Corrigendum to “Machine Learning for the Sensitivity Analysis of a Model of the Cellular Uptake of Nanoparticles for the Treatment of Cancer”

Abstract

S. Iaquinta, S. Khazaie, S. Albanna, et al., “Machine Learning for the Sensitivity Analysis of a Model of the Cellular Uptake of Nanoparticles for the Treatment of Cancer,” International Journal for Numerical Methods in Biomedical Engineering (2024): e3878.

The “PREPRINT” mention was removed from the title of the article.

A new first section, Introduction, has been added, which provides the following clarifications. This addition thus changes the numbering of the subsequent sections.

The cellular uptake of nanoparticles (NPs) is investigated for the purpose of drug delivery. The latter are attached to NPs and are delivered to the cell. These drugs aim, depending on the treatment, at killing or altering the functioning of the cell. Such therapy is currently used and knows an exponential growth for cancer treatment [1]. Hence, it is crucial to properly calibrate the NPs for them to efficiently target the cells and avoid damaging healthy cells. The way of targeting cancer cells can be biochemical or even mechanical. In the second case, the NP is designed to take advantage of significant discrepancies that are observed between the mechanical properties of healthy and cancer cells [2–4]. For instance, comparison between M10 and MCF7 breast cells show that mammalian cancer cells are softer than their healthy counterparts [3, 5–7]. In order to properly understand the phenomena and the parameters that drive mechanically controlled drug delivery, experimental and numerical investigations have been conducted. In our previous works, focused on drug delivery via endocytosis cellular uptake, we proposed a method, based on an existing model of the cellular uptake of NPs [8], for the quantification of the influence of the NP's aspect ratio, NP-cell adhesion and cell membrane tension on the NP's uptake [9]. Then, we presented an enhanced model that accounts for the mechanical response of the cell membrane during the wrapping of the NP by the membrane and we demonstrated that the predictions of the model were altered when considering this mechanical response [10]. Still, the influence of the initial parameters of the system could not be compared to those that represent the mechanical response of the membrane because of the complexity and the computational cost of the approach. As such, the objective of this article is to build a surrogate model in order to evaluate the sensitivity indices (Sobol indices) that describe the influence of the input parameters of the model on its predictions. The outline of this article is the following. Section 2 introduces the model, whose complete description is provided in [9, 10]. Then, the strategies for building a surrogate model, using Kriging, Polynomial Chaos Expansion (PCE), and deep learning approaches, are presented in Section 3. The sensitivity analysis is subsequently conducted in Section 4.

In line 4 of the section Presentation of the model, after the phrase ending with “as detailed in 9”, we have added the following sentences:

The latter is due to the contribution of the energy necessary for bending the membrane around the NP (positive contribution), the energy necessary for stretching the membrane (positive contribution) and the energy released by the adhesion between the NP and the membrane (negative contribution). The equilibrium position of the system is defined as the first (i.e., the one encountered the earliest during the wrapping process) local minimum of potential energy, since it was demonstrated in the literature that there is no external phenomenon capable of making the system leave this position [1].

In the middle of the same paragraph, after “The NP's geometry is”, we added the word “defined.” The beginning of the phrase thus writes: The NP's geometry is defined by its aspect ratio ….

At the end of the fourth line of the subsection Definition of the input dataset (i.e., after the phrase that ends with “via random variables”), we added: Based on general convention in statistics and probability, in our paper capital letters (say X) denote random variables while their values are denoted by lowercase letters (say x). As such, the random variables are defined as follows ….

Five lines afterward, we made this change: The corresponding Probability Density Function (PDF) is provided in [10].

In the next paragraph, after the sixth line (ending with $\tilde{f}\approx 0.97$), we added these sentences:

To understand the physical meaning of these peaks, it is necessary to picture the contributions to the total potential energy of the system that are presented and discussed in detail in our first article [9] and briefly recalled in Section 2. Considering that the dataset contains slightly to highly elongated horizontal and vertical NPs, one can expect the bending energy necessary to wrap the membrane around the NP to vary a lot among the samples. This bending energy depends on the curvature of the NP close to the membrane. For instance, in the case of elongated vertical NPs, an important bending is necessary, which involves a large positive term in the total potential energy of the system, hence pushing the first local minimum close to zero. This is the reason why the first peak can mainly be attributed to elongated vertical NPs, which require a significant and energy-intensive bending of the membrane.

This paragraph ends with a line that has been modified as follows: The final peak at $\tilde{f}\approx 0.03$ pertain to cases where the wrapping is not significantly impeded by highly elongated NPs, or where adhesion parameter is large enough to compensate bending.

In the seventh line of the subsection Kriging and polynomial chaos expansion metamodel, we have added the word “test” in this phrase: Based on these results, 10% of the data set (409 samples) will be used to test the metamodels, and the remaining 90%, that is, 3687 samples, will be used for their training.

In the same subsection, at the end of the second paragraph, we have added “a single mode distribution located at $\tilde{f}\approx 0.03$” to the phrase starting with “In addition, the Kriging model results in.” In the same subsection, the second line after Equation (2), after the phrase ending with “indicated the size of the dataset and Var is the variance,” we have added the following phrases: The predictivity factor ranges between 0 and 1. The closer it is to 1, the more accurate the predictions.

In the subsection Artificial Neural Networks, second paragraph, eighth line, we have modified the phrase as follows: The $Q_2$ values were evaluated using 5-fold cross-validation, each fold containing 736 samples.

In the same paragraph, after the last sentence ending with “regardless of the architecture,” we have added: Still, the architecture (8, 8, 8) provided the most accurate predictions, resulting in $Q_2^{\mathrm{ANN}}=0.85$.

In the section Discussion, we have added this block of text at the beginning: The results of the sensitivity analysis match the preliminary conclusions that was obtained in our previous articles [9, 10]: the NP's aspect ratio is the most important parameter and the variation of the NP-membrane adhesion needs to be taken into account, even though playing a secondary role in the wrapping process. The importance of the NP's aspect ratio, NP-membrane initial adhesion and membrane tension were already well known in the literature, through experimental and numerical investigations [8, 37]. Furthermore, the sensitivity analysis presented in the article was conducted through the Sobol indices, that are commonly used for global sensitivity analyses. Still, the Shapley values are more frequently used in the machine learning community and might appear as more suitable for the investigation of an ANN. Both indices can be used for the interpretation of the sensitivity of the model presented in this article [38]. The results of the Shapley values yielded the same conclusions as those obtained from the Sobol indices. For this reason, it was preferred not to present these results in the article, but they can be found in the Supporting Information Material.

In the same section, after the phrase ending with “Sobol indices predictions has not been quantified,” we added: Furthermore, it is important to note that the Sobol sensitivity indices evaluated in this article provide information on the sensitivity of the ANN model to the input parameters, rather than directly reflecting the sensitivity of the biological phenomenon or the model itself. Therefore, these results should be validated experimentally and can only serve as preliminary guidance.

The Supporting Information section has also been modified as follows: The Python codes containing the model of the cellular uptake of a rigid elliptic NP, along with the routines to get $\tilde{f}$, are available in this Github repository: https://github.com/SarahIaquinta/ANN-sensitivity-adaptive-uptake-of-elliptic-nanoparticles. It also contains the data and methods used to implement and validate the metamodels and the artificial neural networks, as well as algorithms used to compute the sensitivity indices (Sobol indices and Shapley values).

We apologize for the oversight that led to publication of the earlier manuscript version. The corrections included here reflect the final, peer-reviewed content and aim to clarify any misunderstandings.