# Preclinical Pharmacokinetic-Pharmacodynamic Modeling of Antibody Nanoconjugates for Breast Cancer Treatment

Anne Rodallec, Cristina Vaghi, Joseph Ciccolini, Raphaëlle Fanciullino, Sébastien Benzekry

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Anne Rodallec, Cristina Vaghi, Joseph Ciccolini, Raphaëlle Fanciullino, Sébastien Benzekry. Preclinical Pharmacokinetic-Pharmacodynamic Modeling of Antibody Nanoconjugates for Breast Cancer Treatment. 2025. hal-04937059v2

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Distributed under a Creative Commons Attribution 4.0 International License
Preclinical Pharmacokinetic-Pharmacodynamic Modeling of
Antibody Nanoconjugates for Breast Cancer Treatment

Anne Rodallec*¹, Cristina Vaghi*², Joseph Ciccolini¹, Raphaelle Fanciullino¹, and Sebastien
Benzekry†¹

¹COMPutational pharmacology and clinical Oncology, Centre Inria d'Université Côte d'Azur,
Centre de Recherches en Cancérologie de Marseille, Inserm U1068, CNRS UMR7258, Institut
Paoli-Calmettes, Pharmacy faculty, Aix-Marseille University

²MONC, Modélisation Mathématique pour l'Oncologie, Centre Inria de l'Université de
Bordeaux, France

Abstract

Over the past five 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, weekly administration yielded the greatest tumor exposure compared to daily, every-5-day, and every-35-day dosing (mean ± SD tumor AUC:

*These authors contributed equally.

†Corresponding author: sebastien.benzekry@inria.fr
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 [4]. These immunoliposomes, also referred to as antibody nanoconjugates (ANCs), demonstrated strong binding to Her2 receptors on breast cancer cells and showed tumor accumulation even in models with low Her2 expression [5]. However, the exposure-response relationship of ANCs remains poorly understood and requires further investigation. Here, we aim to establish a pharmacokinetic/pharmacodynamic (PK/PD) model to elucidate the mechanisms underlying the activity of these nanoparticles.

Pharmacokinetic-pharmacodynamic (PK/PD) modeling plays a central role in the development of drug delivery systems [6]. PK characterizes the absorption, distribution, metabolism, and elimination of drugs, while PD describes the resulting pharmacological effect, here represented by tumor growth inhibition. By integrating PK and PD into mathematical frameworks, PK/PD models enable a quantitative description of drug exposure, therapeutic effect, and their relationship [7]. Pharmacokinetic models can be classified as noncompartmental or compartmental. Noncompartmental models estimate drug exposure using metrics such as the area under the concentration-time curve (AUC), but they do not provide a mechanistic description of drug kinetics. Compartmental models, in contrast, rely on the empirical assumption that the body can be represented as interconnected compartments (e.g., plasma, organs, or tumor). Drug concentrations within these compartments are described by systems of ordinary differential equations, with parameters such as clearance, tumor uptake, or elimination estimated by fitting experimental data. To account for variability across individuals, nonlinear mixed-effects modeling offers a powerful statistical framework. It allows for robust estimation of population-level parameters while capturing inter-individual variability, thereby providing a comprehensive understanding of both global and individual drug dynamics. Compartmental pharmacokinetic (PK) models are primarily used to characterize drug biodistribution, providing a framework to estimate PK parameters and evaluate dosing strategies [8]. They play a central role in bridging preclinical studies with clinical applications.

Several types of mathematical models have therefore been used to describe the bioavailability of nanoparticles at the tumor site [9, 10, 11]. While a one-compartment model was used to compare active and passive targeting in tumor accumulation [12], intravenous injection studies revealed biphasic plasma concentration declines [13], motivating the use of two-compartment models. In turn, three-compartment models have been applied to estimate tumor vascular permeability and nanoparticle retention [14]. 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 [15, 16]. Dogra *et al.* later extended this approach by formulating a compartmental PBPK model that describes whole-body nanoparticle biodistribution and tumor uptake [17]. 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 [18]. Building on this, Weddell *et al.* developed a detailed PBPK model that integrates
<table><caption>**Table 1.** Treatment scheduling of ANCs. Dose is given in (mg/kg).</caption><thead><tr><th>Drug</th><th>Day 18</th><th>Day 25</th><th>Day 32</th><th>Day 39</th><th>Day 46</th><th>Day 53</th></tr></thead><tbody><tr><td>Docetaxel</td><td>3.3</td><td>5</td><td>5</td><td>5</td><td>5</td><td>3</td></tr><tr><td>Trastuzumab</td><td>1.3</td><td>1.9</td><td>1.9</td><td>1.9</td><td>1.9</td><td>1.15</td></tr></tbody></table>

clinical pharmacokinetics, tumor penetration, and tumor growth inhibition [19].

