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For monoclonal antibody (mAb) drugs, soluble targets may accumulate several thousand fold after binding to the drug. Time course data of mAb and total target is often collected and, although free target is more closely related to clinical effect, it is difficult to measure. Therefore, mathematical models of this data are used to predict target engagement. In this article, a “potency factor” is introduced as an approximation for the model‐predicted target inhibition. This potency factor is defined to be the time‐Averaged Free target concentration to Initial target concentration Ratio (AFIR), and it depends on three key quantities: the average drug concentration at steady state; the binding affinity; and the degree of target accumulation. AFIR provides the intuition for how changes in dosing regimen and binding affinity affect target capture and AFIR can be used to predict the druggability of new targets and the expected benefits of more potent, second‐generation mAbs.
WHAT IS THE CURRENT KNOWLEDGE ON THE TOPIC?
☑ Mathematical models for target mediated drug disposition of mAbs are widely used to guide drug development by predicting the dosing regimen at which a certain threshold of target inhibition is achieved. Although many mathematical analyses of these models have been published, there has not yet been a demonstration for how the key model parameters (like binding affinity and average drug concentration) link to target engagement in repeated dosing scenarios.
WHAT QUESTION DID THIS STUDY ADDRESS?
☑ How do the PK and binding properties of the mAb impact target engagement?
WHAT THIS STUDY ADDS TO OUR KNOWLEDGE
☑ A simple nondimensional potency factor (AFIR) links target engagement to three key quantities: average drug concentration, binding affinity, and total target accumulation.
HOW MIGHT THIS CHANGE DRUG DISCOVERY, DEVELOPMENT, AND/OR THERAPEUTICS?
☑ The AFIR metric provides intuition for standard TMDD models and can be used to rapidly predict the druggability of new targets and the expected benefits of second‐generation, more potent mAbs.
Monoclonal antibodies (mAbs) are one of the fastest growing classes of therapeutic agents with 47 approved as of November 2014 and an expectation of about 4 new approvals per year.1 Unlike small molecules, which have a molecular weight of about 500 Da and that are cleared mainly by the liver and kidneys, mAbs are large molecules with a molecular weight of about 150 kDa and are cleared mainly through cellular uptake followed by proteolytic degradation. Whereas small molecules typically have a half‐life of hours, fully human mAbs exhibit long half‐lives of around 3 weeks due to the FcRn receptor, which binds to the mAb after pinocytosis and rescues it from undergoing lysosomal degradation.2 There are two classes of targets for mAbs: membrane‐bound and soluble. Antibodies with membrane‐bound targets (e.g., trastuzumab/HER2, demosumab/RANKL, nivolumab/programmed cell death protein 1) have an additional route of clearance via receptor‐mediated internalization, which can lead to a nonlinearity in the drug pharmacokinetics (PKs); this phenomenon is known as target‐mediated drug disposition (TMDD). Antibodies with soluble targets (e.g., omalizumab/Immunoglobulin E, bevacizumab/vascular endothelial growth factor, siltuximab/interleukin‐6) often demonstrate significant target accumulation after single (Figure 11 a,b) or repeated dosing (Figure 11 c) because the mAb‐target complex often has a much longer half‐life than the free target molecules.3, 4, 5 Although this accumulation plateaus at large doses, this plateau does not necessarily imply a plateau in efficacy; and increasing the dose after the plateau has been associated with further reduction of the free target concentration,6 greater inhibition of downstream biomarkers,7, 8 and improved efficacy.9, 10
To understand why the plateau in total target concentration does not imply a plateau in efficacy, and to understand how changes in the dose regimen or drug properties may impact target engagement, it is useful to characterize the free and bound concentrations for the drug and target using the model in Figure 2,2, which describes both TMDD and the accumulation of total target during therapy. This model, often referred to as a TMDD model,11, 12 is mathematically more complex than the usual compartmental models for describing the PK of small molecules, because the kinetics of the free drug, free target, and drug‐target complex all need to be considered. Given a dosing regimen and estimates for the parameters governing drug kinetics, target kinetics, and binding affinity, the TMDD model allows one to make predictions for the target inhibition, as shown in Figure 11 where data and model predictions are shown for three antibodies.
