Incorporating target-mediated drug disposition in a minimal physiologically-based pharmacokinetic model for monoclonal antibodies

Abstract

Target-mediated drug disposition (TMDD) usually accounts for nonlinear pharmacokinetics (PK) of drugs whose distribution and/or clearance are affected by their targets owing to high affinity and limited capacity. TMDD is frequently reported for monoclonal antibodies (mAb) for such reason. Minimal physiologically-based pharmacokinetic models (mPBPK), which accommodate the unique PK behaviors of mAb, provide a general approach for analyzing mAbs PK and predicting mAb interstitial concentrations in two groups of tissues. This study assessed the feasibility of incorporating TMDD into mPBPK models to consider target-binding in either plasma (cTMDD) or interstitial fluid (ISF) (pTMDD). The dose-related signature profiles of the pTMDD model reveal a parallel early decay phase, in contrast with the cTMDD model that exhibits a faster initial decline for low doses. The parallel early phase in the pTMDD model is associated with the slow perivascular extravasation of mAb, which restricts the initial decline regardless of interstitial target-mediated elimination. The cTMDD and pTMDD models both preserve the long terminal phase that is typically perceived in conventional two-compartment (2CM) and TMDD models. Having TMDD in ISF impacts the typical relationships between plasma concentrations and receptor occupancy, and between saturation of apparent nonlinear clearance and saturation of receptors. The vascular reflection coefficient (σ_v) was found to affect receptor occupancy in ISF. In the cTMDD model, saturation of nonlinear clearance is equivalent to saturation of receptors. However, in the pTMDD model, they are no longer equal and all parameters pertaining to receptors or receptor binding (R_total, K_D, K_ss, k_int) shifts such relationships. Different TMDD models were utilized in analyzing PK for seven mAbs from digitized literature data. When the target is in plasma, the cTMDD model performed similarly to the 2CM and TMDD models, but with one less system parameter. When the target exists in ISF, the pTMDD functioned well in analyzing only plasma data to reflect interstitial target binding properties. Assigning TMDD consistent with target-expressing tissues is important to obtain reliable characterizations of receptors and receptor binding. The mPBPK model exhibits excellent feasibility in integrating TMDD not only in plasma but also in ISF.

Introduction

Monoclonal antibodies (mAb) have emerged as effective therapeutic agents for a variety of diseases and the mAb market is expected to continue witnessing rapid growth over the next decade [1]. More than 30 antibodies have been approved by the U.S. Food and Drug Administration, and hundreds of candidates are under clinical investigation [2]. The pharmacokinetic (PK) behaviors of mAb are different from small molecules in aspects such as limited vascular permeability, neonatal Fc receptor recycling, and more frequent receptor-mediated nonlinearity [3]. Accommodation of their unique PK properties can assist in better understanding and assessing factors that determine mAb PK and pharmacodynamics (PD). Minimal physiologically-based pharmacokinetic models (mPBPK) offer a simple modeling approach that incorporates physiological elements into PK analysis when only plasma data are available [4]. Our second-generation mPBPK model was developed for mAbs with this modeling concept [5]. We applied this model to extensively survey mAb PK for humans in the literature and demonstrated the feasibility of this model as a general approach in analysis of mAbs with linear PK [6].

The term “target-mediated drug disposition” (TMDD) was introduced by Levy [7] and the modeling framework was established by Mager and Jusko [8]. Drugs exhibiting TMDD often bind with high affinity and to a significant extent (relative to dose) to their target. Such drug-target interaction can be reflected in their PK profiles. TMDD is frequently reported for mAbs with strong target binding affinity and high target abundance [9]. Applying TMDD models in mAb PK analysis can reveal more insights about their targets and target binding dynamics. Although the reductionist feature of mPBPK was emphasized previously [4, 5], the model has flexibility for integrating TMDD mechanisms. The present study demonstrates this feasibility.

Most of the previous TMDD models utilized compartmental models [8–12], and considered target binding in the central compartment (2CM + TMDD) under an assumption that the target is in either blood or tissues that are at rapid equilibrium with blood. For PK analysis of mAbs, this assumption usually yields better parameter identifiability but sometimes gives inconsistent parameter estimates, particularly relating to receptor binding and dynamics [12]. One possible reason is that many mAbs have targets in extravascular space where rapid distributional equilibrium does not occur due to their limited vascular permeability. The “rapid equilibrium” assumption that is commonly applied for small molecules in TMDD models likely causes model misspecifications and yields biased parameters for mAbs when their targets are in extravascular space. Therefore, for these mAbs, building TMDD in the interstitial fluid (ISF) may appropriately reflect and characterize target binding properties. The second-generation mPBPK model provides a simple approach to predict interstitial concentrations of mAbs in two groups of lumped tissues. Thereby, mPBPK models can implement peripheral (interstitial) TMDD features. The present study evaluates this feasibility and further compares model properties where TMDD is present in plasma or ISF. Related issues to be addressed are the influences of perivascular extravasation or vascular reflection coefficients on the interstitial target dynamics and target occupancy, how the interstitial target-mediated elimination affects the apparent nonlinear clearance, and how the perivascular extravasation restricts interstitial target-mediated elimination.

Theoretical

Model structure and simulation

The proposed model structures with TMDD in either plasma (Model A, cTMDD) or ISF (Model B, pTMDD) compartments in the mPBPK model are shown in Fig. 1. The mPBPK model has the same structure and symbol designations as our previous one [5]. Plasma clearance (CL_p) is the only nonspecific clearance in this model because CL_p appears to reflect the most common nonspecific clearance mechanism as found in our recent assessment [6]. In principle, the location of TMDD should be chosen consistent with target-expressing tissues. Here, for the case studies, we considered TMDD in both V_tight and V_leaky, two groups of lumped tissues defined according their endothelium structure [5], as most of the interstitial targets assessed in this study are ubiquitously expressed in various tissues. The same target dynamics and antibody-target binding properties are assumed in two groups of tissues. The target-binding process is approximated by a quasi-steady-state model, where the binding rate is balanced by the sum of the dissociation and internalization rates [13].

