Applications of minimal physiologically-based pharmacokinetic models

Abstract

Conventional mammillary models are frequently used for pharmacokinetic (PK) analysis when only blood or plasma data are available. Such models depend on the quality of the drug disposition data and have vague biological features. An alternative minimal-physiologically-based PK (minimal-PBPK) modeling approach is proposed which inherits and lumps major physiologic attributes from whole-body PBPK models. The body and model are represented as actual blood and tissue usually total body weight) volumes, fractions (f_d) of cardiac output with Fick’s Law of Perfusion, tissue/blood partitioning (K_p), and systemic or intrinsic clearance. Analyzing only blood or plasma concentrations versus time, the minimal-PBPK models parsimoniously generate physiologically-relevant PK parameters which are more easily interpreted than those from mam-millary models. The minimal-PBPK models were applied to four types of therapeutic agents and conditions. The models well captured the human PK profiles of 22 selected beta-lactam antibiotics allowing comparison of fitted and calculated K_p values. Adding a classical hepatic compartment with hepatic blood flow allowed joint fitting of oral and intravenous (IV) data for four hepatic elimination drugs (dihydrocodeine, verapamil, repaglinide, midazolam) providing separate estimates of hepatic intrinsic clearance, non-hepatic clearance, and pre-hepatic bioavailability. The basic model was integrated with allometric scaling principles to simultaneously describe moxifloxacin PK in five species with common K_p and f_d values. A basic model assigning clearance to the tissue compartment well characterized plasma concentrations of six monoclonal antibodies in human subjects, providing good concordance of predictions with expected tissue kinetics. The proposed minimal-PBPK modeling approach offers an alternative and more rational basis for assessing PK than compartmental models.

Introduction

Physiologically-based pharmacokinetic models (PBPK) are more mechanistic than ‘classical’ compartmental models as they incorporate anatomical and physiological knowledge into descriptions of pharmacokinetics (PK) [1]. Building such models, however, requires measurements of drug concentrations in numerous organs and tissues. When only blood or plasma data is available, the convention is to apply noncompartmental (NCA) and/or compartmental analysis [2]. However, mammillary models represent the body as a system of compartments that usually have limited physiologic or anatomic reality. They are highly dependent on the intensity of the blood collection schedule and make no prior use of the physiology of the organism [3, 4]. An intermediate perspective between full PBPK and compartmental models would be helpful. Recirculatory models offer a somewhat more physiologically realistic approach to describe PK in which flows to tissue spaces are considered [5]. These models usually require some empirical adjustments and are not often utilized as they are difficult to implement and interpret [6, 7].

‘Lumping’ is a commonly employed approach to reduce the dimensionality and complexity of whole body PBPK models placing tissues that show similar kinetics together to form fewer compartments [8, 9]. Whole body PBPK models are often successfully represented by a limited number of ‘lumped’ compartments. A PBPK model with only four organs was previously constructed for warfarin [10]. Some organs (muscle, fat, kidney) that exhibited similar kinetics could be further lumped in this model, and the time course of serum warfarin concentrations was mainly supported by nonlinear (liver) and linear tissue distribution components.

The concept of ‘hybrid-PBPK’ models has been utilized in previous studies by using plasma concentration functions with perfusion input to specific organs (mostly liver, kidney, brain, or tumor) [11–13]. This allows more detailed descriptions of drug exposures or disposition for specific tissues. Henthorn et al. [14] employed the term ‘minimal’ compartment model in constructing a parallel channel, lumped-parameter circulatory model for disposition of indocyanine green in dogs.

This report proposes a generalized minimal-PBPK modeling approach which represents the system as a substantially lumped PBPK model. The incorporation of physiological and anatomical knowledge into the minimal-PBPK models allows separation of system- and drug-specific parameters and permits consideration of known physicochemical and metabolic properties in determination of drug PK features (Fig. 1). Applications of the minimal-PBPK models will be shown by characterizing several types of drugs in different situations, revealing better PK insights than conventional mammillary models.

Fig. 1.

Paradigm for constructing PBPK models

Model structures

Figure 2 shows a minimal-PBPK model with two tissue compartments. The model is described using the differential equation structure similar to PBPK models:

$$\frac{{dC}_P}{dt}=\frac{\mathit{Input}}{V_P}+f_{d1}·Q_{CO}·\frac{C_1}{K_{P1}·V_P}+f_{d2}·Q_{CO}·\frac{C_2}{K_{P2}·V_P}-\frac{C_P}{V_P}·(f_{d1}·Q_{CO}+f_{d2}·Q_{CO}+CL)C_P(0)=0$$ (1)

$$\frac{{dC}_1}{dt}=\left(C_P-\frac{C_1}{K_{P1}}\right)·f_{d1}·Q_{CO}/V_1{C}_1(0)=0$$ (2)

$$\frac{{dC}_2}{dt}=\left(C_P-\frac{C_2}{K_{P2}}\right)·f_{d2}·Q_{CO}/V_2{C}_2(0)=0$$ (3)

where C_p is concentration of drug in V_p (blood or plasma volume), C₁ and C₂ are drug concentrations in tissue compartments 1 (V₁) and 2 (V₂), Q_CO is cardiac blood (or plasma) flow, f_d1 and f_d2 are fractions of Q_CO for V₁ and V₂, K_p1 and K_p2 are tissue partition coefficients, and CL is the systemic clearance.