The TMDD model has been used to support many aspects of drug development13 including:
Modelers have sought to develop rules of thumb for predicting how changing the drug properties or dosing regimen will impact measures of drug activity, such as maximum target inhibition or duration of effect of a single dose.18, 19, 20, 21, 22 Thus far, the existing metrics and analyses apply only to single‐dose scenarios, which is of limited value because most mAbs are dosed repeatedly in the clinic.
In this article, a new potency factor based on multiple dosing scenarios is derived. The potency factor is named the Average Free target concentration to Initial target concentration Ratio (AFIR), which depends upon the structural model (Figure 2)2) and the parameters (Table 1) that describe the drug, target, and binding kinetics. AFIR provides an intuitive understanding for how dosing regimen and binding affinity affects target inhibition. Examples showing how this intuition can be used to support drug development decisions are also provided.
In this section, the ratio of steady‐state free target (under regular dosing) to baseline free target is derived. This ratio represents the relative degree of suppression of free target after binding to the drug. The TMDD model commonly used to describe biologics binding to their target is given in Figure 22 and by the equations below. The model parameter descriptions and parameter values for three drugs (omalizumab, bevacizumab, and siltuximab) are provided in Table 1. The model parameters were chosen to match the clinical data from,3, 4, 5 noting that varying and over multiple orders of magnitude while keeping the binding affinity ( ) fixed would yield similar curves. This model describes both subcutaneous and intravenous dosing, distribution of the drug to the peripheral tissue, binding of the drug to target in the serum, synthesis of the target, and elimination of the drug, target, and complex. Distribution, binding, and turnover of the target in peripheral tissue generally are not modeled due to limited available data in the peripheral tissue.
Input Absorption Distribution Binding Elimination
The key quantity of interest in predicting drug effect over time is the ratio of free target to baseline target ( ). This ratio can be written as the product of two ratios as shown below, where is the steady state total target concentration under a repeated dosing regimen of large doses.
To compute the first ratio , the quasi‐equilibrium (QE) approximation is used,23 which assumes that binding and unbinding occurs rapidly compared with other processes, such that the drug, target, and complex are in quasi‐equilibrium.
Substituting the equation for the total target [ ] into Eq. (5) and solving for , gives the equation below. The first approximation holds when D K d and the second approximation holds when , which occurs when the drug is dosed in vast molar excess to the target, as is the case for most mAb drugs in the clinic.
Solving Eq. (7) for steady state when no drug [ ] is present gives . When large amounts of drug are present for a long time during intervals of regular dosing, Eq. (6) shows that very little target is free and, thus and ,24 giving:
At equilibrium, , and the target accumulation ratio ( ) is computed as follows:
For drugs with linear PK that are dosed at regular intervals ( ), the Trough Free target concentration to Initial target concentration Ratio (TFIR) can be computed using the trough drug concentration at steady state , (referred to here as to match the more commonly used nomenclature), which can be written as a sum of exponentials.25
Substituting into Eq. (10) gives:
The AFIR is given in the equation below, where denotes a time at which the drug and target have reached steady state.
In the case of linear PK, recall that the average drug concentration is given by .25 When the drug is given as an infusion at rate , then the steady‐state drug concentration is a constant ( ) giving:
In practice, mAbs are usually dosed every 2–8 weeks and the above equations are approximations rather than exact solutions. It will be shown in the next section that this approximation is often good.
Thus, the average target inhibition (AFIR) depends upon three quantities: the dissociation constant ( ), the target accumulation ratio ( ), and the average drug concentration ( ).
A number of assumptions were made in developing the AFIR metric. When these assumptions do not hold, three alternative formulas for AFIR have been derived in the Supplementary Material.
To gain a better understanding of the AFIR and TFIR ratios and to explore the conditions under which these potency metrics accurately describe the system, a basic sensitivity analysis is performed using the parameters for siltuximab (Figure 3),3), as well as omalizumab and bevacizumab (Supplementary Material). Equation (13) demonstrates that AFIR depends on 8 different parameters; 7 parameters are explored (excluding F because changing F has the same effect as changing Dose) and is also included to confirm that target synthesis does not affect AFIR. Each row of plots in Figure 33 explores the sensitivity of a different quantity: . In each column, is the result of changing one parameter while holding the other parameters fixed. The parameter that is changed and the range over which it is changed is shown at the top of each column. To test the approximations for AFIR and TFIR, a direct calculation of the target inhibition using Eqs. (11) and (14) is compared to a numeric calculation from the model simulation after 2 years of therapy.