Fig. 1.

Model structures of second-generation minimal PBPK models with target-mediated drug disposition in either plasma (cTMDD, Model A) or interstitial fluid (pTMDD, Model B). Symbols and physiological restrictions are defined in Eqs. (1)–(12). The plasma compartment in the left box represents the venous plasma as in full PBPK models but is not applied in this model

The differential equations for Model B are:

$$\begin{array}{l}\frac{{dC}_p}{dt}=\frac{\mathit{Input}}{V_p}+[C_{\mathit{lymph}}·L-C_p·L_1·(1-{\sigma }_1)-C_p·L_2·(1-{\sigma }_2)-{CL}_p·C_p]/V_p\\ IC=0\end{array}$$ (1)

$$\begin{array}{l}\frac{{dA}_{\mathit{tight}\_\mathit{total}}}{dt}=[L_1·(1-{\sigma }_1)·C_p-L_1·(1-{\sigma }_L)·C_{\mathit{tight}\_\mathit{free}}]-{AR}_{\mathit{tight}}·k_{int}·V_{\mathit{tight}}\\ IC=0\end{array}$$ (2)

$$\begin{array}{l}\frac{d{R}_{\mathit{tight}\_\mathit{total}}}{dt}=k_{\mathit{syn}}-(R_{\mathit{tight}\_\mathit{total}}-{AR}_{\mathit{tight}})·k_{\mathit{deg}}-{AR}_{\mathit{tight}}·k_{int}\\ IC=k_{\mathit{syn}}/k_{\mathit{deg}}\end{array}$$ (3)

$$\begin{array}{l}\frac{{dA}_{\mathit{leaky}\_\mathit{total}}}{dt}=[L_2·(1-{\sigma }_2)·C_p-L_2·(1-{\sigma }_L)·C_{\mathit{leaky}\_\mathit{free}}]-{AR}_{\mathit{leaky}}·k_{int}·V_{\mathit{leaky}}\\ IC=0\end{array}$$ (4)

$$\begin{array}{l}\frac{d{R}_{\mathit{leaky}\_\mathit{total}}}{dt}=k_{\mathit{syn}}-(R_{\mathit{leaky}\_\mathit{total}}-{AR}_{\mathit{leaky}})·k_{\mathit{deg}}-{AR}_{\mathit{leaky}}·k_{int}\\ IC=k_{\mathit{syn}}/k_{\mathit{deg}}\end{array}$$ (5)

$$\begin{array}{l}\frac{{dC}_{\mathit{lymph}}}{dt}=[L_1·(1-{\sigma }_L)·C_{\mathit{tight}\_\mathit{free}}+L_2·(1-{\sigma }_L)·C_{\mathit{leaky}\_\mathit{free}}-C_{\mathit{lymph}}·L]/V_{\mathit{lymph}}\\ IC=0\end{array}$$ (6)

where A_{tight_total} and A_{leaky_total} represent total mass of mAb, C_{tight_free} and C_{leaky_free} indicate free concentrations of mAb, R_{tight_total} and R_{leaky_total} refer to total concentrations of target, and AR_tight and AR_leaky are concentrations of drug-target complex in the two groups of lumped tissues V_tight and V_leaky. The IC is the Initial Condition. The V_p is plasma volume, C_lymph is mAb concentrations in lymph, and L₁ and L₂ are lymph flows in two lumped tissues. The σ₁ and σ₂ are vascular reflection coefficients for V_tight and V_leaky, and σ_L is lymphatic reflection coefficient, predefined as 0.2 in this model, as in several previous PBPK models [14]. Rate constants are k_syn for target biosynthesis, k_deg for target degradation, and k_int for antibody-target complex internalization. Considering that TMDD is mostly associated with antibodies that bind with cell membrane receptors, only free mAb is assumed to be collected in lymph and further recycled back to plasma, and the drug-receptor complex is immobile in ISF. The free antibody concentrations are:

$$C_{\mathit{tight}\_\mathit{free}}=0.5·((C_{\mathit{tight}\_\mathit{total}}-K_{ss}-R_{\mathit{tight}\_\mathit{total}})+\sqrt{{(C_{\mathit{tight}\_\mathit{total}}-K_{ss}-R_{\mathit{tight}\_\mathit{total}})}^2+4·C_{\mathit{tight}\_\mathit{total}}·K_{ss}})$$ (7)

$$C_{\mathit{leaky}\_\mathit{free}}=0.5·((C_{\mathit{leaky}\_\mathit{total}}-K_{ss}-R_{\mathit{leaky}\_\mathit{total}})+\sqrt{{(C_{\mathit{leaky}\_\mathit{total}}-K_{ss}-R_{\mathit{leaky}\_\mathit{total}})}^2+4·C_{\mathit{leaky}\_\mathit{total}}·K_{ss}})$$ (8)

The K_ss is a steady-state constant defined by Gibiansky et al. [13] as:

$$K_{ss}=\frac{k_{int}+k_{\mathit{off}}}{k_{on}}$$ (9)

