A Nano-QSAR Model for the Cytotoxicity of Doxorubicin Prodrug Micelles in Multidrug-Resistant Cancer Cells

Abstract

Nano-QSAR modeling provides a cost-effective approach for predicting the toxic effects of nanoparticles by reducing the experimental effort and improving the predictive accuracy. Cytotoxicity is the degree of toxicity of the particles toward the cells. In this study, a nano-QSAR model was developed for estimating the cytotoxicity (pIC₅₀) of doxorubicin prodrug micelles on MCF-7/ADR cells by using degree-based topological indices and physicochemical properties such as particle size and polydispersity index. Doxorubicin is a leading anticancer drug, and using it as a nanomedicine, doxorubicin prodrug loaded micelles, improves the therapeutic efficiency, reduces the side effects, and lowers the multidrug-resistance of tumor cells to doxorubicin. Multiple linear regression analysis was used for the model development. The developed model exhibited a meaningful preliminary correlation, with a R ² value exceeding 0.6, between the topological and physicochemical descriptors and the cytotoxicity. As a proof of concept, this proposed model illustrates the potential significance of topological indices as structural descriptors in nano-QSAR modeling for predicting the cytotoxicity of the prodrug micellar system.

1. Introduction

Nanotechnology is a branch of science that focuses on the modification of atoms and molecules to construct nanostructures that retain unique properties. Innovations in the wide spectrum of nanoscale technologies, referred to as nanomedicines, begin to change approaches to disease diagnosis, treatment, and prevention and pioneer more effective personalized therapies that have the potential to enhance patient outcomes. Nanotechnology based drug delivery systems (NDDS) have drawn considerable attention because of their strategic justification.

Breast cancer is a cancerous growth that originates from healthy mammary gland cells, and chemotherapy is the conventional treatment for such tumors. However, there are certain factors that contribute to chemoresistance, including pharmacological/physiological factors and multidrug-resistance (MDR). The enhanced permeability and retention effect (EPR) of NDDS, due to their size and charge, improves drug solubility, pharmacokinetics, and passive targeting ability of carried drugs. Moreover, modification of NDDS to respond with specific tumor microenvironment cues at the target site, such as changes in pH, temperature, or enzymatic activity, has the potential to improve the tumor-specific drug delivery and retention. Nanotechnology has provided a promising way to overcome MDR, which is the overexpression of the drug efflux transporter P-glycoprotein (P-gp).

Polymeric micelles are among the most promising modalities of drug carriers with demonstrated efficiency in reverse MDR. They have a unique core–shell structure, where the hydrophilic polymeric outer shell can evade the reticuloendothelial system, thereby demonstrating longevity in the circulation of the drug. Hydrophobic drugs in the inner core of the micelles increase the drug solubility. Traditionally, NDDS carry drugs through physical embedding, but to enhance stability and prevent premature drug release, carriers were combined with drugs via covalent bonds using various cross-linking methods, creating prodrugs with the ability to self-assemble into micelles. These micelle polymer prodrugs, which have the properties of both prodrugs and micelles, can alter the biodistribution of the drug, increase its therapeutic effects and reduce the side effects of the drug. −

In this context, performing toxicological tests of prodrug micelles is important, yet complete characterization of all different preparations is extremely laborious. Quantitative structure–activity relationship (QSAR) is a domain of mathematical modeling that studies the relationship between structure and activity using statistical methods and can reduce the need for time and resource-consuming cytotoxicity tests, which are crucial and economically viable. Variable selection, model development, and validation evaluation are the principal steps of QSAR analysis. − A wide range of enhanced and alternative methods have been introduced over the past few years.

The comprehensive study of nanoparticles and an understanding of correlation between their activity and structure are rendered available by nano-QSAR. Li et al. provided a detailed review of nano-QSAR models developed for metallic and metal oxide nanoparticles, addressing descriptor types, modeling algorithms, and validation strategies; emphasized the importance of standardized experimental techniques for dependable model development; and highlighted that the limited data availability continues to be a major challenge in nano-QSAR development. According to Buglak et al., who examined nano-QSAR studies on cytotoxicity for five distinct groups of engineered nanomaterials, including metal oxides, carbon nanotubes, fullerenes, and silica nanoparticles, multiple linear regression is one of the most prominent and interpretable modeling approaches in this domain. Singh and Gupta introduced ensembles learning-based nano-QSAR models for predicting the biological activity of diverse nanomaterials using simple molecular descriptors. This suggests meaningful structure–activity relationships can be established even with small data sets. These models estimate the cytotoxicity of new compounds to identify potential prodrug micelles by employing topological indices from chemical graph theory. Topological indices are the quantitative characteristics that reflect the topology of a covalently connected molecular structure. The in vitro cytotoxicity of prodrug micelles depends on several physicochemical properties in addition to the topological index of the molecular graph of its prodrugs. We could analyze the relationship between the properties of the prodrug micelles and their cytotoxicity by taking them into consideration as variables.