Fig. 2.

Minimal-PBPK model with two tissue compartments. Symbols and physiological restrictions are defined with Eqs. (1–3). The blood compartment in the left box mimics the venous blood as in full PBPK models, but is not utilized in the present model

The key features of this model are the physiological restrictions where V_p is the designated blood or plasma volume, Q_CO is the assigned cardiac output, and:

$$\begin{array}{l}{f}_{d1}+f_{d2}\le 1\\ V_1+V_2+V_p=BW\text{or}\mathit{ECF}\\ (\text{Body}\text{weight}\text{or}\text{extracellular}\text{fluid}\text{volume}).\end{array}$$

This allows V₂ to be treated as a secondary parameter with only estimation of V₁.

The model was further extended by including the liver to describe oral dosing with hepatic first-pass as shown in Fig. 3. The modified model equations are:

$$\begin{array}{l}\frac{{dC}_P}{dt}=\frac{\mathit{Input}}{V_P}+f_{d1}·(Q_{CO}-Q_{\mathit{hep}})·\frac{C_1}{K_P·V_P}\\ +f_{d2}·(Q_{CO}-Q_{\mathit{hep}})·\frac{C_2}{K_P·V_P}\\ +Q_{\mathit{hep}}·(\frac{C_{\mathit{hep}}}{K_P}-C_P)/V_P-C_P·[f_{d1}·(Q_{CO}-Q_{\mathit{hep}})\\ +f_{d2}·(Q_{CO}-Q_{\mathit{hep}})+{CL}_{\mathit{non}-\mathit{hep}}]/V_P{C}_P(0)=0\end{array}$$ (4)

$$\frac{{dC}_1}{dt}=\left(C_P-\frac{C_1}{K_P}\right)·f_{d1}·(Q_{CO}-Q_{\mathit{hep}})/V_1{C}_1(0)=0$$ (5)

$$\frac{{dC}_2}{dt}=\left(C_P-\frac{C_2}{K_P}\right)·f_{d2}·(Q_{CO}-Q_{\mathit{hep}})/V_2{C}_2(0)=0$$ (6)

$$\frac{{dC}_{\mathit{hep}}}{dt}=[F_G·\mathit{Dose}·e^{-k_a·t}+Q_{\mathit{hep}}·\left(C_P-\frac{C_{\mathit{hep}}}{K_P}\right)-\frac{C_{\mathit{hep}}}{K_{P\mathit{intu}}}]/V_{\mathit{hep}}{C}_{\mathit{hep}}(0)=0$$ (7)

where Q_hep is portal vein blood flow, C_hep is drug concentration in liver compartment (V_hep), CL_intu and CL_non-hep are unbound hepatic intrinsic and non-hepatic clearances, and F_G and k_a are pre-hepatic bioavailability and the absorption rate constant. Other symbols represent the same parameters as in Fig. 2. The physiological restrictions of these parameters are:

$$f_{d1}+f_{d2}\le 1\text{and}{V}_1+V_2+V_p+V_{\mathit{hep}}=BW\text{or}\mathit{ECF}$$

Fig. 3.

Minimal-PBPK model extended with the hepatic compartment. Symbols and physiological restrictions are defined in Table 2

The extended model with a hepatic compartment (Fig. 3) can be related to clearance concepts of Rowland et al. [15] who proposed the well-stirred model:

$${CL}_{\mathit{hep}}=\frac{Q_{\mathit{hep}}·f_{up}·{CL}_{\mathit{intu}}}{Q_{\mathit{hep}}+f_{up}·{CL}_{\mathit{intu}}}=Q_{\mathit{hep}}·ER$$ (8)

which often works well for relating hepatic clearance (CL_hep) to intrinsic clearance (CL_intu), fraction unbound in plasma (f_up), and extraction ratio (ER). With addition of F_G and k_a, the minimal-PBPK model can account for both oral and IV profiles. The extended model separates clearances by liver (CL_intu) and other tissues (CL_non-hep). Hepatic clearance (CL_hep) can be calculated from CL_intu that is estimated based upon liver free concentration where 1/K_p is fraction unbound in the liver when f_up = 1.

A basic model was applied to assess the PK of several monoclonal antibodies (mAb) as shown in Fig. 4. Since elimination of mAb mostly occurs in the tissue interstitial space [16], the clearance (CL_T) was assigned to the tissue compartment (Model A) and compared with model with clearance (CL_P) from plasma (Model B). The Model A equations are:

$$\frac{{dC}_p}{dt}=\frac{\mathit{Input}}{V_p}+f_d·Q_{CO}·\left(\frac{C_t}{K_P}-C_p\right)/V_p{C}_p(0)=0$$ (9)

$$\frac{{dC}_t}{dt}=f_d·\frac{Q_{CO}}{V_t}·\left(C_p-\frac{C_t}{K_P}\right)-\frac{C_t}{V_t}·{CL}_T{C}_t(0)=0$$ (10)

where Q_CO is specifically designated as plasma cardiac output for mAb and CL_T was assumed to be relevant with total concentration of mAb in tissue. The physiological restrictions of relevant parameters are: f_d ≤ 1 and V_t + V_p = BW:

Fig. 4.