The k_on and k_off are antibody-receptor association and antibody-receptor dissociation rate constants. The antibody- target complex concentrations are:

$${AR}_{\mathit{tight}}=\frac{R_{\mathit{tight}\_\mathit{total}}·C_{\mathit{tight}\_\mathit{free}}}{K_{ss}+C_{\mathit{tight}\_\mathit{free}}}$$ (10)

$${AR}_{\mathit{leaky}}=\frac{R_{\mathit{leaky}\_\mathit{total}}·C_{\mathit{leaky}\_\mathit{free}}}{K_{ss}+C_{\mathit{leaky}\_\mathit{free}}}$$ (11)

The physiological restrictions are: V_p is actual plasma volume and V_lymph is total lymph volume, and:

$${\sigma }_1<1\text{and}{\sigma }_2<1$$ (12)

$$V_{\mathit{tight}}=0.65·\mathit{ISF}·K_p\text{and}{V}_{\mathit{leaky}}=0.35·\mathit{ISF}·K_p,$$ (13)

where ISF is total volume of system interstitial fluid, and K_p is available fraction of ISF for antibody distribution. The relative fractions of V_tight (0.65) and V_leaky (0.35) to total ISF were calculated based upon the values used in full-PBPK models, as were the fractions of L₁ (0.33) and L₂ (0.67) to L [14, 15]. The physiologic parameters [14, 15] for a 70 kg body weight person are: L = 2.9 L/day, ISF = 15.6 L, V_lymph = 5.2 L, and V_p = 2.6 L. The physiologic parameters for a 2.6 kg body weight monkey are: L = 0.275 L/day, ISF = 0.579 L, V_lymph = 0.193 L, and V_p = 0.0966 L. The physiologic parameters for a 20 g body weight mouse are: L = 0.12 mL/h, ISF = 4.35 mL, V_lymph = 1.7 mL, and V_p = 0.85 mL. Also, K_p = 0.8 for native IgG₁ and 0.4 for native IgG₄. Given the similar isoelectric point (pI) values (in the range of 8–9) of the currently assessed antibodies with native IgG₁ (8.6 for human IgG₁) [16], K_p was set to 0.8 for the following analysis.

Typical plasma concentration versus time profiles were simulated for three conditions when target-binding is assumed present in either plasma or V_leaky or V_tight. A full TMDD model (with k_on and k_off) was applied in the simulations and the differential equations are provided in the Supplementary materials. The parameters used for this simulation are listed in Table 1.

Table 1.

Parameter values employed for model simulations

Parameter	Definition	Value	Unit
σ1	Vascular reflection coefficient for Vtight	0.945	–
σ2	Vascular reflection coefficient for Vleaky	0.687	–
CLp	Systemic clearance	0.001	L/h/kg
kon	Drug receptor association rate	0.100	1/(nM h)
koff	Complex dissociation rate	0.001	1/h
ksyn	Target biosynthesis rate	0.100	nM/h
kdeg	Free target degradation rate	0.010	1/h
kint	Complex internalization rate	0.030	1/h

Receptor occupancy (RO) as a function of σ_v

When targets are in interstitial space, the relationship between plasma concentrations and RO is expected to differ from that when targets in blood. Their relationship would be affected by distribution rate and extent. A simulation was performed to evaluate how interstitial distribution alters the relationship between plasma concentrations and RO. Combining Eqs. (10) and (2) produces:

$$\frac{{dA}_{\mathit{ISF}\_\mathit{total}}}{dt}=L·(1-{\sigma }_V)·C_p-L·(1-{\sigma }_L)·C_{\mathit{ISF}}-\frac{R_{\mathit{total}}·k_{int}·C_{\mathit{ISF}}·V_{\mathit{ISF}}}{K_{ss}+C_{\mathit{ISF}}}$$ (14)

In Eq. (14), σ₁, L₁, and V_tight were replaced with σ_v, L, and V_ISF to represent a general situation. The A_{ISF_total} is total amount of antibody in ISF that could be either A_leaky-total or A_tight-total, or antibody mass in any other tissue ISF. An equilibrium state is obtained when dA_{ISF_total}/dt = 0. This could approximate the situation where antibody concentrations reach steady-state in both plasma and ISF after infusion or multiple-dosing. This approximation factors out “t” (time) to produce an explicit equation evaluating the relationships of other factors. Then,

$$L·(1-{\sigma }_V)·C_p-L·(1-{\sigma }_L)·C_{\mathit{ISF}}-\frac{R_{\mathit{total}}·k_{int}·C_{\mathit{ISF}}·V_{\mathit{ISF}}}{K_{ss}+C_{\mathit{ISF}}}=0$$ (15)

From $RO=\frac{C_{\mathit{ISF}}}{K_{ss}+C_{\mathit{ISF}}}$, yields $C_{\mathit{ISF}}=\frac{RO·K_{ss}}{1-RO}$, and substituting C_ISF in Eq. (15), after rearrangement, generates:

$$RO=\frac{[(1-{\sigma }_V)·C_p+\omega +\upsilon ]-\sqrt{{[(1-{\sigma }_V)·C_p+\omega +\upsilon ]}^2-4\upsilon ·(1-{\sigma }_V)·C_p}}{2\upsilon },$$ (16)

where $\upsilon =R_{\mathit{total}}·k_{int}·\frac{V_{\mathit{ISF}}}{L}$ and ω = K_ss · (1 − σ_L).

The relationship between RO and σ_V was then simulated according to Eq. (16) with a changing value of C_p from 500 to 4,000 nM. The other parameters used in this simulation and for the following analysis of human PK data are: L = 2.9 L/day and V_ISF = 15.6 L for a 70 kg person, σ_L = 0.2, R_total = 10 nM, K_ss = 20 nM, and k_int = 2 h⁻¹.