In this paper, we have considered prodrug micelles of doxorubicin (DOX), a prominent anticancer drug that has been widely used in oncology. The experimental data for DOX prodrug micelles investigated under standardized conditions against MCF-7/ADR cells remain scarce in literature. This exploratory study is built on a carefully curated small data set. The aim of this study is to develop a nano-QSAR framework as a proof of concept, exploring the correlation of degree-based topological indices of DOX prodrugs and physicochemical properties of their corresponding micelles with their cytotoxicity against the MCF-7/ADR cells. Nano-QSAR models are obtained by multiple linear regression analysis. This model is intended to uncover potentially significant structural descriptors and establish a foundation for future nano-QSAR development as larger standardized data sets become available.

2. Materials and Methods

2.1. Selection of Doxorubicin Prodrug Micelles

Prodrug micelle based nanomedicines are a kind of advanced drug delivery system that integrates the advantages of prodrug and micelle technologies to enhance the structure stability, drug loading efficiency, and cellular uptake. DOX is an antibiotic and potent chemotherapy drug, but its usage is limited because of systemic toxicities, especially cardiotoxicity and immunosuppression. The administration of a prodrug of DOX, which is an inactive form of the drug that becomes active after being delivered to the tumor site through specific triggers (such as an enzyme, pH change, or light), helps to expand the blood circulation time of the drug and alter the biodistribution for targeted pharmacokinetics and pharmacodynamics. The micelle structure enhances permeability and retention effect due to their small size. ,

DOX prodrug micelles included in this study were selected based on their (i) effectiveness in overcoming multidrug-resistance, (ii) potent cytotoxicity, which was quantified by the half-maximal inhibitory concentration (IC₅₀) for MCF-7/ADR cells in 48 h using a standard 3-(4,5-dimethylthiazol-2-yl)-2,5-diphenyltetrazolium bromide (MTT) assay, and (iii) well-defined physicochemical properties, namely nanoparticle size and polydispersity index (PDI). The availability of these micelles is currently limited in literature. Table below summarizes the defining characteristics of the curated five micelles, which include particle size, PDI, and cytotoxicity. −

1. Physicochemical & Biological Characterizations of the DOX Prodrug Micelles.

micelle	particle size (nm)	PDI	IC50 (μg/mL)
DOX-VES Micelle	125	0.2	7
DOX-NN-VES Micelle	129	0.2	4
DOX-TPP Micelle	102.4	0.262	8.54
PEG2K-DOX Micelle	91.5	0.14	0.294
TPGS2000-DOX Micelle	24.9	0.259	1.721

2.2. Computation of Degree-Based Topological Indices of Doxorubicin Prodrugs

The topological index is a mathematical method that converts a graph to a real number that can forecast the physiochemical properties and describe the structure of molecules. A micelle is a supramolecular assembly held together by weak noncovalent forces that prevents the prodrug from premature release in the bloodstream. Their physicochemical characteristics, which include self-assembly, core stability, drug loading efficiency, and drug release kinetics, are primarily governed by the covalent structure of the prodrug molecule. The molecular structure of the prodrug is a key determinant of the chemical identity of the micelles. Therefore, the topological index of the DOX prodrugs that regulate cytotoxicity of the selected micelles was examined. In chemical graph theory, a molecular graph illustrates the structure of the molecule, with atoms representing vertices and bonds between the atoms representing edges. Figures , , , and display molecular graphs of the considered DOX prodrugs. Degree-based topological indices were calculated analytically by employing the edge partition approach, which classifies the edges of the molecular graph according to the degree pairs of the end vertices. In this paper, we studied only simple, connected, and undirected molecular graphs, $\mathcal{G}=(V(\mathcal{G}),E(\mathcal{G}))$ , where $V(\mathcal{G})$ is the vertex set and $E(\mathcal{G})$ is the edge set of the graph. The degree of the vertex v is represented by $d_{\mathcal{G}}(v)$ . Table lists the degree-based topological indices used in this paper. −

1.