Basic minimal-PBPK model assigning CL either from tissue (Model A) or plasma compartments (Model B). Symbols and physiological restrictions are defined with Eqs. (9) and (10)

Data analysis

The proposed minimal-PBPK models were applied to several PK data sets that were found in the literature: beta-lactam antibiotics [17–36], hepatic elimination drugs (dihydrocodeine [37], verapamil [38], repaglinide [39], and midazolam [40]), moxifloxacin PK in five species [41], and six mAbs PK in human subjects [42–47]. Concentration–time data were obtained via computer digitalization [48]. Alternative structures of the minimal-PBPK models were assessed such as numbers of tissue compartments, the same or different K_p values, and the same or different f_di values for multiple tissue spaces.

Moxifloxacin PK after intravenous dosing in five species was simultaneously described by integrating allometric scaling principles into the minimal-PBPK model with one tissue compartment. The integrated model was described by:

$$Q_{CO}=0.275·{BW}^{0.75}$$ (11)

$$CL=a·{BW}^b$$ (12)

All species were assumed to have the same fitted f_d and K_p value, while Q_CO and CL were allometrically scaled across species [49, 50] with two estimated parameters for CL (a and b)

Human PK profiles for 6 mAb after intravenous dosing were fitted and compared between minimal-PBPK models with CL from either tissue or plasma compartments (Fig. 4).

Fittings utilized the maximum likelihood method in ADAPT 5 [51]. The variance model was defined as:

$$V_i={({\sigma }_1+{\sigma }_{2}Y(t_i))}^2$$ (13)

where V_i is the variance of the response at the ith time point, t_i is the actual time at the ith time point, and Y(t_i) represents the predicted response at time t_i from the model. Variance parameters σ₁ and σ₂ were estimated together with system parameters during fittings. The goodness-of-fit criteria included visual inspection of the fitted curves, sum of squared residuals, Akaike Information Criterion (AIC), Schwarz Criterion (SC), and Coefficient of Variation (CV) of the estimated parameters.

Results

General evaluation

The first physiological feature of the minimal-PBPK model is the assignment of blood (or plasma) volume as the initial distribution space as is done in full PBPK models. This has worked well in numerous instances and obviates a major problem in mammillary models where early time extrapolations are dependent on the timing and frequency of blood sampling [2, 3]. The assigned blood (or plasma) volume can be more specific with considerations of species, sex, and age differences [52]. The assumed blood volume for man was 5.2 L/70 kg [49].

The second feature is the employment of Fick’s Law of Perfusion for tissue distribution with the capability of tissue drug uptake and release to be either flow- or permeability-limited and limited to passive mechanisms. The f_di·Q_CO parameter, equivalent to distribution clearance, uses an assumed Q_CO (for man, 5.6 L/min) as do PBPK models. The fraction f_di allows for either multiple tissues and/or limitations in either capillary or cellular permeability. The structural or boundary limitation of f_d1 + f_d2 ≤ 1 is needed.

A key aspect of PBPK models is assessment of tissue/plasma partition coefficients by direct measurements. Mammillary models yield a tissue-average K_p value from V_ss/BW [53]. The minimal-PBPK models incorporate one or more K_p values as fitted parameters.

Full PBPK models use actual organ and tissue weights in assessing overall distribution of drug, often requiring a carcass or ‘residual’ weight to account for unsampled sites. The minimal-PBPK models set BW or ECF = V₁ + V₂ + V_p. Here, the values of these parameters are assigned as physical volumes (or masses) of tissues (for man, BW = 70 kg). In certain cases, V₁ (or V₂) can be designated as a specific tissue. For instance, muscle accounts for 40–45 % of total body mass [49]; designating muscle as V₁ (or V₂) along with f_d·Q_CO for its blood flow is feasible. In some situations (beta-lactams), the total distribution space is known to be ECF [54]. Of course, more specific values of all physiological parameters can be matched to actual subject characteristics.

The extended model with hepatic compartment allows separate estimates of CL_intu and CL_non-hep. In certain cases, one clearance can be neglected if the other dominates. If the liver only slightly contributes to overall clearance, then including the hepatic compartment becomes futile and the extended model reduces to the basic model (Fig. 2).

It is feasible to incorporate hepatic intrinsic clearance and renal clearance as in full PBPK models. Tissue intrinsic clearances are easier to match to in vitro observations and have higher scaling potential to other species than systemic clearance [55]. A basic model assigning CL to the tissue compartment may reflect underlying mechanisms of drug elimination and permit estimates of an average tissue ‘intrinsic’ clearance.