Saturation of nonlinear clearance versus saturation of receptor

In Model B, the apparent target-mediated nonlinear clearance (CL_TM) can be derived in a similar manner as the well-stirred hepatic clearance model where plasma apparent clearance is a function of blood flow and hepatic intrinsic clearance [17]. The well-stirred distribution of antibody in interstitial space was based on the fact that antibody has much higher diffusivity in ISF than for perivascular extravasation [18, 19]. The apparent CL_TM is,

$${CL}_{TM}=\frac{L·(1-{\sigma }_V)·\frac{R_{\mathit{total}}·k_{int}·V_{\mathit{ISF}}}{K_{ss}+C_{\mathit{ISF}}}}{L·(1-{\sigma }_L)+\frac{R_{\mathit{total}}·k_{int}·V_{\mathit{ISF}}}{K_{ss}+C_{\mathit{ISF}}}}$$ (17)

This derivation was made assuming a constant R_total, which entails an assumption of k_deg = k_int. This assumption allows a simple derivation of CL_TM, which otherwise may not be easily solved explicitly. The CL_TM will reach its maximum value (CL_{TM_max}) when C_ISF → 0, then,

$${CL}_{TM\_max}=\frac{L·(1-{\sigma }_V)·\frac{R_{\mathit{total}}·k_{int}·V_{\mathit{ISF}}}{K_{ss}}}{L·(1-{\sigma }_L)+\frac{R_{\mathit{total}}·k_{int}·V_{\mathit{ISF}}}{K_{ss}}}$$ (18

For a mAb with nonlinear clearance, the clearance usually increases with a decrease of plasma concentration and a maximum clearance is expected when concentration approaches to zero. The clearance saturation (1 − CL_TM/ CL_{TM_max}) is defined to reflect the remaining fraction of nonlinear clearance at a given concentration. When the target is present in plasma, clearance saturation is equivalent to target saturation (RO) given that antibody concentrations are usually much higher than target concentrations. If the target exists in ISF, the two are no longer equal. Combining Eqs. (17) and (18), their relationship is:

$$1-\frac{{CL}_{TM}}{{CL}_{TM\_max}}=\frac{\omega ·RO}{\omega +\upsilon ·(1-RO)}$$ (19)

where the parameters relating to υ and ω are the same as in Eq. (16). The association between clearance saturation and target saturation was simulated and the factors that influence their relationship were also assessed. The parameters used in this simulation are the same as used for Eq. (16): L = 2.9 L/day and V_ISF = 15.6 L for a 70 kg person, σ_L = 0.2, R_total = 10 nM, K_ss = 20 nM, and k_int = 2 h⁻¹.

Data analysis

The proposed models were applied to nonlinear PK data for seven mAbs that were found in the literature: zalutumumab [20], transzusumab [21], onartuzumab [22], MEDH7945A [23], romosozumab [24], mavrilimumab [25], and efalizumab [26]. Plasma concentrations versus time data for these antibodies were extracted using Digitizer software [27]. Where possible, a wide range of doses were utilized with all data for each mAb fitted jointly. Different models (2CM + TMDD, pTMDD, cTMDD) were compared in analyzing these data. The model structure of 2CM + TMDD is identical as previously proposed [8]. The differential equations for the 2CM + TMDD and cTMDD and all model codes are provided in the Supplementary Materials.

Computer simulations were performed using ADAPT 5 and fittings utilized the maximum likelihood method in ADAPT 5 with naïve pooling data modeling [28]. The variance model was:

$$V_i={(\mathit{intercept}+\mathit{slope}·Y(t_i))}^2$$ (20)

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) is the predicted response at time t_i from the model. Variance parameters intercept and slope were estimated together with system parameters.

Model performance was evaluated by goodness-of-fits, visual inspection, sum of square residuals, Akaike Information Criterion (AIC), Schwarz Criterion (SC), and Coefficient of Variation (CV) of the estimated parameters. Joint fittings were performed for all dose levels of each mAb.

Results

General simulations and evaluations

The simulated plasma concentration versus time profiles with target-binding in either plasma (cTMDD), or V_leaky (pTMDD-V_leaky), or V_tight (pTMDD-V_tight) are shown in Fig. 2. The first two cases exhibit typical nonlinear profiles (concentration-dependent decline slopes) while the last one does not. The PK profiles of the cTMDD model reveal a rapid initial decline phase for low doses, which is similar to the 2CM + TMDD model [8]. In contrast, the plasma concentrations in the pTMDD-V_leaky model decline in parallel for all doses in the early phase. The cTMDD and pTMDD-V_leaky models both exhibit a prolonged terminal phase.

Fig. 2.

Plasma concentration versus time profiles for increasing doses from 2.6 to 1,300 nmol based on target-mediated drug disposition in either plasma (top), V_leaky (middle), or V_tight (bottom) in the mPBPK modeling framework. Differential equations for the simulations are shown in Supplementary Materials and parameter values are listed in Table 1

Additional simulations, assuming extremely high target densities and fast complex internalization rates, indicate that no matter how high the target-mediated clearance, the PK profiles of the pTMDD-V_leaky model never declined at a slope greater than that in the early phase (α-phase) of relatively high doses. This is in contrast with the cTMDD and TMDD + 2CM models where an extremely high decline slope is usually seen for low doses [8]. The early parallel decay phase, as seen in the signature profiles of the pTMDD model, is associated with the slow perivascular extravasation of mAb, which restricts the initial decline regardless of interstitial target-mediated elimination. This is the reason why there was no noticeable nonlinearity when target-binding is in V_tight, as the limited extravasation rate to V_tight (L₁ × (1 − σ₁)) cannot support a significant CL_TM. Based on estimations of average σ₁ and σ₂ values for V_tight and V_leaky [6], the CL_{TM_max} are expected to be around 3.4 mL/kg for V_tight and 54 mL/kg for V_leaky in a 70 kg person. The CL_{TM_max} is expected to be even smaller if targets are exclusively expressed within a limited interstitial space (such as a small solid tumor).