For siltuximab, there was generally good agreement between the theory and the numeric estimates for AFIR and TFIR. However, there is divergence from the theory in each of the AFIR and TFIR plots when and the target does not accumulate to , leading to a lower observed target accumulation ratio ( ) than would be observed for a larger dose. In this case, the calculation described in the Supplementary Material should be used. The inaccuracy of the AFIR equation can be especially pronounced in the limit where the dose and drug concentration approach zero and the theoretical approximations for AFIR approach infinity, even though the true behavior is that AFIR=1. Although the numerically calculated AFIR generally approaches its theoretical value as AFIR falls below 30%, divergence from the theory is observed for very small , (see sensitivity plots of and ) where the quasi‐equilibrium assumption is less accurate because the target elimination rate ( ) is larger than . In this case, drug‐target binding approaches the irreversible‐binding approximation such that further reduction of has no additional benefit. In this scenario, the calculation should be used (see Supplementary Material for details).
A lumped parameter sensitivity analysis was performed in Figure 4;4; the system is reparameterized, replacing rate constants with lumped parameters . The rate constants for the model are then calculated from the lumped parameters as shown:
By parameterizing the system in this way, the effect of changing parameters while keeping AFIR fixed can be examined and minimal impact on the free target dynamics is observed. This can be seen by noting that the AFIR and TFIR plots in Figure 44 are flat. However, there are some exceptions. For large τ (3 months) or large CL (1 L/d), the true AFIR is larger than what is theoretically predicted while the TFIR calculation remains accurate. The inaccuracy in the AFIR theoretical prediction is due to large changes in drug concentration over the dosing interval such that the assumption of a constant drug concentration over the dosing interval leads to inaccuracies. Although high CL=1 L/d is typically not observed for mAbs, infrequent dosing ( months) is sometimes prescribed, as is the case for ustekinumab. As for the basic sensitivity analysis, when , both AFIR and TFIR are higher than predicted by the theory and the irreversible binding approximation should be considered because the quasi‐equilibrium approximation declines in accuracy.
It is instructive to focus on the effect of changing the dose on the total target and free target, as shown in Figure 55 a. Notice that above 1 mg/kg, further increases in dose do not have much impact on the total target accumulation. However, this plateau in total target does not necessarily imply a plateau in free target reduction or efficacy, as demonstrated by the free target curves ( ) and as observed elsewhere.9, 10
In considering the identifiability of the four key parameters governing the target dynamics , it is useful to reparameterize the model as ; T 0 is the target concentration before drug is given; is the total target concentration after the total target reaches its plateau following a large enough dose; governs the rate at which the total target approaches its plateau (see Eq. (8) and ref. 24); and determines the dose needed for the target to approach its plateau (lower means the total target will approach its plateau at lower doses). Thus, all four parameters are identifiable as long as enough measurements are taken and the target assay is sufficiently sensitive.
In practice, the assay for measuring total target is often not sensitive enough to detect the baseline target concentration before the drug is given, as was the case for siltuximab.5, 7 Although the steady‐state plateau and the time scale for reaching it can still be identified, the baseline target level (T 0) is no longer identifiable and this leads to unidentifiability of as well. This can be observed graphically in Figure 55 b, which shows the kinetics after a single dose of siltuximab, where both T 0 and are simultaneously increased while all other parameters are held fixed. The dotted line indicates where a limit of quantification of the total target assay could lie. The sensitivity analysis shows no impact on the profiles for and . For , the blue and gray curves (T 0<0.3 pM and <50 pM) are overlapping indicating that while the quotient is identifiable, only an upper bound for T 0 and can be identified. AFIR is also identifiable as can readily be seen because it can be written as . Thus, if the goal is to predict AFIR, estimation of the ratio may be sufficient and a total target assay that is sensitive enough to measure baseline target levels may not be needed.
All simulations were performed using Matlab R2015a. Code for generating all figures in this article is available in the Supplementary Material.
The key insight from this work is that under many clinically relevant scenarios, AFIR can be estimated using three parameters: the dissociation constant ( ), the target accumulation ( ), and the average drug concentration ( ).
