Throughout the article, the term ’ligand’ refers to both a physiological ligand as well as an exogenous drug ligand.
Detailed model of RME (Model A)
We propose the following detailed model of RME of a ligand as schematically represented in Fig. 1: the ligand L
ex is present in the extracellular space. The ligand reversibly binds to free receptor R
m at the cell membrane with association rate constant k
on to form the ligand–receptor complex RL
m that dissociates with rate constant k
off. The complex is internalized with the rate constant k
interRL forming an endosome. The internalized ligand–receptor complex RL
i is either recycled to the membrane with the rate constant k
recyRL, degraded with the rate constant k
degRL to RL
deg, or dissociates with the rate constant k
break. The dissociation results in the subsequent degradation of the ligand L
deg and the availability of the free receptor R
i inside the cell. Free intracellular receptor R
i is recycled to the membrane with the rate constant k
recyR and free membrane receptor R
m is internalized with the rate constant k
interR. Inside the cell, the receptor R
i is produced with the rate k
synth and degraded with the rate constant k
degR.
Based on the law of mass action, the rates of change for the various molecular species are given by the following system of ordinary differential equations (ODEs):
$$ \hbox{d} L_{\rm ex}/\hbox{d} t = k_{\rm off} \cdot RL_{\rm m} - k_{\rm on}/(V_{\gamma} N_{A}) \cdot R_{\rm m} \cdot L_{\rm ex} $$
(1)
$$ \hbox{d} R_{\rm m}/\hbox{d} t = k_{\rm off} \cdot RL_{\rm m} - k_{\rm on}/(V_{\gamma} N_{A}) \cdot R_{\rm m} \cdot L_{\rm ex} +k_{\rm recyR} \cdot R_{\rm i}- k_{\rm interR} \cdot R_{\rm m} $$
(2)
$$ \hbox{d} RL_{\rm m}/\hbox{d} t = k_{\rm on}/(V_{\gamma} N_{A}) \cdot R_{\rm m} \cdot L_{\rm ex} - k_{\rm off} \cdot RL_{\rm m}- k_{\rm interRL} \cdot RL_{\rm m} + k_{\rm recyRL} \cdot RL_{\rm i} $$
(3)
$$ \hbox{d} RL_{\rm i}/\hbox{d} t = k_{\rm interRL} \cdot RL_{\rm m} - k_{\rm break} \cdot RL_{\rm i} - k_{\rm recyRL} \cdot RL_{\rm i}- k_{\rm degRL} \cdot RL_{\rm i} $$
(4)
$$ \hbox{d} R_{\rm i}/\hbox{d} t = k_{\rm interR} \cdot R_{\rm m} - k_{\rm recyR} \cdot R_{\rm i} + k_{\rm break} \cdot RL_{\rm i} - k_{\rm degR} \cdot R_{\rm i} + k_{\rm synth} $$
(5)
where N
A
is Avogadro’s number and V
γ is the volume of extracellular space per cell. In the above equations, all variables are expressed in number of molecules. All parameters are first-order rate constants in units [1/time] except for k
synth, which is a zero-order rate constant in units [molecules/time], and k
on which is a second-order rate constant in units [1/(concentration × time)]. The factor 1/(V
γ
N
A
) ensures conversion of units from molar concentration to number of molecules. With respect to the receptor, the above equations comprise the following three overall processes (cf. Fig. 1): (1) synthesis and degradation; (2) distribution of the different receptor species within and between the cytoplasm and the cell membrane; and (3) ligand–receptor interaction. With respect to the ligand, its disposition processes consist of the three overall processes: (i) binding to the receptor; (ii) internalization of the ligand–receptor complex; and (iii) intracellular degradation.
Reduced models of RME
One objective of this study is to derive and analyze reduced models of RME that capture the impact of receptor dynamics on the distribution and elimination of a ligand and that still allow for a mechanistic interpretation. While during short time intervals the transient redistribution processes between the different receptor species R
m, RL
m, RL
i and R
i may be of interest, these are usually assumed to be negligible on time scales of interest in pharmacokinetics. Therefore, our approach to reduce the detailed RME model will be based on the assumption that the receptor species R
m, RL
m, RL
i and R
i are in quasi-steady state. In order to finally derive reduced models of RME, it is necessary to make an additional assumption on the time-scale of receptor synthesis and degradation. We distinguish the following two scenarios: (1) the time scale of receptor synthesis and degradation is slow in comparison to the time scale of ligand disposition. In this case, we formally set k
synth = k
degR = k
degRL = 0. As a consequence, the total number of receptors in the system remains constant. Or, (2) the time scale of receptor synthesis and degradation is fast, i.e., comparable to the redistribution processes of the different receptor species. Both scenarios will be used in the following to establish a link between the reduced and the detailed model.