Molecular Graph of DOX-VES Prodrug.

2.

Molecular Graph of DOX-NN-VES Prodrug.

3.

Molecular Graph of DOX-TPP Prodrug.

4.

Molecular Graph of PEG_2K-DOX Prodrug.

5.

Molecular Graph of TPGS2000-DOX Prodrug.

2. Degree-Based Topological Indices.

index	formula
First Zagreb	$$M_1(\mathcal{G})=\sum _{\mathit{uv}\in E(G)}(d_{\mathcal{G}}(u)+d_{\mathcal{G}}(v))$$
Second Zagreb	$$M_2(\mathcal{G})=\sum _{\mathit{uv}\in E(G)}{d}_{\mathcal{G}}(u)d_{\mathcal{G}}(v)$$
Randić	$$R(\mathcal{G})=\sum _{\mathit{uv}\in E(G)}\frac{1}{\sqrt{d_{\mathcal{G}}(u)d_{\mathcal{G}}(v)}}$$
Atom Bond Connectivity	$$\mathit{ABC}(\mathcal{G})=\sum _{\mathit{uv}\in E(G)}\sqrt{\frac{d_{\mathcal{G}}(u)+d_{\mathcal{G}}(v)-2}{d_{\mathcal{G}}(u)d_{\mathcal{G}}(v)}}$$
Geometric Arithmetic	$$\mathit{GA}(\mathcal{G})=\sum _{\mathit{uv}\in E(G)}\frac{2\sqrt{d_{\mathcal{G}}(u)d_{\mathcal{G}}(v)}}{d_{\mathcal{G}}(u)+d_{\mathcal{G}}(v)}$$
Harmonic	$$H(\mathcal{G})=\sum _{\mathit{uv}\in E(G)}\frac{2}{d_{\mathcal{G}}(u)+d_{\mathcal{G}}(v)}$$
Augmented Zagreb	$$\mathit{AZ}(\mathcal{G})=\sum _{\mathit{uv}\in E(G)}{\left(\frac{d_{\mathcal{G}}(u)d_{\mathcal{G}}(v)}{d_{\mathcal{G}}(u)+d_{\mathcal{G}}(v)-2}\right)}^3$$
Forgotten	$$F(\mathcal{G})=\sum _{\mathit{uv}\in E(G)}(d_{\mathcal{G}}{(u)}^2+d_{\mathcal{G}}{(v)}^2)$$
Sum Connectivity	$$\mathit{SC}(\mathcal{G})=\sum _{\mathit{uv}\in E(G)}\frac{1}{\sqrt{d_{\mathcal{G}}(u)+d_{\mathcal{G}}(v)}}$$
Inverse Sum	$$\mathit{IS}(\mathcal{G})=\sum _{\mathit{uv}\in E(G)}\frac{d_{\mathcal{G}}(u)d_{\mathcal{G}}(v)}{d_{\mathcal{G}}(u)+d_{\mathcal{G}}(v)}$$

2.3. Quantitative Structure Analysis of Doxorubicin Prodrug Micelles

In this section, we developed quantitative structure–activity relationship (QSAR) models, specifically designed for nanomaterial systems (nano-QSAR), to correlate the physicochemical properties of DOX prodrug loaded micelles and the topological indices of their associated prodrugs with their cytotoxic potency against MCF-7/ADR breast cancer cells. The aim of nano-QSAR is to statistically determine a relationship between independent variables and dependent variables. In this paper, “descriptors” denote independent variables, while “end points” refer to dependent variables. The main difference between QSAR and nano-QSAR is in the types of chemical entities they model and the particular descriptors used for their characterization. We use the multiple linear regression algorithm for developing the model. The first step in implementing the nano-QSAR model is data collection and data processing.