Beta-lactam antibiotics

The PK of 22 beta-lactam antibiotics were found in the literature and their profiles were analyzed based on the proposed minimal-PBPK models. The final structural model was defined with the same K_p for two tissue compartments for 12 beta-lactam antibiotics. The remaining 10 drugs were described by a model with one tissue compartment. Total volume V_p + V₁ + V₂ was assumed to be ECF in this analysis (volume = 18.2 L) because beta-lactam antibiotics distribute mainly within extracellular fluid [54]. The fitted profiles are shown in Fig. 5 and the parameters are summarized in Table 1. All models captured the observed PK profiles quite well with reasonable CV % for the estimated parameters.

Fig. 5.

Pharmacokinetic profiles of 22 beta-lactams in human subjects

Table 1.

Final estimated pharmacokinetic parameters (CV %) for 22 beta-lactam antibiotics in human subjects

Drugs [ref]	Dose (mg)	Dosing	Model	fup	Kp	fd1	fd2	V1 (L)	CL (L/min)	Vss (L)
Amoxicillin [17]	1,000	IV bolus	IIa	0.82	0.785 (9.25)	0.0180 (26.4)	0.0737 (40.8)	8.26 (14.5)	0.298 (3.73)	15.4 (6.13)
Ampicillin [18]	1,000	INF (30 min)b	II	0.79	2.089 (95.08)	0.00674 (45.8)	0.419 (11.8)	4.05 (19.9)	0.235 (16.9)	32.4 (79.8)
Cefatrizine [19]	260	INF (30 min)	Ic	0.38	0.651 (7.89)	0.0515 (14.5)	–	–	0.231 (2.98)	13.7 (4.88)
Cefazolin [20]	2,000	IV bolus	I	0.11	0.494 (4.69)	0.0229 (10.5)	–	–	0.0747 (1.72)	11.6 (2.59)
Ceforanide [20]	2,100	INF (30 min)	I	0.19	0.484 (2.62)	0.0321 (7.40)	–	–	0.0576 (0.879)	11.5 (1.43)
Dicloxacillin [21]	2,000	INF (30 min)	I	0.04	0.162 (118)	0.00208 (34.0)	–	–	0.0812 (7.57)	7.30 (34.0)
Flucloxacillin [22]	1,000	INF (5 min)	II	0.05	0.445 (10.5)	0.0988 (14.7)	0.00409 (14.6)	6.77 (8.97)	0.141 (1.31)	11.0 (5.53)
Mecillinam [23]	700	INF (15 min)	II	0.78	0.568 (6.21)	0.372 (95.6)	0.00626 (24.2)	8.42 (6.40)	0.200 (2.19)	12.6 (3.65)
Piperacillin [24]	4,000	INF (5 min)	II	0.70	0.547 (4.97)	0.416 (30.7)	0.00311 (18.2)	9.18 (2.83)	0.190 (1.51)	12.3 (2.87)
Temocillin [25]	2,000	IV bolus	II	0.17	1.19 (4.86)	0.0989 (26.4)	0.0325 (9.97)	2.11 (14.9)	0.0396 (3.38)	21.8 (3.77)
Mezlocillin [26]	4,000	INF (5 min)	II	0.65	1.33 (131)	0.895 (29.4)	0.102 (18.8)	12.7 (36.1)	2.49 (18.7)	22.5 (10.1)
Cloxacillin [27]	1,000	INF (5 min)	I	0.05	0.280 (10.8)	0.00798 (18.3)	–	–	0.141 (3.48)	8.84 (4.47)
Methicillin [28]	1,050	INF (15 min)	I	0.65	1.93 (49.9)	1.000 (0.0433)	–	–	0.693 (42.2)	30.3 (41.3)
Benzylpenicillin [29]	600	IV bolus	I	0.32	0.479 (11.4)	0.0351 (18.2)	–	–	0.381 (3.67)	11.4 (6.20)
Ticarcillin [30]	2,000	INF (30 min)	I	0.60	1.60 (6.32)	0.238 (17.6)	–	–	0.391 (3.22)	26.0 (5.06)
Carbenicillin [30]	2,000	INF (30 min)	I	0.50	2.07 (29.3)	1.000 (0.0264)	–	–	0.479 (25.8)	32.1 (24.5)
Azlocillin [32]	4,160	INF (20 min)	II	0.70	0.517 (2.47)	0.0415 (11.8)	0.0755 (37.4)	10.8 (4.66)	0.125 (1.03)	11.9 (1.39)
Cefamandole [33]	2,000	INF (30 min)	I	0.20	0.146 (51.6)	0.00143 (14.9)	–	–	0.149 (4.01)	7.10 (13.8)
Cefonicid [36]	2,100	IV bolus	II	0.18	0.320 (13.9)	0.00306 (20.4)	0.148 (29.2)	7.42 (6.88)	0.0432 (1.26)	9.36 (2.34)
Biapenem [31]	300	INF (30 min)	II	0.96	1.94 (10.7)	0.996 (–)	0.00444 (8.11)	4.72 (7.49)	0.0998 (3.24)	30.4 (8.91)
Cefoxitin [34]	2,100	IV bolus	II	0.33	1.03 (21.6)	0.362 (9.61)	0.0160 (14.5)	7.05 (20.4)	0.390 (3.62)	18.6 (15.6)
Nafcillin [35]	500	IV bolus	II	0.16	1.24 (8.52)	0.979 (0.0529)	0.0203 (0.12)	6.06 (9.00)	0.512 (1.58)	21.4 (6.45)

Assumed 70 kg body weight. All symbols are defined in text

^aMinimal-PBPK model with two tissue compartments

^bIntravenous infusion (duration)

^cMinimal-PBPK model with one tissue compartment

For most beta-lactam antibiotics, tissue distribution rates were much lower than cardiac output (f_d1 + f_d2 <1); indicating permeability-limited uptake.