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, and a Simeoni-resistance model was employed to describe nanoparticle efficacy. The model was calibrated against *in vivo* tumor growth data.

# 2 Material and methods

## 2.1 Data

All the experiment details are provided in [5]. 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 1. Localization and quantification of DiR-fluorescent nanoparticles in tumors were tracked weekly using spectral unmixing (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.

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 $t_{adm,i}$ (for $i = 1, ..., N_{adm}$, where $N_{adm}$ 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 a clearance rate $k_c$. Nanoparticle amount in the tumor is described by the variable $A_t$. The rate of nanoparticle tumor entry from the central
<figure><img src="image_2.png" /><figcaption>**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. Figure from [5]. (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).</figcaption></figure>
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 $k_e$. Denoting by $K_F$ the conversion rate from mg/kg to fluorescence values, the system of ordinary differential equations reads as

$$ \frac{dA_c}{dt} = -k_c A_c - k_{c,t} A_c + \sum_{i}^{N_{adm}} \frac{D_i}{K_F} \delta_D(t - t_{adm,i}), \quad A_c(0) = 0, \quad (1) $$

$$ \frac{dA_t}{dt} = k_{c,t} A_c - k_e A_t, \quad A_t(0) = 0, \quad (2) $$

where $\delta_D$ is the Dirac delta function that satisfies $\delta_D(t \neq 0) = 0$ and $\int_{-\infty}^{\infty} \delta_D(t) dt = 1$.

## 2.3 Pharmacokinetic-pharmacodynamic model: Simeoni-resistance model

The efficacy of the nanoparticles was modeled using a modified Simeoni model [20] to account for the resistance to the drug. The fraction of proliferating cells $x_1$ was assumed to follow the reduced Gompertz model [21] (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 $t_{0,adm}$. 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:

$$ \frac{dx_1}{dt} = \left( k\beta^i - \beta^i \log \left( \frac{x_1}{V_{\text{inj}}} \right) \right) x_1 - e_0 \exp \left( -\frac{t - t_{0,\text{adm}}}{\lambda} \right) A_t(t)x_1, \quad (3a) $$

$$ \frac{dx_2}{dt} = e_0 \exp \left( -\frac{t - t_{0,\text{adm}}}{\lambda} \right) A_t(t)x_1 - k_1 x_2, \quad (3b) $$

$$ \frac{dx_3}{dt} = k_1(x_2 - x_3), \quad (3c) $$

$$ \frac{dx_4}{dt} = k_1(x_3 - x_4), \quad (3d) $$

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

The model was calibrated using the tumor growth data of individuals treated with ANCs and free drugs. 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:

$$ \frac{dA_c}{dt} = -k_c A_c, \quad (4) $$
where the clearance rate $k_c$ was fixed within the population to $2.24 \text{ days}^{-1}$ [22].

## 2.4 Nonlinear mixed effects modeling

The population approach is a particularly appropriate statistical framework to analyze pharmacokinetic and pharmacodynamic data [23]. Monolix 2019 R2 [24] was used to estimate the PK/PD model parameters.

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

$$ \mathbf{Y}_j^i = \mathbf{M}(t_j^i; \boldsymbol{\theta}^i) + \mathbf{e}_j^i, \qquad (5) $$

$$ (6) $$

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

$$ \log \{\boldsymbol{\theta}^i\} = \log \{\boldsymbol{\theta}_{pop}\} + \eta^i, \quad \eta^i \sim \mathcal{N}(0, \omega). $$

Reported values are the $\boldsymbol{\theta}_{pop}$s and inter-individual variability is computed from $\omega$ as coefficients of variations expressed in percents $CV = \omega \times 100$.