2.3.1. Data Preparation and Descriptor Selection

Data regarding nanoparticles evaluated with standardized protocols are limited. In this study, we focus on a carefully curated data set of five distinct DOX prodrug loaded micellar formulations, all examined under identical experimental conditions. A well-defined end point is essential for detailing model prediction. Cytotoxicity data (IC₅₀) were transformed into pIC₅₀ using the formula

$$\mathrm{p}\mathrm{I}{\mathrm{C}}_{50}=-\mathrm{lo}{\mathrm{g}}_{10}(\mathrm{I}{\mathrm{C}}_{50})$$ 1

where IC₅₀ is expressed in μg/mL. The logarithmic transformation is a standard practice that helps in normalizing the distribution and minimizes the influence of outliers. pIC₅₀ values of the selected micelles, which serve as the end point, are presented in Table . The curated data set, although limited in size, provides a validated framework for future expansion as more systematically characterized micelles become available.

3. pIC₅₀ Values of the DOX Prodrug Micelles.

	DOX-VES	DOX-NN-VES	DOX-TPP	PEG2K‑DOX	TPGS2000-DOX
pIC50	–0.8451	–0.6021	–0.9315	0.5317	–0.2358

Data from different sources typically needs to be structured and integrated before further analysis, making preliminary processing essential. Feature selection and reduction are necessary to discard unnecessary or irrelevant descriptors. Various techniques have been employed for feature selection, including genetic algorithms, Pearson and pairwise correlations, principal component analysis, partial least-squares, and genetic function approximation. In this study, feature selection was performed in two stages. First, the Pearson correlation coefficient was computed for each of the ten degree-based topological indices against pIC₅₀ to identify the most significant topological index. To provide a more rigid theoretical basis for descriptor selection, a forward stepwise regression was subsequently performed on the entire descriptor pool using an entry criterion of p < 0.05, where p denotes the statistical significance of the contribution of each descriptor to the model.

The multiple linear regression model accounts for the linear relationships among the descriptor variables, which is one of the most significant differences between the multiple and simple linear regressions. Such relationships, termed “multicollinearity”, can have significant influence on the model coefficients and the estimated accuracy. This section examines multicollinearity. A pairwise correlation matrix of all ten topological indices was developed initially in order to determine strongly correlated descriptors following which only a single topological index was retained. The variance inflation factor (VIF) for the descriptor i, defined as

$${\mathrm{VIF}}_i=\frac{1}{1-R_i^2}$$ 2

where R _i is the coefficient of determination for regressing the descriptor i against the remaining descriptors, was calculated for the final model descriptors to thoroughly validate the absence of concerning multicollinearity. A VIF value below 5 is generally considered acceptable and values approaching 1 indicate negligible multicollinearity.

2.3.2. Multiple Linear Regression Model Development

Developing a multiple linear regression model helps to mathematically describe the relationship between the selected descriptors and the end point cytotoxicity. The advantage of this approach is that it yields a precise and accurate understanding of the influence of each descriptor while controlling the effect of the others. The relationship between the end point z and the descriptors x and y, modeled using multiple linear regression, is represented as

$$z=\mathit{ax}+\mathit{by}+c$$ 3

where a and b are regression coefficients and c is the z-intercept. In this section, we employ MATLAB R2024a, with Statistics and Machine Learning Toolbox, to develop and fit a multiple linear regression model with the selected descriptors after performing feature selection and multicollinearity tests. All possible combinations of the selected topological index and physicochemical descriptors were evaluated and the combination that yield the optimal training fit was retained as the final model. ,

2.3.3. Model Validation and Statistical Analysis

Validation procedures should be assessed for determining the optimal structure and attributes of the nano-QSAR model. Different metrics were used for model validation depending on the type of model. In this paper, we use the coefficient of determination (R ²), root mean squared error (RMSE), mean absolute error (MAE), and F-statistics to evaluate and validate the fitted regression model. R ² ranges from 0 to 1 and is used to evaluate the correlation between the observed outcomes and the model predictions. The average error of the model in predicting the observed outcome is quantified with RMSE and MAE values, which will be lower if the R ² (>0.6) is greater, indicating a better model. A substantial F-statistics indicated the model fits the data compared to the null model. These metrics indicate the training fit of the model. Internal predictive ability of the model was examined by leave-one-out (LOO) cross-validation, and the robustness against chance correlation was assessed using a Y-randomization test. Both LOO cross-validation and Y-randomization were performed using MATLAB R2024a, with no additional toolboxes. The backslash () operator was used to retrain the model using n–1 training points in order to forecast the held-out observation. The predictive performance is measured using Q ², the squared correlation coefficient for prediction. The “perms” function was used to exhaustively permute the response vector, producing all 119 possible reassignments.