The K_p values were also calculated separately according to K_p = f_up/f_ue using the relationship for fraction unbound in interstitial fluid (ISF):

$$f_{ue}=1/(1+E/P·(1-f_{up})/f_{up})$$ (14)

where E/P is the ratio of protein concentrations for ISF: plasma and f_ue was approximated assuming albumin concentration in ISF being half that present in plasma [56]. The estimated K_p obtained by minimal-PBPK modeling correlated roughly (r² = 0.319) with the calculated K_p (Fig. 6). This suggests that the unbound fraction only partly influences the degree to which beta-lactam antibiotics penetrate into interstitial fluids [57]. Other factors may also contribute such as ionization [58], transporters [59], or binding to other macromolecules such as penicillin-binding proteins [60]. Table 1 lists V_ss values calculated by V_ss = V_p + ΣK_pi·V_ti.

Fig. 6.

Correlation between estimated (K_p) and calculated partition coefficients K_p (f_up/f_ue) for 22 beta-lactam antibiotics. The line of identity is shown

Hepatic elimination drugs

Four drugs with hepatic first-pass effects and incomplete apparent bioavailability (Bio) were selected for this analysis: dihydrocodeine [37], verapamil [38], repaglinide [39], and midazolam [40]. The data in the literature included both IV and oral administration. The extended model with the hepatic compartment and with typical Q_hep and V_hep values [49] was used to analyze these data. The assumed Q_hep was 1.45 L/min in man and 9.8 mL/min in rat [49]. As shown in Fig. 7, there is good agreement between the observed data and model-predicted profiles. The estimated parameters are listed in Table 2.

Fig. 7.

Pharmacokinetic profiles for four selected drugs after oral (triangle) and IV (square) dosing

Table 2.

Final estimated pharmacokinetic parameters (CV %) for hepatic extraction drugs

Parameter (units)	Definition	Dihydrocodeine	Verapamil	Repaglinide	Midazolam
Species (Body Weight)	/	Man (70 kg)	Man (70 kg)	Rat (250 g)	Man (70 kg)
Oral dose	/	30 mg	120 mg	0.5 mg/kg	10 mg
IV dose	/	30 mg	15 mg (INF, 15 min)a	0.2 mg/kg	9.6 mg
fd1	Fraction of QCO for V1	0.892 (3.77)	0.903 (1.92)	0.979 (0.00)	0.692 (87.4)
fd2	Fraction of QCO for V2	0.108 (NA)	0.097 (NA)	0.021 (NA)	0.0842 (27.3)
Kp	Partition coefficient	0.940 (4.32)	3.96 (12.5)	0.914 (8.74)	0.655 (9.69)
ka (1/min)	Absorption rate constant	0.00754 (10.3)	0.0150 (12.6)	0.0176 (12.1)	0.218 (154)
CLintu (L/min or mL/min)	Hepatic intrinsic clearance	0.424 (158)	1.34 (59.0)	1.70 (3.48)	0.711 (8.20)
CLhep (L/min or mL/min)	Hepatic clearance	0.328	0.696	1.45	0.477
CLnon-hep (L/min or mL/min)	Non-hepatic clearance	NA	0.214 (28.8)	NA	NA
ERb	Hepatic extraction ratio	0.245	0.520	0.148	0.329
V1 (L or mL)	Tissue compartment 1	46.2 (1.47)	21.3 (13.0)	53.5 (15.2)	24.3 (14.4)
FG	Pre-hepatic bioavailability	0.409 (6.21)	0.553 (2.75)	0.500 (0.236)	0.742 (12.4)
Vss (L or mL)	Steady-state volume	64.5 (3.97)	346 (11.6)	230 (8.23)	46.5 (8.61)
Bioc	Bioavailability based upon NCA	0.266	0.217	0.328	0.453

^aIntravenous infusion (duration)

^bER, calculated according to Eq. (8)

^ccalculated as (AUC_oral/D_oral)/(AUC_iv/D_iv) with AUC from noncompartmental analysis (NCA)

Dihydrocodeine

The PK studies of dihydrocodeine were conducted in healthy subjects by Rowell et al. [37]. The extended model simultaneously captured the PK profiles following oral and IV dosing (Fig. 7). The finding that f_d1 + f_d2 = 1 indicates that dihydrocodeine rapidly diffuses into tissues with uptake restricted by blood flow. Hepatic intrinsic clearance is 0.424 L/min and CL_hep = 0.328 L/min. The hepatic extraction ratio (ER) was calculated according to Eq. 8 as 0.245 and F_G was estimated as 0.409. Thus, overall bio-availability would be about 0.309 based upon F_G·(1-ER), near to the NCA Bio value of 0.266. Dihydrocodeine is known to be extensively metabolized in the liver and the modeling results support this with a negligible estimate of CL_non-hep. The separate estimates of different clearances allow quantitative assessment of their relative contributions to drug elimination and also support an identifiable F_G.

