Predicting the Onset of Nonlinear Pharmacokinetics

Abstract

When analyzing the pharmacokinetics (PK) of drugs, one is often faced with concentration C vs. time curves, which display a sharp transition at a critical concentration C _crit. For C > C _crit, the curve displays linear clearance and for C < C _crit clearance increases in a nonlinear manner as C decreases. Often, it is important to choose a high enough dose such that PK remains linear in order to help ensure that continuous target engagement is achieved throughout the duration of therapy. In this article, we derive a simple expression for C _crit for models involving linear and nonlinear (saturable) clearance, such as Michaelis‐Menten and target‐mediated drug disposition (TMDD) models.

Study Highlights

What is the current knowledge on the topic?

Mathematical models of nonlinear PK and TMDD of mAbs are widely used to guide drug development. Often, it is important to choose a high enough dose such that PK remains linear to help ensure that continual target engagement is achieved throughout the duration of therapy. There has not yet been a demonstration for how the PK/pharmacodynamic parameters impact the onset concentration at which the nonlinearity is observed.

What question did this study address?

How the PK and binding properties of the drug impact the onset of nonlinear PK.

What does this study add to our knowledge?

C _crit was derived and was found to be equal to V _max/Cl for the Michaelis‐Menten model or k _syn/(Cl/V _c) for the TMDD model.

How might this change drug discovery, development, and/or therapeutics?

The C _crit parameter can be used to provide better intuition for how the PK parameters impact drug concentration profiles and, in particular, guide the modeler in understanding where the random effects may need to be added and what parameters of the model are identifiable.

When analyzing the pharmacokinetics (PK) of drugs, one is often faced with concentration vs. time curves which display a sudden increase in the elimination rate below a certain critical concentration (C _crit). Examples of this phenomenon, in which the PK exhibit this nonlinear behavior, are shown in Figure  1 for efalizumab (anti‐CD11a),1 mavrilimumab (antigranulocyte‐macrophage colony‐stimulating factor receptor),2 and romosozumab (antisclerostin).3

Figure 1.

Nonlinear pharmacokinetics for efalizumab, mavrilumab, and romosozumab. Note that for all three drugs, as the concentration drops below the critical concentration (C _crit), the elimination rate suddenly increases.

This nonlinearity shown in Figure  1 results from a combination of two pathways of drug clearance: (i) a linear, nonspecific clearance due to endocytosis; and (ii) a nonlinear, saturable clearance due to internalization of the target receptor:

• At high drug concentrations, the target receptor is saturated and the rate of elimination is governed by the nonspecific clearance.

• At lower drug concentrations, when the target receptor is no longer saturated, both the nonspecific route and the saturable route contribute to drug clearance and, hence, the rate of elimination increases as the concentration decreases.

A common goal when selecting the dose and regimen of a drug is to maintain a drug concentration that stays above the C _crit. There are two reasons for this goal.

1. Ensure lower PK variability in the population, because C _crit may vary between subjects.

2. Ensure that target occupancy remains high, as maintaining drug concentrations above C _crit is thought to be a necessary (although not sufficient) condition to maintain target saturation.4

In this article, we show examples of systems that have a simple critical value for the drug concentration C _crit such that as long as the drug concentration remains above this value, target mediated elimination is saturated and the PK is linear. In Figure  1, C _crit is illustrated; when the concentration falls below C _crit, the elimination rate suddenly increases.

We derive an analytical expression for C _crit for both the Michaelis‐Menten model1, 5 and the target‐mediated drug disposition model (TMDD).6, 7, 8 Both models have two routes of clearance: nonspecific‐linear elimination and saturable‐nonlinear elimination. Both analyses will be discussed in the absence and in the presence of a peripheral compartment. In the presence of a peripheral compartment, the analysis becomes more complex, but it is still transparent in common cases, such as rapid exchange between the two compartments. In all cases, C _crit is independent of drug dose and volume of distribution.