3. Results and Discussion

We summarize the principal findings of the study in this section. The micelles considered in this study were based on their ability to overcome the multidrug-resistant MCF-7/ADR cells. The number of vertices (n) and the number of edges (m) of the prodrug molecular graph of each micelle were obtained from Figures – and it is tabularized in the Table . Degree-based topological indices were computed using the edge partition technique, and the corresponding edge partitions are presented in Table .

4. Number of Vertices and Edges of DOX Prodrugs of Each Micelle.

edge partition	DOX-VES	DOX-NN-VES	DOX-TPP	PEG2K‑DOX	TPGS2000-DOX
n	76	77	63	180	215
m	82	83	70	184	221

5. Edge Partition of DOX Prodrugs of Each Micelle.

edge partition	DOX-VES	DOX-NN-VES	DOX-TPP	PEG2K‑DOX	TPGS2000-DOX
(1,2)	2	2	2	3	2
(1,3)	16	16	8	8	18
(1,4)	2	2	1	1	2
(2,2)	10	11	16	139	143
(2,3)	25	25	21	15	29
(2,4)	5	5	3	2	5
(3,3)	21	21	15	15	21
(3,4)	1	1	4	1	1

3.1. Numerical Results of Degree-Based Topological Indices for Prodrugs

In this subsection, the computation of the degree-based topological indices for the molecular graphs of DOX prodrugs in each micelle is presented. The calculation is performed with the degree-based edge partition of each molecular graph. To avoid the repetition, detailed calculations are presented solely for the first Zagreb index of the prodrug DOX-VES; the others were obtained using the same procedure, and their values are concisely summarized in Table .

6. Values of Degree-Based Topological Indices of DOX Prodrugs.

	DOX-VES	DOX-NN-VES	DOX-TPP	PEG2K‑DOX	TPGS2000-DOX
M 1	408	412	348	786	968
M 2	491	495	429	843	1053
R	35.9145	36.4145	30.3216	88.8596	105.2022
ABC	59.14	59.8471	49.6785	130.4735	157.6466
GA	78.5407	79.5407	67.9769	182.1289	217.1919
H	34.0857	34.5857	29.2762	87.8524	103.1857
AZ	647.7679	655.7679	591.5257	1486.0537	1750.5179
F	1112	1120	938	1754	2248
SC	37.2219	37.7219	31.8537	89.7057	106.5107
IS	94.8143	95.8143	82.6905	192.6810	234.1143

Proposition 3.1. Let $\mathcal{G}$ be the molecular graph of the prodrug DOX-VES. Then, the first Zagreb index of $\mathcal{G}$ is $M_1(\mathcal{G})=408$ .

Proof. Let $\mathcal{G}$ be the molecular graph of the prodrug DOX-VES with $n(V(\mathcal{G}))=76$ and $n(E(\mathcal{G}))=82$ . Then by using the edge partitions given in Table in the definitions of degree-based topological indices in Table , we get the following result.

$$\begin{array}{l}{M}_1(\mathcal{G})=d_{\mathcal{G}}(u)+d_{\mathcal{G}}(v)=2(1+2)+16(1+3)+2(1+4)+10(2+2)+25(2+3)+5(2+4)+21(3+3)+1(3+4)=408\end{array}$$

Following the same edge partition-based procedure, the remaining topological indices for all considered prodrugs are computed. A graphical summary of these computed topological indices was presented in Figure . Each group of bars in the graph represents a DOX prodrug, while different colors indicate the respective degree-based topological indices.

6.

Graphical representation of degree-based topological indices of DOX prodrugs.

3.2. Nano-QSAR Modeling of Micelle Cytotoxicity

Table presents the estimated Pearson correlation coefficient (R) between the degree-based topological indices of DOX prodrugs (Table ) and pIC₅₀ values (Table ) of each micelle. To establish the correlation in this instance, we have employed Microsoft Excel. The highest R value indicates the greater prediction ability of the index with cytotoxicity. According to Table , the Harmonic index (H) shows the strongest correlation with pIC₅₀ with a R value of 0.7927 among the ten topological indices computed. The correlation between each degree-based topological index of the prodrugs with pIC₅₀ is shown graphically in Figure . By analyzing both Table and Figure , we can deduce that the H index appears to be the most significant descriptor for modeling micelle cytotoxicity.

