}. \) The second term, \( {\text{ef}}_{\text{m}}^{\prime} \) or \({\frac{{{\text{CL}}_{\text{int,sec,H}} +{\text{CL}}_{\text{int,met,H}}^{\text{other}} }}{{{\text{CL}}_{\text{int,sec,H}} + {\text{CL}}_{\text{int,met,H}}^{{{\text{P}} \to {\text{Mi}}}}+{\text{CL}}_{\text{int,met,H}}^{\text{other}} }}} \), is the effective recycling coefficient that reduces the metabolic intrinsic clearance of the metabolite in re-forming the precursor, \( {\text{CL}}_{\text{int,met,H}}^{{{\text{Mi}} \to {\text{P}}}} {\text{\{mi\} }} \) (Table 4). The value of \( {\text{ef}}_{\text{m}}^{\prime \prime } \) is less than unity, and a low \( {\text{ef}}_{\text{m}}^{\prime \prime } \) value suggests a pronounced effect of futile cycling on precursor disposition. With futile cycling, \( {\text{ef}}_{\text{m}}^{\prime } \) modifies \( {\text{CL}}_{\text{int,met,H}}^{{{\text{Mi}} \to {\text{P}}}} {\text{\{mi\} }} \) by appearing as a product, thus yielding a lower AUC{mi,P} with a high \( {\text{ef}}_{\text{m}}^{\prime } \). When futile cycling is absent (\( {\text{CL}}_{\text{int,met,H}}^{{{\text{Mi}} \to {\text{P}}}} {\text{\{mi\}}} = 0 \)), \( {\text{ef}}_{\text{m}}^{\prime \prime } \) equals unity.

Fig. 6
figure 6

Schematic depiction of futile cycling between a precursor, P, and its metabolite, Mi. The forward and backward, interconversion intrinsic clearances are \( {\text{CL}}_{\text{int,met,H}}^{{{\text{P}} \to {\text{Mi}}}}\) and \( {\text{CL}}_{\text{int,met,H}}^{{{\text{Mi}} \to {\text{P}}}} {\text{\{mi\} }}\), respectively. Both the drug and metabolite may form other metabolites and are excreted into bile. There is red cell and protein binding of the parent drug and metabolite, shown in liver blood, and also occurs in reservoir blood (modified from Ref. [67], with permission)

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Table 4 Analytical solutions for the area of the curve from time 0 to infinity for the parent drug (AUCR and AUCH) and formed metabolite (AUCR{mi,P} and AUCH{mi,P}) in reservoir and liver tissue, respectively, and cumulative amounts of biliarily excreted parent drug (Ae,bile,∞) and metabolite (Ae,bile,∞{mi,P}) for a drug–metabolite pair in absence (\( \text{CL}_{\rm int,met,H}^{{\rm Mi} \to {\rm P}} \{\text{mi}\} = 0 \)) and presence (\( \text{CL}_{\rm int,met,H}^{{\rm Mi} \to {\rm P}} \{\text{mi}\} > 0 \)) of futile cycling (modified from Ref. [67], with permission)
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The AUC for the precursor in reservoir, AUCR, is highly influenced by the intrinsic clearances of the precursor for basolateral influx (CL Hin ) and efflux (CL Hef ), the metabolic (\( {\text{CL}}_{\text{int,met,H}}^{{{\text{P}} \to {\text{Mi}}}} \)and \( {\text{CL}}_{\text{int,met,H}}^{\text{other}} \)) and secretory (CLint,sec,H) intrinsic clearances as well as \( {\text{ef}}_{\text{m}}^{\prime \prime } \). Analogously, the AUC of the drug in liver, AUCH, is affected by the unbound fraction in liver, fH, and both this and the cumulative excretion of the precursor drug (Ae,bile,∞) are affected by \( {\text{ef}}_{\text{m}}^{\prime \prime } \) when futile cycling occurs (Table 4). The AUC for the formed metabolite in the reservoir, AUCR{mi,P} is dependent on parent drug and additionally, those for metabolite handling (\( {\text{CL}}_{\text{in}}^{\text{H}} {\text{\{mi\}}}, \; {\text{CL}}_{\text{ef}}^{\text{H}} {\text{\{mi\}}},\;{\text{CL}_{\rm int,sec,H}} \{\text{mi}\},\; \text{CL}_{\rm int,met,H}^{{\rm Mi} \to {\rm P}} \{\text{mi}\} \)), metabolite formation (\( {\text{CL}}_{\text{int,met,H}}^{{{\text{P}} \to {\text{Mi}}}} \)), secretory intrinsic (CLint,sec,H) clearances, and the metabolic intrinsic clearance for alternate metabolism (\( {\text{CL}}_{\text{int,met,H}}^{\text{other}} \)). Again, the AUC for metabolite in liver or AUCH{mi,P}, and the cumulative excretion of the metabolite (Ae,bile,∞{mi,P}) are affected by \( {\text{ef}}_{\text{m}}^{\prime } \) when futile cycling occurs. These areas allow the apparent total (CLliver,tot) and excretory (CLliver,ex) clearances, estimated as Dose/AUCR and Ae,bile,∞/AUCR, respectively, with the metabolic clearance (CLliver,met) being estimated by difference (CLliver,tot − CLliver,ex) (Table 5). Notably, \( {\text{ef}}_{\text{m}}^{\prime \prime } \) appears as a product with \( \text{CL}_{\rm {\text{int} ,met,H}}^{{\rm P} \to {\rm Mi}}\) in the solutions of the apparent clearances (Table 5). These relations are much simplified when the transmembrane barrier does not exist, namely, \( {\text{CL}}_{\text{in }}^{\text{H}}= {\text{CL}}_{\text{ef }}^{\text{H}} \gg {\text{Q}}_{\text{H}} \) and CLint,sec,H, \( {\text{ef}}_{\text{m}}^{\prime \prime } {\text{CL}}_{\text{int,met,H}}^{{{\text{P}} \to {\text{Mi}}}}\) and \( {\text{CL}}_{\text{int,met,H}}^{\text{other}} \) (Table 5).