Methods

For modeling the onset of the PK nonlinearity, we focus on the two models that are frequently used by the pharmacometrics community: Michaelis‐Menten and TMDD. A more general theoretical derivation of C _crit for any PK model is beyond the scope of this article. We start by focusing on the simpler case of the one‐compartment model and then extend this analysis to the two‐compartment scenario. The two‐compartment models are shown schematically in Figure  2 below. The corresponding one‐compartment models are identical with the models shown in Figure  2, except that the peripheral compartments (C _p) are removed.

Figure 2.

Structural scheme of the basic model describing (a) Michaelis‐Menten kinetics and (b) target‐mediated drug disposition (TMDD) where a drug C binds a target receptor R which is synthesized by a zero order reaction and degenerates according to a first order reaction. Drug and target form a complex CR, which internalizes through a first‐order reaction.

Michaelis‐Menten elimination

One‐compartment model.

The basic model for linear and nonlinear drug elimination involves a single (central) compartment and a single differential equation for the concentration C of the drug involving linear and saturable elimination side by side:

$$V_c\frac{\mathrm{d}C}{\mathrm{d}t}=-\text{Cl}·C-V_{\max }\frac{C}{C+K_M}.$$ (1)

Here, V _c is the volume of the central compartment, Cl a first order clearance rate, V _max is the maximum rate of saturable elimination and K_M is the Michaelis‐Menten constant. For an initial i.v. bolus dose D, the initial concentration is given by C(0) = D/V_C.

Two‐compartment model.

When the drug is distributed over a central compartment with concentration C _c and a peripheral compartment C _p, with linear and nonlinear elimination occurring only from the central compartment, the PK are described by the system of equations:

$$\left\{\begin{array}{cc}& V_c\frac{{\mathit{\text{dC}}}_c}{\mathit{\text{dt}}}=-\mathit{\text{Cl}}·C_c-V_{\max }\frac{C_c}{C_c+K_M}-Q(C_c-C_p)\hfill \\ & V_p\frac{{\mathit{\text{dC}}}_p}{\mathit{\text{dt}}}=+Q(C_c-C_p)\hfill \end{array}\right.$$ (2)

in which exchange between the compartments is driven by the difference in drug concentration in the two compartments (i.e., by the term Q(C _c –C _p) where Q denotes a nonspecific clearance rate).

If initially, an i.v. bolus dose D is supplied to the central compartment while the peripheral compartment is empty, the initial conditions are here C _c(0) = D/V _c and C _p(0) = 0.

Target‐mediated drug disposition

In its most elementary form, TMDD target kinetics is assumed to follow a zero‐order production rate k _syn and a first‐order elimination rate k _e(R) R. The formation of drug‐target complex CR takes place via a second‐order process k _on C · R, a first‐order loss of the drug‐receptor complex via k _off CR, and an irreversible first‐order elimination process k _e(CR) CR. In Figure  2 this system of reactions is shown schematically.

One‐compartment model.

Mathematically, the one‐compartment TMDD model is described by the system of differential equations below:

$$\left\{\begin{array}{cc}& \frac{\mathit{\text{dC}}}{\mathit{\text{dt}}}=-k_{\mathrm{on}}C·R+k_{\mathrm{off}}CR-k_{e(C)}C;k_{e(C)}=\frac{\mathit{\text{Cl}}}{{\mathit{\text{v}}}_C}\hfill \\ & \frac{\mathit{\text{dR}}}{\mathit{\text{dt}}}=k_{\mathrm{syn}}-k_{\mathrm{on}}C·R+k_{\mathrm{off}}CR-k_{e(R)}R\hfill \\ & \frac{\mathit{\text{dCR}}}{\mathit{\text{dt}}}=k_{\mathrm{on}}C·R-k_{\mathrm{off}}CR-k_{e(CR)}CR\hfill \end{array}\right.$$ (3)

Note that in the absence of drug, the steady‐state target concentration is given by R ₀ = k _syn/k _e(R). Here, we assume that the drug is supplied through an i.v. bolus dose D to the system when it is free from drug and the target is at steady state (i.e., C(0) = D/V _c, R(0) = R ₀ and CR(0) = 0).