In the mPBPK model, vascular reflection (σ_v) is the major factor governing the extravasation process. A higher value of σ_v generally produces lesser interstitial distribution. As shown by the simulated relationship between RO and σ_v at an equilibrium state (dA_{ISF_total}/dt = 0) in Fig. 3, the σ_v negatively influences RO at a given plasma concentration. Zone I indicates the range of σ_v for tissues with fenestrated or discontinuous vascular endothelium (V_leaky) and Zone II represents σ_v range for tissues with continuous vascular endothelium (V_tight). Interestingly, there appears to be a stronger plasma concentration-(or dose-) dependency for lower σ_v values (Zone I) than for the relatively high σ_v values (Zone II) in increasing RO (efficacy). In other words, an improvement of RO (efficacy) may be easier achieved by increasing administered mAb doses for tissues with low σ_v than those with relatively high σ_v. In contrast, tissues with high σ_v exhibit a greater σ-dependency and enhancing extravasation (or reducing σ_v), such as modulating epithelial junctions [29, 30], seems to be a more efficient strategy than increasing doses to make an improvement of RO.

Fig. 3.

Simulated relationships between interstitial receptor occupancy (RO) and vascular reflection coefficient (σ_v) at given plasma antibody concentrations. The simulation is based on Eq. (16) and parameters listed in Table 1. Zone I indicates the range of σ_v for tissues with fenestrated or discontinuous vascular endothelium (V_leaky) and Zone II reflects the range for tissues with continuous vascular endothelium (V_tight)

With an assumption of constant R_total, Fig. 4 displays the relationship between saturation of nonlinear clearance (1 − CL_TM/CL_{TM_max}) and saturation of receptors (RO) with different values of parameters pertaining to receptors and receptor binding properties. When TMDD exists in blood (solid line, Fig. 4), the saturation of nonlinear clearance is equivalent to saturation of targets if one considers that mAb concentrations are normally much higher than their targets. However, as shown in Fig. 4, their relationship is significantly shifted when the target is present in extravascular space and missing the target location in plasma for the TMDD model would lead to a biased inference of RO. Saturation of the apparent nonlinear clearance does not directly imply RO in the pTMDD model. Any parameter pertaining to υ and ω (Eq. (19)) is expected to contribute to the degree of shift. The higher values of R_total and k_int and higher binding affinity (K_ss, K_D) would predict a larger shift. Although σ_v considerably affects interstitial receptor saturation (Fig. 3), it does not seem to contribute to this relationship shift. In general, the higher clearance saturation (1 − CL_TM/ CL_{TM_max}) requires higher plasma drug concentrations. The steep rise (on the right of Fig. 4) indicates that a further improvement of RO needs a large escalation of plasma concentrations (or doses). In other words, when the target is in ISF, further improvement of RO by increasing doses appears to be more challenging for high RO, even though it is important for effective therapeutics. In any case, because of the lower concentrations of mAb in ISF than in plasma, a higher concentration (or dose) of mAb is warranted to achieve the same level of RO as that in plasma.

Fig. 4.

Simulated relationships between saturation of plasma nonlinear clearance (1 − CL_TM/CL_TM-max) and saturation of targets (RO). When the target is in ISF, any parameters that are related to target and target binding contribute to the deviation of their relationship from that with target in blood. The simulation is based on Eq. (19) and parameters listed in Table 1

Case studies

Zalutumumab

Zalutumumab is a human IgG₁ that targets the epidermal growth factor receptor (EGFR). Lammerts van Bueren et al. [20] conducted a PK study of zalutumumab in monkeys. Their analysis revealed nonlinear PK that was attributed to extensive interstitial EGFR binding. The authors successfully applied a two-compartment model with a linear plasma clearance and an EGFR-mediated nonlinear clearance in ISF to simulate the plasma PK profiles. The model simulations showed good agreement with measurements. We extracted the data and applied two models to re-analyze the data: cTMDD and pTMDD (Fig. 1). In both models, k_deg was assumed equal to k_int due to a close estimate of the two parameters in initial fittings. Both models well-captured the PK profiles (Fig. 5) and the estimated parameters are listed in Table 2. However, as discussed in the original paper, EGFR are prevalent in the ISF and the pTMDD model should represent the system more mechanistically. This is reflected by more robust parameter estimates (lower CV%) and slight improvement of fittings (smaller Obj, 233 vs 236) for the pTMDD than for the cTMDD model.

Fig. 5.

Pharmacokinetic profiles of zalutumumab in monkeys. Symbols are observations for the indicated doses and curves are model fittings based on the pTMDD model (solid line) or the cTMDD model (dashed line). Fitted parameters are listed in Table 2

Table 2.