Table 5 Analytical solutions for metabolic (CLliver,met), excretory (CLliver,ex), and total (CLliver,tot) liver clearances in the presence of futile cycling (modified from Ref. [67], with permission)
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To understand how futile cycling kinetics is affected by the transporter function for the excretion of the metabolite and drug, simulations have been conducted to examine the profiles in the liver (AUCH and AUCH{mi,P}) when the transporter activity of MRP2 for the parent drug and metabolite is reduced, using E217G and E23S17G in the perfused rat liver preparation as examples [67]. The extent of change is dramatic when both the precursor and metabolite secretory transporter activities are both diminished (Fig. 7a, b). With loss of biliary secretory function, the net metabolic clearance (forward reaction) is decreased due to increased futile cycling, leading to an apparent decrease in the net metabolic and total clearances for the precursor (Fig. 7c).

Fig. 7
figure 7

Profiles of AUCs for the drug, AUCR (solid line) and metabolite AUCR{mi,P} (dotted line), in reservoir (a), and AUCH (solid line) and AUCH{mi,P}(dotted line) in liver (b), when the secretory function of Mrp2 is reduced (based on relations shown in Table 4). The example was simulated based on perfusion data for Wistar and TR− liver preparations for E217G that undergoes futile cycling with its metabolite E23S17G, both of which are excreted by Mrp2. The metabolic (dotted line), excretory (dashed with dotted line), and total (solid line) drug hepatic clearances with reduction/loss of Mrp2 activities are shown, based on relations shown in Table 5 (c) (from reference [67], with permission)

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Intestine and liver PBPK models

It is well known that metabolism occurs in both the intestine and liver. A drug may form distinctively different metabolites MiI and MiiH within the intestine and liver, respectively (Fig. 8). Each metabolite may be metabolized and/or excreted within its organ of formation and not in other tissues, namely, MiI formed from the intestine may enter the liver but not for further processing, and the same applies to the hepatically formed MiiH. These AUC solutions are very complex, and the ratio of AUCpo/AUCiv or Fsys is FabsFIFH (Table 6), with the FI and FH terms identical to those solved for the intestine and liver (as in Tables 2 and 3). The metabolite areas after oral and intravenous drug dosing are clearly influenced by all of the drug and metabolite parameters. AUCpo{mi,P}/AUCiv{mi,P} for the intestinally formed metabolite, MiI, is affected by Fabs, the separate liver blood flows for the portal vein (QPV) and the hepatic artery (QHA), and expectedly, also by the hepatic intrinsic clearances, the drug influx intrinsic clearances in liver (CL Hin and CL Hef ) and intestine (CL Id1 and CL Id2 ), as well as the renal clearance (CLr). For this hepatically formed metabolite, MiiH, AUCpo{mii,P}/AUCiv{mii,P} is modulated by Fabs, QPV, QHA, the intrinsic metabolic clearance of the intestine, CL Id1 , CL Id2 , CL Hin and CL Hef as well as CLr. The solutions for the AUCpo and AUCiv are found to be simplified considerably if CLr = 0, as are the AUCpo{mii,P}/AUCiv{mii,P} and AUCpo{mii,P}/AUCiv{mii,P} ratios (Table 6).

