Site-of-Action Pharmacokinetic-Pharmacodynamic Modeling

Experiments aimed at characterizing the dose-dependent relationship between drug concentration and tumor size form the backbone of pre-clinical studies in oncology. Typically, the collected time-course measurements are tumor volume and drug concentration in the plasma, which are phenomenologically captured by a simple indirect response model, such as ref. ^{30 (link)}. This correlation might be sufficient for assessing the general dose-response relationship but cannot answer the question of whether the underlying mechanism of action of the drug has been fully engaged.
For this question, we often use pharmacobinding (PB) models^{31 (link)}, which describe the dynamics of the target (such as PD-1 for pembrolizumab), and the reversible binding kinetics between the drug and its target. This allows calculating levels of projected target occupancy, and it is typically expected that if over 90% of the target has been engaged without an effect, then the target may not be the correct one for the selected indication^{31 (link)}. Such PK-PB models can facilitate the development of a mechanistic understanding of the dose-response relationship between the drug and the tumor size.
A step further can be taken with site-of-action models^{32 (link)–34 (link)} that take into account the drug-target dynamics not only in the plasma but also, as the name suggests, at the site of action, such as the TME. These models can vary in degrees of complexity from more mechanistic³⁵ to more detailed physiologically based pharmacokinetic models^{36 (link),37 (link)}. While such models can be used to calculate projected levels of target occupancy in the TME, it is unclear whether these estimates are truly reliable without actually sampling the TME, the question we will be addressing here.
For that, we developed a modified version of a two-compartment site-of-action model which describes drug concentration over time in the central (plasma), peripheral (tissue), and TME compartments. We assume that pharmacobinding occurs in the plasma and TME compartments; while it is possible that some drug-target dynamics occur in the peripheral compartment as well, we assume that it is either negligible with regards to overall dose-response dynamics or cannot be measured; these assumptions can be relaxed if needed.
The model has a standard structure in the plasma compartment, with an assumption of intravenous drug administration that is cleared at a rate $k_{10}$ ; the drug distributes to the peripheral compartment at a rate $\left(V_1/V_2\right)k_{12}$ and back at a rate $\left(V_2/V_1\right)k_{21}$ , where $V_1$ is the volume of distribution in the central compartment, and $V_2$ is the volume of distribution in the peripheral compartment. We assume that the free target $T_p$ is synthesized in the plasma at a rate $k_{\mathit{syn}}$ and, since the model is calibrated to pembrolizumab data whose target PD-1 is membrane-bound, we assume that it is cleared primarily through internalization at a rate $k_{\mathrm{int}P}$ . We also assume reversible binding kinetics between the drug and its target, with the drug-target complex in the plasma forming at a rate $k_{\mathit{on}}$ , dissociating at a rate $k_{\mathit{off}},$ and clearing at the rate $k_{\mathrm{int}P}$ .
The PK-PB dynamics in the TME compartment are largely similar, with several proposed modifications. Firstly, we assume that the rate of drug distribution into the TME is not constant but is a function of the tumor volume, namely, $\left(V_1/\left(x+\delta \right)\right)k_{1T}$ , where $x$ is tumor volume and $\delta$ is introduced to prevent division by zero in the limiting case, where the tumor volume tends to zero. We propose that while $V_1$ and $V_2$ are treated as constant volumes of distribution (as is standard), the volume of distribution into the tumor be treated as variable, thereby capturing the higher or lower distribution of the drug into the TME depending on tumor size. As a consequence of this assumption, we further propose that the rate of target synthesis in the tumor is not constant or at equilibrium as would likely be in the plasma or non-disease compartment, but instead is treated as a function of tumor size. In particular, we assume this rate increases according to a saturating function $\frac{k_{\mathit{synt}}x}{x+K_x}$ , where $k_{\mathit{synt}}$ is the rate of target synthesis in the tumor (which is likely higher than in the plasma), and $K_x$ is the half-maximal concentration of free target $T_{\mathit{TME}}$ in the TME.
