# Optimizing Nanoparticle Design for Selective Targeting of Breast Cancer Cells

Sarah Iaquinta, Shahram Khazaie, Sylvain Fréour, Frédéric Jacquemin

► To cite this version:

Sarah Iaquinta, Shahram Khazaie, Sylvain Fréour, Frédéric Jacquemin. Optimizing Nanoparticle Design for Selective Targeting of Breast Cancer Cells. Medical Engineering & Physics, 2025, 145, pp.104420. 10.1016/j.medengphy.2025.104420. hal-05240533

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https://imt-mines-ales.hal.science/hal-05240533v1

Submitted on 4 Sep 2025

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# Optimizing nanoparticle design for selective targeting of breast cancer cells

Sarah Iaquinta<sup>a,b,*</sup>, Shahram Khazaie<sup>a</sup>, Sylvain Fréour<sup>a</sup>, Frédéric Jacquemin<sup>a</sup>,

## ABSTRACT

Recent efforts in cancer targeting have focused on nanoparticle (NP) drug delivery, yet the complexity of NP uptake makes experimental studies challenging. To streamline this, numerical models help identify key parameters. This study's model, focusing on elliptical NPs, aims to optimize NP aspect ratios for selective uptake by breast cancer cells. Mechanical properties of cells were taken from literature, and the model suggests that non-deformable NPs with aspect ratios between 1/3 and 1/2 are optimal for selective cancer cell internalization. These promising results require experimental validation.

## Keywords:

Cellular uptake

Targeted drug delivery

Cancer therapy

Breast cancer

## 1. Introduction

Breast cancer, the most common cancer globally, accounted for about 2.2 million new cases in 2020, or 12.5% of all diagnoses. Conventional treatments often harm healthy cells, causing discomfort and risking therapeutic failure. Recent research has explored NP delivery systems to target cancer cells specifically, examining how the physical, chemical, and geometric properties of NPs affect their uptake for more precise cancer treatment [1–4]. This paper provides an application of the numerical models of NP cellular uptake developed by [4] and enhanced by incorporating membrane mechanical adaptation from our previous studies [5–7]. Our focus is to determine the optimal NP aspect ratio, $\bar{r}$, that promotes selective internalization by cancer cells while minimizing uptake by healthy cells. To achieve this, we derived input parameters representing the mechanical differences between healthy and cancerous breast cells based on an extensive literature review. The model is then used to assess $\bar{r}$, as illustrated in Fig. 1.

## 2. Model

The model in this study describes the cellular uptake of non-deformable elliptical nanoparticles (NPs) by focusing on the initial wrapping of the NP by the cell membrane. This model, based on our previous work [5] – in which all the hypotheses and mathematical framework are presented, integrates the NP aspect ratio $\bar{r}$, membrane tension $\bar{\sigma}$, and adhesion $\bar{\gamma}$ between the NP and membrane. Both $\bar{\sigma}$ and $\bar{\gamma}$ are dimensionless, normalized by the NP radius and membrane bending rigidity $\kappa$. The model evaluates the system's equilibrium by analyzing

<figure><img src="image_2.png" /><figcaption>Fig. 1. Illustration of the application of the model.</figcaption></figure>

potential energy variation, resulting in the wrapping degree $f$, which indicates the wrapped portion of the NP. Equilibrium is achieved at wrapping degree $\bar{f}$, defined as the value of $f$ at which the first local minimum of total potential energy of the system is reached (see more details on the definition of the equilibrium position in our previous work [5]). This model is suitable for NPs around 100 nm in diameter, assuming the membrane can be treated as flat relative to NP size. The model was enhanced to incorporate mechano-adaptation of the membrane during wrapping, allowing adhesion $\bar{\gamma}$ to follow a sigmoidal evolution in terms of $f$ (see Equation (1)). The motivations for using this 3-parameter evolution law are, together with the mathematical framework, provided in our previous article [6].