By adding the first and the third equations, we obtain an equation for the total amount of drug C _tot = C + CR and, similarly, by adding the second and the third equations we obtain an equation for the total amount of the target, R _tot = R + CR:

$$\left\{\begin{array}{cc}& \frac{{\mathit{\text{dC}}}_{\mathrm{tot}}}{\mathit{\text{dt}}}=-k_{e(C)}C-k_{e(CR)}CR\hfill \\ & \frac{{\mathit{\text{dR}}}_{\mathrm{tot}}}{\mathit{\text{dt}}}=k_{\mathrm{syn}}-k_{e(R)}R-k_{e(CR)}CR\hfill \end{array}\right.$$ (4)

Note that the on‐rate and the off‐rate constants k _on and k _off no longer appear in these equations.

In practice, drug binding and complex internalization is fast, so that after a short time drug, target, and drug‐target complex are in quasi‐steady‐state (QSS)9 (i.e., C, R, and CR) are approximately related through the expressions:

$$CR=R_{\mathrm{tot}}\frac{C}{C+K_{\mathrm{SS}}}\text{and}R=R_{\mathrm{tot}}\frac{K_{\mathrm{SS}}}{C+K_{\mathrm{SS}}}$$ (5)

where

$$K_{\mathrm{SS}}=\frac{k_{\mathrm{off}}+k_{e(CR)}}{k_{\mathrm{on}}}$$ (6)

Two‐compartment model.

The two‐compartment model is similar to the one‐compartment model with the addition of a peripheral compartment with the initial condition C _p(0) = 0.

$$\left\{\begin{array}{cc}& \frac{{\mathit{\text{dC}}}_{\mathrm{tot}}}{\mathit{\text{dt}}}=-k_{e(C)}C-k_{e(CR)}CR-Q(C-C_p)\hfill \\ & \frac{{\mathit{\text{dC}}}_{p\mathrm{tot}}}{\mathit{\text{dt}}}=+Q(C-C_p)\hfill \\ & \frac{{\text{dR}}_{\mathrm{tot}}}{\mathit{\text{dt}}}=k_{\mathrm{syn}}-k_{e(R)}R-k_{e(CR)}CR\hfill \end{array}\right.$$ (7)

Model analysis, fitting, and simulation

A mathematical analysis of the above equations was performed to derive C _crit for each model above. To test the theory, the data from Figure  1 was fit to the two‐compartment TMDD model from Eq.  7 where it was assumed that k _e(R) = k _e(CR). This assumption is typically made when fitting the PK for membrane‐bound targets where it can be especially difficult to estimate k _e(R).10 Good fits can still be achieved even when implementing this constraint.

Sensitivity analyses were performed for the one‐compartment Michaelis‐Menten and TMDD models and the two‐compartment Michaelis‐Menten model. In the sensitivity analysis, one parameter at a time was changed while all other parameters were held fixed, and the calculated C _crit was plotted together with the simulated PK data.

Because dosing of monoclonal antibodies (mAbs) is usually reported in mg/kg, but the binding model requires doses of nmol and concentrations of nM, a dose of 10 mg/kg is converted to nanomoles for the model simulations using the formula below, which assumes a 70 kg patient and a 150 kDa drug (typical antibody).

$$\frac{10\text{mg}}{\text{kg}}·\frac{70\text{kg}}{\text{patient}}·\frac{1\mathrm{g}}{1000\text{mg}}·\frac{1\text{mol}}{150·{10}^3\mathrm{g}}·\frac{{10}^9\text{nmol}}{\text{mol}}=467\text{nmol}$$

Results

Model fit

The model fits to the data are shown in the Supplementary Material and the parameters from the fits that are subsequently used for simulations are listed in Table  1. These parameters were also used to compute the C _crit lines, as described below. Note that the large values for k _e(R) and k _e(CR) for romosozumab and efalizumab are due to the practical unidentifiability of these parameters.11

Table 1.