Additionally, we hypothesize that the apparent rate of drug-target binding in the TME is not necessarily the same as in the plasma, i.e., that $k_{\mathit{onT}}$ may be different from $k_{\mathit{on}}$ . That said, we expect that once the drug-target complex has been formed, the dissociation rate $k_{\mathit{off}}$ will remain the same, as that is more likely to be an intrinsic property^{38 (link)}. Finally, we assume that the tumor grows logistically and is killed as a function of the percent target occupancy in the tumor, which is calculated as $\frac{D{R}_{\mathit{TME}}}{D{R}_{\mathit{TME}}+T_{\mathit{TME}}}$ , where $D{R}_{\mathit{TME}}$ is the concentration of the drug-target complex in the tumor and $T_{\mathit{TME}}$ is the free target in the TME.
The resulting system of equations is as follows: $\left(\begin{array}{c}\hfill \frac{d{D}_p}{dt}=u\left(t\right)-k_{10}{D}_p-k_{on}{D}_P{T}_p+k_{off}{D}_R-k_{12}{D}_p+k_{21}\frac{V_2}{V_1}{D}_t\hfill \\ \hfill -k_{1T}{D}_p+k_{T1}{\frac{x+\delta }{V_1}D}_{TME}\hfill \\ \hfill \frac{d{T}_p}{dt}=k_{syn}-k_{intP}{T}_p-k_{on}{D}_p{T}_P+k_{off}{D}_R\hfill \\ \hfill \frac{d{D}_R}{dt}=k_{on}{D}_p{T}_P-k_{off}{D}_R-k_{intP}{D}_R\hfill \\ \hfill \frac{d{D}_t}{dt}=k_{12}\frac{V_1}{V_2}{D}_p-k_{21}{D}_t\hfill \end{array}\right\}\mathrm{Non}-\mathrm{tumor}\mathrm{compartments}$ $\left(\begin{array}{c}\hfill \frac{d{D}_{TME}}{dt}=k_{1T}{\frac{V_1}{x+\delta }D}_P-k_{T1}{D}_{TME}-k_{onT}{D}_{TME}{T}_{TME}+k_{off}D{R}_{TME}\hfill \\ \hfill \frac{d{T}_{TME}}{dt}=\frac{k_{synt}x}{x+K_x}-k_{int}{T}_{TME}-k_{onT}{D}_{TME}{T}_{TME}+k_{off}D{R}_{TME}\hfill \\ \hfill \frac{dD{R}_{TME}}{dt}=k_{onT}{D}_{TME}{T}_{TME}-k_{off}D{R}_{TME}-k_{int}D{R}_{TME}\hfill \\ \hfill T{O}_{TME}=100\times \frac{D{R}_{TME}}{D{R}_{TME}+T_{TME}}\hfill \\ \hfill \frac{dx}{dt}=rx\left(1-\frac{x}{K}\right)-d\frac{T{O}_{TME}}{T{O}_{TME}+T{O}_{50}}x\hfill \end{array}\right\}\mathrm{TME}\mathrm{compartment}$
The structure of the model is summarized in Fig. 1. Variable definitions, initial conditions, and calibrated parameter values are summarized in Table 1.
The model was calibrated to digitized PK data (Fig. 2A) for pembrolizumab reported in ref. ^{13 (link)} and TGI data reported in ref. ²⁰ . The reason this particular PK dataset was chosen is that it includes measurements of percent TO in the TME, which is typically not available. TGI curves in²⁰ are measured for 2 mg/kg and 10 mg/kg of pembrolizumab, administered on average 3.5 days apart, for C57BL/6 mice implanted with MC38 syngeneic colon adenocarcinoma cells. We further calibrated model parameters to fit the PK-TGI relationship for the dose of 10 mg/kg (Fig. 2B). We also report the projected levels of percent TO in plasma as compared to the TME (Fig. 2C) to emphasize the importance of capturing drug-target dynamics in the TME, as this is where it is expected to drive efficacy.
Model parameterization was validated using untrained data. Figure 2D demonstrates that we were able to successfully recapitulate the PK curves for three doses of 1 mg/kg of pembrolizumab given weekly^{13 (link)}, and Fig. 2E shows that we were able to describe the TGI data for five doses of 2 mg/kg of pembrolizumab given on average every 3.5 days²⁰ . We note that, without the %TO in TME data (Fig. 2F), there was a large number of parameter sets that could recapitulate the PK and TGI equally well, further emphasizing that this piece of data was critical to model parameterization.