Reduced model of saturable distribution into the receptor system and linear degradation (Model B)
The idea in deriving a reduced model of RME is to use the quasi-steady state assumption for the receptor system (RS). This transforms the differential equations (2)–(5) into algebraic equations for R
m, RL
m, RL
i, R
i. For a given number of extracellular ligand molecules L
ex, these algebraic equations can be solved explicitly. This allows us to compute the total number of ligand molecules in the receptor system L
RS = RL
m + RL
i as a function of the extracellular number of ligands L
ex. Based on L
RS, the quasi-steady state number of intracellular ligand–receptor complexes RL
i can be computed, which determines the extent of elimination.
Model B (see Fig. 2) describes the evolution of the total number of ligands L
tot = L
ex + L
RS in form of the following ODE:
$$ \hbox{d} L_{\rm tot}/\hbox{d} t = -k_{\rm deg} L_{\rm RS} \quad\rm{with} $$
(6)
$$ L_{\rm RS} = {\frac{B_{\rm max} L_{\rm ex}}{K_M+L_{\rm ex}}} $$
(7)
$$ L_{\rm ex} ={\frac{1}{2}}\left( L_{\rm tot}-B_{\rm max}-K_{M} + \sqrt{\left( L_{\rm tot}-B_{\rm max}-K_{M}\right)^{2}+4 K_{M}L_{\rm tot}} \right). $$
(8)
The equations comprise three parameters: the maximal ligand binding capacity B
max of the receptor system (in units molecules), the number of extracellular ligand molecules corresponding to a half-maximal binding capacity K
M
(in units molecules), and the degradation rate k
deg (in units 1/time). In this reduced model the combination of saturable distribution and linear degradation results in the overall saturable elimination of the ligand.
For the two scenarios of slow or fast receptor synthesis and degradation, the functional relation between the parameters B
max, K
M
and k
deg and the parameters of the detailed model of RME can be established. In the case of slow receptor synthesis and degradation, it is
$$ B_{\rm max} = R_{0} \cdot {\frac{k_{\rm break}+k_{\rm recyRL}+k_{\rm interRL}} {k_{\rm break}+k_{\rm interRL}+k_{\rm recyRL}+k_{\rm interRL}\cdot k_{\rm break}/k_{\rm recyR}}} $$
(9)
$$ K_M = K_D \cdot {\frac{ V_{\gamma} N_{A} \cdot k_{\rm break} \left(1+{\frac{k_{\rm interRL}}{k_{\rm off}}}+{\frac{k_{\rm recyRL}}{k_{\rm break}}}\right)}{k_{\rm break}+k_{\rm interRL}+k_{\rm recyRL}+k_{\rm interRL}\cdot k_{\rm break}/k_{\rm recyR}}} $$
(10)
$$ k_{\rm deg} = {\frac{k_{\rm break} \cdot k_{\rm interRL}}{k_{\rm interRL}+k_{\rm break}+k_{\rm recyRL}}}, $$
(11)
where R
0 is the total number of receptors and K
D
= k
off/k
on denotes the dissociation constant of the ligand–receptor complex. In the case of fast receptor synthesis and degradation, the relation between the parameters is
$$ B_{\rm max} = {\frac{k_{\rm synth}}{k_{\rm degR}}} \cdot {\frac{ k_{\rm recyR}\cdot(k_{\rm recyRL}+k_{\rm lyso}+k_{\rm interRL})} {k_{\rm interRL}\cdot(k_{\rm lyso}+k_{\rm recyR}\cdot k_{\rm degRL}/k_{\rm degR})}} $$
(12)
$$ K_M = K_D \cdot {\frac{V_{\gamma} N_{A} \cdot k_{\rm interR}\cdot (k_{\rm recyRL}+k_{\rm lyso}+k_{\rm interRL}\cdot k_{\rm lyso}/k_{\rm off})}{k_{\rm interRL}\cdot(k_{\rm lyso}+k_{\rm recyR}\cdot k_{\rm degRL}/k_{\rm degR})}} $$
(13)
$$ k_{\rm deg}={\frac{k_{\rm lyso} \cdot k_{\rm interRL}}{k_{\rm interRL}+k_{\rm lyso}+k_{\rm recyRL}}}, $$
(14)
with k
lyso = k
break + k
degRL.