Parameters for mavrilimumab, efalizumab, and romosozumab, based on fits to data

	Mavrilimumab	Efalizumab	Romosozumab	Units
Weight‐based dose	10	10	10	mg/kg
Equivalent molar dose	4667	4667	4667	nmol
V c	2.8	2.4	2.4	L
V p	5.6	3.6	2.6	L
Cl	0.3	0.46	0.25	L/d
Q	1.7	9.7	0.54	L/d
k syn = V max/V c	2.4	8.5	6.1	nM/d
K ss = KM	1.1	1.2	12	nM
k e(R)	2.2	4400	860	1/d
k e(CR)	2.2	4400	860	1/d
k off	10	–	–	1/d
k on = (k off + k e(CR))/K ss	11	–	–	1/(nM · d)

Cl, clearance; k _e(C), drug elimination rate; k _e(R), receptor elimination rate; K_M, Michaelis‐Menten constant; k _off, dissociation rate; k _on, association rate; Kss, quasi‐steady‐state constant; k _syn, receptor synthesis rate; Q, intercompartmental clearance; V _c, central volume; V _max, maximal rate of saturable (nonlinear) elimination; V _p, peripheral volume.

Michaelis‐Menten elimination

One‐compartment model.

For drug concentrations, which are either large or small with respect to K_M, Eq.  (1) can be approximated by the following simpler equations:

$$\left\{\begin{array}{cc}& V_c\frac{\mathit{\text{dC}}}{\mathit{\text{dt}}}=-\text{Cl}·C-V_{\max }\approx -\text{Cl}·C\text{for}C\gg K_M,\hfill \\ & V_c\frac{\mathit{\text{dC}}}{\mathit{\text{dt}}}=-\left(\mathit{\text{Cl}}+\frac{V_{\max }}{K_M}\right)C\text{for}C\ll K_M.\hfill \end{array}\right.$$ (8)

Thus, on a logarithmic scale, the graph of C(t) will be approximately linear for both large and small values of C; for the range of concentrations in between, the graph curves down connecting the upper linear segment with the lower linear segment (see Figure  3).

Figure 3.

Mavrilimumab: one‐compartment model in Eq.  (1) in which the base parameter values are in Table  1. Each plot shows a sensitivity analysis where the parameter in the title is changed from 0.1‐fold to 10‐fold. The dot shows critical concentration. CL, clearance; V _max, maximal rate of saturable elimination.

Of particular interest is the situation when the rate of elimination increases substantially at low concentrations. According to Eq.  8 , this is the case when:

$$\frac{V_{max}}{K_M}\gg \mathit{\text{Cl}}⟹K_M\ll \frac{V_{\max }}{\mathit{\text{Cl}}}.$$ (9)

Throughout this article, it will be assumed that Eq.  9 holds. The sensitivity analysis for romosozumab in the Supplementary Material demonstrates that for small values of V _max or large values of K_M, the shoulder (where the onset of the PK nonlinearity is observed) disappears entirely and the PK appears linear.

As the concentration drops, the elimination rate increases, and it is important to know at which concentration this transition takes place. We define this critical concentration, C _crit, as the value of C where the rate of elimination doubles from its value at large drug concentrations. This definition was chosen because when the rate of elimination doubles, that means the linear and nonlinear components of elimination contribute equally. C _crit is derived by dividing Eq.  (1) by C, which gives:

$$V_c\frac{d}{\mathit{dt}}log(C)=-\left(\mathit{\text{Cl}}+\frac{V_{\max }}{C+K_M}\right)$$ (10)

It can readily be seen that for large concentrations, the slope is Cl and, thus, the rate of elimination doubles when:

$$\frac{V_{\max }}{C+K_M}=\mathit{\text{Cl}}\text{or}C=\frac{V_{\max }}{\mathit{\text{Cl}}}-K_M$$ (11)

By assumption of Eq.  9 , we may neglect K_M in the right expression in Eq.  11 and, thus, define the critical concentration to be:

$$C_{\mathrm{crit}}\stackrel{\mathrm{def}}{=}\frac{V_{\max }}{\mathit{Cl}}.$$ (12)

Note that neither the drug dose nor the volume V _c of the central compartment enters the definition of C _crit.