$$ \bar{\gamma}(f) = \frac{|\bar{\gamma}_0(\bar{\gamma}_A - 1)|}{1 + e^{-\bar{\gamma}_S(f - \bar{\gamma}_D)}} + \bar{\gamma}_0. \quad (1) $$

<footnote><sup>a</sup> GEM-UMR CNRS 6183, University of Nantes - Centrale Nantes, I.U.T. de Saint-Nazaire, Saint-Nazaire, France</footnote>

<footnote><sup>b</sup> LMGC, IMT Mines Ales, Univ Montpellier, CNRS, Ales, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: sarah.iaquinta@mines-ales.fr (S. Iaquinta).</footnote>
<caption>**Table 1**</caption>

Synthesis of the measured values of the input parameters of the model for healthy and cancer breast cells, according to the literature.

<table><thead><tr><th>Parameter</th><th>Technique</th><th>Value</th><th>Reference</th></tr></thead><tbody><tr><td>κ</td><td>Mass spectroscopy to observe the response of healthy and cancer breast cells to thermal fluctuation</td><td>κ<sup>H</sup> = 4.5 × 10<sup>-19</sup> Nm<br>κ<sup>C</sup> = 1.3 × 10<sup>-19</sup> Nm</td><td>[8]</td></tr><tr><td>E</td><td>Atomic Force Microscopy (AFM) of healthy and cancer breast cell<br>Optical deformability</td><td>E<sup>C</sup> = 0.5 E<sup>H</sup><br>E<sup>C</sup> ∈ [0.2, 0.4] × E<sup>H</sup><br>E<sup>C</sup> ∈ [0.56, 0.71] × E<sup>H</sup><br>E<sup>C</sup> = 0.5 E<sup>H</sup></td><td>[9,10]<br>[11]<br>[12]<br>[13]</td></tr><tr><td>σ</td><td>Mass spectroscopy to observe the response of healthy and cancer breast cells to thermal fluctuation<br>Review on the endocytosis of NPs</td><td>σ<sup>C</sup> = (8.69 ± 1.76) × 10<sup>-4</sup> mN/m</td><td>[8]</td></tr><tr><td>γ<sub>0</sub></td><td>Micro-pipette aspiration between two artificial lipid vesicles<br>Micro-pipette aspiration between two murine sarcoma cells<br>Micro-pipette aspiration between a lipid vesicle and a gold substrate<br>Analytical investigation of the adhesion of a latex sphere to a giant vesicle<br>Review on the endocytosis of NPs</td><td>γ ∈ [0.01, 0.1] mN/m<br>γ ∈ [0.1, 1] mN/m<br>γ ∈ [0.03, 0.07] mN/m<br>γ ∈ [0.1, 5] mN/m</td><td>[14]<br>[15]<br>[16]<br>[17]<br>[2]</td></tr></tbody></table>

In this equation, $\bar{\gamma}_0$ represents the initial adhesion, i.e., $\bar{\gamma}_0 = \bar{\gamma}(f = 0)$, and $\bar{\gamma}_A$ is the amplitude of $\bar{\gamma}$, defined as $\bar{\gamma}_A = \bar{\gamma}(f = 1)/\bar{\gamma}_0$. It corresponds to the ratio of final to initial receptor quantities in the interaction region. The parameter $\bar{\gamma}_D$ is a delay parameter that locates how far from $f = 0.5$ the transition between the initial and the final value of $\bar{\gamma}$ occurs, and $\bar{\gamma}_S$, the curvature parameter, is proportional to the slope of $\bar{\gamma}(f)$ during the transition. It controls the steepness or velocity of the receptor migration with respect to $f$. For further parameter details, see [6]. The wrapping state at equilibrium, $f = \bar{f}$, is classified into three phases. In Phase 1, $\bar{f}$ is minimal at 0.03, representing the lower bound of $f$, where minimal wrapping occurs. Phase 3 indicates full internalization, with the membrane fully enveloping the NP, trapping it inside the cell. Phase 2 includes all intermediate wrapping states where the NP is partially wrapped.

