Preclinical Pharmacokinetic‐Pharmacodynamic Modeling of Antibody Nanoconjugates for Breast Cancer Treatment

ABSTRACT

Over the past 5 years, the number of antibody–drug conjugates (ADCs) entering the oncology market has surged. By contrast, the clinical use of immunoliposomes—nanoparticles that also employ antibodies as targeting agents (i.e., antibody–nanoconjugates, ANCs)—has yet to receive approval. One possible reason is the limited understanding of the relationship between ANC exposure and efficacy. To address this gap, we developed a pharmacokinetic/pharmacodynamic (PK/PD) model for a docetaxel–trastuzumab ANC. PK and PD studies were conducted in MDA‐MB‐231 tumor–bearing mice, divided into three treatment groups: (i) control, (ii) free docetaxel + free trastuzumab, and (iii) docetaxel–trastuzumab ANCs. PK data were obtained by fluorescence imaging and captured systemic and tumor distribution of ANCs, while PD data were based on tumor growth measurements. The PK model was built using data from the ANC group, and the PD model was developed across all three treatment groups, with a leave‐one‐out cross validation strategy for predictive evaluation. Treatment schedule simulations were then performed. Using a population nonlinear mixed‐effects (NLME) approach, the final PK model included a central and a tumor compartment, each with independent elimination rates. Tumor growth dynamics were best described by a reduced Gompertz model coupled with a Simeoni resistance model. This model demonstrated strong predictive performance. Using only data from the first two treatment cycles (4 time points) and Bayesian estimation in the test set, it accurately predicted systemic and tumor ANC concentrations across the subsequent four treatment cycles, with a relative error (RE) below 15%. Prediction precision was further improved (RE < 5%) when data from one additional treatment cycle were incorporated. Similar performance was achieved for PD data, with tumor volume predictions exhibiting RE < 15% and RE < 7.5% when implementing data from two and three treatment cycles, respectively. The model was then applied to simulate alternative treatment schedules. Interestingly, while systemic exposure remained comparable across regimens, every‐5‐day administration yielded the greatest tumor exposure compared to daily, weekly, and every‐35‐day dosing (mean ± SD tumor AUC: 55.4 ± 10.9 mg/kg day vs. 47.6 ± 9.38 mg/kg day, 51.4 ± 11.5 mg/kg day, and 44.6 ± 10.0 mg/kg day, respectively, one‐way ANOVA, p < 0.001), highlighting the importance of dosing frequency. Overall, these results illustrate how PK/PD modeling can refine intra‐tumor PK understanding and support preclinical development of innovative formulations such as ANCs.

Study Highlights

• What is the current knowledge on the topic?

  • Antibody–drug conjugates (ADCs) are already established in clinical practice and several ADCs are approved for the treatment of breast cancer, demonstrating clear clinical benefit. Their success is largely driven by targeted delivery of potent cytotoxic agents through well‐characterized antibodies. In contrast, nanoconjugates (ANCs) have not yet moved beyond the preclinical or early translational stage, despite displaying more favorable pharmacokinetic (PK) properties such as enhanced tumor exposure. One reason is the incomplete understanding of how nanoparticle biodistribution and tumor accumulation translate into therapeutic efficacy. While PK modeling has been extensively applied to free drugs and more lately ADCs, the exposure–response relationship of targeted nanoparticles in vivo is still not well defined.
• What question did this study address?

  • This study aimed to establish a quantitative PK/pharmacodynamic framework to characterize the relationship between biodistribution and antitumor efficacy of an ANC. This approach is applied here in vivo to a docetaxel–trastuzumab immunoliposome developed in our laboratory for breast cancer, with the additional objective of investigating how treatment scheduling influences intratumoral drug exposure, tumor growth inhibition, and efficacy.
• What does this study add to our knowledge?

  • This study provides the community with an openly applicable PK/PD modeling framework tailored to preclinical nanomedicine research. Beyond demonstrating that ANCs achieve higher antitumor efficacy than free drugs due to improved tumor exposure, the model illustrates how modeling can be leveraged in diverse experimental settings combining imaging‐based biodistribution data and tumor growth measurements. In particular, it shows that reliable PK/PD analyses can be performed using in vivo imaging data rather than direct, invasive, and destructive intratumoral dosing measurements. It also supports the use of the reduced Gompertz tumor growth model, which remains underutilized to describe tumor growth with fewer parameters. Finally, the proposed works enable access to key information, such as intratumoral drug exposure and the impact of alternative treatment schedules, that would otherwise be difficult or unethical to obtain experimentally, as they would require repeated tumor excision, extensive animal sacrifice, and substantial time and resource costs.
• How might this change drug discovery, development, and/or therapeutics?