Reduced model of saturable degradation (Model C)
The proposed Model C (see Fig. 2) is a further reduction of Model B. It is based on the additional assumption that the amount of ligand distributed into the receptor system is negligible in comparison to the total amount of ligand molecules, i.e., L
tot = L
ex + L
RS ≈ L
ex. More formally, Model C can be derived from Model B under the assumption
$$ {\frac{B_{\rm max}}{K_M+ L_{\rm ex}}} \ll 1, $$
(15)
which implies L
RS ≪ 1 and thus L
tot ≈ L
ex from Eq. 7. Substituting L
ex by L
tot in Eq. 7 and L
RS into Eq. 6 yields the ODE for the total number of ligand molecules:
$$ \hbox{d} L_{\rm tot} /\hbox{d} t = -{\frac{V_{\rm max}L_{\rm tot}}{K_M+L_{\rm tot}}}. $$
(16)
The model comprises two parameters: the maximal elimination rate of ligand molecules V
max (in units molecules/time) and the number of ligand molecules K
M
, at which the elimination rate is half-maximal. Exploiting the relation
$$ V_{\rm max}=k_{\rm deg}\cdot B_{\rm max}, $$
(17)
we obtain the functional relations between V
max and the parameters of the detailed model of RME (Model A). In the case of slow receptor synthesis and degradation, the functional relationship is given by
$$ V_{\rm max} = R_{0}\cdot {\frac{k_{\rm break} \cdot k_{\rm interRL}} {k_{\rm break} +k_{\rm interRL}+k_{\rm recyRL}+k_{\rm interRL}\cdot k_{\rm break}/k_{\rm recyR}}} $$
(18)
and K
M
is defined as in Eq. 10. In the case of fast receptor synthesis and degradation, it is
$$ V_{\rm max} = {\frac{k_{\rm synth}}{k_{\rm degR}}}\cdot {\frac{k_{\rm lyso} \cdot k_{\rm recyR}}{k_{\rm lyso}+k_{\rm recyR}\cdot k_{\rm degRL}/k_{\rm degR}}} $$
(19)
and K
M
is defined as in Eq. 13.
Integration of RME into compartmental PK models
In order to facilitate the transfer of reduced models of RME into compartmental PK models underlying PK data analysis and for use in the example of therapeutic protein receptor interaction, we explicitly state the system of ODEs for a two-compartment PK model. The model comprises a central compartment (volume V
1 (in units volume) and ligand concentration C
1 (in units mass/volume)) from which linear elimination CLlin (in units volume/time) takes place and a peripheral compartment (volume V
2 and total ligand concentration C
2), where saturable elimination via receptor mediated endocytosis CLRS takes place (see Fig. 3). In the peripheral compartment, we further distinguish between the concentration C
RS within the receptor system and the extracellular concentration C
ex. The inter-compartmental transfer flows are denoted by q
12 and q
21 (in units volume/time).
As in this article we are interested in how to represent RME in PK models, the below mentioned system of ODEs based on the reduced Models B and C represent the proposed structural PK model that can be used for parameter estimation in PK data analysis of nonclinical and clinical trials. The parameter values are determined by performing a fit of the model to the specific in vivo data. Alternatively, the model might be used to scale-up in vitro derived RME parameter values to the in vivo situation (see also Discussion).
If Model B is used to describe the elimination by RME, the system of ODEs is
$$ V_1\cdot {\rm d} C_1/{\rm d} t = q_{21} \cdot C_{\rm ex} - q_{12} \cdot C_1 - \hbox{CL}_{\rm lin} \cdot C_1 + \rm{dosing} $$
(20)
$$ V_2\cdot {\rm d} C_2/{\rm d} t = q_{12} \cdot C_1 - q_{21} \cdot C_{\rm ex} - \hbox{CL}_{\rm RS} \cdot C_{\rm RS}, \hbox{ with} $$
(21)
$$ C_{\rm RS} = {\frac{B_{\rm max} \cdot C_{\rm ex} }{K_M+ C_{\rm ex} }} $$
(22)
$$ C_{\rm ex} ={\frac{1}{2}}\left( C_2-B_{\rm max}-K_M + \sqrt{\left( C_2-B_{\rm max}-K_M \right)^{2}+4 K_M C_2} \right), $$
(23)
where dosing denotes a mass inflow (in units mass/time) of, e.g., an i.v. infusion over a given time. The parameter B
max denotes the total maximal ligand binding capacity in mass per volume or mol per volume, K
M
denotes the concentration at which the binding capacity is half-maximal, CLlin and CLRS denote the total elimination capacities (in units volume/time) in the central and peripheral compartment, respectively. In terms of parameter estimation, the PK model contains eight parameters: V
1, V
2, q
12, q
21, CLlin, CLRS, B
max and K
M
, plus additional variables relating to dosing.