### 3. Model parameters

The model investigates a range of aspect ratios, $\bar{r} \in [1, 6]$, representing typical non-deformable NPs, with $\bar{r}$ used for horizontally oriented NPs and its inverse for vertically oriented ones. The subsequent section reviews literature to determine values for membrane bending rigidity ($\kappa$), membrane tension ($\sigma$), and the parameters governing the adhesion function $\bar{\gamma}(f)$. The conclusions of this review are summarized in Table 1.

#### 3.1. Bending rigidity

The bending rigidity, $\kappa$, was estimated in [8], using mass spectroscopy and thermal fluctuations, reporting $\kappa^H = 4.5 \times 10^{-19}$ Nm for healthy cells and $\kappa^C = 1.3 \times 10^{-19}$ Nm for cancer cells. Note that the superscripts H and C refer to healthy and cancer cells, respectively. To the best of our knowledge, this is the only experimental study in the literature that investigates the bending rigidity of both healthy and cancerous breast cells. However, more data are available for the Young's modulus E, as it is easier to be characterized [18–22]. The Young's modulus is

related to $\kappa$ via $\kappa = Eh^3(1 - v^2)^{-1/12}$, where h is the membrane thickness and v is Poisson's ratio [23]. In this study, we assume that h and v are the same for all cells investigated, leading to $\kappa^C/\kappa^H = E^C/E^H$.

Moreover, atomic force microscopy, commonly used to measure the Young's modulus of cells [24,25], conducted on healthy and cancerous breast cells, yielded $E^H = 2.5$ kPa and $E^C = 1.25$ kPa, thus $E^C = 0.5 E^H$ [9,10]. More, [11] found $E^C \in [0.2, 0.4] \times E^H$, while [12] measured $E^C \in [0.56, 0.71] \times E^H$. These intervals do not overlap, highlighting the variability in measurements.

Furthermore, optical deformability techniques [26] on healthy and cancerous breast cells found that, on average, healthy cells are twice as stiff as cancer cells [13].

#### 3.2. Membrane tension

No studies were found that compare the membrane tension of healthy and cancerous mammary cells. However, [8] measured membrane tension $\sigma$ and bending rigidity $\kappa$ for cancer mammary cells, reporting $\sigma^C = 8.69 \pm 1.76 \times 10^{-2}$ mN/m. Additionally, [2] reported a membrane tension value of $\sigma = 0.05$ mN/m, regardless of the cell type or health status.

#### 3.3. Cellular initial adhesion

No specific data on adhesion between a breast cell membrane and a nanoparticle (NP) was found; only cell-cell and cell-extracellular matrix adhesion values are available. Cell-cell adhesion can be measured through micro-pipette aspiration [26]. However, adhesion values are highly sensitive to experimental conditions [14,15], necessitating consistent settings for accuracy. No studies comparing adhesion between healthy and cancer cells, nor any specific adhesion measurements for breast cells, were identified. Nevertheless, existing adhesion data for other objects offer a useful estimate of the adhesion magnitude, as summarized in Table 1.
<table><caption>**Table 2**

Synthesis of the values used as input parameters for the model.</caption><thead><tr><th>Parameter</th><th>Healthy cell</th><th>Cancer cell</th></tr></thead><tbody><tr><td>a</td><td>100 nm</td><td></td></tr><tr><td>κ</td><td>4.5 × 10<sup>-19</sup> N/m</td><td>1.3 × 10<sup>-19</sup> N/m</td></tr><tr><td>σ</td><td>0.01 mN/m</td><td></td></tr><tr><td>γ</td><td>0.1 mN/m</td><td>0.0625 mN/m</td></tr><tr><td>σ̄</td><td>0.44</td><td>1.54</td></tr><tr><td>γ̄</td><td>4.44</td><td>9.62</td></tr><tr><td>γ̃<sub>A</sub></td><td>1.5</td><td>4</td></tr><tr><td>γ̃<sub>D</sub></td><td>0.2</td><td>-0.2</td></tr><tr><td>γ̃<sub>S</sub></td><td>50</td><td></td></tr></tbody></table>