  • This study provides a model‐informed framework for the preclinical development of antibody–nanoconjugates by quantitatively linking biodistribution to antitumor efficacy. The approach enables prediction of intratumoral drug exposure, optimization of dosing regimens, and comparison of treatment strategies before experimental validation. By supporting rational candidate selection and dose optimization, this framework has the potential to accelerate the development of targeted nanomedicines and facilitate their translation into clinically effective therapies.

1. Introduction

Breast cancer is the most common malignant tumor in women worldwide [1, 2]. To mitigate the toxicity associated with conventional chemotherapeutic agents, nanoparticle‐based drug delivery systems have been developed to selectively target tumor tissue while sparing healthy organs. Among these, anti‐Her2 antibody–drug conjugates (ADCs) have demonstrated remarkable anticancer activity, with improved survival outcomes, even in tumors with low Her2 expression [3], highlighting the therapeutic potential of targeted nanotherapies. This success has stimulated the exploration of alternative nanosystems functionalized with anti‐Her2 antibodies. A prominent example is the immunoliposome, a lipid‐based nanoparticle with well‐established clinical applications due to its favorable pharmacokinetic profile and reduced systemic toxicity [4]. Interestingly, compared to ADCs, immunoliposomes may offer distinct pharmacokinetic advantages (e.g., greater distribution volume, leading to enhanced tumor targeting and accumulation), further supporting their promise as targeted therapies.

In this context, our laboratory has developed an immunoliposome consisting of a liposome encapsulating docetaxel, with trastuzumab conjugated to its surface [5]. These immunoliposomes, also referred to as antibody nanoconjugates (ANCs), demonstrated strong binding to Her2 receptors on breast cancer cells and showed tumor accumulation with stronger effect even in models with low Her2 expression (i.e., MDA‐MB‐231, expressing ${5.10}^3\pm 0.3{.10}^3$ Her2 receptors per cell) [5, 6]. However, the exposure–response relationship of ANCs remains poorly understood and requires further investigation. Here, we aim to develop a pharmacokinetic/pharmacodynamic (PK/PD) model to characterize the relationship between systemic exposure, tumor accumulation, and antitumor activity of these nanoparticles.

Several modeling approaches have previously been proposed to describe the mechanisms underlying the activity of these nanoparticles [7, 8, 9]. While a one‐compartment model was used to compare active and passive targeting in tumor accumulation [10], intravenous injection studies revealed biphasic plasma concentration declines [11], motivating the use of two‐compartment models. In turn, three‐compartment models have been applied to estimate tumor vascular permeability and nanoparticle retention [12]. A further class of more complex models is represented by physiologically based pharmacokinetic (PBPK) models, which explicitly account for physiological processes to capture whole‐body influences on drug distribution. The first PBPK models for nanoparticles—adapted from those developed for free drugs—appeared only recently [13, 14]. Dogra et al. later extended this approach by formulating a compartmental PBPK model that describes whole‐body nanoparticle biodistribution and tumor uptake [15]. Mechanistic models have also been applied to ADCs, such as trastuzumab emtansine (T‐DM1), which combines the monoclonal antibody trastuzumab with the cytotoxic maytansinoid DM1. Singh et al. described its pharmacokinetics with a two‐compartment model incorporating linear elimination from the central compartment, reproducing the biexponential profiles of both trastuzumab and T‐DM1 [16]. Building on this, Weddell et al. developed a detailed PBPK model that integrates clinical pharmacokinetics, tumor penetration, and tumor growth inhibition [17].

Following a similar strategy, here we investigated a two‐compartment model describing nanoparticle exchange between systemic and tumor compartments. In vivo nanoparticle biodistribution was tracked using fluorescence imaging. Nanoparticle efficacy was described using a modified Simeoni model incorporating a time‐dependent resistance term to capture the progressive loss of drug effectiveness over the course of treatment (hereafter referred to as the Simeoni‐resistance model). The model was fitted to in vivo tumor growth data via parameter estimation.