If Model C is used to describe the elimination by RME, the system of ODEs is
$$ V_1\cdot {\rm d} C_1/{\rm d} t = q_{21} \cdot C_2 - q_{12} \cdot C_1 - \hbox{CL}_{\rm lin}\cdot C_1 + \rm{dosing} $$
(24)
$$ V_2\cdot {\rm d} C_2/{\rm d} t = q_{12} \cdot C_1 - q_{21} \cdot C_2 - {\frac{V_{\rm max} \cdot C_2}{K_M + C_2}}, $$
(25)
where V
max denotes the total maximal elimination (in units mass/time), and all remaining parameters are defined as above. In terms of parameter estimation, the PK model contains seven parameters: V
1, V
2, q
12, q
21, CLlin, V
max and K
M
, in addition to the parameters relating to dosing.
Nonlinear PK caused by RME
In this section, we investigate the extent of nonlinearity in the context of the Michaelis–Menten model defined in Eqs. 24 and 25. We aim to examine the effect of drug and cell properties on the nonlinearity of the pharmacokinetics, e.g., different drug affinities to the receptor (different k
on and k
off values) or different rates of internalization and recycling of the drug in different cells.
In the chosen setting of the two-compartment PK model (cf. Eqs. 24 and 25, the total clearance CLtot is given by
$$ \hbox{CL}_{\rm tot} = \hbox{CL}_{\rm lin} + \hbox{CL}_{\rm RS}= \hbox{CL}_{\rm lin} + \frac{V_{\rm max}} {K_M + C}, $$
(26)
where C denotes the relevant ligand concentration in the RME compartment (e.g., C
2 in Eq. 25). While the linear clearance is constant, the clearance attributed to RME varies between V
max/K
M
for small ligand concentrations and 0 for high ligand concentrations. Therefore, we consider the quotient V
max/K
M
as a measure of the extent of nonlinearity, i.e., the increase in total clearance for small ligand concentrations.
In order to jointly analyze the slow and the fast receptor synthesis and degradation scenario, we set
$$ R_{0} = R_{\rm m}+R_{\rm i} = {\frac{k_{\rm synth}}{k_{\rm degR}}}\cdot \left( 1+ {\frac{k_{\rm interR}}{k_{\rm recyR}}}\right) $$
(27)
and replace the quotient k
synth/k
degR in Eq. 19 by R
0/(1 + k
interR/k
recyR) according to Eq. 27. Moreover, we extend the definition of k
lyso to the slow scenario by setting k
lyso = k
break in this case (note: for the fast scenario k
lyso = k
break + k
degRL). Then, the extent of nonlinearity for both, the fast and the slow scenario, is given by
$$ {\frac{V_{\rm max}}{K_M}} = {\frac{R_{0}}{V_{\gamma} N_{A}}}\cdot {\frac{k_{\rm on}}{{\frac{k_{\rm off}}{k_{\rm interRL}}}\left(1+{\frac{k_{\rm recyRL}}{k_{\rm lyso}}}\right)+1}}\cdot \left( {\frac{1}{\left(1+{\frac{k_{\rm interR}}{k_{\rm recyR}}}\right)\left( {\frac{k_{\rm interR}}{k_{\rm recyR}}}\right)}}\right)^{p}, $$
(28)
where p = 0 for the slow scenario and p = 1 for the fast scenario. The above equation allows us to study in detail the influence of the various parameters on the extent of nonlinearity.
It can be inferred from Table 1 that ligand-specific, receptor system-specific as well as mixed parameters influence the extent of nonlinearity of the PK: nonlinearity increases for higher affinity drugs (k
on) and cell types, which have a higher receptor concentration at the surface of the cell membrane (R
0, k
recyR) and faster degradation processes (k
lyso). In contrast, higher values of k
off, k
recyRL and higher k
interR, k
degR will decrease the extent of nonlinearity by resulting in a lower number of intracellular ligand receptor complexes, free receptor molecules, or a smaller number of receptor molecules at the cell surface membrane.
Table 1 Contribution of the different parameters to the extent of nonlinearity
In order to more clearly highlight the contribution of the dissociation constant K
D
, we also give the following alternative representation of Eq. 28:
$$ {\frac{V_{\rm max}}{K_M}} ={\frac{R_{0}}{V_{\gamma} N_{A}}}\cdot {\frac{1}{K_D}} \cdot {\frac{1}{{\frac{1}{k_{\rm interRL}}}\left(1+{\frac{k_{\rm recyRL}}{k_{\rm lyso}}}\right)+{\frac{1}{k_{\rm off}}}}}\cdot \left( {\frac{1} {\left(1+{\frac{k_{\rm interR}}{k_{\rm recyR}}}\right)\left( {\frac{k_{\rm interR}}{k_{\rm recyR}}}\right)}}\right)^{p}. $$
(29)
As can be inferred from the above relation, the extent of nonlinearity can be very different for ligands with the same dissociation constant K
D
, but different absolute values of k
off. The difference depends on the relative magnitude of the two terms in the first denominator in Eq. 29, i.e., 1/k
off to 1/k
interRL · (1 + k
recyRL/k
lyso).