Pharmacokinetic parameters (CV%, coefficient of variation) estimated using minimal PBPK models with target-mediated drug disposition

Case [Reference]		1 [20]		2[21]	3[22]	4[23]		5[24]	6[25]		7[26]
Antibody		Zalutumumab		Traszusumab	Onartuzumab	MEHD7945A		Romosozumab	Mavrilimumab		Efalizumab
Model		pTMDD	cTMDD	pTMDD	pTMDD	pTMDD		pTMDD	cTMDD	2CM + TMDD	cTMDD
Parameter/units		Monkey		Man	Monkey	Mouse	Monkey	Man	Man		Man
σ1	–	0.952 (1.11)	0.843 (182)	0.95 (−)	0.99 (<0.01)	0.99 (<0.01)	0.823 (7.11)	0.99 (<0.01)	0.95 (−)	/	0.95 (−)
σ2	–	0.546 (2.91)	0.709 (112)	0.512 (6.51)	0.715 (0.64)	0.260 (27.3)	0.718 (5.75)	0.712 (3.70)	0.769 (0.473)	/	0.870 (5.26)
CLp	ml/h/kg	0.298 (6.75)	0.333 (5.46)	0.092 (23.5)	0.456 (2.78)	0.170 (38.4)	0.438 (11.3)	0.128 (9.16)	0.141 (2.49)	0.149 (6.73)	0.254 (10.2)
Kss	nM	0.376 (74.3)	1.79 (74.5)	0.99 (25.8)	1.61 (76.9)	206 (48.6)	0.094 (620)	1.31 (51.0)	0.930 (18.0)	0.634 (221)	25.5 (653)
ksyn	nM/h	0.310 (36.1)	0.973 (21.9)	0.376 (11.4)	0.318 (24.9)	3.47 (46.2)	0.679 (24.0)	0.172 (25.9)	0.094 (2.47)	0.0104 (6.73)	0.160 (56.8)
kdeg	10−2/h	0.243 (39.2)	0.754 (22.7)	1.17 (19.4)	0.629 (35)	1.51 (34.2)	1.45 (91.2)	1.21 (60.3)	8.16 (4.71)	9.67 (8.39)	0.535 (66.5)
kint	10−2/h	0.243 (39.2)	0.754 (22.7)	1.17 (19.4)	0.611 (34)	1.51 (34.2)	1.45 (91.2)	0.624 (24.4)	1.18 × 102 (−)	1.18 × 102 (−)	0.312 (464)
Vc	L									2.43 (0.246)
Vp	L									4.20 (9.94)
CLd	L/h/70kg									0.052 (2.52)

Traszusumab

Tukoda et al. [21] evaluated the PK of traszusumab, another humanized IgG₁ targeting HER2 after ascending doses were given to patients with metastatic breast cancer. The plasma PK profiles showed nonlinear behavior with dose-dependent terminal half-lives. The mean plasma data were analyzed with the pTMDD model. The value of k_deg was assumed equal to k_int. The model captured the PK profile well (Fig. 6) with reasonable parameter estimates (Table 2).

Fig. 6.

Pharmacokinetic profiles of traszusumab in patients with metastatic breast cancer. Symbols are observations for the listed doses and curves are model fittings using the pTMDD model. Fitted parameters are listed in Table 2

Onartuzumab

Onartuzumab is a monovalent (one-armed) humanized IgG₁ directed against the hepatocyte growth factor receptor (c-Met) with potential antineoplastic activity. Xiang et al. [22] found nonlinear PK of onartuzumab over a dose range of 0.5–30 mg/kg in monkeys. Onartuzumab exhibited a parallel early decline phase for all doses followed by concentration-dependent elimination. The data were fitted with the pTMDD model. The model nicely captured the PK profiles (Fig. 7) and resulted in parameters with reasonable precision (Table 2).

Fig. 7.

Pharmacokinetic profiles of onartuzumab in monkey. Symbols are observations for the listed doses and curves are model fittings using the pTMDD model. Fitted parameters are listed in Table 2

MEHD7945A

MEHD7945A is a human IgG₁ with dual binding specificity that is designed to target both HER3 and EGFR. Kamath et al. [23] assessed its PK in SCID beige mice and monkeys and described the PK using a two-compartment model with linear and nonlinear plasma clearances. In principle, a two-target-mediated drug disposition model is mechanistically preferred for MEHD7945A PK analysis. However, the separate contribution of each target to the nonlinear plasma profile could not be successfully identified in our initial analysis, particularly when only based on the mean plasma data. The pTMDD model with one target was then applied and the model captured the PK profiles well in both species (Fig. 8). The parameter estimates are listed in Table 2. According to our analysis, monkeys seem to have higher binding affinity than mice, suggested by a much lower K_ss value (0.094 vs 206 nM).

Fig. 8.

Pharmacokinetic profiles of MEHD7945A in mice (top) and monkeys (bottom). Symbols are observations for the listed doses and curves are model fittings using the pTMDD model. Fitted parameters are listed in Table 2

Romosozumab

Romosozumab is a humanized IgG₂ that binds to sclerostin, a protein secreted by bone cells, which inhibits bone formation. A first-in-human study by Padhi et al. [24] investigated the PK of romosozumab in healthy men and postmenopausal women. Nonlinear PK was observed, which is likely due to sclerostin-mediated disposition and elimination. The cTMDD model was tested first. Even though this model captured the profile well, it predicted a plasma basal sclerostin concentration (k_syn/k_deg = 19.4 nM) much higher than experimental measurements (~0.05 nM) [31]. Such low plasma sclerostin concentrations cannot adequately support a large nonlinear clearance if sclerostin-mediated elimination is the reason for non-linear PK, suggesting that sclerostin-mediated elimination may primarily occur in extravascular space, a relatively large pool and/or a pool with high sclerostin expression. The pTMDD model seems to be a reasonable option. The pTMDD model captured the profile well (Fig. 9) and yielded reasonable parameter estimates (Table 2). The estimated basal sclerostin concentration in interstitial fluid was about 14.2 nM, which is lower than the plasma predicted value, but still much higher than plasma measurement values. Given that sclerostin is primarily produced and secreted in tissues, it is likely that tissue concentrations are higher than in plasma. With more data, it might be possible to include joint plasma and ISF TMDD models.

Fig. 9.