2. Material and Methods

2.1. Data

All the experiment details are provided in [6]. Distribution studies were performed on MDA‐MB‐231 bearing mice. 6 week‐old female BALB/c nude mice were orthotopically grafted (mammary fat pad) with 80,000 MDA‐MB‐231 cells in 60% Matrigel. After 18 days, mice were divided into groups according to the treatment. Here, we consider mice treated with ANC (n = 10 animals) to analyze the both the pharmacokinetics and pharmacodynamics, and mice treated with free docetaxel plus free trastuzumab to compare the pharmacodynamics of the different therapies. Treatment was administered intravenously (i.e., retro‐orbital) once a Week for 6 consecutive weeks. Drug administration is detailed in Table S1. Localization and quantification of DiR‐fluorescent nanoparticles in tumors were tracked weekly using spectral unmixing, which enables the separation of the three distinct signals: animal autofluorescence, tumor cell fluorescence, and ANC fluorescence (see Figure 1A). Imaging was performed on an IVIS Spectrum imager equipped with Living Image 4.2.1 software. DiR‐labeled nanoparticles were excited at 745 nm and emission was recorded from 780 to 840 nm. The spaghetti plot of the nanoparticle fluorescence of the ANC group is shown in Figure 1B.

FIGURE 1.

ANCs biodistribution and tumor growth kinetics. (A) Example of fluorescence imaging to quantify immunoliposome accumulation in tumors: Tumor localization and nanoparticle distribution in the entire body, 30 days after tumor inoculation (B) Spaghetti plot of the nanoparticle distribution in the entire body (top) and in the tumor (bottom). Vertical black dotted lines correspond to the days of nanoparticle administration. (C) Tumor growth measured by fluorescence (left) and caliper (right).

Tumor growth was initially monitored using fluorescence imaging, while starting from Day 44 caliper measurements were employed because of loss of dTomato expression in vivo. Using nonlinear regression and assuming that the fluorescence signal is proportional to volume, we transformed fluorescence data to volume data. The spaghetti plot of tumor growth is shown in Figure 1B.

2.2. Two‐Compartments Pharmacokinetics Model

We assumed that the body is composed of two compartments: the central compartment and the tumor. A schematic of the model is provided in Figure 2A. The injected dose $D_i$ at day $t_{\mathrm{adm},\mathrm{i}}$ (for $i=1,\dots ,N_{\mathrm{adm}}$, where $N_{\mathrm{adm}}$ is the total number of drug administration) enters into the systemic circulation, where nanoparticle amount is denoted by $A_c$. Here, nanoparticles are eliminated by the macrophages, liver, spleen and to a lower extent kidneys with rate $k_c$. Nanoparticle amount in the tumor is described by the variable $A_t$. The rate of nanoparticle tumor entry from the central compartment is denoted by $k_{c,t}$ and we assumed that NPs could not return in the vessels once they had extravasated into the malignant tissue. Moreover, nanoparticles might be cleared or uptaken by the tumor cells with a rate $k_{\mathrm{e}}$. Denoting by $K_F$ the conversion rate from mg/kg to fluorescence values, the system of ordinary differential equations reads as:

FIGURE 2.

Schematic of the two‐compartment pharmacokinetic (PK) model and Simeoni‐resistance pharmacodynamic model. (A) PK model. The nanoparticle amounts in the central and in the tumor compartments are denoted, respectively, by $A_c$ and $A_t$. The injected dose enters into the central compartment. The parameter $k_{c,t}$ drive the nanoparticle exchange between the tumor compartment and the central compartment. Nanoparticles are eliminated by the body using the microconstant $k_c$ and by the tumor using the microconstant $k_{\mathrm{e}}$. (B) Simeoni‐resistance model. Proliferating cells follow a Gompertzian growth; $k_1$ is the first‐order rate constant of transit; $e_0$ is the measure of the drug potency; $\lambda$ denotes the resistance.

$$\frac{d{A}_c}{\mathit{dt}}=-k_c{A}_c-k_{c,t}{A}_c+\sum _i^{N_{\mathrm{adm}}}\frac{D_i}{K_F}{\delta }_D\left(t-t_{\mathrm{adm},\mathrm{i}}\right),A_c\left(0\right)=0,$$ (1)

$$\begin{array}{c}\frac{d{A}_t}{\mathit{dt}}=k_{c,t}{A}_c-k_{\mathrm{e}}{A}_t,A_t\left(0\right)=0,\end{array}$$ (2)

where ${\delta }_D$ is the Dirac delta function that satisfies ${\delta }_D\left(t\ne 0\right)=0$ and ${\delta }_D\left(t=0\right)=1$.