Fig. 8
figure 8

A whole body PBPK model depicting the intestine and liver as tissues for metabolite formation and sequential metabolism. Separate metabolites may be formed by the intestine and liver, respectively, and the formed metabolites can distribute into the alternate organ but is only eliminated by the formation organ and kidney. Or, the same primary metabolite is formed in both liver and intestine. Both the drug and metabolite are excreted by the kidneys (modified from reference [51], with permission)

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Table 6 Area under the curves for parent drug and formed metabolite under conditions of intestinal and liver metabolism, each forming a different metabolite, MiI in intestine and MiiH in liver, with and without renal drug excretion (modified from reference [40], with permission)
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The more common occurrence is when the same metabolite Mi is formed in both the intestine and the liver. The intestinally formed metabolite may enter the liver for further processing; the same applies to the hepatically formed metabolite, which can enter the intestine for further sequential processing. In this scenario, solutions for the drug AUCs and metabolite (po and iv) are too bulky to be simplified into presentable formats. The AUC ratio of the precursor, Fsys, the product of Fabs, FI, and FH, is identical to that found in Table 6, regardless of whether CLr = 0 or > 0. However, the AUCpo{mi,P}, AUCiv{mi,P} or the ratio are not in presentable forms, and, from the clusters of the solution, one can readily come to the conclusion that these are different from the former cases (intestine, liver, and intestine and liver forming different metabolites, Tables 2, 3, and 6) [40]. From the analyses involving different metabolite formation organs, the outcome of different solutions for AUC{mi,P} point to the importance of knowing which are metabolite formation organs, and which are metabolite metabolism organs, whether competing pathways exist within the same organ for both the drug and metabolite, and if competing eliminating organs are present. Since multiple metabolite formation organs are likely to be present, and since the formed, phase I metabolite usually undergoes sequential, phase II metabolism within the same or other organs, the solutions for these scenarios, though existing, are not readily presentable and are unlikely to be useful. Hence, for metabolite kinetics, it can be deduced readily that there is a need for stating the underlying assumptions on organs for drug and the metabolite formation as well as removal to expand our understanding of the factors that affect the AUC{mi,P}. Indeed, there is a need to consider not only the elimination organs but also the differential transport barriers or transporters.

PBPK metabolite modeling

While there is no simple solution for AUC{mi,P} in complicated situations, modeling and simulation of sequential metabolism data offers an alternate solution [68]. This type of approach has been performed for the sequential metabolism of codeine to morphine and morphine-3-glucuronide (M3G). The tissue partitioning coefficients, calculated according to Rodgers and Rowland [2, 3], were used to optimize parameters for transport and metabolism to correlate to literature data after the oral (po) and intravenous (iv) administration to man and the rat in PBPK models that describe the N-demethylation of codeine and glucuronidation of morphine in the intestinal and liver. It becomes apparent that the whole body PBPK model consisting of the SFM and not the TM for the intestine, together with liver metabolism and renal excretion, is superior, when ratios of [AUCM3G/AUCmorphine]po and [AUCM3G/AUCmorphine]iv are used as discriminators [68].

The PBPK model is well suited for investigation of metabolite toxicity and risk assessment as it gives more a more accurate estimate of the kinetics of sequential metabolism, particularly for environmental chemicals or metabolites in human or animal models [69–71], as exemplified by the trichloroethylene metabolites [72]. In a mouse PBPK model, Sweeney et al. [73] correlated the occurrence of liver carcinomas and adenomas to the hepatic exposure (AUC) of trichloroacetic acid, a toxic metabolite of perchloroethylene (a solvent commonly utilized in dry cleaning). In another example, the resultant toxic effects of the interaction between trichloroethylene (TCE) and 1,1-dichloroethylene (1,1-DCE) were well predicted using PK/PD modeling [74]. Table 7 highlights other studies in which PBPK modeling of metabolites was applied for the purpose of safety assessment.

Table 7 Metabolite kinetics with PBPK modeling
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Conclusions

The solution for AUC{mi,P} is unique to define the circumstances for metabolite formation and the competing pathways for elimination organs. The AUCpo{mi,P}/AUCiv{mi,P} is useful for (Fabs) only when intestine or liver is the sole drug elimination organ; the ratio does not yield Fsys. For other cases, the ratio is not very useful. In the case of futile cycling, apical transporter activity modulates the AUC for drug and metabolite, and the net metabolism of drug when the metabolite and/or drug are excreted. The PBPK model, encompassing all the involved kinetic factors, is a useful tool to study transporter-enzyme interplay and to predict DDI, and can be a useful tool in risk assessment.