Pharmacokinetic profiles of romosozumab in healthy men and postmenopausal women. Symbols are observations for the listed doses and curves are model fittings using the pTMDD model. Fitted parameters are listed in Table 2

Mavrilimumab

Mavrilimumab is full human IgG₁ targeting granulocyte– macrophage colony-stimulating factor receptor. A first-inhuman study by Bemester et al. [25] assessed the PK of mavrilimumab in subjects with rheumatoid arthritis. Profound nonlinearity was observed over a wide dose range of 0.01–10 mg/kg. Wang et al. further analyzed the data with a model consisting of a 2CM model with both linear and nonlinear clearance (target-mediated) in the central compartment (2CM + TMDD). The mean PK data were digitized and reanalyzed in our study with the cTMDD as well as 2CM models. Both models captured the profiles well and produced comparable parameter values (Table 2). The cTMDD model adequately describes the plasma data (Fig. 10) with one less systemic parameter than the 2CM + TMDD model (ΔObj<0.1).

Fig. 10.

Pharmacokinetic profiles of mavrilimumab in human subjects with rheumatoid arthritis. Symbols are observations for the listed doses and curves are model fittings using the cTMDD model. Fitted parameters are listed in Table 2

Efalizumab

Efalizumab is a humanized IgG₁ that binds to the α-subunit of LFA-1 (lymphocyte function-associated antigen-1). Efalizumab was approved for use in psoriasis but was later withdrawn from the market due to fatal brain infections. The PK data of efalizumab from a number of clinical studies was reviewed and analyzed by Joshi et al. [26] with a 2CM + TMDD model. We re-analyzed the PK data with both cTMDD and pTMDD models. As shown in Fig. 11, the cTMDD model well-captured all of the plasma profiles, but the pTMDD model failed to account for the profiles at low doses. This example further demonstrated that the apparent CL_TM cannot go larger than distribution clearance for the pTMDD model and the slope of plasma profiles at any time cannot be higher than that in the early decline phase (α-phase) at relatively high doses. This model preference is consistent with LFA-1 having expression sites on lymphocytes that are largely circulated in blood.

Fig. 11.

Pharmacokinetic profiles of efalizumab in humans. Symbols are observations for the listed doses and curves are model fittings based on the cTMDD model (top) or the pTMDD model (bottom). The inadequate prediction of the low dose profiles using the pTMDD model is associated with an important feature of the model that the apparent nonlinear clearance (CL_TM) (Eq. (17)) is always smaller than distribution clearance (ΣL·(1 − σ_v)). Fitted parameters are listed in Table 2

Discussion

A minimal PBPK model has been developed specifically for mAb PK analysis utilizing only plasma data to characterize mAb disposition and predict ISF concentrations in two groups of lumped tissues. The feasibility of incorporating TMDD mechanisms into mPBPK models was assessed in this study to consider target-binding in either plasma (cTMDD) or ISF (pTMDD). The simulated signature profiles reveal a parallel early decline phase for the pTMDD model, in contrast with the cTMDD and TMDD + 2CM models where an extremely rapid initial decline is usually observed for low doses [8]. The cTMDD and pTMDD models both preserve the prolonged terminal phase that is typically found in 2CM + TMDD models. The case studies further demonstrated that the mPBPK model has feasibility in enacting TMDD in either plasma or ISF, in accordance with where targets are primarily expressed. This extends the traditional TMDD modeling framework to consider target-binding not only in vascular space, but also in extravascular space.

The TMDD models depict the drug-target binding process as the key component for the nonlinear PK analysis. When only plasma data are available, the convention is to integrate TMDD with the 2CM model with target-binding in the central compartment [8]. Parameter identifiability has been the major concern for the conventional 2CM + TMDD model if TMDD is assigned to a peripheral compartment when only plasma data are available. The mPBPK model accommodates the unique distribution features of mAb, such as distribution limited to ISF, vascular reflection coefficients (σ₁ and σ₂) governing the initial distribution, and the lymph reflection coefficient (σ_L = 0.2) predefined. These physiological assignments and constraints allow mPBPK models to give reasonable predictions of mAb concentrations in ISF and appear to support an identifiable pTMDD model (Cases 1–5). The appropriate assignment of target binding sites would anticipate more realistic characterizations of target dynamics and target binding properties. When the target is in plasma, the cTMDD model showed similar model performance to the conventional 2CM + TMDD model, but with one less system parameter (Case 6). The pTMDD model gave more precise and reliable estimations than the cTMDD model when targets are primarily in ISF (Cases 1 and 5). In contrast, the cTMDD model performed better than the pTMDD model when targets are mostly in plasma (Case 7). The study only assessed TMDD in either plasma or ISF. But other situations, such as TMDD in V_tight or V_leaky alone, or in both plasma and ISF, are also feasible in mPBPK models. Thus, the mPBPK model appears to be able to handle TMDD in accordance with the target-expressing tissues. It is important to assign TMDD at suitable sites, as it anticipates more reasonable evaluations of receptor binding properties and makes more reliable predictions of RO and the related pharmacodynamics.

When targets are present in plasma (Model A, cTMDD), the mAb in plasma are directly associated with target binding. Characterization of target-mediated clearance is independent of mAb interstitial distribution. The higher target density and faster mAb-target complex internalization usually predict greater target-mediated plasma clearance, irrespective of distribution rate. However, when targets are in ISF (Model B, pTMDD), the mAb in ISF rather than that in plasma are related to target binding. The plasma concentrations, upon which our model-fitting is usually based, no longer directly and faithfully reflect target binding processes. In this case, the apparent CL_TM is greatly restricted by extravasation and the value of CL_TM will become a function of extravasation rate and target-mediated interstitial clearance (Eq. (17)). The statement that higher target density and faster complex internalization lead to larger plasma clearance will no longer hold particularly when the target-mediated elimination rate is faster than the extravasation rate, wherein the apparent nonlinear clearance will be mainly defined by extravasation rate (L × (1 − σ_v)). Thus, an important feature of the pTMDD model is that the apparent nonlinear clearance (CL_TM) (Eq. (17)) is always smaller than distribution clearance (ΣL · (1 − σ_v)). This is reflected in plasma profiles as the slope of plasma decline is always lower than that in the early phase (α-phase) no matter how low are the concentrations (or dose) and how large is the interstitial target-mediated elimination (Figs. 2, 5, 6, 7, 8, 9). This is in contrast to the cTMDD model where an extreme high decline slope is usually observed at low concentrations (or dose) (Figs. 2, 10, 11).