2.3. Pharmacokinetic‐Pharmacodynamic Model: Simeoni‐Resistance Model

The efficacy of the nanoparticles was modeled using a modified Simeoni model [18] to account for the resistance to the drug. The fraction of proliferating cells $x_1$ was assumed to follow the reduced Gompertz model [19] (Figure 2B). The parameters $\beta$ and $k$ characterize the unperturbed tumor growth. After nanoparticle administration, the effect of the compound was considered to be proportional to the amount of nanoparticles in the tumor compartment through the constant parameter $e_0$. To account for a progressive development of resistance, we assumed that the efficacy of the drug decreases exponentially in time according to $\lambda$ starting from the day of the first administration $t_{0,\mathrm{adm}}$ [20]. A four‐transit compartment model was used to model the delay of the death of tumor cells with respect to the exposure. The cell fractions in each compartment $x_2$, $x_3$ and $x_4$ denote progressive stages of damage at the end of which they die. The dynamic of the transit of cells from a compartment to the next one is denoted by the transit rate constant $k_1$. To summarize, the model reads as:

$$\begin{array}{c}\frac{d{x}_1}{\mathit{dt}}=\left(k{\beta }^i-{\beta }^i\log \left(\frac{x_1}{V_{\mathrm{inj}}}\right)\right)x_1-e_0\exp \left(-\frac{t-t_{0,\mathrm{adm}}}{\lambda }{\mathbb{1}}_{t>t_{0,\mathrm{adm}}}\right)A_t\left(t\right)x_1,\end{array}$$ (3a)

$$\begin{array}{c}\frac{d{x}_2}{\mathit{dt}}=e_0\exp \left(-\frac{t-t_{0,\mathrm{adm}}}{\lambda }{\mathbb{1}}_{t>t_{0,\mathrm{adm}}}\right)A_t\left(t\right)x_1-k_1{x}_2,\end{array}$$ (3b)

$$\begin{array}{c}\frac{d{x}_3}{\mathit{dt}}=k_1\left(x_2-x_3\right),\end{array}$$ (3c)

$$\begin{array}{rrr}\frac{d{x}_4}{\mathit{dt}}& =k_1\left(x_3-x_4\right),& \end{array}$$ (3d)

$$V=x_1+x_2+x_3+x_4.$$ (3e)

The model parameters were identified using the tumor growth data of animals treated with ANCs and free drugs, using likelihood maximization from a nonlinear mixed effects statistical framework (see below). We used the two compartment PK model to evaluate the exposure $A_t$ of the nanoparticles. To study the efficacy of the free drugs, we assumed the exposure of free docetaxel to be equal to the plasma amount of docetaxel, which dynamic was described by a one compartment model:

$$\frac{d{A}_c}{\mathit{dt}}=-k_c{A}_c,$$ (4)

where the microconstant $k_c$ was fixed within the population to 2.24 days⁻¹ [21].

2.4. Nonlinear Mixed Effects Modeling

Denoting by ${\mathrm{Y}}_j^i$ the vector of (pharmacokinetics or pharmacodynamics) observations of subject $i$ at time $t_j^i$, the following observation error model is assumed:

$$\begin{array}{c}{\mathrm{Y}}_j^i=M\left(t_j^i;{\mathit{\theta }}^i\right)+{\mathit{e}}_j^i,\end{array}$$ (5)

where $M\left(t_j^i;{\mathit{\theta }}^i\right)$ is the value of the structural model evaluated at time $t_j^i$, ${\mathit{\theta }}^i$ is the vector of the parameters of subject $i$ and $e_j^i$ is the residual error, which can be constant ($e_j^i=\sigma {\epsilon }_j^i$) or proportional ($e_j^i=\mathit{\sigma M}\left(t_j^i;{\mathit{\theta }}^i\right){\epsilon }_j^i$), with ${\epsilon }_j^i\sim \mathcal{N}\left(0,1\right)$. The individual parameters ${\mathit{\theta }}^i$ are assumed to follow lognormal distributions:

$$\log \left\{{\mathit{\theta }}^i\right\}=\log \left\{{\mathit{\theta }}_{\mathit{pop}}\right\}+{\eta }^i,{\eta }^i\sim \mathcal{N}\left(0,\omega \right)$$ (6)

Reported values are the elements of ${\mathit{\theta }}_{\mathit{pop}}$ and their $\omega$ values.