In Figure  3, we show simulations of concentration graphs based on Eq.  (1) for data from mavrilimumab in which Cl = 0.3 L/d, V _max = 6.7 nmol/d, and K _M = 1.1 nM (Table  1). Then, according to Eq.  12 , C _crit = 22 nM. Note that here C _crit is large compared to K _M so that for the base values condition Eq.  9 is satisfied.

Figure  3 shows how C _crit, computed from Eq.  12 and indicated by a dot, changes when one of the parameters involved in the model is varied. In addition, it shows where C _crit fits in the concentration‐time curve of the drug. The graph for the largest parameter value is drawn in red and the graph for the smallest is drawn in magenta.

In the first panel in Figure  3, in which the dose is varied while keeping C(0) > C _crit, it is shown how both numerically and geometrically C _crit corresponds to the kink in the graph, irrespective of the dose value. According to Eq.  12 , as V _max increases, C _crit moves up, while as Cl increases C _crit moves down, and this is observed in the second and third panels. Because C _crit does not depend on either V _c or K _M, it does not move in the fourth and the fifth panels.

Two‐compartment model.

As we shall see, for initial drug concentrations that are large enough, the switch from linear to nonlinear drug elimination is apparent in this situation as well, and it is possible to identify a critical drug concentration C _crit. Here, we only derive the formula for C _crit for the special case when drug exchange between the two compartments is fast relative to the nonspecific clearance (Cl) and the saturable clearance (V _max), such as when:

$$Q\gg \text{Cl}\text{and}Q\gg V_{\max }.$$ (13)

Under these conditions, it is found that the two concentrations, C _c and C _p, rapidly coalesce (i.e., C _c(t)–C _p(t) → 0), and C _crit in both the central and peripheral compartments is given by the same formula as for the one‐compartment model (i.e., by Eq.  12 ):

$$C_{\mathrm{crit}}=\frac{V_{\max }}{\text{Cl}}.$$ (14)

The details of the derivation of Eq.  14 are provided further below.

In Figure  4, we show simulations for the two‐compartment Michaelis‐Menten model for parameter values, which are varied around the base parameters listed in Table  1, as indicated in the headings of the different panels. The graphs in the different panels are comparable to those shown in Figure  3 for the one‐compartment model, except that many show a weak point of inflection below the C _crit.

Figure 4.

Mavrilimumab two‐compartment Michaelis‐Menten model, in which the base parameter values are listed in Table  1, and each plot shows a sensitivity analysis in which the parameter in the title is changed from 0.1‐fold to 10‐fold. The dot shows critical concentration. CL, clearance; V _max, maximal rate of metabolism.

Evidently, the location of C _crit on the graphs, as defined by Eq.  14 , has not changed much from what was seen in Figure  3. This is remarkable because the condition in Eq.  13 is not satisfied by the base values of the parameters. Specifically, the value of the intercompartmental clearance Q is not large compared to V _max. On the other hand, Q is indeed large compared to the nonspecific clearance Cl, as is also borne out by the different graphs shown in Figure  4, which exhibit a clear two‐phase structure during the initial development.

Rapid exchange between central and peripheral compartments

Suppose that Q ≫ Cl and Q ≫ V _max. Then, when we divide the system of Eq.  (2) by Q, and scale the time variable according to τ = Qt, we obtain, with x (τ) = C _c(τ/Q) and y (τ) = C _p(τ/Q),

$$\left\{\begin{array}{cc}& V_c\frac{\mathit{\text{dx}}}{\mathit{\text{d}}{\displaystyle \mathrm{\tau }}}=-ϵ·x-(x-y)-\nu (\tau ),\hfill \\ & V_p\frac{\mathit{\text{dy}}}{\mathit{\text{d}}{\displaystyle \mathrm{\tau }}}=x-y\hfill \end{array}\right.$$ (15)

where

$$ϵ=\frac{\text{Cl}}{Q}\ll 1\text{and}\nu (\tau )=\frac{V_{\max }}{Q}\frac{C_c}{C_c+K_M}\ll 1$$ (16)