## 2.5 Individual predictions of the response to the therapy

Bayesian inference was used to predict the response to the therapy of the animals. The animals were split into two groups, the training set and the test set. The leave-one-out cross validation was used to validate our methodology. At each iteration, the parameters distribution of the population composed by $N-1$ individuals (training set) was computed and used as prior to predict the response to the therapy of the excluded subject $i$ (test set). The animals in the training set were pooled together to learn the population parameters of the PKPD model using Monolix. Then, this information was used to make predictions of an animal $i$ in the test set based only on a subset of observations $p$. First, the empirical Bayesian estimates (EBEs) of the PK model parameters were computed based on the population parameters obtained with the training set and on the first $p$ observations of the animal $i$. Then, the individual PD model parameters were computed using the same approach.

The prediction accuracy was defined by the relative error for each animal $i$ in the test set, which was defined as:

$$ \mathrm{RE}_p^i = \sum_{t=1}^{n_i} \left| \frac{y_j^i - f(t_j^i; \boldsymbol{\theta}_p^i)}{\sigma_p y_j^i} \right|, \qquad (7) $$

where $y_j^i$ is the actual observation, $f(t_j^i; \boldsymbol{\theta}_p^i)$ is the model evaluated at time $t_j^i$, $\boldsymbol{\theta}_p^i$ and $\sigma_p$ are the model parameter values and error model parameters, respectively, obtained by fitting the model to the first $p$ observations of animal $i$, $n_i$ is the total number of observations of animal $i$ and $p$ is the number of observations which were considered to compute the EBEs. Three data subsets were evaluated,
<figure><img src="image_3.png" /><figcaption>**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 $k_{c,t}$ drives the nanoparticle exchange between the tumor compartment and the central compartment. Nanoparticles are eliminated by the body with a clearance rate $k_c$ and by the tumor with a rate $k_e$. (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.</figcaption></figure>

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 [25]. 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, weekly, every 5 days, or once every 5 weeks (Table 2). At the selected dose, we ensured not to exceed the reported maximal tolerated dose (MTD) [26] while also testing the most commonly used regimen in the literature (Schedule 2).
<table><caption>Table 2. Simulated scheduling treatment of docetaxel administered intravenously as ANCs over 6 weeks.</caption><thead><tr><th>Simulated Scheduling</th><th>Total (mg/kg)</th><th>dose</th><th>Number of administrations</th><th>Dose of docetaxel per administration (mg/kg)</th><th>Time between administration (Days)</th></tr></thead><tbody><tr><td>1</td><td>70</td><td>35</td><td></td><td>2</td><td>1</td></tr><tr><td>2</td><td>70</td><td>5</td><td></td><td>14</td><td>7</td></tr><tr><td>3</td><td>70</td><td>7</td><td></td><td>10</td><td>5</td></tr><tr><td>4</td><td>70</td><td>2</td><td></td><td>35</td><td>5x7</td></tr></tbody></table>

# 3 Results

## 3.1 Pharmacokinetic model

Since docetaxel was the cytotoxic drug encapsulated in the nanoparticles, the injected dose $D$ was calibrated based on the docetaxel administration. Because trastuzumab was given at a sub-therapeutic dose and only for targeting, it was omitted from the PK model. Moreover, we assumed the conversion rate $K_F$ and the tumor elimination rate $k_e$ to have fixed values within the population, while all other parameters were assumed to have fixed and random effects within the population. 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 3. Good parameter identifiability was found in the analysis.
<table><caption>**Table 3.** Population pharmacokinetic parameters for ANCs. $\sigma_c$, $\sigma_t$ and are the dimensionless error model parameters of the central and tumor compartments, respectively. R.S.E. = relative standard errors of the estimates. IIV = interindividual variability is expressed as coefficient of variation (%).</caption><thead><tr><th>Parameter</th><th>Fixed effects (R.S.E. (%))</th><th>IIV</th></tr></thead><tbody><tr><td>K<sub>F</sub> (mg/kg · (phot./s))</td><td>7.67e-11 (5.1)</td><td>-</td></tr><tr><td>k<sub>c,t</sub> (day<sup>-1</sup>)</td><td>0.1 (9.25)</td><td>2.06</td></tr><tr><td>k<sub>c</sub> (day<sup>-1</sup>)</td><td>0.0212 (64.5)</td><td>37.3</td></tr><tr><td>k<sub>e</sub> (day<sup>-1</sup>)</td><td>1.17 (2.37)</td><td>-</td></tr><tr><td>σ<sub>c</sub> (-)</td><td>0.333 (8.17)</td><td>-</td></tr><tr><td>σ<sub>t</sub> (-)</td><td>0.466 (9.22)</td><td>-</td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>**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 of 6.25% and 9.38% for the central compartment (A) and tumor compartment (B), respectively.</figcaption></figure>