Population parameter values were estimated by likelihood maximization using the Stochastic Approximation of the Expectation–Maximization algorithm. Individual parameters were identified using the maximum a posteriori estimate (empirical Bayes estimates). These calculations were performed using Monolix R2 [22].

2.5. Individual Predictions of the Response to the Therapy

We evaluated the ability of the model to forecast individual future trajectories—(i.e., the systemic and tumor drug amounts) and tumor volume—from $p$ early measurements. For prediction in the animal $i$, we used the population of $N-1$ animals with $i$ excluded to define the population prior distribution of the parameters. This was done to prevent using the observations to be predicted in the data used to make the prediction. This prior was then used to perform Bayesian inference (maximum a posteriori estimator) from the $p$ first measurements in animal $i$, leading to an estimate of the individual parameters ${\widehat{\mathit{\theta }}}_p^i$. This was performed sequentially for the PK and PD models.

To evaluate the prediction accuracy, we defined the relative error:

$${\mathrm{RE}}_p^i=\sum _{t=1}^{n_i}\left|\frac{y_j^i-f\left(t_j^i;{\widehat{\mathit{\theta }}}_p^i\right)}{\sigma y_j^i}\right|,$$ (7)

where $y_j^i$ is the actual observation, $f\left(t_j^i;{\widehat{\mathit{\theta }}}_p^i\right)$ is the model evaluated at time $t_j^i$ (either PK or PD).

Three settings were considered, incorporating observations up to 31, 46, and 53 days after tumor inoculation (corresponding to 2, 4, and 5 treatment cycles, respectively). The relative error was calculated for the two PK variables (systemic and tumor ANC amount) and for the tumor volume to evaluate the predictive performance of the three models.

2.6. In silico Exploration for Treatment Scheduling

The final PK/PD model obtained from the totality of data was used for scheduling exploration [23]. The areas under the curves (AUCs) of the systemic and tumor concentrations of docetaxel for the ANCs' group, as well as the associated tumor growth, were simulated in 100 animals. Briefly, a total dose of 70 mg/kg of docetaxel was simulated in 100 mice treated with ANCs over 5 weeks, administered daily, every‐5‐days, weekly, or once every 5 weeks (Table 1). At the selected dose, we ensured not to exceed the reported maximal tolerated dose (MTD) [24] while also testing the most commonly used regimen in the literature (Schedule 2).

TABLE 1.

Simulated treatment schedules of intravenously administered docetaxel ANCs over 6 weeks.

Scheduling	Total dose (mg/kg)	Nb. administrations	Dose per administration (mg/kg)	Time between administrations (days)
1	70	35	2	1
2	70	5	14	7
3	70	7	10	5
4	70	2	35	$5\times 7$

3. Results

3.1. Pharmacokinetic Model

Because trastuzumab was administered at a sub‐therapeutic dose solely for targeting purposes in this HER2‐low model, it was omitted from the PK model; consequently, the injected dose $D$ was parameterized based on docetaxel administration only, the cytotoxic drug encapsulated in the nanoparticles. Moreover, we assumed the conversion rate $K_F$ and the tumor elimination parameter $k_{\mathrm{e}}$ to have fixed values within the population, while all other parameters were assumed to have both fixed and random effects. For the former, it was justified by its definition, which is not subject‐dependent. For the latter, first explorations with random effects led to unidentifiability. Eventually, we assumed a proportional error model $e_j^i$ for both observations in the central and tumor compartments, and denoted by ${\sigma }_c$ and ${\sigma }_t$ the error model parameters of the two compartments.

After the estimation of the PK parameters, the individual empirical Bayesian estimates were computed. They were incorporated in the tumor growth volume dataset to determine the pharmacodynamic parameters of the nanoparticles.