Verapamil

Hepatic metabolism is the predominant route of elimination for verapamil and almost no intact drug is excreted according to previous studies [61]. The model supported this with a small estimate of CL_non-hep as 0.214 L/min. Figure 7 shows the fitted results. CL_intu is 1.34 L/min, CL_hep = 0.696 L/min, and ER was 0.520. High hepatic extraction resulted in low systemic bioavailability (26.6 %), in line with a previous study and the NCA Bio of 0.217 [62]. The high estimate of f_d1 + f_d2 indicates that the permeability of verapamil across capillary and cellular membranes was very high and distribution rate is primarily controlled by blood flow.

Repaglinide

This PK study of repaglinide was conducted in rats. Figure 7 shows the fitted results. A previous study indicated that repaglinide is mainly excreted via bile into feces and hepatic clearance dominates the overall elimination [63]. Our analysis confirmed this. The F_G was 0.5, multiplying 1-ER (0.148), gives an overall bioavailability of 0.426. The high estimate of f_d1 + f_d2 suggested that distribution rate is mainly controlled by blood flow.

Midazolam

As a probe of CYP450 3A4/5, midazolam is subject to extensive first-pass metabolism in both intestine and liver [64]. Several approaches have evaluated the relative contributions of intestinal and hepatic metabolism to the overall first-pass effect [65, 66]. The fitted profiles are shown in Fig. 7. Our modeling gave estimates of their contributions. The F_G was 0.742 and liver ER is 0.329, indicating comparable first-pass effects from intestine and liver. Such first-pass effect was also observed in previous studies [65, 66]. The K_p was about 0.655, consistent with an average PBPK literature value [67]. Hepatic and intestinal clearance accounts for most drug elimination as no other clearance was detected in our analysis.

Inter-species PK of moxifloxacin

As shown in Fig. 8, the PK of moxifloxacin in five species was assessed simultaneously by the integrated minimal-PBPK model. The estimated parameters are listed in Table 3. The low CV % values for the parameter estimates indicate good model performance although fittings for some of the digitized data are only approximate. Moxifloxacin appears to efficiently penetrate into tissues with blood flow as the rate-limiting step in the distribution process for the five species as shown by the high f_d value of 1.0. The exponent (b) of the allometric equation for CL was 0.591, close to a reported value [68]. The K_p was 2.70, indicating moderate tissue partitioning and consistent with average tissue concentration in rats [69]. The initial misfit of the last several points for man observed in Fig. 8 may indicate that of CL does not precisely follow a conventional allometric relationship (Eq. 12). The fitting of human data was improved when b was increased. It is well appreciated that CL in man sometimes needs modifications from expectations of preclinical data. The main point of this case was to show a global mPBPK approach to assess f_d and K_p, as well as scale CL across species.

Fig. 8.

Pharmacokinetic profiles for moxifloxacin in five species. The dotted line provides fitting with an adjusted b value for man

Table 3.

Pharmacokinetic parameters of moxifloxacin across five species

Species	Body Wt, kg	Dose mg/kg
Mouse	0.02	9.2
Rat	0.25	9.2
Monkey	5	2.8
Dog	10	2.8
Man	70	1.2

Parameter	Definition	Value	CV%
fd	Fraction of QCO for V1	1.00	< 0.01
Kp	Partition coefficient	2.70	7.57
a	Normalization constant for CL	0.014	5.75
b	Scaling exponent for CL	0.591	2.18
bhum	Scaling exponent for CL in man	0.691	3.07

Monoclonal antibodies

The clinical PK studies of six mAb were selected for this analysis: Hu12F6mu [42], lexatumumab [43], MEDI-528 [44], MEDI-563 [45], CR002 [46] and pateclizumab [47]. All mAb selected in this analysis showed linear PK that is usually analyzed with a 2CM mammillary model. The fitted profiles and simulated tissue concentrations are shown in Fig. 9 and the parameters are listed in Table 4 for models with tissue (Model A) versus plasma clearance (Model B). The models captured the observed PK profiles quite well with precise estimates of parameters. The tissue: plasma exposure ratios (AUC ratios) are reflected by the K_p values, which ranged 0.0475–0.0664 with Model A, and ranged 0.0330–0.0429 with Model B. Studies in mice have shown the muscle: serum AUC ratio of an antiplatelet antibody 7E3 to be 0.038, while other tissues had AUC ratios of 0.125 ± 0.036 [70]. Since muscle is about 60 % of BW, the mass-average AUC ratio should be about 0.07. A study in patients observed similar tissue: plasma AUC ratios of ¹¹¹ln-labeled ZCEO25 IgG [71]. Thus, Model A appears somewhat superior to Model B. Both models yielded low estimates of f_d, and Model A yielded f_d in the range of 1.1–11.1 × 10⁻⁴. This gives a vascular permeability (f_d·Q_CO) in the range of 0.017–0.17, close to a permeability value of 0.00567 L/h/70 kg (PS_L + PS_S) employed in a PBPK model by Baxter et al. [71]. mAbs are subject to extremely low capillary permeability [72] and net tissue uptake is further diminished by FcRn recycling [73]. Predictions of functional tissue concentrations of mAb should preferably employ interstitial fluid volume for V_t rather than BW − V_p owing to their limited cellular uptake except when a cellular target or receptor exists [71]. These models could be readily adapted to handle target-mediated disposition in plasma [74] or in tissue [75].