Cellular adhesion is primarily driven by the glycocalyx [27,28]. In breast cancer cells, the glycocalyx contains fewer integrins compared to healthy cells [29–31], with integrins being the major protein family present in the glycocalyx [32]. Therefore, we assume that integrins make up 3/4 of the proteins and that their presence is reduced by 50% in cancer cells. Hence:

$$ \gamma^C = \left( \frac{1}{2} * \frac{3}{4} + \frac{1}{4} \right) \gamma^H = 0.625 \gamma^H \quad (2) $$

### 3.4. Evolution of the adhesion

The phenomenon of adhesion adaptation has been observed, yet its experimental characterization remains incomplete. Consequently, there are no measurements of adhesion variation during the wrapping process of NPs. However, studies on the kinetic segregation of receptors in the glycocalyx suggest an increase in the concentration of integrins near regions in contact with extracellular penetrants [33,34]. This effect appears to be more pronounced in cancer cells, though it has not been quantified.

### 3.5. Values for the input parameters of the model

First, we used the bending stiffness values estimated by [8], i.e., $κ^H = 4.5 \times 10^{-19}$ N/m and $κ^C = 1.3 \times 10^{-19}$ N/m. For membrane tension, we assumed an intermediate value for both healthy and cancer cells, i.e., $σ^H = σ^C = 0.01$ mN/m, as no precise data were available to differentiate them. The adhesion of healthy breast cells was set to $γ^H = 0.1$ mN/m, and for cancer cells, $γ^C = 0.0625$ mN/m from Equation (2).

Thus, the dimensionless parameters $σ̄$ and $γ̄$ are calculated using $σ̄ = 2σa^2/κ$ and $γ̄ = 2γa^2/κ$, where a is the radius of the NP, set to 100 nm. This results in $σ̄^H = 0.44$, $σ̄^C = 1.54$, $γ̄^H = 4.44$, and $γ̄^C = 9.62$.

Although data on membrane adaptation are lacking, we assume that cancer cell membranes are generally more reactive due to their adaptive behavior during proliferation. To emphasize this distinction, we selected extreme values within the domains of definition of $γ_A$, $γ_D$, and $γ_S$. Specifically, $γ_A^H = 1.5$, $γ_A^C = 4$, $γ_D^H = 0.2$, and $γ_D^C = -0.2$. The value of $γ_S$ was set to 50 for both cases, as shown in our previous studies to have no significant effect on cellular uptake predictions [6,7]. The input parameter values are summarized in Table 2.

## 4. Results

The study evaluates NP wrapping based on aspect ratio $\bar{r}$, identifying distinct uptake behaviors (Fig. 2).

For $\bar{r} < 1/3$, neither healthy nor cancer cells wrap the NP. In the range [1/3, 1/2], NPs remain unwrapped by healthy cells (phase 1) but are fully wrapped by cancer cells (phase 3), indicating selectivity. For $\bar{r}$ between 1/2 and 1.5, NPs are fully wrapped by both cell types. In the range [1.5, 5], NPs are fully wrapped by cancer cells but partially by healthy cells. Finally, for $\bar{r}$ between 5 and 6, NPs are partially wrapped by both cell types. These results are summarized in Table 3.