This simple formula provides intuition for how changing the dosing regimen or improving the binding affinity of the drug would be expected to alter target inhibition: doubling the dose, halving the dosing interval, or halving the dissociation constant by using a higher affinity drug would all reduce the free target concentration by 50%. Although target accumulation plateaus at large doses, the AFIR formula shows that further increasing the dose continues to reduce the free target levels (as illustrated in Figure 55 a), which could then potentially lead to greater biomarker inhibition,5, 8 and efficacy.9, 10
Although target occupancy (ratio of bound target to total target) provides another metric to assess target engagement, this metric is misleading for soluble targets because it does not account for target accumulation. For example, consider a scenario where there is 99% target occupancy together with a 100‐fold accumulation of target. In this scenario, no reduction in the absolute concentration free target has been achieved and, thus, achieving 99% target occupancy would not be expected to provide clinical benefit. Thus, for soluble targets, AFIR is a preferred metric of target inhibition.
The AFIR potency metric allows for a rapid assessment of new drugs without requiring extensive simulation. Specifically, the formula allows the drug developer to quantify how a second generation drug with better PK properties or higher binding affinity could lead to improved target inhibition. Alternatively, AFIR could be used to identify dosing regimens that allow for less frequent dosing, ultimately leading to a reduction in the number of injections, the number of visits to the doctor, and the cost of goods.
In designing a phase II dose‐finding study, if one has an idea of what level of target inhibition is required, AFIR provides a means to identify the largest dose to be given. When designing preclinical studies, the AFIR metric indicates that it is possible to predict target inhibition without estimating all parameters of the TMDD model. In particular, it is sufficient to have an estimate of the dissociation constant and the expected target accumulation ratio . The target accumulation can be predicted either by first measuring the baseline target concentration and then measuring the total target after a large dose, or by computing the ratio of estimates of the half‐life of the target and the complex. Furthermore, if one has rich time‐course data for the total target dynamics, even if the assay is not sensitive enough to estimate the baseline target concentration T 0, it may be reasonable to fix the baseline level at a value from the literature; although this will affect the estimate for , it will not impact the AFIR prediction. The overarching principle is that because the AFIR metric is a lumped parameter, it is not necessary to estimate each individual rate constant of the model to make predictions for target engagement.
When interpreting the AFIR metric, one must ensure that the following conditions hold:
Each of these terms is derived in the Supplementary Material, although performance of the metrics has not yet been numerically explored. It is also important to recognize that the TMDD model analyzed here is a simplification and leaves out many physiological processes such as:
Implicit in this work is the assumption that the model prediction of free target based on total drug and total target data is accurate, because usually free target data is not available. Thus far, the free target prediction has only been validated for omalizumab.16 Others tried to test this prediction with siltuximab, but because dosing of siltuximab led to an immediate input of IL‐6 into the blood, a more complex model was needed to describe the system and many possible free target levels and AFIR ratios were consistent with the data.34 Publication of other systems in which both total and free target are measured would help to validate the assumption that the model prediction for free target is generally accurate.
Furthermore, target engagement is only the first pharmacodynamic step toward achieving efficacy, and thus AFIR must be interpreted with caution. Even though the AFIR equation demonstrates that the free target concentration will continue to decline with increasing dose, there may be a threshold (e.g., 5%) such that further reduction in AFIR no longer leads to improved clinical efficacy. Understanding this process can be aided with collection of a downstream pharmacodynamic biomarker (e.g., C‐reactive protein and neutrophil levels for anti‐IL‐6 therapy7, 35). Finally, AFIR should not be applied to agonists like TGN141216 or blinatumomab,36 where only small amounts of target binding ( ) are necessary.
In summary, the AFIR potency metric has been derived. This metric predicts the target inhibition at steady state under a repeated dosing regimen using three quantities: the average drug concentration at steady state, the dissociation constant, and the target accumulation ratio. AFIR provides intuition for how various physiological properties of the system impact target engagement. In addition, AFIR has been used for rapid assessment of the druggability of new targets and exploration of the binding affinity, half‐life, bioavailability, and dosing regimen needed for a second‐generation drug to achieve comparable or superior efficacy to a marketed compound.
A.S. analyzed the data and performed the research. A.S. and R.R. wrote the manuscript.
The authors would like to thank Kostas Biliouris, Bruce Gomes, Mick Looby, Phil Lowe, Mark Milton, Bert Peletier, Brian Stoll, Jean‐Louis Steimer, Karthik Subramanian, and Wenping Wang for many helpful discussions.
A.S. and R.R. are employed by Novartis Institute for BioMedical Research.