Adding the two equations we find for the total amount of drug:

$$A(\tau )\stackrel{\mathrm{def}}{=}{V}_{c}x(\tau )+V_{p}x(\tau ),$$ (17)

the conservation law:

$$\frac{\mathrm{d}}{\mathrm{d}\tau }(V_{c}x+V_{p}y)=-ϵ·x-\nu \approx 0$$ (18)

Thus, there is very little loss of drug: on the fast time scale τ = O(1) (i.e., t = O(1/Q)). Similarly, by dividing the equations by their respective volumes and then subtracting the two equations we obtain:

$$\frac{\mathrm{d}}{\mathrm{d}{\displaystyle \tau }}(x-y)\approx -\left(\frac{1}{V_c}+\frac{1}{V_p}\right)(x-y)$$ (19)

so that C _c(t) – C _p(t) → 0 on the fast time scale as well. Therefore, within a very short time, we may write A(t) = (V _c + V _p)C _c(t).

We now go back to Eq.  18 and multiply it by Q so that the original time variable t is returned. In light of Eq.  19 , this yields the following equation:

$$V_{\mathrm{ss}}\frac{\mathrm{d}{C}_c}{\mathrm{d}t}=-\text{Cl}·C_c-V_{\max }\frac{C_c}{C_c+K_M}$$ (20)

$$V_{\mathrm{ss}}=V_c+V_p$$ (21)

where V _ss is the volume of distribution at steady state.

Following the arguments given for the one‐compartment model, we find that the formula for C _crit, as given by Eq.  12 , has not changed.

Target‐mediated drug disposition

One‐compartment model.

Suppose that the drug concentration is large compared to the dissociation constant (i.e., C(t) ≫ K _SS over a period 0 < t < t ₀). Then Eq.  5 implies that during this interval we have:

$$CR\approx R_{\mathrm{tot}}\text{and}R\approx 0$$ (22)

With these equalities, the system in Eq.  (4) can be approximated by the following pair of equations:

$$\left\{\begin{array}{cc}& \frac{\mathit{\text{d}}}{\mathit{\text{dt}}}(C+R_{\mathrm{tot}})=-k_{e(C)}C-k_{e(CR)}{R}_{\mathrm{tot}}\hfill \\ & \frac{{\mathit{\text{dR}}}_{\mathrm{tot}}}{\mathit{\text{dt}}}=k_{\mathrm{syn}}-k_{e(CR)}{R}_{\mathrm{tot}}\hfill \end{array}\right.$$ (23)

Thus, when we subtract the second equation in Eq.  23 from the first we obtain for 0 < t < t ₀

$$\frac{\mathrm{d}C}{\mathrm{d}t}=-k_{e(C)}C-k_{\mathrm{syn}}$$ (24)

or, when we multiply by V _c,

$$V_c\frac{\mathrm{d}C}{\mathrm{d}t}=-\text{Cl}·C-V_{\max }\text{where}{V}_{\max }=V_c{k}_{\mathrm{syn}}$$ (25)

This equation is formally the same as the first equation in the system of Eq.  8 . Following the reasoning used in the Michaelis‐Menten elimination Subsection to derive C _crit, we assume, as in Eq.  9 , that:

$$\frac{V_{\max }}{\text{Cl}}\gg K_{\mathrm{ss}}\text{or}\frac{k_{\mathrm{syn}}}{k_{e(C)}}\gg K_{\mathrm{ss}}$$ (26)

and show that

$$C_{\mathrm{crit}}=\frac{V_{\max }}{\text{Cl}}=\frac{k_{\mathrm{syn}}}{k_{e(C)}}$$ (27)

Thus, by assuming rapid drug binding and complex internalization, drug clearance can be seen as the sum of linear and Michaelis‐Menten type elimination resulting in the same expression for the C _crit.

In Figure  5, we show simulations for the one‐compartment Michaelis‐Menten and TMDD models and we see that for mavrilimumab, a typical antibody for which Eq.  26 is satisfied, simulations give corresponding results and, as we have seen in Figures  3 and 4 for the Michaelis‐Menten model, C _crit fits snugly in the arm of the concentration curves.