In the pTMDD model, the extravasation-limited CL_TM will shift the traditional relationships between plasma concentrations and RO (Fig. 3), and between saturation of CL_TM and RO (Fig. 4). The extravasation properties, such as σ_v, play essential roles in shaping these relationships. It is generally anticipated that the larger σ_v predicts lower ISF concentrations, and subsequently produces smaller RO. For mAbs with limited accessibility (high σ_v) to target tissues, improving extravasation (or reducing σ_v) seems to be more efficient than increasing doses to achieve a higher RO (Fig. 3, Zone II). This conclusion may assist decision-making in mAb development. In addition, in the pTMDD model, saturation of clearance (1 − CL_TM/CL_TM-max) does not directly imply the saturation of target (RO). The larger target-mediated elimination in ISF would create greater deviations between the clearance saturation and target saturation, and make it more difficult to enhance RO by simply increasing doses (Fig. 4).

Selection between pTMDD and cTMDD models should be made according to target expression sites. The model performance in comparing pTMDD and cTMDD models in analyzing the same data set sometimes implies the location where the targets primarily mediate the nonlinear clearance. In considering and comparing the extravasation rates of many mAbs [6], the maximal nonlinear clearance (CL_{total, low dose} − CL_{linear, high dose}) that is obtained in Noncompartmental Analysis may also infer the target sites and imply the degree of antibody accessibility to such target sites. If a target is exclusively expressed in a certain tissue (such as a solid tumor), then the limited accessibility to this tissue may not produce a sufficiently high plasma CL_TM and the CL_{TM_max} in this situation may somewhat reflect the degree of antibody accessibility into this tissue. Therefore, higher nonlinear clearances are frequently associated with targets in either plasma (Case 6 and 7) or tissues with high vascular permeability.

Comparisons between 2CM + TMDD and Michaelis– Menten (M–M) models have been made [32]. The 2CM + TMDD model approximates to the M–M model when total target concentrations are constant (dR_total/ dt = 0) and free drug concentrations are relatively higher than target concentrations (R_total · K_D/(K_D + C)² ≪ 1). To satisfy the second condition, the plasma concentration should be high enough to obscure the early rapid decline at low doses, which is typically seen in 2CM + TMDD signature profiles [8]. Then, the M–M model would emerge as a comparable approach in describing the data at relatively high plasma concentrations. When the target is in ISF (pTMDD), the early rapid decline phase does not exist at low doses due to the restricted extravasation, but the feature of a relatively long terminal phase is preserved. Therefore, when the PK data does not reveal a long terminal phase (Cases 2, 4, 5), the M–M model may provide comparable curve fittings as the pTMDD model, but the parameter interpretation is complicated (results not included). Most importantly, a chance to explore the interstitial target and target binding properties will be wasted if a M – M model is chosen for this type of PK data analysis. The M–M model cannot capture a long terminal phase as shown in Cases 1 and 3. Systematic comparisons between the pTMDD, cTMDD, and M–M models may be helpful for further model selection of new mAbs.

When targets are primarily in plasma, antibodies are directly exposed to targeted antigens. However, when targets are in ISF, the extravasated antibody has limited presentation to antigens. Only part of the interstitial space is actually accessible for antibody distribution [33, 34]. The limited accessibility is associated with size and steric exclusion [35], irregular vascularization [36], and target-mediated elimination [37, 38]. These factors may also cause spatial heterogeneity in distribution [39]. The limited accessibility and spatial heterogeneity in ISF distribution would affect the actual fraction of targets presented to antibody. Of note, this would challenge the pTMDD model which simply assumes an instantaneous and homogenous distribution in ISF. Such assumption may create a virtual dilution of the receptor pool and result in a relatively lower estimate of target concentrations and probably slower complex internalization. Thus, the parameters derived from this pTMDD model should be considered with respect to the target accessibility and distribution heterogeneity of antibody in ISF. When tissue data and/or more prior knowledge are available, the pTMDD model may be able to accommodate these distribution complexities. However, when analyzing only plasma data, defining target-expressing tissues (total ISF) and the accessible fraction in ISF become imperative for the pTMDD model to obtain reasonable characterization of receptor binding and dynamics. Any bias in defining the available ISF volume would result in the deviation of parameter estimates pertaining to receptor and receptor binding.

In conclusion, the mPBPK models show flexibility in handling TMDD models in either plasma (cTMDD) or ISF (pTMDD). The simulated profiles of the pTMDD model reveal a parallel early decline phase due to the restricted extravasation, which is in contrast with typical signature profiles of the cTMDD model with a rapid early decline for low doses. Both models preserve the feature of the long terminal phase as seen with the conventional TMDD + 2CM model. Although more investigations are warranted, the mPBPK model seems to support a pTMDD model without parameter identifiability issues. Having TMDD in ISF would shift the typical relationships between plasma concentration and RO and saturation of nonlinear clearance and RO. The importance of finding the target-expressing tissues and accessible fraction in the tissues is important for the pTMDD model to obtain reasonable characterizations of receptors and receptor binding properties.