## 3.2 Pharmacodynamic model

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

$$y_j^i = V(t_j^i; \boldsymbol{\theta}^i) + \sigma_V V(t_j^i; \boldsymbol{\theta}^i) \varepsilon_j^i, \quad (8)$$
where $\theta^i$ is the set of parameters of the tumor growth model $V(t)$, $\sigma_v$ is the error model parameter, and $\varepsilon_j^i \sim N(0, 1)$ is the residual error.

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 [21]. Assuming that the carrying capacity of the MDA-MB-231 cell line in nude mice is $K = 2600 \text{ mm}^3$ and the volume of injected cells is $V_{\text{inj}} = 0.08 \text{ mm}^3$, it was determined as follows:

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

The time of the first injection $t_{0,\text{adm}}$ was fixed to 18 days. Finally, the transit rate constant of the Simeoni model was determined according to the results from [27] ($k_1 = 0.523 \text{ day}^{-1}$, $\omega_{k_1} = 0.141 \text{ day}^{-1}$).

Parameters of the pharmacodynamic model are provided in Table 4. 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.

<table><caption>**Table 4.** Population pharmacodynamic parameters for ANCs and free drugs. $\sigma_v$ is the dimensionless error model parameters of the volume data. R.S.E. = relative standard errors of the estimates. IIV = interindividual variability expressed as coefficient of variation (%).</caption><thead><tr><th rowspan="2">Parameter</th><th colspan="2">ANCs</th><th colspan="2">Free drugs</th></tr><tr><th>Estimate (R.S.E. (%))</th><th>IIV</th><th>Estimate (R.S.E. (%))</th><th>IIV</th></tr></thead><tbody><tr><td>β (day<sup>-1</sup>)</td><td>0.0565 (6.49)</td><td>3.57</td><td>0.0705 (5.37)</td><td>2.29</td></tr><tr><td>e<sub>0</sub> (day · mg/kg)<sup>-1</sup></td><td>0.256 (2.88)</td><td>-</td><td>0.196 (0.726)</td><td>-</td></tr><tr><td>λ (day)</td><td>3.34 (98.6)</td><td>0.49</td><td>4.71 (61)</td><td>0.09</td></tr><tr><td>σ<sub>v</sub> (-)</td><td>0.231 (9.84)</td><td>-</td><td>0.282 (9.57)</td><td>-</td></tr></tbody></table>

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).

## 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
<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /><figcaption>**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.</figcaption></figure>
(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.

## 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), weekly (scheduling 2), every 5 days (scheduling 3) and every 5 weeks (scheduling 4). While simulated systemic AUC showed no difference, weekly 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 versus 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.

# 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 [28, 29], while in the tumor, clearance can result from high interstitial fluid velocity driving particles outward or from lymphatic drainage [30]. 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 [31].

This approach allowed us to determine pharmacokinetic parameters for the ANCs (Table 3), as well as the optimal treatment regimen (14 mg/kg eq. docetaxel, weekly 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,
<figure><img src="image_11.png" /><figcaption>**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.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>**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).</figcaption></figure>
and the resulting pain.

Beyond the assumptions of a stable fluorescent signal and the absence of early nanoparticle leakage, 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 clearance rate $k_c$ of the NPs, although it has been proven that stealth liposomes improve significantly drug pharmacokinetics [9]. 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 [32]. 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 [33]. 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 [34]. 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 $\lambda$.

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.

Declaration of generative AI and AI-assisted technologies in the manuscript preparation
process

During the preparation of this manuscript, the authors used ChatGPT to assist with language polishing and generate the drawings for the graphical abstract. After using this tool, the authors reviewed and edited the content as needed and take full responsibility for the content of the published article.
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