<figure><img src="image_3.png" /><figcaption>**Fig. 2.** Evolution of the wrapping phase of the NP in terms of its aspect ratio $\bar{r}$, for both healthy (solid blue line) and cancer (dotted orange line) breast cells.</figcaption></figure>

<table><caption>**Table 3**

Wrapping phases of the NP in terms of its aspect ratio $\bar{r}$, for both healthy and cancer breast cells.</caption><thead><tr><th>Value of r̄</th><th>Healthy cell</th><th>Cancer cell</th></tr></thead><tbody><tr><td>r̄ ∈ [1/6, 1/3]</td><td>phase 1: no wrapping</td><td></td></tr><tr><td>r̄ ∈ [1/3, 1/2]</td><td>phase 1: no wrapping</td><td>phase 3: full wrapping</td></tr><tr><td>r̄ ∈ [1/2, 1.5]</td><td>phase 3: full wrapping</td><td></td></tr><tr><td>r̄ ∈ [1.5, 5]</td><td>phase 2: partial wrapping</td><td>phase 3: full wrapping</td></tr><tr><td>r̄ ∈ [5, 6]</td><td>phase 2: partial wrapping</td><td></td></tr></tbody></table>

The variation of the total potential energy during the wrapping of NPs with $\bar{r} \in \{1/4, 1/2.5, 4\}$ by healthy and cancer cells is illustrated in Fig. 3, where the location of $\bar{r}$ is highlighted.

From these results, it can be concluded that using NPs with $\bar{r}$ in the range [1/3, 1/2] increases the likelihood of selective internalization by cancer cells only.

## 5. Discussion

The study faced challenges due to limited and varied input data, making comparisons difficult. Assumptions were made about membrane properties between healthy and cancer cells, despite potential differences in composition. Certain parameters, such as those affecting adhesion evolution during wrapping, could not be determined but are considered minimally influential based on the sensitivity analysis that we conducted on this model and that is presented in [7]. However, the model is highly sensitive to the initial adhesion value, $\bar{γ}_0$, which is uncertain. An uncertainty analysis could refine the $\bar{r}$ range for selective NP uptake by cancer cells, reducing uncertainty in the results.

## 6. Conclusion

This study developed a numerical model to simulate the cellular uptake of elliptical elastic NPs in breast cancer, aiming to optimize the NP aspect ratio for selective cancer cell entry. Model parameters were derived from a comprehensive review of experimental studies. Although breast cancer was chosen for its extensive documentation, data limitations persist.

The input parameter values were selected from limited studies, providing only approximate values, but consistent results suggest reliability. The model identified an optimal range of NP aspect ratios, [1/3, 1/2], where NPs are selectively fully wrapped by cancer cells, avoiding healthy cells. Aspect ratios within [1/2, 1.5] should be avoided, as they can be internalized by both cancerous and healthy cells, lacking selectivity.

The findings suggest that NP aspect ratio can be optimized to enhance selective internalization by breast cancer cells, though experimental validation is needed to confirm these results.
<figure><img src="image_4.png" /><figcaption>Fig. 3. Variation of $\overline{\Delta E}$ during the wrapping of NPs with (a) $\bar{r} = 1/4$, (b) $\bar{r} = 1/2.5$ and (c) $\bar{r} = 4$. Solid blue (resp. dotted orange) lines represent healthy (resp. cancer) breast cells. The colored markers correspond the equilibrium positions ($f = \bar{f}$).</figcaption></figure>

## CRediT authorship contribution statement

S.I. designed the model and the computational framework, analyzed the data and wrote the manuscript draft. S.K. carried out the implementation and wrote the manuscript draft. S.F. and F.J. conceived the study and were in charge of overall direction and planning, and reviewed the manuscript.

## Ethical approval

Not required.

## Funding

This work was supported by the French National Research Agency (Grant/Award Number: ANR-16-INDEX-0007), via i-SITE NExT, and CNRS (French National Centre for Scientific Research) via the ENAMEL project (Défi Mécanobiologie).

## Declaration of competing interest

Nothing to declare.

## Acknowledgements

The authors express their gratitude to Dr. Christophe Blanquart and Prof. Elena Ishow for their invaluable long-term collaboration and support.

A CC-BY public copyright license has been applied by the authors to the present document and will be applied to all subsequent versions, in accordance with the grant's open access conditions.

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