7. Correlation Coefficient (R) between Degree-Based Topological Indices of the Prodrugs and pIC₅₀ of the Selected Micelles.

	M 1	M 2	R	ABC	GA	H	AZ	F	SC	IS
R value	0.7462	0.7196	0.7875	0.7701	0.7790	0.7927	0.7823	0.6886	0.7841	0.7582

7.

Graphical representation of the correlation coefficients of each topological index with pIC₅₀.

To establish a more robust framework for descriptor selection, a forward stepwise regression (entry criterion p < 0.05) was performed on the whole topological indices collection. The corresponding F-statistics and p-values are summarized in Table . The H index yielded the minimum p-value among all indices (F = 5.0724, p = 0.1097), although none satisfied the p < 0.05 entry threshold, and the stepwise regression, therefore, terminated without admitting any descriptor. Due to the small sample size (n = 5), the stepwise regression was constrained in statistical power. Therefore, the correlation-based selection of the H index is maintained as the primary topological descriptor, in accordance with established nano-QSAR practice for small exploratory data sets. Before model development, the pairwise correlation matrix of all ten topological indices was evaluated. Figure illustrates all pairs had |R| > 0.95, confirming that the topological indices contain highly similar structural information. As a result, only a single topological index was considered as a descriptor, and the H index was selected since it had the highest correlation with pIC₅₀.

8. Entry Statistics (F Statistics, p-value) for the Degree-Based Topological Indices Evaluated at Step 1 of the Forward Stepwise Regression.

	M1	M2	R	ABC	GA	H	AZ	F	SC	IS
F Statistics	3.7686	3.2217	4.8984	4.3721	4.6319	5.0724	4.7311	2.7048	4.7875	4.0577
p-value	0.1475	0.1705	0.1138	0.1277	0.1205	0.1097	0.1179	0.1986	0.1165	0.1374

8.

Pairwise correlation Matrix of all topological indices.

The main emphasis of this study is the application of topological indices; thus, the H index was considered a mandatory descriptor. Every combination of H index, including H alone, H and particle size, and H and PDI, was evaluated. The combination of the H index and PDI provided the best training fit with R ² = 0.9365, RMSE = 0.1341, MAE = 0.1144, and F-statistics = 14.75 and was selected as the final model. The variance inflation factor (VIF) was used to quantify the multicollinearity between these descriptors. The maximum VIF value obtained is 1.02, which is considerably below the accepted limit of 5 and confirms that there is no concerning multicollinearity in the final regression model. The inclusion of PDI as the second descriptor is justified by the correlation matrix of H index, particle size, and PDI illustrated in Figure , which confirms that the H index and particle size exhibit a strong correlation with R = −0.85, while the H index and PDI show only a weak correlation of R = −0.13.

9.

Correlation Matrix of Harmonic Index, Particle size and PDI.

The multiple linear regression nano-QSAR model developed for the cytotoxicity (pIC₅₀) of doxorubicin prodrug micelles by employing the descriptors, harmonic index of prodrugs (H) and polydispersity index of micelles (PDI), is presented as follows

$$\mathrm{pI}{\mathrm{C}}_{50}=0.0123H-6.5978\mathrm{PDI}+0.273$$ 4

This derived equation can explain the quantitative contribution of H index and PDI to pIC₅₀. The H index captures the branching connectivity of the prodrug molecular graph, which is closely related to the self-assembly of prodrug molecules in an aqueous environment. Similar branching-sensitive connectivity indices have already been used to predict micellar self-assembly behavior, including critical micelle concentration, in structurally related amphiphilic systems. , This gives reasonable grounds to support the plausibility of a similar relationship between prodrug topology and micellar self-assembly here. The positive regression coefficient for H indicates a positive correlation, and the substantial negative coefficient for PDI shows a strong inverse relation with pIC₅₀, suggesting the prodrugs with greater molecular connectivity and the micelles with more uniform nanoparticles exhibit greater cytotoxicity against MCF-7/ADR cells. Uniform micelles exhibit more consistent cellular uptake and drug release behavior.