Fig. 9.

Pharmacokinetic profiles for six mAbs given to human subjects and simulated tissue concentrations

Table 4.

Pharmacokinetic parameters (CV %) for six monoclonal antibodies (mAb) in human subjects

mAb [Ref]	Dosea	Model A: CL from tissue				Model B: CL from plasma
		fd (−)	Kp (−)	CLT (L/h)	${\mathit{AUC}}_R^{\text{b}}$	fd (−)	Kp (−)	CLP (L/h)	${\mathit{AUC}}_R^{\text{b}}$
Hu12F6mu [42]	10 mg	0.00111 (15.3)	0.0522 (8.86)	0.412 (9.18)	0.045	0.000936 (18.1)	0.0429 (6.61)	0.0187 (2.83)	0.042
Lexatumumab [43]	10 mg/kg	0.000692 (25.0)	0.0475 (19.9)	0.404 (24.6)	0.037	0.000562 (26.8)	0.0346 (15.2)	0.0162 (7.62)	0.032
MEDI-528 [44]	9 mg/kg	0.000156 (10.9)	0.0562 (9.62)	0.126 (9.48)	0.042	0.000123 (13.5)	0.0330 (5.17)	0.00553 (1.79)	0.031
MEDI-563 [45]	3 mg/kg	0.000398 (7.69)	0.0519 (5.30)	0.289 (3.80)	0.041	0.000320 (9.56)	0.0337 (3.56)	0.0121 (1.45)	0.033
CR002 [46]	30 mg/kg	0.000110 (31.2)	0.0664 (29.8)	0.106 (23.1)	0.046	0.0000779 (43.6)	0.0333 (20.7)	0.005 (6.58)	0.031
Pateclizumab [47]	5 mg/kg	0.000310 (14.5)	0.0607 (12.6)	0.241 (9.58)	0.045	0.000231 (19.7)	0.0368 (8.82)	0.0112 (2.63)	0.035

^aAssumed 70 kg body weight

^bAUC_tissue: AUC_serum ratio

Discussion

Full PBPK models use the modeling paradigm depicted in Fig. 1 melding all known body and drug properties to either simulate expected concentration–time profiles or perform fittings of available data. Blood flows and organ/tissue weights are largely taken from literature sources [49]. The PBPK models thus either confirm basic expectations about drug disposition or reveal further complexities where investigators add new model features based on experimental data and/or reasonable assumptions.

Minimal PBPK models offer similar possibilities with the constraints of only assessing blood or plasma concentration versus time data and yielding tissue-average drug concentrations. Our case studies with an array of drugs and disposition profiles allowed for reasonable fittings of data with model features predicated on the physicochemical and metabolic properties of a variety of drugs but providing outcomes with low dimensionality and ease of interpretation. As with full PBPK models, there is flexibility in adjusting major model components to accommodate specific drugs and conditions. The parameter estimates can be interpreted with more realism than compartmental models.

The purpose of this report is to demonstrate how minimal PBPK models can be envisioned and applied. The data were all digitized and thus analysis results will be imperfect. Owing to differences in assumptions, the fittings and parameters may differ somewhat from use of compartment models.

As with full PBPK models, blood or plasma volume, cardiac output, and total body space (BW or ECF) are assumed physiological components. For more extensive data sets, these parameters could be specified for subjects of varying ages, sexes, and body weights [76]. This provides freedom to allow the data to reveal f_d, K_p, and CL values, parameters of special interest in PK. Most PK profiles are intrinsically bi- or tri-exponential allowing fitting of 4 or 6 parameters, both for compartmental models [2] as well as with minimal-PBPK models. Both approaches involve ‘lumping’ perspectives. When these models do not function well, there are likely added complexities which are not encompassed within the basic model structures and which need exploration.

Full PBPK, minimal-PBPK, and compartmental models have the expectation that CL = Dose_IV/AUC when elimination occurs from blood or plasma. Compartmental models have the weakness that limited early blood sampling can miss high initial drug concentrations [77] and lead to deviations in estimates of the central compartment volume and cause bias of AUC prediction. For other clearance mechanisms, the minimal-PBPK models can easily include the liver for assessment of first-pass effects when jointly fitting oral and IV data (Fig. 3) or tissue clearance can be implemented (Fig. 4).