Diagnostic plots of the PK model are shown in Figure 3. The ANCs provided good fits, except for the third injection (Day 32), which we hypothesize was affected by a fluorochrome encapsulation artifact, as no biological explanation has been identified. Since this ANC formulation must be prepared extemporaneously, it is possible that this specific batch contained a higher fluorochrome load. Parameter estimates are provided in Table S2. Good parameter identifiability was found in the analysis.

FIGURE 3.

Diagnostic plots of the pharmacokinetic model obtained for 500 simulations. Visual predictive check (left), prediction distribution (center), and observations versus predictions (right) of the nanoparticle fluorescence in the central compartment (A) and tumor compartment (B). Vertical dashed lines are the days of nanoparticle administration. The outlier proportion was 6.25% and 9.38% for the central compartment (A) and tumor compartment (B) respectively.

3.2. Pharmacodynamic Model

For the volume data, the observation model reads as follows:

$$y_j^i=V\left(t_j^i;{\mathit{\theta }}^i\right)+{\sigma }_{V}V\left(t_j^i;{\mathit{\theta }}^i\right){\epsilon }_j^i,$$ (8)

where ${\mathit{\theta }}^i$ is the set of parameters of the tumor growth model $V\left(t\right)$, ${\sigma }_{\mathrm{V}}$ is the error model parameter, and ${\epsilon }_j^i\sim \mathcal{N}\left(0,1\right)$ is the residual error. A proportional error model was used as the closest approximation available in Monolix of the error model determined experimentally for tumor volume measurements [25].

In Equation (3), we fixed some parameters which were known from previous experiments or from literature. The reduced Gompertz model parameter $k$ was considered to have fixed effects and we estimated it from previous experiments [19]. In this previous work, we also demonstrated that the carrying capacity of the MDA‐MB‐231 cell line in nude mice is $K=$ 2600 mm³ and the volume of injected cells is $V_{\mathrm{inj}}=0.08$ mm³, it was determined as follows:

$$k=\log \left(\frac{K}{V_{\mathrm{inj}}}\right)\approx 10.4.$$

The time of the first injection $t_{0,\mathrm{adm}}$ was 18 days. Finally, the transit rate constant of the Simeoni model was determined according to the results from [26] ($k_1=0.523$ day⁻¹, ${\omega }_{k_1}=0.141$).

Parameters of the pharmacodynamic model are provided in Table S3. We compared the efficacy of the ANCs with the free drugs (docetaxel + trastuzumab). We noticed that the drug potency $e_0$ of the ANCs was higher than that of the free drugs. The resistance parameter $\lambda$ showed high uncertainty since the standard error of the estimate was found to be 98.6%. This could be due to a lack of identifiability of the model due to the limited information on the resistance to the ANCs and free drugs. Future studies could assess this feature in a more detailed manner.

Under these conditions, goodness‐of‐fit was observed, and the model could capture inter‐animal variability for the two treatments of docetaxel, as ANC (Figure 4A) and as free drug (Figure 4B), but also for unperturbed tumor growth (Figure 4C). Inter‐animal variability was also observed at the individual level, underlying the effect of the PK on the PD and confirming the ability of the model to fit tumor growth for three animals chosen randomly (Figure 4D).

FIGURE 4.

Pharmacodynamic of ANCs and free drugs. (A) prediction distributions of the Simeoni‐resistance model fitted to the ANCs dataset (A), free drugs (B) and control (C). (D) impact of the ANCs pharmacokinetic on tumor growth: Individual systemic and tumor amount of docetaxel (top and center, respectively) and tumor volume (bottom). $A_c$ and $A_t$: Amount of drug in the circulating and tumor compartments, respectively.

3.3. Individual Predictions of the Response to the Therapy

The predictive performance of the model was evaluated on three subsets of observations incorporating data up to 31, 46, and 53 days after tumor inoculation. This corresponded to 4 (27%), 9 (60%), and 11 (73%) observations for the PK analysis and 4 (33%), 6 (50%), and 8 (67%) observations for the PD analysis, respectively (Figure 5A). The results showed that good accuracy of individual prediction (i.e., low relative error (RE)) was achieved using only four observations for systemic and tumor ANC concentrations, while eight observations were required for tumor growth. This validates the model's suitability for exploring treatment scheduling. Increasing number of observations from 4 to 5 allowed to highly lower the uncertainty as shown by the shaded areas (prediction intervals) of Figure 5A and found in the values of RE in Figure 5B. Increasing the number of observations substantially reduced prediction uncertainty, as shown by the shaded areas in Figure 5A and supported by the RE values in Figure 5B, with five observations identified as the optimal number.