3.2.1. Validation of the Nano-QSAR Model for Cytotoxicity

LOO cross-validation was used to evaluate the internal predictive ability of the developed model. The negative Q ² yielded by the LOO cross-validation reflected the statistical instability caused by the small data set. To evaluate whether the observed structure–activity relationship could have occurred randomly, a Y-randomization test was performed. As shown in Figure , across all 119 permutations of the response vector, the randomized models had a mean R ² of 0.4963 and a maximum R ² of 0.9943, compared with R ² = 0.9365 for the true model. Only 9 of 119 permutations matched or exceeded the true R ² (empirical p = 0.076), and the cR _p value of 0.642 exceeds the commonly used threshold of 0.5. This suggests the model outperforms most randomly permuted models, though the small sample size (n = 5) limits the statistical power of the test, consistent with the negative Q ² result above.

10.

Y-randomization test showing the distribution of R ² values from 119 randomly permuted models compared to the R ² value of true model.

The performance and validity of the developed nano-QSAR model were further evaluated through the effect leverage plot and the predicted vs actual plot. The effect leverage plot presented as Figure is a regression diagnostic plot used in multiple linear regression models to ascertain the unique impact of a specific descriptor has on the end point, after accounting for all other descriptors included in the model. This single plot is an aggregated visualization of the model’s overall fit. The slope of the red line represents the combined, adjusted effect of both of the descriptors working together. Overall model is considered significant if the 95% confidence bounds (dotted red lines) for the fitted model do not overlap with the horizontal blue zero line (the null model, where the slope is 0). In this plot, the dotted lines are entirely within the positive Y region and the negative Y region, respectively, separated by the data points. This means the model’s result is statistically significant since the actual slope is positive, not zero. The visual data in the plot confirms that the aggregated effect of the descriptors has a strong, positive, and statistically significant relationship with pIC₅₀ values.

11.

Multiple Linear regression model fit

The predicted vs actual plot, shown in Figure , evaluates the model performance and is used to understand how well the regression model makes predictions for different end points. The dashed red line in the plot represents the perfect prediction line y = x, and the points closely aligned to this line indicate a satisfactory correlation between the actual and predicted values of pIC₅₀.

12.

Graphical representation of the predicted values from the multiple linear regression model vs actual values of pIC₅₀.

4. Conclusion

The future of nanomedicine depends on the methodical design of nanotechnologies like prodrug micelles based on a detailed and thorough understanding of biological processes. Since doxorubicin cancer therapy exhibits some limitations, such as a lack of solubility, poor biodistribution, and nonspecific effects that result in cardiac and renal damage, the development of a DOX prodrug micellar system is a promising strategy to overcome these limitations. In this study, multiple linear regression based nano-QSAR models have been proposed for predicting the cytotoxicity of DOX prodrug micelles against the multidrug-resistant breast cancer cell line MCF-7/ADR using H index of its prodrug and the PDI of the micelles. The H index showed the strongest correlation with pIC₅₀, confirming it as the most significant topological descriptor for this model. The developed model exhibits a high apparent training fit with a R ² of 0.9365 and low training errors, and the positive regression coefficient for the H index alongside the negative coefficient for PDI collectively suggest that the systematic variation in the DOX prodrug molecular structure and the uniformity in size of the resulting micelles together influence their cytotoxic potency. The VIF value of 1.0163 confirms the absence of multicollinearity between the final descriptors.

The primary constraint of the present model is the limited data set of five prodrug micelles, which is restricted by the availability of systematically characterized DOX prodrug micelles with particle size, PDI, and MCF-7/ADR cytotoxicity data under identical experimental conditions. Multiple linear regression with two variables and five data points carries the risk of overfitting, and the negative LOO cross-validation Q ² reflects this limitation. Therefore, the proposed model should be interpreted as a proof-of-concept demonstration of the applicability of degree-based topological indices in nano-QSAR modeling of prodrug micellar systems rather than a completely validated predictive tool. As the prodrug micelle nanomedicine develops and standardized experimental reporting advances, expanding the data set will be a priority for future work to improve the statistical robustness and generalizability of the model. This present framework establishes a basis for the future systematic design and development of DOX prodrug micelle nanomedicines with improved efficiency, reduced side effects, and better multidrug-resistance overcoming capability for breast cancer treatment.

All data generated or analyzed during this study are included in this published article.

S.K.: Conceptualization; writing original draft; validation; investigation; visualization; and methodology. R.S.: Supervision, conceptualization, and validation.

The authors declare no competing financial interest.