Figure 5.

Mavrilimumab one‐compartment Michaelis‐Menten model (MM; solid line) and target‐mediated drug disposition (TMDD; dashed line) model, in which the base parameter values are in Table  1 and each plot shows a sensitivity analysis where the parameters in the title is changed from 0.1‐fold to 10‐fold. The dot shows critical concentration. For the TMDD model, k _syn = V _max/V _c, Kss = K_M

In the first five graphs, in which simulations for the Michaelis‐Menten model and the TMDD model are shown together (the Michaelis‐Menten model is dashed and the TMDD is solid), the Michaelis‐Menten model curves lie below the TMDD curves. This can be understood by comparing the equations for C _tot and R _tot in Eqs.  (4) and 23 and remembering that CR < R _tot.

The similarity between the Michaelis‐Menten and TMDD models has been reported elsewhere.10, 11 Comparable findings were observed for the two‐compartment model as well. Note that the similarity of the Michaelis‐Menten and TMDD models may not apply for other biologics, such as bispecific target engagers or cytokines where the doses are much lower.7, 12

Discussion

In practice, C _crit is a useful quantity. We list a few applications below.

One application of C _crit is in providing an initial estimate for V _max or k _syn from a graphical inspection of the data. For large, i.v. doses, Cl can be estimated by noncompartmental analysis by computing the area under the curve (AUC)_0–∞ for a single dose and then computing Cl = Dose/AUC_0‐∞. Then, C _crit can be estimated from a graph of the PK and Eq.  14 is then applied to get V _max = C _crit · Cl or k _syn = C _crit · Cl/V _c.

Another application is that when fitting a population PK model, there may be patients who have similar PK above C _crit (and, thus, similar Cl) but different C _crit levels. When this is observed, we have found that a random effect on V _max was needed to capture the intersubject variability in the low concentration data.

Finally, C _crit is also useful in understanding the identifiability of TMDD models when fit to data of the type shown in Figure  1. Here, it has been shown that once the linear PK parameters have been identified (say from high‐dose data) k _syn is the only remaining parameter needed to determine C _crit. Then, given k _syn, K _SS is the only remaining parameter needed to determine the slope of the nonlinearity. Thus, these two parameters are often estimable from the nonlinear PK alone. However, other parameters of the TMDD model, such as the receptor density R ₀ = k _syn/k _e(R) (often assumed to be constant with k _e(R) = k _e(CR)) is often not identifiable with PK data alone, as previously reported.10, 11

One key assumption for the C _crit estimate to apply is that initially, the drug concentration is large so that the nonlinear elimination term initially has a negligible contribution to the PK curve. If this assumption does not hold, there may be no kink and hence no C _crit. In addition, there may be a kink, but if the dose is low enough such that the nonlinear component already contributes significantly to the elimination, C _crit as calculated here will not describe the kink. In the extreme case that Cl → 0 for a drug with only saturable elimination, C _crit → ∞ and is essentially unobservable. Thus C _crit as defined here is no longer of practical value in scenarios where the saturable route of elimination dominates for all observable drug concentrations, as is the case for drugs that are given at low doses, such as bispecific target engagers and cytokines. To describe the kink in the PK curve in these scenarios (shown in more detail in the Supplementary Material ) an alternative definition of C _crit would be needed. C _crit is of its greatest utility for mAbs (or other drugs) given at sufficiently high doses, such that both linear and nonlinear elimination phases are observable.

In summary, when developing antagonists, it is often the goal to pick a dosing regimen where the drug concentration stays above C _crit. In this article, we have developed a simple formula for this critical concentration: C _crit = V _max/Cl = k _syn/k _e(C).

Source of Funding

No funding was received for this work.

Conflict of Interest

Andrew Stein is employed by Novartis Institute for BioMedical Research.

Author Contributions

A.M.S. and L.A.P. wrote the manuscript. A.M.S. and L.A.P. performed the research. A.M.S. and L.A.P. analyzed the data.

Supporting information