FIGURE 5.

Example of an individual prediction depending on the number of implemented observations. (A) Prediction of systemic and tumor concentration of docetaxel for ANCs and associated tumor growth for three subsets of observations implementing data up to 31 (top line), 46 (middle line) and 53 (bottom line) days after tumor inoculation. Median (blue line) and 90% confidence interval (blue area) of the individual fits were obtained by evaluating the model on the subset of observations (black dots). Red dashed line and area represent the median and 90% confidence interval of the predictions. Red stars are the observations that were not considered to fit the model. (B) Evaluation of the relative error for prediction of systemic and tumor concentration of docetaxel for ANCs and associated tumor growth.

3.4. In silico Exploration for Treatment Scheduling

Four new treatment schedules of the same total dose were investigated over 5 weeks for daily (scheduling 1), every‐5‐day (scheduling 2), weekly (scheduling 3), and every 5 weeks administrations (scheduling 4). While simulated systemic AUC showed no difference, every‐5‐day dosing allowed higher tumor exposure compared to any other schedules (i.e., mean ± SD of tumor AUC was 55.4 ± 10.9 mg/kg day for scheduling 2 vs. 47.6 ± 9.38 mg/kg day, 51.4 ± 11.5 mg/kg day, and 44.6 ± 10.0 mg/kg day for scheduling 1, 3, and 4, respectively, one‐way ANOVA, p < 0.001) (Figure 6A). However, this difference in tumor accumulation did not translate into a reduction in tumor growth (Figure 6B), probably because of the high‐inter‐animal variability.

FIGURE 6.

Simulations of new administration and prediction error obtained for 100 animals. (A) Area under the curve (AUC) of the systemic (left) and tumor (right) concentration of docetaxel; (B) simulations of the amount of drug in the systemic compartment (left), in the tumor compartment (center), and tumor growth inhibition simulations (right); (C) prediction error. All schedules delivered a total dose of 70 mg/kg over 35 days: Daily (Schedule 1), weekly (Schedule 2), every 5 days (Schedule 3), and once every 35 days (Schedule 4).

4. Discussion

We analyzed the circulating and tumor pharmacokinetics of nanoparticles using a two‐compartment model based on fluorescence imaging, which quantified nanoparticle distribution. We assumed that nanoparticle fluorescence reflects their amount in the tumor interstitium and systemic circulation, respectively. This assumption guided the structure of the PK model (Figure 2A), which describes the exchanges between a central compartment (systemic circulation) and a tumor compartment through a parameter dependent on permeability and surface area (i.e., $k_{c,t}$). Nanoparticles can be cleared from both compartments: in the circulation, elimination may occur via interactions with the macrophage system, liver, or kidney [27, 28], while in the tumor, clearance can result from high interstitial fluid velocity driving particles outward or from lymphatic drainage [29]. Looking at the PD aspect of the model, the delay in docetaxel effect was described using a Simeoni‐resistance model, which assumes that drug potency decreases exponentially over time. Such resistance has been previously reported in breast cancer and may result from mechanisms such as alterations in drug targets (e.g., microtubules) or increased drug efflux [30].

This approach allowed us to determine pharmacokinetic parameters for the ANCs (Table S2), as well as the optimal treatment regimen (14 mg/kg eq. docetaxel, every‐5‐day for 5 weeks) (Figure 6) while addressing ethical concerns in research, since mathematical modeling reduces reliance on animal experiments and fluorescence imaging, being non‐destructive, enables longitudinal monitoring in the same animals. Our model also allows experimentation refinement by accurately predicting later time points from a limited number of early measurements (Figure 5A). This ability to anticipate tumor progression makes it possible to stop experiments earlier, thereby saving time and cost, but more importantly, refining animal experimentation by sparing animals the burden of large tumor volumes, associated metastases, and the resulting pain.