Compartmental models (with plasma CL) yield a whole-body distribution coefficient, K_p = V_ss/BW [53]. The same parameter can be generated from both full and minimal PBPK models from V_ss = V_p + ΣK_pi · V_ti. The minimal-PBPK models provide one or two K_p values depending on the number of exponential phases in the data. This number and value(s) are determined in a trial-and-error fitting approach. Of advantage for interpretation, this parameter is directly revealed as one of the primary ‘independent’ variables for each drug. These K_p values reflect either fractional tissue spaces [78] and/or plasma and tissue binding and can be compared among congeneric drugs (beta-lactams, Table 1) or among species (moxifloxacin, Table 3). Scaling of animal data to man is readily feasible with minimal-PBPK models using aspects of allometry for select model components as was done for moxifloxacin.

The minimal-PBPK models are attractive in utilizing the well-known Fick’s Law of Perfusion. The employment of the fractional distribution (f_d) parameter not only accounts for organ/tissue ‘lumping’, but allows for inaccuracies or variability in Q_CO as well as flexibility for distribution to be controlled by permeability as revealed by small values of f_d. The value of f_d·Q_CO is equivalent to distribution clearance (CL_D) as proposed by Stec and Atkinson [79] who adapted earlier concepts from the Kety–Renkin–Crone equation [80–83]:

$${CL}_D=Q·(1-e^{-PS/Q})$$ (15)

where Q is blood flow and PS is a permeability/surface area coefficient. The function allows the value of CL_D → Q when PS ≫ Q and CL_D → PS when PS ≫ Q: Thus small values of f_d depict permeability-limited tissue access and reflect operation of Fick’s Law of Diffusion. However, within the context of the minimal-PBPK paradigm, the f_d values may represent either differing blood distribution to tissues/organs and/or flow/permeability. Of course, the presence of transporters will complicate such assessment [84].

It has been somewhat surprising that full PBPK models commonly work well with use of blood flow to various organs and tissues with occasional introduction of PS values for interstitial-cell water distribution [85]. It has been long appreciated that capillary permeability varies with molecular size producing very slow tissue access for large molecules such as plasma proteins [86, 87]. More frequent model assessment should be done based on permeability-limited access for various molecules. The minimal-PBPK models provide f_d values allowing for a full range of PS and Q values to be operative for tissue distribution.

The parameters of minimal-PBPK models have a simple and logical basis allowing ease of fitting and interpretation. Preliminary NCA analysis can provide estimates of CL from Dose/AUC and K_p from V_ss/BW = MRT·CL/BW where MRT is Mean Residence Time. Values of f_d must fall between 0 and 1 and both f_d and K_p are dimensionless. The value of V₁ must fall with BW − V_p ≥ V₁. Thus, most model parameters have natural physiological limits with restricted parameter spaces allowing for robust parameter estimates. The opposite extreme exists with finding rate constants in compartmental models which are non-intuitive and range widely.

The minimal-PBPK models are highly flexible allowing for most of the modeling adjustments found in applying full-PBPK models. An infusion rate constant or Bischoff g(t) input function [88] may be needed to mimic very early circulatory kinetics [6] rather than assuming Initial Conditions of Dose/V_p. The preferred organ perfusion medium is arterial rather than venous blood [1–3], although this is seldom feasible in human studies. Our modeling employed venous drug concentrations which is also not uncommon in enacting full PBPK models.

The model with hepatic clearance involved use of the well-stirred model and assumptions that the K_p value of liver was the same as other tissues and intrinsic clearance acts on unbound drug in liver. Thus the calculated parameters would differ if another hepatic model was applied (e.g. parallel-tube), if differential tissue binding occurs, and if facilitated uptake or nonrestrictive metabolism was present [89, 90]. Most of our conditions are commonly applied in full PBPK models.

The site of drug administration (IV, IM, SC, oral) can be more relevant than in compartmental models as shown with the extended model (Fig. 3). Nonlinearity in CL or K_p can be added when the range of doses, drug properties, and data profiles allow. As demonstrated with our cases, greater modeling flexibility is gained when studies include diverse dosing (Fig. 7), species (Fig. 8), or experimental conditions (Figs. 5, 9).

The present demonstration of various applications of minimal PBPK modeling utilized digitized data from various literature sources. Thus the parameter estimates are approximate but clearly reflect the feasibility of this modeling approach. However, as a fifth example of minimal-PBPK modeling with more comprehensive metrics, Grimsrud et al. [91] successfully applied a model similar to that in Fig. 2 to assess detomidine PK after IV and IM doses in horses using a population approach with specific physiological parameters for each horse. A physiologic muscle compartment was employed for V₂ as the site of IM drug administration.

Application of full PBPK models remains the gold standard in PK, but requires physicochemical, physiologic, animal tissue, and in vitro metabolic measurements along with appreciable modeling efforts. Minimal-PBPK models offer a sensible compromise when only blood or plasma data are available. They allow the insightful pharmacokineticist great liberties in evaluating data and providing parameters of practical value.