Among the limitations of this work, we would first like to highlight our choice to use an indirect fluorescence‐based quantification approach, which does not distinguish between intact nanoparticles, nanoparticle residues or released dye within the tumor microenvironment. Consequently, the imaging data used here should not be interpreted as an absolute quantification of intratumoral drug concentration, but rather as a semi‐quantitative surrogate of nanoparticle‐associated signal distribution over time. While this method provides access to longitudinal and ethically favorable in vivo data, it may potentially lead to misinterpretation if appropriate controls and prior validations are not carefully performed. For the data implemented here [6], these controls were conducted and resulted in what we considered an acceptable level of leakage and experimental uncertainty (i.e., a background signal below 15% applied to a maximum leaked fraction of 20%), corresponding to an estimated error comparable to the analytical variability generally accepted for validated quantification methods.

Beyond this assumption, additional limitations are inherent to the model presented here. First of all, we considered the nanoparticle amount of the systemic compartment to be proportional to the nanoparticle fluorescence of the entire body. However, we did not take into account that immunoliposomes might be absorbed by other organs (such as the liver, spleen, or kidneys) and remain fluorescent. This could explain the low values of the microconstant $k_c$ of the NPs, although it has been proven that stealth liposomes improve significantly drug pharmacokinetics [7]. We did not take into account temporal variations of the clearance rate, although while it was not clearly observed in our data, it might be possible that the macrophage system recognizes and eliminates faster the nanoparticles after the first injections. Especially, ANCs might be cleared faster since they have an antibody coated on the surface as well as pegylation [31]. Moreover, we assumed that the NP fluorescence at the tumor compartment is proportional to the amount of nanoparticle in the tumor interstitium, without considering the cell internalization. This was due to a lack of information on the internalization properties of the nanoparticles but could be integrated into future works. The parameter that drives the extravasation of liposomes from the vascular network to the tumor interstitium depends on the effective permeability of the vessel walls and the vascular surface per unit volume $S/V$. We have considered the parameter $k_{c,t}$ to be constant in time, while it has been observed that tumor vascularization evolves with tumor growth [32]. This could explain why the model was not able to describe the data at the earliest measurements.

As a future perspective, a physiologically‐based pharmacokinetic (PBPK) model could mechanistically describe nanoparticle biodistribution and provide insights into clearance. We have previously pursued some efforts in this direction by developing a partial differential equation model for intra‐tumor drug transport, although unlinked to the circulating concentration [33]. In our study, the PK model fitted systemic circulation data better than tumor data, likely due to the absence of mechanistic detail on tumor size and perfusion. Incorporating these factors could improve tumor predictions. Moreover, a mechanistic approach could better characterize the observed docetaxel resistance, as reflected by the high uncertainty of the parameter $\lambda$.

Another important perspective of this work, beyond efficacy, would indeed be to address toxicity‐related questions, which remain one of the major barriers limiting nanoparticle clinical translation. In this context, PBPK‐based approaches could help predict and quantify nanoparticle accumulation within critical organs, particularly the heart, which is of specific concern for HER2‐targeted therapies. Such mechanistic modeling could therefore contribute to a better balance between efficacy and safety while reducing the need for extensive animal experimentation.

Overall, our model enables an ethical characterization of ANC systemic and intratumor distribution, as well as the identification of optimal administration regimens, which could accelerate the clinical translation of these targeted nanomedicines, still lacking approval despite strong preclinical evidence of safety and efficacy.

Author Contributions

Anne Rodallec, Cristina Vaghi, and Sebastien Benzekry wrote the manuscript; Sebastien Benzekry and Raphaelle Fanciullino designed the research; Anne Rodallec, Cristina Vaghi, and Sebastien Benzekry performed the research; Sebastien Benzekry, Raphaelle Fanciullino, and Joseph Ciccolini analyzed the data.

Funding

This work was supported by a grant from the French government, managed by the National Research Agency (ANR), under the France 2030 program within the DIGPHAT project, reference ANR‐22‐PESN‐0017.

Conflicts of Interest

The authors declare no conflicts of interest.

Supporting information

Table S1: Treatment scheduling of ANCs. Dose is given in (mg/kg).

Table S2: Population pharmacokinetic parameters for ANCs. σc, σt and are the dimensionless error model parameters of the central and tumor compartments, respectively. R.S.E., relative standard errors of the estimates; ω, random effect.

Table S3: Population pharmacodynamic parameters for ANCs and free drugs. σv is the dimension‐ less error model parameters of the volume data. R.S.E., relative standard errors of the estimates; ω, random effect.